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Applets:Physical Signal & Equivalent Lowpass Signal

2018-08-28T14:10:56Z

Xiaohan: /* $x_{\rm TP}(t)$–Representation of a Sum of Three Harmonic Oscillations */

{{LntAppletLinkEn|analPhysSignal}}

==Applet Description==
 
This applet shows the relationship between the physical bandpass signal $x(t)$ and the associated equivalent low-pass signal $x_{\rm TP}(t)$. It is assumed that the bandpass signal $x(t)$ has a frequency-discrete spectrum $X(f)$:
:$$x(t) = x_{\rm U}(t) + x_{\rm T}(t) + x_{\rm O}(t) = A_{\rm U}\cdot \cos\left(2\pi f_{\rm U}\cdot t- \varphi_{\rm U}\right)+A_{\rm T}\cdot \cos\left(2\pi f_{\rm T}\cdot t- \varphi_{\rm T}\right)+A_{\rm O}\cdot \cos\left(2\pi f_{\rm O}\cdot t- \varphi_{\rm O}\right). $$
The physical signal $x(t)$ is thus composed of three harmonic oscillations, a constellation that can be found, for example, in the ''Double-sideband Amplitude Modulation''
*of the message signal $x_{\rm N}(t) = A_{\rm N}\cdot \cos\left(2\pi f_{\rm N}\cdot t- \varphi_{\rm N}\right)$   ⇒   in German:   '''N'''achrichtensignal
*with the carrier signal $x_{\rm T}(t) = A_{\rm T}\cdot \cos\left(2\pi f_{\rm T}\cdot t - \varphi_{\rm T}\right)$   ⇒   in German:   '''T'''rägersignal.

The nomenclature is also adapted to this case:
* $x_{\rm O}(t)$ denotes the &bdquo;upper sideband”   (in German:   '''O'''beres Seitenband) with the amplitude $A_{\rm O}= A_{\rm N}/2$, the frequency $f_{\rm O} = f_{\rm T} + f_{\rm N}$ and the phase $\varphi_{\rm O} = \varphi_{\rm T} + \varphi_{\rm N}$.
*Similarly, for the &bdquo;lower sideband”   (in German:   '''U'''nteres Seitenband) $x_{\rm U}(t)$ with $f_{\rm U} = f_{\rm T} - f_{\rm N}$, $A_{\rm U}= A_{\rm O}$ and $\varphi_{\rm U} = -\varphi_{\rm O}$.

The associated equivalent low-pass–signal is $f_{\rm O}\hspace{0.01cm}' = f_{\rm O}- f_{\rm T} > 0$,   $f_{\rm U}\hspace{0.01cm}' = f_{\rm U}- f_{\rm T} < 0$  and  $f_{\rm T}\hspace{0.01cm}' = 0$:

:$$x_{\rm TP}(t) = x_\text{TP, T}(t) + x_\text{TP, O}(t) + x_\text{TP, U}(t) = A_{\rm T}\cdot {\rm e}^{-{\rm j} \varphi_{\rm T} } \hspace{0.1cm}+ \hspace{0.1cm} A_{\rm O}\cdot {\rm e}^{-{\rm j} \varphi_{\rm O} } \cdot {\rm e}^{ {\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm O}\hspace{0.01cm}'\hspace{0.05cm}\cdot \hspace{0.05cm}t}\hspace{0.1cm}+ \hspace{0.1cm}
A_{\rm U}\cdot {\rm e}^{-{\rm j} \varphi_{\rm U} } \cdot {\rm e}^{ {\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm U}\hspace{0.01cm}'\hspace{0.05cm}\cdot \hspace{0.05cm}t} . $$

[[File:Ortskurve_1.png|right|frame|Equivalent low-pass signal currently $t=0$ for cosinusoidal carrier   ⇒   $\varphi_{\rm T} = 0$]]
The program shows $x_{\rm TP}(t)$ as the vectorial sum of three rotation pointers as a violet dot (see figure for start time $t=0$ and cosinusoidal carrier):

*The (red) pointer of the carrier $x_\text{TP, T}(t)$ with the length $A_{\rm T}$ and the zero phase position $\varphi_{\rm T}=0$ is fixed in the complex plane. So it applies to all times $t$:   $x_{\rm TP}(t)= A_{\rm T}\cdot {\rm e}^{-{\rm j} \varphi_{\rm T} }$.

*The (blue) pointer of the upper sideband $x_\text{TP, O}(t)$ with the length $A_{\rm O}$ and the zero phase position $\varphi_{\rm O}$ rotates at the angular velocity $2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm O}\hspace{0.01cm}'$ in mathematically positive direction (one revolution in time $1/f_{\rm O}\hspace{0.01cm}')$.

*The (green) pointer of the lower sideband $x_{\rm U+}(t)$ with the length $A_{\rm U}$ and the zero phase position $\varphi_{\rm U}$ rotates at the angular velocity $2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm U}\hspace{0.01cm}'$, because of $f_{\rm U}\hspace{0.01cm}'<0$ counterclockwise.

*With $f_{\rm U}\hspace{0.01cm}' = -f_{\rm O}\hspace{0.01cm}'$ the blue and green pointers will spin at the same speed but in different directions. Also, if $A_{\rm O} = A_{\rm U}$ and $\varphi_{\rm U} = -\varphi_{\rm O}$, then $x_{\rm TP}(t)$ moves on a straight line with a incline of $\varphi_{\rm T}$.

''Note:''   In the figure $\varphi_{\rm O} = +30^\circ$. From this follows for the start time $t=0$ the angle of the blue pointer (USB, blue pointer) with respect to the coordinate system:   $\phi_{\rm O} = -\varphi_{\rm O} = -30^\circ$. Likewise, the zero phase position $\varphi_{\rm U} = -30^\circ$ of the lower sideband (LSB, green pointer) follows for the phase angle to be considered in the complex plane:   $\phi_{\rm U} = +30^\circ$.

The temporal process of $x_{\rm TP}(t)$ is also referred to below as '''locus'''. The relationship between $x_{\rm TP}(t)$ and the physical bandpass signal $x(t)$ is given in the section and the associated analytic signal is $x_+(t)$ :

:$$x_{\rm TP}(t) = x_{\rm +}(t)\cdot {\rm e}^{-{\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm}2 \pi \cdot f_{\rm T}\cdot \hspace{0.05cm}t},$$
:$$x_{\rm +}(t) = x_{\rm TP}(t)\cdot {\rm e}^{+{\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm}2 \pi \cdot f_{\rm T}\cdot \hspace{0.05cm}t}.$$

[[Applets:Physikalisches_Signal_%26_Äquivalentes_TP-Signal|'''German description''']]

==Theoretical Background==
 
===Description of Bandpass Signals===
[[File:Zeigerdiagramm_1a.png|right|frame|bandpass spectrum $X(f)$ |class=fit]]
We consider '''bandpass signals''' $x(t)$ with the property that their spectra $X(f)$ are not in the range around the frequency $f=0$, but around a carrier frequency $f_{\rm T}$. In most cases it can also be assumed that the bandwidth is $B \ll f_{\rm T}$.

The figure shows such a bandpass spectrum $X(f)$. Assuming that the associated $x(t)$ is a physical signal and thus real, the spectral function $X(f)$ has a symmetry with respect to the frequency $f = 0$, if $x(t)$ is an even function   ⇒   $x(-t)=x(t)$, $X(f)$ is real and even.

Beside the physical signal $x(t)\ \circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\, \ X(f)$, one can also use the following descriptions of bandpass signals:
*the analytic signal $x_+(t)\ \circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\, \ X_+(f)$, see applet [[Applets:Physikalisches_Signal_%26_Analytisches_Signal|Physikalisches Signal & Analytisches Signal]],
*the equivalent low-pass–signal $x_{\rm TP}(t)\ \circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\, \ X_{\rm TP}(f)$, see next page
 

===Spectral Functions of the Analytic and the Equivalent Low-Pass Signal===

The '''analytic signal''' $x_+(t)$ belonging to the physical signal $x(t)$ is the time function whose spectrum fulfills the following property:
[[File:Ortskurve_2.png|right|frame|spectral functions $X_+(f)$ and $X_{\rm TP}(f)$ |class=fit]]
:$$X_+(f)=\big[1+{\rm sign}(f)\big] \cdot X(f) = \left\{ {2 \cdot
X(f) \; \hspace{0.2cm}\rm for\hspace{0.2cm} {\it f} > 0, \atop {\,\,\,\, \rm 0 \; \hspace{0.9cm}\rm for\hspace{0.2cm} {\it f} < 0.} }\right.$$

The ''Signum function'' is for positive values of $f$ equal to $+1$ and for negative $f$ values equal to $-1$.
*The (double-sided) limit returns $\sign(0) = 0$.
*The index „+” should make it clear that $X_+(f)$ only has parts at positive frequencies.

From the graph you can see the calculation rule for $X_+(f)$: The actual BP spectrum $X(f)$ becomes
*doubled at the positive frequencies, and
*set to zero at the negative frequencies.

Due to the asymmetry of $X_+(f)$ with respect to the frequency $f = 0$, it can already be said that the time function $x_+(t)$ except for a trivial special case $x_+(t)= 0 \ \ \circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\,\ \ X_+(f)= 0$ is always complex.

The spectrum $X_{\rm TP}(f)$ of the equivalent low-pass signal is obtained by shifting $X_+(f)$ to the left by the carrier frequency $f_{\rm T}$:
:$$X_{\rm TP}(f)= X_+(f+f_{\rm T}).$$

In the time domain this operation corresponds to the multiplication of $x_{\rm +}(t)$ with the complex exponential function with negative exponent:
:$$x_{\rm TP}(t) = x_{\rm +}(t)\cdot {\rm e}^{-{\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm}2 \pi \cdot f_{\rm T}\cdot \hspace{0.05cm}t}.$$

It can be seen that $x_{\rm TP}(t)$ is generally complex. But if $X_+(f)$ is symmetric about the carrier frequency $f_{\rm T}$, $X_{\rm TP}(f)$ is symmetric about the frequency $f=0$ and there is accordingly real time function $x_{\rm TP}(t)$.

===$x_{\rm TP}(t)$–Representation of a Sum of Three Harmonic Oscillations===

In our applet, we always assume a set of three rotating pointers. The physical signal is:
:$$x(t) = x_{\rm U}(t) + x_{\rm T}(t) + x_{\rm O}(t) = A_{\rm U}\cdot \cos\left(2\pi f_{\rm U}\cdot t- \varphi_{\rm U}\right)+A_{\rm T}\cdot \cos\left(2\pi f_{\rm T}\cdot t- \varphi_{\rm T}\right)+A_{\rm O}\cdot \cos\left(2\pi f_{\rm O}\cdot t- \varphi_{\rm O}\right). $$
* Each of the three harmonic oscillations $x_{\rm T}(t)$, $x_{\rm U}(t)$ and $x_{\rm O}(t)$ is represented by an amplitude $(A)$, a frequency $(f)$ and a phase value $(\varphi)$.
*The indices are based on the modulation method [[Modulationsverfahren/Zweiseitenband-Amplitudenmodulation|double sideband amplitude modulation]]. &bdquo;T” stands for &bdquo;carrier”, &bdquo;U” for &bdquo;lower sideband” and &bdquo;O” for &bdquo;upper Sideband”. Similarly, $f_{\rm U} < f_{\rm T}$ and $f_{\rm O} > f_{\rm T}$. There are no restrictions for the amplitudes and phases.

The associated equivalent low-pass signal is with $f_{\rm O}\hspace{0.01cm}' = f_{\rm O}- f_{\rm T} > 0$,   $f_{\rm U}\hspace{0.01cm}' = f_{\rm U}- f_{\rm T} < 0$  and  $f_{\rm T}\hspace{0.01cm}' = 0$:

:$$x_{\rm TP}(t) = x_\text{TP, T}(t) + x_\text{TP, O}(t) + x_\text{TP, U}(t) = A_{\rm T}\cdot {\rm e}^{-{\rm j} \varphi_{\rm T} } \hspace{0.1cm}+ \hspace{0.1cm} A_{\rm O}\cdot {\rm e}^{-{\rm j} \varphi_{\rm O} } \cdot {\rm e}^{ {\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm O}\hspace{0.01cm}'\hspace{0.05cm}\cdot \hspace{0.05cm}t}\hspace{0.1cm}+ \hspace{0.1cm}
A_{\rm U}\cdot {\rm e}^{-{\rm j} \varphi_{\rm U} } \cdot {\rm e}^{ {\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm U}\hspace{0.01cm}'\hspace{0.05cm}\cdot \hspace{0.05cm}t} . $$

{{GraueBox|TEXT=
$\text{Example 1:}$ 
The constellation given here results, for example, in the [[Modulationsverfahren/Zweiseitenband-Amplitudenmodulation#AM-Signale_und_-Spektren_bei_harmonischem_Eingangssignal|double sideband amplitude modulation]] of the message signal $x_{\rm N}(t) = A_{\rm N}\cdot \cos\left(2\pi f_{\rm N}\cdot t- \varphi_{\rm N}\right)$ with the carrier signal $x_{\rm T}(t) = A_{\rm T}\cdot \cos\left(2\pi f_{\rm T}\cdot t - \varphi_{\rm T}\right)$. This is discussed frequently in the exercises.

[[File:Zeigerdiagramm_abzug4.png|center|frame|Spectrum $X_{\rm TP}(f)$ of the equivalent low–pass signal for different phase constellations |class=fit]]

There are some limitations to the program parameters in this approach:
* The frequencies are always $f\hspace{0.05cm}'_{\rm O} = f_{\rm N}$ and $f\hspace{0.05cm}'_{\rm U} = -f_{\rm N}$.
*Without distortion, the amplitude of the sidebands is $A_{\rm O}= A_{\rm U}= A_{\rm N}/2$.
*The respective phase relationships can be seen in the following graphic.

}}

===Representation of the Equivalent Low-Pass Signal by Magnitude and Phase===

The generally complex equivalent low-pass signal
:$$x_{\rm TP}(t) = a(t) \cdot {\rm e}^{ {\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm} \phi(t) }$$
can be split into a magnitude function $a(t)$ and a phase function $\phi(t)$ according to the equation given here, where:
:$$a(t) = \vert x_{\rm TP}(t)\vert = \sqrt{ {\rm Re}^2\big [x_{\rm TP}(t)\big ] + {\rm Im}^2\big [x_{\rm TP}(t)\big ] }\hspace{0.05cm},$$
:$$\phi(t) = \text{arc }x_{\rm TP}(t) = \arctan \frac{{\rm Im}\big [x_{\rm TP}(t)\big ]}{{\rm Re}\big [x_{\rm TP}(t)\big ]}.$$

The reason for this is that a bandpass–signal $x(t)$ is usually described by the equivalent low-pass signal $x_{\rm TP}(t)$ that the functions $a(t)$ and $\phi(t)$ are interpretable in both representations:
*The amount $a(t)$ of the equivalent low-pass signal $x_{\rm TP}(t)$ indicates the (time-dependent) envelope of $x(t)$.
*The phase $\phi(t)$ of $x_{\rm TP}(t)$ denotes the location of the zero crossings of $x(t)$, where:
:–   For $\phi(t)>0$ the zero crossing is earlier than its nominal position   ⇒   the signal is leading here.
:–  When $\phi(t)<0$, the zero crossing is later than its target position   ⇒   the signal is trailing here.

{{GraueBox|TEXT=
$\text{Example 2:}$ 
The graph is intended to illustrate this relationship, assuming $A_{\rm U} > A_{\rm O}$   ⇒   the green pointer (for the lower sideband) is longer than the blue pointer (upper sideband). This is a snapshot at time $t_0$:

[[File:Ortskurve_3_neu.png|center|frame|bandpass spectrum $X(f)$ |class=fit]]

*For these system parameters, the top of the pointer cluster $x_{\rm TP}(t)$ – that is, the geometric sum of red, blue and green pointers – on an ellipse.
* The amount $a(t_0) = \vert x_{\rm TP}(t_0) \vert$ is drawn in black in the left-hand diagram and the phase value $\phi(t_0) = \text{arc }x_{\rm TP}(t_0) > 0$ is indicated in brown color.
*In the graph on the right, the amount $a(t_0) = \vert x_{\rm TP}(t_0) \vert$ of the equivalent low-pass signal indicates the envelope of the physical signal $x(t)$.
* At $\phi(t) \equiv 0$, all zero crossings of $x(t)$ would occur at equidistant intervals. Because of $\phi(t_0) > 0$, the signal is leading at the time $t_0$, that is: the zero crossings come earlier than the grid dictates. }}

==Exercises==
[[File:Exercises_verzerrungen.png|right]]
*First select the task number.
*A task description is displayed.
*Parameter values are adjusted.
*Solution after pressing &bdquo;Hide solition”.

The number &bdquo;0” will reset the program and output a text with the further explanation of the applet.

In the following, $\rm Green$ denotes the lower sideband   ⇒   $\big (A_{\rm U}, f_{\rm U}, \varphi_{\rm U}\big )$,  
$\rm Red$ the carrier   ⇒   $\big (A_{\rm T}, f_{\rm T}, \varphi_{\rm T}\big )$ and
$\rm Blue$ the upper sideband   ⇒   $\big (A_{\rm O}, f_{\rm O}, \varphi_{\rm O}\big )$.

{{BlaueBox|TEXT=
'''(1)'''   Let   $\text{Red:} \hspace{0.15cm} A_{\rm T} = 1\ \text{V}, f_{\rm T} = 100 \ \text{kHz}, \varphi_{\rm T} = 0^\circ$,   $\text{Green:} \hspace{0.15cm} A_{\rm U} = 0.4 \text{V}, \ f_{\rm U} = 80 \ \text{kHz}, \varphi_{\rm U} = -90^\circ$,   $\text{Blue:} \hspace{0.15cm} A_{\rm O} = 0.4\ \text{V}, f_{\rm O} = 120 \ \text{kHz}, \varphi_{\rm O} = 90^\circ$.

:Consider and interpret the equivalent low-pass signal $x_{\rm TP}(t)$ and the physical signal $x(t)$. Which period $T_0$ recognizable?}}

:: The equivalent low-pass signal $x_{\rm TP}(t)$ takes from $x_{\rm TP}(t=0)=1\ \text{V}$ on the real axis values between $0.2\ \text{V}$ and $1.8\ \text{V}$   ⇒   phase $\phi(t) \equiv 0$.  The amount $|x_{\rm TP}(t)|$ indicates the envelope $a(t)$ of the physical signal $x(t)$. It holds $A_{\rm N} = 0.8\ \text{V}$ and $f_{\rm N} = 20\ \text{kHz}$:   $a(t) = A_{\rm T}+ A_{\rm N} \cdot \sin(2\pi\cdot f_{\rm N} \cdot t)$.  Both $x_{\rm TP}(t)$ and $x(t)$ are periodically with the period $T_0 = 1/f_{\rm N} = 50\ \rm µ s$.

{{BlaueBox|TEXT=
'''(2)'''   How do the ratios change to '''(1)''' with $f_{\rm U} = 99 \ \text{kHz}$ and $f_{\rm O} = 101 \ \text{kHz}$ ? How could $x(t)$ have arisen?}}

:: For the envelope $a(t)$ of the signal $x(t)$ we still have $a(t) = A_{\rm T}+ A_{\rm N} \cdot \sin(2\pi\cdot f_{\rm N} \cdot t)$, but now $f_{\rm N} = 1\ \text{kHz}$. Even though it can not be recognized:  $x_{\rm TP}(t)$ and $x(t)$ are still periodically:   $T_0 = 1\ \rm ms$. Example: Double sideband Amplitude modulation '''(DSB–AM)''' of a sine signal with cosine carrier.

{{BlaueBox|TEXT=
'''(3)'''   Which settings have to be changed from '''(2)''' in order to arrive at the DSB–AM of a cosine signal with sine carrier. What changes over '''(2)'''?}}

::The carrier phase must be changed to $\varphi_{\rm T} = 90^\circ$   ⇒   sine carrier. Similarly, $\varphi_{\rm O} =\varphi_{\rm U} =\varphi_{\rm T} = 90^\circ$ must be set   ⇒   cosinusoidal message  The locus now lies on the imaginary axis  ⇒   $\phi(t) \equiv -90^\circ$. At the beginning $x_{\rm TP}(t=0)= - {\rm j} \cdot 1.8 \ \text{V}$.

{{BlaueBox|TEXT=
'''(4)'''   Now let   $\text{Red:} \hspace{0.15cm} A_{\rm T} = 1\ \text{V}, \ f_{\rm T} = 100 \ \text{kHz}, \ \varphi_{\rm T} = 0^\circ$,   $\text{Green:} \hspace{0.15cm} A_{\rm U} = 0.4 \text{V}, \ f_{\rm U} = 80 \ \text{kHz}, \ \varphi_{\rm U} = 0^\circ$,   $\text{Blue:} \hspace{0.15cm} A_{\rm O} = 0.4\ \text{V}, \ f_{\rm O} = 120 \ \text{kHz}, \ \varphi_{\rm O} = 0^\circ$.

:What are the characteristics of this system &bdquo;DSB–AM, where the message signal and carrier are respectively cosinusoidal”? What is the degree of modulation $m$?}}

:: The equivalent low-pass signal $x_{\rm TP}(t)$ takes from $x_{\rm TP}(t=0)=1.8\ \text{V}$ on the real axis values between $0.2\ \text{V}$ and $1.8\ \text{V}$   ⇒   phase $\phi(t) \equiv 0$.  Except for the start state $x_{\rm TP}(t=0)$ same behavior as at the setting '''(1)'''. The degree of modulation is $m = 0.8$.

{{BlaueBox|TEXT=
'''(5)'''   The parameters are still valid according to '''(4)''' with the exception of $A_{\rm T}= 0.6 \text{V}$. What is the degree of modulation $m$? What are the consequences?}}

:: There is now a DSB–AM with modulation degree $m = 1.333$. For $m > 1$, the simpler [[Modulationsverfahren/Hüllkurvendemodulation|envelope demodulation]] is not applicable, since the phase function $\phi(t) \in \{ 0, \ \pm 180^\circ\}$ is no more constant and the envelope $a(t)$ no more matches the message signal. Rather, the complex [[Modulationsverfahren/Synchrondemodulation|synchronous demodulation]] must be used. Envelope detection would produce nonlinear distortions.

{{BlaueBox|TEXT=
'''(6)'''   The parameters are still valid according to '''(4)''' or '''(5)''' with the exception from $A_{\rm T}= 0$ on   ⇒   $m \to \infty$. Which modulation method is described in this way?}}

::It is a '''DSB–AM without carrier''' and a synchronous demodulation is required. The equivalent low-pass signal $x_{\rm TP}(t)$ is on the real axis, but not only in the right half-plane. Thus, the phase function $\phi(t) \in \{ 0, \ \pm 180^\circ\}$, also applies here, which means that envelope demodulation is not applicable.

{{BlaueBox|TEXT=
'''(7)'''   Now let   $\text{Red:} \hspace{0.15cm} A_{\rm T} = 1\ \text{V}, f_{\rm T} = 100 \ \text{kHz}, \varphi_{\rm T} = 0^\circ$,   $\text{Green:} \hspace{0.15cm} A_{\rm U} = 0, \ f_{\rm U} = 80 \ \text{kHz}, \varphi_{\rm U} = -90^\circ$,   $\text{Blue:} \hspace{0.15cm} A_{\rm O} = 0.8\ \text{V}, f_{\rm O} = 120 \ \text{kHz}, \varphi_{\rm O} = 90^\circ$.

:Which constellation is described here? Which characteristics of this procedure can be recognized from the graphic?}}

::It is a [[Modulationsverfahren/Einseitenbandmodulation|single-sideband modulation]] '''(SSB–AM)''', more specifically an '''OSB–AM''': the red carrier is fixed, the green pointer missing and the blue pointer (OSB) turns counterclockwise. The degree of modulation is $\mu = 0.8$ (in the case of SSB we denote the degree of modulation with $\mu$ instead of $m$). The carrier signal is cosinusoidal and the message signal sinusoidal. The locus is a circle. $x_{\rm TP}(t)$ moves in a mathematically positive direction. Because of $\phi(t) \ne \text{const.}$ the envelope demodulation is not applicable here:  This can be seen by the fact that the envelope $a(t)$ is not cosinusoidal. Rather, the lower half-wave is sharper than the upper one   ⇒   strong linear distortions.

{{BlaueBox|TEXT=
'''(8)'''   The parameters are still valid according to '''(7)''' with the exception of $A_{\rm O}= 0$ and $A_{\rm U}= 0.8 \text{ V}$. What differences arise opposite '''(7)'''?}}

::Now it is a '''LSB–AM''': The red carrier is fixed, the blue pointer is missing and the green pointer (LSB) rotates clockwise. All other statements of '''(7)''' apply here as well.

{{BlaueBox|TEXT=
'''(9)'''   The parameters according to '''(7)''' are still valid with the exception of $A_{\rm O} = 0.2 \text{ V} \ne A_{\rm U} = 0.4 \text{ V} $. What are the differences from '''(7)'''?}}

::The locus $x_{\rm TP}(t)$ is not a horizontal straight line, but an ellipse with the real part between $0.4 \text{ V}$ and $1.6 \text{ V}$ and the imaginary part in the range $\pm 0.2 \text{ V}$. Because of $\phi(t) \ne \text{const.}$ , Envelope demodulation would lead to non-linear distortions here too. The constellation simulated here describes the situation of '''(4)''', namely a DSB–AM with modulation degree $m = 0.8$, where the upper sideband is reduced to $50\%$ due to channel attenuation.

==Applet Manual==
[[File:Ortskurve_abzug3.png|right]]
* The red parameters $(A_{\rm T}, \ f_{\rm T}, \ \varphi_{\rm T})$ and the red pointer mark the ''Carrier''. (German: '''T'''räger).
* The green parameters $(A_{\rm U}, \ f_{\rm U} < f_{\rm T}, \ \varphi_{\rm U})$ mark the ''Lower sideband''. (German: '''U'''ntere Seitenband).
* The blue parameters $(A_{\rm O}, \ f_{\rm O} > f_{\rm T}, \ \varphi_{\rm O})$ mark the '' Upper sideband''. (German: '''O'''bere Seitenband).
* The red pointer does not turn.
* The green pointer rotates in a mathematically negative direction (clockwise).
* The blue pointer turns counterclockwise.

Meaning of the letters in the adjacent graphic:

    '''(A)'''     Plot of the equivalent low-pass Signal $x_{\rm TP}(t)$

    '''(B)'''     Plot of the physical signal $x(t)$

    '''(C)'''     Parameter input via slider:   amplitudes, frequencies, phase values

    '''(D)'''     Control elements:   Start – Step – Pause/Continue – Reset

    '''(E)'''     Speed of animation:   &bdquo;Speed”   ⇒   Values: 1, 2 oder 3

    '''(F)'''     &bdquo;Trace”   ⇒   On or Off, trace of equivalent low-pass Signal   $x_{\rm TP}(t)$

    '''(G)'''     Numerical output:   time $t$, the signal values  ${\rm Re}[x_{\rm TP}(t)]$  and  ${\rm Im}[x_{\rm TP}(t)]$,

$\text{}\hspace{4.2cm}$   envelope $a(t) = |x_{\rm TP}(t)|$  and  phase $\phi(t) = {\rm arc} \ x_{\rm TP}(t)$

    '''(H)'''     Variations for the graphical representation

$\hspace{1.5cm}$Zoom–Functions &bdquo;$+$” (Enlarge), &bdquo;$-$” (Decrease) and $\rm o$ (Reset to default)

$\hspace{1.5cm}$Move with &bdquo;$\leftarrow$” (Section to the left, ordinate to the right), &bdquo;$\uparrow$” &bdquo;$\downarrow$” &bdquo;$\rightarrow$”

    '''(I)'''     Experiment section:   Task selection and task

    '''(J)'''     Experiment section:  solution
 

==About the Authors==
This interactive calculation was designed and realized at the [http://www.lnt.ei.tum.de/startseite Lehrstuhl für Nachrichtentechnik] of the [https://www.tum.de/ Technical University of Munich] .
*The original version was created in 2005 by [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Studierende#Ji_Li_.28Bachelorarbeit_EI_2003.2C_Diplomarbeit_EI_2005.29|Ji Li]] as part of her Diploma thesis using &bdquo;FlashMX–Actionscript” (Supervisor: [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Mitarbeiter_und_Dozenten#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28am_LNT_seit_1974.29|Günter Söder]]).
*In 2018 this Applet was redesigned and updated to &bdquo;HTML5” by [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Studierende#Xiaohan_Liu_.28Bachelorarbeit_2018.29|Xiaohan Liu]] as part of her Bachelor's thesis (Supervisor: [[Biografien_und_Bibliografien/Beteiligte_der_Professur_Leitungsgebundene_%C3%9Cbertragungstechnik#Tasn.C3.A1d_Kernetzky.2C_M.Sc._.28bei_L.C3.9CT_seit_2014.29|Tasnád Kernetzky]]).

==Once again: Open Applet in new Tab==

{{LntAppletLinkEn|physAnSignal}}

[[Category:Applets|^Verzerrungen^]]

LNTwww:Applets

2018-08-28T14:09:41Z

Xiaohan:

{{BlaueBox|TEXT=
Derzeit sind erst sechs interaktive Applets in deutscher Sprache und drei in englischer Sprache auf HTML 5 umgesetzt.
*Die im letzten Menüpunkt $\text{alte Interaktionsmodule (SWF)}$ zusammengefassten Anwendungen müssen noch konvertiert werden.
*In der jetzigen Form sind diese für viele Systeme (Smartphones, Apple, Linux, ...) ungeeignet.
*Weitere Hinweise hierzu im grauen Kasten beim letzten Menüpunkt.}}


{{Collapse|ID=signald|TITEL='''zum Buch &bdquo;Signaldarstellung”'''|TEXT=
* [[Applets:Frequenzgang|Frequenzgang & Impulsantwort]]
* [[Applets:Impulse und Spektren|Impulse & Spektren]]
* [[Applets:Periodendauer|Periodendauer periodischer Signale]]
* [[Applets:Physikalisches_Signal_und_Analytisches_Signal|Physikalisches Signal & Analytisches Signal]]
* [[Applets:Physikalisches_Signal_und_Äquivalentes_TP-Signal|Physikalisches Signal und Äquivalentes TP-Signal]]
}}

{{Collapse|ID=lzs|TITEL='''zum Buch &bdquo;Lineare zeitinvariante Systeme”'''|TEXT=
* [[Applets:Dämpfung_von_Kupferkabeln|Dämpfung von Kupferkabeln]]
* [[Applets:Frequenzgang|Frequenzgang & Impulsantwort]]
*[[Applets:Lineare_Verzerrungen_periodischer Signale|Lineare Verzerrungen periodischer Signale]]
}}

{{Collapse|ID=stosi|TITEL='''zum Buch &bdquo;Stochastische Signaltheorie”'''|TEXT=
* [[Applets:Gaußsche_Fehlerfunktionen|Komplementäre Gaußsche Fehlerfunktionen, '''noch in Bearbeitung''']]
*[[Applets:Gegenüberstellung_Binomial-_und_Poissonverteilung|Binomial– und Poissonverteilung]]
}}

{{Collapse|ID=infot|TITEL='''zum Buch &bdquo;Informationstheorie”'''|TEXT=
*[[Applets:Gegenüberstellung_Binomial-_und_Poissonverteilung|Binomial– und Poissonverteilung]]
}}

{{Collapse|ID=modula|TITEL='''zum Buch &bdquo;Modulationsverfahren”'''|TEXT=
* [[Applets:Besselfunktionen_erster_Art_(neues_Applet)|Besselfunktionen erster Art]]
* [[Applets:Frequenzgang|Frequenzgang & Impulsantwort]]
*[[Applets:Lineare_Verzerrungen_periodischer Signale|Lineare Verzerrungen periodischer Signale]]
}}

{{Collapse|ID=digsig|TITEL='''zum Buch &bdquo;Digitalsignalübertragung”'''|TEXT=
* [[Applets:Dämpfung_von_Kupferkabeln|Dämpfung von Kupferkabeln]]
* [[Applets:Frequenzgang|Frequenzgang & Impulsantwort]]
*[[Applets:Gegenüberstellung_Binomial-_und_Poissonverteilung|Binomial– und Poissonverteilung]]
* [[Applets:Impulse und Spektren|Impulse & Spektren]]
}}

{{Collapse|ID=mobcomm|TITEL='''zum Buch &bdquo;Mobile Kommunikation”'''|TEXT=
* [[Applets:Frequenzgang|Frequenzgang & Impulsantwort]]
}}

{{Collapse|ID=chancod|TITEL='''zum Buch &bdquo;Kanalcodierung”'''|TEXT=
*[[Applets:Gegenüberstellung_Binomial-_und_Poissonverteilung|Binomial– und Poissonverteilung]]
}}

{{Collapse|ID=nachrbeisp|TITEL='''zum Buch &bdquo;Beispiele von Nachrichtensysteme”'''|TEXT=
* [[Applets:Dämpfung_von_Kupferkabeln|Dämpfung von Kupferkabeln]]
}}

{{Collapse|ID=englisch|TITEL='''English versions'''|TEXT=
*[[Applets:Attenuation_of_Copper_Cables|Attenuation of Copper Cables]]
*[[Applets:Bessel_Functions_of_the_First_Kind|Bessel Functions of the First Kind]]
*[[Applets:Binomial_and_Poisson_Distribution_(Applet)|Binomial and Poisson Distribution]]
*[[Applets:Linear_Distortions_of_Periodic_Signals|Linear Distortions of Periodic Signals]]
*[[Applets:Physical_Signal_&_Analytic_Signal|Physical Signal & Analytic Signal]]
*[[Applets:Physical_Signal_%26_Equivalent_Lowpass_Signal|Physical Signal & Equivalent Low-pass Signal]]
}}

{{Collapse|ID=alte|TITEL='''alte Interaktionsmodule (SWF) - noch zu programmieren'''|TEXT=
{{GraueBox|TEXT=
*Die nachfolgend aufgeführten Anwendungen sind in der jetzigen Form für viele Systeme (Smartphones, Apple, Linux, ...) ungeeignet.
*Sie können derzeit noch unter $\rm Windows$ verwendet werden, wenn der $\text{Adobe Flash Player}$ installiert ist.
*Wir weisen Sie allerdings darauf hin, dass es hinsichtlich dieser Installation '''Sicherheitsbedenken''' gibt.
*Wir werden zeitnah die verbliebenen $\text{SWF–Module}$ nach $\text{HTML 5}$ konvertieren.
*Sollten Sie im Bachelorstudiengang (BSEI) an der TU München studieren, so bieten wir Ihnen gerne eine entsprechende IP-Arbeit an.}}

'''zu &bdquo;Signaldarstellung” und weitere Bücher:'''
* [[Applets:Abtastung|Abtastung analoger Signale & Signalrekonstruktion]]
* [[Applets:Diskrete_Fouriertransformation_(Applet)|Diskrete Fouriertransformation]]
* [[Ortskurve_–_Darstellung_des_äquivalenten_Tiefpass-Signals_(Applet)|Ortskurve – Darstellung des äquivalenten Tiefpass-Signals]] (in Bearbeitung)
* [[Applets:Zeigerdiagramm_–_Darstellung_des_analytischen_Signals_(Applet)|Zeigerdiagramm – Darstellung des analytischen Signals]] (in Bearbeitung)
* [[Applets:Graphische_Faltung|Zur Verdeutlichung der grafischen Faltung]]

 '''zu &bdquo;LZI-Systeme” und weitere Bücher:'''
* [[Applets:Kabeldaempfung|Dämpfung von Kupferkabeln]] (bereits konvertiert)
* [[Applets:Bandbegrenzung|Einfluss einer Bandbegrenzung auf Sprache und Musik]]
* [[Applets:Laplace|Kausale Systeme & Laplacetransformation]]
* [[Applets:Verzerrungen|Lineare Verzerrungen periodischer Signale]] (bereits konvertiert)
* [[Applets:Laufzeit|Phasenlaufzeit & Gruppenlaufzeit]]
* [[Applets:Zeitverhalten_von_Kupferkabeln|Zeitverhalten von Kupferkabeln]]
* [[Applets:Graphische_Faltung|Zur Verdeutlichung der grafischen Faltung]]

 '''zu &bdquo;Stochastische Signale” und weitere Bücher:'''
* [[Applets:Digitalfilter|Digitales Filter]]
* [[Applets:Markovketten|Ereigniswahrscheinlichkeit einer Markovkette erster Ordnung]]
* [[Applets:Poisson_poisson|Ereigniswahrscheinlichkeiten der Poissonverteilung]] (bereits konvertiert)
* [[Applets:Binom_Binom|Ereigniswahrscheinlichkeiten der Binomialverteilung]] (bereits konvertiert)
* [[Applets:Gegenüberestellung_von_Binomialverteilung_vs._Poissonverteilung|Gegenüberstellung Binomialverteilung vs. Poissonverteilung]] (bereits konvertiert)
* [[Applets:QFunction|Komplementäre Gaußsche Fehlerfunktionen]] (in Bearbeitung)
* [[Applets:Korrelation|Korrelationskoeffizient & Regressionsgerade]]
* [[Applets:2D_Gauss|WDF und VTF bei Gaußschen 2D-Zufallsgrößen]]
* [[Applets:WDF_VTF|WDF, VTF und Momente spezieller Verteilungen]]
* [[Applets:Matched_Filter|Zur Verdeutlichung des Matched-Filters]]

 '''zu &bdquo;Informationstheorie” und weitere Bücher:'''
* [[Applets:Abtastung|Abtastung analoger Signale & Signalrekonstruktion]]
* [[Applets:Bandbegrenzung|Einfluss einer Bandbegrenzung auf Sprache und Musik]]
* [[Applets:Quellenentropie|Entropien von Nachrichtenquellen]]
* [[Applets:Binom_Binom|Ereigniswahrscheinlichkeiten der Binomialverteilung]] (bereits konvertiert)
* [[Applets:Markovketten|Ereigniswahrscheinlichkeit einer Markovkette erster Ordnung]]
* [[Applets:QFunction|Komplementäre Gaußsche Fehlerfunktionen]] (in Bearbeitung)
* [[Applets:Lempel-Ziv-Welch|Lempel-Ziv-Welch-Algorithmen]]
* [[Applets:Sprachcodecs|Qualität verschiedener Sprachcodecs]]
* [[Applets:Huffman_Shannon_Fano|Shannon-Fano- & Huffman-Codierung]]
*[[Applets:Pseudoternaercodierung|Signale, AKF und LDS der Pseudoternärcodierung ]]
* [[Applets:Transinformation|Transinformation zwischen diskreten Zufallsgrößen]]
* [[Applets:2D_Gauss|WDF und VTF bei Gaußschen 2D-Zufallsgrößen]]
* [[Applets:WDF_VTF|WDF, VTF und Momente spezieller Verteilungen]]

 '''zu &bdquo;Modulationsverfahren” und weitere Bücher:'''
* [[Applets:Abtastung|Abtastung analoger Signale & Signalrekonstruktion]]
* [[Applets:Besselfunktion|Besselfunktion erster Art und n-ter Ordnung]]
* [[Applets:DMT|Discrete Multitone Transmission]]
* [[Applets:Diskrete_Fouriertransformation_(Applet)|Diskrete Fouriertransformation]]
* [[Applets:Synchrondemodulator|Eigenschaften des Synchrondemodulators bei ZSB und ESB]]
* [[Applets:Bandbegrenzung|Einfluss einer Bandbegrenzung auf Sprache und Musik]]
* [[Applets:Frequency_Shift_Keying_%26_Continuous_Phase_Modulation|Frequency Shift Keying & Continuous Phase Modulation]]
* [[Applets:QFunction|Komplementäre Gaußsche Fehlerfunktionen]] (in Bearbeitung)
* [[Applets:Verzerrungen|Lineare Verzerrungen periodischer Signale]] (bereits konvertiert)
* [[Applets:OFDM|OFDM - Spektrum & Signale]]
* [[Ortskurve_–_Darstellung_des_äquivalenten_Tiefpass-Signals_(Applet)|Ortskurve – Darstellung des äquivalenten Tiefpass-Signals]] (in Bearbeitung)
* [[Applets:OVSF-Codes|OVSF-Codes]]
* [[Applets:QPSK|QPSK und Offset–QPSK]]
* [[Applets:DMT-Prinzip|Prinzip der Discrete Multitone Transmission]]
* [[Applets:Prinzip_der_Quadratur-Amplitudenmodulation_(Applet)|Prinzip der Quadratur–Amplitudenmodulation]]
* [[Applets:Zeigerdiagramm_–_Darstellung_des_analytischen_Signals_(Applet)|Zeigerdiagramm – Darstellung des analytischen Signals]] (in Bearbeitung)
* [[Applets:Walsh|Zur Erzeugung von Walsh-Funktionen]]
* [[Applets:Rauschen|Rauschen bei AM und WM - '''eigentlich ein Lernvideo''']]

 '''zu &bdquo;Digitalsignalübertragung” und weitere Bücher:'''
* [[Applets:Abtastung|Abtastung analoger Signale & Signalrekonstruktion]]
* [[Applets:Augendiagramm|Augendiagramm & Augenöffnung]]
* [[Applets:Kabeldaempfung|Dämpfung von Kupferkabeln]] (bereits konvertiert)
* [[Applets:Bandbegrenzung|Einfluss einer Bandbegrenzung auf Sprache und Musik]]
*[[Applets:Entscheidungsrückkopplung|Entscheidungsrückkopplung]]
* [[Applets:Gegenüberestellung_von_Binomialverteilung_vs._Poissonverteilung|Gegenüberstellung Binomialverteilung vs. Poissonverteilung]] (bereits konvertiert)
* [[Applets:Gram-Schmidt-Verfahren|Gram-Schmidt-Verfahren]]
* [[Applets:QFunction|Komplementäre Gaußsche Fehlerfunktionen]] (in Bearbeitung)
* [[Applets:Lineare_Nyquistentzerrung|Lineare Nyquistentzerrung]]
* [[Applets:MPSK_%26_Union-Bound(Applet)|Mehrstufige PSK & Union Bound]]
* [[Applets:On-Off-Keying|Nichtkohärentes On-Off-Keying]]
*[[Applets:Entscheidungsregionen|Optimale Entscheidungsregionen]]
* [[Applets:Prinzip_der_Quadratur-Amplitudenmodulation_(Applet)|Prinzip der Quadratur–Amplitudenmodulation]]
*[[Applets:4B3T-Codes|Prinzip der 4B3T-Codierung]]
*[[Applets:Pseudoternaercodierung|Signale, AKF und LDS der Pseudoternärcodierung ]]
* [[Applets:Fehlerwahrscheinlichkeit|Symbolfehlerwahrscheinlichkeit von Digitalsystemen]]
*[[Applets:Viterbi|Viterbi-Empfänger für einen Vorläufer]]
* [[Applets:2D_Gauss|WDF und VTF bei Gaußschen 2D-Zufallsgrößen]]
* [[Applets:Zeitverhalten_von_Kupferkabeln|Zeitverhalten von Kupferkabeln]]
* [[Applets:Graphische_Faltung|Zur Verdeutlichung der grafischen Faltung]]
* [[Applets:Matched_Filter|Zur Verdeutlichung des Matched-Filters]]
*[[Applets:2D_Laplace|Zweidimensionale Laplaceverteilung]]

 '''zu &bdquo;Mobile Kommunikation” und weitere Bücher:'''
* [[Applets:Frequenzselektivitaet|Auswirkungen des Mehrwegeempfangs]]
* [[Applets:Dopplereffekt|Zur Verdeutlichung des Dopplereffekts]]

 '''zu &bdquo;Beispiele von Nachrichtensystemen” und weitere Bücher:'''
* [[Applets:Sprachcodecs|Qualität verschiedener Sprachcodecs]]

}}

Applets:Physical Signal & Equivalent Lowpass Signal

2018-08-28T09:33:48Z

Xiaohan: /* Applet Manual */

{{LntAppletLinkEn|analPhysSignal}}

==Applet Description==
 
This applet shows the relationship between the physical bandpass signal $x(t)$ and the associated equivalent low-pass signal $x_{\rm TP}(t)$. It is assumed that the bandpass signal $x(t)$ has a frequency-discrete spectrum $X(f)$:
:$$x(t) = x_{\rm U}(t) + x_{\rm T}(t) + x_{\rm O}(t) = A_{\rm U}\cdot \cos\left(2\pi f_{\rm U}\cdot t- \varphi_{\rm U}\right)+A_{\rm T}\cdot \cos\left(2\pi f_{\rm T}\cdot t- \varphi_{\rm T}\right)+A_{\rm O}\cdot \cos\left(2\pi f_{\rm O}\cdot t- \varphi_{\rm O}\right). $$
The physical signal $x(t)$ is thus composed of three harmonic oscillations, a constellation that can be found, for example, in the ''Double-sideband Amplitude Modulation''
*of the message signal $x_{\rm N}(t) = A_{\rm N}\cdot \cos\left(2\pi f_{\rm N}\cdot t- \varphi_{\rm N}\right)$   ⇒   in German:   '''N'''achrichtensignal
*with the carrier signal $x_{\rm T}(t) = A_{\rm T}\cdot \cos\left(2\pi f_{\rm T}\cdot t - \varphi_{\rm T}\right)$   ⇒   in German:   '''T'''rägersignal.

The nomenclature is also adapted to this case:
* $x_{\rm O}(t)$ denotes the &bdquo;upper sideband”   (in German:   '''O'''beres Seitenband) with the amplitude $A_{\rm O}= A_{\rm N}/2$, the frequency $f_{\rm O} = f_{\rm T} + f_{\rm N}$ and the phase $\varphi_{\rm O} = \varphi_{\rm T} + \varphi_{\rm N}$.
*Similarly, for the &bdquo;lower sideband”   (in German:   '''U'''nteres Seitenband) $x_{\rm U}(t)$ with $f_{\rm U} = f_{\rm T} - f_{\rm N}$, $A_{\rm U}= A_{\rm O}$ and $\varphi_{\rm U} = -\varphi_{\rm O}$.

The associated equivalent low-pass–signal is $f_{\rm O}\hspace{0.01cm}' = f_{\rm O}- f_{\rm T} > 0$,   $f_{\rm U}\hspace{0.01cm}' = f_{\rm U}- f_{\rm T} < 0$  and  $f_{\rm T}\hspace{0.01cm}' = 0$:

:$$x_{\rm TP}(t) = x_\text{TP, T}(t) + x_\text{TP, O}(t) + x_\text{TP, U}(t) = A_{\rm T}\cdot {\rm e}^{-{\rm j} \varphi_{\rm T} } \hspace{0.1cm}+ \hspace{0.1cm} A_{\rm O}\cdot {\rm e}^{-{\rm j} \varphi_{\rm O} } \cdot {\rm e}^{ {\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm O}\hspace{0.01cm}'\hspace{0.05cm}\cdot \hspace{0.05cm}t}\hspace{0.1cm}+ \hspace{0.1cm}
A_{\rm U}\cdot {\rm e}^{-{\rm j} \varphi_{\rm U} } \cdot {\rm e}^{ {\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm U}\hspace{0.01cm}'\hspace{0.05cm}\cdot \hspace{0.05cm}t} . $$

[[File:Ortskurve_1.png|right|frame|Equivalent low-pass signal currently $t=0$ for cosinusoidal carrier   ⇒   $\varphi_{\rm T} = 0$]]
The program shows $x_{\rm TP}(t)$ as the vectorial sum of three rotation pointers as a violet dot (see figure for start time $t=0$ and cosinusoidal carrier):

*The (red) pointer of the carrier $x_\text{TP, T}(t)$ with the length $A_{\rm T}$ and the zero phase position $\varphi_{\rm T}=0$ is fixed in the complex plane. So it applies to all times $t$:   $x_{\rm TP}(t)= A_{\rm T}\cdot {\rm e}^{-{\rm j} \varphi_{\rm T} }$.

*The (blue) pointer of the upper sideband $x_\text{TP, O}(t)$ with the length $A_{\rm O}$ and the zero phase position $\varphi_{\rm O}$ rotates at the angular velocity $2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm O}\hspace{0.01cm}'$ in mathematically positive direction (one revolution in time $1/f_{\rm O}\hspace{0.01cm}')$.

*The (green) pointer of the lower sideband $x_{\rm U+}(t)$ with the length $A_{\rm U}$ and the zero phase position $\varphi_{\rm U}$ rotates at the angular velocity $2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm U}\hspace{0.01cm}'$, because of $f_{\rm U}\hspace{0.01cm}'<0$ counterclockwise.

*With $f_{\rm U}\hspace{0.01cm}' = -f_{\rm O}\hspace{0.01cm}'$ the blue and green pointers will spin at the same speed but in different directions. Also, if $A_{\rm O} = A_{\rm U}$ and $\varphi_{\rm U} = -\varphi_{\rm O}$, then $x_{\rm TP}(t)$ moves on a straight line with a incline of $\varphi_{\rm T}$.

''Note:''   In the figure $\varphi_{\rm O} = +30^\circ$. From this follows for the start time $t=0$ the angle of the blue pointer (USB, blue pointer) with respect to the coordinate system:   $\phi_{\rm O} = -\varphi_{\rm O} = -30^\circ$. Likewise, the zero phase position $\varphi_{\rm U} = -30^\circ$ of the lower sideband (LSB, green pointer) follows for the phase angle to be considered in the complex plane:   $\phi_{\rm U} = +30^\circ$.

The temporal process of $x_{\rm TP}(t)$ is also referred to below as '''locus'''. The relationship between $x_{\rm TP}(t)$ and the physical bandpass signal $x(t)$ is given in the section and the associated analytic signal is $x_+(t)$ :

:$$x_{\rm TP}(t) = x_{\rm +}(t)\cdot {\rm e}^{-{\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm}2 \pi \cdot f_{\rm T}\cdot \hspace{0.05cm}t},$$
:$$x_{\rm +}(t) = x_{\rm TP}(t)\cdot {\rm e}^{+{\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm}2 \pi \cdot f_{\rm T}\cdot \hspace{0.05cm}t}.$$

[[Applets:Physikalisches_Signal_%26_Äquivalentes_TP-Signal|'''German description''']]

==Theoretical Background==
 
===Description of Bandpass Signals===
[[File:Zeigerdiagramm_1a.png|right|frame|bandpass spectrum $X(f)$ |class=fit]]
We consider '''bandpass signals''' $x(t)$ with the property that their spectra $X(f)$ are not in the range around the frequency $f=0$, but around a carrier frequency $f_{\rm T}$. In most cases it can also be assumed that the bandwidth is $B \ll f_{\rm T}$.

The figure shows such a bandpass spectrum $X(f)$. Assuming that the associated $x(t)$ is a physical signal and thus real, the spectral function $X(f)$ has a symmetry with respect to the frequency $f = 0$, if $x(t)$ is an even function   ⇒   $x(-t)=x(t)$, $X(f)$ is real and even.

Beside the physical signal $x(t)\ \circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\, \ X(f)$, one can also use the following descriptions of bandpass signals:
*the analytic signal $x_+(t)\ \circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\, \ X_+(f)$, see applet [[Applets:Physikalisches_Signal_%26_Analytisches_Signal|Physikalisches Signal & Analytisches Signal]],
*the equivalent low-pass–signal $x_{\rm TP}(t)\ \circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\, \ X_{\rm TP}(f)$, see next page
 

===Spectral Functions of the Analytic and the Equivalent Low-Pass Signal===

The '''analytic signal''' $x_+(t)$ belonging to the physical signal $x(t)$ is the time function whose spectrum fulfills the following property:
[[File:Ortskurve_2.png|right|frame|spectral functions $X_+(f)$ and $X_{\rm TP}(f)$ |class=fit]]
:$$X_+(f)=\big[1+{\rm sign}(f)\big] \cdot X(f) = \left\{ {2 \cdot
X(f) \; \hspace{0.2cm}\rm for\hspace{0.2cm} {\it f} > 0, \atop {\,\,\,\, \rm 0 \; \hspace{0.9cm}\rm for\hspace{0.2cm} {\it f} < 0.} }\right.$$

The ''Signum function'' is for positive values of $f$ equal to $+1$ and for negative $f$ values equal to $-1$.
*The (double-sided) limit returns $\sign(0) = 0$.
*The index „+” should make it clear that $X_+(f)$ only has parts at positive frequencies.

From the graph you can see the calculation rule for $X_+(f)$: The actual BP spectrum $X(f)$ becomes
*doubled at the positive frequencies, and
*set to zero at the negative frequencies.

Due to the asymmetry of $X_+(f)$ with respect to the frequency $f = 0$, it can already be said that the time function $x_+(t)$ except for a trivial special case $x_+(t)= 0 \ \ \circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\,\ \ X_+(f)= 0$ is always complex.

The spectrum $X_{\rm TP}(f)$ of the equivalent low-pass signal is obtained by shifting $X_+(f)$ to the left by the carrier frequency $f_{\rm T}$:
:$$X_{\rm TP}(f)= X_+(f+f_{\rm T}).$$

In the time domain this operation corresponds to the multiplication of $x_{\rm +}(t)$ with the complex exponential function with negative exponent:
:$$x_{\rm TP}(t) = x_{\rm +}(t)\cdot {\rm e}^{-{\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm}2 \pi \cdot f_{\rm T}\cdot \hspace{0.05cm}t}.$$

It can be seen that $x_{\rm TP}(t)$ is generally complex. But if $X_+(f)$ is symmetric about the carrier frequency $f_{\rm T}$, $X_{\rm TP}(f)$ is symmetric about the frequency $f=0$ and there is accordingly real time function $x_{\rm TP}(t)$.

===$x_{\rm TP}(t)$–Representation of a Sum of Three Harmonic Oscillations===

In our applet, we always assume a set of three rotating pointers. The physical signal is:
:$$x(t) = x_{\rm U}(t) + x_{\rm T}(t) + x_{\rm O}(t) = A_{\rm U}\cdot \cos\left(2\pi f_{\rm U}\cdot t- \varphi_{\rm U}\right)+A_{\rm T}\cdot \cos\left(2\pi f_{\rm T}\cdot t- \varphi_{\rm T}\right)+A_{\rm O}\cdot \cos\left(2\pi f_{\rm O}\cdot t- \varphi_{\rm O}\right). $$
* Each of the three harmonic oscillations $x_{\rm T}(t)$, $x_{\rm U}(t)$ and $x_{\rm O}(t)$ is represented by an amplitude $(A)$, a frequency $(f)$ and a phase value $(\varphi)$.
*The indices are based on the modulation method [[Modulationsverfahren/Zweiseitenband-Amplitudenmodulation|double sideband amplitude modulation]]. &bdquo;T” stands for &bdquo;carrier”, &bdquo;U” for &bdquo;lower sideband” and &bdquo;O” for &bdquo;upper Sideband”. Similarly, $f_{\rm U} < f_{\rm T}$ and $f_{\rm O} > f_{\rm T}$. There are no restrictions for the amplitudes and phases.

The associated equivalent low-pass signal is with $f_{\rm O}\hspace{0.01cm}' = f_{\rm O}- f_{\rm T} > 0$,   $f_{\rm U}\hspace{0.01cm}' = f_{\rm U}- f_{\rm T} < 0$  and  $f_{\rm T}\hspace{0.01cm}' = 0$:

:$$x_{\rm TP}(t) = x_\text{TP, T}(t) + x_\text{TP, O}(t) + x_\text{TP, U}(t) = A_{\rm T}\cdot {\rm e}^{-{\rm j} \varphi_{\rm T} } \hspace{0.1cm}+ \hspace{0.1cm} A_{\rm O}\cdot {\rm e}^{-{\rm j} \varphi_{\rm O} } \cdot {\rm e}^{ {\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm O}\hspace{0.01cm}'\hspace{0.05cm}\cdot \hspace{0.05cm}t}\hspace{0.1cm}+ \hspace{0.1cm}
A_{\rm U}\cdot {\rm e}^{-{\rm j} \varphi_{\rm U} } \cdot {\rm e}^{ {\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm U}\hspace{0.01cm}'\hspace{0.05cm}\cdot \hspace{0.05cm}t} . $$

{{GraueBox|TEXT=
$\text{Example 1:}$ 
The constellation given here results, for example, in the [[Modulationsverfahren/Zweiseitenband-Amplitudenmodulation#AM-Signale_und_-Spektren_bei_harmonischem_Eingangssignal|double sideband amplitude modulation]] of the message signal $x_{\rm N}(t) = A_{\rm N}\cdot \cos\left(2\pi f_{\rm N}\cdot t- \varphi_{\rm N}\right)$ with the carrier signal $x_{\rm T}(t) = A_{\rm T}\cdot \cos\left(2\pi f_{\rm T}\cdot t - \varphi_{\rm T}\right)$. This is discussed frequently in the exercises.

[[File:Ortskurve_4.png|center|frame|Spectrum $X_{\rm TP}(f)$ of the equivalent low–pass signal for different phase constellations |class=fit]]

There are some limitations to the program parameters in this approach:
* The frequencies are always $f\hspace{0.05cm}'_{\rm O} = f_{\rm N}$ and $f\hspace{0.05cm}'_{\rm U} = -f_{\rm N}$.
*Without distortion, the amplitude of the sidebands is $A_{\rm O}= A_{\rm U}= A_{\rm N}/2$.
*The respective phase relationships can be seen in the following graphic.

}}

===Representation of the Equivalent Low-Pass Signal by Magnitude and Phase===

The generally complex equivalent low-pass signal
:$$x_{\rm TP}(t) = a(t) \cdot {\rm e}^{ {\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm} \phi(t) }$$
can be split into a magnitude function $a(t)$ and a phase function $\phi(t)$ according to the equation given here, where:
:$$a(t) = \vert x_{\rm TP}(t)\vert = \sqrt{ {\rm Re}^2\big [x_{\rm TP}(t)\big ] + {\rm Im}^2\big [x_{\rm TP}(t)\big ] }\hspace{0.05cm},$$
:$$\phi(t) = \text{arc }x_{\rm TP}(t) = \arctan \frac{{\rm Im}\big [x_{\rm TP}(t)\big ]}{{\rm Re}\big [x_{\rm TP}(t)\big ]}.$$

The reason for this is that a bandpass–signal $x(t)$ is usually described by the equivalent low-pass signal $x_{\rm TP}(t)$ that the functions $a(t)$ and $\phi(t)$ are interpretable in both representations:
*The amount $a(t)$ of the equivalent low-pass signal $x_{\rm TP}(t)$ indicates the (time-dependent) envelope of $x(t)$.
*The phase $\phi(t)$ of $x_{\rm TP}(t)$ denotes the location of the zero crossings of $x(t)$, where:
:–   For $\phi(t)>0$ the zero crossing is earlier than its nominal position   ⇒   the signal is leading here.
:–  When $\phi(t)<0$, the zero crossing is later than its target position   ⇒   the signal is trailing here.

{{GraueBox|TEXT=
$\text{Example 2:}$ 
The graph is intended to illustrate this relationship, assuming $A_{\rm U} > A_{\rm O}$   ⇒   the green pointer (for the lower sideband) is longer than the blue pointer (upper sideband). This is a snapshot at time $t_0$:

[[File:Ortskurve_3_neu.png|center|frame|bandpass spectrum $X(f)$ |class=fit]]

*For these system parameters, the top of the pointer cluster $x_{\rm TP}(t)$ – that is, the geometric sum of red, blue and green pointers – on an ellipse.
* The amount $a(t_0) = \vert x_{\rm TP}(t_0) \vert$ is drawn in black in the left-hand diagram and the phase value $\phi(t_0) = \text{arc }x_{\rm TP}(t_0) > 0$ is indicated in brown color.
*In the graph on the right, the amount $a(t_0) = \vert x_{\rm TP}(t_0) \vert$ of the equivalent low-pass signal indicates the envelope of the physical signal $x(t)$.
* At $\phi(t) \equiv 0$, all zero crossings of $x(t)$ would occur at equidistant intervals. Because of $\phi(t_0) > 0$, the signal is leading at the time $t_0$, that is: the zero crossings come earlier than the grid dictates. }}

==Exercises==
[[File:Exercises_verzerrungen.png|right]]
*First select the task number.
*A task description is displayed.
*Parameter values are adjusted.
*Solution after pressing &bdquo;Hide solition”.

The number &bdquo;0” will reset the program and output a text with the further explanation of the applet.

In the following, $\rm Green$ denotes the lower sideband   ⇒   $\big (A_{\rm U}, f_{\rm U}, \varphi_{\rm U}\big )$,  
$\rm Red$ the carrier   ⇒   $\big (A_{\rm T}, f_{\rm T}, \varphi_{\rm T}\big )$ and
$\rm Blue$ the upper sideband   ⇒   $\big (A_{\rm O}, f_{\rm O}, \varphi_{\rm O}\big )$.

{{BlaueBox|TEXT=
'''(1)'''   Let   $\text{Red:} \hspace{0.15cm} A_{\rm T} = 1\ \text{V}, f_{\rm T} = 100 \ \text{kHz}, \varphi_{\rm T} = 0^\circ$,   $\text{Green:} \hspace{0.15cm} A_{\rm U} = 0.4 \text{V}, \ f_{\rm U} = 80 \ \text{kHz}, \varphi_{\rm U} = -90^\circ$,   $\text{Blue:} \hspace{0.15cm} A_{\rm O} = 0.4\ \text{V}, f_{\rm O} = 120 \ \text{kHz}, \varphi_{\rm O} = 90^\circ$.

:Consider and interpret the equivalent low-pass signal $x_{\rm TP}(t)$ and the physical signal $x(t)$. Which period $T_0$ recognizable?}}

:: The equivalent low-pass signal $x_{\rm TP}(t)$ takes from $x_{\rm TP}(t=0)=1\ \text{V}$ on the real axis values between $0.2\ \text{V}$ and $1.8\ \text{V}$   ⇒   phase $\phi(t) \equiv 0$.  The amount $|x_{\rm TP}(t)|$ indicates the envelope $a(t)$ of the physical signal $x(t)$. It holds $A_{\rm N} = 0.8\ \text{V}$ and $f_{\rm N} = 20\ \text{kHz}$:   $a(t) = A_{\rm T}+ A_{\rm N} \cdot \sin(2\pi\cdot f_{\rm N} \cdot t)$.  Both $x_{\rm TP}(t)$ and $x(t)$ are periodically with the period $T_0 = 1/f_{\rm N} = 50\ \rm µ s$.

{{BlaueBox|TEXT=
'''(2)'''   How do the ratios change to '''(1)''' with $f_{\rm U} = 99 \ \text{kHz}$ and $f_{\rm O} = 101 \ \text{kHz}$ ? How could $x(t)$ have arisen?}}

:: For the envelope $a(t)$ of the signal $x(t)$ we still have $a(t) = A_{\rm T}+ A_{\rm N} \cdot \sin(2\pi\cdot f_{\rm N} \cdot t)$, but now $f_{\rm N} = 1\ \text{kHz}$. Even though it can not be recognized:  $x_{\rm TP}(t)$ and $x(t)$ are still periodically:   $T_0 = 1\ \rm ms$. Example: Double sideband Amplitude modulation '''(DSB–AM)''' of a sine signal with cosine carrier.

{{BlaueBox|TEXT=
'''(3)'''   Which settings have to be changed from '''(2)''' in order to arrive at the DSB–AM of a cosine signal with sine carrier. What changes over '''(2)'''?}}

::The carrier phase must be changed to $\varphi_{\rm T} = 90^\circ$   ⇒   sine carrier. Similarly, $\varphi_{\rm O} =\varphi_{\rm U} =\varphi_{\rm T} = 90^\circ$ must be set   ⇒   cosinusoidal message  The locus now lies on the imaginary axis  ⇒   $\phi(t) \equiv -90^\circ$. At the beginning $x_{\rm TP}(t=0)= - {\rm j} \cdot 1.8 \ \text{V}$.

{{BlaueBox|TEXT=
'''(4)'''   Now let   $\text{Red:} \hspace{0.15cm} A_{\rm T} = 1\ \text{V}, \ f_{\rm T} = 100 \ \text{kHz}, \ \varphi_{\rm T} = 0^\circ$,   $\text{Green:} \hspace{0.15cm} A_{\rm U} = 0.4 \text{V}, \ f_{\rm U} = 80 \ \text{kHz}, \ \varphi_{\rm U} = 0^\circ$,   $\text{Blue:} \hspace{0.15cm} A_{\rm O} = 0.4\ \text{V}, \ f_{\rm O} = 120 \ \text{kHz}, \ \varphi_{\rm O} = 0^\circ$.

:What are the characteristics of this system &bdquo;DSB–AM, where the message signal and carrier are respectively cosinusoidal”? What is the degree of modulation $m$?}}

:: The equivalent low-pass signal $x_{\rm TP}(t)$ takes from $x_{\rm TP}(t=0)=1.8\ \text{V}$ on the real axis values between $0.2\ \text{V}$ and $1.8\ \text{V}$   ⇒   phase $\phi(t) \equiv 0$.  Except for the start state $x_{\rm TP}(t=0)$ same behavior as at the setting '''(1)'''. The degree of modulation is $m = 0.8$.

{{BlaueBox|TEXT=
'''(5)'''   The parameters are still valid according to '''(4)''' with the exception of $A_{\rm T}= 0.6 \text{V}$. What is the degree of modulation $m$? What are the consequences?}}

:: There is now a DSB–AM with modulation degree $m = 1.333$. For $m > 1$, the simpler [[Modulationsverfahren/Hüllkurvendemodulation|envelope demodulation]] is not applicable, since the phase function $\phi(t) \in \{ 0, \ \pm 180^\circ\}$ is no more constant and the envelope $a(t)$ no more matches the message signal. Rather, the complex [[Modulationsverfahren/Synchrondemodulation|synchronous demodulation]] must be used. Envelope detection would produce nonlinear distortions.

{{BlaueBox|TEXT=
'''(6)'''   The parameters are still valid according to '''(4)''' or '''(5)''' with the exception from $A_{\rm T}= 0$ on   ⇒   $m \to \infty$. Which modulation method is described in this way?}}

::It is a '''DSB–AM without carrier''' and a synchronous demodulation is required. The equivalent low-pass signal $x_{\rm TP}(t)$ is on the real axis, but not only in the right half-plane. Thus, the phase function $\phi(t) \in \{ 0, \ \pm 180^\circ\}$, also applies here, which means that envelope demodulation is not applicable.

{{BlaueBox|TEXT=
'''(7)'''   Now let   $\text{Red:} \hspace{0.15cm} A_{\rm T} = 1\ \text{V}, f_{\rm T} = 100 \ \text{kHz}, \varphi_{\rm T} = 0^\circ$,   $\text{Green:} \hspace{0.15cm} A_{\rm U} = 0, \ f_{\rm U} = 80 \ \text{kHz}, \varphi_{\rm U} = -90^\circ$,   $\text{Blue:} \hspace{0.15cm} A_{\rm O} = 0.8\ \text{V}, f_{\rm O} = 120 \ \text{kHz}, \varphi_{\rm O} = 90^\circ$.

:Which constellation is described here? Which characteristics of this procedure can be recognized from the graphic?}}

::It is a [[Modulationsverfahren/Einseitenbandmodulation|single-sideband modulation]] '''(SSB–AM)''', more specifically an '''OSB–AM''': the red carrier is fixed, the green pointer missing and the blue pointer (OSB) turns counterclockwise. The degree of modulation is $\mu = 0.8$ (in the case of SSB we denote the degree of modulation with $\mu$ instead of $m$). The carrier signal is cosinusoidal and the message signal sinusoidal. The locus is a circle. $x_{\rm TP}(t)$ moves in a mathematically positive direction. Because of $\phi(t) \ne \text{const.}$ the envelope demodulation is not applicable here:  This can be seen by the fact that the envelope $a(t)$ is not cosinusoidal. Rather, the lower half-wave is sharper than the upper one   ⇒   strong linear distortions.

{{BlaueBox|TEXT=
'''(8)'''   The parameters are still valid according to '''(7)''' with the exception of $A_{\rm O}= 0$ and $A_{\rm U}= 0.8 \text{ V}$. What differences arise opposite '''(7)'''?}}

::Now it is a '''LSB–AM''': The red carrier is fixed, the blue pointer is missing and the green pointer (LSB) rotates clockwise. All other statements of '''(7)''' apply here as well.

{{BlaueBox|TEXT=
'''(9)'''   The parameters according to '''(7)''' are still valid with the exception of $A_{\rm O} = 0.2 \text{ V} \ne A_{\rm U} = 0.4 \text{ V} $. What are the differences from '''(7)'''?}}

::The locus $x_{\rm TP}(t)$ is not a horizontal straight line, but an ellipse with the real part between $0.4 \text{ V}$ and $1.6 \text{ V}$ and the imaginary part in the range $\pm 0.2 \text{ V}$. Because of $\phi(t) \ne \text{const.}$ , Envelope demodulation would lead to non-linear distortions here too. The constellation simulated here describes the situation of '''(4)''', namely a DSB–AM with modulation degree $m = 0.8$, where the upper sideband is reduced to $50\%$ due to channel attenuation.

==Applet Manual==
[[File:Ortskurve_abzug3.png|right]]
* The red parameters $(A_{\rm T}, \ f_{\rm T}, \ \varphi_{\rm T})$ and the red pointer mark the ''Carrier''. (German: '''T'''räger).
* The green parameters $(A_{\rm U}, \ f_{\rm U} < f_{\rm T}, \ \varphi_{\rm U})$ mark the ''Lower sideband''. (German: '''U'''ntere Seitenband).
* The blue parameters $(A_{\rm O}, \ f_{\rm O} > f_{\rm T}, \ \varphi_{\rm O})$ mark the '' Upper sideband''. (German: '''O'''bere Seitenband).
* The red pointer does not turn.
* The green pointer rotates in a mathematically negative direction (clockwise).
* The blue pointer turns counterclockwise.

Meaning of the letters in the adjacent graphic:

    '''(A)'''     Plot of the equivalent low-pass Signal $x_{\rm TP}(t)$

    '''(B)'''     Plot of the physical signal $x(t)$

    '''(C)'''     Parameter input via slider:   amplitudes, frequencies, phase values

    '''(D)'''     Control elements:   Start – Step – Pause/Continue – Reset

    '''(E)'''     Speed of animation:   &bdquo;Speed”   ⇒   Values: 1, 2 oder 3

    '''(F)'''     &bdquo;Trace”   ⇒   On or Off, trace of equivalent low-pass Signal   $x_{\rm TP}(t)$

    '''(G)'''     Numerical output:   time $t$, the signal values  ${\rm Re}[x_{\rm TP}(t)]$  and  ${\rm Im}[x_{\rm TP}(t)]$,

$\text{}\hspace{4.2cm}$   envelope $a(t) = |x_{\rm TP}(t)|$  and  phase $\phi(t) = {\rm arc} \ x_{\rm TP}(t)$

    '''(H)'''     Variations for the graphical representation

$\hspace{1.5cm}$Zoom–Functions &bdquo;$+$” (Enlarge), &bdquo;$-$” (Decrease) and $\rm o$ (Reset to default)

$\hspace{1.5cm}$Move with &bdquo;$\leftarrow$” (Section to the left, ordinate to the right), &bdquo;$\uparrow$” &bdquo;$\downarrow$” &bdquo;$\rightarrow$”

    '''(I)'''     Experiment section:   Task selection and task

    '''(J)'''     Experiment section:  solution
 

==About the Authors==
This interactive calculation was designed and realized at the [http://www.lnt.ei.tum.de/startseite Lehrstuhl für Nachrichtentechnik] of the [https://www.tum.de/ Technical University of Munich] .
*The original version was created in 2005 by [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Studierende#Ji_Li_.28Bachelorarbeit_EI_2003.2C_Diplomarbeit_EI_2005.29|Ji Li]] as part of her Diploma thesis using &bdquo;FlashMX–Actionscript” (Supervisor: [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Mitarbeiter_und_Dozenten#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28am_LNT_seit_1974.29|Günter Söder]]).
*In 2018 this Applet was redesigned and updated to &bdquo;HTML5” by [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Studierende#Xiaohan_Liu_.28Bachelorarbeit_2018.29|Xiaohan Liu]] as part of her Bachelor's thesis (Supervisor: [[Biografien_und_Bibliografien/Beteiligte_der_Professur_Leitungsgebundene_%C3%9Cbertragungstechnik#Tasn.C3.A1d_Kernetzky.2C_M.Sc._.28bei_L.C3.9CT_seit_2014.29|Tasnád Kernetzky]]).

==Once again: Open Applet in new Tab==

{{LntAppletLinkEn|physAnSignal}}

[[Category:Applets|^Verzerrungen^]]

Applets:Physical Signal & Equivalent Lowpass Signal

2018-08-28T09:26:22Z

Xiaohan: /* Applet Description */

{{LntAppletLinkEn|analPhysSignal}}

==Applet Description==
 
This applet shows the relationship between the physical bandpass signal $x(t)$ and the associated equivalent low-pass signal $x_{\rm TP}(t)$. It is assumed that the bandpass signal $x(t)$ has a frequency-discrete spectrum $X(f)$:
:$$x(t) = x_{\rm U}(t) + x_{\rm T}(t) + x_{\rm O}(t) = A_{\rm U}\cdot \cos\left(2\pi f_{\rm U}\cdot t- \varphi_{\rm U}\right)+A_{\rm T}\cdot \cos\left(2\pi f_{\rm T}\cdot t- \varphi_{\rm T}\right)+A_{\rm O}\cdot \cos\left(2\pi f_{\rm O}\cdot t- \varphi_{\rm O}\right). $$
The physical signal $x(t)$ is thus composed of three harmonic oscillations, a constellation that can be found, for example, in the ''Double-sideband Amplitude Modulation''
*of the message signal $x_{\rm N}(t) = A_{\rm N}\cdot \cos\left(2\pi f_{\rm N}\cdot t- \varphi_{\rm N}\right)$   ⇒   in German:   '''N'''achrichtensignal
*with the carrier signal $x_{\rm T}(t) = A_{\rm T}\cdot \cos\left(2\pi f_{\rm T}\cdot t - \varphi_{\rm T}\right)$   ⇒   in German:   '''T'''rägersignal.

The nomenclature is also adapted to this case:
* $x_{\rm O}(t)$ denotes the &bdquo;upper sideband”   (in German:   '''O'''beres Seitenband) with the amplitude $A_{\rm O}= A_{\rm N}/2$, the frequency $f_{\rm O} = f_{\rm T} + f_{\rm N}$ and the phase $\varphi_{\rm O} = \varphi_{\rm T} + \varphi_{\rm N}$.
*Similarly, for the &bdquo;lower sideband”   (in German:   '''U'''nteres Seitenband) $x_{\rm U}(t)$ with $f_{\rm U} = f_{\rm T} - f_{\rm N}$, $A_{\rm U}= A_{\rm O}$ and $\varphi_{\rm U} = -\varphi_{\rm O}$.

The associated equivalent low-pass–signal is $f_{\rm O}\hspace{0.01cm}' = f_{\rm O}- f_{\rm T} > 0$,   $f_{\rm U}\hspace{0.01cm}' = f_{\rm U}- f_{\rm T} < 0$  and  $f_{\rm T}\hspace{0.01cm}' = 0$:

:$$x_{\rm TP}(t) = x_\text{TP, T}(t) + x_\text{TP, O}(t) + x_\text{TP, U}(t) = A_{\rm T}\cdot {\rm e}^{-{\rm j} \varphi_{\rm T} } \hspace{0.1cm}+ \hspace{0.1cm} A_{\rm O}\cdot {\rm e}^{-{\rm j} \varphi_{\rm O} } \cdot {\rm e}^{ {\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm O}\hspace{0.01cm}'\hspace{0.05cm}\cdot \hspace{0.05cm}t}\hspace{0.1cm}+ \hspace{0.1cm}
A_{\rm U}\cdot {\rm e}^{-{\rm j} \varphi_{\rm U} } \cdot {\rm e}^{ {\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm U}\hspace{0.01cm}'\hspace{0.05cm}\cdot \hspace{0.05cm}t} . $$

[[File:Ortskurve_1.png|right|frame|Equivalent low-pass signal currently $t=0$ for cosinusoidal carrier   ⇒   $\varphi_{\rm T} = 0$]]
The program shows $x_{\rm TP}(t)$ as the vectorial sum of three rotation pointers as a violet dot (see figure for start time $t=0$ and cosinusoidal carrier):

*The (red) pointer of the carrier $x_\text{TP, T}(t)$ with the length $A_{\rm T}$ and the zero phase position $\varphi_{\rm T}=0$ is fixed in the complex plane. So it applies to all times $t$:   $x_{\rm TP}(t)= A_{\rm T}\cdot {\rm e}^{-{\rm j} \varphi_{\rm T} }$.

*The (blue) pointer of the upper sideband $x_\text{TP, O}(t)$ with the length $A_{\rm O}$ and the zero phase position $\varphi_{\rm O}$ rotates at the angular velocity $2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm O}\hspace{0.01cm}'$ in mathematically positive direction (one revolution in time $1/f_{\rm O}\hspace{0.01cm}')$.

*The (green) pointer of the lower sideband $x_{\rm U+}(t)$ with the length $A_{\rm U}$ and the zero phase position $\varphi_{\rm U}$ rotates at the angular velocity $2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm U}\hspace{0.01cm}'$, because of $f_{\rm U}\hspace{0.01cm}'<0$ counterclockwise.

*With $f_{\rm U}\hspace{0.01cm}' = -f_{\rm O}\hspace{0.01cm}'$ the blue and green pointers will spin at the same speed but in different directions. Also, if $A_{\rm O} = A_{\rm U}$ and $\varphi_{\rm U} = -\varphi_{\rm O}$, then $x_{\rm TP}(t)$ moves on a straight line with a incline of $\varphi_{\rm T}$.

''Note:''   In the figure $\varphi_{\rm O} = +30^\circ$. From this follows for the start time $t=0$ the angle of the blue pointer (USB, blue pointer) with respect to the coordinate system:   $\phi_{\rm O} = -\varphi_{\rm O} = -30^\circ$. Likewise, the zero phase position $\varphi_{\rm U} = -30^\circ$ of the lower sideband (LSB, green pointer) follows for the phase angle to be considered in the complex plane:   $\phi_{\rm U} = +30^\circ$.

The temporal process of $x_{\rm TP}(t)$ is also referred to below as '''locus'''. The relationship between $x_{\rm TP}(t)$ and the physical bandpass signal $x(t)$ is given in the section and the associated analytic signal is $x_+(t)$ :

:$$x_{\rm TP}(t) = x_{\rm +}(t)\cdot {\rm e}^{-{\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm}2 \pi \cdot f_{\rm T}\cdot \hspace{0.05cm}t},$$
:$$x_{\rm +}(t) = x_{\rm TP}(t)\cdot {\rm e}^{+{\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm}2 \pi \cdot f_{\rm T}\cdot \hspace{0.05cm}t}.$$

[[Applets:Physikalisches_Signal_%26_Äquivalentes_TP-Signal|'''German description''']]

==Theoretical Background==
 
===Description of Bandpass Signals===
[[File:Zeigerdiagramm_1a.png|right|frame|bandpass spectrum $X(f)$ |class=fit]]
We consider '''bandpass signals''' $x(t)$ with the property that their spectra $X(f)$ are not in the range around the frequency $f=0$, but around a carrier frequency $f_{\rm T}$. In most cases it can also be assumed that the bandwidth is $B \ll f_{\rm T}$.

The figure shows such a bandpass spectrum $X(f)$. Assuming that the associated $x(t)$ is a physical signal and thus real, the spectral function $X(f)$ has a symmetry with respect to the frequency $f = 0$, if $x(t)$ is an even function   ⇒   $x(-t)=x(t)$, $X(f)$ is real and even.

Beside the physical signal $x(t)\ \circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\, \ X(f)$, one can also use the following descriptions of bandpass signals:
*the analytic signal $x_+(t)\ \circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\, \ X_+(f)$, see applet [[Applets:Physikalisches_Signal_%26_Analytisches_Signal|Physikalisches Signal & Analytisches Signal]],
*the equivalent low-pass–signal $x_{\rm TP}(t)\ \circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\, \ X_{\rm TP}(f)$, see next page
 

===Spectral Functions of the Analytic and the Equivalent Low-Pass Signal===

The '''analytic signal''' $x_+(t)$ belonging to the physical signal $x(t)$ is the time function whose spectrum fulfills the following property:
[[File:Ortskurve_2.png|right|frame|spectral functions $X_+(f)$ and $X_{\rm TP}(f)$ |class=fit]]
:$$X_+(f)=\big[1+{\rm sign}(f)\big] \cdot X(f) = \left\{ {2 \cdot
X(f) \; \hspace{0.2cm}\rm for\hspace{0.2cm} {\it f} > 0, \atop {\,\,\,\, \rm 0 \; \hspace{0.9cm}\rm for\hspace{0.2cm} {\it f} < 0.} }\right.$$

The ''Signum function'' is for positive values of $f$ equal to $+1$ and for negative $f$ values equal to $-1$.
*The (double-sided) limit returns $\sign(0) = 0$.
*The index „+” should make it clear that $X_+(f)$ only has parts at positive frequencies.

From the graph you can see the calculation rule for $X_+(f)$: The actual BP spectrum $X(f)$ becomes
*doubled at the positive frequencies, and
*set to zero at the negative frequencies.

Due to the asymmetry of $X_+(f)$ with respect to the frequency $f = 0$, it can already be said that the time function $x_+(t)$ except for a trivial special case $x_+(t)= 0 \ \ \circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\,\ \ X_+(f)= 0$ is always complex.

The spectrum $X_{\rm TP}(f)$ of the equivalent low-pass signal is obtained by shifting $X_+(f)$ to the left by the carrier frequency $f_{\rm T}$:
:$$X_{\rm TP}(f)= X_+(f+f_{\rm T}).$$

In the time domain this operation corresponds to the multiplication of $x_{\rm +}(t)$ with the complex exponential function with negative exponent:
:$$x_{\rm TP}(t) = x_{\rm +}(t)\cdot {\rm e}^{-{\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm}2 \pi \cdot f_{\rm T}\cdot \hspace{0.05cm}t}.$$

It can be seen that $x_{\rm TP}(t)$ is generally complex. But if $X_+(f)$ is symmetric about the carrier frequency $f_{\rm T}$, $X_{\rm TP}(f)$ is symmetric about the frequency $f=0$ and there is accordingly real time function $x_{\rm TP}(t)$.

===$x_{\rm TP}(t)$–Representation of a Sum of Three Harmonic Oscillations===

In our applet, we always assume a set of three rotating pointers. The physical signal is:
:$$x(t) = x_{\rm U}(t) + x_{\rm T}(t) + x_{\rm O}(t) = A_{\rm U}\cdot \cos\left(2\pi f_{\rm U}\cdot t- \varphi_{\rm U}\right)+A_{\rm T}\cdot \cos\left(2\pi f_{\rm T}\cdot t- \varphi_{\rm T}\right)+A_{\rm O}\cdot \cos\left(2\pi f_{\rm O}\cdot t- \varphi_{\rm O}\right). $$
* Each of the three harmonic oscillations $x_{\rm T}(t)$, $x_{\rm U}(t)$ and $x_{\rm O}(t)$ is represented by an amplitude $(A)$, a frequency $(f)$ and a phase value $(\varphi)$.
*The indices are based on the modulation method [[Modulationsverfahren/Zweiseitenband-Amplitudenmodulation|double sideband amplitude modulation]]. &bdquo;T” stands for &bdquo;carrier”, &bdquo;U” for &bdquo;lower sideband” and &bdquo;O” for &bdquo;upper Sideband”. Similarly, $f_{\rm U} < f_{\rm T}$ and $f_{\rm O} > f_{\rm T}$. There are no restrictions for the amplitudes and phases.

The associated equivalent low-pass signal is with $f_{\rm O}\hspace{0.01cm}' = f_{\rm O}- f_{\rm T} > 0$,   $f_{\rm U}\hspace{0.01cm}' = f_{\rm U}- f_{\rm T} < 0$  and  $f_{\rm T}\hspace{0.01cm}' = 0$:

:$$x_{\rm TP}(t) = x_\text{TP, T}(t) + x_\text{TP, O}(t) + x_\text{TP, U}(t) = A_{\rm T}\cdot {\rm e}^{-{\rm j} \varphi_{\rm T} } \hspace{0.1cm}+ \hspace{0.1cm} A_{\rm O}\cdot {\rm e}^{-{\rm j} \varphi_{\rm O} } \cdot {\rm e}^{ {\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm O}\hspace{0.01cm}'\hspace{0.05cm}\cdot \hspace{0.05cm}t}\hspace{0.1cm}+ \hspace{0.1cm}
A_{\rm U}\cdot {\rm e}^{-{\rm j} \varphi_{\rm U} } \cdot {\rm e}^{ {\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm U}\hspace{0.01cm}'\hspace{0.05cm}\cdot \hspace{0.05cm}t} . $$

{{GraueBox|TEXT=
$\text{Example 1:}$ 
The constellation given here results, for example, in the [[Modulationsverfahren/Zweiseitenband-Amplitudenmodulation#AM-Signale_und_-Spektren_bei_harmonischem_Eingangssignal|double sideband amplitude modulation]] of the message signal $x_{\rm N}(t) = A_{\rm N}\cdot \cos\left(2\pi f_{\rm N}\cdot t- \varphi_{\rm N}\right)$ with the carrier signal $x_{\rm T}(t) = A_{\rm T}\cdot \cos\left(2\pi f_{\rm T}\cdot t - \varphi_{\rm T}\right)$. This is discussed frequently in the exercises.

[[File:Ortskurve_4.png|center|frame|Spectrum $X_{\rm TP}(f)$ of the equivalent low–pass signal for different phase constellations |class=fit]]

There are some limitations to the program parameters in this approach:
* The frequencies are always $f\hspace{0.05cm}'_{\rm O} = f_{\rm N}$ and $f\hspace{0.05cm}'_{\rm U} = -f_{\rm N}$.
*Without distortion, the amplitude of the sidebands is $A_{\rm O}= A_{\rm U}= A_{\rm N}/2$.
*The respective phase relationships can be seen in the following graphic.

}}

===Representation of the Equivalent Low-Pass Signal by Magnitude and Phase===

The generally complex equivalent low-pass signal
:$$x_{\rm TP}(t) = a(t) \cdot {\rm e}^{ {\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm} \phi(t) }$$
can be split into a magnitude function $a(t)$ and a phase function $\phi(t)$ according to the equation given here, where:
:$$a(t) = \vert x_{\rm TP}(t)\vert = \sqrt{ {\rm Re}^2\big [x_{\rm TP}(t)\big ] + {\rm Im}^2\big [x_{\rm TP}(t)\big ] }\hspace{0.05cm},$$
:$$\phi(t) = \text{arc }x_{\rm TP}(t) = \arctan \frac{{\rm Im}\big [x_{\rm TP}(t)\big ]}{{\rm Re}\big [x_{\rm TP}(t)\big ]}.$$

The reason for this is that a bandpass–signal $x(t)$ is usually described by the equivalent low-pass signal $x_{\rm TP}(t)$ that the functions $a(t)$ and $\phi(t)$ are interpretable in both representations:
*The amount $a(t)$ of the equivalent low-pass signal $x_{\rm TP}(t)$ indicates the (time-dependent) envelope of $x(t)$.
*The phase $\phi(t)$ of $x_{\rm TP}(t)$ denotes the location of the zero crossings of $x(t)$, where:
:–   For $\phi(t)>0$ the zero crossing is earlier than its nominal position   ⇒   the signal is leading here.
:–  When $\phi(t)<0$, the zero crossing is later than its target position   ⇒   the signal is trailing here.

{{GraueBox|TEXT=
$\text{Example 2:}$ 
The graph is intended to illustrate this relationship, assuming $A_{\rm U} > A_{\rm O}$   ⇒   the green pointer (for the lower sideband) is longer than the blue pointer (upper sideband). This is a snapshot at time $t_0$:

[[File:Ortskurve_3_neu.png|center|frame|bandpass spectrum $X(f)$ |class=fit]]

*For these system parameters, the top of the pointer cluster $x_{\rm TP}(t)$ – that is, the geometric sum of red, blue and green pointers – on an ellipse.
* The amount $a(t_0) = \vert x_{\rm TP}(t_0) \vert$ is drawn in black in the left-hand diagram and the phase value $\phi(t_0) = \text{arc }x_{\rm TP}(t_0) > 0$ is indicated in brown color.
*In the graph on the right, the amount $a(t_0) = \vert x_{\rm TP}(t_0) \vert$ of the equivalent low-pass signal indicates the envelope of the physical signal $x(t)$.
* At $\phi(t) \equiv 0$, all zero crossings of $x(t)$ would occur at equidistant intervals. Because of $\phi(t_0) > 0$, the signal is leading at the time $t_0$, that is: the zero crossings come earlier than the grid dictates. }}

==Exercises==
[[File:Exercises_verzerrungen.png|right]]
*First select the task number.
*A task description is displayed.
*Parameter values are adjusted.
*Solution after pressing &bdquo;Hide solition”.

The number &bdquo;0” will reset the program and output a text with the further explanation of the applet.

In the following, $\rm Green$ denotes the lower sideband   ⇒   $\big (A_{\rm U}, f_{\rm U}, \varphi_{\rm U}\big )$,  
$\rm Red$ the carrier   ⇒   $\big (A_{\rm T}, f_{\rm T}, \varphi_{\rm T}\big )$ and
$\rm Blue$ the upper sideband   ⇒   $\big (A_{\rm O}, f_{\rm O}, \varphi_{\rm O}\big )$.

{{BlaueBox|TEXT=
'''(1)'''   Let   $\text{Red:} \hspace{0.15cm} A_{\rm T} = 1\ \text{V}, f_{\rm T} = 100 \ \text{kHz}, \varphi_{\rm T} = 0^\circ$,   $\text{Green:} \hspace{0.15cm} A_{\rm U} = 0.4 \text{V}, \ f_{\rm U} = 80 \ \text{kHz}, \varphi_{\rm U} = -90^\circ$,   $\text{Blue:} \hspace{0.15cm} A_{\rm O} = 0.4\ \text{V}, f_{\rm O} = 120 \ \text{kHz}, \varphi_{\rm O} = 90^\circ$.

:Consider and interpret the equivalent low-pass signal $x_{\rm TP}(t)$ and the physical signal $x(t)$. Which period $T_0$ recognizable?}}

:: The equivalent low-pass signal $x_{\rm TP}(t)$ takes from $x_{\rm TP}(t=0)=1\ \text{V}$ on the real axis values between $0.2\ \text{V}$ and $1.8\ \text{V}$   ⇒   phase $\phi(t) \equiv 0$.  The amount $|x_{\rm TP}(t)|$ indicates the envelope $a(t)$ of the physical signal $x(t)$. It holds $A_{\rm N} = 0.8\ \text{V}$ and $f_{\rm N} = 20\ \text{kHz}$:   $a(t) = A_{\rm T}+ A_{\rm N} \cdot \sin(2\pi\cdot f_{\rm N} \cdot t)$.  Both $x_{\rm TP}(t)$ and $x(t)$ are periodically with the period $T_0 = 1/f_{\rm N} = 50\ \rm µ s$.

{{BlaueBox|TEXT=
'''(2)'''   How do the ratios change to '''(1)''' with $f_{\rm U} = 99 \ \text{kHz}$ and $f_{\rm O} = 101 \ \text{kHz}$ ? How could $x(t)$ have arisen?}}

:: For the envelope $a(t)$ of the signal $x(t)$ we still have $a(t) = A_{\rm T}+ A_{\rm N} \cdot \sin(2\pi\cdot f_{\rm N} \cdot t)$, but now $f_{\rm N} = 1\ \text{kHz}$. Even though it can not be recognized:  $x_{\rm TP}(t)$ and $x(t)$ are still periodically:   $T_0 = 1\ \rm ms$. Example: Double sideband Amplitude modulation '''(DSB–AM)''' of a sine signal with cosine carrier.

{{BlaueBox|TEXT=
'''(3)'''   Which settings have to be changed from '''(2)''' in order to arrive at the DSB–AM of a cosine signal with sine carrier. What changes over '''(2)'''?}}

::The carrier phase must be changed to $\varphi_{\rm T} = 90^\circ$   ⇒   sine carrier. Similarly, $\varphi_{\rm O} =\varphi_{\rm U} =\varphi_{\rm T} = 90^\circ$ must be set   ⇒   cosinusoidal message  The locus now lies on the imaginary axis  ⇒   $\phi(t) \equiv -90^\circ$. At the beginning $x_{\rm TP}(t=0)= - {\rm j} \cdot 1.8 \ \text{V}$.

{{BlaueBox|TEXT=
'''(4)'''   Now let   $\text{Red:} \hspace{0.15cm} A_{\rm T} = 1\ \text{V}, \ f_{\rm T} = 100 \ \text{kHz}, \ \varphi_{\rm T} = 0^\circ$,   $\text{Green:} \hspace{0.15cm} A_{\rm U} = 0.4 \text{V}, \ f_{\rm U} = 80 \ \text{kHz}, \ \varphi_{\rm U} = 0^\circ$,   $\text{Blue:} \hspace{0.15cm} A_{\rm O} = 0.4\ \text{V}, \ f_{\rm O} = 120 \ \text{kHz}, \ \varphi_{\rm O} = 0^\circ$.

:What are the characteristics of this system &bdquo;DSB–AM, where the message signal and carrier are respectively cosinusoidal”? What is the degree of modulation $m$?}}

:: The equivalent low-pass signal $x_{\rm TP}(t)$ takes from $x_{\rm TP}(t=0)=1.8\ \text{V}$ on the real axis values between $0.2\ \text{V}$ and $1.8\ \text{V}$   ⇒   phase $\phi(t) \equiv 0$.  Except for the start state $x_{\rm TP}(t=0)$ same behavior as at the setting '''(1)'''. The degree of modulation is $m = 0.8$.

{{BlaueBox|TEXT=
'''(5)'''   The parameters are still valid according to '''(4)''' with the exception of $A_{\rm T}= 0.6 \text{V}$. What is the degree of modulation $m$? What are the consequences?}}

:: There is now a DSB–AM with modulation degree $m = 1.333$. For $m > 1$, the simpler [[Modulationsverfahren/Hüllkurvendemodulation|envelope demodulation]] is not applicable, since the phase function $\phi(t) \in \{ 0, \ \pm 180^\circ\}$ is no more constant and the envelope $a(t)$ no more matches the message signal. Rather, the complex [[Modulationsverfahren/Synchrondemodulation|synchronous demodulation]] must be used. Envelope detection would produce nonlinear distortions.

{{BlaueBox|TEXT=
'''(6)'''   The parameters are still valid according to '''(4)''' or '''(5)''' with the exception from $A_{\rm T}= 0$ on   ⇒   $m \to \infty$. Which modulation method is described in this way?}}

::It is a '''DSB–AM without carrier''' and a synchronous demodulation is required. The equivalent low-pass signal $x_{\rm TP}(t)$ is on the real axis, but not only in the right half-plane. Thus, the phase function $\phi(t) \in \{ 0, \ \pm 180^\circ\}$, also applies here, which means that envelope demodulation is not applicable.

{{BlaueBox|TEXT=
'''(7)'''   Now let   $\text{Red:} \hspace{0.15cm} A_{\rm T} = 1\ \text{V}, f_{\rm T} = 100 \ \text{kHz}, \varphi_{\rm T} = 0^\circ$,   $\text{Green:} \hspace{0.15cm} A_{\rm U} = 0, \ f_{\rm U} = 80 \ \text{kHz}, \varphi_{\rm U} = -90^\circ$,   $\text{Blue:} \hspace{0.15cm} A_{\rm O} = 0.8\ \text{V}, f_{\rm O} = 120 \ \text{kHz}, \varphi_{\rm O} = 90^\circ$.

:Which constellation is described here? Which characteristics of this procedure can be recognized from the graphic?}}

::It is a [[Modulationsverfahren/Einseitenbandmodulation|single-sideband modulation]] '''(SSB–AM)''', more specifically an '''OSB–AM''': the red carrier is fixed, the green pointer missing and the blue pointer (OSB) turns counterclockwise. The degree of modulation is $\mu = 0.8$ (in the case of SSB we denote the degree of modulation with $\mu$ instead of $m$). The carrier signal is cosinusoidal and the message signal sinusoidal. The locus is a circle. $x_{\rm TP}(t)$ moves in a mathematically positive direction. Because of $\phi(t) \ne \text{const.}$ the envelope demodulation is not applicable here:  This can be seen by the fact that the envelope $a(t)$ is not cosinusoidal. Rather, the lower half-wave is sharper than the upper one   ⇒   strong linear distortions.

{{BlaueBox|TEXT=
'''(8)'''   The parameters are still valid according to '''(7)''' with the exception of $A_{\rm O}= 0$ and $A_{\rm U}= 0.8 \text{ V}$. What differences arise opposite '''(7)'''?}}

::Now it is a '''LSB–AM''': The red carrier is fixed, the blue pointer is missing and the green pointer (LSB) rotates clockwise. All other statements of '''(7)''' apply here as well.

{{BlaueBox|TEXT=
'''(9)'''   The parameters according to '''(7)''' are still valid with the exception of $A_{\rm O} = 0.2 \text{ V} \ne A_{\rm U} = 0.4 \text{ V} $. What are the differences from '''(7)'''?}}

::The locus $x_{\rm TP}(t)$ is not a horizontal straight line, but an ellipse with the real part between $0.4 \text{ V}$ and $1.6 \text{ V}$ and the imaginary part in the range $\pm 0.2 \text{ V}$. Because of $\phi(t) \ne \text{const.}$ , Envelope demodulation would lead to non-linear distortions here too. The constellation simulated here describes the situation of '''(4)''', namely a DSB–AM with modulation degree $m = 0.8$, where the upper sideband is reduced to $50\%$ due to channel attenuation.

==Applet Manual==
[[File:Ortskurve_abzug3.png|right]]
* The red parameters $(A_{\rm T}, \ f_{\rm T}, \ \varphi_{\rm T})$ and the red pointer mark the ''Carrier''. (German: '''T'''räger).
* The green parameters $(A_{\rm U}, \ f_{\rm U} < f_{\rm T}, \ \varphi_{\rm U})$ mark the ''Lower sideband''. (German: '''U'''ntere Seitenband).
* The blue parameters $(A_{\rm O}, \ f_{\rm O} > f_{\rm T}, \ \varphi_{\rm O})$ mark the '' Upper sideband''. (German: '''O'''bere Seitenband).
* The red pointer does not turn.
* The green pointer rotates in a mathematically negative direction (clockwise).
* The blue pointer turns counterclockwise.

Meaning of the letters in the adjacent graphic:

    '''(A)'''     Plot of the equivalent low-pass Signal $x_{\rm TP}(t)$

    '''(B)'''     Plot of the physical signal $x(t)$

    '''(C)'''     Parameter input via slider:   amplitudes, frequencies, phase values

    '''(D)'''     Control elements:   Start – Step – Pause/Continue – Reset

    '''(E)'''     Speed of animation:   &bdquo;Speed”   ⇒   Values: 1, 2 oder 3

    '''(F)'''     &bdquo;Trace”   ⇒   On or Off, trace of equivalent low-pass Signal   $x_{\rm TP}(t)$

    '''(G)'''     Numerical output:   time $t$, the signal values  ${\rm Re}[x_{\rm TP}(t)]$  and  ${\rm Im}[x_{\rm TP}(t)]$,

$\text{}\hspace{4.2cm}$   envelope $a(t) = |x_{\rm TP}(t)|$  and  phase $\phi(t) = {\rm arc} \ x_{\rm TP}(t)$

    '''(H)'''     Variations for the graphical representation

$\hspace{1.5cm}$Zoom–Functions &bdquo;$+$” (Enlarge), &bdquo;$-$” (Decrease) und $\rm o$ (Reset to default)

$\hspace{1.5cm}$Move with &bdquo;$\leftarrow$” (Section to the left, ordinate to the right), &bdquo;$\uparrow$” &bdquo;$\downarrow$” &bdquo;$\rightarrow$”

    '''(I)'''     Experiment section:   Task selection and task

    '''(J)'''     Experiment section:  solution
 

==About the Authors==
This interactive calculation was designed and realized at the [http://www.lnt.ei.tum.de/startseite Lehrstuhl für Nachrichtentechnik] of the [https://www.tum.de/ Technical University of Munich] .
*The original version was created in 2005 by [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Studierende#Ji_Li_.28Bachelorarbeit_EI_2003.2C_Diplomarbeit_EI_2005.29|Ji Li]] as part of her Diploma thesis using &bdquo;FlashMX–Actionscript” (Supervisor: [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Mitarbeiter_und_Dozenten#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28am_LNT_seit_1974.29|Günter Söder]]).
*In 2018 this Applet was redesigned and updated to &bdquo;HTML5” by [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Studierende#Xiaohan_Liu_.28Bachelorarbeit_2018.29|Xiaohan Liu]] as part of her Bachelor's thesis (Supervisor: [[Biografien_und_Bibliografien/Beteiligte_der_Professur_Leitungsgebundene_%C3%9Cbertragungstechnik#Tasn.C3.A1d_Kernetzky.2C_M.Sc._.28bei_L.C3.9CT_seit_2014.29|Tasnád Kernetzky]]).

==Once again: Open Applet in new Tab==

{{LntAppletLinkEn|physAnSignal}}

[[Category:Applets|^Verzerrungen^]]

Applets:Physical Signal & Equivalent Lowpass Signal

2018-08-28T09:22:08Z

Xiaohan: /* Applet Description */

{{LntAppletLinkEn|analPhysSignal}}

==Applet Description==
 
This applet shows the relationship between the physical bandpass signal $x(t)$ and the associated equivalent low-pass signal $x_{\rm TP}(t)$. It is assumed that the bandpass signal $x(t)$ has a frequency-discrete spectrum $X(f)$:
:$$x(t) = x_{\rm U}(t) + x_{\rm T}(t) + x_{\rm O}(t) = A_{\rm U}\cdot \cos\left(2\pi f_{\rm U}\cdot t- \varphi_{\rm U}\right)+A_{\rm T}\cdot \cos\left(2\pi f_{\rm T}\cdot t- \varphi_{\rm T}\right)+A_{\rm O}\cdot \cos\left(2\pi f_{\rm O}\cdot t- \varphi_{\rm O}\right). $$
The physical signal $x(t)$ is thus composed of three harmonic oscillations, a constellation that can be found, for example, in the ''Double-sideband Amplitude Modulation''
*of the message signal $x_{\rm N}(t) = A_{\rm N}\cdot \cos\left(2\pi f_{\rm N}\cdot t- \varphi_{\rm N}\right)$   ⇒   in German:   '''N'''achrichtensignal
*with the carrier signal $x_{\rm T}(t) = A_{\rm T}\cdot \cos\left(2\pi f_{\rm T}\cdot t - \varphi_{\rm T}\right)$   ⇒   in German:   '''T'''rägersignal.

The nomenclature is also adapted to this case:
* $x_{\rm O}(t)$ denotes the &bdquo;upper sideband”   (in German:   '''O'''beres Seitenband) with the amplitude $A_{\rm O}= A_{\rm N}/2$, the frequency $f_{\rm O} = f_{\rm T} + f_{\rm N}$ and the phase $\varphi_{\rm O} = \varphi_{\rm T} + \varphi_{\rm N}$.
*Similarly, for the &bdquo;lower sideband”   (in German:   '''U'''nteres Seitenband) $x_{\rm U}(t)$ with $f_{\rm U} = f_{\rm T} - f_{\rm N}$, $A_{\rm U}= A_{\rm O}$ and $\varphi_{\rm U} = -\varphi_{\rm O}$.

The associated equivalent low-pass–signal is $f_{\rm O}\hspace{0.01cm}' = f_{\rm O}- f_{\rm T} > 0$,   $f_{\rm U}\hspace{0.01cm}' = f_{\rm U}- f_{\rm T} < 0$  and  $f_{\rm T}\hspace{0.01cm}' = 0$:

:$$x_{\rm TP}(t) = x_\text{TP, T}(t) + x_\text{TP, O}(t) + x_\text{TP, U}(t) = A_{\rm T}\cdot {\rm e}^{-{\rm j} \varphi_{\rm T} } \hspace{0.1cm}+ \hspace{0.1cm} A_{\rm O}\cdot {\rm e}^{-{\rm j} \varphi_{\rm O} } \cdot {\rm e}^{ {\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm O}\hspace{0.01cm}'\hspace{0.05cm}\cdot \hspace{0.05cm}t}\hspace{0.1cm}+ \hspace{0.1cm}
A_{\rm U}\cdot {\rm e}^{-{\rm j} \varphi_{\rm U} } \cdot {\rm e}^{ {\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm U}\hspace{0.01cm}'\hspace{0.05cm}\cdot \hspace{0.05cm}t} . $$

[[File:Ortskurve_1.png|right|frame|Equivalent low-pass signal currently $t=0$ for cosinusoidal carrier   ⇒   $\varphi_{\rm T} = 0$]]
The program shows $x_{\rm TP}(t)$ as the vectorial sum of three rotation pointers as a violet dot (see figure for start time $t=0$ and cosinusoidal carrier):

*The (red) pointer of the carrier $x_\text{TP, T}(t)$ with the length $A_{\rm T}$ and the zero phase position $\varphi_{\rm T}=0$ is fixed in the complex plane. So it applies to all times $t$:   $x_{\rm TP}(t)= A_{\rm T}\cdot {\rm e}^{-{\rm j} \varphi_{\rm T} }$.

*The (blue) pointer of the upper sideband $x_\text{TP, O}(t)$ with the length $A_{\rm O}$ and the zero phase position $\varphi_{\rm O}$ rotates at the angular velocity $2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm O}\hspace{0.01cm}'$ in mathematically positive direction (one revolution in time $1/f_{\rm O}\hspace{0.01cm}')$.

*The (green) pointer of the lower sideband $x_{\rm U+}(t)$ with the length $A_{\rm U}$ and the zero phase position $\varphi_{\rm U}$ rotates at the angular velocity $2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm U}\hspace{0.01cm}'$, because of $f_{\rm U}\hspace{0.01cm}'<0$ counterclockwise.

*With $f_{\rm U}\hspace{0.01cm}' = -f_{\rm O}\hspace{0.01cm}'$ the blue and green pointers will spin at the same speed but in different directions. Also, if $A_{\rm O} = A_{\rm U}$ and $\varphi_{\rm U} = -\varphi_{\rm O}$, then $x_{\rm TP}(t)$ moves on a straight line with a incline of $\varphi_{\rm T}$.

''Note:''   In the figure $\varphi_{\rm O} = +30^\circ$. From this follows for the start time $t=0$ the angle of the blue pointer (OSB) with respect to the coordinate system:   $\phi_{\rm O} = -\varphi_{\rm O} = -30^\circ$. Likewise, the zero phase position $\varphi_{\rm U} = -30^\circ$ of the lower sideband (USB, grüner Zeiger) follows for the phase angle to be considered in the complex plane:   $\phi_{\rm U} = +30^\circ$.

The temporal process of $x_{\rm TP}(t)$ is also referred to below as '''locus'''. The relationship between $x_{\rm TP}(t)$ and the physical bandpass signal $x(t)$ is given in the section and the associated analytic signal is $x_+(t)$ :

:$$x_{\rm TP}(t) = x_{\rm +}(t)\cdot {\rm e}^{-{\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm}2 \pi \cdot f_{\rm T}\cdot \hspace{0.05cm}t},$$
:$$x_{\rm +}(t) = x_{\rm TP}(t)\cdot {\rm e}^{+{\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm}2 \pi \cdot f_{\rm T}\cdot \hspace{0.05cm}t}.$$

[[Applets:Physikalisches_Signal_%26_Äquivalentes_TP-Signal|'''German description''']]

==Theoretical Background==
 
===Description of Bandpass Signals===
[[File:Zeigerdiagramm_1a.png|right|frame|bandpass spectrum $X(f)$ |class=fit]]
We consider '''bandpass signals''' $x(t)$ with the property that their spectra $X(f)$ are not in the range around the frequency $f=0$, but around a carrier frequency $f_{\rm T}$. In most cases it can also be assumed that the bandwidth is $B \ll f_{\rm T}$.

The figure shows such a bandpass spectrum $X(f)$. Assuming that the associated $x(t)$ is a physical signal and thus real, the spectral function $X(f)$ has a symmetry with respect to the frequency $f = 0$, if $x(t)$ is an even function   ⇒   $x(-t)=x(t)$, $X(f)$ is real and even.

Beside the physical signal $x(t)\ \circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\, \ X(f)$, one can also use the following descriptions of bandpass signals:
*the analytic signal $x_+(t)\ \circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\, \ X_+(f)$, see applet [[Applets:Physikalisches_Signal_%26_Analytisches_Signal|Physikalisches Signal & Analytisches Signal]],
*the equivalent low-pass–signal $x_{\rm TP}(t)\ \circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\, \ X_{\rm TP}(f)$, see next page
 

===Spectral Functions of the Analytic and the Equivalent Low-Pass Signal===

The '''analytic signal''' $x_+(t)$ belonging to the physical signal $x(t)$ is the time function whose spectrum fulfills the following property:
[[File:Ortskurve_2.png|right|frame|spectral functions $X_+(f)$ and $X_{\rm TP}(f)$ |class=fit]]
:$$X_+(f)=\big[1+{\rm sign}(f)\big] \cdot X(f) = \left\{ {2 \cdot
X(f) \; \hspace{0.2cm}\rm for\hspace{0.2cm} {\it f} > 0, \atop {\,\,\,\, \rm 0 \; \hspace{0.9cm}\rm for\hspace{0.2cm} {\it f} < 0.} }\right.$$

The ''Signum function'' is for positive values of $f$ equal to $+1$ and for negative $f$ values equal to $-1$.
*The (double-sided) limit returns $\sign(0) = 0$.
*The index „+” should make it clear that $X_+(f)$ only has parts at positive frequencies.

From the graph you can see the calculation rule for $X_+(f)$: The actual BP spectrum $X(f)$ becomes
*doubled at the positive frequencies, and
*set to zero at the negative frequencies.

Due to the asymmetry of $X_+(f)$ with respect to the frequency $f = 0$, it can already be said that the time function $x_+(t)$ except for a trivial special case $x_+(t)= 0 \ \ \circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\,\ \ X_+(f)= 0$ is always complex.

The spectrum $X_{\rm TP}(f)$ of the equivalent low-pass signal is obtained by shifting $X_+(f)$ to the left by the carrier frequency $f_{\rm T}$:
:$$X_{\rm TP}(f)= X_+(f+f_{\rm T}).$$

In the time domain this operation corresponds to the multiplication of $x_{\rm +}(t)$ with the complex exponential function with negative exponent:
:$$x_{\rm TP}(t) = x_{\rm +}(t)\cdot {\rm e}^{-{\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm}2 \pi \cdot f_{\rm T}\cdot \hspace{0.05cm}t}.$$

It can be seen that $x_{\rm TP}(t)$ is generally complex. But if $X_+(f)$ is symmetric about the carrier frequency $f_{\rm T}$, $X_{\rm TP}(f)$ is symmetric about the frequency $f=0$ and there is accordingly real time function $x_{\rm TP}(t)$.

===$x_{\rm TP}(t)$–Representation of a Sum of Three Harmonic Oscillations===

In our applet, we always assume a set of three rotating pointers. The physical signal is:
:$$x(t) = x_{\rm U}(t) + x_{\rm T}(t) + x_{\rm O}(t) = A_{\rm U}\cdot \cos\left(2\pi f_{\rm U}\cdot t- \varphi_{\rm U}\right)+A_{\rm T}\cdot \cos\left(2\pi f_{\rm T}\cdot t- \varphi_{\rm T}\right)+A_{\rm O}\cdot \cos\left(2\pi f_{\rm O}\cdot t- \varphi_{\rm O}\right). $$
* Each of the three harmonic oscillations $x_{\rm T}(t)$, $x_{\rm U}(t)$ and $x_{\rm O}(t)$ is represented by an amplitude $(A)$, a frequency $(f)$ and a phase value $(\varphi)$.
*The indices are based on the modulation method [[Modulationsverfahren/Zweiseitenband-Amplitudenmodulation|double sideband amplitude modulation]]. &bdquo;T” stands for &bdquo;carrier”, &bdquo;U” for &bdquo;lower sideband” and &bdquo;O” for &bdquo;upper Sideband”. Similarly, $f_{\rm U} < f_{\rm T}$ and $f_{\rm O} > f_{\rm T}$. There are no restrictions for the amplitudes and phases.

The associated equivalent low-pass signal is with $f_{\rm O}\hspace{0.01cm}' = f_{\rm O}- f_{\rm T} > 0$,   $f_{\rm U}\hspace{0.01cm}' = f_{\rm U}- f_{\rm T} < 0$  and  $f_{\rm T}\hspace{0.01cm}' = 0$:

:$$x_{\rm TP}(t) = x_\text{TP, T}(t) + x_\text{TP, O}(t) + x_\text{TP, U}(t) = A_{\rm T}\cdot {\rm e}^{-{\rm j} \varphi_{\rm T} } \hspace{0.1cm}+ \hspace{0.1cm} A_{\rm O}\cdot {\rm e}^{-{\rm j} \varphi_{\rm O} } \cdot {\rm e}^{ {\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm O}\hspace{0.01cm}'\hspace{0.05cm}\cdot \hspace{0.05cm}t}\hspace{0.1cm}+ \hspace{0.1cm}
A_{\rm U}\cdot {\rm e}^{-{\rm j} \varphi_{\rm U} } \cdot {\rm e}^{ {\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm U}\hspace{0.01cm}'\hspace{0.05cm}\cdot \hspace{0.05cm}t} . $$

{{GraueBox|TEXT=
$\text{Example 1:}$ 
The constellation given here results, for example, in the [[Modulationsverfahren/Zweiseitenband-Amplitudenmodulation#AM-Signale_und_-Spektren_bei_harmonischem_Eingangssignal|double sideband amplitude modulation]] of the message signal $x_{\rm N}(t) = A_{\rm N}\cdot \cos\left(2\pi f_{\rm N}\cdot t- \varphi_{\rm N}\right)$ with the carrier signal $x_{\rm T}(t) = A_{\rm T}\cdot \cos\left(2\pi f_{\rm T}\cdot t - \varphi_{\rm T}\right)$. This is discussed frequently in the exercises.

[[File:Ortskurve_4.png|center|frame|Spectrum $X_{\rm TP}(f)$ of the equivalent low–pass signal for different phase constellations |class=fit]]

There are some limitations to the program parameters in this approach:
* The frequencies are always $f\hspace{0.05cm}'_{\rm O} = f_{\rm N}$ and $f\hspace{0.05cm}'_{\rm U} = -f_{\rm N}$.
*Without distortion, the amplitude of the sidebands is $A_{\rm O}= A_{\rm U}= A_{\rm N}/2$.
*The respective phase relationships can be seen in the following graphic.

}}

===Representation of the Equivalent Low-Pass Signal by Magnitude and Phase===

The generally complex equivalent low-pass signal
:$$x_{\rm TP}(t) = a(t) \cdot {\rm e}^{ {\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm} \phi(t) }$$
can be split into a magnitude function $a(t)$ and a phase function $\phi(t)$ according to the equation given here, where:
:$$a(t) = \vert x_{\rm TP}(t)\vert = \sqrt{ {\rm Re}^2\big [x_{\rm TP}(t)\big ] + {\rm Im}^2\big [x_{\rm TP}(t)\big ] }\hspace{0.05cm},$$
:$$\phi(t) = \text{arc }x_{\rm TP}(t) = \arctan \frac{{\rm Im}\big [x_{\rm TP}(t)\big ]}{{\rm Re}\big [x_{\rm TP}(t)\big ]}.$$

The reason for this is that a bandpass–signal $x(t)$ is usually described by the equivalent low-pass signal $x_{\rm TP}(t)$ that the functions $a(t)$ and $\phi(t)$ are interpretable in both representations:
*The amount $a(t)$ of the equivalent low-pass signal $x_{\rm TP}(t)$ indicates the (time-dependent) envelope of $x(t)$.
*The phase $\phi(t)$ of $x_{\rm TP}(t)$ denotes the location of the zero crossings of $x(t)$, where:
:–   For $\phi(t)>0$ the zero crossing is earlier than its nominal position   ⇒   the signal is leading here.
:–  When $\phi(t)<0$, the zero crossing is later than its target position   ⇒   the signal is trailing here.

{{GraueBox|TEXT=
$\text{Example 2:}$ 
The graph is intended to illustrate this relationship, assuming $A_{\rm U} > A_{\rm O}$   ⇒   the green pointer (for the lower sideband) is longer than the blue pointer (upper sideband). This is a snapshot at time $t_0$:

[[File:Ortskurve_3_neu.png|center|frame|bandpass spectrum $X(f)$ |class=fit]]

*For these system parameters, the top of the pointer cluster $x_{\rm TP}(t)$ – that is, the geometric sum of red, blue and green pointers – on an ellipse.
* The amount $a(t_0) = \vert x_{\rm TP}(t_0) \vert$ is drawn in black in the left-hand diagram and the phase value $\phi(t_0) = \text{arc }x_{\rm TP}(t_0) > 0$ is indicated in brown color.
*In the graph on the right, the amount $a(t_0) = \vert x_{\rm TP}(t_0) \vert$ of the equivalent low-pass signal indicates the envelope of the physical signal $x(t)$.
* At $\phi(t) \equiv 0$, all zero crossings of $x(t)$ would occur at equidistant intervals. Because of $\phi(t_0) > 0$, the signal is leading at the time $t_0$, that is: the zero crossings come earlier than the grid dictates. }}

==Exercises==
[[File:Exercises_verzerrungen.png|right]]
*First select the task number.
*A task description is displayed.
*Parameter values are adjusted.
*Solution after pressing &bdquo;Hide solition”.

The number &bdquo;0” will reset the program and output a text with the further explanation of the applet.

In the following, $\rm Green$ denotes the lower sideband   ⇒   $\big (A_{\rm U}, f_{\rm U}, \varphi_{\rm U}\big )$,  
$\rm Red$ the carrier   ⇒   $\big (A_{\rm T}, f_{\rm T}, \varphi_{\rm T}\big )$ and
$\rm Blue$ the upper sideband   ⇒   $\big (A_{\rm O}, f_{\rm O}, \varphi_{\rm O}\big )$.

{{BlaueBox|TEXT=
'''(1)'''   Let   $\text{Red:} \hspace{0.15cm} A_{\rm T} = 1\ \text{V}, f_{\rm T} = 100 \ \text{kHz}, \varphi_{\rm T} = 0^\circ$,   $\text{Green:} \hspace{0.15cm} A_{\rm U} = 0.4 \text{V}, \ f_{\rm U} = 80 \ \text{kHz}, \varphi_{\rm U} = -90^\circ$,   $\text{Blue:} \hspace{0.15cm} A_{\rm O} = 0.4\ \text{V}, f_{\rm O} = 120 \ \text{kHz}, \varphi_{\rm O} = 90^\circ$.

:Consider and interpret the equivalent low-pass signal $x_{\rm TP}(t)$ and the physical signal $x(t)$. Which period $T_0$ recognizable?}}

:: The equivalent low-pass signal $x_{\rm TP}(t)$ takes from $x_{\rm TP}(t=0)=1\ \text{V}$ on the real axis values between $0.2\ \text{V}$ and $1.8\ \text{V}$   ⇒   phase $\phi(t) \equiv 0$.  The amount $|x_{\rm TP}(t)|$ indicates the envelope $a(t)$ of the physical signal $x(t)$. It holds $A_{\rm N} = 0.8\ \text{V}$ and $f_{\rm N} = 20\ \text{kHz}$:   $a(t) = A_{\rm T}+ A_{\rm N} \cdot \sin(2\pi\cdot f_{\rm N} \cdot t)$.  Both $x_{\rm TP}(t)$ and $x(t)$ are periodically with the period $T_0 = 1/f_{\rm N} = 50\ \rm µ s$.

{{BlaueBox|TEXT=
'''(2)'''   How do the ratios change to '''(1)''' with $f_{\rm U} = 99 \ \text{kHz}$ and $f_{\rm O} = 101 \ \text{kHz}$ ? How could $x(t)$ have arisen?}}

:: For the envelope $a(t)$ of the signal $x(t)$ we still have $a(t) = A_{\rm T}+ A_{\rm N} \cdot \sin(2\pi\cdot f_{\rm N} \cdot t)$, but now $f_{\rm N} = 1\ \text{kHz}$. Even though it can not be recognized:  $x_{\rm TP}(t)$ and $x(t)$ are still periodically:   $T_0 = 1\ \rm ms$. Example: Double sideband Amplitude modulation '''(DSB–AM)''' of a sine signal with cosine carrier.

{{BlaueBox|TEXT=
'''(3)'''   Which settings have to be changed from '''(2)''' in order to arrive at the DSB–AM of a cosine signal with sine carrier. What changes over '''(2)'''?}}

::The carrier phase must be changed to $\varphi_{\rm T} = 90^\circ$   ⇒   sine carrier. Similarly, $\varphi_{\rm O} =\varphi_{\rm U} =\varphi_{\rm T} = 90^\circ$ must be set   ⇒   cosinusoidal message  The locus now lies on the imaginary axis  ⇒   $\phi(t) \equiv -90^\circ$. At the beginning $x_{\rm TP}(t=0)= - {\rm j} \cdot 1.8 \ \text{V}$.

{{BlaueBox|TEXT=
'''(4)'''   Now let   $\text{Red:} \hspace{0.15cm} A_{\rm T} = 1\ \text{V}, \ f_{\rm T} = 100 \ \text{kHz}, \ \varphi_{\rm T} = 0^\circ$,   $\text{Green:} \hspace{0.15cm} A_{\rm U} = 0.4 \text{V}, \ f_{\rm U} = 80 \ \text{kHz}, \ \varphi_{\rm U} = 0^\circ$,   $\text{Blue:} \hspace{0.15cm} A_{\rm O} = 0.4\ \text{V}, \ f_{\rm O} = 120 \ \text{kHz}, \ \varphi_{\rm O} = 0^\circ$.

:What are the characteristics of this system &bdquo;DSB–AM, where the message signal and carrier are respectively cosinusoidal”? What is the degree of modulation $m$?}}

:: The equivalent low-pass signal $x_{\rm TP}(t)$ takes from $x_{\rm TP}(t=0)=1.8\ \text{V}$ on the real axis values between $0.2\ \text{V}$ and $1.8\ \text{V}$   ⇒   phase $\phi(t) \equiv 0$.  Except for the start state $x_{\rm TP}(t=0)$ same behavior as at the setting '''(1)'''. The degree of modulation is $m = 0.8$.

{{BlaueBox|TEXT=
'''(5)'''   The parameters are still valid according to '''(4)''' with the exception of $A_{\rm T}= 0.6 \text{V}$. What is the degree of modulation $m$? What are the consequences?}}

:: There is now a DSB–AM with modulation degree $m = 1.333$. For $m > 1$, the simpler [[Modulationsverfahren/Hüllkurvendemodulation|envelope demodulation]] is not applicable, since the phase function $\phi(t) \in \{ 0, \ \pm 180^\circ\}$ is no more constant and the envelope $a(t)$ no more matches the message signal. Rather, the complex [[Modulationsverfahren/Synchrondemodulation|synchronous demodulation]] must be used. Envelope detection would produce nonlinear distortions.

{{BlaueBox|TEXT=
'''(6)'''   The parameters are still valid according to '''(4)''' or '''(5)''' with the exception from $A_{\rm T}= 0$ on   ⇒   $m \to \infty$. Which modulation method is described in this way?}}

::It is a '''DSB–AM without carrier''' and a synchronous demodulation is required. The equivalent low-pass signal $x_{\rm TP}(t)$ is on the real axis, but not only in the right half-plane. Thus, the phase function $\phi(t) \in \{ 0, \ \pm 180^\circ\}$, also applies here, which means that envelope demodulation is not applicable.

{{BlaueBox|TEXT=
'''(7)'''   Now let   $\text{Red:} \hspace{0.15cm} A_{\rm T} = 1\ \text{V}, f_{\rm T} = 100 \ \text{kHz}, \varphi_{\rm T} = 0^\circ$,   $\text{Green:} \hspace{0.15cm} A_{\rm U} = 0, \ f_{\rm U} = 80 \ \text{kHz}, \varphi_{\rm U} = -90^\circ$,   $\text{Blue:} \hspace{0.15cm} A_{\rm O} = 0.8\ \text{V}, f_{\rm O} = 120 \ \text{kHz}, \varphi_{\rm O} = 90^\circ$.

:Which constellation is described here? Which characteristics of this procedure can be recognized from the graphic?}}

::It is a [[Modulationsverfahren/Einseitenbandmodulation|single-sideband modulation]] '''(SSB–AM)''', more specifically an '''OSB–AM''': the red carrier is fixed, the green pointer missing and the blue pointer (OSB) turns counterclockwise. The degree of modulation is $\mu = 0.8$ (in the case of SSB we denote the degree of modulation with $\mu$ instead of $m$). The carrier signal is cosinusoidal and the message signal sinusoidal. The locus is a circle. $x_{\rm TP}(t)$ moves in a mathematically positive direction. Because of $\phi(t) \ne \text{const.}$ the envelope demodulation is not applicable here:  This can be seen by the fact that the envelope $a(t)$ is not cosinusoidal. Rather, the lower half-wave is sharper than the upper one   ⇒   strong linear distortions.

{{BlaueBox|TEXT=
'''(8)'''   The parameters are still valid according to '''(7)''' with the exception of $A_{\rm O}= 0$ and $A_{\rm U}= 0.8 \text{ V}$. What differences arise opposite '''(7)'''?}}

::Now it is a '''LSB–AM''': The red carrier is fixed, the blue pointer is missing and the green pointer (LSB) rotates clockwise. All other statements of '''(7)''' apply here as well.

{{BlaueBox|TEXT=
'''(9)'''   The parameters according to '''(7)''' are still valid with the exception of $A_{\rm O} = 0.2 \text{ V} \ne A_{\rm U} = 0.4 \text{ V} $. What are the differences from '''(7)'''?}}

::The locus $x_{\rm TP}(t)$ is not a horizontal straight line, but an ellipse with the real part between $0.4 \text{ V}$ and $1.6 \text{ V}$ and the imaginary part in the range $\pm 0.2 \text{ V}$. Because of $\phi(t) \ne \text{const.}$ , Envelope demodulation would lead to non-linear distortions here too. The constellation simulated here describes the situation of '''(4)''', namely a DSB–AM with modulation degree $m = 0.8$, where the upper sideband is reduced to $50\%$ due to channel attenuation.

==Applet Manual==
[[File:Ortskurve_abzug3.png|right]]
* The red parameters $(A_{\rm T}, \ f_{\rm T}, \ \varphi_{\rm T})$ and the red pointer mark the ''Carrier''. (German: '''T'''räger).
* The green parameters $(A_{\rm U}, \ f_{\rm U} < f_{\rm T}, \ \varphi_{\rm U})$ mark the ''Lower sideband''. (German: '''U'''ntere Seitenband).
* The blue parameters $(A_{\rm O}, \ f_{\rm O} > f_{\rm T}, \ \varphi_{\rm O})$ mark the '' Upper sideband''. (German: '''O'''bere Seitenband).
* The red pointer does not turn.
* The green pointer rotates in a mathematically negative direction (clockwise).
* The blue pointer turns counterclockwise.

Meaning of the letters in the adjacent graphic:

    '''(A)'''     Plot of the equivalent low-pass Signal $x_{\rm TP}(t)$

    '''(B)'''     Plot of the physical signal $x(t)$

    '''(C)'''     Parameter input via slider:   amplitudes, frequencies, phase values

    '''(D)'''     Control elements:   Start – Step – Pause/Continue – Reset

    '''(E)'''     Speed of animation:   &bdquo;Speed”   ⇒   Values: 1, 2 oder 3

    '''(F)'''     &bdquo;Trace”   ⇒   On or Off, trace of equivalent low-pass Signal   $x_{\rm TP}(t)$

    '''(G)'''     Numerical output:   time $t$, the signal values  ${\rm Re}[x_{\rm TP}(t)]$  and  ${\rm Im}[x_{\rm TP}(t)]$,

$\text{}\hspace{4.2cm}$   envelope $a(t) = |x_{\rm TP}(t)|$  and  phase $\phi(t) = {\rm arc} \ x_{\rm TP}(t)$

    '''(H)'''     Variations for the graphical representation

$\hspace{1.5cm}$Zoom–Functions &bdquo;$+$” (Enlarge), &bdquo;$-$” (Decrease) und $\rm o$ (Reset to default)

$\hspace{1.5cm}$Move with &bdquo;$\leftarrow$” (Section to the left, ordinate to the right), &bdquo;$\uparrow$” &bdquo;$\downarrow$” &bdquo;$\rightarrow$”

    '''(I)'''     Experiment section:   Task selection and task

    '''(J)'''     Experiment section:  solution
 

==About the Authors==
This interactive calculation was designed and realized at the [http://www.lnt.ei.tum.de/startseite Lehrstuhl für Nachrichtentechnik] of the [https://www.tum.de/ Technical University of Munich] .
*The original version was created in 2005 by [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Studierende#Ji_Li_.28Bachelorarbeit_EI_2003.2C_Diplomarbeit_EI_2005.29|Ji Li]] as part of her Diploma thesis using &bdquo;FlashMX–Actionscript” (Supervisor: [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Mitarbeiter_und_Dozenten#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28am_LNT_seit_1974.29|Günter Söder]]).
*In 2018 this Applet was redesigned and updated to &bdquo;HTML5” by [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Studierende#Xiaohan_Liu_.28Bachelorarbeit_2018.29|Xiaohan Liu]] as part of her Bachelor's thesis (Supervisor: [[Biografien_und_Bibliografien/Beteiligte_der_Professur_Leitungsgebundene_%C3%9Cbertragungstechnik#Tasn.C3.A1d_Kernetzky.2C_M.Sc._.28bei_L.C3.9CT_seit_2014.29|Tasnád Kernetzky]]).

==Once again: Open Applet in new Tab==

{{LntAppletLinkEn|physAnSignal}}

[[Category:Applets|^Verzerrungen^]]

Applets:Physical Signal & Equivalent Lowpass Signal

2018-08-26T17:16:10Z

Xiaohan: /* Exercises */

{{LntAppletLinkEn|analPhysSignal}}

==Applet Description==
 
This applet shows the relationship between the physical bandpass signal $x(t)$ and the associated equivalent low-pass signal $x_{\rm TP}(t)$. It is assumed that the bandpass signal $x(t)$ has a frequency-discrete spectrum $X(f)$:
:$$x(t) = x_{\rm U}(t) + x_{\rm T}(t) + x_{\rm O}(t) = A_{\rm U}\cdot \cos\left(2\pi f_{\rm U}\cdot t- \varphi_{\rm U}\right)+A_{\rm T}\cdot \cos\left(2\pi f_{\rm T}\cdot t- \varphi_{\rm T}\right)+A_{\rm O}\cdot \cos\left(2\pi f_{\rm O}\cdot t- \varphi_{\rm O}\right). $$
The physical signal $x(t)$ is thus composed of three harmonic oscillations, a constellation that can be found, for example, in the ''Double-sideband Amplitude Modulation''
*of the message signal $x_{\rm N}(t) = A_{\rm N}\cdot \cos\left(2\pi f_{\rm N}\cdot t- \varphi_{\rm N}\right)$   ⇒   in German:   '''N'''achrichtensignal
*with the carrier signal $x_{\rm T}(t) = A_{\rm T}\cdot \cos\left(2\pi f_{\rm T}\cdot t - \varphi_{\rm T}\right)$   ⇒   in German:   '''T'''rägersignal.

The nomenclature is also adapted to this case:
* $x_{\rm O}(t)$ denotes the &bdquo;upper sideband”   (in German:   '''O'''beres Seitenband) with the amplitude $A_{\rm O}= A_{\rm N}/2$, the frequency $f_{\rm O} = f_{\rm T} + f_{\rm N}$ and the phase $\varphi_{\rm O} = \varphi_{\rm T} + \varphi_{\rm N}$.
*Similarly, for the &bdquo;lower sideband”   (in German:   '''U'''nteres Seitenband) $x_{\rm U}(t)$ with $f_{\rm U} = f_{\rm T} - f_{\rm N}$, $A_{\rm U}= A_{\rm O}$ and $\varphi_{\rm U} = -\varphi_{\rm O}$.

The associated equivalent low-pass–signal is $f_{\rm O}\hspace{0.01cm}' = f_{\rm O}- f_{\rm T} > 0$,   $f_{\rm U}\hspace{0.01cm}' = f_{\rm U}- f_{\rm T} < 0$  and  $f_{\rm T}\hspace{0.01cm}' = 0$:

:$$x_{\rm TP}(t) = x_\text{TP, T}(t) + x_\text{TP, O}(t) + x_\text{TP, U}(t) = A_{\rm T}\cdot {\rm e}^{-{\rm j} \varphi_{\rm T} } \hspace{0.1cm}+ \hspace{0.1cm} A_{\rm O}\cdot {\rm e}^{-{\rm j} \varphi_{\rm O} } \cdot {\rm e}^{ {\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm O}\hspace{0.01cm}'\hspace{0.05cm}\cdot \hspace{0.05cm}t}\hspace{0.1cm}+ \hspace{0.1cm}
A_{\rm U}\cdot {\rm e}^{-{\rm j} \varphi_{\rm U} } \cdot {\rm e}^{ {\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm U}\hspace{0.01cm}'\hspace{0.05cm}\cdot \hspace{0.05cm}t} . $$

[[File:Ortskurve_1.png|right|frame|Equivalent low-pass signal currently $t=0$ for cosinusoidal carrier   ⇒   $\varphi_{\rm T} = 0$]]
The program shows $x_{\rm TP}(t)$ as the vectorial sum of three rotation pointers as a violet dot (see figure for start time $t=0$ and cosinusoidal carrier):

*The (red) pointer of the carrier $x_\text{TP, T}(t)$ with the length $A_{\rm T}$ and the zero phase position $\varphi_{\rm T}=0$ is fixed in the complex plane. So it applies to all times $t$:   $x_{\rm TP}(t)= A_{\rm T}\cdot {\rm e}^{-{\rm j} \varphi_{\rm T} }$.

*The (blue) pointer of the upper sideband $x_\text{TP, O}(t)$ with the length $A_{\rm O}$ and the zero phase position $\varphi_{\rm O}$ rotates at the angular velocity $2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm O}\hspace{0.01cm}'$ in mathematically positive direction (one revolution in time $1/f_{\rm O}\hspace{0.01cm}')$.

*The (green) pointer of the lower sideband $x_{\rm U+}(t)$ with the length $A_{\rm U}$ and the zero phase position $\varphi_{\rm U}$ rotates at the angular velocity $2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm U}\hspace{0.01cm}'$, because of $f_{\rm U}\hspace{0.01cm}'<0$ counterclockwise.

*With $f_{\rm U}\hspace{0.01cm}' = -f_{\rm O}\hspace{0.01cm}'$ the blue and green pointers will spin at the same speed but in different directions. Also, if $A_{\rm O} = A_{\rm U}$ and $\varphi_{\rm U} = -\varphi_{\rm O}$, then $x_{\rm TP}(t)$ moves on a straight line with a incline of $\varphi_{\rm T}$.

''Note:''   In the figure $\varphi_{\rm O} = +30^\circ$. From this follows for the start time $t=0$ the angle of the blue pointer (OSB) with respect to the coordinate system:   $\phi_{\rm O} = -\varphi_{\rm O} = -30^\circ$. Likewise, the null phase position $\varphi_{\rm U} = -30^\circ$ of the lower sideband (USB, grüner Zeiger) follows for the phase angle to be considered in the complex plane:   $\phi_{\rm U} = +30^\circ$.

The temporal process of $x_{\rm TP}(t)$ is also referred to below as '''locus'''. The relationship between $x_{\rm TP}(t)$ and the physical bandpass signal $x(t)$ is given in the section and the associated analytic signal is $x_+(t)$ :

:$$x_{\rm TP}(t) = x_{\rm +}(t)\cdot {\rm e}^{-{\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm}2 \pi \cdot f_{\rm T}\cdot \hspace{0.05cm}t},$$
:$$x_{\rm +}(t) = x_{\rm TP}(t)\cdot {\rm e}^{+{\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm}2 \pi \cdot f_{\rm T}\cdot \hspace{0.05cm}t}.$$

[[Applets:Physikalisches_Signal_%26_Äquivalentes_TP-Signal|'''German description''']]

==Theoretical Background==
 
===Description of Bandpass Signals===
[[File:Zeigerdiagramm_1a.png|right|frame|bandpass spectrum $X(f)$ |class=fit]]
We consider '''bandpass signals''' $x(t)$ with the property that their spectra $X(f)$ are not in the range around the frequency $f=0$, but around a carrier frequency $f_{\rm T}$. In most cases it can also be assumed that the bandwidth is $B \ll f_{\rm T}$.

The figure shows such a bandpass spectrum $X(f)$. Assuming that the associated $x(t)$ is a physical signal and thus real, the spectral function $X(f)$ has a symmetry with respect to the frequency $f = 0$, if $x(t)$ is an even function   ⇒   $x(-t)=x(t)$, $X(f)$ is real and even.

Beside the physical signal $x(t)\ \circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\, \ X(f)$, one can also use the following descriptions of bandpass signals:
*the analytic signal $x_+(t)\ \circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\, \ X_+(f)$, see applet [[Applets:Physikalisches_Signal_%26_Analytisches_Signal|Physikalisches Signal & Analytisches Signal]],
*the equivalent low-pass–signal $x_{\rm TP}(t)\ \circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\, \ X_{\rm TP}(f)$, see next page
 

===Spectral Functions of the Analytic and the Equivalent Low-Pass Signal===

The '''analytic signal''' $x_+(t)$ belonging to the physical signal $x(t)$ is the time function whose spectrum fulfills the following property:
[[File:Ortskurve_2.png|right|frame|spectral functions $X_+(f)$ and $X_{\rm TP}(f)$ |class=fit]]
:$$X_+(f)=\big[1+{\rm sign}(f)\big] \cdot X(f) = \left\{ {2 \cdot
X(f) \; \hspace{0.2cm}\rm for\hspace{0.2cm} {\it f} > 0, \atop {\,\,\,\, \rm 0 \; \hspace{0.9cm}\rm for\hspace{0.2cm} {\it f} < 0.} }\right.$$

The ''Signum function'' is for positive values of $f$ equal to $+1$ and for negative $f$ values equal to $-1$.
*The (double-sided) limit returns $\sign(0) = 0$.
*The index „+” should make it clear that $X_+(f)$ only has parts at positive frequencies.

From the graph you can see the calculation rule for $X_+(f)$: The actual BP spectrum $X(f)$ becomes
*doubled at the positive frequencies, and
*set to zero at the negative frequencies.

Due to the asymmetry of $X_+(f)$ with respect to the frequency $f = 0$, it can already be said that the time function $x_+(t)$ except for a trivial special case $x_+(t)= 0 \ \ \circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\,\ \ X_+(f)= 0$ is always complex.

The spectrum $X_{\rm TP}(f)$ of the equivalent low-pass signal is obtained by shifting $X_+(f)$ to the left by the carrier frequency $f_{\rm T}$:
:$$X_{\rm TP}(f)= X_+(f+f_{\rm T}).$$

In the time domain this operation corresponds to the multiplication of $x_{\rm +}(t)$ with the complex exponential function with negative exponent:
:$$x_{\rm TP}(t) = x_{\rm +}(t)\cdot {\rm e}^{-{\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm}2 \pi \cdot f_{\rm T}\cdot \hspace{0.05cm}t}.$$

It can be seen that $x_{\rm TP}(t)$ is generally complex. But if $X_+(f)$ is symmetric about the carrier frequency $f_{\rm T}$, $X_{\rm TP}(f)$ is symmetric about the frequency $f=0$ and there is accordingly real time function $x_{\rm TP}(t)$.

===$x_{\rm TP}(t)$–Representation of a Sum of Three Harmonic Oscillations===

In our applet, we always assume a set of three rotating pointers. The physical signal is:
:$$x(t) = x_{\rm U}(t) + x_{\rm T}(t) + x_{\rm O}(t) = A_{\rm U}\cdot \cos\left(2\pi f_{\rm U}\cdot t- \varphi_{\rm U}\right)+A_{\rm T}\cdot \cos\left(2\pi f_{\rm T}\cdot t- \varphi_{\rm T}\right)+A_{\rm O}\cdot \cos\left(2\pi f_{\rm O}\cdot t- \varphi_{\rm O}\right). $$
* Each of the three harmonic oscillations $x_{\rm T}(t)$, $x_{\rm U}(t)$ and $x_{\rm O}(t)$ is represented by an amplitude $(A)$, a frequency $(f)$ and a phase value $(\varphi)$.
*The indices are based on the modulation method [[Modulationsverfahren/Zweiseitenband-Amplitudenmodulation|double sideband amplitude modulation]]. &bdquo;T” stands for &bdquo;carrier”, &bdquo;U” for &bdquo;lower sideband” and &bdquo;O” for &bdquo;upper Sideband”. Similarly, $f_{\rm U} < f_{\rm T}$ and $f_{\rm O} > f_{\rm T}$. There are no restrictions for the amplitudes and phases.

The associated equivalent low-pass signal is with $f_{\rm O}\hspace{0.01cm}' = f_{\rm O}- f_{\rm T} > 0$,   $f_{\rm U}\hspace{0.01cm}' = f_{\rm U}- f_{\rm T} < 0$  and  $f_{\rm T}\hspace{0.01cm}' = 0$:

:$$x_{\rm TP}(t) = x_\text{TP, T}(t) + x_\text{TP, O}(t) + x_\text{TP, U}(t) = A_{\rm T}\cdot {\rm e}^{-{\rm j} \varphi_{\rm T} } \hspace{0.1cm}+ \hspace{0.1cm} A_{\rm O}\cdot {\rm e}^{-{\rm j} \varphi_{\rm O} } \cdot {\rm e}^{ {\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm O}\hspace{0.01cm}'\hspace{0.05cm}\cdot \hspace{0.05cm}t}\hspace{0.1cm}+ \hspace{0.1cm}
A_{\rm U}\cdot {\rm e}^{-{\rm j} \varphi_{\rm U} } \cdot {\rm e}^{ {\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm U}\hspace{0.01cm}'\hspace{0.05cm}\cdot \hspace{0.05cm}t} . $$

{{GraueBox|TEXT=
$\text{Example 1:}$ 
The constellation given here results, for example, in the [[Modulationsverfahren/Zweiseitenband-Amplitudenmodulation#AM-Signale_und_-Spektren_bei_harmonischem_Eingangssignal|double sideband amplitude modulation]] of the message signal $x_{\rm N}(t) = A_{\rm N}\cdot \cos\left(2\pi f_{\rm N}\cdot t- \varphi_{\rm N}\right)$ with the carrier signal $x_{\rm T}(t) = A_{\rm T}\cdot \cos\left(2\pi f_{\rm T}\cdot t - \varphi_{\rm T}\right)$. This is discussed frequently in the exercises.

[[File:Ortskurve_4.png|center|frame|Spectrum $X_{\rm TP}(f)$ of the equivalent low–pass signal for different phase constellations |class=fit]]

There are some limitations to the program parameters in this approach:
* The frequencies are always $f\hspace{0.05cm}'_{\rm O} = f_{\rm N}$ and $f\hspace{0.05cm}'_{\rm U} = -f_{\rm N}$.
*Without distortion, the amplitude of the sidebands is $A_{\rm O}= A_{\rm U}= A_{\rm N}/2$.
*The respective phase relationships can be seen in the following graphic.

}}

===Representation of the Equivalent Low-Pass Signal by Magnitude and Phase===

The generally complex equivalent low-pass signal
:$$x_{\rm TP}(t) = a(t) \cdot {\rm e}^{ {\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm} \phi(t) }$$
can be split into a magnitude function $a(t)$ and a phase function $\phi(t)$ according to the equation given here, where:
:$$a(t) = \vert x_{\rm TP}(t)\vert = \sqrt{ {\rm Re}^2\big [x_{\rm TP}(t)\big ] + {\rm Im}^2\big [x_{\rm TP}(t)\big ] }\hspace{0.05cm},$$
:$$\phi(t) = \text{arc }x_{\rm TP}(t) = \arctan \frac{{\rm Im}\big [x_{\rm TP}(t)\big ]}{{\rm Re}\big [x_{\rm TP}(t)\big ]}.$$

The reason for this is that a bandpass–signal $x(t)$ is usually described by the equivalent low-pass signal $x_{\rm TP}(t)$ that the functions $a(t)$ and $\phi(t)$ are interpretable in both representations:
*The amount $a(t)$ of the equivalent low-pass signal $x_{\rm TP}(t)$ indicates the (time-dependent) envelope of $x(t)$.
*The phase $\phi(t)$ of $x_{\rm TP}(t)$ denotes the location of the zero crossings of $x(t)$, where:
:–   For $\phi(t)>0$ the zero crossing is earlier than its nominal position   ⇒   the signal is leading here.
:–  When $\phi(t)<0$, the zero crossing is later than its target position   ⇒   the signal is trailing here.

{{GraueBox|TEXT=
$\text{Example 2:}$ 
The graph is intended to illustrate this relationship, assuming $A_{\rm U} > A_{\rm O}$   ⇒   the green pointer (for the lower sideband) is longer than the blue pointer (upper sideband). This is a snapshot at time $t_0$:

[[File:Ortskurve_3_neu.png|center|frame|bandpass spectrum $X(f)$ |class=fit]]

*For these system parameters, the top of the pointer cluster $x_{\rm TP}(t)$ – that is, the geometric sum of red, blue and green pointers – on an ellipse.
* The amount $a(t_0) = \vert x_{\rm TP}(t_0) \vert$ is drawn in black in the left-hand diagram and the phase value $\phi(t_0) = \text{arc }x_{\rm TP}(t_0) > 0$ is indicated in brown color.
*In the graph on the right, the amount $a(t_0) = \vert x_{\rm TP}(t_0) \vert$ of the equivalent low-pass signal indicates the envelope of the physical signal $x(t)$.
* At $\phi(t) \equiv 0$, all zero crossings of $x(t)$ would occur at equidistant intervals. Because of $\phi(t_0) > 0$, the signal is leading at the time $t_0$, that is: the zero crossings come earlier than the grid dictates. }}

==Exercises==
[[File:Exercises_verzerrungen.png|right]]
*First select the task number.
*A task description is displayed.
*Parameter values are adjusted.
*Solution after pressing &bdquo;Hide solition”.

The number &bdquo;0” will reset the program and output a text with the further explanation of the applet.

In the following, $\rm Green$ denotes the lower sideband   ⇒   $\big (A_{\rm U}, f_{\rm U}, \varphi_{\rm U}\big )$,  
$\rm Red$ the carrier   ⇒   $\big (A_{\rm T}, f_{\rm T}, \varphi_{\rm T}\big )$ and
$\rm Blue$ the upper sideband   ⇒   $\big (A_{\rm O}, f_{\rm O}, \varphi_{\rm O}\big )$.

{{BlaueBox|TEXT=
'''(1)'''   Let   $\text{Red:} \hspace{0.15cm} A_{\rm T} = 1\ \text{V}, f_{\rm T} = 100 \ \text{kHz}, \varphi_{\rm T} = 0^\circ$,   $\text{Green:} \hspace{0.15cm} A_{\rm U} = 0.4 \text{V}, \ f_{\rm U} = 80 \ \text{kHz}, \varphi_{\rm U} = -90^\circ$,   $\text{Blue:} \hspace{0.15cm} A_{\rm O} = 0.4\ \text{V}, f_{\rm O} = 120 \ \text{kHz}, \varphi_{\rm O} = 90^\circ$.

:Consider and interpret the equivalent low-pass signal $x_{\rm TP}(t)$ and the physical signal $x(t)$. Which period $T_0$ recognizable?}}

:: The equivalent low-pass signal $x_{\rm TP}(t)$ takes from $x_{\rm TP}(t=0)=1\ \text{V}$ on the real axis values between $0.2\ \text{V}$ and $1.8\ \text{V}$   ⇒   phase $\phi(t) \equiv 0$.  The amount $|x_{\rm TP}(t)|$ indicates the envelope $a(t)$ of the physical signal $x(t)$. It holds $A_{\rm N} = 0.8\ \text{V}$ and $f_{\rm N} = 20\ \text{kHz}$:   $a(t) = A_{\rm T}+ A_{\rm N} \cdot \sin(2\pi\cdot f_{\rm N} \cdot t)$.  Both $x_{\rm TP}(t)$ and $x(t)$ are periodically with the period $T_0 = 1/f_{\rm N} = 50\ \rm µ s$.

{{BlaueBox|TEXT=
'''(2)'''   How do the ratios change to '''(1)''' with $f_{\rm U} = 99 \ \text{kHz}$ and $f_{\rm O} = 101 \ \text{kHz}$ ? How could $x(t)$ have arisen?}}

:: For the envelope $a(t)$ of the signal $x(t)$ we still have $a(t) = A_{\rm T}+ A_{\rm N} \cdot \sin(2\pi\cdot f_{\rm N} \cdot t)$, but now $f_{\rm N} = 1\ \text{kHz}$. Even though it can not be recognized:  $x_{\rm TP}(t)$ and $x(t)$ are still periodically:   $T_0 = 1\ \rm ms$. Example: Double sideband Amplitude modulation '''(DSB–AM)''' of a sine signal with cosine carrier.

{{BlaueBox|TEXT=
'''(3)'''   Which settings have to be changed from '''(2)''' in order to arrive at the DSB–AM of a cosine signal with sine carrier. What changes over '''(2)'''?}}

::The carrier phase must be changed to $\varphi_{\rm T} = 90^\circ$   ⇒   sine carrier. Similarly, $\varphi_{\rm O} =\varphi_{\rm U} =\varphi_{\rm T} = 90^\circ$ must be set   ⇒   cosinusoidal message  The locus now lies on the imaginary axis  ⇒   $\phi(t) \equiv -90^\circ$. At the beginning $x_{\rm TP}(t=0)= - {\rm j} \cdot 1.8 \ \text{V}$.

{{BlaueBox|TEXT=
'''(4)'''   Now let   $\text{Red:} \hspace{0.15cm} A_{\rm T} = 1\ \text{V}, \ f_{\rm T} = 100 \ \text{kHz}, \ \varphi_{\rm T} = 0^\circ$,   $\text{Green:} \hspace{0.15cm} A_{\rm U} = 0.4 \text{V}, \ f_{\rm U} = 80 \ \text{kHz}, \ \varphi_{\rm U} = 0^\circ$,   $\text{Blue:} \hspace{0.15cm} A_{\rm O} = 0.4\ \text{V}, \ f_{\rm O} = 120 \ \text{kHz}, \ \varphi_{\rm O} = 0^\circ$.

:What are the characteristics of this system &bdquo;DSB–AM, where the message signal and carrier are respectively cosinusoidal”? What is the degree of modulation $m$?}}

:: The equivalent low-pass signal $x_{\rm TP}(t)$ takes from $x_{\rm TP}(t=0)=1.8\ \text{V}$ on the real axis values between $0.2\ \text{V}$ and $1.8\ \text{V}$   ⇒   phase $\phi(t) \equiv 0$.  Except for the start state $x_{\rm TP}(t=0)$ same behavior as at the setting '''(1)'''. The degree of modulation is $m = 0.8$.

{{BlaueBox|TEXT=
'''(5)'''   The parameters are still valid according to '''(4)''' with the exception of $A_{\rm T}= 0.6 \text{V}$. What is the degree of modulation $m$? What are the consequences?}}

:: There is now a DSB–AM with modulation degree $m = 1.333$. For $m > 1$, the simpler [[Modulationsverfahren/Hüllkurvendemodulation|envelope demodulation]] is not applicable, since the phase function $\phi(t) \in \{ 0, \ \pm 180^\circ\}$ is no more constant and the envelope $a(t)$ no more matches the message signal. Rather, the complex [[Modulationsverfahren/Synchrondemodulation|synchronous demodulation]] must be used. Envelope detection would produce nonlinear distortions.

{{BlaueBox|TEXT=
'''(6)'''   The parameters are still valid according to '''(4)''' or '''(5)''' with the exception from $A_{\rm T}= 0$ on   ⇒   $m \to \infty$. Which modulation method is described in this way?}}

::It is a '''DSB–AM without carrier''' and a synchronous demodulation is required. The equivalent low-pass signal $x_{\rm TP}(t)$ is on the real axis, but not only in the right half-plane. Thus, the phase function $\phi(t) \in \{ 0, \ \pm 180^\circ\}$, also applies here, which means that envelope demodulation is not applicable.

{{BlaueBox|TEXT=
'''(7)'''   Now let   $\text{Red:} \hspace{0.15cm} A_{\rm T} = 1\ \text{V}, f_{\rm T} = 100 \ \text{kHz}, \varphi_{\rm T} = 0^\circ$,   $\text{Green:} \hspace{0.15cm} A_{\rm U} = 0, \ f_{\rm U} = 80 \ \text{kHz}, \varphi_{\rm U} = -90^\circ$,   $\text{Blue:} \hspace{0.15cm} A_{\rm O} = 0.8\ \text{V}, f_{\rm O} = 120 \ \text{kHz}, \varphi_{\rm O} = 90^\circ$.

:Which constellation is described here? Which characteristics of this procedure can be recognized from the graphic?}}

::It is a [[Modulationsverfahren/Einseitenbandmodulation|single-sideband modulation]] '''(SSB–AM)''', more specifically an '''OSB–AM''': the red carrier is fixed, the green pointer missing and the blue pointer (OSB) turns counterclockwise. The degree of modulation is $\mu = 0.8$ (in the case of SSB we denote the degree of modulation with $\mu$ instead of $m$). The carrier signal is cosinusoidal and the message signal sinusoidal. The locus is a circle. $x_{\rm TP}(t)$ moves in a mathematically positive direction. Because of $\phi(t) \ne \text{const.}$ the envelope demodulation is not applicable here:  This can be seen by the fact that the envelope $a(t)$ is not cosinusoidal. Rather, the lower half-wave is sharper than the upper one   ⇒   strong linear distortions.

{{BlaueBox|TEXT=
'''(8)'''   The parameters are still valid according to '''(7)''' with the exception of $A_{\rm O}= 0$ and $A_{\rm U}= 0.8 \text{ V}$. What differences arise opposite '''(7)'''?}}

::Now it is a '''LSB–AM''': The red carrier is fixed, the blue pointer is missing and the green pointer (LSB) rotates clockwise. All other statements of '''(7)''' apply here as well.

{{BlaueBox|TEXT=
'''(9)'''   The parameters according to '''(7)''' are still valid with the exception of $A_{\rm O} = 0.2 \text{ V} \ne A_{\rm U} = 0.4 \text{ V} $. What are the differences from '''(7)'''?}}

::The locus $x_{\rm TP}(t)$ is not a horizontal straight line, but an ellipse with the real part between $0.4 \text{ V}$ and $1.6 \text{ V}$ and the imaginary part in the range $\pm 0.2 \text{ V}$. Because of $\phi(t) \ne \text{const.}$ , Envelope demodulation would lead to non-linear distortions here too. The constellation simulated here describes the situation of '''(4)''', namely a DSB–AM with modulation degree $m = 0.8$, where the upper sideband is reduced to $50\%$ due to channel attenuation.

==Applet Manual==
[[File:Ortskurve_abzug3.png|right]]
* The red parameters $(A_{\rm T}, \ f_{\rm T}, \ \varphi_{\rm T})$ and the red pointer mark the ''Carrier''. (German: '''T'''räger).
* The green parameters $(A_{\rm U}, \ f_{\rm U} < f_{\rm T}, \ \varphi_{\rm U})$ mark the ''Lower sideband''. (German: '''U'''ntere Seitenband).
* The blue parameters $(A_{\rm O}, \ f_{\rm O} > f_{\rm T}, \ \varphi_{\rm O})$ mark the '' Upper sideband''. (German: '''O'''bere Seitenband).
* The red pointer does not turn.
* The green pointer rotates in a mathematically negative direction (clockwise).
* The blue pointer turns counterclockwise.

Meaning of the letters in the adjacent graphic:

    '''(A)'''     Plot of the equivalent low-pass Signal $x_{\rm TP}(t)$

    '''(B)'''     Plot of the physical signal $x(t)$

    '''(C)'''     Parameter input via slider:   amplitudes, frequencies, phase values

    '''(D)'''     Control elements:   Start – Step – Pause/Continue – Reset

    '''(E)'''     Speed of animation:   &bdquo;Speed”   ⇒   Values: 1, 2 oder 3

    '''(F)'''     &bdquo;Trace”   ⇒   On or Off, trace of equivalent low-pass Signal   $x_{\rm TP}(t)$

    '''(G)'''     Numerical output:   time $t$, the signal values  ${\rm Re}[x_{\rm TP}(t)]$  and  ${\rm Im}[x_{\rm TP}(t)]$,

$\text{}\hspace{4.2cm}$   envelope $a(t) = |x_{\rm TP}(t)|$  and  phase $\phi(t) = {\rm arc} \ x_{\rm TP}(t)$

    '''(H)'''     Variations for the graphical representation

$\hspace{1.5cm}$Zoom–Functions &bdquo;$+$” (Enlarge), &bdquo;$-$” (Decrease) und $\rm o$ (Reset to default)

$\hspace{1.5cm}$Move with &bdquo;$\leftarrow$” (Section to the left, ordinate to the right), &bdquo;$\uparrow$” &bdquo;$\downarrow$” &bdquo;$\rightarrow$”

    '''(I)'''     Experiment section:   Task selection and task

    '''(J)'''     Experiment section:  solution
 

==About the Authors==
This interactive calculation was designed and realized at the [http://www.lnt.ei.tum.de/startseite Lehrstuhl für Nachrichtentechnik] of the [https://www.tum.de/ Technical University of Munich] .
*The original version was created in 2005 by [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Studierende#Ji_Li_.28Bachelorarbeit_EI_2003.2C_Diplomarbeit_EI_2005.29|Ji Li]] as part of her Diploma thesis using &bdquo;FlashMX–Actionscript” (Supervisor: [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Mitarbeiter_und_Dozenten#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28am_LNT_seit_1974.29|Günter Söder]]).
*In 2018 this Applet was redesigned and updated to &bdquo;HTML5” by [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Studierende#Xiaohan_Liu_.28Bachelorarbeit_2018.29|Xiaohan Liu]] as part of her Bachelor's thesis (Supervisor: [[Biografien_und_Bibliografien/Beteiligte_der_Professur_Leitungsgebundene_%C3%9Cbertragungstechnik#Tasn.C3.A1d_Kernetzky.2C_M.Sc._.28bei_L.C3.9CT_seit_2014.29|Tasnád Kernetzky]]).

==Once again: Open Applet in new Tab==

{{LntAppletLinkEn|physAnSignal}}

[[Category:Applets|^Verzerrungen^]]

Applets:Bessel Functions of the First Kind

2018-08-26T16:27:58Z

Xiaohan: /* Applications of the Bessel Functions */

{{LntAppletLink|bessel}}

==Applet Description==
 
This applet allows the calculation and graphical representation of the Bessel functions of the first kind and $n$–th order according to the series representation:

:$${\rm J}_n (x) = \sum\limits_{k=0}^{\infty}\frac{(-1)^k \cdot (x/2)^{n \hspace{0.05cm} + \hspace{0.05cm} 2
\hspace{0.02cm}\cdot \hspace{0.05cm}k}}{k! \cdot (n+k)!} \hspace{0.05cm}.$$

*The functions ${\rm J}_n (x)$ can be represented graphically for the order $n=0$ to $n=9$ in different colors.
*The left output provides the function values ${\rm J}_0 (x = x_1)$, ... , ${\rm J}_9 (x = x_1)$ for a slider-settable value $x_1$ in the range $0 \le x_1 \le 15$ with increment $0.5$.
*The right output provides the function values ${\rm J}_0 (x = x_2)$, ... , ${\rm J}_9 (x = x_2)$ for a slider-settable value $x_2$ (same range and value) Increment like on the left).

[[Applets:Besselfunktionen_erster_Art_(neues_Applet)|'''German description''']]

==Theoretical Background==
 
===General Information about the Bessel Functions===
Bessel functions (or cylinder functions) are solutions of the Bessel differential equation of the form

:$$x^2 \cdot \frac{ {\rm d}^2}{{\rm d}x^2}\ {\rm J}_n (x) \ + \ x \cdot \frac{ {\rm d}}{{\rm d}x}\ {\rm J}_n (x) \ + \ (x^2 - n^2)
\cdot {\rm J}_n (x)= 0. $$

This is an ordinary second-order linear differential equation. The parameter $ n $ is usually integer, also in this program. These mathematical functions, introduced by [https://de.wikipedia.org/wiki/Friedrich_Wilhelm_Bessel Friedrich Wilhelm Bessel] in 1844, can also be represented in closed form as integrals:

:$${\rm J}_n (x) = \frac{1}{2\pi}\cdot \int_{-\pi}^{+\pi} {{\rm e}^{\hspace{0.05cm}{\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}[\hspace{0.05cm}x \hspace{0.05cm}\cdot \hspace{0.05cm}\sin(\alpha) -\hspace{0.05cm} n \hspace{0.05cm}\cdot \hspace{0.05cm}\alpha \hspace{0.05cm}]}}\hspace{0.1cm}{\rm d}\alpha
\hspace{0.05cm}.$$

The functions ${\rm J}_n (x)$ belong to the class of Bessel functions of the first kind (German:   ''Besselfunktionen erster Art''). The parameter $n$ is called the ''Order''.

''Annotation:''   There are a number of modifications of the Bessel functions, including the second kind Bessel functions named ${\rm Y}_n (x)$. For integer $n$, ${\rm Y}_n (x)$ can be replaced by ${\rm J}_n (x)$–functions. However, in this applet, only the first kind Bessel functions are   ⇒   ${\rm J}_n (x)$ is considered.
 

===Properties of the Bessel Functions===

{{BlaueBox|TEXT=
$\text{Property (A):}$   If the function values for $n = 0$ and $n = 1$ are known, then the Bessel function for $n ≥ 2$ can be determined iteratively:
:$${\rm J}_n (x) ={2 \cdot (n-1)}/{x} \cdot {\rm J}_{n-1} (x) - {\rm J}_{n-2} (x) \hspace{0.05cm}.$$
}}

{{GraueBox|TEXT=
$\text{Example (A):}$   Let ${\rm J}_0 (x = 2) = 0.22389$ and ${\rm J}_1 (x= 2) = 0.57672$. From this it can be calculated iteratively:
:$${\rm J}_2 (x= 2) ={2 \cdot 1}/{2} \cdot {\rm J}_{1} (x= 2) - {\rm J}_{0} (x= 2) = 0.57672 - 0.22389 = \hspace{0.15cm}\underline{0.35283}\hspace{0.05cm},$$
:$${\rm J}_3 (x= 2) ={2 \cdot 2}/{2} \cdot {\rm J}_{2} (x= 2) - {\rm J}_{1} (x= 2) = 2 \cdot 0.35283 - 0.57672 = \hspace{0.15cm}\underline{0.12894}\hspace{0.05cm},$$
:$${\rm J}_4 (x= 2) ={2 \cdot 3}/{2} \cdot {\rm J}_{3} (x= 2) - {\rm J}_{2} (x= 2) = 3 \cdot 0.12894 - 0.35283 = \hspace{0.15cm}\underline{0.03400}\hspace{0.05cm}.$$
}}

{{BlaueBox|TEXT=
$\text{Property (B):}$   The symmetry relationship applies ${\rm J}_{–n}(x) = (–1)^n · {\rm J}_n(x)$:
:$${\rm J}_{-1}(x) = - {\rm J}_{1}(x), \hspace{0.3cm}{\rm J}_{-2}(x) = {\rm J}_{2}(x), \hspace{0.3cm}{\rm J}_{-3}(x) = - {\rm J}_{3}(x).$$
}}

{{GraueBox|TEXT=
$\text{Example (B):}$   For the spectrum of the analytic signal, the following applies to [[Modulationsverfahren/Phasenmodulation_(PM)#Spektralfunktion_eines_phasenmodulierten_Sinussignals|phase modulation of a sinusoidal signal]]:
[[File:Mod_T_3_1_S4_version2.png|right|frame|Spectrum of the analytic signal in phase modulation]]
:$$S_{\rm +}(f) = A_{\rm T} \cdot \sum_{n = - \infty}^{+\infty}{\rm J}_n (\eta) \cdot \delta \big[f - (f_{\rm T}+ n \cdot f_{\rm N})\big]\hspace{0.05cm}.$$
Denote this
*$f_{\rm T}$ the carrier frequency,
*$f_{\rm N}$ the message frequency,
* $A_{\rm T}$ the carrier amplitude.

The parameter of the Bessel functions in this application is the modulation index $\eta$.

Based on the graphic, the following statements are possible:
*$S_+(f)$ consists here of infinitely many discrete lines at a distance of $f_{\rm N}$.
*It is thus in principle infinitely extended.
*The weights of the spectral lines at $f_{\rm T} + n · f_{\rm N}$ ($n$ integer) are determined by the modulation index $η$ over the Bessel functions ${\rm J}_n(η)$.
*The spectral lines are real in the case of a sinusoidal source signal and a cosinusoidal carrier and symmetrical about $f_{\rm T}$ for even $n$.
*In the case of odd $n$, a change of sign corresponding to $\text{Property (B)}$ must be taken into account.
*The phase modulation of another phase oscillation of source and / or carrier signal provides the same magnitude spectrum.}}
 

===Applications of the Bessel Functions===

The applications of the Bessel functions in nature and engineering are diverse and play an important role in physics, for example:
* Investigation of natural vibrations of cylindrical resonators,
* Solution of the radial Schrödinger-equation,
* Sound pressure amplitudes of low-viscosity rotational flows,
* Heat conduction in cylindrical bodies,
* Scattering problem of a grid,
* Dynamics of oscillating bodies,
* Angular resolution.

The Bessel functions are counted among the special functions because of their many applications in mathematical physics.

In the following, we restrict ourselves to a few areas that are addressed in our tutorial $\rm LNTwww$.

'''In the English original'''
Electromagnetic waves in a cylindrical waveguide
Pressure amplitudes of inviscid rotational flows
Heat conduction in a cylindrical object
Modes of vibration of a thin circular (or annular) acoustic membrane (such as a drum or other membranophone)
Diffusion problems on a lattice
Solutions to the radial Schrödinger equation (in spherical and cylindrical coordinates) for a free particle
Solving for patterns of acoustical radiation
Frequency-dependent friction in circular pipelines
Dynamics of floating bodies
Angular resolution
'''The Ende'''

{{GraueBox|TEXT=
$\text{Example (C):} \hspace{0.5cm} \text{Use in Spectral Analysis} \ \Rightarrow \ \text{Kaiser-Bessel filter}$

The '''spectral leakage effect''' is the falsification of the spectrum of a periodic and thereby time unlimited signal due to the implicit time limitation of the discrete Fourier transform (DFT). This will be done, for example, by a spectrum analyzer
* in the time signal not existing frequency components simulated, and/or
* actually existing spectral components are obscured by sidelobes.

The task of the [[Signaldarstellung/Spektralanalyse|spectral analysis]] is to limit the influence of the ''spectral leakage effect'' by providing suitable window functions.

Such a window function provides, for example, the Kaiser–Bessel window   ⇒   see section [[Signaldarstellung/Spektralanalyse#Spezielle_Fensterfunktionen|Special Window Functions]]. Its time-discrete window function with the Bessel function zero order   ⇒   ${\rm J}_0(x)$, the parameter $\alpha=3.5$ and the window length $N$:
:$$w_\nu = \frac{ {\rm J}_0\big(\pi \cdot \alpha \cdot \sqrt{1 - (2\nu/N)^2}\big)}{ {\rm J}_0\big(\pi \cdot \alpha \big)}.$$
On the page [[Signaldarstellung/Spektralanalyse#G.C3.BCtekriterien_von_Fensterfunktionen|Quality Criteria of Window Functions]] and other the parameters of the Kaiser-Bessel window are given:
*Conveniently, the large are &bdquo;minimum distance between the main lobe and side lobes” and the desired small &bdquo;maximum scaling error”.
*Due to the very large &bdquo;equivalent noise width” the Kaiser-Bessel window cuts in the most important comparison criterion &bdquo;maximum process loss” but worse than the established Hamming and Hanning windows.}}

{{GraueBox|TEXT=
$\text{Example (D):} \hspace{0.5cm} \text{Rice-Fading Channel Model}$

The [[Mobile_Kommunikation/Wahrscheinlichkeitsdichte_des_Rayleigh%E2%80%93Fadings#Allgemeine_Beschreibung_des_Mobilfunkkanals| Rayleigh - Distribution]] describes the mobile channel on the assumption that there is no direct path and thus the multiplicative factor $z(t) = x(t) + {\rm j} \cdot y(t)$ is composed solely of diffusely scattered components.

In the case of a direct component (English: Line of Sight, LoS) in the model for the mean-free Gaussian processes $x(t)$ and $y(t)$ one has to add equal components $x_0$ and/or $y_0$ :

[[File:P ID2126 Mob T 1 4 S1 v3.png|right|frame|Rice-Fading channel model|class=fit]]
$$\hspace{0.2cm}x(t) \hspace{0.1cm} \Rightarrow \hspace{0.1cm} x(t) +x_0 \hspace{0.05cm}, \hspace{0.2cm} y(t) \hspace{0.1cm} \Rightarrow \hspace{0.1cm} y(t) +y_0\hspace{0.05cm},$$

$$\hspace{0.2cm}z(t) = x(t) + {\rm j} \cdot y(t) \hspace{0.1cm} \Rightarrow \hspace{0.1cm} z(t) +z_0 \hspace{0.05cm},\hspace{0.2cm}
z_0 = x_0 + {\rm j} \cdot y_0\hspace{0.05cm}.$$

The graph shows the ''Rice–Fading channel model''. It can be summarized as follows:
*The real part $x(t)$ is Gaussianized with mean $x_0$ and variance $\sigma ^2$.
*The imaginary part $y(t)$ is also Gaussian (mean $y_0$, equal variance $\sigma ^2$) and independent of $x(t)$. 

*For $z_0 \ne 0$, the amount $\vert z(t)\vert$ rice is distributed, from which the term &bdquo;Rice–Fading” arises.

*To simplify the notation, we set $\vert z(t)\vert = a(t)$. For $a < 0$ , the amounts - WDF are $f_a(a) \equiv 0$, for $a \ge 0$ the following equation holds, where ${\rm I_0}(x)$ is the modified Besselfunktion zeroth order denotes:
:$$f_a(a) = \frac{a}{\sigma^2} \cdot {\rm e}^{ - (a^2 + \vert z_0 \vert ^2)/(2\sigma^2)} \cdot {\rm I}_0 \left [ \frac{a \cdot \vert z_0 \vert}{\sigma^2} \right ] \hspace{0.5cm}\text{with}\hspace{0.5cm}{\rm I }_0 (u) = {\rm J }_0 ({\rm j} \cdot u) =
\sum_{k = 0}^{\infty} \frac{ (u/2)^{2k} }{k! \cdot \Gamma (k+1)}
\hspace{0.05cm}.$$
*Between the modified Bessel function and the traditional Bessel function ${\rm I_0}(x)$ – in each case of first kind – So the connection exists ${\rm I }_0 (u) = {\rm J }_0 ({\rm j} \cdot u)$.}}

{{GraueBox|TEXT=
$\text{Example (E):} \hspace{0.5cm} \text{Analysis of the Frequency Spectrum of Frequency Modulated Signals}$

$\text{Example (B)}$ has already been shown that the angle modulation of a harmonic oscillation of the frequency $f_{\rm N}$ leads to a line spectrum. The spectral lines are around the carrier frequency $f_{\rm T}$ bei $f_{\rm T} + n \cdot f_{\rm N}$ with $n \in \{ \ \text{...}, -2, -1, \ 0, +1, +2, \text{...} \ \}$. The weights of the Dirac lines are ${\rm J }_n(\eta)$, depending on the modulation index $\eta$.

[[File:P_ID1095__Mod_T_3_2_S4_neu.png|center|frame|Discrete spectra with phase modulation (left) and frequency modulation (right)]]

The graph shows the magnitude spectrum $\vert S_{\rm +}(f) \vert$ of the analytic signal in phase modulation (PM) and frequency modulation (FM), two different forms of angle modulation (WM). Bessellines with values less than $0.03$ are neglected in both cases.

For the upper half of the image, the modulator parameters are chosen so that for each $f_{\rm N} = 5 \ \rm kHz$ here is a Bessel spectrum with the modulation index $η = 1.5$. Disregarding the phase relationships, the same spectra and the same signals result for both systems.

The lower graphics apply with otherwise identical settings for the same message frequency $f_{\rm N} = 3 \ \rm kHz$. One notices:
*In phase modulation, the spectral function is narrower than $f_{\rm N} = 5 \ \rm kHz$, because the distance of the Bessel lines is now only $3 \ \rm kHz$. Since the modulation index of PM is independent of $f_{\rm N}$, the same bessel weights result as in $f_{\rm N} = 5 \ \rm kHz$.
*Also in the frequency modulation, the Bessel lines now occur at a distance of $3 \ \rm kHz$. However, since FM has a modulation index inversely proportional to $f_{\rm N}$, there are now more significantly more Bessel lines at the bottom due to the larger modulation index $η = 2.5$ than in the upper right (for $η = 1.5$ valid) chart.}}

'''Das folgende Kapitel muss noch angepasst werden!'''

==Applet Manual==
[[File:Bessel_abzug3.png|left|600px]]
    '''(A)'''     Sum formula of the Bessel functions ${\rm J}_n(x)$

    '''(B)'''     Selection of the order $n$ for the graphical representation

    '''(C)'''     Plot of the Bessel functions

    '''(D)'''     Variation of the graphic representation

$\hspace{1.5cm}$&bdquo;$+$” (Enlarge),

$\hspace{1.5cm}$ &bdquo;$-$” (Decrease)

$\hspace{1.5cm}$ &bdquo;$\rm o$” (Reset to default)

$\hspace{1.5cm}$ &bdquo;$\leftarrow$” (pushed to the left), usw.

    '''(E)'''     Selection of the abscissa value $x_1$ for the left numeric output

    '''(F)'''     Numerical output of the Bessel function values ${\rm J}_n(x_1)$

    '''(G)'''     Selection of the abscissa value $x_2$ for the right numeric output

    '''(F)'''     Numerical output of the Bessel function values ${\rm J}_n(x_2)$
 

==About the Authors==
This interactive calculation was designed and realized at the [http://www.lnt.ei.tum.de/startseite Lehrstuhl für Nachrichtentechnik] of the [https://www.tum.de/ Technical University of Munich] .
*The original version was created in 2005 by [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Studierende#Markus_Elsberger_.28Diplomarbeit_LB_2006.29|Markus Elsberger]] and [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Studierende#Slim_Lamine_.28Studienarbeit_EI_2006.29|Slim Lamine]] as part of thesis using &bdquo;FlashMX–Actionscript” (Supervisor: [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Mitarbeiter_und_Dozenten#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28am_LNT_seit_1974.29|Günter Söder]]).
*In 2018 this Applet was redesigned and updated to &bdquo;HTML5” by [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Studierende#Xiaohan_Liu_.28Bachelorarbeit_2018.29|Xiaohan Liu]] as part of her Bachelor's thesis (Supervisor: [[Biografien_und_Bibliografien/Beteiligte_der_Professur_Leitungsgebundene_%C3%9Cbertragungstechnik#Tasn.C3.A1d_Kernetzky.2C_M.Sc._.28bei_L.C3.9CT_seit_2014.29|Tasnád Kernetzky]]).

==Nochmalige Aufrufmöglichkeit des Applets in neuem Fenster==

{{LntAppletLink|bessel}}

Applets:Bessel Functions of the First Kind

2018-08-26T15:06:12Z

Xiaohan: /* Properties of the Bessel Functions */

{{LntAppletLink|bessel}}

==Applet Description==
 
This applet allows the calculation and graphical representation of the Bessel functions of the first kind and $n$–th order according to the series representation:

:$${\rm J}_n (x) = \sum\limits_{k=0}^{\infty}\frac{(-1)^k \cdot (x/2)^{n \hspace{0.05cm} + \hspace{0.05cm} 2
\hspace{0.02cm}\cdot \hspace{0.05cm}k}}{k! \cdot (n+k)!} \hspace{0.05cm}.$$

*The functions ${\rm J}_n (x)$ can be represented graphically for the order $n=0$ to $n=9$ in different colors.
*The left output provides the function values ${\rm J}_0 (x = x_1)$, ... , ${\rm J}_9 (x = x_1)$ for a slider-settable value $x_1$ in the range $0 \le x_1 \le 15$ with increment $0.5$.
*The right output provides the function values ${\rm J}_0 (x = x_2)$, ... , ${\rm J}_9 (x = x_2)$ for a slider-settable value $x_2$ (same range and value) Increment like on the left).

[[Applets:Besselfunktionen_erster_Art_(neues_Applet)|'''German description''']]

==Theoretical Background==
 
===General Information about the Bessel Functions===
Bessel functions (or cylinder functions) are solutions of the Bessel differential equation of the form

:$$x^2 \cdot \frac{ {\rm d}^2}{{\rm d}x^2}\ {\rm J}_n (x) \ + \ x \cdot \frac{ {\rm d}}{{\rm d}x}\ {\rm J}_n (x) \ + \ (x^2 - n^2)
\cdot {\rm J}_n (x)= 0. $$

This is an ordinary second-order linear differential equation. The parameter $ n $ is usually integer, also in this program. These mathematical functions, introduced by [https://de.wikipedia.org/wiki/Friedrich_Wilhelm_Bessel Friedrich Wilhelm Bessel] in 1844, can also be represented in closed form as integrals:

:$${\rm J}_n (x) = \frac{1}{2\pi}\cdot \int_{-\pi}^{+\pi} {{\rm e}^{\hspace{0.05cm}{\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}[\hspace{0.05cm}x \hspace{0.05cm}\cdot \hspace{0.05cm}\sin(\alpha) -\hspace{0.05cm} n \hspace{0.05cm}\cdot \hspace{0.05cm}\alpha \hspace{0.05cm}]}}\hspace{0.1cm}{\rm d}\alpha
\hspace{0.05cm}.$$

The functions ${\rm J}_n (x)$ belong to the class of Bessel functions of the first kind (German:   ''Besselfunktionen erster Art''). The parameter $n$ is called the ''Order''.

''Annotation:''   There are a number of modifications of the Bessel functions, including the second kind Bessel functions named ${\rm Y}_n (x)$. For integer $n$, ${\rm Y}_n (x)$ can be replaced by ${\rm J}_n (x)$–functions. However, in this applet, only the first kind Bessel functions are   ⇒   ${\rm J}_n (x)$ is considered.
 

===Properties of the Bessel Functions===

{{BlaueBox|TEXT=
$\text{Property (A):}$   If the function values for $n = 0$ and $n = 1$ are known, then the Bessel function for $n ≥ 2$ can be determined iteratively:
:$${\rm J}_n (x) ={2 \cdot (n-1)}/{x} \cdot {\rm J}_{n-1} (x) - {\rm J}_{n-2} (x) \hspace{0.05cm}.$$
}}

{{GraueBox|TEXT=
$\text{Example (A):}$   Let ${\rm J}_0 (x = 2) = 0.22389$ and ${\rm J}_1 (x= 2) = 0.57672$. From this it can be calculated iteratively:
:$${\rm J}_2 (x= 2) ={2 \cdot 1}/{2} \cdot {\rm J}_{1} (x= 2) - {\rm J}_{0} (x= 2) = 0.57672 - 0.22389 = \hspace{0.15cm}\underline{0.35283}\hspace{0.05cm},$$
:$${\rm J}_3 (x= 2) ={2 \cdot 2}/{2} \cdot {\rm J}_{2} (x= 2) - {\rm J}_{1} (x= 2) = 2 \cdot 0.35283 - 0.57672 = \hspace{0.15cm}\underline{0.12894}\hspace{0.05cm},$$
:$${\rm J}_4 (x= 2) ={2 \cdot 3}/{2} \cdot {\rm J}_{3} (x= 2) - {\rm J}_{2} (x= 2) = 3 \cdot 0.12894 - 0.35283 = \hspace{0.15cm}\underline{0.03400}\hspace{0.05cm}.$$
}}

{{BlaueBox|TEXT=
$\text{Property (B):}$   The symmetry relationship applies ${\rm J}_{–n}(x) = (–1)^n · {\rm J}_n(x)$:
:$${\rm J}_{-1}(x) = - {\rm J}_{1}(x), \hspace{0.3cm}{\rm J}_{-2}(x) = {\rm J}_{2}(x), \hspace{0.3cm}{\rm J}_{-3}(x) = - {\rm J}_{3}(x).$$
}}

{{GraueBox|TEXT=
$\text{Example (B):}$   For the spectrum of the analytic signal, the following applies to [[Modulationsverfahren/Phasenmodulation_(PM)#Spektralfunktion_eines_phasenmodulierten_Sinussignals|phase modulation of a sinusoidal signal]]:
[[File:Mod_T_3_1_S4_version2.png|right|frame|Spectrum of the analytic signal in phase modulation]]
:$$S_{\rm +}(f) = A_{\rm T} \cdot \sum_{n = - \infty}^{+\infty}{\rm J}_n (\eta) \cdot \delta \big[f - (f_{\rm T}+ n \cdot f_{\rm N})\big]\hspace{0.05cm}.$$
Denote this
*$f_{\rm T}$ the carrier frequency,
*$f_{\rm N}$ the message frequency,
* $A_{\rm T}$ the carrier amplitude.

The parameter of the Bessel functions in this application is the modulation index $\eta$.

Based on the graphic, the following statements are possible:
*$S_+(f)$ consists here of infinitely many discrete lines at a distance of $f_{\rm N}$.
*It is thus in principle infinitely extended.
*The weights of the spectral lines at $f_{\rm T} + n · f_{\rm N}$ ($n$ integer) are determined by the modulation index $η$ over the Bessel functions ${\rm J}_n(η)$.
*The spectral lines are real in the case of a sinusoidal source signal and a cosinusoidal carrier and symmetrical about $f_{\rm T}$ for even $n$.
*In the case of odd $n$, a change of sign corresponding to $\text{Property (B)}$ must be taken into account.
*The phase modulation of another phase oscillation of source and / or carrier signal provides the same magnitude spectrum.}}
 

===Applications of the Bessel Functions===

The applications of the Bessel functions in nature–and engineering are diverse and play an important role in physics, for example:
* Examination of natural vibrations of cylindrical resonators,
* Solution of the radial Schrödinger-equation,
* Sound pressure amplitudes of low-viscosity rotational flows,
* Heat conduction in cylindrical bodies,
* Scattering problem of a grid,
* Dynamics of oscillating bodies,
* Angular resolution.

The Bessel functions are counted among the special functions because of their many applications in mathematical physics.

In the following, we restrict ourselves to a few areas that are addressed in our tutorial $\rm LNTwww$.

'''In the English original'''
Electromagnetic waves in a cylindrical waveguide
Pressure amplitudes of inviscid rotational flows
Heat conduction in a cylindrical object
Modes of vibration of a thin circular (or annular) acoustic membrane (such as a drum or other membranophone)
Diffusion problems on a lattice
Solutions to the radial Schrödinger equation (in spherical and cylindrical coordinates) for a free particle
Solving for patterns of acoustical radiation
Frequency-dependent friction in circular pipelines
Dynamics of floating bodies
Angular resolution
'''The Ende'''

{{GraueBox|TEXT=
$\text{Example (C):} \hspace{0.5cm} \text{Use in spectral analysis} \ \Rightarrow \ \text{Kaiser-Bessel filter}$

The '''spectral leakage effect''' is the falsification of the spectrum of a periodic and hence temporally unlimited signal due to the implicit time limitation of the discrete Fourier transform (DFT). This will be done, for example, by a spectrum analyzer
* in the time signal not existing frequency components simulated, and/or
* actually existing spectral components are obscured by sidelobes.

The task of the [[Signaldarstellung/Spektralanalyse|spectral analysis]] is to limit the influence of the ''spectral leakage effect'' by providing suitable window functions.

Such a window function provides, for example, the Kaiser–Bessel–window   ⇒   see section [[Signaldarstellung/Spektralanalyse#Spezielle_Fensterfunktionen|Special Window Functions]]. Its time-discrete Fenser function reads with the Bessel function zero order   ⇒   ${\rm J}_0(x)$, the parameter $\alpha=3.5$ and the window length $N$:
:$$w_\nu = \frac{ {\rm J}_0\big(\pi \cdot \alpha \cdot \sqrt{1 - (2\nu/N)^2}\big)}{ {\rm J}_0\big(\pi \cdot \alpha \big)}.$$
On the page [[Signaldarstellung/Spektralanalyse#G.C3.BCtekriterien_von_Fensterfunktionen|Quality Criteria of Window Functions]] and other the characteristics of the Kaiser-Bessel-window are given:
*Conveniently, the large is &bdquo;minimum distance between the main lobe and side lobes” and the desired small &bdquo;maximum scaling error”.
*Due to the very large &bdquo;equivalent noise width” the Kaiser-Bessel window cuts in the main comparison criterion &bdquo;maximum process loss” but worse off than the established Hamming– and Hanning–windows.}}

{{GraueBox|TEXT=
$\text{Example (D):} \hspace{0.5cm} \text{Rice-Fading channel model}$

The [[Mobile_Kommunikation/Wahrscheinlichkeitsdichte_des_Rayleigh%E2%80%93Fadings#Allgemeine_Beschreibung_des_Mobilfunkkanals| Rayleigh - Distribution]] describes the mobile channel assuming that there is no direct path, and thus the multiplicative factor $z(t) = x(t) + {\rm j} \cdot y(t)$ is solely diffuse composed of scattered components.

In the case of a direct component (English: Line of Sight, LoS) in the model for the mean-free Gaussian processes $x(t)$ and $y(t)$ one has to add equal components $x_0$ and/or $y_0$ :

[[File:P ID2126 Mob T 1 4 S1 v3.png|right|frame|Rice-Fading channel model|class=fit]]
$$\hspace{0.2cm}x(t) \hspace{0.1cm} \Rightarrow \hspace{0.1cm} x(t) +x_0 \hspace{0.05cm}, \hspace{0.2cm} y(t) \hspace{0.1cm} \Rightarrow \hspace{0.1cm} y(t) +y_0\hspace{0.05cm},$$

$$\hspace{0.2cm}z(t) = x(t) + {\rm j} \cdot y(t) \hspace{0.1cm} \Rightarrow \hspace{0.1cm} z(t) +z_0 \hspace{0.05cm},\hspace{0.2cm}
z_0 = x_0 + {\rm j} \cdot y_0\hspace{0.05cm}.$$

The graph shows the ''Rice–Fading–Channel model''. It can be summarized as follows:
*The real part $x(t)$ is Gaussianized with mean $x_0$ and variance $\sigma ^2$.
*The imaginary part $y(t)$ is also Gaussian (mean $y_0$, equal variance $\sigma ^2$) and independent of $x(t)$. 

*For $z_0 \ne 0$, the amount $\vert z(t)\vert$ rice is distributed, from which the term &bdquo;Rice–Fading” arises.

*To simplify the notation, we set $\vert z(t)\vert = a(t)$. For $a < 0$ , the amounts - WDF are $f_a(a) \equiv 0$, for $a \ge 0$ the following equation holds, where ${\rm I_0}(x)$ is the modified Besselfunktion zeroth order denotes:
:$$f_a(a) = \frac{a}{\sigma^2} \cdot {\rm e}^{ - (a^2 + \vert z_0 \vert ^2)/(2\sigma^2)} \cdot {\rm I}_0 \left [ \frac{a \cdot \vert z_0 \vert}{\sigma^2} \right ] \hspace{0.5cm}\text{with}\hspace{0.5cm}{\rm I }_0 (u) = {\rm J }_0 ({\rm j} \cdot u) =
\sum_{k = 0}^{\infty} \frac{ (u/2)^{2k} }{k! \cdot \Gamma (k+1)}
\hspace{0.05cm}.$$
*Between the modified Bessel function and the traditional Bessel function ${\rm I_0}(x)$ – each first kind – So the connection exists ${\rm I }_0 (u) = {\rm J }_0 ({\rm j} \cdot u)$.}}

{{GraueBox|TEXT=
$\text{Example (E):} \hspace{0.5cm} \text{Analysis of the frequency spectrum of frequency modulated signals}$

$\text{Example (B)}$ has already shown that the angle modulation of a harmonic oscillation of the frequency $f_{\rm N}$ leads to a line spectrum. The spectral lines are around the carrier frequency $f_{\rm T}$ bei $f_{\rm T} + n \cdot f_{\rm N}$ with $n \in \{ \ \text{...}, -2, -1, \ 0, +1, +2, \text{...} \ \}$. The weights of the Dirac lines are ${\rm J }_n(\eta)$, depending on the modulation index $\eta$.

[[File:P_ID1095__Mod_T_3_2_S4_neu.png|center|frame|Discrete spectra with phase modulation (left) and frequency modulation (right)]]

The graph shows the magnitude spectrum $\vert S_{\rm +}(f) \vert$ of the phase modulation (PM) and frequency modulation (FM) analytic signal, two different forms of angle modulation (WM). Bessellines with values less than $0.03$ are neglected in both cases.

For the upper half of the image, the modulator parameters are chosen so that for each $f_{\rm N} = 5 \ \rm kHz$ here is a Bessel spectrum with the modulation index $η = 1.5$. Disregarding the phase relationships, the same spectra and the same signals result for both systems.

The lower graphics are valid for the same message frequency $f_{\rm N} = 3 \ \rm kHz$. One notices:
*In phase modulation, the spectral function is narrower than $f_{\rm N} = 5 \ \rm kHz$, because the distance of the Bessel lines is now only $3 \ \rm kHz$. Since the modulation index of PM is independent of $f_{\rm N}$, the same bessel weights result as in $f_{\rm N} = 5 \ \rm kHz$.
*Also in the frequency modulation, the Bessel lines now appear at a distance of $3 \ \rm kHz$. However, since FM has a modulation index inversely proportional to $f_{\rm N}$, there is now much more Besselline lines than in the upper right (for $η = 1.5$ valid) chart due to the larger modulation index $η = 2.5$.}}

'''Das folgende Kapitel muss noch angepasst werden!'''

==Applet Manual==
[[File:Bessel_abzug3.png|left|600px]]
    '''(A)'''     Sum formula of the Bessel functions ${\rm J}_n(x)$

    '''(B)'''     Selection of the order $n$ for the graphical representation

    '''(C)'''     Plot of the Bessel functions

    '''(D)'''     Variation of the graphic representation

$\hspace{1.5cm}$&bdquo;$+$” (Enlarge),

$\hspace{1.5cm}$ &bdquo;$-$” (Decrease)

$\hspace{1.5cm}$ &bdquo;$\rm o$” (Reset to default)

$\hspace{1.5cm}$ &bdquo;$\leftarrow$” (pushed to the left), usw.

    '''(E)'''     Selection of the abscissa value $x_1$ for the left numeric output

    '''(F)'''     Numerical output of the Bessel function values ${\rm J}_n(x_1)$

    '''(G)'''     Selection of the abscissa value $x_2$ for the right numeric output

    '''(F)'''     Numerical output of the Bessel function values ${\rm J}_n(x_2)$
 

==About the Authors==
This interactive calculation was designed and realized at the [http://www.lnt.ei.tum.de/startseite Lehrstuhl für Nachrichtentechnik] of the [https://www.tum.de/ Technical University of Munich] .
*The original version was created in 2005 by [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Studierende#Markus_Elsberger_.28Diplomarbeit_LB_2006.29|Markus Elsberger]] and [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Studierende#Slim_Lamine_.28Studienarbeit_EI_2006.29|Slim Lamine]] as part of thesis using &bdquo;FlashMX–Actionscript” (Supervisor: [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Mitarbeiter_und_Dozenten#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28am_LNT_seit_1974.29|Günter Söder]]).
*In 2018 this Applet was redesigned and updated to &bdquo;HTML5” by [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Studierende#Xiaohan_Liu_.28Bachelorarbeit_2018.29|Xiaohan Liu]] as part of her Bachelor's thesis (Supervisor: [[Biografien_und_Bibliografien/Beteiligte_der_Professur_Leitungsgebundene_%C3%9Cbertragungstechnik#Tasn.C3.A1d_Kernetzky.2C_M.Sc._.28bei_L.C3.9CT_seit_2014.29|Tasnád Kernetzky]]).

==Nochmalige Aufrufmöglichkeit des Applets in neuem Fenster==

{{LntAppletLink|bessel}}

Applets:Bessel Functions of the First Kind

2018-08-26T14:44:51Z

Xiaohan: /* Applications of the Bessel Functions */

{{LntAppletLink|bessel}}

==Applet Description==
 
This applet allows the calculation and graphical representation of the Bessel functions of the first kind and $n$–th order according to the series representation:

:$${\rm J}_n (x) = \sum\limits_{k=0}^{\infty}\frac{(-1)^k \cdot (x/2)^{n \hspace{0.05cm} + \hspace{0.05cm} 2
\hspace{0.02cm}\cdot \hspace{0.05cm}k}}{k! \cdot (n+k)!} \hspace{0.05cm}.$$

*The functions ${\rm J}_n (x)$ can be represented graphically for the order $n=0$ to $n=9$ in different colors.
*The left output provides the function values ${\rm J}_0 (x = x_1)$, ... , ${\rm J}_9 (x = x_1)$ for a slider-settable value $x_1$ in the range $0 \le x_1 \le 15$ with increment $0.5$.
*The right output provides the function values ${\rm J}_0 (x = x_2)$, ... , ${\rm J}_9 (x = x_2)$ for a slider-settable value $x_2$ (same range and value) Increment like on the left).

[[Applets:Besselfunktionen_erster_Art_(neues_Applet)|'''German description''']]

==Theoretical Background==
 
===General Information about the Bessel Functions===
Bessel functions (or cylinder functions) are solutions of the Bessel differential equation of the form

:$$x^2 \cdot \frac{ {\rm d}^2}{{\rm d}x^2}\ {\rm J}_n (x) \ + \ x \cdot \frac{ {\rm d}}{{\rm d}x}\ {\rm J}_n (x) \ + \ (x^2 - n^2)
\cdot {\rm J}_n (x)= 0. $$

This is an ordinary second-order linear differential equation. The parameter $ n $ is usually integer, also in this program. These mathematical functions, introduced by [https://de.wikipedia.org/wiki/Friedrich_Wilhelm_Bessel Friedrich Wilhelm Bessel] in 1844, can also be represented in closed form as integrals:

:$${\rm J}_n (x) = \frac{1}{2\pi}\cdot \int_{-\pi}^{+\pi} {{\rm e}^{\hspace{0.05cm}{\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}[\hspace{0.05cm}x \hspace{0.05cm}\cdot \hspace{0.05cm}\sin(\alpha) -\hspace{0.05cm} n \hspace{0.05cm}\cdot \hspace{0.05cm}\alpha \hspace{0.05cm}]}}\hspace{0.1cm}{\rm d}\alpha
\hspace{0.05cm}.$$

The functions ${\rm J}_n (x)$ belong to the class of Bessel functions of the first kind (German:   ''Besselfunktionen erster Art''). The parameter $n$ is called the ''Order''.

''Annotation:''   There are a number of modifications of the Bessel functions, including the second kind Bessel functions named ${\rm Y}_n (x)$. For integer $n$, ${\rm Y}_n (x)$ can be replaced by ${\rm J}_n (x)$–functions. However, in this applet, only the first kind Bessel functions are   ⇒   ${\rm J}_n (x)$ is considered.
 

===Properties of the Bessel Functions===

{{BlaueBox|TEXT=
$\text{Property (A):}$   If the function values for $n = 0$ and $n = 1$ are known, then the Bessel function for $n ≥ 2$ can be determined iteratively:
:$${\rm J}_n (x) ={2 \cdot (n-1)}/{x} \cdot {\rm J}_{n-1} (x) - {\rm J}_{n-2} (x) \hspace{0.05cm}.$$
}}

{{GraueBox|TEXT=
$\text{Example (A):}$   Consider ${\rm J}_0 (x = 2) = 0.22389$ and ${\rm J}_1 (x= 2) = 0.57672$. From this it can be calculated iteratively:
:$${\rm J}_2 (x= 2) ={2 \cdot 1}/{2} \cdot {\rm J}_{1} (x= 2) - {\rm J}_{0} (x= 2) = 0.57672 - 0.22389 = \hspace{0.15cm}\underline{0.35283}\hspace{0.05cm},$$
:$${\rm J}_3 (x= 2) ={2 \cdot 2}/{2} \cdot {\rm J}_{2} (x= 2) - {\rm J}_{1} (x= 2) = 2 \cdot 0.35283 - 0.57672 = \hspace{0.15cm}\underline{0.12894}\hspace{0.05cm},$$
:$${\rm J}_4 (x= 2) ={2 \cdot 3}/{2} \cdot {\rm J}_{3} (x= 2) - {\rm J}_{2} (x= 2) = 3 \cdot 0.12894 - 0.35283 = \hspace{0.15cm}\underline{0.03400}\hspace{0.05cm}.$$
}}

{{BlaueBox|TEXT=
$\text{Property (B):}$   The symmetry relationship applies ${\rm J}_{–n}(x) = (–1)^n · {\rm J}_n(x)$:
:$${\rm J}_{-1}(x) = - {\rm J}_{1}(x), \hspace{0.3cm}{\rm J}_{-2}(x) = {\rm J}_{2}(x), \hspace{0.3cm}{\rm J}_{-3}(x) = - {\rm J}_{3}(x).$$
}}

{{GraueBox|TEXT=
$\text{Example (B):}$   For the spectrum of the analytic signal, in [[Modulationsverfahren/Phasenmodulation_(PM)#Spektralfunktion_eines_phasenmodulierten_Sinussignals|phase modulation of a sinusoidal signal]]:
[[File:Mod_T_3_1_S4_version2.png|right|frame|Spectrum of the analytic signal in phase modulation]]
:$$S_{\rm +}(f) = A_{\rm T} \cdot \sum_{n = - \infty}^{+\infty}{\rm J}_n (\eta) \cdot \delta \big[f - (f_{\rm T}+ n \cdot f_{\rm N})\big]\hspace{0.05cm}.$$
Denote this
*$f_{\rm T}$ the carrier frequency,
*$f_{\rm N}$ the message frequency,
* $A_{\rm T}$ the carrier amplitude.

The parameter of the Bessel functions in this application is the modulation index $\eta$.

Based on the graphic, the following statements are possible:
*Here $S_+(f)$ consists of an infinite number of discrete lines at a distance of $f_{\rm N}$.
*It is thus in principle infinitely extended.
*The weights of the spectral lines at $f_{\rm T} + n · f_{\rm N}$ ($n$ integer) are determined by the modulation index $η$ over the Bessel functions ${\rm J}_n(η)$.
*The spectral lines are real in the case of a sinusoidal source signal and a cosinusoidal carrier and symmetric about $f_{\rm T}$ for even $n$.
*For odd $n$ , a sign change according to the $\text{Property (B)}$ has to be considered.
*The phase modulation of another phase oscillation of source and / or carrier signal provides the same magnitude spectrum.}}
 

===Applications of the Bessel Functions===

The applications of the Bessel functions in nature–and engineering are diverse and play an important role in physics, for example:
* Examination of natural vibrations of cylindrical resonators,
* Solution of the radial Schrödinger-equation,
* Sound pressure amplitudes of low-viscosity rotational flows,
* Heat conduction in cylindrical bodies,
* Scattering problem of a grid,
* Dynamics of oscillating bodies,
* Angular resolution.

The Bessel functions are counted among the special functions because of their many applications in mathematical physics.

In the following, we restrict ourselves to a few areas that are addressed in our tutorial $\rm LNTwww$.

'''In the English original'''
Electromagnetic waves in a cylindrical waveguide
Pressure amplitudes of inviscid rotational flows
Heat conduction in a cylindrical object
Modes of vibration of a thin circular (or annular) acoustic membrane (such as a drum or other membranophone)
Diffusion problems on a lattice
Solutions to the radial Schrödinger equation (in spherical and cylindrical coordinates) for a free particle
Solving for patterns of acoustical radiation
Frequency-dependent friction in circular pipelines
Dynamics of floating bodies
Angular resolution
'''The Ende'''

{{GraueBox|TEXT=
$\text{Example (C):} \hspace{0.5cm} \text{Use in spectral analysis} \ \Rightarrow \ \text{Kaiser-Bessel filter}$

The '''spectral leakage effect''' is the falsification of the spectrum of a periodic and hence temporally unlimited signal due to the implicit time limitation of the discrete Fourier transform (DFT). This will be done, for example, by a spectrum analyzer
* in the time signal not existing frequency components simulated, and/or
* actually existing spectral components are obscured by sidelobes.

The task of the [[Signaldarstellung/Spektralanalyse|spectral analysis]] is to limit the influence of the ''spectral leakage effect'' by providing suitable window functions.

Such a window function provides, for example, the Kaiser–Bessel–window   ⇒   see section [[Signaldarstellung/Spektralanalyse#Spezielle_Fensterfunktionen|Special Window Functions]]. Its time-discrete Fenser function reads with the Bessel function zero order   ⇒   ${\rm J}_0(x)$, the parameter $\alpha=3.5$ and the window length $N$:
:$$w_\nu = \frac{ {\rm J}_0\big(\pi \cdot \alpha \cdot \sqrt{1 - (2\nu/N)^2}\big)}{ {\rm J}_0\big(\pi \cdot \alpha \big)}.$$
On the page [[Signaldarstellung/Spektralanalyse#G.C3.BCtekriterien_von_Fensterfunktionen|Quality Criteria of Window Functions]] and other the characteristics of the Kaiser-Bessel-window are given:
*Conveniently, the large is &bdquo;minimum distance between the main lobe and side lobes” and the desired small &bdquo;maximum scaling error”.
*Due to the very large &bdquo;equivalent noise width” the Kaiser-Bessel window cuts in the main comparison criterion &bdquo;maximum process loss” but worse off than the established Hamming– and Hanning–windows.}}

{{GraueBox|TEXT=
$\text{Example (D):} \hspace{0.5cm} \text{Rice-Fading channel model}$

The [[Mobile_Kommunikation/Wahrscheinlichkeitsdichte_des_Rayleigh%E2%80%93Fadings#Allgemeine_Beschreibung_des_Mobilfunkkanals| Rayleigh - Distribution]] describes the mobile channel assuming that there is no direct path, and thus the multiplicative factor $z(t) = x(t) + {\rm j} \cdot y(t)$ is solely diffuse composed of scattered components.

In the case of a direct component (English: Line of Sight, LoS) in the model for the mean-free Gaussian processes $x(t)$ and $y(t)$ one has to add equal components $x_0$ and/or $y_0$ :

[[File:P ID2126 Mob T 1 4 S1 v3.png|right|frame|Rice-Fading channel model|class=fit]]
$$\hspace{0.2cm}x(t) \hspace{0.1cm} \Rightarrow \hspace{0.1cm} x(t) +x_0 \hspace{0.05cm}, \hspace{0.2cm} y(t) \hspace{0.1cm} \Rightarrow \hspace{0.1cm} y(t) +y_0\hspace{0.05cm},$$

$$\hspace{0.2cm}z(t) = x(t) + {\rm j} \cdot y(t) \hspace{0.1cm} \Rightarrow \hspace{0.1cm} z(t) +z_0 \hspace{0.05cm},\hspace{0.2cm}
z_0 = x_0 + {\rm j} \cdot y_0\hspace{0.05cm}.$$

The graph shows the ''Rice–Fading–Channel model''. It can be summarized as follows:
*The real part $x(t)$ is Gaussianized with mean $x_0$ and variance $\sigma ^2$.
*The imaginary part $y(t)$ is also Gaussian (mean $y_0$, equal variance $\sigma ^2$) and independent of $x(t)$. 

*For $z_0 \ne 0$, the amount $\vert z(t)\vert$ rice is distributed, from which the term &bdquo;Rice–Fading” arises.

*To simplify the notation, we set $\vert z(t)\vert = a(t)$. For $a < 0$ , the amounts - WDF are $f_a(a) \equiv 0$, for $a \ge 0$ the following equation holds, where ${\rm I_0}(x)$ is the modified Besselfunktion zeroth order denotes:
:$$f_a(a) = \frac{a}{\sigma^2} \cdot {\rm e}^{ - (a^2 + \vert z_0 \vert ^2)/(2\sigma^2)} \cdot {\rm I}_0 \left [ \frac{a \cdot \vert z_0 \vert}{\sigma^2} \right ] \hspace{0.5cm}\text{with}\hspace{0.5cm}{\rm I }_0 (u) = {\rm J }_0 ({\rm j} \cdot u) =
\sum_{k = 0}^{\infty} \frac{ (u/2)^{2k} }{k! \cdot \Gamma (k+1)}
\hspace{0.05cm}.$$
*Between the modified Bessel function and the traditional Bessel function ${\rm I_0}(x)$ – each first kind – So the connection exists ${\rm I }_0 (u) = {\rm J }_0 ({\rm j} \cdot u)$.}}

{{GraueBox|TEXT=
$\text{Example (E):} \hspace{0.5cm} \text{Analysis of the frequency spectrum of frequency modulated signals}$

$\text{Example (B)}$ has already shown that the angle modulation of a harmonic oscillation of the frequency $f_{\rm N}$ leads to a line spectrum. The spectral lines are around the carrier frequency $f_{\rm T}$ bei $f_{\rm T} + n \cdot f_{\rm N}$ with $n \in \{ \ \text{...}, -2, -1, \ 0, +1, +2, \text{...} \ \}$. The weights of the Dirac lines are ${\rm J }_n(\eta)$, depending on the modulation index $\eta$.

[[File:P_ID1095__Mod_T_3_2_S4_neu.png|center|frame|Discrete spectra with phase modulation (left) and frequency modulation (right)]]

The graph shows the magnitude spectrum $\vert S_{\rm +}(f) \vert$ of the phase modulation (PM) and frequency modulation (FM) analytic signal, two different forms of angle modulation (WM). Bessellines with values less than $0.03$ are neglected in both cases.

For the upper half of the image, the modulator parameters are chosen so that for each $f_{\rm N} = 5 \ \rm kHz$ here is a Bessel spectrum with the modulation index $η = 1.5$. Disregarding the phase relationships, the same spectra and the same signals result for both systems.

The lower graphics are valid for the same message frequency $f_{\rm N} = 3 \ \rm kHz$. One notices:
*In phase modulation, the spectral function is narrower than $f_{\rm N} = 5 \ \rm kHz$, because the distance of the Bessel lines is now only $3 \ \rm kHz$. Since the modulation index of PM is independent of $f_{\rm N}$, the same bessel weights result as in $f_{\rm N} = 5 \ \rm kHz$.
*Also in the frequency modulation, the Bessel lines now appear at a distance of $3 \ \rm kHz$. However, since FM has a modulation index inversely proportional to $f_{\rm N}$, there is now much more Besselline lines than in the upper right (for $η = 1.5$ valid) chart due to the larger modulation index $η = 2.5$.}}

'''Das folgende Kapitel muss noch angepasst werden!'''

==Applet Manual==
[[File:Bessel_abzug3.png|left|600px]]
    '''(A)'''     Sum formula of the Bessel functions ${\rm J}_n(x)$

    '''(B)'''     Selection of the order $n$ for the graphical representation

    '''(C)'''     Plot of the Bessel functions

    '''(D)'''     Variation of the graphic representation

$\hspace{1.5cm}$&bdquo;$+$” (Enlarge),

$\hspace{1.5cm}$ &bdquo;$-$” (Decrease)

$\hspace{1.5cm}$ &bdquo;$\rm o$” (Reset to default)

$\hspace{1.5cm}$ &bdquo;$\leftarrow$” (pushed to the left), usw.

    '''(E)'''     Selection of the abscissa value $x_1$ for the left numeric output

    '''(F)'''     Numerical output of the Bessel function values ${\rm J}_n(x_1)$

    '''(G)'''     Selection of the abscissa value $x_2$ for the right numeric output

    '''(F)'''     Numerical output of the Bessel function values ${\rm J}_n(x_2)$
 

==About the Authors==
This interactive calculation was designed and realized at the [http://www.lnt.ei.tum.de/startseite Lehrstuhl für Nachrichtentechnik] of the [https://www.tum.de/ Technical University of Munich] .
*The original version was created in 2005 by [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Studierende#Markus_Elsberger_.28Diplomarbeit_LB_2006.29|Markus Elsberger]] and [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Studierende#Slim_Lamine_.28Studienarbeit_EI_2006.29|Slim Lamine]] as part of thesis using &bdquo;FlashMX–Actionscript” (Supervisor: [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Mitarbeiter_und_Dozenten#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28am_LNT_seit_1974.29|Günter Söder]]).
*In 2018 this Applet was redesigned and updated to &bdquo;HTML5” by [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Studierende#Xiaohan_Liu_.28Bachelorarbeit_2018.29|Xiaohan Liu]] as part of her Bachelor's thesis (Supervisor: [[Biografien_und_Bibliografien/Beteiligte_der_Professur_Leitungsgebundene_%C3%9Cbertragungstechnik#Tasn.C3.A1d_Kernetzky.2C_M.Sc._.28bei_L.C3.9CT_seit_2014.29|Tasnád Kernetzky]]).

==Nochmalige Aufrufmöglichkeit des Applets in neuem Fenster==

{{LntAppletLink|bessel}}

Applets:Bessel Functions of the First Kind

2018-08-26T14:42:28Z

Xiaohan: /* General Information about the Bessel Functions */

{{LntAppletLink|bessel}}

==Applet Description==
 
This applet allows the calculation and graphical representation of the Bessel functions of the first kind and $n$–th order according to the series representation:

:$${\rm J}_n (x) = \sum\limits_{k=0}^{\infty}\frac{(-1)^k \cdot (x/2)^{n \hspace{0.05cm} + \hspace{0.05cm} 2
\hspace{0.02cm}\cdot \hspace{0.05cm}k}}{k! \cdot (n+k)!} \hspace{0.05cm}.$$

*The functions ${\rm J}_n (x)$ can be represented graphically for the order $n=0$ to $n=9$ in different colors.
*The left output provides the function values ${\rm J}_0 (x = x_1)$, ... , ${\rm J}_9 (x = x_1)$ for a slider-settable value $x_1$ in the range $0 \le x_1 \le 15$ with increment $0.5$.
*The right output provides the function values ${\rm J}_0 (x = x_2)$, ... , ${\rm J}_9 (x = x_2)$ for a slider-settable value $x_2$ (same range and value) Increment like on the left).

[[Applets:Besselfunktionen_erster_Art_(neues_Applet)|'''German description''']]

==Theoretical Background==
 
===General Information about the Bessel Functions===
Bessel functions (or cylinder functions) are solutions of the Bessel differential equation of the form

:$$x^2 \cdot \frac{ {\rm d}^2}{{\rm d}x^2}\ {\rm J}_n (x) \ + \ x \cdot \frac{ {\rm d}}{{\rm d}x}\ {\rm J}_n (x) \ + \ (x^2 - n^2)
\cdot {\rm J}_n (x)= 0. $$

This is an ordinary second-order linear differential equation. The parameter $ n $ is usually integer, also in this program. These mathematical functions, introduced by [https://de.wikipedia.org/wiki/Friedrich_Wilhelm_Bessel Friedrich Wilhelm Bessel] in 1844, can also be represented in closed form as integrals:

:$${\rm J}_n (x) = \frac{1}{2\pi}\cdot \int_{-\pi}^{+\pi} {{\rm e}^{\hspace{0.05cm}{\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}[\hspace{0.05cm}x \hspace{0.05cm}\cdot \hspace{0.05cm}\sin(\alpha) -\hspace{0.05cm} n \hspace{0.05cm}\cdot \hspace{0.05cm}\alpha \hspace{0.05cm}]}}\hspace{0.1cm}{\rm d}\alpha
\hspace{0.05cm}.$$

The functions ${\rm J}_n (x)$ belong to the class of Bessel functions of the first kind (German:   ''Besselfunktionen erster Art''). The parameter $n$ is called the ''Order''.

''Annotation:''   There are a number of modifications of the Bessel functions, including the second kind Bessel functions named ${\rm Y}_n (x)$. For integer $n$, ${\rm Y}_n (x)$ can be replaced by ${\rm J}_n (x)$–functions. However, in this applet, only the first kind Bessel functions are   ⇒   ${\rm J}_n (x)$ is considered.
 

===Properties of the Bessel Functions===

{{BlaueBox|TEXT=
$\text{Property (A):}$   If the function values for $n = 0$ and $n = 1$ are known, then the Bessel function for $n ≥ 2$ can be determined iteratively:
:$${\rm J}_n (x) ={2 \cdot (n-1)}/{x} \cdot {\rm J}_{n-1} (x) - {\rm J}_{n-2} (x) \hspace{0.05cm}.$$
}}

{{GraueBox|TEXT=
$\text{Example (A):}$   Consider ${\rm J}_0 (x = 2) = 0.22389$ and ${\rm J}_1 (x= 2) = 0.57672$. From this it can be calculated iteratively:
:$${\rm J}_2 (x= 2) ={2 \cdot 1}/{2} \cdot {\rm J}_{1} (x= 2) - {\rm J}_{0} (x= 2) = 0.57672 - 0.22389 = \hspace{0.15cm}\underline{0.35283}\hspace{0.05cm},$$
:$${\rm J}_3 (x= 2) ={2 \cdot 2}/{2} \cdot {\rm J}_{2} (x= 2) - {\rm J}_{1} (x= 2) = 2 \cdot 0.35283 - 0.57672 = \hspace{0.15cm}\underline{0.12894}\hspace{0.05cm},$$
:$${\rm J}_4 (x= 2) ={2 \cdot 3}/{2} \cdot {\rm J}_{3} (x= 2) - {\rm J}_{2} (x= 2) = 3 \cdot 0.12894 - 0.35283 = \hspace{0.15cm}\underline{0.03400}\hspace{0.05cm}.$$
}}

{{BlaueBox|TEXT=
$\text{Property (B):}$   The symmetry relationship applies ${\rm J}_{–n}(x) = (–1)^n · {\rm J}_n(x)$:
:$${\rm J}_{-1}(x) = - {\rm J}_{1}(x), \hspace{0.3cm}{\rm J}_{-2}(x) = {\rm J}_{2}(x), \hspace{0.3cm}{\rm J}_{-3}(x) = - {\rm J}_{3}(x).$$
}}

{{GraueBox|TEXT=
$\text{Example (B):}$   For the spectrum of the analytic signal, in [[Modulationsverfahren/Phasenmodulation_(PM)#Spektralfunktion_eines_phasenmodulierten_Sinussignals|phase modulation of a sinusoidal signal]]:
[[File:Mod_T_3_1_S4_version2.png|right|frame|Spectrum of the analytic signal in phase modulation]]
:$$S_{\rm +}(f) = A_{\rm T} \cdot \sum_{n = - \infty}^{+\infty}{\rm J}_n (\eta) \cdot \delta \big[f - (f_{\rm T}+ n \cdot f_{\rm N})\big]\hspace{0.05cm}.$$
Denote this
*$f_{\rm T}$ the carrier frequency,
*$f_{\rm N}$ the message frequency,
* $A_{\rm T}$ the carrier amplitude.

The parameter of the Bessel functions in this application is the modulation index $\eta$.

Based on the graphic, the following statements are possible:
*Here $S_+(f)$ consists of an infinite number of discrete lines at a distance of $f_{\rm N}$.
*It is thus in principle infinitely extended.
*The weights of the spectral lines at $f_{\rm T} + n · f_{\rm N}$ ($n$ integer) are determined by the modulation index $η$ over the Bessel functions ${\rm J}_n(η)$.
*The spectral lines are real in the case of a sinusoidal source signal and a cosinusoidal carrier and symmetric about $f_{\rm T}$ for even $n$.
*For odd $n$ , a sign change according to the $\text{Property (B)}$ has to be considered.
*The phase modulation of another phase oscillation of source and / or carrier signal provides the same magnitude spectrum.}}
 

===Applications of the Bessel Functions===

The applications of the Bessel functions in nature–and engineering are diverse and play an important role in physics, for example:
* Examination of natural vibrations of cylindrical resonators,
* Solution of the radial Schrödinger-equation,
* Sound pressure amplitudes of low-viscosity rotational flows,
* Heat conduction in cylindrical bodies,
* Scattering problem of a grid,
* Dynamics of oscillating bodies,
* Angular resolution.

The Bessel functions are counted among the special functions because of their many applications in mathematical physics.

In the following, we restrict ourselves to a few areas that are addressed in our tutorial $\rm LNTwww$.

'''In the English original'''
Electromagnetic waves in a cylindrical waveguide
Pressure amplitudes of inviscid rotational flows
Heat conduction in a cylindrical object
Modes of vibration of a thin circular (or annular) acoustic membrane (such as a drum or other membranophone)
Diffusion problems on a lattice
Solutions to the radial Schrödinger equation (in spherical and cylindrical coordinates) for a free particle
Solving for patterns of acoustical radiation
Frequency-dependent friction in circular pipelines
Dynamics of floating bodies
Angular resolution
'''The Ende'''

{{GraueBox|TEXT=
$\text{Example (C):} \hspace{0.5cm} \text{Use in spectral analysis} \ \Rightarrow \ \text{Kaiser-Bessel filter}$

The '''spectral leakage effect''' is the falsification of the spectrum of a periodic and hence temporally unlimited signal due to the implicit time limitation of the discrete Fourier transform (DFT). This will be done, for example, by a spectrum analyzer
* in the time signal not existing frequency components simulated, and/or
* actually existing spectral components are obscured by sidelobes.

The task of the [[Signaldarstellung/Spektralanalyse|spectral analysis]] is to limit the influence of the ''spectral leakage effect'' by providing suitable window functions.

Such a window function provides, for example, the Kaiser–Bessel–window   ⇒   see section [[Signaldarstellung/Spektralanalyse#Spezielle_Fensterfunktionen|Special Window Functions]]. Its time-discrete Fenser function reads with the Bessel function zero order   ⇒   ${\rm J}_0(x)$, the parameter $\alpha=3.5$ and the window length $N$:
:$$w_\nu = \frac{ {\rm J}_0\big(\pi \cdot \alpha \cdot \sqrt{1 - (2\nu/N)^2}\big)}{ {\rm J}_0\big(\pi \cdot \alpha \big)}.$$
On the page [[Signaldarstellung/Spektralanalyse#G.C3.BCtekriterien_von_Fensterfunktionen|Quality Criteria of Window Functions]] and other the characteristics of the Kaiser-Bessel-window are given:
*Conveniently, the large is &bdquo;minimum distance between the main lobe and side lobes” and the desired small &bdquo;maximum scaling error”.
*Due to the very large &bdquo;equivalent noise width” the Kaiser-Bessel window cuts in the main comparison criterion &bdquo;maximum process loss” but worse off than the established Hamming– and Hanning–windows.}}

{{GraueBox|TEXT=
$\text{Example (D):} \hspace{0.5cm} \text{Rice-Fading channel model}$

The [[Mobile_Kommunikation/Wahrscheinlichkeitsdichte_des_Rayleigh%E2%80%93Fadings#Allgemeine_Beschreibung_des_Mobilfunkkanals| Rayleigh - Distribution]] describes the mobile channel assuming that there is no direct path, and thus the multiplicative factor $z(t) = x(t) + {\rm j} \cdot y(t)$ is solely diffuse composed of scattered components.

In the case of a direct component (English: Line of Sight, LoS) in the model for the mean-free Gaussian processes $x(t)$ and $y(t)$ one has to add equal components $x_0$ and/or $y_0$ :

[[File:P ID2126 Mob T 1 4 S1 v3.png|right|frame|Rice-Fading channel model|class=fit]]
$$\hspace{0.2cm}x(t) \hspace{0.1cm} \Rightarrow \hspace{0.1cm} x(t) +x_0 \hspace{0.05cm}, \hspace{0.2cm} y(t) \hspace{0.1cm} \Rightarrow \hspace{0.1cm} y(t) +y_0\hspace{0.05cm},$$

$$\hspace{0.2cm}z(t) = x(t) + {\rm j} \cdot y(t) \hspace{0.1cm} \Rightarrow \hspace{0.1cm} z(t) +z_0 \hspace{0.05cm},\hspace{0.2cm}
z_0 = x_0 + {\rm j} \cdot y_0\hspace{0.05cm}.$$

The graph shows the ''Rice–Fading–Channel model''. It can be summarized as follows:
*The real part $x(t)$ is Gaussianized with mean $x_0$ and variance $\sigma ^2$.
*The imaginary part $y(t)$ is also Gaussian (mean $y_0$, equal variance $\sigma ^2$) and independent of $x(t)$. 

*For $z_0 \ne 0$, the amount $\vert z(t)\vert$ rice is distributed, from which the term &bdquo;Rice–Fading” arises.

*To simplify the notation, we set $\vert z(t)\vert = a(t)$. For $a < 0$ , the amounts - WDF are $f_a(a) \equiv 0$, for $a \ge 0$ the following equation holds, where ${\rm I_0}(x)$ is the modified Besselfunktion zeroth order denotes:
:$$f_a(a) = \frac{a}{\sigma^2} \cdot {\rm e}^{ - (a^2 + \vert z_0 \vert ^2)/(2\sigma^2)} \cdot {\rm I}_0 \left [ \frac{a \cdot \vert z_0 \vert}{\sigma^2} \right ] \hspace{0.5cm}\text{with}\hspace{0.5cm}{\rm I }_0 (u) = {\rm J }_0 ({\rm j} \cdot u) =
\sum_{k = 0}^{\infty} \frac{ (u/2)^{2k} }{k! \cdot \Gamma (k+1)}
\hspace{0.05cm}.$$
*Between the modified Bessel function and the traditional Bessel function ${\rm I_0}(x)$ – each first type – So the connection exists ${\rm I }_0 (u) = {\rm J }_0 ({\rm j} \cdot u)$.}}

{{GraueBox|TEXT=
$\text{Example (E):} \hspace{0.5cm} \text{Analysis of the frequency spectrum of frequency modulated signals}$

$\text{Example (B)}$ has already shown that the angle modulation of a harmonic oscillation of the frequency $f_{\rm N}$ leads to a line spectrum. The spectral lines are around the carrier frequency $f_{\rm T}$ bei $f_{\rm T} + n \cdot f_{\rm N}$ with $n \in \{ \ \text{...}, -2, -1, \ 0, +1, +2, \text{...} \ \}$. The weights of the Dirac lines are ${\rm J }_n(\eta)$, depending on the modulation index $\eta$.

[[File:P_ID1095__Mod_T_3_2_S4_neu.png|center|frame|Discrete spectra with phase modulation (left) and frequency modulation (right)]]

The graph shows the magnitude spectrum $\vert S_{\rm +}(f) \vert$ of the phase modulation (PM) and frequency modulation (FM) analytic signal, two different forms of angle modulation (WM). Bessellines with values less than $0.03$ are neglected in both cases.

For the upper half of the image, the modulator parameters are chosen so that for each $f_{\rm N} = 5 \ \rm kHz$ here is a Bessel spectrum with the modulation index $η = 1.5$. Disregarding the phase relationships, the same spectra and the same signals result for both systems.

The lower graphics are valid for the same message frequency $f_{\rm N} = 3 \ \rm kHz$. One notices:
*In phase modulation, the spectral function is narrower than $f_{\rm N} = 5 \ \rm kHz$, because the distance of the Bessel lines is now only $3 \ \rm kHz$. Since the modulation index of PM is independent of $f_{\rm N}$, the same bessel weights result as in $f_{\rm N} = 5 \ \rm kHz$.
*Also in the frequency modulation, the Bessel lines now appear at a distance of $3 \ \rm kHz$. However, since FM has a modulation index inversely proportional to $f_{\rm N}$, there is now much more Besselline lines than in the upper right (for $η = 1.5$ valid) chart due to the larger modulation index $η = 2.5$.}}

'''Das folgende Kapitel muss noch angepasst werden!'''

==Applet Manual==
[[File:Bessel_abzug3.png|left|600px]]
    '''(A)'''     Sum formula of the Bessel functions ${\rm J}_n(x)$

    '''(B)'''     Selection of the order $n$ for the graphical representation

    '''(C)'''     Plot of the Bessel functions

    '''(D)'''     Variation of the graphic representation

$\hspace{1.5cm}$&bdquo;$+$” (Enlarge),

$\hspace{1.5cm}$ &bdquo;$-$” (Decrease)

$\hspace{1.5cm}$ &bdquo;$\rm o$” (Reset to default)

$\hspace{1.5cm}$ &bdquo;$\leftarrow$” (pushed to the left), usw.

    '''(E)'''     Selection of the abscissa value $x_1$ for the left numeric output

    '''(F)'''     Numerical output of the Bessel function values ${\rm J}_n(x_1)$

    '''(G)'''     Selection of the abscissa value $x_2$ for the right numeric output

    '''(F)'''     Numerical output of the Bessel function values ${\rm J}_n(x_2)$
 

==About the Authors==
This interactive calculation was designed and realized at the [http://www.lnt.ei.tum.de/startseite Lehrstuhl für Nachrichtentechnik] of the [https://www.tum.de/ Technical University of Munich] .
*The original version was created in 2005 by [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Studierende#Markus_Elsberger_.28Diplomarbeit_LB_2006.29|Markus Elsberger]] and [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Studierende#Slim_Lamine_.28Studienarbeit_EI_2006.29|Slim Lamine]] as part of thesis using &bdquo;FlashMX–Actionscript” (Supervisor: [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Mitarbeiter_und_Dozenten#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28am_LNT_seit_1974.29|Günter Söder]]).
*In 2018 this Applet was redesigned and updated to &bdquo;HTML5” by [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Studierende#Xiaohan_Liu_.28Bachelorarbeit_2018.29|Xiaohan Liu]] as part of her Bachelor's thesis (Supervisor: [[Biografien_und_Bibliografien/Beteiligte_der_Professur_Leitungsgebundene_%C3%9Cbertragungstechnik#Tasn.C3.A1d_Kernetzky.2C_M.Sc._.28bei_L.C3.9CT_seit_2014.29|Tasnád Kernetzky]]).

==Nochmalige Aufrufmöglichkeit des Applets in neuem Fenster==

{{LntAppletLink|bessel}}

Applets:Bessel Functions of the First Kind

2018-08-26T14:41:04Z

Xiaohan: /* General Information about the Bessel Functions */

{{LntAppletLink|bessel}}

==Applet Description==
 
This applet allows the calculation and graphical representation of the Bessel functions of the first kind and $n$–th order according to the series representation:

:$${\rm J}_n (x) = \sum\limits_{k=0}^{\infty}\frac{(-1)^k \cdot (x/2)^{n \hspace{0.05cm} + \hspace{0.05cm} 2
\hspace{0.02cm}\cdot \hspace{0.05cm}k}}{k! \cdot (n+k)!} \hspace{0.05cm}.$$

*The functions ${\rm J}_n (x)$ can be represented graphically for the order $n=0$ to $n=9$ in different colors.
*The left output provides the function values ${\rm J}_0 (x = x_1)$, ... , ${\rm J}_9 (x = x_1)$ for a slider-settable value $x_1$ in the range $0 \le x_1 \le 15$ with increment $0.5$.
*The right output provides the function values ${\rm J}_0 (x = x_2)$, ... , ${\rm J}_9 (x = x_2)$ for a slider-settable value $x_2$ (same range and value) Increment like on the left).

[[Applets:Besselfunktionen_erster_Art_(neues_Applet)|'''German description''']]

==Theoretical Background==
 
===General Information about the Bessel Functions===
Bessel functions (or cylinder functions) are solutions of the Bessel differential equation of the form

:$$x^2 \cdot \frac{ {\rm d}^2}{{\rm d}x^2}\ {\rm J}_n (x) \ + \ x \cdot \frac{ {\rm d}}{{\rm d}x}\ {\rm J}_n (x) \ + \ (x^2 - n^2)
\cdot {\rm J}_n (x)= 0. $$

This is an ordinary second-order linear differential equation. The parameter $ n $ is usually integer, also in this program. These mathematical functions, introduced by [https://de.wikipedia.org/wiki/Friedrich_Wilhelm_Bessel Friedrich Wilhelm Bessel] in 1844, can also be represented in closed form as integrals:

:$${\rm J}_n (x) = \frac{1}{2\pi}\cdot \int_{-\pi}^{+\pi} {{\rm e}^{\hspace{0.05cm}{\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}[\hspace{0.05cm}x \hspace{0.05cm}\cdot \hspace{0.05cm}\sin(\alpha) -\hspace{0.05cm} n \hspace{0.05cm}\cdot \hspace{0.05cm}\alpha \hspace{0.05cm}]}}\hspace{0.1cm}{\rm d}\alpha
\hspace{0.05cm}.$$

The functions ${\rm J}_n (x)$ belong to the class of Bessel functions of the first kind (German:   ''Besselfunktionen erster Art''). The parameter $n$ is called the ''Order''.

''Annotation:''   There are a number of modifications of the Bessel functions, including the second-order Bessel functions named ${\rm Y}_n (x)$. For integer $n$, ${\rm Y}_n (x)$ can be replaced by ${\rm J}_n (x)$–functions. However, in this applet, only the first kind Bessel functions are   ⇒   ${\rm J}_n (x)$ is considered.
 

===Properties of the Bessel Functions===

{{BlaueBox|TEXT=
$\text{Property (A):}$   If the function values for $n = 0$ and $n = 1$ are known, then the Bessel function for $n ≥ 2$ can be determined iteratively:
:$${\rm J}_n (x) ={2 \cdot (n-1)}/{x} \cdot {\rm J}_{n-1} (x) - {\rm J}_{n-2} (x) \hspace{0.05cm}.$$
}}

{{GraueBox|TEXT=
$\text{Example (A):}$   Consider ${\rm J}_0 (x = 2) = 0.22389$ and ${\rm J}_1 (x= 2) = 0.57672$. From this it can be calculated iteratively:
:$${\rm J}_2 (x= 2) ={2 \cdot 1}/{2} \cdot {\rm J}_{1} (x= 2) - {\rm J}_{0} (x= 2) = 0.57672 - 0.22389 = \hspace{0.15cm}\underline{0.35283}\hspace{0.05cm},$$
:$${\rm J}_3 (x= 2) ={2 \cdot 2}/{2} \cdot {\rm J}_{2} (x= 2) - {\rm J}_{1} (x= 2) = 2 \cdot 0.35283 - 0.57672 = \hspace{0.15cm}\underline{0.12894}\hspace{0.05cm},$$
:$${\rm J}_4 (x= 2) ={2 \cdot 3}/{2} \cdot {\rm J}_{3} (x= 2) - {\rm J}_{2} (x= 2) = 3 \cdot 0.12894 - 0.35283 = \hspace{0.15cm}\underline{0.03400}\hspace{0.05cm}.$$
}}

{{BlaueBox|TEXT=
$\text{Property (B):}$   The symmetry relationship applies ${\rm J}_{–n}(x) = (–1)^n · {\rm J}_n(x)$:
:$${\rm J}_{-1}(x) = - {\rm J}_{1}(x), \hspace{0.3cm}{\rm J}_{-2}(x) = {\rm J}_{2}(x), \hspace{0.3cm}{\rm J}_{-3}(x) = - {\rm J}_{3}(x).$$
}}

{{GraueBox|TEXT=
$\text{Example (B):}$   For the spectrum of the analytic signal, in [[Modulationsverfahren/Phasenmodulation_(PM)#Spektralfunktion_eines_phasenmodulierten_Sinussignals|phase modulation of a sinusoidal signal]]:
[[File:Mod_T_3_1_S4_version2.png|right|frame|Spectrum of the analytic signal in phase modulation]]
:$$S_{\rm +}(f) = A_{\rm T} \cdot \sum_{n = - \infty}^{+\infty}{\rm J}_n (\eta) \cdot \delta \big[f - (f_{\rm T}+ n \cdot f_{\rm N})\big]\hspace{0.05cm}.$$
Denote this
*$f_{\rm T}$ the carrier frequency,
*$f_{\rm N}$ the message frequency,
* $A_{\rm T}$ the carrier amplitude.

The parameter of the Bessel functions in this application is the modulation index $\eta$.

Based on the graphic, the following statements are possible:
*Here $S_+(f)$ consists of an infinite number of discrete lines at a distance of $f_{\rm N}$.
*It is thus in principle infinitely extended.
*The weights of the spectral lines at $f_{\rm T} + n · f_{\rm N}$ ($n$ integer) are determined by the modulation index $η$ over the Bessel functions ${\rm J}_n(η)$.
*The spectral lines are real in the case of a sinusoidal source signal and a cosinusoidal carrier and symmetric about $f_{\rm T}$ for even $n$.
*For odd $n$ , a sign change according to the $\text{Property (B)}$ has to be considered.
*The phase modulation of another phase oscillation of source and / or carrier signal provides the same magnitude spectrum.}}
 

===Applications of the Bessel Functions===

The applications of the Bessel functions in nature–and engineering are diverse and play an important role in physics, for example:
* Examination of natural vibrations of cylindrical resonators,
* Solution of the radial Schrödinger-equation,
* Sound pressure amplitudes of low-viscosity rotational flows,
* Heat conduction in cylindrical bodies,
* Scattering problem of a grid,
* Dynamics of oscillating bodies,
* Angular resolution.

The Bessel functions are counted among the special functions because of their many applications in mathematical physics.

In the following, we restrict ourselves to a few areas that are addressed in our tutorial $\rm LNTwww$.

'''In the English original'''
Electromagnetic waves in a cylindrical waveguide
Pressure amplitudes of inviscid rotational flows
Heat conduction in a cylindrical object
Modes of vibration of a thin circular (or annular) acoustic membrane (such as a drum or other membranophone)
Diffusion problems on a lattice
Solutions to the radial Schrödinger equation (in spherical and cylindrical coordinates) for a free particle
Solving for patterns of acoustical radiation
Frequency-dependent friction in circular pipelines
Dynamics of floating bodies
Angular resolution
'''The Ende'''

{{GraueBox|TEXT=
$\text{Example (C):} \hspace{0.5cm} \text{Use in spectral analysis} \ \Rightarrow \ \text{Kaiser-Bessel filter}$

The '''spectral leakage effect''' is the falsification of the spectrum of a periodic and hence temporally unlimited signal due to the implicit time limitation of the discrete Fourier transform (DFT). This will be done, for example, by a spectrum analyzer
* in the time signal not existing frequency components simulated, and/or
* actually existing spectral components are obscured by sidelobes.

The task of the [[Signaldarstellung/Spektralanalyse|spectral analysis]] is to limit the influence of the ''spectral leakage effect'' by providing suitable window functions.

Such a window function provides, for example, the Kaiser–Bessel–window   ⇒   see section [[Signaldarstellung/Spektralanalyse#Spezielle_Fensterfunktionen|Special Window Functions]]. Its time-discrete Fenser function reads with the Bessel function zero order   ⇒   ${\rm J}_0(x)$, the parameter $\alpha=3.5$ and the window length $N$:
:$$w_\nu = \frac{ {\rm J}_0\big(\pi \cdot \alpha \cdot \sqrt{1 - (2\nu/N)^2}\big)}{ {\rm J}_0\big(\pi \cdot \alpha \big)}.$$
On the page [[Signaldarstellung/Spektralanalyse#G.C3.BCtekriterien_von_Fensterfunktionen|Quality Criteria of Window Functions]] and other the characteristics of the Kaiser-Bessel-window are given:
*Conveniently, the large is &bdquo;minimum distance between the main lobe and side lobes” and the desired small &bdquo;maximum scaling error”.
*Due to the very large &bdquo;equivalent noise width” the Kaiser-Bessel window cuts in the main comparison criterion &bdquo;maximum process loss” but worse off than the established Hamming– and Hanning–windows.}}

{{GraueBox|TEXT=
$\text{Example (D):} \hspace{0.5cm} \text{Rice-Fading channel model}$

The [[Mobile_Kommunikation/Wahrscheinlichkeitsdichte_des_Rayleigh%E2%80%93Fadings#Allgemeine_Beschreibung_des_Mobilfunkkanals| Rayleigh - Distribution]] describes the mobile channel assuming that there is no direct path, and thus the multiplicative factor $z(t) = x(t) + {\rm j} \cdot y(t)$ is solely diffuse composed of scattered components.

In the case of a direct component (English: Line of Sight, LoS) in the model for the mean-free Gaussian processes $x(t)$ and $y(t)$ one has to add equal components $x_0$ and/or $y_0$ :

[[File:P ID2126 Mob T 1 4 S1 v3.png|right|frame|Rice-Fading channel model|class=fit]]
$$\hspace{0.2cm}x(t) \hspace{0.1cm} \Rightarrow \hspace{0.1cm} x(t) +x_0 \hspace{0.05cm}, \hspace{0.2cm} y(t) \hspace{0.1cm} \Rightarrow \hspace{0.1cm} y(t) +y_0\hspace{0.05cm},$$

$$\hspace{0.2cm}z(t) = x(t) + {\rm j} \cdot y(t) \hspace{0.1cm} \Rightarrow \hspace{0.1cm} z(t) +z_0 \hspace{0.05cm},\hspace{0.2cm}
z_0 = x_0 + {\rm j} \cdot y_0\hspace{0.05cm}.$$

The graph shows the ''Rice–Fading–Channel model''. It can be summarized as follows:
*The real part $x(t)$ is Gaussianized with mean $x_0$ and variance $\sigma ^2$.
*The imaginary part $y(t)$ is also Gaussian (mean $y_0$, equal variance $\sigma ^2$) and independent of $x(t)$. 

*For $z_0 \ne 0$, the amount $\vert z(t)\vert$ rice is distributed, from which the term &bdquo;Rice–Fading” arises.

*To simplify the notation, we set $\vert z(t)\vert = a(t)$. For $a < 0$ , the amounts - WDF are $f_a(a) \equiv 0$, for $a \ge 0$ the following equation holds, where ${\rm I_0}(x)$ is the modified Besselfunktion zeroth order denotes:
:$$f_a(a) = \frac{a}{\sigma^2} \cdot {\rm e}^{ - (a^2 + \vert z_0 \vert ^2)/(2\sigma^2)} \cdot {\rm I}_0 \left [ \frac{a \cdot \vert z_0 \vert}{\sigma^2} \right ] \hspace{0.5cm}\text{with}\hspace{0.5cm}{\rm I }_0 (u) = {\rm J }_0 ({\rm j} \cdot u) =
\sum_{k = 0}^{\infty} \frac{ (u/2)^{2k} }{k! \cdot \Gamma (k+1)}
\hspace{0.05cm}.$$
*Between the modified Bessel function and the traditional Bessel function ${\rm I_0}(x)$ – each first type – So the connection exists ${\rm I }_0 (u) = {\rm J }_0 ({\rm j} \cdot u)$.}}

{{GraueBox|TEXT=
$\text{Example (E):} \hspace{0.5cm} \text{Analysis of the frequency spectrum of frequency modulated signals}$

$\text{Example (B)}$ has already shown that the angle modulation of a harmonic oscillation of the frequency $f_{\rm N}$ leads to a line spectrum. The spectral lines are around the carrier frequency $f_{\rm T}$ bei $f_{\rm T} + n \cdot f_{\rm N}$ with $n \in \{ \ \text{...}, -2, -1, \ 0, +1, +2, \text{...} \ \}$. The weights of the Dirac lines are ${\rm J }_n(\eta)$, depending on the modulation index $\eta$.

[[File:P_ID1095__Mod_T_3_2_S4_neu.png|center|frame|Discrete spectra with phase modulation (left) and frequency modulation (right)]]

The graph shows the magnitude spectrum $\vert S_{\rm +}(f) \vert$ of the phase modulation (PM) and frequency modulation (FM) analytic signal, two different forms of angle modulation (WM). Bessellines with values less than $0.03$ are neglected in both cases.

For the upper half of the image, the modulator parameters are chosen so that for each $f_{\rm N} = 5 \ \rm kHz$ here is a Bessel spectrum with the modulation index $η = 1.5$. Disregarding the phase relationships, the same spectra and the same signals result for both systems.

The lower graphics are valid for the same message frequency $f_{\rm N} = 3 \ \rm kHz$. One notices:
*In phase modulation, the spectral function is narrower than $f_{\rm N} = 5 \ \rm kHz$, because the distance of the Bessel lines is now only $3 \ \rm kHz$. Since the modulation index of PM is independent of $f_{\rm N}$, the same bessel weights result as in $f_{\rm N} = 5 \ \rm kHz$.
*Also in the frequency modulation, the Bessel lines now appear at a distance of $3 \ \rm kHz$. However, since FM has a modulation index inversely proportional to $f_{\rm N}$, there is now much more Besselline lines than in the upper right (for $η = 1.5$ valid) chart due to the larger modulation index $η = 2.5$.}}

'''Das folgende Kapitel muss noch angepasst werden!'''

==Applet Manual==
[[File:Bessel_abzug3.png|left|600px]]
    '''(A)'''     Sum formula of the Bessel functions ${\rm J}_n(x)$

    '''(B)'''     Selection of the order $n$ for the graphical representation

    '''(C)'''     Plot of the Bessel functions

    '''(D)'''     Variation of the graphic representation

$\hspace{1.5cm}$&bdquo;$+$” (Enlarge),

$\hspace{1.5cm}$ &bdquo;$-$” (Decrease)

$\hspace{1.5cm}$ &bdquo;$\rm o$” (Reset to default)

$\hspace{1.5cm}$ &bdquo;$\leftarrow$” (pushed to the left), usw.

    '''(E)'''     Selection of the abscissa value $x_1$ for the left numeric output

    '''(F)'''     Numerical output of the Bessel function values ${\rm J}_n(x_1)$

    '''(G)'''     Selection of the abscissa value $x_2$ for the right numeric output

    '''(F)'''     Numerical output of the Bessel function values ${\rm J}_n(x_2)$
 

==About the Authors==
This interactive calculation was designed and realized at the [http://www.lnt.ei.tum.de/startseite Lehrstuhl für Nachrichtentechnik] of the [https://www.tum.de/ Technical University of Munich] .
*The original version was created in 2005 by [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Studierende#Markus_Elsberger_.28Diplomarbeit_LB_2006.29|Markus Elsberger]] and [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Studierende#Slim_Lamine_.28Studienarbeit_EI_2006.29|Slim Lamine]] as part of thesis using &bdquo;FlashMX–Actionscript” (Supervisor: [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Mitarbeiter_und_Dozenten#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28am_LNT_seit_1974.29|Günter Söder]]).
*In 2018 this Applet was redesigned and updated to &bdquo;HTML5” by [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Studierende#Xiaohan_Liu_.28Bachelorarbeit_2018.29|Xiaohan Liu]] as part of her Bachelor's thesis (Supervisor: [[Biografien_und_Bibliografien/Beteiligte_der_Professur_Leitungsgebundene_%C3%9Cbertragungstechnik#Tasn.C3.A1d_Kernetzky.2C_M.Sc._.28bei_L.C3.9CT_seit_2014.29|Tasnád Kernetzky]]).

==Nochmalige Aufrufmöglichkeit des Applets in neuem Fenster==

{{LntAppletLink|bessel}}

Applets:Bessel Functions of the First Kind

2018-08-26T14:26:28Z

Xiaohan: /* Applet Description */

{{LntAppletLink|bessel}}

==Applet Description==
 
This applet allows the calculation and graphical representation of the Bessel functions of the first kind and $n$–th order according to the series representation:

:$${\rm J}_n (x) = \sum\limits_{k=0}^{\infty}\frac{(-1)^k \cdot (x/2)^{n \hspace{0.05cm} + \hspace{0.05cm} 2
\hspace{0.02cm}\cdot \hspace{0.05cm}k}}{k! \cdot (n+k)!} \hspace{0.05cm}.$$

*The functions ${\rm J}_n (x)$ can be represented graphically for the order $n=0$ to $n=9$ in different colors.
*The left output provides the function values ${\rm J}_0 (x = x_1)$, ... , ${\rm J}_9 (x = x_1)$ for a slider-settable value $x_1$ in the range $0 \le x_1 \le 15$ with increment $0.5$.
*The right output provides the function values ${\rm J}_0 (x = x_2)$, ... , ${\rm J}_9 (x = x_2)$ for a slider-settable value $x_2$ (same range and value) Increment like on the left).

[[Applets:Besselfunktionen_erster_Art_(neues_Applet)|'''German description''']]

==Theoretical Background==
 
===General Information about the Bessel Functions===
Bessel functions (or cylinder functions) are solutions of the Bessel differential equation of the form

:$$x^2 \cdot \frac{ {\rm d}^2}{{\rm d}x^2}\ {\rm J}_n (x) \ + \ x \cdot \frac{ {\rm d}}{{\rm d}x}\ {\rm J}_n (x) \ + \ (x^2 - n^2)
\cdot {\rm J}_n (x)= 0. $$

This is an ordinary second-order linear differential equation. The parameter $ n $ is usually integer, as in this program. These mathematical functions, which were introduced in 1844 by [https://de.wikipedia.org/wiki/Friedrich_Wilhelm_Bessel Friedrich Wilhelm Bessel], can also be represented in closed form as integrals:

:$${\rm J}_n (x) = \frac{1}{2\pi}\cdot \int_{-\pi}^{+\pi} {{\rm e}^{\hspace{0.05cm}{\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}[\hspace{0.05cm}x \hspace{0.05cm}\cdot \hspace{0.05cm}\sin(\alpha) -\hspace{0.05cm} n \hspace{0.05cm}\cdot \hspace{0.05cm}\alpha \hspace{0.05cm}]}}\hspace{0.1cm}{\rm d}\alpha
\hspace{0.05cm}.$$

The functions ${\rm J}_n (x)$ belong to the class of Bessel functions of the first kind (German:   ''Besselfunktionen erster Art''). The parameter $n$ is called the ''Order''.

''Annotation:''   There are a number of modifications of the Bessel functions, including the second-order Bessel functions named ${\rm Y}_n (x)$. For integer $n$, ${\rm Y}_n (x)$ can be replaced by ${\rm J}_n (x)$–functions. However, in this applet, only the first-order Bessel functions are   ⇒   ${\rm J}_n (x)$ is considered.
 
===Properties of the Bessel Functions===

{{BlaueBox|TEXT=
$\text{Property (A):}$   If the function values for $n = 0$ and $n = 1$ are known, then the Bessel function for $n ≥ 2$ can be determined iteratively:
:$${\rm J}_n (x) ={2 \cdot (n-1)}/{x} \cdot {\rm J}_{n-1} (x) - {\rm J}_{n-2} (x) \hspace{0.05cm}.$$
}}

{{GraueBox|TEXT=
$\text{Example (A):}$   Consider ${\rm J}_0 (x = 2) = 0.22389$ and ${\rm J}_1 (x= 2) = 0.57672$. From this it can be calculated iteratively:
:$${\rm J}_2 (x= 2) ={2 \cdot 1}/{2} \cdot {\rm J}_{1} (x= 2) - {\rm J}_{0} (x= 2) = 0.57672 - 0.22389 = \hspace{0.15cm}\underline{0.35283}\hspace{0.05cm},$$
:$${\rm J}_3 (x= 2) ={2 \cdot 2}/{2} \cdot {\rm J}_{2} (x= 2) - {\rm J}_{1} (x= 2) = 2 \cdot 0.35283 - 0.57672 = \hspace{0.15cm}\underline{0.12894}\hspace{0.05cm},$$
:$${\rm J}_4 (x= 2) ={2 \cdot 3}/{2} \cdot {\rm J}_{3} (x= 2) - {\rm J}_{2} (x= 2) = 3 \cdot 0.12894 - 0.35283 = \hspace{0.15cm}\underline{0.03400}\hspace{0.05cm}.$$
}}

{{BlaueBox|TEXT=
$\text{Property (B):}$   The symmetry relationship applies ${\rm J}_{–n}(x) = (–1)^n · {\rm J}_n(x)$:
:$${\rm J}_{-1}(x) = - {\rm J}_{1}(x), \hspace{0.3cm}{\rm J}_{-2}(x) = {\rm J}_{2}(x), \hspace{0.3cm}{\rm J}_{-3}(x) = - {\rm J}_{3}(x).$$
}}

{{GraueBox|TEXT=
$\text{Example (B):}$   For the spectrum of the analytic signal, in [[Modulationsverfahren/Phasenmodulation_(PM)#Spektralfunktion_eines_phasenmodulierten_Sinussignals|phase modulation of a sinusoidal signal]]:
[[File:Mod_T_3_1_S4_version2.png|right|frame|Spectrum of the analytic signal in phase modulation]]
:$$S_{\rm +}(f) = A_{\rm T} \cdot \sum_{n = - \infty}^{+\infty}{\rm J}_n (\eta) \cdot \delta \big[f - (f_{\rm T}+ n \cdot f_{\rm N})\big]\hspace{0.05cm}.$$
Denote this
*$f_{\rm T}$ the carrier frequency,
*$f_{\rm N}$ the message frequency,
* $A_{\rm T}$ the carrier amplitude.

The parameter of the Bessel functions in this application is the modulation index $\eta$.

Based on the graphic, the following statements are possible:
*Here $S_+(f)$ consists of an infinite number of discrete lines at a distance of $f_{\rm N}$.
*It is thus in principle infinitely extended.
*The weights of the spectral lines at $f_{\rm T} + n · f_{\rm N}$ ($n$ integer) are determined by the modulation index $η$ over the Bessel functions ${\rm J}_n(η)$.
*The spectral lines are real in the case of a sinusoidal source signal and a cosinusoidal carrier and symmetric about $f_{\rm T}$ for even $n$.
*For odd $n$ , a sign change according to the $\text{Property (B)}$ has to be considered.
*The phase modulation of another phase oscillation of source and / or carrier signal provides the same magnitude spectrum.}}
 

===Applications of the Bessel Functions===

The applications of the Bessel functions in nature–and engineering are diverse and play an important role in physics, for example:
* Examination of natural vibrations of cylindrical resonators,
* Solution of the radial Schrödinger-equation,
* Sound pressure amplitudes of low-viscosity rotational flows,
* Heat conduction in cylindrical bodies,
* Scattering problem of a grid,
* Dynamics of oscillating bodies,
* Angular resolution.

The Bessel functions are counted among the special functions because of their many applications in mathematical physics.

In the following, we restrict ourselves to a few areas that are addressed in our tutorial $\rm LNTwww$.

'''In the English original'''
Electromagnetic waves in a cylindrical waveguide
Pressure amplitudes of inviscid rotational flows
Heat conduction in a cylindrical object
Modes of vibration of a thin circular (or annular) acoustic membrane (such as a drum or other membranophone)
Diffusion problems on a lattice
Solutions to the radial Schrödinger equation (in spherical and cylindrical coordinates) for a free particle
Solving for patterns of acoustical radiation
Frequency-dependent friction in circular pipelines
Dynamics of floating bodies
Angular resolution
'''The Ende'''

{{GraueBox|TEXT=
$\text{Example (C):} \hspace{0.5cm} \text{Use in spectral analysis} \ \Rightarrow \ \text{Kaiser-Bessel filter}$

The '''spectral leakage effect''' is the falsification of the spectrum of a periodic and hence temporally unlimited signal due to the implicit time limitation of the discrete Fourier transform (DFT). This will be done, for example, by a spectrum analyzer
* in the time signal not existing frequency components simulated, and/or
* actually existing spectral components are obscured by sidelobes.

The task of the [[Signaldarstellung/Spektralanalyse|spectral analysis]] is to limit the influence of the ''spectral leakage effect'' by providing suitable window functions.

Such a window function provides, for example, the Kaiser–Bessel–window   ⇒   see section [[Signaldarstellung/Spektralanalyse#Spezielle_Fensterfunktionen|Special Window Functions]]. Its time-discrete Fenser function reads with the Bessel function zero order   ⇒   ${\rm J}_0(x)$, the parameter $\alpha=3.5$ and the window length $N$:
:$$w_\nu = \frac{ {\rm J}_0\big(\pi \cdot \alpha \cdot \sqrt{1 - (2\nu/N)^2}\big)}{ {\rm J}_0\big(\pi \cdot \alpha \big)}.$$
On the page [[Signaldarstellung/Spektralanalyse#G.C3.BCtekriterien_von_Fensterfunktionen|Quality Criteria of Window Functions]] and other the characteristics of the Kaiser-Bessel-window are given:
*Conveniently, the large is &bdquo;minimum distance between the main lobe and side lobes” and the desired small &bdquo;maximum scaling error”.
*Due to the very large &bdquo;equivalent noise width” the Kaiser-Bessel window cuts in the main comparison criterion &bdquo;maximum process loss” but worse off than the established Hamming– and Hanning–windows.}}

{{GraueBox|TEXT=
$\text{Example (D):} \hspace{0.5cm} \text{Rice-Fading channel model}$

The [[Mobile_Kommunikation/Wahrscheinlichkeitsdichte_des_Rayleigh%E2%80%93Fadings#Allgemeine_Beschreibung_des_Mobilfunkkanals| Rayleigh - Distribution]] describes the mobile channel assuming that there is no direct path, and thus the multiplicative factor $z(t) = x(t) + {\rm j} \cdot y(t)$ is solely diffuse composed of scattered components.

In the case of a direct component (English: Line of Sight, LoS) in the model for the mean-free Gaussian processes $x(t)$ and $y(t)$ one has to add equal components $x_0$ and/or $y_0$ :

[[File:P ID2126 Mob T 1 4 S1 v3.png|right|frame|Rice-Fading channel model|class=fit]]
$$\hspace{0.2cm}x(t) \hspace{0.1cm} \Rightarrow \hspace{0.1cm} x(t) +x_0 \hspace{0.05cm}, \hspace{0.2cm} y(t) \hspace{0.1cm} \Rightarrow \hspace{0.1cm} y(t) +y_0\hspace{0.05cm},$$

$$\hspace{0.2cm}z(t) = x(t) + {\rm j} \cdot y(t) \hspace{0.1cm} \Rightarrow \hspace{0.1cm} z(t) +z_0 \hspace{0.05cm},\hspace{0.2cm}
z_0 = x_0 + {\rm j} \cdot y_0\hspace{0.05cm}.$$

The graph shows the ''Rice–Fading–Channel model''. It can be summarized as follows:
*The real part $x(t)$ is Gaussianized with mean $x_0$ and variance $\sigma ^2$.
*The imaginary part $y(t)$ is also Gaussian (mean $y_0$, equal variance $\sigma ^2$) and independent of $x(t)$. 

*For $z_0 \ne 0$, the amount $\vert z(t)\vert$ rice is distributed, from which the term &bdquo;Rice–Fading” arises.

*To simplify the notation, we set $\vert z(t)\vert = a(t)$. For $a < 0$ , the amounts - WDF are $f_a(a) \equiv 0$, for $a \ge 0$ the following equation holds, where ${\rm I_0}(x)$ is the modified Besselfunktion zeroth order denotes:
:$$f_a(a) = \frac{a}{\sigma^2} \cdot {\rm e}^{ - (a^2 + \vert z_0 \vert ^2)/(2\sigma^2)} \cdot {\rm I}_0 \left [ \frac{a \cdot \vert z_0 \vert}{\sigma^2} \right ] \hspace{0.5cm}\text{with}\hspace{0.5cm}{\rm I }_0 (u) = {\rm J }_0 ({\rm j} \cdot u) =
\sum_{k = 0}^{\infty} \frac{ (u/2)^{2k} }{k! \cdot \Gamma (k+1)}
\hspace{0.05cm}.$$
*Between the modified Bessel function and the traditional Bessel function ${\rm I_0}(x)$ – each first type – So the connection exists ${\rm I }_0 (u) = {\rm J }_0 ({\rm j} \cdot u)$.}}

{{GraueBox|TEXT=
$\text{Example (E):} \hspace{0.5cm} \text{Analysis of the frequency spectrum of frequency modulated signals}$

$\text{Example (B)}$ has already shown that the angle modulation of a harmonic oscillation of the frequency $f_{\rm N}$ leads to a line spectrum. The spectral lines are around the carrier frequency $f_{\rm T}$ bei $f_{\rm T} + n \cdot f_{\rm N}$ with $n \in \{ \ \text{...}, -2, -1, \ 0, +1, +2, \text{...} \ \}$. The weights of the Dirac lines are ${\rm J }_n(\eta)$, depending on the modulation index $\eta$.

[[File:P_ID1095__Mod_T_3_2_S4_neu.png|center|frame|Discrete spectra with phase modulation (left) and frequency modulation (right)]]

The graph shows the magnitude spectrum $\vert S_{\rm +}(f) \vert$ of the phase modulation (PM) and frequency modulation (FM) analytic signal, two different forms of angle modulation (WM). Bessellines with values less than $0.03$ are neglected in both cases.

For the upper half of the image, the modulator parameters are chosen so that for each $f_{\rm N} = 5 \ \rm kHz$ here is a Bessel spectrum with the modulation index $η = 1.5$. Disregarding the phase relationships, the same spectra and the same signals result for both systems.

The lower graphics are valid for the same message frequency $f_{\rm N} = 3 \ \rm kHz$. One notices:
*In phase modulation, the spectral function is narrower than $f_{\rm N} = 5 \ \rm kHz$, because the distance of the Bessel lines is now only $3 \ \rm kHz$. Since the modulation index of PM is independent of $f_{\rm N}$, the same bessel weights result as in $f_{\rm N} = 5 \ \rm kHz$.
*Also in the frequency modulation, the Bessel lines now appear at a distance of $3 \ \rm kHz$. However, since FM has a modulation index inversely proportional to $f_{\rm N}$, there is now much more Besselline lines than in the upper right (for $η = 1.5$ valid) chart due to the larger modulation index $η = 2.5$.}}

'''Das folgende Kapitel muss noch angepasst werden!'''

==Applet Manual==
[[File:Bessel_abzug3.png|left|600px]]
    '''(A)'''     Sum formula of the Bessel functions ${\rm J}_n(x)$

    '''(B)'''     Selection of the order $n$ for the graphical representation

    '''(C)'''     Plot of the Bessel functions

    '''(D)'''     Variation of the graphic representation

$\hspace{1.5cm}$&bdquo;$+$” (Enlarge),

$\hspace{1.5cm}$ &bdquo;$-$” (Decrease)

$\hspace{1.5cm}$ &bdquo;$\rm o$” (Reset to default)

$\hspace{1.5cm}$ &bdquo;$\leftarrow$” (pushed to the left), usw.

    '''(E)'''     Selection of the abscissa value $x_1$ for the left numeric output

    '''(F)'''     Numerical output of the Bessel function values ${\rm J}_n(x_1)$

    '''(G)'''     Selection of the abscissa value $x_2$ for the right numeric output

    '''(F)'''     Numerical output of the Bessel function values ${\rm J}_n(x_2)$
 

==About the Authors==
This interactive calculation was designed and realized at the [http://www.lnt.ei.tum.de/startseite Lehrstuhl für Nachrichtentechnik] of the [https://www.tum.de/ Technical University of Munich] .
*The original version was created in 2005 by [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Studierende#Markus_Elsberger_.28Diplomarbeit_LB_2006.29|Markus Elsberger]] and [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Studierende#Slim_Lamine_.28Studienarbeit_EI_2006.29|Slim Lamine]] as part of thesis using &bdquo;FlashMX–Actionscript” (Supervisor: [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Mitarbeiter_und_Dozenten#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28am_LNT_seit_1974.29|Günter Söder]]).
*In 2018 this Applet was redesigned and updated to &bdquo;HTML5” by [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Studierende#Xiaohan_Liu_.28Bachelorarbeit_2018.29|Xiaohan Liu]] as part of her Bachelor's thesis (Supervisor: [[Biografien_und_Bibliografien/Beteiligte_der_Professur_Leitungsgebundene_%C3%9Cbertragungstechnik#Tasn.C3.A1d_Kernetzky.2C_M.Sc._.28bei_L.C3.9CT_seit_2014.29|Tasnád Kernetzky]]).

==Nochmalige Aufrufmöglichkeit des Applets in neuem Fenster==

{{LntAppletLink|bessel}}

Applets:Physical Signal & Equivalent Lowpass Signal

2018-08-26T10:10:00Z

Xiaohan: /* Exercises */

{{LntAppletLinkEn|analPhysSignal}}

==Applet Description==
 
This applet shows the relationship between the physical bandpass signal $x(t)$ and the associated equivalent low-pass signal $x_{\rm TP}(t)$. It is assumed that the bandpass signal $x(t)$ has a frequency-discrete spectrum $X(f)$:
:$$x(t) = x_{\rm U}(t) + x_{\rm T}(t) + x_{\rm O}(t) = A_{\rm U}\cdot \cos\left(2\pi f_{\rm U}\cdot t- \varphi_{\rm U}\right)+A_{\rm T}\cdot \cos\left(2\pi f_{\rm T}\cdot t- \varphi_{\rm T}\right)+A_{\rm O}\cdot \cos\left(2\pi f_{\rm O}\cdot t- \varphi_{\rm O}\right). $$
The physical signal $x(t)$ is thus composed of three harmonic oscillations, a constellation that can be found, for example, in the ''Double-sideband Amplitude Modulation''
*of the message signal $x_{\rm N}(t) = A_{\rm N}\cdot \cos\left(2\pi f_{\rm N}\cdot t- \varphi_{\rm N}\right)$   ⇒   in German:   '''N'''achrichtensignal
*with the carrier signal $x_{\rm T}(t) = A_{\rm T}\cdot \cos\left(2\pi f_{\rm T}\cdot t - \varphi_{\rm T}\right)$   ⇒   in German:   '''T'''rägersignal.

The nomenclature is also adapted to this case:
* $x_{\rm O}(t)$ denotes the &bdquo;upper sideband”   (in German:   '''O'''beres Seitenband) with the amplitude $A_{\rm O}= A_{\rm N}/2$, the frequency $f_{\rm O} = f_{\rm T} + f_{\rm N}$ and the phase $\varphi_{\rm O} = \varphi_{\rm T} + \varphi_{\rm N}$.
*Similarly, for the &bdquo;lower sideband”   (in German:   '''U'''nteres Seitenband) $x_{\rm U}(t)$ with $f_{\rm U} = f_{\rm T} - f_{\rm N}$, $A_{\rm U}= A_{\rm O}$ and $\varphi_{\rm U} = -\varphi_{\rm O}$.

The associated equivalent low-pass–signal is $f_{\rm O}\hspace{0.01cm}' = f_{\rm O}- f_{\rm T} > 0$,   $f_{\rm U}\hspace{0.01cm}' = f_{\rm U}- f_{\rm T} < 0$  and  $f_{\rm T}\hspace{0.01cm}' = 0$:

:$$x_{\rm TP}(t) = x_\text{TP, T}(t) + x_\text{TP, O}(t) + x_\text{TP, U}(t) = A_{\rm T}\cdot {\rm e}^{-{\rm j} \varphi_{\rm T} } \hspace{0.1cm}+ \hspace{0.1cm} A_{\rm O}\cdot {\rm e}^{-{\rm j} \varphi_{\rm O} } \cdot {\rm e}^{ {\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm O}\hspace{0.01cm}'\hspace{0.05cm}\cdot \hspace{0.05cm}t}\hspace{0.1cm}+ \hspace{0.1cm}
A_{\rm U}\cdot {\rm e}^{-{\rm j} \varphi_{\rm U} } \cdot {\rm e}^{ {\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm U}\hspace{0.01cm}'\hspace{0.05cm}\cdot \hspace{0.05cm}t} . $$

[[File:Ortskurve_1.png|right|frame|Equivalent low-pass signal currently $t=0$ for cosinusoidal carrier   ⇒   $\varphi_{\rm T} = 0$]]
The program shows $x_{\rm TP}(t)$ as the vectorial sum of three rotation pointers as a violet dot (see figure for start time $t=0$ and cosinusoidal carrier):

*The (red) pointer of the carrier $x_\text{TP, T}(t)$ with the length $A_{\rm T}$ and the zero phase position $\varphi_{\rm T}=0$ is fixed in the complex plane. So it applies to all times $t$:   $x_{\rm TP}(t)= A_{\rm T}\cdot {\rm e}^{-{\rm j} \varphi_{\rm T} }$.

*The (blue) pointer of the upper sideband $x_\text{TP, O}(t)$ with the length $A_{\rm O}$ and the zero phase position $\varphi_{\rm O}$ rotates at the angular velocity $2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm O}\hspace{0.01cm}'$ in mathematically positive direction (one revolution in time $1/f_{\rm O}\hspace{0.01cm}')$.

*The (green) pointer of the lower sideband $x_{\rm U+}(t)$ with the length $A_{\rm U}$ and the zero phase position $\varphi_{\rm U}$ rotates at the angular velocity $2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm U}\hspace{0.01cm}'$, because of $f_{\rm U}\hspace{0.01cm}'<0$ counterclockwise.

*With $f_{\rm U}\hspace{0.01cm}' = -f_{\rm O}\hspace{0.01cm}'$ the blue and green pointers will spin at the same speed but in different directions. Also, if $A_{\rm O} = A_{\rm U}$ and $\varphi_{\rm U} = -\varphi_{\rm O}$, then $x_{\rm TP}(t)$ moves on a straight line with a incline of $\varphi_{\rm T}$.

''Note:''   In the figure $\varphi_{\rm O} = +30^\circ$. From this follows for the start time $t=0$ the angle of the blue pointer (OSB) with respect to the coordinate system:   $\phi_{\rm O} = -\varphi_{\rm O} = -30^\circ$. Likewise, the null phase position $\varphi_{\rm U} = -30^\circ$ of the lower sideband (USB, grüner Zeiger) follows for the phase angle to be considered in the complex plane:   $\phi_{\rm U} = +30^\circ$.

The temporal process of $x_{\rm TP}(t)$ is also referred to below as '''locus'''. The relationship between $x_{\rm TP}(t)$ and the physical bandpass signal $x(t)$ is given in the section and the associated analytic signal is $x_+(t)$ :

:$$x_{\rm TP}(t) = x_{\rm +}(t)\cdot {\rm e}^{-{\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm}2 \pi \cdot f_{\rm T}\cdot \hspace{0.05cm}t},$$
:$$x_{\rm +}(t) = x_{\rm TP}(t)\cdot {\rm e}^{+{\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm}2 \pi \cdot f_{\rm T}\cdot \hspace{0.05cm}t}.$$

[[Applets:Physikalisches_Signal_%26_Äquivalentes_TP-Signal|'''German description''']]

==Theoretical Background==
 
===Description of Bandpass Signals===
[[File:Zeigerdiagramm_1a.png|right|frame|bandpass spectrum $X(f)$ |class=fit]]
We consider '''bandpass signals''' $x(t)$ with the property that their spectra $X(f)$ are not in the range around the frequency $f=0$, but around a carrier frequency $f_{\rm T}$. In most cases it can also be assumed that the bandwidth is $B \ll f_{\rm T}$.

The figure shows such a bandpass spectrum $X(f)$. Assuming that the associated $x(t)$ is a physical signal and thus real, the spectral function $X(f)$ has a symmetry with respect to the frequency $f = 0$, if $x(t)$ is an even function   ⇒   $x(-t)=x(t)$, $X(f)$ is real and even.

Beside the physical signal $x(t)\ \circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\, \ X(f)$, one can also use the following descriptions of bandpass signals:
*the analytic signal $x_+(t)\ \circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\, \ X_+(f)$, see applet [[Applets:Physikalisches_Signal_%26_Analytisches_Signal|Physikalisches Signal & Analytisches Signal]],
*the equivalent low-pass–signal $x_{\rm TP}(t)\ \circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\, \ X_{\rm TP}(f)$, see next page
 

===Spectral Functions of the Analytic and the Equivalent Low-Pass Signal===

The '''analytic signal''' $x_+(t)$ belonging to the physical signal $x(t)$ is the time function whose spectrum fulfills the following property:
[[File:Ortskurve_2.png|right|frame|spectral functions $X_+(f)$ and $X_{\rm TP}(f)$ |class=fit]]
:$$X_+(f)=\big[1+{\rm sign}(f)\big] \cdot X(f) = \left\{ {2 \cdot
X(f) \; \hspace{0.2cm}\rm for\hspace{0.2cm} {\it f} > 0, \atop {\,\,\,\, \rm 0 \; \hspace{0.9cm}\rm for\hspace{0.2cm} {\it f} < 0.} }\right.$$

The ''Signum function'' is for positive values of $f$ equal to $+1$ and for negative $f$ values equal to $-1$.
*The (double-sided) limit returns $\sign(0) = 0$.
*The index „+” should make it clear that $X_+(f)$ only has parts at positive frequencies.

From the graph you can see the calculation rule for $X_+(f)$: The actual BP spectrum $X(f)$ becomes
*doubled at the positive frequencies, and
*set to zero at the negative frequencies.

Due to the asymmetry of $X_+(f)$ with respect to the frequency $f = 0$, it can already be said that the time function $x_+(t)$ except for a trivial special case $x_+(t)= 0 \ \ \circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\,\ \ X_+(f)= 0$ is always complex.

The spectrum $X_{\rm TP}(f)$ of the equivalent low-pass signal is obtained by shifting $X_+(f)$ to the left by the carrier frequency $f_{\rm T}$:
:$$X_{\rm TP}(f)= X_+(f+f_{\rm T}).$$

In the time domain this operation corresponds to the multiplication of $x_{\rm +}(t)$ with the complex exponential function with negative exponent:
:$$x_{\rm TP}(t) = x_{\rm +}(t)\cdot {\rm e}^{-{\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm}2 \pi \cdot f_{\rm T}\cdot \hspace{0.05cm}t}.$$

It can be seen that $x_{\rm TP}(t)$ is generally complex. But if $X_+(f)$ is symmetric about the carrier frequency $f_{\rm T}$, $X_{\rm TP}(f)$ is symmetric about the frequency $f=0$ and there is accordingly real time function $x_{\rm TP}(t)$.

===$x_{\rm TP}(t)$–Representation of a Sum of Three Harmonic Oscillations===

In our applet, we always assume a set of three rotating pointers. The physical signal is:
:$$x(t) = x_{\rm U}(t) + x_{\rm T}(t) + x_{\rm O}(t) = A_{\rm U}\cdot \cos\left(2\pi f_{\rm U}\cdot t- \varphi_{\rm U}\right)+A_{\rm T}\cdot \cos\left(2\pi f_{\rm T}\cdot t- \varphi_{\rm T}\right)+A_{\rm O}\cdot \cos\left(2\pi f_{\rm O}\cdot t- \varphi_{\rm O}\right). $$
* Each of the three harmonic oscillations $x_{\rm T}(t)$, $x_{\rm U}(t)$ and $x_{\rm O}(t)$ is represented by an amplitude $(A)$, a frequency $(f)$ and a phase value $(\varphi)$.
*The indices are based on the modulation method [[Modulationsverfahren/Zweiseitenband-Amplitudenmodulation|double sideband amplitude modulation]]. &bdquo;T” stands for &bdquo;carrier”, &bdquo;U” for &bdquo;lower sideband” and &bdquo;O” for &bdquo;upper Sideband”. Similarly, $f_{\rm U} < f_{\rm T}$ and $f_{\rm O} > f_{\rm T}$. There are no restrictions for the amplitudes and phases.

The associated equivalent low-pass signal is with $f_{\rm O}\hspace{0.01cm}' = f_{\rm O}- f_{\rm T} > 0$,   $f_{\rm U}\hspace{0.01cm}' = f_{\rm U}- f_{\rm T} < 0$  and  $f_{\rm T}\hspace{0.01cm}' = 0$:

:$$x_{\rm TP}(t) = x_\text{TP, T}(t) + x_\text{TP, O}(t) + x_\text{TP, U}(t) = A_{\rm T}\cdot {\rm e}^{-{\rm j} \varphi_{\rm T} } \hspace{0.1cm}+ \hspace{0.1cm} A_{\rm O}\cdot {\rm e}^{-{\rm j} \varphi_{\rm O} } \cdot {\rm e}^{ {\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm O}\hspace{0.01cm}'\hspace{0.05cm}\cdot \hspace{0.05cm}t}\hspace{0.1cm}+ \hspace{0.1cm}
A_{\rm U}\cdot {\rm e}^{-{\rm j} \varphi_{\rm U} } \cdot {\rm e}^{ {\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm U}\hspace{0.01cm}'\hspace{0.05cm}\cdot \hspace{0.05cm}t} . $$

{{GraueBox|TEXT=
$\text{Example 1:}$ 
The constellation given here results, for example, in the [[Modulationsverfahren/Zweiseitenband-Amplitudenmodulation#AM-Signale_und_-Spektren_bei_harmonischem_Eingangssignal|double sideband amplitude modulation]] of the message signal $x_{\rm N}(t) = A_{\rm N}\cdot \cos\left(2\pi f_{\rm N}\cdot t- \varphi_{\rm N}\right)$ with the carrier signal $x_{\rm T}(t) = A_{\rm T}\cdot \cos\left(2\pi f_{\rm T}\cdot t - \varphi_{\rm T}\right)$. This is discussed frequently in the exercises.

[[File:Ortskurve_4.png|center|frame|Spectrum $X_{\rm TP}(f)$ of the equivalent low–pass signal for different phase constellations |class=fit]]

There are some limitations to the program parameters in this approach:
* The frequencies are always $f\hspace{0.05cm}'_{\rm O} = f_{\rm N}$ and $f\hspace{0.05cm}'_{\rm U} = -f_{\rm N}$.
*Without distortion, the amplitude of the sidebands is $A_{\rm O}= A_{\rm U}= A_{\rm N}/2$.
*The respective phase relationships can be seen in the following graphic.

}}

===Representation of the Equivalent Low-Pass Signal by Magnitude and Phase===

The generally complex equivalent low-pass signal
:$$x_{\rm TP}(t) = a(t) \cdot {\rm e}^{ {\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm} \phi(t) }$$
can be split into a magnitude function $a(t)$ and a phase function $\phi(t)$ according to the equation given here, where:
:$$a(t) = \vert x_{\rm TP}(t)\vert = \sqrt{ {\rm Re}^2\big [x_{\rm TP}(t)\big ] + {\rm Im}^2\big [x_{\rm TP}(t)\big ] }\hspace{0.05cm},$$
:$$\phi(t) = \text{arc }x_{\rm TP}(t) = \arctan \frac{{\rm Im}\big [x_{\rm TP}(t)\big ]}{{\rm Re}\big [x_{\rm TP}(t)\big ]}.$$

The reason for this is that a bandpass–signal $x(t)$ is usually described by the equivalent low-pass signal $x_{\rm TP}(t)$ that the functions $a(t)$ and $\phi(t)$ are interpretable in both representations:
*The amount $a(t)$ of the equivalent low-pass signal $x_{\rm TP}(t)$ indicates the (time-dependent) envelope of $x(t)$.
*The phase $\phi(t)$ of $x_{\rm TP}(t)$ denotes the location of the zero crossings of $x(t)$, where:
:–   For $\phi(t)>0$ the zero crossing is earlier than its nominal position   ⇒   the signal is leading here.
:–  When $\phi(t)<0$, the zero crossing is later than its target position   ⇒   the signal is trailing here.

{{GraueBox|TEXT=
$\text{Example 2:}$ 
The graph is intended to illustrate this relationship, assuming $A_{\rm U} > A_{\rm O}$   ⇒   the green pointer (for the lower sideband) is longer than the blue pointer (upper sideband). This is a snapshot at time $t_0$:

[[File:Ortskurve_3_neu.png|center|frame|bandpass spectrum $X(f)$ |class=fit]]

*For these system parameters, the top of the pointer cluster $x_{\rm TP}(t)$ – that is, the geometric sum of red, blue and green pointers – on an ellipse.
* The amount $a(t_0) = \vert x_{\rm TP}(t_0) \vert$ is drawn in black in the left-hand diagram and the phase value $\phi(t_0) = \text{arc }x_{\rm TP}(t_0) > 0$ is indicated in brown color.
*In the graph on the right, the amount $a(t_0) = \vert x_{\rm TP}(t_0) \vert$ of the equivalent low-pass signal indicates the envelope of the physical signal $x(t)$.
* At $\phi(t) \equiv 0$, all zero crossings of $x(t)$ would occur at equidistant intervals. Because of $\phi(t_0) > 0$, the signal is leading at the time $t_0$, that is: the zero crossings come earlier than the grid dictates. }}

==Exercises==
[[File:Exercises_verzerrungen.png|right]]
*First select the task number.
*A task description is displayed.
*Parameter values are adjusted.
*Solution after pressing &bdquo;Hide solition”.

The number &bdquo;0” will reset the program and output a text with the further explanation of the applet.

In the following, $\rm Green$ denotes the lower sideband   ⇒   $\big (A_{\rm U}, f_{\rm U}, \varphi_{\rm U}\big )$,  
$\rm Red$ the carrier   ⇒   $\big (A_{\rm T}, f_{\rm T}, \varphi_{\rm T}\big )$ and
$\rm Blue$ the upper sideband   ⇒   $\big (A_{\rm O}, f_{\rm O}, \varphi_{\rm O}\big )$.

{{BlaueBox|TEXT=
'''(1)'''   Let   $\text{Red:} \hspace{0.15cm} A_{\rm T} = 1\ \text{V}, f_{\rm T} = 100 \ \text{kHz}, \varphi_{\rm T} = 0^\circ$,   $\text{Green:} \hspace{0.15cm} A_{\rm U} = 0.4 \text{V}, \ f_{\rm U} = 80 \ \text{kHz}, \varphi_{\rm U} = -90^\circ$,   $\text{Blue:} \hspace{0.15cm} A_{\rm O} = 0.4\ \text{V}, f_{\rm O} = 120 \ \text{kHz}, \varphi_{\rm O} = 90^\circ$.

:Consider and interpret the equivalent low-pass signal $x_{\rm TP}(t)$ and the physical signal $x(t)$. Which period $T_0$ recognizable?}}

:: The equivalent low-pass signal $x_{\rm TP}(t)$ takes from $x_{\rm TP}(t=0)=1\ \text{V}$ on the real axis values between $0.2\ \text{V}$ and $1.8\ \text{V}$   ⇒   phase $\phi(t) \equiv 0$.  The amount $|x_{\rm TP}(t)|$ indicates the envelope $a(t)$ of the physical signal $x(t)$. It holds $A_{\rm N} = 0.8\ \text{V}$ and $f_{\rm N} = 20\ \text{kHz}$:   $a(t) = A_{\rm T}+ A_{\rm N} \cdot \sin(2\pi\cdot f_{\rm N} \cdot t)$.  Both $x_{\rm TP}(t)$ and $x(t)$ are periodically with the period $T_0 = 1/f_{\rm N} = 50\ \rm µ s$.

{{BlaueBox|TEXT=
'''(2)'''   How do the ratios change to '''(1)''' with $f_{\rm U} = 99 \ \text{kHz}$ and $f_{\rm O} = 101 \ \text{kHz}$ ? How could $x(t)$ have arisen?}}

:: For the envelope $a(t)$ of the signal $x(t)$ we still have $a(t) = A_{\rm T}+ A_{\rm N} \cdot \sin(2\pi\cdot f_{\rm N} \cdot t)$, but now $f_{\rm N} = 1\ \text{kHz}$. Even though it can not be recognized:  $x_{\rm TP}(t)$ and $x(t)$ are still periodically:   $T_0 = 1\ \rm ms$. Example: Double sideband Amplitude modulation '''(DSB–AM)''' of a sine signal with cosine–carrier.

{{BlaueBox|TEXT=
'''(3)'''   Which settings have to be changed from '''(2)''' in order to arrive at the DSB–AM of a cosine signal with sine carrier. What changes over '''(2)'''?}}

::The carrier phase must be changed to $\varphi_{\rm T} = 90^\circ$   ⇒   sine carrier. Similarly, $\varphi_{\rm O} =\varphi_{\rm U} =\varphi_{\rm T} = 90^\circ$ must be set   ⇒   cosinusoidal message  The locus now lies on the imaginary axis  ⇒   $\phi(t) \equiv -90^\circ$. At the beginning $x_{\rm TP}(t=0)= - {\rm j} \cdot 1.8 \ \text{V}$.

{{BlaueBox|TEXT=
'''(4)'''   Now let   $\text{Red:} \hspace{0.15cm} A_{\rm T} = 1\ \text{V}, \ f_{\rm T} = 100 \ \text{kHz}, \ \varphi_{\rm T} = 0^\circ$,   $\text{Green:} \hspace{0.15cm} A_{\rm U} = 0.4 \text{V}, \ f_{\rm U} = 80 \ \text{kHz}, \ \varphi_{\rm U} = 0^\circ$,   $\text{Blue:} \hspace{0.15cm} A_{\rm O} = 0.4\ \text{V}, \ f_{\rm O} = 120 \ \text{kHz}, \ \varphi_{\rm O} = 0^\circ$.

:What are the characteristics of this system &bdquo;DSB–AM, where the message signal and carrier are respectively cosinusoidal”? What is the degree of modulation $m$?}}

:: The equivalent low-pass signal $x_{\rm TP}(t)$ takes from $x_{\rm TP}(t=0)=1.8\ \text{V}$ on the real axis values between $0.2\ \text{V}$ and $1.8\ \text{V}$   ⇒   phase $\phi(t) \equiv 0$.  Except for the start state $x_{\rm TP}(t=0)$ same behavior as at the setting '''(1)'''. The degree of modulation is $m = 0.8$.

{{BlaueBox|TEXT=
'''(5)'''   The parameters are still valid according to '''(4)''' with the exception of $A_{\rm T}= 0.6 \text{V}$. What is the degree of modulation $m$? What are the consequences?}}

:: There is now a DSB–AM with modulation degree $m = 1.333$. For $m > 1$, the simpler [[Modulationsverfahren/Hüllkurvendemodulation|envelope demodulation]] is not applicable, since the phase function $\phi(t) \in \{ 0, \ \pm 180^\circ\}$ is no more constant and the envelope $a(t)$ no more matches the message signal. Rather, the complex [[Modulationsverfahren/Synchrondemodulation|synchronous demodulation]] must be used. Envelope detection would produce nonlinear distortions.

{{BlaueBox|TEXT=
'''(6)'''   The parameters are still valid according to '''(4)''' or '''(5)''' with the exception from $A_{\rm T}= 0$ on   ⇒   $m \to \infty$. Which modulation method is described in this way?}}

::It is a '''DSB–AM without carrier''' and a synchronous demodulation is required. The equivalent low-pass signal $x_{\rm TP}(t)$ is on the real axis, but not only in the right half-plane. Thus, the phase function $\phi(t) \in \{ 0, \ \pm 180^\circ\}$, also applies here, which means that envelope demodulation is not applicable.

{{BlaueBox|TEXT=
'''(7)'''   Now let   $\text{Red:} \hspace{0.15cm} A_{\rm T} = 1\ \text{V}, f_{\rm T} = 100 \ \text{kHz}, \varphi_{\rm T} = 0^\circ$,   $\text{Green:} \hspace{0.15cm} A_{\rm U} = 0, \ f_{\rm U} = 80 \ \text{kHz}, \varphi_{\rm U} = -90^\circ$,   $\text{Blue:} \hspace{0.15cm} A_{\rm O} = 0.8\ \text{V}, f_{\rm O} = 120 \ \text{kHz}, \varphi_{\rm O} = 90^\circ$.

:Which constellation is described here? Which characteristics of this procedure can be recognized from the graphic?}}

::It is a [[Modulationsverfahren/Einseitenbandmodulation|single-sideband modulation]] '''(SSB–AM)''', more specifically an '''OSB–AM''': the red carrier is fixed, the green pointer missing and the blue pointer (OSB) turns counterclockwise. The degree of modulation is $\mu = 0.8$ (in the case of SSB we denote the degree of modulation with $\mu$ instead of $m$). The carrier signal is cosinusoidal and the message signal sinusoidal. The locus is a circle. $x_{\rm TP}(t)$ moves in a mathematically positive direction. Because of $\phi(t) \ne \text{const.}$ the envelope demodulation is not applicable here:  This can be seen by the fact that the envelope $a(t)$ is not cosinusoidal. Rather, the lower half-wave is sharper than the upper one   ⇒   strong linear distortions.

{{BlaueBox|TEXT=
'''(8)'''   The parameters are still valid according to '''(7)''' with the exception of $A_{\rm O}= 0$ and $A_{\rm U}= 0.8 \text{ V}$. What differences arise opposite '''(7)'''?}}

::Now it is a '''LSB–AM''': The red carrier is fixed, the blue pointer is missing and the green pointer (LSB) rotates clockwise. All other statements of '''(7)''' apply here as well.

{{BlaueBox|TEXT=
'''(9)'''   The parameters according to '''(7)''' are still valid with the exception of $A_{\rm O} = 0.2 \text{ V} \ne A_{\rm U} = 0.4 \text{ V} $. What are the differences from '''(7)'''?}}

::The locus $x_{\rm TP}(t)$ is not a horizontal straight line, but an ellipse with the real part between $0.4 \text{ V}$ and $1.6 \text{ V}$ and the imaginary part in the range $\pm 0.2 \text{ V}$. Because of $\phi(t) \ne \text{const.}$ , Envelope demodulation would lead to non-linear distortions here too. The constellation simulated here describes the situation of '''(4)''', namely a DSB–AM with modulation degree $m = 0.8$, where the upper sideband is reduced to $50\%$ due to channel attenuation.

==Applet Manual==
[[File:Ortskurve_abzug3.png|right]]
* The red parameters $(A_{\rm T}, \ f_{\rm T}, \ \varphi_{\rm T})$ and the red pointer mark the ''Carrier''. (German: '''T'''räger).
* The green parameters $(A_{\rm U}, \ f_{\rm U} < f_{\rm T}, \ \varphi_{\rm U})$ mark the ''Lower sideband''. (German: '''U'''ntere Seitenband).
* The blue parameters $(A_{\rm O}, \ f_{\rm O} > f_{\rm T}, \ \varphi_{\rm O})$ mark the '' Upper sideband''. (German: '''O'''bere Seitenband).
* The red pointer does not turn.
* The green pointer rotates in a mathematically negative direction (clockwise).
* The blue pointer turns counterclockwise.

Meaning of the letters in the adjacent graphic:

    '''(A)'''     Plot of the equivalent low-pass Signal $x_{\rm TP}(t)$

    '''(B)'''     Plot of the physical signal $x(t)$

    '''(C)'''     Parameter input via slider:   amplitudes, frequencies, phase values

    '''(D)'''     Control elements:   Start – Step – Pause/Continue – Reset

    '''(E)'''     Speed of animation:   &bdquo;Speed”   ⇒   Values: 1, 2 oder 3

    '''(F)'''     &bdquo;Trace”   ⇒   On or Off, trace of equivalent low-pass Signal   $x_{\rm TP}(t)$

    '''(G)'''     Numerical output:   time $t$, the signal values  ${\rm Re}[x_{\rm TP}(t)]$  and  ${\rm Im}[x_{\rm TP}(t)]$,

$\text{}\hspace{4.2cm}$   envelope $a(t) = |x_{\rm TP}(t)|$  and  phase $\phi(t) = {\rm arc} \ x_{\rm TP}(t)$

    '''(H)'''     Variations for the graphical representation

$\hspace{1.5cm}$Zoom–Functions &bdquo;$+$” (Enlarge), &bdquo;$-$” (Decrease) und $\rm o$ (Reset to default)

$\hspace{1.5cm}$Move with &bdquo;$\leftarrow$” (Section to the left, ordinate to the right), &bdquo;$\uparrow$” &bdquo;$\downarrow$” &bdquo;$\rightarrow$”

    '''(I)'''     Experiment section:   Task selection and task

    '''(J)'''     Experiment section:  solution
 

==About the Authors==
This interactive calculation was designed and realized at the [http://www.lnt.ei.tum.de/startseite Lehrstuhl für Nachrichtentechnik] of the [https://www.tum.de/ Technical University of Munich] .
*The original version was created in 2005 by [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Studierende#Ji_Li_.28Bachelorarbeit_EI_2003.2C_Diplomarbeit_EI_2005.29|Ji Li]] as part of her Diploma thesis using &bdquo;FlashMX–Actionscript” (Supervisor: [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Mitarbeiter_und_Dozenten#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28am_LNT_seit_1974.29|Günter Söder]]).
*In 2018 this Applet was redesigned and updated to &bdquo;HTML5” by [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Studierende#Xiaohan_Liu_.28Bachelorarbeit_2018.29|Xiaohan Liu]] as part of her Bachelor's thesis (Supervisor: [[Biografien_und_Bibliografien/Beteiligte_der_Professur_Leitungsgebundene_%C3%9Cbertragungstechnik#Tasn.C3.A1d_Kernetzky.2C_M.Sc._.28bei_L.C3.9CT_seit_2014.29|Tasnád Kernetzky]]).

==Once again: Open Applet in new Tab==

{{LntAppletLinkEn|physAnSignal}}

[[Category:Applets|^Verzerrungen^]]

Applets:Physical Signal & Equivalent Lowpass Signal

2018-08-26T10:00:27Z

Xiaohan: /* Exercises */

{{LntAppletLinkEn|analPhysSignal}}

==Applet Description==
 
This applet shows the relationship between the physical bandpass signal $x(t)$ and the associated equivalent low-pass signal $x_{\rm TP}(t)$. It is assumed that the bandpass signal $x(t)$ has a frequency-discrete spectrum $X(f)$:
:$$x(t) = x_{\rm U}(t) + x_{\rm T}(t) + x_{\rm O}(t) = A_{\rm U}\cdot \cos\left(2\pi f_{\rm U}\cdot t- \varphi_{\rm U}\right)+A_{\rm T}\cdot \cos\left(2\pi f_{\rm T}\cdot t- \varphi_{\rm T}\right)+A_{\rm O}\cdot \cos\left(2\pi f_{\rm O}\cdot t- \varphi_{\rm O}\right). $$
The physical signal $x(t)$ is thus composed of three harmonic oscillations, a constellation that can be found, for example, in the ''Double-sideband Amplitude Modulation''
*of the message signal $x_{\rm N}(t) = A_{\rm N}\cdot \cos\left(2\pi f_{\rm N}\cdot t- \varphi_{\rm N}\right)$   ⇒   in German:   '''N'''achrichtensignal
*with the carrier signal $x_{\rm T}(t) = A_{\rm T}\cdot \cos\left(2\pi f_{\rm T}\cdot t - \varphi_{\rm T}\right)$   ⇒   in German:   '''T'''rägersignal.

The nomenclature is also adapted to this case:
* $x_{\rm O}(t)$ denotes the &bdquo;upper sideband”   (in German:   '''O'''beres Seitenband) with the amplitude $A_{\rm O}= A_{\rm N}/2$, the frequency $f_{\rm O} = f_{\rm T} + f_{\rm N}$ and the phase $\varphi_{\rm O} = \varphi_{\rm T} + \varphi_{\rm N}$.
*Similarly, for the &bdquo;lower sideband”   (in German:   '''U'''nteres Seitenband) $x_{\rm U}(t)$ with $f_{\rm U} = f_{\rm T} - f_{\rm N}$, $A_{\rm U}= A_{\rm O}$ and $\varphi_{\rm U} = -\varphi_{\rm O}$.

The associated equivalent low-pass–signal is $f_{\rm O}\hspace{0.01cm}' = f_{\rm O}- f_{\rm T} > 0$,   $f_{\rm U}\hspace{0.01cm}' = f_{\rm U}- f_{\rm T} < 0$  and  $f_{\rm T}\hspace{0.01cm}' = 0$:

:$$x_{\rm TP}(t) = x_\text{TP, T}(t) + x_\text{TP, O}(t) + x_\text{TP, U}(t) = A_{\rm T}\cdot {\rm e}^{-{\rm j} \varphi_{\rm T} } \hspace{0.1cm}+ \hspace{0.1cm} A_{\rm O}\cdot {\rm e}^{-{\rm j} \varphi_{\rm O} } \cdot {\rm e}^{ {\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm O}\hspace{0.01cm}'\hspace{0.05cm}\cdot \hspace{0.05cm}t}\hspace{0.1cm}+ \hspace{0.1cm}
A_{\rm U}\cdot {\rm e}^{-{\rm j} \varphi_{\rm U} } \cdot {\rm e}^{ {\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm U}\hspace{0.01cm}'\hspace{0.05cm}\cdot \hspace{0.05cm}t} . $$

[[File:Ortskurve_1.png|right|frame|Equivalent low-pass signal currently $t=0$ for cosinusoidal carrier   ⇒   $\varphi_{\rm T} = 0$]]
The program shows $x_{\rm TP}(t)$ as the vectorial sum of three rotation pointers as a violet dot (see figure for start time $t=0$ and cosinusoidal carrier):

*The (red) pointer of the carrier $x_\text{TP, T}(t)$ with the length $A_{\rm T}$ and the zero phase position $\varphi_{\rm T}=0$ is fixed in the complex plane. So it applies to all times $t$:   $x_{\rm TP}(t)= A_{\rm T}\cdot {\rm e}^{-{\rm j} \varphi_{\rm T} }$.

*The (blue) pointer of the upper sideband $x_\text{TP, O}(t)$ with the length $A_{\rm O}$ and the zero phase position $\varphi_{\rm O}$ rotates at the angular velocity $2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm O}\hspace{0.01cm}'$ in mathematically positive direction (one revolution in time $1/f_{\rm O}\hspace{0.01cm}')$.

*The (green) pointer of the lower sideband $x_{\rm U+}(t)$ with the length $A_{\rm U}$ and the zero phase position $\varphi_{\rm U}$ rotates at the angular velocity $2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm U}\hspace{0.01cm}'$, because of $f_{\rm U}\hspace{0.01cm}'<0$ counterclockwise.

*With $f_{\rm U}\hspace{0.01cm}' = -f_{\rm O}\hspace{0.01cm}'$ the blue and green pointers will spin at the same speed but in different directions. Also, if $A_{\rm O} = A_{\rm U}$ and $\varphi_{\rm U} = -\varphi_{\rm O}$, then $x_{\rm TP}(t)$ moves on a straight line with a incline of $\varphi_{\rm T}$.

''Note:''   In the figure $\varphi_{\rm O} = +30^\circ$. From this follows for the start time $t=0$ the angle of the blue pointer (OSB) with respect to the coordinate system:   $\phi_{\rm O} = -\varphi_{\rm O} = -30^\circ$. Likewise, the null phase position $\varphi_{\rm U} = -30^\circ$ of the lower sideband (USB, grüner Zeiger) follows for the phase angle to be considered in the complex plane:   $\phi_{\rm U} = +30^\circ$.

The temporal process of $x_{\rm TP}(t)$ is also referred to below as '''locus'''. The relationship between $x_{\rm TP}(t)$ and the physical bandpass signal $x(t)$ is given in the section and the associated analytic signal is $x_+(t)$ :

:$$x_{\rm TP}(t) = x_{\rm +}(t)\cdot {\rm e}^{-{\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm}2 \pi \cdot f_{\rm T}\cdot \hspace{0.05cm}t},$$
:$$x_{\rm +}(t) = x_{\rm TP}(t)\cdot {\rm e}^{+{\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm}2 \pi \cdot f_{\rm T}\cdot \hspace{0.05cm}t}.$$

[[Applets:Physikalisches_Signal_%26_Äquivalentes_TP-Signal|'''German description''']]

==Theoretical Background==
 
===Description of Bandpass Signals===
[[File:Zeigerdiagramm_1a.png|right|frame|bandpass spectrum $X(f)$ |class=fit]]
We consider '''bandpass signals''' $x(t)$ with the property that their spectra $X(f)$ are not in the range around the frequency $f=0$, but around a carrier frequency $f_{\rm T}$. In most cases it can also be assumed that the bandwidth is $B \ll f_{\rm T}$.

The figure shows such a bandpass spectrum $X(f)$. Assuming that the associated $x(t)$ is a physical signal and thus real, the spectral function $X(f)$ has a symmetry with respect to the frequency $f = 0$, if $x(t)$ is an even function   ⇒   $x(-t)=x(t)$, $X(f)$ is real and even.

Beside the physical signal $x(t)\ \circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\, \ X(f)$, one can also use the following descriptions of bandpass signals:
*the analytic signal $x_+(t)\ \circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\, \ X_+(f)$, see applet [[Applets:Physikalisches_Signal_%26_Analytisches_Signal|Physikalisches Signal & Analytisches Signal]],
*the equivalent low-pass–signal $x_{\rm TP}(t)\ \circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\, \ X_{\rm TP}(f)$, see next page
 

===Spectral Functions of the Analytic and the Equivalent Low-Pass Signal===

The '''analytic signal''' $x_+(t)$ belonging to the physical signal $x(t)$ is the time function whose spectrum fulfills the following property:
[[File:Ortskurve_2.png|right|frame|spectral functions $X_+(f)$ and $X_{\rm TP}(f)$ |class=fit]]
:$$X_+(f)=\big[1+{\rm sign}(f)\big] \cdot X(f) = \left\{ {2 \cdot
X(f) \; \hspace{0.2cm}\rm for\hspace{0.2cm} {\it f} > 0, \atop {\,\,\,\, \rm 0 \; \hspace{0.9cm}\rm for\hspace{0.2cm} {\it f} < 0.} }\right.$$

The ''Signum function'' is for positive values of $f$ equal to $+1$ and for negative $f$ values equal to $-1$.
*The (double-sided) limit returns $\sign(0) = 0$.
*The index „+” should make it clear that $X_+(f)$ only has parts at positive frequencies.

From the graph you can see the calculation rule for $X_+(f)$: The actual BP spectrum $X(f)$ becomes
*doubled at the positive frequencies, and
*set to zero at the negative frequencies.

Due to the asymmetry of $X_+(f)$ with respect to the frequency $f = 0$, it can already be said that the time function $x_+(t)$ except for a trivial special case $x_+(t)= 0 \ \ \circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\,\ \ X_+(f)= 0$ is always complex.

The spectrum $X_{\rm TP}(f)$ of the equivalent low-pass signal is obtained by shifting $X_+(f)$ to the left by the carrier frequency $f_{\rm T}$:
:$$X_{\rm TP}(f)= X_+(f+f_{\rm T}).$$

In the time domain this operation corresponds to the multiplication of $x_{\rm +}(t)$ with the complex exponential function with negative exponent:
:$$x_{\rm TP}(t) = x_{\rm +}(t)\cdot {\rm e}^{-{\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm}2 \pi \cdot f_{\rm T}\cdot \hspace{0.05cm}t}.$$

It can be seen that $x_{\rm TP}(t)$ is generally complex. But if $X_+(f)$ is symmetric about the carrier frequency $f_{\rm T}$, $X_{\rm TP}(f)$ is symmetric about the frequency $f=0$ and there is accordingly real time function $x_{\rm TP}(t)$.

===$x_{\rm TP}(t)$–Representation of a Sum of Three Harmonic Oscillations===

In our applet, we always assume a set of three rotating pointers. The physical signal is:
:$$x(t) = x_{\rm U}(t) + x_{\rm T}(t) + x_{\rm O}(t) = A_{\rm U}\cdot \cos\left(2\pi f_{\rm U}\cdot t- \varphi_{\rm U}\right)+A_{\rm T}\cdot \cos\left(2\pi f_{\rm T}\cdot t- \varphi_{\rm T}\right)+A_{\rm O}\cdot \cos\left(2\pi f_{\rm O}\cdot t- \varphi_{\rm O}\right). $$
* Each of the three harmonic oscillations $x_{\rm T}(t)$, $x_{\rm U}(t)$ and $x_{\rm O}(t)$ is represented by an amplitude $(A)$, a frequency $(f)$ and a phase value $(\varphi)$.
*The indices are based on the modulation method [[Modulationsverfahren/Zweiseitenband-Amplitudenmodulation|double sideband amplitude modulation]]. &bdquo;T” stands for &bdquo;carrier”, &bdquo;U” for &bdquo;lower sideband” and &bdquo;O” for &bdquo;upper Sideband”. Similarly, $f_{\rm U} < f_{\rm T}$ and $f_{\rm O} > f_{\rm T}$. There are no restrictions for the amplitudes and phases.

The associated equivalent low-pass signal is with $f_{\rm O}\hspace{0.01cm}' = f_{\rm O}- f_{\rm T} > 0$,   $f_{\rm U}\hspace{0.01cm}' = f_{\rm U}- f_{\rm T} < 0$  and  $f_{\rm T}\hspace{0.01cm}' = 0$:

:$$x_{\rm TP}(t) = x_\text{TP, T}(t) + x_\text{TP, O}(t) + x_\text{TP, U}(t) = A_{\rm T}\cdot {\rm e}^{-{\rm j} \varphi_{\rm T} } \hspace{0.1cm}+ \hspace{0.1cm} A_{\rm O}\cdot {\rm e}^{-{\rm j} \varphi_{\rm O} } \cdot {\rm e}^{ {\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm O}\hspace{0.01cm}'\hspace{0.05cm}\cdot \hspace{0.05cm}t}\hspace{0.1cm}+ \hspace{0.1cm}
A_{\rm U}\cdot {\rm e}^{-{\rm j} \varphi_{\rm U} } \cdot {\rm e}^{ {\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm U}\hspace{0.01cm}'\hspace{0.05cm}\cdot \hspace{0.05cm}t} . $$

{{GraueBox|TEXT=
$\text{Example 1:}$ 
The constellation given here results, for example, in the [[Modulationsverfahren/Zweiseitenband-Amplitudenmodulation#AM-Signale_und_-Spektren_bei_harmonischem_Eingangssignal|double sideband amplitude modulation]] of the message signal $x_{\rm N}(t) = A_{\rm N}\cdot \cos\left(2\pi f_{\rm N}\cdot t- \varphi_{\rm N}\right)$ with the carrier signal $x_{\rm T}(t) = A_{\rm T}\cdot \cos\left(2\pi f_{\rm T}\cdot t - \varphi_{\rm T}\right)$. This is discussed frequently in the exercises.

[[File:Ortskurve_4.png|center|frame|Spectrum $X_{\rm TP}(f)$ of the equivalent low–pass signal for different phase constellations |class=fit]]

There are some limitations to the program parameters in this approach:
* The frequencies are always $f\hspace{0.05cm}'_{\rm O} = f_{\rm N}$ and $f\hspace{0.05cm}'_{\rm U} = -f_{\rm N}$.
*Without distortion, the amplitude of the sidebands is $A_{\rm O}= A_{\rm U}= A_{\rm N}/2$.
*The respective phase relationships can be seen in the following graphic.

}}

===Representation of the Equivalent Low-Pass Signal by Magnitude and Phase===

The generally complex equivalent low-pass signal
:$$x_{\rm TP}(t) = a(t) \cdot {\rm e}^{ {\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm} \phi(t) }$$
can be split into a magnitude function $a(t)$ and a phase function $\phi(t)$ according to the equation given here, where:
:$$a(t) = \vert x_{\rm TP}(t)\vert = \sqrt{ {\rm Re}^2\big [x_{\rm TP}(t)\big ] + {\rm Im}^2\big [x_{\rm TP}(t)\big ] }\hspace{0.05cm},$$
:$$\phi(t) = \text{arc }x_{\rm TP}(t) = \arctan \frac{{\rm Im}\big [x_{\rm TP}(t)\big ]}{{\rm Re}\big [x_{\rm TP}(t)\big ]}.$$

The reason for this is that a bandpass–signal $x(t)$ is usually described by the equivalent low-pass signal $x_{\rm TP}(t)$ that the functions $a(t)$ and $\phi(t)$ are interpretable in both representations:
*The amount $a(t)$ of the equivalent low-pass signal $x_{\rm TP}(t)$ indicates the (time-dependent) envelope of $x(t)$.
*The phase $\phi(t)$ of $x_{\rm TP}(t)$ denotes the location of the zero crossings of $x(t)$, where:
:–   For $\phi(t)>0$ the zero crossing is earlier than its nominal position   ⇒   the signal is leading here.
:–  When $\phi(t)<0$, the zero crossing is later than its target position   ⇒   the signal is trailing here.

{{GraueBox|TEXT=
$\text{Example 2:}$ 
The graph is intended to illustrate this relationship, assuming $A_{\rm U} > A_{\rm O}$   ⇒   the green pointer (for the lower sideband) is longer than the blue pointer (upper sideband). This is a snapshot at time $t_0$:

[[File:Ortskurve_3_neu.png|center|frame|bandpass spectrum $X(f)$ |class=fit]]

*For these system parameters, the top of the pointer cluster $x_{\rm TP}(t)$ – that is, the geometric sum of red, blue and green pointers – on an ellipse.
* The amount $a(t_0) = \vert x_{\rm TP}(t_0) \vert$ is drawn in black in the left-hand diagram and the phase value $\phi(t_0) = \text{arc }x_{\rm TP}(t_0) > 0$ is indicated in brown color.
*In the graph on the right, the amount $a(t_0) = \vert x_{\rm TP}(t_0) \vert$ of the equivalent low-pass signal indicates the envelope of the physical signal $x(t)$.
* At $\phi(t) \equiv 0$, all zero crossings of $x(t)$ would occur at equidistant intervals. Because of $\phi(t_0) > 0$, the signal is leading at the time $t_0$, that is: the zero crossings come earlier than the grid dictates. }}

==Exercises==
[[File:Exercises_verzerrungen.png|right]]
*First select the task number.
*A task description is displayed.
*Parameter values are adjusted.
*Solution after pressing &bdquo;Hide solition”.

The number &bdquo;0” will reset the program and output a text with the further explanation of the applet.

In the following, $\rm Green$ denotes the lower sideband   ⇒   $\big (A_{\rm U}, f_{\rm U}, \varphi_{\rm U}\big )$,  
$\rm Red$ the carrier   ⇒   $\big (A_{\rm T}, f_{\rm T}, \varphi_{\rm T}\big )$ and
$\rm Blue$ the upper sideband   ⇒   $\big (A_{\rm O}, f_{\rm O}, \varphi_{\rm O}\big )$.

{{BlaueBox|TEXT=
'''(1)'''   Let   $\text{Red:} \hspace{0.15cm} A_{\rm T} = 1\ \text{V}, f_{\rm T} = 100 \ \text{kHz}, \varphi_{\rm T} = 0^\circ$,   $\text{Green:} \hspace{0.15cm} A_{\rm U} = 0.4 \text{V}, \ f_{\rm U} = 80 \ \text{kHz}, \varphi_{\rm U} = -90^\circ$,   $\text{Blue:} \hspace{0.15cm} A_{\rm O} = 0.4\ \text{V}, f_{\rm O} = 120 \ \text{kHz}, \varphi_{\rm O} = 90^\circ$.

:Consider and interpret the equivalent low-pass signal $x_{\rm TP}(t)$ and the physical signal $x(t)$. Which period $T_0$ recognizable?}}

:: The equivalent low-pass signal $x_{\rm TP}(t)$ takes from $x_{\rm TP}(t=0)=1\ \text{V}$ on the real axis values between $0.2\ \text{V}$ and $1.8\ \text{V}$   ⇒   phase $\phi(t) \equiv 0$.  The amount $|x_{\rm TP}(t)|$ indicates the envelope $a(t)$ of the physical signal $x(t)$. It holds $A_{\rm N} = 0.8\ \text{V}$ and $f_{\rm N} = 20\ \text{kHz}$:   $a(t) = A_{\rm T}+ A_{\rm N} \cdot \sin(2\pi\cdot f_{\rm N} \cdot t)$.  Both $x_{\rm TP}(t)$ and $x(t)$ are periodically with the period $T_0 = 1/f_{\rm N} = 50\ \rm µ s$.

{{BlaueBox|TEXT=
'''(2)'''   How do the ratios change to '''(1)''' with $f_{\rm U} = 99 \ \text{kHz}$ and $f_{\rm O} = 101 \ \text{kHz}$ ? How could $x(t)$ have arisen?}}

:: For the envelope $a(t)$ of the signal $x(t)$ we still have $a(t) = A_{\rm T}+ A_{\rm N} \cdot \sin(2\pi\cdot f_{\rm N} \cdot t)$, but now $f_{\rm N} = 1\ \text{kHz}$. Even though it can not be recognized:  $x_{\rm TP}(t)$ and $x(t)$ are still periodically:   $T_0 = 1\ \rm ms$. Example: Double sideband Amplitude modulation '''(DSB–AM)''' of a sine signal with cosine–carrier.

{{BlaueBox|TEXT=
'''(3)'''   Which settings have to be changed from '''(2)''' in order to arrive at the DSB–AM of a cosine signal with sine carrier. What changes over '''(2)'''?}}

::The carrier phase must be changed to $\varphi_{\rm T} = 90^\circ$   ⇒   sine carrier. Similarly, $\varphi_{\rm O} =\varphi_{\rm U} =\varphi_{\rm T} = 90^\circ$ must be set   ⇒   cosinusoidal message  The locus now lies on the imaginary axis  ⇒   $\phi(t) \equiv -90^\circ$. At the beginning $x_{\rm TP}(t=0)= - {\rm j} \cdot 1.8 \ \text{V}$.

{{BlaueBox|TEXT=
'''(4)'''   Now let   $\text{Red:} \hspace{0.15cm} A_{\rm T} = 1\ \text{V}, \ f_{\rm T} = 100 \ \text{kHz}, \ \varphi_{\rm T} = 0^\circ$,   $\text{Green:} \hspace{0.15cm} A_{\rm U} = 0.4 \text{V}, \ f_{\rm U} = 80 \ \text{kHz}, \ \varphi_{\rm U} = 0^\circ$,   $\text{Blue:} \hspace{0.15cm} A_{\rm O} = 0.4\ \text{V}, \ f_{\rm O} = 120 \ \text{kHz}, \ \varphi_{\rm O} = 0^\circ$.

:What are the characteristics of this system &bdquo;DSB–AM, where the message signal and carrier are respectively cosinusoidal”? What is the degree of modulation $m$?}}

:: The equivalent low-pass signal $x_{\rm TP}(t)$ takes from $x_{\rm TP}(t=0)=1.8\ \text{V}$ on the real axis values between $0.2\ \text{V}$ and $1.8\ \text{V}$   ⇒   phase $\phi(t) \equiv 0$.  Except for the start state $x_{\rm TP}(t=0)$ same behavior as at the setting '''(1)'''. The degree of modulation is $m = 0.8$.

{{BlaueBox|TEXT=
'''(5)'''   The parameters are still valid according to '''(4)''' with the exception of $A_{\rm T}= 0.6 \text{V}$. What is the degree of modulation $m$? What are the consequences?}}

:: There is now a DSB–AM with modulation degree $m = 1.333$. For $m > 1$, the simpler [[Modulationsverfahren/Hüllkurvendemodulation|envelope demodulation]] is not applicable, since the phase function $\phi(t) \in \{ 0, \ \pm 180^\circ\}$ is no more constant and the envelope $a(t)$ no more matches the message signal. Rather, the complex [[Modulationsverfahren/Synchrondemodulation|synchronous demodulation]] must be used. Envelope detection would produce nonlinear distortions.

{{BlaueBox|TEXT=
'''(6)'''   The parameters are still valid according to '''(4)''' or '''(5)''' with the exception from $A_{\rm T}= 0$ on   ⇒   $m \to \infty$. Which modulation method is described in this way?}}

::It is a '''DSB–AM without carrier''' and a synchronous demodulation is required. The equivalent low-pass signal $x_{\rm TP}(t)$ is on the real axis, but not only in the right half-plane. Thus, the phase function $\phi(t) \in \{ 0, \ \pm 180^\circ\}$, also applies here, which means that envelope demodulation is not applicable.

{{BlaueBox|TEXT=
'''(7)'''   Now let   $\text{Red:} \hspace{0.15cm} A_{\rm T} = 1\ \text{V}, f_{\rm T} = 100 \ \text{kHz}, \varphi_{\rm T} = 0^\circ$,   $\text{Green:} \hspace{0.15cm} A_{\rm U} = 0, \ f_{\rm U} = 80 \ \text{kHz}, \varphi_{\rm U} = -90^\circ$,   $\text{Blue:} \hspace{0.15cm} A_{\rm O} = 0.8\ \text{V}, f_{\rm O} = 120 \ \text{kHz}, \varphi_{\rm O} = 90^\circ$.

:Which constellation is described here? Which characteristics of this procedure can be recognized from the graphic?}}

::It is a [[Modulationsverfahren/Einseitenbandmodulation|single-sideband modulation]] '''(SSB–AM)''', more specifically an '''OSB–AM''': the red carrier is fixed, the green pointer missing and the blue pointer (OSB) turns counterclockwise. The degree of modulation is $\mu = 0.8$ (in the case of SSB we denote the degree of modulation with $\mu$ instead of $m$). The carrier signal is cosinusoidal and the message signal sinusoidal. The locus is a circle. $x_{\rm TP}(t)$ moves in a mathematically positive direction. Because of $\phi(t) \ne \text{const.}$ The envelope demodulation is not applicable here:  This can be seen by the fact that the envelope $a(t)$ is not cosinusoidal. Rather, the lower half-wave is sharper than the upper   ⇒   strong linear distortions.

{{BlaueBox|TEXT=
'''(8)'''   The parameters are still valid according to '''(7)''' with the exception of $A_{\rm O}= 0$ and $A_{\rm U}= 0.8 \text{ V}$. What differences arise opposite '''(7)'''?}}

::Now it is a '''USB–AM''': The red carrier is fixed, the blue pointer is missing and the green pointer (LSB) rotates clockwise. All other statements of '''(7)''' apply here as well.

{{BlaueBox|TEXT=
'''(9)'''   The parameters according to '''(7)''' are still valid with the exception of $A_{\rm O} = 0.2 \text{ V} \ne A_{\rm U} = 0.4 \text{ V} $. What are the differences from '''(7)'''?}}

::The locus $x_{\rm TP}(t)$ is not a horizontal straight line, but an ellipse with the real part between $0.4 \text{ V}$ and $1.6 \text{ V}$ and the imaginary part in the range $\pm 0.2 \text{ V}$. Because of $\phi(t) \ne \text{const.}$ , Envelope demodulation would lead to non-linear distortions here too. The constellation simulated here describes the situation of '''(4)''', namely a ZSB–AM with modulation degree $m = 0.8$, where the upper sideband is reduced to $50\%$ due to channel attenuation.

==Applet Manual==
[[File:Ortskurve_abzug3.png|right]]
* The red parameters $(A_{\rm T}, \ f_{\rm T}, \ \varphi_{\rm T})$ and the red pointer mark the ''Carrier''. (German: '''T'''räger).
* The green parameters $(A_{\rm U}, \ f_{\rm U} < f_{\rm T}, \ \varphi_{\rm U})$ mark the ''Lower sideband''. (German: '''U'''ntere Seitenband).
* The blue parameters $(A_{\rm O}, \ f_{\rm O} > f_{\rm T}, \ \varphi_{\rm O})$ mark the '' Upper sideband''. (German: '''O'''bere Seitenband).
* The red pointer does not turn.
* The green pointer rotates in a mathematically negative direction (clockwise).
* The blue pointer turns counterclockwise.

Meaning of the letters in the adjacent graphic:

    '''(A)'''     Plot of the equivalent low-pass Signal $x_{\rm TP}(t)$

    '''(B)'''     Plot of the physical signal $x(t)$

    '''(C)'''     Parameter input via slider:   amplitudes, frequencies, phase values

    '''(D)'''     Control elements:   Start – Step – Pause/Continue – Reset

    '''(E)'''     Speed of animation:   &bdquo;Speed”   ⇒   Values: 1, 2 oder 3

    '''(F)'''     &bdquo;Trace”   ⇒   On or Off, trace of equivalent low-pass Signal   $x_{\rm TP}(t)$

    '''(G)'''     Numerical output:   time $t$, the signal values  ${\rm Re}[x_{\rm TP}(t)]$  and  ${\rm Im}[x_{\rm TP}(t)]$,

$\text{}\hspace{4.2cm}$   envelope $a(t) = |x_{\rm TP}(t)|$  and  phase $\phi(t) = {\rm arc} \ x_{\rm TP}(t)$

    '''(H)'''     Variations for the graphical representation

$\hspace{1.5cm}$Zoom–Functions &bdquo;$+$” (Enlarge), &bdquo;$-$” (Decrease) und $\rm o$ (Reset to default)

$\hspace{1.5cm}$Move with &bdquo;$\leftarrow$” (Section to the left, ordinate to the right), &bdquo;$\uparrow$” &bdquo;$\downarrow$” &bdquo;$\rightarrow$”

    '''(I)'''     Experiment section:   Task selection and task

    '''(J)'''     Experiment section:  solution
 

==About the Authors==
This interactive calculation was designed and realized at the [http://www.lnt.ei.tum.de/startseite Lehrstuhl für Nachrichtentechnik] of the [https://www.tum.de/ Technical University of Munich] .
*The original version was created in 2005 by [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Studierende#Ji_Li_.28Bachelorarbeit_EI_2003.2C_Diplomarbeit_EI_2005.29|Ji Li]] as part of her Diploma thesis using &bdquo;FlashMX–Actionscript” (Supervisor: [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Mitarbeiter_und_Dozenten#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28am_LNT_seit_1974.29|Günter Söder]]).
*In 2018 this Applet was redesigned and updated to &bdquo;HTML5” by [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Studierende#Xiaohan_Liu_.28Bachelorarbeit_2018.29|Xiaohan Liu]] as part of her Bachelor's thesis (Supervisor: [[Biografien_und_Bibliografien/Beteiligte_der_Professur_Leitungsgebundene_%C3%9Cbertragungstechnik#Tasn.C3.A1d_Kernetzky.2C_M.Sc._.28bei_L.C3.9CT_seit_2014.29|Tasnád Kernetzky]]).

==Once again: Open Applet in new Tab==

{{LntAppletLinkEn|physAnSignal}}

[[Category:Applets|^Verzerrungen^]]

Applets:Physical Signal & Equivalent Lowpass Signal

2018-08-25T21:31:28Z

Xiaohan: /* Applet Manual */

{{LntAppletLinkEn|analPhysSignal}}

==Applet Description==
 
This applet shows the relationship between the physical bandpass signal $x(t)$ and the associated equivalent low-pass signal $x_{\rm TP}(t)$. It is assumed that the bandpass signal $x(t)$ has a frequency-discrete spectrum $X(f)$:
:$$x(t) = x_{\rm U}(t) + x_{\rm T}(t) + x_{\rm O}(t) = A_{\rm U}\cdot \cos\left(2\pi f_{\rm U}\cdot t- \varphi_{\rm U}\right)+A_{\rm T}\cdot \cos\left(2\pi f_{\rm T}\cdot t- \varphi_{\rm T}\right)+A_{\rm O}\cdot \cos\left(2\pi f_{\rm O}\cdot t- \varphi_{\rm O}\right). $$
The physical signal $x(t)$ is thus composed of three harmonic oscillations, a constellation that can be found, for example, in the ''Double-sideband Amplitude Modulation''
*of the message signal $x_{\rm N}(t) = A_{\rm N}\cdot \cos\left(2\pi f_{\rm N}\cdot t- \varphi_{\rm N}\right)$   ⇒   in German:   '''N'''achrichtensignal
*with the carrier signal $x_{\rm T}(t) = A_{\rm T}\cdot \cos\left(2\pi f_{\rm T}\cdot t - \varphi_{\rm T}\right)$   ⇒   in German:   '''T'''rägersignal.

The nomenclature is also adapted to this case:
* $x_{\rm O}(t)$ denotes the &bdquo;upper sideband”   (in German:   '''O'''beres Seitenband) with the amplitude $A_{\rm O}= A_{\rm N}/2$, the frequency $f_{\rm O} = f_{\rm T} + f_{\rm N}$ and the phase $\varphi_{\rm O} = \varphi_{\rm T} + \varphi_{\rm N}$.
*Similarly, for the &bdquo;lower sideband”   (in German:   '''U'''nteres Seitenband) $x_{\rm U}(t)$ with $f_{\rm U} = f_{\rm T} - f_{\rm N}$, $A_{\rm U}= A_{\rm O}$ and $\varphi_{\rm U} = -\varphi_{\rm O}$.

The associated equivalent low-pass–signal is $f_{\rm O}\hspace{0.01cm}' = f_{\rm O}- f_{\rm T} > 0$,   $f_{\rm U}\hspace{0.01cm}' = f_{\rm U}- f_{\rm T} < 0$  and  $f_{\rm T}\hspace{0.01cm}' = 0$:

:$$x_{\rm TP}(t) = x_\text{TP, T}(t) + x_\text{TP, O}(t) + x_\text{TP, U}(t) = A_{\rm T}\cdot {\rm e}^{-{\rm j} \varphi_{\rm T} } \hspace{0.1cm}+ \hspace{0.1cm} A_{\rm O}\cdot {\rm e}^{-{\rm j} \varphi_{\rm O} } \cdot {\rm e}^{ {\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm O}\hspace{0.01cm}'\hspace{0.05cm}\cdot \hspace{0.05cm}t}\hspace{0.1cm}+ \hspace{0.1cm}
A_{\rm U}\cdot {\rm e}^{-{\rm j} \varphi_{\rm U} } \cdot {\rm e}^{ {\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm U}\hspace{0.01cm}'\hspace{0.05cm}\cdot \hspace{0.05cm}t} . $$

[[File:Ortskurve_1.png|right|frame|Equivalent low-pass signal currently $t=0$ for cosinusoidal carrier   ⇒   $\varphi_{\rm T} = 0$]]
The program shows $x_{\rm TP}(t)$ as the vectorial sum of three rotation pointers as a violet dot (see figure for start time $t=0$ and cosinusoidal carrier):

*The (red) pointer of the carrier $x_\text{TP, T}(t)$ with the length $A_{\rm T}$ and the zero phase position $\varphi_{\rm T}=0$ is fixed in the complex plane. So it applies to all times $t$:   $x_{\rm TP}(t)= A_{\rm T}\cdot {\rm e}^{-{\rm j} \varphi_{\rm T} }$.

*The (blue) pointer of the upper sideband $x_\text{TP, O}(t)$ with the length $A_{\rm O}$ and the zero phase position $\varphi_{\rm O}$ rotates at the angular velocity $2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm O}\hspace{0.01cm}'$ in mathematically positive direction (one revolution in time $1/f_{\rm O}\hspace{0.01cm}')$.

*The (green) pointer of the lower sideband $x_{\rm U+}(t)$ with the length $A_{\rm U}$ and the zero phase position $\varphi_{\rm U}$ rotates at the angular velocity $2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm U}\hspace{0.01cm}'$, because of $f_{\rm U}\hspace{0.01cm}'<0$ counterclockwise.

*With $f_{\rm U}\hspace{0.01cm}' = -f_{\rm O}\hspace{0.01cm}'$ the blue and green pointers will spin at the same speed but in different directions. Also, if $A_{\rm O} = A_{\rm U}$ and $\varphi_{\rm U} = -\varphi_{\rm O}$, then $x_{\rm TP}(t)$ moves on a straight line with a incline of $\varphi_{\rm T}$.

''Note:''   In the figure $\varphi_{\rm O} = +30^\circ$. From this follows for the start time $t=0$ the angle of the blue pointer (OSB) with respect to the coordinate system:   $\phi_{\rm O} = -\varphi_{\rm O} = -30^\circ$. Likewise, the null phase position $\varphi_{\rm U} = -30^\circ$ of the lower sideband (USB, grüner Zeiger) follows for the phase angle to be considered in the complex plane:   $\phi_{\rm U} = +30^\circ$.

The temporal process of $x_{\rm TP}(t)$ is also referred to below as '''locus'''. The relationship between $x_{\rm TP}(t)$ and the physical bandpass signal $x(t)$ is given in the section and the associated analytic signal is $x_+(t)$ :

:$$x_{\rm TP}(t) = x_{\rm +}(t)\cdot {\rm e}^{-{\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm}2 \pi \cdot f_{\rm T}\cdot \hspace{0.05cm}t},$$
:$$x_{\rm +}(t) = x_{\rm TP}(t)\cdot {\rm e}^{+{\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm}2 \pi \cdot f_{\rm T}\cdot \hspace{0.05cm}t}.$$

[[Applets:Physikalisches_Signal_%26_Äquivalentes_TP-Signal|'''German description''']]

==Theoretical Background==
 
===Description of Bandpass Signals===
[[File:Zeigerdiagramm_1a.png|right|frame|bandpass spectrum $X(f)$ |class=fit]]
We consider '''bandpass signals''' $x(t)$ with the property that their spectra $X(f)$ are not in the range around the frequency $f=0$, but around a carrier frequency $f_{\rm T}$. In most cases it can also be assumed that the bandwidth is $B \ll f_{\rm T}$.

The figure shows such a bandpass spectrum $X(f)$. Assuming that the associated $x(t)$ is a physical signal and thus real, the spectral function $X(f)$ has a symmetry with respect to the frequency $f = 0$, if $x(t)$ is an even function   ⇒   $x(-t)=x(t)$, $X(f)$ is real and even.

Beside the physical signal $x(t)\ \circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\, \ X(f)$, one can also use the following descriptions of bandpass signals:
*the analytic signal $x_+(t)\ \circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\, \ X_+(f)$, see applet [[Applets:Physikalisches_Signal_%26_Analytisches_Signal|Physikalisches Signal & Analytisches Signal]],
*the equivalent low-pass–signal $x_{\rm TP}(t)\ \circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\, \ X_{\rm TP}(f)$, see next page
 

===Spectral Functions of the Analytic and the Equivalent Low-Pass Signal===

The '''analytic signal''' $x_+(t)$ belonging to the physical signal $x(t)$ is the time function whose spectrum fulfills the following property:
[[File:Ortskurve_2.png|right|frame|spectral functions $X_+(f)$ and $X_{\rm TP}(f)$ |class=fit]]
:$$X_+(f)=\big[1+{\rm sign}(f)\big] \cdot X(f) = \left\{ {2 \cdot
X(f) \; \hspace{0.2cm}\rm for\hspace{0.2cm} {\it f} > 0, \atop {\,\,\,\, \rm 0 \; \hspace{0.9cm}\rm for\hspace{0.2cm} {\it f} < 0.} }\right.$$

The ''Signum function'' is for positive values of $f$ equal to $+1$ and for negative $f$ values equal to $-1$.
*The (double-sided) limit returns $\sign(0) = 0$.
*The index „+” should make it clear that $X_+(f)$ only has parts at positive frequencies.

From the graph you can see the calculation rule for $X_+(f)$: The actual BP spectrum $X(f)$ becomes
*doubled at the positive frequencies, and
*set to zero at the negative frequencies.

Due to the asymmetry of $X_+(f)$ with respect to the frequency $f = 0$, it can already be said that the time function $x_+(t)$ except for a trivial special case $x_+(t)= 0 \ \ \circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\,\ \ X_+(f)= 0$ is always complex.

The spectrum $X_{\rm TP}(f)$ of the equivalent low-pass signal is obtained by shifting $X_+(f)$ to the left by the carrier frequency $f_{\rm T}$:
:$$X_{\rm TP}(f)= X_+(f+f_{\rm T}).$$

In the time domain this operation corresponds to the multiplication of $x_{\rm +}(t)$ with the complex exponential function with negative exponent:
:$$x_{\rm TP}(t) = x_{\rm +}(t)\cdot {\rm e}^{-{\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm}2 \pi \cdot f_{\rm T}\cdot \hspace{0.05cm}t}.$$

It can be seen that $x_{\rm TP}(t)$ is generally complex. But if $X_+(f)$ is symmetric about the carrier frequency $f_{\rm T}$, $X_{\rm TP}(f)$ is symmetric about the frequency $f=0$ and there is accordingly real time function $x_{\rm TP}(t)$.

===$x_{\rm TP}(t)$–Representation of a Sum of Three Harmonic Oscillations===

In our applet, we always assume a set of three rotating pointers. The physical signal is:
:$$x(t) = x_{\rm U}(t) + x_{\rm T}(t) + x_{\rm O}(t) = A_{\rm U}\cdot \cos\left(2\pi f_{\rm U}\cdot t- \varphi_{\rm U}\right)+A_{\rm T}\cdot \cos\left(2\pi f_{\rm T}\cdot t- \varphi_{\rm T}\right)+A_{\rm O}\cdot \cos\left(2\pi f_{\rm O}\cdot t- \varphi_{\rm O}\right). $$
* Each of the three harmonic oscillations $x_{\rm T}(t)$, $x_{\rm U}(t)$ and $x_{\rm O}(t)$ is represented by an amplitude $(A)$, a frequency $(f)$ and a phase value $(\varphi)$.
*The indices are based on the modulation method [[Modulationsverfahren/Zweiseitenband-Amplitudenmodulation|double sideband amplitude modulation]]. &bdquo;T” stands for &bdquo;carrier”, &bdquo;U” for &bdquo;lower sideband” and &bdquo;O” for &bdquo;upper Sideband”. Similarly, $f_{\rm U} < f_{\rm T}$ and $f_{\rm O} > f_{\rm T}$. There are no restrictions for the amplitudes and phases.

The associated equivalent low-pass signal is with $f_{\rm O}\hspace{0.01cm}' = f_{\rm O}- f_{\rm T} > 0$,   $f_{\rm U}\hspace{0.01cm}' = f_{\rm U}- f_{\rm T} < 0$  and  $f_{\rm T}\hspace{0.01cm}' = 0$:

:$$x_{\rm TP}(t) = x_\text{TP, T}(t) + x_\text{TP, O}(t) + x_\text{TP, U}(t) = A_{\rm T}\cdot {\rm e}^{-{\rm j} \varphi_{\rm T} } \hspace{0.1cm}+ \hspace{0.1cm} A_{\rm O}\cdot {\rm e}^{-{\rm j} \varphi_{\rm O} } \cdot {\rm e}^{ {\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm O}\hspace{0.01cm}'\hspace{0.05cm}\cdot \hspace{0.05cm}t}\hspace{0.1cm}+ \hspace{0.1cm}
A_{\rm U}\cdot {\rm e}^{-{\rm j} \varphi_{\rm U} } \cdot {\rm e}^{ {\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm U}\hspace{0.01cm}'\hspace{0.05cm}\cdot \hspace{0.05cm}t} . $$

{{GraueBox|TEXT=
$\text{Example 1:}$ 
The constellation given here results, for example, in the [[Modulationsverfahren/Zweiseitenband-Amplitudenmodulation#AM-Signale_und_-Spektren_bei_harmonischem_Eingangssignal|double sideband amplitude modulation]] of the message signal $x_{\rm N}(t) = A_{\rm N}\cdot \cos\left(2\pi f_{\rm N}\cdot t- \varphi_{\rm N}\right)$ with the carrier signal $x_{\rm T}(t) = A_{\rm T}\cdot \cos\left(2\pi f_{\rm T}\cdot t - \varphi_{\rm T}\right)$. This is discussed frequently in the exercises.

[[File:Ortskurve_4.png|center|frame|Spectrum $X_{\rm TP}(f)$ of the equivalent low–pass signal for different phase constellations |class=fit]]

There are some limitations to the program parameters in this approach:
* The frequencies are always $f\hspace{0.05cm}'_{\rm O} = f_{\rm N}$ and $f\hspace{0.05cm}'_{\rm U} = -f_{\rm N}$.
*Without distortion, the amplitude of the sidebands is $A_{\rm O}= A_{\rm U}= A_{\rm N}/2$.
*The respective phase relationships can be seen in the following graphic.

}}

===Representation of the Equivalent Low-Pass Signal by Magnitude and Phase===

The generally complex equivalent low-pass signal
:$$x_{\rm TP}(t) = a(t) \cdot {\rm e}^{ {\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm} \phi(t) }$$
can be split into a magnitude function $a(t)$ and a phase function $\phi(t)$ according to the equation given here, where:
:$$a(t) = \vert x_{\rm TP}(t)\vert = \sqrt{ {\rm Re}^2\big [x_{\rm TP}(t)\big ] + {\rm Im}^2\big [x_{\rm TP}(t)\big ] }\hspace{0.05cm},$$
:$$\phi(t) = \text{arc }x_{\rm TP}(t) = \arctan \frac{{\rm Im}\big [x_{\rm TP}(t)\big ]}{{\rm Re}\big [x_{\rm TP}(t)\big ]}.$$

The reason for this is that a bandpass–signal $x(t)$ is usually described by the equivalent low-pass signal $x_{\rm TP}(t)$ that the functions $a(t)$ and $\phi(t)$ are interpretable in both representations:
*The amount $a(t)$ of the equivalent low-pass signal $x_{\rm TP}(t)$ indicates the (time-dependent) envelope of $x(t)$.
*The phase $\phi(t)$ of $x_{\rm TP}(t)$ denotes the location of the zero crossings of $x(t)$, where:
:–   For $\phi(t)>0$ the zero crossing is earlier than its nominal position   ⇒   the signal is leading here.
:–  When $\phi(t)<0$, the zero crossing is later than its target position   ⇒   the signal is trailing here.

{{GraueBox|TEXT=
$\text{Example 2:}$ 
The graph is intended to illustrate this relationship, assuming $A_{\rm U} > A_{\rm O}$   ⇒   the green pointer (for the lower sideband) is longer than the blue pointer (upper sideband). This is a snapshot at time $t_0$:

[[File:Ortskurve_3_neu.png|center|frame|bandpass spectrum $X(f)$ |class=fit]]

*For these system parameters, the top of the pointer cluster $x_{\rm TP}(t)$ – that is, the geometric sum of red, blue and green pointers – on an ellipse.
* The amount $a(t_0) = \vert x_{\rm TP}(t_0) \vert$ is drawn in black in the left-hand diagram and the phase value $\phi(t_0) = \text{arc }x_{\rm TP}(t_0) > 0$ is indicated in brown color.
*In the graph on the right, the amount $a(t_0) = \vert x_{\rm TP}(t_0) \vert$ of the equivalent low-pass signal indicates the envelope of the physical signal $x(t)$.
* At $\phi(t) \equiv 0$, all zero crossings of $x(t)$ would occur at equidistant intervals. Because of $\phi(t_0) > 0$, the signal is leading at the time $t_0$, that is: the zero crossings come earlier than the grid dictates. }}

==Exercises==
[[File:Exercises_verzerrungen.png|right]]
*First select the task number.
*A task description is displayed.
*Parameter values are adjusted.
*Solution after pressing &bdquo;Hide solition”.

The number &bdquo;0” will reset the program and output a text with the further explanation of the applet.

In the following, $\rm Green$ denotes the lower sideband   ⇒   $\big (A_{\rm U}, f_{\rm U}, \varphi_{\rm U}\big )$,  
$\rm Red$ the carrier   ⇒   $\big (A_{\rm T}, f_{\rm T}, \varphi_{\rm T}\big )$ and
$\rm Blue$ the upper sideband   ⇒   $\big (A_{\rm O}, f_{\rm O}, \varphi_{\rm O}\big )$.

{{BlaueBox|TEXT=
'''(1)'''   Let   $\text{Red:} \hspace{0.15cm} A_{\rm T} = 1\ \text{V}, f_{\rm T} = 100 \ \text{kHz}, \varphi_{\rm T} = 0^\circ$,   $\text{Green:} \hspace{0.15cm} A_{\rm U} = 0.4 \text{V}, \ f_{\rm U} = 80 \ \text{kHz}, \varphi_{\rm U} = -90^\circ$,   $\text{Blue:} \hspace{0.15cm} A_{\rm O} = 0.4\ \text{V}, f_{\rm O} = 120 \ \text{kHz}, \varphi_{\rm O} = 90^\circ$.

:Consider and interpret the equivalent low-pass signal $x_{\rm TP}(t)$ and the physical signal $x(t)$. Which period $T_0$ does one recognize?}}

:: The equivalent low-pass signal $x_{\rm TP}(t)$ takes from $x_{\rm TP}(t=0)=1\ \text{V}$ on the real axis values between $0.2\ \text{V}$ and $1.8\ \text{V}$   ⇒   phase $\phi(t) \equiv 0$.  The amount $|x_{\rm TP}(t)|$ indicates the envelope $a(t)$ of the physical signal $x(t)$. It holds $A_{\rm N} = 0.8\ \text{V}$ and $f_{\rm N} = 20\ \text{kHz}$:   $a(t) = A_{\rm T}+ A_{\rm N} \cdot \sin(2\pi\cdot f_{\rm N} \cdot t)$.  Both $x_{\rm TP}(t)$ and $x(t)$ are periodic with the period $T_0 = 1/f_{\rm N} = 50\ \rm µ s$.

{{BlaueBox|TEXT=
'''(2)'''   How do the ratios change to '''(1)''' with $f_{\rm U} = 99 \ \text{kHz}$ and $f_{\rm O} = 101 \ \text{kHz}$ ? How could $x(t)$ have arisen?}}

:: For the envelope $a(t)$ of the signal $x(t)$ we still have $a(t) = A_{\rm T}+ A_{\rm N} \cdot \sin(2\pi\cdot f_{\rm N} \cdot t)$, but now $f_{\rm N} = 1\ \text{kHz}$. Even though it can not be recognized:  $x_{\rm TP}(t)$ and $x(t)$ are still periodic:   $T_0 = 1\ \rm ms$. Example: Double sideband - Amplitude modulation '''(ZSB–AM)''' of a sine signal with cosine–carrier.

{{BlaueBox|TEXT=
'''(3)'''   Which settings have to be changed from '''(2)''' in order to arrive at the ZSB–AM of a cosine signal with sine–carrier. What changes over '''(2)'''?}}

::The carrier phase must be changed to $\varphi_{\rm T} = 90^\circ$   ⇒   sine–carrier. Similarly, $\varphi_{\rm O} =\varphi_{\rm U} =\varphi_{\rm T} = 90^\circ$ must be set   ⇒   cosine-shaped message  The locus now lies on the imaginary axis  ⇒   $\phi(t) \equiv -90^\circ$. At the beginning we have $x_{\rm TP}(t=0)= - {\rm j} \cdot 1.8 \ \text{V}$.

{{BlaueBox|TEXT=
'''(4)'''   Now let   $\text{Red:} \hspace{0.15cm} A_{\rm T} = 1\ \text{V}, \ f_{\rm T} = 100 \ \text{kHz}, \ \varphi_{\rm T} = 0^\circ$,   $\text{Green:} \hspace{0.15cm} A_{\rm U} = 0.4 \text{V}, \ f_{\rm U} = 80 \ \text{kHz}, \ \varphi_{\rm U} = 0^\circ$,   $\text{Blue:} \hspace{0.15cm} A_{\rm O} = 0.4\ \text{V}, \ f_{\rm O} = 120 \ \text{kHz}, \ \varphi_{\rm O} = 0^\circ$.

:What are the characteristics of this system &bdquo;ZSB–AM, where the message signal and carrier are respectively cosinusoidal”? What is the modulation depth $m$?}}

:: The equivalent low-pass signal $x_{\rm TP}(t)$ takes from $x_{\rm TP}(t=0)=1.8\ \text{V}$ on the real axis values between $0.2\ \text{V}$ and $1.8\ \text{V}$   ⇒   phase $\phi(t) \equiv 0$.  Except for the start state $x_{\rm TP}(t=0)$ same behavior as with the setting '''(1)'''. The modulation depth is $m = 0.8$.

{{BlaueBox|TEXT=
'''(5)'''   The parameters are still valid according to '''(4)''' with the exception of $A_{\rm T}= 0.6 \text{V}$. How big is the degree of modulation $m$? What are the consequences?}}

:: There is now a ZSB–AM with modulation degree $m = 1.333$. For $m > 1$, the simpler [[Modulationsverfahren/Hüllkurvendemodulation|envelope demodulation]] is not applicable, since the phase function $\phi(t) \in \{ 0, \ \pm 180^\circ\}$ is no longer constant and the envelope $a(t)$ no longer matches the message signal. Rather, the more complex [[Modulationsverfahren/Synchrondemodulation|synchronous demodulation]] must be used. Envelope detection would produce nonlinear distortions.

{{BlaueBox|TEXT=
'''(6)'''   The parameters are still valid according to '''(4)''' or '''(5)''' with the exception from $A_{\rm T}= 0$ on   ⇒   $m \to \infty$. Which modulation method is described in this way?}}

::It is a '''ZSB–AM without carrier''' and requires synchronous demodulation. The equivalent low-pass signal $x_{\rm TP}(t)$ is on the real axis, but not only in the right half-plane. Thus, the phase function $\phi(t) \in \{ 0, \ \pm 180^\circ\}$, also applies here, which means that envelope demodulation is not applicable.

{{BlaueBox|TEXT=
'''(7)'''   Now let   $\text{Red:} \hspace{0.15cm} A_{\rm T} = 1\ \text{V}, f_{\rm T} = 100 \ \text{kHz}, \varphi_{\rm T} = 0^\circ$,   $\text{Green:} \hspace{0.15cm} A_{\rm U} = 0, \ f_{\rm U} = 80 \ \text{kHz}, \varphi_{\rm U} = -90^\circ$,   $\text{Blue:} \hspace{0.15cm} A_{\rm O} = 0.8\ \text{V}, f_{\rm O} = 120 \ \text{kHz}, \varphi_{\rm O} = 90^\circ$.

:Which constellation is described here? Which characteristics of this procedure can be recognized from the graphic?}}

::It is a [[Modulationsverfahren/Einseitenbandmodulation|single-sideband modulation]] '''(ESB–AM)''', more specifically an '''OSB–AM''': the red carrier is fixed, the green pointer missing and the blue pointer (OSB) turns counterclockwise. The degree of modulation is $\mu = 0.8$ (in the case of ESB we denote the degree of modulation with $\mu$ instead of $m$). The carrier signal is cosinusoidal and the message signal sinusoidal. The locus is a circle. $x_{\rm TP}(t)$ moves in a mathematically positive direction. Because of $\phi(t) \ne \text{const.}$, The envelope demodulation is not applicable here:  This can be seen by the fact that the envelope $a(t)$ is not cosinusoidal. Rather, the lower half-wave is sharper than the upper   ⇒   strong linear distortions.

{{BlaueBox|TEXT=
'''(8)'''   The parameters are still valid according to '''(7)''' with the exception of $A_{\rm O}= 0$ and $A_{\rm U}= 0.8 \text{ V}$. What differences arise opposite '''(7)'''?}}

::Now it is a '''USB–AM''': The red carrier is fixed, the blue pointer is missing and the green pointer (USB) rotates clockwise. All other statements of '''(7)''' apply here as well.

{{BlaueBox|TEXT=
'''(9)'''   The parameters according to '''(7)''' are still valid with the exception of $A_{\rm O} = 0.2 \text{ V} \ne A_{\rm U} = 0.4 \text{ V} $. What are the differences from '''(7)'''?}}

::The locus $x_{\rm TP}(t)$ is not a horizontal straight line, but an ellipse with the real part between $0.4 \text{ V}$ and $1.6 \text{ V}$ and the imaginary part in the range $\pm 0.2 \text{ V}$. Because of $\phi(t) \ne \text{const.}$ , Envelope demodulation would lead to non-linear distortions here too. The constellation simulated here describes the situation of '''(4)''', namely a ZSB–AM with modulation degree $m = 0.8$, where the upper sideband is reduced to $50\%$ due to channel attenuation.

==Applet Manual==
[[File:Ortskurve_abzug3.png|right]]
* The red parameters $(A_{\rm T}, \ f_{\rm T}, \ \varphi_{\rm T})$ and the red pointer mark the ''Carrier''. (German: '''T'''räger).
* The green parameters $(A_{\rm U}, \ f_{\rm U} < f_{\rm T}, \ \varphi_{\rm U})$ mark the ''Lower sideband''. (German: '''U'''ntere Seitenband).
* The blue parameters $(A_{\rm O}, \ f_{\rm O} > f_{\rm T}, \ \varphi_{\rm O})$ mark the '' Upper sideband''. (German: '''O'''bere Seitenband).
* The red pointer does not turn.
* The green pointer rotates in a mathematically negative direction (clockwise).
* The blue pointer turns counterclockwise.

Meaning of the letters in the adjacent graphic:

    '''(A)'''     Plot of the equivalent low-pass Signal $x_{\rm TP}(t)$

    '''(B)'''     Plot of the physical signal $x(t)$

    '''(C)'''     Parameter input via slider:   amplitudes, frequencies, phase values

    '''(D)'''     Control elements:   Start – Step – Pause/Continue – Reset

    '''(E)'''     Speed of animation:   &bdquo;Speed”   ⇒   Values: 1, 2 oder 3

    '''(F)'''     &bdquo;Trace”   ⇒   On or Off, trace of equivalent low-pass Signal   $x_{\rm TP}(t)$

    '''(G)'''     Numerical output:   time $t$, the signal values  ${\rm Re}[x_{\rm TP}(t)]$  and  ${\rm Im}[x_{\rm TP}(t)]$,

$\text{}\hspace{4.2cm}$   envelope $a(t) = |x_{\rm TP}(t)|$  and  phase $\phi(t) = {\rm arc} \ x_{\rm TP}(t)$

    '''(H)'''     Variations for the graphical representation

$\hspace{1.5cm}$Zoom–Functions &bdquo;$+$” (Enlarge), &bdquo;$-$” (Decrease) und $\rm o$ (Reset to default)

$\hspace{1.5cm}$Move with &bdquo;$\leftarrow$” (Section to the left, ordinate to the right), &bdquo;$\uparrow$” &bdquo;$\downarrow$” &bdquo;$\rightarrow$”

    '''(I)'''     Experiment section:   Task selection and task

    '''(J)'''     Experiment section:  solution
 

==About the Authors==
This interactive calculation was designed and realized at the [http://www.lnt.ei.tum.de/startseite Lehrstuhl für Nachrichtentechnik] of the [https://www.tum.de/ Technical University of Munich] .
*The original version was created in 2005 by [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Studierende#Ji_Li_.28Bachelorarbeit_EI_2003.2C_Diplomarbeit_EI_2005.29|Ji Li]] as part of her Diploma thesis using &bdquo;FlashMX–Actionscript” (Supervisor: [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Mitarbeiter_und_Dozenten#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28am_LNT_seit_1974.29|Günter Söder]]).
*In 2018 this Applet was redesigned and updated to &bdquo;HTML5” by [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Studierende#Xiaohan_Liu_.28Bachelorarbeit_2018.29|Xiaohan Liu]] as part of her Bachelor's thesis (Supervisor: [[Biografien_und_Bibliografien/Beteiligte_der_Professur_Leitungsgebundene_%C3%9Cbertragungstechnik#Tasn.C3.A1d_Kernetzky.2C_M.Sc._.28bei_L.C3.9CT_seit_2014.29|Tasnád Kernetzky]]).

==Once again: Open Applet in new Tab==

{{LntAppletLinkEn|physAnSignal}}

[[Category:Applets|^Verzerrungen^]]

Applets:Physical Signal & Equivalent Lowpass Signal

2018-08-25T17:25:07Z

Xiaohan: /* $x_{\rm TP}(t)$–Representation of a Sum of Three Harmonic Oscillations */

{{LntAppletLinkEn|analPhysSignal}}

==Applet Description==
 
This applet shows the relationship between the physical bandpass signal $x(t)$ and the associated equivalent low-pass signal $x_{\rm TP}(t)$. It is assumed that the bandpass signal $x(t)$ has a frequency-discrete spectrum $X(f)$:
:$$x(t) = x_{\rm U}(t) + x_{\rm T}(t) + x_{\rm O}(t) = A_{\rm U}\cdot \cos\left(2\pi f_{\rm U}\cdot t- \varphi_{\rm U}\right)+A_{\rm T}\cdot \cos\left(2\pi f_{\rm T}\cdot t- \varphi_{\rm T}\right)+A_{\rm O}\cdot \cos\left(2\pi f_{\rm O}\cdot t- \varphi_{\rm O}\right). $$
The physical signal $x(t)$ is thus composed of three harmonic oscillations, a constellation that can be found, for example, in the ''Double-sideband Amplitude Modulation''
*of the message signal $x_{\rm N}(t) = A_{\rm N}\cdot \cos\left(2\pi f_{\rm N}\cdot t- \varphi_{\rm N}\right)$   ⇒   in German:   '''N'''achrichtensignal
*with the carrier signal $x_{\rm T}(t) = A_{\rm T}\cdot \cos\left(2\pi f_{\rm T}\cdot t - \varphi_{\rm T}\right)$   ⇒   in German:   '''T'''rägersignal.

The nomenclature is also adapted to this case:
* $x_{\rm O}(t)$ denotes the &bdquo;upper sideband”   (in German:   '''O'''beres Seitenband) with the amplitude $A_{\rm O}= A_{\rm N}/2$, the frequency $f_{\rm O} = f_{\rm T} + f_{\rm N}$ and the phase $\varphi_{\rm O} = \varphi_{\rm T} + \varphi_{\rm N}$.
*Similarly, for the &bdquo;lower sideband”   (in German:   '''U'''nteres Seitenband) $x_{\rm U}(t)$ with $f_{\rm U} = f_{\rm T} - f_{\rm N}$, $A_{\rm U}= A_{\rm O}$ and $\varphi_{\rm U} = -\varphi_{\rm O}$.

The associated equivalent low-pass–signal is $f_{\rm O}\hspace{0.01cm}' = f_{\rm O}- f_{\rm T} > 0$,   $f_{\rm U}\hspace{0.01cm}' = f_{\rm U}- f_{\rm T} < 0$  and  $f_{\rm T}\hspace{0.01cm}' = 0$:

:$$x_{\rm TP}(t) = x_\text{TP, T}(t) + x_\text{TP, O}(t) + x_\text{TP, U}(t) = A_{\rm T}\cdot {\rm e}^{-{\rm j} \varphi_{\rm T} } \hspace{0.1cm}+ \hspace{0.1cm} A_{\rm O}\cdot {\rm e}^{-{\rm j} \varphi_{\rm O} } \cdot {\rm e}^{ {\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm O}\hspace{0.01cm}'\hspace{0.05cm}\cdot \hspace{0.05cm}t}\hspace{0.1cm}+ \hspace{0.1cm}
A_{\rm U}\cdot {\rm e}^{-{\rm j} \varphi_{\rm U} } \cdot {\rm e}^{ {\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm U}\hspace{0.01cm}'\hspace{0.05cm}\cdot \hspace{0.05cm}t} . $$

[[File:Ortskurve_1.png|right|frame|Equivalent low-pass signal currently $t=0$ for cosinusoidal carrier   ⇒   $\varphi_{\rm T} = 0$]]
The program shows $x_{\rm TP}(t)$ as the vectorial sum of three rotation pointers as a violet dot (see figure for start time $t=0$ and cosinusoidal carrier):

*The (red) pointer of the carrier $x_\text{TP, T}(t)$ with the length $A_{\rm T}$ and the zero phase position $\varphi_{\rm T}=0$ is fixed in the complex plane. So it applies to all times $t$:   $x_{\rm TP}(t)= A_{\rm T}\cdot {\rm e}^{-{\rm j} \varphi_{\rm T} }$.

*The (blue) pointer of the upper sideband $x_\text{TP, O}(t)$ with the length $A_{\rm O}$ and the zero phase position $\varphi_{\rm O}$ rotates at the angular velocity $2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm O}\hspace{0.01cm}'$ in mathematically positive direction (one revolution in time $1/f_{\rm O}\hspace{0.01cm}')$.

*The (green) pointer of the lower sideband $x_{\rm U+}(t)$ with the length $A_{\rm U}$ and the zero phase position $\varphi_{\rm U}$ rotates at the angular velocity $2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm U}\hspace{0.01cm}'$, because of $f_{\rm U}\hspace{0.01cm}'<0$ counterclockwise.

*With $f_{\rm U}\hspace{0.01cm}' = -f_{\rm O}\hspace{0.01cm}'$ the blue and green pointers will spin at the same speed but in different directions. Also, if $A_{\rm O} = A_{\rm U}$ and $\varphi_{\rm U} = -\varphi_{\rm O}$, then $x_{\rm TP}(t)$ moves on a straight line with a incline of $\varphi_{\rm T}$.

''Note:''   In the figure $\varphi_{\rm O} = +30^\circ$. From this follows for the start time $t=0$ the angle of the blue pointer (OSB) with respect to the coordinate system:   $\phi_{\rm O} = -\varphi_{\rm O} = -30^\circ$. Likewise, the null phase position $\varphi_{\rm U} = -30^\circ$ of the lower sideband (USB, grüner Zeiger) follows for the phase angle to be considered in the complex plane:   $\phi_{\rm U} = +30^\circ$.

The temporal process of $x_{\rm TP}(t)$ is also referred to below as '''locus'''. The relationship between $x_{\rm TP}(t)$ and the physical bandpass signal $x(t)$ is given in the section and the associated analytic signal is $x_+(t)$ :

:$$x_{\rm TP}(t) = x_{\rm +}(t)\cdot {\rm e}^{-{\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm}2 \pi \cdot f_{\rm T}\cdot \hspace{0.05cm}t},$$
:$$x_{\rm +}(t) = x_{\rm TP}(t)\cdot {\rm e}^{+{\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm}2 \pi \cdot f_{\rm T}\cdot \hspace{0.05cm}t}.$$

[[Applets:Physikalisches_Signal_%26_Äquivalentes_TP-Signal|'''German description''']]

==Theoretical Background==
 
===Description of Bandpass Signals===
[[File:Zeigerdiagramm_1a.png|right|frame|bandpass spectrum $X(f)$ |class=fit]]
We consider '''bandpass signals''' $x(t)$ with the property that their spectra $X(f)$ are not in the range around the frequency $f=0$, but around a carrier frequency $f_{\rm T}$. In most cases it can also be assumed that the bandwidth is $B \ll f_{\rm T}$.

The figure shows such a bandpass spectrum $X(f)$. Assuming that the associated $x(t)$ is a physical signal and thus real, the spectral function $X(f)$ has a symmetry with respect to the frequency $f = 0$, if $x(t)$ is an even function   ⇒   $x(-t)=x(t)$, $X(f)$ is real and even.

Beside the physical signal $x(t)\ \circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\, \ X(f)$, one can also use the following descriptions of bandpass signals:
*the analytic signal $x_+(t)\ \circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\, \ X_+(f)$, see applet [[Applets:Physikalisches_Signal_%26_Analytisches_Signal|Physikalisches Signal & Analytisches Signal]],
*the equivalent low-pass–signal $x_{\rm TP}(t)\ \circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\, \ X_{\rm TP}(f)$, see next page
 

===Spectral Functions of the Analytic and the Equivalent Low-Pass Signal===

The '''analytic signal''' $x_+(t)$ belonging to the physical signal $x(t)$ is the time function whose spectrum fulfills the following property:
[[File:Ortskurve_2.png|right|frame|spectral functions $X_+(f)$ and $X_{\rm TP}(f)$ |class=fit]]
:$$X_+(f)=\big[1+{\rm sign}(f)\big] \cdot X(f) = \left\{ {2 \cdot
X(f) \; \hspace{0.2cm}\rm for\hspace{0.2cm} {\it f} > 0, \atop {\,\,\,\, \rm 0 \; \hspace{0.9cm}\rm for\hspace{0.2cm} {\it f} < 0.} }\right.$$

The ''Signum function'' is for positive values of $f$ equal to $+1$ and for negative $f$ values equal to $-1$.
*The (double-sided) limit returns $\sign(0) = 0$.
*The index „+” should make it clear that $X_+(f)$ only has parts at positive frequencies.

From the graph you can see the calculation rule for $X_+(f)$: The actual BP spectrum $X(f)$ becomes
*doubled at the positive frequencies, and
*set to zero at the negative frequencies.

Due to the asymmetry of $X_+(f)$ with respect to the frequency $f = 0$, it can already be said that the time function $x_+(t)$ except for a trivial special case $x_+(t)= 0 \ \ \circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\,\ \ X_+(f)= 0$ is always complex.

The spectrum $X_{\rm TP}(f)$ of the equivalent low-pass signal is obtained by shifting $X_+(f)$ to the left by the carrier frequency $f_{\rm T}$:
:$$X_{\rm TP}(f)= X_+(f+f_{\rm T}).$$

In the time domain this operation corresponds to the multiplication of $x_{\rm +}(t)$ with the complex exponential function with negative exponent:
:$$x_{\rm TP}(t) = x_{\rm +}(t)\cdot {\rm e}^{-{\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm}2 \pi \cdot f_{\rm T}\cdot \hspace{0.05cm}t}.$$

It can be seen that $x_{\rm TP}(t)$ is generally complex. But if $X_+(f)$ is symmetric about the carrier frequency $f_{\rm T}$, $X_{\rm TP}(f)$ is symmetric about the frequency $f=0$ and there is accordingly real time function $x_{\rm TP}(t)$.

===$x_{\rm TP}(t)$–Representation of a Sum of Three Harmonic Oscillations===

In our applet, we always assume a set of three rotating pointers. The physical signal is:
:$$x(t) = x_{\rm U}(t) + x_{\rm T}(t) + x_{\rm O}(t) = A_{\rm U}\cdot \cos\left(2\pi f_{\rm U}\cdot t- \varphi_{\rm U}\right)+A_{\rm T}\cdot \cos\left(2\pi f_{\rm T}\cdot t- \varphi_{\rm T}\right)+A_{\rm O}\cdot \cos\left(2\pi f_{\rm O}\cdot t- \varphi_{\rm O}\right). $$
* Each of the three harmonic oscillations $x_{\rm T}(t)$, $x_{\rm U}(t)$ and $x_{\rm O}(t)$ is represented by an amplitude $(A)$, a frequency $(f)$ and a phase value $(\varphi)$.
*The indices are based on the modulation method [[Modulationsverfahren/Zweiseitenband-Amplitudenmodulation|double sideband amplitude modulation]]. &bdquo;T” stands for &bdquo;carrier”, &bdquo;U” for &bdquo;lower sideband” and &bdquo;O” for &bdquo;upper Sideband”. Similarly, $f_{\rm U} < f_{\rm T}$ and $f_{\rm O} > f_{\rm T}$. There are no restrictions for the amplitudes and phases.

The associated equivalent low-pass signal is with $f_{\rm O}\hspace{0.01cm}' = f_{\rm O}- f_{\rm T} > 0$,   $f_{\rm U}\hspace{0.01cm}' = f_{\rm U}- f_{\rm T} < 0$  and  $f_{\rm T}\hspace{0.01cm}' = 0$:

:$$x_{\rm TP}(t) = x_\text{TP, T}(t) + x_\text{TP, O}(t) + x_\text{TP, U}(t) = A_{\rm T}\cdot {\rm e}^{-{\rm j} \varphi_{\rm T} } \hspace{0.1cm}+ \hspace{0.1cm} A_{\rm O}\cdot {\rm e}^{-{\rm j} \varphi_{\rm O} } \cdot {\rm e}^{ {\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm O}\hspace{0.01cm}'\hspace{0.05cm}\cdot \hspace{0.05cm}t}\hspace{0.1cm}+ \hspace{0.1cm}
A_{\rm U}\cdot {\rm e}^{-{\rm j} \varphi_{\rm U} } \cdot {\rm e}^{ {\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm U}\hspace{0.01cm}'\hspace{0.05cm}\cdot \hspace{0.05cm}t} . $$

{{GraueBox|TEXT=
$\text{Example 1:}$ 
The constellation given here results, for example, in the [[Modulationsverfahren/Zweiseitenband-Amplitudenmodulation#AM-Signale_und_-Spektren_bei_harmonischem_Eingangssignal|double sideband amplitude modulation]] of the message signal $x_{\rm N}(t) = A_{\rm N}\cdot \cos\left(2\pi f_{\rm N}\cdot t- \varphi_{\rm N}\right)$ with the carrier signal $x_{\rm T}(t) = A_{\rm T}\cdot \cos\left(2\pi f_{\rm T}\cdot t - \varphi_{\rm T}\right)$. This is discussed frequently in the exercises.

[[File:Ortskurve_4.png|center|frame|Spectrum $X_{\rm TP}(f)$ of the equivalent low–pass signal for different phase constellations |class=fit]]

There are some limitations to the program parameters in this approach:
* The frequencies are always $f\hspace{0.05cm}'_{\rm O} = f_{\rm N}$ and $f\hspace{0.05cm}'_{\rm U} = -f_{\rm N}$.
*Without distortion, the amplitude of the sidebands is $A_{\rm O}= A_{\rm U}= A_{\rm N}/2$.
*The respective phase relationships can be seen in the following graphic.

}}

===Representation of the Equivalent Low-Pass Signal by Magnitude and Phase===

The generally complex equivalent low-pass signal
:$$x_{\rm TP}(t) = a(t) \cdot {\rm e}^{ {\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm} \phi(t) }$$
can be split into a magnitude function $a(t)$ and a phase function $\phi(t)$ according to the equation given here, where:
:$$a(t) = \vert x_{\rm TP}(t)\vert = \sqrt{ {\rm Re}^2\big [x_{\rm TP}(t)\big ] + {\rm Im}^2\big [x_{\rm TP}(t)\big ] }\hspace{0.05cm},$$
:$$\phi(t) = \text{arc }x_{\rm TP}(t) = \arctan \frac{{\rm Im}\big [x_{\rm TP}(t)\big ]}{{\rm Re}\big [x_{\rm TP}(t)\big ]}.$$

The reason for this is that a bandpass–signal $x(t)$ is usually described by the equivalent low-pass signal $x_{\rm TP}(t)$ that the functions $a(t)$ and $\phi(t)$ are interpretable in both representations:
*The amount $a(t)$ of the equivalent low-pass signal $x_{\rm TP}(t)$ indicates the (time-dependent) envelope of $x(t)$.
*The phase $\phi(t)$ of $x_{\rm TP}(t)$ denotes the location of the zero crossings of $x(t)$, where:
:–   For $\phi(t)>0$ the zero crossing is earlier than its nominal position   ⇒   the signal is leading here.
:–  When $\phi(t)<0$, the zero crossing is later than its target position   ⇒   the signal is trailing here.

{{GraueBox|TEXT=
$\text{Example 2:}$ 
The graph is intended to illustrate this relationship, assuming $A_{\rm U} > A_{\rm O}$   ⇒   the green pointer (for the lower sideband) is longer than the blue pointer (upper sideband). This is a snapshot at time $t_0$:

[[File:Ortskurve_3_neu.png|center|frame|bandpass spectrum $X(f)$ |class=fit]]

*For these system parameters, the top of the pointer cluster $x_{\rm TP}(t)$ – that is, the geometric sum of red, blue and green pointers – on an ellipse.
* The amount $a(t_0) = \vert x_{\rm TP}(t_0) \vert$ is drawn in black in the left-hand diagram and the phase value $\phi(t_0) = \text{arc }x_{\rm TP}(t_0) > 0$ is indicated in brown color.
*In the graph on the right, the amount $a(t_0) = \vert x_{\rm TP}(t_0) \vert$ of the equivalent low-pass signal indicates the envelope of the physical signal $x(t)$.
* At $\phi(t) \equiv 0$, all zero crossings of $x(t)$ would occur at equidistant intervals. Because of $\phi(t_0) > 0$, the signal is leading at the time $t_0$, that is: the zero crossings come earlier than the grid dictates. }}

==Exercises==
[[File:Exercises_verzerrungen.png|right]]
*First select the task number.
*A task description is displayed.
*Parameter values are adjusted.
*Solution after pressing &bdquo;Hide solition”.

The number &bdquo;0” will reset the program and output a text with the further explanation of the applet.

In the following, $\rm Green$ denotes the lower sideband   ⇒   $\big (A_{\rm U}, f_{\rm U}, \varphi_{\rm U}\big )$,  
$\rm Red$ the carrier   ⇒   $\big (A_{\rm T}, f_{\rm T}, \varphi_{\rm T}\big )$ and
$\rm Blue$ the upper sideband   ⇒   $\big (A_{\rm O}, f_{\rm O}, \varphi_{\rm O}\big )$.

{{BlaueBox|TEXT=
'''(1)'''   Let   $\text{Red:} \hspace{0.15cm} A_{\rm T} = 1\ \text{V}, f_{\rm T} = 100 \ \text{kHz}, \varphi_{\rm T} = 0^\circ$,   $\text{Green:} \hspace{0.15cm} A_{\rm U} = 0.4 \text{V}, \ f_{\rm U} = 80 \ \text{kHz}, \varphi_{\rm U} = -90^\circ$,   $\text{Blue:} \hspace{0.15cm} A_{\rm O} = 0.4\ \text{V}, f_{\rm O} = 120 \ \text{kHz}, \varphi_{\rm O} = 90^\circ$.

:Consider and interpret the equivalent low-pass signal $x_{\rm TP}(t)$ and the physical signal $x(t)$. Which period $T_0$ does one recognize?}}

:: The equivalent low-pass signal $x_{\rm TP}(t)$ takes from $x_{\rm TP}(t=0)=1\ \text{V}$ on the real axis values between $0.2\ \text{V}$ and $1.8\ \text{V}$   ⇒   phase $\phi(t) \equiv 0$.  The amount $|x_{\rm TP}(t)|$ indicates the envelope $a(t)$ of the physical signal $x(t)$. It holds $A_{\rm N} = 0.8\ \text{V}$ and $f_{\rm N} = 20\ \text{kHz}$:   $a(t) = A_{\rm T}+ A_{\rm N} \cdot \sin(2\pi\cdot f_{\rm N} \cdot t)$.  Both $x_{\rm TP}(t)$ and $x(t)$ are periodic with the period $T_0 = 1/f_{\rm N} = 50\ \rm µ s$.

{{BlaueBox|TEXT=
'''(2)'''   How do the ratios change to '''(1)''' with $f_{\rm U} = 99 \ \text{kHz}$ and $f_{\rm O} = 101 \ \text{kHz}$ ? How could $x(t)$ have arisen?}}

:: For the envelope $a(t)$ of the signal $x(t)$ we still have $a(t) = A_{\rm T}+ A_{\rm N} \cdot \sin(2\pi\cdot f_{\rm N} \cdot t)$, but now $f_{\rm N} = 1\ \text{kHz}$. Even though it can not be recognized:  $x_{\rm TP}(t)$ and $x(t)$ are still periodic:   $T_0 = 1\ \rm ms$. Example: Double sideband - Amplitude modulation '''(ZSB–AM)''' of a sine signal with cosine–carrier.

{{BlaueBox|TEXT=
'''(3)'''   Which settings have to be changed from '''(2)''' in order to arrive at the ZSB–AM of a cosine signal with sine–carrier. What changes over '''(2)'''?}}

::The carrier phase must be changed to $\varphi_{\rm T} = 90^\circ$   ⇒   sine–carrier. Similarly, $\varphi_{\rm O} =\varphi_{\rm U} =\varphi_{\rm T} = 90^\circ$ must be set   ⇒   cosine-shaped message  The locus now lies on the imaginary axis  ⇒   $\phi(t) \equiv -90^\circ$. At the beginning we have $x_{\rm TP}(t=0)= - {\rm j} \cdot 1.8 \ \text{V}$.

{{BlaueBox|TEXT=
'''(4)'''   Now let   $\text{Red:} \hspace{0.15cm} A_{\rm T} = 1\ \text{V}, \ f_{\rm T} = 100 \ \text{kHz}, \ \varphi_{\rm T} = 0^\circ$,   $\text{Green:} \hspace{0.15cm} A_{\rm U} = 0.4 \text{V}, \ f_{\rm U} = 80 \ \text{kHz}, \ \varphi_{\rm U} = 0^\circ$,   $\text{Blue:} \hspace{0.15cm} A_{\rm O} = 0.4\ \text{V}, \ f_{\rm O} = 120 \ \text{kHz}, \ \varphi_{\rm O} = 0^\circ$.

:What are the characteristics of this system &bdquo;ZSB–AM, where the message signal and carrier are respectively cosinusoidal”? What is the modulation depth $m$?}}

:: The equivalent low-pass signal $x_{\rm TP}(t)$ takes from $x_{\rm TP}(t=0)=1.8\ \text{V}$ on the real axis values between $0.2\ \text{V}$ and $1.8\ \text{V}$   ⇒   phase $\phi(t) \equiv 0$.  Except for the start state $x_{\rm TP}(t=0)$ same behavior as with the setting '''(1)'''. The modulation depth is $m = 0.8$.

{{BlaueBox|TEXT=
'''(5)'''   The parameters are still valid according to '''(4)''' with the exception of $A_{\rm T}= 0.6 \text{V}$. How big is the degree of modulation $m$? What are the consequences?}}

:: There is now a ZSB–AM with modulation degree $m = 1.333$. For $m > 1$, the simpler [[Modulationsverfahren/Hüllkurvendemodulation|envelope demodulation]] is not applicable, since the phase function $\phi(t) \in \{ 0, \ \pm 180^\circ\}$ is no longer constant and the envelope $a(t)$ no longer matches the message signal. Rather, the more complex [[Modulationsverfahren/Synchrondemodulation|synchronous demodulation]] must be used. Envelope detection would produce nonlinear distortions.

{{BlaueBox|TEXT=
'''(6)'''   The parameters are still valid according to '''(4)''' or '''(5)''' with the exception from $A_{\rm T}= 0$ on   ⇒   $m \to \infty$. Which modulation method is described in this way?}}

::It is a '''ZSB–AM without carrier''' and requires synchronous demodulation. The equivalent low-pass signal $x_{\rm TP}(t)$ is on the real axis, but not only in the right half-plane. Thus, the phase function $\phi(t) \in \{ 0, \ \pm 180^\circ\}$, also applies here, which means that envelope demodulation is not applicable.

{{BlaueBox|TEXT=
'''(7)'''   Now let   $\text{Red:} \hspace{0.15cm} A_{\rm T} = 1\ \text{V}, f_{\rm T} = 100 \ \text{kHz}, \varphi_{\rm T} = 0^\circ$,   $\text{Green:} \hspace{0.15cm} A_{\rm U} = 0, \ f_{\rm U} = 80 \ \text{kHz}, \varphi_{\rm U} = -90^\circ$,   $\text{Blue:} \hspace{0.15cm} A_{\rm O} = 0.8\ \text{V}, f_{\rm O} = 120 \ \text{kHz}, \varphi_{\rm O} = 90^\circ$.

:Which constellation is described here? Which characteristics of this procedure can be recognized from the graphic?}}

::It is a [[Modulationsverfahren/Einseitenbandmodulation|single-sideband modulation]] '''(ESB–AM)''', more specifically an '''OSB–AM''': the red carrier is fixed, the green pointer missing and the blue pointer (OSB) turns counterclockwise. The degree of modulation is $\mu = 0.8$ (in the case of ESB we denote the degree of modulation with $\mu$ instead of $m$). The carrier signal is cosinusoidal and the message signal sinusoidal. The locus is a circle. $x_{\rm TP}(t)$ moves in a mathematically positive direction. Because of $\phi(t) \ne \text{const.}$, The envelope demodulation is not applicable here:  This can be seen by the fact that the envelope $a(t)$ is not cosinusoidal. Rather, the lower half-wave is sharper than the upper   ⇒   strong linear distortions.

{{BlaueBox|TEXT=
'''(8)'''   The parameters are still valid according to '''(7)''' with the exception of $A_{\rm O}= 0$ and $A_{\rm U}= 0.8 \text{ V}$. What differences arise opposite '''(7)'''?}}

::Now it is a '''USB–AM''': The red carrier is fixed, the blue pointer is missing and the green pointer (USB) rotates clockwise. All other statements of '''(7)''' apply here as well.

{{BlaueBox|TEXT=
'''(9)'''   The parameters according to '''(7)''' are still valid with the exception of $A_{\rm O} = 0.2 \text{ V} \ne A_{\rm U} = 0.4 \text{ V} $. What are the differences from '''(7)'''?}}

::The locus $x_{\rm TP}(t)$ is not a horizontal straight line, but an ellipse with the real part between $0.4 \text{ V}$ and $1.6 \text{ V}$ and the imaginary part in the range $\pm 0.2 \text{ V}$. Because of $\phi(t) \ne \text{const.}$ , Envelope demodulation would lead to non-linear distortions here too. The constellation simulated here describes the situation of '''(4)''', namely a ZSB–AM with modulation degree $m = 0.8$, where the upper sideband is reduced to $50\%$ due to channel attenuation.

==Applet Manual==
[[File:Ortskurve_abzug3.png|right]]
* The red parameters $(A_{\rm T}, \ f_{\rm T}, \ \varphi_{\rm T})$ and the red pointer mark the ''Carrier''. (German: '''T'''räger).
* The green parameters $(A_{\rm U}, \ f_{\rm U} < f_{\rm T}, \ \varphi_{\rm U})$ mark the ''Lower sideband''. (German: '''U'''ntere Seitenband).
* The blue parameters $(A_{\rm O}, \ f_{\rm O} > f_{\rm T}, \ \varphi_{\rm O})$ mark the '' Upper sideband''. (German: '''O'''bere Seitenband).
* The red pointer does not turn.
* The green pointer rotates in a mathematically negative direction (clockwise).
* The blue pointer turns counterclockwise.

Meaning of the letters in the adjacent graphic:

    '''(A)'''     Plot of the equivalent low-pass Signal $x_{\rm TP}(t)$

    '''(B)'''     Plot of the physical signal $x(t)$

    '''(C)'''     Parameter input via slider:   amplitudes, frequencies, phase values

    '''(D)'''     Control elements:   Start – Step – Pause/Continue – Reset

    '''(E)'''     Speed of animation:   &bdquo;Speed”   ⇒   Values: 1, 2 oder 3

    '''(F)'''     &bdquo;Trace”   ⇒   On or Off, trace of equivalent low-pass Signal   $x_{\rm TP}(t)$

    '''(G)'''     Numerical output:   time $t$, the signal values  ${\rm Re}[x_{\rm TP}(t)]$  and  ${\rm Im}[x_{\rm TP}(t)]$,

$\text{}\hspace{4.2cm}$   Hüllkurve $a(t) = |x_{\rm TP}(t)|$  und  Phase $\phi(t) = {\rm arc} \ x_{\rm TP}(t)$

    '''(H)'''     Variations for the graphical representation

$\hspace{1.5cm}$Zoom–Functions &bdquo;$+$” (Enlarge), &bdquo;$-$” (Decrease) und $\rm o$ (Reset to default)

$\hspace{1.5cm}$Move with &bdquo;$\leftarrow$” (Section to the left, ordinate to the right), &bdquo;$\uparrow$” &bdquo;$\downarrow$” &bdquo;$\rightarrow$”

    '''(I)'''     Experiment section:   Task selection and task

    '''(J)'''     Experiment section:  solution
 

==About the Authors==
This interactive calculation was designed and realized at the [http://www.lnt.ei.tum.de/startseite Lehrstuhl für Nachrichtentechnik] of the [https://www.tum.de/ Technical University of Munich] .
*The original version was created in 2005 by [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Studierende#Ji_Li_.28Bachelorarbeit_EI_2003.2C_Diplomarbeit_EI_2005.29|Ji Li]] as part of her Diploma thesis using &bdquo;FlashMX–Actionscript” (Supervisor: [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Mitarbeiter_und_Dozenten#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28am_LNT_seit_1974.29|Günter Söder]]).
*In 2018 this Applet was redesigned and updated to &bdquo;HTML5” by [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Studierende#Xiaohan_Liu_.28Bachelorarbeit_2018.29|Xiaohan Liu]] as part of her Bachelor's thesis (Supervisor: [[Biografien_und_Bibliografien/Beteiligte_der_Professur_Leitungsgebundene_%C3%9Cbertragungstechnik#Tasn.C3.A1d_Kernetzky.2C_M.Sc._.28bei_L.C3.9CT_seit_2014.29|Tasnád Kernetzky]]).

==Once again: Open Applet in new Tab==

{{LntAppletLinkEn|physAnSignal}}

[[Category:Applets|^Verzerrungen^]]

Applets:Physical Signal & Equivalent Lowpass Signal

2018-08-25T16:59:14Z

Xiaohan: /* Applet Description */

{{LntAppletLinkEn|analPhysSignal}}

==Applet Description==
 
This applet shows the relationship between the physical bandpass signal $x(t)$ and the associated equivalent low-pass signal $x_{\rm TP}(t)$. It is assumed that the bandpass signal $x(t)$ has a frequency-discrete spectrum $X(f)$:
:$$x(t) = x_{\rm U}(t) + x_{\rm T}(t) + x_{\rm O}(t) = A_{\rm U}\cdot \cos\left(2\pi f_{\rm U}\cdot t- \varphi_{\rm U}\right)+A_{\rm T}\cdot \cos\left(2\pi f_{\rm T}\cdot t- \varphi_{\rm T}\right)+A_{\rm O}\cdot \cos\left(2\pi f_{\rm O}\cdot t- \varphi_{\rm O}\right). $$
The physical signal $x(t)$ is thus composed of three harmonic oscillations, a constellation that can be found, for example, in the ''Double-sideband Amplitude Modulation''
*of the message signal $x_{\rm N}(t) = A_{\rm N}\cdot \cos\left(2\pi f_{\rm N}\cdot t- \varphi_{\rm N}\right)$   ⇒   in German:   '''N'''achrichtensignal
*with the carrier signal $x_{\rm T}(t) = A_{\rm T}\cdot \cos\left(2\pi f_{\rm T}\cdot t - \varphi_{\rm T}\right)$   ⇒   in German:   '''T'''rägersignal.

The nomenclature is also adapted to this case:
* $x_{\rm O}(t)$ denotes the &bdquo;upper sideband”   (in German:   '''O'''beres Seitenband) with the amplitude $A_{\rm O}= A_{\rm N}/2$, the frequency $f_{\rm O} = f_{\rm T} + f_{\rm N}$ and the phase $\varphi_{\rm O} = \varphi_{\rm T} + \varphi_{\rm N}$.
*Similarly, for the &bdquo;lower sideband”   (in German:   '''U'''nteres Seitenband) $x_{\rm U}(t)$ with $f_{\rm U} = f_{\rm T} - f_{\rm N}$, $A_{\rm U}= A_{\rm O}$ and $\varphi_{\rm U} = -\varphi_{\rm O}$.

The associated equivalent low-pass–signal is $f_{\rm O}\hspace{0.01cm}' = f_{\rm O}- f_{\rm T} > 0$,   $f_{\rm U}\hspace{0.01cm}' = f_{\rm U}- f_{\rm T} < 0$  and  $f_{\rm T}\hspace{0.01cm}' = 0$:

:$$x_{\rm TP}(t) = x_\text{TP, T}(t) + x_\text{TP, O}(t) + x_\text{TP, U}(t) = A_{\rm T}\cdot {\rm e}^{-{\rm j} \varphi_{\rm T} } \hspace{0.1cm}+ \hspace{0.1cm} A_{\rm O}\cdot {\rm e}^{-{\rm j} \varphi_{\rm O} } \cdot {\rm e}^{ {\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm O}\hspace{0.01cm}'\hspace{0.05cm}\cdot \hspace{0.05cm}t}\hspace{0.1cm}+ \hspace{0.1cm}
A_{\rm U}\cdot {\rm e}^{-{\rm j} \varphi_{\rm U} } \cdot {\rm e}^{ {\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm U}\hspace{0.01cm}'\hspace{0.05cm}\cdot \hspace{0.05cm}t} . $$

[[File:Ortskurve_1.png|right|frame|Equivalent low-pass signal currently $t=0$ for cosinusoidal carrier   ⇒   $\varphi_{\rm T} = 0$]]
The program shows $x_{\rm TP}(t)$ as the vectorial sum of three rotation pointers as a violet dot (see figure for start time $t=0$ and cosinusoidal carrier):

*The (red) pointer of the carrier $x_\text{TP, T}(t)$ with the length $A_{\rm T}$ and the zero phase position $\varphi_{\rm T}=0$ is fixed in the complex plane. So it applies to all times $t$:   $x_{\rm TP}(t)= A_{\rm T}\cdot {\rm e}^{-{\rm j} \varphi_{\rm T} }$.

*The (blue) pointer of the upper sideband $x_\text{TP, O}(t)$ with the length $A_{\rm O}$ and the zero phase position $\varphi_{\rm O}$ rotates at the angular velocity $2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm O}\hspace{0.01cm}'$ in mathematically positive direction (one revolution in time $1/f_{\rm O}\hspace{0.01cm}')$.

*The (green) pointer of the lower sideband $x_{\rm U+}(t)$ with the length $A_{\rm U}$ and the zero phase position $\varphi_{\rm U}$ rotates at the angular velocity $2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm U}\hspace{0.01cm}'$, because of $f_{\rm U}\hspace{0.01cm}'<0$ counterclockwise.

*With $f_{\rm U}\hspace{0.01cm}' = -f_{\rm O}\hspace{0.01cm}'$ the blue and green pointers will spin at the same speed but in different directions. Also, if $A_{\rm O} = A_{\rm U}$ and $\varphi_{\rm U} = -\varphi_{\rm O}$, then $x_{\rm TP}(t)$ moves on a straight line with a incline of $\varphi_{\rm T}$.

''Note:''   In the figure $\varphi_{\rm O} = +30^\circ$. From this follows for the start time $t=0$ the angle of the blue pointer (OSB) with respect to the coordinate system:   $\phi_{\rm O} = -\varphi_{\rm O} = -30^\circ$. Likewise, the null phase position $\varphi_{\rm U} = -30^\circ$ of the lower sideband (USB, grüner Zeiger) follows for the phase angle to be considered in the complex plane:   $\phi_{\rm U} = +30^\circ$.

The temporal process of $x_{\rm TP}(t)$ is also referred to below as '''locus'''. The relationship between $x_{\rm TP}(t)$ and the physical bandpass signal $x(t)$ is given in the section and the associated analytic signal is $x_+(t)$ :

:$$x_{\rm TP}(t) = x_{\rm +}(t)\cdot {\rm e}^{-{\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm}2 \pi \cdot f_{\rm T}\cdot \hspace{0.05cm}t},$$
:$$x_{\rm +}(t) = x_{\rm TP}(t)\cdot {\rm e}^{+{\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm}2 \pi \cdot f_{\rm T}\cdot \hspace{0.05cm}t}.$$

[[Applets:Physikalisches_Signal_%26_Äquivalentes_TP-Signal|'''German description''']]

==Theoretical Background==
 
===Description of Bandpass Signals===
[[File:Zeigerdiagramm_1a.png|right|frame|bandpass spectrum $X(f)$ |class=fit]]
We consider '''bandpass signals''' $x(t)$ with the property that their spectra $X(f)$ are not in the range around the frequency $f=0$, but around a carrier frequency $f_{\rm T}$. In most cases it can also be assumed that the bandwidth is $B \ll f_{\rm T}$.

The figure shows such a bandpass spectrum $X(f)$. Assuming that the associated $x(t)$ is a physical signal and thus real, the spectral function $X(f)$ has a symmetry with respect to the frequency $f = 0$, if $x(t)$ is an even function   ⇒   $x(-t)=x(t)$, $X(f)$ is real and even.

Beside the physical signal $x(t)\ \circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\, \ X(f)$, one can also use the following descriptions of bandpass signals:
*the analytic signal $x_+(t)\ \circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\, \ X_+(f)$, see applet [[Applets:Physikalisches_Signal_%26_Analytisches_Signal|Physikalisches Signal & Analytisches Signal]],
*the equivalent low-pass–signal $x_{\rm TP}(t)\ \circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\, \ X_{\rm TP}(f)$, see next page
 

===Spectral Functions of the Analytic and the Equivalent Low-Pass Signal===

The '''analytic signal''' $x_+(t)$ belonging to the physical signal $x(t)$ is the time function whose spectrum fulfills the following property:
[[File:Ortskurve_2.png|right|frame|spectral functions $X_+(f)$ and $X_{\rm TP}(f)$ |class=fit]]
:$$X_+(f)=\big[1+{\rm sign}(f)\big] \cdot X(f) = \left\{ {2 \cdot
X(f) \; \hspace{0.2cm}\rm for\hspace{0.2cm} {\it f} > 0, \atop {\,\,\,\, \rm 0 \; \hspace{0.9cm}\rm for\hspace{0.2cm} {\it f} < 0.} }\right.$$

The ''Signum function'' is for positive values of $f$ equal to $+1$ and for negative $f$ values equal to $-1$.
*The (double-sided) limit returns $\sign(0) = 0$.
*The index „+” should make it clear that $X_+(f)$ only has parts at positive frequencies.

From the graph you can see the calculation rule for $X_+(f)$: The actual BP spectrum $X(f)$ becomes
*doubled at the positive frequencies, and
*set to zero at the negative frequencies.

Due to the asymmetry of $X_+(f)$ with respect to the frequency $f = 0$, it can already be said that the time function $x_+(t)$ except for a trivial special case $x_+(t)= 0 \ \ \circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\,\ \ X_+(f)= 0$ is always complex.

The spectrum $X_{\rm TP}(f)$ of the equivalent low-pass signal is obtained by shifting $X_+(f)$ to the left by the carrier frequency $f_{\rm T}$:
:$$X_{\rm TP}(f)= X_+(f+f_{\rm T}).$$

In the time domain this operation corresponds to the multiplication of $x_{\rm +}(t)$ with the complex exponential function with negative exponent:
:$$x_{\rm TP}(t) = x_{\rm +}(t)\cdot {\rm e}^{-{\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm}2 \pi \cdot f_{\rm T}\cdot \hspace{0.05cm}t}.$$

It can be seen that $x_{\rm TP}(t)$ is generally complex. But if $X_+(f)$ is symmetric about the carrier frequency $f_{\rm T}$, $X_{\rm TP}(f)$ is symmetric about the frequency $f=0$ and there is accordingly real time function $x_{\rm TP}(t)$.

===$x_{\rm TP}(t)$–Representation of a Sum of Three Harmonic Oscillations===

In our applet, we always assume a set of three rotating pointers. The physical signal is:
:$$x(t) = x_{\rm U}(t) + x_{\rm T}(t) + x_{\rm O}(t) = A_{\rm U}\cdot \cos\left(2\pi f_{\rm U}\cdot t- \varphi_{\rm U}\right)+A_{\rm T}\cdot \cos\left(2\pi f_{\rm T}\cdot t- \varphi_{\rm T}\right)+A_{\rm O}\cdot \cos\left(2\pi f_{\rm O}\cdot t- \varphi_{\rm O}\right). $$
* Each of the three harmonic oscillations $x_{\rm T}(t)$, $x_{\rm U}(t)$ and $x_{\rm O}(t)$ is represented by an amplitude $(A)$, a frequency $(f)$ and a phase value $(\varphi)$.
*The indices are based on the modulation method [[Modulationsverfahren/Zweiseitenband-Amplitudenmodulation|double sideband –amplitude modulation]]. &bdquo;T” stands for &bdquo;carrier”, &bdquo;U” for &bdquo;lowei sideband” and &bdquo;O” for &bdquo;upper Sideband”. Similarly, $f_{\rm U} < f_{\rm T}$ and $f_{\rm O} > f_{\rm T}$. There are no restrictions for the amplitudes and phases.

The associated equivalent low-pass signal is with $f_{\rm O}\hspace{0.01cm}' = f_{\rm O}- f_{\rm T} > 0$,   $f_{\rm U}\hspace{0.01cm}' = f_{\rm U}- f_{\rm T} < 0$  and  $f_{\rm T}\hspace{0.01cm}' = 0$:

:$$x_{\rm TP}(t) = x_\text{TP, T}(t) + x_\text{TP, O}(t) + x_\text{TP, U}(t) = A_{\rm T}\cdot {\rm e}^{-{\rm j} \varphi_{\rm T} } \hspace{0.1cm}+ \hspace{0.1cm} A_{\rm O}\cdot {\rm e}^{-{\rm j} \varphi_{\rm O} } \cdot {\rm e}^{ {\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm O}\hspace{0.01cm}'\hspace{0.05cm}\cdot \hspace{0.05cm}t}\hspace{0.1cm}+ \hspace{0.1cm}
A_{\rm U}\cdot {\rm e}^{-{\rm j} \varphi_{\rm U} } \cdot {\rm e}^{ {\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm U}\hspace{0.01cm}'\hspace{0.05cm}\cdot \hspace{0.05cm}t} . $$

{{GraueBox|TEXT=
$\text{Example 1:}$ 
The constellation given here results, for example, in the [[Modulationsverfahren/Zweiseitenband-Amplitudenmodulation#AM-Signale_und_-Spektren_bei_harmonischem_Eingangssignal|double sideband amplitude modulation]] of the message signal $x_{\rm N}(t) = A_{\rm N}\cdot \cos\left(2\pi f_{\rm N}\cdot t- \varphi_{\rm N}\right)$ with the carrier signal $x_{\rm T}(t) = A_{\rm T}\cdot \cos\left(2\pi f_{\rm T}\cdot t - \varphi_{\rm T}\right)$. This is discussed frequently in the exercises.

[[File:Ortskurve_4.png|center|frame|Spectrum $X_{\rm TP}(f)$ of the equivalent low–pass signal for different phase constellations |class=fit]]

There are some limitations to the program parameters in this approach:
* The frequencies are always $f\hspace{0.05cm}'_{\rm O} = f_{\rm N}$ and $f\hspace{0.05cm}'_{\rm U} = -f_{\rm N}$.
*Without distortion, the amplitude of the sidebands is $A_{\rm O}= A_{\rm U}= A_{\rm N}/2$.
*The respective phase relationships can be seen from the graphic.

}}

===Representation of the Equivalent Low-Pass Signal by Magnitude and Phase===

The generally complex equivalent low-pass signal
:$$x_{\rm TP}(t) = a(t) \cdot {\rm e}^{ {\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm} \phi(t) }$$
can be split into a magnitude function $a(t)$ and a phase function $\phi(t)$ according to the equation given here, where:
:$$a(t) = \vert x_{\rm TP}(t)\vert = \sqrt{ {\rm Re}^2\big [x_{\rm TP}(t)\big ] + {\rm Im}^2\big [x_{\rm TP}(t)\big ] }\hspace{0.05cm},$$
:$$\phi(t) = \text{arc }x_{\rm TP}(t) = \arctan \frac{{\rm Im}\big [x_{\rm TP}(t)\big ]}{{\rm Re}\big [x_{\rm TP}(t)\big ]}.$$

The reason for this is that a bandpass–signal $x(t)$ is usually described by the equivalent low-pass signal $x_{\rm TP}(t)$ that the functions $a(t)$ and $\phi(t)$ are interpretable in both representations:
*The amount $a(t)$ of the equivalent low-pass signal $x_{\rm TP}(t)$ indicates the (time-dependent) envelope of $x(t)$.
*The phase $\phi(t)$ of $x_{\rm TP}(t)$ denotes the location of the zero crossings of $x(t)$, where:
:–   For $\phi(t)>0$ the zero crossing is earlier than its nominal position   ⇒   the signal is leading here.
:–  When $\phi(t)<0$, the zero crossing is later than its target position   ⇒   the signal is trailing here.

{{GraueBox|TEXT=
$\text{Example 2:}$ 
The graph is intended to illustrate this relationship, assuming $A_{\rm U} > A_{\rm O}$   ⇒   the green pointer (for the lower sideband) is longer than the blue pointer (upper sideband). This is a snapshot at time $t_0$:

[[File:Ortskurve_3_neu.png|center|frame|bandpass spectrum $X(f)$ |class=fit]]

*For these system parameters, the top of the pointer cluster $x_{\rm TP}(t)$ – that is, the geometric sum of red, blue and green pointers – on an ellipse.
* The amount $a(t_0) = \vert x_{\rm TP}(t_0) \vert$ is drawn in black in the left-hand diagram and the phase value $\phi(t_0) = \text{arc }x_{\rm TP}(t_0) > 0$ is indicated in brown color.
*In the graph on the right, the amount $a(t_0) = \vert x_{\rm TP}(t_0) \vert$ of the equivalent low-pass signal indicates the envelope of the physical signal $x(t)$.
* At $\phi(t) \equiv 0$, all zero crossings of $x(t)$ would occur at equidistant intervals. Because of $\phi(t_0) > 0$, the signal is leading at the time $t_0$, that is: the zero crossings come earlier than the grid dictates. }}

==Exercises==
[[File:Exercises_verzerrungen.png|right]]
*First select the task number.
*A task description is displayed.
*Parameter values are adjusted.
*Solution after pressing &bdquo;Hide solition”.

The number &bdquo;0” will reset the program and output a text with the further explanation of the applet.

In the following, $\rm Green$ denotes the lower sideband   ⇒   $\big (A_{\rm U}, f_{\rm U}, \varphi_{\rm U}\big )$,  
$\rm Red$ the carrier   ⇒   $\big (A_{\rm T}, f_{\rm T}, \varphi_{\rm T}\big )$ and
$\rm Blue$ the upper sideband   ⇒   $\big (A_{\rm O}, f_{\rm O}, \varphi_{\rm O}\big )$.

{{BlaueBox|TEXT=
'''(1)'''   Let   $\text{Red:} \hspace{0.15cm} A_{\rm T} = 1\ \text{V}, f_{\rm T} = 100 \ \text{kHz}, \varphi_{\rm T} = 0^\circ$,   $\text{Green:} \hspace{0.15cm} A_{\rm U} = 0.4 \text{V}, \ f_{\rm U} = 80 \ \text{kHz}, \varphi_{\rm U} = -90^\circ$,   $\text{Blue:} \hspace{0.15cm} A_{\rm O} = 0.4\ \text{V}, f_{\rm O} = 120 \ \text{kHz}, \varphi_{\rm O} = 90^\circ$.

:Consider and interpret the equivalent low-pass signal $x_{\rm TP}(t)$ and the physical signal $x(t)$. Which period $T_0$ does one recognize?}}

:: The equivalent low-pass signal $x_{\rm TP}(t)$ takes from $x_{\rm TP}(t=0)=1\ \text{V}$ on the real axis values between $0.2\ \text{V}$ and $1.8\ \text{V}$   ⇒   phase $\phi(t) \equiv 0$.  The amount $|x_{\rm TP}(t)|$ indicates the envelope $a(t)$ of the physical signal $x(t)$. It holds $A_{\rm N} = 0.8\ \text{V}$ and $f_{\rm N} = 20\ \text{kHz}$:   $a(t) = A_{\rm T}+ A_{\rm N} \cdot \sin(2\pi\cdot f_{\rm N} \cdot t)$.  Both $x_{\rm TP}(t)$ and $x(t)$ are periodic with the period $T_0 = 1/f_{\rm N} = 50\ \rm µ s$.

{{BlaueBox|TEXT=
'''(2)'''   How do the ratios change to '''(1)''' with $f_{\rm U} = 99 \ \text{kHz}$ and $f_{\rm O} = 101 \ \text{kHz}$ ? How could $x(t)$ have arisen?}}

:: For the envelope $a(t)$ of the signal $x(t)$ we still have $a(t) = A_{\rm T}+ A_{\rm N} \cdot \sin(2\pi\cdot f_{\rm N} \cdot t)$, but now $f_{\rm N} = 1\ \text{kHz}$. Even though it can not be recognized:  $x_{\rm TP}(t)$ and $x(t)$ are still periodic:   $T_0 = 1\ \rm ms$. Example: Double sideband - Amplitude modulation '''(ZSB–AM)''' of a sine signal with cosine–carrier.

{{BlaueBox|TEXT=
'''(3)'''   Which settings have to be changed from '''(2)''' in order to arrive at the ZSB–AM of a cosine signal with sine–carrier. What changes over '''(2)'''?}}

::The carrier phase must be changed to $\varphi_{\rm T} = 90^\circ$   ⇒   sine–carrier. Similarly, $\varphi_{\rm O} =\varphi_{\rm U} =\varphi_{\rm T} = 90^\circ$ must be set   ⇒   cosine-shaped message  The locus now lies on the imaginary axis  ⇒   $\phi(t) \equiv -90^\circ$. At the beginning we have $x_{\rm TP}(t=0)= - {\rm j} \cdot 1.8 \ \text{V}$.

{{BlaueBox|TEXT=
'''(4)'''   Now let   $\text{Red:} \hspace{0.15cm} A_{\rm T} = 1\ \text{V}, \ f_{\rm T} = 100 \ \text{kHz}, \ \varphi_{\rm T} = 0^\circ$,   $\text{Green:} \hspace{0.15cm} A_{\rm U} = 0.4 \text{V}, \ f_{\rm U} = 80 \ \text{kHz}, \ \varphi_{\rm U} = 0^\circ$,   $\text{Blue:} \hspace{0.15cm} A_{\rm O} = 0.4\ \text{V}, \ f_{\rm O} = 120 \ \text{kHz}, \ \varphi_{\rm O} = 0^\circ$.

:What are the characteristics of this system &bdquo;ZSB–AM, where the message signal and carrier are respectively cosinusoidal”? What is the modulation depth $m$?}}

:: The equivalent low-pass signal $x_{\rm TP}(t)$ takes from $x_{\rm TP}(t=0)=1.8\ \text{V}$ on the real axis values between $0.2\ \text{V}$ and $1.8\ \text{V}$   ⇒   phase $\phi(t) \equiv 0$.  Except for the start state $x_{\rm TP}(t=0)$ same behavior as with the setting '''(1)'''. The modulation depth is $m = 0.8$.

{{BlaueBox|TEXT=
'''(5)'''   The parameters are still valid according to '''(4)''' with the exception of $A_{\rm T}= 0.6 \text{V}$. How big is the degree of modulation $m$? What are the consequences?}}

:: There is now a ZSB–AM with modulation degree $m = 1.333$. For $m > 1$, the simpler [[Modulationsverfahren/Hüllkurvendemodulation|envelope demodulation]] is not applicable, since the phase function $\phi(t) \in \{ 0, \ \pm 180^\circ\}$ is no longer constant and the envelope $a(t)$ no longer matches the message signal. Rather, the more complex [[Modulationsverfahren/Synchrondemodulation|synchronous demodulation]] must be used. Envelope detection would produce nonlinear distortions.

{{BlaueBox|TEXT=
'''(6)'''   The parameters are still valid according to '''(4)''' or '''(5)''' with the exception from $A_{\rm T}= 0$ on   ⇒   $m \to \infty$. Which modulation method is described in this way?}}

::It is a '''ZSB–AM without carrier''' and requires synchronous demodulation. The equivalent low-pass signal $x_{\rm TP}(t)$ is on the real axis, but not only in the right half-plane. Thus, the phase function $\phi(t) \in \{ 0, \ \pm 180^\circ\}$, also applies here, which means that envelope demodulation is not applicable.

{{BlaueBox|TEXT=
'''(7)'''   Now let   $\text{Red:} \hspace{0.15cm} A_{\rm T} = 1\ \text{V}, f_{\rm T} = 100 \ \text{kHz}, \varphi_{\rm T} = 0^\circ$,   $\text{Green:} \hspace{0.15cm} A_{\rm U} = 0, \ f_{\rm U} = 80 \ \text{kHz}, \varphi_{\rm U} = -90^\circ$,   $\text{Blue:} \hspace{0.15cm} A_{\rm O} = 0.8\ \text{V}, f_{\rm O} = 120 \ \text{kHz}, \varphi_{\rm O} = 90^\circ$.

:Which constellation is described here? Which characteristics of this procedure can be recognized from the graphic?}}

::It is a [[Modulationsverfahren/Einseitenbandmodulation|single-sideband modulation]] '''(ESB–AM)''', more specifically an '''OSB–AM''': the red carrier is fixed, the green pointer missing and the blue pointer (OSB) turns counterclockwise. The degree of modulation is $\mu = 0.8$ (in the case of ESB we denote the degree of modulation with $\mu$ instead of $m$). The carrier signal is cosinusoidal and the message signal sinusoidal. The locus is a circle. $x_{\rm TP}(t)$ moves in a mathematically positive direction. Because of $\phi(t) \ne \text{const.}$, The envelope demodulation is not applicable here:  This can be seen by the fact that the envelope $a(t)$ is not cosinusoidal. Rather, the lower half-wave is sharper than the upper   ⇒   strong linear distortions.

{{BlaueBox|TEXT=
'''(8)'''   The parameters are still valid according to '''(7)''' with the exception of $A_{\rm O}= 0$ and $A_{\rm U}= 0.8 \text{ V}$. What differences arise opposite '''(7)'''?}}

::Now it is a '''USB–AM''': The red carrier is fixed, the blue pointer is missing and the green pointer (USB) rotates clockwise. All other statements of '''(7)''' apply here as well.

{{BlaueBox|TEXT=
'''(9)'''   The parameters according to '''(7)''' are still valid with the exception of $A_{\rm O} = 0.2 \text{ V} \ne A_{\rm U} = 0.4 \text{ V} $. What are the differences from '''(7)'''?}}

::The locus $x_{\rm TP}(t)$ is not a horizontal straight line, but an ellipse with the real part between $0.4 \text{ V}$ and $1.6 \text{ V}$ and the imaginary part in the range $\pm 0.2 \text{ V}$. Because of $\phi(t) \ne \text{const.}$ , Envelope demodulation would lead to non-linear distortions here too. The constellation simulated here describes the situation of '''(4)''', namely a ZSB–AM with modulation degree $m = 0.8$, where the upper sideband is reduced to $50\%$ due to channel attenuation.

==Applet Manual==
[[File:Ortskurve_abzug3.png|right]]
* The red parameters $(A_{\rm T}, \ f_{\rm T}, \ \varphi_{\rm T})$ and the red pointer mark the ''Carrier''. (German: '''T'''räger).
* The green parameters $(A_{\rm U}, \ f_{\rm U} < f_{\rm T}, \ \varphi_{\rm U})$ mark the ''Lower sideband''. (German: '''U'''ntere Seitenband).
* The blue parameters $(A_{\rm O}, \ f_{\rm O} > f_{\rm T}, \ \varphi_{\rm O})$ mark the '' Upper sideband''. (German: '''O'''bere Seitenband).
* The red pointer does not turn.
* The green pointer rotates in a mathematically negative direction (clockwise).
* The blue pointer turns counterclockwise.

Meaning of the letters in the adjacent graphic:

    '''(A)'''     Plot of the equivalent low-pass Signal $x_{\rm TP}(t)$

    '''(B)'''     Plot of the physical signal $x(t)$

    '''(C)'''     Parameter input via slider:   amplitudes, frequencies, phase values

    '''(D)'''     Control elements:   Start – Step – Pause/Continue – Reset

    '''(E)'''     Speed of animation:   &bdquo;Speed”   ⇒   Values: 1, 2 oder 3

    '''(F)'''     &bdquo;Trace”   ⇒   On or Off, trace of equivalent low-pass Signal   $x_{\rm TP}(t)$

    '''(G)'''     Numerical output:   time $t$, the signal values  ${\rm Re}[x_{\rm TP}(t)]$  and  ${\rm Im}[x_{\rm TP}(t)]$,

$\text{}\hspace{4.2cm}$   Hüllkurve $a(t) = |x_{\rm TP}(t)|$  und  Phase $\phi(t) = {\rm arc} \ x_{\rm TP}(t)$

    '''(H)'''     Variations for the graphical representation

$\hspace{1.5cm}$Zoom–Functions &bdquo;$+$” (Enlarge), &bdquo;$-$” (Decrease) und $\rm o$ (Reset to default)

$\hspace{1.5cm}$Move with &bdquo;$\leftarrow$” (Section to the left, ordinate to the right), &bdquo;$\uparrow$” &bdquo;$\downarrow$” &bdquo;$\rightarrow$”

    '''(I)'''     Experiment section:   Task selection and task

    '''(J)'''     Experiment section:  solution
 

==About the Authors==
This interactive calculation was designed and realized at the [http://www.lnt.ei.tum.de/startseite Lehrstuhl für Nachrichtentechnik] of the [https://www.tum.de/ Technical University of Munich] .
*The original version was created in 2005 by [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Studierende#Ji_Li_.28Bachelorarbeit_EI_2003.2C_Diplomarbeit_EI_2005.29|Ji Li]] as part of her Diploma thesis using &bdquo;FlashMX–Actionscript” (Supervisor: [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Mitarbeiter_und_Dozenten#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28am_LNT_seit_1974.29|Günter Söder]]).
*In 2018 this Applet was redesigned and updated to &bdquo;HTML5” by [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Studierende#Xiaohan_Liu_.28Bachelorarbeit_2018.29|Xiaohan Liu]] as part of her Bachelor's thesis (Supervisor: [[Biografien_und_Bibliografien/Beteiligte_der_Professur_Leitungsgebundene_%C3%9Cbertragungstechnik#Tasn.C3.A1d_Kernetzky.2C_M.Sc._.28bei_L.C3.9CT_seit_2014.29|Tasnád Kernetzky]]).

==Once again: Open Applet in new Tab==

{{LntAppletLinkEn|physAnSignal}}

[[Category:Applets|^Verzerrungen^]]

Applets:Physical Signal & Equivalent Lowpass Signal

2018-08-25T16:55:51Z

Xiaohan: /* Applet Description */

{{LntAppletLinkEn|analPhysSignal}}

==Applet Description==
 
This applet shows the relationship between the physical bandpass signal $x(t)$ and the associated equivalent low-pass signal $x_{\rm TP}(t)$. It is assumed that the bandpass signal $x(t)$ has a frequency-discrete spectrum $X(f)$:
:$$x(t) = x_{\rm U}(t) + x_{\rm T}(t) + x_{\rm O}(t) = A_{\rm U}\cdot \cos\left(2\pi f_{\rm U}\cdot t- \varphi_{\rm U}\right)+A_{\rm T}\cdot \cos\left(2\pi f_{\rm T}\cdot t- \varphi_{\rm T}\right)+A_{\rm O}\cdot \cos\left(2\pi f_{\rm O}\cdot t- \varphi_{\rm O}\right). $$
The physical signal $x(t)$ is thus composed of three harmonic oscillations, a constellation that can be found, for example, in the ''Double-sideband Amplitude Modulation''
*of the message signal $x_{\rm N}(t) = A_{\rm N}\cdot \cos\left(2\pi f_{\rm N}\cdot t- \varphi_{\rm N}\right)$   ⇒   in German:   '''N'''achrichtensignal
*with the carrier signal $x_{\rm T}(t) = A_{\rm T}\cdot \cos\left(2\pi f_{\rm T}\cdot t - \varphi_{\rm T}\right)$   ⇒   in German:   '''T'''rägersignal.

The nomenclature is also adapted to this case:
* $x_{\rm O}(t)$ denotes the &bdquo;upper sideband”   (in German:   '''O'''beres Seitenband) with the amplitude $A_{\rm O}= A_{\rm N}/2$, the frequency $f_{\rm O} = f_{\rm T} + f_{\rm N}$ and the phase $\varphi_{\rm O} = \varphi_{\rm T} + \varphi_{\rm N}$.
*Similarly, for the &bdquo;lower sideband”   (in German:   '''U'''nteres Seitenband) $x_{\rm U}(t)$ with $f_{\rm U} = f_{\rm T} - f_{\rm N}$, $A_{\rm U}= A_{\rm O}$ and $\varphi_{\rm U} = -\varphi_{\rm O}$.

The associated equivalent low-pass–signal is $f_{\rm O}\hspace{0.01cm}' = f_{\rm O}- f_{\rm T} > 0$,   $f_{\rm U}\hspace{0.01cm}' = f_{\rm U}- f_{\rm T} < 0$  and  $f_{\rm T}\hspace{0.01cm}' = 0$:

:$$x_{\rm TP}(t) = x_\text{TP, T}(t) + x_\text{TP, O}(t) + x_\text{TP, U}(t) = A_{\rm T}\cdot {\rm e}^{-{\rm j} \varphi_{\rm T} } \hspace{0.1cm}+ \hspace{0.1cm} A_{\rm O}\cdot {\rm e}^{-{\rm j} \varphi_{\rm O} } \cdot {\rm e}^{ {\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm O}\hspace{0.01cm}'\hspace{0.05cm}\cdot \hspace{0.05cm}t}\hspace{0.1cm}+ \hspace{0.1cm}
A_{\rm U}\cdot {\rm e}^{-{\rm j} \varphi_{\rm U} } \cdot {\rm e}^{ {\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm U}\hspace{0.01cm}'\hspace{0.05cm}\cdot \hspace{0.05cm}t} . $$

[[File:Ortskurve_1.png|right|frame|Equivalent lowpass–signal currently $t=0$ for cosinusoidal carrier   ⇒   $\varphi_{\rm T} = 0$]]
The program shows $x_{\rm TP}(t)$ as the vectorial sum of three rotation pointers as a violet dot (see figure for start time $t=0$ and cosinusoidal carrier):

*The (red) pointer of the carrier $x_\text{TP, T}(t)$ with the length $A_{\rm T}$ and the zero phase position $\varphi_{\rm T}=0$ is fixed in the complex plane. So it applies to all times $t$:   $x_{\rm TP}(t)= A_{\rm T}\cdot {\rm e}^{-{\rm j} \varphi_{\rm T} }$.

*The (blue) pointer of the upper sideband $x_\text{TP, O}(t)$ with the length $A_{\rm O}$ and the zero phase position $\varphi_{\rm O}$ rotates at the angular velocity $2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm O}\hspace{0.01cm}'$ in mathematically positive direction (one revolution in time $1/f_{\rm O}\hspace{0.01cm}')$.

*The (green) pointer of the lower sideband $x_{\rm U+}(t)$ with the length $A_{\rm U}$ and the zero phase position $\varphi_{\rm U}$ rotates at the angular velocity $2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm U}\hspace{0.01cm}'$, because of $f_{\rm U}\hspace{0.01cm}'<0$ counterclockwise.

*With $f_{\rm U}\hspace{0.01cm}' = -f_{\rm O}\hspace{0.01cm}'$ the blue and green pointers will spin at the same speed but in different directions. Also, if $A_{\rm O} = A_{\rm U}$ and $\varphi_{\rm U} = -\varphi_{\rm O}$, then $x_{\rm TP}(t)$ moves on a straight line with a incline of $\varphi_{\rm T}$.

''Note:''   In the figure $\varphi_{\rm O} = +30^\circ$. From this follows for the start time $t=0$ the angle of the blue pointer (OSB) with respect to the coordinate system:   $\phi_{\rm O} = -\varphi_{\rm O} = -30^\circ$. Likewise, the null phase position $\varphi_{\rm U} = -30^\circ$ of the lower sideband (USB, grüner Zeiger) follows for the phase angle to be considered in the complex plane:   $\phi_{\rm U} = +30^\circ$.

The temporal process of $x_{\rm TP}(t)$ is also referred to below as '''locus'''. The relationship between $x_{\rm TP}(t)$ and the physical bandpass signal $x(t)$ is given in the section and the associated analytic signal is $x_+(t)$ :

:$$x_{\rm TP}(t) = x_{\rm +}(t)\cdot {\rm e}^{-{\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm}2 \pi \cdot f_{\rm T}\cdot \hspace{0.05cm}t},$$
:$$x_{\rm +}(t) = x_{\rm TP}(t)\cdot {\rm e}^{+{\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm}2 \pi \cdot f_{\rm T}\cdot \hspace{0.05cm}t}.$$

[[Applets:Physikalisches_Signal_%26_Äquivalentes_TP-Signal|'''German description''']]

==Theoretical Background==
 
===Description of Bandpass Signals===
[[File:Zeigerdiagramm_1a.png|right|frame|bandpass spectrum $X(f)$ |class=fit]]
We consider '''bandpass signals''' $x(t)$ with the property that their spectra $X(f)$ are not in the range around the frequency $f=0$, but around a carrier frequency $f_{\rm T}$. In most cases it can also be assumed that the bandwidth is $B \ll f_{\rm T}$.

The figure shows such a bandpass spectrum $X(f)$. Assuming that the associated $x(t)$ is a physical signal and thus real, the spectral function $X(f)$ has a symmetry with respect to the frequency $f = 0$, if $x(t)$ is an even function   ⇒   $x(-t)=x(t)$, $X(f)$ is real and even.

Beside the physical signal $x(t)\ \circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\, \ X(f)$, one can also use the following descriptions of bandpass signals:
*the analytic signal $x_+(t)\ \circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\, \ X_+(f)$, see applet [[Applets:Physikalisches_Signal_%26_Analytisches_Signal|Physikalisches Signal & Analytisches Signal]],
*the equivalent low-pass–signal $x_{\rm TP}(t)\ \circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\, \ X_{\rm TP}(f)$, see next page
 

===Spectral Functions of the Analytic and the Equivalent Low-Pass Signal===

The '''analytic signal''' $x_+(t)$ belonging to the physical signal $x(t)$ is the time function whose spectrum fulfills the following property:
[[File:Ortskurve_2.png|right|frame|spectral functions $X_+(f)$ and $X_{\rm TP}(f)$ |class=fit]]
:$$X_+(f)=\big[1+{\rm sign}(f)\big] \cdot X(f) = \left\{ {2 \cdot
X(f) \; \hspace{0.2cm}\rm for\hspace{0.2cm} {\it f} > 0, \atop {\,\,\,\, \rm 0 \; \hspace{0.9cm}\rm for\hspace{0.2cm} {\it f} < 0.} }\right.$$

The ''Signum function'' is for positive values of $f$ equal to $+1$ and for negative $f$ values equal to $-1$.
*The (double-sided) limit returns $\sign(0) = 0$.
*The index „+” should make it clear that $X_+(f)$ only has parts at positive frequencies.

From the graph you can see the calculation rule for $X_+(f)$: The actual BP spectrum $X(f)$ becomes
*doubled at the positive frequencies, and
*set to zero at the negative frequencies.

Due to the asymmetry of $X_+(f)$ with respect to the frequency $f = 0$, it can already be said that the time function $x_+(t)$ except for a trivial special case $x_+(t)= 0 \ \ \circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\,\ \ X_+(f)= 0$ is always complex.

The spectrum $X_{\rm TP}(f)$ of the equivalent low-pass signal is obtained by shifting $X_+(f)$ to the left by the carrier frequency $f_{\rm T}$:
:$$X_{\rm TP}(f)= X_+(f+f_{\rm T}).$$

In the time domain this operation corresponds to the multiplication of $x_{\rm +}(t)$ with the complex exponential function with negative exponent:
:$$x_{\rm TP}(t) = x_{\rm +}(t)\cdot {\rm e}^{-{\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm}2 \pi \cdot f_{\rm T}\cdot \hspace{0.05cm}t}.$$

It can be seen that $x_{\rm TP}(t)$ is generally complex. But if $X_+(f)$ is symmetric about the carrier frequency $f_{\rm T}$, $X_{\rm TP}(f)$ is symmetric about the frequency $f=0$ and there is accordingly real time function $x_{\rm TP}(t)$.

===$x_{\rm TP}(t)$–Representation of a Sum of Three Harmonic Oscillations===

In our applet, we always assume a set of three rotating pointers. The physical signal is:
:$$x(t) = x_{\rm U}(t) + x_{\rm T}(t) + x_{\rm O}(t) = A_{\rm U}\cdot \cos\left(2\pi f_{\rm U}\cdot t- \varphi_{\rm U}\right)+A_{\rm T}\cdot \cos\left(2\pi f_{\rm T}\cdot t- \varphi_{\rm T}\right)+A_{\rm O}\cdot \cos\left(2\pi f_{\rm O}\cdot t- \varphi_{\rm O}\right). $$
* Each of the three harmonic oscillations $x_{\rm T}(t)$, $x_{\rm U}(t)$ and $x_{\rm O}(t)$ is represented by an amplitude $(A)$, a frequency $(f)$ and a phase value $(\varphi)$.
*The indices are based on the modulation method [[Modulationsverfahren/Zweiseitenband-Amplitudenmodulation|double sideband –amplitude modulation]]. &bdquo;T” stands for &bdquo;carrier”, &bdquo;U” for &bdquo;lowei sideband” and &bdquo;O” for &bdquo;upper Sideband”. Similarly, $f_{\rm U} < f_{\rm T}$ and $f_{\rm O} > f_{\rm T}$. There are no restrictions for the amplitudes and phases.

The associated equivalent low-pass signal is with $f_{\rm O}\hspace{0.01cm}' = f_{\rm O}- f_{\rm T} > 0$,   $f_{\rm U}\hspace{0.01cm}' = f_{\rm U}- f_{\rm T} < 0$  and  $f_{\rm T}\hspace{0.01cm}' = 0$:

:$$x_{\rm TP}(t) = x_\text{TP, T}(t) + x_\text{TP, O}(t) + x_\text{TP, U}(t) = A_{\rm T}\cdot {\rm e}^{-{\rm j} \varphi_{\rm T} } \hspace{0.1cm}+ \hspace{0.1cm} A_{\rm O}\cdot {\rm e}^{-{\rm j} \varphi_{\rm O} } \cdot {\rm e}^{ {\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm O}\hspace{0.01cm}'\hspace{0.05cm}\cdot \hspace{0.05cm}t}\hspace{0.1cm}+ \hspace{0.1cm}
A_{\rm U}\cdot {\rm e}^{-{\rm j} \varphi_{\rm U} } \cdot {\rm e}^{ {\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm U}\hspace{0.01cm}'\hspace{0.05cm}\cdot \hspace{0.05cm}t} . $$

{{GraueBox|TEXT=
$\text{Example 1:}$ 
The constellation given here results, for example, in the [[Modulationsverfahren/Zweiseitenband-Amplitudenmodulation#AM-Signale_und_-Spektren_bei_harmonischem_Eingangssignal|double sideband amplitude modulation]] of the message signal $x_{\rm N}(t) = A_{\rm N}\cdot \cos\left(2\pi f_{\rm N}\cdot t- \varphi_{\rm N}\right)$ with the carrier signal $x_{\rm T}(t) = A_{\rm T}\cdot \cos\left(2\pi f_{\rm T}\cdot t - \varphi_{\rm T}\right)$. This is discussed frequently in the exercises.

[[File:Ortskurve_4.png|center|frame|Spectrum $X_{\rm TP}(f)$ of the equivalent low–pass signal for different phase constellations |class=fit]]

There are some limitations to the program parameters in this approach:
* The frequencies are always $f\hspace{0.05cm}'_{\rm O} = f_{\rm N}$ and $f\hspace{0.05cm}'_{\rm U} = -f_{\rm N}$.
*Without distortion, the amplitude of the sidebands is $A_{\rm O}= A_{\rm U}= A_{\rm N}/2$.
*The respective phase relationships can be seen from the graphic.

}}

===Representation of the Equivalent Low-Pass Signal by Magnitude and Phase===

The generally complex equivalent low-pass signal
:$$x_{\rm TP}(t) = a(t) \cdot {\rm e}^{ {\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm} \phi(t) }$$
can be split into a magnitude function $a(t)$ and a phase function $\phi(t)$ according to the equation given here, where:
:$$a(t) = \vert x_{\rm TP}(t)\vert = \sqrt{ {\rm Re}^2\big [x_{\rm TP}(t)\big ] + {\rm Im}^2\big [x_{\rm TP}(t)\big ] }\hspace{0.05cm},$$
:$$\phi(t) = \text{arc }x_{\rm TP}(t) = \arctan \frac{{\rm Im}\big [x_{\rm TP}(t)\big ]}{{\rm Re}\big [x_{\rm TP}(t)\big ]}.$$

The reason for this is that a bandpass–signal $x(t)$ is usually described by the equivalent low-pass signal $x_{\rm TP}(t)$ that the functions $a(t)$ and $\phi(t)$ are interpretable in both representations:
*The amount $a(t)$ of the equivalent low-pass signal $x_{\rm TP}(t)$ indicates the (time-dependent) envelope of $x(t)$.
*The phase $\phi(t)$ of $x_{\rm TP}(t)$ denotes the location of the zero crossings of $x(t)$, where:
:–   For $\phi(t)>0$ the zero crossing is earlier than its nominal position   ⇒   the signal is leading here.
:–  When $\phi(t)<0$, the zero crossing is later than its target position   ⇒   the signal is trailing here.

{{GraueBox|TEXT=
$\text{Example 2:}$ 
The graph is intended to illustrate this relationship, assuming $A_{\rm U} > A_{\rm O}$   ⇒   the green pointer (for the lower sideband) is longer than the blue pointer (upper sideband). This is a snapshot at time $t_0$:

[[File:Ortskurve_3_neu.png|center|frame|bandpass spectrum $X(f)$ |class=fit]]

*For these system parameters, the top of the pointer cluster $x_{\rm TP}(t)$ – that is, the geometric sum of red, blue and green pointers – on an ellipse.
* The amount $a(t_0) = \vert x_{\rm TP}(t_0) \vert$ is drawn in black in the left-hand diagram and the phase value $\phi(t_0) = \text{arc }x_{\rm TP}(t_0) > 0$ is indicated in brown color.
*In the graph on the right, the amount $a(t_0) = \vert x_{\rm TP}(t_0) \vert$ of the equivalent low-pass signal indicates the envelope of the physical signal $x(t)$.
* At $\phi(t) \equiv 0$, all zero crossings of $x(t)$ would occur at equidistant intervals. Because of $\phi(t_0) > 0$, the signal is leading at the time $t_0$, that is: the zero crossings come earlier than the grid dictates. }}

==Exercises==
[[File:Exercises_verzerrungen.png|right]]
*First select the task number.
*A task description is displayed.
*Parameter values are adjusted.
*Solution after pressing &bdquo;Hide solition”.

The number &bdquo;0” will reset the program and output a text with the further explanation of the applet.

In the following, $\rm Green$ denotes the lower sideband   ⇒   $\big (A_{\rm U}, f_{\rm U}, \varphi_{\rm U}\big )$,  
$\rm Red$ the carrier   ⇒   $\big (A_{\rm T}, f_{\rm T}, \varphi_{\rm T}\big )$ and
$\rm Blue$ the upper sideband   ⇒   $\big (A_{\rm O}, f_{\rm O}, \varphi_{\rm O}\big )$.

{{BlaueBox|TEXT=
'''(1)'''   Let   $\text{Red:} \hspace{0.15cm} A_{\rm T} = 1\ \text{V}, f_{\rm T} = 100 \ \text{kHz}, \varphi_{\rm T} = 0^\circ$,   $\text{Green:} \hspace{0.15cm} A_{\rm U} = 0.4 \text{V}, \ f_{\rm U} = 80 \ \text{kHz}, \varphi_{\rm U} = -90^\circ$,   $\text{Blue:} \hspace{0.15cm} A_{\rm O} = 0.4\ \text{V}, f_{\rm O} = 120 \ \text{kHz}, \varphi_{\rm O} = 90^\circ$.

:Consider and interpret the equivalent low-pass signal $x_{\rm TP}(t)$ and the physical signal $x(t)$. Which period $T_0$ does one recognize?}}

:: The equivalent low-pass signal $x_{\rm TP}(t)$ takes from $x_{\rm TP}(t=0)=1\ \text{V}$ on the real axis values between $0.2\ \text{V}$ and $1.8\ \text{V}$   ⇒   phase $\phi(t) \equiv 0$.  The amount $|x_{\rm TP}(t)|$ indicates the envelope $a(t)$ of the physical signal $x(t)$. It holds $A_{\rm N} = 0.8\ \text{V}$ and $f_{\rm N} = 20\ \text{kHz}$:   $a(t) = A_{\rm T}+ A_{\rm N} \cdot \sin(2\pi\cdot f_{\rm N} \cdot t)$.  Both $x_{\rm TP}(t)$ and $x(t)$ are periodic with the period $T_0 = 1/f_{\rm N} = 50\ \rm µ s$.

{{BlaueBox|TEXT=
'''(2)'''   How do the ratios change to '''(1)''' with $f_{\rm U} = 99 \ \text{kHz}$ and $f_{\rm O} = 101 \ \text{kHz}$ ? How could $x(t)$ have arisen?}}

:: For the envelope $a(t)$ of the signal $x(t)$ we still have $a(t) = A_{\rm T}+ A_{\rm N} \cdot \sin(2\pi\cdot f_{\rm N} \cdot t)$, but now $f_{\rm N} = 1\ \text{kHz}$. Even though it can not be recognized:  $x_{\rm TP}(t)$ and $x(t)$ are still periodic:   $T_0 = 1\ \rm ms$. Example: Double sideband - Amplitude modulation '''(ZSB–AM)''' of a sine signal with cosine–carrier.

{{BlaueBox|TEXT=
'''(3)'''   Which settings have to be changed from '''(2)''' in order to arrive at the ZSB–AM of a cosine signal with sine–carrier. What changes over '''(2)'''?}}

::The carrier phase must be changed to $\varphi_{\rm T} = 90^\circ$   ⇒   sine–carrier. Similarly, $\varphi_{\rm O} =\varphi_{\rm U} =\varphi_{\rm T} = 90^\circ$ must be set   ⇒   cosine-shaped message  The locus now lies on the imaginary axis  ⇒   $\phi(t) \equiv -90^\circ$. At the beginning we have $x_{\rm TP}(t=0)= - {\rm j} \cdot 1.8 \ \text{V}$.

{{BlaueBox|TEXT=
'''(4)'''   Now let   $\text{Red:} \hspace{0.15cm} A_{\rm T} = 1\ \text{V}, \ f_{\rm T} = 100 \ \text{kHz}, \ \varphi_{\rm T} = 0^\circ$,   $\text{Green:} \hspace{0.15cm} A_{\rm U} = 0.4 \text{V}, \ f_{\rm U} = 80 \ \text{kHz}, \ \varphi_{\rm U} = 0^\circ$,   $\text{Blue:} \hspace{0.15cm} A_{\rm O} = 0.4\ \text{V}, \ f_{\rm O} = 120 \ \text{kHz}, \ \varphi_{\rm O} = 0^\circ$.

:What are the characteristics of this system &bdquo;ZSB–AM, where the message signal and carrier are respectively cosinusoidal”? What is the modulation depth $m$?}}

:: The equivalent low-pass signal $x_{\rm TP}(t)$ takes from $x_{\rm TP}(t=0)=1.8\ \text{V}$ on the real axis values between $0.2\ \text{V}$ and $1.8\ \text{V}$   ⇒   phase $\phi(t) \equiv 0$.  Except for the start state $x_{\rm TP}(t=0)$ same behavior as with the setting '''(1)'''. The modulation depth is $m = 0.8$.

{{BlaueBox|TEXT=
'''(5)'''   The parameters are still valid according to '''(4)''' with the exception of $A_{\rm T}= 0.6 \text{V}$. How big is the degree of modulation $m$? What are the consequences?}}

:: There is now a ZSB–AM with modulation degree $m = 1.333$. For $m > 1$, the simpler [[Modulationsverfahren/Hüllkurvendemodulation|envelope demodulation]] is not applicable, since the phase function $\phi(t) \in \{ 0, \ \pm 180^\circ\}$ is no longer constant and the envelope $a(t)$ no longer matches the message signal. Rather, the more complex [[Modulationsverfahren/Synchrondemodulation|synchronous demodulation]] must be used. Envelope detection would produce nonlinear distortions.

{{BlaueBox|TEXT=
'''(6)'''   The parameters are still valid according to '''(4)''' or '''(5)''' with the exception from $A_{\rm T}= 0$ on   ⇒   $m \to \infty$. Which modulation method is described in this way?}}

::It is a '''ZSB–AM without carrier''' and requires synchronous demodulation. The equivalent low-pass signal $x_{\rm TP}(t)$ is on the real axis, but not only in the right half-plane. Thus, the phase function $\phi(t) \in \{ 0, \ \pm 180^\circ\}$, also applies here, which means that envelope demodulation is not applicable.

{{BlaueBox|TEXT=
'''(7)'''   Now let   $\text{Red:} \hspace{0.15cm} A_{\rm T} = 1\ \text{V}, f_{\rm T} = 100 \ \text{kHz}, \varphi_{\rm T} = 0^\circ$,   $\text{Green:} \hspace{0.15cm} A_{\rm U} = 0, \ f_{\rm U} = 80 \ \text{kHz}, \varphi_{\rm U} = -90^\circ$,   $\text{Blue:} \hspace{0.15cm} A_{\rm O} = 0.8\ \text{V}, f_{\rm O} = 120 \ \text{kHz}, \varphi_{\rm O} = 90^\circ$.

:Which constellation is described here? Which characteristics of this procedure can be recognized from the graphic?}}

::It is a [[Modulationsverfahren/Einseitenbandmodulation|single-sideband modulation]] '''(ESB–AM)''', more specifically an '''OSB–AM''': the red carrier is fixed, the green pointer missing and the blue pointer (OSB) turns counterclockwise. The degree of modulation is $\mu = 0.8$ (in the case of ESB we denote the degree of modulation with $\mu$ instead of $m$). The carrier signal is cosinusoidal and the message signal sinusoidal. The locus is a circle. $x_{\rm TP}(t)$ moves in a mathematically positive direction. Because of $\phi(t) \ne \text{const.}$, The envelope demodulation is not applicable here:  This can be seen by the fact that the envelope $a(t)$ is not cosinusoidal. Rather, the lower half-wave is sharper than the upper   ⇒   strong linear distortions.

{{BlaueBox|TEXT=
'''(8)'''   The parameters are still valid according to '''(7)''' with the exception of $A_{\rm O}= 0$ and $A_{\rm U}= 0.8 \text{ V}$. What differences arise opposite '''(7)'''?}}

::Now it is a '''USB–AM''': The red carrier is fixed, the blue pointer is missing and the green pointer (USB) rotates clockwise. All other statements of '''(7)''' apply here as well.

{{BlaueBox|TEXT=
'''(9)'''   The parameters according to '''(7)''' are still valid with the exception of $A_{\rm O} = 0.2 \text{ V} \ne A_{\rm U} = 0.4 \text{ V} $. What are the differences from '''(7)'''?}}

::The locus $x_{\rm TP}(t)$ is not a horizontal straight line, but an ellipse with the real part between $0.4 \text{ V}$ and $1.6 \text{ V}$ and the imaginary part in the range $\pm 0.2 \text{ V}$. Because of $\phi(t) \ne \text{const.}$ , Envelope demodulation would lead to non-linear distortions here too. The constellation simulated here describes the situation of '''(4)''', namely a ZSB–AM with modulation degree $m = 0.8$, where the upper sideband is reduced to $50\%$ due to channel attenuation.

==Applet Manual==
[[File:Ortskurve_abzug3.png|right]]
* The red parameters $(A_{\rm T}, \ f_{\rm T}, \ \varphi_{\rm T})$ and the red pointer mark the ''Carrier''. (German: '''T'''räger).
* The green parameters $(A_{\rm U}, \ f_{\rm U} < f_{\rm T}, \ \varphi_{\rm U})$ mark the ''Lower sideband''. (German: '''U'''ntere Seitenband).
* The blue parameters $(A_{\rm O}, \ f_{\rm O} > f_{\rm T}, \ \varphi_{\rm O})$ mark the '' Upper sideband''. (German: '''O'''bere Seitenband).
* The red pointer does not turn.
* The green pointer rotates in a mathematically negative direction (clockwise).
* The blue pointer turns counterclockwise.

Meaning of the letters in the adjacent graphic:

    '''(A)'''     Plot of the equivalent low-pass Signal $x_{\rm TP}(t)$

    '''(B)'''     Plot of the physical signal $x(t)$

    '''(C)'''     Parameter input via slider:   amplitudes, frequencies, phase values

    '''(D)'''     Control elements:   Start – Step – Pause/Continue – Reset

    '''(E)'''     Speed of animation:   &bdquo;Speed”   ⇒   Values: 1, 2 oder 3

    '''(F)'''     &bdquo;Trace”   ⇒   On or Off, trace of equivalent low-pass Signal   $x_{\rm TP}(t)$

    '''(G)'''     Numerical output:   time $t$, the signal values  ${\rm Re}[x_{\rm TP}(t)]$  and  ${\rm Im}[x_{\rm TP}(t)]$,

$\text{}\hspace{4.2cm}$   Hüllkurve $a(t) = |x_{\rm TP}(t)|$  und  Phase $\phi(t) = {\rm arc} \ x_{\rm TP}(t)$

    '''(H)'''     Variations for the graphical representation

$\hspace{1.5cm}$Zoom–Functions &bdquo;$+$” (Enlarge), &bdquo;$-$” (Decrease) und $\rm o$ (Reset to default)

$\hspace{1.5cm}$Move with &bdquo;$\leftarrow$” (Section to the left, ordinate to the right), &bdquo;$\uparrow$” &bdquo;$\downarrow$” &bdquo;$\rightarrow$”

    '''(I)'''     Experiment section:   Task selection and task

    '''(J)'''     Experiment section:  solution
 

==About the Authors==
This interactive calculation was designed and realized at the [http://www.lnt.ei.tum.de/startseite Lehrstuhl für Nachrichtentechnik] of the [https://www.tum.de/ Technical University of Munich] .
*The original version was created in 2005 by [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Studierende#Ji_Li_.28Bachelorarbeit_EI_2003.2C_Diplomarbeit_EI_2005.29|Ji Li]] as part of her Diploma thesis using &bdquo;FlashMX–Actionscript” (Supervisor: [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Mitarbeiter_und_Dozenten#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28am_LNT_seit_1974.29|Günter Söder]]).
*In 2018 this Applet was redesigned and updated to &bdquo;HTML5” by [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Studierende#Xiaohan_Liu_.28Bachelorarbeit_2018.29|Xiaohan Liu]] as part of her Bachelor's thesis (Supervisor: [[Biografien_und_Bibliografien/Beteiligte_der_Professur_Leitungsgebundene_%C3%9Cbertragungstechnik#Tasn.C3.A1d_Kernetzky.2C_M.Sc._.28bei_L.C3.9CT_seit_2014.29|Tasnád Kernetzky]]).

==Once again: Open Applet in new Tab==

{{LntAppletLinkEn|physAnSignal}}

[[Category:Applets|^Verzerrungen^]]

Applets:Physical Signal & Equivalent Lowpass Signal

2018-08-25T16:37:34Z

Xiaohan: /* Applet Description */

{{LntAppletLinkEn|analPhysSignal}}

==Applet Description==
 
This applet shows the relationship between the physical bandpass signal $x(t)$ and the associated equivalent low-pass signal $x_{\rm TP}(t)$. It is assumed that the bandpass signal $x(t)$ has a frequency-discrete spectrum $X(f)$:
:$$x(t) = x_{\rm U}(t) + x_{\rm T}(t) + x_{\rm O}(t) = A_{\rm U}\cdot \cos\left(2\pi f_{\rm U}\cdot t- \varphi_{\rm U}\right)+A_{\rm T}\cdot \cos\left(2\pi f_{\rm T}\cdot t- \varphi_{\rm T}\right)+A_{\rm O}\cdot \cos\left(2\pi f_{\rm O}\cdot t- \varphi_{\rm O}\right). $$
The physical signal $x(t)$ is thus composed of three harmonic oscillations, a constellation that can be found, for example, in the ''Double-sideband Amplitude Modulation''
*of the message signal $x_{\rm N}(t) = A_{\rm N}\cdot \cos\left(2\pi f_{\rm N}\cdot t- \varphi_{\rm N}\right)$   ⇒   in German:   '''N'''achrichtensignal
*with the carrier signal $x_{\rm T}(t) = A_{\rm T}\cdot \cos\left(2\pi f_{\rm T}\cdot t - \varphi_{\rm T}\right)$   ⇒   in German:   '''T'''rägersignal.

The nomenclature is also adapted to this case:
* $x_{\rm O}(t)$ denotes the &bdquo;upper sideband”   (in German:   '''O'''beres Seitenband) with the amplitude $A_{\rm O}= A_{\rm N}/2$, the frequency $f_{\rm O} = f_{\rm T} + f_{\rm N}$ and the phase $\varphi_{\rm O} = \varphi_{\rm T} + \varphi_{\rm N}$.
*Similarly, for the &bdquo;lower sideband”   (in German:   '''U'''nteres Seitenband) $x_{\rm U}(t)$ with $f_{\rm U} = f_{\rm T} - f_{\rm N}$, $A_{\rm U}= A_{\rm O}$ and $\varphi_{\rm U} = -\varphi_{\rm O}$.

The associated equivalent low-pass–signal is $f_{\rm O}\hspace{0.01cm}' = f_{\rm O}- f_{\rm T} > 0$,   $f_{\rm U}\hspace{0.01cm}' = f_{\rm U}- f_{\rm T} < 0$  and  $f_{\rm T}\hspace{0.01cm}' = 0$:

:$$x_{\rm TP}(t) = x_\text{TP, T}(t) + x_\text{TP, O}(t) + x_\text{TP, U}(t) = A_{\rm T}\cdot {\rm e}^{-{\rm j} \varphi_{\rm T} } \hspace{0.1cm}+ \hspace{0.1cm} A_{\rm O}\cdot {\rm e}^{-{\rm j} \varphi_{\rm O} } \cdot {\rm e}^{ {\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm O}\hspace{0.01cm}'\hspace{0.05cm}\cdot \hspace{0.05cm}t}\hspace{0.1cm}+ \hspace{0.1cm}
A_{\rm U}\cdot {\rm e}^{-{\rm j} \varphi_{\rm U} } \cdot {\rm e}^{ {\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm U}\hspace{0.01cm}'\hspace{0.05cm}\cdot \hspace{0.05cm}t} . $$

[[File:Ortskurve_1.png|right|frame|Equivalent lowpass–signal currently $t=0$ for cosinusoidal carrier   ⇒   $\varphi_{\rm T} = 0$]]
The program shows $x_{\rm TP}(t)$ as the vectorial sum of three rotation pointers as a violet dot (see figure for start time $t=0$ and cosinusoidal carrier):

*The (red) pointer of the carrier $x_\text{TP, T}(t)$ with the length $A_{\rm T}$ and the zero phase position $\varphi_{\rm T}=0$ is fixed in the complex plane. So it applies to all times $t$:   $x_{\rm TP}(t)= A_{\rm T}\cdot {\rm e}^{-{\rm j} \varphi_{\rm T} }$.

*The (blue) pointer of the upper sideband $x_\text{TP, O}(t)$ with the length $A_{\rm O}$ and the zero phase position $\varphi_{\rm O}$ rotates at the angular velocity $2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm O}\hspace{0.01cm}'$ in mathematically positive direction (one revolution in time $1/f_{\rm O}\hspace{0.01cm}')$.

*The (green) pointer of the lower sideband $x_{\rm U+}(t)$ with the length $A_{\rm U}$ and the zero phase position $\varphi_{\rm U}$ rotates at the angular velocity $2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm U}\hspace{0.01cm}'$, because of $f_{\rm U}\hspace{0.01cm}'<0$ counterclockwise.

*With $f_{\rm U}\hspace{0.01cm}' = -f_{\rm O}\hspace{0.01cm}'$ the blue and green pointers will spin at the same speed but in different directions. Also, if $A_{\rm O} = A_{\rm U}$ and $\varphi_{\rm U} = -\varphi_{\rm O}$, then $x_{\rm TP}(t)$ moves on a straight line with a incline of $\varphi_{\rm T}$.

''Note:''   In the figure $\varphi_{\rm O} = +30^\circ$. From this follows for the start time $t=0$ the angle of the blue pointer (OSB) with respect to the coordinate system:   $\phi_{\rm O} = -\varphi_{\rm O} = -30^\circ$. Likewise, the null phase position $\varphi_{\rm U} = -30^\circ$ of the lower sideband (USB, grüner Zeiger) follows for the phase angle to be considered in the complex plane:   $\phi_{\rm U} = +30^\circ$.

The temporal process of $x_{\rm TP}(t)$ is also referred to below as '''locus'''. The relationship between $x_{\rm TP}(t)$ and the physical bandpass –signal $x(t)$ is given in the section and the associated analytic signal is $x_+(t)$ :

:$$x_{\rm TP}(t) = x_{\rm +}(t)\cdot {\rm e}^{-{\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm}2 \pi \cdot f_{\rm T}\cdot \hspace{0.05cm}t},$$
:$$x_{\rm +}(t) = x_{\rm TP}(t)\cdot {\rm e}^{+{\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm}2 \pi \cdot f_{\rm T}\cdot \hspace{0.05cm}t}.$$

[[Applets:Physikalisches_Signal_%26_Äquivalentes_TP-Signal|'''German description''']]

==Theoretical Background==
 
===Description of Bandpass Signals===
[[File:Zeigerdiagramm_1a.png|right|frame|bandpass spectrum $X(f)$ |class=fit]]
We consider '''bandpass signals''' $x(t)$ with the property that their spectra $X(f)$ are not in the range around the frequency $f=0$, but around a carrier frequency $f_{\rm T}$. In most cases it can also be assumed that the bandwidth is $B \ll f_{\rm T}$.

The figure shows such a bandpass spectrum $X(f)$. Assuming that the associated $x(t)$ is a physical signal and thus real, the spectral function $X(f)$ has a symmetry with respect to the frequency $f = 0$, if $x(t)$ is an even function   ⇒   $x(-t)=x(t)$, $X(f)$ is real and even.

Beside the physical signal $x(t)\ \circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\, \ X(f)$, one can also use the following descriptions of bandpass signals:
*the analytic signal $x_+(t)\ \circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\, \ X_+(f)$, see applet [[Applets:Physikalisches_Signal_%26_Analytisches_Signal|Physikalisches Signal & Analytisches Signal]],
*the equivalent low-pass–signal $x_{\rm TP}(t)\ \circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\, \ X_{\rm TP}(f)$, see next page
 

===Spectral Functions of the Analytic and the Equivalent Low-Pass Signal===

The '''analytic signal''' $x_+(t)$ belonging to the physical signal $x(t)$ is the time function whose spectrum fulfills the following property:
[[File:Ortskurve_2.png|right|frame|spectral functions $X_+(f)$ and $X_{\rm TP}(f)$ |class=fit]]
:$$X_+(f)=\big[1+{\rm sign}(f)\big] \cdot X(f) = \left\{ {2 \cdot
X(f) \; \hspace{0.2cm}\rm for\hspace{0.2cm} {\it f} > 0, \atop {\,\,\,\, \rm 0 \; \hspace{0.9cm}\rm for\hspace{0.2cm} {\it f} < 0.} }\right.$$

The ''Signum function'' is for positive values of $f$ equal to $+1$ and for negative $f$ values equal to $-1$.
*The (double-sided) limit returns $\sign(0) = 0$.
*The index „+” should make it clear that $X_+(f)$ only has parts at positive frequencies.

From the graph you can see the calculation rule for $X_+(f)$: The actual BP spectrum $X(f)$ becomes
*doubled at the positive frequencies, and
*set to zero at the negative frequencies.

Due to the asymmetry of $X_+(f)$ with respect to the frequency $f = 0$, it can already be said that the time function $x_+(t)$ except for a trivial special case $x_+(t)= 0 \ \ \circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\,\ \ X_+(f)= 0$ is always complex.

The spectrum $X_{\rm TP}(f)$ of the equivalent low-pass signal is obtained by shifting $X_+(f)$ to the left by the carrier frequency $f_{\rm T}$:
:$$X_{\rm TP}(f)= X_+(f+f_{\rm T}).$$

In the time domain this operation corresponds to the multiplication of $x_{\rm +}(t)$ with the complex exponential function with negative exponent:
:$$x_{\rm TP}(t) = x_{\rm +}(t)\cdot {\rm e}^{-{\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm}2 \pi \cdot f_{\rm T}\cdot \hspace{0.05cm}t}.$$

It can be seen that $x_{\rm TP}(t)$ is generally complex. But if $X_+(f)$ is symmetric about the carrier frequency $f_{\rm T}$, $X_{\rm TP}(f)$ is symmetric about the frequency $f=0$ and there is accordingly real time function $x_{\rm TP}(t)$.

===$x_{\rm TP}(t)$–Representation of a Sum of Three Harmonic Oscillations===

In our applet, we always assume a set of three rotating pointers. The physical signal is:
:$$x(t) = x_{\rm U}(t) + x_{\rm T}(t) + x_{\rm O}(t) = A_{\rm U}\cdot \cos\left(2\pi f_{\rm U}\cdot t- \varphi_{\rm U}\right)+A_{\rm T}\cdot \cos\left(2\pi f_{\rm T}\cdot t- \varphi_{\rm T}\right)+A_{\rm O}\cdot \cos\left(2\pi f_{\rm O}\cdot t- \varphi_{\rm O}\right). $$
* Each of the three harmonic oscillations $x_{\rm T}(t)$, $x_{\rm U}(t)$ and $x_{\rm O}(t)$ is represented by an amplitude $(A)$, a frequency $(f)$ and a phase value $(\varphi)$.
*The indices are based on the modulation method [[Modulationsverfahren/Zweiseitenband-Amplitudenmodulation|double sideband –amplitude modulation]]. &bdquo;T” stands for &bdquo;carrier”, &bdquo;U” for &bdquo;lowei sideband” and &bdquo;O” for &bdquo;upper Sideband”. Similarly, $f_{\rm U} < f_{\rm T}$ and $f_{\rm O} > f_{\rm T}$. There are no restrictions for the amplitudes and phases.

The associated equivalent low-pass signal is with $f_{\rm O}\hspace{0.01cm}' = f_{\rm O}- f_{\rm T} > 0$,   $f_{\rm U}\hspace{0.01cm}' = f_{\rm U}- f_{\rm T} < 0$  and  $f_{\rm T}\hspace{0.01cm}' = 0$:

:$$x_{\rm TP}(t) = x_\text{TP, T}(t) + x_\text{TP, O}(t) + x_\text{TP, U}(t) = A_{\rm T}\cdot {\rm e}^{-{\rm j} \varphi_{\rm T} } \hspace{0.1cm}+ \hspace{0.1cm} A_{\rm O}\cdot {\rm e}^{-{\rm j} \varphi_{\rm O} } \cdot {\rm e}^{ {\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm O}\hspace{0.01cm}'\hspace{0.05cm}\cdot \hspace{0.05cm}t}\hspace{0.1cm}+ \hspace{0.1cm}
A_{\rm U}\cdot {\rm e}^{-{\rm j} \varphi_{\rm U} } \cdot {\rm e}^{ {\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm U}\hspace{0.01cm}'\hspace{0.05cm}\cdot \hspace{0.05cm}t} . $$

{{GraueBox|TEXT=
$\text{Example 1:}$ 
The constellation given here results, for example, in the [[Modulationsverfahren/Zweiseitenband-Amplitudenmodulation#AM-Signale_und_-Spektren_bei_harmonischem_Eingangssignal|double sideband amplitude modulation]] of the message signal $x_{\rm N}(t) = A_{\rm N}\cdot \cos\left(2\pi f_{\rm N}\cdot t- \varphi_{\rm N}\right)$ with the carrier signal $x_{\rm T}(t) = A_{\rm T}\cdot \cos\left(2\pi f_{\rm T}\cdot t - \varphi_{\rm T}\right)$. This is discussed frequently in the exercises.

[[File:Ortskurve_4.png|center|frame|Spectrum $X_{\rm TP}(f)$ of the equivalent low–pass signal for different phase constellations |class=fit]]

There are some limitations to the program parameters in this approach:
* The frequencies are always $f\hspace{0.05cm}'_{\rm O} = f_{\rm N}$ and $f\hspace{0.05cm}'_{\rm U} = -f_{\rm N}$.
*Without distortion, the amplitude of the sidebands is $A_{\rm O}= A_{\rm U}= A_{\rm N}/2$.
*The respective phase relationships can be seen from the graphic.

}}

===Representation of the Equivalent Low-Pass Signal by Magnitude and Phase===

The generally complex equivalent low-pass signal
:$$x_{\rm TP}(t) = a(t) \cdot {\rm e}^{ {\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm} \phi(t) }$$
can be split into a magnitude function $a(t)$ and a phase function $\phi(t)$ according to the equation given here, where:
:$$a(t) = \vert x_{\rm TP}(t)\vert = \sqrt{ {\rm Re}^2\big [x_{\rm TP}(t)\big ] + {\rm Im}^2\big [x_{\rm TP}(t)\big ] }\hspace{0.05cm},$$
:$$\phi(t) = \text{arc }x_{\rm TP}(t) = \arctan \frac{{\rm Im}\big [x_{\rm TP}(t)\big ]}{{\rm Re}\big [x_{\rm TP}(t)\big ]}.$$

The reason for this is that a bandpass–signal $x(t)$ is usually described by the equivalent low-pass signal $x_{\rm TP}(t)$ that the functions $a(t)$ and $\phi(t)$ are interpretable in both representations:
*The amount $a(t)$ of the equivalent low-pass signal $x_{\rm TP}(t)$ indicates the (time-dependent) envelope of $x(t)$.
*The phase $\phi(t)$ of $x_{\rm TP}(t)$ denotes the location of the zero crossings of $x(t)$, where:
:–   For $\phi(t)>0$ the zero crossing is earlier than its nominal position   ⇒   the signal is leading here.
:–  When $\phi(t)<0$, the zero crossing is later than its target position   ⇒   the signal is trailing here.

{{GraueBox|TEXT=
$\text{Example 2:}$ 
The graph is intended to illustrate this relationship, assuming $A_{\rm U} > A_{\rm O}$   ⇒   the green pointer (for the lower sideband) is longer than the blue pointer (upper sideband). This is a snapshot at time $t_0$:

[[File:Ortskurve_3_neu.png|center|frame|bandpass spectrum $X(f)$ |class=fit]]

*For these system parameters, the top of the pointer cluster $x_{\rm TP}(t)$ – that is, the geometric sum of red, blue and green pointers – on an ellipse.
* The amount $a(t_0) = \vert x_{\rm TP}(t_0) \vert$ is drawn in black in the left-hand diagram and the phase value $\phi(t_0) = \text{arc }x_{\rm TP}(t_0) > 0$ is indicated in brown color.
*In the graph on the right, the amount $a(t_0) = \vert x_{\rm TP}(t_0) \vert$ of the equivalent low-pass signal indicates the envelope of the physical signal $x(t)$.
* At $\phi(t) \equiv 0$, all zero crossings of $x(t)$ would occur at equidistant intervals. Because of $\phi(t_0) > 0$, the signal is leading at the time $t_0$, that is: the zero crossings come earlier than the grid dictates. }}

==Exercises==
[[File:Exercises_verzerrungen.png|right]]
*First select the task number.
*A task description is displayed.
*Parameter values are adjusted.
*Solution after pressing &bdquo;Hide solition”.

The number &bdquo;0” will reset the program and output a text with the further explanation of the applet.

In the following, $\rm Green$ denotes the lower sideband   ⇒   $\big (A_{\rm U}, f_{\rm U}, \varphi_{\rm U}\big )$,  
$\rm Red$ the carrier   ⇒   $\big (A_{\rm T}, f_{\rm T}, \varphi_{\rm T}\big )$ and
$\rm Blue$ the upper sideband   ⇒   $\big (A_{\rm O}, f_{\rm O}, \varphi_{\rm O}\big )$.

{{BlaueBox|TEXT=
'''(1)'''   Let   $\text{Red:} \hspace{0.15cm} A_{\rm T} = 1\ \text{V}, f_{\rm T} = 100 \ \text{kHz}, \varphi_{\rm T} = 0^\circ$,   $\text{Green:} \hspace{0.15cm} A_{\rm U} = 0.4 \text{V}, \ f_{\rm U} = 80 \ \text{kHz}, \varphi_{\rm U} = -90^\circ$,   $\text{Blue:} \hspace{0.15cm} A_{\rm O} = 0.4\ \text{V}, f_{\rm O} = 120 \ \text{kHz}, \varphi_{\rm O} = 90^\circ$.

:Consider and interpret the equivalent low-pass signal $x_{\rm TP}(t)$ and the physical signal $x(t)$. Which period $T_0$ does one recognize?}}

:: The equivalent low-pass signal $x_{\rm TP}(t)$ takes from $x_{\rm TP}(t=0)=1\ \text{V}$ on the real axis values between $0.2\ \text{V}$ and $1.8\ \text{V}$   ⇒   phase $\phi(t) \equiv 0$.  The amount $|x_{\rm TP}(t)|$ indicates the envelope $a(t)$ of the physical signal $x(t)$. It holds $A_{\rm N} = 0.8\ \text{V}$ and $f_{\rm N} = 20\ \text{kHz}$:   $a(t) = A_{\rm T}+ A_{\rm N} \cdot \sin(2\pi\cdot f_{\rm N} \cdot t)$.  Both $x_{\rm TP}(t)$ and $x(t)$ are periodic with the period $T_0 = 1/f_{\rm N} = 50\ \rm µ s$.

{{BlaueBox|TEXT=
'''(2)'''   How do the ratios change to '''(1)''' with $f_{\rm U} = 99 \ \text{kHz}$ and $f_{\rm O} = 101 \ \text{kHz}$ ? How could $x(t)$ have arisen?}}

:: For the envelope $a(t)$ of the signal $x(t)$ we still have $a(t) = A_{\rm T}+ A_{\rm N} \cdot \sin(2\pi\cdot f_{\rm N} \cdot t)$, but now $f_{\rm N} = 1\ \text{kHz}$. Even though it can not be recognized:  $x_{\rm TP}(t)$ and $x(t)$ are still periodic:   $T_0 = 1\ \rm ms$. Example: Double sideband - Amplitude modulation '''(ZSB–AM)''' of a sine signal with cosine–carrier.

{{BlaueBox|TEXT=
'''(3)'''   Which settings have to be changed from '''(2)''' in order to arrive at the ZSB–AM of a cosine signal with sine–carrier. What changes over '''(2)'''?}}

::The carrier phase must be changed to $\varphi_{\rm T} = 90^\circ$   ⇒   sine–carrier. Similarly, $\varphi_{\rm O} =\varphi_{\rm U} =\varphi_{\rm T} = 90^\circ$ must be set   ⇒   cosine-shaped message  The locus now lies on the imaginary axis  ⇒   $\phi(t) \equiv -90^\circ$. At the beginning we have $x_{\rm TP}(t=0)= - {\rm j} \cdot 1.8 \ \text{V}$.

{{BlaueBox|TEXT=
'''(4)'''   Now let   $\text{Red:} \hspace{0.15cm} A_{\rm T} = 1\ \text{V}, \ f_{\rm T} = 100 \ \text{kHz}, \ \varphi_{\rm T} = 0^\circ$,   $\text{Green:} \hspace{0.15cm} A_{\rm U} = 0.4 \text{V}, \ f_{\rm U} = 80 \ \text{kHz}, \ \varphi_{\rm U} = 0^\circ$,   $\text{Blue:} \hspace{0.15cm} A_{\rm O} = 0.4\ \text{V}, \ f_{\rm O} = 120 \ \text{kHz}, \ \varphi_{\rm O} = 0^\circ$.

:What are the characteristics of this system &bdquo;ZSB–AM, where the message signal and carrier are respectively cosinusoidal”? What is the modulation depth $m$?}}

:: The equivalent low-pass signal $x_{\rm TP}(t)$ takes from $x_{\rm TP}(t=0)=1.8\ \text{V}$ on the real axis values between $0.2\ \text{V}$ and $1.8\ \text{V}$   ⇒   phase $\phi(t) \equiv 0$.  Except for the start state $x_{\rm TP}(t=0)$ same behavior as with the setting '''(1)'''. The modulation depth is $m = 0.8$.

{{BlaueBox|TEXT=
'''(5)'''   The parameters are still valid according to '''(4)''' with the exception of $A_{\rm T}= 0.6 \text{V}$. How big is the degree of modulation $m$? What are the consequences?}}

:: There is now a ZSB–AM with modulation degree $m = 1.333$. For $m > 1$, the simpler [[Modulationsverfahren/Hüllkurvendemodulation|envelope demodulation]] is not applicable, since the phase function $\phi(t) \in \{ 0, \ \pm 180^\circ\}$ is no longer constant and the envelope $a(t)$ no longer matches the message signal. Rather, the more complex [[Modulationsverfahren/Synchrondemodulation|synchronous demodulation]] must be used. Envelope detection would produce nonlinear distortions.

{{BlaueBox|TEXT=
'''(6)'''   The parameters are still valid according to '''(4)''' or '''(5)''' with the exception from $A_{\rm T}= 0$ on   ⇒   $m \to \infty$. Which modulation method is described in this way?}}

::It is a '''ZSB–AM without carrier''' and requires synchronous demodulation. The equivalent low-pass signal $x_{\rm TP}(t)$ is on the real axis, but not only in the right half-plane. Thus, the phase function $\phi(t) \in \{ 0, \ \pm 180^\circ\}$, also applies here, which means that envelope demodulation is not applicable.

{{BlaueBox|TEXT=
'''(7)'''   Now let   $\text{Red:} \hspace{0.15cm} A_{\rm T} = 1\ \text{V}, f_{\rm T} = 100 \ \text{kHz}, \varphi_{\rm T} = 0^\circ$,   $\text{Green:} \hspace{0.15cm} A_{\rm U} = 0, \ f_{\rm U} = 80 \ \text{kHz}, \varphi_{\rm U} = -90^\circ$,   $\text{Blue:} \hspace{0.15cm} A_{\rm O} = 0.8\ \text{V}, f_{\rm O} = 120 \ \text{kHz}, \varphi_{\rm O} = 90^\circ$.

:Which constellation is described here? Which characteristics of this procedure can be recognized from the graphic?}}

::It is a [[Modulationsverfahren/Einseitenbandmodulation|single-sideband modulation]] '''(ESB–AM)''', more specifically an '''OSB–AM''': the red carrier is fixed, the green pointer missing and the blue pointer (OSB) turns counterclockwise. The degree of modulation is $\mu = 0.8$ (in the case of ESB we denote the degree of modulation with $\mu$ instead of $m$). The carrier signal is cosinusoidal and the message signal sinusoidal. The locus is a circle. $x_{\rm TP}(t)$ moves in a mathematically positive direction. Because of $\phi(t) \ne \text{const.}$, The envelope demodulation is not applicable here:  This can be seen by the fact that the envelope $a(t)$ is not cosinusoidal. Rather, the lower half-wave is sharper than the upper   ⇒   strong linear distortions.

{{BlaueBox|TEXT=
'''(8)'''   The parameters are still valid according to '''(7)''' with the exception of $A_{\rm O}= 0$ and $A_{\rm U}= 0.8 \text{ V}$. What differences arise opposite '''(7)'''?}}

::Now it is a '''USB–AM''': The red carrier is fixed, the blue pointer is missing and the green pointer (USB) rotates clockwise. All other statements of '''(7)''' apply here as well.

{{BlaueBox|TEXT=
'''(9)'''   The parameters according to '''(7)''' are still valid with the exception of $A_{\rm O} = 0.2 \text{ V} \ne A_{\rm U} = 0.4 \text{ V} $. What are the differences from '''(7)'''?}}

::The locus $x_{\rm TP}(t)$ is not a horizontal straight line, but an ellipse with the real part between $0.4 \text{ V}$ and $1.6 \text{ V}$ and the imaginary part in the range $\pm 0.2 \text{ V}$. Because of $\phi(t) \ne \text{const.}$ , Envelope demodulation would lead to non-linear distortions here too. The constellation simulated here describes the situation of '''(4)''', namely a ZSB–AM with modulation degree $m = 0.8$, where the upper sideband is reduced to $50\%$ due to channel attenuation.

==Applet Manual==
[[File:Ortskurve_abzug3.png|right]]
* The red parameters $(A_{\rm T}, \ f_{\rm T}, \ \varphi_{\rm T})$ and the red pointer mark the ''Carrier''. (German: '''T'''räger).
* The green parameters $(A_{\rm U}, \ f_{\rm U} < f_{\rm T}, \ \varphi_{\rm U})$ mark the ''Lower sideband''. (German: '''U'''ntere Seitenband).
* The blue parameters $(A_{\rm O}, \ f_{\rm O} > f_{\rm T}, \ \varphi_{\rm O})$ mark the '' Upper sideband''. (German: '''O'''bere Seitenband).
* The red pointer does not turn.
* The green pointer rotates in a mathematically negative direction (clockwise).
* The blue pointer turns counterclockwise.

Meaning of the letters in the adjacent graphic:

    '''(A)'''     Plot of the equivalent low-pass Signal $x_{\rm TP}(t)$

    '''(B)'''     Plot of the physical signal $x(t)$

    '''(C)'''     Parameter input via slider:   amplitudes, frequencies, phase values

    '''(D)'''     Control elements:   Start – Step – Pause/Continue – Reset

    '''(E)'''     Speed of animation:   &bdquo;Speed”   ⇒   Values: 1, 2 oder 3

    '''(F)'''     &bdquo;Trace”   ⇒   On or Off, trace of equivalent low-pass Signal   $x_{\rm TP}(t)$

    '''(G)'''     Numerical output:   time $t$, the signal values  ${\rm Re}[x_{\rm TP}(t)]$  and  ${\rm Im}[x_{\rm TP}(t)]$,

$\text{}\hspace{4.2cm}$   Hüllkurve $a(t) = |x_{\rm TP}(t)|$  und  Phase $\phi(t) = {\rm arc} \ x_{\rm TP}(t)$

    '''(H)'''     Variations for the graphical representation

$\hspace{1.5cm}$Zoom–Functions &bdquo;$+$” (Enlarge), &bdquo;$-$” (Decrease) und $\rm o$ (Reset to default)

$\hspace{1.5cm}$Move with &bdquo;$\leftarrow$” (Section to the left, ordinate to the right), &bdquo;$\uparrow$” &bdquo;$\downarrow$” &bdquo;$\rightarrow$”

    '''(I)'''     Experiment section:   Task selection and task

    '''(J)'''     Experiment section:  solution
 

==About the Authors==
This interactive calculation was designed and realized at the [http://www.lnt.ei.tum.de/startseite Lehrstuhl für Nachrichtentechnik] of the [https://www.tum.de/ Technical University of Munich] .
*The original version was created in 2005 by [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Studierende#Ji_Li_.28Bachelorarbeit_EI_2003.2C_Diplomarbeit_EI_2005.29|Ji Li]] as part of her Diploma thesis using &bdquo;FlashMX–Actionscript” (Supervisor: [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Mitarbeiter_und_Dozenten#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28am_LNT_seit_1974.29|Günter Söder]]).
*In 2018 this Applet was redesigned and updated to &bdquo;HTML5” by [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Studierende#Xiaohan_Liu_.28Bachelorarbeit_2018.29|Xiaohan Liu]] as part of her Bachelor's thesis (Supervisor: [[Biografien_und_Bibliografien/Beteiligte_der_Professur_Leitungsgebundene_%C3%9Cbertragungstechnik#Tasn.C3.A1d_Kernetzky.2C_M.Sc._.28bei_L.C3.9CT_seit_2014.29|Tasnád Kernetzky]]).

==Once again: Open Applet in new Tab==

{{LntAppletLinkEn|physAnSignal}}

[[Category:Applets|^Verzerrungen^]]

Applets:Physical Signal & Equivalent Lowpass Signal

2018-08-25T16:30:30Z

Xiaohan: /* Applet Description */

{{LntAppletLinkEn|analPhysSignal}}

==Applet Description==
 
This applet shows the relationship between the physical bandpass signal $x(t)$ and the associated analytic signal $x_+(t)$. It is assumed that the bandpass signal $x(t)$ has a frequency-discrete spectrum $X(f)$:
:$$x(t) = x_{\rm U}(t) + x_{\rm T}(t) + x_{\rm O}(t) = A_{\rm U}\cdot \cos\left(2\pi f_{\rm U}\cdot t- \varphi_{\rm U}\right)+A_{\rm T}\cdot \cos\left(2\pi f_{\rm T}\cdot t- \varphi_{\rm T}\right)+A_{\rm O}\cdot \cos\left(2\pi f_{\rm O}\cdot t- \varphi_{\rm O}\right). $$
The physical signal $x(t)$ is thus composed of three harmonic oscillations, a constellation that can be found, for example, in the ''Double-sideband Amplitude Modulation''
*of the message signal $x_{\rm N}(t) = A_{\rm N}\cdot \cos\left(2\pi f_{\rm N}\cdot t- \varphi_{\rm N}\right)$   ⇒   in German:   '''N'''achrichtensignal
*with the carrier signal $x_{\rm T}(t) = A_{\rm T}\cdot \cos\left(2\pi f_{\rm T}\cdot t - \varphi_{\rm T}\right)$   ⇒   in German:   '''T'''rägersignal.

The nomenclature is also adapted to this case:
* $x_{\rm O}(t)$ denotes the &bdquo;upper sideband”   (in German:   '''O'''beres Seitenband) with the amplitude $A_{\rm O}= A_{\rm N}/2$, the frequency $f_{\rm O} = f_{\rm T} + f_{\rm N}$ and the phase $\varphi_{\rm O} = \varphi_{\rm T} + \varphi_{\rm N}$.
*Similarly, for the &bdquo;lower sideband”   (in German:   '''U'''nteres Seitenband) $x_{\rm U}(t)$ with $f_{\rm U} = f_{\rm T} - f_{\rm N}$, $A_{\rm U}= A_{\rm O}$ and $\varphi_{\rm U} = -\varphi_{\rm O}$.

The associated equivalent low-pass–signal is $f_{\rm O}\hspace{0.01cm}' = f_{\rm O}- f_{\rm T} > 0$,   $f_{\rm U}\hspace{0.01cm}' = f_{\rm U}- f_{\rm T} < 0$  and  $f_{\rm T}\hspace{0.01cm}' = 0$:

:$$x_{\rm TP}(t) = x_\text{TP, T}(t) + x_\text{TP, O}(t) + x_\text{TP, U}(t) = A_{\rm T}\cdot {\rm e}^{-{\rm j} \varphi_{\rm T} } \hspace{0.1cm}+ \hspace{0.1cm} A_{\rm O}\cdot {\rm e}^{-{\rm j} \varphi_{\rm O} } \cdot {\rm e}^{ {\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm O}\hspace{0.01cm}'\hspace{0.05cm}\cdot \hspace{0.05cm}t}\hspace{0.1cm}+ \hspace{0.1cm}
A_{\rm U}\cdot {\rm e}^{-{\rm j} \varphi_{\rm U} } \cdot {\rm e}^{ {\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm U}\hspace{0.01cm}'\hspace{0.05cm}\cdot \hspace{0.05cm}t} . $$

[[File:Ortskurve_1.png|right|frame|Equivalent lowpass–signal currently $t=0$ for cosinusoidal carrier   ⇒   $\varphi_{\rm T} = 0$]]
The program shows $x_{\rm TP}(t)$ as the vectorial sum of three rotation pointers as a violet dot (see figure for start time $t=0$ and cosinusoidal carrier):

*The (red) pointer of the carrier $x_\text{TP, T}(t)$ with the length $A_{\rm T}$ and the zero phase position $\varphi_{\rm T}=0$ is fixed in the complex plane. So it applies to all times $t$:   $x_{\rm TP}(t)= A_{\rm T}\cdot {\rm e}^{-{\rm j} \varphi_{\rm T} }$.

*The (blue) pointer of the upper sideband $x_\text{TP, O}(t)$ with the length $A_{\rm O}$ and the zero phase position $\varphi_{\rm O}$ rotates at the angular velocity $2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm O}\hspace{0.01cm}'$ in mathematically positive direction (one revolution in time $1/f_{\rm O}\hspace{0.01cm}')$.

*The (green) pointer of the lower sideband $x_{\rm U+}(t)$ with the length $A_{\rm U}$ and the zero phase position $\varphi_{\rm U}$ rotates at the angular velocity $2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm U}\hspace{0.01cm}'$, because of $f_{\rm U}\hspace{0.01cm}'<0$ counterclockwise.

*With $f_{\rm U}\hspace{0.01cm}' = -f_{\rm O}\hspace{0.01cm}'$ the blue and green pointers will spin at the same speed but in different directions. Also, if $A_{\rm O} = A_{\rm U}$ and $\varphi_{\rm U} = -\varphi_{\rm O}$, then $x_{\rm TP}(t)$ moves on a straight line with a incline of $\varphi_{\rm T}$.

''Note:''   In the figure $\varphi_{\rm O} = +30^\circ$. From this follows for the start time $t=0$ the angle of the blue pointer (OSB) with respect to the coordinate system:   $\phi_{\rm O} = -\varphi_{\rm O} = -30^\circ$. Likewise, the null phase position $\varphi_{\rm U} = -30^\circ$ of the lower sideband (USB, grüner Zeiger) follows for the phase angle to be considered in the complex plane:   $\phi_{\rm U} = +30^\circ$.

The temporal process of $x_{\rm TP}(t)$ is also referred to below as '''locus'''. The relationship between $x_{\rm TP}(t)$ and the physical bandpass –signal $x(t)$ is given in the section and the associated analytic signal is $x_+(t)$ :

:$$x_{\rm TP}(t) = x_{\rm +}(t)\cdot {\rm e}^{-{\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm}2 \pi \cdot f_{\rm T}\cdot \hspace{0.05cm}t},$$
:$$x_{\rm +}(t) = x_{\rm TP}(t)\cdot {\rm e}^{+{\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm}2 \pi \cdot f_{\rm T}\cdot \hspace{0.05cm}t}.$$

[[Applets:Physikalisches_Signal_%26_Äquivalentes_TP-Signal|'''German description''']]

==Theoretical Background==
 
===Description of Bandpass Signals===
[[File:Zeigerdiagramm_1a.png|right|frame|bandpass spectrum $X(f)$ |class=fit]]
We consider '''bandpass signals''' $x(t)$ with the property that their spectra $X(f)$ are not in the range around the frequency $f=0$, but around a carrier frequency $f_{\rm T}$. In most cases it can also be assumed that the bandwidth is $B \ll f_{\rm T}$.

The figure shows such a bandpass spectrum $X(f)$. Assuming that the associated $x(t)$ is a physical signal and thus real, the spectral function $X(f)$ has a symmetry with respect to the frequency $f = 0$, if $x(t)$ is an even function   ⇒   $x(-t)=x(t)$, $X(f)$ is real and even.

Beside the physical signal $x(t)\ \circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\, \ X(f)$, one can also use the following descriptions of bandpass signals:
*the analytic signal $x_+(t)\ \circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\, \ X_+(f)$, see applet [[Applets:Physikalisches_Signal_%26_Analytisches_Signal|Physikalisches Signal & Analytisches Signal]],
*the equivalent low-pass–signal $x_{\rm TP}(t)\ \circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\, \ X_{\rm TP}(f)$, see next page
 

===Spectral Functions of the Analytic and the Equivalent Low-Pass Signal===

The '''analytic signal''' $x_+(t)$ belonging to the physical signal $x(t)$ is the time function whose spectrum fulfills the following property:
[[File:Ortskurve_2.png|right|frame|spectral functions $X_+(f)$ and $X_{\rm TP}(f)$ |class=fit]]
:$$X_+(f)=\big[1+{\rm sign}(f)\big] \cdot X(f) = \left\{ {2 \cdot
X(f) \; \hspace{0.2cm}\rm for\hspace{0.2cm} {\it f} > 0, \atop {\,\,\,\, \rm 0 \; \hspace{0.9cm}\rm for\hspace{0.2cm} {\it f} < 0.} }\right.$$

The ''Signum function'' is for positive values of $f$ equal to $+1$ and for negative $f$ values equal to $-1$.
*The (double-sided) limit returns $\sign(0) = 0$.
*The index „+” should make it clear that $X_+(f)$ only has parts at positive frequencies.

From the graph you can see the calculation rule for $X_+(f)$: The actual BP spectrum $X(f)$ becomes
*doubled at the positive frequencies, and
*set to zero at the negative frequencies.

Due to the asymmetry of $X_+(f)$ with respect to the frequency $f = 0$, it can already be said that the time function $x_+(t)$ except for a trivial special case $x_+(t)= 0 \ \ \circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\,\ \ X_+(f)= 0$ is always complex.

The spectrum $X_{\rm TP}(f)$ of the equivalent low-pass signal is obtained by shifting $X_+(f)$ to the left by the carrier frequency $f_{\rm T}$:
:$$X_{\rm TP}(f)= X_+(f+f_{\rm T}).$$

In the time domain this operation corresponds to the multiplication of $x_{\rm +}(t)$ with the complex exponential function with negative exponent:
:$$x_{\rm TP}(t) = x_{\rm +}(t)\cdot {\rm e}^{-{\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm}2 \pi \cdot f_{\rm T}\cdot \hspace{0.05cm}t}.$$

It can be seen that $x_{\rm TP}(t)$ is generally complex. But if $X_+(f)$ is symmetric about the carrier frequency $f_{\rm T}$, $X_{\rm TP}(f)$ is symmetric about the frequency $f=0$ and there is accordingly real time function $x_{\rm TP}(t)$.

===$x_{\rm TP}(t)$–Representation of a Sum of Three Harmonic Oscillations===

In our applet, we always assume a set of three rotating pointers. The physical signal is:
:$$x(t) = x_{\rm U}(t) + x_{\rm T}(t) + x_{\rm O}(t) = A_{\rm U}\cdot \cos\left(2\pi f_{\rm U}\cdot t- \varphi_{\rm U}\right)+A_{\rm T}\cdot \cos\left(2\pi f_{\rm T}\cdot t- \varphi_{\rm T}\right)+A_{\rm O}\cdot \cos\left(2\pi f_{\rm O}\cdot t- \varphi_{\rm O}\right). $$
* Each of the three harmonic oscillations $x_{\rm T}(t)$, $x_{\rm U}(t)$ and $x_{\rm O}(t)$ is represented by an amplitude $(A)$, a frequency $(f)$ and a phase value $(\varphi)$.
*The indices are based on the modulation method [[Modulationsverfahren/Zweiseitenband-Amplitudenmodulation|double sideband –amplitude modulation]]. &bdquo;T” stands for &bdquo;carrier”, &bdquo;U” for &bdquo;lowei sideband” and &bdquo;O” for &bdquo;upper Sideband”. Similarly, $f_{\rm U} < f_{\rm T}$ and $f_{\rm O} > f_{\rm T}$. There are no restrictions for the amplitudes and phases.

The associated equivalent low-pass signal is with $f_{\rm O}\hspace{0.01cm}' = f_{\rm O}- f_{\rm T} > 0$,   $f_{\rm U}\hspace{0.01cm}' = f_{\rm U}- f_{\rm T} < 0$  and  $f_{\rm T}\hspace{0.01cm}' = 0$:

:$$x_{\rm TP}(t) = x_\text{TP, T}(t) + x_\text{TP, O}(t) + x_\text{TP, U}(t) = A_{\rm T}\cdot {\rm e}^{-{\rm j} \varphi_{\rm T} } \hspace{0.1cm}+ \hspace{0.1cm} A_{\rm O}\cdot {\rm e}^{-{\rm j} \varphi_{\rm O} } \cdot {\rm e}^{ {\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm O}\hspace{0.01cm}'\hspace{0.05cm}\cdot \hspace{0.05cm}t}\hspace{0.1cm}+ \hspace{0.1cm}
A_{\rm U}\cdot {\rm e}^{-{\rm j} \varphi_{\rm U} } \cdot {\rm e}^{ {\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm U}\hspace{0.01cm}'\hspace{0.05cm}\cdot \hspace{0.05cm}t} . $$

{{GraueBox|TEXT=
$\text{Example 1:}$ 
The constellation given here results, for example, in the [[Modulationsverfahren/Zweiseitenband-Amplitudenmodulation#AM-Signale_und_-Spektren_bei_harmonischem_Eingangssignal|double sideband amplitude modulation]] of the message signal $x_{\rm N}(t) = A_{\rm N}\cdot \cos\left(2\pi f_{\rm N}\cdot t- \varphi_{\rm N}\right)$ with the carrier signal $x_{\rm T}(t) = A_{\rm T}\cdot \cos\left(2\pi f_{\rm T}\cdot t - \varphi_{\rm T}\right)$. This is discussed frequently in the exercises.

[[File:Ortskurve_4.png|center|frame|Spectrum $X_{\rm TP}(f)$ of the equivalent low–pass signal for different phase constellations |class=fit]]

There are some limitations to the program parameters in this approach:
* The frequencies are always $f\hspace{0.05cm}'_{\rm O} = f_{\rm N}$ and $f\hspace{0.05cm}'_{\rm U} = -f_{\rm N}$.
*Without distortion, the amplitude of the sidebands is $A_{\rm O}= A_{\rm U}= A_{\rm N}/2$.
*The respective phase relationships can be seen from the graphic.

}}

===Representation of the Equivalent Low-Pass Signal by Magnitude and Phase===

The generally complex equivalent low-pass signal
:$$x_{\rm TP}(t) = a(t) \cdot {\rm e}^{ {\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm} \phi(t) }$$
can be split into a magnitude function $a(t)$ and a phase function $\phi(t)$ according to the equation given here, where:
:$$a(t) = \vert x_{\rm TP}(t)\vert = \sqrt{ {\rm Re}^2\big [x_{\rm TP}(t)\big ] + {\rm Im}^2\big [x_{\rm TP}(t)\big ] }\hspace{0.05cm},$$
:$$\phi(t) = \text{arc }x_{\rm TP}(t) = \arctan \frac{{\rm Im}\big [x_{\rm TP}(t)\big ]}{{\rm Re}\big [x_{\rm TP}(t)\big ]}.$$

The reason for this is that a bandpass–signal $x(t)$ is usually described by the equivalent low-pass signal $x_{\rm TP}(t)$ that the functions $a(t)$ and $\phi(t)$ are interpretable in both representations:
*The amount $a(t)$ of the equivalent low-pass signal $x_{\rm TP}(t)$ indicates the (time-dependent) envelope of $x(t)$.
*The phase $\phi(t)$ of $x_{\rm TP}(t)$ denotes the location of the zero crossings of $x(t)$, where:
:–   For $\phi(t)>0$ the zero crossing is earlier than its nominal position   ⇒   the signal is leading here.
:–  When $\phi(t)<0$, the zero crossing is later than its target position   ⇒   the signal is trailing here.

{{GraueBox|TEXT=
$\text{Example 2:}$ 
The graph is intended to illustrate this relationship, assuming $A_{\rm U} > A_{\rm O}$   ⇒   the green pointer (for the lower sideband) is longer than the blue pointer (upper sideband). This is a snapshot at time $t_0$:

[[File:Ortskurve_3_neu.png|center|frame|bandpass spectrum $X(f)$ |class=fit]]

*For these system parameters, the top of the pointer cluster $x_{\rm TP}(t)$ – that is, the geometric sum of red, blue and green pointers – on an ellipse.
* The amount $a(t_0) = \vert x_{\rm TP}(t_0) \vert$ is drawn in black in the left-hand diagram and the phase value $\phi(t_0) = \text{arc }x_{\rm TP}(t_0) > 0$ is indicated in brown color.
*In the graph on the right, the amount $a(t_0) = \vert x_{\rm TP}(t_0) \vert$ of the equivalent low-pass signal indicates the envelope of the physical signal $x(t)$.
* At $\phi(t) \equiv 0$, all zero crossings of $x(t)$ would occur at equidistant intervals. Because of $\phi(t_0) > 0$, the signal is leading at the time $t_0$, that is: the zero crossings come earlier than the grid dictates. }}

==Exercises==
[[File:Exercises_verzerrungen.png|right]]
*First select the task number.
*A task description is displayed.
*Parameter values are adjusted.
*Solution after pressing &bdquo;Hide solition”.

The number &bdquo;0” will reset the program and output a text with the further explanation of the applet.

In the following, $\rm Green$ denotes the lower sideband   ⇒   $\big (A_{\rm U}, f_{\rm U}, \varphi_{\rm U}\big )$,  
$\rm Red$ the carrier   ⇒   $\big (A_{\rm T}, f_{\rm T}, \varphi_{\rm T}\big )$ and
$\rm Blue$ the upper sideband   ⇒   $\big (A_{\rm O}, f_{\rm O}, \varphi_{\rm O}\big )$.

{{BlaueBox|TEXT=
'''(1)'''   Let   $\text{Red:} \hspace{0.15cm} A_{\rm T} = 1\ \text{V}, f_{\rm T} = 100 \ \text{kHz}, \varphi_{\rm T} = 0^\circ$,   $\text{Green:} \hspace{0.15cm} A_{\rm U} = 0.4 \text{V}, \ f_{\rm U} = 80 \ \text{kHz}, \varphi_{\rm U} = -90^\circ$,   $\text{Blue:} \hspace{0.15cm} A_{\rm O} = 0.4\ \text{V}, f_{\rm O} = 120 \ \text{kHz}, \varphi_{\rm O} = 90^\circ$.

:Consider and interpret the equivalent low-pass signal $x_{\rm TP}(t)$ and the physical signal $x(t)$. Which period $T_0$ does one recognize?}}

:: The equivalent low-pass signal $x_{\rm TP}(t)$ takes from $x_{\rm TP}(t=0)=1\ \text{V}$ on the real axis values between $0.2\ \text{V}$ and $1.8\ \text{V}$   ⇒   phase $\phi(t) \equiv 0$.  The amount $|x_{\rm TP}(t)|$ indicates the envelope $a(t)$ of the physical signal $x(t)$. It holds $A_{\rm N} = 0.8\ \text{V}$ and $f_{\rm N} = 20\ \text{kHz}$:   $a(t) = A_{\rm T}+ A_{\rm N} \cdot \sin(2\pi\cdot f_{\rm N} \cdot t)$.  Both $x_{\rm TP}(t)$ and $x(t)$ are periodic with the period $T_0 = 1/f_{\rm N} = 50\ \rm µ s$.

{{BlaueBox|TEXT=
'''(2)'''   How do the ratios change to '''(1)''' with $f_{\rm U} = 99 \ \text{kHz}$ and $f_{\rm O} = 101 \ \text{kHz}$ ? How could $x(t)$ have arisen?}}

:: For the envelope $a(t)$ of the signal $x(t)$ we still have $a(t) = A_{\rm T}+ A_{\rm N} \cdot \sin(2\pi\cdot f_{\rm N} \cdot t)$, but now $f_{\rm N} = 1\ \text{kHz}$. Even though it can not be recognized:  $x_{\rm TP}(t)$ and $x(t)$ are still periodic:   $T_0 = 1\ \rm ms$. Example: Double sideband - Amplitude modulation '''(ZSB–AM)''' of a sine signal with cosine–carrier.

{{BlaueBox|TEXT=
'''(3)'''   Which settings have to be changed from '''(2)''' in order to arrive at the ZSB–AM of a cosine signal with sine–carrier. What changes over '''(2)'''?}}

::The carrier phase must be changed to $\varphi_{\rm T} = 90^\circ$   ⇒   sine–carrier. Similarly, $\varphi_{\rm O} =\varphi_{\rm U} =\varphi_{\rm T} = 90^\circ$ must be set   ⇒   cosine-shaped message  The locus now lies on the imaginary axis  ⇒   $\phi(t) \equiv -90^\circ$. At the beginning we have $x_{\rm TP}(t=0)= - {\rm j} \cdot 1.8 \ \text{V}$.

{{BlaueBox|TEXT=
'''(4)'''   Now let   $\text{Red:} \hspace{0.15cm} A_{\rm T} = 1\ \text{V}, \ f_{\rm T} = 100 \ \text{kHz}, \ \varphi_{\rm T} = 0^\circ$,   $\text{Green:} \hspace{0.15cm} A_{\rm U} = 0.4 \text{V}, \ f_{\rm U} = 80 \ \text{kHz}, \ \varphi_{\rm U} = 0^\circ$,   $\text{Blue:} \hspace{0.15cm} A_{\rm O} = 0.4\ \text{V}, \ f_{\rm O} = 120 \ \text{kHz}, \ \varphi_{\rm O} = 0^\circ$.

:What are the characteristics of this system &bdquo;ZSB–AM, where the message signal and carrier are respectively cosinusoidal”? What is the modulation depth $m$?}}

:: The equivalent low-pass signal $x_{\rm TP}(t)$ takes from $x_{\rm TP}(t=0)=1.8\ \text{V}$ on the real axis values between $0.2\ \text{V}$ and $1.8\ \text{V}$   ⇒   phase $\phi(t) \equiv 0$.  Except for the start state $x_{\rm TP}(t=0)$ same behavior as with the setting '''(1)'''. The modulation depth is $m = 0.8$.

{{BlaueBox|TEXT=
'''(5)'''   The parameters are still valid according to '''(4)''' with the exception of $A_{\rm T}= 0.6 \text{V}$. How big is the degree of modulation $m$? What are the consequences?}}

:: There is now a ZSB–AM with modulation degree $m = 1.333$. For $m > 1$, the simpler [[Modulationsverfahren/Hüllkurvendemodulation|envelope demodulation]] is not applicable, since the phase function $\phi(t) \in \{ 0, \ \pm 180^\circ\}$ is no longer constant and the envelope $a(t)$ no longer matches the message signal. Rather, the more complex [[Modulationsverfahren/Synchrondemodulation|synchronous demodulation]] must be used. Envelope detection would produce nonlinear distortions.

{{BlaueBox|TEXT=
'''(6)'''   The parameters are still valid according to '''(4)''' or '''(5)''' with the exception from $A_{\rm T}= 0$ on   ⇒   $m \to \infty$. Which modulation method is described in this way?}}

::It is a '''ZSB–AM without carrier''' and requires synchronous demodulation. The equivalent low-pass signal $x_{\rm TP}(t)$ is on the real axis, but not only in the right half-plane. Thus, the phase function $\phi(t) \in \{ 0, \ \pm 180^\circ\}$, also applies here, which means that envelope demodulation is not applicable.

{{BlaueBox|TEXT=
'''(7)'''   Now let   $\text{Red:} \hspace{0.15cm} A_{\rm T} = 1\ \text{V}, f_{\rm T} = 100 \ \text{kHz}, \varphi_{\rm T} = 0^\circ$,   $\text{Green:} \hspace{0.15cm} A_{\rm U} = 0, \ f_{\rm U} = 80 \ \text{kHz}, \varphi_{\rm U} = -90^\circ$,   $\text{Blue:} \hspace{0.15cm} A_{\rm O} = 0.8\ \text{V}, f_{\rm O} = 120 \ \text{kHz}, \varphi_{\rm O} = 90^\circ$.

:Which constellation is described here? Which characteristics of this procedure can be recognized from the graphic?}}

::It is a [[Modulationsverfahren/Einseitenbandmodulation|single-sideband modulation]] '''(ESB–AM)''', more specifically an '''OSB–AM''': the red carrier is fixed, the green pointer missing and the blue pointer (OSB) turns counterclockwise. The degree of modulation is $\mu = 0.8$ (in the case of ESB we denote the degree of modulation with $\mu$ instead of $m$). The carrier signal is cosinusoidal and the message signal sinusoidal. The locus is a circle. $x_{\rm TP}(t)$ moves in a mathematically positive direction. Because of $\phi(t) \ne \text{const.}$, The envelope demodulation is not applicable here:  This can be seen by the fact that the envelope $a(t)$ is not cosinusoidal. Rather, the lower half-wave is sharper than the upper   ⇒   strong linear distortions.

{{BlaueBox|TEXT=
'''(8)'''   The parameters are still valid according to '''(7)''' with the exception of $A_{\rm O}= 0$ and $A_{\rm U}= 0.8 \text{ V}$. What differences arise opposite '''(7)'''?}}

::Now it is a '''USB–AM''': The red carrier is fixed, the blue pointer is missing and the green pointer (USB) rotates clockwise. All other statements of '''(7)''' apply here as well.

{{BlaueBox|TEXT=
'''(9)'''   The parameters according to '''(7)''' are still valid with the exception of $A_{\rm O} = 0.2 \text{ V} \ne A_{\rm U} = 0.4 \text{ V} $. What are the differences from '''(7)'''?}}

::The locus $x_{\rm TP}(t)$ is not a horizontal straight line, but an ellipse with the real part between $0.4 \text{ V}$ and $1.6 \text{ V}$ and the imaginary part in the range $\pm 0.2 \text{ V}$. Because of $\phi(t) \ne \text{const.}$ , Envelope demodulation would lead to non-linear distortions here too. The constellation simulated here describes the situation of '''(4)''', namely a ZSB–AM with modulation degree $m = 0.8$, where the upper sideband is reduced to $50\%$ due to channel attenuation.

==Applet Manual==
[[File:Ortskurve_abzug3.png|right]]
* The red parameters $(A_{\rm T}, \ f_{\rm T}, \ \varphi_{\rm T})$ and the red pointer mark the ''Carrier''. (German: '''T'''räger).
* The green parameters $(A_{\rm U}, \ f_{\rm U} < f_{\rm T}, \ \varphi_{\rm U})$ mark the ''Lower sideband''. (German: '''U'''ntere Seitenband).
* The blue parameters $(A_{\rm O}, \ f_{\rm O} > f_{\rm T}, \ \varphi_{\rm O})$ mark the '' Upper sideband''. (German: '''O'''bere Seitenband).
* The red pointer does not turn.
* The green pointer rotates in a mathematically negative direction (clockwise).
* The blue pointer turns counterclockwise.

Meaning of the letters in the adjacent graphic:

    '''(A)'''     Plot of the equivalent low-pass Signal $x_{\rm TP}(t)$

    '''(B)'''     Plot of the physical signal $x(t)$

    '''(C)'''     Parameter input via slider:   amplitudes, frequencies, phase values

    '''(D)'''     Control elements:   Start – Step – Pause/Continue – Reset

    '''(E)'''     Speed of animation:   &bdquo;Speed”   ⇒   Values: 1, 2 oder 3

    '''(F)'''     &bdquo;Trace”   ⇒   On or Off, trace of equivalent low-pass Signal   $x_{\rm TP}(t)$

    '''(G)'''     Numerical output:   time $t$, the signal values  ${\rm Re}[x_{\rm TP}(t)]$  and  ${\rm Im}[x_{\rm TP}(t)]$,

$\text{}\hspace{4.2cm}$   Hüllkurve $a(t) = |x_{\rm TP}(t)|$  und  Phase $\phi(t) = {\rm arc} \ x_{\rm TP}(t)$

    '''(H)'''     Variations for the graphical representation

$\hspace{1.5cm}$Zoom–Functions &bdquo;$+$” (Enlarge), &bdquo;$-$” (Decrease) und $\rm o$ (Reset to default)

$\hspace{1.5cm}$Move with &bdquo;$\leftarrow$” (Section to the left, ordinate to the right), &bdquo;$\uparrow$” &bdquo;$\downarrow$” &bdquo;$\rightarrow$”

    '''(I)'''     Experiment section:   Task selection and task

    '''(J)'''     Experiment section:  solution
 

==About the Authors==
This interactive calculation was designed and realized at the [http://www.lnt.ei.tum.de/startseite Lehrstuhl für Nachrichtentechnik] of the [https://www.tum.de/ Technical University of Munich] .
*The original version was created in 2005 by [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Studierende#Ji_Li_.28Bachelorarbeit_EI_2003.2C_Diplomarbeit_EI_2005.29|Ji Li]] as part of her Diploma thesis using &bdquo;FlashMX–Actionscript” (Supervisor: [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Mitarbeiter_und_Dozenten#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28am_LNT_seit_1974.29|Günter Söder]]).
*In 2018 this Applet was redesigned and updated to &bdquo;HTML5” by [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Studierende#Xiaohan_Liu_.28Bachelorarbeit_2018.29|Xiaohan Liu]] as part of her Bachelor's thesis (Supervisor: [[Biografien_und_Bibliografien/Beteiligte_der_Professur_Leitungsgebundene_%C3%9Cbertragungstechnik#Tasn.C3.A1d_Kernetzky.2C_M.Sc._.28bei_L.C3.9CT_seit_2014.29|Tasnád Kernetzky]]).

==Once again: Open Applet in new Tab==

{{LntAppletLinkEn|physAnSignal}}

[[Category:Applets|^Verzerrungen^]]

Applets:Physical Signal & Equivalent Lowpass Signal

2018-08-25T16:06:50Z

Xiaohan: /* Applet Description */

{{LntAppletLinkEn|analPhysSignal}}

==Applet Description==
 
This applet shows the relationship between the physical bandpass signal $x(t)$ and the associated equivalent low-pass signal $x_{\rm TP}(t)$. The starting point is always a bandpass signal $x(t)$ with a frequency-discrete spectrum $X(f)$:
:$$x(t) = x_{\rm T}(t) + x_{\rm O}(t) + x_{\rm U}(t) = A_{\rm T}\cdot \cos\left(2\pi f_{\rm T}\cdot t- \varphi_{\rm T}\right)+A_{\rm O}\cdot \cos\left(2\pi f_{\rm O}\cdot t- \varphi_{\rm O}\right)+ A_{\rm U}\cdot \cos\left(2\pi f_{\rm U}\cdot t- \varphi_{\rm U}\right). $$
The physical signal $x(t)$ is thus composed of three [[Signaldarstellung/Harmonische_Schwingung|harmonic oscillations]], a constellation that can be found, for example, in the [[Modulationsverfahren/Zweiseitenband-Amplitudenmodulation#AM-Signale_und_-Spektren_bei_harmonischem_Eingangssignal|double-sideband amplitude modulation]] of the message signal $x_{\rm N}(t) = A_{\rm N}\cdot \cos\left(2\pi f_{\rm N}\cdot t- \varphi_{\rm N}\right)$ with the carrier signal $x_{\rm T}(t) = A_{\rm T}\cdot \cos\left(2\pi f_{\rm T}\cdot t - \varphi_{\rm T}\right)$ returns. The nomenclature is also adapted to this case:
* $x_{\rm O}(t)$ denotes the &bdquo;upper sideband” with the amplitude $A_{\rm O}= A_{\rm N}/2$, the frequency $f_{\rm O} = f_{\rm T} + f_{\rm N}$ and the phase $\varphi_{\rm O} = \varphi_{\rm T} + \varphi_{\rm N}$.
*Similarly, for the &bdquo;lower Sideband” $x_{\rm U}(t)$ with $f_{\rm U} = f_{\rm T} + f_{\rm N}$, $A_{\rm U}= A_{\rm O}$ and $\varphi_{\rm U} = -\varphi_{\rm O}$.

The corresponding equivalent low-pass–signal is $f_{\rm O}\hspace{0.01cm}' = f_{\rm O}- f_{\rm T} > 0$,   $f_{\rm U}\hspace{0.01cm}' = f_{\rm U}- f_{\rm T} < 0$  and  $f_{\rm T}\hspace{0.01cm}' = 0$:

:$$x_{\rm TP}(t) = x_\text{TP, T}(t) + x_\text{TP, O}(t) + x_\text{TP, U}(t) = A_{\rm T}\cdot {\rm e}^{-{\rm j} \varphi_{\rm T} } \hspace{0.1cm}+ \hspace{0.1cm} A_{\rm O}\cdot {\rm e}^{-{\rm j} \varphi_{\rm O} } \cdot {\rm e}^{ {\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm O}\hspace{0.01cm}'\hspace{0.05cm}\cdot \hspace{0.05cm}t}\hspace{0.1cm}+ \hspace{0.1cm}
A_{\rm U}\cdot {\rm e}^{-{\rm j} \varphi_{\rm U} } \cdot {\rm e}^{ {\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm U}\hspace{0.01cm}'\hspace{0.05cm}\cdot \hspace{0.05cm}t} . $$

[[File:Ortskurve_1.png|right|frame|Equivalent lowpass–signal currently $t=0$ for cosinusoidal carrier   ⇒   $\varphi_{\rm T} = 0$]]
The program shows $x_{\rm TP}(t)$ as the vectorial sum of three rotation pointers as a violet point (see example graph for start time $t=0$ and cosinusoidal carrier):

*The (red) pointer of the carrier $x_\text{TP, T}(t)$ with the length $A_{\rm T}$ and the zero phase position $\varphi_{\rm T}=0$ is fixed in the complex plane. So it applies to all times $t$:   $x_{\rm TP}(t)= A_{\rm T}\cdot {\rm e}^{-{\rm j} \varphi_{\rm T} }$.

*The (blue) pointer of the upper sideband $x_\text{TP, O}(t)$ with the length $A_{\rm O}$ and the zero phase position $\varphi_{\rm O}$ rotates at the angular velocity $2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm O}\hspace{0.01cm}'$ in mathematically positive direction (one revolution in time $1/f_{\rm O}\hspace{0.01cm}')$.

*The (green) pointer of the lower sideband $x_{\rm U+}(t)$ with the length $A_{\rm U}$ and the zero phase position $\varphi_{\rm U}$ rotates at the angular velocity $2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm U}\hspace{0.01cm}'$, because of $f_{\rm U}\hspace{0.01cm}'<0$ counterclockwise.

*With $f_{\rm U}\hspace{0.01cm}' = -f_{\rm O}\hspace{0.01cm}'$ the blue and green pointers will spin at the same speed but in different directions. Also, if $A_{\rm O} = A_{\rm U}$ and $\varphi_{\rm U} = -\varphi_{\rm O}$, then $x_{\rm TP}(t)$ moves on a straight line with a incline of $\varphi_{\rm T}$.

''Hinweis:''   The graph applies to $\varphi_{\rm O} = +30^\circ$. From this follows for the start time $t=0$ the angle of the blue pointer (OSB) with respect to the coordinate system:   $\phi_{\rm O} = -\varphi_{\rm O} = -30^\circ$. Likewise, the null phase position $\varphi_{\rm U} = -30^\circ$ of the lower sideband (USB, grüner Zeiger) follows for the phase angle to be considered in the complex plane:   $\phi_{\rm U} = +30^\circ$.

The temporal process of $x_{\rm TP}(t)$ is also referred to below as '''locus'''. The relationship between $x_{\rm TP}(t)$ and the physical bandpass –signal $x(t)$ is given in the section and the associated analytic signal is $x_+(t)$ :

:$$x_{\rm TP}(t) = x_{\rm +}(t)\cdot {\rm e}^{-{\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm}2 \pi \cdot f_{\rm T}\cdot \hspace{0.05cm}t},$$
:$$x_{\rm +}(t) = x_{\rm TP}(t)\cdot {\rm e}^{+{\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm}2 \pi \cdot f_{\rm T}\cdot \hspace{0.05cm}t}.$$

[[Applets:Physikalisches_Signal_%26_Äquivalentes_TP-Signal|'''German description''']]

==Theoretical Background==
 
===Description of Bandpass Signals===
[[File:Zeigerdiagramm_1a.png|right|frame|bandpass spectrum $X(f)$ |class=fit]]
We consider '''bandpass signals''' $x(t)$ with the property that their spectra $X(f)$ are not in the range around the frequency $f=0$, but around a carrier frequency $f_{\rm T}$. In most cases it can also be assumed that the bandwidth is $B \ll f_{\rm T}$.

The figure shows such a bandpass spectrum $X(f)$. Assuming that the associated $x(t)$ is a physical signal and thus real, the spectral function $X(f)$ has a symmetry with respect to the frequency $f = 0$, if $x(t)$ is an even function   ⇒   $x(-t)=x(t)$, $X(f)$ is real and even.

Beside the physical signal $x(t)\ \circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\, \ X(f)$, one can also use the following descriptions of bandpass signals:
*the analytic signal $x_+(t)\ \circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\, \ X_+(f)$, see applet [[Applets:Physikalisches_Signal_%26_Analytisches_Signal|Physikalisches Signal & Analytisches Signal]],
*the equivalent low-pass–signal $x_{\rm TP}(t)\ \circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\, \ X_{\rm TP}(f)$, see next page
 

===Spectral Functions of the Analytic and the Equivalent Low-Pass Signal===

The '''analytic signal''' $x_+(t)$ belonging to the physical signal $x(t)$ is the time function whose spectrum fulfills the following property:
[[File:Ortskurve_2.png|right|frame|spectral functions $X_+(f)$ and $X_{\rm TP}(f)$ |class=fit]]
:$$X_+(f)=\big[1+{\rm sign}(f)\big] \cdot X(f) = \left\{ {2 \cdot
X(f) \; \hspace{0.2cm}\rm for\hspace{0.2cm} {\it f} > 0, \atop {\,\,\,\, \rm 0 \; \hspace{0.9cm}\rm for\hspace{0.2cm} {\it f} < 0.} }\right.$$

The ''Signum function'' is for positive values of $f$ equal to $+1$ and for negative $f$ values equal to $-1$.
*The (double-sided) limit returns $\sign(0) = 0$.
*The index „+” should make it clear that $X_+(f)$ only has parts at positive frequencies.

From the graph you can see the calculation rule for $X_+(f)$: The actual BP spectrum $X(f)$ becomes
*doubled at the positive frequencies, and
*set to zero at the negative frequencies.

Due to the asymmetry of $X_+(f)$ with respect to the frequency $f = 0$, it can already be said that the time function $x_+(t)$ except for a trivial special case $x_+(t)= 0 \ \ \circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\,\ \ X_+(f)= 0$ is always complex.

The spectrum $X_{\rm TP}(f)$ of the equivalent low-pass signal is obtained by shifting $X_+(f)$ to the left by the carrier frequency $f_{\rm T}$:
:$$X_{\rm TP}(f)= X_+(f+f_{\rm T}).$$

In the time domain this operation corresponds to the multiplication of $x_{\rm +}(t)$ with the complex exponential function with negative exponent:
:$$x_{\rm TP}(t) = x_{\rm +}(t)\cdot {\rm e}^{-{\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm}2 \pi \cdot f_{\rm T}\cdot \hspace{0.05cm}t}.$$

It can be seen that $x_{\rm TP}(t)$ is generally complex. But if $X_+(f)$ is symmetric about the carrier frequency $f_{\rm T}$, $X_{\rm TP}(f)$ is symmetric about the frequency $f=0$ and there is accordingly real time function $x_{\rm TP}(t)$.

===$x_{\rm TP}(t)$–Representation of a Sum of Three Harmonic Oscillations===

In our applet, we always assume a set of three rotating pointers. The physical signal is:
:$$x(t) = x_{\rm U}(t) + x_{\rm T}(t) + x_{\rm O}(t) = A_{\rm U}\cdot \cos\left(2\pi f_{\rm U}\cdot t- \varphi_{\rm U}\right)+A_{\rm T}\cdot \cos\left(2\pi f_{\rm T}\cdot t- \varphi_{\rm T}\right)+A_{\rm O}\cdot \cos\left(2\pi f_{\rm O}\cdot t- \varphi_{\rm O}\right). $$
* Each of the three harmonic oscillations $x_{\rm T}(t)$, $x_{\rm U}(t)$ and $x_{\rm O}(t)$ is represented by an amplitude $(A)$, a frequency $(f)$ and a phase value $(\varphi)$.
*The indices are based on the modulation method [[Modulationsverfahren/Zweiseitenband-Amplitudenmodulation|double sideband –amplitude modulation]]. &bdquo;T” stands for &bdquo;carrier”, &bdquo;U” for &bdquo;lowei sideband” and &bdquo;O” for &bdquo;upper Sideband”. Similarly, $f_{\rm U} < f_{\rm T}$ and $f_{\rm O} > f_{\rm T}$. There are no restrictions for the amplitudes and phases.

The associated equivalent low-pass signal is with $f_{\rm O}\hspace{0.01cm}' = f_{\rm O}- f_{\rm T} > 0$,   $f_{\rm U}\hspace{0.01cm}' = f_{\rm U}- f_{\rm T} < 0$  and  $f_{\rm T}\hspace{0.01cm}' = 0$:

:$$x_{\rm TP}(t) = x_\text{TP, T}(t) + x_\text{TP, O}(t) + x_\text{TP, U}(t) = A_{\rm T}\cdot {\rm e}^{-{\rm j} \varphi_{\rm T} } \hspace{0.1cm}+ \hspace{0.1cm} A_{\rm O}\cdot {\rm e}^{-{\rm j} \varphi_{\rm O} } \cdot {\rm e}^{ {\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm O}\hspace{0.01cm}'\hspace{0.05cm}\cdot \hspace{0.05cm}t}\hspace{0.1cm}+ \hspace{0.1cm}
A_{\rm U}\cdot {\rm e}^{-{\rm j} \varphi_{\rm U} } \cdot {\rm e}^{ {\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm U}\hspace{0.01cm}'\hspace{0.05cm}\cdot \hspace{0.05cm}t} . $$

{{GraueBox|TEXT=
$\text{Example 1:}$ 
The constellation given here results, for example, in the [[Modulationsverfahren/Zweiseitenband-Amplitudenmodulation#AM-Signale_und_-Spektren_bei_harmonischem_Eingangssignal|double sideband amplitude modulation]] of the message signal $x_{\rm N}(t) = A_{\rm N}\cdot \cos\left(2\pi f_{\rm N}\cdot t- \varphi_{\rm N}\right)$ with the carrier signal $x_{\rm T}(t) = A_{\rm T}\cdot \cos\left(2\pi f_{\rm T}\cdot t - \varphi_{\rm T}\right)$. This is discussed frequently in the exercises.

[[File:Ortskurve_4.png|center|frame|Spectrum $X_{\rm TP}(f)$ of the equivalent low–pass signal for different phase constellations |class=fit]]

There are some limitations to the program parameters in this approach:
* The frequencies are always $f\hspace{0.05cm}'_{\rm O} = f_{\rm N}$ and $f\hspace{0.05cm}'_{\rm U} = -f_{\rm N}$.
*Without distortion, the amplitude of the sidebands is $A_{\rm O}= A_{\rm U}= A_{\rm N}/2$.
*The respective phase relationships can be seen from the graphic.

}}

===Representation of the Equivalent Low-Pass Signal by Magnitude and Phase===

The generally complex equivalent low-pass signal
:$$x_{\rm TP}(t) = a(t) \cdot {\rm e}^{ {\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm} \phi(t) }$$
can be split into a magnitude function $a(t)$ and a phase function $\phi(t)$ according to the equation given here, where:
:$$a(t) = \vert x_{\rm TP}(t)\vert = \sqrt{ {\rm Re}^2\big [x_{\rm TP}(t)\big ] + {\rm Im}^2\big [x_{\rm TP}(t)\big ] }\hspace{0.05cm},$$
:$$\phi(t) = \text{arc }x_{\rm TP}(t) = \arctan \frac{{\rm Im}\big [x_{\rm TP}(t)\big ]}{{\rm Re}\big [x_{\rm TP}(t)\big ]}.$$

The reason for this is that a bandpass–signal $x(t)$ is usually described by the equivalent low-pass signal $x_{\rm TP}(t)$ that the functions $a(t)$ and $\phi(t)$ are interpretable in both representations:
*The amount $a(t)$ of the equivalent low-pass signal $x_{\rm TP}(t)$ indicates the (time-dependent) envelope of $x(t)$.
*The phase $\phi(t)$ of $x_{\rm TP}(t)$ denotes the location of the zero crossings of $x(t)$, where:
:–   For $\phi(t)>0$ the zero crossing is earlier than its nominal position   ⇒   the signal is leading here.
:–  When $\phi(t)<0$, the zero crossing is later than its target position   ⇒   the signal is trailing here.

{{GraueBox|TEXT=
$\text{Example 2:}$ 
The graph is intended to illustrate this relationship, assuming $A_{\rm U} > A_{\rm O}$   ⇒   the green pointer (for the lower sideband) is longer than the blue pointer (upper sideband). This is a snapshot at time $t_0$:

[[File:Ortskurve_3_neu.png|center|frame|bandpass spectrum $X(f)$ |class=fit]]

*For these system parameters, the top of the pointer cluster $x_{\rm TP}(t)$ – that is, the geometric sum of red, blue and green pointers – on an ellipse.
* The amount $a(t_0) = \vert x_{\rm TP}(t_0) \vert$ is drawn in black in the left-hand diagram and the phase value $\phi(t_0) = \text{arc }x_{\rm TP}(t_0) > 0$ is indicated in brown color.
*In the graph on the right, the amount $a(t_0) = \vert x_{\rm TP}(t_0) \vert$ of the equivalent low-pass signal indicates the envelope of the physical signal $x(t)$.
* At $\phi(t) \equiv 0$, all zero crossings of $x(t)$ would occur at equidistant intervals. Because of $\phi(t_0) > 0$, the signal is leading at the time $t_0$, that is: the zero crossings come earlier than the grid dictates. }}

==Exercises==
[[File:Exercises_verzerrungen.png|right]]
*First select the task number.
*A task description is displayed.
*Parameter values are adjusted.
*Solution after pressing &bdquo;Hide solition”.

The number &bdquo;0” will reset the program and output a text with the further explanation of the applet.

In the following, $\rm Green$ denotes the lower sideband   ⇒   $\big (A_{\rm U}, f_{\rm U}, \varphi_{\rm U}\big )$,  
$\rm Red$ the carrier   ⇒   $\big (A_{\rm T}, f_{\rm T}, \varphi_{\rm T}\big )$ and
$\rm Blue$ the upper sideband   ⇒   $\big (A_{\rm O}, f_{\rm O}, \varphi_{\rm O}\big )$.

{{BlaueBox|TEXT=
'''(1)'''   Let   $\text{Red:} \hspace{0.15cm} A_{\rm T} = 1\ \text{V}, f_{\rm T} = 100 \ \text{kHz}, \varphi_{\rm T} = 0^\circ$,   $\text{Green:} \hspace{0.15cm} A_{\rm U} = 0.4 \text{V}, \ f_{\rm U} = 80 \ \text{kHz}, \varphi_{\rm U} = -90^\circ$,   $\text{Blue:} \hspace{0.15cm} A_{\rm O} = 0.4\ \text{V}, f_{\rm O} = 120 \ \text{kHz}, \varphi_{\rm O} = 90^\circ$.

:Consider and interpret the equivalent low-pass signal $x_{\rm TP}(t)$ and the physical signal $x(t)$. Which period $T_0$ does one recognize?}}

:: The equivalent low-pass signal $x_{\rm TP}(t)$ takes from $x_{\rm TP}(t=0)=1\ \text{V}$ on the real axis values between $0.2\ \text{V}$ and $1.8\ \text{V}$   ⇒   phase $\phi(t) \equiv 0$.  The amount $|x_{\rm TP}(t)|$ indicates the envelope $a(t)$ of the physical signal $x(t)$. It holds $A_{\rm N} = 0.8\ \text{V}$ and $f_{\rm N} = 20\ \text{kHz}$:   $a(t) = A_{\rm T}+ A_{\rm N} \cdot \sin(2\pi\cdot f_{\rm N} \cdot t)$.  Both $x_{\rm TP}(t)$ and $x(t)$ are periodic with the period $T_0 = 1/f_{\rm N} = 50\ \rm µ s$.

{{BlaueBox|TEXT=
'''(2)'''   How do the ratios change to '''(1)''' with $f_{\rm U} = 99 \ \text{kHz}$ and $f_{\rm O} = 101 \ \text{kHz}$ ? How could $x(t)$ have arisen?}}

:: For the envelope $a(t)$ of the signal $x(t)$ we still have $a(t) = A_{\rm T}+ A_{\rm N} \cdot \sin(2\pi\cdot f_{\rm N} \cdot t)$, but now $f_{\rm N} = 1\ \text{kHz}$. Even though it can not be recognized:  $x_{\rm TP}(t)$ and $x(t)$ are still periodic:   $T_0 = 1\ \rm ms$. Example: Double sideband - Amplitude modulation '''(ZSB–AM)''' of a sine signal with cosine–carrier.

{{BlaueBox|TEXT=
'''(3)'''   Which settings have to be changed from '''(2)''' in order to arrive at the ZSB–AM of a cosine signal with sine–carrier. What changes over '''(2)'''?}}

::The carrier phase must be changed to $\varphi_{\rm T} = 90^\circ$   ⇒   sine–carrier. Similarly, $\varphi_{\rm O} =\varphi_{\rm U} =\varphi_{\rm T} = 90^\circ$ must be set   ⇒   cosine-shaped message  The locus now lies on the imaginary axis  ⇒   $\phi(t) \equiv -90^\circ$. At the beginning we have $x_{\rm TP}(t=0)= - {\rm j} \cdot 1.8 \ \text{V}$.

{{BlaueBox|TEXT=
'''(4)'''   Now let   $\text{Red:} \hspace{0.15cm} A_{\rm T} = 1\ \text{V}, \ f_{\rm T} = 100 \ \text{kHz}, \ \varphi_{\rm T} = 0^\circ$,   $\text{Green:} \hspace{0.15cm} A_{\rm U} = 0.4 \text{V}, \ f_{\rm U} = 80 \ \text{kHz}, \ \varphi_{\rm U} = 0^\circ$,   $\text{Blue:} \hspace{0.15cm} A_{\rm O} = 0.4\ \text{V}, \ f_{\rm O} = 120 \ \text{kHz}, \ \varphi_{\rm O} = 0^\circ$.

:What are the characteristics of this system &bdquo;ZSB–AM, where the message signal and carrier are respectively cosinusoidal”? What is the modulation depth $m$?}}

:: The equivalent low-pass signal $x_{\rm TP}(t)$ takes from $x_{\rm TP}(t=0)=1.8\ \text{V}$ on the real axis values between $0.2\ \text{V}$ and $1.8\ \text{V}$   ⇒   phase $\phi(t) \equiv 0$.  Except for the start state $x_{\rm TP}(t=0)$ same behavior as with the setting '''(1)'''. The modulation depth is $m = 0.8$.

{{BlaueBox|TEXT=
'''(5)'''   The parameters are still valid according to '''(4)''' with the exception of $A_{\rm T}= 0.6 \text{V}$. How big is the degree of modulation $m$? What are the consequences?}}

:: There is now a ZSB–AM with modulation degree $m = 1.333$. For $m > 1$, the simpler [[Modulationsverfahren/Hüllkurvendemodulation|envelope demodulation]] is not applicable, since the phase function $\phi(t) \in \{ 0, \ \pm 180^\circ\}$ is no longer constant and the envelope $a(t)$ no longer matches the message signal. Rather, the more complex [[Modulationsverfahren/Synchrondemodulation|synchronous demodulation]] must be used. Envelope detection would produce nonlinear distortions.

{{BlaueBox|TEXT=
'''(6)'''   The parameters are still valid according to '''(4)''' or '''(5)''' with the exception from $A_{\rm T}= 0$ on   ⇒   $m \to \infty$. Which modulation method is described in this way?}}

::It is a '''ZSB–AM without carrier''' and requires synchronous demodulation. The equivalent low-pass signal $x_{\rm TP}(t)$ is on the real axis, but not only in the right half-plane. Thus, the phase function $\phi(t) \in \{ 0, \ \pm 180^\circ\}$, also applies here, which means that envelope demodulation is not applicable.

{{BlaueBox|TEXT=
'''(7)'''   Now let   $\text{Red:} \hspace{0.15cm} A_{\rm T} = 1\ \text{V}, f_{\rm T} = 100 \ \text{kHz}, \varphi_{\rm T} = 0^\circ$,   $\text{Green:} \hspace{0.15cm} A_{\rm U} = 0, \ f_{\rm U} = 80 \ \text{kHz}, \varphi_{\rm U} = -90^\circ$,   $\text{Blue:} \hspace{0.15cm} A_{\rm O} = 0.8\ \text{V}, f_{\rm O} = 120 \ \text{kHz}, \varphi_{\rm O} = 90^\circ$.

:Which constellation is described here? Which characteristics of this procedure can be recognized from the graphic?}}

::It is a [[Modulationsverfahren/Einseitenbandmodulation|single-sideband modulation]] '''(ESB–AM)''', more specifically an '''OSB–AM''': the red carrier is fixed, the green pointer missing and the blue pointer (OSB) turns counterclockwise. The degree of modulation is $\mu = 0.8$ (in the case of ESB we denote the degree of modulation with $\mu$ instead of $m$). The carrier signal is cosinusoidal and the message signal sinusoidal. The locus is a circle. $x_{\rm TP}(t)$ moves in a mathematically positive direction. Because of $\phi(t) \ne \text{const.}$, The envelope demodulation is not applicable here:  This can be seen by the fact that the envelope $a(t)$ is not cosinusoidal. Rather, the lower half-wave is sharper than the upper   ⇒   strong linear distortions.

{{BlaueBox|TEXT=
'''(8)'''   The parameters are still valid according to '''(7)''' with the exception of $A_{\rm O}= 0$ and $A_{\rm U}= 0.8 \text{ V}$. What differences arise opposite '''(7)'''?}}

::Now it is a '''USB–AM''': The red carrier is fixed, the blue pointer is missing and the green pointer (USB) rotates clockwise. All other statements of '''(7)''' apply here as well.

{{BlaueBox|TEXT=
'''(9)'''   The parameters according to '''(7)''' are still valid with the exception of $A_{\rm O} = 0.2 \text{ V} \ne A_{\rm U} = 0.4 \text{ V} $. What are the differences from '''(7)'''?}}

::The locus $x_{\rm TP}(t)$ is not a horizontal straight line, but an ellipse with the real part between $0.4 \text{ V}$ and $1.6 \text{ V}$ and the imaginary part in the range $\pm 0.2 \text{ V}$. Because of $\phi(t) \ne \text{const.}$ , Envelope demodulation would lead to non-linear distortions here too. The constellation simulated here describes the situation of '''(4)''', namely a ZSB–AM with modulation degree $m = 0.8$, where the upper sideband is reduced to $50\%$ due to channel attenuation.

==Applet Manual==
[[File:Ortskurve_abzug3.png|right]]
* The red parameters $(A_{\rm T}, \ f_{\rm T}, \ \varphi_{\rm T})$ and the red pointer mark the ''Carrier''. (German: '''T'''räger).
* The green parameters $(A_{\rm U}, \ f_{\rm U} < f_{\rm T}, \ \varphi_{\rm U})$ mark the ''Lower sideband''. (German: '''U'''ntere Seitenband).
* The blue parameters $(A_{\rm O}, \ f_{\rm O} > f_{\rm T}, \ \varphi_{\rm O})$ mark the '' Upper sideband''. (German: '''O'''bere Seitenband).
* The red pointer does not turn.
* The green pointer rotates in a mathematically negative direction (clockwise).
* The blue pointer turns counterclockwise.

Meaning of the letters in the adjacent graphic:

    '''(A)'''     Plot of the equivalent low-pass Signal $x_{\rm TP}(t)$

    '''(B)'''     Plot of the physical signal $x(t)$

    '''(C)'''     Parameter input via slider:   amplitudes, frequencies, phase values

    '''(D)'''     Control elements:   Start – Step – Pause/Continue – Reset

    '''(E)'''     Speed of animation:   &bdquo;Speed”   ⇒   Values: 1, 2 oder 3

    '''(F)'''     &bdquo;Trace”   ⇒   On or Off, trace of equivalent low-pass Signal   $x_{\rm TP}(t)$

    '''(G)'''     Numerical output:   time $t$, the signal values  ${\rm Re}[x_{\rm TP}(t)]$  and  ${\rm Im}[x_{\rm TP}(t)]$,

$\text{}\hspace{4.2cm}$   Hüllkurve $a(t) = |x_{\rm TP}(t)|$  und  Phase $\phi(t) = {\rm arc} \ x_{\rm TP}(t)$

    '''(H)'''     Variations for the graphical representation

$\hspace{1.5cm}$Zoom–Functions &bdquo;$+$” (Enlarge), &bdquo;$-$” (Decrease) und $\rm o$ (Reset to default)

$\hspace{1.5cm}$Move with &bdquo;$\leftarrow$” (Section to the left, ordinate to the right), &bdquo;$\uparrow$” &bdquo;$\downarrow$” &bdquo;$\rightarrow$”

    '''(I)'''     Experiment section:   Task selection and task

    '''(J)'''     Experiment section:  solution
 

==About the Authors==
This interactive calculation was designed and realized at the [http://www.lnt.ei.tum.de/startseite Lehrstuhl für Nachrichtentechnik] of the [https://www.tum.de/ Technical University of Munich] .
*The original version was created in 2005 by [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Studierende#Ji_Li_.28Bachelorarbeit_EI_2003.2C_Diplomarbeit_EI_2005.29|Ji Li]] as part of her Diploma thesis using &bdquo;FlashMX–Actionscript” (Supervisor: [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Mitarbeiter_und_Dozenten#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28am_LNT_seit_1974.29|Günter Söder]]).
*In 2018 this Applet was redesigned and updated to &bdquo;HTML5” by [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Studierende#Xiaohan_Liu_.28Bachelorarbeit_2018.29|Xiaohan Liu]] as part of her Bachelor's thesis (Supervisor: [[Biografien_und_Bibliografien/Beteiligte_der_Professur_Leitungsgebundene_%C3%9Cbertragungstechnik#Tasn.C3.A1d_Kernetzky.2C_M.Sc._.28bei_L.C3.9CT_seit_2014.29|Tasnád Kernetzky]]).

==Once again: Open Applet in new Tab==

{{LntAppletLinkEn|physAnSignal}}

[[Category:Applets|^Verzerrungen^]]

Applets:Physical Signal & Equivalent Lowpass Signal

2018-08-25T15:32:06Z

Xiaohan: /* Description of Bandpass Signals */

{{LntAppletLinkEn|analPhysSignal}}

==Applet Description==
 
This applet shows the relationship between the physical bandpass signal $x(t)$ and the associated equivalent low-pass signal $x_{\rm TP}(t)$. The starting point is always a bandpass signal $x(t)$ with a frequency-discrete spectrum $X(f)$:
:$$x(t) = x_{\rm T}(t) + x_{\rm O}(t) + x_{\rm U}(t) = A_{\rm T}\cdot \cos\left(2\pi f_{\rm T}\cdot t- \varphi_{\rm T}\right)+A_{\rm O}\cdot \cos\left(2\pi f_{\rm O}\cdot t- \varphi_{\rm O}\right)+ A_{\rm U}\cdot \cos\left(2\pi f_{\rm U}\cdot t- \varphi_{\rm U}\right). $$
The physical signal $x(t)$ is thus composed of three [[Signaldarstellung/Harmonische_Schwingung|harmonic oscillations]],which results in a constellation, for example, in the [[Modulationsverfahren/Zweiseitenband-Amplitudenmodulation#AM-Signale_und_-Spektren_bei_harmonischem_Eingangssignal|double-sideband amplitude modulation]] of the message signal $x_{\rm N}(t) = A_{\rm N}\cdot \cos\left(2\pi f_{\rm N}\cdot t- \varphi_{\rm N}\right)$ with the carrier signal $x_{\rm T}(t) = A_{\rm T}\cdot \cos\left(2\pi f_{\rm T}\cdot t - \varphi_{\rm T}\right)$ returns. The nomenclature is also adapted to this case:
* $x_{\rm O}(t)$ denotes the &bdquo;upper sideband” with the amplitude $A_{\rm O}= A_{\rm N}/2$, the frequency $f_{\rm O} = f_{\rm T} + f_{\rm N}$ and the phase $\varphi_{\rm O} = \varphi_{\rm T} + \varphi_{\rm N}$.
*Similarly, for the &bdquo;lower Sideband” $x_{\rm U}(t)$ with $f_{\rm U} = f_{\rm T} + f_{\rm N}$, $A_{\rm U}= A_{\rm O}$ and $\varphi_{\rm U} = -\varphi_{\rm O}$.

The corresponding equivalent low-pass–signal is $f_{\rm O}\hspace{0.01cm}' = f_{\rm O}- f_{\rm T} > 0$,   $f_{\rm U}\hspace{0.01cm}' = f_{\rm U}- f_{\rm T} < 0$  and  $f_{\rm T}\hspace{0.01cm}' = 0$:

:$$x_{\rm TP}(t) = x_\text{TP, T}(t) + x_\text{TP, O}(t) + x_\text{TP, U}(t) = A_{\rm T}\cdot {\rm e}^{-{\rm j} \varphi_{\rm T} } \hspace{0.1cm}+ \hspace{0.1cm} A_{\rm O}\cdot {\rm e}^{-{\rm j} \varphi_{\rm O} } \cdot {\rm e}^{ {\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm O}\hspace{0.01cm}'\hspace{0.05cm}\cdot \hspace{0.05cm}t}\hspace{0.1cm}+ \hspace{0.1cm}
A_{\rm U}\cdot {\rm e}^{-{\rm j} \varphi_{\rm U} } \cdot {\rm e}^{ {\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm U}\hspace{0.01cm}'\hspace{0.05cm}\cdot \hspace{0.05cm}t} . $$

[[File:Ortskurve_1.png|right|frame|Equivalent lowpass–signal currently $t=0$ for cosinusoidal carrier   ⇒   $\varphi_{\rm T} = 0$]]
The program shows $x_{\rm TP}(t)$ as the vectorial sum of three rotation pointers as a violet point (see example graph for start time $t=0$ and cosinusoidal carrier):

*The (red) pointer of the carrier $x_\text{TP, T}(t)$ with the length $A_{\rm T}$ and the zero phase position $\varphi_{\rm T}=0$ is fixed in the complex plane. So it applies to all times $t$:   $x_{\rm TP}(t)= A_{\rm T}\cdot {\rm e}^{-{\rm j} \varphi_{\rm T} }$.

*The (blue) pointer of the upper sideband $x_\text{TP, O}(t)$ with the length $A_{\rm O}$ and the zero phase position $\varphi_{\rm O}$ rotates at the angular velocity $2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm O}\hspace{0.01cm}'$ in mathematically positive direction (one revolution in time $1/f_{\rm O}\hspace{0.01cm}')$.

*The (green) pointer of the lower sideband $x_{\rm U+}(t)$ with the length $A_{\rm U}$ and the zero phase position $\varphi_{\rm U}$ rotates at the angular velocity $2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm U}\hspace{0.01cm}'$, because of $f_{\rm U}\hspace{0.01cm}'<0$ counterclockwise.

*With $f_{\rm U}\hspace{0.01cm}' = -f_{\rm O}\hspace{0.01cm}'$ the blue and green pointers will spin at the same speed but in different directions. Also, if $A_{\rm O} = A_{\rm U}$ and $\varphi_{\rm U} = -\varphi_{\rm O}$, then $x_{\rm TP}(t)$ moves on a straight line with a incline of $\varphi_{\rm T}$.

''Hinweis:''   The graph applies to $\varphi_{\rm O} = +30^\circ$. From this follows for the start time $t=0$ the angle of the blue pointer (OSB) with respect to the coordinate system:   $\phi_{\rm O} = -\varphi_{\rm O} = -30^\circ$. Likewise, the null phase position $\varphi_{\rm U} = -30^\circ$ of the lower sideband (USB, grüner Zeiger) follows for the phase angle to be considered in the complex plane:   $\phi_{\rm U} = +30^\circ$.

The temporal process of $x_{\rm TP}(t)$ is also referred to below as '''locus'''. The relationship between $x_{\rm TP}(t)$ and the physical bandpass –signal $x(t)$ is given in the section and the associated analytic signal is $x_+(t)$ :

:$$x_{\rm TP}(t) = x_{\rm +}(t)\cdot {\rm e}^{-{\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm}2 \pi \cdot f_{\rm T}\cdot \hspace{0.05cm}t},$$
:$$x_{\rm +}(t) = x_{\rm TP}(t)\cdot {\rm e}^{+{\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm}2 \pi \cdot f_{\rm T}\cdot \hspace{0.05cm}t}.$$

[[Applets:Physikalisches_Signal_%26_Äquivalentes_TP-Signal|'''German description''']]

==Theoretical Background==
 
===Description of Bandpass Signals===
[[File:Zeigerdiagramm_1a.png|right|frame|bandpass spectrum $X(f)$ |class=fit]]
We consider '''bandpass signals''' $x(t)$ with the property that their spectra $X(f)$ are not in the range around the frequency $f=0$, but around a carrier frequency $f_{\rm T}$. In most cases it can also be assumed that the bandwidth is $B \ll f_{\rm T}$.

The figure shows such a bandpass spectrum $X(f)$. Assuming that the associated $x(t)$ is a physical signal and thus real, the spectral function $X(f)$ has a symmetry with respect to the frequency $f = 0$, if $x(t)$ is an even function   ⇒   $x(-t)=x(t)$, $X(f)$ is real and even.

Beside the physical signal $x(t)\ \circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\, \ X(f)$, one can also use the following descriptions of bandpass signals:
*the analytic signal $x_+(t)\ \circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\, \ X_+(f)$, see applet [[Applets:Physikalisches_Signal_%26_Analytisches_Signal|Physikalisches Signal & Analytisches Signal]],
*the equivalent low-pass–signal $x_{\rm TP}(t)\ \circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\, \ X_{\rm TP}(f)$, see next page
 

===Spectral Functions of the Analytic and the Equivalent Low-Pass Signal===

The '''analytic signal''' $x_+(t)$ belonging to the physical signal $x(t)$ is the time function whose spectrum fulfills the following property:
[[File:Ortskurve_2.png|right|frame|spectral functions $X_+(f)$ and $X_{\rm TP}(f)$ |class=fit]]
:$$X_+(f)=\big[1+{\rm sign}(f)\big] \cdot X(f) = \left\{ {2 \cdot
X(f) \; \hspace{0.2cm}\rm for\hspace{0.2cm} {\it f} > 0, \atop {\,\,\,\, \rm 0 \; \hspace{0.9cm}\rm for\hspace{0.2cm} {\it f} < 0.} }\right.$$

The ''Signum function'' is for positive values of $f$ equal to $+1$ and for negative $f$ values equal to $-1$.
*The (double-sided) limit returns $\sign(0) = 0$.
*The index „+” should make it clear that $X_+(f)$ only has parts at positive frequencies.

From the graph you can see the calculation rule for $X_+(f)$: The actual BP spectrum $X(f)$ becomes
*doubled at the positive frequencies, and
*set to zero at the negative frequencies.

Due to the asymmetry of $X_+(f)$ with respect to the frequency $f = 0$, it can already be said that the time function $x_+(t)$ except for a trivial special case $x_+(t)= 0 \ \ \circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\,\ \ X_+(f)= 0$ is always complex.

The spectrum $X_{\rm TP}(f)$ of the equivalent low-pass signal is obtained by shifting $X_+(f)$ to the left by the carrier frequency $f_{\rm T}$:
:$$X_{\rm TP}(f)= X_+(f+f_{\rm T}).$$

In the time domain this operation corresponds to the multiplication of $x_{\rm +}(t)$ with the complex exponential function with negative exponent:
:$$x_{\rm TP}(t) = x_{\rm +}(t)\cdot {\rm e}^{-{\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm}2 \pi \cdot f_{\rm T}\cdot \hspace{0.05cm}t}.$$

It can be seen that $x_{\rm TP}(t)$ is generally complex. But if $X_+(f)$ is symmetric about the carrier frequency $f_{\rm T}$, $X_{\rm TP}(f)$ is symmetric about the frequency $f=0$ and there is accordingly real time function $x_{\rm TP}(t)$.

===$x_{\rm TP}(t)$–Representation of a Sum of Three Harmonic Oscillations===

In our applet, we always assume a set of three rotating pointers. The physical signal is:
:$$x(t) = x_{\rm U}(t) + x_{\rm T}(t) + x_{\rm O}(t) = A_{\rm U}\cdot \cos\left(2\pi f_{\rm U}\cdot t- \varphi_{\rm U}\right)+A_{\rm T}\cdot \cos\left(2\pi f_{\rm T}\cdot t- \varphi_{\rm T}\right)+A_{\rm O}\cdot \cos\left(2\pi f_{\rm O}\cdot t- \varphi_{\rm O}\right). $$
* Each of the three harmonic oscillations $x_{\rm T}(t)$, $x_{\rm U}(t)$ and $x_{\rm O}(t)$ is represented by an amplitude $(A)$, a frequency $(f)$ and a phase value $(\varphi)$.
*The indices are based on the modulation method [[Modulationsverfahren/Zweiseitenband-Amplitudenmodulation|double sideband –amplitude modulation]]. &bdquo;T” stands for &bdquo;carrier”, &bdquo;U” for &bdquo;lowei sideband” and &bdquo;O” for &bdquo;upper Sideband”. Similarly, $f_{\rm U} < f_{\rm T}$ and $f_{\rm O} > f_{\rm T}$. There are no restrictions for the amplitudes and phases.

The associated equivalent low-pass signal is with $f_{\rm O}\hspace{0.01cm}' = f_{\rm O}- f_{\rm T} > 0$,   $f_{\rm U}\hspace{0.01cm}' = f_{\rm U}- f_{\rm T} < 0$  and  $f_{\rm T}\hspace{0.01cm}' = 0$:

:$$x_{\rm TP}(t) = x_\text{TP, T}(t) + x_\text{TP, O}(t) + x_\text{TP, U}(t) = A_{\rm T}\cdot {\rm e}^{-{\rm j} \varphi_{\rm T} } \hspace{0.1cm}+ \hspace{0.1cm} A_{\rm O}\cdot {\rm e}^{-{\rm j} \varphi_{\rm O} } \cdot {\rm e}^{ {\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm O}\hspace{0.01cm}'\hspace{0.05cm}\cdot \hspace{0.05cm}t}\hspace{0.1cm}+ \hspace{0.1cm}
A_{\rm U}\cdot {\rm e}^{-{\rm j} \varphi_{\rm U} } \cdot {\rm e}^{ {\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm U}\hspace{0.01cm}'\hspace{0.05cm}\cdot \hspace{0.05cm}t} . $$

{{GraueBox|TEXT=
$\text{Example 1:}$ 
The constellation given here results, for example, in the [[Modulationsverfahren/Zweiseitenband-Amplitudenmodulation#AM-Signale_und_-Spektren_bei_harmonischem_Eingangssignal|double sideband amplitude modulation]] of the message signal $x_{\rm N}(t) = A_{\rm N}\cdot \cos\left(2\pi f_{\rm N}\cdot t- \varphi_{\rm N}\right)$ with the carrier signal $x_{\rm T}(t) = A_{\rm T}\cdot \cos\left(2\pi f_{\rm T}\cdot t - \varphi_{\rm T}\right)$. This is discussed frequently in the exercises.

[[File:Ortskurve_4.png|center|frame|Spectrum $X_{\rm TP}(f)$ of the equivalent low–pass signal for different phase constellations |class=fit]]

There are some limitations to the program parameters in this approach:
* The frequencies are always $f\hspace{0.05cm}'_{\rm O} = f_{\rm N}$ and $f\hspace{0.05cm}'_{\rm U} = -f_{\rm N}$.
*Without distortion, the amplitude of the sidebands is $A_{\rm O}= A_{\rm U}= A_{\rm N}/2$.
*The respective phase relationships can be seen from the graphic.

}}

===Representation of the Equivalent Low-Pass Signal by Magnitude and Phase===

The generally complex equivalent low-pass signal
:$$x_{\rm TP}(t) = a(t) \cdot {\rm e}^{ {\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm} \phi(t) }$$
can be split into a magnitude function $a(t)$ and a phase function $\phi(t)$ according to the equation given here, where:
:$$a(t) = \vert x_{\rm TP}(t)\vert = \sqrt{ {\rm Re}^2\big [x_{\rm TP}(t)\big ] + {\rm Im}^2\big [x_{\rm TP}(t)\big ] }\hspace{0.05cm},$$
:$$\phi(t) = \text{arc }x_{\rm TP}(t) = \arctan \frac{{\rm Im}\big [x_{\rm TP}(t)\big ]}{{\rm Re}\big [x_{\rm TP}(t)\big ]}.$$

The reason for this is that a bandpass–signal $x(t)$ is usually described by the equivalent low-pass signal $x_{\rm TP}(t)$ that the functions $a(t)$ and $\phi(t)$ are interpretable in both representations:
*The amount $a(t)$ of the equivalent low-pass signal $x_{\rm TP}(t)$ indicates the (time-dependent) envelope of $x(t)$.
*The phase $\phi(t)$ of $x_{\rm TP}(t)$ denotes the location of the zero crossings of $x(t)$, where:
:–   For $\phi(t)>0$ the zero crossing is earlier than its nominal position   ⇒   the signal is leading here.
:–  When $\phi(t)<0$, the zero crossing is later than its target position   ⇒   the signal is trailing here.

{{GraueBox|TEXT=
$\text{Example 2:}$ 
The graph is intended to illustrate this relationship, assuming $A_{\rm U} > A_{\rm O}$   ⇒   the green pointer (for the lower sideband) is longer than the blue pointer (upper sideband). This is a snapshot at time $t_0$:

[[File:Ortskurve_3_neu.png|center|frame|bandpass spectrum $X(f)$ |class=fit]]

*For these system parameters, the top of the pointer cluster $x_{\rm TP}(t)$ – that is, the geometric sum of red, blue and green pointers – on an ellipse.
* The amount $a(t_0) = \vert x_{\rm TP}(t_0) \vert$ is drawn in black in the left-hand diagram and the phase value $\phi(t_0) = \text{arc }x_{\rm TP}(t_0) > 0$ is indicated in brown color.
*In the graph on the right, the amount $a(t_0) = \vert x_{\rm TP}(t_0) \vert$ of the equivalent low-pass signal indicates the envelope of the physical signal $x(t)$.
* At $\phi(t) \equiv 0$, all zero crossings of $x(t)$ would occur at equidistant intervals. Because of $\phi(t_0) > 0$, the signal is leading at the time $t_0$, that is: the zero crossings come earlier than the grid dictates. }}

==Exercises==
[[File:Exercises_verzerrungen.png|right]]
*First select the task number.
*A task description is displayed.
*Parameter values are adjusted.
*Solution after pressing &bdquo;Hide solition”.

The number &bdquo;0” will reset the program and output a text with the further explanation of the applet.

In the following, $\rm Green$ denotes the lower sideband   ⇒   $\big (A_{\rm U}, f_{\rm U}, \varphi_{\rm U}\big )$,  
$\rm Red$ the carrier   ⇒   $\big (A_{\rm T}, f_{\rm T}, \varphi_{\rm T}\big )$ and
$\rm Blue$ the upper sideband   ⇒   $\big (A_{\rm O}, f_{\rm O}, \varphi_{\rm O}\big )$.

{{BlaueBox|TEXT=
'''(1)'''   Let   $\text{Red:} \hspace{0.15cm} A_{\rm T} = 1\ \text{V}, f_{\rm T} = 100 \ \text{kHz}, \varphi_{\rm T} = 0^\circ$,   $\text{Green:} \hspace{0.15cm} A_{\rm U} = 0.4 \text{V}, \ f_{\rm U} = 80 \ \text{kHz}, \varphi_{\rm U} = -90^\circ$,   $\text{Blue:} \hspace{0.15cm} A_{\rm O} = 0.4\ \text{V}, f_{\rm O} = 120 \ \text{kHz}, \varphi_{\rm O} = 90^\circ$.

:Consider and interpret the equivalent low-pass signal $x_{\rm TP}(t)$ and the physical signal $x(t)$. Which period $T_0$ does one recognize?}}

:: The equivalent low-pass signal $x_{\rm TP}(t)$ takes from $x_{\rm TP}(t=0)=1\ \text{V}$ on the real axis values between $0.2\ \text{V}$ and $1.8\ \text{V}$   ⇒   phase $\phi(t) \equiv 0$.  The amount $|x_{\rm TP}(t)|$ indicates the envelope $a(t)$ of the physical signal $x(t)$. It holds $A_{\rm N} = 0.8\ \text{V}$ and $f_{\rm N} = 20\ \text{kHz}$:   $a(t) = A_{\rm T}+ A_{\rm N} \cdot \sin(2\pi\cdot f_{\rm N} \cdot t)$.  Both $x_{\rm TP}(t)$ and $x(t)$ are periodic with the period $T_0 = 1/f_{\rm N} = 50\ \rm µ s$.

{{BlaueBox|TEXT=
'''(2)'''   How do the ratios change to '''(1)''' with $f_{\rm U} = 99 \ \text{kHz}$ and $f_{\rm O} = 101 \ \text{kHz}$ ? How could $x(t)$ have arisen?}}

:: For the envelope $a(t)$ of the signal $x(t)$ we still have $a(t) = A_{\rm T}+ A_{\rm N} \cdot \sin(2\pi\cdot f_{\rm N} \cdot t)$, but now $f_{\rm N} = 1\ \text{kHz}$. Even though it can not be recognized:  $x_{\rm TP}(t)$ and $x(t)$ are still periodic:   $T_0 = 1\ \rm ms$. Example: Double sideband - Amplitude modulation '''(ZSB–AM)''' of a sine signal with cosine–carrier.

{{BlaueBox|TEXT=
'''(3)'''   Which settings have to be changed from '''(2)''' in order to arrive at the ZSB–AM of a cosine signal with sine–carrier. What changes over '''(2)'''?}}

::The carrier phase must be changed to $\varphi_{\rm T} = 90^\circ$   ⇒   sine–carrier. Similarly, $\varphi_{\rm O} =\varphi_{\rm U} =\varphi_{\rm T} = 90^\circ$ must be set   ⇒   cosine-shaped message  The locus now lies on the imaginary axis  ⇒   $\phi(t) \equiv -90^\circ$. At the beginning we have $x_{\rm TP}(t=0)= - {\rm j} \cdot 1.8 \ \text{V}$.

{{BlaueBox|TEXT=
'''(4)'''   Now let   $\text{Red:} \hspace{0.15cm} A_{\rm T} = 1\ \text{V}, \ f_{\rm T} = 100 \ \text{kHz}, \ \varphi_{\rm T} = 0^\circ$,   $\text{Green:} \hspace{0.15cm} A_{\rm U} = 0.4 \text{V}, \ f_{\rm U} = 80 \ \text{kHz}, \ \varphi_{\rm U} = 0^\circ$,   $\text{Blue:} \hspace{0.15cm} A_{\rm O} = 0.4\ \text{V}, \ f_{\rm O} = 120 \ \text{kHz}, \ \varphi_{\rm O} = 0^\circ$.

:What are the characteristics of this system &bdquo;ZSB–AM, where the message signal and carrier are respectively cosinusoidal”? What is the modulation depth $m$?}}

:: The equivalent low-pass signal $x_{\rm TP}(t)$ takes from $x_{\rm TP}(t=0)=1.8\ \text{V}$ on the real axis values between $0.2\ \text{V}$ and $1.8\ \text{V}$   ⇒   phase $\phi(t) \equiv 0$.  Except for the start state $x_{\rm TP}(t=0)$ same behavior as with the setting '''(1)'''. The modulation depth is $m = 0.8$.

{{BlaueBox|TEXT=
'''(5)'''   The parameters are still valid according to '''(4)''' with the exception of $A_{\rm T}= 0.6 \text{V}$. How big is the degree of modulation $m$? What are the consequences?}}

:: There is now a ZSB–AM with modulation degree $m = 1.333$. For $m > 1$, the simpler [[Modulationsverfahren/Hüllkurvendemodulation|envelope demodulation]] is not applicable, since the phase function $\phi(t) \in \{ 0, \ \pm 180^\circ\}$ is no longer constant and the envelope $a(t)$ no longer matches the message signal. Rather, the more complex [[Modulationsverfahren/Synchrondemodulation|synchronous demodulation]] must be used. Envelope detection would produce nonlinear distortions.

{{BlaueBox|TEXT=
'''(6)'''   The parameters are still valid according to '''(4)''' or '''(5)''' with the exception from $A_{\rm T}= 0$ on   ⇒   $m \to \infty$. Which modulation method is described in this way?}}

::It is a '''ZSB–AM without carrier''' and requires synchronous demodulation. The equivalent low-pass signal $x_{\rm TP}(t)$ is on the real axis, but not only in the right half-plane. Thus, the phase function $\phi(t) \in \{ 0, \ \pm 180^\circ\}$, also applies here, which means that envelope demodulation is not applicable.

{{BlaueBox|TEXT=
'''(7)'''   Now let   $\text{Red:} \hspace{0.15cm} A_{\rm T} = 1\ \text{V}, f_{\rm T} = 100 \ \text{kHz}, \varphi_{\rm T} = 0^\circ$,   $\text{Green:} \hspace{0.15cm} A_{\rm U} = 0, \ f_{\rm U} = 80 \ \text{kHz}, \varphi_{\rm U} = -90^\circ$,   $\text{Blue:} \hspace{0.15cm} A_{\rm O} = 0.8\ \text{V}, f_{\rm O} = 120 \ \text{kHz}, \varphi_{\rm O} = 90^\circ$.

:Which constellation is described here? Which characteristics of this procedure can be recognized from the graphic?}}

::It is a [[Modulationsverfahren/Einseitenbandmodulation|single-sideband modulation]] '''(ESB–AM)''', more specifically an '''OSB–AM''': the red carrier is fixed, the green pointer missing and the blue pointer (OSB) turns counterclockwise. The degree of modulation is $\mu = 0.8$ (in the case of ESB we denote the degree of modulation with $\mu$ instead of $m$). The carrier signal is cosinusoidal and the message signal sinusoidal. The locus is a circle. $x_{\rm TP}(t)$ moves in a mathematically positive direction. Because of $\phi(t) \ne \text{const.}$, The envelope demodulation is not applicable here:  This can be seen by the fact that the envelope $a(t)$ is not cosinusoidal. Rather, the lower half-wave is sharper than the upper   ⇒   strong linear distortions.

{{BlaueBox|TEXT=
'''(8)'''   The parameters are still valid according to '''(7)''' with the exception of $A_{\rm O}= 0$ and $A_{\rm U}= 0.8 \text{ V}$. What differences arise opposite '''(7)'''?}}

::Now it is a '''USB–AM''': The red carrier is fixed, the blue pointer is missing and the green pointer (USB) rotates clockwise. All other statements of '''(7)''' apply here as well.

{{BlaueBox|TEXT=
'''(9)'''   The parameters according to '''(7)''' are still valid with the exception of $A_{\rm O} = 0.2 \text{ V} \ne A_{\rm U} = 0.4 \text{ V} $. What are the differences from '''(7)'''?}}

::The locus $x_{\rm TP}(t)$ is not a horizontal straight line, but an ellipse with the real part between $0.4 \text{ V}$ and $1.6 \text{ V}$ and the imaginary part in the range $\pm 0.2 \text{ V}$. Because of $\phi(t) \ne \text{const.}$ , Envelope demodulation would lead to non-linear distortions here too. The constellation simulated here describes the situation of '''(4)''', namely a ZSB–AM with modulation degree $m = 0.8$, where the upper sideband is reduced to $50\%$ due to channel attenuation.

==Applet Manual==
[[File:Ortskurve_abzug3.png|right]]
* The red parameters $(A_{\rm T}, \ f_{\rm T}, \ \varphi_{\rm T})$ and the red pointer mark the ''Carrier''. (German: '''T'''räger).
* The green parameters $(A_{\rm U}, \ f_{\rm U} < f_{\rm T}, \ \varphi_{\rm U})$ mark the ''Lower sideband''. (German: '''U'''ntere Seitenband).
* The blue parameters $(A_{\rm O}, \ f_{\rm O} > f_{\rm T}, \ \varphi_{\rm O})$ mark the '' Upper sideband''. (German: '''O'''bere Seitenband).
* The red pointer does not turn.
* The green pointer rotates in a mathematically negative direction (clockwise).
* The blue pointer turns counterclockwise.

Meaning of the letters in the adjacent graphic:

    '''(A)'''     Plot of the equivalent low-pass Signal $x_{\rm TP}(t)$

    '''(B)'''     Plot of the physical signal $x(t)$

    '''(C)'''     Parameter input via slider:   amplitudes, frequencies, phase values

    '''(D)'''     Control elements:   Start – Step – Pause/Continue – Reset

    '''(E)'''     Speed of animation:   &bdquo;Speed”   ⇒   Values: 1, 2 oder 3

    '''(F)'''     &bdquo;Trace”   ⇒   On or Off, trace of equivalent low-pass Signal   $x_{\rm TP}(t)$

    '''(G)'''     Numerical output:   time $t$, the signal values  ${\rm Re}[x_{\rm TP}(t)]$  and  ${\rm Im}[x_{\rm TP}(t)]$,

$\text{}\hspace{4.2cm}$   Hüllkurve $a(t) = |x_{\rm TP}(t)|$  und  Phase $\phi(t) = {\rm arc} \ x_{\rm TP}(t)$

    '''(H)'''     Variations for the graphical representation

$\hspace{1.5cm}$Zoom–Functions &bdquo;$+$” (Enlarge), &bdquo;$-$” (Decrease) und $\rm o$ (Reset to default)

$\hspace{1.5cm}$Move with &bdquo;$\leftarrow$” (Section to the left, ordinate to the right), &bdquo;$\uparrow$” &bdquo;$\downarrow$” &bdquo;$\rightarrow$”

    '''(I)'''     Experiment section:   Task selection and task

    '''(J)'''     Experiment section:  solution
 

==About the Authors==
This interactive calculation was designed and realized at the [http://www.lnt.ei.tum.de/startseite Lehrstuhl für Nachrichtentechnik] of the [https://www.tum.de/ Technical University of Munich] .
*The original version was created in 2005 by [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Studierende#Ji_Li_.28Bachelorarbeit_EI_2003.2C_Diplomarbeit_EI_2005.29|Ji Li]] as part of her Diploma thesis using &bdquo;FlashMX–Actionscript” (Supervisor: [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Mitarbeiter_und_Dozenten#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28am_LNT_seit_1974.29|Günter Söder]]).
*In 2018 this Applet was redesigned and updated to &bdquo;HTML5” by [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Studierende#Xiaohan_Liu_.28Bachelorarbeit_2018.29|Xiaohan Liu]] as part of her Bachelor's thesis (Supervisor: [[Biografien_und_Bibliografien/Beteiligte_der_Professur_Leitungsgebundene_%C3%9Cbertragungstechnik#Tasn.C3.A1d_Kernetzky.2C_M.Sc._.28bei_L.C3.9CT_seit_2014.29|Tasnád Kernetzky]]).

==Once again: Open Applet in new Tab==

{{LntAppletLinkEn|physAnSignal}}

[[Category:Applets|^Verzerrungen^]]

Applets:Physical Signal & Equivalent Lowpass Signal

2018-08-25T15:27:46Z

Xiaohan:

{{LntAppletLinkEn|analPhysSignal}}

==Applet Description==
 
This applet shows the relationship between the physical bandpass signal $x(t)$ and the associated equivalent low-pass signal $x_{\rm TP}(t)$. The starting point is always a bandpass signal $x(t)$ with a frequency-discrete spectrum $X(f)$:
:$$x(t) = x_{\rm T}(t) + x_{\rm O}(t) + x_{\rm U}(t) = A_{\rm T}\cdot \cos\left(2\pi f_{\rm T}\cdot t- \varphi_{\rm T}\right)+A_{\rm O}\cdot \cos\left(2\pi f_{\rm O}\cdot t- \varphi_{\rm O}\right)+ A_{\rm U}\cdot \cos\left(2\pi f_{\rm U}\cdot t- \varphi_{\rm U}\right). $$
The physical signal $x(t)$ is thus composed of three [[Signaldarstellung/Harmonische_Schwingung|harmonic oscillations]],which results in a constellation, for example, in the [[Modulationsverfahren/Zweiseitenband-Amplitudenmodulation#AM-Signale_und_-Spektren_bei_harmonischem_Eingangssignal|double-sideband amplitude modulation]] of the message signal $x_{\rm N}(t) = A_{\rm N}\cdot \cos\left(2\pi f_{\rm N}\cdot t- \varphi_{\rm N}\right)$ with the carrier signal $x_{\rm T}(t) = A_{\rm T}\cdot \cos\left(2\pi f_{\rm T}\cdot t - \varphi_{\rm T}\right)$ returns. The nomenclature is also adapted to this case:
* $x_{\rm O}(t)$ denotes the &bdquo;upper sideband” with the amplitude $A_{\rm O}= A_{\rm N}/2$, the frequency $f_{\rm O} = f_{\rm T} + f_{\rm N}$ and the phase $\varphi_{\rm O} = \varphi_{\rm T} + \varphi_{\rm N}$.
*Similarly, for the &bdquo;lower Sideband” $x_{\rm U}(t)$ with $f_{\rm U} = f_{\rm T} + f_{\rm N}$, $A_{\rm U}= A_{\rm O}$ and $\varphi_{\rm U} = -\varphi_{\rm O}$.

The corresponding equivalent low-pass–signal is $f_{\rm O}\hspace{0.01cm}' = f_{\rm O}- f_{\rm T} > 0$,   $f_{\rm U}\hspace{0.01cm}' = f_{\rm U}- f_{\rm T} < 0$  and  $f_{\rm T}\hspace{0.01cm}' = 0$:

:$$x_{\rm TP}(t) = x_\text{TP, T}(t) + x_\text{TP, O}(t) + x_\text{TP, U}(t) = A_{\rm T}\cdot {\rm e}^{-{\rm j} \varphi_{\rm T} } \hspace{0.1cm}+ \hspace{0.1cm} A_{\rm O}\cdot {\rm e}^{-{\rm j} \varphi_{\rm O} } \cdot {\rm e}^{ {\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm O}\hspace{0.01cm}'\hspace{0.05cm}\cdot \hspace{0.05cm}t}\hspace{0.1cm}+ \hspace{0.1cm}
A_{\rm U}\cdot {\rm e}^{-{\rm j} \varphi_{\rm U} } \cdot {\rm e}^{ {\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm U}\hspace{0.01cm}'\hspace{0.05cm}\cdot \hspace{0.05cm}t} . $$

[[File:Ortskurve_1.png|right|frame|Equivalent lowpass–signal currently $t=0$ for cosinusoidal carrier   ⇒   $\varphi_{\rm T} = 0$]]
The program shows $x_{\rm TP}(t)$ as the vectorial sum of three rotation pointers as a violet point (see example graph for start time $t=0$ and cosinusoidal carrier):

*The (red) pointer of the carrier $x_\text{TP, T}(t)$ with the length $A_{\rm T}$ and the zero phase position $\varphi_{\rm T}=0$ is fixed in the complex plane. So it applies to all times $t$:   $x_{\rm TP}(t)= A_{\rm T}\cdot {\rm e}^{-{\rm j} \varphi_{\rm T} }$.

*The (blue) pointer of the upper sideband $x_\text{TP, O}(t)$ with the length $A_{\rm O}$ and the zero phase position $\varphi_{\rm O}$ rotates at the angular velocity $2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm O}\hspace{0.01cm}'$ in mathematically positive direction (one revolution in time $1/f_{\rm O}\hspace{0.01cm}')$.

*The (green) pointer of the lower sideband $x_{\rm U+}(t)$ with the length $A_{\rm U}$ and the zero phase position $\varphi_{\rm U}$ rotates at the angular velocity $2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm U}\hspace{0.01cm}'$, because of $f_{\rm U}\hspace{0.01cm}'<0$ counterclockwise.

*With $f_{\rm U}\hspace{0.01cm}' = -f_{\rm O}\hspace{0.01cm}'$ the blue and green pointers will spin at the same speed but in different directions. Also, if $A_{\rm O} = A_{\rm U}$ and $\varphi_{\rm U} = -\varphi_{\rm O}$, then $x_{\rm TP}(t)$ moves on a straight line with a incline of $\varphi_{\rm T}$.

''Hinweis:''   The graph applies to $\varphi_{\rm O} = +30^\circ$. From this follows for the start time $t=0$ the angle of the blue pointer (OSB) with respect to the coordinate system:   $\phi_{\rm O} = -\varphi_{\rm O} = -30^\circ$. Likewise, the null phase position $\varphi_{\rm U} = -30^\circ$ of the lower sideband (USB, grüner Zeiger) follows for the phase angle to be considered in the complex plane:   $\phi_{\rm U} = +30^\circ$.

The temporal process of $x_{\rm TP}(t)$ is also referred to below as '''locus'''. The relationship between $x_{\rm TP}(t)$ and the physical bandpass –signal $x(t)$ is given in the section and the associated analytic signal is $x_+(t)$ :

:$$x_{\rm TP}(t) = x_{\rm +}(t)\cdot {\rm e}^{-{\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm}2 \pi \cdot f_{\rm T}\cdot \hspace{0.05cm}t},$$
:$$x_{\rm +}(t) = x_{\rm TP}(t)\cdot {\rm e}^{+{\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm}2 \pi \cdot f_{\rm T}\cdot \hspace{0.05cm}t}.$$

[[Applets:Physikalisches_Signal_%26_Äquivalentes_TP-Signal|'''German description''']]

==Theoretical Background==
 
===Description of Bandpass Signals===
[[File:Zeigerdiagramm_1a.png|right|frame|bandpass–spectrum $X(f)$ |class=fit]]
We consider '''bandpass signals''' $x(t)$ with the property that their spectra $X(f)$ are not in the range around the frequency $f=0$, but around a carrier frequency $f_{\rm T}$. In most cases it can also be assumed that the bandwidth is $B \ll f_{\rm T}$.

The figure shows such a bandpass spectrum $X(f)$. Assuming that the associated $x(t)$ is a physical signal and thus real, the spectral function $X(f)$ has a symmetry with respect to the frequency $f = 0$, if $x(t)$ is an even function   ⇒   $x(-t)=x(t)$, $X(f)$ is real and even.

Beside the physical signal $x(t)\ \circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\, \ X(f)$, one can also use the following descriptions of bandpass signals:
*the analytic signal $x_+(t)\ \circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\, \ X_+(f)$, see applet [[Applets:Physikalisches_Signal_%26_Analytisches_Signal|Physikalisches Signal & Analytisches Signal]],
*the equivalent low-pass–signal $x_{\rm TP}(t)\ \circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\, \ X_{\rm TP}(f)$, see next page
 

===Spectral Functions of the Analytic and the Equivalent Low-Pass Signal===

The '''analytic signal''' $x_+(t)$ belonging to the physical signal $x(t)$ is the time function whose spectrum fulfills the following property:
[[File:Ortskurve_2.png|right|frame|spectral functions $X_+(f)$ and $X_{\rm TP}(f)$ |class=fit]]
:$$X_+(f)=\big[1+{\rm sign}(f)\big] \cdot X(f) = \left\{ {2 \cdot
X(f) \; \hspace{0.2cm}\rm for\hspace{0.2cm} {\it f} > 0, \atop {\,\,\,\, \rm 0 \; \hspace{0.9cm}\rm for\hspace{0.2cm} {\it f} < 0.} }\right.$$

The ''Signum function'' is for positive values of $f$ equal to $+1$ and for negative $f$ values equal to $-1$.
*The (double-sided) limit returns $\sign(0) = 0$.
*The index „+” should make it clear that $X_+(f)$ only has parts at positive frequencies.

From the graph you can see the calculation rule for $X_+(f)$: The actual BP spectrum $X(f)$ becomes
*doubled at the positive frequencies, and
*set to zero at the negative frequencies.

Due to the asymmetry of $X_+(f)$ with respect to the frequency $f = 0$, it can already be said that the time function $x_+(t)$ except for a trivial special case $x_+(t)= 0 \ \ \circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\,\ \ X_+(f)= 0$ is always complex.

The spectrum $X_{\rm TP}(f)$ of the equivalent low-pass signal is obtained by shifting $X_+(f)$ to the left by the carrier frequency $f_{\rm T}$:
:$$X_{\rm TP}(f)= X_+(f+f_{\rm T}).$$

In the time domain this operation corresponds to the multiplication of $x_{\rm +}(t)$ with the complex exponential function with negative exponent:
:$$x_{\rm TP}(t) = x_{\rm +}(t)\cdot {\rm e}^{-{\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm}2 \pi \cdot f_{\rm T}\cdot \hspace{0.05cm}t}.$$

It can be seen that $x_{\rm TP}(t)$ is generally complex. But if $X_+(f)$ is symmetric about the carrier frequency $f_{\rm T}$, $X_{\rm TP}(f)$ is symmetric about the frequency $f=0$ and there is accordingly real time function $x_{\rm TP}(t)$.

===$x_{\rm TP}(t)$–Representation of a Sum of Three Harmonic Oscillations===

In our applet, we always assume a set of three rotating pointers. The physical signal is:
:$$x(t) = x_{\rm U}(t) + x_{\rm T}(t) + x_{\rm O}(t) = A_{\rm U}\cdot \cos\left(2\pi f_{\rm U}\cdot t- \varphi_{\rm U}\right)+A_{\rm T}\cdot \cos\left(2\pi f_{\rm T}\cdot t- \varphi_{\rm T}\right)+A_{\rm O}\cdot \cos\left(2\pi f_{\rm O}\cdot t- \varphi_{\rm O}\right). $$
* Each of the three harmonic oscillations $x_{\rm T}(t)$, $x_{\rm U}(t)$ and $x_{\rm O}(t)$ is represented by an amplitude $(A)$, a frequency $(f)$ and a phase value $(\varphi)$.
*The indices are based on the modulation method [[Modulationsverfahren/Zweiseitenband-Amplitudenmodulation|double sideband –amplitude modulation]]. &bdquo;T” stands for &bdquo;carrier”, &bdquo;U” for &bdquo;lowei sideband” and &bdquo;O” for &bdquo;upper Sideband”. Similarly, $f_{\rm U} < f_{\rm T}$ and $f_{\rm O} > f_{\rm T}$. There are no restrictions for the amplitudes and phases.

The associated equivalent low-pass signal is with $f_{\rm O}\hspace{0.01cm}' = f_{\rm O}- f_{\rm T} > 0$,   $f_{\rm U}\hspace{0.01cm}' = f_{\rm U}- f_{\rm T} < 0$  and  $f_{\rm T}\hspace{0.01cm}' = 0$:

:$$x_{\rm TP}(t) = x_\text{TP, T}(t) + x_\text{TP, O}(t) + x_\text{TP, U}(t) = A_{\rm T}\cdot {\rm e}^{-{\rm j} \varphi_{\rm T} } \hspace{0.1cm}+ \hspace{0.1cm} A_{\rm O}\cdot {\rm e}^{-{\rm j} \varphi_{\rm O} } \cdot {\rm e}^{ {\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm O}\hspace{0.01cm}'\hspace{0.05cm}\cdot \hspace{0.05cm}t}\hspace{0.1cm}+ \hspace{0.1cm}
A_{\rm U}\cdot {\rm e}^{-{\rm j} \varphi_{\rm U} } \cdot {\rm e}^{ {\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm U}\hspace{0.01cm}'\hspace{0.05cm}\cdot \hspace{0.05cm}t} . $$

{{GraueBox|TEXT=
$\text{Example 1:}$ 
The constellation given here results, for example, in the [[Modulationsverfahren/Zweiseitenband-Amplitudenmodulation#AM-Signale_und_-Spektren_bei_harmonischem_Eingangssignal|double sideband amplitude modulation]] of the message signal $x_{\rm N}(t) = A_{\rm N}\cdot \cos\left(2\pi f_{\rm N}\cdot t- \varphi_{\rm N}\right)$ with the carrier signal $x_{\rm T}(t) = A_{\rm T}\cdot \cos\left(2\pi f_{\rm T}\cdot t - \varphi_{\rm T}\right)$. This is discussed frequently in the exercises.

[[File:Ortskurve_4.png|center|frame|Spectrum $X_{\rm TP}(f)$ of the equivalent low–pass signal for different phase constellations |class=fit]]

There are some limitations to the program parameters in this approach:
* The frequencies are always $f\hspace{0.05cm}'_{\rm O} = f_{\rm N}$ and $f\hspace{0.05cm}'_{\rm U} = -f_{\rm N}$.
*Without distortion, the amplitude of the sidebands is $A_{\rm O}= A_{\rm U}= A_{\rm N}/2$.
*The respective phase relationships can be seen from the graphic.

}}

===Representation of the Equivalent Low-Pass Signal by Magnitude and Phase===

The generally complex equivalent low-pass signal
:$$x_{\rm TP}(t) = a(t) \cdot {\rm e}^{ {\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm} \phi(t) }$$
can be split into a magnitude function $a(t)$ and a phase function $\phi(t)$ according to the equation given here, where:
:$$a(t) = \vert x_{\rm TP}(t)\vert = \sqrt{ {\rm Re}^2\big [x_{\rm TP}(t)\big ] + {\rm Im}^2\big [x_{\rm TP}(t)\big ] }\hspace{0.05cm},$$
:$$\phi(t) = \text{arc }x_{\rm TP}(t) = \arctan \frac{{\rm Im}\big [x_{\rm TP}(t)\big ]}{{\rm Re}\big [x_{\rm TP}(t)\big ]}.$$

The reason for this is that a bandpass–signal $x(t)$ is usually described by the equivalent low-pass signal $x_{\rm TP}(t)$ that the functions $a(t)$ and $\phi(t)$ are interpretable in both representations:
*The amount $a(t)$ of the equivalent low-pass signal $x_{\rm TP}(t)$ indicates the (time-dependent) envelope of $x(t)$.
*The phase $\phi(t)$ of $x_{\rm TP}(t)$ denotes the location of the zero crossings of $x(t)$, where:
:–   For $\phi(t)>0$ the zero crossing is earlier than its nominal position   ⇒   the signal is leading here.
:–  When $\phi(t)<0$, the zero crossing is later than its target position   ⇒   the signal is trailing here.

{{GraueBox|TEXT=
$\text{Example 2:}$ 
The graph is intended to illustrate this relationship, assuming $A_{\rm U} > A_{\rm O}$   ⇒   the green pointer (for the lower sideband) is longer than the blue pointer (upper sideband). This is a snapshot at time $t_0$:

[[File:Ortskurve_3_neu.png|center|frame|bandpass spectrum $X(f)$ |class=fit]]

*For these system parameters, the top of the pointer cluster $x_{\rm TP}(t)$ – that is, the geometric sum of red, blue and green pointers – on an ellipse.
* The amount $a(t_0) = \vert x_{\rm TP}(t_0) \vert$ is drawn in black in the left-hand diagram and the phase value $\phi(t_0) = \text{arc }x_{\rm TP}(t_0) > 0$ is indicated in brown color.
*In the graph on the right, the amount $a(t_0) = \vert x_{\rm TP}(t_0) \vert$ of the equivalent low-pass signal indicates the envelope of the physical signal $x(t)$.
* At $\phi(t) \equiv 0$, all zero crossings of $x(t)$ would occur at equidistant intervals. Because of $\phi(t_0) > 0$, the signal is leading at the time $t_0$, that is: the zero crossings come earlier than the grid dictates. }}

==Exercises==
[[File:Exercises_verzerrungen.png|right]]
*First select the task number.
*A task description is displayed.
*Parameter values are adjusted.
*Solution after pressing &bdquo;Hide solition”.

The number &bdquo;0” will reset the program and output a text with the further explanation of the applet.

In the following, $\rm Green$ denotes the lower sideband   ⇒   $\big (A_{\rm U}, f_{\rm U}, \varphi_{\rm U}\big )$,  
$\rm Red$ the carrier   ⇒   $\big (A_{\rm T}, f_{\rm T}, \varphi_{\rm T}\big )$ and
$\rm Blue$ the upper sideband   ⇒   $\big (A_{\rm O}, f_{\rm O}, \varphi_{\rm O}\big )$.

{{BlaueBox|TEXT=
'''(1)'''   Let   $\text{Red:} \hspace{0.15cm} A_{\rm T} = 1\ \text{V}, f_{\rm T} = 100 \ \text{kHz}, \varphi_{\rm T} = 0^\circ$,   $\text{Green:} \hspace{0.15cm} A_{\rm U} = 0.4 \text{V}, \ f_{\rm U} = 80 \ \text{kHz}, \varphi_{\rm U} = -90^\circ$,   $\text{Blue:} \hspace{0.15cm} A_{\rm O} = 0.4\ \text{V}, f_{\rm O} = 120 \ \text{kHz}, \varphi_{\rm O} = 90^\circ$.

:Consider and interpret the equivalent low-pass signal $x_{\rm TP}(t)$ and the physical signal $x(t)$. Which period $T_0$ does one recognize?}}

:: The equivalent low-pass signal $x_{\rm TP}(t)$ takes from $x_{\rm TP}(t=0)=1\ \text{V}$ on the real axis values between $0.2\ \text{V}$ and $1.8\ \text{V}$   ⇒   phase $\phi(t) \equiv 0$.  The amount $|x_{\rm TP}(t)|$ indicates the envelope $a(t)$ of the physical signal $x(t)$. It holds $A_{\rm N} = 0.8\ \text{V}$ and $f_{\rm N} = 20\ \text{kHz}$:   $a(t) = A_{\rm T}+ A_{\rm N} \cdot \sin(2\pi\cdot f_{\rm N} \cdot t)$.  Both $x_{\rm TP}(t)$ and $x(t)$ are periodic with the period $T_0 = 1/f_{\rm N} = 50\ \rm µ s$.

{{BlaueBox|TEXT=
'''(2)'''   How do the ratios change to '''(1)''' with $f_{\rm U} = 99 \ \text{kHz}$ and $f_{\rm O} = 101 \ \text{kHz}$ ? How could $x(t)$ have arisen?}}

:: For the envelope $a(t)$ of the signal $x(t)$ we still have $a(t) = A_{\rm T}+ A_{\rm N} \cdot \sin(2\pi\cdot f_{\rm N} \cdot t)$, but now $f_{\rm N} = 1\ \text{kHz}$. Even though it can not be recognized:  $x_{\rm TP}(t)$ and $x(t)$ are still periodic:   $T_0 = 1\ \rm ms$. Example: Double sideband - Amplitude modulation '''(ZSB–AM)''' of a sine signal with cosine–carrier.

{{BlaueBox|TEXT=
'''(3)'''   Which settings have to be changed from '''(2)''' in order to arrive at the ZSB–AM of a cosine signal with sine–carrier. What changes over '''(2)'''?}}

::The carrier phase must be changed to $\varphi_{\rm T} = 90^\circ$   ⇒   sine–carrier. Similarly, $\varphi_{\rm O} =\varphi_{\rm U} =\varphi_{\rm T} = 90^\circ$ must be set   ⇒   cosine-shaped message  The locus now lies on the imaginary axis  ⇒   $\phi(t) \equiv -90^\circ$. At the beginning we have $x_{\rm TP}(t=0)= - {\rm j} \cdot 1.8 \ \text{V}$.

{{BlaueBox|TEXT=
'''(4)'''   Now let   $\text{Red:} \hspace{0.15cm} A_{\rm T} = 1\ \text{V}, \ f_{\rm T} = 100 \ \text{kHz}, \ \varphi_{\rm T} = 0^\circ$,   $\text{Green:} \hspace{0.15cm} A_{\rm U} = 0.4 \text{V}, \ f_{\rm U} = 80 \ \text{kHz}, \ \varphi_{\rm U} = 0^\circ$,   $\text{Blue:} \hspace{0.15cm} A_{\rm O} = 0.4\ \text{V}, \ f_{\rm O} = 120 \ \text{kHz}, \ \varphi_{\rm O} = 0^\circ$.

:What are the characteristics of this system &bdquo;ZSB–AM, where the message signal and carrier are respectively cosinusoidal”? What is the modulation depth $m$?}}

:: The equivalent low-pass signal $x_{\rm TP}(t)$ takes from $x_{\rm TP}(t=0)=1.8\ \text{V}$ on the real axis values between $0.2\ \text{V}$ and $1.8\ \text{V}$   ⇒   phase $\phi(t) \equiv 0$.  Except for the start state $x_{\rm TP}(t=0)$ same behavior as with the setting '''(1)'''. The modulation depth is $m = 0.8$.

{{BlaueBox|TEXT=
'''(5)'''   The parameters are still valid according to '''(4)''' with the exception of $A_{\rm T}= 0.6 \text{V}$. How big is the degree of modulation $m$? What are the consequences?}}

:: There is now a ZSB–AM with modulation degree $m = 1.333$. For $m > 1$, the simpler [[Modulationsverfahren/Hüllkurvendemodulation|envelope demodulation]] is not applicable, since the phase function $\phi(t) \in \{ 0, \ \pm 180^\circ\}$ is no longer constant and the envelope $a(t)$ no longer matches the message signal. Rather, the more complex [[Modulationsverfahren/Synchrondemodulation|synchronous demodulation]] must be used. Envelope detection would produce nonlinear distortions.

{{BlaueBox|TEXT=
'''(6)'''   The parameters are still valid according to '''(4)''' or '''(5)''' with the exception from $A_{\rm T}= 0$ on   ⇒   $m \to \infty$. Which modulation method is described in this way?}}

::It is a '''ZSB–AM without carrier''' and requires synchronous demodulation. The equivalent low-pass signal $x_{\rm TP}(t)$ is on the real axis, but not only in the right half-plane. Thus, the phase function $\phi(t) \in \{ 0, \ \pm 180^\circ\}$, also applies here, which means that envelope demodulation is not applicable.

{{BlaueBox|TEXT=
'''(7)'''   Now let   $\text{Red:} \hspace{0.15cm} A_{\rm T} = 1\ \text{V}, f_{\rm T} = 100 \ \text{kHz}, \varphi_{\rm T} = 0^\circ$,   $\text{Green:} \hspace{0.15cm} A_{\rm U} = 0, \ f_{\rm U} = 80 \ \text{kHz}, \varphi_{\rm U} = -90^\circ$,   $\text{Blue:} \hspace{0.15cm} A_{\rm O} = 0.8\ \text{V}, f_{\rm O} = 120 \ \text{kHz}, \varphi_{\rm O} = 90^\circ$.

:Which constellation is described here? Which characteristics of this procedure can be recognized from the graphic?}}

::It is a [[Modulationsverfahren/Einseitenbandmodulation|single-sideband modulation]] '''(ESB–AM)''', more specifically an '''OSB–AM''': the red carrier is fixed, the green pointer missing and the blue pointer (OSB) turns counterclockwise. The degree of modulation is $\mu = 0.8$ (in the case of ESB we denote the degree of modulation with $\mu$ instead of $m$). The carrier signal is cosinusoidal and the message signal sinusoidal. The locus is a circle. $x_{\rm TP}(t)$ moves in a mathematically positive direction. Because of $\phi(t) \ne \text{const.}$, The envelope demodulation is not applicable here:  This can be seen by the fact that the envelope $a(t)$ is not cosinusoidal. Rather, the lower half-wave is sharper than the upper   ⇒   strong linear distortions.

{{BlaueBox|TEXT=
'''(8)'''   The parameters are still valid according to '''(7)''' with the exception of $A_{\rm O}= 0$ and $A_{\rm U}= 0.8 \text{ V}$. What differences arise opposite '''(7)'''?}}

::Now it is a '''USB–AM''': The red carrier is fixed, the blue pointer is missing and the green pointer (USB) rotates clockwise. All other statements of '''(7)''' apply here as well.

{{BlaueBox|TEXT=
'''(9)'''   The parameters according to '''(7)''' are still valid with the exception of $A_{\rm O} = 0.2 \text{ V} \ne A_{\rm U} = 0.4 \text{ V} $. What are the differences from '''(7)'''?}}

::The locus $x_{\rm TP}(t)$ is not a horizontal straight line, but an ellipse with the real part between $0.4 \text{ V}$ and $1.6 \text{ V}$ and the imaginary part in the range $\pm 0.2 \text{ V}$. Because of $\phi(t) \ne \text{const.}$ , Envelope demodulation would lead to non-linear distortions here too. The constellation simulated here describes the situation of '''(4)''', namely a ZSB–AM with modulation degree $m = 0.8$, where the upper sideband is reduced to $50\%$ due to channel attenuation.

==Applet Manual==
[[File:Ortskurve_abzug3.png|right]]
* The red parameters $(A_{\rm T}, \ f_{\rm T}, \ \varphi_{\rm T})$ and the red pointer mark the ''Carrier''. (German: '''T'''räger).
* The green parameters $(A_{\rm U}, \ f_{\rm U} < f_{\rm T}, \ \varphi_{\rm U})$ mark the ''Lower sideband''. (German: '''U'''ntere Seitenband).
* The blue parameters $(A_{\rm O}, \ f_{\rm O} > f_{\rm T}, \ \varphi_{\rm O})$ mark the '' Upper sideband''. (German: '''O'''bere Seitenband).
* The red pointer does not turn.
* The green pointer rotates in a mathematically negative direction (clockwise).
* The blue pointer turns counterclockwise.

Meaning of the letters in the adjacent graphic:

    '''(A)'''     Plot of the equivalent low-pass Signal $x_{\rm TP}(t)$

    '''(B)'''     Plot of the physical signal $x(t)$

    '''(C)'''     Parameter input via slider:   amplitudes, frequencies, phase values

    '''(D)'''     Control elements:   Start – Step – Pause/Continue – Reset

    '''(E)'''     Speed of animation:   &bdquo;Speed”   ⇒   Values: 1, 2 oder 3

    '''(F)'''     &bdquo;Trace”   ⇒   On or Off, trace of equivalent low-pass Signal   $x_{\rm TP}(t)$

    '''(G)'''     Numerical output:   time $t$, the signal values  ${\rm Re}[x_{\rm TP}(t)]$  and  ${\rm Im}[x_{\rm TP}(t)]$,

$\text{}\hspace{4.2cm}$   Hüllkurve $a(t) = |x_{\rm TP}(t)|$  und  Phase $\phi(t) = {\rm arc} \ x_{\rm TP}(t)$

    '''(H)'''     Variations for the graphical representation

$\hspace{1.5cm}$Zoom–Functions &bdquo;$+$” (Enlarge), &bdquo;$-$” (Decrease) und $\rm o$ (Reset to default)

$\hspace{1.5cm}$Move with &bdquo;$\leftarrow$” (Section to the left, ordinate to the right), &bdquo;$\uparrow$” &bdquo;$\downarrow$” &bdquo;$\rightarrow$”

    '''(I)'''     Experiment section:   Task selection and task

    '''(J)'''     Experiment section:  solution
 

==About the Authors==
This interactive calculation was designed and realized at the [http://www.lnt.ei.tum.de/startseite Lehrstuhl für Nachrichtentechnik] of the [https://www.tum.de/ Technical University of Munich] .
*The original version was created in 2005 by [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Studierende#Ji_Li_.28Bachelorarbeit_EI_2003.2C_Diplomarbeit_EI_2005.29|Ji Li]] as part of her Diploma thesis using &bdquo;FlashMX–Actionscript” (Supervisor: [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Mitarbeiter_und_Dozenten#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28am_LNT_seit_1974.29|Günter Söder]]).
*In 2018 this Applet was redesigned and updated to &bdquo;HTML5” by [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Studierende#Xiaohan_Liu_.28Bachelorarbeit_2018.29|Xiaohan Liu]] as part of her Bachelor's thesis (Supervisor: [[Biografien_und_Bibliografien/Beteiligte_der_Professur_Leitungsgebundene_%C3%9Cbertragungstechnik#Tasn.C3.A1d_Kernetzky.2C_M.Sc._.28bei_L.C3.9CT_seit_2014.29|Tasnád Kernetzky]]).

==Once again: Open Applet in new Tab==

{{LntAppletLinkEn|physAnSignal}}

[[Category:Applets|^Verzerrungen^]]

Applets:Physikalisches Signal & Äquivalentes TP-Signal

2018-08-25T15:11:37Z

Xiaohan: /* Spektralfunktionen des analytischen und des äquivalenten TP–Signals */

{{LntAppletLink|physAnTPSignal}}

==Programmbeschreibung==
 
Dieses Applet zeigt den Zusammenhang zwischen dem physikalischen Bandpass–Signal $x(t)$ und dem dazugehörigen äquivalenten Tiefpass–Signal $x_{\rm TP}(t)$. Ausgegangen wird stets von einem Bandpass–Signal $x(t)$ mit frequenzdiskretem Spektrum $X(f)$:
:$$x(t) = x_{\rm T}(t) + x_{\rm O}(t) + x_{\rm U}(t) = A_{\rm T}\cdot \cos\left(2\pi f_{\rm T}\cdot t- \varphi_{\rm T}\right)+A_{\rm O}\cdot \cos\left(2\pi f_{\rm O}\cdot t- \varphi_{\rm O}\right)+ A_{\rm U}\cdot \cos\left(2\pi f_{\rm U}\cdot t- \varphi_{\rm U}\right). $$
Das physikalische Signal $x(t)$ setzt sich also aus drei [[Signaldarstellung/Harmonische_Schwingung|harmonischen Schwingungen]] zusammen, einer Konstellation, die sich zum Beispiel bei der [[Modulationsverfahren/Zweiseitenband-Amplitudenmodulation#AM-Signale_und_-Spektren_bei_harmonischem_Eingangssignal|Zweiseitenband-Amplitudenmodulation]] des Nachrichtensignals $x_{\rm N}(t) = A_{\rm N}\cdot \cos\left(2\pi f_{\rm N}\cdot t- \varphi_{\rm N}\right)$ mit dem Trägersignal $x_{\rm T}(t) = A_{\rm T}\cdot \cos\left(2\pi f_{\rm T}\cdot t - \varphi_{\rm T}\right)$ ergibt. Die Nomenklatur ist ebenfalls an diesen Fall angepasst:
* $x_{\rm O}(t)$ bezeichnet das &bdquo;Obere Seitenband” mit der Amplitude $A_{\rm O}= A_{\rm N}/2$, der Frequenz $f_{\rm O} = f_{\rm T} + f_{\rm N}$ und der Phase $\varphi_{\rm O} = \varphi_{\rm T} + \varphi_{\rm N}$.
*Entsprechend gilt für das &bdquo;Untere Seitenband” $x_{\rm U}(t)$ mit $f_{\rm U} = f_{\rm T} - f_{\rm N}$, $A_{\rm U}= A_{\rm O}$ und $\varphi_{\rm U} = -\varphi_{\rm O}$.

Das dazugehörige äquivalente Tiefpass–Signal lautet mit $f_{\rm O}\hspace{0.01cm}' = f_{\rm O}- f_{\rm T} > 0$,   $f_{\rm U}\hspace{0.01cm}' = f_{\rm U}- f_{\rm T} < 0$  und  $f_{\rm T}\hspace{0.01cm}' = 0$:

:$$x_{\rm TP}(t) = x_\text{TP, T}(t) + x_\text{TP, O}(t) + x_\text{TP, U}(t) = A_{\rm T}\cdot {\rm e}^{-{\rm j} \varphi_{\rm T} } \hspace{0.1cm}+ \hspace{0.1cm} A_{\rm O}\cdot {\rm e}^{-{\rm j} \varphi_{\rm O} } \cdot {\rm e}^{ {\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm O}\hspace{0.01cm}'\hspace{0.05cm}\cdot \hspace{0.05cm}t}\hspace{0.1cm}+ \hspace{0.1cm}
A_{\rm U}\cdot {\rm e}^{-{\rm j} \varphi_{\rm U} } \cdot {\rm e}^{ {\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm U}\hspace{0.01cm}'\hspace{0.05cm}\cdot \hspace{0.05cm}t} . $$

[[File:Ortskurve_1.png|right|frame|Äquivalentes TP–Signal zur Zeit $t=0$ bei cosinusförmigem Träger   ⇒   $\varphi_{\rm T} = 0$]]
Im Programm dargestellt wird $x_{\rm TP}(t)$ als vektorielle Summe dreier Drehzeiger als violetter Punkt (siehe beispielhafte Grafik für den Startzeitpunkt $t=0$ und cosinusförmigem Träger):

*Der (rote) Zeiger des Trägers $x_\text{TP, T}(t)$ mit der Länge $A_{\rm T}$ und der Nullphasenlage $\varphi_{\rm T} = 0$ liegt in der komplexen Ebene fest. Es gilt also für alle Zeiten $t$:   $x_{\rm TP}(t)= A_{\rm T}\cdot {\rm e}^{-{\rm j} \varphi_{\rm T} }$.

*Der (blaue) Zeiger des Oberen Seitenbandes $x_\text{TP, O}(t)$ mit der Länge $A_{\rm O}$ und der Nullphasenlage $\varphi_{\rm O}$ dreht mit der Winkelgeschwindigkeit $2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm O}\hspace{0.01cm}'$ in mathematisch positiver Richtung (eine Umdrehung in der Zeit $1/f_{\rm O}\hspace{0.01cm}')$.

*Der (grüne) Zeiger des Unteren Seitenbandes $x_{\rm U+}(t)$ mit der Länge $A_{\rm U}$ und der Nullphasenlage $\varphi_{\rm U}$ dreht mit der Winkelgeschwindigkeit $2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm U}\hspace{0.01cm}'$, wegen $f_{\rm U}\hspace{0.01cm}'<0$ im Uhrzeigersinn (mathematisch negative Richtung).

*Mit $f_{\rm U}\hspace{0.01cm}' = -f_{\rm O}\hspace{0.01cm}'$ drehen der blaue und der grüne Zeiger gleich schnell, aber in unterschiedlichen Richtungen. Gilt zudem $A_{\rm O} = A_{\rm U}$ und $\varphi_{\rm U} = -\varphi_{\rm O}$, so bewegt sich $x_{\rm TP}(t)$ auf einer Geraden mit einer Neigung von $\varphi_{\rm T}$.

''Hinweis:''   Die Grafik gilt für $\varphi_{\rm O} = +30^\circ$. Daraus folgt für den Startzeitpunkt $t=0$ der Winkel des blauen Zeigers (OSB) gegenüber dem Koordinatensystem:   $\phi_{\rm O} = -\varphi_{\rm O} = -30^\circ$. Ebenso folgt aus der Nullphasennlage $\varphi_{\rm U} = -30^\circ$ des unteren Seitenbandes (USB, grüner Zeiger) für den in der komplexen Ebene zu berücksichtigenden Phasenwinkel:   $\phi_{\rm U} = +30^\circ$.

Den zeitlichen Verlauf von $x_{\rm TP}(t)$ bezeichnen wir im Folgenden auch als '''Ortskurve'''. Der Zusammenhang zwischen $x_{\rm TP}(t)$ und dem physikalischen Bandpass–Signal $x(t)$ wird im Abschnitt [[???]] angegeben. Der Zusammenhang zwischen $x_{\rm TP}(t)$ und dem dazugehörigen analytischen Signal $x_+(t)$ lautet:

:$$x_{\rm TP}(t) = x_{\rm +}(t)\cdot {\rm e}^{-{\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm}2 \pi \cdot f_{\rm T}\cdot \hspace{0.05cm}t},$$
:$$x_{\rm +}(t) = x_{\rm TP}(t)\cdot {\rm e}^{+{\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm}2 \pi \cdot f_{\rm T}\cdot \hspace{0.05cm}t}.$$

[[Applets:Physical_Signal_%26_Equivalent_Low-pass_Signal|'''Englische Beschreibung''']]

==Theoretischer Hintergrund==
 
===Beschreibungsmöglichkeiten von Bandpass-Signalen===
[[File:Zeigerdiagramm_1a.png|right|frame|Bandpass–Spektrum $X(f)$ |class=fit]]
Wir betrachten hier '''Bandpass-Signale''' $x(t)$ mit der Eigenschaft, dass deren Spektren $X(f)$ nicht im Bereich um die Frequenz $f = 0$ liegen, sondern um eine Trägerfrequenz $f_{\rm T}$. Meist kann auch davon ausgegangen werden, dass die Bandbreite $B \ll f_{\rm T}$ ist.

Die Grafik zeigt ein solches Bandpass–Spektrum $X(f)$. Unter der Annahme, dass das zugehörige $x(t)$ ein physikalisches Signal und damit reell ist, ergibt sich für die Spektralfunktion $X(f)$ eine Symmetrie bezüglich der Frequenz $f = 0$. Ist $x(t)$ eine gerade Funktion   ⇒   $x(-t)=x(+t)$, so ist auch $X(f)$ reell und gerade.

Neben dem physikalischen Signal $x(t)\ \circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\, \ X(f)$ verwendet man zur Beschreibung von Bandpass-Signalen gleichermaßen:
*das analytische Signal $x_+(t)\ \circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\, \ X_+(f)$, siehe Applet [[Applets:Physikalisches_Signal_%26_Analytisches_Signal|Physikalisches Signal & Analytisches Signal]],
*das äquivalente Tiefpass–Signal $x_{\rm TP}(t)\ \circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\, \ X_{\rm TP}(f)$, wie im nächsten Unterabschnitt beschrieben.
 
===Spektralfunktionen des analytischen und des äquivalenten TP–Signals===

Das zum physikalischen Signal $x(t)$ gehörige '''analytische Signal''' $x_+(t)$ ist diejenige Zeitfunktion, deren Spektrum folgende Eigenschaft erfüllt:
[[File:Ortskurve_2.png|right|frame|Spektralfunktionen $X_+(f)$ und $X_{\rm TP}(f)$ |class=fit]]
:$$X_+(f)=\big[1+{\rm sign}(f)\big] \cdot X(f) = \left\{ {2 \cdot
X(f) \; \hspace{0.2cm}\rm f\ddot{u}r\hspace{0.2cm} {\it f} > 0, \atop {\,\,\,\, \rm 0 \; \hspace{0.9cm}\rm f\ddot{u}r\hspace{0.2cm} {\it f} < 0.} }\right.$$

Die so genannte ''Signumfunktion'' ist dabei für positive Werte von $f$ gleich $+1$ und für negative $f$–Werte gleich $-1$.
*Der (beidseitige) Grenzwert liefert $\sign(0) = 0$.
*Der Index „+” soll deutlich machen, dass $X_+(f)$ nur Anteile bei positiven Frequenzen besitzt.

Aus der Grafik erkennt man die Berechnungsvorschrift für $X_+(f)$: Das tatsächliche BP–Spektrum $X(f)$ wird
*bei den positiven Frequenzen verdoppelt, und
*bei den negativen Frequenzen zu Null gesetzt.

Aufgrund der Unsymmetrie von $X_+(f)$ bezüglich der Frequenz $f = 0$ kann man bereits jetzt schon sagen, dass die Zeitfunktion $x_+(t)$ bis auf einen trivialen Sonderfall $x_+(t)= 0 \ \ \circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\,\ \ X_+(f)= 0$ stets komplex ist.

Zum Spektrum $X_{\rm TP}(f)$ des äquivalenten TP–Signals kommt man, indem man $X_+(f)$ um die Trägerfrequenz $f_{\rm T}$ nach links verschiebt:
:$$X_{\rm TP}(f)= X_+(f+f_{\rm T}).$$

Im Zeitbereich entspricht diese Operation der Multiplkation von $x_{\rm +}(t)$ mit der komplexen Exponentialfunktion mit negativem Exponenten:
:$$x_{\rm TP}(t) = x_{\rm +}(t)\cdot {\rm e}^{-{\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm}2 \pi \cdot f_{\rm T}\cdot \hspace{0.05cm}t}.$$

Man erkennt, dass $x_{\rm TP}(t)$ im Allgemeinen komplexwertig ist. Ist aber $X_+(f)$ symmetrisch um die Trägerfrequenz $f_{\rm T}$, so ist $X_{\rm TP}(f)$ symmetrisch um die Frequenz $f=0$ und es ergibt sich dementsprechend eine reelle Zeitfunktion $x_{\rm TP}(t)$.

===$x_{\rm TP}(t)$–Darstellung einer Summe aus drei harmonischen Schwingungen===

In unserem Applet setzen wir stets einen Zeigerverbund aus drei Drehzeigern voraus. Das physikalische Signal lautet:
:$$x(t) = x_{\rm U}(t) + x_{\rm T}(t) + x_{\rm O}(t) = A_{\rm U}\cdot \cos\left(2\pi f_{\rm U}\cdot t- \varphi_{\rm U}\right)+A_{\rm T}\cdot \cos\left(2\pi f_{\rm T}\cdot t- \varphi_{\rm T}\right)+A_{\rm O}\cdot \cos\left(2\pi f_{\rm O}\cdot t- \varphi_{\rm O}\right). $$
* Jede der drei harmonischen Schwingungen harmonischen Schwingungen $x_{\rm T}(t)$, $x_{\rm U}(t)$ und $x_{\rm O}(t)$ wird durch eine Amplitude $(A)$, eine Frequenz $(f)$ und einen Phasenwert $(\varphi)$ charakterisiert.
*Die Indizes sind an das Modulationsverfahren [[Modulationsverfahren/Zweiseitenband-Amplitudenmodulation|Zweiseitenband–Amplitudenmodulation]] angelehnt. &bdquo;T” steht für &bdquo;Träger”, &bdquo;U” für &bdquo;Unteres Seitenband” und &bdquo;O” für &bdquo;Oberes Seitenband”. Entsprechend gilt stets $f_{\rm U} < f_{\rm T}$ und $f_{\rm O} > f_{\rm T}$. Für die Amplituden und Phasen gibt es keine Einschränkungen.

Das dazugehörige äquivalente Tiefpass–Signal lautet mit $f_{\rm O}\hspace{0.01cm}' = f_{\rm O}- f_{\rm T} > 0$,   $f_{\rm U}\hspace{0.01cm}' = f_{\rm U}- f_{\rm T} < 0$  und  $f_{\rm T}\hspace{0.01cm}' = 0$:

:$$x_{\rm TP}(t) = x_\text{TP, T}(t) + x_\text{TP, O}(t) + x_\text{TP, U}(t) = A_{\rm T}\cdot {\rm e}^{-{\rm j} \varphi_{\rm T} } \hspace{0.1cm}+ \hspace{0.1cm} A_{\rm O}\cdot {\rm e}^{-{\rm j} \varphi_{\rm O} } \cdot {\rm e}^{ {\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm O}\hspace{0.01cm}'\hspace{0.05cm}\cdot \hspace{0.05cm}t}\hspace{0.1cm}+ \hspace{0.1cm}
A_{\rm U}\cdot {\rm e}^{-{\rm j} \varphi_{\rm U} } \cdot {\rm e}^{ {\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm U}\hspace{0.01cm}'\hspace{0.05cm}\cdot \hspace{0.05cm}t} . $$

{{GraueBox|TEXT=
$\text{Beispiel 1:}$ 
Die hier angegebene Konstellation ergibt sich zum Beispiel bei der [[Modulationsverfahren/Zweiseitenband-Amplitudenmodulation#AM-Signale_und_-Spektren_bei_harmonischem_Eingangssignal|Zweiseitenband-Amplitudenmodulation]] des Nachrichtensignals $x_{\rm N}(t) = A_{\rm N}\cdot \cos\left(2\pi f_{\rm N}\cdot t- \varphi_{\rm N}\right)$ mit dem Trägersignal $x_{\rm T}(t) = A_{\rm T}\cdot \cos\left(2\pi f_{\rm T}\cdot t - \varphi_{\rm T}\right)$. Hierauf wird in der Versuchsdurchführung häufiger eingegangen.

[[File:Ortskurve_4.png|center|frame|Spektum $X_{\rm TP}(f)$ des äquivalenten TP–Signals für verschiedene Phasenkonstellationen |class=fit]]

Bei dieser Betrachtungsweise gibt es einige Einschränkungen bezüglich der Programmparameter:
* Für die Frequenzen gelte stets $f\hspace{0.05cm}'_{\rm O} = f_{\rm N}$ und $f\hspace{0.05cm}'_{\rm U} = -f_{\rm N}$.
*Ohne Verzerrungen sind die Amplitude der Seitenbänder $A_{\rm O}= A_{\rm U}= A_{\rm N}/2$.
*Die jeweiligen Phasenverhältnisse können der Grafik entnommen werden.

}}

===Darstellung des äquivalenten TP–Signals nach Betrag und Phase===

Das im Allgemeinen komplexwertige äquivalenten TP–Signal
:$$x_{\rm TP}(t) = a(t) \cdot {\rm e}^{ {\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm} \phi(t) }$$
kann entsprechend der hier angegebenen Gleichung in eine Betragsfunktion $a(t)$ und eine Phasenfunktion $\phi(t)$ aufgespalten werden, wobei gilt:
:$$a(t) = \vert x_{\rm TP}(t)\vert = \sqrt{ {\rm Re}^2\big [x_{\rm TP}(t)\big ] + {\rm Im}^2\big [x_{\rm TP}(t)\big ] }\hspace{0.05cm},$$
:$$\phi(t) = \text{arc }x_{\rm TP}(t) = \arctan \frac{{\rm Im}\big [x_{\rm TP}(t)\big ]}{{\rm Re}\big [x_{\rm TP}(t)\big ]}.$$

Der Grund dafür, dass man ein Bandpass–Signal $x(t)$ meist durch das äquivalente TP–Signal $x_{\rm TP}(t)$ beschreibt ist, dass die Funktionen $a(t)$ und $\phi(t)$ in beiden Darstellungen interpretierbar sind:
*Der Betrag $a(t)$ des äquivalentes TP–Signals $x_{\rm TP}(t)$ gibt die (zeitabhängige) Hüllkurve von $x(t)$ an.
*Die Phase $\phi(t)$ von $x_{\rm TP}(t)$ kennzeichnet die Lage der Nulldurchgänge von $x(t)$, wobei gilt:
:–   Bei $\phi(t)>0$ ist der Nulldurchgang früher als seine Solllage   ⇒   das Signal ist hier vorlaufend.
:–  Bei $\phi(t)<0$ ist der Nulldurchgang später als seine Solllage   ⇒   das Signal ist hier nachlaufend.

{{GraueBox|TEXT=
$\text{Beispiel 2:}$ 
Die Grafik soll diesen Zusammenhang verdeutlichen, wobei $A_{\rm U} > A_{\rm O}$ vorausgesetzt ist   ⇒   der grüne Zeiger (für das untere Seitenband) ist länger als der blaue Zeiger (oberes Seitenband). Es handelt sich um eine Momentaufnahme zum Zeitpunkt $t_0$:

[[File:Ortskurve_3_neu.png|center|frame|Bandpass–Spektrum $X(f)$ |class=fit]]

*Bei diesen Systemparametern liegt die Spitze des Zeigerverbundes $x_{\rm TP}(t)$ – also die geometrisch Summe aus rotem, blauem und grünem Zeiger – auf einer Ellipse.
* In der linken Grafik schwarz eingezeichnet ist der Betrag $a(t_0) = \vert x_{\rm TP}(t_0) \vert$ und in brauner Farbe angedeutet ist der Phasenwert $\phi(t_0) = \text{arc }x_{\rm TP}(t_0) > 0.$
*In der rechten Grafik gibt der Betrag $a(t_0) = \vert x_{\rm TP}(t_0) \vert$ des äquivalenten TP–Signals die Hüllkurve des physikalischen Signals $x(t)$ an.
* Bei $\phi(t) \equiv 0$ würden alle Nulldurchgänge von $x(t)$ in äquidistenten Abständen auftreten. Wegen $\phi(t_0) > 0$ ist zum Zeitpunkt $t_0$ das Signal vorlaufend, das heißt: Die Nulldurchgänge kommen früher, als es das Raster vorgibt. }}

==Versuchsdurchführung==
[[File:Zeigerdiagramm_aufgabe_2.png|right]]
*Wählen Sie zunächst die Aufgabennummer.
*Eine Aufgabenbeschreibung wird angezeigt.
*Parameterwerte sind angepasst.
*Lösung nach Drücken von &bdquo;Hide solition”.

Mit der Nummer &bdquo;0” wird auf die gleichen Einstellung wie beim Programmstart zurückgesetzt und es wird ein Text mit weiteren Erläuterungen zum Applet ausgegeben.

Im Folgenden bezeichnet $\rm Grün$ das Untere Seitenband   ⇒   $\big (A_{\rm U}, f_{\rm U}, \varphi_{\rm U}\big )$,  
$\rm Rot$ den Träger   ⇒   $\big (A_{\rm T}, f_{\rm T}, \varphi_{\rm T}\big )$ und
$\rm Blau$ das Obere Seitenband   ⇒   $\big (A_{\rm O}, f_{\rm O}, \varphi_{\rm O}\big )$.

{{BlaueBox|TEXT=
'''(1)'''   Es gelte   $\text{Rot:} \hspace{0.15cm} A_{\rm T} = 1\ \text{V}, f_{\rm T} = 100 \ \text{kHz}, \varphi_{\rm T} = 0^\circ$,   $\text{Grün:} \hspace{0.15cm} A_{\rm U} = 0.4 \text{V}, \ f_{\rm U} = 80 \ \text{kHz}, \varphi_{\rm U} = -90^\circ$,   $\text{Blau:} \hspace{0.15cm} A_{\rm O} = 0.4\ \text{V}, f_{\rm O} = 120 \ \text{kHz}, \varphi_{\rm O} = 90^\circ$.

:Betrachten und interpretieren Sie das äquivalente TP–Signal $x_{\rm TP}(t)$ und das physikalische Signal $x(t)$. Welche Periodendauer $T_0$ erkennt man?}}

:: Das äquivalente TP–Signal $x_{\rm TP}(t)$ nimmt ausgehend von $x_{\rm TP}(t=0)=1\ \text{V}$ auf der reellen Achse Werte zwischen $0.2\ \text{V}$ und $1.8\ \text{V}$ an   ⇒   Phase $\phi(t) \equiv 0$.  Der Betrag $|x_{\rm TP}(t)|$ gibt die Hüllkurve $a(t)$ des physikalischen Signals $x(t)$ an. Es gilt mit $A_{\rm N} = 0.8\ \text{V}$ und $f_{\rm N} = 20\ \text{kHz}$:   $a(t) = A_{\rm T}+ A_{\rm N} \cdot \sin(2\pi\cdot f_{\rm N} \cdot t)$.  Sowohl $x_{\rm TP}(t)$ als auch $x(t)$ sind periodisch mit der Periodendauer $T_0 = 1/f_{\rm N} = 50\ \rm µ s$.

{{BlaueBox|TEXT=
'''(2)'''   Wie ändern sich die Verhältnisse gegenüber '''(1)''' mit $f_{\rm U} = 99 \ \text{kHz}$ und $f_{\rm O} = 101 \ \text{kHz}$ ? Wie könnte $x(t)$ entstanden sein?}}

:: Für die Hüllkurve $a(t)$ des Signals $x(t)$ gilt weiterhin $a(t) = A_{\rm T}+ A_{\rm N} \cdot \sin(2\pi\cdot f_{\rm N} \cdot t)$, aber nun mit $f_{\rm N} = 1\ \text{kHz}$. Auch wenn es nicht zu erkennen ist:  $x_{\rm TP}(t)$ und $x(t)$ sind weiterhin periodisch:   $T_0 = 1\ \rm ms$. Beispiel: Zweiseitenband–Amplitudenmodulation '''(ZSB–AM)''' eines Sinussignals mit Cosinus–Träger.

{{BlaueBox|TEXT=
'''(3)'''   Welche Einstellungen müssen gegenüber '''(2)''' geändert werden, um zur ZSB–AM eines Cosinussignals mit Sinus–Träger zu gelangen. Was ändert sich gegenüber '''(2)'''?}}

::Die Trägerphase muss auf $\varphi_{\rm T} = 90^\circ$ geändert werden   ⇒   Sinus–Träger. Ebenso muss $\varphi_{\rm O} =\varphi_{\rm U} =\varphi_{\rm T} = 90^\circ$ eingestellt werden   ⇒   cosinusförmige Nachricht  Die Ortskurve liegt nun auf der imaginären Achse  ⇒   $\phi(t) \equiv -90^\circ$. Zu Beginn gilt $x_{\rm TP}(t=0)= - {\rm j} \cdot 1.8 \ \text{V}$.

{{BlaueBox|TEXT=
'''(4)'''   Nun gelte   $\text{Rot:} \hspace{0.15cm} A_{\rm T} = 1\ \text{V}, \ f_{\rm T} = 100 \ \text{kHz}, \ \varphi_{\rm T} = 0^\circ$,   $\text{Grün:} \hspace{0.15cm} A_{\rm U} = 0.4 \text{V}, \ f_{\rm U} = 80 \ \text{kHz}, \ \varphi_{\rm U} = 0^\circ$,   $\text{Blau:} \hspace{0.15cm} A_{\rm O} = 0.4\ \text{V}, \ f_{\rm O} = 120 \ \text{kHz}, \ \varphi_{\rm O} = 0^\circ$.

:Welche Eigenschaften weist dieses System &bdquo;ZSB–AM, wobei Nachrichtensignal und Träger jeweils cosinusförmig” auf? Wie groß ist der Modulationsgrad $m$?}}

:: Das äquivalente TP–Signal $x_{\rm TP}(t)$ nimmt ausgehend von $x_{\rm TP}(t=0)=1.8\ \text{V}$ auf der reellen Achse Werte zwischen $0.2\ \text{V}$ und $1.8\ \text{V}$ an   ⇒   Phase $\phi(t) \equiv 0$.  Bis auf den Startzustand $x_{\rm TP}(t=0)$ gleiches Verhalten wie bei der Einstellung '''(1)'''. Der Modulationsgrad ist jeweils $m = 0.8$.

{{BlaueBox|TEXT=
'''(5)'''   Es gelten weiter die Parameter gemäß '''(4)''' mit Ausnahme von $A_{\rm T}= 0.6 \text{V}$. Wie groß ist nun der Modulationsgrad $m$? Welche Konsequenzen hat das?}}

:: Es liegt nun eine ZSB–AM mit Modulationsgrad $m = 1.333$ vor. Bei $m > 1$ ist die einfachere [[Modulationsverfahren/Hüllkurvendemodulation|Hüllkurvendemodulation]] nicht anwendbar, da nun die Phasenfunktion $\phi(t) \in \{ 0, \ \pm 180^\circ\}$ nicht mehr konstant ist und die Hüllkurve $a(t)$ nicht mehr mit dem Nachrichtensignal übereinstimmt. Vielmehr muss die aufwändigere [[Modulationsverfahren/Synchrondemodulation|Synchrondemodulation]] verwendet werden. Bei Hüllkurvendemodulation käme es zu nichtlinearen Verzerrungen.

{{BlaueBox|TEXT=
'''(6)'''   Es gelten weiter die Parameter gemäß '''(4)''' bzw. '''(5)''' mit Ausnahme von $A_{\rm T}= 0$ an   ⇒   $m \to \infty$. Welches Modulationsverfahren wird so beschrieben?}}

::Es handelt sich um eine '''ZSB–AM ohne Träger''' und es ist eine eine Synchrondemodulation erforderlich. Das äquivalente TP–Signal $x_{\rm TP}(t)$ liegt zwar auf der reellen Achse, aber nicht nur in der rechten Halbebene. Damit gilt auch hier für die Phasenfunktion $\phi(t) \in \{ 0, \ \pm 180^\circ\}$, wodurch Hüllkurvendemodulation nicht anwendbar ist.

{{BlaueBox|TEXT=
'''(7)'''   Nun gelte   $\text{Rot:} \hspace{0.15cm} A_{\rm T} = 1\ \text{V}, f_{\rm T} = 100 \ \text{kHz}, \varphi_{\rm T} = 0^\circ$,   $\text{Grün:} \hspace{0.15cm} A_{\rm U} = 0, \ f_{\rm U} = 80 \ \text{kHz}, \varphi_{\rm U} = -90^\circ$,   $\text{Blau:} \hspace{0.15cm} A_{\rm O} = 0.8\ \text{V}, f_{\rm O} = 120 \ \text{kHz}, \varphi_{\rm O} = 90^\circ$.

:Welches Konstellation wird hiermit beschrieben? Welche Eigenschaften dieses Verfahrens erkennt man aus der Grafik?}}

::Es handelt es sich um eine [[Modulationsverfahren/Einseitenbandmodulation|Einseitenbandmodulation]] '''(ESB–AM)''', genauer gesagt um eine '''OSB–AM''': Der rote Träger liegt fest, der grüne Zeiger fehlt und der blaue Zeiger (OSB) dreht entgegen dem Uhrzeigersinn. Der Modulationsgrad ist $\mu = 0.8$ (bei ESB bezeichnen wir den Modulationsgrad mit $\mu$ anstelle von $m$). Das Trägersignal ist cosinusförmig und das Nachrichtensignal sinusförmig. Die Ortskurve ist ein Kreis. $x_{\rm TP}(t)$ bewegt sich darauf in mathematisch positiver Richtung. Wegen $\phi(t) \ne \text{const.}$ ist auch hier die Hüllkurvendemodulation nicht anwendbar:  Dies erkennt man daran, dass die Hüllkurve $a(t)$ nicht cosinusförmig ist. Vielmehr ist die untere Halbwelle spitzer als die obere   ⇒   starke lineare Verzerrungen.

{{BlaueBox|TEXT=
'''(8)'''   Es gelten weiter die Parameter gemäß '''(7)''' mit Ausnahme von $A_{\rm O}= 0$ und $A_{\rm U}= 0.8 \text{V}$. Welche Unterschiede ergeben sich gegenüber '''(7)'''?}}

::Nun handelt es sich um eine '''USB–AM''': Der rote Träger liegt fest, der blaue Zeiger fehlt und der grüne Zeiger (USB) dreht im Uhrzeigersinn. Alle anderen Aussagen von '''(7)''' treffen auch hier zu.

{{BlaueBox|TEXT=
'''(9)'''   Es gelten weiter die Parameter gemäß '''(7)''' mit Ausnahme von $A_{\rm O} = 0.2 \text{ V} \ne A_{\rm U} = 0.4 \text{ V} $. Welche Unterschiede ergeben sich gegenüber '''(7)'''?}}

::Die Ortskurve $x_{\rm TP}(t)$ ist nun keine horizontale Gerade, sondern eine Ellipse mit dem Realteil zwischen $0.4 \text{ V}$ und $1.6 \text{ V}$ sowie dem Imaginärteil im Bereich $\pm 0.2 \text{ V}$. Wegen $\phi(t) \ne \text{const.}$ würde auch hier die Hüllkurvendemodulation zu nichtlinearen Verzerrungen führen Die hier simulierte Konstellation beschreibt die Situation von '''(4)''', nämlich eine ZSB–AM mit Modulationsgrad $m = 0.8$, wobei das obere Seitenband aufgrund der Kanaldämpfung auf $50\%$ reduziert wird.

==Zur Handhabung des Applets==
 
[[File:Ortskurve_abzug3.png|right]]
* Die roten Parameter $(A_{\rm T}, \ f_{\rm T}, \ \varphi_{\rm T})$ und der rote Zeiger kennzeichnen den '''T'''räger.
* Die grünen Parameter $(A_{\rm U}, \ f_{\rm U} < f_{\rm T}, \ \varphi_{\rm U})$ kennzeichnen das '''U'''ntere Seitenband.
* Die blauen Parameter $(A_{\rm O}, \ f_{\rm O} > f_{\rm T}, \ \varphi_{\rm O})$ kennzeichnen das '''O'''bere Seitenband.
* Der rote Zeiger dreht nicht.
* Der grüne Zeiger dreht in mathematisch negativer Richtung (im Uhrzeigersinn).
* Der blaue Zeiger dreht entgegen dem Uhrzeigersinn.

Bedeutung der Buchstaben in nebenstehender Grafik:

    '''(A)'''     Grafikfeld für das äquivalente TP–Signal $x_{\rm TP}(t)$

    '''(B)'''     Grafikfeld für das physikalische Signal $x(t)$

    '''(C)'''     Parametereingabe per Slider:   Amplituden, Frequenzen, Phasenwerte

    '''(D)'''     Bedienelemente:   Start – Step – Pause/Continue – Reset

    '''(E)'''     Geschwindigkeit der Animation:   &bdquo;Speed”   ⇒   Werte: 1, 2 oder 3

    '''(F)'''     &bdquo;Trace”   ⇒   Ein oder Aus, Spur des äquivalenten TP–Signals   $x_{\rm TP}(t)$

    '''(G)'''     Numerikausgabe:   Zeit $t$, Signalwerte  ${\rm Re}[x_{\rm TP}(t)]$  und  ${\rm Im}[x_{\rm TP}(t)]$,

$\text{}\hspace{4.2cm}$   Hüllkurve $a(t) = |x_{\rm TP}(t)|$  und  Phase $\phi(t) = {\rm arc} \ x_{\rm TP}(t)$

    '''(H)'''     Variationsmöglichkeiten für die grafische Darstellung

$\hspace{1.5cm}$Zoom–Funktionen &bdquo;$+$” (Vergrößern), &bdquo;$-$” (Verkleinern) und $\rm o$ (Zurücksetzen)

$\hspace{1.5cm}$Verschieben mit &bdquo;$\leftarrow$” (Ausschnitt nach links, Ordinate nach rechts), &bdquo;$\uparrow$” &bdquo;$\downarrow$” &bdquo;$\rightarrow$”

    '''(I)'''     Bereich für die Versuchsdurchführung:  Aufgabenauswahl und Aufgabenstellung

    '''(J)'''     Bereich für die Versuchsdurchführung:  Musterlösung
 

==Über die Autoren==
Dieses interaktive Berechnungstool wurde am [http://www.lnt.ei.tum.de/startseite Lehrstuhl für Nachrichtentechnik] der [https://www.tum.de/ Technischen Universität München] konzipiert und realisiert.
*Die erste Version wurde 2005 von [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Studierende#Ji_Li_.28Bachelorarbeit_EI_2003.2C_Diplomarbeit_EI_2005.29|Ji Li]] im Rahmen ihrer Diplomarbeit mit &bdquo;FlashMX–Actionscript” erstellt (Betreuer: [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Mitarbeiter_und_Dozenten#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28am_LNT_seit_1974.29|Günter Söder]]).
*2018 wurde dieses Programm von [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Studierende#Xiaohan_Liu_.28Bachelorarbeit_2018.29|Xiaohan Liu]] im Rahmen ihrer Bachelorarbeit (Betreuer: [[Biografien_und_Bibliografien/Beteiligte_der_Professur_Leitungsgebundene_%C3%9Cbertragungstechnik#Tasn.C3.A1d_Kernetzky.2C_M.Sc._.28bei_L.C3.9CT_seit_2014.29|Tasnád Kernetzky]]) neu gestaltet und erweitert.

==Nochmalige Aufrufmöglichkeit des Applets in neuem Fenster==

{{LntAppletLink|physAnTPSignal}}

[[Category:Applets|^Verzerrungen^]]

Applets:Physikalisches Signal & Äquivalentes TP-Signal

2018-08-25T15:02:19Z

Xiaohan: /* Programmbeschreibung */

Applets:Physical Signal & Equivalent Lowpass Signal

2018-08-25T13:56:17Z

Xiaohan:

{{LntAppletLinkEn|analPhysSignal}}

==Applet Description==
 
This applet shows the relationship between the physical bandpass–signal $x(t)$ and the associated equivalent low pass–signal $x_{\rm TP}(t)$. The starting point is always a bandpass–signal $x(t)$ with a frequency-discrete spectrum $X(f)$:
:$$x(t) = x_{\rm T}(t) + x_{\rm O}(t) + x_{\rm U}(t) = A_{\rm T}\cdot \cos\left(2\pi f_{\rm T}\cdot t- \varphi_{\rm T}\right)+A_{\rm O}\cdot \cos\left(2\pi f_{\rm O}\cdot t- \varphi_{\rm O}\right)+ A_{\rm U}\cdot \cos\left(2\pi f_{\rm U}\cdot t- \varphi_{\rm U}\right). $$
The physical signal $x(t)$ is thus composed of three [[Signaldarstellung/Harmonische_Schwingung|harmonic oscillations]], a constellation which, for example, in the [[Modulationsverfahren/Zweiseitenband-Amplitudenmodulation#AM-Signale_und_-Spektren_bei_harmonischem_Eingangssignal|double-sideband amplitude modulation]] of the message signal $x_{\rm N}(t) = A_{\rm N}\cdot \cos\left(2\pi f_{\rm N}\cdot t- \varphi_{\rm N}\right)$ with the carrier signal $x_{\rm T}(t) = A_{\rm T}\cdot \cos\left(2\pi f_{\rm T}\cdot t - \varphi_{\rm T}\right)$ returns. The nomenclature is also adapted to this case:
* $x_{\rm O}(t)$ denotes the &bdquo;upper sideband” with the amplitude $A_{\rm O}= A_{\rm N}/2$, the frequency $f_{\rm O} = f_{\rm T} + f_{\rm N}$ and the phase $\varphi_{\rm O} = \varphi_{\rm T} + \varphi_{\rm N}$.
*Similarly, for the &bdquo;lower Sideband” $x_{\rm U}(t)$ with $f_{\rm U} = f_{\rm T} + f_{\rm N}$, $A_{\rm U}= A_{\rm O}$ and $\varphi_{\rm U} = -\varphi_{\rm O}$.

The corresponding equivalent low-pass–signal is $f_{\rm O}\hspace{0.01cm}' = f_{\rm O}- f_{\rm T} > 0$,   $f_{\rm U}\hspace{0.01cm}' = f_{\rm U}- f_{\rm T} < 0$  and  $f_{\rm T}\hspace{0.01cm}' = 0$:

:$$x_{\rm TP}(t) = x_\text{TP, T}(t) + x_\text{TP, O}(t) + x_\text{TP, U}(t) = A_{\rm T}\cdot {\rm e}^{-{\rm j} \varphi_{\rm T} } \hspace{0.1cm}+ \hspace{0.1cm} A_{\rm O}\cdot {\rm e}^{-{\rm j} \varphi_{\rm O} } \cdot {\rm e}^{ {\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm O}\hspace{0.01cm}'\hspace{0.05cm}\cdot \hspace{0.05cm}t}\hspace{0.1cm}+ \hspace{0.1cm}
A_{\rm U}\cdot {\rm e}^{-{\rm j} \varphi_{\rm U} } \cdot {\rm e}^{ {\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm U}\hspace{0.01cm}'\hspace{0.05cm}\cdot \hspace{0.05cm}t} . $$

[[File:Ortskurve_1.png|right|frame|Equivalent lowpass–signal currently $t=0$ for cosinusoidal carrier   ⇒   $\varphi_{\rm T} = 0$]]
The program shows $x_{\rm TP}(t)$ as the vectorial sum of three rotation pointers as a violet point (see example graph for start time $t=0$ and cosinusoidal support):

*The (red) pointer of the carrier $x_\text{TP, T}(t)$ with the length $A_{\rm T}$ and the zero phase position $\varphi_{\rm T}=0$ lies in the complex plane firmly. So it applies to all times $t$:   $x_{\rm TP}(t)= A_{\rm T}\cdot {\rm e}^{-{\rm j} \varphi_{\rm T} }$.

*The (blue) pointer of the upper sideband $x_\text{TP, O}(t)$ with the length $A_{\rm O}$ and the zero phase position $\varphi_{\rm O}$ rotates at the angular velocity $2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm O}\hspace{0.01cm}'$ in mathematically positive direction (one revolution in time $1/f_{\rm O}\hspace{0.01cm}')$.

*The (green) pointer of the lower sideband $x_{\rm U+}(t)$ with the length $A_{\rm U}$ and the zero phase position $\varphi_{\rm U}$ rotates at the angular velocity $2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm U}\hspace{0.01cm}'$, because of $f_{\rm U}\hspace{0.01cm}'<0$ counterclockwise.

*With $f_{\rm U}\hspace{0.01cm}' = -f_{\rm O}\hspace{0.01cm}'$ the blue and green pointers will spin at the same speed but in different directions. Also, if $A_{\rm O} = A_{\rm U}$ and $\varphi_{\rm U} = -\varphi_{\rm O}$, then $x_{\rm TP}(t)$ on a line with a slope of $\varphi_{\rm T}$.

''Hinweis:''   The graphic applies to $\varphi_{\rm O} = +30^\circ$. From this follows for the start time $t=0$ the angle of the blue pointer (OSB) with respect to the coordinate system:   $\phi_{\rm O} = -\varphi_{\rm O} = -30^\circ$. Likewise, the null phantom $\varphi_{\rm U} = -30^\circ$ of the lower sideband (USB, grüner Zeiger) follows for the phase angle to be considered in the complex plane:   $\phi_{\rm U} = +30^\circ$.

The temporal course of $x_{\rm TP}(t)$ is also referred to below as '''locus'''. The relationship between $x_{\rm TP}(t)$ and the physical bandpass –signal $x(t)$ is given in the section [[???]] and the associated analytic signal is $x_+(t)$ :

:$$x_{\rm TP}(t) = x_{\rm +}(t)\cdot {\rm e}^{-{\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm}2 \pi \cdot f_{\rm T}\cdot \hspace{0.05cm}t},$$
:$$x_{\rm +}(t) = x_{\rm TP}(t)\cdot {\rm e}^{+{\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm}2 \pi \cdot f_{\rm T}\cdot \hspace{0.05cm}t}.$$

[[Applets:Physikalisches_Signal_%26_Äquivalentes_TP-Signal|'''German description''']]

==Theoretical Background==
 
===Description of Bandpass Signals===
[[File:Zeigerdiagramm_1a.png|right|frame|bandpass–spectrum $X(f)$ |class=fit]]
We consider '''bandpass signals''' $x(t)$ with the property that their spectra $X(f)$ are not in the range around the frequency $f=0$, but around a carrier frequency $f_{\rm T}$. In most cases it can also be assumed that the bandwidth is $B \ll f_{\rm T}$.

The figure shows such a bandpass spectrum $X(f)$. Assuming that the associated $x(t)$ is a physical signal and thus real, the spectral function $X(f)$ has a symmetry with respect to the frequency $f = 0$, if $x(t)$ is an even function   ⇒   $x(-t)=x(t)$, $X(f)$ is real and even.

Beside the physical signal $x(t)\ \circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\, \ X(f)$, one can also use the following descriptions of bandpass signals:
*the analytic signal $x_+(t)\ \circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\, \ X_+(f)$, see applet [[Applets:Physikalisches_Signal_%26_Analytisches_Signal|Physikalisches Signal & Analytisches Signal]],
*the equivalent low-pass–signal $x_{\rm TP}(t)\ \circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\, \ X_{\rm TP}(f)$, see next page
 

===Spectral Functions of the Analytic and the Equivalent Low-Pass Signal===

The '''analytische Signal''' $x_+(t)$ belonging to the physical signal $x(t)$ is the time function whose spectrum fulfills the following property:
[[File:Ortskurve_2.png|right|frame|spectral functions $X_+(f)$ and $X_{\rm TP}(f)$ |class=fit]]
:$$X_+(f)=\big[1+{\rm sign}(f)\big] \cdot X(f) = \left\{ {2 \cdot
X(f) \; \hspace{0.2cm}\rm for\hspace{0.2cm} {\it f} > 0, \atop {\,\,\,\, \rm 0 \; \hspace{0.9cm}\rm for\hspace{0.2cm} {\it f} < 0.} }\right.$$

The ''Signumfunktion'' is for positive values of $f$ equal to $+1$ and for negative $f$ values equal to $-1$.
*The (double-sided) limit returns $\sign(0) = 0$.
*The index „+” should make it clear that $X_+(f)$ only has parts at positive frequencies.

From the graph you can see the calculation rule for $X_+(f)$: The actual BP spectrum $X(f)$ becomes
*doubled at the positive frequencies, and
*set to zero at the negative frequencies.

Due to the asymmetry of $X_+(f)$ with respect to the frequency $f = 0$, it can already be said that the time function $x_+(t)$ except for a trivial special case $x_+(t)= 0 \ \ \circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\,\ \ X_+(f)= 0$ is always complex.

The spectrum $X_{\rm TP}(f)$ of the equivalent low-pass signal is obtained by shifting $X_+(f)$ to the left by the carrier frequency $f_{\rm T}$:
:$$X_{\rm TP}(f)= X_+(f+f_{\rm T}).$$

In the time domain this operation corresponds to the multiplication of $x_{\rm +}(t)$ with the complex exponential function with negative exponent:
:$$x_{\rm TP}(t) = x_{\rm +}(t)\cdot {\rm e}^{-{\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm}2 \pi \cdot f_{\rm T}\cdot \hspace{0.05cm}t}.$$

It can be seen that $x_{\rm TP}(t)$ s generally complex. But if $X_+(f)$ is symmetric about the carrier frequency $f_{\rm T}$, $X_{\rm TP}(f)$ is symmetric about the frequency $f=0$ and the result is accordingly real time function $x_{\rm TP}(t)$.

===$x_{\rm TP}(t)$–Representation of a Sum of Three Harmonic Oscillations===

In our applet, we always assume a set of three rotating pointers. The physical signal is:
:$$x(t) = x_{\rm U}(t) + x_{\rm T}(t) + x_{\rm O}(t) = A_{\rm U}\cdot \cos\left(2\pi f_{\rm U}\cdot t- \varphi_{\rm U}\right)+A_{\rm T}\cdot \cos\left(2\pi f_{\rm T}\cdot t- \varphi_{\rm T}\right)+A_{\rm O}\cdot \cos\left(2\pi f_{\rm O}\cdot t- \varphi_{\rm O}\right). $$
* Each of the three harmonic oscillations $x_{\rm T}(t)$, $x_{\rm U}(t)$ and $x_{\rm O}(t)$ is represented by an amplitude $(A)$, a frequency $(f)$ and a phase value $(\varphi)$.
*The indices are based on the modulation method [[Modulationsverfahren/Zweiseitenband-Amplitudenmodulation|double sideband –amplitude modulation]]. &bdquo;T” stands for &bdquo;carrier”, &bdquo;U” for &bdquo;lowei sideband” and &bdquo;O” for &bdquo;upper Sideband”. Similarly, $f_{\rm U} < f_{\rm T}$ and $f_{\rm O} > f_{\rm T}$. There are no restrictions for the amplitudes and phases.

The associated equivalent low-pass signal is with $f_{\rm O}\hspace{0.01cm}' = f_{\rm O}- f_{\rm T} > 0$,   $f_{\rm U}\hspace{0.01cm}' = f_{\rm U}- f_{\rm T} < 0$  and  $f_{\rm T}\hspace{0.01cm}' = 0$:

:$$x_{\rm TP}(t) = x_\text{TP, T}(t) + x_\text{TP, O}(t) + x_\text{TP, U}(t) = A_{\rm T}\cdot {\rm e}^{-{\rm j} \varphi_{\rm T} } \hspace{0.1cm}+ \hspace{0.1cm} A_{\rm O}\cdot {\rm e}^{-{\rm j} \varphi_{\rm O} } \cdot {\rm e}^{ {\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm O}\hspace{0.01cm}'\hspace{0.05cm}\cdot \hspace{0.05cm}t}\hspace{0.1cm}+ \hspace{0.1cm}
A_{\rm U}\cdot {\rm e}^{-{\rm j} \varphi_{\rm U} } \cdot {\rm e}^{ {\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm U}\hspace{0.01cm}'\hspace{0.05cm}\cdot \hspace{0.05cm}t} . $$

{{GraueBox|TEXT=
$\text{Example 1:}$ 
The constellation given here results, for example, in the [[Modulationsverfahren/Zweiseitenband-Amplitudenmodulation#AM-Signale_und_-Spektren_bei_harmonischem_Eingangssignal|double sideband amplitude modulation]] of the message signal $x_{\rm N}(t) = A_{\rm N}\cdot \cos\left(2\pi f_{\rm N}\cdot t- \varphi_{\rm N}\right)$ with the carrier signal $x_{\rm T}(t) = A_{\rm T}\cdot \cos\left(2\pi f_{\rm T}\cdot t - \varphi_{\rm T}\right)$. This is discussed more frequently in the experimental procedure.

[[File:Ortskurve_4.png|center|frame|Spectrum $X_{\rm TP}(f)$ of the equivalent low–pass signal for different phase constellations |class=fit]]

There are some limitations to the program parameters in this approach:
* The frequencies are always $f\hspace{0.05cm}'_{\rm O} = f_{\rm N}$ and $f\hspace{0.05cm}'_{\rm U} = -f_{\rm N}$.
*Without distortion, the amplitude of the sidebands is $A_{\rm O}= A_{\rm U}= A_{\rm N}/2$.
*The respective phase relationships can be seen from the graph.

}}

===Representation of the Equivalent Low-Pass Signal by Magnitude and Phase===

The generally complex valued equivalent low-pass signal
:$$x_{\rm TP}(t) = a(t) \cdot {\rm e}^{ {\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm} \phi(t) }$$
can be split into a magnitude function $a(t)$ and a phase function $\phi(t)$ according to the equation given here, where:
:$$a(t) = \vert x_{\rm TP}(t)\vert = \sqrt{ {\rm Re}^2\big [x_{\rm TP}(t)\big ] + {\rm Im}^2\big [x_{\rm TP}(t)\big ] }\hspace{0.05cm},$$
:$$\phi(t) = \text{arc }x_{\rm TP}(t) = \arctan \frac{{\rm Im}\big [x_{\rm TP}(t)\big ]}{{\rm Re}\big [x_{\rm TP}(t)\big ]}.$$

The reason that a bandpass–signal $x(t)$ is usually described by the equivalent low-pass signal $x_{\rm TP}(t)$ is that the functions $a(t)$ and $\phi(t)$ are interpretable in both representations:
*The amount $a(t)$ of the equivalent low-pass signal $x_{\rm TP}(t)$ indicates the (time-dependent) envelope of $x(t)$.
*The phase $\phi(t)$ of $x_{\rm TP}(t)$ denotes the location of the zero crossings of $x(t)$, where:
:–   For $\phi(t)>0$ the zero crossing is earlier than its nominal position   ⇒   the signal is leading here.
:–  When $\phi(t)<0$, the zero crossing is later than its target position   ⇒   the signal is trailing here.

{{GraueBox|TEXT=
$\text{Example 2:}$ 
The graph is intended to illustrate this relationship, assuming $A_{\rm U} > A_{\rm O}$   ⇒   the green pointer (for the lower sideband) is longer than the blue pointer (upper sideband). This is a snapshot at time $t_0$:

[[File:Ortskurve_3_neu.png|center|frame|bandpass–Spectrum $X(f)$ |class=fit]]

*For these system parameters, the top of the pointer cluster $x_{\rm TP}(t)$ – that is, the geometric sum of red, blue and green pointers – on an ellipse.
* The amount $a(t_0) = \vert x_{\rm TP}(t_0) \vert$ is drawn in black in the left-hand diagram and the phase value $\phi(t_0) = \text{arc }x_{\rm TP}(t_0) > 0$ is indicated in brown color.
*In the graph on the right, the amount $a(t_0) = \vert x_{\rm TP}(t_0) \vert$ of the equivalent low-pass signal indicates the envelope of the physical signal $x(t)$.
* At $\phi(t) \equiv 0$, all zero crossings of $x(t)$ would occur at equidistant intervals. Because of $\phi(t_0) > 0$, the signal is leading at the time $t_0$, that is: the zero crossings come earlier than the grid dictates. }}

==Exercises==
[[File:Exercises_verzerrungen.png|right]]
*First select the task number.
*A task description is displayed.
*Parameter values are adjusted.
*Solution after pressing &bdquo;Hide solition”.

The number &bdquo;0” will reset the program and output a text with the further explanation of the applet.

In the following, $\rm Green$ denotes the lower sideband   ⇒   $\big (A_{\rm U}, f_{\rm U}, \varphi_{\rm U}\big )$,  
$\rm Red$ the carrier   ⇒   $\big (A_{\rm T}, f_{\rm T}, \varphi_{\rm T}\big )$ and
$\rm Blue$ the upper sideband   ⇒   $\big (A_{\rm O}, f_{\rm O}, \varphi_{\rm O}\big )$.

{{BlaueBox|TEXT=
'''(1)'''   Let   $\text{Red:} \hspace{0.15cm} A_{\rm T} = 1\ \text{V}, f_{\rm T} = 100 \ \text{kHz}, \varphi_{\rm T} = 0^\circ$,   $\text{Green:} \hspace{0.15cm} A_{\rm U} = 0.4 \text{V}, \ f_{\rm U} = 80 \ \text{kHz}, \varphi_{\rm U} = -90^\circ$,   $\text{Blue:} \hspace{0.15cm} A_{\rm O} = 0.4\ \text{V}, f_{\rm O} = 120 \ \text{kHz}, \varphi_{\rm O} = 90^\circ$.

:Consider and interpret the equivalent low-pass signal $x_{\rm TP}(t)$ and the physical signal $x(t)$. Which period $T_0$ does one recognize?}}

:: The equivalent low-pass signal $x_{\rm TP}(t)$ takes from $x_{\rm TP}(t=0)=1\ \text{V}$ on the real axis values between $0.2\ \text{V}$ and $1.8\ \text{V}$   ⇒   phase $\phi(t) \equiv 0$.  The amount $|x_{\rm TP}(t)|$ indicates the envelope $a(t)$ of the physical signal $x(t)$. It holds $A_{\rm N} = 0.8\ \text{V}$ and $f_{\rm N} = 20\ \text{kHz}$:   $a(t) = A_{\rm T}+ A_{\rm N} \cdot \sin(2\pi\cdot f_{\rm N} \cdot t)$.  Both $x_{\rm TP}(t)$ and $x(t)$ are periodic with the period $T_0 = 1/f_{\rm N} = 50\ \rm µ s$.

{{BlaueBox|TEXT=
'''(2)'''   How do the ratios change to '''(1)''' with $f_{\rm U} = 99 \ \text{kHz}$ and $f_{\rm O} = 101 \ \text{kHz}$ ? How could $x(t)$ have arisen?}}

:: For the envelope $a(t)$ of the signal $x(t)$ we still have $a(t) = A_{\rm T}+ A_{\rm N} \cdot \sin(2\pi\cdot f_{\rm N} \cdot t)$, but now $f_{\rm N} = 1\ \text{kHz}$. Even though it can not be recognized:  $x_{\rm TP}(t)$ and $x(t)$ are still periodic:   $T_0 = 1\ \rm ms$. Example: Double sideband - Amplitude modulation '''(ZSB–AM)''' of a sine signal with cosine–carrier.

{{BlaueBox|TEXT=
'''(3)'''   Which settings have to be changed from '''(2)''' in order to arrive at the ZSB–AM of a cosine signal with sine–carrier. What changes over '''(2)'''?}}

::The carrier phase must be changed to $\varphi_{\rm T} = 90^\circ$   ⇒   sine–carrier. Similarly, $\varphi_{\rm O} =\varphi_{\rm U} =\varphi_{\rm T} = 90^\circ$ must be set   ⇒   cosine-shaped message  The locus now lies on the imaginary axis  ⇒   $\phi(t) \equiv -90^\circ$. At the beginning we have $x_{\rm TP}(t=0)= - {\rm j} \cdot 1.8 \ \text{V}$.

{{BlaueBox|TEXT=
'''(4)'''   Now let   $\text{Red:} \hspace{0.15cm} A_{\rm T} = 1\ \text{V}, \ f_{\rm T} = 100 \ \text{kHz}, \ \varphi_{\rm T} = 0^\circ$,   $\text{Green:} \hspace{0.15cm} A_{\rm U} = 0.4 \text{V}, \ f_{\rm U} = 80 \ \text{kHz}, \ \varphi_{\rm U} = 0^\circ$,   $\text{Blue:} \hspace{0.15cm} A_{\rm O} = 0.4\ \text{V}, \ f_{\rm O} = 120 \ \text{kHz}, \ \varphi_{\rm O} = 0^\circ$.

:What are the characteristics of this system &bdquo;ZSB–AM, where the message signal and carrier are respectively cosinusoidal”? What is the modulation depth $m$?}}

:: The equivalent low-pass signal $x_{\rm TP}(t)$ takes from $x_{\rm TP}(t=0)=1.8\ \text{V}$ on the real axis values between $0.2\ \text{V}$ and $1.8\ \text{V}$   ⇒   phase $\phi(t) \equiv 0$.  Except for the start state $x_{\rm TP}(t=0)$ same behavior as with the setting '''(1)'''. The modulation depth is $m = 0.8$.

{{BlaueBox|TEXT=
'''(5)'''   The parameters are still valid according to '''(4)''' with the exception of $A_{\rm T}= 0.6 \text{V}$. How big is the degree of modulation $m$? What are the consequences?}}

:: There is now a ZSB–AM with modulation degree $m = 1.333$. For $m > 1$, the simpler [[Modulationsverfahren/Hüllkurvendemodulation|envelope demodulation]] is not applicable, since the phase function $\phi(t) \in \{ 0, \ \pm 180^\circ\}$ is no longer constant and the envelope $a(t)$ no longer matches the message signal. Rather, the more complex [[Modulationsverfahren/Synchrondemodulation|synchronous demodulation]] must be used. Envelope detection would produce nonlinear distortions.

{{BlaueBox|TEXT=
'''(6)'''   The parameters are still valid according to '''(4)''' or '''(5)''' with the exception from $A_{\rm T}= 0$ on   ⇒   $m \to \infty$. Which modulation method is described in this way?}}

::It is a '''ZSB–AM without carrier''' and requires synchronous demodulation. The equivalent low-pass signal $x_{\rm TP}(t)$ is on the real axis, but not only in the right half-plane. Thus, the phase function $\phi(t) \in \{ 0, \ \pm 180^\circ\}$, also applies here, which means that envelope demodulation is not applicable.

{{BlaueBox|TEXT=
'''(7)'''   Now let   $\text{Red:} \hspace{0.15cm} A_{\rm T} = 1\ \text{V}, f_{\rm T} = 100 \ \text{kHz}, \varphi_{\rm T} = 0^\circ$,   $\text{Green:} \hspace{0.15cm} A_{\rm U} = 0, \ f_{\rm U} = 80 \ \text{kHz}, \varphi_{\rm U} = -90^\circ$,   $\text{Blue:} \hspace{0.15cm} A_{\rm O} = 0.8\ \text{V}, f_{\rm O} = 120 \ \text{kHz}, \varphi_{\rm O} = 90^\circ$.

:Which constellation is described here? Which characteristics of this procedure can be recognized from the graphic?}}

::It is a [[Modulationsverfahren/Einseitenbandmodulation|single-sideband modulation]] '''(ESB–AM)''', more specifically an '''OSB–AM''': the red carrier is fixed, the green pointer missing and the blue pointer (OSB) turns counterclockwise. The degree of modulation is $\mu = 0.8$ (in the case of ESB we denote the degree of modulation with $\mu$ instead of $m$). The carrier signal is cosinusoidal and the message signal sinusoidal. The locus is a circle. $x_{\rm TP}(t)$ moves in a mathematically positive direction. Because of $\phi(t) \ne \text{const.}$, The envelope demodulation is not applicable here:  This can be seen by the fact that the envelope $a(t)$ is not cosinusoidal. Rather, the lower half-wave is sharper than the upper   ⇒   strong linear distortions.

{{BlaueBox|TEXT=
'''(8)'''   The parameters are still valid according to '''(7)''' with the exception of $A_{\rm O}= 0$ and $A_{\rm U}= 0.8 \text{ V}$. What differences arise opposite '''(7)'''?}}

::Now it is a '''USB–AM''': The red carrier is fixed, the blue pointer is missing and the green pointer (USB) rotates clockwise. All other statements of '''(7)''' apply here as well.

{{BlaueBox|TEXT=
'''(9)'''   The parameters according to '''(7)''' are still valid with the exception of $A_{\rm O} = 0.2 \text{ V} \ne A_{\rm U} = 0.4 \text{ V} $. What are the differences from '''(7)'''?}}

::The locus $x_{\rm TP}(t)$ is not a horizontal straight line, but an ellipse with the real part between $0.4 \text{ V}$ and $1.6 \text{ V}$ and the imaginary part in the range $\pm 0.2 \text{ V}$. Because of $\phi(t) \ne \text{const.}$ , Envelope demodulation would lead to non-linear distortions here too. The constellation simulated here describes the situation of '''(4)''', namely a ZSB–AM with modulation degree $m = 0.8$, where the upper sideband is reduced to $50\%$ due to channel attenuation.

==Applet Manual==
[[File:Ortskurve_abzug3.png|right]]
* The red parameters $(A_{\rm T}, \ f_{\rm T}, \ \varphi_{\rm T})$ and the red pointer mark the ''Carrier''. (German: '''T'''räger).
* The green parameters $(A_{\rm U}, \ f_{\rm U} < f_{\rm T}, \ \varphi_{\rm U})$ mark the ''Lower sideband''. (German: '''U'''ntere Seitenband).
* The blue parameters $(A_{\rm O}, \ f_{\rm O} > f_{\rm T}, \ \varphi_{\rm O})$ mark the '' Upper sideband''. (German: '''O'''bere Seitenband).
* The red pointer does not turn.
* The green pointer rotates in a mathematically negative direction (clockwise).
* The blue pointer turns counterclockwise.

Meaning of the letters in the adjacent graphic:

    '''(A)'''     Plot of the equivalent low-pass Signal $x_{\rm TP}(t)$

    '''(B)'''     Plot of the physical signal $x(t)$

    '''(C)'''     Parameter input via slider:   amplitudes, frequencies, phase values

    '''(D)'''     Control elements:   Start – Step – Pause/Continue – Reset

    '''(E)'''     Speed of animation:   &bdquo;Speed”   ⇒   Values: 1, 2 oder 3

    '''(F)'''     &bdquo;Trace”   ⇒   On or Off, trace of equivalent low-pass Signal   $x_{\rm TP}(t)$

    '''(G)'''     Numerical output:   time $t$, the signal values  ${\rm Re}[x_{\rm TP}(t)]$  and  ${\rm Im}[x_{\rm TP}(t)]$,

$\text{}\hspace{4.2cm}$   Hüllkurve $a(t) = |x_{\rm TP}(t)|$  und  Phase $\phi(t) = {\rm arc} \ x_{\rm TP}(t)$

    '''(H)'''     Variations for the graphical representation

$\hspace{1.5cm}$Zoom–Functions &bdquo;$+$” (Enlarge), &bdquo;$-$” (Decrease) und $\rm o$ (Reset to default)

$\hspace{1.5cm}$Move with &bdquo;$\leftarrow$” (Section to the left, ordinate to the right), &bdquo;$\uparrow$” &bdquo;$\downarrow$” &bdquo;$\rightarrow$”

    '''(I)'''     Experiment section:   Task selection and task

    '''(J)'''     Experiment section:  solution
 

==About the Authors==
This interactive calculation was designed and realized at the [http://www.lnt.ei.tum.de/startseite Lehrstuhl für Nachrichtentechnik] of the [https://www.tum.de/ Technical University of Munich] .
*The original version was created in 2005 by [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Studierende#Ji_Li_.28Bachelorarbeit_EI_2003.2C_Diplomarbeit_EI_2005.29|Ji Li]] as part of her Diploma thesis using &bdquo;FlashMX–Actionscript” (Supervisor: [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Mitarbeiter_und_Dozenten#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28am_LNT_seit_1974.29|Günter Söder]]).
*In 2018 this Applet was redesigned and updated to &bdquo;HTML5” by [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Studierende#Xiaohan_Liu_.28Bachelorarbeit_2018.29|Xiaohan Liu]] as part of her Bachelor's thesis (Supervisor: [[Biografien_und_Bibliografien/Beteiligte_der_Professur_Leitungsgebundene_%C3%9Cbertragungstechnik#Tasn.C3.A1d_Kernetzky.2C_M.Sc._.28bei_L.C3.9CT_seit_2014.29|Tasnád Kernetzky]]).

==Once again: Open Applet in new Tab==

{{LntAppletLinkEn|physAnSignal}}

[[Category:Applets|^Verzerrungen^]]

Applets:Physikalisches Signal & Äquivalentes TP-Signal

2018-08-25T13:53:58Z

Xiaohan: /* Programmbeschreibung */

{{LntAppletLink|physAnTPSignal}}

==Programmbeschreibung==
 
Dieses Applet zeigt den Zusammenhang zwischen dem physikalischen Bandpass–Signal $x(t)$ und dem dazugehörigen äquivalenten Tiefpass–Signal $x_{\rm TP}(t)$. Ausgegangen wird stets von einem Bandpass–Signal $x(t)$ mit frequenzdiskretem Spektrum $X(f)$:
:$$x(t) = x_{\rm T}(t) + x_{\rm O}(t) + x_{\rm U}(t) = A_{\rm T}\cdot \cos\left(2\pi f_{\rm T}\cdot t- \varphi_{\rm T}\right)+A_{\rm O}\cdot \cos\left(2\pi f_{\rm O}\cdot t- \varphi_{\rm O}\right)+ A_{\rm U}\cdot \cos\left(2\pi f_{\rm U}\cdot t- \varphi_{\rm U}\right). $$
Das physikalische Signal $x(t)$ setzt sich also aus drei [[Signaldarstellung/Harmonische_Schwingung|harmonischen Schwingungen]] zusammen, einer Konstellation, die sich zum Beispiel bei der [[Modulationsverfahren/Zweiseitenband-Amplitudenmodulation#AM-Signale_und_-Spektren_bei_harmonischem_Eingangssignal|Zweiseitenband-Amplitudenmodulation]] des Nachrichtensignals $x_{\rm N}(t) = A_{\rm N}\cdot \cos\left(2\pi f_{\rm N}\cdot t- \varphi_{\rm N}\right)$ mit dem Trägersignal $x_{\rm T}(t) = A_{\rm T}\cdot \cos\left(2\pi f_{\rm T}\cdot t - \varphi_{\rm T}\right)$ ergibt. Die Nomenklatur ist ebenfalls an diesen Fall angepasst:
* $x_{\rm O}(t)$ bezeichnet das &bdquo;Obere Seitenband” mit der Amplitude $A_{\rm O}= A_{\rm N}/2$, der Frequenz $f_{\rm O} = f_{\rm T} + f_{\rm N}$ und der Phase $\varphi_{\rm O} = \varphi_{\rm T} + \varphi_{\rm N}$.
*Entsprechend gilt für das &bdquo;Untere Seitenband” $x_{\rm U}(t)$ mit $f_{\rm U} = f_{\rm T} - f_{\rm N}$, $A_{\rm U}= A_{\rm O}$ und $\varphi_{\rm U} = -\varphi_{\rm O}$.

Das dazugehörige äquivalente Tiefpass–Signal lautet mit $f_{\rm O}\hspace{0.01cm}' = f_{\rm O}- f_{\rm T} > 0$,   $f_{\rm U}\hspace{0.01cm}' = f_{\rm U}- f_{\rm T} < 0$  und  $f_{\rm T}\hspace{0.01cm}' = 0$:

:$$x_{\rm TP}(t) = x_\text{TP, T}(t) + x_\text{TP, O}(t) + x_\text{TP, U}(t) = A_{\rm T}\cdot {\rm e}^{-{\rm j} \varphi_{\rm T} } \hspace{0.1cm}+ \hspace{0.1cm} A_{\rm O}\cdot {\rm e}^{-{\rm j} \varphi_{\rm O} } \cdot {\rm e}^{ {\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm O}\hspace{0.01cm}'\hspace{0.05cm}\cdot \hspace{0.05cm}t}\hspace{0.1cm}+ \hspace{0.1cm}
A_{\rm U}\cdot {\rm e}^{-{\rm j} \varphi_{\rm U} } \cdot {\rm e}^{ {\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm U}\hspace{0.01cm}'\hspace{0.05cm}\cdot \hspace{0.05cm}t} . $$

[[File:Ortskurve_1.png|right|frame|Äquivalentes TP–Signal zur Zeit $t=0$ bei cosinusförmigem Träger   ⇒   $\varphi_{\rm T} = 0$]]
Im Programm dargestellt wird $x_{\rm TP}(t)$ als vektorielle Summe dreier Drehzeiger als violetter Punkt (siehe beispielhafte Grafik für den Startzeitpunkt $t=0$ und cosinusförmigem Träger):

*Der (rote) Zeiger des Trägers $x_\text{TP, T}(t)$ mit der Länge $A_{\rm T}$ und der Nullphasenlage $\varphi_{\rm T} = 0$ liegt in der komplexen Ebene fest. Es gilt also für alle Zeiten $t$:   $x_{\rm TP}(t)= A_{\rm T}\cdot {\rm e}^{-{\rm j} \varphi_{\rm T} }$.

*Der (blaue) Zeiger des Oberen Seitenbandes $x_\text{TP, O}(t)$ mit der Länge $A_{\rm O}$ und der Nullphasenlage $\varphi_{\rm O}$ dreht mit der Winkelgeschwindigkeit $2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm O}\hspace{0.01cm}'$ in mathematisch positiver Richtung (eine Umdrehung in der Zeit $1/f_{\rm O}\hspace{0.01cm}')$.

*Der (grüne) Zeiger des Unteren Seitenbandes $x_{\rm U+}(t)$ mit der Länge $A_{\rm U}$ und der Nullphasenlage $\varphi_{\rm U}$ dreht mit der Winkelgeschwindigkeit $2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm U}\hspace{0.01cm}'$, wegen $f_{\rm U}\hspace{0.01cm}'<0$ im Uhrzeigersinn (mathematisch negative Richtung).

*Mit $f_{\rm U}\hspace{0.01cm}' = -f_{\rm O}\hspace{0.01cm}'$ drehen der blaue und der grüne Zeiger gleich schnell, aber in unterschiedlichen Richtungen. Gilt zudem $A_{\rm O} = A_{\rm U}$ und $\varphi_{\rm U} = -\varphi_{\rm O}$, so bewegt sich $x_{\rm TP}(t)$ auf einer Geraden mit einer Neigung von $\varphi_{\rm T}$.

''Hinweis:''   Die Grafik gilt für $\varphi_{\rm O} = +30^\circ$. Daraus folgt für den Startzeitpunkt $t=0$ der Winkel des blauen Zeigers (OSB) gegenüber dem Koordinatensystem:   $\phi_{\rm O} = -\varphi_{\rm O} = -30^\circ$. Ebenso folgt aus der Nullphanlage $\varphi_{\rm U} = -30^\circ$ des unteren Seitenbandes (USB, grüner Zeiger) für den in der komplexen Ebene zu berücksichtigenden Phasenwinkel:   $\phi_{\rm U} = +30^\circ$.

Den zeitlichen Verlauf von $x_{\rm TP}(t)$ bezeichnen wir im Folgenden auch als '''Ortskurve'''. Der Zusammenhang zwischen $x_{\rm TP}(t)$ und dem physikalischen Bandpass–Signal $x(t)$ wird im Abschnitt [[???]] angegeben. Der Zusammenhang zwischen $x_{\rm TP}(t)$ und dem dazugehörigen analytischen Signal $x_+(t)$ lautet:

:$$x_{\rm TP}(t) = x_{\rm +}(t)\cdot {\rm e}^{-{\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm}2 \pi \cdot f_{\rm T}\cdot \hspace{0.05cm}t},$$
:$$x_{\rm +}(t) = x_{\rm TP}(t)\cdot {\rm e}^{+{\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm}2 \pi \cdot f_{\rm T}\cdot \hspace{0.05cm}t}.$$

[[Applets:Physical_Signal_%26_Equivalent_Low-pass_Signal|'''Englische Beschreibung''']]

==Theoretischer Hintergrund==
 
===Beschreibungsmöglichkeiten von Bandpass-Signalen===
[[File:Zeigerdiagramm_1a.png|right|frame|Bandpass–Spektrum $X(f)$ |class=fit]]
Wir betrachten hier '''Bandpass-Signale''' $x(t)$ mit der Eigenschaft, dass deren Spektren $X(f)$ nicht im Bereich um die Frequenz $f = 0$ liegen, sondern um eine Trägerfrequenz $f_{\rm T}$. Meist kann auch davon ausgegangen werden, dass die Bandbreite $B \ll f_{\rm T}$ ist.

Die Grafik zeigt ein solches Bandpass–Spektrum $X(f)$. Unter der Annahme, dass das zugehörige $x(t)$ ein physikalisches Signal und damit reell ist, ergibt sich für die Spektralfunktion $X(f)$ eine Symmetrie bezüglich der Frequenz $f = 0$. Ist $x(t)$ eine gerade Funktion   ⇒   $x(-t)=x(+t)$, so ist auch $X(f)$ reell und gerade.

Neben dem physikalischen Signal $x(t)\ \circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\, \ X(f)$ verwendet man zur Beschreibung von Bandpass-Signalen gleichermaßen:
*das analytische Signal $x_+(t)\ \circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\, \ X_+(f)$, siehe Applet [[Applets:Physikalisches_Signal_%26_Analytisches_Signal|Physikalisches Signal & Analytisches Signal]],
*das äquivalente Tiefpass–Signal $x_{\rm TP}(t)\ \circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\, \ X_{\rm TP}(f)$, wie im nächsten Unterabschnitt beschrieben.
 
===Spektralfunktionen des analytischen und des äquivalenten TP–Signals===

Das zum physikalischen Signal $x(t)$ gehörige '''analytische Signal''' $x_+(t)$ ist diejenige Zeitfunktion, deren Spektrum folgende Eigenschaft erfüllt:
[[File:Ortskurve_2.png|right|frame|Spektralfunktionen $X_+(f)$ und $X_{\rm TP}(f)$ |class=fit]]
:$$X_+(f)=\big[1+{\rm sign}(f)\big] \cdot X(f) = \left\{ {2 \cdot
X(f) \; \hspace{0.2cm}\rm f\ddot{u}r\hspace{0.2cm} {\it f} > 0, \atop {\,\,\,\, \rm 0 \; \hspace{0.9cm}\rm f\ddot{u}r\hspace{0.2cm} {\it f} < 0.} }\right.$$

Die so genannte ''Signumfunktion'' ist dabei für positive Werte von $f$ gleich $+1$ und für negative $f$–Werte gleich $-1$.
*Der (beidseitige) Grenzwert liefert $\sign(0) = 0$.
*Der Index „+” soll deutlich machen, dass $X_+(f)$ nur Anteile bei positiven Frequenzen besitzt.

Aus der Grafik erkennt man die Berechnungsvorschrift für $X_+(f)$: Das tatsächliche BP–Spektrum $X(f)$ wird
*bei den positiven Frequenzen verdoppelt, und
*bei den negativen Frequenzen zu Null gesetzt.

Aufgrund der Unsymmetrie von $X_+(f)$ bezüglich der Frequenz $f = 0$ kann man bereits jetzt schon sagen, dass die Zeitfunktion $x_+(t)$ bis auf einen trivialen Sonderfall $x_+(t)= 0 \ \ \circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\,\ \ X_+(f)= 0$ stets komplex ist.

Zum Spektrum $X_{\rm TP}(f)$ des äquivalenten TP–Signals kommt man, indem man $X_+(f)$ um die Trägerfrequenz $f_{\rm T}$ nach links verschiebt:
:$$X_{\rm TP}(f)= X_+(f+f_{\rm T}).$$

Im Zeitbereich entspricht diese Operation der Multiplkation von $x_{\rm +}(t)$ mit der komplexen Exponentialfunktion mit negativem Exponenten:
:$$x_{\rm TP}(t) = x_{\rm +}(t)\cdot {\rm e}^{-{\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm}2 \pi \cdot f_{\rm T}\cdot \hspace{0.05cm}t}.$$

Man erkennt, dass $x_{\rm TP}(t)$ im Allgemeinen komplexwertig ist. Ist aber $X_+(f)$ symmetrisch um die Trägerfrequenz $f_{\rm T}$, so ist $X_{\rm TP}(f)$ symmetrisch um die Frequenz $f=0$ und es ergibt sich dementsprechend eine reelle Zeitfunktion $x_{\rm TP}(t)$.

===$x_{\rm TP}(t)$–Darstellung einer Summe aus drei harmonischen Schwingungen===

In unserem Applet setzen wir stets einen Zeigerverbund aus drei Drehzeigern voraus. Das physikalische Signal lautet:
:$$x(t) = x_{\rm U}(t) + x_{\rm T}(t) + x_{\rm O}(t) = A_{\rm U}\cdot \cos\left(2\pi f_{\rm U}\cdot t- \varphi_{\rm U}\right)+A_{\rm T}\cdot \cos\left(2\pi f_{\rm T}\cdot t- \varphi_{\rm T}\right)+A_{\rm O}\cdot \cos\left(2\pi f_{\rm O}\cdot t- \varphi_{\rm O}\right). $$
* Jede der drei harmonischen Schwingungen harmonischen Schwingungen $x_{\rm T}(t)$, $x_{\rm U}(t)$ und $x_{\rm O}(t)$ wird durch eine Amplitude $(A)$, eine Frequenz $(f)$ und einen Phasenwert $(\varphi)$ charakterisiert.
*Die Indizes sind an das Modulationsverfahren [[Modulationsverfahren/Zweiseitenband-Amplitudenmodulation|Zweiseitenband–Amplitudenmodulation]] angelehnt. &bdquo;T” steht für &bdquo;Träger”, &bdquo;U” für &bdquo;Unteres Seitenband” und &bdquo;O” für &bdquo;Oberes Seitenband”. Entsprechend gilt stets $f_{\rm U} < f_{\rm T}$ und $f_{\rm O} > f_{\rm T}$. Für die Amplituden und Phasen gibt es keine Einschränkungen.

Das dazugehörige äquivalente Tiefpass–Signal lautet mit $f_{\rm O}\hspace{0.01cm}' = f_{\rm O}- f_{\rm T} > 0$,   $f_{\rm U}\hspace{0.01cm}' = f_{\rm U}- f_{\rm T} < 0$  und  $f_{\rm T}\hspace{0.01cm}' = 0$:

:$$x_{\rm TP}(t) = x_\text{TP, T}(t) + x_\text{TP, O}(t) + x_\text{TP, U}(t) = A_{\rm T}\cdot {\rm e}^{-{\rm j} \varphi_{\rm T} } \hspace{0.1cm}+ \hspace{0.1cm} A_{\rm O}\cdot {\rm e}^{-{\rm j} \varphi_{\rm O} } \cdot {\rm e}^{ {\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm O}\hspace{0.01cm}'\hspace{0.05cm}\cdot \hspace{0.05cm}t}\hspace{0.1cm}+ \hspace{0.1cm}
A_{\rm U}\cdot {\rm e}^{-{\rm j} \varphi_{\rm U} } \cdot {\rm e}^{ {\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm U}\hspace{0.01cm}'\hspace{0.05cm}\cdot \hspace{0.05cm}t} . $$

{{GraueBox|TEXT=
$\text{Beispiel 1:}$ 
Die hier angegebene Konstellation ergibt sich zum Beispiel bei der [[Modulationsverfahren/Zweiseitenband-Amplitudenmodulation#AM-Signale_und_-Spektren_bei_harmonischem_Eingangssignal|Zweiseitenband-Amplitudenmodulation]] des Nachrichtensignals $x_{\rm N}(t) = A_{\rm N}\cdot \cos\left(2\pi f_{\rm N}\cdot t- \varphi_{\rm N}\right)$ mit dem Trägersignal $x_{\rm T}(t) = A_{\rm T}\cdot \cos\left(2\pi f_{\rm T}\cdot t - \varphi_{\rm T}\right)$. Hierauf wird in der Versuchsdurchführung häufiger eingegangen.

[[File:Ortskurve_4.png|center|frame|Spektum $X_{\rm TP}(f)$ des äquivalenten TP–Signals für verschiedene Phasenkonstellationen |class=fit]]

Bei dieser Betrachtungsweise gibt es einige Einschränkungen bezüglich der Programmparameter:
* Für die Frequenzen gelte stets $f\hspace{0.05cm}'_{\rm O} = f_{\rm N}$ und $f\hspace{0.05cm}'_{\rm U} = -f_{\rm N}$.
*Ohne Verzerrungen sind die Amplitude der Seitenbänder $A_{\rm O}= A_{\rm U}= A_{\rm N}/2$.
*Die jeweiligen Phasenverhältnisse können der Grafik entnommen werden.

}}

===Darstellung des äquivalenten TP–Signals nach Betrag und Phase===

Das im Allgemeinen komplexwertige äquivalenten TP–Signal
:$$x_{\rm TP}(t) = a(t) \cdot {\rm e}^{ {\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm} \phi(t) }$$
kann entsprechend der hier angegebenen Gleichung in eine Betragsfunktion $a(t)$ und eine Phasenfunktion $\phi(t)$ aufgespalten werden, wobei gilt:
:$$a(t) = \vert x_{\rm TP}(t)\vert = \sqrt{ {\rm Re}^2\big [x_{\rm TP}(t)\big ] + {\rm Im}^2\big [x_{\rm TP}(t)\big ] }\hspace{0.05cm},$$
:$$\phi(t) = \text{arc }x_{\rm TP}(t) = \arctan \frac{{\rm Im}\big [x_{\rm TP}(t)\big ]}{{\rm Re}\big [x_{\rm TP}(t)\big ]}.$$

Der Grund dafür, dass man ein Bandpass–Signal $x(t)$ meist durch das äquivalente TP–Signal $x_{\rm TP}(t)$ beschreibt ist, dass die Funktionen $a(t)$ und $\phi(t)$ in beiden Darstellungen interpretierbar sind:
*Der Betrag $a(t)$ des äquivalentes TP–Signals $x_{\rm TP}(t)$ gibt die (zeitabhängige) Hüllkurve von $x(t)$ an.
*Die Phase $\phi(t)$ von $x_{\rm TP}(t)$ kennzeichnet die Lage der Nulldurchgänge von $x(t)$, wobei gilt:
:–   Bei $\phi(t)>0$ ist der Nulldurchgang früher als seine Solllage   ⇒   das Signal ist hier vorlaufend.
:–  Bei $\phi(t)<0$ ist der Nulldurchgang später als seine Solllage   ⇒   das Signal ist hier nachlaufend.

{{GraueBox|TEXT=
$\text{Beispiel 2:}$ 
Die Grafik soll diesen Zusammenhang verdeutlichen, wobei $A_{\rm U} > A_{\rm O}$ vorausgesetzt ist   ⇒   der grüne Zeiger (für das untere Seitenband) ist länger als der blaue Zeiger (oberes Seitenband). Es handelt sich um eine Momentaufnahme zum Zeitpunkt $t_0$:

[[File:Ortskurve_3_neu.png|center|frame|Bandpass–Spektrum $X(f)$ |class=fit]]

*Bei diesen Systemparametern liegt die Spitze des Zeigerverbundes $x_{\rm TP}(t)$ – also die geometrisch Summe aus rotem, blauem und grünem Zeiger – auf einer Ellipse.
* In der linken Grafik schwarz eingezeichnet ist der Betrag $a(t_0) = \vert x_{\rm TP}(t_0) \vert$ und in brauner Farbe angedeutet ist der Phasenwert $\phi(t_0) = \text{arc }x_{\rm TP}(t_0) > 0.$
*In der rechten Grafik gibt der Betrag $a(t_0) = \vert x_{\rm TP}(t_0) \vert$ des äquivalenten TP–Signals die Hüllkurve des physikalischen Signals $x(t)$ an.
* Bei $\phi(t) \equiv 0$ würden alle Nulldurchgänge von $x(t)$ in äquidistenten Abständen auftreten. Wegen $\phi(t_0) > 0$ ist zum Zeitpunkt $t_0$ das Signal vorlaufend, das heißt: Die Nulldurchgänge kommen früher, als es das Raster vorgibt. }}

==Versuchsdurchführung==
[[File:Zeigerdiagramm_aufgabe_2.png|right]]
*Wählen Sie zunächst die Aufgabennummer.
*Eine Aufgabenbeschreibung wird angezeigt.
*Parameterwerte sind angepasst.
*Lösung nach Drücken von &bdquo;Hide solition”.

Mit der Nummer &bdquo;0” wird auf die gleichen Einstellung wie beim Programmstart zurückgesetzt und es wird ein Text mit weiteren Erläuterungen zum Applet ausgegeben.

Im Folgenden bezeichnet $\rm Grün$ das Untere Seitenband   ⇒   $\big (A_{\rm U}, f_{\rm U}, \varphi_{\rm U}\big )$,  
$\rm Rot$ den Träger   ⇒   $\big (A_{\rm T}, f_{\rm T}, \varphi_{\rm T}\big )$ und
$\rm Blau$ das Obere Seitenband   ⇒   $\big (A_{\rm O}, f_{\rm O}, \varphi_{\rm O}\big )$.

{{BlaueBox|TEXT=
'''(1)'''   Es gelte   $\text{Rot:} \hspace{0.15cm} A_{\rm T} = 1\ \text{V}, f_{\rm T} = 100 \ \text{kHz}, \varphi_{\rm T} = 0^\circ$,   $\text{Grün:} \hspace{0.15cm} A_{\rm U} = 0.4 \text{V}, \ f_{\rm U} = 80 \ \text{kHz}, \varphi_{\rm U} = -90^\circ$,   $\text{Blau:} \hspace{0.15cm} A_{\rm O} = 0.4\ \text{V}, f_{\rm O} = 120 \ \text{kHz}, \varphi_{\rm O} = 90^\circ$.

:Betrachten und interpretieren Sie das äquivalente TP–Signal $x_{\rm TP}(t)$ und das physikalische Signal $x(t)$. Welche Periodendauer $T_0$ erkennt man?}}

:: Das äquivalente TP–Signal $x_{\rm TP}(t)$ nimmt ausgehend von $x_{\rm TP}(t=0)=1\ \text{V}$ auf der reellen Achse Werte zwischen $0.2\ \text{V}$ und $1.8\ \text{V}$ an   ⇒   Phase $\phi(t) \equiv 0$.  Der Betrag $|x_{\rm TP}(t)|$ gibt die Hüllkurve $a(t)$ des physikalischen Signals $x(t)$ an. Es gilt mit $A_{\rm N} = 0.8\ \text{V}$ und $f_{\rm N} = 20\ \text{kHz}$:   $a(t) = A_{\rm T}+ A_{\rm N} \cdot \sin(2\pi\cdot f_{\rm N} \cdot t)$.  Sowohl $x_{\rm TP}(t)$ als auch $x(t)$ sind periodisch mit der Periodendauer $T_0 = 1/f_{\rm N} = 50\ \rm µ s$.

{{BlaueBox|TEXT=
'''(2)'''   Wie ändern sich die Verhältnisse gegenüber '''(1)''' mit $f_{\rm U} = 99 \ \text{kHz}$ und $f_{\rm O} = 101 \ \text{kHz}$ ? Wie könnte $x(t)$ entstanden sein?}}

:: Für die Hüllkurve $a(t)$ des Signals $x(t)$ gilt weiterhin $a(t) = A_{\rm T}+ A_{\rm N} \cdot \sin(2\pi\cdot f_{\rm N} \cdot t)$, aber nun mit $f_{\rm N} = 1\ \text{kHz}$. Auch wenn es nicht zu erkennen ist:  $x_{\rm TP}(t)$ und $x(t)$ sind weiterhin periodisch:   $T_0 = 1\ \rm ms$. Beispiel: Zweiseitenband–Amplitudenmodulation '''(ZSB–AM)''' eines Sinussignals mit Cosinus–Träger.

{{BlaueBox|TEXT=
'''(3)'''   Welche Einstellungen müssen gegenüber '''(2)''' geändert werden, um zur ZSB–AM eines Cosinussignals mit Sinus–Träger zu gelangen. Was ändert sich gegenüber '''(2)'''?}}

::Die Trägerphase muss auf $\varphi_{\rm T} = 90^\circ$ geändert werden   ⇒   Sinus–Träger. Ebenso muss $\varphi_{\rm O} =\varphi_{\rm U} =\varphi_{\rm T} = 90^\circ$ eingestellt werden   ⇒   cosinusförmige Nachricht  Die Ortskurve liegt nun auf der imaginären Achse  ⇒   $\phi(t) \equiv -90^\circ$. Zu Beginn gilt $x_{\rm TP}(t=0)= - {\rm j} \cdot 1.8 \ \text{V}$.

{{BlaueBox|TEXT=
'''(4)'''   Nun gelte   $\text{Rot:} \hspace{0.15cm} A_{\rm T} = 1\ \text{V}, \ f_{\rm T} = 100 \ \text{kHz}, \ \varphi_{\rm T} = 0^\circ$,   $\text{Grün:} \hspace{0.15cm} A_{\rm U} = 0.4 \text{V}, \ f_{\rm U} = 80 \ \text{kHz}, \ \varphi_{\rm U} = 0^\circ$,   $\text{Blau:} \hspace{0.15cm} A_{\rm O} = 0.4\ \text{V}, \ f_{\rm O} = 120 \ \text{kHz}, \ \varphi_{\rm O} = 0^\circ$.

:Welche Eigenschaften weist dieses System &bdquo;ZSB–AM, wobei Nachrichtensignal und Träger jeweils cosinusförmig” auf? Wie groß ist der Modulationsgrad $m$?}}

:: Das äquivalente TP–Signal $x_{\rm TP}(t)$ nimmt ausgehend von $x_{\rm TP}(t=0)=1.8\ \text{V}$ auf der reellen Achse Werte zwischen $0.2\ \text{V}$ und $1.8\ \text{V}$ an   ⇒   Phase $\phi(t) \equiv 0$.  Bis auf den Startzustand $x_{\rm TP}(t=0)$ gleiches Verhalten wie bei der Einstellung '''(1)'''. Der Modulationsgrad ist jeweils $m = 0.8$.

{{BlaueBox|TEXT=
'''(5)'''   Es gelten weiter die Parameter gemäß '''(4)''' mit Ausnahme von $A_{\rm T}= 0.6 \text{V}$. Wie groß ist nun der Modulationsgrad $m$? Welche Konsequenzen hat das?}}

:: Es liegt nun eine ZSB–AM mit Modulationsgrad $m = 1.333$ vor. Bei $m > 1$ ist die einfachere [[Modulationsverfahren/Hüllkurvendemodulation|Hüllkurvendemodulation]] nicht anwendbar, da nun die Phasenfunktion $\phi(t) \in \{ 0, \ \pm 180^\circ\}$ nicht mehr konstant ist und die Hüllkurve $a(t)$ nicht mehr mit dem Nachrichtensignal übereinstimmt. Vielmehr muss die aufwändigere [[Modulationsverfahren/Synchrondemodulation|Synchrondemodulation]] verwendet werden. Bei Hüllkurvendemodulation käme es zu nichtlinearen Verzerrungen.

{{BlaueBox|TEXT=
'''(6)'''   Es gelten weiter die Parameter gemäß '''(4)''' bzw. '''(5)''' mit Ausnahme von $A_{\rm T}= 0$ an   ⇒   $m \to \infty$. Welches Modulationsverfahren wird so beschrieben?}}

::Es handelt sich um eine '''ZSB–AM ohne Träger''' und es ist eine eine Synchrondemodulation erforderlich. Das äquivalente TP–Signal $x_{\rm TP}(t)$ liegt zwar auf der reellen Achse, aber nicht nur in der rechten Halbebene. Damit gilt auch hier für die Phasenfunktion $\phi(t) \in \{ 0, \ \pm 180^\circ\}$, wodurch Hüllkurvendemodulation nicht anwendbar ist.

{{BlaueBox|TEXT=
'''(7)'''   Nun gelte   $\text{Rot:} \hspace{0.15cm} A_{\rm T} = 1\ \text{V}, f_{\rm T} = 100 \ \text{kHz}, \varphi_{\rm T} = 0^\circ$,   $\text{Grün:} \hspace{0.15cm} A_{\rm U} = 0, \ f_{\rm U} = 80 \ \text{kHz}, \varphi_{\rm U} = -90^\circ$,   $\text{Blau:} \hspace{0.15cm} A_{\rm O} = 0.8\ \text{V}, f_{\rm O} = 120 \ \text{kHz}, \varphi_{\rm O} = 90^\circ$.

:Welches Konstellation wird hiermit beschrieben? Welche Eigenschaften dieses Verfahrens erkennt man aus der Grafik?}}

::Es handelt es sich um eine [[Modulationsverfahren/Einseitenbandmodulation|Einseitenbandmodulation]] '''(ESB–AM)''', genauer gesagt um eine '''OSB–AM''': Der rote Träger liegt fest, der grüne Zeiger fehlt und der blaue Zeiger (OSB) dreht entgegen dem Uhrzeigersinn. Der Modulationsgrad ist $\mu = 0.8$ (bei ESB bezeichnen wir den Modulationsgrad mit $\mu$ anstelle von $m$). Das Trägersignal ist cosinusförmig und das Nachrichtensignal sinusförmig. Die Ortskurve ist ein Kreis. $x_{\rm TP}(t)$ bewegt sich darauf in mathematisch positiver Richtung. Wegen $\phi(t) \ne \text{const.}$ ist auch hier die Hüllkurvendemodulation nicht anwendbar:  Dies erkennt man daran, dass die Hüllkurve $a(t)$ nicht cosinusförmig ist. Vielmehr ist die untere Halbwelle spitzer als die obere   ⇒   starke lineare Verzerrungen.

{{BlaueBox|TEXT=
'''(8)'''   Es gelten weiter die Parameter gemäß '''(7)''' mit Ausnahme von $A_{\rm O}= 0$ und $A_{\rm U}= 0.8 \text{V}$. Welche Unterschiede ergeben sich gegenüber '''(7)'''?}}

::Nun handelt es sich um eine '''USB–AM''': Der rote Träger liegt fest, der blaue Zeiger fehlt und der grüne Zeiger (USB) dreht im Uhrzeigersinn. Alle anderen Aussagen von '''(7)''' treffen auch hier zu.

{{BlaueBox|TEXT=
'''(9)'''   Es gelten weiter die Parameter gemäß '''(7)''' mit Ausnahme von $A_{\rm O} = 0.2 \text{ V} \ne A_{\rm U} = 0.4 \text{ V} $. Welche Unterschiede ergeben sich gegenüber '''(7)'''?}}

::Die Ortskurve $x_{\rm TP}(t)$ ist nun keine horizontale Gerade, sondern eine Ellipse mit dem Realteil zwischen $0.4 \text{ V}$ und $1.6 \text{ V}$ sowie dem Imaginärteil im Bereich $\pm 0.2 \text{ V}$. Wegen $\phi(t) \ne \text{const.}$ würde auch hier die Hüllkurvendemodulation zu nichtlinearen Verzerrungen führen Die hier simulierte Konstellation beschreibt die Situation von '''(4)''', nämlich eine ZSB–AM mit Modulationsgrad $m = 0.8$, wobei das obere Seitenband aufgrund der Kanaldämpfung auf $50\%$ reduziert wird.

==Zur Handhabung des Applets==
 
[[File:Ortskurve_abzug3.png|right]]
* Die roten Parameter $(A_{\rm T}, \ f_{\rm T}, \ \varphi_{\rm T})$ und der rote Zeiger kennzeichnen den '''T'''räger.
* Die grünen Parameter $(A_{\rm U}, \ f_{\rm U} < f_{\rm T}, \ \varphi_{\rm U})$ kennzeichnen das '''U'''ntere Seitenband.
* Die blauen Parameter $(A_{\rm O}, \ f_{\rm O} > f_{\rm T}, \ \varphi_{\rm O})$ kennzeichnen das '''O'''bere Seitenband.
* Der rote Zeiger dreht nicht.
* Der grüne Zeiger dreht in mathematisch negativer Richtung (im Uhrzeigersinn).
* Der blaue Zeiger dreht entgegen dem Uhrzeigersinn.

Bedeutung der Buchstaben in nebenstehender Grafik:

    '''(A)'''     Grafikfeld für das äquivalente TP–Signal $x_{\rm TP}(t)$

    '''(B)'''     Grafikfeld für das physikalische Signal $x(t)$

    '''(C)'''     Parametereingabe per Slider:   Amplituden, Frequenzen, Phasenwerte

    '''(D)'''     Bedienelemente:   Start – Step – Pause/Continue – Reset

    '''(E)'''     Geschwindigkeit der Animation:   &bdquo;Speed”   ⇒   Werte: 1, 2 oder 3

    '''(F)'''     &bdquo;Trace”   ⇒   Ein oder Aus, Spur des äquivalenten TP–Signals   $x_{\rm TP}(t)$

    '''(G)'''     Numerikausgabe:   Zeit $t$, Signalwerte  ${\rm Re}[x_{\rm TP}(t)]$  und  ${\rm Im}[x_{\rm TP}(t)]$,

$\text{}\hspace{4.2cm}$   Hüllkurve $a(t) = |x_{\rm TP}(t)|$  und  Phase $\phi(t) = {\rm arc} \ x_{\rm TP}(t)$

    '''(H)'''     Variationsmöglichkeiten für die grafische Darstellung

$\hspace{1.5cm}$Zoom–Funktionen &bdquo;$+$” (Vergrößern), &bdquo;$-$” (Verkleinern) und $\rm o$ (Zurücksetzen)

$\hspace{1.5cm}$Verschieben mit &bdquo;$\leftarrow$” (Ausschnitt nach links, Ordinate nach rechts), &bdquo;$\uparrow$” &bdquo;$\downarrow$” &bdquo;$\rightarrow$”

    '''(I)'''     Bereich für die Versuchsdurchführung:  Aufgabenauswahl und Aufgabenstellung

    '''(J)'''     Bereich für die Versuchsdurchführung:  Musterlösung
 

==Über die Autoren==
Dieses interaktive Berechnungstool wurde am [http://www.lnt.ei.tum.de/startseite Lehrstuhl für Nachrichtentechnik] der [https://www.tum.de/ Technischen Universität München] konzipiert und realisiert.
*Die erste Version wurde 2005 von [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Studierende#Ji_Li_.28Bachelorarbeit_EI_2003.2C_Diplomarbeit_EI_2005.29|Ji Li]] im Rahmen ihrer Diplomarbeit mit &bdquo;FlashMX–Actionscript” erstellt (Betreuer: [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Mitarbeiter_und_Dozenten#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28am_LNT_seit_1974.29|Günter Söder]]).
*2018 wurde dieses Programm von [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Studierende#Xiaohan_Liu_.28Bachelorarbeit_2018.29|Xiaohan Liu]] im Rahmen ihrer Bachelorarbeit (Betreuer: [[Biografien_und_Bibliografien/Beteiligte_der_Professur_Leitungsgebundene_%C3%9Cbertragungstechnik#Tasn.C3.A1d_Kernetzky.2C_M.Sc._.28bei_L.C3.9CT_seit_2014.29|Tasnád Kernetzky]]) neu gestaltet und erweitert.

==Nochmalige Aufrufmöglichkeit des Applets in neuem Fenster==

{{LntAppletLink|physAnTPSignal}}

[[Category:Applets|^Verzerrungen^]]

Applets:Bessel Functions of the First Kind

2018-08-25T13:41:33Z

Xiaohan: /* Applet Description */

{{LntAppletLink|bessel}}

==Applet Description==
 
This applet allows the calculation and graphical representation of the Bessel functions of the first kind and $n$–th order according to the series representation:

:$${\rm J}_n (x) = \sum\limits_{k=0}^{\infty}\frac{(-1)^k \cdot (x/2)^{n \hspace{0.05cm} + \hspace{0.05cm} 2
\hspace{0.02cm}\cdot \hspace{0.05cm}k}}{k! \cdot (n+k)!} \hspace{0.05cm}.$$

*Graphically represented, the functions ${\rm J}_n (x)$ for the order $n=0$ to $n=9$ can become different colors.
*The left-hand output provides the function values ${\rm J}_0 (x = x_1)$, ... , ${\rm J}_9 (x = x_1)$ for a slider-settable value $x_1$ in the range $0 \le x_1 \le 15$ with increment $0.5$.
*The right hand side returns the function values ${\rm J}_0 (x = x_2)$, ... , ${\rm J}_9 (x = x_2)$ for a slider-settable value $x_2$ (same range and value) Increment as on the left).

[[Applets:Besselfunktionen_erster_Art_(neues_Applet)|'''German description''']]

==Theoretical Background==
 
===General Information about the Bessel Functions===
Bessel functions (or cylinder functions) are solutions of the Bessel differential equation of the form

:$$x^2 \cdot \frac{ {\rm d}^2}{{\rm d}x^2}\ {\rm J}_n (x) \ + \ x \cdot \frac{ {\rm d}}{{\rm d}x}\ {\rm J}_n (x) \ + \ (x^2 - n^2)
\cdot {\rm J}_n (x)= 0. $$

This is an ordinary second-order linear differential equation. The parameter $ n $ is usually integer, as in this program. These mathematical functions, which were introduced in 1844 by [https://de.wikipedia.org/wiki/Friedrich_Wilhelm_Bessel Friedrich Wilhelm Bessel], can also be represented in closed form as integrals:

:$${\rm J}_n (x) = \frac{1}{2\pi}\cdot \int_{-\pi}^{+\pi} {{\rm e}^{\hspace{0.05cm}{\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}[\hspace{0.05cm}x \hspace{0.05cm}\cdot \hspace{0.05cm}\sin(\alpha) -\hspace{0.05cm} n \hspace{0.05cm}\cdot \hspace{0.05cm}\alpha \hspace{0.05cm}]}}\hspace{0.1cm}{\rm d}\alpha
\hspace{0.05cm}.$$

The functions ${\rm J}_n (x)$ belong to the class of Bessel functions of the first kind (German:   ''Besselfunktionen erster Art''). The parameter $n$ is called the ''Order''.

''Annotation:''   There are a number of modifications of the Bessel functions, including the second-order Bessel functions named ${\rm Y}_n (x)$. For integer $n$, ${\rm Y}_n (x)$ can be replaced by ${\rm J}_n (x)$–functions. However, in this applet, only the first-order Bessel functions are   ⇒   ${\rm J}_n (x)$ is considered.
 
===Properties of the Bessel Functions===

{{BlaueBox|TEXT=
$\text{Property (A):}$   If the function values for $n = 0$ and $n = 1$ are known, then the Bessel function for $n ≥ 2$ can be determined iteratively:
:$${\rm J}_n (x) ={2 \cdot (n-1)}/{x} \cdot {\rm J}_{n-1} (x) - {\rm J}_{n-2} (x) \hspace{0.05cm}.$$
}}

{{GraueBox|TEXT=
$\text{Example (A):}$   Consider ${\rm J}_0 (x = 2) = 0.22389$ and ${\rm J}_1 (x= 2) = 0.57672$. From this it can be calculated iteratively:
:$${\rm J}_2 (x= 2) ={2 \cdot 1}/{2} \cdot {\rm J}_{1} (x= 2) - {\rm J}_{0} (x= 2) = 0.57672 - 0.22389 = \hspace{0.15cm}\underline{0.35283}\hspace{0.05cm},$$
:$${\rm J}_3 (x= 2) ={2 \cdot 2}/{2} \cdot {\rm J}_{2} (x= 2) - {\rm J}_{1} (x= 2) = 2 \cdot 0.35283 - 0.57672 = \hspace{0.15cm}\underline{0.12894}\hspace{0.05cm},$$
:$${\rm J}_4 (x= 2) ={2 \cdot 3}/{2} \cdot {\rm J}_{3} (x= 2) - {\rm J}_{2} (x= 2) = 3 \cdot 0.12894 - 0.35283 = \hspace{0.15cm}\underline{0.03400}\hspace{0.05cm}.$$
}}

{{BlaueBox|TEXT=
$\text{Property (B):}$   The symmetry relationship applies ${\rm J}_{–n}(x) = (–1)^n · {\rm J}_n(x)$:
:$${\rm J}_{-1}(x) = - {\rm J}_{1}(x), \hspace{0.3cm}{\rm J}_{-2}(x) = {\rm J}_{2}(x), \hspace{0.3cm}{\rm J}_{-3}(x) = - {\rm J}_{3}(x).$$
}}

{{GraueBox|TEXT=
$\text{Example (B):}$   For the spectrum of the analytic signal, in [[Modulationsverfahren/Phasenmodulation_(PM)#Spektralfunktion_eines_phasenmodulierten_Sinussignals|phase modulation of a sinusoidal signal]]:
[[File:Mod_T_3_1_S4_version2.png|right|frame|Spectrum of the analytic signal in phase modulation]]
:$$S_{\rm +}(f) = A_{\rm T} \cdot \sum_{n = - \infty}^{+\infty}{\rm J}_n (\eta) \cdot \delta \big[f - (f_{\rm T}+ n \cdot f_{\rm N})\big]\hspace{0.05cm}.$$
Denote this
*$f_{\rm T}$ the carrier frequency,
*$f_{\rm N}$ the message frequency,
* $A_{\rm T}$ the carrier amplitude.

The parameter of the Bessel functions in this application is the modulation index $\eta$.

Based on the graphic, the following statements are possible:
*Here $S_+(f)$ consists of an infinite number of discrete lines at a distance of $f_{\rm N}$.
*It is thus in principle infinitely extended.
*The weights of the spectral lines at $f_{\rm T} + n · f_{\rm N}$ ($n$ integer) are determined by the modulation index $η$ over the Bessel functions ${\rm J}_n(η)$.
*The spectral lines are real in the case of a sinusoidal source signal and a cosinusoidal carrier and symmetric about $f_{\rm T}$ for even $n$.
*For odd $n$ , a sign change according to the $\text{Property (B)}$ has to be considered.
*The phase modulation of another phase oscillation of source and / or carrier signal provides the same magnitude spectrum.}}
 

===Applications of the Bessel Functions===

The applications of the Bessel functions in nature–and engineering are diverse and play an important role in physics, for example:
* Examination of natural vibrations of cylindrical resonators,
* Solution of the radial Schrödinger-equation,
* Sound pressure amplitudes of low-viscosity rotational flows,
* Heat conduction in cylindrical bodies,
* Scattering problem of a grid,
* Dynamics of oscillating bodies,
* Angular resolution.

The Bessel functions are counted among the special functions because of their many applications in mathematical physics.

In the following, we restrict ourselves to a few areas that are addressed in our tutorial $\rm LNTwww$.

'''In the English original'''
Electromagnetic waves in a cylindrical waveguide
Pressure amplitudes of inviscid rotational flows
Heat conduction in a cylindrical object
Modes of vibration of a thin circular (or annular) acoustic membrane (such as a drum or other membranophone)
Diffusion problems on a lattice
Solutions to the radial Schrödinger equation (in spherical and cylindrical coordinates) for a free particle
Solving for patterns of acoustical radiation
Frequency-dependent friction in circular pipelines
Dynamics of floating bodies
Angular resolution
'''The Ende'''

{{GraueBox|TEXT=
$\text{Example (C):} \hspace{0.5cm} \text{Use in spectral analysis} \ \Rightarrow \ \text{Kaiser-Bessel filter}$

The '''spectral leakage effect''' is the falsification of the spectrum of a periodic and hence temporally unlimited signal due to the implicit time limitation of the discrete Fourier transform (DFT). This will be done, for example, by a spectrum analyzer
* in the time signal not existing frequency components simulated, and/or
* actually existing spectral components are obscured by sidelobes.

The task of the [[Signaldarstellung/Spektralanalyse|spectral analysis]] is to limit the influence of the ''spectral leakage effect'' by providing suitable window functions.

Such a window function provides, for example, the Kaiser–Bessel–window   ⇒   see section [[Signaldarstellung/Spektralanalyse#Spezielle_Fensterfunktionen|Special Window Functions]]. Its time-discrete Fenser function reads with the Bessel function zero order   ⇒   ${\rm J}_0(x)$, the parameter $\alpha=3.5$ and the window length $N$:
:$$w_\nu = \frac{ {\rm J}_0\big(\pi \cdot \alpha \cdot \sqrt{1 - (2\nu/N)^2}\big)}{ {\rm J}_0\big(\pi \cdot \alpha \big)}.$$
On the page [[Signaldarstellung/Spektralanalyse#G.C3.BCtekriterien_von_Fensterfunktionen|Quality Criteria of Window Functions]] and other the characteristics of the Kaiser-Bessel-window are given:
*Conveniently, the large is &bdquo;minimum distance between the main lobe and side lobes” and the desired small &bdquo;maximum scaling error”.
*Due to the very large &bdquo;equivalent noise width” the Kaiser-Bessel window cuts in the main comparison criterion &bdquo;maximum process loss” but worse off than the established Hamming– and Hanning–windows.}}

{{GraueBox|TEXT=
$\text{Example (D):} \hspace{0.5cm} \text{Rice-Fading channel model}$

The [[Mobile_Kommunikation/Wahrscheinlichkeitsdichte_des_Rayleigh%E2%80%93Fadings#Allgemeine_Beschreibung_des_Mobilfunkkanals| Rayleigh - Distribution]] describes the mobile channel assuming that there is no direct path, and thus the multiplicative factor $z(t) = x(t) + {\rm j} \cdot y(t)$ is solely diffuse composed of scattered components.

In the case of a direct component (English: Line of Sight, LoS) in the model for the mean-free Gaussian processes $x(t)$ and $y(t)$ one has to add equal components $x_0$ and/or $y_0$ :

[[File:P ID2126 Mob T 1 4 S1 v3.png|right|frame|Rice-Fading channel model|class=fit]]
$$\hspace{0.2cm}x(t) \hspace{0.1cm} \Rightarrow \hspace{0.1cm} x(t) +x_0 \hspace{0.05cm}, \hspace{0.2cm} y(t) \hspace{0.1cm} \Rightarrow \hspace{0.1cm} y(t) +y_0\hspace{0.05cm},$$

$$\hspace{0.2cm}z(t) = x(t) + {\rm j} \cdot y(t) \hspace{0.1cm} \Rightarrow \hspace{0.1cm} z(t) +z_0 \hspace{0.05cm},\hspace{0.2cm}
z_0 = x_0 + {\rm j} \cdot y_0\hspace{0.05cm}.$$

The graph shows the ''Rice–Fading–Channel model''. It can be summarized as follows:
*The real part $x(t)$ is Gaussianized with mean $x_0$ and variance $\sigma ^2$.
*The imaginary part $y(t)$ is also Gaussian (mean $y_0$, equal variance $\sigma ^2$) and independent of $x(t)$. 

*For $z_0 \ne 0$, the amount $\vert z(t)\vert$ rice is distributed, from which the term &bdquo;Rice–Fading” arises.

*To simplify the notation, we set $\vert z(t)\vert = a(t)$. For $a < 0$ , the amounts - WDF are $f_a(a) \equiv 0$, for $a \ge 0$ the following equation holds, where ${\rm I_0}(x)$ is the modified Besselfunktion zeroth order denotes:
:$$f_a(a) = \frac{a}{\sigma^2} \cdot {\rm e}^{ - (a^2 + \vert z_0 \vert ^2)/(2\sigma^2)} \cdot {\rm I}_0 \left [ \frac{a \cdot \vert z_0 \vert}{\sigma^2} \right ] \hspace{0.5cm}\text{with}\hspace{0.5cm}{\rm I }_0 (u) = {\rm J }_0 ({\rm j} \cdot u) =
\sum_{k = 0}^{\infty} \frac{ (u/2)^{2k} }{k! \cdot \Gamma (k+1)}
\hspace{0.05cm}.$$
*Between the modified Bessel function and the traditional Bessel function ${\rm I_0}(x)$ – each first type – So the connection exists ${\rm I }_0 (u) = {\rm J }_0 ({\rm j} \cdot u)$.}}

{{GraueBox|TEXT=
$\text{Example (E):} \hspace{0.5cm} \text{Analysis of the frequency spectrum of frequency modulated signals}$

$\text{Example (B)}$ has already shown that the angle modulation of a harmonic oscillation of the frequency $f_{\rm N}$ leads to a line spectrum. The spectral lines are around the carrier frequency $f_{\rm T}$ bei $f_{\rm T} + n \cdot f_{\rm N}$ with $n \in \{ \ \text{...}, -2, -1, \ 0, +1, +2, \text{...} \ \}$. The weights of the Dirac lines are ${\rm J }_n(\eta)$, depending on the modulation index $\eta$.

[[File:P_ID1095__Mod_T_3_2_S4_neu.png|center|frame|Discrete spectra with phase modulation (left) and frequency modulation (right)]]

The graph shows the magnitude spectrum $\vert S_{\rm +}(f) \vert$ of the phase modulation (PM) and frequency modulation (FM) analytic signal, two different forms of angle modulation (WM). Bessellines with values less than $0.03$ are neglected in both cases.

For the upper half of the image, the modulator parameters are chosen so that for each $f_{\rm N} = 5 \ \rm kHz$ here is a Bessel spectrum with the modulation index $η = 1.5$. Disregarding the phase relationships, the same spectra and the same signals result for both systems.

The lower graphics are valid for the same message frequency $f_{\rm N} = 3 \ \rm kHz$. One notices:
*In phase modulation, the spectral function is narrower than $f_{\rm N} = 5 \ \rm kHz$, because the distance of the Bessel lines is now only $3 \ \rm kHz$. Since the modulation index of PM is independent of $f_{\rm N}$, the same bessel weights result as in $f_{\rm N} = 5 \ \rm kHz$.
*Also in the frequency modulation, the Bessel lines now appear at a distance of $3 \ \rm kHz$. However, since FM has a modulation index inversely proportional to $f_{\rm N}$, there is now much more Besselline lines than in the upper right (for $η = 1.5$ valid) chart due to the larger modulation index $η = 2.5$.}}

'''Das folgende Kapitel muss noch angepasst werden!'''

==Applet Manual==
[[File:Bessel_abzug3.png|left|600px]]
    '''(A)'''     Sum formula of the Bessel functions ${\rm J}_n(x)$

    '''(B)'''     Selection of the order $n$ for the graphical representation

    '''(C)'''     Plot of the Bessel functions

    '''(D)'''     Variation of the graphic representation

$\hspace{1.5cm}$&bdquo;$+$” (Enlarge),

$\hspace{1.5cm}$ &bdquo;$-$” (Decrease)

$\hspace{1.5cm}$ &bdquo;$\rm o$” (Reset to default)

$\hspace{1.5cm}$ &bdquo;$\leftarrow$” (pushed to the left), usw.

    '''(E)'''     Selection of the abscissa value $x_1$ for the left numeric output

    '''(F)'''     Numerical output of the Bessel function values ${\rm J}_n(x_1)$

    '''(G)'''     Selection of the abscissa value $x_2$ for the right numeric output

    '''(F)'''     Numerical output of the Bessel function values ${\rm J}_n(x_2)$
 

==About the Authors==
This interactive calculation was designed and realized at the [http://www.lnt.ei.tum.de/startseite Lehrstuhl für Nachrichtentechnik] of the [https://www.tum.de/ Technical University of Munich] .
*The original version was created in 2005 by [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Studierende#Markus_Elsberger_.28Diplomarbeit_LB_2006.29|Markus Elsberger]] and [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Studierende#Slim_Lamine_.28Studienarbeit_EI_2006.29|Slim Lamine]] as part of thesis using &bdquo;FlashMX–Actionscript” (Supervisor: [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Mitarbeiter_und_Dozenten#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28am_LNT_seit_1974.29|Günter Söder]]).
*In 2018 this Applet was redesigned and updated to &bdquo;HTML5” by [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Studierende#Xiaohan_Liu_.28Bachelorarbeit_2018.29|Xiaohan Liu]] as part of her Bachelor's thesis (Supervisor: [[Biografien_und_Bibliografien/Beteiligte_der_Professur_Leitungsgebundene_%C3%9Cbertragungstechnik#Tasn.C3.A1d_Kernetzky.2C_M.Sc._.28bei_L.C3.9CT_seit_2014.29|Tasnád Kernetzky]]).

==Nochmalige Aufrufmöglichkeit des Applets in neuem Fenster==

{{LntAppletLink|bessel}}

Applets:Bessel Functions of the First Kind

2018-08-25T13:39:46Z

Xiaohan:

{{LntAppletLink|bessel}}

==Applet Description==
 
This applet allows the calculation and graphical representation of the Bessel functions of the first kind and $n$–th order according to the series representation:

:$${\rm J}_n (x) = \sum\limits_{k=0}^{\infty}\frac{(-1)^k \cdot (x/2)^{n \hspace{0.05cm} + \hspace{0.05cm} 2
\hspace{0.02cm}\cdot \hspace{0.05cm}k}}{k! \cdot (n+k)!} \hspace{0.05cm}.$$

*Graphically represented, the functions ${\rm J}_n (x)$ for the order $n=0$ to $n=9$ can become different colors.
*The left-hand output provides the function values ${\rm J}_0 (x = x_1)$, ... , ${\rm J}_9 (x = x_1)$ for a slider-settable value $x_1$ in the range $0 \le x_1 \le 15$ with increment $0.5$.
*The right hand side returns the function values ${\rm J}_0 (x = x_2)$, ... , ${\rm J}_9 (x = x_2)$ for a slider-settable value $x_2$ (same range and value) Increment as on the left).

[[Applets:Applets:Besselfunktionen_erster_Art|'''German description''']]

==Theoretical Background==
 
===General Information about the Bessel Functions===
Bessel functions (or cylinder functions) are solutions of the Bessel differential equation of the form

:$$x^2 \cdot \frac{ {\rm d}^2}{{\rm d}x^2}\ {\rm J}_n (x) \ + \ x \cdot \frac{ {\rm d}}{{\rm d}x}\ {\rm J}_n (x) \ + \ (x^2 - n^2)
\cdot {\rm J}_n (x)= 0. $$

This is an ordinary second-order linear differential equation. The parameter $ n $ is usually integer, as in this program. These mathematical functions, which were introduced in 1844 by [https://de.wikipedia.org/wiki/Friedrich_Wilhelm_Bessel Friedrich Wilhelm Bessel], can also be represented in closed form as integrals:

:$${\rm J}_n (x) = \frac{1}{2\pi}\cdot \int_{-\pi}^{+\pi} {{\rm e}^{\hspace{0.05cm}{\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}[\hspace{0.05cm}x \hspace{0.05cm}\cdot \hspace{0.05cm}\sin(\alpha) -\hspace{0.05cm} n \hspace{0.05cm}\cdot \hspace{0.05cm}\alpha \hspace{0.05cm}]}}\hspace{0.1cm}{\rm d}\alpha
\hspace{0.05cm}.$$

The functions ${\rm J}_n (x)$ belong to the class of Bessel functions of the first kind (German:   ''Besselfunktionen erster Art''). The parameter $n$ is called the ''Order''.

''Annotation:''   There are a number of modifications of the Bessel functions, including the second-order Bessel functions named ${\rm Y}_n (x)$. For integer $n$, ${\rm Y}_n (x)$ can be replaced by ${\rm J}_n (x)$–functions. However, in this applet, only the first-order Bessel functions are   ⇒   ${\rm J}_n (x)$ is considered.
 
===Properties of the Bessel Functions===

{{BlaueBox|TEXT=
$\text{Property (A):}$   If the function values for $n = 0$ and $n = 1$ are known, then the Bessel function for $n ≥ 2$ can be determined iteratively:
:$${\rm J}_n (x) ={2 \cdot (n-1)}/{x} \cdot {\rm J}_{n-1} (x) - {\rm J}_{n-2} (x) \hspace{0.05cm}.$$
}}

{{GraueBox|TEXT=
$\text{Example (A):}$   Consider ${\rm J}_0 (x = 2) = 0.22389$ and ${\rm J}_1 (x= 2) = 0.57672$. From this it can be calculated iteratively:
:$${\rm J}_2 (x= 2) ={2 \cdot 1}/{2} \cdot {\rm J}_{1} (x= 2) - {\rm J}_{0} (x= 2) = 0.57672 - 0.22389 = \hspace{0.15cm}\underline{0.35283}\hspace{0.05cm},$$
:$${\rm J}_3 (x= 2) ={2 \cdot 2}/{2} \cdot {\rm J}_{2} (x= 2) - {\rm J}_{1} (x= 2) = 2 \cdot 0.35283 - 0.57672 = \hspace{0.15cm}\underline{0.12894}\hspace{0.05cm},$$
:$${\rm J}_4 (x= 2) ={2 \cdot 3}/{2} \cdot {\rm J}_{3} (x= 2) - {\rm J}_{2} (x= 2) = 3 \cdot 0.12894 - 0.35283 = \hspace{0.15cm}\underline{0.03400}\hspace{0.05cm}.$$
}}

{{BlaueBox|TEXT=
$\text{Property (B):}$   The symmetry relationship applies ${\rm J}_{–n}(x) = (–1)^n · {\rm J}_n(x)$:
:$${\rm J}_{-1}(x) = - {\rm J}_{1}(x), \hspace{0.3cm}{\rm J}_{-2}(x) = {\rm J}_{2}(x), \hspace{0.3cm}{\rm J}_{-3}(x) = - {\rm J}_{3}(x).$$
}}

{{GraueBox|TEXT=
$\text{Example (B):}$   For the spectrum of the analytic signal, in [[Modulationsverfahren/Phasenmodulation_(PM)#Spektralfunktion_eines_phasenmodulierten_Sinussignals|phase modulation of a sinusoidal signal]]:
[[File:Mod_T_3_1_S4_version2.png|right|frame|Spectrum of the analytic signal in phase modulation]]
:$$S_{\rm +}(f) = A_{\rm T} \cdot \sum_{n = - \infty}^{+\infty}{\rm J}_n (\eta) \cdot \delta \big[f - (f_{\rm T}+ n \cdot f_{\rm N})\big]\hspace{0.05cm}.$$
Denote this
*$f_{\rm T}$ the carrier frequency,
*$f_{\rm N}$ the message frequency,
* $A_{\rm T}$ the carrier amplitude.

The parameter of the Bessel functions in this application is the modulation index $\eta$.

Based on the graphic, the following statements are possible:
*Here $S_+(f)$ consists of an infinite number of discrete lines at a distance of $f_{\rm N}$.
*It is thus in principle infinitely extended.
*The weights of the spectral lines at $f_{\rm T} + n · f_{\rm N}$ ($n$ integer) are determined by the modulation index $η$ over the Bessel functions ${\rm J}_n(η)$.
*The spectral lines are real in the case of a sinusoidal source signal and a cosinusoidal carrier and symmetric about $f_{\rm T}$ for even $n$.
*For odd $n$ , a sign change according to the $\text{Property (B)}$ has to be considered.
*The phase modulation of another phase oscillation of source and / or carrier signal provides the same magnitude spectrum.}}
 

===Applications of the Bessel Functions===

The applications of the Bessel functions in nature–and engineering are diverse and play an important role in physics, for example:
* Examination of natural vibrations of cylindrical resonators,
* Solution of the radial Schrödinger-equation,
* Sound pressure amplitudes of low-viscosity rotational flows,
* Heat conduction in cylindrical bodies,
* Scattering problem of a grid,
* Dynamics of oscillating bodies,
* Angular resolution.

The Bessel functions are counted among the special functions because of their many applications in mathematical physics.

In the following, we restrict ourselves to a few areas that are addressed in our tutorial $\rm LNTwww$.

'''In the English original'''
Electromagnetic waves in a cylindrical waveguide
Pressure amplitudes of inviscid rotational flows
Heat conduction in a cylindrical object
Modes of vibration of a thin circular (or annular) acoustic membrane (such as a drum or other membranophone)
Diffusion problems on a lattice
Solutions to the radial Schrödinger equation (in spherical and cylindrical coordinates) for a free particle
Solving for patterns of acoustical radiation
Frequency-dependent friction in circular pipelines
Dynamics of floating bodies
Angular resolution
'''The Ende'''

{{GraueBox|TEXT=
$\text{Example (C):} \hspace{0.5cm} \text{Use in spectral analysis} \ \Rightarrow \ \text{Kaiser-Bessel filter}$

The '''spectral leakage effect''' is the falsification of the spectrum of a periodic and hence temporally unlimited signal due to the implicit time limitation of the discrete Fourier transform (DFT). This will be done, for example, by a spectrum analyzer
* in the time signal not existing frequency components simulated, and/or
* actually existing spectral components are obscured by sidelobes.

The task of the [[Signaldarstellung/Spektralanalyse|spectral analysis]] is to limit the influence of the ''spectral leakage effect'' by providing suitable window functions.

Such a window function provides, for example, the Kaiser–Bessel–window   ⇒   see section [[Signaldarstellung/Spektralanalyse#Spezielle_Fensterfunktionen|Special Window Functions]]. Its time-discrete Fenser function reads with the Bessel function zero order   ⇒   ${\rm J}_0(x)$, the parameter $\alpha=3.5$ and the window length $N$:
:$$w_\nu = \frac{ {\rm J}_0\big(\pi \cdot \alpha \cdot \sqrt{1 - (2\nu/N)^2}\big)}{ {\rm J}_0\big(\pi \cdot \alpha \big)}.$$
On the page [[Signaldarstellung/Spektralanalyse#G.C3.BCtekriterien_von_Fensterfunktionen|Quality Criteria of Window Functions]] and other the characteristics of the Kaiser-Bessel-window are given:
*Conveniently, the large is &bdquo;minimum distance between the main lobe and side lobes” and the desired small &bdquo;maximum scaling error”.
*Due to the very large &bdquo;equivalent noise width” the Kaiser-Bessel window cuts in the main comparison criterion &bdquo;maximum process loss” but worse off than the established Hamming– and Hanning–windows.}}

{{GraueBox|TEXT=
$\text{Example (D):} \hspace{0.5cm} \text{Rice-Fading channel model}$

The [[Mobile_Kommunikation/Wahrscheinlichkeitsdichte_des_Rayleigh%E2%80%93Fadings#Allgemeine_Beschreibung_des_Mobilfunkkanals| Rayleigh - Distribution]] describes the mobile channel assuming that there is no direct path, and thus the multiplicative factor $z(t) = x(t) + {\rm j} \cdot y(t)$ is solely diffuse composed of scattered components.

In the case of a direct component (English: Line of Sight, LoS) in the model for the mean-free Gaussian processes $x(t)$ and $y(t)$ one has to add equal components $x_0$ and/or $y_0$ :

[[File:P ID2126 Mob T 1 4 S1 v3.png|right|frame|Rice-Fading channel model|class=fit]]
$$\hspace{0.2cm}x(t) \hspace{0.1cm} \Rightarrow \hspace{0.1cm} x(t) +x_0 \hspace{0.05cm}, \hspace{0.2cm} y(t) \hspace{0.1cm} \Rightarrow \hspace{0.1cm} y(t) +y_0\hspace{0.05cm},$$

$$\hspace{0.2cm}z(t) = x(t) + {\rm j} \cdot y(t) \hspace{0.1cm} \Rightarrow \hspace{0.1cm} z(t) +z_0 \hspace{0.05cm},\hspace{0.2cm}
z_0 = x_0 + {\rm j} \cdot y_0\hspace{0.05cm}.$$

The graph shows the ''Rice–Fading–Channel model''. It can be summarized as follows:
*The real part $x(t)$ is Gaussianized with mean $x_0$ and variance $\sigma ^2$.
*The imaginary part $y(t)$ is also Gaussian (mean $y_0$, equal variance $\sigma ^2$) and independent of $x(t)$. 

*For $z_0 \ne 0$, the amount $\vert z(t)\vert$ rice is distributed, from which the term &bdquo;Rice–Fading” arises.

*To simplify the notation, we set $\vert z(t)\vert = a(t)$. For $a < 0$ , the amounts - WDF are $f_a(a) \equiv 0$, for $a \ge 0$ the following equation holds, where ${\rm I_0}(x)$ is the modified Besselfunktion zeroth order denotes:
:$$f_a(a) = \frac{a}{\sigma^2} \cdot {\rm e}^{ - (a^2 + \vert z_0 \vert ^2)/(2\sigma^2)} \cdot {\rm I}_0 \left [ \frac{a \cdot \vert z_0 \vert}{\sigma^2} \right ] \hspace{0.5cm}\text{with}\hspace{0.5cm}{\rm I }_0 (u) = {\rm J }_0 ({\rm j} \cdot u) =
\sum_{k = 0}^{\infty} \frac{ (u/2)^{2k} }{k! \cdot \Gamma (k+1)}
\hspace{0.05cm}.$$
*Between the modified Bessel function and the traditional Bessel function ${\rm I_0}(x)$ – each first type – So the connection exists ${\rm I }_0 (u) = {\rm J }_0 ({\rm j} \cdot u)$.}}

{{GraueBox|TEXT=
$\text{Example (E):} \hspace{0.5cm} \text{Analysis of the frequency spectrum of frequency modulated signals}$

$\text{Example (B)}$ has already shown that the angle modulation of a harmonic oscillation of the frequency $f_{\rm N}$ leads to a line spectrum. The spectral lines are around the carrier frequency $f_{\rm T}$ bei $f_{\rm T} + n \cdot f_{\rm N}$ with $n \in \{ \ \text{...}, -2, -1, \ 0, +1, +2, \text{...} \ \}$. The weights of the Dirac lines are ${\rm J }_n(\eta)$, depending on the modulation index $\eta$.

[[File:P_ID1095__Mod_T_3_2_S4_neu.png|center|frame|Discrete spectra with phase modulation (left) and frequency modulation (right)]]

The graph shows the magnitude spectrum $\vert S_{\rm +}(f) \vert$ of the phase modulation (PM) and frequency modulation (FM) analytic signal, two different forms of angle modulation (WM). Bessellines with values less than $0.03$ are neglected in both cases.

For the upper half of the image, the modulator parameters are chosen so that for each $f_{\rm N} = 5 \ \rm kHz$ here is a Bessel spectrum with the modulation index $η = 1.5$. Disregarding the phase relationships, the same spectra and the same signals result for both systems.

The lower graphics are valid for the same message frequency $f_{\rm N} = 3 \ \rm kHz$. One notices:
*In phase modulation, the spectral function is narrower than $f_{\rm N} = 5 \ \rm kHz$, because the distance of the Bessel lines is now only $3 \ \rm kHz$. Since the modulation index of PM is independent of $f_{\rm N}$, the same bessel weights result as in $f_{\rm N} = 5 \ \rm kHz$.
*Also in the frequency modulation, the Bessel lines now appear at a distance of $3 \ \rm kHz$. However, since FM has a modulation index inversely proportional to $f_{\rm N}$, there is now much more Besselline lines than in the upper right (for $η = 1.5$ valid) chart due to the larger modulation index $η = 2.5$.}}

'''Das folgende Kapitel muss noch angepasst werden!'''

==Applet Manual==
[[File:Bessel_abzug3.png|left|600px]]
    '''(A)'''     Sum formula of the Bessel functions ${\rm J}_n(x)$

    '''(B)'''     Selection of the order $n$ for the graphical representation

    '''(C)'''     Plot of the Bessel functions

    '''(D)'''     Variation of the graphic representation

$\hspace{1.5cm}$&bdquo;$+$” (Enlarge),

$\hspace{1.5cm}$ &bdquo;$-$” (Decrease)

$\hspace{1.5cm}$ &bdquo;$\rm o$” (Reset to default)

$\hspace{1.5cm}$ &bdquo;$\leftarrow$” (pushed to the left), usw.

    '''(E)'''     Selection of the abscissa value $x_1$ for the left numeric output

    '''(F)'''     Numerical output of the Bessel function values ${\rm J}_n(x_1)$

    '''(G)'''     Selection of the abscissa value $x_2$ for the right numeric output

    '''(F)'''     Numerical output of the Bessel function values ${\rm J}_n(x_2)$
 

==About the Authors==
This interactive calculation was designed and realized at the [http://www.lnt.ei.tum.de/startseite Lehrstuhl für Nachrichtentechnik] of the [https://www.tum.de/ Technical University of Munich] .
*The original version was created in 2005 by [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Studierende#Markus_Elsberger_.28Diplomarbeit_LB_2006.29|Markus Elsberger]] and [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Studierende#Slim_Lamine_.28Studienarbeit_EI_2006.29|Slim Lamine]] as part of thesis using &bdquo;FlashMX–Actionscript” (Supervisor: [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Mitarbeiter_und_Dozenten#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28am_LNT_seit_1974.29|Günter Söder]]).
*In 2018 this Applet was redesigned and updated to &bdquo;HTML5” by [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Studierende#Xiaohan_Liu_.28Bachelorarbeit_2018.29|Xiaohan Liu]] as part of her Bachelor's thesis (Supervisor: [[Biografien_und_Bibliografien/Beteiligte_der_Professur_Leitungsgebundene_%C3%9Cbertragungstechnik#Tasn.C3.A1d_Kernetzky.2C_M.Sc._.28bei_L.C3.9CT_seit_2014.29|Tasnád Kernetzky]]).

==Nochmalige Aufrufmöglichkeit des Applets in neuem Fenster==

{{LntAppletLink|bessel}}

Applets:Bessel Functions of the First Kind

2018-08-25T13:38:15Z

Xiaohan:

{{LntAppletLink|bessel}}

==Applet Description==
 
This applet allows the calculation and graphical representation of the Bessel functions of the first kind and $n$–th order according to the series representation:

:$${\rm J}_n (x) = \sum\limits_{k=0}^{\infty}\frac{(-1)^k \cdot (x/2)^{n \hspace{0.05cm} + \hspace{0.05cm} 2
\hspace{0.02cm}\cdot \hspace{0.05cm}k}}{k! \cdot (n+k)!} \hspace{0.05cm}.$$

*Graphically represented, the functions ${\rm J}_n (x)$ for the order $n=0$ to $n=9$ can become different colors.
*The left-hand output provides the function values ${\rm J}_0 (x = x_1)$, ... , ${\rm J}_9 (x = x_1)$ for a slider-settable value $x_1$ in the range $0 \le x_1 \le 15$ with increment $0.5$.
*The right hand side returns the function values ${\rm J}_0 (x = x_2)$, ... , ${\rm J}_9 (x = x_2)$ for a slider-settable value $x_2$ (same range and value) Increment as on the left).

[[Applets:Applets:Besselfunktionen_erster_Art_(neues_Applet)|'''German description''']]

==Theoretical Background==
 
===General Information about the Bessel Functions===
Bessel functions (or cylinder functions) are solutions of the Bessel differential equation of the form

:$$x^2 \cdot \frac{ {\rm d}^2}{{\rm d}x^2}\ {\rm J}_n (x) \ + \ x \cdot \frac{ {\rm d}}{{\rm d}x}\ {\rm J}_n (x) \ + \ (x^2 - n^2)
\cdot {\rm J}_n (x)= 0. $$

This is an ordinary second-order linear differential equation. The parameter $ n $ is usually integer, as in this program. These mathematical functions, which were introduced in 1844 by [https://de.wikipedia.org/wiki/Friedrich_Wilhelm_Bessel Friedrich Wilhelm Bessel], can also be represented in closed form as integrals:

:$${\rm J}_n (x) = \frac{1}{2\pi}\cdot \int_{-\pi}^{+\pi} {{\rm e}^{\hspace{0.05cm}{\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}[\hspace{0.05cm}x \hspace{0.05cm}\cdot \hspace{0.05cm}\sin(\alpha) -\hspace{0.05cm} n \hspace{0.05cm}\cdot \hspace{0.05cm}\alpha \hspace{0.05cm}]}}\hspace{0.1cm}{\rm d}\alpha
\hspace{0.05cm}.$$

The functions ${\rm J}_n (x)$ belong to the class of Bessel functions of the first kind (German:   ''Besselfunktionen erster Art''). The parameter $n$ is called the ''Order''.

''Annotation:''   There are a number of modifications of the Bessel functions, including the second-order Bessel functions named ${\rm Y}_n (x)$. For integer $n$, ${\rm Y}_n (x)$ can be replaced by ${\rm J}_n (x)$–functions. However, in this applet, only the first-order Bessel functions are   ⇒   ${\rm J}_n (x)$ is considered.
 
===Properties of the Bessel Functions===

{{BlaueBox|TEXT=
$\text{Property (A):}$   If the function values for $n = 0$ and $n = 1$ are known, then the Bessel function for $n ≥ 2$ can be determined iteratively:
:$${\rm J}_n (x) ={2 \cdot (n-1)}/{x} \cdot {\rm J}_{n-1} (x) - {\rm J}_{n-2} (x) \hspace{0.05cm}.$$
}}

{{GraueBox|TEXT=
$\text{Example (A):}$   Consider ${\rm J}_0 (x = 2) = 0.22389$ and ${\rm J}_1 (x= 2) = 0.57672$. From this it can be calculated iteratively:
:$${\rm J}_2 (x= 2) ={2 \cdot 1}/{2} \cdot {\rm J}_{1} (x= 2) - {\rm J}_{0} (x= 2) = 0.57672 - 0.22389 = \hspace{0.15cm}\underline{0.35283}\hspace{0.05cm},$$
:$${\rm J}_3 (x= 2) ={2 \cdot 2}/{2} \cdot {\rm J}_{2} (x= 2) - {\rm J}_{1} (x= 2) = 2 \cdot 0.35283 - 0.57672 = \hspace{0.15cm}\underline{0.12894}\hspace{0.05cm},$$
:$${\rm J}_4 (x= 2) ={2 \cdot 3}/{2} \cdot {\rm J}_{3} (x= 2) - {\rm J}_{2} (x= 2) = 3 \cdot 0.12894 - 0.35283 = \hspace{0.15cm}\underline{0.03400}\hspace{0.05cm}.$$
}}

{{BlaueBox|TEXT=
$\text{Property (B):}$   The symmetry relationship applies ${\rm J}_{–n}(x) = (–1)^n · {\rm J}_n(x)$:
:$${\rm J}_{-1}(x) = - {\rm J}_{1}(x), \hspace{0.3cm}{\rm J}_{-2}(x) = {\rm J}_{2}(x), \hspace{0.3cm}{\rm J}_{-3}(x) = - {\rm J}_{3}(x).$$
}}

{{GraueBox|TEXT=
$\text{Example (B):}$   For the spectrum of the analytic signal, in [[Modulationsverfahren/Phasenmodulation_(PM)#Spektralfunktion_eines_phasenmodulierten_Sinussignals|phase modulation of a sinusoidal signal]]:
[[File:Mod_T_3_1_S4_version2.png|right|frame|Spectrum of the analytic signal in phase modulation]]
:$$S_{\rm +}(f) = A_{\rm T} \cdot \sum_{n = - \infty}^{+\infty}{\rm J}_n (\eta) \cdot \delta \big[f - (f_{\rm T}+ n \cdot f_{\rm N})\big]\hspace{0.05cm}.$$
Denote this
*$f_{\rm T}$ the carrier frequency,
*$f_{\rm N}$ the message frequency,
* $A_{\rm T}$ the carrier amplitude.

The parameter of the Bessel functions in this application is the modulation index $\eta$.

Based on the graphic, the following statements are possible:
*Here $S_+(f)$ consists of an infinite number of discrete lines at a distance of $f_{\rm N}$.
*It is thus in principle infinitely extended.
*The weights of the spectral lines at $f_{\rm T} + n · f_{\rm N}$ ($n$ integer) are determined by the modulation index $η$ over the Bessel functions ${\rm J}_n(η)$.
*The spectral lines are real in the case of a sinusoidal source signal and a cosinusoidal carrier and symmetric about $f_{\rm T}$ for even $n$.
*For odd $n$ , a sign change according to the $\text{Property (B)}$ has to be considered.
*The phase modulation of another phase oscillation of source and / or carrier signal provides the same magnitude spectrum.}}
 

===Applications of the Bessel Functions===

The applications of the Bessel functions in nature–and engineering are diverse and play an important role in physics, for example:
* Examination of natural vibrations of cylindrical resonators,
* Solution of the radial Schrödinger-equation,
* Sound pressure amplitudes of low-viscosity rotational flows,
* Heat conduction in cylindrical bodies,
* Scattering problem of a grid,
* Dynamics of oscillating bodies,
* Angular resolution.

The Bessel functions are counted among the special functions because of their many applications in mathematical physics.

In the following, we restrict ourselves to a few areas that are addressed in our tutorial $\rm LNTwww$.

'''In the English original'''
Electromagnetic waves in a cylindrical waveguide
Pressure amplitudes of inviscid rotational flows
Heat conduction in a cylindrical object
Modes of vibration of a thin circular (or annular) acoustic membrane (such as a drum or other membranophone)
Diffusion problems on a lattice
Solutions to the radial Schrödinger equation (in spherical and cylindrical coordinates) for a free particle
Solving for patterns of acoustical radiation
Frequency-dependent friction in circular pipelines
Dynamics of floating bodies
Angular resolution
'''The Ende'''

{{GraueBox|TEXT=
$\text{Example (C):} \hspace{0.5cm} \text{Use in spectral analysis} \ \Rightarrow \ \text{Kaiser-Bessel filter}$

The '''spectral leakage effect''' is the falsification of the spectrum of a periodic and hence temporally unlimited signal due to the implicit time limitation of the discrete Fourier transform (DFT). This will be done, for example, by a spectrum analyzer
* in the time signal not existing frequency components simulated, and/or
* actually existing spectral components are obscured by sidelobes.

The task of the [[Signaldarstellung/Spektralanalyse|spectral analysis]] is to limit the influence of the ''spectral leakage effect'' by providing suitable window functions.

Such a window function provides, for example, the Kaiser–Bessel–window   ⇒   see section [[Signaldarstellung/Spektralanalyse#Spezielle_Fensterfunktionen|Special Window Functions]]. Its time-discrete Fenser function reads with the Bessel function zero order   ⇒   ${\rm J}_0(x)$, the parameter $\alpha=3.5$ and the window length $N$:
:$$w_\nu = \frac{ {\rm J}_0\big(\pi \cdot \alpha \cdot \sqrt{1 - (2\nu/N)^2}\big)}{ {\rm J}_0\big(\pi \cdot \alpha \big)}.$$
On the page [[Signaldarstellung/Spektralanalyse#G.C3.BCtekriterien_von_Fensterfunktionen|Quality Criteria of Window Functions]] and other the characteristics of the Kaiser-Bessel-window are given:
*Conveniently, the large is &bdquo;minimum distance between the main lobe and side lobes” and the desired small &bdquo;maximum scaling error”.
*Due to the very large &bdquo;equivalent noise width” the Kaiser-Bessel window cuts in the main comparison criterion &bdquo;maximum process loss” but worse off than the established Hamming– and Hanning–windows.}}

{{GraueBox|TEXT=
$\text{Example (D):} \hspace{0.5cm} \text{Rice-Fading channel model}$

The [[Mobile_Kommunikation/Wahrscheinlichkeitsdichte_des_Rayleigh%E2%80%93Fadings#Allgemeine_Beschreibung_des_Mobilfunkkanals| Rayleigh - Distribution]] describes the mobile channel assuming that there is no direct path, and thus the multiplicative factor $z(t) = x(t) + {\rm j} \cdot y(t)$ is solely diffuse composed of scattered components.

In the case of a direct component (English: Line of Sight, LoS) in the model for the mean-free Gaussian processes $x(t)$ and $y(t)$ one has to add equal components $x_0$ and/or $y_0$ :

[[File:P ID2126 Mob T 1 4 S1 v3.png|right|frame|Rice-Fading channel model|class=fit]]
$$\hspace{0.2cm}x(t) \hspace{0.1cm} \Rightarrow \hspace{0.1cm} x(t) +x_0 \hspace{0.05cm}, \hspace{0.2cm} y(t) \hspace{0.1cm} \Rightarrow \hspace{0.1cm} y(t) +y_0\hspace{0.05cm},$$

$$\hspace{0.2cm}z(t) = x(t) + {\rm j} \cdot y(t) \hspace{0.1cm} \Rightarrow \hspace{0.1cm} z(t) +z_0 \hspace{0.05cm},\hspace{0.2cm}
z_0 = x_0 + {\rm j} \cdot y_0\hspace{0.05cm}.$$

The graph shows the ''Rice–Fading–Channel model''. It can be summarized as follows:
*The real part $x(t)$ is Gaussianized with mean $x_0$ and variance $\sigma ^2$.
*The imaginary part $y(t)$ is also Gaussian (mean $y_0$, equal variance $\sigma ^2$) and independent of $x(t)$. 

*For $z_0 \ne 0$, the amount $\vert z(t)\vert$ rice is distributed, from which the term &bdquo;Rice–Fading” arises.

*To simplify the notation, we set $\vert z(t)\vert = a(t)$. For $a < 0$ , the amounts - WDF are $f_a(a) \equiv 0$, for $a \ge 0$ the following equation holds, where ${\rm I_0}(x)$ is the modified Besselfunktion zeroth order denotes:
:$$f_a(a) = \frac{a}{\sigma^2} \cdot {\rm e}^{ - (a^2 + \vert z_0 \vert ^2)/(2\sigma^2)} \cdot {\rm I}_0 \left [ \frac{a \cdot \vert z_0 \vert}{\sigma^2} \right ] \hspace{0.5cm}\text{with}\hspace{0.5cm}{\rm I }_0 (u) = {\rm J }_0 ({\rm j} \cdot u) =
\sum_{k = 0}^{\infty} \frac{ (u/2)^{2k} }{k! \cdot \Gamma (k+1)}
\hspace{0.05cm}.$$
*Between the modified Bessel function and the traditional Bessel function ${\rm I_0}(x)$ – each first type – So the connection exists ${\rm I }_0 (u) = {\rm J }_0 ({\rm j} \cdot u)$.}}

{{GraueBox|TEXT=
$\text{Example (E):} \hspace{0.5cm} \text{Analysis of the frequency spectrum of frequency modulated signals}$

$\text{Example (B)}$ has already shown that the angle modulation of a harmonic oscillation of the frequency $f_{\rm N}$ leads to a line spectrum. The spectral lines are around the carrier frequency $f_{\rm T}$ bei $f_{\rm T} + n \cdot f_{\rm N}$ with $n \in \{ \ \text{...}, -2, -1, \ 0, +1, +2, \text{...} \ \}$. The weights of the Dirac lines are ${\rm J }_n(\eta)$, depending on the modulation index $\eta$.

[[File:P_ID1095__Mod_T_3_2_S4_neu.png|center|frame|Discrete spectra with phase modulation (left) and frequency modulation (right)]]

The graph shows the magnitude spectrum $\vert S_{\rm +}(f) \vert$ of the phase modulation (PM) and frequency modulation (FM) analytic signal, two different forms of angle modulation (WM). Bessellines with values less than $0.03$ are neglected in both cases.

For the upper half of the image, the modulator parameters are chosen so that for each $f_{\rm N} = 5 \ \rm kHz$ here is a Bessel spectrum with the modulation index $η = 1.5$. Disregarding the phase relationships, the same spectra and the same signals result for both systems.

The lower graphics are valid for the same message frequency $f_{\rm N} = 3 \ \rm kHz$. One notices:
*In phase modulation, the spectral function is narrower than $f_{\rm N} = 5 \ \rm kHz$, because the distance of the Bessel lines is now only $3 \ \rm kHz$. Since the modulation index of PM is independent of $f_{\rm N}$, the same bessel weights result as in $f_{\rm N} = 5 \ \rm kHz$.
*Also in the frequency modulation, the Bessel lines now appear at a distance of $3 \ \rm kHz$. However, since FM has a modulation index inversely proportional to $f_{\rm N}$, there is now much more Besselline lines than in the upper right (for $η = 1.5$ valid) chart due to the larger modulation index $η = 2.5$.}}

'''Das folgende Kapitel muss noch angepasst werden!'''

==Applet Manual==
[[File:Bessel_abzug3.png|left|600px]]
    '''(A)'''     Sum formula of the Bessel functions ${\rm J}_n(x)$

    '''(B)'''     Selection of the order $n$ for the graphical representation

    '''(C)'''     Plot of the Bessel functions

    '''(D)'''     Variation of the graphic representation

$\hspace{1.5cm}$&bdquo;$+$” (Enlarge),

$\hspace{1.5cm}$ &bdquo;$-$” (Decrease)

$\hspace{1.5cm}$ &bdquo;$\rm o$” (Reset to default)

$\hspace{1.5cm}$ &bdquo;$\leftarrow$” (pushed to the left), usw.

    '''(E)'''     Selection of the abscissa value $x_1$ for the left numeric output

    '''(F)'''     Numerical output of the Bessel function values ${\rm J}_n(x_1)$

    '''(G)'''     Selection of the abscissa value $x_2$ for the right numeric output

    '''(F)'''     Numerical output of the Bessel function values ${\rm J}_n(x_2)$
 

==About the Authors==
This interactive calculation was designed and realized at the [http://www.lnt.ei.tum.de/startseite Lehrstuhl für Nachrichtentechnik] of the [https://www.tum.de/ Technical University of Munich] .
*The original version was created in 2005 by [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Studierende#Markus_Elsberger_.28Diplomarbeit_LB_2006.29|Markus Elsberger]] and [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Studierende#Slim_Lamine_.28Studienarbeit_EI_2006.29|Slim Lamine]] as part of thesis using &bdquo;FlashMX–Actionscript” (Supervisor: [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Mitarbeiter_und_Dozenten#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28am_LNT_seit_1974.29|Günter Söder]]).
*In 2018 this Applet was redesigned and updated to &bdquo;HTML5” by [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Studierende#Xiaohan_Liu_.28Bachelorarbeit_2018.29|Xiaohan Liu]] as part of her Bachelor's thesis (Supervisor: [[Biografien_und_Bibliografien/Beteiligte_der_Professur_Leitungsgebundene_%C3%9Cbertragungstechnik#Tasn.C3.A1d_Kernetzky.2C_M.Sc._.28bei_L.C3.9CT_seit_2014.29|Tasnád Kernetzky]]).

==Nochmalige Aufrufmöglichkeit des Applets in neuem Fenster==

{{LntAppletLink|bessel}}

Applets:Bessel Functions of the First Kind

2018-08-25T13:36:21Z

Xiaohan:

{{LntAppletLink|bessel}}

==Applet Description==
 
This applet allows the calculation and graphical representation of the Bessel functions of the first kind and $n$–th order according to the series representation:

:$${\rm J}_n (x) = \sum\limits_{k=0}^{\infty}\frac{(-1)^k \cdot (x/2)^{n \hspace{0.05cm} + \hspace{0.05cm} 2
\hspace{0.02cm}\cdot \hspace{0.05cm}k}}{k! \cdot (n+k)!} \hspace{0.05cm}.$$

*Graphically represented, the functions ${\rm J}_n (x)$ for the order $n=0$ to $n=9$ can become different colors.
*The left-hand output provides the function values ${\rm J}_0 (x = x_1)$, ... , ${\rm J}_9 (x = x_1)$ for a slider-settable value $x_1$ in the range $0 \le x_1 \le 15$ with increment $0.5$.
*The right hand side returns the function values ${\rm J}_0 (x = x_2)$, ... , ${\rm J}_9 (x = x_2)$ for a slider-settable value $x_2$ (same range and value) Increment as on the left).

[[Applets:Physical_Signal_%26_Analytical_Signal|'''German description''']]

==Theoretical Background==
 
===General Information about the Bessel Functions===
Bessel functions (or cylinder functions) are solutions of the Bessel differential equation of the form

:$$x^2 \cdot \frac{ {\rm d}^2}{{\rm d}x^2}\ {\rm J}_n (x) \ + \ x \cdot \frac{ {\rm d}}{{\rm d}x}\ {\rm J}_n (x) \ + \ (x^2 - n^2)
\cdot {\rm J}_n (x)= 0. $$

This is an ordinary second-order linear differential equation. The parameter $ n $ is usually integer, as in this program. These mathematical functions, which were introduced in 1844 by [https://de.wikipedia.org/wiki/Friedrich_Wilhelm_Bessel Friedrich Wilhelm Bessel], can also be represented in closed form as integrals:

:$${\rm J}_n (x) = \frac{1}{2\pi}\cdot \int_{-\pi}^{+\pi} {{\rm e}^{\hspace{0.05cm}{\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}[\hspace{0.05cm}x \hspace{0.05cm}\cdot \hspace{0.05cm}\sin(\alpha) -\hspace{0.05cm} n \hspace{0.05cm}\cdot \hspace{0.05cm}\alpha \hspace{0.05cm}]}}\hspace{0.1cm}{\rm d}\alpha
\hspace{0.05cm}.$$

The functions ${\rm J}_n (x)$ belong to the class of Bessel functions of the first kind (German:   ''Besselfunktionen erster Art''). The parameter $n$ is called the ''Order''.

''Annotation:''   There are a number of modifications of the Bessel functions, including the second-order Bessel functions named ${\rm Y}_n (x)$. For integer $n$, ${\rm Y}_n (x)$ can be replaced by ${\rm J}_n (x)$–functions. However, in this applet, only the first-order Bessel functions are   ⇒   ${\rm J}_n (x)$ is considered.
 
===Properties of the Bessel Functions===

{{BlaueBox|TEXT=
$\text{Property (A):}$   If the function values for $n = 0$ and $n = 1$ are known, then the Bessel function for $n ≥ 2$ can be determined iteratively:
:$${\rm J}_n (x) ={2 \cdot (n-1)}/{x} \cdot {\rm J}_{n-1} (x) - {\rm J}_{n-2} (x) \hspace{0.05cm}.$$
}}

{{GraueBox|TEXT=
$\text{Example (A):}$   Consider ${\rm J}_0 (x = 2) = 0.22389$ and ${\rm J}_1 (x= 2) = 0.57672$. From this it can be calculated iteratively:
:$${\rm J}_2 (x= 2) ={2 \cdot 1}/{2} \cdot {\rm J}_{1} (x= 2) - {\rm J}_{0} (x= 2) = 0.57672 - 0.22389 = \hspace{0.15cm}\underline{0.35283}\hspace{0.05cm},$$
:$${\rm J}_3 (x= 2) ={2 \cdot 2}/{2} \cdot {\rm J}_{2} (x= 2) - {\rm J}_{1} (x= 2) = 2 \cdot 0.35283 - 0.57672 = \hspace{0.15cm}\underline{0.12894}\hspace{0.05cm},$$
:$${\rm J}_4 (x= 2) ={2 \cdot 3}/{2} \cdot {\rm J}_{3} (x= 2) - {\rm J}_{2} (x= 2) = 3 \cdot 0.12894 - 0.35283 = \hspace{0.15cm}\underline{0.03400}\hspace{0.05cm}.$$
}}

{{BlaueBox|TEXT=
$\text{Property (B):}$   The symmetry relationship applies ${\rm J}_{–n}(x) = (–1)^n · {\rm J}_n(x)$:
:$${\rm J}_{-1}(x) = - {\rm J}_{1}(x), \hspace{0.3cm}{\rm J}_{-2}(x) = {\rm J}_{2}(x), \hspace{0.3cm}{\rm J}_{-3}(x) = - {\rm J}_{3}(x).$$
}}

{{GraueBox|TEXT=
$\text{Example (B):}$   For the spectrum of the analytic signal, in [[Modulationsverfahren/Phasenmodulation_(PM)#Spektralfunktion_eines_phasenmodulierten_Sinussignals|phase modulation of a sinusoidal signal]]:
[[File:Mod_T_3_1_S4_version2.png|right|frame|Spectrum of the analytic signal in phase modulation]]
:$$S_{\rm +}(f) = A_{\rm T} \cdot \sum_{n = - \infty}^{+\infty}{\rm J}_n (\eta) \cdot \delta \big[f - (f_{\rm T}+ n \cdot f_{\rm N})\big]\hspace{0.05cm}.$$
Denote this
*$f_{\rm T}$ the carrier frequency,
*$f_{\rm N}$ the message frequency,
* $A_{\rm T}$ the carrier amplitude.

The parameter of the Bessel functions in this application is the modulation index $\eta$.

Based on the graphic, the following statements are possible:
*Here $S_+(f)$ consists of an infinite number of discrete lines at a distance of $f_{\rm N}$.
*It is thus in principle infinitely extended.
*The weights of the spectral lines at $f_{\rm T} + n · f_{\rm N}$ ($n$ integer) are determined by the modulation index $η$ over the Bessel functions ${\rm J}_n(η)$.
*The spectral lines are real in the case of a sinusoidal source signal and a cosinusoidal carrier and symmetric about $f_{\rm T}$ for even $n$.
*For odd $n$ , a sign change according to the $\text{Property (B)}$ has to be considered.
*The phase modulation of another phase oscillation of source and / or carrier signal provides the same magnitude spectrum.}}
 

===Applications of the Bessel Functions===

The applications of the Bessel functions in nature–and engineering are diverse and play an important role in physics, for example:
* Examination of natural vibrations of cylindrical resonators,
* Solution of the radial Schrödinger-equation,
* Sound pressure amplitudes of low-viscosity rotational flows,
* Heat conduction in cylindrical bodies,
* Scattering problem of a grid,
* Dynamics of oscillating bodies,
* Angular resolution.

The Bessel functions are counted among the special functions because of their many applications in mathematical physics.

In the following, we restrict ourselves to a few areas that are addressed in our tutorial $\rm LNTwww$.

'''In the English original'''
Electromagnetic waves in a cylindrical waveguide
Pressure amplitudes of inviscid rotational flows
Heat conduction in a cylindrical object
Modes of vibration of a thin circular (or annular) acoustic membrane (such as a drum or other membranophone)
Diffusion problems on a lattice
Solutions to the radial Schrödinger equation (in spherical and cylindrical coordinates) for a free particle
Solving for patterns of acoustical radiation
Frequency-dependent friction in circular pipelines
Dynamics of floating bodies
Angular resolution
'''The Ende'''

{{GraueBox|TEXT=
$\text{Example (C):} \hspace{0.5cm} \text{Use in spectral analysis} \ \Rightarrow \ \text{Kaiser-Bessel filter}$

The '''spectral leakage effect''' is the falsification of the spectrum of a periodic and hence temporally unlimited signal due to the implicit time limitation of the discrete Fourier transform (DFT). This will be done, for example, by a spectrum analyzer
* in the time signal not existing frequency components simulated, and/or
* actually existing spectral components are obscured by sidelobes.

The task of the [[Signaldarstellung/Spektralanalyse|spectral analysis]] is to limit the influence of the ''spectral leakage effect'' by providing suitable window functions.

Such a window function provides, for example, the Kaiser–Bessel–window   ⇒   see section [[Signaldarstellung/Spektralanalyse#Spezielle_Fensterfunktionen|Special Window Functions]]. Its time-discrete Fenser function reads with the Bessel function zero order   ⇒   ${\rm J}_0(x)$, the parameter $\alpha=3.5$ and the window length $N$:
:$$w_\nu = \frac{ {\rm J}_0\big(\pi \cdot \alpha \cdot \sqrt{1 - (2\nu/N)^2}\big)}{ {\rm J}_0\big(\pi \cdot \alpha \big)}.$$
On the page [[Signaldarstellung/Spektralanalyse#G.C3.BCtekriterien_von_Fensterfunktionen|Quality Criteria of Window Functions]] and other the characteristics of the Kaiser-Bessel-window are given:
*Conveniently, the large is &bdquo;minimum distance between the main lobe and side lobes” and the desired small &bdquo;maximum scaling error”.
*Due to the very large &bdquo;equivalent noise width” the Kaiser-Bessel window cuts in the main comparison criterion &bdquo;maximum process loss” but worse off than the established Hamming– and Hanning–windows.}}

{{GraueBox|TEXT=
$\text{Example (D):} \hspace{0.5cm} \text{Rice-Fading channel model}$

The [[Mobile_Kommunikation/Wahrscheinlichkeitsdichte_des_Rayleigh%E2%80%93Fadings#Allgemeine_Beschreibung_des_Mobilfunkkanals| Rayleigh - Distribution]] describes the mobile channel assuming that there is no direct path, and thus the multiplicative factor $z(t) = x(t) + {\rm j} \cdot y(t)$ is solely diffuse composed of scattered components.

In the case of a direct component (English: Line of Sight, LoS) in the model for the mean-free Gaussian processes $x(t)$ and $y(t)$ one has to add equal components $x_0$ and/or $y_0$ :

[[File:P ID2126 Mob T 1 4 S1 v3.png|right|frame|Rice-Fading channel model|class=fit]]
$$\hspace{0.2cm}x(t) \hspace{0.1cm} \Rightarrow \hspace{0.1cm} x(t) +x_0 \hspace{0.05cm}, \hspace{0.2cm} y(t) \hspace{0.1cm} \Rightarrow \hspace{0.1cm} y(t) +y_0\hspace{0.05cm},$$

$$\hspace{0.2cm}z(t) = x(t) + {\rm j} \cdot y(t) \hspace{0.1cm} \Rightarrow \hspace{0.1cm} z(t) +z_0 \hspace{0.05cm},\hspace{0.2cm}
z_0 = x_0 + {\rm j} \cdot y_0\hspace{0.05cm}.$$

The graph shows the ''Rice–Fading–Channel model''. It can be summarized as follows:
*The real part $x(t)$ is Gaussianized with mean $x_0$ and variance $\sigma ^2$.
*The imaginary part $y(t)$ is also Gaussian (mean $y_0$, equal variance $\sigma ^2$) and independent of $x(t)$. 

*For $z_0 \ne 0$, the amount $\vert z(t)\vert$ rice is distributed, from which the term &bdquo;Rice–Fading” arises.

*To simplify the notation, we set $\vert z(t)\vert = a(t)$. For $a < 0$ , the amounts - WDF are $f_a(a) \equiv 0$, for $a \ge 0$ the following equation holds, where ${\rm I_0}(x)$ is the modified Besselfunktion zeroth order denotes:
:$$f_a(a) = \frac{a}{\sigma^2} \cdot {\rm e}^{ - (a^2 + \vert z_0 \vert ^2)/(2\sigma^2)} \cdot {\rm I}_0 \left [ \frac{a \cdot \vert z_0 \vert}{\sigma^2} \right ] \hspace{0.5cm}\text{with}\hspace{0.5cm}{\rm I }_0 (u) = {\rm J }_0 ({\rm j} \cdot u) =
\sum_{k = 0}^{\infty} \frac{ (u/2)^{2k} }{k! \cdot \Gamma (k+1)}
\hspace{0.05cm}.$$
*Between the modified Bessel function and the traditional Bessel function ${\rm I_0}(x)$ – each first type – So the connection exists ${\rm I }_0 (u) = {\rm J }_0 ({\rm j} \cdot u)$.}}

{{GraueBox|TEXT=
$\text{Example (E):} \hspace{0.5cm} \text{Analysis of the frequency spectrum of frequency modulated signals}$

$\text{Example (B)}$ has already shown that the angle modulation of a harmonic oscillation of the frequency $f_{\rm N}$ leads to a line spectrum. The spectral lines are around the carrier frequency $f_{\rm T}$ bei $f_{\rm T} + n \cdot f_{\rm N}$ with $n \in \{ \ \text{...}, -2, -1, \ 0, +1, +2, \text{...} \ \}$. The weights of the Dirac lines are ${\rm J }_n(\eta)$, depending on the modulation index $\eta$.

[[File:P_ID1095__Mod_T_3_2_S4_neu.png|center|frame|Discrete spectra with phase modulation (left) and frequency modulation (right)]]

The graph shows the magnitude spectrum $\vert S_{\rm +}(f) \vert$ of the phase modulation (PM) and frequency modulation (FM) analytic signal, two different forms of angle modulation (WM). Bessellines with values less than $0.03$ are neglected in both cases.

For the upper half of the image, the modulator parameters are chosen so that for each $f_{\rm N} = 5 \ \rm kHz$ here is a Bessel spectrum with the modulation index $η = 1.5$. Disregarding the phase relationships, the same spectra and the same signals result for both systems.

The lower graphics are valid for the same message frequency $f_{\rm N} = 3 \ \rm kHz$. One notices:
*In phase modulation, the spectral function is narrower than $f_{\rm N} = 5 \ \rm kHz$, because the distance of the Bessel lines is now only $3 \ \rm kHz$. Since the modulation index of PM is independent of $f_{\rm N}$, the same bessel weights result as in $f_{\rm N} = 5 \ \rm kHz$.
*Also in the frequency modulation, the Bessel lines now appear at a distance of $3 \ \rm kHz$. However, since FM has a modulation index inversely proportional to $f_{\rm N}$, there is now much more Besselline lines than in the upper right (for $η = 1.5$ valid) chart due to the larger modulation index $η = 2.5$.}}

'''Das folgende Kapitel muss noch angepasst werden!'''

==Applet Manual==
[[File:Bessel_abzug3.png|left|600px]]
    '''(A)'''     Sum formula of the Bessel functions ${\rm J}_n(x)$

    '''(B)'''     Selection of the order $n$ for the graphical representation

    '''(C)'''     Plot of the Bessel functions

    '''(D)'''     Variation of the graphic representation

$\hspace{1.5cm}$&bdquo;$+$” (Enlarge),

$\hspace{1.5cm}$ &bdquo;$-$” (Decrease)

$\hspace{1.5cm}$ &bdquo;$\rm o$” (Reset to default)

$\hspace{1.5cm}$ &bdquo;$\leftarrow$” (pushed to the left), usw.

    '''(E)'''     Selection of the abscissa value $x_1$ for the left numeric output

    '''(F)'''     Numerical output of the Bessel function values ${\rm J}_n(x_1)$

    '''(G)'''     Selection of the abscissa value $x_2$ for the right numeric output

    '''(F)'''     Numerical output of the Bessel function values ${\rm J}_n(x_2)$
 

==About the Authors==
This interactive calculation was designed and realized at the [http://www.lnt.ei.tum.de/startseite Lehrstuhl für Nachrichtentechnik] of the [https://www.tum.de/ Technical University of Munich] .
*The original version was created in 2005 by [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Studierende#Markus_Elsberger_.28Diplomarbeit_LB_2006.29|Markus Elsberger]] and [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Studierende#Slim_Lamine_.28Studienarbeit_EI_2006.29|Slim Lamine]] as part of thesis using &bdquo;FlashMX–Actionscript” (Supervisor: [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Mitarbeiter_und_Dozenten#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28am_LNT_seit_1974.29|Günter Söder]]).
*In 2018 this Applet was redesigned and updated to &bdquo;HTML5” by [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Studierende#Xiaohan_Liu_.28Bachelorarbeit_2018.29|Xiaohan Liu]] as part of her Bachelor's thesis (Supervisor: [[Biografien_und_Bibliografien/Beteiligte_der_Professur_Leitungsgebundene_%C3%9Cbertragungstechnik#Tasn.C3.A1d_Kernetzky.2C_M.Sc._.28bei_L.C3.9CT_seit_2014.29|Tasnád Kernetzky]]).

==Nochmalige Aufrufmöglichkeit des Applets in neuem Fenster==

{{LntAppletLink|bessel}}

Applets:Bessel Functions of the First Kind

2018-08-24T08:51:32Z

Xiaohan: /* Applet Manual */

{{LntAppletLink|bessel}}

==Applet Description==
 
This applet allows the calculation and graphical representation of the Bessel functions of the first kind and $n$–th order according to the series representation:

:$${\rm J}_n (x) = \sum\limits_{k=0}^{\infty}\frac{(-1)^k \cdot (x/2)^{n \hspace{0.05cm} + \hspace{0.05cm} 2
\hspace{0.02cm}\cdot \hspace{0.05cm}k}}{k! \cdot (n+k)!} \hspace{0.05cm}.$$

*Graphically represented, the functions ${\rm J}_n (x)$ for the order $n=0$ to $n=9$ can become different colors.
*The left-hand output provides the function values ${\rm J}_0 (x = x_1)$, ... , ${\rm J}_9 (x = x_1)$ for a slider-settable value $x_1$ in the range $0 \le x_1 \le 15$ with increment $0.5$.
*The right hand side returns the function values ${\rm J}_0 (x = x_2)$, ... , ${\rm J}_9 (x = x_2)$ for a slider-settable value $x_2$ (same range and value) Increment as on the left).

[[Applets:Physical_Signal_%26_Analytical_Signal|'''German description''']]

==Theoretical Background==
 
===General Information about the Bessel Functions===
Bessel functions (or cylinder functions) are solutions of the Bessel differential equation of the form

:$$x^2 \cdot \frac{ {\rm d}^2}{{\rm d}x^2}\ {\rm J}_n (x) \ + \ x \cdot \frac{ {\rm d}}{{\rm d}x}\ {\rm J}_n (x) \ + \ (x^2 - n^2)
\cdot {\rm J}_n (x)= 0. $$

This is an ordinary second-order linear differential equation. The parameter $ n $ is usually integer, as in this program. These mathematical functions, which were introduced in 1844 by [https://de.wikipedia.org/wiki/Friedrich_Wilhelm_Bessel Friedrich Wilhelm Bessel], can also be represented in closed form as integrals:

:$${\rm J}_n (x) = \frac{1}{2\pi}\cdot \int_{-\pi}^{+\pi} {{\rm e}^{\hspace{0.05cm}{\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}[\hspace{0.05cm}x \hspace{0.05cm}\cdot \hspace{0.05cm}\sin(\alpha) -\hspace{0.05cm} n \hspace{0.05cm}\cdot \hspace{0.05cm}\alpha \hspace{0.05cm}]}}\hspace{0.1cm}{\rm d}\alpha
\hspace{0.05cm}.$$

The functions ${\rm J}_n (x)$ belong to the class of Bessel functions of the first kind (German:   ''Besselfunktionen erster Art''). The parameter $n$ is called the ''Order''.

''Annotation:''   There are a number of modifications of the Bessel functions, including the second-order Bessel functions named ${\rm Y}_n (x)$. For integer $n$, ${\rm Y}_n (x)$ can be replaced by ${\rm J}_n (x)$–functions. However, in this applet, only the first-order Bessel functions are   ⇒   ${\rm J}_n (x)$ is considered.
 
===Properties of the Bessel Functions===

{{BlaueBox|TEXT=
$\text{Property (A):}$   If the function values for $n = 0$ and $n = 1$ are known, then the Bessel function for $n ≥ 2$ can be determined iteratively:
:$${\rm J}_n (x) ={2 \cdot (n-1)}/{x} \cdot {\rm J}_{n-1} (x) - {\rm J}_{n-2} (x) \hspace{0.05cm}.$$
}}

{{GraueBox|TEXT=
$\text{Example (A):}$   Consider ${\rm J}_0 (x = 2) = 0.22389$ and ${\rm J}_1 (x= 2) = 0.57672$. From this it can be calculated iteratively:
:$${\rm J}_2 (x= 2) ={2 \cdot 1}/{2} \cdot {\rm J}_{1} (x= 2) - {\rm J}_{0} (x= 2) = 0.57672 - 0.22389 = \hspace{0.15cm}\underline{0.35283}\hspace{0.05cm},$$
:$${\rm J}_3 (x= 2) ={2 \cdot 2}/{2} \cdot {\rm J}_{2} (x= 2) - {\rm J}_{1} (x= 2) = 2 \cdot 0.35283 - 0.57672 = \hspace{0.15cm}\underline{0.12894}\hspace{0.05cm},$$
:$${\rm J}_4 (x= 2) ={2 \cdot 3}/{2} \cdot {\rm J}_{3} (x= 2) - {\rm J}_{2} (x= 2) = 3 \cdot 0.12894 - 0.35283 = \hspace{0.15cm}\underline{0.03400}\hspace{0.05cm}.$$
}}

{{BlaueBox|TEXT=
$\text{Property (B):}$   The symmetry relationship applies ${\rm J}_{–n}(x) = (–1)^n · {\rm J}_n(x)$:
:$${\rm J}_{-1}(x) = - {\rm J}_{1}(x), \hspace{0.3cm}{\rm J}_{-2}(x) = {\rm J}_{2}(x), \hspace{0.3cm}{\rm J}_{-3}(x) = - {\rm J}_{3}(x).$$
}}

{{GraueBox|TEXT=
$\text{Example (B):}$   For the spectrum of the analytic signal, in [[Modulationsverfahren/Phasenmodulation_(PM)#Spektralfunktion_eines_phasenmodulierten_Sinussignals|phase modulation of a sinusoidal signal]]:
[[File:Mod_T_3_1_S4_version2.png|right|frame|Spectrum of the analytic signal in phase modulation]]
:$$S_{\rm +}(f) = A_{\rm T} \cdot \sum_{n = - \infty}^{+\infty}{\rm J}_n (\eta) \cdot \delta \big[f - (f_{\rm T}+ n \cdot f_{\rm N})\big]\hspace{0.05cm}.$$
Denote this
*$f_{\rm T}$ the carrier frequency,
*$f_{\rm N}$ the message frequency,
* $A_{\rm T}$ the carrier amplitude.

The parameter of the Bessel functions in this application is the modulation index $\eta$.

Based on the graphic, the following statements are possible:
*Here $S_+(f)$ consists of an infinite number of discrete lines at a distance of $f_{\rm N}$.
*It is thus in principle infinitely extended.
*The weights of the spectral lines at $f_{\rm T} + n · f_{\rm N}$ ($n$ integer) are determined by the modulation index $η$ over the Bessel functions ${\rm J}_n(η)$.
*The spectral lines are real in the case of a sinusoidal source signal and a cosinusoidal carrier and symmetric about $f_{\rm T}$ for even $n$.
*For odd $n$ , a sign change according to the $\text{Property (B)}$ has to be considered.
*The phase modulation of another phase oscillation of source and / or carrier signal provides the same magnitude spectrum.}}
 

===Applications of the Bessel Functions===

The applications of the Bessel functions in nature–and engineering are diverse and play an important role in physics, for example:
* Examination of natural vibrations of cylindrical resonators,
* Solution of the radial Schrödinger-equation,
* Sound pressure amplitudes of low-viscosity rotational flows,
* Heat conduction in cylindrical bodies,
* Scattering problem of a grid,
* Dynamics of oscillating bodies,
* Angular resolution.

The Bessel functions are counted among the special functions because of their many applications in mathematical physics.

In the following, we restrict ourselves to a few areas that are addressed in our tutorial $\rm LNTwww$.

'''In the English original'''
Electromagnetic waves in a cylindrical waveguide
Pressure amplitudes of inviscid rotational flows
Heat conduction in a cylindrical object
Modes of vibration of a thin circular (or annular) acoustic membrane (such as a drum or other membranophone)
Diffusion problems on a lattice
Solutions to the radial Schrödinger equation (in spherical and cylindrical coordinates) for a free particle
Solving for patterns of acoustical radiation
Frequency-dependent friction in circular pipelines
Dynamics of floating bodies
Angular resolution
'''The Ende'''

{{GraueBox|TEXT=
$\text{Example (C):} \hspace{0.5cm} \text{Use in spectral analysis} \ \Rightarrow \ \text{Kaiser-Bessel filter}$

The '''spectral leakage effect''' is the falsification of the spectrum of a periodic and hence temporally unlimited signal due to the implicit time limitation of the discrete Fourier transform (DFT). This will be done, for example, by a spectrum analyzer
* in the time signal not existing frequency components simulated, and/or
* actually existing spectral components are obscured by sidelobes.

The task of the [[Signaldarstellung/Spektralanalyse|spectral analysis]] is to limit the influence of the ''spectral leakage effect'' by providing suitable window functions.

Such a window function provides, for example, the Kaiser–Bessel–window   ⇒   see section [[Signaldarstellung/Spektralanalyse#Spezielle_Fensterfunktionen|Special Window Functions]]. Its time-discrete Fenser function reads with the Bessel function zero order   ⇒   ${\rm J}_0(x)$, the parameter $\alpha=3.5$ and the window length $N$:
:$$w_\nu = \frac{ {\rm J}_0\big(\pi \cdot \alpha \cdot \sqrt{1 - (2\nu/N)^2}\big)}{ {\rm J}_0\big(\pi \cdot \alpha \big)}.$$
On the page [[Signaldarstellung/Spektralanalyse#G.C3.BCtekriterien_von_Fensterfunktionen|Quality Criteria of Window Functions]] and other the characteristics of the Kaiser-Bessel-window are given:
*Conveniently, the large is &bdquo;minimum distance between the main lobe and side lobes” and the desired small &bdquo;maximum scaling error”.
*Due to the very large &bdquo;equivalent noise width” the Kaiser-Bessel window cuts in the main comparison criterion &bdquo;maximum process loss” but worse off than the established Hamming– and Hanning–windows.}}

{{GraueBox|TEXT=
$\text{Example (D):} \hspace{0.5cm} \text{Rice-Fading channel model}$

The [[Mobile_Kommunikation/Wahrscheinlichkeitsdichte_des_Rayleigh%E2%80%93Fadings#Allgemeine_Beschreibung_des_Mobilfunkkanals| Rayleigh - Distribution]] describes the mobile channel assuming that there is no direct path, and thus the multiplicative factor $z(t) = x(t) + {\rm j} \cdot y(t)$ is solely diffuse composed of scattered components.

In the case of a direct component (English: Line of Sight, LoS) in the model for the mean-free Gaussian processes $x(t)$ and $y(t)$ one has to add equal components $x_0$ and/or $y_0$ :

[[File:P ID2126 Mob T 1 4 S1 v3.png|right|frame|Rice-Fading channel model|class=fit]]
$$\hspace{0.2cm}x(t) \hspace{0.1cm} \Rightarrow \hspace{0.1cm} x(t) +x_0 \hspace{0.05cm}, \hspace{0.2cm} y(t) \hspace{0.1cm} \Rightarrow \hspace{0.1cm} y(t) +y_0\hspace{0.05cm},$$

$$\hspace{0.2cm}z(t) = x(t) + {\rm j} \cdot y(t) \hspace{0.1cm} \Rightarrow \hspace{0.1cm} z(t) +z_0 \hspace{0.05cm},\hspace{0.2cm}
z_0 = x_0 + {\rm j} \cdot y_0\hspace{0.05cm}.$$

The graph shows the ''Rice–Fading–Channel model''. It can be summarized as follows:
*The real part $x(t)$ is Gaussianized with mean $x_0$ and variance $\sigma ^2$.
*The imaginary part $y(t)$ is also Gaussian (mean $y_0$, equal variance $\sigma ^2$) and independent of $x(t)$. 

*For $z_0 \ne 0$, the amount $\vert z(t)\vert$ rice is distributed, from which the term &bdquo;Rice–Fading” arises.

*To simplify the notation, we set $\vert z(t)\vert = a(t)$. For $a < 0$ , the amounts - WDF are $f_a(a) \equiv 0$, for $a \ge 0$ the following equation holds, where ${\rm I_0}(x)$ is the modified Besselfunktion zeroth order denotes:
:$$f_a(a) = \frac{a}{\sigma^2} \cdot {\rm e}^{ - (a^2 + \vert z_0 \vert ^2)/(2\sigma^2)} \cdot {\rm I}_0 \left [ \frac{a \cdot \vert z_0 \vert}{\sigma^2} \right ] \hspace{0.5cm}\text{with}\hspace{0.5cm}{\rm I }_0 (u) = {\rm J }_0 ({\rm j} \cdot u) =
\sum_{k = 0}^{\infty} \frac{ (u/2)^{2k} }{k! \cdot \Gamma (k+1)}
\hspace{0.05cm}.$$
*Between the modified Bessel function and the traditional Bessel function ${\rm I_0}(x)$ – each first type – So the connection exists ${\rm I }_0 (u) = {\rm J }_0 ({\rm j} \cdot u)$.}}

{{GraueBox|TEXT=
$\text{Example (E):} \hspace{0.5cm} \text{Analysis of the frequency spectrum of frequency modulated signals}$

$\text{Example (B)}$ has already shown that the angle modulation of a harmonic oscillation of the frequency $f_{\rm N}$ leads to a line spectrum. The spectral lines are around the carrier frequency $f_{\rm T}$ bei $f_{\rm T} + n \cdot f_{\rm N}$ with $n \in \{ \ \text{...}, -2, -1, \ 0, +1, +2, \text{...} \ \}$. The weights of the Dirac lines are ${\rm J }_n(\eta)$, depending on the modulation index $\eta$.

[[File:P_ID1095__Mod_T_3_2_S4_neu.png|center|frame|Discrete spectra with phase modulation (left) and frequency modulation (right)]]

The graph shows the magnitude spectrum $\vert S_{\rm +}(f) \vert$ of the phase modulation (PM) and frequency modulation (FM) analytical signal, two different forms of angle modulation (WM). Bessellines with values less than $0.03$ are neglected in both cases.

For the upper half of the image, the modulator parameters are chosen so that for each $f_{\rm N} = 5 \ \rm kHz$ here is a Bessel spectrum with the modulation index $η = 1.5$. Disregarding the phase relationships, the same spectra and the same signals result for both systems.

The lower graphics are valid for the same message frequency $f_{\rm N} = 3 \ \rm kHz$. One notices:
*In phase modulation, the spectral function is narrower than $f_{\rm N} = 5 \ \rm kHz$, because the distance of the Bessel lines is now only $3 \ \rm kHz$. Since the modulation index of PM is independent of $f_{\rm N}$, the same bessel weights result as in $f_{\rm N} = 5 \ \rm kHz$.
*Also in the frequency modulation, the Bessel lines now appear at a distance of $3 \ \rm kHz$. However, since FM has a modulation index inversely proportional to $f_{\rm N}$, there is now much more Besselline lines than in the upper right (for $η = 1.5$ valid) chart due to the larger modulation index $η = 2.5$.}}

'''Das folgende Kapitel muss noch angepasst werden!'''

==Applet Manual==
[[File:Bessel_abzug3.png|left|600px]]
    '''(A)'''     Sum formula of the Bessel functions ${\rm J}_n(x)$

    '''(B)'''     Selection of the order $n$ for the graphical representation

    '''(C)'''     Plot of the Bessel functions

    '''(D)'''     Variation of the graphic representation

$\hspace{1.5cm}$&bdquo;$+$” (Enlarge),

$\hspace{1.5cm}$ &bdquo;$-$” (Decrease)

$\hspace{1.5cm}$ &bdquo;$\rm o$” (Reset to default)

$\hspace{1.5cm}$ &bdquo;$\leftarrow$” (pushed to the left), usw.

    '''(E)'''     Selection of the abscissa value $x_1$ for the left numeric output

    '''(F)'''     Numerical output of the Bessel function values ${\rm J}_n(x_1)$

    '''(G)'''     Selection of the abscissa value $x_2$ for the right numeric output

    '''(F)'''     Numerical output of the Bessel function values ${\rm J}_n(x_2)$
 

==About the Authors==
This interactive calculation was designed and realized at the [http://www.lnt.ei.tum.de/startseite Lehrstuhl für Nachrichtentechnik] of the [https://www.tum.de/ Technical University of Munich] .
*The original version was created in 2005 by [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Studierende#Markus_Elsberger_.28Diplomarbeit_LB_2006.29|Markus Elsberger]] and [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Studierende#Slim_Lamine_.28Studienarbeit_EI_2006.29|Slim Lamine]] as part of thesis using &bdquo;FlashMX–Actionscript” (Supervisor: [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Mitarbeiter_und_Dozenten#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28am_LNT_seit_1974.29|Günter Söder]]).
*In 2018 this Applet was redesigned and updated to &bdquo;HTML5” by [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Studierende#Xiaohan_Liu_.28Bachelorarbeit_2018.29|Xiaohan Liu]] as part of her Bachelor's thesis (Supervisor: [[Biografien_und_Bibliografien/Beteiligte_der_Professur_Leitungsgebundene_%C3%9Cbertragungstechnik#Tasn.C3.A1d_Kernetzky.2C_M.Sc._.28bei_L.C3.9CT_seit_2014.29|Tasnád Kernetzky]]).

==Nochmalige Aufrufmöglichkeit des Applets in neuem Fenster==

{{LntAppletLink|bessel}}

Applets:Physical Signal & Analytic Signal

2018-08-24T08:41:18Z

Xiaohan: /* Applet Manual */

{{LntAppletLinkEn|physAnSignal}}

==Applet Description==
 
This applet shows the relationship between the physical bandpass signal $x(t)$ and the associated analytical signal $x_+(t)$. It is assumed that the bandpass signal $x(t)$ has a frequency-discrete spectrum $X(f)$:
:$$x(t) = x_{\rm U}(t) + x_{\rm T}(t) + x_{\rm O}(t) = A_{\rm U}\cdot \cos\left(2\pi f_{\rm U}\cdot t- \varphi_{\rm U}\right)+A_{\rm T}\cdot \cos\left(2\pi f_{\rm T}\cdot t- \varphi_{\rm T}\right)+A_{\rm O}\cdot \cos\left(2\pi f_{\rm O}\cdot t- \varphi_{\rm O}\right). $$
The physical signal $x(t)$ is thus composed of three harmonic oscillations, a constellation that can be found, for example, in the ''Double-sideband Amplitude Modulation''
*of the message signal $x_{\rm N}(t) = A_{\rm N}\cdot \cos\left(2\pi f_{\rm N}\cdot t- \varphi_{\rm N}\right)$   ⇒   in German:   '''N'''achrichtensignal
*with the carrier signal $x_{\rm T}(t) = A_{\rm T}\cdot \cos\left(2\pi f_{\rm T}\cdot t - \varphi_{\rm T}\right)$   ⇒   in German:   '''T'''rägersignal.

The nomenclature is also adapted to this case:
* $x_{\rm O}(t)$ denotes the &bdquo;upper sideband”   (in German:   '''O'''beres Seitenband) with the amplitude $A_{\rm O}= A_{\rm N}/2$, the frequency $f_{\rm O} = f_{\rm T} + f_{\rm N}$ and the phase $\varphi_{\rm O} = \varphi_{\rm T} + \varphi_{\rm N}$.
*Similarly, for the &bdquo;lower sideband”   (in German:   '''U'''nteres Seitenband) $x_{\rm U}(t)$ with $f_{\rm U} = f_{\rm T} - f_{\rm N}$, $A_{\rm U}= A_{\rm O}$ and $\varphi_{\rm U} = -\varphi_{\rm O}$.

The associated analytical signal is:

:$$x_+(t) = x_{\rm U+}(t) + x_{\rm T+}(t) + x_{\rm O+}(t) = A_{\rm U}\cdot {\rm e}^{\hspace{0.05cm}{\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}(2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm U}\hspace{0.05cm}\cdot \hspace{0.05cm}t- \varphi_{\rm U})}
\hspace{0.1cm}+ \hspace{0.1cm}A_{\rm T}\cdot {\rm e}^{\hspace{0.05cm}{\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}(2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm T}\hspace{0.05cm}\cdot \hspace{0.05cm}t- \varphi_{\rm T})}
\hspace{0.1cm}+\hspace{0.1cm} A_{\rm O}\cdot {\rm e}^{\hspace{0.05cm}{\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}(2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm O}\hspace{0.05cm}\cdot \hspace{0.05cm}t- \varphi_{\rm O})}. $$

[[File:Zeigerdiagramm_2a_version2.png|right|frame|Analytical signal at the time $t=0$]]
The program displays $x_+(t)$ as the vectorial sum of three rotating pointers (all with counterclockwise) as a violet dot (see figure for start time $t=0$):

*The (red) pointer of the carrier $x_{\rm T+}(t)$ with length $A_{\rm T}$ and zero phase position $\varphi_{\rm T} = 0$ rotates at constant angular velocity $2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm T}$ (one revolution in time $1/f_{\rm T})$.

*The (blue) pointer of the upper sideband $x_{\rm O+}(t)$ with length $A_{\rm O}$ and zero phase position $\varphi_{\rm O}$ rotates at the angular velocity $2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm O}$, which is slightly faster than $x_{\rm T+}(t)$.

*The (green) pointer of the lower sideband $x_{\rm U+}(t)$ with length $A_{\rm U}$ and zero phase position $\varphi_{\rm U}$ rotates at the angular velocity $2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm U}$, which is slightly slower than $x_{\rm T+}(t)$.

The time trace of $x_+(t)$ is also referred to below as ''Pointer Diagram''. The relationship between the physical bandpass signal $x(t)$ and the associated analytical signal $x_+(t)$ is:

:$$x(t) = {\rm Re}\big [x_+(t)\big ].$$

''Note:''   In the figure $\varphi_{\rm O} = +30^\circ$. This leads to the angle with respect to the coordinate system at $t=0$:   $\phi_{\rm O}=-\varphi_{\rm O}=-30^\circ$. Similarly, the null phase angle $\varphi_{\rm U}=-30^\circ$ of the lower sideband leads to the phase angle to be considered in the complex plane:   $\phi_{\rm U}=+30^\circ$.

[[Applets:Physikalisches_Signal_%26_Analytisches_Signal|'''German Description''']]

==Theoretical Background==
 
===Description of Bandpass Signals===
[[File:Zeigerdiagramm_1a.png|right|frame|Bandpass spectrum $X(f)$ |class=fit]]
We consider '''bandpass signals''' $x(t)$ with the property that their spectra $X(f)$ are not in the range around the frequency $f=0$, but around a carrier frequency $f_{\rm T}$. In most cases it can also be assumed that the bandwidth is $B \ll f_{\rm T}$.

The figure shows such a bandpass spectrum $X(f)$. Assuming that the associated $x(t)$ is a physical signal and thus real, the spectral function $X(f)$ has a symmetry with respect to the frequency $f = 0$, if $x(t)$ is an even function   ⇒   $x(-t)=x(t)$, $X(f)$ is real and even.

Besides the physical signal $x(t)\ \circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\, \ X(f)$, one can also use the following descriptions of bandpass signals:
*the analytical signal $x_+(t)\ \circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\, \ X_+(f)$, see next page,
*the equivalent lowpass signal   (in German:   äquivalentes '''T'''ief '''P'''ass–Signal) $x_{\rm TP}(t)\ \circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\, \ X_{\rm TP}(f)$, see Applet [[Applets:Physical_Signal_%26_Equivalent_Low-pass_Signal|Physical Signal & Equivalent Lowpass signal]].
 

===Analytical Signal – Frequency Domain===

The '''analytical signal''' $x_+(t)$ belonging to the physical signal $x(t)$ is the time function whose spectrum fulfills the following property:
[[File:Zeigerdiagramm_3a.png|right|frame|Construction of the spectral function $X_+(f)$ |class=fit]]
:$$X_+(f)=\big[1+{\rm sign}(f)\big] \cdot X(f) = \left\{ {2 \cdot
X(f) \; \hspace{0.2cm}\rm for\hspace{0.2cm} {\it f} > 0, \atop {\,\,\,\, \rm 0 \; \hspace{0.9cm}\rm for\hspace{0.2cm} {\it f} < 0.} }\right.$$

The ''signum function'' is for positive values of $f$ equal to $+1$ and for negative $f$ values equal to $-1$.
* The (double-sided) limit returns $\sign(0)=0$.
* The index „+” should make it clear that $X_+(f)$ only has parts at positive frequencies.

From the graph you can see the calculation rule for $X_+(f)$:

The actual bandpass spectrum $X(f)$ becomes
* doubled at the positive frequencies, and
* set to zero at the negative frequencies.

Due to the asymmetry of $X_+(f)$ with respect to the frequency $f=0$, it can already be said that the time function $x_+(t)$ except for a trivial special case $x_+(t)=0 \ \circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\,X_+(f)=0$ is always complex.
 

===Analytical Signal – Time Domain===
At this point, it is necessary to briefly discuss another spectral transformation.

{{BlaueBox|TEXT=
$\text{Definition:}$ 
For the '''Hilbert transform''' $ {\rm H}\left\{x(t)\right\}$ of a time function $x(t)$ we have:

:$$y(t) = {\rm H}\left\{x(t)\right\} = \frac{1}{ {\rm \pi} } \cdot
\hspace{0.03cm}\int_{-\infty}^{+\infty}\frac{x(\tau)}{ {t -
\tau} }\hspace{0.15cm} {\rm d}\tau.$$

This particular integral is not solvable in a simple, conventional way, but must be evaluated using the [https://en.wikipedia.org/wiki/Cauchy_principal_value Cauchy principal value theorem].

Accordingly, in the frequency domain:
:$$Y(f) = {\rm -j \cdot sign}(f) \cdot X(f) \hspace{0.05cm} .$$}}

The above result can be summarized with this definition as follows:
* The analytical signal $x_+(t)$ is obtained from the physical bandpass signal $x(t)$ by adding an imaginary part to $x(t)$ according to the Hilbert transform:

:$$x_+(t) = x(t)+{\rm j} \cdot {\rm H}\left\{x(t)\right\} .$$

*$\text{H}\{x(t)\}$ disappears only for the case $x(t) = \rm const.$   ⇒   the same signal. For all other signal forms, the analytic signal $x_+(t)$ is complex.

* From the analytical signal $x_+(t)$, the physical bandpass signal can be easily determined by the following operation:
:$$x(t) = {\rm Re}\big[x_+(t)\big] .$$

{{GraueBox|TEXT=
$\text{Example 1:}$  The principle of the Hilbert transformation should be further clarified by the following graphic:
*After the left representation $\rm(A)$ one gets from the physical signal $x(t)$ to the analytical signal $x_+(t)$, by adding an imaginary part ${\rm j} \cdot y(t)$.
*Here $y(t) = {\rm H}\left\{x(t)\right\}$ is a real time function that can be indicated in the spectral domain by multiplying the spectrum $X(f)$ with ${\rm - j} \cdot \sign(f)$.

[[File:P_ID2729__Sig_T_4_2_S2b_neu.png|center|frame|To clarify the Hilbert transform]]

The right representation $\rm(B)$ is equivalent to $\rm(A)$. Now $x_+(t) = x(t) + z(t)$ stand with the purely imaginary function $z(t)$. A comparison of the two figures shows that in fact $z(t) = {\rm j} \cdot y(t)$.}}
 

===Representation of the Harmonic Oscillation as an Analytical Signal===

The spectral function $X(f)$ of a harmonic oscillation $x(t) = A\cdot\text{cos}(2\pi f_{\rm T}\cdot t - \varphi)$ is known to consist of two Dirac functions at the frequencies
* $+f_{\rm T}$ with the complex weight $A/2 \cdot \text{e}^{-\text{j}\hspace{0.05cm}\varphi}$,
* $-f_{\rm T}$ with the complex weight $A/2 \cdot \text{e}^{+\text{j}\hspace{0.05cm}\varphi}$.

Thus, the spectrum of the analytical signal (that is, without the Dirac function at the frequency $f =-f_{\rm T}$, but doubling at $f =+f_{\rm T}$):

:$$X_+(f) = A \cdot {\rm e}^{-{\rm j} \hspace{0.05cm}\varphi}\cdot\delta (f - f_{\rm
T}) .$$

The associated time function is obtained by applying the Displacement Law:

:$$x_+(t) = x_{\rm U+}(t) + x_{\rm T+}(t) + x_{\rm O+}(t) = A_{\rm U}\cdot {\rm e}^{\hspace{0.05cm}{\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}(2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm U}\hspace{0.05cm}\cdot \hspace{0.05cm}t- \varphi_{\rm U})}
\hspace{0.1cm}+ \hspace{0.1cm}A_{\rm T}\cdot {\rm e}^{\hspace{0.05cm}{\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}(2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm T}\hspace{0.05cm}\cdot \hspace{0.05cm}t- \varphi_{\rm T})}
\hspace{0.1cm}+\hspace{0.1cm} A_{\rm O}\cdot {\rm e}^{\hspace{0.05cm}{\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}(2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm O}\hspace{0.05cm}\cdot \hspace{0.05cm}t- \varphi_{\rm O})}. $$

This equation describes a pointer rotating at constant angular velocity $\omega_{\rm T} = 2\pi f_{\rm T}$.

{{GraueBox|TEXT=
$\text{Example 2:}$  Here the coordinate system is rotated by $90^\circ$ (real part up, imaginary part to the left) contrary to the usual representation.

[[File:P_ID712__Sig_T_4_2_S3.png|center|frame|Pointer diagram of a harmonic oscillation]]

Based on this graphic, the following statements are possible:
* At time $t = 0$, the pointer of length $A$ (signal amplitude) lies with the angle $-\varphi$ in the complex plane. In the example shown, $\varphi=45^\circ$.
* For times $t>0$, the constant angular velocity vector $\omega_{\rm T}$ rotates in a mathematically positive direction, that is, counterclockwise.
* The tip of the pointer is thus always on a circle with radius $A$ and needs exactly the time $T_0$, i.e. the period of the harmonic oscillation $x(t)$ for one revolution.
* The projection of the analytical signal $x_+(t)$ on the real axis, marked by red dots, gives the instantaneous values of $x(t)$.}}
 

===Analytical Signal Representation of a Sum of Three Harmonic Oscillations===

In our applet, we always assume a set of three rotating pointers. The physical signal is:
:$$x(t) = x_{\rm U}(t) + x_{\rm T}(t) + x_{\rm O}(t) = A_{\rm U}\cdot \cos\left(2\pi f_{\rm U}\cdot t- \varphi_{\rm U}\right)+A_{\rm T}\cdot \cos\left(2\pi f_{\rm T}\cdot t- \varphi_{\rm T}\right)+A_{\rm O}\cdot \cos\left(2\pi f_{\rm O}\cdot t- \varphi_{\rm O}\right). $$
* Each of the three harmonic oscillations $x_{\rm T}(t)$, $x_{\rm U}(t)$ and $x_{\rm O}(t)$ is represented by an amplitude $(A)$, a frequency $(f)$ and a phase value $(\varphi)$.
*The indices are based on the ''Double-sideband Amplitude Modulation'' method. &bdquo;T” stands for &bdquo;carrier”, &bdquo;U” for &bdquo;lower sideband” and &bdquo;O” for &bdquo;upper Sideband”.
*Accordingly, $f_{\rm U} < f_{\rm T}$ and $f_{\rm O} > f_{\rm T}$. There are no restrictions for the amplitudes and phases.

The associated analytical signal is:
:$$x_+(t) = x_{\rm U+}(t) + x_{\rm T+}(t) + x_{\rm O+}(t) = A_{\rm U}\cdot {\rm e}^{\hspace{0.05cm}{\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}(2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm U}\hspace{0.05cm}\cdot \hspace{0.05cm}t- \varphi_{\rm U})}
\hspace{0.1cm}+ \hspace{0.1cm}A_{\rm T}\cdot {\rm e}^{\hspace{0.05cm}{\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}(2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm T}\hspace{0.05cm}\cdot \hspace{0.05cm}t- \varphi_{\rm T})}
\hspace{0.1cm}+\hspace{0.1cm} A_{\rm O}\cdot {\rm e}^{\hspace{0.05cm}{\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}(2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm O}\hspace{0.05cm}\cdot \hspace{0.05cm}t- \varphi_{\rm O})}. $$

{{GraueBox|TEXT=
$\text{Example 3:}$ 
Shown the constellation arises i.e. in the [https://en.wikipedia.org/wiki/Sideband Double-sideband Amplitude Modulation] (with carrier) of the message signal $x_{\rm N}(t) = A_{\rm N}\cdot \cos\left(2\pi f_{\rm N}\cdot t- \varphi_{\rm N}\right)$ with the carrier signal $x_{\rm T}(t) = A_{\rm T}\cdot \cos\left(2\pi f_{\rm T}\cdot t - \varphi_{\rm T}\right)$. This is discussed frequently in the Exercises.

There are some limitations to the program parameters in this approach:
* For the frequencies, it always applies $f_{\rm O} = f_{\rm T} + f_{\rm N}$ and $f_{\rm U} = f_{\rm T} - f_{\rm N}$.

*Without distortions the amplitude of the sidebands are $A_{\rm O}= A_{\rm U}= A_{\rm N}/2$.
*The respective phase relationships can be seen in the following graphic.

[[File:Zeigerdiagramm_2_neu.png|center|frame|Spectrum $X_+(f)$ of the analytical signal for different phase constellations |class=fit]]}}

==Exercises==
[[File:Zeigerdiagramm_aufgabe_2.png|right]]
*First select the task number.
*A task description is displayed.
*Parameter values are adjusted.
*Solution after pressing &bdquo;Hide solition”.

The number &bdquo;0” will reset the program and output a text with further explanation of the applet.
 
In the following, $\rm Green$ denotes the lower sideband   ⇒   $\big (A_{\rm U}, f_{\rm U}, \varphi_{\rm U}\big )$,  
$\rm Red$ the carrier   ⇒   $\big (A_{\rm T}, f_{\rm T}, \varphi_{\rm T}\big )$ and
$\rm Blue$ the upper sideband   ⇒   $\big (A_{\rm O}, f_{\rm O}, \varphi_{\rm O}\big )$.
{{BlaueBox|TEXT=
'''(1)'''   Consider and interpret the analytical signal $x_+(t)$ for $\text{Red:} \hspace{0.15cm} A_{\rm T} = 1.5\ \text{V}, \ f_{\rm T} = 50 \ \text{kHz}, \ \varphi_{\rm T} = 0^\circ$, $A_{\rm U} = A_{\rm O} = 0$.

:Which signal values $x_+(t)$ result for $t = 0$, $t = 5 \ \rm µ s$ and $t = 20 \ \rm µ s$? What are the corresponding signal values for $x(t)$? }}

:: For a cosine signal, let $x_+(t= 0) = A_{\rm T} = 1.5\ \text{V}$. Then $x_+(t)$ rotates in a mathematically positive direction (one revolution per period $T_0 = 1/f_{\rm T}$):

:: $x_+(t= 20 \ {\rm µ s}) = x_+(t= 0) = 1.5\ \text{V}\hspace{0.3cm}\Rightarrow\hspace{0.3cm}x(t= 20 \ {\rm µ s}) = 1.5\ \text{V,}$
:: $x_+(t= 5 \ {\rm µ s}) = {\rm j} \cdot 1.5\ \text{V}\hspace{0.3cm}\Rightarrow\hspace{0.3cm}x(t= 5 \ {\rm µ s}) = {\rm Re}[x_+(t= 5 \ {\rm µ s})] = 0$.

{{BlaueBox|TEXT=
'''(2)'''   How do the ratios change for $\text{Red:} \hspace{0.15cm} A_{\rm T} = 1.0\ \text{V}, \ f_{\rm T} = 100 \ \text{kHz}, \ \varphi_{\rm T} = 90^\circ$?}}

::The signal $x(t)$ is now a sine signal with a smaller amplitude. The analytical signal now starts because of $\varphi_{\rm T} = 90^\circ$   ⇒   $\phi_{\rm T} = -90^\circ$ at $x_+(t= 0) = -{\rm j} \cdot A_{\rm T}$. After that, $x_+(t)$ rotates again in a mathematically positive direction, but twice as fast because of $T_0 = 10 \ \rm µ s$ as in $\rm (1)$.

{{BlaueBox|TEXT=
'''(3)'''   Now   $\text{Red:} \hspace{0.15cm} A_{\rm T} = 1\ \text{V}, \ f_{\rm T} = 100 \ \text{kHz}, \ \varphi_{\rm T} = 0^\circ$,   $\text{Green:} \hspace{0.15cm} A_{\rm U} = 0.4\ \text{V}, \ f_{\rm U} = 80 \ \text{kHz}, \ \varphi_{\rm U} = 0^\circ$,   $\text{Blue:} \hspace{0.15cm} A_{\rm O} = 0.4\ \text{V}, \ f_{\rm O} = 120 \ \text{kHz}, \ \varphi_{\rm O} = 0^\circ$.

:Consider and interpret the physical signal $x(t)$ and the analytical signal $x_+(t)$.}}

::The Signal $x(t)$ results in the [https://en.wikipedia.org/wiki/Sideband Double-sideband Amplitude Modulation] (DSB–AM) of the message signal $A_{\rm N}\cdot \cos\left(2\pi f_{\rm N}\cdot t\right)$ with $A_{\rm N} = 0.8\ \text{V}$, $f_{\rm N} = 20\ \text{kHz}$. The carrier $x_{\rm T}(t)$ with $f_{\rm T} = 100\ \text{kHz}$ is also cosinusoidal. The degree of modulation is $m = A_{\rm N}/A_{\rm T} = 0.8$ and the period $T_{\rm 0} = 50\ \text{µs}$.

::In the phase diagram, the (red) carrier rotates faster than the (green) lower sideband and slower than the (blue) upper sideband. The analytical signal $x_+(t)$ results as the geometric sum of the three rotating pointers. It seems that the blue pointer is leading the carrier and the green pointer is following the carrier.

{{BlaueBox|TEXT=
'''(4)'''   The settings of task '''(3)''' still apply. Which signal values are obtained at $t=0$, $t=2.5 \ \rm µ s$, $t= 5 \ \rm µ s$ and $t=10 \ \rm µ s$? }}

::At time $t=0$, all pointers are in the direction of the real axis, so that $x(t=0) = {\rm Re}\big [x+(t= 0)\big] = A_{\rm U} + A_{\rm T} + A_{\rm O} = 1.8\ \text{V}$.

::Until the time $t=2.5 \ \rm µ s$, the red carrier has rotated by $90^\circ$, the blue one by $108^\circ$ and the green one by $72^\circ$. We have $x(t=2.5 \ \rm µ s) = {\rm Re}\big [x_+(t= 2.5 \ \rm µ s)\big] = 0$, because now the pointer group points in the direction of the imaginary axis. The other sought signal values are $x(t=5 \ \rm µ s) = {\rm Re}\big [x_+(t= 5 \ \rm µ s)\big] = -1.647\ \text{V}$ and $x(t=10 \ \rm µ s) = {\rm Re}\big [x_+(t= 10 \ \rm µ s)\big] = 1.247\ \text{V}$.
::For $x_+(t)$ a spiral shape results, alternating with a smaller radius and then with a larger radius.

{{BlaueBox|TEXT=
'''(5)'''   How should the phase parameters $\varphi_{\rm T}$, $\varphi_{\rm U}$ and $\varphi_{\rm O}$ be set if both the carrier $x_{\rm T}(t)$ and the message signal $x_{\rm N}(t)$ are sinusoidal?}}

::The parameter selection $\varphi_{\rm T} = \varphi_{\rm U} = \varphi_{\rm O}=90^\circ$ describes the signals $x_{\rm T}(t) = A_{\rm T}\cdot \sin\left(2\pi f_{\rm T}\cdot t\right)$ and $x_{\rm N}(t) = A_{\rm N}\cdot \cos\left(2\pi f_{\rm N}\cdot t\right)$. If, in addition, the message $x_{\rm N}(t)$ is sinusoidal, then $\varphi_{\rm O}=\varphi_{\rm T} - 90^\circ = 0$ and $\varphi_{\rm U}=\varphi_{\rm T} + 90^\circ = 180^\circ$ must be set.

{{BlaueBox|TEXT=
'''(6)'''   The settings of task '''(3)''' apply except $A_{\rm T} = 0.6\ \text{V}$. Which modulation method is described here?

: What are the consequences of this? What changes with $A_{\rm T} = 0$? }}

::It is a [https://en.wikipedia.org/wiki/Sideband Double-sideband Amplitude Modulation] (DSB–AM with carrier) with the modulation degree $m=0.8/0.6 = 1.333$. For $m > 1$, however, [https://www.radio-electronics.com/info/rf-technology-design/am-reception/synchronous-demodulator-demodulation-detector.php Synchronous Demodulation] is required. [https://en.wikipedia.org/wiki/Envelope_detector Envelope Detection] no longer works. One reason for this is that now the zero crossings of $x(t)$ are no longer equidistant from $5\ \rm µ s$   ⇒   additional phase modulation.

::With $A_{\rm T} = 0$   ⇒   $m \to \infty$ results in a [https://en.wikipedia.org/wiki/Double-sideband_suppressed-carrier_transmission ''DSB–AM without carrier'']. For this, one also needs coherent demodulation.

{{BlaueBox|TEXT=
'''(7)'''     Now let   $\text{Red:} \hspace{0.15cm} A_{\rm T} = 1\ \text{V}, \ f_{\rm T} = 100 \ \text{kHz}, \ \varphi_{\rm T} = 0^\circ$,   $\text{Green:} \hspace{0.15cm} A_{\rm U} = 0$,   $\text{Blue:} \hspace{0.15cm} A_{\rm O} = 0.8\ \text{V}, \ f_{\rm O} = 120 \ \text{kHz}, \ \varphi_{\rm O} = 90^\circ$.

:Which constellation is described here? Which figure is given for the equivalent low-pass signal $x_{\rm TP}(t)$?   ⇒   &bdquo;locus”? What changes with $A_{\rm U} = 0.8\ \text{V}$ and $A_{\rm O} = 0$?}}

::In both cases, it is a [https://en.wikipedia.org/wiki/Single-sideband_modulation Single-sideband Amplitude Modulation] (SSB–AM) with the modulation degree $\mu = 0.8$ (in SSB we denote the degree of modulation with $\mu$ instead of $m$). The carrier signal is cosinusoidal and the message signal is sinusoidal. The equivalent low-pass signal $x_{\rm TP}(t)$ has a circular course in the complex plane.

:: $A_{\rm O} = 0.8\ \text{V}$, $A_{\rm U} = 0$ is an OSB modulation. The green pointer is missing and the blue pointer rotates faster compared to the red carrier.

:: $A_{\rm U} = 0.8\ \text{V}$, $A_{\rm O} = 0$ is a USB modulation. The blue pointer is missing and the green pointer rotates slower compared to the red carrier.

{{BlaueBox|TEXT=
'''(8)'''   Now let   $\text{Red:} \hspace{0.05cm} A_{\rm T} = 1\ \text{V}, \ f_{\rm T} = 100 \ \text{kHz}, \ \varphi_{\rm T} = 0^\circ$,   $\text{Green:} \hspace{0.05cm} A_{\rm U} = 0.4\ \text{V}, \ f_{\rm U} = 80 \ \text{kHz}, \ \varphi_{\rm U} = -90^\circ$,   $\text{Blue:} \hspace{0.05cm} A_{\rm O} = 0.2\ \text{V}, \ f_{\rm O} = 120 \ \text{kHz}, \ \varphi_{\rm O} = +90^\circ$.

:Which constellation could be described here? Which shape results for the equivalent lowpass signal $x_{\rm TP}(t)$?}}

::It could be a DSB–AM of a sinusoidal signal with cosinusoidal carrier and modulation degree $m=0.8$, in which the upper sideband is attenuated by a factor of 2. The equivalent lowpass signal $x_{\rm TP}(t)$ has an elliptical trace in the complex plane.

==Applet Manual==
 
[[File:Zeigerdiagramm_abzug.png|right]]

* The red parameters $(A_{\rm T}, \ f_{\rm T}, \ \varphi_{\rm T})$ and the red pointer mark the ''Carrier'' (German:   '''T'''räger).
* The green parameters $(A_{\rm U}, \ f_{\rm U} < f_{\rm T}, \ \varphi_{\rm U})$ mark the ''Lower sideband'' (German:  '''U'''nteres Seitenband).
* The blue parameters $(A_{\rm O}, \ f_{\rm O} > f_{\rm T}, \ \varphi_{\rm O})$ mark the ''Upper sideband'' (German:  '''O'''beres Seitenband).
*All pointers rotate in a mathematically positive direction (counterclockwise).

 
Meaning of the letters in the adjacent graphic:

    '''(A)'''     Plot of the analytical signal $x_{\rm +}(t)$

    '''(B)'''     Plot of the physical signal $x(t)$

    '''(C)'''     Parameter input via slider: amplitudes, frequencies, phase values

    '''(D)'''     Control elements:   Start – Step – Pause/Continue – Reset

    '''(E)'''     Speed of animation:   &bdquo;Speed”   ⇒   Values: 1, 2, 3

    '''(F)'''     &bdquo;Trace”   ⇒   On or Off, trace of complex signal values $x_{\rm +}(t)$

    '''(G)'''     Numerical output of the time $t$ and the signal values  ${\rm Re}[x_{\rm +}(t)] = x(t)$  and  ${\rm Im}[x_{\rm +}(t)]$

    '''(H)'''     Variations for the graphical representation

$\hspace{1.5cm}$Zoom–Functions &bdquo;$+$” (Enlarge), &bdquo;$-$” (Decrease) and $\rm o$ (Reset to default)

$\hspace{1.5cm}$Move with &bdquo;$\leftarrow$” (Section to the left, ordinate to the right), &bdquo;$\uparrow$” &bdquo;$\downarrow$” and &bdquo;$\rightarrow$”

    '''(I)'''     Experiment section:  Task selection and task

    '''(J)'''     Experiment section:  solution
 

==About the Authors==
This interactive calculation was designed and realized at the [http://www.lnt.ei.tum.de/startseite Lehrstuhl für Nachrichtentechnik] of the [https://www.tum.de/ Technical University of Munich] .
*The original version was created in 2005 by [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Studierende#Ji_Li_.28Bachelorarbeit_EI_2003.2C_Diplomarbeit_EI_2005.29|Ji Li]] as part of her Diploma thesis using &bdquo;FlashMX–Actionscript” (Supervisor: [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Mitarbeiter_und_Dozenten#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28am_LNT_seit_1974.29|Günter Söder]]).
*In 2018 this Applet was redesigned and updated to &bdquo;HTML5” by [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Studierende#Xiaohan_Liu_.28Bachelorarbeit_2018.29|Xiaohan Liu]] as part of her Bachelor's thesis (Supervisor: [[Biografien_und_Bibliografien/Beteiligte_der_Professur_Leitungsgebundene_%C3%9Cbertragungstechnik#Tasn.C3.A1d_Kernetzky.2C_M.Sc._.28bei_L.C3.9CT_seit_2014.29|Tasnád Kernetzky]]).

==Once again: Open Applet in new Tab==

{{LntAppletLinkEn|physAnSignal}}

[[Category:Applets|^Verzerrungen^]]

Applets:Physical Signal & Analytic Signal

2018-08-24T08:34:25Z

Xiaohan: /* Applet Manual */

{{LntAppletLinkEn|physAnSignal}}

==Applet Description==
 
This applet shows the relationship between the physical bandpass signal $x(t)$ and the associated analytical signal $x_+(t)$. It is assumed that the bandpass signal $x(t)$ has a frequency-discrete spectrum $X(f)$:
:$$x(t) = x_{\rm U}(t) + x_{\rm T}(t) + x_{\rm O}(t) = A_{\rm U}\cdot \cos\left(2\pi f_{\rm U}\cdot t- \varphi_{\rm U}\right)+A_{\rm T}\cdot \cos\left(2\pi f_{\rm T}\cdot t- \varphi_{\rm T}\right)+A_{\rm O}\cdot \cos\left(2\pi f_{\rm O}\cdot t- \varphi_{\rm O}\right). $$
The physical signal $x(t)$ is thus composed of three harmonic oscillations, a constellation that can be found, for example, in the ''Double-sideband Amplitude Modulation''
*of the message signal $x_{\rm N}(t) = A_{\rm N}\cdot \cos\left(2\pi f_{\rm N}\cdot t- \varphi_{\rm N}\right)$   ⇒   in German:   '''N'''achrichtensignal
*with the carrier signal $x_{\rm T}(t) = A_{\rm T}\cdot \cos\left(2\pi f_{\rm T}\cdot t - \varphi_{\rm T}\right)$   ⇒   in German:   '''T'''rägersignal.

The nomenclature is also adapted to this case:
* $x_{\rm O}(t)$ denotes the &bdquo;upper sideband”   (in German:   '''O'''beres Seitenband) with the amplitude $A_{\rm O}= A_{\rm N}/2$, the frequency $f_{\rm O} = f_{\rm T} + f_{\rm N}$ and the phase $\varphi_{\rm O} = \varphi_{\rm T} + \varphi_{\rm N}$.
*Similarly, for the &bdquo;lower sideband”   (in German:   '''U'''nteres Seitenband) $x_{\rm U}(t)$ with $f_{\rm U} = f_{\rm T} - f_{\rm N}$, $A_{\rm U}= A_{\rm O}$ and $\varphi_{\rm U} = -\varphi_{\rm O}$.

The associated analytical signal is:

:$$x_+(t) = x_{\rm U+}(t) + x_{\rm T+}(t) + x_{\rm O+}(t) = A_{\rm U}\cdot {\rm e}^{\hspace{0.05cm}{\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}(2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm U}\hspace{0.05cm}\cdot \hspace{0.05cm}t- \varphi_{\rm U})}
\hspace{0.1cm}+ \hspace{0.1cm}A_{\rm T}\cdot {\rm e}^{\hspace{0.05cm}{\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}(2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm T}\hspace{0.05cm}\cdot \hspace{0.05cm}t- \varphi_{\rm T})}
\hspace{0.1cm}+\hspace{0.1cm} A_{\rm O}\cdot {\rm e}^{\hspace{0.05cm}{\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}(2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm O}\hspace{0.05cm}\cdot \hspace{0.05cm}t- \varphi_{\rm O})}. $$

[[File:Zeigerdiagramm_2a_version2.png|right|frame|Analytical signal at the time $t=0$]]
The program displays $x_+(t)$ as the vectorial sum of three rotating pointers (all with counterclockwise) as a violet dot (see figure for start time $t=0$):

*The (red) pointer of the carrier $x_{\rm T+}(t)$ with length $A_{\rm T}$ and zero phase position $\varphi_{\rm T} = 0$ rotates at constant angular velocity $2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm T}$ (one revolution in time $1/f_{\rm T})$.

*The (blue) pointer of the upper sideband $x_{\rm O+}(t)$ with length $A_{\rm O}$ and zero phase position $\varphi_{\rm O}$ rotates at the angular velocity $2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm O}$, which is slightly faster than $x_{\rm T+}(t)$.

*The (green) pointer of the lower sideband $x_{\rm U+}(t)$ with length $A_{\rm U}$ and zero phase position $\varphi_{\rm U}$ rotates at the angular velocity $2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm U}$, which is slightly slower than $x_{\rm T+}(t)$.

The time trace of $x_+(t)$ is also referred to below as ''Pointer Diagram''. The relationship between the physical bandpass signal $x(t)$ and the associated analytical signal $x_+(t)$ is:

:$$x(t) = {\rm Re}\big [x_+(t)\big ].$$

''Note:''   In the figure $\varphi_{\rm O} = +30^\circ$. This leads to the angle with respect to the coordinate system at $t=0$:   $\phi_{\rm O}=-\varphi_{\rm O}=-30^\circ$. Similarly, the null phase angle $\varphi_{\rm U}=-30^\circ$ of the lower sideband leads to the phase angle to be considered in the complex plane:   $\phi_{\rm U}=+30^\circ$.

[[Applets:Physikalisches_Signal_%26_Analytisches_Signal|'''German Description''']]

==Theoretical Background==
 
===Description of Bandpass Signals===
[[File:Zeigerdiagramm_1a.png|right|frame|Bandpass spectrum $X(f)$ |class=fit]]
We consider '''bandpass signals''' $x(t)$ with the property that their spectra $X(f)$ are not in the range around the frequency $f=0$, but around a carrier frequency $f_{\rm T}$. In most cases it can also be assumed that the bandwidth is $B \ll f_{\rm T}$.

The figure shows such a bandpass spectrum $X(f)$. Assuming that the associated $x(t)$ is a physical signal and thus real, the spectral function $X(f)$ has a symmetry with respect to the frequency $f = 0$, if $x(t)$ is an even function   ⇒   $x(-t)=x(t)$, $X(f)$ is real and even.

Besides the physical signal $x(t)\ \circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\, \ X(f)$, one can also use the following descriptions of bandpass signals:
*the analytical signal $x_+(t)\ \circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\, \ X_+(f)$, see next page,
*the equivalent lowpass signal   (in German:   äquivalentes '''T'''ief '''P'''ass–Signal) $x_{\rm TP}(t)\ \circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\, \ X_{\rm TP}(f)$, see Applet [[Applets:Physical_Signal_%26_Equivalent_Low-pass_Signal|Physical Signal & Equivalent Lowpass signal]].
 

===Analytical Signal – Frequency Domain===

The '''analytical signal''' $x_+(t)$ belonging to the physical signal $x(t)$ is the time function whose spectrum fulfills the following property:
[[File:Zeigerdiagramm_3a.png|right|frame|Construction of the spectral function $X_+(f)$ |class=fit]]
:$$X_+(f)=\big[1+{\rm sign}(f)\big] \cdot X(f) = \left\{ {2 \cdot
X(f) \; \hspace{0.2cm}\rm for\hspace{0.2cm} {\it f} > 0, \atop {\,\,\,\, \rm 0 \; \hspace{0.9cm}\rm for\hspace{0.2cm} {\it f} < 0.} }\right.$$

The ''signum function'' is for positive values of $f$ equal to $+1$ and for negative $f$ values equal to $-1$.
* The (double-sided) limit returns $\sign(0)=0$.
* The index „+” should make it clear that $X_+(f)$ only has parts at positive frequencies.

From the graph you can see the calculation rule for $X_+(f)$:

The actual bandpass spectrum $X(f)$ becomes
* doubled at the positive frequencies, and
* set to zero at the negative frequencies.

Due to the asymmetry of $X_+(f)$ with respect to the frequency $f=0$, it can already be said that the time function $x_+(t)$ except for a trivial special case $x_+(t)=0 \ \circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\,X_+(f)=0$ is always complex.
 

===Analytical Signal – Time Domain===
At this point, it is necessary to briefly discuss another spectral transformation.

{{BlaueBox|TEXT=
$\text{Definition:}$ 
For the '''Hilbert transform''' $ {\rm H}\left\{x(t)\right\}$ of a time function $x(t)$ we have:

:$$y(t) = {\rm H}\left\{x(t)\right\} = \frac{1}{ {\rm \pi} } \cdot
\hspace{0.03cm}\int_{-\infty}^{+\infty}\frac{x(\tau)}{ {t -
\tau} }\hspace{0.15cm} {\rm d}\tau.$$

This particular integral is not solvable in a simple, conventional way, but must be evaluated using the [https://en.wikipedia.org/wiki/Cauchy_principal_value Cauchy principal value theorem].

Accordingly, in the frequency domain:
:$$Y(f) = {\rm -j \cdot sign}(f) \cdot X(f) \hspace{0.05cm} .$$}}

The above result can be summarized with this definition as follows:
* The analytical signal $x_+(t)$ is obtained from the physical bandpass signal $x(t)$ by adding an imaginary part to $x(t)$ according to the Hilbert transform:

:$$x_+(t) = x(t)+{\rm j} \cdot {\rm H}\left\{x(t)\right\} .$$

*$\text{H}\{x(t)\}$ disappears only for the case $x(t) = \rm const.$   ⇒   the same signal. For all other signal forms, the analytic signal $x_+(t)$ is complex.

* From the analytical signal $x_+(t)$, the physical bandpass signal can be easily determined by the following operation:
:$$x(t) = {\rm Re}\big[x_+(t)\big] .$$

{{GraueBox|TEXT=
$\text{Example 1:}$  The principle of the Hilbert transformation should be further clarified by the following graphic:
*After the left representation $\rm(A)$ one gets from the physical signal $x(t)$ to the analytical signal $x_+(t)$, by adding an imaginary part ${\rm j} \cdot y(t)$.
*Here $y(t) = {\rm H}\left\{x(t)\right\}$ is a real time function that can be indicated in the spectral domain by multiplying the spectrum $X(f)$ with ${\rm - j} \cdot \sign(f)$.

[[File:P_ID2729__Sig_T_4_2_S2b_neu.png|center|frame|To clarify the Hilbert transform]]

The right representation $\rm(B)$ is equivalent to $\rm(A)$. Now $x_+(t) = x(t) + z(t)$ stand with the purely imaginary function $z(t)$. A comparison of the two figures shows that in fact $z(t) = {\rm j} \cdot y(t)$.}}
 

===Representation of the Harmonic Oscillation as an Analytical Signal===

The spectral function $X(f)$ of a harmonic oscillation $x(t) = A\cdot\text{cos}(2\pi f_{\rm T}\cdot t - \varphi)$ is known to consist of two Dirac functions at the frequencies
* $+f_{\rm T}$ with the complex weight $A/2 \cdot \text{e}^{-\text{j}\hspace{0.05cm}\varphi}$,
* $-f_{\rm T}$ with the complex weight $A/2 \cdot \text{e}^{+\text{j}\hspace{0.05cm}\varphi}$.

Thus, the spectrum of the analytical signal (that is, without the Dirac function at the frequency $f =-f_{\rm T}$, but doubling at $f =+f_{\rm T}$):

:$$X_+(f) = A \cdot {\rm e}^{-{\rm j} \hspace{0.05cm}\varphi}\cdot\delta (f - f_{\rm
T}) .$$

The associated time function is obtained by applying the Displacement Law:

:$$x_+(t) = x_{\rm U+}(t) + x_{\rm T+}(t) + x_{\rm O+}(t) = A_{\rm U}\cdot {\rm e}^{\hspace{0.05cm}{\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}(2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm U}\hspace{0.05cm}\cdot \hspace{0.05cm}t- \varphi_{\rm U})}
\hspace{0.1cm}+ \hspace{0.1cm}A_{\rm T}\cdot {\rm e}^{\hspace{0.05cm}{\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}(2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm T}\hspace{0.05cm}\cdot \hspace{0.05cm}t- \varphi_{\rm T})}
\hspace{0.1cm}+\hspace{0.1cm} A_{\rm O}\cdot {\rm e}^{\hspace{0.05cm}{\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}(2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm O}\hspace{0.05cm}\cdot \hspace{0.05cm}t- \varphi_{\rm O})}. $$

This equation describes a pointer rotating at constant angular velocity $\omega_{\rm T} = 2\pi f_{\rm T}$.

{{GraueBox|TEXT=
$\text{Example 2:}$  Here the coordinate system is rotated by $90^\circ$ (real part up, imaginary part to the left) contrary to the usual representation.

[[File:P_ID712__Sig_T_4_2_S3.png|center|frame|Pointer diagram of a harmonic oscillation]]

Based on this graphic, the following statements are possible:
* At time $t = 0$, the pointer of length $A$ (signal amplitude) lies with the angle $-\varphi$ in the complex plane. In the example shown, $\varphi=45^\circ$.
* For times $t>0$, the constant angular velocity vector $\omega_{\rm T}$ rotates in a mathematically positive direction, that is, counterclockwise.
* The tip of the pointer is thus always on a circle with radius $A$ and needs exactly the time $T_0$, i.e. the period of the harmonic oscillation $x(t)$ for one revolution.
* The projection of the analytical signal $x_+(t)$ on the real axis, marked by red dots, gives the instantaneous values of $x(t)$.}}
 

===Analytical Signal Representation of a Sum of Three Harmonic Oscillations===

In our applet, we always assume a set of three rotating pointers. The physical signal is:
:$$x(t) = x_{\rm U}(t) + x_{\rm T}(t) + x_{\rm O}(t) = A_{\rm U}\cdot \cos\left(2\pi f_{\rm U}\cdot t- \varphi_{\rm U}\right)+A_{\rm T}\cdot \cos\left(2\pi f_{\rm T}\cdot t- \varphi_{\rm T}\right)+A_{\rm O}\cdot \cos\left(2\pi f_{\rm O}\cdot t- \varphi_{\rm O}\right). $$
* Each of the three harmonic oscillations $x_{\rm T}(t)$, $x_{\rm U}(t)$ and $x_{\rm O}(t)$ is represented by an amplitude $(A)$, a frequency $(f)$ and a phase value $(\varphi)$.
*The indices are based on the ''Double-sideband Amplitude Modulation'' method. &bdquo;T” stands for &bdquo;carrier”, &bdquo;U” for &bdquo;lower sideband” and &bdquo;O” for &bdquo;upper Sideband”.
*Accordingly, $f_{\rm U} < f_{\rm T}$ and $f_{\rm O} > f_{\rm T}$. There are no restrictions for the amplitudes and phases.

The associated analytical signal is:
:$$x_+(t) = x_{\rm U+}(t) + x_{\rm T+}(t) + x_{\rm O+}(t) = A_{\rm U}\cdot {\rm e}^{\hspace{0.05cm}{\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}(2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm U}\hspace{0.05cm}\cdot \hspace{0.05cm}t- \varphi_{\rm U})}
\hspace{0.1cm}+ \hspace{0.1cm}A_{\rm T}\cdot {\rm e}^{\hspace{0.05cm}{\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}(2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm T}\hspace{0.05cm}\cdot \hspace{0.05cm}t- \varphi_{\rm T})}
\hspace{0.1cm}+\hspace{0.1cm} A_{\rm O}\cdot {\rm e}^{\hspace{0.05cm}{\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}(2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm O}\hspace{0.05cm}\cdot \hspace{0.05cm}t- \varphi_{\rm O})}. $$

{{GraueBox|TEXT=
$\text{Example 3:}$ 
Shown the constellation arises i.e. in the [https://en.wikipedia.org/wiki/Sideband Double-sideband Amplitude Modulation] (with carrier) of the message signal $x_{\rm N}(t) = A_{\rm N}\cdot \cos\left(2\pi f_{\rm N}\cdot t- \varphi_{\rm N}\right)$ with the carrier signal $x_{\rm T}(t) = A_{\rm T}\cdot \cos\left(2\pi f_{\rm T}\cdot t - \varphi_{\rm T}\right)$. This is discussed frequently in the Exercises.

There are some limitations to the program parameters in this approach:
* For the frequencies, it always applies $f_{\rm O} = f_{\rm T} + f_{\rm N}$ and $f_{\rm U} = f_{\rm T} - f_{\rm N}$.

*Without distortions the amplitude of the sidebands are $A_{\rm O}= A_{\rm U}= A_{\rm N}/2$.
*The respective phase relationships can be seen in the following graphic.

[[File:Zeigerdiagramm_2_neu.png|center|frame|Spectrum $X_+(f)$ of the analytical signal for different phase constellations |class=fit]]}}

==Exercises==
[[File:Zeigerdiagramm_aufgabe_2.png|right]]
*First select the task number.
*A task description is displayed.
*Parameter values are adjusted.
*Solution after pressing &bdquo;Hide solition”.

The number &bdquo;0” will reset the program and output a text with further explanation of the applet.
 
In the following, $\rm Green$ denotes the lower sideband   ⇒   $\big (A_{\rm U}, f_{\rm U}, \varphi_{\rm U}\big )$,  
$\rm Red$ the carrier   ⇒   $\big (A_{\rm T}, f_{\rm T}, \varphi_{\rm T}\big )$ and
$\rm Blue$ the upper sideband   ⇒   $\big (A_{\rm O}, f_{\rm O}, \varphi_{\rm O}\big )$.
{{BlaueBox|TEXT=
'''(1)'''   Consider and interpret the analytical signal $x_+(t)$ for $\text{Red:} \hspace{0.15cm} A_{\rm T} = 1.5\ \text{V}, \ f_{\rm T} = 50 \ \text{kHz}, \ \varphi_{\rm T} = 0^\circ$, $A_{\rm U} = A_{\rm O} = 0$.

:Which signal values $x_+(t)$ result for $t = 0$, $t = 5 \ \rm µ s$ and $t = 20 \ \rm µ s$? What are the corresponding signal values for $x(t)$? }}

:: For a cosine signal, let $x_+(t= 0) = A_{\rm T} = 1.5\ \text{V}$. Then $x_+(t)$ rotates in a mathematically positive direction (one revolution per period $T_0 = 1/f_{\rm T}$):

:: $x_+(t= 20 \ {\rm µ s}) = x_+(t= 0) = 1.5\ \text{V}\hspace{0.3cm}\Rightarrow\hspace{0.3cm}x(t= 20 \ {\rm µ s}) = 1.5\ \text{V,}$
:: $x_+(t= 5 \ {\rm µ s}) = {\rm j} \cdot 1.5\ \text{V}\hspace{0.3cm}\Rightarrow\hspace{0.3cm}x(t= 5 \ {\rm µ s}) = {\rm Re}[x_+(t= 5 \ {\rm µ s})] = 0$.

{{BlaueBox|TEXT=
'''(2)'''   How do the ratios change for $\text{Red:} \hspace{0.15cm} A_{\rm T} = 1.0\ \text{V}, \ f_{\rm T} = 100 \ \text{kHz}, \ \varphi_{\rm T} = 90^\circ$?}}

::The signal $x(t)$ is now a sine signal with a smaller amplitude. The analytical signal now starts because of $\varphi_{\rm T} = 90^\circ$   ⇒   $\phi_{\rm T} = -90^\circ$ at $x_+(t= 0) = -{\rm j} \cdot A_{\rm T}$. After that, $x_+(t)$ rotates again in a mathematically positive direction, but twice as fast because of $T_0 = 10 \ \rm µ s$ as in $\rm (1)$.

{{BlaueBox|TEXT=
'''(3)'''   Now   $\text{Red:} \hspace{0.15cm} A_{\rm T} = 1\ \text{V}, \ f_{\rm T} = 100 \ \text{kHz}, \ \varphi_{\rm T} = 0^\circ$,   $\text{Green:} \hspace{0.15cm} A_{\rm U} = 0.4\ \text{V}, \ f_{\rm U} = 80 \ \text{kHz}, \ \varphi_{\rm U} = 0^\circ$,   $\text{Blue:} \hspace{0.15cm} A_{\rm O} = 0.4\ \text{V}, \ f_{\rm O} = 120 \ \text{kHz}, \ \varphi_{\rm O} = 0^\circ$.

:Consider and interpret the physical signal $x(t)$ and the analytical signal $x_+(t)$.}}

::The Signal $x(t)$ results in the [https://en.wikipedia.org/wiki/Sideband Double-sideband Amplitude Modulation] (DSB–AM) of the message signal $A_{\rm N}\cdot \cos\left(2\pi f_{\rm N}\cdot t\right)$ with $A_{\rm N} = 0.8\ \text{V}$, $f_{\rm N} = 20\ \text{kHz}$. The carrier $x_{\rm T}(t)$ with $f_{\rm T} = 100\ \text{kHz}$ is also cosinusoidal. The degree of modulation is $m = A_{\rm N}/A_{\rm T} = 0.8$ and the period $T_{\rm 0} = 50\ \text{µs}$.

::In the phase diagram, the (red) carrier rotates faster than the (green) lower sideband and slower than the (blue) upper sideband. The analytical signal $x_+(t)$ results as the geometric sum of the three rotating pointers. It seems that the blue pointer is leading the carrier and the green pointer is following the carrier.

{{BlaueBox|TEXT=
'''(4)'''   The settings of task '''(3)''' still apply. Which signal values are obtained at $t=0$, $t=2.5 \ \rm µ s$, $t= 5 \ \rm µ s$ and $t=10 \ \rm µ s$? }}

::At time $t=0$, all pointers are in the direction of the real axis, so that $x(t=0) = {\rm Re}\big [x+(t= 0)\big] = A_{\rm U} + A_{\rm T} + A_{\rm O} = 1.8\ \text{V}$.

::Until the time $t=2.5 \ \rm µ s$, the red carrier has rotated by $90^\circ$, the blue one by $108^\circ$ and the green one by $72^\circ$. We have $x(t=2.5 \ \rm µ s) = {\rm Re}\big [x_+(t= 2.5 \ \rm µ s)\big] = 0$, because now the pointer group points in the direction of the imaginary axis. The other sought signal values are $x(t=5 \ \rm µ s) = {\rm Re}\big [x_+(t= 5 \ \rm µ s)\big] = -1.647\ \text{V}$ and $x(t=10 \ \rm µ s) = {\rm Re}\big [x_+(t= 10 \ \rm µ s)\big] = 1.247\ \text{V}$.
::For $x_+(t)$ a spiral shape results, alternating with a smaller radius and then with a larger radius.

{{BlaueBox|TEXT=
'''(5)'''   How should the phase parameters $\varphi_{\rm T}$, $\varphi_{\rm U}$ and $\varphi_{\rm O}$ be set if both the carrier $x_{\rm T}(t)$ and the message signal $x_{\rm N}(t)$ are sinusoidal?}}

::The parameter selection $\varphi_{\rm T} = \varphi_{\rm U} = \varphi_{\rm O}=90^\circ$ describes the signals $x_{\rm T}(t) = A_{\rm T}\cdot \sin\left(2\pi f_{\rm T}\cdot t\right)$ and $x_{\rm N}(t) = A_{\rm N}\cdot \cos\left(2\pi f_{\rm N}\cdot t\right)$. If, in addition, the message $x_{\rm N}(t)$ is sinusoidal, then $\varphi_{\rm O}=\varphi_{\rm T} - 90^\circ = 0$ and $\varphi_{\rm U}=\varphi_{\rm T} + 90^\circ = 180^\circ$ must be set.

{{BlaueBox|TEXT=
'''(6)'''   The settings of task '''(3)''' apply except $A_{\rm T} = 0.6\ \text{V}$. Which modulation method is described here?

: What are the consequences of this? What changes with $A_{\rm T} = 0$? }}

::It is a [https://en.wikipedia.org/wiki/Sideband Double-sideband Amplitude Modulation] (DSB–AM with carrier) with the modulation degree $m=0.8/0.6 = 1.333$. For $m > 1$, however, [https://www.radio-electronics.com/info/rf-technology-design/am-reception/synchronous-demodulator-demodulation-detector.php Synchronous Demodulation] is required. [https://en.wikipedia.org/wiki/Envelope_detector Envelope Detection] no longer works. One reason for this is that now the zero crossings of $x(t)$ are no longer equidistant from $5\ \rm µ s$   ⇒   additional phase modulation.

::With $A_{\rm T} = 0$   ⇒   $m \to \infty$ results in a [https://en.wikipedia.org/wiki/Double-sideband_suppressed-carrier_transmission ''DSB–AM without carrier'']. For this, one also needs coherent demodulation.

{{BlaueBox|TEXT=
'''(7)'''     Now let   $\text{Red:} \hspace{0.15cm} A_{\rm T} = 1\ \text{V}, \ f_{\rm T} = 100 \ \text{kHz}, \ \varphi_{\rm T} = 0^\circ$,   $\text{Green:} \hspace{0.15cm} A_{\rm U} = 0$,   $\text{Blue:} \hspace{0.15cm} A_{\rm O} = 0.8\ \text{V}, \ f_{\rm O} = 120 \ \text{kHz}, \ \varphi_{\rm O} = 90^\circ$.

:Which constellation is described here? Which figure is given for the equivalent low-pass signal $x_{\rm TP}(t)$?   ⇒   &bdquo;locus”? What changes with $A_{\rm U} = 0.8\ \text{V}$ and $A_{\rm O} = 0$?}}

::In both cases, it is a [https://en.wikipedia.org/wiki/Single-sideband_modulation Single-sideband Amplitude Modulation] (SSB–AM) with the modulation degree $\mu = 0.8$ (in SSB we denote the degree of modulation with $\mu$ instead of $m$). The carrier signal is cosinusoidal and the message signal is sinusoidal. The equivalent low-pass signal $x_{\rm TP}(t)$ has a circular course in the complex plane.

:: $A_{\rm O} = 0.8\ \text{V}$, $A_{\rm U} = 0$ is an OSB modulation. The green pointer is missing and the blue pointer rotates faster compared to the red carrier.

:: $A_{\rm U} = 0.8\ \text{V}$, $A_{\rm O} = 0$ is a USB modulation. The blue pointer is missing and the green pointer rotates slower compared to the red carrier.

{{BlaueBox|TEXT=
'''(8)'''   Now let   $\text{Red:} \hspace{0.05cm} A_{\rm T} = 1\ \text{V}, \ f_{\rm T} = 100 \ \text{kHz}, \ \varphi_{\rm T} = 0^\circ$,   $\text{Green:} \hspace{0.05cm} A_{\rm U} = 0.4\ \text{V}, \ f_{\rm U} = 80 \ \text{kHz}, \ \varphi_{\rm U} = -90^\circ$,   $\text{Blue:} \hspace{0.05cm} A_{\rm O} = 0.2\ \text{V}, \ f_{\rm O} = 120 \ \text{kHz}, \ \varphi_{\rm O} = +90^\circ$.

:Which constellation could be described here? Which shape results for the equivalent lowpass signal $x_{\rm TP}(t)$?}}

::It could be a DSB–AM of a sinusoidal signal with cosinusoidal carrier and modulation degree $m=0.8$, in which the upper sideband is attenuated by a factor of 2. The equivalent lowpass signal $x_{\rm TP}(t)$ has an elliptical trace in the complex plane.

==Applet Manual==
 
[[File:Zeigerdiagramm_abzug.png|right]]

* The red parameters $(A_{\rm T}, \ f_{\rm T}, \ \varphi_{\rm T})$ and the red pointer mark the ''Carrier'' (German:   '''T'''räger).
* The green parameters $(A_{\rm U}, \ f_{\rm U} < f_{\rm T}, \ \varphi_{\rm U})$ mark the ''Lower sideband'' (German:  '''U'''nteres Seitenband).
* The blue parameters $(A_{\rm O}, \ f_{\rm O} > f_{\rm T}, \ \varphi_{\rm O})$ mark the ''Upper sideband'' (German:  '''O'''beres Seitenband).
*All pointers rotate in a mathematically positive direction (counterclockwise).

 
Meaning of the letters in the adjacent graphic:

    '''(A)'''     Plot of the analytical signal $x_{\rm +}(t)$

    '''(B)'''     Plot of for the physical signal $x(t)$

    '''(C)'''     Parameter input via slider: amplitudes, frequencies, phase values

    '''(D)'''     Control elements:   Start – Step – Pause/Continue – Reset

    '''(E)'''     Speed of animation:   &bdquo;Speed”   ⇒   Values: 1, 2, 3

    '''(F)'''     &bdquo;Trace”   ⇒   On or Off, trace of complex signal values $x_{\rm +}(t)$

    '''(G)'''     Numerical output of the time $t$ and the signal values  ${\rm Re}[x_{\rm +}(t)] = x(t)$  and  ${\rm Im}[x_{\rm +}(t)]$

    '''(H)'''     Variations for the graphical representation

$\hspace{1.5cm}$Zoom–Functions &bdquo;$+$” (Enlarge), &bdquo;$-$” (Decrease) and $\rm o$ (Reset to default)

$\hspace{1.5cm}$Move with &bdquo;$\leftarrow$” (Section to the left, ordinate to the right), &bdquo;$\uparrow$” &bdquo;$\downarrow$” and &bdquo;$\rightarrow$”

    '''(I)'''     Experiment section:  Task selection and task

    '''(J)'''     Experiment section:  solution
 

==About the Authors==
This interactive calculation was designed and realized at the [http://www.lnt.ei.tum.de/startseite Lehrstuhl für Nachrichtentechnik] of the [https://www.tum.de/ Technical University of Munich] .
*The original version was created in 2005 by [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Studierende#Ji_Li_.28Bachelorarbeit_EI_2003.2C_Diplomarbeit_EI_2005.29|Ji Li]] as part of her Diploma thesis using &bdquo;FlashMX–Actionscript” (Supervisor: [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Mitarbeiter_und_Dozenten#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28am_LNT_seit_1974.29|Günter Söder]]).
*In 2018 this Applet was redesigned and updated to &bdquo;HTML5” by [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Studierende#Xiaohan_Liu_.28Bachelorarbeit_2018.29|Xiaohan Liu]] as part of her Bachelor's thesis (Supervisor: [[Biografien_und_Bibliografien/Beteiligte_der_Professur_Leitungsgebundene_%C3%9Cbertragungstechnik#Tasn.C3.A1d_Kernetzky.2C_M.Sc._.28bei_L.C3.9CT_seit_2014.29|Tasnád Kernetzky]]).

==Once again: Open Applet in new Tab==

{{LntAppletLinkEn|physAnSignal}}

[[Category:Applets|^Verzerrungen^]]

Applets:Bessel Functions of the First Kind

2018-08-23T16:58:06Z

Xiaohan: /* Applet Manual */

{{LntAppletLink|bessel}}

==Applet Description==
 
This applet allows the calculation and graphical representation of the Bessel functions of the first kind and $n$–th order according to the series representation:

:$${\rm J}_n (x) = \sum\limits_{k=0}^{\infty}\frac{(-1)^k \cdot (x/2)^{n \hspace{0.05cm} + \hspace{0.05cm} 2
\hspace{0.02cm}\cdot \hspace{0.05cm}k}}{k! \cdot (n+k)!} \hspace{0.05cm}.$$

*Graphically represented, the functions ${\rm J}_n (x)$ for the order $n=0$ to $n=9$ can become different colors.
*The left-hand output provides the function values ${\rm J}_0 (x = x_1)$, ... , ${\rm J}_9 (x = x_1)$ for a slider-settable value $x_1$ in the range $0 \le x_1 \le 15$ with increment $0.5$.
*The right hand side returns the function values ${\rm J}_0 (x = x_2)$, ... , ${\rm J}_9 (x = x_2)$ for a slider-settable value $x_2$ (same range and value) Increment as on the left).

[[Applets:Physical_Signal_%26_Analytical_Signal|'''German description''']]

==Theoretical Background==
 
===General Information about the Bessel Functions===
Bessel functions (or cylinder functions) are solutions of the Bessel differential equation of the form

:$$x^2 \cdot \frac{ {\rm d}^2}{{\rm d}x^2}\ {\rm J}_n (x) \ + \ x \cdot \frac{ {\rm d}}{{\rm d}x}\ {\rm J}_n (x) \ + \ (x^2 - n^2)
\cdot {\rm J}_n (x)= 0. $$

This is an ordinary second-order linear differential equation. The parameter $ n $ is usually integer, as in this program. These mathematical functions, which were introduced in 1844 by [https://de.wikipedia.org/wiki/Friedrich_Wilhelm_Bessel Friedrich Wilhelm Bessel], can also be represented in closed form as integrals:

:$${\rm J}_n (x) = \frac{1}{2\pi}\cdot \int_{-\pi}^{+\pi} {{\rm e}^{\hspace{0.05cm}{\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}[\hspace{0.05cm}x \hspace{0.05cm}\cdot \hspace{0.05cm}\sin(\alpha) -\hspace{0.05cm} n \hspace{0.05cm}\cdot \hspace{0.05cm}\alpha \hspace{0.05cm}]}}\hspace{0.1cm}{\rm d}\alpha
\hspace{0.05cm}.$$

The functions ${\rm J}_n (x)$ belong to the class of Bessel functions of the first kind (German:   ''Besselfunktionen erster Art''). The parameter $n$ is called the ''Order''.

''Annotation:''   There are a number of modifications of the Bessel functions, including the second-order Bessel functions named ${\rm Y}_n (x)$. For integer $n$, ${\rm Y}_n (x)$ can be replaced by ${\rm J}_n (x)$–functions. However, in this applet, only the first-order Bessel functions are   ⇒   ${\rm J}_n (x)$ is considered.
 
===Properties of the Bessel Functions===

{{BlaueBox|TEXT=
$\text{Property (A):}$   If the function values for $n = 0$ and $n = 1$ are known, then the Bessel function for $n ≥ 2$ can be determined iteratively:
:$${\rm J}_n (x) ={2 \cdot (n-1)}/{x} \cdot {\rm J}_{n-1} (x) - {\rm J}_{n-2} (x) \hspace{0.05cm}.$$
}}

{{GraueBox|TEXT=
$\text{Example (A):}$   Consider ${\rm J}_0 (x = 2) = 0.22389$ and ${\rm J}_1 (x= 2) = 0.57672$. From this it can be calculated iteratively:
:$${\rm J}_2 (x= 2) ={2 \cdot 1}/{2} \cdot {\rm J}_{1} (x= 2) - {\rm J}_{0} (x= 2) = 0.57672 - 0.22389 = \hspace{0.15cm}\underline{0.35283}\hspace{0.05cm},$$
:$${\rm J}_3 (x= 2) ={2 \cdot 2}/{2} \cdot {\rm J}_{2} (x= 2) - {\rm J}_{1} (x= 2) = 2 \cdot 0.35283 - 0.57672 = \hspace{0.15cm}\underline{0.12894}\hspace{0.05cm},$$
:$${\rm J}_4 (x= 2) ={2 \cdot 3}/{2} \cdot {\rm J}_{3} (x= 2) - {\rm J}_{2} (x= 2) = 3 \cdot 0.12894 - 0.35283 = \hspace{0.15cm}\underline{0.03400}\hspace{0.05cm}.$$
}}

{{BlaueBox|TEXT=
$\text{Property (B):}$   The symmetry relationship applies ${\rm J}_{–n}(x) = (–1)^n · {\rm J}_n(x)$:
:$${\rm J}_{-1}(x) = - {\rm J}_{1}(x), \hspace{0.3cm}{\rm J}_{-2}(x) = {\rm J}_{2}(x), \hspace{0.3cm}{\rm J}_{-3}(x) = - {\rm J}_{3}(x).$$
}}

{{GraueBox|TEXT=
$\text{Example (B):}$   For the spectrum of the analytic signal, in [[Modulationsverfahren/Phasenmodulation_(PM)#Spektralfunktion_eines_phasenmodulierten_Sinussignals|phase modulation of a sinusoidal signal]]:
[[File:Mod_T_3_1_S4_version2.png|right|frame|Spectrum of the analytic signal in phase modulation]]
:$$S_{\rm +}(f) = A_{\rm T} \cdot \sum_{n = - \infty}^{+\infty}{\rm J}_n (\eta) \cdot \delta \big[f - (f_{\rm T}+ n \cdot f_{\rm N})\big]\hspace{0.05cm}.$$
Denote this
*$f_{\rm T}$ the carrier frequency,
*$f_{\rm N}$ the message frequency,
* $A_{\rm T}$ the carrier amplitude.

The parameter of the Bessel functions in this application is the modulation index $\eta$.

Based on the graphic, the following statements are possible:
*Here $S_+(f)$ consists of an infinite number of discrete lines at a distance of $f_{\rm N}$.
*It is thus in principle infinitely extended.
*The weights of the spectral lines at $f_{\rm T} + n · f_{\rm N}$ ($n$ integer) are determined by the modulation index $η$ over the Bessel functions ${\rm J}_n(η)$.
*The spectral lines are real in the case of a sinusoidal source signal and a cosinusoidal carrier and symmetric about $f_{\rm T}$ for even $n$.
*For odd $n$ , a sign change according to the $\text{Property (B)}$ has to be considered.
*The phase modulation of another phase oscillation of source and / or carrier signal provides the same magnitude spectrum.}}
 

===Applications of the Bessel Functions===

The applications of the Bessel functions in nature–and engineering are diverse and play an important role in physics, for example:
* Examination of natural vibrations of cylindrical resonators,
* Solution of the radial Schrödinger-equation,
* Sound pressure amplitudes of low-viscosity rotational flows,
* Heat conduction in cylindrical bodies,
* Scattering problem of a grid,
* Dynamics of oscillating bodies,
* Angular resolution.

The Bessel functions are counted among the special functions because of their many applications in mathematical physics.

In the following, we restrict ourselves to a few areas that are addressed in our tutorial $\rm LNTwww$.

'''In the English original'''
Electromagnetic waves in a cylindrical waveguide
Pressure amplitudes of inviscid rotational flows
Heat conduction in a cylindrical object
Modes of vibration of a thin circular (or annular) acoustic membrane (such as a drum or other membranophone)
Diffusion problems on a lattice
Solutions to the radial Schrödinger equation (in spherical and cylindrical coordinates) for a free particle
Solving for patterns of acoustical radiation
Frequency-dependent friction in circular pipelines
Dynamics of floating bodies
Angular resolution
'''The Ende'''

{{GraueBox|TEXT=
$\text{Example (C):} \hspace{0.5cm} \text{Use in spectral analysis} \ \Rightarrow \ \text{Kaiser-Bessel filter}$

The '''spectral leakage effect''' is the falsification of the spectrum of a periodic and hence temporally unlimited signal due to the implicit time limitation of the discrete Fourier transform (DFT). This will be done, for example, by a spectrum analyzer
* in the time signal not existing frequency components simulated, and/or
* actually existing spectral components are obscured by sidelobes.

The task of the [[Signaldarstellung/Spektralanalyse|spectral analysis]] is to limit the influence of the ''spectral leakage effect'' by providing suitable window functions.

Such a window function provides, for example, the Kaiser–Bessel–window   ⇒   see section [[Signaldarstellung/Spektralanalyse#Spezielle_Fensterfunktionen|Special Window Functions]]. Its time-discrete Fenser function reads with the Bessel function zero order   ⇒   ${\rm J}_0(x)$, the parameter $\alpha=3.5$ and the window length $N$:
:$$w_\nu = \frac{ {\rm J}_0\big(\pi \cdot \alpha \cdot \sqrt{1 - (2\nu/N)^2}\big)}{ {\rm J}_0\big(\pi \cdot \alpha \big)}.$$
On the page [[Signaldarstellung/Spektralanalyse#G.C3.BCtekriterien_von_Fensterfunktionen|Quality Criteria of Window Functions]] and other the characteristics of the Kaiser-Bessel-window are given:
*Conveniently, the large is &bdquo;minimum distance between the main lobe and side lobes” and the desired small &bdquo;maximum scaling error”.
*Due to the very large &bdquo;equivalent noise width” the Kaiser-Bessel window cuts in the main comparison criterion &bdquo;maximum process loss” but worse off than the established Hamming– and Hanning–windows.}}

{{GraueBox|TEXT=
$\text{Example (D):} \hspace{0.5cm} \text{Rice-Fading channel model}$

The [[Mobile_Kommunikation/Wahrscheinlichkeitsdichte_des_Rayleigh%E2%80%93Fadings#Allgemeine_Beschreibung_des_Mobilfunkkanals| Rayleigh - Distribution]] describes the mobile channel assuming that there is no direct path, and thus the multiplicative factor $z(t) = x(t) + {\rm j} \cdot y(t)$ is solely diffuse composed of scattered components.

In the case of a direct component (English: Line of Sight, LoS) in the model for the mean-free Gaussian processes $x(t)$ and $y(t)$ one has to add equal components $x_0$ and/or $y_0$ :

[[File:P ID2126 Mob T 1 4 S1 v3.png|right|frame|Rice-Fading channel model|class=fit]]
$$\hspace{0.2cm}x(t) \hspace{0.1cm} \Rightarrow \hspace{0.1cm} x(t) +x_0 \hspace{0.05cm}, \hspace{0.2cm} y(t) \hspace{0.1cm} \Rightarrow \hspace{0.1cm} y(t) +y_0\hspace{0.05cm},$$

$$\hspace{0.2cm}z(t) = x(t) + {\rm j} \cdot y(t) \hspace{0.1cm} \Rightarrow \hspace{0.1cm} z(t) +z_0 \hspace{0.05cm},\hspace{0.2cm}
z_0 = x_0 + {\rm j} \cdot y_0\hspace{0.05cm}.$$

The graph shows the ''Rice–Fading–Channel model''. It can be summarized as follows:
*The real part $x(t)$ is Gaussianized with mean $x_0$ and variance $\sigma ^2$.
*The imaginary part $y(t)$ is also Gaussian (mean $y_0$, equal variance $\sigma ^2$) and independent of $x(t)$. 

*For $z_0 \ne 0$, the amount $\vert z(t)\vert$ rice is distributed, from which the term &bdquo;Rice–Fading” arises.

*To simplify the notation, we set $\vert z(t)\vert = a(t)$. For $a < 0$ , the amounts - WDF are $f_a(a) \equiv 0$, for $a \ge 0$ the following equation holds, where ${\rm I_0}(x)$ is the modified Besselfunktion zeroth order denotes:
:$$f_a(a) = \frac{a}{\sigma^2} \cdot {\rm e}^{ - (a^2 + \vert z_0 \vert ^2)/(2\sigma^2)} \cdot {\rm I}_0 \left [ \frac{a \cdot \vert z_0 \vert}{\sigma^2} \right ] \hspace{0.5cm}\text{with}\hspace{0.5cm}{\rm I }_0 (u) = {\rm J }_0 ({\rm j} \cdot u) =
\sum_{k = 0}^{\infty} \frac{ (u/2)^{2k} }{k! \cdot \Gamma (k+1)}
\hspace{0.05cm}.$$
*Between the modified Bessel function and the traditional Bessel function ${\rm I_0}(x)$ – each first type – So the connection exists ${\rm I }_0 (u) = {\rm J }_0 ({\rm j} \cdot u)$.}}

{{GraueBox|TEXT=
$\text{Example (E):} \hspace{0.5cm} \text{Analysis of the frequency spectrum of frequency modulated signals}$

$\text{Example (B)}$ has already shown that the angle modulation of a harmonic oscillation of the frequency $f_{\rm N}$ leads to a line spectrum. The spectral lines are around the carrier frequency $f_{\rm T}$ bei $f_{\rm T} + n \cdot f_{\rm N}$ with $n \in \{ \ \text{...}, -2, -1, \ 0, +1, +2, \text{...} \ \}$. The weights of the Dirac lines are ${\rm J }_n(\eta)$, depending on the modulation index $\eta$.

[[File:P_ID1095__Mod_T_3_2_S4_neu.png|center|frame|Discrete spectra with phase modulation (left) and frequency modulation (right)]]

The graph shows the magnitude spectrum $\vert S_{\rm +}(f) \vert$ of the phase modulation (PM) and frequency modulation (FM) analytical signal, two different forms of angle modulation (WM). Bessellines with values less than $0.03$ are neglected in both cases.

For the upper half of the image, the modulator parameters are chosen so that for each $f_{\rm N} = 5 \ \rm kHz$ here is a Bessel spectrum with the modulation index $η = 1.5$. Disregarding the phase relationships, the same spectra and the same signals result for both systems.

The lower graphics are valid for the same message frequency $f_{\rm N} = 3 \ \rm kHz$. One notices:
*In phase modulation, the spectral function is narrower than $f_{\rm N} = 5 \ \rm kHz$, because the distance of the Bessel lines is now only $3 \ \rm kHz$. Since the modulation index of PM is independent of $f_{\rm N}$, the same bessel weights result as in $f_{\rm N} = 5 \ \rm kHz$.
*Also in the frequency modulation, the Bessel lines now appear at a distance of $3 \ \rm kHz$. However, since FM has a modulation index inversely proportional to $f_{\rm N}$, there is now much more Besselline lines than in the upper right (for $η = 1.5$ valid) chart due to the larger modulation index $η = 2.5$.}}

'''Das folgende Kapitel muss noch angepasst werden!'''

==Applet Manual==
[[File:Bessel_abzug3.png|left|600px]]
    '''(A)'''     Sum formula of the Bessel functions ${\rm J}_n(x)$

    '''(B)'''     Selection of the order $n$ for the graphical representation

    '''(C)'''     Graphical representation of the Bessel functions

    '''(D)'''     Variation of the graphic representation

$\hspace{1.5cm}$&bdquo;$+$” (Enlarge),

$\hspace{1.5cm}$ &bdquo;$-$” (Decrease)

$\hspace{1.5cm}$ &bdquo;$\rm o$” (Reset to default)

$\hspace{1.5cm}$ &bdquo;$\leftarrow$” (pushed to the left), usw.

    '''(E)'''     Selection of the abscissa value $x_1$ for the left numeric output

    '''(F)'''     Numerical output of the Bessel function values ${\rm J}_n(x_1)$

    '''(G)'''     Selection of the abscissa value $x_2$ for the right numeric output

    '''(F)'''     Numerical output of the Bessel function values ${\rm J}_n(x_2)$
 

==About the Authors==
This interactive calculation was designed and realized at the [http://www.lnt.ei.tum.de/startseite Lehrstuhl für Nachrichtentechnik] of the [https://www.tum.de/ Technical University of Munich] .
*The original version was created in 2005 by [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Studierende#Markus_Elsberger_.28Diplomarbeit_LB_2006.29|Markus Elsberger]] and [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Studierende#Slim_Lamine_.28Studienarbeit_EI_2006.29|Slim Lamine]] as part of thesis using &bdquo;FlashMX–Actionscript” (Supervisor: [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Mitarbeiter_und_Dozenten#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28am_LNT_seit_1974.29|Günter Söder]]).
*In 2018 this Applet was redesigned and updated to &bdquo;HTML5” by [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Studierende#Xiaohan_Liu_.28Bachelorarbeit_2018.29|Xiaohan Liu]] as part of her Bachelor's thesis (Supervisor: [[Biografien_und_Bibliografien/Beteiligte_der_Professur_Leitungsgebundene_%C3%9Cbertragungstechnik#Tasn.C3.A1d_Kernetzky.2C_M.Sc._.28bei_L.C3.9CT_seit_2014.29|Tasnád Kernetzky]]).

==Nochmalige Aufrufmöglichkeit des Applets in neuem Fenster==

{{LntAppletLink|bessel}}

Applets:Physical Signal & Analytic Signal

2018-08-22T17:20:01Z

Xiaohan: /* Exercises */

{{LntAppletLinkEn|physAnSignal}}

==Applet Description==
 
This applet shows the relationship between the physical bandpass signal $x(t)$ and the associated analytical signal $x_+(t)$. It is assumed that the bandpass signal $x(t)$ has a frequency-discrete spectrum $X(f)$:
:$$x(t) = x_{\rm U}(t) + x_{\rm T}(t) + x_{\rm O}(t) = A_{\rm U}\cdot \cos\left(2\pi f_{\rm U}\cdot t- \varphi_{\rm U}\right)+A_{\rm T}\cdot \cos\left(2\pi f_{\rm T}\cdot t- \varphi_{\rm T}\right)+A_{\rm O}\cdot \cos\left(2\pi f_{\rm O}\cdot t- \varphi_{\rm O}\right). $$
The physical signal $x(t)$ is thus composed of three harmonic oscillations, a constellation that can be found, for example, in the ''Double-sideband Amplitude Modulation''
*of the message signal $x_{\rm N}(t) = A_{\rm N}\cdot \cos\left(2\pi f_{\rm N}\cdot t- \varphi_{\rm N}\right)$   ⇒   in German:   '''N'''achrichtensignal
*with the carrier signal $x_{\rm T}(t) = A_{\rm T}\cdot \cos\left(2\pi f_{\rm T}\cdot t - \varphi_{\rm T}\right)$   ⇒   in German:   '''T'''rägersignal.

The nomenclature is also adapted to this case:
* $x_{\rm O}(t)$ denotes the &bdquo;upper sideband”   (in German:   '''O'''beres Seitenband) with the amplitude $A_{\rm O}= A_{\rm N}/2$, the frequency $f_{\rm O} = f_{\rm T} + f_{\rm N}$ and the phase $\varphi_{\rm O} = \varphi_{\rm T} + \varphi_{\rm N}$.
*Similarly, for the &bdquo;lower sideband”   (in German:   '''U'''nteres Seitenband) $x_{\rm U}(t)$ with $f_{\rm U} = f_{\rm T} - f_{\rm N}$, $A_{\rm U}= A_{\rm O}$ and $\varphi_{\rm U} = -\varphi_{\rm O}$.

The associated analytical signal is:

:$$x_+(t) = x_{\rm U+}(t) + x_{\rm T+}(t) + x_{\rm O+}(t) = A_{\rm U}\cdot {\rm e}^{\hspace{0.05cm}{\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}(2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm U}\hspace{0.05cm}\cdot \hspace{0.05cm}t- \varphi_{\rm U})}
\hspace{0.1cm}+ \hspace{0.1cm}A_{\rm T}\cdot {\rm e}^{\hspace{0.05cm}{\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}(2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm T}\hspace{0.05cm}\cdot \hspace{0.05cm}t- \varphi_{\rm T})}
\hspace{0.1cm}+\hspace{0.1cm} A_{\rm O}\cdot {\rm e}^{\hspace{0.05cm}{\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}(2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm O}\hspace{0.05cm}\cdot \hspace{0.05cm}t- \varphi_{\rm O})}. $$

[[File:Zeigerdiagramm_2a_version2.png|right|frame|Analytical signal at the time $t=0$]]
The program displays $x_+(t)$ as the vectorial sum of three rotating pointers (all with counterclockwise) as a violet dot (see figure for start time $t=0$):

*The (red) pointer of the carrier $x_{\rm T+}(t)$ with length $A_{\rm T}$ and zero phase position $\varphi_{\rm T} = 0$ rotates at constant angular velocity $2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm T}$ (one revolution in time $1/f_{\rm T})$.

*The (blue) pointer of the upper sideband $x_{\rm O+}(t)$ with length $A_{\rm O}$ and zero phase position $\varphi_{\rm O}$ rotates at the angular velocity $2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm O}$, which is slightly faster than $x_{\rm T+}(t)$.

*The (green) pointer of the lower sideband $x_{\rm U+}(t)$ with length $A_{\rm U}$ and zero phase position $\varphi_{\rm U}$ rotates at the angular velocity $2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm U}$, which is slightly slower than $x_{\rm T+}(t)$.

The time trace of $x_+(t)$ is also referred to below as ''Pointer Diagram''. The relationship between the physical bandpass signal $x(t)$ and the associated analytical signal $x_+(t)$ is:

:$$x(t) = {\rm Re}\big [x_+(t)\big ].$$

''Note:''   In the figure $\varphi_{\rm O} = +30^\circ$. This leads to the angle with respect to the coordinate system at $t=0$:   $\phi_{\rm O}=-\varphi_{\rm O}=-30^\circ$. Similarly, the null phase angle $\varphi_{\rm U}=-30^\circ$ of the lower sideband leads to the phase angle to be considered in the complex plane:   $\phi_{\rm U}=+30^\circ$.

[[Applets:Physikalisches_Signal_%26_Analytisches_Signal|'''German Description''']]

==Theoretical Background==
 
===Description of Bandpass Signals===
[[File:Zeigerdiagramm_1a.png|right|frame|Bandpass spectrum $X(f)$ |class=fit]]
We consider '''bandpass signals''' $x(t)$ with the property that their spectra $X(f)$ are not in the range around the frequency $f=0$, but around a carrier frequency $f_{\rm T}$. In most cases it can also be assumed that the bandwidth is $B \ll f_{\rm T}$.

The figure shows such a bandpass spectrum $X(f)$. Assuming that the associated $x(t)$ is a physical signal and thus real, the spectral function $X(f)$ has a symmetry with respect to the frequency $f = 0$, if $x(t)$ is an even function   ⇒   $x(-t)=x(t)$, $X(f)$ is real and even.

Besides the physical signal $x(t)\ \circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\, \ X(f)$, one can also use the following descriptions of bandpass signals:
*the analytical signal $x_+(t)\ \circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\, \ X_+(f)$, see next page,
*the equivalent lowpass signal   (in German:   äquivalentes '''T'''ief '''P'''ass–Signal) $x_{\rm TP}(t)\ \circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\, \ X_{\rm TP}(f)$, see Applet [[Applets:Physical_Signal_%26_Equivalent_Low-pass_Signal|Physical Signal & Equivalent Lowpass signal]].
 

===Analytical Signal – Frequency Domain===

The '''analytical signal''' $x_+(t)$ belonging to the physical signal $x(t)$ is the time function whose spectrum fulfills the following property:
[[File:Zeigerdiagramm_3a.png|right|frame|Construction of the spectral function $X_+(f)$ |class=fit]]
:$$X_+(f)=\big[1+{\rm sign}(f)\big] \cdot X(f) = \left\{ {2 \cdot
X(f) \; \hspace{0.2cm}\rm for\hspace{0.2cm} {\it f} > 0, \atop {\,\,\,\, \rm 0 \; \hspace{0.9cm}\rm for\hspace{0.2cm} {\it f} < 0.} }\right.$$

The ''signum function'' is for positive values of $f$ equal to $+1$ and for negative $f$ values equal to $-1$.
* The (double-sided) limit returns $\sign(0)=0$.
* The index „+” should make it clear that $X_+(f)$ only has parts at positive frequencies.

From the graph you can see the calculation rule for $X_+(f)$:

The actual bandpass spectrum $X(f)$ becomes
* doubled at the positive frequencies, and
* set to zero at the negative frequencies.

Due to the asymmetry of $X_+(f)$ with respect to the frequency $f=0$, it can already be said that the time function $x_+(t)$ except for a trivial special case $x_+(t)=0 \ \circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\,X_+(f)=0$ is always complex.
 

===Analytical Signal – Time Domain===
At this point, it is necessary to briefly discuss another spectral transformation.

{{BlaueBox|TEXT=
$\text{Definition:}$ 
For the '''Hilbert transform''' $ {\rm H}\left\{x(t)\right\}$ of a time function $x(t)$ we have:

:$$y(t) = {\rm H}\left\{x(t)\right\} = \frac{1}{ {\rm \pi} } \cdot
\hspace{0.03cm}\int_{-\infty}^{+\infty}\frac{x(\tau)}{ {t -
\tau} }\hspace{0.15cm} {\rm d}\tau.$$

This particular integral is not solvable in a simple, conventional way, but must be evaluated using the [https://en.wikipedia.org/wiki/Cauchy_principal_value Cauchy principal value theorem].

Accordingly, in the frequency domain:
:$$Y(f) = {\rm -j \cdot sign}(f) \cdot X(f) \hspace{0.05cm} .$$}}

The above result can be summarized with this definition as follows:
* The analytical signal $x_+(t)$ is obtained from the physical bandpass signal $x(t)$ by adding an imaginary part to $x(t)$ according to the Hilbert transform:

:$$x_+(t) = x(t)+{\rm j} \cdot {\rm H}\left\{x(t)\right\} .$$

*$\text{H}\{x(t)\}$ disappears only for the case $x(t) = \rm const.$   ⇒   the same signal. For all other signal forms, the analytic signal $x_+(t)$ is complex.

* From the analytical signal $x_+(t)$, the physical bandpass signal can be easily determined by the following operation:
:$$x(t) = {\rm Re}\big[x_+(t)\big] .$$

{{GraueBox|TEXT=
$\text{Example 1:}$  The principle of the Hilbert transformation should be further clarified by the following graphic:
*After the left representation $\rm(A)$ one gets from the physical signal $x(t)$ to the analytical signal $x_+(t)$, by adding an imaginary part ${\rm j} \cdot y(t)$.
*Here $y(t) = {\rm H}\left\{x(t)\right\}$ is a real time function that can be indicated in the spectral domain by multiplying the spectrum $X(f)$ with ${\rm - j} \cdot \sign(f)$.

[[File:P_ID2729__Sig_T_4_2_S2b_neu.png|center|frame|To clarify the Hilbert transform]]

The right representation $\rm(B)$ is equivalent to $\rm(A)$. Now $x_+(t) = x(t) + z(t)$ stand with the purely imaginary function $z(t)$. A comparison of the two figures shows that in fact $z(t) = {\rm j} \cdot y(t)$.}}
 

===Representation of the Harmonic Oscillation as an Analytical Signal===

The spectral function $X(f)$ of a harmonic oscillation $x(t) = A\cdot\text{cos}(2\pi f_{\rm T}\cdot t - \varphi)$ is known to consist of two Dirac functions at the frequencies
* $+f_{\rm T}$ with the complex weight $A/2 \cdot \text{e}^{-\text{j}\hspace{0.05cm}\varphi}$,
* $-f_{\rm T}$ with the complex weight $A/2 \cdot \text{e}^{+\text{j}\hspace{0.05cm}\varphi}$.

Thus, the spectrum of the analytical signal (that is, without the Dirac function at the frequency $f =-f_{\rm T}$, but doubling at $f =+f_{\rm T}$):

:$$X_+(f) = A \cdot {\rm e}^{-{\rm j} \hspace{0.05cm}\varphi}\cdot\delta (f - f_{\rm
T}) .$$

The associated time function is obtained by applying the Displacement Law:

:$$x_+(t) = x_{\rm U+}(t) + x_{\rm T+}(t) + x_{\rm O+}(t) = A_{\rm U}\cdot {\rm e}^{\hspace{0.05cm}{\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}(2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm U}\hspace{0.05cm}\cdot \hspace{0.05cm}t- \varphi_{\rm U})}
\hspace{0.1cm}+ \hspace{0.1cm}A_{\rm T}\cdot {\rm e}^{\hspace{0.05cm}{\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}(2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm T}\hspace{0.05cm}\cdot \hspace{0.05cm}t- \varphi_{\rm T})}
\hspace{0.1cm}+\hspace{0.1cm} A_{\rm O}\cdot {\rm e}^{\hspace{0.05cm}{\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}(2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm O}\hspace{0.05cm}\cdot \hspace{0.05cm}t- \varphi_{\rm O})}. $$

This equation describes a pointer rotating at constant angular velocity $\omega_{\rm T} = 2\pi f_{\rm T}$.

{{GraueBox|TEXT=
$\text{Example 2:}$  Here the coordinate system is rotated by $90^\circ$ (real part up, imaginary part to the left) contrary to the usual representation.

[[File:P_ID712__Sig_T_4_2_S3.png|center|frame|Pointer diagram of a harmonic oscillation]]

Based on this graphic, the following statements are possible:
* At time $t = 0$, the pointer of length $A$ (signal amplitude) lies with the angle $-\varphi$ in the complex plane. In the example shown, $\varphi=45^\circ$.
* For times $t>0$, the constant angular velocity vector $\omega_{\rm T}$ rotates in a mathematically positive direction, that is, counterclockwise.
* The tip of the pointer is thus always on a circle with radius $A$ and needs exactly the time $T_0$, i.e. the period of the harmonic oscillation $x(t)$ for one revolution.
* The projection of the analytical signal $x_+(t)$ on the real axis, marked by red dots, gives the instantaneous values of $x(t)$.}}
 

===Analytical Signal Representation of a Sum of Three Harmonic Oscillations===

In our applet, we always assume a set of three rotating pointers. The physical signal is:
:$$x(t) = x_{\rm U}(t) + x_{\rm T}(t) + x_{\rm O}(t) = A_{\rm U}\cdot \cos\left(2\pi f_{\rm U}\cdot t- \varphi_{\rm U}\right)+A_{\rm T}\cdot \cos\left(2\pi f_{\rm T}\cdot t- \varphi_{\rm T}\right)+A_{\rm O}\cdot \cos\left(2\pi f_{\rm O}\cdot t- \varphi_{\rm O}\right). $$
* Each of the three harmonic oscillations $x_{\rm T}(t)$, $x_{\rm U}(t)$ and $x_{\rm O}(t)$ is represented by an amplitude $(A)$, a frequency $(f)$ and a phase value $(\varphi)$.
*The indices are based on the ''Double-sideband Amplitude Modulation'' method. &bdquo;T” stands for &bdquo;carrier”, &bdquo;U” for &bdquo;lower sideband” and &bdquo;O” for &bdquo;upper Sideband”.
*Accordingly, $f_{\rm U} < f_{\rm T}$ and $f_{\rm O} > f_{\rm T}$. There are no restrictions for the amplitudes and phases.

The associated analytical signal is:
:$$x_+(t) = x_{\rm U+}(t) + x_{\rm T+}(t) + x_{\rm O+}(t) = A_{\rm U}\cdot {\rm e}^{\hspace{0.05cm}{\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}(2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm U}\hspace{0.05cm}\cdot \hspace{0.05cm}t- \varphi_{\rm U})}
\hspace{0.1cm}+ \hspace{0.1cm}A_{\rm T}\cdot {\rm e}^{\hspace{0.05cm}{\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}(2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm T}\hspace{0.05cm}\cdot \hspace{0.05cm}t- \varphi_{\rm T})}
\hspace{0.1cm}+\hspace{0.1cm} A_{\rm O}\cdot {\rm e}^{\hspace{0.05cm}{\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}(2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm O}\hspace{0.05cm}\cdot \hspace{0.05cm}t- \varphi_{\rm O})}. $$

{{GraueBox|TEXT=
$\text{Example 3:}$ 
Shown the constellation arises i.e. in the [https://en.wikipedia.org/wiki/Sideband Double-sideband Amplitude Modulation] (with carrier) of the message signal $x_{\rm N}(t) = A_{\rm N}\cdot \cos\left(2\pi f_{\rm N}\cdot t- \varphi_{\rm N}\right)$ with the carrier signal $x_{\rm T}(t) = A_{\rm T}\cdot \cos\left(2\pi f_{\rm T}\cdot t - \varphi_{\rm T}\right)$. This is discussed frequently in the Exercises.

There are some limitations to the program parameters in this approach:
* For the frequencies, it always applies $f_{\rm O} = f_{\rm T} + f_{\rm N}$ and $f_{\rm U} = f_{\rm T} - f_{\rm N}$.

*Without distortions the amplitude of the sidebands are $A_{\rm O}= A_{\rm U}= A_{\rm N}/2$.
*The respective phase relationships can be seen in the following graphic.

[[File:Zeigerdiagramm_2_neu.png|center|frame|Spectrum $X_+(f)$ of the analytical signal for different phase constellations |class=fit]]}}

==Exercises==
[[File:Zeigerdiagramm_aufgabe_2.png|right]]
*First select the task number.
*A task description is displayed.
*Parameter values are adjusted.
*Solution after pressing &bdquo;Hide solition”.

The number &bdquo;0” will reset the program and output a text with further explanation of the applet.
 
In the following, $\rm Green$ denotes the lower sideband   ⇒   $\big (A_{\rm U}, f_{\rm U}, \varphi_{\rm U}\big )$,  
$\rm Red$ the carrier   ⇒   $\big (A_{\rm T}, f_{\rm T}, \varphi_{\rm T}\big )$ and
$\rm Blue$ the upper sideband   ⇒   $\big (A_{\rm O}, f_{\rm O}, \varphi_{\rm O}\big )$.
{{BlaueBox|TEXT=
'''(1)'''   Consider and interpret the analytical signal $x_+(t)$ for $\text{Red:} \hspace{0.15cm} A_{\rm T} = 1.5\ \text{V}, \ f_{\rm T} = 50 \ \text{kHz}, \ \varphi_{\rm T} = 0^\circ$, $A_{\rm U} = A_{\rm O} = 0$.

:Which signal values $x_+(t)$ result for $t = 0$, $t = 5 \ \rm µ s$ and $t = 20 \ \rm µ s$? What are the corresponding signal values for $x(t)$? }}

:: For a cosine signal, let $x_+(t= 0) = A_{\rm T} = 1.5\ \text{V}$. Then $x_+(t)$ rotates in a mathematically positive direction (one revolution per period $T_0 = 1/f_{\rm T}$):

:: $x_+(t= 20 \ {\rm µ s}) = x_+(t= 0) = 1.5\ \text{V}\hspace{0.3cm}\Rightarrow\hspace{0.3cm}x(t= 20 \ {\rm µ s}) = 1.5\ \text{V,}$
:: $x_+(t= 5 \ {\rm µ s}) = {\rm j} \cdot 1.5\ \text{V}\hspace{0.3cm}\Rightarrow\hspace{0.3cm}x(t= 5 \ {\rm µ s}) = {\rm Re}[x_+(t= 5 \ {\rm µ s})] = 0$.

{{BlaueBox|TEXT=
'''(2)'''   How do the ratios change for $\text{Red:} \hspace{0.15cm} A_{\rm T} = 1.0\ \text{V}, \ f_{\rm T} = 100 \ \text{kHz}, \ \varphi_{\rm T} = 90^\circ$?}}

::The signal $x(t)$ is now a sine signal with a smaller amplitude. The analytical signal now starts because of $\varphi_{\rm T} = 90^\circ$   ⇒   $\phi_{\rm T} = -90^\circ$ at $x_+(t= 0) = -{\rm j} \cdot A_{\rm T}$. After that, $x_+(t)$ rotates again in a mathematically positive direction, but twice as fast because of $T_0 = 10 \ \rm µ s$ as in $\rm (1)$.

{{BlaueBox|TEXT=
'''(3)'''   Now   $\text{Red:} \hspace{0.15cm} A_{\rm T} = 1\ \text{V}, \ f_{\rm T} = 100 \ \text{kHz}, \ \varphi_{\rm T} = 0^\circ$,   $\text{Green:} \hspace{0.15cm} A_{\rm U} = 0.4\ \text{V}, \ f_{\rm U} = 80 \ \text{kHz}, \ \varphi_{\rm U} = 0^\circ$,   $\text{Blue:} \hspace{0.15cm} A_{\rm O} = 0.4\ \text{V}, \ f_{\rm O} = 120 \ \text{kHz}, \ \varphi_{\rm O} = 0^\circ$.

:Consider and interpret the physical signal $x(t)$ and the analytical signal $x_+(t)$.}}

::The Signal $x(t)$ results in the [https://en.wikipedia.org/wiki/Sideband Double-sideband Amplitude Modulation] (DSB–AM) of the message signal $A_{\rm N}\cdot \cos\left(2\pi f_{\rm N}\cdot t\right)$ with $A_{\rm N} = 0.8\ \text{V}$, $f_{\rm N} = 20\ \text{kHz}$. The carrier $x_{\rm T}(t)$ with $f_{\rm T} = 100\ \text{kHz}$ is also cosinusoidal. The degree of modulation is $m = A_{\rm N}/A_{\rm T} = 0.8$ and the period $T_{\rm 0} = 50\ \text{µs}$.

::In the phase diagram, the (red) carrier rotates faster than the (green) lower sideband and slower than the (blue) upper sideband. The analytical signal $x_+(t)$ results as the geometric sum of the three rotating pointers. It seems that the blue pointer is leading the carrier and the green pointer is following the carrier.

{{BlaueBox|TEXT=
'''(4)'''   The settings of task '''(3)''' still apply. Which signal values are obtained at $t=0$, $t=2.5 \ \rm µ s$, $t= 5 \ \rm µ s$ and $t=10 \ \rm µ s$? }}

::At time $t=0$, all pointers are in the direction of the real axis, so that $x(t=0) = {\rm Re}\big [x+(t= 0)\big] = A_{\rm U} + A_{\rm T} + A_{\rm O} = 1.8\ \text{V}$.

::Until the time $t=2.5 \ \rm µ s$, the red carrier has rotated by $90^\circ$, the blue one by $108^\circ$ and the green one by $72^\circ$. We have $x(t=2.5 \ \rm µ s) = {\rm Re}\big [x_+(t= 2.5 \ \rm µ s)\big] = 0$, because now the pointer group points in the direction of the imaginary axis. The other sought signal values are $x(t=5 \ \rm µ s) = {\rm Re}\big [x_+(t= 5 \ \rm µ s)\big] = -1.647\ \text{V}$ and $x(t=10 \ \rm µ s) = {\rm Re}\big [x_+(t= 10 \ \rm µ s)\big] = 1.247\ \text{V}$.
::For $x_+(t)$ a spiral shape results, alternating with a smaller radius and then with a larger radius.

{{BlaueBox|TEXT=
'''(5)'''   How should the phase parameters $\varphi_{\rm T}$, $\varphi_{\rm U}$ and $\varphi_{\rm O}$ be set if both the carrier $x_{\rm T}(t)$ and the message signal $x_{\rm N}(t)$ are sinusoidal?}}

::The parameter selection $\varphi_{\rm T} = \varphi_{\rm U} = \varphi_{\rm O}=90^\circ$ describes the signals $x_{\rm T}(t) = A_{\rm T}\cdot \sin\left(2\pi f_{\rm T}\cdot t\right)$ and $x_{\rm N}(t) = A_{\rm N}\cdot \cos\left(2\pi f_{\rm N}\cdot t\right)$. If, in addition, the message $x_{\rm N}(t)$ is sinusoidal, then $\varphi_{\rm O}=\varphi_{\rm T} - 90^\circ = 0$ and $\varphi_{\rm U}=\varphi_{\rm T} + 90^\circ = 180^\circ$ must be set.

{{BlaueBox|TEXT=
'''(6)'''   The settings of task '''(3)''' apply except $A_{\rm T} = 0.6\ \text{V}$. Which modulation method is described here?

: What are the consequences of this? What changes with $A_{\rm T} = 0$? }}

::It is a [https://en.wikipedia.org/wiki/Sideband Double-sideband Amplitude Modulation] (DSB–AM with carrier) with the modulation degree $m=0.8/0.6 = 1.333$. For $m > 1$, however, [https://www.radio-electronics.com/info/rf-technology-design/am-reception/synchronous-demodulator-demodulation-detector.php Synchronous Demodulation] is required. [https://en.wikipedia.org/wiki/Envelope_detector Envelope Detection] no longer works. One reason for this is that now the zero crossings of $x(t)$ are no longer equidistant from $5\ \rm µ s$   ⇒   additional phase modulation.

::With $A_{\rm T} = 0$   ⇒   $m \to \infty$ results in a [https://en.wikipedia.org/wiki/Double-sideband_suppressed-carrier_transmission ''DSB–AM without carrier'']. For this, one also needs coherent demodulation.

{{BlaueBox|TEXT=
'''(7)'''     Now let   $\text{Red:} \hspace{0.15cm} A_{\rm T} = 1\ \text{V}, \ f_{\rm T} = 100 \ \text{kHz}, \ \varphi_{\rm T} = 0^\circ$,   $\text{Green:} \hspace{0.15cm} A_{\rm U} = 0$,   $\text{Blue:} \hspace{0.15cm} A_{\rm O} = 0.8\ \text{V}, \ f_{\rm O} = 120 \ \text{kHz}, \ \varphi_{\rm O} = 90^\circ$.

:Which constellation is described here? Which figure is given for the equivalent low-pass signal $x_{\rm TP}(t)$?   ⇒   &bdquo;locus”? What changes with $A_{\rm U} = 0.8\ \text{V}$ and $A_{\rm O} = 0$?}}

::In both cases, it is a [https://en.wikipedia.org/wiki/Single-sideband_modulation Single-sideband Amplitude Modulation] (SSB–AM) with the modulation degree $\mu = 0.8$ (in SSB we denote the degree of modulation with $\mu$ instead of $m$). The carrier signal is cosinusoidal and the message signal is sinusoidal. The equivalent low-pass signal $x_{\rm TP}(t)$ has a circular course in the complex plane.

:: $A_{\rm O} = 0.8\ \text{V}$, $A_{\rm U} = 0$ is an OSB modulation. The green pointer is missing and the blue pointer rotates faster compared to the red carrier.

:: $A_{\rm U} = 0.8\ \text{V}$, $A_{\rm O} = 0$ is a USB modulation. The blue pointer is missing and the green pointer rotates slower compared to the red carrier.

{{BlaueBox|TEXT=
'''(8)'''   Now let   $\text{Red:} \hspace{0.05cm} A_{\rm T} = 1\ \text{V}, \ f_{\rm T} = 100 \ \text{kHz}, \ \varphi_{\rm T} = 0^\circ$,   $\text{Green:} \hspace{0.05cm} A_{\rm U} = 0.4\ \text{V}, \ f_{\rm U} = 80 \ \text{kHz}, \ \varphi_{\rm U} = -90^\circ$,   $\text{Blue:} \hspace{0.05cm} A_{\rm O} = 0.2\ \text{V}, \ f_{\rm O} = 120 \ \text{kHz}, \ \varphi_{\rm O} = +90^\circ$.

:Which constellation could be described here? Which shape results for the equivalent lowpass signal $x_{\rm TP}(t)$?}}

::It could be a DSB–AM of a sinusoidal signal with cosinusoidal carrier and modulation degree $m=0.8$, in which the upper sideband is attenuated by a factor of 2. The equivalent lowpass signal $x_{\rm TP}(t)$ has an elliptical trace in the complex plane.

==Applet Manual==
 
[[File:Zeigerdiagramm_abzug.png|right]]

* The red parameters $(A_{\rm T}, \ f_{\rm T}, \ \varphi_{\rm T})$ and the red pointer mark the ''Carrier'' (German:   '''T'''räger).
* The green parameters $(A_{\rm U}, \ f_{\rm U} < f_{\rm T}, \ \varphi_{\rm U})$ mark the ''Lower sideband'' (German:  '''U'''nteres Seitenband).
* The blue parameters $(A_{\rm O}, \ f_{\rm O} > f_{\rm T}, \ \varphi_{\rm O})$ mark the mark the ''Upper sideband'' (German:  '''O'''beres Seitenband).
*All pointers rotate in a mathematically positive direction (counterclockwise).

 
Meaning of the letters in the adjacent graphic:

    '''(A)'''     Plot of the analytical signal $x_{\rm +}(t)$

    '''(B)'''     Plot of for the physical signal $x(t)$

    '''(C)'''     Parameter input via slider: amplitudes, frequencies, phase values

    '''(D)'''     Control elements:   Start – Step – Pause/Continue – Reset

    '''(E)'''     Speed of animation:   &bdquo;Speed”   ⇒   Values: 1, 2, 3

    '''(F)'''     &bdquo;Trace”   ⇒   On or Off, trace of complex signal values $x_{\rm +}(t)$

    '''(G)'''     Numerical output of the time $t$ and the signal values  ${\rm Re}[x_{\rm +}(t)] = x(t)$  and  ${\rm Im}[x_{\rm +}(t)]$

    '''(H)'''     Variations for the graphical representation

$\hspace{1.5cm}$Zoom–Functions &bdquo;$+$” (Enlarge), &bdquo;$-$” (Decrease) and $\rm o$ (Reset to default)

$\hspace{1.5cm}$Move with &bdquo;$\leftarrow$” (Section to the left, ordinate to the right), &bdquo;$\uparrow$” &bdquo;$\downarrow$” and &bdquo;$\rightarrow$”

    '''(I)'''     Experiment section:  Task selection and task

    '''(J)'''     Experiment section:  solution
 

==About the Authors==
This interactive calculation was designed and realized at the [http://www.lnt.ei.tum.de/startseite Lehrstuhl für Nachrichtentechnik] of the [https://www.tum.de/ Technical University of Munich] .
*The original version was created in 2005 by [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Studierende#Ji_Li_.28Bachelorarbeit_EI_2003.2C_Diplomarbeit_EI_2005.29|Ji Li]] as part of her Diploma thesis using &bdquo;FlashMX–Actionscript” (Supervisor: [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Mitarbeiter_und_Dozenten#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28am_LNT_seit_1974.29|Günter Söder]]).
*In 2018 this Applet was redesigned and updated to &bdquo;HTML5” by [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Studierende#Xiaohan_Liu_.28Bachelorarbeit_2018.29|Xiaohan Liu]] as part of her Bachelor's thesis (Supervisor: [[Biografien_und_Bibliografien/Beteiligte_der_Professur_Leitungsgebundene_%C3%9Cbertragungstechnik#Tasn.C3.A1d_Kernetzky.2C_M.Sc._.28bei_L.C3.9CT_seit_2014.29|Tasnád Kernetzky]]).

==Once again: Open Applet in new Tab==

{{LntAppletLinkEn|physAnSignal}}

[[Category:Applets|^Verzerrungen^]]

Applets:Physical Signal & Analytic Signal

2018-08-22T16:11:43Z

Xiaohan: /* Theoretical Background */

{{LntAppletLinkEn|physAnSignal}}

==Applet Description==
 
This applet shows the relationship between the physical bandpass signal $x(t)$ and the associated analytical signal $x_+(t)$. It is assumed that the bandpass signal $x(t)$ has a frequency-discrete spectrum $X(f)$:
:$$x(t) = x_{\rm U}(t) + x_{\rm T}(t) + x_{\rm O}(t) = A_{\rm U}\cdot \cos\left(2\pi f_{\rm U}\cdot t- \varphi_{\rm U}\right)+A_{\rm T}\cdot \cos\left(2\pi f_{\rm T}\cdot t- \varphi_{\rm T}\right)+A_{\rm O}\cdot \cos\left(2\pi f_{\rm O}\cdot t- \varphi_{\rm O}\right). $$
The physical signal $x(t)$ is thus composed of three harmonic oscillations, a constellation that can be found, for example, in the ''Double-sideband Amplitude Modulation''
*of the message signal $x_{\rm N}(t) = A_{\rm N}\cdot \cos\left(2\pi f_{\rm N}\cdot t- \varphi_{\rm N}\right)$   ⇒   in German:   '''N'''achrichtensignal
*with the carrier signal $x_{\rm T}(t) = A_{\rm T}\cdot \cos\left(2\pi f_{\rm T}\cdot t - \varphi_{\rm T}\right)$   ⇒   in German:   '''T'''rägersignal.

The nomenclature is also adapted to this case:
* $x_{\rm O}(t)$ denotes the &bdquo;upper sideband”   (in German:   '''O'''beres Seitenband) with the amplitude $A_{\rm O}= A_{\rm N}/2$, the frequency $f_{\rm O} = f_{\rm T} + f_{\rm N}$ and the phase $\varphi_{\rm O} = \varphi_{\rm T} + \varphi_{\rm N}$.
*Similarly, for the &bdquo;lower sideband”   (in German:   '''U'''nteres Seitenband) $x_{\rm U}(t)$ with $f_{\rm U} = f_{\rm T} - f_{\rm N}$, $A_{\rm U}= A_{\rm O}$ and $\varphi_{\rm U} = -\varphi_{\rm O}$.

The associated analytical signal is:

:$$x_+(t) = x_{\rm U+}(t) + x_{\rm T+}(t) + x_{\rm O+}(t) = A_{\rm U}\cdot {\rm e}^{\hspace{0.05cm}{\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}(2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm U}\hspace{0.05cm}\cdot \hspace{0.05cm}t- \varphi_{\rm U})}
\hspace{0.1cm}+ \hspace{0.1cm}A_{\rm T}\cdot {\rm e}^{\hspace{0.05cm}{\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}(2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm T}\hspace{0.05cm}\cdot \hspace{0.05cm}t- \varphi_{\rm T})}
\hspace{0.1cm}+\hspace{0.1cm} A_{\rm O}\cdot {\rm e}^{\hspace{0.05cm}{\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}(2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm O}\hspace{0.05cm}\cdot \hspace{0.05cm}t- \varphi_{\rm O})}. $$

[[File:Zeigerdiagramm_2a_version2.png|right|frame|Analytical signal at the time $t=0$]]
The program displays $x_+(t)$ as the vectorial sum of three rotating pointers (all with counterclockwise) as a violet dot (see figure for start time $t=0$):

*The (red) pointer of the carrier $x_{\rm T+}(t)$ with length $A_{\rm T}$ and zero phase position $\varphi_{\rm T} = 0$ rotates at constant angular velocity $2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm T}$ (one revolution in time $1/f_{\rm T})$.

*The (blue) pointer of the upper sideband $x_{\rm O+}(t)$ with length $A_{\rm O}$ and zero phase position $\varphi_{\rm O}$ rotates at the angular velocity $2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm O}$, which is slightly faster than $x_{\rm T+}(t)$.

*The (green) pointer of the lower sideband $x_{\rm U+}(t)$ with length $A_{\rm U}$ and zero phase position $\varphi_{\rm U}$ rotates at the angular velocity $2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm U}$, which is slightly slower than $x_{\rm T+}(t)$.

The time trace of $x_+(t)$ is also referred to below as ''Pointer Diagram''. The relationship between the physical bandpass signal $x(t)$ and the associated analytical signal $x_+(t)$ is:

:$$x(t) = {\rm Re}\big [x_+(t)\big ].$$

''Note:''   In the figure $\varphi_{\rm O} = +30^\circ$. This leads to the angle with respect to the coordinate system at $t=0$:   $\phi_{\rm O}=-\varphi_{\rm O}=-30^\circ$. Similarly, the null phase angle $\varphi_{\rm U}=-30^\circ$ of the lower sideband leads to the phase angle to be considered in the complex plane:   $\phi_{\rm U}=+30^\circ$.

[[Applets:Physikalisches_Signal_%26_Analytisches_Signal|'''German Description''']]

==Theoretical Background==
 
===Description of Bandpass Signals===
[[File:Zeigerdiagramm_1a.png|right|frame|Bandpass spectrum $X(f)$ |class=fit]]
We consider '''bandpass signals''' $x(t)$ with the property that their spectra $X(f)$ are not in the range around the frequency $f=0$, but around a carrier frequency $f_{\rm T}$. In most cases it can also be assumed that the bandwidth is $B \ll f_{\rm T}$.

The figure shows such a bandpass spectrum $X(f)$. Assuming that the associated $x(t)$ is a physical signal and thus real, the spectral function $X(f)$ has a symmetry with respect to the frequency $f = 0$, if $x(t)$ is an even function   ⇒   $x(-t)=x(t)$, $X(f)$ is real and even.

Besides the physical signal $x(t)\ \circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\, \ X(f)$, one can also use the following descriptions of bandpass signals:
*the analytical signal $x_+(t)\ \circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\, \ X_+(f)$, see next page,
*the equivalent lowpass signal   (in German:   äquivalentes '''T'''ief '''P'''ass–Signal) $x_{\rm TP}(t)\ \circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\, \ X_{\rm TP}(f)$, see Applet [[Applets:Physical_Signal_%26_Equivalent_Low-pass_Signal|Physical Signal & Equivalent Lowpass signal]].
 

===Analytical Signal – Frequency Domain===

The '''analytical signal''' $x_+(t)$ belonging to the physical signal $x(t)$ is the time function whose spectrum fulfills the following property:
[[File:Zeigerdiagramm_3a.png|right|frame|Construction of the spectral function $X_+(f)$ |class=fit]]
:$$X_+(f)=\big[1+{\rm sign}(f)\big] \cdot X(f) = \left\{ {2 \cdot
X(f) \; \hspace{0.2cm}\rm for\hspace{0.2cm} {\it f} > 0, \atop {\,\,\,\, \rm 0 \; \hspace{0.9cm}\rm for\hspace{0.2cm} {\it f} < 0.} }\right.$$

The ''signum function'' is for positive values of $f$ equal to $+1$ and for negative $f$ values equal to $-1$.
* The (double-sided) limit returns $\sign(0)=0$.
* The index „+” should make it clear that $X_+(f)$ only has parts at positive frequencies.

From the graph you can see the calculation rule for $X_+(f)$:

The actual bandpass spectrum $X(f)$ becomes
* doubled at the positive frequencies, and
* set to zero at the negative frequencies.

Due to the asymmetry of $X_+(f)$ with respect to the frequency $f=0$, it can already be said that the time function $x_+(t)$ except for a trivial special case $x_+(t)=0 \ \circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\,X_+(f)=0$ is always complex.
 

===Analytical Signal – Time Domain===
At this point, it is necessary to briefly discuss another spectral transformation.

{{BlaueBox|TEXT=
$\text{Definition:}$ 
For the '''Hilbert transform''' $ {\rm H}\left\{x(t)\right\}$ of a time function $x(t)$ we have:

:$$y(t) = {\rm H}\left\{x(t)\right\} = \frac{1}{ {\rm \pi} } \cdot
\hspace{0.03cm}\int_{-\infty}^{+\infty}\frac{x(\tau)}{ {t -
\tau} }\hspace{0.15cm} {\rm d}\tau.$$

This particular integral is not solvable in a simple, conventional way, but must be evaluated using the [https://en.wikipedia.org/wiki/Cauchy_principal_value Cauchy principal value theorem].

Accordingly, in the frequency domain:
:$$Y(f) = {\rm -j \cdot sign}(f) \cdot X(f) \hspace{0.05cm} .$$}}

The above result can be summarized with this definition as follows:
* The analytical signal $x_+(t)$ is obtained from the physical bandpass signal $x(t)$ by adding an imaginary part to $x(t)$ according to the Hilbert transform:

:$$x_+(t) = x(t)+{\rm j} \cdot {\rm H}\left\{x(t)\right\} .$$

*$\text{H}\{x(t)\}$ disappears only for the case $x(t) = \rm const.$   ⇒   the same signal. For all other signal forms, the analytic signal $x_+(t)$ is complex.

* From the analytical signal $x_+(t)$, the physical bandpass signal can be easily determined by the following operation:
:$$x(t) = {\rm Re}\big[x_+(t)\big] .$$

{{GraueBox|TEXT=
$\text{Example 1:}$  The principle of the Hilbert transformation should be further clarified by the following graphic:
*After the left representation $\rm(A)$ one gets from the physical signal $x(t)$ to the analytical signal $x_+(t)$, by adding an imaginary part ${\rm j} \cdot y(t)$.
*Here $y(t) = {\rm H}\left\{x(t)\right\}$ is a real time function that can be indicated in the spectral domain by multiplying the spectrum $X(f)$ with ${\rm - j} \cdot \sign(f)$.

[[File:P_ID2729__Sig_T_4_2_S2b_neu.png|center|frame|To clarify the Hilbert transform]]

The right representation $\rm(B)$ is equivalent to $\rm(A)$. Now $x_+(t) = x(t) + z(t)$ stand with the purely imaginary function $z(t)$. A comparison of the two figures shows that in fact $z(t) = {\rm j} \cdot y(t)$.}}
 

===Representation of the Harmonic Oscillation as an Analytical Signal===

The spectral function $X(f)$ of a harmonic oscillation $x(t) = A\cdot\text{cos}(2\pi f_{\rm T}\cdot t - \varphi)$ is known to consist of two Dirac functions at the frequencies
* $+f_{\rm T}$ with the complex weight $A/2 \cdot \text{e}^{-\text{j}\hspace{0.05cm}\varphi}$,
* $-f_{\rm T}$ with the complex weight $A/2 \cdot \text{e}^{+\text{j}\hspace{0.05cm}\varphi}$.

Thus, the spectrum of the analytical signal (that is, without the Dirac function at the frequency $f =-f_{\rm T}$, but doubling at $f =+f_{\rm T}$):

:$$X_+(f) = A \cdot {\rm e}^{-{\rm j} \hspace{0.05cm}\varphi}\cdot\delta (f - f_{\rm
T}) .$$

The associated time function is obtained by applying the Displacement Law:

:$$x_+(t) = x_{\rm U+}(t) + x_{\rm T+}(t) + x_{\rm O+}(t) = A_{\rm U}\cdot {\rm e}^{\hspace{0.05cm}{\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}(2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm U}\hspace{0.05cm}\cdot \hspace{0.05cm}t- \varphi_{\rm U})}
\hspace{0.1cm}+ \hspace{0.1cm}A_{\rm T}\cdot {\rm e}^{\hspace{0.05cm}{\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}(2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm T}\hspace{0.05cm}\cdot \hspace{0.05cm}t- \varphi_{\rm T})}
\hspace{0.1cm}+\hspace{0.1cm} A_{\rm O}\cdot {\rm e}^{\hspace{0.05cm}{\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}(2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm O}\hspace{0.05cm}\cdot \hspace{0.05cm}t- \varphi_{\rm O})}. $$

This equation describes a pointer rotating at constant angular velocity $\omega_{\rm T} = 2\pi f_{\rm T}$.

{{GraueBox|TEXT=
$\text{Example 2:}$  Here the coordinate system is rotated by $90^\circ$ (real part up, imaginary part to the left) contrary to the usual representation.

[[File:P_ID712__Sig_T_4_2_S3.png|center|frame|Pointer diagram of a harmonic oscillation]]

Based on this graphic, the following statements are possible:
* At time $t = 0$, the pointer of length $A$ (signal amplitude) lies with the angle $-\varphi$ in the complex plane. In the example shown, $\varphi=45^\circ$.
* For times $t>0$, the constant angular velocity vector $\omega_{\rm T}$ rotates in a mathematically positive direction, that is, counterclockwise.
* The tip of the pointer is thus always on a circle with radius $A$ and needs exactly the time $T_0$, i.e. the period of the harmonic oscillation $x(t)$ for one revolution.
* The projection of the analytical signal $x_+(t)$ on the real axis, marked by red dots, gives the instantaneous values of $x(t)$.}}
 

===Analytical Signal Representation of a Sum of Three Harmonic Oscillations===

In our applet, we always assume a set of three rotating pointers. The physical signal is:
:$$x(t) = x_{\rm U}(t) + x_{\rm T}(t) + x_{\rm O}(t) = A_{\rm U}\cdot \cos\left(2\pi f_{\rm U}\cdot t- \varphi_{\rm U}\right)+A_{\rm T}\cdot \cos\left(2\pi f_{\rm T}\cdot t- \varphi_{\rm T}\right)+A_{\rm O}\cdot \cos\left(2\pi f_{\rm O}\cdot t- \varphi_{\rm O}\right). $$
* Each of the three harmonic oscillations $x_{\rm T}(t)$, $x_{\rm U}(t)$ and $x_{\rm O}(t)$ is represented by an amplitude $(A)$, a frequency $(f)$ and a phase value $(\varphi)$.
*The indices are based on the ''Double-sideband Amplitude Modulation'' method. &bdquo;T” stands for &bdquo;carrier”, &bdquo;U” for &bdquo;lower sideband” and &bdquo;O” for &bdquo;upper Sideband”.
*Accordingly, $f_{\rm U} < f_{\rm T}$ and $f_{\rm O} > f_{\rm T}$. There are no restrictions for the amplitudes and phases.

The associated analytical signal is:
:$$x_+(t) = x_{\rm U+}(t) + x_{\rm T+}(t) + x_{\rm O+}(t) = A_{\rm U}\cdot {\rm e}^{\hspace{0.05cm}{\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}(2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm U}\hspace{0.05cm}\cdot \hspace{0.05cm}t- \varphi_{\rm U})}
\hspace{0.1cm}+ \hspace{0.1cm}A_{\rm T}\cdot {\rm e}^{\hspace{0.05cm}{\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}(2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm T}\hspace{0.05cm}\cdot \hspace{0.05cm}t- \varphi_{\rm T})}
\hspace{0.1cm}+\hspace{0.1cm} A_{\rm O}\cdot {\rm e}^{\hspace{0.05cm}{\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}(2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm O}\hspace{0.05cm}\cdot \hspace{0.05cm}t- \varphi_{\rm O})}. $$

{{GraueBox|TEXT=
$\text{Example 3:}$ 
Shown the constellation arises i.e. in the [https://en.wikipedia.org/wiki/Sideband Double-sideband Amplitude Modulation] (with carrier) of the message signal $x_{\rm N}(t) = A_{\rm N}\cdot \cos\left(2\pi f_{\rm N}\cdot t- \varphi_{\rm N}\right)$ with the carrier signal $x_{\rm T}(t) = A_{\rm T}\cdot \cos\left(2\pi f_{\rm T}\cdot t - \varphi_{\rm T}\right)$. This is discussed frequently in the Exercises.

There are some limitations to the program parameters in this approach:
* For the frequencies, it always applies $f_{\rm O} = f_{\rm T} + f_{\rm N}$ and $f_{\rm U} = f_{\rm T} - f_{\rm N}$.

*Without distortions the amplitude of the sidebands are $A_{\rm O}= A_{\rm U}= A_{\rm N}/2$.
*The respective phase relationships can be seen in the following graphic.

[[File:Zeigerdiagramm_2_neu.png|center|frame|Spectrum $X_+(f)$ of the analytical signal for different phase constellations |class=fit]]}}

==Exercises==
[[File:Zeigerdiagramm_aufgabe_2.png|right]]
*First select the task number.
*A task description is displayed.
*Parameter values are adjusted.
*Solution after pressing &bdquo;Hide solition”.

The number &bdquo;0” will reset the program and output a text with further explanation of the applet.
 
In the following, $\rm Green$ denotes the lower sideband   ⇒   $\big (A_{\rm U}, f_{\rm U}, \varphi_{\rm U}\big )$,  
$\rm Red$ the carrier   ⇒   $\big (A_{\rm T}, f_{\rm T}, \varphi_{\rm T}\big )$ and
$\rm Blue$ the upper sideband   ⇒   $\big (A_{\rm O}, f_{\rm O}, \varphi_{\rm O}\big )$.
{{BlaueBox|TEXT=
'''(1)'''   Consider and interpret the analytical signal $x_+(t)$ for $\text{Red:} \hspace{0.15cm} A_{\rm T} = 1.5\ \text{V}, \ f_{\rm T} = 50 \ \text{kHz}, \ \varphi_{\rm T} = 0^\circ$, $A_{\rm U} = A_{\rm O} = 0$.

:Which signal values $x_+(t)$ result for $t = 0$, $t = 5 \ \rm µ s$ and $t = 20 \ \rm µ s$? What are the corresponding signal values for $x(t)$? }}

:: For a cosine signal, let $x_+(t= 0) = A_{\rm T} = 1.5\ \text{V}$. Then $x_+(t)$ rotates in a mathematically positive direction (one revolution per period $T_0 = 1/f_{\rm T}$):

:: $x_+(t= 20 \ {\rm µ s}) = x_+(t= 0) = 1.5\ \text{V}\hspace{0.3cm}\Rightarrow\hspace{0.3cm}x(t= 20 \ {\rm µ s}) = 1.5\ \text{V,}$
:: $x_+(t= 5 \ {\rm µ s}) = {\rm j} \cdot 1.5\ \text{V}\hspace{0.3cm}\Rightarrow\hspace{0.3cm}x(t= 5 \ {\rm µ s}) = {\rm Re}[x_+(t= 5 \ {\rm µ s})] = 0$.

{{BlaueBox|TEXT=
'''(2)'''   How do the ratios change for $\text{Red:} \hspace{0.15cm} A_{\rm T} = 1.0\ \text{V}, \ f_{\rm T} = 100 \ \text{kHz}, \ \varphi_{\rm T} = 90^\circ$?}}

::The signal $x(t)$ is now a sine signal with a smaller amplitude. The analytical signal now starts because of $\varphi_{\rm T} = 90^\circ$   ⇒   $\phi_{\rm T} = -90^\circ$ at $x_+(t= 0) = -{\rm j} \cdot A_{\rm T}$. After that, $x_+(t)$ rotates again in a mathematically positive direction, but twice as fast because of $T_0 = 10 \ \rm µ s$ as in $\rm (1)$.

{{BlaueBox|TEXT=
'''(3)'''   Now   $\text{Red:} \hspace{0.15cm} A_{\rm T} = 1\ \text{V}, \ f_{\rm T} = 100 \ \text{kHz}, \ \varphi_{\rm T} = 0^\circ$,   $\text{Green:} \hspace{0.15cm} A_{\rm U} = 0.4\ \text{V}, \ f_{\rm U} = 80 \ \text{kHz}, \ \varphi_{\rm U} = 0^\circ$,   $\text{Blue:} \hspace{0.15cm} A_{\rm O} = 0.4\ \text{V}, \ f_{\rm O} = 120 \ \text{kHz}, \ \varphi_{\rm O} = 0^\circ$.

:Consider and interpret the physical signal $x(t)$ and the analytical signal $x_+(t)$.}}

::The Signal $x(t)$ results in the [https://en.wikipedia.org/wiki/Sideband Double-sideband Amplitude Modulation] (DSB–AM) of the message signal $A_{\rm N}\cdot \cos\left(2\pi f_{\rm N}\cdot t\right)$ with $A_{\rm N} = 0.8\ \text{V}$, $f_{\rm N} = 20\ \text{kHz}$. The carrier $x_{\rm T}(t)$ with $f_{\rm T} = 100\ \text{kHz}$ is also cosinusoidal. The degree of modulation is $m = A_{\rm N}/A_{\rm T} = 0.8$ and the period $T_{\rm 0} = 50\ \text{µs}$.

::In the phase diagram, the (red) carrier rotates faster than the (green) lower sideband and slower than the (blue) upper sideband. The analytical signal $x_+(t)$ results as the geometric sum of the three rotating pointers. It seems that the blue pointer is leading the carrier and the green pointer is following the carrier.

{{BlaueBox|TEXT=
'''(4)'''   The settings of task '''(3)''' still apply. Which signal values are obtained at $t=0$, $t=2.5 \ \rm µ s$, $t= 5 \ \rm µ s$ and $t=10 \ \rm µ s$? }}

::At time $t=0$, all pointers are in the direction of the real axis, so that $x(t=0) = {\rm Re}\big [x+(t= 0)\big] = A_{\rm U} + A_{\rm T} + A_{\rm O} = 1.8\ \text{V}$.

::Until the time $t=2.5 \ \rm µ s$, the red carrier has rotated by $90^\circ$, the blue one by $108^\circ$ and the green one by $72^\circ$. We have $x(t=2.5 \ \rm µ s) = {\rm Re}\big [x_+(t= 2.5 \ \rm µ s)\big] = 0$, because now the pointer group points in the direction of the imaginary axis. The other sought signal values are $x(t=5 \ \rm µ s) = {\rm Re}\big [x_+(t= 5 \ \rm µ s)\big] = -1.647\ \text{V}$ and $x(t=10 \ \rm µ s) = {\rm Re}\big [x_+(t= 10 \ \rm µ s)\big] = 1.247\ \text{V}$.
::For $x_+(t)$ a spiral shape results, alternating with a smaller radius and then with a larger radius.

{{BlaueBox|TEXT=
'''(5)'''   How should the phase parameters $\varphi_{\rm T}$, $\varphi_{\rm U}$ and $\varphi_{\rm O}$ be set if both the carrier $x_{\rm T}(t)$ and the message signal $x_{\rm N}(t)$ are sinusoidal?}}

::The parameter selection $\varphi_{\rm T} = \varphi_{\rm U} = \varphi_{\rm O}=90^\circ$ describes the signals $x_{\rm T}(t) = A_{\rm T}\cdot \sin\left(2\pi f_{\rm T}\cdot t\right)$ and $x_{\rm N}(t) = A_{\rm N}\cdot \cos\left(2\pi f_{\rm N}\cdot t\right)$. If, in addition, the message $x_{\rm N}(t)$ is sinusoidal, then $\varphi_{\rm O}=\varphi_{\rm T} - 90^\circ = 0$ and $\varphi_{\rm U}=\varphi_{\rm T} + 90^\circ = 180^\circ$ must be set.

{{BlaueBox|TEXT=
'''(6)'''   The settings of task '''(3)''' apply except $A_{\rm T} = 0.6\ \text{V}$. Which modulation method is described here?

: What are the consequences of this? What changes with $A_{\rm T} = 0$? }}

::It is a [https://en.wikipedia.org/wiki/Sideband Double-sideband Amplitude Modulation] (DSB–AM with carrier) with the modulation degree $m=0.8/0.6 = 1.333$. For $m > 1$, however, [https://www.radio-electronics.com/info/rf-technology-design/am-reception/synchronous-demodulator-demodulation-detector.php Synchronous Demodulation] is required. [https://en.wikipedia.org/wiki/Envelope_detector Envelope Detection] no longer works.

::With $A_{\rm T} = 0$   ⇒   $m \to \infty$ results in a [https://en.wikipedia.org/wiki/Double-sideband_suppressed-carrier_transmission ''DSB–AM suppressed carrier'']. For this, one also needs coherent demodulation.

{{BlaueBox|TEXT=
'''(7)'''     Now let   $\text{Red:} \hspace{0.15cm} A_{\rm T} = 1\ \text{V}, \ f_{\rm T} = 100 \ \text{kHz}, \ \varphi_{\rm T} = 0^\circ$,   $\text{Green:} \hspace{0.15cm} A_{\rm U} = 0$,   $\text{Blue:} \hspace{0.15cm} A_{\rm O} = 0.8\ \text{V}, \ f_{\rm O} = 120 \ \text{kHz}, \ \varphi_{\rm O} = 90^\circ$.

:Which constellation is described here? What changes with $A_{\rm U} = 0.8\ \text{V}$ and $A_{\rm O} = 0$?}}

::In both cases, it is a [https://en.wikipedia.org/wiki/Single-sideband_modulation Single-sideband Amplitude Modulation] (SSB–AM) with the modulation degree $\mu = 0.8$ (in SSB we denote the degree of modulation with $\mu$ instead of $m$). The carrier signal is cosinusoidal and the message signal is sinusoidal.

:: $A_{\rm O} = 0.8\ \text{V}$, $A_{\rm U} = 0$ is an OSB modulation. The green pointer is missing and the blue pointer rotates faster compared to the red carrier.

:: $A_{\rm U} = 0.8\ \text{V}$, $A_{\rm O} = 0$ is a USB modulation. The blue pointer is missing and the green pointer rotates slower compared to the red carrier.

{{BlaueBox|TEXT=
'''(8)'''   Now let   $\text{Red:} \hspace{0.05cm} A_{\rm T} = 1\ \text{V}, \ f_{\rm T} = 100 \ \text{kHz}, \ \varphi_{\rm T} = 0^\circ$,   $\text{Green:} \hspace{0.05cm} A_{\rm U} = 0.4\ \text{V}, \ f_{\rm U} = 80 \ \text{kHz}, \ \varphi_{\rm U} = -90^\circ$,   $\text{Blue:} \hspace{0.05cm} A_{\rm O} = 0.2\ \text{V}, \ f_{\rm O} = 120 \ \text{kHz}, \ \varphi_{\rm O} = +90^\circ$.

:Which constellation could be described here? Which shape results for the equivalent lowpass signal $x_{\rm TP}(t)$?}}

::It could be a DSB–AM of a sinusoidal signal with cosinusoidal carrier and modulation degree $m=0.8$, in which the upper sideband is attenuated by a factor of 2. The equivalent lowpass signal $x_{\rm TP}(t)$ has an elliptical trace in the complex plane.

==Applet Manual==
 
[[File:Zeigerdiagramm_abzug.png|right]]

* The red parameters $(A_{\rm T}, \ f_{\rm T}, \ \varphi_{\rm T})$ and the red pointer mark the ''Carrier'' (German:   '''T'''räger).
* The green parameters $(A_{\rm U}, \ f_{\rm U} < f_{\rm T}, \ \varphi_{\rm U})$ mark the ''Lower sideband'' (German:  '''U'''nteres Seitenband).
* The blue parameters $(A_{\rm O}, \ f_{\rm O} > f_{\rm T}, \ \varphi_{\rm O})$ mark the mark the ''Upper sideband'' (German:  '''O'''beres Seitenband).
*All pointers rotate in a mathematically positive direction (counterclockwise).

 
Meaning of the letters in the adjacent graphic:

    '''(A)'''     Plot of the analytical signal $x_{\rm +}(t)$

    '''(B)'''     Plot of for the physical signal $x(t)$

    '''(C)'''     Parameter input via slider: amplitudes, frequencies, phase values

    '''(D)'''     Control elements:   Start – Step – Pause/Continue – Reset

    '''(E)'''     Speed of animation:   &bdquo;Speed”   ⇒   Values: 1, 2, 3

    '''(F)'''     &bdquo;Trace”   ⇒   On or Off, trace of complex signal values $x_{\rm +}(t)$

    '''(G)'''     Numerical output of the time $t$ and the signal values  ${\rm Re}[x_{\rm +}(t)] = x(t)$  and  ${\rm Im}[x_{\rm +}(t)]$

    '''(H)'''     Variations for the graphical representation

$\hspace{1.5cm}$Zoom–Functions &bdquo;$+$” (Enlarge), &bdquo;$-$” (Decrease) and $\rm o$ (Reset to default)

$\hspace{1.5cm}$Move with &bdquo;$\leftarrow$” (Section to the left, ordinate to the right), &bdquo;$\uparrow$” &bdquo;$\downarrow$” and &bdquo;$\rightarrow$”

    '''(I)'''     Experiment section:  Task selection and task

    '''(J)'''     Experiment section:  solution
 

==About the Authors==
This interactive calculation was designed and realized at the [http://www.lnt.ei.tum.de/startseite Lehrstuhl für Nachrichtentechnik] of the [https://www.tum.de/ Technical University of Munich] .
*The original version was created in 2005 by [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Studierende#Ji_Li_.28Bachelorarbeit_EI_2003.2C_Diplomarbeit_EI_2005.29|Ji Li]] as part of her Diploma thesis using &bdquo;FlashMX–Actionscript” (Supervisor: [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Mitarbeiter_und_Dozenten#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28am_LNT_seit_1974.29|Günter Söder]]).
*In 2018 this Applet was redesigned and updated to &bdquo;HTML5” by [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Studierende#Xiaohan_Liu_.28Bachelorarbeit_2018.29|Xiaohan Liu]] as part of her Bachelor's thesis (Supervisor: [[Biografien_und_Bibliografien/Beteiligte_der_Professur_Leitungsgebundene_%C3%9Cbertragungstechnik#Tasn.C3.A1d_Kernetzky.2C_M.Sc._.28bei_L.C3.9CT_seit_2014.29|Tasnád Kernetzky]]).

==Once again: Open Applet in new Tab==

{{LntAppletLinkEn|physAnSignal}}

[[Category:Applets|^Verzerrungen^]]

Applets:Bessel Functions of the First Kind

2018-08-22T14:52:31Z

Xiaohan: /* Zur Handhabung des Applets */

{{LntAppletLink|bessel}}

==Applet Description==
 
This applet allows the calculation and graphical representation of the Bessel functions of the first kind and $n$–th order according to the series representation:

:$${\rm J}_n (x) = \sum\limits_{k=0}^{\infty}\frac{(-1)^k \cdot (x/2)^{n \hspace{0.05cm} + \hspace{0.05cm} 2
\hspace{0.02cm}\cdot \hspace{0.05cm}k}}{k! \cdot (n+k)!} \hspace{0.05cm}.$$

*Graphically represented, the functions ${\rm J}_n (x)$ for the order $n=0$ to $n=9$ can become different colors.
*The left-hand output provides the function values ${\rm J}_0 (x = x_1)$, ... , ${\rm J}_9 (x = x_1)$ for a slider-settable value $x_1$ in the range $0 \le x_1 \le 15$ with increment $0.5$.
*The right hand side returns the function values ${\rm J}_0 (x = x_2)$, ... , ${\rm J}_9 (x = x_2)$ for a slider-settable value $x_2$ (same range and value) Increment as on the left).

[[Applets:Physical_Signal_%26_Analytical_Signal|'''German description''']]

==Theoretical Background==
 
===General Information about the Bessel Functions===
Bessel functions (or cylinder functions) are solutions of the Bessel differential equation of the form

:$$x^2 \cdot \frac{ {\rm d}^2}{{\rm d}x^2}\ {\rm J}_n (x) \ + \ x \cdot \frac{ {\rm d}}{{\rm d}x}\ {\rm J}_n (x) \ + \ (x^2 - n^2)
\cdot {\rm J}_n (x)= 0. $$

This is an ordinary second-order linear differential equation. The parameter $ n $ is usually integer, as in this program. These mathematical functions, which were introduced in 1844 by [https://de.wikipedia.org/wiki/Friedrich_Wilhelm_Bessel Friedrich Wilhelm Bessel], can also be represented in closed form as integrals:

:$${\rm J}_n (x) = \frac{1}{2\pi}\cdot \int_{-\pi}^{+\pi} {{\rm e}^{\hspace{0.05cm}{\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}[\hspace{0.05cm}x \hspace{0.05cm}\cdot \hspace{0.05cm}\sin(\alpha) -\hspace{0.05cm} n \hspace{0.05cm}\cdot \hspace{0.05cm}\alpha \hspace{0.05cm}]}}\hspace{0.1cm}{\rm d}\alpha
\hspace{0.05cm}.$$

The functions ${\rm J}_n (x)$ belong to the class of Bessel functions of the first kind (German:   ''Besselfunktionen erster Art''). The parameter $n$ is called the ''Order''.

''Annotation:''   There are a number of modifications of the Bessel functions, including the second-order Bessel functions named ${\rm Y}_n (x)$. For integer $n$, ${\rm Y}_n (x)$ can be replaced by ${\rm J}_n (x)$–functions. However, in this applet, only the first-order Bessel functions are   ⇒   ${\rm J}_n (x)$ is considered.
 
===Properties of the Bessel Functions===

{{BlaueBox|TEXT=
$\text{Property (A):}$   If the function values for $n = 0$ and $n = 1$ are known, then the Bessel function for $n ≥ 2$ can be determined iteratively:
:$${\rm J}_n (x) ={2 \cdot (n-1)}/{x} \cdot {\rm J}_{n-1} (x) - {\rm J}_{n-2} (x) \hspace{0.05cm}.$$
}}

{{GraueBox|TEXT=
$\text{Example (A):}$   Consider ${\rm J}_0 (x = 2) = 0.22389$ and ${\rm J}_1 (x= 2) = 0.57672$. From this it can be calculated iteratively:
:$${\rm J}_2 (x= 2) ={2 \cdot 1}/{2} \cdot {\rm J}_{1} (x= 2) - {\rm J}_{0} (x= 2) = 0.57672 - 0.22389 = \hspace{0.15cm}\underline{0.35283}\hspace{0.05cm},$$
:$${\rm J}_3 (x= 2) ={2 \cdot 2}/{2} \cdot {\rm J}_{2} (x= 2) - {\rm J}_{1} (x= 2) = 2 \cdot 0.35283 - 0.57672 = \hspace{0.15cm}\underline{0.12894}\hspace{0.05cm},$$
:$${\rm J}_4 (x= 2) ={2 \cdot 3}/{2} \cdot {\rm J}_{3} (x= 2) - {\rm J}_{2} (x= 2) = 3 \cdot 0.12894 - 0.35283 = \hspace{0.15cm}\underline{0.03400}\hspace{0.05cm}.$$
}}

{{BlaueBox|TEXT=
$\text{Property (B):}$   The symmetry relationship applies ${\rm J}_{–n}(x) = (–1)^n · {\rm J}_n(x)$:
:$${\rm J}_{-1}(x) = - {\rm J}_{1}(x), \hspace{0.3cm}{\rm J}_{-2}(x) = {\rm J}_{2}(x), \hspace{0.3cm}{\rm J}_{-3}(x) = - {\rm J}_{3}(x).$$
}}

{{GraueBox|TEXT=
$\text{Example (B):}$   For the spectrum of the analytic signal, in [[Modulationsverfahren/Phasenmodulation_(PM)#Spektralfunktion_eines_phasenmodulierten_Sinussignals|phase modulation of a sinusoidal signal]]:
[[File:Mod_T_3_1_S4_version2.png|right|frame|Spectrum of the analytic signal in phase modulation]]
:$$S_{\rm +}(f) = A_{\rm T} \cdot \sum_{n = - \infty}^{+\infty}{\rm J}_n (\eta) \cdot \delta \big[f - (f_{\rm T}+ n \cdot f_{\rm N})\big]\hspace{0.05cm}.$$
Denote this
*$f_{\rm T}$ the carrier frequency,
*$f_{\rm N}$ the message frequency,
* $A_{\rm T}$ the carrier amplitude.

The parameter of the Bessel functions in this application is the modulation index $\eta$.

Based on the graphic, the following statements are possible:
*Here $S_+(f)$ consists of an infinite number of discrete lines at a distance of $f_{\rm N}$.
*It is thus in principle infinitely extended.
*The weights of the spectral lines at $f_{\rm T} + n · f_{\rm N}$ ($n$ integer) are determined by the modulation index $η$ over the Bessel functions ${\rm J}_n(η)$.
*The spectral lines are real in the case of a sinusoidal source signal and a cosinusoidal carrier and symmetric about $f_{\rm T}$ for even $n$.
*For odd $n$ , a sign change according to the $\text{Property (B)}$ has to be considered.
*The phase modulation of another phase oscillation of source and / or carrier signal provides the same magnitude spectrum.}}
 

===Applications of the Bessel Functions===

The applications of the Bessel functions in nature–and engineering are diverse and play an important role in physics, for example:
* Examination of natural vibrations of cylindrical resonators,
* Solution of the radial Schrödinger-equation,
* Sound pressure amplitudes of low-viscosity rotational flows,
* Heat conduction in cylindrical bodies,
* Scattering problem of a grid,
* Dynamics of oscillating bodies,
* Angular resolution.

The Bessel functions are counted among the special functions because of their many applications in mathematical physics.

In the following, we restrict ourselves to a few areas that are addressed in our tutorial $\rm LNTwww$.

'''In the English original'''
Electromagnetic waves in a cylindrical waveguide
Pressure amplitudes of inviscid rotational flows
Heat conduction in a cylindrical object
Modes of vibration of a thin circular (or annular) acoustic membrane (such as a drum or other membranophone)
Diffusion problems on a lattice
Solutions to the radial Schrödinger equation (in spherical and cylindrical coordinates) for a free particle
Solving for patterns of acoustical radiation
Frequency-dependent friction in circular pipelines
Dynamics of floating bodies
Angular resolution
'''The Ende'''

{{GraueBox|TEXT=
$\text{Example (C):} \hspace{0.5cm} \text{Use in spectral analysis} \ \Rightarrow \ \text{Kaiser-Bessel filter}$

The '''spectral leakage effect''' is the falsification of the spectrum of a periodic and hence temporally unlimited signal due to the implicit time limitation of the discrete Fourier transform (DFT). This will be done, for example, by a spectrum analyzer
* in the time signal not existing frequency components simulated, and/or
* actually existing spectral components are obscured by sidelobes.

The task of the [[Signaldarstellung/Spektralanalyse|spectral analysis]] is to limit the influence of the ''spectral leakage effect'' by providing suitable window functions.

Such a window function provides, for example, the Kaiser–Bessel–window   ⇒   see section [[Signaldarstellung/Spektralanalyse#Spezielle_Fensterfunktionen|Special Window Functions]]. Its time-discrete Fenser function reads with the Bessel function zero order   ⇒   ${\rm J}_0(x)$, the parameter $\alpha=3.5$ and the window length $N$:
:$$w_\nu = \frac{ {\rm J}_0\big(\pi \cdot \alpha \cdot \sqrt{1 - (2\nu/N)^2}\big)}{ {\rm J}_0\big(\pi \cdot \alpha \big)}.$$
On the page [[Signaldarstellung/Spektralanalyse#G.C3.BCtekriterien_von_Fensterfunktionen|Quality Criteria of Window Functions]] and other the characteristics of the Kaiser-Bessel-window are given:
*Conveniently, the large is &bdquo;minimum distance between the main lobe and side lobes” and the desired small &bdquo;maximum scaling error”.
*Due to the very large &bdquo;equivalent noise width” the Kaiser-Bessel window cuts in the main comparison criterion &bdquo;maximum process loss” but worse off than the established Hamming– and Hanning–windows.}}

{{GraueBox|TEXT=
$\text{Example (D):} \hspace{0.5cm} \text{Rice-Fading channel model}$

The [[Mobile_Kommunikation/Wahrscheinlichkeitsdichte_des_Rayleigh%E2%80%93Fadings#Allgemeine_Beschreibung_des_Mobilfunkkanals| Rayleigh - Distribution]] describes the mobile channel assuming that there is no direct path, and thus the multiplicative factor $z(t) = x(t) + {\rm j} \cdot y(t)$ is solely diffuse composed of scattered components.

In the case of a direct component (English: Line of Sight, LoS) in the model for the mean-free Gaussian processes $x(t)$ and $y(t)$ one has to add equal components $x_0$ and/or $y_0$ :

[[File:P ID2126 Mob T 1 4 S1 v3.png|right|frame|Rice-Fading channel model|class=fit]]
$$\hspace{0.2cm}x(t) \hspace{0.1cm} \Rightarrow \hspace{0.1cm} x(t) +x_0 \hspace{0.05cm}, \hspace{0.2cm} y(t) \hspace{0.1cm} \Rightarrow \hspace{0.1cm} y(t) +y_0\hspace{0.05cm},$$

$$\hspace{0.2cm}z(t) = x(t) + {\rm j} \cdot y(t) \hspace{0.1cm} \Rightarrow \hspace{0.1cm} z(t) +z_0 \hspace{0.05cm},\hspace{0.2cm}
z_0 = x_0 + {\rm j} \cdot y_0\hspace{0.05cm}.$$

The graph shows the ''Rice–Fading–Channel model''. It can be summarized as follows:
*The real part $x(t)$ is Gaussianized with mean $x_0$ and variance $\sigma ^2$.
*The imaginary part $y(t)$ is also Gaussian (mean $y_0$, equal variance $\sigma ^2$) and independent of $x(t)$. 

*For $z_0 \ne 0$, the amount $\vert z(t)\vert$ rice is distributed, from which the term &bdquo;Rice–Fading” arises.

*To simplify the notation, we set $\vert z(t)\vert = a(t)$. For $a < 0$ , the amounts - WDF are $f_a(a) \equiv 0$, for $a \ge 0$ the following equation holds, where ${\rm I_0}(x)$ is the modified Besselfunktion zeroth order denotes:
:$$f_a(a) = \frac{a}{\sigma^2} \cdot {\rm e}^{ - (a^2 + \vert z_0 \vert ^2)/(2\sigma^2)} \cdot {\rm I}_0 \left [ \frac{a \cdot \vert z_0 \vert}{\sigma^2} \right ] \hspace{0.5cm}\text{with}\hspace{0.5cm}{\rm I }_0 (u) = {\rm J }_0 ({\rm j} \cdot u) =
\sum_{k = 0}^{\infty} \frac{ (u/2)^{2k} }{k! \cdot \Gamma (k+1)}
\hspace{0.05cm}.$$
*Between the modified Bessel function and the traditional Bessel function ${\rm I_0}(x)$ – each first type – So the connection exists ${\rm I }_0 (u) = {\rm J }_0 ({\rm j} \cdot u)$.}}

{{GraueBox|TEXT=
$\text{Example (E):} \hspace{0.5cm} \text{Analysis of the frequency spectrum of frequency modulated signals}$

$\text{Example (B)}$ has already shown that the angle modulation of a harmonic oscillation of the frequency $f_{\rm N}$ leads to a line spectrum. The spectral lines are around the carrier frequency $f_{\rm T}$ bei $f_{\rm T} + n \cdot f_{\rm N}$ with $n \in \{ \ \text{...}, -2, -1, \ 0, +1, +2, \text{...} \ \}$. The weights of the Dirac lines are ${\rm J }_n(\eta)$, depending on the modulation index $\eta$.

[[File:P_ID1095__Mod_T_3_2_S4_neu.png|center|frame|Discrete spectra with phase modulation (left) and frequency modulation (right)]]

The graph shows the magnitude spectrum $\vert S_{\rm +}(f) \vert$ of the phase modulation (PM) and frequency modulation (FM) analytical signal, two different forms of angle modulation (WM). Bessellines with values less than $0.03$ are neglected in both cases.

For the upper half of the image, the modulator parameters are chosen so that for each $f_{\rm N} = 5 \ \rm kHz$ here is a Bessel spectrum with the modulation index $η = 1.5$. Disregarding the phase relationships, the same spectra and the same signals result for both systems.

The lower graphics are valid for the same message frequency $f_{\rm N} = 3 \ \rm kHz$. One notices:
*In phase modulation, the spectral function is narrower than $f_{\rm N} = 5 \ \rm kHz$, because the distance of the Bessel lines is now only $3 \ \rm kHz$. Since the modulation index of PM is independent of $f_{\rm N}$, the same bessel weights result as in $f_{\rm N} = 5 \ \rm kHz$.
*Also in the frequency modulation, the Bessel lines now appear at a distance of $3 \ \rm kHz$. However, since FM has a modulation index inversely proportional to $f_{\rm N}$, there is now much more Besselline lines than in the upper right (for $η = 1.5$ valid) chart due to the larger modulation index $η = 2.5$.}}

'''Das folgende Kapitel muss noch angepasst werden!'''

==Applet Manual==
[[File:Handhabung_binomial.png|left|600px]]
    '''(A)'''     Vorauswahl für blauen Parametersatz

    '''(B)'''     Parametereingabe $I$ und $p$ per Slider

    '''(C)'''     Vorauswahl für roten Parametersatz

    '''(D)'''     Parametereingabe $\lambda$ per Slider

    '''(E)'''     Graphische Darstellung der Verteilungen

    '''(F)'''     Momentenausgabe für blauen Parametersatz

    '''(G)'''     Momentenausgabe für roten Parametersatz

    '''(H)'''     Variation der grafischen Darstellung

$\hspace{1.5cm}$&bdquo;$+$” (Vergrößern),

$\hspace{1.5cm}$ &bdquo;$-$” (Verkleinern)

$\hspace{1.5cm}$ &bdquo;$\rm o$” (Zurücksetzen)

$\hspace{1.5cm}$ &bdquo;$\leftarrow$” (Verschieben nach links), usw.

    '''( I )'''     Ausgabe von ${\rm Pr} (z = \mu)$ und ${\rm Pr} (z \le \mu)$

    '''(J)'''     Bereich für die Versuchsdurchführung
 
 '''Andere Möglichkeiten zur Variation der grafischen Darstellung''':
*Gedrückte Shifttaste und Scrollen: Zoomen im Koordinatensystem,
*Gedrückte Shifttaste und linke Maustaste: Verschieben des Koordinatensystems.

==About the Authors==
This interactive calculation was designed and realized at the [http://www.lnt.ei.tum.de/startseite Lehrstuhl für Nachrichtentechnik] of the [https://www.tum.de/ Technical University of Munich] .
*The original version was created in 2005 by [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Studierende#Markus_Elsberger_.28Diplomarbeit_LB_2006.29|Markus Elsberger]] and [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Studierende#Slim_Lamine_.28Studienarbeit_EI_2006.29|Slim Lamine]] as part of thesis using &bdquo;FlashMX–Actionscript” (Supervisor: [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Mitarbeiter_und_Dozenten#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28am_LNT_seit_1974.29|Günter Söder]]).
*In 2018 this Applet was redesigned and updated to &bdquo;HTML5” by [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Studierende#Xiaohan_Liu_.28Bachelorarbeit_2018.29|Xiaohan Liu]] as part of her Bachelor's thesis (Supervisor: [[Biografien_und_Bibliografien/Beteiligte_der_Professur_Leitungsgebundene_%C3%9Cbertragungstechnik#Tasn.C3.A1d_Kernetzky.2C_M.Sc._.28bei_L.C3.9CT_seit_2014.29|Tasnád Kernetzky]]).

==Nochmalige Aufrufmöglichkeit des Applets in neuem Fenster==

{{LntAppletLink|bessel}}

Applets:Bessel Functions of the First Kind

2018-08-22T14:51:06Z

Xiaohan: /* Theoretical Background */

{{LntAppletLink|bessel}}

==Applet Description==
 
This applet allows the calculation and graphical representation of the Bessel functions of the first kind and $n$–th order according to the series representation:

:$${\rm J}_n (x) = \sum\limits_{k=0}^{\infty}\frac{(-1)^k \cdot (x/2)^{n \hspace{0.05cm} + \hspace{0.05cm} 2
\hspace{0.02cm}\cdot \hspace{0.05cm}k}}{k! \cdot (n+k)!} \hspace{0.05cm}.$$

*Graphically represented, the functions ${\rm J}_n (x)$ for the order $n=0$ to $n=9$ can become different colors.
*The left-hand output provides the function values ${\rm J}_0 (x = x_1)$, ... , ${\rm J}_9 (x = x_1)$ for a slider-settable value $x_1$ in the range $0 \le x_1 \le 15$ with increment $0.5$.
*The right hand side returns the function values ${\rm J}_0 (x = x_2)$, ... , ${\rm J}_9 (x = x_2)$ for a slider-settable value $x_2$ (same range and value) Increment as on the left).

[[Applets:Physical_Signal_%26_Analytical_Signal|'''German description''']]

==Theoretical Background==
 
===General Information about the Bessel Functions===
Bessel functions (or cylinder functions) are solutions of the Bessel differential equation of the form

:$$x^2 \cdot \frac{ {\rm d}^2}{{\rm d}x^2}\ {\rm J}_n (x) \ + \ x \cdot \frac{ {\rm d}}{{\rm d}x}\ {\rm J}_n (x) \ + \ (x^2 - n^2)
\cdot {\rm J}_n (x)= 0. $$

This is an ordinary second-order linear differential equation. The parameter $ n $ is usually integer, as in this program. These mathematical functions, which were introduced in 1844 by [https://de.wikipedia.org/wiki/Friedrich_Wilhelm_Bessel Friedrich Wilhelm Bessel], can also be represented in closed form as integrals:

:$${\rm J}_n (x) = \frac{1}{2\pi}\cdot \int_{-\pi}^{+\pi} {{\rm e}^{\hspace{0.05cm}{\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}[\hspace{0.05cm}x \hspace{0.05cm}\cdot \hspace{0.05cm}\sin(\alpha) -\hspace{0.05cm} n \hspace{0.05cm}\cdot \hspace{0.05cm}\alpha \hspace{0.05cm}]}}\hspace{0.1cm}{\rm d}\alpha
\hspace{0.05cm}.$$

The functions ${\rm J}_n (x)$ belong to the class of Bessel functions of the first kind (German:   ''Besselfunktionen erster Art''). The parameter $n$ is called the ''Order''.

''Annotation:''   There are a number of modifications of the Bessel functions, including the second-order Bessel functions named ${\rm Y}_n (x)$. For integer $n$, ${\rm Y}_n (x)$ can be replaced by ${\rm J}_n (x)$–functions. However, in this applet, only the first-order Bessel functions are   ⇒   ${\rm J}_n (x)$ is considered.
 
===Properties of the Bessel Functions===

{{BlaueBox|TEXT=
$\text{Property (A):}$   If the function values for $n = 0$ and $n = 1$ are known, then the Bessel function for $n ≥ 2$ can be determined iteratively:
:$${\rm J}_n (x) ={2 \cdot (n-1)}/{x} \cdot {\rm J}_{n-1} (x) - {\rm J}_{n-2} (x) \hspace{0.05cm}.$$
}}

{{GraueBox|TEXT=
$\text{Example (A):}$   Consider ${\rm J}_0 (x = 2) = 0.22389$ and ${\rm J}_1 (x= 2) = 0.57672$. From this it can be calculated iteratively:
:$${\rm J}_2 (x= 2) ={2 \cdot 1}/{2} \cdot {\rm J}_{1} (x= 2) - {\rm J}_{0} (x= 2) = 0.57672 - 0.22389 = \hspace{0.15cm}\underline{0.35283}\hspace{0.05cm},$$
:$${\rm J}_3 (x= 2) ={2 \cdot 2}/{2} \cdot {\rm J}_{2} (x= 2) - {\rm J}_{1} (x= 2) = 2 \cdot 0.35283 - 0.57672 = \hspace{0.15cm}\underline{0.12894}\hspace{0.05cm},$$
:$${\rm J}_4 (x= 2) ={2 \cdot 3}/{2} \cdot {\rm J}_{3} (x= 2) - {\rm J}_{2} (x= 2) = 3 \cdot 0.12894 - 0.35283 = \hspace{0.15cm}\underline{0.03400}\hspace{0.05cm}.$$
}}

{{BlaueBox|TEXT=
$\text{Property (B):}$   The symmetry relationship applies ${\rm J}_{–n}(x) = (–1)^n · {\rm J}_n(x)$:
:$${\rm J}_{-1}(x) = - {\rm J}_{1}(x), \hspace{0.3cm}{\rm J}_{-2}(x) = {\rm J}_{2}(x), \hspace{0.3cm}{\rm J}_{-3}(x) = - {\rm J}_{3}(x).$$
}}

{{GraueBox|TEXT=
$\text{Example (B):}$   For the spectrum of the analytic signal, in [[Modulationsverfahren/Phasenmodulation_(PM)#Spektralfunktion_eines_phasenmodulierten_Sinussignals|phase modulation of a sinusoidal signal]]:
[[File:Mod_T_3_1_S4_version2.png|right|frame|Spectrum of the analytic signal in phase modulation]]
:$$S_{\rm +}(f) = A_{\rm T} \cdot \sum_{n = - \infty}^{+\infty}{\rm J}_n (\eta) \cdot \delta \big[f - (f_{\rm T}+ n \cdot f_{\rm N})\big]\hspace{0.05cm}.$$
Denote this
*$f_{\rm T}$ the carrier frequency,
*$f_{\rm N}$ the message frequency,
* $A_{\rm T}$ the carrier amplitude.

The parameter of the Bessel functions in this application is the modulation index $\eta$.

Based on the graphic, the following statements are possible:
*Here $S_+(f)$ consists of an infinite number of discrete lines at a distance of $f_{\rm N}$.
*It is thus in principle infinitely extended.
*The weights of the spectral lines at $f_{\rm T} + n · f_{\rm N}$ ($n$ integer) are determined by the modulation index $η$ over the Bessel functions ${\rm J}_n(η)$.
*The spectral lines are real in the case of a sinusoidal source signal and a cosinusoidal carrier and symmetric about $f_{\rm T}$ for even $n$.
*For odd $n$ , a sign change according to the $\text{Property (B)}$ has to be considered.
*The phase modulation of another phase oscillation of source and / or carrier signal provides the same magnitude spectrum.}}
 

===Applications of the Bessel Functions===

The applications of the Bessel functions in nature–and engineering are diverse and play an important role in physics, for example:
* Examination of natural vibrations of cylindrical resonators,
* Solution of the radial Schrödinger-equation,
* Sound pressure amplitudes of low-viscosity rotational flows,
* Heat conduction in cylindrical bodies,
* Scattering problem of a grid,
* Dynamics of oscillating bodies,
* Angular resolution.

The Bessel functions are counted among the special functions because of their many applications in mathematical physics.

In the following, we restrict ourselves to a few areas that are addressed in our tutorial $\rm LNTwww$.

'''In the English original'''
Electromagnetic waves in a cylindrical waveguide
Pressure amplitudes of inviscid rotational flows
Heat conduction in a cylindrical object
Modes of vibration of a thin circular (or annular) acoustic membrane (such as a drum or other membranophone)
Diffusion problems on a lattice
Solutions to the radial Schrödinger equation (in spherical and cylindrical coordinates) for a free particle
Solving for patterns of acoustical radiation
Frequency-dependent friction in circular pipelines
Dynamics of floating bodies
Angular resolution
'''The Ende'''

{{GraueBox|TEXT=
$\text{Example (C):} \hspace{0.5cm} \text{Use in spectral analysis} \ \Rightarrow \ \text{Kaiser-Bessel filter}$

The '''spectral leakage effect''' is the falsification of the spectrum of a periodic and hence temporally unlimited signal due to the implicit time limitation of the discrete Fourier transform (DFT). This will be done, for example, by a spectrum analyzer
* in the time signal not existing frequency components simulated, and/or
* actually existing spectral components are obscured by sidelobes.

The task of the [[Signaldarstellung/Spektralanalyse|spectral analysis]] is to limit the influence of the ''spectral leakage effect'' by providing suitable window functions.

Such a window function provides, for example, the Kaiser–Bessel–window   ⇒   see section [[Signaldarstellung/Spektralanalyse#Spezielle_Fensterfunktionen|Special Window Functions]]. Its time-discrete Fenser function reads with the Bessel function zero order   ⇒   ${\rm J}_0(x)$, the parameter $\alpha=3.5$ and the window length $N$:
:$$w_\nu = \frac{ {\rm J}_0\big(\pi \cdot \alpha \cdot \sqrt{1 - (2\nu/N)^2}\big)}{ {\rm J}_0\big(\pi \cdot \alpha \big)}.$$
On the page [[Signaldarstellung/Spektralanalyse#G.C3.BCtekriterien_von_Fensterfunktionen|Quality Criteria of Window Functions]] and other the characteristics of the Kaiser-Bessel-window are given:
*Conveniently, the large is &bdquo;minimum distance between the main lobe and side lobes” and the desired small &bdquo;maximum scaling error”.
*Due to the very large &bdquo;equivalent noise width” the Kaiser-Bessel window cuts in the main comparison criterion &bdquo;maximum process loss” but worse off than the established Hamming– and Hanning–windows.}}

{{GraueBox|TEXT=
$\text{Example (D):} \hspace{0.5cm} \text{Rice-Fading channel model}$

The [[Mobile_Kommunikation/Wahrscheinlichkeitsdichte_des_Rayleigh%E2%80%93Fadings#Allgemeine_Beschreibung_des_Mobilfunkkanals| Rayleigh - Distribution]] describes the mobile channel assuming that there is no direct path, and thus the multiplicative factor $z(t) = x(t) + {\rm j} \cdot y(t)$ is solely diffuse composed of scattered components.

In the case of a direct component (English: Line of Sight, LoS) in the model for the mean-free Gaussian processes $x(t)$ and $y(t)$ one has to add equal components $x_0$ and/or $y_0$ :

[[File:P ID2126 Mob T 1 4 S1 v3.png|right|frame|Rice-Fading channel model|class=fit]]
$$\hspace{0.2cm}x(t) \hspace{0.1cm} \Rightarrow \hspace{0.1cm} x(t) +x_0 \hspace{0.05cm}, \hspace{0.2cm} y(t) \hspace{0.1cm} \Rightarrow \hspace{0.1cm} y(t) +y_0\hspace{0.05cm},$$

$$\hspace{0.2cm}z(t) = x(t) + {\rm j} \cdot y(t) \hspace{0.1cm} \Rightarrow \hspace{0.1cm} z(t) +z_0 \hspace{0.05cm},\hspace{0.2cm}
z_0 = x_0 + {\rm j} \cdot y_0\hspace{0.05cm}.$$

The graph shows the ''Rice–Fading–Channel model''. It can be summarized as follows:
*The real part $x(t)$ is Gaussianized with mean $x_0$ and variance $\sigma ^2$.
*The imaginary part $y(t)$ is also Gaussian (mean $y_0$, equal variance $\sigma ^2$) and independent of $x(t)$. 

*For $z_0 \ne 0$, the amount $\vert z(t)\vert$ rice is distributed, from which the term &bdquo;Rice–Fading” arises.

*To simplify the notation, we set $\vert z(t)\vert = a(t)$. For $a < 0$ , the amounts - WDF are $f_a(a) \equiv 0$, for $a \ge 0$ the following equation holds, where ${\rm I_0}(x)$ is the modified Besselfunktion zeroth order denotes:
:$$f_a(a) = \frac{a}{\sigma^2} \cdot {\rm e}^{ - (a^2 + \vert z_0 \vert ^2)/(2\sigma^2)} \cdot {\rm I}_0 \left [ \frac{a \cdot \vert z_0 \vert}{\sigma^2} \right ] \hspace{0.5cm}\text{with}\hspace{0.5cm}{\rm I }_0 (u) = {\rm J }_0 ({\rm j} \cdot u) =
\sum_{k = 0}^{\infty} \frac{ (u/2)^{2k} }{k! \cdot \Gamma (k+1)}
\hspace{0.05cm}.$$
*Between the modified Bessel function and the traditional Bessel function ${\rm I_0}(x)$ – each first type – So the connection exists ${\rm I }_0 (u) = {\rm J }_0 ({\rm j} \cdot u)$.}}

{{GraueBox|TEXT=
$\text{Example (E):} \hspace{0.5cm} \text{Analysis of the frequency spectrum of frequency modulated signals}$

$\text{Example (B)}$ has already shown that the angle modulation of a harmonic oscillation of the frequency $f_{\rm N}$ leads to a line spectrum. The spectral lines are around the carrier frequency $f_{\rm T}$ bei $f_{\rm T} + n \cdot f_{\rm N}$ with $n \in \{ \ \text{...}, -2, -1, \ 0, +1, +2, \text{...} \ \}$. The weights of the Dirac lines are ${\rm J }_n(\eta)$, depending on the modulation index $\eta$.

[[File:P_ID1095__Mod_T_3_2_S4_neu.png|center|frame|Discrete spectra with phase modulation (left) and frequency modulation (right)]]

The graph shows the magnitude spectrum $\vert S_{\rm +}(f) \vert$ of the phase modulation (PM) and frequency modulation (FM) analytical signal, two different forms of angle modulation (WM). Bessellines with values less than $0.03$ are neglected in both cases.

For the upper half of the image, the modulator parameters are chosen so that for each $f_{\rm N} = 5 \ \rm kHz$ here is a Bessel spectrum with the modulation index $η = 1.5$. Disregarding the phase relationships, the same spectra and the same signals result for both systems.

The lower graphics are valid for the same message frequency $f_{\rm N} = 3 \ \rm kHz$. One notices:
*In phase modulation, the spectral function is narrower than $f_{\rm N} = 5 \ \rm kHz$, because the distance of the Bessel lines is now only $3 \ \rm kHz$. Since the modulation index of PM is independent of $f_{\rm N}$, the same bessel weights result as in $f_{\rm N} = 5 \ \rm kHz$.
*Also in the frequency modulation, the Bessel lines now appear at a distance of $3 \ \rm kHz$. However, since FM has a modulation index inversely proportional to $f_{\rm N}$, there is now much more Besselline lines than in the upper right (for $η = 1.5$ valid) chart due to the larger modulation index $η = 2.5$.}}

'''Das folgende Kapitel muss noch angepasst werden!'''

==Zur Handhabung des Applets==
[[File:Handhabung_binomial.png|left|600px]]
    '''(A)'''     Vorauswahl für blauen Parametersatz

    '''(B)'''     Parametereingabe $I$ und $p$ per Slider

    '''(C)'''     Vorauswahl für roten Parametersatz

    '''(D)'''     Parametereingabe $\lambda$ per Slider

    '''(E)'''     Graphische Darstellung der Verteilungen

    '''(F)'''     Momentenausgabe für blauen Parametersatz

    '''(G)'''     Momentenausgabe für roten Parametersatz

    '''(H)'''     Variation der grafischen Darstellung

$\hspace{1.5cm}$&bdquo;$+$” (Vergrößern),

$\hspace{1.5cm}$ &bdquo;$-$” (Verkleinern)

$\hspace{1.5cm}$ &bdquo;$\rm o$” (Zurücksetzen)

$\hspace{1.5cm}$ &bdquo;$\leftarrow$” (Verschieben nach links), usw.

    '''( I )'''     Ausgabe von ${\rm Pr} (z = \mu)$ und ${\rm Pr} (z \le \mu)$

    '''(J)'''     Bereich für die Versuchsdurchführung
 
 '''Andere Möglichkeiten zur Variation der grafischen Darstellung''':
*Gedrückte Shifttaste und Scrollen: Zoomen im Koordinatensystem,
*Gedrückte Shifttaste und linke Maustaste: Verschieben des Koordinatensystems.

==About the Authors==
This interactive calculation was designed and realized at the [http://www.lnt.ei.tum.de/startseite Lehrstuhl für Nachrichtentechnik] of the [https://www.tum.de/ Technical University of Munich] .
*The original version was created in 2005 by [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Studierende#Markus_Elsberger_.28Diplomarbeit_LB_2006.29|Markus Elsberger]] and [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Studierende#Slim_Lamine_.28Studienarbeit_EI_2006.29|Slim Lamine]] as part of thesis using &bdquo;FlashMX–Actionscript” (Supervisor: [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Mitarbeiter_und_Dozenten#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28am_LNT_seit_1974.29|Günter Söder]]).
*In 2018 this Applet was redesigned and updated to &bdquo;HTML5” by [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Studierende#Xiaohan_Liu_.28Bachelorarbeit_2018.29|Xiaohan Liu]] as part of her Bachelor's thesis (Supervisor: [[Biografien_und_Bibliografien/Beteiligte_der_Professur_Leitungsgebundene_%C3%9Cbertragungstechnik#Tasn.C3.A1d_Kernetzky.2C_M.Sc._.28bei_L.C3.9CT_seit_2014.29|Tasnád Kernetzky]]).

==Nochmalige Aufrufmöglichkeit des Applets in neuem Fenster==

{{LntAppletLink|bessel}}

Applets:Bessel Functions of the First Kind

2018-08-21T21:16:00Z

Xiaohan: /* About the Authors */

{{LntAppletLink|bessel}}

==Applet Description==
 
This applet allows the calculation and graphical representation of the Bessel functions of the first kind and $n$–th order according to the series representation:

:$${\rm J}_n (x) = \sum\limits_{k=0}^{\infty}\frac{(-1)^k \cdot (x/2)^{n \hspace{0.05cm} + \hspace{0.05cm} 2
\hspace{0.02cm}\cdot \hspace{0.05cm}k}}{k! \cdot (n+k)!} \hspace{0.05cm}.$$

*Graphically represented, the functions ${\rm J}_n (x)$ for the order $n=0$ to $n=9$ can become different colors.
*The left-hand output provides the function values ${\rm J}_0 (x = x_1)$, ... , ${\rm J}_9 (x = x_1)$ for a slider-settable value $x_1$ in the range $0 \le x_1 \le 15$ with increment $0.5$.
*The right hand side returns the function values ${\rm J}_0 (x = x_2)$, ... , ${\rm J}_9 (x = x_2)$ for a slider-settable value $x_2$ (same range and value) Increment as on the left).

[[Applets:Physical_Signal_%26_Analytical_Signal|'''German description''']]

==Theoretical Background==
 
===General Information about the Bessel Functions===
Bessel functions (or cylinder functions) are solutions of the Bessel differential equation of the form

:$$x^2 \cdot \frac{ {\rm d}^2}{{\rm d}x^2}\ {\rm J}_n (x) \ + \ x \cdot \frac{ {\rm d}}{{\rm d}x}\ {\rm J}_n (x) \ + \ (x^2 - n^2)
\cdot {\rm J}_n (x)= 0. $$

This is an ordinary second-order linear differential equation. The parameter $ n $ is usually integer, as in this program. These mathematical functions, which were introduced in 1844 by [https://de.wikipedia.org/wiki/Friedrich_Wilhelm_Bessel Friedrich Wilhelm Bessel], can also be represented in closed form as integrals:

:$${\rm J}_n (x) = \frac{1}{2\pi}\cdot \int_{-\pi}^{+\pi} {{\rm e}^{\hspace{0.05cm}{\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}[\hspace{0.05cm}x \hspace{0.05cm}\cdot \hspace{0.05cm}\sin(\alpha) -\hspace{0.05cm} n \hspace{0.05cm}\cdot \hspace{0.05cm}\alpha \hspace{0.05cm}]}}\hspace{0.1cm}{\rm d}\alpha
\hspace{0.05cm}.$$

The functions ${\rm J}_n (x)$ belong to the class of Bessel functions of the first kind (German:   ''Besselfunktionen erster Art''). The parameter $n$ is called the ''Order''.

''Annotation:''   There are a number of modifications of the Bessel functions, including the second-order Bessel functions named ${\rm Y}_n (x)$. For integer $n$, ${\rm Y}_n (x)$ can be replaced by ${\rm J}_n (x)$–functions. However, in this applet, only the first-order Bessel functions are   ⇒   ${\rm J}_n (x)$ is considered.
 
===Properties of the Bessel Functions===

{{BlaueBox|TEXT=
$\text{Property (A):}$   If the function values for $n = 0$ and $n = 1$ are known, then the Bessel function for $n ≥ 2$ can be determined iteratively:
:$${\rm J}_n (x) ={2 \cdot (n-1)}/{x} \cdot {\rm J}_{n-1} (x) - {\rm J}_{n-2} (x) \hspace{0.05cm}.$$
}}

{{GraueBox|TEXT=
$\text{Example (A):}$   Consider ${\rm J}_0 (x = 2) = 0.22389$ and ${\rm J}_1 (x= 2) = 0.57672$. From this it can be calculated iteratively:
:$${\rm J}_2 (x= 2) ={2 \cdot 1}/{2} \cdot {\rm J}_{1} (x= 2) - {\rm J}_{0} (x= 2) = 0.57672 - 0.22389 = \hspace{0.15cm}\underline{0.35283}\hspace{0.05cm},$$
:$${\rm J}_3 (x= 2) ={2 \cdot 2}/{2} \cdot {\rm J}_{2} (x= 2) - {\rm J}_{1} (x= 2) = 2 \cdot 0.35283 - 0.57672 = \hspace{0.15cm}\underline{0.12894}\hspace{0.05cm},$$
:$${\rm J}_4 (x= 2) ={2 \cdot 3}/{2} \cdot {\rm J}_{3} (x= 2) - {\rm J}_{2} (x= 2) = 3 \cdot 0.12894 - 0.35283 = \hspace{0.15cm}\underline{0.03400}\hspace{0.05cm}.$$
}}

{{BlaueBox|TEXT=
$\text{Property (B):}$   The symmetry relationship applies ${\rm J}_{–n}(x) = (–1)^n · {\rm J}_n(x)$:
:$${\rm J}_{-1}(x) = - {\rm J}_{1}(x), \hspace{0.3cm}{\rm J}_{-2}(x) = {\rm J}_{2}(x), \hspace{0.3cm}{\rm J}_{-3}(x) = - {\rm J}_{3}(x).$$
}}

{{GraueBox|TEXT=
$\text{Example (B):}$   For the spectrum of the analytic signal, in [[Modulationsverfahren/Phasenmodulation_(PM)#Spektralfunktion_eines_phasenmodulierten_Sinussignals|phase modulation of a sinusoidal signal]]:
[[File:Mod_T_3_1_S4_version2.png|right|frame|Spectrum of the analytic signal in phase modulation]]
:$$S_{\rm +}(f) = A_{\rm T} \cdot \sum_{n = - \infty}^{+\infty}{\rm J}_n (\eta) \cdot \delta \big[f - (f_{\rm T}+ n \cdot f_{\rm N})\big]\hspace{0.05cm}.$$
Denote this
*$f_{\rm T}$ the carrier frequency,
*$f_{\rm N}$ the message frequency,
* $A_{\rm T}$ the carrier amplitude.

The parameter of the Bessel functions in this application is the modulation index $\eta$.

Based on the graphic, the following statements are possible:
*Here $S_+(f)$ consists of an infinite number of discrete lines at a distance of $f_{\rm N}$.
*It is thus in principle infinitely extended.
*The weights of the spectral lines at $f_{\rm T} + n · f_{\rm N}$ ($n$ integer) are determined by the modulation index $η$ over the Bessel functions ${\rm J}_n(η)$.
*The spectral lines are real in the case of a sinusoidal source signal and a cosinusoidal carrier and symmetric about $f_{\rm T}$ for even $n$.
*For odd $n$ , a sign change according to the $\text{Property (B)}$ has to be considered.
*The phase modulation of another phase oscillation of source and / or carrier signal provides the same magnitude spectrum.}}
 

===Applications of the Bessel Functions===

The applications of the Bessel functions in nature–and engineering are diverse and play an important role in physics, for example:
* Examination of natural vibrations of cylindrical resonators,
* Solution of the radial Schrödinger-equation,
* Sound pressure amplitudes of low-viscosity rotational flows,
* Heat conduction in cylindrical bodies,
* Scattering problem of a grid,
* Dynamics of oscillating bodies,
* Angular resolution.

The Bessel functions are counted among the special functions because of their many applications in mathematical physics.

In the following, we restrict ourselves to a few areas that are addressed in our tutorial $\rm LNTwww$.

'''In the English original'''
Electromagnetic waves in a cylindrical waveguide
Pressure amplitudes of inviscid rotational flows
Heat conduction in a cylindrical object
Modes of vibration of a thin circular (or annular) acoustic membrane (such as a drum or other membranophone)
Diffusion problems on a lattice
Solutions to the radial Schrödinger equation (in spherical and cylindrical coordinates) for a free particle
Solving for patterns of acoustical radiation
Frequency-dependent friction in circular pipelines
Dynamics of floating bodies
Angular resolution
'''The Ende'''

{{GraueBox|TEXT=
$\text{Example (C):} \hspace{0.5cm} \text{Use in spectral analysis} \ \Rightarrow \ \text{Kaiser-Bessel filter}$

The '''spectral leakage effect''' is the falsification of the spectrum of a periodic and hence temporally unlimited signal due to the implicit time limitation of the discrete Fourier transform (DFT). This will be done, for example, by a spectrum analyzer
* in the time signal not existing frequency components simulated, and / or
* actually existing spectral components are obscured by sidelobes.

The task of the [[Signaldarstellung/Spektralanalyse|spectral analysis]] is to limit the influence of the ''spectral leakage effect'' by providing suitable window functions.

Such a window function provides, for example, the Kaiser–Bessel–window   ⇒   see section [[Signaldarstellung/Spektralanalyse#Spezielle_Fensterfunktionen|Special Window Functions]]. Its time-discrete Fenser function reads with the Bessel function zero order   ⇒   ${\rm J}_0(x)$, the parameter $\alpha=3.5$ and the window length $N$:
:$$w_\nu = \frac{ {\rm J}_0\big(\pi \cdot \alpha \cdot \sqrt{1 - (2\nu/N)^2}\big)}{ {\rm J}_0\big(\pi \cdot \alpha \big)}.$$
On the page [[Signaldarstellung/Spektralanalyse#G.C3.BCtekriterien_von_Fensterfunktionen|Quality Criteria of Window Functions]] and other the characteristics of the Kaiser-Bessel-window are given:
*Conveniently, the large is &bdquo;minimum distance between the main lobe and side lobes” and the desired small &bdquo;maximum scaling error”.
*Due to the very large &bdquo;equivalent noise width” the Kaiser-Bessel window cuts in the main comparison criterion &bdquo;maximum process loss” but worse off than the established Hamming– and Hanning–windows.}}

{{GraueBox|TEXT=
$\text{Example (D):} \hspace{0.5cm} \text{Rice-Fading channel model}$

The [[Mobile_Kommunikation/Wahrscheinlichkeitsdichte_des_Rayleigh%E2%80%93Fadings#Allgemeine_Beschreibung_des_Mobilfunkkanals| Rayleigh - Distribution]] describes the mobile channel assuming that there is no direct path, and thus the multiplicative factor $z(t) = x(t) + {\rm j} \cdot y(t)$ is solely diffuse composed of scattered components.

In the case of a direct component (English: Line of Sight, LoS) in the model for the mean-free Gaussian processes $x(t)$ and $y(t)$ one has to add equal components $x_0$ and/or $y_0$ :

[[File:P ID2126 Mob T 1 4 S1 v3.png|right|frame|Rice-Fading channel model|class=fit]]
$$\hspace{0.2cm}x(t) \hspace{0.1cm} \Rightarrow \hspace{0.1cm} x(t) +x_0 \hspace{0.05cm}, \hspace{0.2cm} y(t) \hspace{0.1cm} \Rightarrow \hspace{0.1cm} y(t) +y_0\hspace{0.05cm},$$

$$\hspace{0.2cm}z(t) = x(t) + {\rm j} \cdot y(t) \hspace{0.1cm} \Rightarrow \hspace{0.1cm} z(t) +z_0 \hspace{0.05cm},\hspace{0.2cm}
z_0 = x_0 + {\rm j} \cdot y_0\hspace{0.05cm}.$$

The graph shows the ''Rice–Fading–Channel model''. It can be summarized as follows:
*The real part $x(t)$ is Gaussianized with mean $x_0$ and variance $\sigma ^2$.
*The imaginary part $y(t)$ is also Gaussian (mean $y_0$, equal variance $\sigma ^2$) and independent of $x(t)$. 

*For $z_0 \ne 0$, the amount $\vert z(t)\vert$ rice is distributed, from which the term &bdquo;Rice–Fading” arises.

*To simplify the notation, we set $\vert z(t)\vert = a(t)$. For $a < 0$ , the amounts - WDF are $f_a(a) \equiv 0$, for $a \ge 0$ the following equation holds, where ${\rm I_0}(x)$ is the modified Besselfunktion zeroth order denotes:
:$$f_a(a) = \frac{a}{\sigma^2} \cdot {\rm e}^{ - (a^2 + \vert z_0 \vert ^2)/(2\sigma^2)} \cdot {\rm I}_0 \left [ \frac{a \cdot \vert z_0 \vert}{\sigma^2} \right ] \hspace{0.5cm}\text{with}\hspace{0.5cm}{\rm I }_0 (u) = {\rm J }_0 ({\rm j} \cdot u) =
\sum_{k = 0}^{\infty} \frac{ (u/2)^{2k} }{k! \cdot \Gamma (k+1)}
\hspace{0.05cm}.$$
*Between the modified Bessel function and the traditional Bessel function ${\rm I_0}(x)$ – each first type – So the connection exists ${\rm I }_0 (u) = {\rm J }_0 ({\rm j} \cdot u)$.}}

{{GraueBox|TEXT=
$\text{Example (E):} \hspace{0.5cm} \text{Analysis of the frequency spectrum of frequency modulated signals}$

$\text{Example (B)}$ has already shown that the angle modulation of a harmonic oscillation of the frequency $f_{\rm N}$ leads to a line spectrum. The spectral lines are around the carrier frequency $f_{\rm T}$ bei $f_{\rm T} + n \cdot f_{\rm N}$ with $n \in \{ \ \text{...}, -2, -1, \ 0, +1, +2, \text{...} \ \}$. The weights of the Dirac lines are ${\rm J }_n(\eta)$, depending on the modulation index $\eta$.

[[File:P_ID1095__Mod_T_3_2_S4_neu.png|center|frame|Discrete spectra with phase modulation (left) and frequency modulation (right)]]

The graph shows the magnitude spectrum $\vert S_{\rm +}(f) \vert$ of the phase modulation (PM) and frequency modulation (FM) analytical signal, two different forms of angle modulation (WM). Bessellines with values less than $0.03$ are neglected in both cases.

For the upper half of the image, the modulator parameters are chosen so that for each $f_{\rm N} = 5 \ \rm kHz$ here is a Bessel spectrum with the modulation index $η = 1.5$. Disregarding the phase relationships, the same spectra and the same signals result for both systems.

The lower graphics are valid for the same message frequency $f_{\rm N} = 3 \ \rm kHz$. One notices:
*In phase modulation, the spectral function is narrower than $f_{\rm N} = 5 \ \rm kHz$, because the distance of the Bessel lines is now only $3 \ \rm kHz$. Since the modulation index of PM is independent of $f_{\rm N}$, the same bessel weights result as in $f_{\rm N} = 5 \ \rm kHz$.
*Also in the frequency modulation, the Bessel lines now appear at a distance of $3 \ \rm kHz$. However, since FM has a modulation index inversely proportional to $f_{\rm N}$, there is now much more Besselline lines than in the upper right (for $η = 1.5$ valid) chart due to the larger modulation index $η = 2.5$.}}

'''Das folgende Kapitel muss noch angepasst werden!'''

==Zur Handhabung des Applets==
[[File:Handhabung_binomial.png|left|600px]]
    '''(A)'''     Vorauswahl für blauen Parametersatz

    '''(B)'''     Parametereingabe $I$ und $p$ per Slider

    '''(C)'''     Vorauswahl für roten Parametersatz

    '''(D)'''     Parametereingabe $\lambda$ per Slider

    '''(E)'''     Graphische Darstellung der Verteilungen

    '''(F)'''     Momentenausgabe für blauen Parametersatz

    '''(G)'''     Momentenausgabe für roten Parametersatz

    '''(H)'''     Variation der grafischen Darstellung

$\hspace{1.5cm}$&bdquo;$+$” (Vergrößern),

$\hspace{1.5cm}$ &bdquo;$-$” (Verkleinern)

$\hspace{1.5cm}$ &bdquo;$\rm o$” (Zurücksetzen)

$\hspace{1.5cm}$ &bdquo;$\leftarrow$” (Verschieben nach links), usw.

    '''( I )'''     Ausgabe von ${\rm Pr} (z = \mu)$ und ${\rm Pr} (z \le \mu)$

    '''(J)'''     Bereich für die Versuchsdurchführung
 
 '''Andere Möglichkeiten zur Variation der grafischen Darstellung''':
*Gedrückte Shifttaste und Scrollen: Zoomen im Koordinatensystem,
*Gedrückte Shifttaste und linke Maustaste: Verschieben des Koordinatensystems.

==About the Authors==
This interactive calculation was designed and realized at the [http://www.lnt.ei.tum.de/startseite Lehrstuhl für Nachrichtentechnik] of the [https://www.tum.de/ Technical University of Munich] .
*The original version was created in 2005 by [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Studierende#Markus_Elsberger_.28Diplomarbeit_LB_2006.29|Markus Elsberger]] and [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Studierende#Slim_Lamine_.28Studienarbeit_EI_2006.29|Slim Lamine]] as part of thesis using &bdquo;FlashMX–Actionscript” (Supervisor: [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Mitarbeiter_und_Dozenten#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28am_LNT_seit_1974.29|Günter Söder]]).
*In 2018 this Applet was redesigned and updated to &bdquo;HTML5” by [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Studierende#Xiaohan_Liu_.28Bachelorarbeit_2018.29|Xiaohan Liu]] as part of her Bachelor's thesis (Supervisor: [[Biografien_und_Bibliografien/Beteiligte_der_Professur_Leitungsgebundene_%C3%9Cbertragungstechnik#Tasn.C3.A1d_Kernetzky.2C_M.Sc._.28bei_L.C3.9CT_seit_2014.29|Tasnád Kernetzky]]).

==Nochmalige Aufrufmöglichkeit des Applets in neuem Fenster==

{{LntAppletLink|bessel}}

Applets:Bessel Functions of the First Kind

2018-08-21T21:15:44Z

Xiaohan: /* Über die Autoren */

{{LntAppletLink|bessel}}

==Applet Description==
 
This applet allows the calculation and graphical representation of the Bessel functions of the first kind and $n$–th order according to the series representation:

:$${\rm J}_n (x) = \sum\limits_{k=0}^{\infty}\frac{(-1)^k \cdot (x/2)^{n \hspace{0.05cm} + \hspace{0.05cm} 2
\hspace{0.02cm}\cdot \hspace{0.05cm}k}}{k! \cdot (n+k)!} \hspace{0.05cm}.$$

*Graphically represented, the functions ${\rm J}_n (x)$ for the order $n=0$ to $n=9$ can become different colors.
*The left-hand output provides the function values ${\rm J}_0 (x = x_1)$, ... , ${\rm J}_9 (x = x_1)$ for a slider-settable value $x_1$ in the range $0 \le x_1 \le 15$ with increment $0.5$.
*The right hand side returns the function values ${\rm J}_0 (x = x_2)$, ... , ${\rm J}_9 (x = x_2)$ for a slider-settable value $x_2$ (same range and value) Increment as on the left).

[[Applets:Physical_Signal_%26_Analytical_Signal|'''German description''']]

==Theoretical Background==
 
===General Information about the Bessel Functions===
Bessel functions (or cylinder functions) are solutions of the Bessel differential equation of the form

:$$x^2 \cdot \frac{ {\rm d}^2}{{\rm d}x^2}\ {\rm J}_n (x) \ + \ x \cdot \frac{ {\rm d}}{{\rm d}x}\ {\rm J}_n (x) \ + \ (x^2 - n^2)
\cdot {\rm J}_n (x)= 0. $$

This is an ordinary second-order linear differential equation. The parameter $ n $ is usually integer, as in this program. These mathematical functions, which were introduced in 1844 by [https://de.wikipedia.org/wiki/Friedrich_Wilhelm_Bessel Friedrich Wilhelm Bessel], can also be represented in closed form as integrals:

:$${\rm J}_n (x) = \frac{1}{2\pi}\cdot \int_{-\pi}^{+\pi} {{\rm e}^{\hspace{0.05cm}{\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}[\hspace{0.05cm}x \hspace{0.05cm}\cdot \hspace{0.05cm}\sin(\alpha) -\hspace{0.05cm} n \hspace{0.05cm}\cdot \hspace{0.05cm}\alpha \hspace{0.05cm}]}}\hspace{0.1cm}{\rm d}\alpha
\hspace{0.05cm}.$$

The functions ${\rm J}_n (x)$ belong to the class of Bessel functions of the first kind (German:   ''Besselfunktionen erster Art''). The parameter $n$ is called the ''Order''.

''Annotation:''   There are a number of modifications of the Bessel functions, including the second-order Bessel functions named ${\rm Y}_n (x)$. For integer $n$, ${\rm Y}_n (x)$ can be replaced by ${\rm J}_n (x)$–functions. However, in this applet, only the first-order Bessel functions are   ⇒   ${\rm J}_n (x)$ is considered.
 
===Properties of the Bessel Functions===

{{BlaueBox|TEXT=
$\text{Property (A):}$   If the function values for $n = 0$ and $n = 1$ are known, then the Bessel function for $n ≥ 2$ can be determined iteratively:
:$${\rm J}_n (x) ={2 \cdot (n-1)}/{x} \cdot {\rm J}_{n-1} (x) - {\rm J}_{n-2} (x) \hspace{0.05cm}.$$
}}

{{GraueBox|TEXT=
$\text{Example (A):}$   Consider ${\rm J}_0 (x = 2) = 0.22389$ and ${\rm J}_1 (x= 2) = 0.57672$. From this it can be calculated iteratively:
:$${\rm J}_2 (x= 2) ={2 \cdot 1}/{2} \cdot {\rm J}_{1} (x= 2) - {\rm J}_{0} (x= 2) = 0.57672 - 0.22389 = \hspace{0.15cm}\underline{0.35283}\hspace{0.05cm},$$
:$${\rm J}_3 (x= 2) ={2 \cdot 2}/{2} \cdot {\rm J}_{2} (x= 2) - {\rm J}_{1} (x= 2) = 2 \cdot 0.35283 - 0.57672 = \hspace{0.15cm}\underline{0.12894}\hspace{0.05cm},$$
:$${\rm J}_4 (x= 2) ={2 \cdot 3}/{2} \cdot {\rm J}_{3} (x= 2) - {\rm J}_{2} (x= 2) = 3 \cdot 0.12894 - 0.35283 = \hspace{0.15cm}\underline{0.03400}\hspace{0.05cm}.$$
}}

{{BlaueBox|TEXT=
$\text{Property (B):}$   The symmetry relationship applies ${\rm J}_{–n}(x) = (–1)^n · {\rm J}_n(x)$:
:$${\rm J}_{-1}(x) = - {\rm J}_{1}(x), \hspace{0.3cm}{\rm J}_{-2}(x) = {\rm J}_{2}(x), \hspace{0.3cm}{\rm J}_{-3}(x) = - {\rm J}_{3}(x).$$
}}

{{GraueBox|TEXT=
$\text{Example (B):}$   For the spectrum of the analytic signal, in [[Modulationsverfahren/Phasenmodulation_(PM)#Spektralfunktion_eines_phasenmodulierten_Sinussignals|phase modulation of a sinusoidal signal]]:
[[File:Mod_T_3_1_S4_version2.png|right|frame|Spectrum of the analytic signal in phase modulation]]
:$$S_{\rm +}(f) = A_{\rm T} \cdot \sum_{n = - \infty}^{+\infty}{\rm J}_n (\eta) \cdot \delta \big[f - (f_{\rm T}+ n \cdot f_{\rm N})\big]\hspace{0.05cm}.$$
Denote this
*$f_{\rm T}$ the carrier frequency,
*$f_{\rm N}$ the message frequency,
* $A_{\rm T}$ the carrier amplitude.

The parameter of the Bessel functions in this application is the modulation index $\eta$.

Based on the graphic, the following statements are possible:
*Here $S_+(f)$ consists of an infinite number of discrete lines at a distance of $f_{\rm N}$.
*It is thus in principle infinitely extended.
*The weights of the spectral lines at $f_{\rm T} + n · f_{\rm N}$ ($n$ integer) are determined by the modulation index $η$ over the Bessel functions ${\rm J}_n(η)$.
*The spectral lines are real in the case of a sinusoidal source signal and a cosinusoidal carrier and symmetric about $f_{\rm T}$ for even $n$.
*For odd $n$ , a sign change according to the $\text{Property (B)}$ has to be considered.
*The phase modulation of another phase oscillation of source and / or carrier signal provides the same magnitude spectrum.}}
 

===Applications of the Bessel Functions===

The applications of the Bessel functions in nature–and engineering are diverse and play an important role in physics, for example:
* Examination of natural vibrations of cylindrical resonators,
* Solution of the radial Schrödinger-equation,
* Sound pressure amplitudes of low-viscosity rotational flows,
* Heat conduction in cylindrical bodies,
* Scattering problem of a grid,
* Dynamics of oscillating bodies,
* Angular resolution.

The Bessel functions are counted among the special functions because of their many applications in mathematical physics.

In the following, we restrict ourselves to a few areas that are addressed in our tutorial $\rm LNTwww$.

'''In the English original'''
Electromagnetic waves in a cylindrical waveguide
Pressure amplitudes of inviscid rotational flows
Heat conduction in a cylindrical object
Modes of vibration of a thin circular (or annular) acoustic membrane (such as a drum or other membranophone)
Diffusion problems on a lattice
Solutions to the radial Schrödinger equation (in spherical and cylindrical coordinates) for a free particle
Solving for patterns of acoustical radiation
Frequency-dependent friction in circular pipelines
Dynamics of floating bodies
Angular resolution
'''The Ende'''

{{GraueBox|TEXT=
$\text{Example (C):} \hspace{0.5cm} \text{Use in spectral analysis} \ \Rightarrow \ \text{Kaiser-Bessel filter}$

The '''spectral leakage effect''' is the falsification of the spectrum of a periodic and hence temporally unlimited signal due to the implicit time limitation of the discrete Fourier transform (DFT). This will be done, for example, by a spectrum analyzer
* in the time signal not existing frequency components simulated, and / or
* actually existing spectral components are obscured by sidelobes.

The task of the [[Signaldarstellung/Spektralanalyse|spectral analysis]] is to limit the influence of the ''spectral leakage effect'' by providing suitable window functions.

Such a window function provides, for example, the Kaiser–Bessel–window   ⇒   see section [[Signaldarstellung/Spektralanalyse#Spezielle_Fensterfunktionen|Special Window Functions]]. Its time-discrete Fenser function reads with the Bessel function zero order   ⇒   ${\rm J}_0(x)$, the parameter $\alpha=3.5$ and the window length $N$:
:$$w_\nu = \frac{ {\rm J}_0\big(\pi \cdot \alpha \cdot \sqrt{1 - (2\nu/N)^2}\big)}{ {\rm J}_0\big(\pi \cdot \alpha \big)}.$$
On the page [[Signaldarstellung/Spektralanalyse#G.C3.BCtekriterien_von_Fensterfunktionen|Quality Criteria of Window Functions]] and other the characteristics of the Kaiser-Bessel-window are given:
*Conveniently, the large is &bdquo;minimum distance between the main lobe and side lobes” and the desired small &bdquo;maximum scaling error”.
*Due to the very large &bdquo;equivalent noise width” the Kaiser-Bessel window cuts in the main comparison criterion &bdquo;maximum process loss” but worse off than the established Hamming– and Hanning–windows.}}

{{GraueBox|TEXT=
$\text{Example (D):} \hspace{0.5cm} \text{Rice-Fading channel model}$

The [[Mobile_Kommunikation/Wahrscheinlichkeitsdichte_des_Rayleigh%E2%80%93Fadings#Allgemeine_Beschreibung_des_Mobilfunkkanals| Rayleigh - Distribution]] describes the mobile channel assuming that there is no direct path, and thus the multiplicative factor $z(t) = x(t) + {\rm j} \cdot y(t)$ is solely diffuse composed of scattered components.

In the case of a direct component (English: Line of Sight, LoS) in the model for the mean-free Gaussian processes $x(t)$ and $y(t)$ one has to add equal components $x_0$ and/or $y_0$ :

[[File:P ID2126 Mob T 1 4 S1 v3.png|right|frame|Rice-Fading channel model|class=fit]]
$$\hspace{0.2cm}x(t) \hspace{0.1cm} \Rightarrow \hspace{0.1cm} x(t) +x_0 \hspace{0.05cm}, \hspace{0.2cm} y(t) \hspace{0.1cm} \Rightarrow \hspace{0.1cm} y(t) +y_0\hspace{0.05cm},$$

$$\hspace{0.2cm}z(t) = x(t) + {\rm j} \cdot y(t) \hspace{0.1cm} \Rightarrow \hspace{0.1cm} z(t) +z_0 \hspace{0.05cm},\hspace{0.2cm}
z_0 = x_0 + {\rm j} \cdot y_0\hspace{0.05cm}.$$

The graph shows the ''Rice–Fading–Channel model''. It can be summarized as follows:
*The real part $x(t)$ is Gaussianized with mean $x_0$ and variance $\sigma ^2$.
*The imaginary part $y(t)$ is also Gaussian (mean $y_0$, equal variance $\sigma ^2$) and independent of $x(t)$. 

*For $z_0 \ne 0$, the amount $\vert z(t)\vert$ rice is distributed, from which the term &bdquo;Rice–Fading” arises.

*To simplify the notation, we set $\vert z(t)\vert = a(t)$. For $a < 0$ , the amounts - WDF are $f_a(a) \equiv 0$, for $a \ge 0$ the following equation holds, where ${\rm I_0}(x)$ is the modified Besselfunktion zeroth order denotes:
:$$f_a(a) = \frac{a}{\sigma^2} \cdot {\rm e}^{ - (a^2 + \vert z_0 \vert ^2)/(2\sigma^2)} \cdot {\rm I}_0 \left [ \frac{a \cdot \vert z_0 \vert}{\sigma^2} \right ] \hspace{0.5cm}\text{with}\hspace{0.5cm}{\rm I }_0 (u) = {\rm J }_0 ({\rm j} \cdot u) =
\sum_{k = 0}^{\infty} \frac{ (u/2)^{2k} }{k! \cdot \Gamma (k+1)}
\hspace{0.05cm}.$$
*Between the modified Bessel function and the traditional Bessel function ${\rm I_0}(x)$ – each first type – So the connection exists ${\rm I }_0 (u) = {\rm J }_0 ({\rm j} \cdot u)$.}}

{{GraueBox|TEXT=
$\text{Example (E):} \hspace{0.5cm} \text{Analysis of the frequency spectrum of frequency modulated signals}$

$\text{Example (B)}$ has already shown that the angle modulation of a harmonic oscillation of the frequency $f_{\rm N}$ leads to a line spectrum. The spectral lines are around the carrier frequency $f_{\rm T}$ bei $f_{\rm T} + n \cdot f_{\rm N}$ with $n \in \{ \ \text{...}, -2, -1, \ 0, +1, +2, \text{...} \ \}$. The weights of the Dirac lines are ${\rm J }_n(\eta)$, depending on the modulation index $\eta$.

[[File:P_ID1095__Mod_T_3_2_S4_neu.png|center|frame|Discrete spectra with phase modulation (left) and frequency modulation (right)]]

The graph shows the magnitude spectrum $\vert S_{\rm +}(f) \vert$ of the phase modulation (PM) and frequency modulation (FM) analytical signal, two different forms of angle modulation (WM). Bessellines with values less than $0.03$ are neglected in both cases.

For the upper half of the image, the modulator parameters are chosen so that for each $f_{\rm N} = 5 \ \rm kHz$ here is a Bessel spectrum with the modulation index $η = 1.5$. Disregarding the phase relationships, the same spectra and the same signals result for both systems.

The lower graphics are valid for the same message frequency $f_{\rm N} = 3 \ \rm kHz$. One notices:
*In phase modulation, the spectral function is narrower than $f_{\rm N} = 5 \ \rm kHz$, because the distance of the Bessel lines is now only $3 \ \rm kHz$. Since the modulation index of PM is independent of $f_{\rm N}$, the same bessel weights result as in $f_{\rm N} = 5 \ \rm kHz$.
*Also in the frequency modulation, the Bessel lines now appear at a distance of $3 \ \rm kHz$. However, since FM has a modulation index inversely proportional to $f_{\rm N}$, there is now much more Besselline lines than in the upper right (for $η = 1.5$ valid) chart due to the larger modulation index $η = 2.5$.}}

'''Das folgende Kapitel muss noch angepasst werden!'''

==Zur Handhabung des Applets==
[[File:Handhabung_binomial.png|left|600px]]
    '''(A)'''     Vorauswahl für blauen Parametersatz

    '''(B)'''     Parametereingabe $I$ und $p$ per Slider

    '''(C)'''     Vorauswahl für roten Parametersatz

    '''(D)'''     Parametereingabe $\lambda$ per Slider

    '''(E)'''     Graphische Darstellung der Verteilungen

    '''(F)'''     Momentenausgabe für blauen Parametersatz

    '''(G)'''     Momentenausgabe für roten Parametersatz

    '''(H)'''     Variation der grafischen Darstellung

$\hspace{1.5cm}$&bdquo;$+$” (Vergrößern),

$\hspace{1.5cm}$ &bdquo;$-$” (Verkleinern)

$\hspace{1.5cm}$ &bdquo;$\rm o$” (Zurücksetzen)

$\hspace{1.5cm}$ &bdquo;$\leftarrow$” (Verschieben nach links), usw.

    '''( I )'''     Ausgabe von ${\rm Pr} (z = \mu)$ und ${\rm Pr} (z \le \mu)$

    '''(J)'''     Bereich für die Versuchsdurchführung
 
 '''Andere Möglichkeiten zur Variation der grafischen Darstellung''':
*Gedrückte Shifttaste und Scrollen: Zoomen im Koordinatensystem,
*Gedrückte Shifttaste und linke Maustaste: Verschieben des Koordinatensystems.

==About the Authors==
This interactive calculation was designed and realized at the [http://www.lnt.ei.tum.de/startseite Lehrstuhl für Nachrichtentechnik] of the [https://www.tum.de/ Technical University of Munich] .
*The original version was created in 2005 by [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Studierende#Markus_Elsberger_.28Diplomarbeit_LB_2006.29|Markus Elsberger]] and [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Studierende#Slim_Lamine_.28Studienarbeit_EI_2006.29|Slim Lamine]] as part of thesis using &bdquo;FlashMX–Actionscript” (Supervisor: [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Mitarbeiter_und_Dozenten#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28am_LNT_seit_1974.29|Günter Söder]]).
**In 2018 this Applet was redesigned and updated to &bdquo;HTML5” by [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Studierende#Xiaohan_Liu_.28Bachelorarbeit_2018.29|Xiaohan Liu]] as part of her Bachelor's thesis (Supervisor: [[Biografien_und_Bibliografien/Beteiligte_der_Professur_Leitungsgebundene_%C3%9Cbertragungstechnik#Tasn.C3.A1d_Kernetzky.2C_M.Sc._.28bei_L.C3.9CT_seit_2014.29|Tasnád Kernetzky]]).

==Nochmalige Aufrufmöglichkeit des Applets in neuem Fenster==

{{LntAppletLink|bessel}}

Applets:Bessel Functions of the First Kind

2018-08-21T21:02:43Z

Xiaohan: /* Applications of the Bessel Functions */

{{LntAppletLink|bessel}}

==Applet Description==
 
This applet allows the calculation and graphical representation of the Bessel functions of the first kind and $n$–th order according to the series representation:

:$${\rm J}_n (x) = \sum\limits_{k=0}^{\infty}\frac{(-1)^k \cdot (x/2)^{n \hspace{0.05cm} + \hspace{0.05cm} 2
\hspace{0.02cm}\cdot \hspace{0.05cm}k}}{k! \cdot (n+k)!} \hspace{0.05cm}.$$

*Graphically represented, the functions ${\rm J}_n (x)$ for the order $n=0$ to $n=9$ can become different colors.
*The left-hand output provides the function values ${\rm J}_0 (x = x_1)$, ... , ${\rm J}_9 (x = x_1)$ for a slider-settable value $x_1$ in the range $0 \le x_1 \le 15$ with increment $0.5$.
*The right hand side returns the function values ${\rm J}_0 (x = x_2)$, ... , ${\rm J}_9 (x = x_2)$ for a slider-settable value $x_2$ (same range and value) Increment as on the left).

[[Applets:Physical_Signal_%26_Analytical_Signal|'''German description''']]

==Theoretical Background==
 
===General Information about the Bessel Functions===
Bessel functions (or cylinder functions) are solutions of the Bessel differential equation of the form

:$$x^2 \cdot \frac{ {\rm d}^2}{{\rm d}x^2}\ {\rm J}_n (x) \ + \ x \cdot \frac{ {\rm d}}{{\rm d}x}\ {\rm J}_n (x) \ + \ (x^2 - n^2)
\cdot {\rm J}_n (x)= 0. $$

This is an ordinary second-order linear differential equation. The parameter $ n $ is usually integer, as in this program. These mathematical functions, which were introduced in 1844 by [https://de.wikipedia.org/wiki/Friedrich_Wilhelm_Bessel Friedrich Wilhelm Bessel], can also be represented in closed form as integrals:

:$${\rm J}_n (x) = \frac{1}{2\pi}\cdot \int_{-\pi}^{+\pi} {{\rm e}^{\hspace{0.05cm}{\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}[\hspace{0.05cm}x \hspace{0.05cm}\cdot \hspace{0.05cm}\sin(\alpha) -\hspace{0.05cm} n \hspace{0.05cm}\cdot \hspace{0.05cm}\alpha \hspace{0.05cm}]}}\hspace{0.1cm}{\rm d}\alpha
\hspace{0.05cm}.$$

The functions ${\rm J}_n (x)$ belong to the class of Bessel functions of the first kind (German:   ''Besselfunktionen erster Art''). The parameter $n$ is called the ''Order''.

''Annotation:''   There are a number of modifications of the Bessel functions, including the second-order Bessel functions named ${\rm Y}_n (x)$. For integer $n$, ${\rm Y}_n (x)$ can be replaced by ${\rm J}_n (x)$–functions. However, in this applet, only the first-order Bessel functions are   ⇒   ${\rm J}_n (x)$ is considered.
 
===Properties of the Bessel Functions===

{{BlaueBox|TEXT=
$\text{Property (A):}$   If the function values for $n = 0$ and $n = 1$ are known, then the Bessel function for $n ≥ 2$ can be determined iteratively:
:$${\rm J}_n (x) ={2 \cdot (n-1)}/{x} \cdot {\rm J}_{n-1} (x) - {\rm J}_{n-2} (x) \hspace{0.05cm}.$$
}}

{{GraueBox|TEXT=
$\text{Example (A):}$   Consider ${\rm J}_0 (x = 2) = 0.22389$ and ${\rm J}_1 (x= 2) = 0.57672$. From this it can be calculated iteratively:
:$${\rm J}_2 (x= 2) ={2 \cdot 1}/{2} \cdot {\rm J}_{1} (x= 2) - {\rm J}_{0} (x= 2) = 0.57672 - 0.22389 = \hspace{0.15cm}\underline{0.35283}\hspace{0.05cm},$$
:$${\rm J}_3 (x= 2) ={2 \cdot 2}/{2} \cdot {\rm J}_{2} (x= 2) - {\rm J}_{1} (x= 2) = 2 \cdot 0.35283 - 0.57672 = \hspace{0.15cm}\underline{0.12894}\hspace{0.05cm},$$
:$${\rm J}_4 (x= 2) ={2 \cdot 3}/{2} \cdot {\rm J}_{3} (x= 2) - {\rm J}_{2} (x= 2) = 3 \cdot 0.12894 - 0.35283 = \hspace{0.15cm}\underline{0.03400}\hspace{0.05cm}.$$
}}

{{BlaueBox|TEXT=
$\text{Property (B):}$   The symmetry relationship applies ${\rm J}_{–n}(x) = (–1)^n · {\rm J}_n(x)$:
:$${\rm J}_{-1}(x) = - {\rm J}_{1}(x), \hspace{0.3cm}{\rm J}_{-2}(x) = {\rm J}_{2}(x), \hspace{0.3cm}{\rm J}_{-3}(x) = - {\rm J}_{3}(x).$$
}}

{{GraueBox|TEXT=
$\text{Example (B):}$   For the spectrum of the analytic signal, in [[Modulationsverfahren/Phasenmodulation_(PM)#Spektralfunktion_eines_phasenmodulierten_Sinussignals|phase modulation of a sinusoidal signal]]:
[[File:Mod_T_3_1_S4_version2.png|right|frame|Spectrum of the analytic signal in phase modulation]]
:$$S_{\rm +}(f) = A_{\rm T} \cdot \sum_{n = - \infty}^{+\infty}{\rm J}_n (\eta) \cdot \delta \big[f - (f_{\rm T}+ n \cdot f_{\rm N})\big]\hspace{0.05cm}.$$
Denote this
*$f_{\rm T}$ the carrier frequency,
*$f_{\rm N}$ the message frequency,
* $A_{\rm T}$ the carrier amplitude.

The parameter of the Bessel functions in this application is the modulation index $\eta$.

Based on the graphic, the following statements are possible:
*Here $S_+(f)$ consists of an infinite number of discrete lines at a distance of $f_{\rm N}$.
*It is thus in principle infinitely extended.
*The weights of the spectral lines at $f_{\rm T} + n · f_{\rm N}$ ($n$ integer) are determined by the modulation index $η$ over the Bessel functions ${\rm J}_n(η)$.
*The spectral lines are real in the case of a sinusoidal source signal and a cosinusoidal carrier and symmetric about $f_{\rm T}$ for even $n$.
*For odd $n$ , a sign change according to the $\text{Property (B)}$ has to be considered.
*The phase modulation of another phase oscillation of source and / or carrier signal provides the same magnitude spectrum.}}
 

===Applications of the Bessel Functions===

The applications of the Bessel functions in nature–and engineering are diverse and play an important role in physics, for example:
* Examination of natural vibrations of cylindrical resonators,
* Solution of the radial Schrödinger-equation,
* Sound pressure amplitudes of low-viscosity rotational flows,
* Heat conduction in cylindrical bodies,
* Scattering problem of a grid,
* Dynamics of oscillating bodies,
* Angular resolution.

The Bessel functions are counted among the special functions because of their many applications in mathematical physics.

In the following, we restrict ourselves to a few areas that are addressed in our tutorial $\rm LNTwww$.

'''In the English original'''
Electromagnetic waves in a cylindrical waveguide
Pressure amplitudes of inviscid rotational flows
Heat conduction in a cylindrical object
Modes of vibration of a thin circular (or annular) acoustic membrane (such as a drum or other membranophone)
Diffusion problems on a lattice
Solutions to the radial Schrödinger equation (in spherical and cylindrical coordinates) for a free particle
Solving for patterns of acoustical radiation
Frequency-dependent friction in circular pipelines
Dynamics of floating bodies
Angular resolution
'''The Ende'''

{{GraueBox|TEXT=
$\text{Example (C):} \hspace{0.5cm} \text{Use in spectral analysis} \ \Rightarrow \ \text{Kaiser-Bessel filter}$

The '''spectral leakage effect''' is the falsification of the spectrum of a periodic and hence temporally unlimited signal due to the implicit time limitation of the discrete Fourier transform (DFT). This will be done, for example, by a spectrum analyzer
* in the time signal not existing frequency components simulated, and / or
* actually existing spectral components are obscured by sidelobes.

The task of the [[Signaldarstellung/Spektralanalyse|spectral analysis]] is to limit the influence of the ''spectral leakage effect'' by providing suitable window functions.

Such a window function provides, for example, the Kaiser–Bessel–window   ⇒   see section [[Signaldarstellung/Spektralanalyse#Spezielle_Fensterfunktionen|Special Window Functions]]. Its time-discrete Fenser function reads with the Bessel function zero order   ⇒   ${\rm J}_0(x)$, the parameter $\alpha=3.5$ and the window length $N$:
:$$w_\nu = \frac{ {\rm J}_0\big(\pi \cdot \alpha \cdot \sqrt{1 - (2\nu/N)^2}\big)}{ {\rm J}_0\big(\pi \cdot \alpha \big)}.$$
On the page [[Signaldarstellung/Spektralanalyse#G.C3.BCtekriterien_von_Fensterfunktionen|Quality Criteria of Window Functions]] and other the characteristics of the Kaiser-Bessel-window are given:
*Conveniently, the large is &bdquo;minimum distance between the main lobe and side lobes” and the desired small &bdquo;maximum scaling error”.
*Due to the very large &bdquo;equivalent noise width” the Kaiser-Bessel window cuts in the main comparison criterion &bdquo;maximum process loss” but worse off than the established Hamming– and Hanning–windows.}}

{{GraueBox|TEXT=
$\text{Example (D):} \hspace{0.5cm} \text{Rice-Fading channel model}$

The [[Mobile_Kommunikation/Wahrscheinlichkeitsdichte_des_Rayleigh%E2%80%93Fadings#Allgemeine_Beschreibung_des_Mobilfunkkanals| Rayleigh - Distribution]] describes the mobile channel assuming that there is no direct path, and thus the multiplicative factor $z(t) = x(t) + {\rm j} \cdot y(t)$ is solely diffuse composed of scattered components.

In the case of a direct component (English: Line of Sight, LoS) in the model for the mean-free Gaussian processes $x(t)$ and $y(t)$ one has to add equal components $x_0$ and/or $y_0$ :

[[File:P ID2126 Mob T 1 4 S1 v3.png|right|frame|Rice-Fading channel model|class=fit]]
$$\hspace{0.2cm}x(t) \hspace{0.1cm} \Rightarrow \hspace{0.1cm} x(t) +x_0 \hspace{0.05cm}, \hspace{0.2cm} y(t) \hspace{0.1cm} \Rightarrow \hspace{0.1cm} y(t) +y_0\hspace{0.05cm},$$

$$\hspace{0.2cm}z(t) = x(t) + {\rm j} \cdot y(t) \hspace{0.1cm} \Rightarrow \hspace{0.1cm} z(t) +z_0 \hspace{0.05cm},\hspace{0.2cm}
z_0 = x_0 + {\rm j} \cdot y_0\hspace{0.05cm}.$$

The graph shows the ''Rice–Fading–Channel model''. It can be summarized as follows:
*The real part $x(t)$ is Gaussianized with mean $x_0$ and variance $\sigma ^2$.
*The imaginary part $y(t)$ is also Gaussian (mean $y_0$, equal variance $\sigma ^2$) and independent of $x(t)$. 

*For $z_0 \ne 0$, the amount $\vert z(t)\vert$ rice is distributed, from which the term &bdquo;Rice–Fading” arises.

*To simplify the notation, we set $\vert z(t)\vert = a(t)$. For $a < 0$ , the amounts - WDF are $f_a(a) \equiv 0$, for $a \ge 0$ the following equation holds, where ${\rm I_0}(x)$ is the modified Besselfunktion zeroth order denotes:
:$$f_a(a) = \frac{a}{\sigma^2} \cdot {\rm e}^{ - (a^2 + \vert z_0 \vert ^2)/(2\sigma^2)} \cdot {\rm I}_0 \left [ \frac{a \cdot \vert z_0 \vert}{\sigma^2} \right ] \hspace{0.5cm}\text{with}\hspace{0.5cm}{\rm I }_0 (u) = {\rm J }_0 ({\rm j} \cdot u) =
\sum_{k = 0}^{\infty} \frac{ (u/2)^{2k} }{k! \cdot \Gamma (k+1)}
\hspace{0.05cm}.$$
*Between the modified Bessel function and the traditional Bessel function ${\rm I_0}(x)$ – each first type – So the connection exists ${\rm I }_0 (u) = {\rm J }_0 ({\rm j} \cdot u)$.}}

{{GraueBox|TEXT=
$\text{Example (E):} \hspace{0.5cm} \text{Analysis of the frequency spectrum of frequency modulated signals}$

$\text{Example (B)}$ has already shown that the angle modulation of a harmonic oscillation of the frequency $f_{\rm N}$ leads to a line spectrum. The spectral lines are around the carrier frequency $f_{\rm T}$ bei $f_{\rm T} + n \cdot f_{\rm N}$ with $n \in \{ \ \text{...}, -2, -1, \ 0, +1, +2, \text{...} \ \}$. The weights of the Dirac lines are ${\rm J }_n(\eta)$, depending on the modulation index $\eta$.

[[File:P_ID1095__Mod_T_3_2_S4_neu.png|center|frame|Discrete spectra with phase modulation (left) and frequency modulation (right)]]

The graph shows the magnitude spectrum $\vert S_{\rm +}(f) \vert$ of the phase modulation (PM) and frequency modulation (FM) analytical signal, two different forms of angle modulation (WM). Bessellines with values less than $0.03$ are neglected in both cases.

For the upper half of the image, the modulator parameters are chosen so that for each $f_{\rm N} = 5 \ \rm kHz$ here is a Bessel spectrum with the modulation index $η = 1.5$. Disregarding the phase relationships, the same spectra and the same signals result for both systems.

The lower graphics are valid for the same message frequency $f_{\rm N} = 3 \ \rm kHz$. One notices:
*In phase modulation, the spectral function is narrower than $f_{\rm N} = 5 \ \rm kHz$, because the distance of the Bessel lines is now only $3 \ \rm kHz$. Since the modulation index of PM is independent of $f_{\rm N}$, the same bessel weights result as in $f_{\rm N} = 5 \ \rm kHz$.
*Also in the frequency modulation, the Bessel lines now appear at a distance of $3 \ \rm kHz$. However, since FM has a modulation index inversely proportional to $f_{\rm N}$, there is now much more Besselline lines than in the upper right (for $η = 1.5$ valid) chart due to the larger modulation index $η = 2.5$.}}

'''Das folgende Kapitel muss noch angepasst werden!'''

==Zur Handhabung des Applets==
[[File:Handhabung_binomial.png|left|600px]]
    '''(A)'''     Vorauswahl für blauen Parametersatz

    '''(B)'''     Parametereingabe $I$ und $p$ per Slider

    '''(C)'''     Vorauswahl für roten Parametersatz

    '''(D)'''     Parametereingabe $\lambda$ per Slider

    '''(E)'''     Graphische Darstellung der Verteilungen

    '''(F)'''     Momentenausgabe für blauen Parametersatz

    '''(G)'''     Momentenausgabe für roten Parametersatz

    '''(H)'''     Variation der grafischen Darstellung

$\hspace{1.5cm}$&bdquo;$+$” (Vergrößern),

$\hspace{1.5cm}$ &bdquo;$-$” (Verkleinern)

$\hspace{1.5cm}$ &bdquo;$\rm o$” (Zurücksetzen)

$\hspace{1.5cm}$ &bdquo;$\leftarrow$” (Verschieben nach links), usw.

    '''( I )'''     Ausgabe von ${\rm Pr} (z = \mu)$ und ${\rm Pr} (z \le \mu)$

    '''(J)'''     Bereich für die Versuchsdurchführung
 
 '''Andere Möglichkeiten zur Variation der grafischen Darstellung''':
*Gedrückte Shifttaste und Scrollen: Zoomen im Koordinatensystem,
*Gedrückte Shifttaste und linke Maustaste: Verschieben des Koordinatensystems.

==Über die Autoren==
Dieses interaktive Berechnungstool wurde am [http://www.lnt.ei.tum.de/startseite Lehrstuhl für Nachrichtentechnik] der [https://www.tum.de/ Technischen Universität München] konzipiert und realisiert.
*Die erste Version wurde 2006 von [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Studierende#Markus_Elsberger_.28Diplomarbeit_LB_2006.29|Markus Elsberger]] und [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Studierende#Slim_Lamine_.28Studienarbeit_EI_2006.29|Slim Lamine]] im Rahmen von Abschlussarbeiten mit &bdquo;FlashMX–Actionscript” erstellt (Betreuer: [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Mitarbeiter_und_Dozenten#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28am_LNT_seit_1974.29|Günter Söder]]).
*2018 wurde das Programm von [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Studierende#Xiaohan_Liu_.28Bachelorarbeit_2018.29|Xiaohan Liu]] (Bachelorarbeit, Betreuer: [[Biografien_und_Bibliografien/Beteiligte_der_Professur_Leitungsgebundene_%C3%9Cbertragungstechnik#Tasn.C3.A1d_Kernetzky.2C_M.Sc._.28bei_L.C3.9CT_seit_2014.29|Tasnád Kernetzky]] ) auf &bdquo;HTML5” umgesetzt.

==Nochmalige Aufrufmöglichkeit des Applets in neuem Fenster==

{{LntAppletLink|bessel}}

Applets:Bessel Functions of the First Kind

2018-08-21T15:57:58Z

Xiaohan: /* Anwendungen der Besselfunktionen */

{{LntAppletLink|bessel}}

==Applet Description==
 
This applet allows the calculation and graphical representation of the Bessel functions of the first kind and $n$–th order according to the series representation:

:$${\rm J}_n (x) = \sum\limits_{k=0}^{\infty}\frac{(-1)^k \cdot (x/2)^{n \hspace{0.05cm} + \hspace{0.05cm} 2
\hspace{0.02cm}\cdot \hspace{0.05cm}k}}{k! \cdot (n+k)!} \hspace{0.05cm}.$$

*Graphically represented, the functions ${\rm J}_n (x)$ for the order $n=0$ to $n=9$ can become different colors.
*The left-hand output provides the function values ${\rm J}_0 (x = x_1)$, ... , ${\rm J}_9 (x = x_1)$ for a slider-settable value $x_1$ in the range $0 \le x_1 \le 15$ with increment $0.5$.
*The right hand side returns the function values ${\rm J}_0 (x = x_2)$, ... , ${\rm J}_9 (x = x_2)$ for a slider-settable value $x_2$ (same range and value) Increment as on the left).

[[Applets:Physical_Signal_%26_Analytical_Signal|'''German description''']]

==Theoretical Background==
 
===General Information about the Bessel Functions===
Bessel functions (or cylinder functions) are solutions of the Bessel differential equation of the form

:$$x^2 \cdot \frac{ {\rm d}^2}{{\rm d}x^2}\ {\rm J}_n (x) \ + \ x \cdot \frac{ {\rm d}}{{\rm d}x}\ {\rm J}_n (x) \ + \ (x^2 - n^2)
\cdot {\rm J}_n (x)= 0. $$

This is an ordinary second-order linear differential equation. The parameter $ n $ is usually integer, as in this program. These mathematical functions, which were introduced in 1844 by [https://de.wikipedia.org/wiki/Friedrich_Wilhelm_Bessel Friedrich Wilhelm Bessel], can also be represented in closed form as integrals:

:$${\rm J}_n (x) = \frac{1}{2\pi}\cdot \int_{-\pi}^{+\pi} {{\rm e}^{\hspace{0.05cm}{\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}[\hspace{0.05cm}x \hspace{0.05cm}\cdot \hspace{0.05cm}\sin(\alpha) -\hspace{0.05cm} n \hspace{0.05cm}\cdot \hspace{0.05cm}\alpha \hspace{0.05cm}]}}\hspace{0.1cm}{\rm d}\alpha
\hspace{0.05cm}.$$

The functions ${\rm J}_n (x)$ belong to the class of Bessel functions of the first kind (German:   ''Besselfunktionen erster Art''). The parameter $n$ is called the ''Order''.

''Annotation:''   There are a number of modifications of the Bessel functions, including the second-order Bessel functions named ${\rm Y}_n (x)$. For integer $n$, ${\rm Y}_n (x)$ can be replaced by ${\rm J}_n (x)$–functions. However, in this applet, only the first-order Bessel functions are   ⇒   ${\rm J}_n (x)$ is considered.
 
===Properties of the Bessel Functions===

{{BlaueBox|TEXT=
$\text{Property (A):}$   If the function values for $n = 0$ and $n = 1$ are known, then the Bessel function for $n ≥ 2$ can be determined iteratively:
:$${\rm J}_n (x) ={2 \cdot (n-1)}/{x} \cdot {\rm J}_{n-1} (x) - {\rm J}_{n-2} (x) \hspace{0.05cm}.$$
}}

{{GraueBox|TEXT=
$\text{Example (A):}$   Consider ${\rm J}_0 (x = 2) = 0.22389$ and ${\rm J}_1 (x= 2) = 0.57672$. From this it can be calculated iteratively:
:$${\rm J}_2 (x= 2) ={2 \cdot 1}/{2} \cdot {\rm J}_{1} (x= 2) - {\rm J}_{0} (x= 2) = 0.57672 - 0.22389 = \hspace{0.15cm}\underline{0.35283}\hspace{0.05cm},$$
:$${\rm J}_3 (x= 2) ={2 \cdot 2}/{2} \cdot {\rm J}_{2} (x= 2) - {\rm J}_{1} (x= 2) = 2 \cdot 0.35283 - 0.57672 = \hspace{0.15cm}\underline{0.12894}\hspace{0.05cm},$$
:$${\rm J}_4 (x= 2) ={2 \cdot 3}/{2} \cdot {\rm J}_{3} (x= 2) - {\rm J}_{2} (x= 2) = 3 \cdot 0.12894 - 0.35283 = \hspace{0.15cm}\underline{0.03400}\hspace{0.05cm}.$$
}}

{{BlaueBox|TEXT=
$\text{Property (B):}$   The symmetry relationship applies ${\rm J}_{–n}(x) = (–1)^n · {\rm J}_n(x)$:
:$${\rm J}_{-1}(x) = - {\rm J}_{1}(x), \hspace{0.3cm}{\rm J}_{-2}(x) = {\rm J}_{2}(x), \hspace{0.3cm}{\rm J}_{-3}(x) = - {\rm J}_{3}(x).$$
}}

{{GraueBox|TEXT=
$\text{Example (B):}$   For the spectrum of the analytic signal, in [[Modulationsverfahren/Phasenmodulation_(PM)#Spektralfunktion_eines_phasenmodulierten_Sinussignals|phase modulation of a sinusoidal signal]]:
[[File:Mod_T_3_1_S4_version2.png|right|frame|Spectrum of the analytic signal in phase modulation]]
:$$S_{\rm +}(f) = A_{\rm T} \cdot \sum_{n = - \infty}^{+\infty}{\rm J}_n (\eta) \cdot \delta \big[f - (f_{\rm T}+ n \cdot f_{\rm N})\big]\hspace{0.05cm}.$$
Denote this
*$f_{\rm T}$ the carrier frequency,
*$f_{\rm N}$ the message frequency,
* $A_{\rm T}$ the carrier amplitude.

The parameter of the Bessel functions in this application is the modulation index $\eta$.

Based on the graphic, the following statements are possible:
*Here $S_+(f)$ consists of an infinite number of discrete lines at a distance of $f_{\rm N}$.
*It is thus in principle infinitely extended.
*The weights of the spectral lines at $f_{\rm T} + n · f_{\rm N}$ ($n$ integer) are determined by the modulation index $η$ over the Bessel functions ${\rm J}_n(η)$.
*The spectral lines are real in the case of a sinusoidal source signal and a cosinusoidal carrier and symmetric about $f_{\rm T}$ for even $n$.
*For odd $n$ , a sign change according to the $\text{Property (B)}$ has to be considered.
*The phase modulation of another phase oscillation of source and / or carrier signal provides the same magnitude spectrum.}}
 

===Applications of the Bessel Functions===

The applications of the Bessel functions in nature–and engineering are diverse and play an important role in physics, for example:
* Examination of natural vibrations of cylindrical resonators,
* Solution of the radial Schrödinger-equation,
* Sound pressure amplitudes of low-viscosity rotational flows,
* Heat conduction in cylindrical bodies,
* Scattering problem of a grid,
* Dynamics of oscillating bodies,
* Angular resolution.

The Bessel functions are counted among the special functions because of their many applications in mathematical physics.

In the following, we restrict ourselves to a few areas that are addressed in our tutorial $\rm LNTwww$.

'''In the English original'''
Electromagnetic waves in a cylindrical waveguide
Pressure amplitudes of inviscid rotational flows
Heat conduction in a cylindrical object
Modes of vibration of a thin circular (or annular) acoustic membrane (such as a drum or other membranophone)
Diffusion problems on a lattice
Solutions to the radial Schrödinger equation (in spherical and cylindrical coordinates) for a free particle
Solving for patterns of acoustical radiation
Frequency-dependent friction in circular pipelines
Dynamics of floating bodies
Angular resolution
'''The Ende'''

{{GraueBox|TEXT=
$\text{Example (C):} \hspace{0.5cm} \text{Use in spectral analysis} \ \Rightarrow \ \text{Kaiser-Bessel filter}$

The '''spectral leakage effect''' is the falsification of the spectrum of a periodic and hence temporally unlimited signal due to the implicit time limitation of the discrete Fourier transform (DFT). This will be done, for example, by a spectrum analyzer
* in the time signal not existing frequency components simulated, and / or
* actually existing spectral components are obscured by sidelobes.

The task of the [[Signaldarstellung/Spektralanalyse|spectral analysis]] is to limit the influence of the ''spectral leakage effect'' by providing suitable window functions.

Such a window function provides, for example, the Kaiser–Bessel–window   ⇒   see section [[Signaldarstellung/Spektralanalyse#Spezielle_Fensterfunktionen|Special Window Functions]]. Its time-discrete Fenser function reads with the Bessel function zero order   ⇒   ${\rm J}_0(x)$, the parameter $\alpha=3.5$ and the window length $N$:
:$$w_\nu = \frac{ {\rm J}_0\big(\pi \cdot \alpha \cdot \sqrt{1 - (2\nu/N)^2}\big)}{ {\rm J}_0\big(\pi \cdot \alpha \big)}.$$
Auf der Seite [[Signaldarstellung/Spektralanalyse#G.C3.BCtekriterien_von_Fensterfunktionen|Gütekriterien von Fensterfunktionen]] sind u.a. die Kenngrößen des Kaiser–Bessel–Fensters angegeben:
*Günstig sind der große &bdquo;Minimale Abstand zwischen Hauptkeule und Seitenkeulen” und der gewünscht kleine &bdquo;Maximale Skalierungsfehler”.
*Aufgrund der sehr großen &bdquo;Äquivalenten Rauschbreite” schneidet das Kaiser–Bessel–Fenster im wichtigsten Vergleichskriterium &bdquo;Maximaler Prozessverlust” doch schlechter ab als die etablierten Hamming– und Hanning–Fenster.}}

{{GraueBox|TEXT=
$\text{Beispiel (D):} \hspace{0.5cm} \text{Rice-Fading-Kanalmodell}$

Die [[Mobile_Kommunikation/Wahrscheinlichkeitsdichte_des_Rayleigh%E2%80%93Fadings#Allgemeine_Beschreibung_des_Mobilfunkkanals| Rayleigh–Verteilung]] beschreibt den Mobilfunkkanal unter der Annahme, dass kein direkter Pfad vorhanden ist und sich somit der multiplikative Faktor $z(t) = x(t) + {\rm j} \cdot y(t)$ allein aus diffus gestreuten Komponenten zusammensetzt.

Bei Vorhandensein einer Direktkomponente (englisch: Line of Sight, LoS) muss man im Modell zu den mittelwertfreien Gaußprozessen $x(t)$ und $y(t)$ noch Gleichkomponenten $x_0$ und/oder $y_0$ hinzufügen:

[[File:P ID2126 Mob T 1 4 S1 v3.png|right|frame|Rice-Fading-Kanalmodell|class=fit]]
$$\hspace{0.2cm}x(t) \hspace{0.1cm} \Rightarrow \hspace{0.1cm} x(t) +x_0 \hspace{0.05cm}, \hspace{0.2cm} y(t) \hspace{0.1cm} \Rightarrow \hspace{0.1cm} y(t) +y_0\hspace{0.05cm},$$

$$\hspace{0.2cm}z(t) = x(t) + {\rm j} \cdot y(t) \hspace{0.1cm} \Rightarrow \hspace{0.1cm} z(t) +z_0 \hspace{0.05cm},\hspace{0.2cm}
z_0 = x_0 + {\rm j} \cdot y_0\hspace{0.05cm}.$$

Die Grafik zeigt das ''Rice–Fading–Kanalmodell''. Es lässt sich wie folgt zusammenfassen:
*Der Realteil $x(t)$ ist gaußverteilt mit Mittelwert $x_0$ und Varianz $\sigma ^2$.
*Der Imaginärteil $y(t)$ ist ebenfalls gaußverteilt (Mittelwert $y_0$, gleiche Varianz $\sigma ^2$) sowie unabhängig von $x(t)$. 

*Für $z_0 \ne 0$ ist der Betrag $\vert z(t)\vert$ riceverteilt, woraus die Bezeichnung &bdquo;Rice–Fading” herrührt.

*Zur Vereinfachung der Schreibweise setzen wir $\vert z(t)\vert = a(t)$. Für $a < 0$ ist die Betrags–WDF $f_a(a) \equiv 0$, für $a \ge 0$ gilt folgende Gleichung, wobei ${\rm I_0}(x)$ die modifizierte Besselfunktion nullter Ordnung bezeichnet:
:$$f_a(a) = \frac{a}{\sigma^2} \cdot {\rm e}^{ - (a^2 + \vert z_0 \vert ^2)/(2\sigma^2)} \cdot {\rm I}_0 \left [ \frac{a \cdot \vert z_0 \vert}{\sigma^2} \right ] \hspace{0.5cm}\text{mit}\hspace{0.5cm}{\rm I }_0 (u) = {\rm J }_0 ({\rm j} \cdot u) =
\sum_{k = 0}^{\infty} \frac{ (u/2)^{2k} }{k! \cdot \Gamma (k+1)}
\hspace{0.05cm}.$$
*Zwischen der modifizierten Besselfunktion und der herkömmlichen Besselfunktion ${\rm I_0}(x)$ – jeweils erster Art – besteht also der Zusammenhang ${\rm I }_0 (u) = {\rm J }_0 ({\rm j} \cdot u)$.}}

{{GraueBox|TEXT=
$\text{Beispiel (E):} \hspace{0.5cm} \text{Analyse des Frequenzspektrums von frequenzmodulierten Signalen}$

Im $\text{Beispiel (B)}$ wurde bereits gezeigt, dass die Winkelmodulation einer harmonischen Schwingung der Frequenz $f_{\rm N}$ zu einem Linienspektrum führt. Die Spektrallinien liegen um die Trägerfrequenz $f_{\rm T}$ bei $f_{\rm T} + n \cdot f_{\rm N}$ mit $n \in \{ \ \text{...}, -2, -1, \ 0, +1, +2, \text{...} \ \}$. Die Gewichte der Diraclinien sind ${\rm J }_n(\eta)$, abhängig vom Modulationsindex $\eta$.

[[File:P_ID1095__Mod_T_3_2_S4_neu.png|center|frame|Diskrete Spektren bei Phasenmodulation (links) und Frequenzmodulation (rechts)]]

Die Grafik zeigt das Betragsspektrum $\vert S_{\rm +}(f) \vert$ des analytischen Signals bei Phasenmodulation (PM) und Frequenzmodulation (FM), zwei unterschiedliche Formen der Winkelmodulation (WM). Bessellinien mit Werten kleiner als $0.03$ sind hierbei in beiden Fällen vernachlässigt.

Für die obere Bildhälfte sind die Modulatorparameter so gewählt, dass sich für $f_{\rm N} = 5 \ \rm kHz$ jeweils ein Besselspektrum mit dem Modulationsindex $η = 1.5$ ergibt. Lässt man die Phasenbeziehungen außer Acht, so ergeben sich für beide Systeme gleiche Spektren und gleiche Signale.

Die unteren Grafiken gelten bei sonst gleichen Einstellungen für die Nachrichtenfrequenz $f_{\rm N} = 3 \ \rm kHz$. Man erkennt:
*Bei der Phasenmodulation ergibt sich gegenüber $f_{\rm N} = 5 \ \rm kHz$ eine schmalere Spektralfunktion, da nun der Abstand der Bessellinien nur mehr $3 \ \rm kHz$ beträgt. Da bei PM der Modulationsindex unabhängig von $f_{\rm N}$ ist, ergeben sich die gleichen Besselgewichte wie bei $f_{\rm N} = 5 \ \rm kHz$.
*Auch bei der Frequenzmodulation treten nun die Bessellinien im Abstand von $3 \ \rm kHz$ auf. Da aber bei FM der Modulationsindex umgekehrt proportional zu $f_{\rm N}$ ist, gibt es nun unten aufgrund des größeren Modulationsindex $η = 2.5$ deutlich mehr Bessellinien als im rechten oberen (für $η = 1.5$ gültigen) Diagramm. }}

'''Das folgende Kapitel muss noch angepasst werden!'''

==Zur Handhabung des Applets==
[[File:Handhabung_binomial.png|left|600px]]
    '''(A)'''     Vorauswahl für blauen Parametersatz

    '''(B)'''     Parametereingabe $I$ und $p$ per Slider

    '''(C)'''     Vorauswahl für roten Parametersatz

    '''(D)'''     Parametereingabe $\lambda$ per Slider

    '''(E)'''     Graphische Darstellung der Verteilungen

    '''(F)'''     Momentenausgabe für blauen Parametersatz

    '''(G)'''     Momentenausgabe für roten Parametersatz

    '''(H)'''     Variation der grafischen Darstellung

$\hspace{1.5cm}$&bdquo;$+$” (Vergrößern),

$\hspace{1.5cm}$ &bdquo;$-$” (Verkleinern)

$\hspace{1.5cm}$ &bdquo;$\rm o$” (Zurücksetzen)

$\hspace{1.5cm}$ &bdquo;$\leftarrow$” (Verschieben nach links), usw.

    '''( I )'''     Ausgabe von ${\rm Pr} (z = \mu)$ und ${\rm Pr} (z \le \mu)$

    '''(J)'''     Bereich für die Versuchsdurchführung
 
 '''Andere Möglichkeiten zur Variation der grafischen Darstellung''':
*Gedrückte Shifttaste und Scrollen: Zoomen im Koordinatensystem,
*Gedrückte Shifttaste und linke Maustaste: Verschieben des Koordinatensystems.

==Über die Autoren==
Dieses interaktive Berechnungstool wurde am [http://www.lnt.ei.tum.de/startseite Lehrstuhl für Nachrichtentechnik] der [https://www.tum.de/ Technischen Universität München] konzipiert und realisiert.
*Die erste Version wurde 2006 von [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Studierende#Markus_Elsberger_.28Diplomarbeit_LB_2006.29|Markus Elsberger]] und [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Studierende#Slim_Lamine_.28Studienarbeit_EI_2006.29|Slim Lamine]] im Rahmen von Abschlussarbeiten mit &bdquo;FlashMX–Actionscript” erstellt (Betreuer: [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Mitarbeiter_und_Dozenten#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28am_LNT_seit_1974.29|Günter Söder]]).
*2018 wurde das Programm von [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Studierende#Xiaohan_Liu_.28Bachelorarbeit_2018.29|Xiaohan Liu]] (Bachelorarbeit, Betreuer: [[Biografien_und_Bibliografien/Beteiligte_der_Professur_Leitungsgebundene_%C3%9Cbertragungstechnik#Tasn.C3.A1d_Kernetzky.2C_M.Sc._.28bei_L.C3.9CT_seit_2014.29|Tasnád Kernetzky]] ) auf &bdquo;HTML5” umgesetzt.

==Nochmalige Aufrufmöglichkeit des Applets in neuem Fenster==

{{LntAppletLink|bessel}}

Applets:Bessel Functions of the First Kind

2018-08-21T15:37:34Z

Xiaohan: /* Eigenschaften der Besselfunktionen */

{{LntAppletLink|bessel}}

==Applet Description==
 
This applet allows the calculation and graphical representation of the Bessel functions of the first kind and $n$–th order according to the series representation:

:$${\rm J}_n (x) = \sum\limits_{k=0}^{\infty}\frac{(-1)^k \cdot (x/2)^{n \hspace{0.05cm} + \hspace{0.05cm} 2
\hspace{0.02cm}\cdot \hspace{0.05cm}k}}{k! \cdot (n+k)!} \hspace{0.05cm}.$$

*Graphically represented, the functions ${\rm J}_n (x)$ for the order $n=0$ to $n=9$ can become different colors.
*The left-hand output provides the function values ${\rm J}_0 (x = x_1)$, ... , ${\rm J}_9 (x = x_1)$ for a slider-settable value $x_1$ in the range $0 \le x_1 \le 15$ with increment $0.5$.
*The right hand side returns the function values ${\rm J}_0 (x = x_2)$, ... , ${\rm J}_9 (x = x_2)$ for a slider-settable value $x_2$ (same range and value) Increment as on the left).

[[Applets:Physical_Signal_%26_Analytical_Signal|'''German description''']]

==Theoretical Background==
 
===General Information about the Bessel Functions===
Bessel functions (or cylinder functions) are solutions of the Bessel differential equation of the form

:$$x^2 \cdot \frac{ {\rm d}^2}{{\rm d}x^2}\ {\rm J}_n (x) \ + \ x \cdot \frac{ {\rm d}}{{\rm d}x}\ {\rm J}_n (x) \ + \ (x^2 - n^2)
\cdot {\rm J}_n (x)= 0. $$

This is an ordinary second-order linear differential equation. The parameter $ n $ is usually integer, as in this program. These mathematical functions, which were introduced in 1844 by [https://de.wikipedia.org/wiki/Friedrich_Wilhelm_Bessel Friedrich Wilhelm Bessel], can also be represented in closed form as integrals:

:$${\rm J}_n (x) = \frac{1}{2\pi}\cdot \int_{-\pi}^{+\pi} {{\rm e}^{\hspace{0.05cm}{\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}[\hspace{0.05cm}x \hspace{0.05cm}\cdot \hspace{0.05cm}\sin(\alpha) -\hspace{0.05cm} n \hspace{0.05cm}\cdot \hspace{0.05cm}\alpha \hspace{0.05cm}]}}\hspace{0.1cm}{\rm d}\alpha
\hspace{0.05cm}.$$

The functions ${\rm J}_n (x)$ belong to the class of Bessel functions of the first kind (German:   ''Besselfunktionen erster Art''). The parameter $n$ is called the ''Order''.

''Annotation:''   There are a number of modifications of the Bessel functions, including the second-order Bessel functions named ${\rm Y}_n (x)$. For integer $n$, ${\rm Y}_n (x)$ can be replaced by ${\rm J}_n (x)$–functions. However, in this applet, only the first-order Bessel functions are   ⇒   ${\rm J}_n (x)$ is considered.
 
===Properties of the Bessel Functions===

{{BlaueBox|TEXT=
$\text{Property (A):}$   If the function values for $n = 0$ and $n = 1$ are known, then the Bessel function for $n ≥ 2$ can be determined iteratively:
:$${\rm J}_n (x) ={2 \cdot (n-1)}/{x} \cdot {\rm J}_{n-1} (x) - {\rm J}_{n-2} (x) \hspace{0.05cm}.$$
}}

{{GraueBox|TEXT=
$\text{Example (A):}$   Consider ${\rm J}_0 (x = 2) = 0.22389$ and ${\rm J}_1 (x= 2) = 0.57672$. From this it can be calculated iteratively:
:$${\rm J}_2 (x= 2) ={2 \cdot 1}/{2} \cdot {\rm J}_{1} (x= 2) - {\rm J}_{0} (x= 2) = 0.57672 - 0.22389 = \hspace{0.15cm}\underline{0.35283}\hspace{0.05cm},$$
:$${\rm J}_3 (x= 2) ={2 \cdot 2}/{2} \cdot {\rm J}_{2} (x= 2) - {\rm J}_{1} (x= 2) = 2 \cdot 0.35283 - 0.57672 = \hspace{0.15cm}\underline{0.12894}\hspace{0.05cm},$$
:$${\rm J}_4 (x= 2) ={2 \cdot 3}/{2} \cdot {\rm J}_{3} (x= 2) - {\rm J}_{2} (x= 2) = 3 \cdot 0.12894 - 0.35283 = \hspace{0.15cm}\underline{0.03400}\hspace{0.05cm}.$$
}}

{{BlaueBox|TEXT=
$\text{Property (B):}$   The symmetry relationship applies ${\rm J}_{–n}(x) = (–1)^n · {\rm J}_n(x)$:
:$${\rm J}_{-1}(x) = - {\rm J}_{1}(x), \hspace{0.3cm}{\rm J}_{-2}(x) = {\rm J}_{2}(x), \hspace{0.3cm}{\rm J}_{-3}(x) = - {\rm J}_{3}(x).$$
}}

{{GraueBox|TEXT=
$\text{Example (B):}$   For the spectrum of the analytic signal, in [[Modulationsverfahren/Phasenmodulation_(PM)#Spektralfunktion_eines_phasenmodulierten_Sinussignals|phase modulation of a sinusoidal signal]]:
[[File:Mod_T_3_1_S4_version2.png|right|frame|Spectrum of the analytic signal in phase modulation]]
:$$S_{\rm +}(f) = A_{\rm T} \cdot \sum_{n = - \infty}^{+\infty}{\rm J}_n (\eta) \cdot \delta \big[f - (f_{\rm T}+ n \cdot f_{\rm N})\big]\hspace{0.05cm}.$$
Denote this
*$f_{\rm T}$ the carrier frequency,
*$f_{\rm N}$ the message frequency,
* $A_{\rm T}$ the carrier amplitude.

The parameter of the Bessel functions in this application is the modulation index $\eta$.

Based on the graphic, the following statements are possible:
*Here $S_+(f)$ consists of an infinite number of discrete lines at a distance of $f_{\rm N}$.
*It is thus in principle infinitely extended.
*The weights of the spectral lines at $f_{\rm T} + n · f_{\rm N}$ ($n$ integer) are determined by the modulation index $η$ over the Bessel functions ${\rm J}_n(η)$.
*The spectral lines are real in the case of a sinusoidal source signal and a cosinusoidal carrier and symmetric about $f_{\rm T}$ for even $n$.
*For odd $n$ , a sign change according to the $\text{Property (B)}$ has to be considered.
*The phase modulation of another phase oscillation of source and / or carrier signal provides the same magnitude spectrum.}}
 

===Anwendungen der Besselfunktionen===

Die Anwendungen der Besselfunktionen in den Natur– und Ingenieurswissenschaften sind vielfältig und spielen eine wichtige Rolle in der Physik, zum Beispiel:
*Untersuchung von Eigenschwingungen von zylindrischen Resonatoren,
*Lösung der radialen Schrödinger–Gleichung,
*Schalldruckamplituden von dünnflüssgigen Rotationsströmen,
*Wärmeleitung in zylindrischen Körpern,
*Streuungsproblem eines Gitters,
*Dynamik von Schwingkörpern,
*Winkelauflösung.

Man zählt die Besselfunktionen wegen ihrer vielfältigen Anwendungen in der mathematischen Physik zu den speziellen Funktionen.

Wir beschränken uns im Folgenden auf einige Gebiete, die in unserem Lerntutorial $\rm LNTwww$ angesprochen werden.

'''Im enlischen Original'''
Electromagnetic waves in a cylindrical waveguide
Pressure amplitudes of inviscid rotational flows
Heat conduction in a cylindrical object
Modes of vibration of a thin circular (or annular) acoustic membrane (such as a drum or other membranophone)
Diffusion problems on a lattice
Solutions to the radial Schrödinger equation (in spherical and cylindrical coordinates) for a free particle
Solving for patterns of acoustical radiation
Frequency-dependent friction in circular pipelines
Dynamics of floating bodies
Angular resolution
'''Ende'''

{{GraueBox|TEXT=
$\text{Beispiel (C):} \hspace{0.5cm} \text{Einsatz in der Spektralanalyse} \ \Rightarrow \ \text{Kaiser-Bessel-Filter}$

Als '''spektralen Leckeffekt''' bezeichnet man die Verfälschung des Spektrums eines periodischen und damit zeitlich unbegrenzten Signals aufgrund der impliziten Zeitbegrenzung der Diskreten Fouriertransformation (DFT). Dadurch werden zum Beispiel von einem Spektrumanalyzer
*im Zeitsignal nicht vorhandene Frequenzanteile vorgetäuscht, und/oder
*tatsächlich vorhandene Spektralkomponenten durch Seitenkeulen verdeckt.

Aufgabe der [[Signaldarstellung/Spektralanalyse|Spektralanalyse]] ist es, durch die Bereitstellung geeigneter Fensterfunktionen den Einfluss des ''spektralen Leckeffektes'' zu begrenzen.

Eine solche Fensterfunktion liefert zum Beispiel das Kaiser–Bessel–Fenster   ⇒   siehe Abschnitt [[Signaldarstellung/Spektralanalyse#Spezielle_Fensterfunktionen|Spezielle Fensterfunktionen]]. Dessen zeitdiskrete Fenserfunktion lautet mit der Besselfunktion nullter Ordnung   ⇒   ${\rm J}_0(x)$, dem Parameter $\alpha=3.5$ und der Fensterlänge $N$:
:$$w_\nu = \frac{ {\rm J}_0\big(\pi \cdot \alpha \cdot \sqrt{1 - (2\nu/N)^2}\big)}{ {\rm J}_0\big(\pi \cdot \alpha \big)}.$$
Auf der Seite [[Signaldarstellung/Spektralanalyse#G.C3.BCtekriterien_von_Fensterfunktionen|Gütekriterien von Fensterfunktionen]] sind u.a. die Kenngrößen des Kaiser–Bessel–Fensters angegeben:
*Günstig sind der große &bdquo;Minimale Abstand zwischen Hauptkeule und Seitenkeulen” und der gewünscht kleine &bdquo;Maximale Skalierungsfehler”.
*Aufgrund der sehr großen &bdquo;Äquivalenten Rauschbreite” schneidet das Kaiser–Bessel–Fenster im wichtigsten Vergleichskriterium &bdquo;Maximaler Prozessverlust” doch schlechter ab als die etablierten Hamming– und Hanning–Fenster.}}

{{GraueBox|TEXT=
$\text{Beispiel (D):} \hspace{0.5cm} \text{Rice-Fading-Kanalmodell}$

Die [[Mobile_Kommunikation/Wahrscheinlichkeitsdichte_des_Rayleigh%E2%80%93Fadings#Allgemeine_Beschreibung_des_Mobilfunkkanals| Rayleigh–Verteilung]] beschreibt den Mobilfunkkanal unter der Annahme, dass kein direkter Pfad vorhanden ist und sich somit der multiplikative Faktor $z(t) = x(t) + {\rm j} \cdot y(t)$ allein aus diffus gestreuten Komponenten zusammensetzt.

Bei Vorhandensein einer Direktkomponente (englisch: Line of Sight, LoS) muss man im Modell zu den mittelwertfreien Gaußprozessen $x(t)$ und $y(t)$ noch Gleichkomponenten $x_0$ und/oder $y_0$ hinzufügen:

[[File:P ID2126 Mob T 1 4 S1 v3.png|right|frame|Rice-Fading-Kanalmodell|class=fit]]
$$\hspace{0.2cm}x(t) \hspace{0.1cm} \Rightarrow \hspace{0.1cm} x(t) +x_0 \hspace{0.05cm}, \hspace{0.2cm} y(t) \hspace{0.1cm} \Rightarrow \hspace{0.1cm} y(t) +y_0\hspace{0.05cm},$$

$$\hspace{0.2cm}z(t) = x(t) + {\rm j} \cdot y(t) \hspace{0.1cm} \Rightarrow \hspace{0.1cm} z(t) +z_0 \hspace{0.05cm},\hspace{0.2cm}
z_0 = x_0 + {\rm j} \cdot y_0\hspace{0.05cm}.$$

Die Grafik zeigt das ''Rice–Fading–Kanalmodell''. Es lässt sich wie folgt zusammenfassen:
*Der Realteil $x(t)$ ist gaußverteilt mit Mittelwert $x_0$ und Varianz $\sigma ^2$.
*Der Imaginärteil $y(t)$ ist ebenfalls gaußverteilt (Mittelwert $y_0$, gleiche Varianz $\sigma ^2$) sowie unabhängig von $x(t)$. 

*Für $z_0 \ne 0$ ist der Betrag $\vert z(t)\vert$ riceverteilt, woraus die Bezeichnung &bdquo;Rice–Fading” herrührt.

*Zur Vereinfachung der Schreibweise setzen wir $\vert z(t)\vert = a(t)$. Für $a < 0$ ist die Betrags–WDF $f_a(a) \equiv 0$, für $a \ge 0$ gilt folgende Gleichung, wobei ${\rm I_0}(x)$ die modifizierte Besselfunktion nullter Ordnung bezeichnet:
:$$f_a(a) = \frac{a}{\sigma^2} \cdot {\rm e}^{ - (a^2 + \vert z_0 \vert ^2)/(2\sigma^2)} \cdot {\rm I}_0 \left [ \frac{a \cdot \vert z_0 \vert}{\sigma^2} \right ] \hspace{0.5cm}\text{mit}\hspace{0.5cm}{\rm I }_0 (u) = {\rm J }_0 ({\rm j} \cdot u) =
\sum_{k = 0}^{\infty} \frac{ (u/2)^{2k} }{k! \cdot \Gamma (k+1)}
\hspace{0.05cm}.$$
*Zwischen der modifizierten Besselfunktion und der herkömmlichen Besselfunktion ${\rm I_0}(x)$ – jeweils erster Art – besteht also der Zusammenhang ${\rm I }_0 (u) = {\rm J }_0 ({\rm j} \cdot u)$.}}

{{GraueBox|TEXT=
$\text{Beispiel (E):} \hspace{0.5cm} \text{Analyse des Frequenzspektrums von frequenzmodulierten Signalen}$

Im $\text{Beispiel (B)}$ wurde bereits gezeigt, dass die Winkelmodulation einer harmonischen Schwingung der Frequenz $f_{\rm N}$ zu einem Linienspektrum führt. Die Spektrallinien liegen um die Trägerfrequenz $f_{\rm T}$ bei $f_{\rm T} + n \cdot f_{\rm N}$ mit $n \in \{ \ \text{...}, -2, -1, \ 0, +1, +2, \text{...} \ \}$. Die Gewichte der Diraclinien sind ${\rm J }_n(\eta)$, abhängig vom Modulationsindex $\eta$.

[[File:P_ID1095__Mod_T_3_2_S4_neu.png|center|frame|Diskrete Spektren bei Phasenmodulation (links) und Frequenzmodulation (rechts)]]

Die Grafik zeigt das Betragsspektrum $\vert S_{\rm +}(f) \vert$ des analytischen Signals bei Phasenmodulation (PM) und Frequenzmodulation (FM), zwei unterschiedliche Formen der Winkelmodulation (WM). Bessellinien mit Werten kleiner als $0.03$ sind hierbei in beiden Fällen vernachlässigt.

Für die obere Bildhälfte sind die Modulatorparameter so gewählt, dass sich für $f_{\rm N} = 5 \ \rm kHz$ jeweils ein Besselspektrum mit dem Modulationsindex $η = 1.5$ ergibt. Lässt man die Phasenbeziehungen außer Acht, so ergeben sich für beide Systeme gleiche Spektren und gleiche Signale.

Die unteren Grafiken gelten bei sonst gleichen Einstellungen für die Nachrichtenfrequenz $f_{\rm N} = 3 \ \rm kHz$. Man erkennt:
*Bei der Phasenmodulation ergibt sich gegenüber $f_{\rm N} = 5 \ \rm kHz$ eine schmalere Spektralfunktion, da nun der Abstand der Bessellinien nur mehr $3 \ \rm kHz$ beträgt. Da bei PM der Modulationsindex unabhängig von $f_{\rm N}$ ist, ergeben sich die gleichen Besselgewichte wie bei $f_{\rm N} = 5 \ \rm kHz$.
*Auch bei der Frequenzmodulation treten nun die Bessellinien im Abstand von $3 \ \rm kHz$ auf. Da aber bei FM der Modulationsindex umgekehrt proportional zu $f_{\rm N}$ ist, gibt es nun unten aufgrund des größeren Modulationsindex $η = 2.5$ deutlich mehr Bessellinien als im rechten oberen (für $η = 1.5$ gültigen) Diagramm. }}

'''Das folgende Kapitel muss noch angepasst werden!'''

==Zur Handhabung des Applets==
[[File:Handhabung_binomial.png|left|600px]]
    '''(A)'''     Vorauswahl für blauen Parametersatz

    '''(B)'''     Parametereingabe $I$ und $p$ per Slider

    '''(C)'''     Vorauswahl für roten Parametersatz

    '''(D)'''     Parametereingabe $\lambda$ per Slider

    '''(E)'''     Graphische Darstellung der Verteilungen

    '''(F)'''     Momentenausgabe für blauen Parametersatz

    '''(G)'''     Momentenausgabe für roten Parametersatz

    '''(H)'''     Variation der grafischen Darstellung

$\hspace{1.5cm}$&bdquo;$+$” (Vergrößern),

$\hspace{1.5cm}$ &bdquo;$-$” (Verkleinern)

$\hspace{1.5cm}$ &bdquo;$\rm o$” (Zurücksetzen)

$\hspace{1.5cm}$ &bdquo;$\leftarrow$” (Verschieben nach links), usw.

    '''( I )'''     Ausgabe von ${\rm Pr} (z = \mu)$ und ${\rm Pr} (z \le \mu)$

    '''(J)'''     Bereich für die Versuchsdurchführung
 
 '''Andere Möglichkeiten zur Variation der grafischen Darstellung''':
*Gedrückte Shifttaste und Scrollen: Zoomen im Koordinatensystem,
*Gedrückte Shifttaste und linke Maustaste: Verschieben des Koordinatensystems.

==Über die Autoren==
Dieses interaktive Berechnungstool wurde am [http://www.lnt.ei.tum.de/startseite Lehrstuhl für Nachrichtentechnik] der [https://www.tum.de/ Technischen Universität München] konzipiert und realisiert.
*Die erste Version wurde 2006 von [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Studierende#Markus_Elsberger_.28Diplomarbeit_LB_2006.29|Markus Elsberger]] und [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Studierende#Slim_Lamine_.28Studienarbeit_EI_2006.29|Slim Lamine]] im Rahmen von Abschlussarbeiten mit &bdquo;FlashMX–Actionscript” erstellt (Betreuer: [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Mitarbeiter_und_Dozenten#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28am_LNT_seit_1974.29|Günter Söder]]).
*2018 wurde das Programm von [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Studierende#Xiaohan_Liu_.28Bachelorarbeit_2018.29|Xiaohan Liu]] (Bachelorarbeit, Betreuer: [[Biografien_und_Bibliografien/Beteiligte_der_Professur_Leitungsgebundene_%C3%9Cbertragungstechnik#Tasn.C3.A1d_Kernetzky.2C_M.Sc._.28bei_L.C3.9CT_seit_2014.29|Tasnád Kernetzky]] ) auf &bdquo;HTML5” umgesetzt.

==Nochmalige Aufrufmöglichkeit des Applets in neuem Fenster==

{{LntAppletLink|bessel}}

Applets:Bessel Functions of the First Kind

2018-08-16T14:04:12Z

Xiaohan: /* Theoretischer Hintergrund */

{{LntAppletLink|bessel}}

==Applet Description==
 
This applet allows the calculation and graphical representation of the Bessel functions of the first kind and $n$–th order according to the series representation:

:$${\rm J}_n (x) = \sum\limits_{k=0}^{\infty}\frac{(-1)^k \cdot (x/2)^{n \hspace{0.05cm} + \hspace{0.05cm} 2
\hspace{0.02cm}\cdot \hspace{0.05cm}k}}{k! \cdot (n+k)!} \hspace{0.05cm}.$$

*Graphically represented, the functions ${\rm J}_n (x)$ for the order $n=0$ to $n=9$ can become different colors.
*The left-hand output provides the function values ${\rm J}_0 (x = x_1)$, ... , ${\rm J}_9 (x = x_1)$ for a slider-settable value $x_1$ in the range $0 \le x_1 \le 15$ with increment $0.5$.
*The right hand side returns the function values ${\rm J}_0 (x = x_2)$, ... , ${\rm J}_9 (x = x_2)$ for a slider-settable value $x_2$ (same range and value) Increment as on the left).

[[Applets:Physical_Signal_%26_Analytical_Signal|'''German description''']]

==Theoretical Background==
 
===General Information about the Bessel Functions===
Bessel functions (or cylinder functions) are solutions of the Bessel differential equation of the form

:$$x^2 \cdot \frac{ {\rm d}^2}{{\rm d}x^2}\ {\rm J}_n (x) \ + \ x \cdot \frac{ {\rm d}}{{\rm d}x}\ {\rm J}_n (x) \ + \ (x^2 - n^2)
\cdot {\rm J}_n (x)= 0. $$

This is an ordinary second-order linear differential equation. The parameter $ n $ is usually integer, as in this program. These mathematical functions, which were introduced in 1844 by [https://de.wikipedia.org/wiki/Friedrich_Wilhelm_Bessel Friedrich Wilhelm Bessel], can also be represented in closed form as integrals:

:$${\rm J}_n (x) = \frac{1}{2\pi}\cdot \int_{-\pi}^{+\pi} {{\rm e}^{\hspace{0.05cm}{\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}[\hspace{0.05cm}x \hspace{0.05cm}\cdot \hspace{0.05cm}\sin(\alpha) -\hspace{0.05cm} n \hspace{0.05cm}\cdot \hspace{0.05cm}\alpha \hspace{0.05cm}]}}\hspace{0.1cm}{\rm d}\alpha
\hspace{0.05cm}.$$

The functions ${\rm J}_n (x)$ belong to the class of Bessel functions of the first kind (German:   ''Besselfunktionen erster Art''). The parameter $n$ is called the ''Order''.

''Annotation:''   There are a number of modifications of the Bessel functions, including the second-order Bessel functions named ${\rm Y}_n (x)$. For integer $n$, ${\rm Y}_n (x)$ can be replaced by ${\rm J}_n (x)$–functions. However, in this applet, only the first-order Bessel functions are   ⇒   ${\rm J}_n (x)$ is considered.
 
===Eigenschaften der Besselfunktionen===

{{BlaueBox|TEXT=
$\text{Eigenschaft (A):}$   Sind die Funktionswerte für $n = 0$ und $n = 1$ bekannt, so können daraus die Besselfunktionen für $n ≥ 2$ iterativ ermittelt werden:
:$${\rm J}_n (x) ={2 \cdot (n-1)}/{x} \cdot {\rm J}_{n-1} (x) - {\rm J}_{n-2} (x) \hspace{0.05cm}.$$
}}

{{GraueBox|TEXT=
$\text{Beispiel (A):}$   Es gelte ${\rm J}_0 (x = 2) = 0.22389$ und ${\rm J}_1 (x= 2) = 0.57672$. Daraus können iterativ berechnet werden:
:$${\rm J}_2 (x= 2) ={2 \cdot 1}/{2} \cdot {\rm J}_{1} (x= 2) - {\rm J}_{0} (x= 2) = 0.57672 - 0.22389 = \hspace{0.15cm}\underline{0.35283}\hspace{0.05cm},$$
:$${\rm J}_3 (x= 2) ={2 \cdot 2}/{2} \cdot {\rm J}_{2} (x= 2) - {\rm J}_{1} (x= 2) = 2 \cdot 0.35283 - 0.57672 = \hspace{0.15cm}\underline{0.12894}\hspace{0.05cm},$$
:$${\rm J}_4 (x= 2) ={2 \cdot 3}/{2} \cdot {\rm J}_{3} (x= 2) - {\rm J}_{2} (x= 2) = 3 \cdot 0.12894 - 0.35283 = \hspace{0.15cm}\underline{0.03400}\hspace{0.05cm}.$$
}}

{{BlaueBox|TEXT=
$\text{Eigenschaft (B):}$   Es gilt die Symmetriebeziehung ${\rm J}_{–n}(x) = (–1)^n · {\rm J}_n(x)$:
:$${\rm J}_{-1}(x) = - {\rm J}_{1}(x), \hspace{0.3cm}{\rm J}_{-2}(x) = {\rm J}_{2}(x), \hspace{0.3cm}{\rm J}_{-3}(x) = - {\rm J}_{3}(x).$$
}}

{{GraueBox|TEXT=
$\text{Beispiel (B):}$   Für das Spektrum des analytischen Signals gilt bei [[Modulationsverfahren/Phasenmodulation_(PM)#Spektralfunktion_eines_phasenmodulierten_Sinussignals|Phasenmodulation eines Sinussignals]]:
[[File:Mod_T_3_1_S4_version2.png|right|frame|Spektrum des analytischen Signals bei Phasenmodulation]]
:$$S_{\rm +}(f) = A_{\rm T} \cdot \sum_{n = - \infty}^{+\infty}{\rm J}_n (\eta) \cdot \delta \big[f - (f_{\rm T}+ n \cdot f_{\rm N})\big]\hspace{0.05cm}.$$
Hierbei bezeichnen
*$f_{\rm T}$ die Trägerfrequenz,
*$f_{\rm N}$ die Nachrichtenfrequenz,
* $A_{\rm T}$ die Trägeramplitude.

Der Parameter der Besselfunktionen ist bei dieser Anwendung der Modulationsindex $\eta$.

Anhand der Grafik sind folgende Aussagen möglich:
*$S_+(f)$ besteht hier aus unendlich vielen diskreten Linien im Abstand von $f_{\rm N}$.
*Es ist somit prinzipiell unendlich weit ausgedehnt.
*Die Gewichte der Spektrallinien bei $f_{\rm T} + n · f_{\rm N}$ ($n$ ganzzahlig) sind durch den Modulationsindex $η$ über die Besselfunktionen ${\rm J}_n(η)$ festgelegt.
*Die Spektrallinien sind bei sinusförmigem Quellensignal und cosinusförmigem Träger reell und für gerades $n$ symmetrisch um $f_{\rm T}$.
*Bei ungeradem $n$ ist ein Vorzeichenwechsel entsprechend der $\text{Eigenschaft (B)}$ zu berücksichtigen.
*Die Phasenmodulation einer Schwingung mit anderer Phase von Quellen– und/oder Trägersignal liefert das gleiche Betragsspektrum.}}
 
===Anwendungen der Besselfunktionen===

Die Anwendungen der Besselfunktionen in den Natur– und Ingenieurswissenschaften sind vielfältig und spielen eine wichtige Rolle in der Physik, zum Beispiel:
*Untersuchung von Eigenschwingungen von zylindrischen Resonatoren,
*Lösung der radialen Schrödinger–Gleichung,
*Schalldruckamplituden von dünnflüssgigen Rotationsströmen,
*Wärmeleitung in zylindrischen Körpern,
*Streuungsproblem eines Gitters,
*Dynamik von Schwingkörpern,
*Winkelauflösung.

Man zählt die Besselfunktionen wegen ihrer vielfältigen Anwendungen in der mathematischen Physik zu den speziellen Funktionen.

Wir beschränken uns im Folgenden auf einige Gebiete, die in unserem Lerntutorial $\rm LNTwww$ angesprochen werden.

'''Im enlischen Original'''
Electromagnetic waves in a cylindrical waveguide
Pressure amplitudes of inviscid rotational flows
Heat conduction in a cylindrical object
Modes of vibration of a thin circular (or annular) acoustic membrane (such as a drum or other membranophone)
Diffusion problems on a lattice
Solutions to the radial Schrödinger equation (in spherical and cylindrical coordinates) for a free particle
Solving for patterns of acoustical radiation
Frequency-dependent friction in circular pipelines
Dynamics of floating bodies
Angular resolution
'''Ende'''

{{GraueBox|TEXT=
$\text{Beispiel (C):} \hspace{0.5cm} \text{Einsatz in der Spektralanalyse} \ \Rightarrow \ \text{Kaiser-Bessel-Filter}$

Als '''spektralen Leckeffekt''' bezeichnet man die Verfälschung des Spektrums eines periodischen und damit zeitlich unbegrenzten Signals aufgrund der impliziten Zeitbegrenzung der Diskreten Fouriertransformation (DFT). Dadurch werden zum Beispiel von einem Spektrumanalyzer
*im Zeitsignal nicht vorhandene Frequenzanteile vorgetäuscht, und/oder
*tatsächlich vorhandene Spektralkomponenten durch Seitenkeulen verdeckt.

Aufgabe der [[Signaldarstellung/Spektralanalyse|Spektralanalyse]] ist es, durch die Bereitstellung geeigneter Fensterfunktionen den Einfluss des ''spektralen Leckeffektes'' zu begrenzen.

Eine solche Fensterfunktion liefert zum Beispiel das Kaiser–Bessel–Fenster   ⇒   siehe Abschnitt [[Signaldarstellung/Spektralanalyse#Spezielle_Fensterfunktionen|Spezielle Fensterfunktionen]]. Dessen zeitdiskrete Fenserfunktion lautet mit der Besselfunktion nullter Ordnung   ⇒   ${\rm J}_0(x)$, dem Parameter $\alpha=3.5$ und der Fensterlänge $N$:
:$$w_\nu = \frac{ {\rm J}_0\big(\pi \cdot \alpha \cdot \sqrt{1 - (2\nu/N)^2}\big)}{ {\rm J}_0\big(\pi \cdot \alpha \big)}.$$
Auf der Seite [[Signaldarstellung/Spektralanalyse#G.C3.BCtekriterien_von_Fensterfunktionen|Gütekriterien von Fensterfunktionen]] sind u.a. die Kenngrößen des Kaiser–Bessel–Fensters angegeben:
*Günstig sind der große &bdquo;Minimale Abstand zwischen Hauptkeule und Seitenkeulen” und der gewünscht kleine &bdquo;Maximale Skalierungsfehler”.
*Aufgrund der sehr großen &bdquo;Äquivalenten Rauschbreite” schneidet das Kaiser–Bessel–Fenster im wichtigsten Vergleichskriterium &bdquo;Maximaler Prozessverlust” doch schlechter ab als die etablierten Hamming– und Hanning–Fenster.}}

{{GraueBox|TEXT=
$\text{Beispiel (D):} \hspace{0.5cm} \text{Rice-Fading-Kanalmodell}$

Die [[Mobile_Kommunikation/Wahrscheinlichkeitsdichte_des_Rayleigh%E2%80%93Fadings#Allgemeine_Beschreibung_des_Mobilfunkkanals| Rayleigh–Verteilung]] beschreibt den Mobilfunkkanal unter der Annahme, dass kein direkter Pfad vorhanden ist und sich somit der multiplikative Faktor $z(t) = x(t) + {\rm j} \cdot y(t)$ allein aus diffus gestreuten Komponenten zusammensetzt.

Bei Vorhandensein einer Direktkomponente (englisch: Line of Sight, LoS) muss man im Modell zu den mittelwertfreien Gaußprozessen $x(t)$ und $y(t)$ noch Gleichkomponenten $x_0$ und/oder $y_0$ hinzufügen:

[[File:P ID2126 Mob T 1 4 S1 v3.png|right|frame|Rice-Fading-Kanalmodell|class=fit]]
$$\hspace{0.2cm}x(t) \hspace{0.1cm} \Rightarrow \hspace{0.1cm} x(t) +x_0 \hspace{0.05cm}, \hspace{0.2cm} y(t) \hspace{0.1cm} \Rightarrow \hspace{0.1cm} y(t) +y_0\hspace{0.05cm},$$

$$\hspace{0.2cm}z(t) = x(t) + {\rm j} \cdot y(t) \hspace{0.1cm} \Rightarrow \hspace{0.1cm} z(t) +z_0 \hspace{0.05cm},\hspace{0.2cm}
z_0 = x_0 + {\rm j} \cdot y_0\hspace{0.05cm}.$$

Die Grafik zeigt das ''Rice–Fading–Kanalmodell''. Es lässt sich wie folgt zusammenfassen:
*Der Realteil $x(t)$ ist gaußverteilt mit Mittelwert $x_0$ und Varianz $\sigma ^2$.
*Der Imaginärteil $y(t)$ ist ebenfalls gaußverteilt (Mittelwert $y_0$, gleiche Varianz $\sigma ^2$) sowie unabhängig von $x(t)$. 

*Für $z_0 \ne 0$ ist der Betrag $\vert z(t)\vert$ riceverteilt, woraus die Bezeichnung &bdquo;Rice–Fading” herrührt.

*Zur Vereinfachung der Schreibweise setzen wir $\vert z(t)\vert = a(t)$. Für $a < 0$ ist die Betrags–WDF $f_a(a) \equiv 0$, für $a \ge 0$ gilt folgende Gleichung, wobei ${\rm I_0}(x)$ die modifizierte Besselfunktion nullter Ordnung bezeichnet:
:$$f_a(a) = \frac{a}{\sigma^2} \cdot {\rm e}^{ - (a^2 + \vert z_0 \vert ^2)/(2\sigma^2)} \cdot {\rm I}_0 \left [ \frac{a \cdot \vert z_0 \vert}{\sigma^2} \right ] \hspace{0.5cm}\text{mit}\hspace{0.5cm}{\rm I }_0 (u) = {\rm J }_0 ({\rm j} \cdot u) =
\sum_{k = 0}^{\infty} \frac{ (u/2)^{2k} }{k! \cdot \Gamma (k+1)}
\hspace{0.05cm}.$$
*Zwischen der modifizierten Besselfunktion und der herkömmlichen Besselfunktion ${\rm I_0}(x)$ – jeweils erster Art – besteht also der Zusammenhang ${\rm I }_0 (u) = {\rm J }_0 ({\rm j} \cdot u)$.}}

{{GraueBox|TEXT=
$\text{Beispiel (E):} \hspace{0.5cm} \text{Analyse des Frequenzspektrums von frequenzmodulierten Signalen}$

Im $\text{Beispiel (B)}$ wurde bereits gezeigt, dass die Winkelmodulation einer harmonischen Schwingung der Frequenz $f_{\rm N}$ zu einem Linienspektrum führt. Die Spektrallinien liegen um die Trägerfrequenz $f_{\rm T}$ bei $f_{\rm T} + n \cdot f_{\rm N}$ mit $n \in \{ \ \text{...}, -2, -1, \ 0, +1, +2, \text{...} \ \}$. Die Gewichte der Diraclinien sind ${\rm J }_n(\eta)$, abhängig vom Modulationsindex $\eta$.

[[File:P_ID1095__Mod_T_3_2_S4_neu.png|center|frame|Diskrete Spektren bei Phasenmodulation (links) und Frequenzmodulation (rechts)]]

Die Grafik zeigt das Betragsspektrum $\vert S_{\rm +}(f) \vert$ des analytischen Signals bei Phasenmodulation (PM) und Frequenzmodulation (FM), zwei unterschiedliche Formen der Winkelmodulation (WM). Bessellinien mit Werten kleiner als $0.03$ sind hierbei in beiden Fällen vernachlässigt.

Für die obere Bildhälfte sind die Modulatorparameter so gewählt, dass sich für $f_{\rm N} = 5 \ \rm kHz$ jeweils ein Besselspektrum mit dem Modulationsindex $η = 1.5$ ergibt. Lässt man die Phasenbeziehungen außer Acht, so ergeben sich für beide Systeme gleiche Spektren und gleiche Signale.

Die unteren Grafiken gelten bei sonst gleichen Einstellungen für die Nachrichtenfrequenz $f_{\rm N} = 3 \ \rm kHz$. Man erkennt:
*Bei der Phasenmodulation ergibt sich gegenüber $f_{\rm N} = 5 \ \rm kHz$ eine schmalere Spektralfunktion, da nun der Abstand der Bessellinien nur mehr $3 \ \rm kHz$ beträgt. Da bei PM der Modulationsindex unabhängig von $f_{\rm N}$ ist, ergeben sich die gleichen Besselgewichte wie bei $f_{\rm N} = 5 \ \rm kHz$.
*Auch bei der Frequenzmodulation treten nun die Bessellinien im Abstand von $3 \ \rm kHz$ auf. Da aber bei FM der Modulationsindex umgekehrt proportional zu $f_{\rm N}$ ist, gibt es nun unten aufgrund des größeren Modulationsindex $η = 2.5$ deutlich mehr Bessellinien als im rechten oberen (für $η = 1.5$ gültigen) Diagramm. }}

'''Das folgende Kapitel muss noch angepasst werden!'''

==Zur Handhabung des Applets==
[[File:Handhabung_binomial.png|left|600px]]
    '''(A)'''     Vorauswahl für blauen Parametersatz

    '''(B)'''     Parametereingabe $I$ und $p$ per Slider

    '''(C)'''     Vorauswahl für roten Parametersatz

    '''(D)'''     Parametereingabe $\lambda$ per Slider

    '''(E)'''     Graphische Darstellung der Verteilungen

    '''(F)'''     Momentenausgabe für blauen Parametersatz

    '''(G)'''     Momentenausgabe für roten Parametersatz

    '''(H)'''     Variation der grafischen Darstellung

$\hspace{1.5cm}$&bdquo;$+$” (Vergrößern),

$\hspace{1.5cm}$ &bdquo;$-$” (Verkleinern)

$\hspace{1.5cm}$ &bdquo;$\rm o$” (Zurücksetzen)

$\hspace{1.5cm}$ &bdquo;$\leftarrow$” (Verschieben nach links), usw.

    '''( I )'''     Ausgabe von ${\rm Pr} (z = \mu)$ und ${\rm Pr} (z \le \mu)$

    '''(J)'''     Bereich für die Versuchsdurchführung
 
 '''Andere Möglichkeiten zur Variation der grafischen Darstellung''':
*Gedrückte Shifttaste und Scrollen: Zoomen im Koordinatensystem,
*Gedrückte Shifttaste und linke Maustaste: Verschieben des Koordinatensystems.

==Über die Autoren==
Dieses interaktive Berechnungstool wurde am [http://www.lnt.ei.tum.de/startseite Lehrstuhl für Nachrichtentechnik] der [https://www.tum.de/ Technischen Universität München] konzipiert und realisiert.
*Die erste Version wurde 2006 von [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Studierende#Markus_Elsberger_.28Diplomarbeit_LB_2006.29|Markus Elsberger]] und [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Studierende#Slim_Lamine_.28Studienarbeit_EI_2006.29|Slim Lamine]] im Rahmen von Abschlussarbeiten mit &bdquo;FlashMX–Actionscript” erstellt (Betreuer: [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Mitarbeiter_und_Dozenten#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28am_LNT_seit_1974.29|Günter Söder]]).
*2018 wurde das Programm von [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Studierende#Xiaohan_Liu_.28Bachelorarbeit_2018.29|Xiaohan Liu]] (Bachelorarbeit, Betreuer: [[Biografien_und_Bibliografien/Beteiligte_der_Professur_Leitungsgebundene_%C3%9Cbertragungstechnik#Tasn.C3.A1d_Kernetzky.2C_M.Sc._.28bei_L.C3.9CT_seit_2014.29|Tasnád Kernetzky]] ) auf &bdquo;HTML5” umgesetzt.

==Nochmalige Aufrufmöglichkeit des Applets in neuem Fenster==

{{LntAppletLink|bessel}}

Applets:Bessel Functions of the First Kind

2018-08-16T13:51:34Z

Xiaohan: /* Programmbeschreibung */

{{LntAppletLink|bessel}}

==Applet Description==
 
This applet allows the calculation and graphical representation of the Bessel functions of the first kind and $n$–th order according to the series representation:

:$${\rm J}_n (x) = \sum\limits_{k=0}^{\infty}\frac{(-1)^k \cdot (x/2)^{n \hspace{0.05cm} + \hspace{0.05cm} 2
\hspace{0.02cm}\cdot \hspace{0.05cm}k}}{k! \cdot (n+k)!} \hspace{0.05cm}.$$

*Graphically represented, the functions ${\rm J}_n (x)$ for the order $n=0$ to $n=9$ can become different colors.
*The left-hand output provides the function values ${\rm J}_0 (x = x_1)$, ... , ${\rm J}_9 (x = x_1)$ for a slider-settable value $x_1$ in the range $0 \le x_1 \le 15$ with increment $0.5$.
*The right hand side returns the function values ${\rm J}_0 (x = x_2)$, ... , ${\rm J}_9 (x = x_2)$ for a slider-settable value $x_2$ (same range and value) Increment as on the left).

[[Applets:Physical_Signal_%26_Analytical_Signal|'''German description''']]

==Theoretischer Hintergrund==
 
===Allgemeines zu den Besselfunktionen===
Besselfunktionen (oder auch Zylinderfunktionen) sind Lösungen der Besselschen Differentialgleichung der Form

:$$x^2 \cdot \frac{ {\rm d}^2}{{\rm d}x^2}\ {\rm J}_n (x) \ + \ x \cdot \frac{ {\rm d}}{{\rm d}x}\ {\rm J}_n (x) \ + \ (x^2 - n^2)
\cdot {\rm J}_n (x)= 0. $$

Hierbei handelt es sich um eine gewöhnliche lineare Differentialgleichung zweiter Ordnung. Der Parameter $n$ ist meistens ganzzahlig, so auch in diesem Programm. Diese bereits 1844 von [https://de.wikipedia.org/wiki/Friedrich_Wilhelm_Bessel Friedrich Wilhelm Bessel] eingeführten mathematischen Funktionen können auch in geschlossener Form als Integrale dargestellt werden:

:$${\rm J}_n (x) = \frac{1}{2\pi}\cdot \int_{-\pi}^{+\pi} {{\rm e}^{\hspace{0.05cm}{\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}[\hspace{0.05cm}x \hspace{0.05cm}\cdot \hspace{0.05cm}\sin(\alpha) -\hspace{0.05cm} n \hspace{0.05cm}\cdot \hspace{0.05cm}\alpha \hspace{0.05cm}]}}\hspace{0.1cm}{\rm d}\alpha
\hspace{0.05cm}.$$

Die Funktionen ${\rm J}_n (x)$ gehören zur Klasse der Besselfunktionen erster Art (englisch:   ''Bessel Functions of the First Kind''). Den Parameter $n$ nennt man die ''Ordnung''.

''Anmerkung:''   Es gibt eine Vielzahl von Abwandlungen der Besselfunktionen, unter anderem die mit ${\rm Y}_n (x)$ benannten Besselfunktionen zweiter Art. Für ganzzahliges $n$ lässt sich ${\rm Y}_n (x)$ durch ${\rm J}_n (x)$–Funktionen ausdrücken. In diesem Applet werden jedoch nur die Besselfunktionen erster Art   ⇒   ${\rm J}_n (x)$ betrachtet.
 
===Eigenschaften der Besselfunktionen===

{{BlaueBox|TEXT=
$\text{Eigenschaft (A):}$   Sind die Funktionswerte für $n = 0$ und $n = 1$ bekannt, so können daraus die Besselfunktionen für $n ≥ 2$ iterativ ermittelt werden:
:$${\rm J}_n (x) ={2 \cdot (n-1)}/{x} \cdot {\rm J}_{n-1} (x) - {\rm J}_{n-2} (x) \hspace{0.05cm}.$$
}}

{{GraueBox|TEXT=
$\text{Beispiel (A):}$   Es gelte ${\rm J}_0 (x = 2) = 0.22389$ und ${\rm J}_1 (x= 2) = 0.57672$. Daraus können iterativ berechnet werden:
:$${\rm J}_2 (x= 2) ={2 \cdot 1}/{2} \cdot {\rm J}_{1} (x= 2) - {\rm J}_{0} (x= 2) = 0.57672 - 0.22389 = \hspace{0.15cm}\underline{0.35283}\hspace{0.05cm},$$
:$${\rm J}_3 (x= 2) ={2 \cdot 2}/{2} \cdot {\rm J}_{2} (x= 2) - {\rm J}_{1} (x= 2) = 2 \cdot 0.35283 - 0.57672 = \hspace{0.15cm}\underline{0.12894}\hspace{0.05cm},$$
:$${\rm J}_4 (x= 2) ={2 \cdot 3}/{2} \cdot {\rm J}_{3} (x= 2) - {\rm J}_{2} (x= 2) = 3 \cdot 0.12894 - 0.35283 = \hspace{0.15cm}\underline{0.03400}\hspace{0.05cm}.$$
}}

{{BlaueBox|TEXT=
$\text{Eigenschaft (B):}$   Es gilt die Symmetriebeziehung ${\rm J}_{–n}(x) = (–1)^n · {\rm J}_n(x)$:
:$${\rm J}_{-1}(x) = - {\rm J}_{1}(x), \hspace{0.3cm}{\rm J}_{-2}(x) = {\rm J}_{2}(x), \hspace{0.3cm}{\rm J}_{-3}(x) = - {\rm J}_{3}(x).$$
}}

{{GraueBox|TEXT=
$\text{Beispiel (B):}$   Für das Spektrum des analytischen Signals gilt bei [[Modulationsverfahren/Phasenmodulation_(PM)#Spektralfunktion_eines_phasenmodulierten_Sinussignals|Phasenmodulation eines Sinussignals]]:
[[File:Mod_T_3_1_S4_version2.png|right|frame|Spektrum des analytischen Signals bei Phasenmodulation]]
:$$S_{\rm +}(f) = A_{\rm T} \cdot \sum_{n = - \infty}^{+\infty}{\rm J}_n (\eta) \cdot \delta \big[f - (f_{\rm T}+ n \cdot f_{\rm N})\big]\hspace{0.05cm}.$$
Hierbei bezeichnen
*$f_{\rm T}$ die Trägerfrequenz,
*$f_{\rm N}$ die Nachrichtenfrequenz,
* $A_{\rm T}$ die Trägeramplitude.

Der Parameter der Besselfunktionen ist bei dieser Anwendung der Modulationsindex $\eta$.

Anhand der Grafik sind folgende Aussagen möglich:
*$S_+(f)$ besteht hier aus unendlich vielen diskreten Linien im Abstand von $f_{\rm N}$.
*Es ist somit prinzipiell unendlich weit ausgedehnt.
*Die Gewichte der Spektrallinien bei $f_{\rm T} + n · f_{\rm N}$ ($n$ ganzzahlig) sind durch den Modulationsindex $η$ über die Besselfunktionen ${\rm J}_n(η)$ festgelegt.
*Die Spektrallinien sind bei sinusförmigem Quellensignal und cosinusförmigem Träger reell und für gerades $n$ symmetrisch um $f_{\rm T}$.
*Bei ungeradem $n$ ist ein Vorzeichenwechsel entsprechend der $\text{Eigenschaft (B)}$ zu berücksichtigen.
*Die Phasenmodulation einer Schwingung mit anderer Phase von Quellen– und/oder Trägersignal liefert das gleiche Betragsspektrum.}}
 
===Anwendungen der Besselfunktionen===

Die Anwendungen der Besselfunktionen in den Natur– und Ingenieurswissenschaften sind vielfältig und spielen eine wichtige Rolle in der Physik, zum Beispiel:
*Untersuchung von Eigenschwingungen von zylindrischen Resonatoren,
*Lösung der radialen Schrödinger–Gleichung,
*Schalldruckamplituden von dünnflüssgigen Rotationsströmen,
*Wärmeleitung in zylindrischen Körpern,
*Streuungsproblem eines Gitters,
*Dynamik von Schwingkörpern,
*Winkelauflösung.

Man zählt die Besselfunktionen wegen ihrer vielfältigen Anwendungen in der mathematischen Physik zu den speziellen Funktionen.

Wir beschränken uns im Folgenden auf einige Gebiete, die in unserem Lerntutorial $\rm LNTwww$ angesprochen werden.

'''Im enlischen Original'''
Electromagnetic waves in a cylindrical waveguide
Pressure amplitudes of inviscid rotational flows
Heat conduction in a cylindrical object
Modes of vibration of a thin circular (or annular) acoustic membrane (such as a drum or other membranophone)
Diffusion problems on a lattice
Solutions to the radial Schrödinger equation (in spherical and cylindrical coordinates) for a free particle
Solving for patterns of acoustical radiation
Frequency-dependent friction in circular pipelines
Dynamics of floating bodies
Angular resolution
'''Ende'''

{{GraueBox|TEXT=
$\text{Beispiel (C):} \hspace{0.5cm} \text{Einsatz in der Spektralanalyse} \ \Rightarrow \ \text{Kaiser-Bessel-Filter}$

Als '''spektralen Leckeffekt''' bezeichnet man die Verfälschung des Spektrums eines periodischen und damit zeitlich unbegrenzten Signals aufgrund der impliziten Zeitbegrenzung der Diskreten Fouriertransformation (DFT). Dadurch werden zum Beispiel von einem Spektrumanalyzer
*im Zeitsignal nicht vorhandene Frequenzanteile vorgetäuscht, und/oder
*tatsächlich vorhandene Spektralkomponenten durch Seitenkeulen verdeckt.

Aufgabe der [[Signaldarstellung/Spektralanalyse|Spektralanalyse]] ist es, durch die Bereitstellung geeigneter Fensterfunktionen den Einfluss des ''spektralen Leckeffektes'' zu begrenzen.

Eine solche Fensterfunktion liefert zum Beispiel das Kaiser–Bessel–Fenster   ⇒   siehe Abschnitt [[Signaldarstellung/Spektralanalyse#Spezielle_Fensterfunktionen|Spezielle Fensterfunktionen]]. Dessen zeitdiskrete Fenserfunktion lautet mit der Besselfunktion nullter Ordnung   ⇒   ${\rm J}_0(x)$, dem Parameter $\alpha=3.5$ und der Fensterlänge $N$:
:$$w_\nu = \frac{ {\rm J}_0\big(\pi \cdot \alpha \cdot \sqrt{1 - (2\nu/N)^2}\big)}{ {\rm J}_0\big(\pi \cdot \alpha \big)}.$$
Auf der Seite [[Signaldarstellung/Spektralanalyse#G.C3.BCtekriterien_von_Fensterfunktionen|Gütekriterien von Fensterfunktionen]] sind u.a. die Kenngrößen des Kaiser–Bessel–Fensters angegeben:
*Günstig sind der große &bdquo;Minimale Abstand zwischen Hauptkeule und Seitenkeulen” und der gewünscht kleine &bdquo;Maximale Skalierungsfehler”.
*Aufgrund der sehr großen &bdquo;Äquivalenten Rauschbreite” schneidet das Kaiser–Bessel–Fenster im wichtigsten Vergleichskriterium &bdquo;Maximaler Prozessverlust” doch schlechter ab als die etablierten Hamming– und Hanning–Fenster.}}

{{GraueBox|TEXT=
$\text{Beispiel (D):} \hspace{0.5cm} \text{Rice-Fading-Kanalmodell}$

Die [[Mobile_Kommunikation/Wahrscheinlichkeitsdichte_des_Rayleigh%E2%80%93Fadings#Allgemeine_Beschreibung_des_Mobilfunkkanals| Rayleigh–Verteilung]] beschreibt den Mobilfunkkanal unter der Annahme, dass kein direkter Pfad vorhanden ist und sich somit der multiplikative Faktor $z(t) = x(t) + {\rm j} \cdot y(t)$ allein aus diffus gestreuten Komponenten zusammensetzt.

Bei Vorhandensein einer Direktkomponente (englisch: Line of Sight, LoS) muss man im Modell zu den mittelwertfreien Gaußprozessen $x(t)$ und $y(t)$ noch Gleichkomponenten $x_0$ und/oder $y_0$ hinzufügen:

[[File:P ID2126 Mob T 1 4 S1 v3.png|right|frame|Rice-Fading-Kanalmodell|class=fit]]
$$\hspace{0.2cm}x(t) \hspace{0.1cm} \Rightarrow \hspace{0.1cm} x(t) +x_0 \hspace{0.05cm}, \hspace{0.2cm} y(t) \hspace{0.1cm} \Rightarrow \hspace{0.1cm} y(t) +y_0\hspace{0.05cm},$$

$$\hspace{0.2cm}z(t) = x(t) + {\rm j} \cdot y(t) \hspace{0.1cm} \Rightarrow \hspace{0.1cm} z(t) +z_0 \hspace{0.05cm},\hspace{0.2cm}
z_0 = x_0 + {\rm j} \cdot y_0\hspace{0.05cm}.$$

Die Grafik zeigt das ''Rice–Fading–Kanalmodell''. Es lässt sich wie folgt zusammenfassen:
*Der Realteil $x(t)$ ist gaußverteilt mit Mittelwert $x_0$ und Varianz $\sigma ^2$.
*Der Imaginärteil $y(t)$ ist ebenfalls gaußverteilt (Mittelwert $y_0$, gleiche Varianz $\sigma ^2$) sowie unabhängig von $x(t)$. 

*Für $z_0 \ne 0$ ist der Betrag $\vert z(t)\vert$ riceverteilt, woraus die Bezeichnung &bdquo;Rice–Fading” herrührt.

*Zur Vereinfachung der Schreibweise setzen wir $\vert z(t)\vert = a(t)$. Für $a < 0$ ist die Betrags–WDF $f_a(a) \equiv 0$, für $a \ge 0$ gilt folgende Gleichung, wobei ${\rm I_0}(x)$ die modifizierte Besselfunktion nullter Ordnung bezeichnet:
:$$f_a(a) = \frac{a}{\sigma^2} \cdot {\rm e}^{ - (a^2 + \vert z_0 \vert ^2)/(2\sigma^2)} \cdot {\rm I}_0 \left [ \frac{a \cdot \vert z_0 \vert}{\sigma^2} \right ] \hspace{0.5cm}\text{mit}\hspace{0.5cm}{\rm I }_0 (u) = {\rm J }_0 ({\rm j} \cdot u) =
\sum_{k = 0}^{\infty} \frac{ (u/2)^{2k} }{k! \cdot \Gamma (k+1)}
\hspace{0.05cm}.$$
*Zwischen der modifizierten Besselfunktion und der herkömmlichen Besselfunktion ${\rm I_0}(x)$ – jeweils erster Art – besteht also der Zusammenhang ${\rm I }_0 (u) = {\rm J }_0 ({\rm j} \cdot u)$.}}

{{GraueBox|TEXT=
$\text{Beispiel (E):} \hspace{0.5cm} \text{Analyse des Frequenzspektrums von frequenzmodulierten Signalen}$

Im $\text{Beispiel (B)}$ wurde bereits gezeigt, dass die Winkelmodulation einer harmonischen Schwingung der Frequenz $f_{\rm N}$ zu einem Linienspektrum führt. Die Spektrallinien liegen um die Trägerfrequenz $f_{\rm T}$ bei $f_{\rm T} + n \cdot f_{\rm N}$ mit $n \in \{ \ \text{...}, -2, -1, \ 0, +1, +2, \text{...} \ \}$. Die Gewichte der Diraclinien sind ${\rm J }_n(\eta)$, abhängig vom Modulationsindex $\eta$.

[[File:P_ID1095__Mod_T_3_2_S4_neu.png|center|frame|Diskrete Spektren bei Phasenmodulation (links) und Frequenzmodulation (rechts)]]

Die Grafik zeigt das Betragsspektrum $\vert S_{\rm +}(f) \vert$ des analytischen Signals bei Phasenmodulation (PM) und Frequenzmodulation (FM), zwei unterschiedliche Formen der Winkelmodulation (WM). Bessellinien mit Werten kleiner als $0.03$ sind hierbei in beiden Fällen vernachlässigt.

Für die obere Bildhälfte sind die Modulatorparameter so gewählt, dass sich für $f_{\rm N} = 5 \ \rm kHz$ jeweils ein Besselspektrum mit dem Modulationsindex $η = 1.5$ ergibt. Lässt man die Phasenbeziehungen außer Acht, so ergeben sich für beide Systeme gleiche Spektren und gleiche Signale.

Die unteren Grafiken gelten bei sonst gleichen Einstellungen für die Nachrichtenfrequenz $f_{\rm N} = 3 \ \rm kHz$. Man erkennt:
*Bei der Phasenmodulation ergibt sich gegenüber $f_{\rm N} = 5 \ \rm kHz$ eine schmalere Spektralfunktion, da nun der Abstand der Bessellinien nur mehr $3 \ \rm kHz$ beträgt. Da bei PM der Modulationsindex unabhängig von $f_{\rm N}$ ist, ergeben sich die gleichen Besselgewichte wie bei $f_{\rm N} = 5 \ \rm kHz$.
*Auch bei der Frequenzmodulation treten nun die Bessellinien im Abstand von $3 \ \rm kHz$ auf. Da aber bei FM der Modulationsindex umgekehrt proportional zu $f_{\rm N}$ ist, gibt es nun unten aufgrund des größeren Modulationsindex $η = 2.5$ deutlich mehr Bessellinien als im rechten oberen (für $η = 1.5$ gültigen) Diagramm. }}

'''Das folgende Kapitel muss noch angepasst werden!'''

==Zur Handhabung des Applets==
[[File:Handhabung_binomial.png|left|600px]]
    '''(A)'''     Vorauswahl für blauen Parametersatz

    '''(B)'''     Parametereingabe $I$ und $p$ per Slider

    '''(C)'''     Vorauswahl für roten Parametersatz

    '''(D)'''     Parametereingabe $\lambda$ per Slider

    '''(E)'''     Graphische Darstellung der Verteilungen

    '''(F)'''     Momentenausgabe für blauen Parametersatz

    '''(G)'''     Momentenausgabe für roten Parametersatz

    '''(H)'''     Variation der grafischen Darstellung

$\hspace{1.5cm}$&bdquo;$+$” (Vergrößern),

$\hspace{1.5cm}$ &bdquo;$-$” (Verkleinern)

$\hspace{1.5cm}$ &bdquo;$\rm o$” (Zurücksetzen)

$\hspace{1.5cm}$ &bdquo;$\leftarrow$” (Verschieben nach links), usw.

    '''( I )'''     Ausgabe von ${\rm Pr} (z = \mu)$ und ${\rm Pr} (z \le \mu)$

    '''(J)'''     Bereich für die Versuchsdurchführung
 
 '''Andere Möglichkeiten zur Variation der grafischen Darstellung''':
*Gedrückte Shifttaste und Scrollen: Zoomen im Koordinatensystem,
*Gedrückte Shifttaste und linke Maustaste: Verschieben des Koordinatensystems.

==Über die Autoren==
Dieses interaktive Berechnungstool wurde am [http://www.lnt.ei.tum.de/startseite Lehrstuhl für Nachrichtentechnik] der [https://www.tum.de/ Technischen Universität München] konzipiert und realisiert.
*Die erste Version wurde 2006 von [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Studierende#Markus_Elsberger_.28Diplomarbeit_LB_2006.29|Markus Elsberger]] und [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Studierende#Slim_Lamine_.28Studienarbeit_EI_2006.29|Slim Lamine]] im Rahmen von Abschlussarbeiten mit &bdquo;FlashMX–Actionscript” erstellt (Betreuer: [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Mitarbeiter_und_Dozenten#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28am_LNT_seit_1974.29|Günter Söder]]).
*2018 wurde das Programm von [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Studierende#Xiaohan_Liu_.28Bachelorarbeit_2018.29|Xiaohan Liu]] (Bachelorarbeit, Betreuer: [[Biografien_und_Bibliografien/Beteiligte_der_Professur_Leitungsgebundene_%C3%9Cbertragungstechnik#Tasn.C3.A1d_Kernetzky.2C_M.Sc._.28bei_L.C3.9CT_seit_2014.29|Tasnád Kernetzky]] ) auf &bdquo;HTML5” umgesetzt.

==Nochmalige Aufrufmöglichkeit des Applets in neuem Fenster==

{{LntAppletLink|bessel}}

Applets:Physical Signal & Analytic Signal

2018-08-13T08:49:26Z

Xiaohan: /* Exercises */

{{LntAppletLink|physAnSignal}}

==Applet Description==
 
This applet shows the relationship between the physical bandpass signal $x(t)$ and the associated analytical signal $x_+(t)$. It is assumed that the bandpass signal $x(t)$ has a frequency-discrete spectrum $X(f)$:
:$$x(t) = x_{\rm U}(t) + x_{\rm T}(t) + x_{\rm O}(t) = A_{\rm U}\cdot \cos\left(2\pi f_{\rm U}\cdot t- \varphi_{\rm U}\right)+A_{\rm T}\cdot \cos\left(2\pi f_{\rm T}\cdot t- \varphi_{\rm T}\right)+A_{\rm O}\cdot \cos\left(2\pi f_{\rm O}\cdot t- \varphi_{\rm O}\right). $$
The physical signal $x(t)$ is thus composed of three harmonic oscillations, a constellation that can be found, for example, in the ''Double-sideband Amplitude Modulation''
*of the message signal $x_{\rm N}(t) = A_{\rm N}\cdot \cos\left(2\pi f_{\rm N}\cdot t- \varphi_{\rm N}\right)$   ⇒   in German:   '''N'''achrichtensignal
*with the carrier signal $x_{\rm T}(t) = A_{\rm T}\cdot \cos\left(2\pi f_{\rm T}\cdot t - \varphi_{\rm T}\right)$   ⇒   in German:   '''T'''rägersignal.

The nomenclature is also adapted to this case:
* $x_{\rm O}(t)$ denotes the &bdquo;upper sideband”   (in German:   '''O'''beres Seitenband) with the amplitude $A_{\rm O}= A_{\rm N}/2$, the frequency $f_{\rm O} = f_{\rm T} + f_{\rm N}$ and the phase $\varphi_{\rm O} = \varphi_{\rm T} + \varphi_{\rm N}$.
*Similarly, for the &bdquo;lower sideband”   (in German:   '''U'''nteres Seitenband) $x_{\rm U}(t)$ with $f_{\rm U} = f_{\rm T} - f_{\rm N}$, $A_{\rm U}= A_{\rm O}$ and $\varphi_{\rm U} = -\varphi_{\rm O}$.

The associated analytical signal is:

:$$x_+(t) = x_{\rm U+}(t) + x_{\rm T+}(t) + x_{\rm O+}(t) = A_{\rm U}\cdot {\rm e}^{\hspace{0.05cm}{\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}(2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm U}\hspace{0.05cm}\cdot \hspace{0.05cm}t- \varphi_{\rm U})}
\hspace{0.1cm}+ \hspace{0.1cm}A_{\rm T}\cdot {\rm e}^{\hspace{0.05cm}{\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}(2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm T}\hspace{0.05cm}\cdot \hspace{0.05cm}t- \varphi_{\rm T})}
\hspace{0.1cm}+\hspace{0.1cm} A_{\rm O}\cdot {\rm e}^{\hspace{0.05cm}{\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}(2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm O}\hspace{0.05cm}\cdot \hspace{0.05cm}t- \varphi_{\rm O})}. $$

[[File:Zeigerdiagramm_2a_version2.png|right|frame|Analytical signal at the time $t=0$]]
The program displays $x_+(t)$ as the vectorial sum of three rotating pointers (all with counterclockwise) as a violet dot (see figure for start time $t=0$):

*The (red) pointer of the carrier $x_{\rm T+}(t)$ with length $A_{\rm T}$ and zero phase position $\varphi_{\rm T} = 0$ rotates at constant angular velocity $2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm T}$ (one revolution in time $1/f_{\rm T})$.

*The (blue) pointer of the upper sideband $x_{\rm O+}(t)$ with length $A_{\rm O}$ and zero phase position $\varphi_{\rm O}$ rotates at the angular velocity $2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm O}$, which is slightly faster than $x_{\rm T+}(t)$.

*The (green) pointer of the lower sideband $x_{\rm U+}(t)$ with length $A_{\rm U}$ and zero phase position $\varphi_{\rm U}$ rotates at the angular velocity $2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm U}$, which is slightly slower than $x_{\rm T+}(t)$.

The time trace of $x_+(t)$ is also referred to below as ''Pointer Diagram''. The relationship between the physical bandpass signal $x(t)$ and the associated analytical signal $x_+(t)$ is:

:$$x(t) = {\rm Re}\big [x_+(t)\big ].$$

''Note:''   In the figure $\varphi_{\rm O} = +30^\circ$. This leads to the angle with respect to the coordinate system at $t=0$:   $\phi_{\rm O}=-\varphi_{\rm O}=-30^\circ$. Similarly, the null phase angle $\varphi_{\rm U}=-30^\circ$ of the lower sideband leads to the phase angle to be considered in the complex plane:   $\phi_{\rm U}=+30^\circ$.

[[Applets:Physikalisches_Signal_%26_Analytisches_Signal|'''German Description''']]

==Theoretical Background==
 
===Description of Bandpass Signals===
[[File:Zeigerdiagramm_1a.png|right|frame|Bandpass spectrum $X(f)$ |class=fit]]
We consider '''bandpass signals''' $x(t)$ with the property that their spectra $X(f)$ are not in the range around the frequency $f=0$, but around a carrier frequency $f_{\rm T}$. In most cases it can also be assumed that the bandwidth is $B \ll f_{\rm T}$.

The figure shows such a bandpass spectrum $X(f)$. Assuming that the associated $x(t)$ is a physical signal and thus real, the spectral function $X(f)$ has a symmetry with respect to the frequency $f = 0$, if $x(t)$ is an even function   ⇒   $x(-t)=x(t)$, $X(f)$ is real and even.

Besides the physical signal $x(t)\ \circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\, \ X(f)$, one can also use the following descriptions of bandpass signals:
*the analytical signal $x_+(t)\ \circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\, \ X_+(f)$, see next page,
*the equivalent lowpass signal   (in German:   äquivalentes '''T'''ief '''P'''ass–Signal) $x_{\rm TP}(t)\ \circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\, \ X_{\rm TP}(f)$, see Applet [[Applets:Physical_Signal_%26_Equivalent_Low-pass_Signal|Physical Signal & Equivalent Lowpass signal]].
 

===Analytical Signal – Frequency Domain===

The '''analytical signal''' $x_+(t)$ belonging to the physical signal $x(t)$ is the time function whose spectrum fulfills the following property:
[[File:Zeigerdiagramm_3a.png|right|frame|Construction of the spectral function $X_+(f)$ |class=fit]]
:$$X_+(f)=\big[1+{\rm sign}(f)\big] \cdot X(f) = \left\{ {2 \cdot
X(f) \; \hspace{0.2cm}\rm for\hspace{0.2cm} {\it f} > 0, \atop {\,\,\,\, \rm 0 \; \hspace{0.9cm}\rm for\hspace{0.2cm} {\it f} < 0.} }\right.$$

The ''signum function'' is for positive values of $f$ equal to $+1$ and for negative $f$ values equal to $-1$.
* The (double-sided) limit returns $\sign(0)=0$.
* The index „+” should make it clear that $X_+(f)$ only has parts at positive frequencies.

From the graph you can see the calculation rule for $X_+(f)$:

The actual bandpass spectrum $X(f)$ becomes
* doubled at the positive frequencies, and
* set to zero at the negative frequencies.

Due to the asymmetry of $X_+(f)$ with respect to the frequency $f=0$, it can already be said that the time function $x_+(t)$ except for a trivial special case $x_+(t)=0 \ \circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\,X_+(f)=0$ is always complex.
 

===Analytical Signal – Time Domain===
At this point, it is necessary to briefly discuss another spectral transformation.

{{BlaueBox|TEXT=
$\text{Definition:}$ 
For the '''Hilbert transform''' $ {\rm H}\left\{x(t)\right\}$ of a time function $x(t)$ we have:

:$$y(t) = {\rm H}\left\{x(t)\right\} = \frac{1}{ {\rm \pi} } \cdot
\hspace{0.03cm}\int_{-\infty}^{+\infty}\frac{x(\tau)}{ {t -
\tau} }\hspace{0.15cm} {\rm d}\tau.$$

This particular integral is not solvable in a simple, conventional way, but must be evaluated using the [https://en.wikipedia.org/wiki/Cauchy_principal_value Cauchy principal value theorem].

Accordingly, in the frequency domain:
:$$Y(f) = {\rm -j \cdot sign}(f) \cdot X(f) \hspace{0.05cm} .$$}}

The above result can be summarized with this definition as follows:
* The analytical signal $x_+(t)$ is obtained from the physical bandpass signal $x(t)$ by adding an imaginary part to $x(t)$ according to the Hilbert transform:

:$$x_+(t) = x(t)+{\rm j} \cdot {\rm H}\left\{x(t)\right\} .$$

*$\text{H}\{x(t)\}$ disappears only for the case $x(t) = \rm const.$. For all other signal forms, the analytic signal $x_+(t)$ is complex.

* From the analytical signal $x_+(t)$, the physical bandpass signal can be easily determined by the following operation:
:$$x(t) = {\rm Re}\big[x_+(t)\big] .$$

{{GraueBox|TEXT=
$\text{Example 1:}$  The principle of the Hilbert transformation should be further clarified by the following graphic:
*After the left representation $\rm(A)$ one gets from the physical signal $x(t)$ to the analytical signal $x_+(t)$, by adding an imaginary part ${\rm j} \cdot y(t)$.
*Here $y(t) = {\rm H}\left\{x(t)\right\}$ is a real time function that can be indicated in the spectral domain by multiplying the spectrum $X(f)$ with ${\rm - j} \cdot \sign(f)$.

[[File:P_ID2729__Sig_T_4_2_S2b_neu.png|center|frame|To clarify the Hilbert transform]]

The right representation $\rm(B)$ is equivalent to $\rm(A)$. Now $x_+(t) = x(t) + z(t)$ stand with the purely imaginary function $z(t)$. A comparison of the two figures shows that in fact $z(t) = {\rm j} \cdot y(t)$.}}
 

===Representation of the Harmonic Oscillation as an Analytical Signal===

The spectral function $X(f)$ of a harmonic oscillation $x(t) = A\cdot\text{cos}(2\pi f_{\rm T}\cdot t - \varphi)$ is known to consist of two Dirac functions at the frequencies
* $+f_{\rm T}$ with the complex weight $A/2 \cdot \text{e}^{-\text{j}\hspace{0.05cm}\varphi}$,
* $-f_{\rm T}$ with the complex weight $A/2 \cdot \text{e}^{+\text{j}\hspace{0.05cm}\varphi}$.

Thus, the spectrum of the analytical signal (that is, without the Dirac function at the frequency $f =-f_{\rm T}$, but doubling at $f =+f_{\rm T}$):

:$$X_+(f) = A \cdot {\rm e}^{-{\rm j} \hspace{0.05cm}\varphi}\cdot\delta (f - f_{\rm
T}) .$$

The associated time function is obtained by applying the Displacement Law:

:$$x_+(t) = x_{\rm U+}(t) + x_{\rm T+}(t) + x_{\rm O+}(t) = A_{\rm U}\cdot {\rm e}^{\hspace{0.05cm}{\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}(2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm U}\hspace{0.05cm}\cdot \hspace{0.05cm}t- \varphi_{\rm U})}
\hspace{0.1cm}+ \hspace{0.1cm}A_{\rm T}\cdot {\rm e}^{\hspace{0.05cm}{\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}(2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm T}\hspace{0.05cm}\cdot \hspace{0.05cm}t- \varphi_{\rm T})}
\hspace{0.1cm}+\hspace{0.1cm} A_{\rm O}\cdot {\rm e}^{\hspace{0.05cm}{\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}(2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm O}\hspace{0.05cm}\cdot \hspace{0.05cm}t- \varphi_{\rm O})}. $$

This equation describes a pointer rotating at constant angular velocity $\omega_{\rm T} = 2\pi f_{\rm T}$.

{{GraueBox|TEXT=
$\text{Example 2:}$  Here the coordinate system is rotated by $90^\circ$ (real part up, imaginary part to the left) contrary to the usual representation.

[[File:P_ID712__Sig_T_4_2_S3.png|center|frame|Pointer diagram of a harmonic oscillation]]

Based on this graphic, the following statements are possible:
* At time $t = 0$, the pointer of length $A$ (signal amplitude) lies with the angle $-\varphi$ in the complex plane. In the example shown, $\varphi=45^\circ$.
* For times $t>0$, the constant angular velocity vector $\omega_{\rm T}$ rotates in a mathematically positive direction, that is, counterclockwise.
* The tip of the pointer is thus always on a circle with radius $A$ and needs exactly the time $T_0$, i.e. the period of the harmonic oscillation $x(t)$ for one revolution.
* The projection of the analytical signal $x_+(t)$ on the real axis, marked by red dots, gives the instantaneous values of $x(t)$.}}
 

===Analytical Signal Representation of a Sum of Three Harmonic Oscillations===

In our applet, we always assume a set of three rotating pointers. The physical signal is:
:$$x(t) = x_{\rm U}(t) + x_{\rm T}(t) + x_{\rm O}(t) = A_{\rm U}\cdot \cos\left(2\pi f_{\rm U}\cdot t- \varphi_{\rm U}\right)+A_{\rm T}\cdot \cos\left(2\pi f_{\rm T}\cdot t- \varphi_{\rm T}\right)+A_{\rm O}\cdot \cos\left(2\pi f_{\rm O}\cdot t- \varphi_{\rm O}\right). $$
* Each of the three harmonic oscillations $x_{\rm T}(t)$, $x_{\rm U}(t)$ and $x_{\rm O}(t)$ is represented by an amplitude $(A)$, a frequency $(f)$ and a phase value $(\varphi)$.
*The indices are based on the ''Double-sideband Amplitude Modulation'' method. &bdquo;T” stands for &bdquo;carrier”, &bdquo;U” for &bdquo;lower sideband” and &bdquo;O” for &bdquo;Upper Sideband”.
*Accordingly, $f_{\rm U} < f_{\rm T}$ and $f_{\rm O} > f_{\rm T}$. There are no restrictions for the amplitudes and phases.

The associated analytical signal is:
:$$x_+(t) = x_{\rm U+}(t) + x_{\rm T+}(t) + x_{\rm O+}(t) = A_{\rm U}\cdot {\rm e}^{\hspace{0.05cm}{\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}(2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm U}\hspace{0.05cm}\cdot \hspace{0.05cm}t- \varphi_{\rm U})}
\hspace{0.1cm}+ \hspace{0.1cm}A_{\rm T}\cdot {\rm e}^{\hspace{0.05cm}{\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}(2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm T}\hspace{0.05cm}\cdot \hspace{0.05cm}t- \varphi_{\rm T})}
\hspace{0.1cm}+\hspace{0.1cm} A_{\rm O}\cdot {\rm e}^{\hspace{0.05cm}{\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}(2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm O}\hspace{0.05cm}\cdot \hspace{0.05cm}t- \varphi_{\rm O})}. $$

{{GraueBox|TEXT=
$\text{Example 3:}$ 
Shown the constellation arises i.e. in the [https://en.wikipedia.org/wiki/Sideband Double-sideband Amplitude Modulation] of the message signal $x_{\rm N}(t) = A_{\rm N}\cdot \cos\left(2\pi f_{\rm N}\cdot t- \varphi_{\rm N}\right)$ with the carrier signal $x_{\rm T}(t) = A_{\rm T}\cdot \cos\left(2\pi f_{\rm T}\cdot t - \varphi_{\rm T}\right)$. This is discussed frequently in the Exercises.

There are some limitations to the program parameters in this approach:
* For the frequencies, it always applies $f_{\rm O} = f_{\rm T} + f_{\rm N}$ und $f_{\rm U} = f_{\rm T} - f_{\rm N}$.

*Without distortions the amplitude of the sidebands are $A_{\rm O}= A_{\rm U}= A_{\rm N}/2$.
*The respective phase relationships can be seen in the following graphic.

[[File:Zeigerdiagramm_2_neu.png|center|frame|Spectrum $X_+(f)$ of the analytical signal for different phase constellations |class=fit]]}}

==Exercises==
[[File:Zeigerdiagramm_aufgabe_2.png|right]]
*First select the task number.
*A task description is displayed.
*Parameter values are adjusted.
*Solution after pressing &bdquo;Hide solition”.

The number &bdquo;0” will reset the program and output a text with further explanation of the applet.
 
In the following, $\rm Green$ denotes the lower sideband   ⇒   $\big (A_{\rm U}, f_{\rm U}, \varphi_{\rm U}\big )$,  
$\rm Red$ the carrier   ⇒   $\big (A_{\rm T}, f_{\rm T}, \varphi_{\rm T}\big )$ and
$\rm Blue$ the upper sideband   ⇒   $\big (A_{\rm O}, f_{\rm O}, \varphi_{\rm O}\big )$.
{{BlaueBox|TEXT=
'''(1)'''   Consider and interpret the analytical signal $x_+(t)$ for $\text{Red:} \hspace{0.15cm} A_{\rm T} = 1.5\ \text{V}, \ f_{\rm T} = 50 \ \text{kHz}, \ \varphi_{\rm T} = 0^\circ$, $A_{\rm U} = A_{\rm O} = 0$.

:Which signal values $x_+(t)$ result for $t = 0$, $t = 5 \ \rm µ s$ and $t = 20 \ \rm µ s$? What are the corresponding signal values for $x(t)$? }}

:: For a cosine signal, let $x_+(t= 0) = A_{\rm T} = 1.5\ \text{V}$. Then $x_+(t)$ rotates in a mathematically positive direction (one revolution per period $T_0 = 1/f_{\rm T}$):

:: $x_+(t= 20 \ {\rm µ s}) = x_+(t= 0) = 1.5\ \text{V}\hspace{0.3cm}\Rightarrow\hspace{0.3cm}x(t= 20 \ {\rm µ s}) = 1.5\ \text{V,}$
:: $x_+(t= 5 \ {\rm µ s}) = {\rm j} \cdot 1.5\ \text{V}\hspace{0.3cm}\Rightarrow\hspace{0.3cm}x(t= 5 \ {\rm µ s}) = {\rm Re}[x_+(t= 5 \ {\rm µ s})] = 0$.

{{BlaueBox|TEXT=
'''(2)'''   How do the ratios change for $\text{Red:} \hspace{0.15cm} A_{\rm T} = 1.0\ \text{V}, \ f_{\rm T} = 100 \ \text{kHz}, \ \varphi_{\rm T} = 90^\circ$?}}

::The signal $x(t)$ is now a sine signal with a smaller amplitude. The analytical signal now starts because of $\varphi_{\rm T} = 90^\circ$   ⇒   $\phi_{\rm T} = -90^\circ$ at $x_+(t= 0) = -{\rm j} \cdot A_{\rm T}$. After that, $x_+(t)$ rotates again in a mathematically positive direction, but twice as fast because of $T_0 = 10 \ \rm µ s$ as in $\rm (1)$.

{{BlaueBox|TEXT=
'''(3)'''   Now   $\text{Red:} \hspace{0.15cm} A_{\rm T} = 1\ \text{V}, \ f_{\rm T} = 100 \ \text{kHz}, \ \varphi_{\rm T} = 0^\circ$,   $\text{Green:} \hspace{0.15cm} A_{\rm U} = 0.4\ \text{V}, \ f_{\rm U} = 80 \ \text{kHz}, \ \varphi_{\rm U} = 0^\circ$,   $\text{Blue:} \hspace{0.15cm} A_{\rm O} = 0.4\ \text{V}, \ f_{\rm O} = 120 \ \text{kHz}, \ \varphi_{\rm O} = 0^\circ$.

:Consider and interpret the physical signal $x(t)$ and the analytical signal $x_+(t)$.}}

::The Signal $x(t)$ results in the [https://en.wikipedia.org/wiki/Sideband Double-sideband Amplitude Modulation] (DSB–AM) of the message signal $A_{\rm N}\cdot \cos\left(2\pi f_{\rm N}\cdot t\right)$ with $A_{\rm N} = 0.8\ \text{V}$, $f_{\rm N} = 20\ \text{kHz}$. The carrier $x_{\rm T}(t)$ with $f_{\rm T} = 100\ \text{kHz}$ is also cosinusoidal. The degree of modulation is $m = A_{\rm N}/A_{\rm T} = 0.8$ and the period $T_{\rm 0} = 50\ \text{µs}$.

::In the phase diagram, the (red) carrier rotates faster than the (green) lower sideband and slower than the (blue) upper sideband. The analytical signal $x_+(t)$ results as the geometric sum of the three rotating pointers. It seems that the blue pointer is leading the carrier and the green pointer is following the carrier.

{{BlaueBox|TEXT=
'''(4)'''   The settings of task '''(3)''' still apply. Which signal values are obtained at $t=0$, $t=2.5 \ \rm µ s$, $t= 5 \ \rm µ s$ and $t=10 \ \rm µ s$? }}

::At time $t=0$, all pointers are in the direction of the real axis, so that $x(t=0) = {\rm Re}\big [x+(t= 0)\big] = A_{\rm U} + A_{\rm T} + A_{\rm O} = 1.8\ \text{V}$.

::Until the time $t=2.5 \ \rm µ s$, the red carrier has rotated by $90^\circ$, the blue one by $108^\circ$ and the green one by $72^\circ$. We have $x(t=2.5 \ \rm µ s) = {\rm Re}\big [x_+(t= 2.5 \ \rm µ s)\big] = 0$, because now the pointer group points in the direction of the imaginary axis. The other sought signal values are $x(t=5 \ \rm µ s) = {\rm Re}\big [x_+(t= 5 \ \rm µ s)\big] = -1.647\ \text{V}$ and $x(t=10 \ \rm µ s) = {\rm Re}\big [x_+(t= 10 \ \rm µ s)\big] = 1.247\ \text{V}$.
::For $x_+(t)$ a spiral shape results, alternating with a smaller radius and then with a larger radius.

{{BlaueBox|TEXT=
'''(5)'''   How should the phase parameters $\varphi_{\rm T}$, $\varphi_{\rm U}$ and $\varphi_{\rm O}$ be set if both the carrier $x_{\rm T}(t)$ and the message signal $x_{\rm N}(t)$ are sinusoidal?}}

::The parameter selection $\varphi_{\rm T} = \varphi_{\rm U} = \varphi_{\rm O}=90^\circ$ describes the signals $x_{\rm T}(t) = A_{\rm T}\cdot \sin\left(2\pi f_{\rm T}\cdot t\right)$ and $x_{\rm N}(t) = A_{\rm N}\cdot \cos\left(2\pi f_{\rm N}\cdot t\right)$. If, in addition, the message $x_{\rm N}(t)$ is sinusoidal, then $\varphi_{\rm O}=\varphi_{\rm T} - 90^\circ = 0$ and $\varphi_{\rm U}=\varphi_{\rm T} + 90^\circ = 180^\circ$ must be set.

{{BlaueBox|TEXT=
'''(6)'''   The settings of task '''(3)''' apply except $A_{\rm T} = 0.6\ \text{V}$. Which modulation method is described here?

: What are the consequences of this? What changes with $A_{\rm T} = 0$? }}

::It is a [https://en.wikipedia.org/wiki/Sideband Double-sideband Amplitude Modulation] (DSB–AM with carrier) with the modulation degree $m=0.8/0.6 = 1.333$. For $m > 1$, however, [https://www.radio-electronics.com/info/rf-technology-design/am-reception/synchronous-demodulator-demodulation-detector.php Synchronous Demodulation] is required. [https://en.wikipedia.org/wiki/Envelope_detector Envelope Detection] no longer works.

::With $A_{\rm T} = 0$   ⇒   $m \to \infty$ results in a [https://en.wikipedia.org/wiki/Double-sideband_suppressed-carrier_transmission ''DSB–AM suppressed carrier'']. For this, one also needs coherent demodulation.

{{BlaueBox|TEXT=
'''(7)'''     Now let   $\text{Red:} \hspace{0.15cm} A_{\rm T} = 1\ \text{V}, \ f_{\rm T} = 100 \ \text{kHz}, \ \varphi_{\rm T} = 0^\circ$,   $\text{Green:} \hspace{0.15cm} A_{\rm U} = 0$,   $\text{Blue:} \hspace{0.15cm} A_{\rm O} = 0.8\ \text{V}, \ f_{\rm O} = 120 \ \text{kHz}, \ \varphi_{\rm O} = 90^\circ$.

:Which constellation is described here? What changes with $A_{\rm U} = 0.8\ \text{V}$ and $A_{\rm O} = 0$?}}

::In both cases, it is a [https://en.wikipedia.org/wiki/Single-sideband_modulation Single-sideband Amplitude Modulation] (SSB–AM) with the modulation degree $\mu = 0.8$ (in SSB we denote the degree of modulation with $\mu$ instead of $m$). The carrier signal is cosinusoidal and the message signal is sinusoidal.

:: $A_{\rm O} = 0.8\ \text{V}$, $A_{\rm U} = 0$ is an OSB modulation. The green pointer is missing and the blue pointer rotates faster compared to the red carrier.

:: $A_{\rm U} = 0.8\ \text{V}$, $A_{\rm O} = 0$ is a USB modulation. The blue pointer is missing and the green pointer rotates slower compared to the red carrier.

{{BlaueBox|TEXT=
'''(8)'''   Now let   $\text{Red:} \hspace{0.05cm} A_{\rm T} = 1\ \text{V}, \ f_{\rm T} = 100 \ \text{kHz}, \ \varphi_{\rm T} = 0^\circ$,   $\text{Green:} \hspace{0.05cm} A_{\rm U} = 0.4\ \text{V}, \ f_{\rm U} = 80 \ \text{kHz}, \ \varphi_{\rm U} = -90^\circ$,   $\text{Blue:} \hspace{0.05cm} A_{\rm O} = 0.2\ \text{V}, \ f_{\rm O} = 120 \ \text{kHz}, \ \varphi_{\rm O} = +90^\circ$.

:Which constellation could be described here? Which shape results for the equivalent lowpass signal $x_{\rm TP}(t)$?}}

::It could be a DSB–AM of a sinusoidal signal with cosinusoidal carrier and modulation degree $m=0.8$, in which the upper sideband is attenuated by a factor of 2. The equivalent lowpass signal $x_{\rm TP}(t)$ has an elliptical trace in the complex plane.

==Applet Manual==
 
[[File:Zeigerdiagramm_abzug.png|right]]

* The red parameters $(A_{\rm T}, \ f_{\rm T}, \ \varphi_{\rm T})$ and the red pointer mark the ''Carrier'' (German:   '''T'''räger).
* The green parameters $(A_{\rm U}, \ f_{\rm U} < f_{\rm T}, \ \varphi_{\rm U})$ mark the ''Lower sideband'' (German:  '''U'''nteres Seitenband).
* The blue parameters $(A_{\rm O}, \ f_{\rm O} > f_{\rm T}, \ \varphi_{\rm O})$ mark the mark the ''Upper sideband'' (German:  '''O'''beres Seitenband).
*All pointers rotate in a mathematically positive direction (counterclockwise).

 
Meaning of the letters in the adjacent graphic:

    '''(A)'''     Plot of the analytical signal $x_{\rm +}(t)$

    '''(B)'''     Plot of for the physical signal $x(t)$

    '''(C)'''     Parameter input via slider: amplitudes, frequencies, phase values

    '''(D)'''     Control elements:   Start – Step – Pause/Continue – Reset

    '''(E)'''     Speed of animation:   &bdquo;Speed”   ⇒   Values: 1, 2, 3

    '''(F)'''     &bdquo;Trace”   ⇒   On or Off, trace of complex signal values $x_{\rm +}(t)$

    '''(G)'''     Numerical output of the time $t$ and the signal values  ${\rm Re}[x_{\rm +}(t)] = x(t)$  and  ${\rm Im}[x_{\rm +}(t)]$

    '''(H)'''     Variations for the graphical representation

$\hspace{1.5cm}$Zoom–Functions &bdquo;$+$” (Enlarge), &bdquo;$-$” (Decrease) and $\rm o$ (Reset to default)

$\hspace{1.5cm}$Move with &bdquo;$\leftarrow$” (Section to the left, ordinate to the right), &bdquo;$\uparrow$” &bdquo;$\downarrow$” and &bdquo;$\rightarrow$”

    '''(I)'''     Experiment section:  Task selection and task

    '''(J)'''     Experiment section:  solution
 

==About the Authors==
This interactive calculation was designed and realized at the [http://www.lnt.ei.tum.de/startseite Lehrstuhl für Nachrichtentechnik] of the [https://www.tum.de/ Technical University of Munich] .
*The original version was created in 2005 by [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Studierende#Ji_Li_.28Bachelorarbeit_EI_2003.2C_Diplomarbeit_EI_2005.29|Ji Li]] as part of her Diploma thesis using &bdquo;FlashMX–Actionscript” (Supervisor: [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Mitarbeiter_und_Dozenten#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28am_LNT_seit_1974.29|Günter Söder]]).
*In 2018 this Applet was redesigned and updated to &bdquo;HTML5” by [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Studierende#Xiaohan_Liu_.28Bachelorarbeit_2018.29|Xiaohan Liu]] as part of her Bachelor's thesis (Supervisor: [[Biografien_und_Bibliografien/Beteiligte_der_Professur_Leitungsgebundene_%C3%9Cbertragungstechnik#Tasn.C3.A1d_Kernetzky.2C_M.Sc._.28bei_L.C3.9CT_seit_2014.29|Tasnád Kernetzky]]).

==Once again: Open Applet in new Tab==

{{LntAppletLink|physAnSignal}}

[[Category:Applets|^Verzerrungen^]]

Applets:Physical Signal & Analytic Signal

2018-08-10T14:12:40Z

Xiaohan: /* About the Authors */

{{LntAppletLink|physAnSignal}}

==Applet Description==
 
This applet shows the relationship between the physical bandpass signal $x(t)$ and the associated analytic signal $x_+(t)$. It is assumed that the bandpass signal $x(t)$ has a frequency-discrete spectrum $X(f)$:
:$$x(t) = x_{\rm U}(t) + x_{\rm T}(t) + x_{\rm O}(t) = A_{\rm U}\cdot \cos\left(2\pi f_{\rm U}\cdot t- \varphi_{\rm U}\right)+A_{\rm T}\cdot \cos\left(2\pi f_{\rm T}\cdot t- \varphi_{\rm T}\right)+A_{\rm O}\cdot \cos\left(2\pi f_{\rm O}\cdot t- \varphi_{\rm O}\right). $$
The physical signal $x(t)$ is thus composed of three [[Signaldarstellung/Harmonische_Schwingung|harmonic oscillations]], a constellation that can be found, for example, in the ''Double-sideband Amplitude Modulation''
*of the message signal $x_{\rm N}(t) = A_{\rm N}\cdot \cos\left(2\pi f_{\rm N}\cdot t- \varphi_{\rm N}\right)$   ⇒   in German:   '''N'''achrichtensignal
*with the carrier signal $x_{\rm T}(t) = A_{\rm T}\cdot \cos\left(2\pi f_{\rm T}\cdot t - \varphi_{\rm T}\right)$   ⇒   in German:   '''T'''rägersignal.

The nomenclature is also adapted to this case:
* $x_{\rm O}(t)$ denotes the &bdquo;upper sideband”   (in German:   '''O'''beres Seitenband) with the amplitude $A_{\rm O}= A_{\rm N}/2$, the frequency $f_{\rm O} = f_{\rm T} + f_{\rm N}$ and the phase $\varphi_{\rm O} = \varphi_{\rm T} + \varphi_{\rm N}$.
*Similarly, for the &bdquo;lower sideband”   (in German:   '''U'''nteres Seitenband) $x_{\rm U}(t)$ with $f_{\rm U} = f_{\rm T} + f_{\rm N}$, $A_{\rm U}= A_{\rm O}$ and $\varphi_{\rm U} = -\varphi_{\rm O}$.

The associated analytical signal is:

:$$x_+(t) = x_{\rm U+}(t) + x_{\rm T+}(t) + x_{\rm O+}(t) = A_{\rm U}\cdot {\rm e}^{\hspace{0.05cm}{\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}(2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm U}\hspace{0.05cm}\cdot \hspace{0.05cm}t- \varphi_{\rm U})}
\hspace{0.1cm}+ \hspace{0.1cm}A_{\rm T}\cdot {\rm e}^{\hspace{0.05cm}{\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}(2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm T}\hspace{0.05cm}\cdot \hspace{0.05cm}t- \varphi_{\rm T})}
\hspace{0.1cm}+\hspace{0.1cm} A_{\rm O}\cdot {\rm e}^{\hspace{0.05cm}{\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}(2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm O}\hspace{0.05cm}\cdot \hspace{0.05cm}t- \varphi_{\rm O})}. $$

[[File:Zeigerdiagramm_2a_version2.png|right|frame|Analytical signal at the time $t=0$]]
The program displays $x_+(t)$ as the vectorial sum of three rotating pointers (all with counterclockwise) as a violet dot (see figure for start time $t=0$):

*The (red) pointer of the carrier $x_{\rm T+}(t)$ with length $A_{\rm T}$ and zero phase position $\varphi_{\rm T} = 0$ rotates at constant angular velocity $2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm T}$ (one revolution in time $1/f_{\rm T})$.

*The (blue) pointer of the upper sideband $x_{\rm O+}(t)$ with length $A_{\rm O}$ and zero phase position $\varphi_{\rm O}$ rotates at the angular velocity $2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm O}$, which is slightly faster than $x_{\rm T+}(t)$.

*The (green) pointer of the lower sideband $x_{\rm U+}(t)$ with length $A_{\rm U}$ and zero phase position $\varphi_{\rm U}$ rotates at the angular velocity $2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm U}$, which is slightly faster than $x_{\rm T+}(t)$.

The time trace of $x_+(t)$ is also referred to below as '''Pointer Diagram'''. The relationship between the physical bandpass signal $x(t)$ and the associated analytic signal $x_+(t)$ is:

:$$x(t) = {\rm Re}\big [x_+(t)\big ].$$

''Note:''   In the figure $\varphi_{\rm O} = +30^\circ$. This leads to the angle with respect to the coordinate system at $t=0$:   $\phi_{\rm O}=-\varphi_{\rm O}=-30^\circ$. Similarly, the null phase angle $\varphi_{\rm U}=-30^\circ$ of the lower sideband leads to the phase angle to be considered in the complex plane:   $\phi_{\rm U}=+30^\circ$.

[[Applets:Physikalisches_Signal_%26_Analytisches_Signal|'''German description''']]

==Theoretical Background==
 
===Description of Bandpass Signals===
[[File:Zeigerdiagramm_1a.png|right|frame|Bandpass–spectrum $X(f)$ |class=fit]]
We consider '''bandpass signals''' $x(t)$ with the property that their spectra $X(f)$ are not in the range around the frequency $f=0$, but around a carrier frequency $f_{\rm T}$. In most cases it can also be assumed that the bandwidth is $B \ll f_{\rm T}$.

The figure shows such a bandpass spectrum $X(f)$. Assuming that the associated $x(t)$ is a physical signal and thus real, the spectral function $X(f)$ has a symmetry with respect to the frequency $f = 0$, if $x(t)$ is an even function   ⇒   $x(-t)=x(t)$, $X(f)$ is real and even.

Besides the physical signal $x(t)\ \circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\, \ X(f)$, one can also use the following descriptions of Bandpass signals:
*the analytic signal $x_+(t)\ \circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\, \ X_+(f)$, see next page,
*the equivalent low-pass –signal   (in German:   äquivalentes '''T'''ief'''P'''ass–Signal) $x_{\rm TP}(t)\ \circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\, \ X_{\rm TP}(f)$, see Applet [[Applets:Physical_Signal_%26_Equivalent_Low-pass_Signal|Physical Signal & Equivalent Low–pass signal]].
 

===Analytical Signal – Frequency Domain===

The '''analytical signal''' $x_+(t)$ belonging to the physical signal $x(t)$ is the time function whose spectrum fulfills the following property:
[[File:Zeigerdiagramm_3a.png|right|frame|Construction of the spectral function $X_+(f)$ |class=fit]]
:$$X_+(f)=\big[1+{\rm sign}(f)\big] \cdot X(f) = \left\{ {2 \cdot
X(f) \; \hspace{0.2cm}\rm for\hspace{0.2cm} {\it f} > 0, \atop {\,\,\,\, \rm 0 \; \hspace{0.9cm}\rm for\hspace{0.2cm} {\it f} < 0.} }\right.$$

The ''signum function'' is for positive values of $f$ equal to $+1$ and for negative $f$ values equal to $-1$.
* The (double-sided) limit returns $\sign(0)=0$.
* The index „+” should make it clear that $X_+(f)$ only has parts at positive frequencies.

From the graph you can see the calculation rule for $X_+(f)$:

The actual band pass spectrum $X(f)$ becomes
* doubled at the positive frequencies, and
* set to zero at the negative frequencies.

Due to the asymmetry of $X_+(f)$ with respect to the frequency $f=0$, it can already be said that the time function $x_+(t)$ except for a trivial special case $x_+(t)=0 \ \circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\,X_+(f)=0$ is always complex.
 

===Analytical Signal – Time Domain===
At this point, it is necessary to briefly discuss another spectral transformation.

{{BlaueBox|TEXT=
$\text{Definition:}$ 
For the '''Hilbert transformed ''' $ {\rm H}\left\{x(t)\right\}$ of a time function $x(t)$ we have::

:$$y(t) = {\rm H}\left\{x(t)\right\} = \frac{1}{ {\rm \pi} } \cdot
\hspace{0.03cm}\int_{-\infty}^{+\infty}\frac{x(\tau)}{ {t -
\tau} }\hspace{0.15cm} {\rm d}\tau.$$

This particular integral is not solvable in a simple, conventional way, but must be evaluated using the [https://de.wikipedia.org/wiki/Cauchyscher_Hauptwert Cauchy principal value theorem].

Accordingly, in the frequency domain:
:$$Y(f) = {\rm -j \cdot sign}(f) \cdot X(f) \hspace{0.05cm} .$$}}

The above result can be summarized with this definition as follows:
* The analytic signal $x_+(t)$ is obtained from the physical bandpass signal $x(t)$ by adding an imaginary part to $x(t)$ according to the Hilbert transform:

:$$x_+(t) = x(t)+{\rm j} \cdot {\rm H}\left\{x(t)\right\} .$$

*$\text{H}\{x(t)\}$ disappears only for the case $x(t) = \rm const.$. For all other signal forms, the analytic signal $x_+(t)$ is complex.

* From the analytic signal $x_+(t)$, the physical bandpass signal can be easily determined by the following operation:
:$$x(t) = {\rm Re}\big[x_+(t)\big] .$$

{{GraueBox|TEXT=
$\text{Example 1:}$  The principle of the Hilbert transformation should be further clarified by the following graphic:
*After the left representation $\rm(A)$ one gets from the physical signal $x(t)$ to the analytic signal $x_+(t)$, by adding an imaginary part ${\rm j} \cdot y(t)$.
*Here $y(t) = {\rm H}\left\{x(t)\right\}$ is a real time function that can be indicated in the spectral domain by multiplying the spectrum $X(f)$ with ${\rm - j} \cdot \sign(f)$.

[[File:P_ID2729__Sig_T_4_2_S2b_neu.png|center|frame|To clarify the Hilbert transform]]

The right representation $\rm(B)$ is equivalent to $\rm(A)$. Now $x_+(t) = x(t) + z(t)$ stand with the purely imaginary function $z(t)$. A comparison of the two pictures shows that in fact $z(t) = {\rm j} \cdot y(t)$.}}
 

===Representation of the Harmonic Oscillation as an Analytical Signal===

The spectral function $X(f)$ of a harmonic oscillation $x(t) = A\cdot\text{cos}(2\pi f_{\rm T}\cdot t - \varphi)$ is known to consist of two dirac functions in the frequencies
* $+f_{\rm T}$ with the complex weight $A/2 \cdot \text{e}^{-\text{j}\hspace{0.05cm}\varphi}$,
* $-f_{\rm T}$ with the complex weight $A/2 \cdot \text{e}^{+\text{j}\hspace{0.05cm}\varphi}$.

Thus, the spectrum of the analytic signal (that is, without the Dirac function at the frequency $f =-f_{\rm T}$, but doubling at $f =+f_{\rm T}$):

:$$X_+(f) = A \cdot {\rm e}^{-{\rm j} \hspace{0.05cm}\varphi}\cdot\delta (f - f_{\rm
T}) .$$

The associated time function is obtained by applying the [[Signaldarstellung/Gesetzmäßigkeiten_der_Fouriertransformation#Verschiebungssatz|Displacement law]]:

:$$x_+(t) = x_{\rm U+}(t) + x_{\rm T+}(t) + x_{\rm O+}(t) = A_{\rm U}\cdot {\rm e}^{\hspace{0.05cm}{\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}(2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm U}\hspace{0.05cm}\cdot \hspace{0.05cm}t- \varphi_{\rm U})}
\hspace{0.1cm}+ \hspace{0.1cm}A_{\rm T}\cdot {\rm e}^{\hspace{0.05cm}{\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}(2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm T}\hspace{0.05cm}\cdot \hspace{0.05cm}t- \varphi_{\rm T})}
\hspace{0.1cm}+\hspace{0.1cm} A_{\rm O}\cdot {\rm e}^{\hspace{0.05cm}{\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}(2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm O}\hspace{0.05cm}\cdot \hspace{0.05cm}t- \varphi_{\rm O})}. $$

This equation describes a pointer rotating at constant angular velocity $\omega_{\rm T} = 2\pi f_{\rm T}$.

{{GraueBox|TEXT=
$\text{Example 2:}$  Here the coordinate system is rotated by $90^\circ$ (real part up, imaginary part to the left) contrary to the usual representation.

[[File:P_ID712__Sig_T_4_2_S3.png|center|frame|Pointer diagram of a harmonic oscillation]]

Based on this graphic, the following statements are possible:
* At the start time $t = 0$, the pointer of length $A$ (signal amplitude) lies with the angle $-\varphi$ in the complex plane. In the example shown, $\varphi=45^\circ$.
* For times $t>0$, the constant angular velocity vector $\omega_{\rm T}$ rotates in a mathematically positive direction, that is, counterclockwise.
* The tip of the pointer is thus always on a circle with radius $A$ and needs exactly the time $T_0$, i.e. the period of the harmonic oscillation $x(t)$ for one revolution.
* The projection of the analytic signal $x_+(t)$ on the real axis, marked by red dots, gives the instantaneous values of $x(t)$.}}
 

===Analytical Signal Representation of a Sum of Three Harmonic Oscillations===

In our applet, we always assume a set of three rotating pointers. The physical signal is:
:$$x(t) = x_{\rm U}(t) + x_{\rm T}(t) + x_{\rm O}(t) = A_{\rm U}\cdot \cos\left(2\pi f_{\rm U}\cdot t- \varphi_{\rm U}\right)+A_{\rm T}\cdot \cos\left(2\pi f_{\rm T}\cdot t- \varphi_{\rm T}\right)+A_{\rm O}\cdot \cos\left(2\pi f_{\rm O}\cdot t- \varphi_{\rm O}\right). $$
* Each of the three harmonic oscillations $x_{\rm T}(t)$, $x_{\rm U}(t)$ and $x_{\rm O}(t)$ is represented by an amplitude $(A)$, a frequency $(f)$ and a phase value $(\varphi)$.
*The indices are based on the [[Modulationsverfahren/Zweiseitenband-Amplitudenmodulation|Double sideband Amplitude Modulation]] method. &bdquo;T” stands for &bdquo;carrier”, &bdquo;U” for &bdquo;lower sideband” and &bdquo;O” for &bdquo;Upper Sideband”.
*Accordingly, $f_{\rm U} < f_{\rm T}$ and $f_{\rm O} > f_{\rm T}$. There are no restrictions for the amplitudes and phases.

The associated analytical signal is:
:$$x_+(t) = x_{\rm U+}(t) + x_{\rm T+}(t) + x_{\rm O+}(t) = A_{\rm U}\cdot {\rm e}^{\hspace{0.05cm}{\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}(2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm U}\hspace{0.05cm}\cdot \hspace{0.05cm}t- \varphi_{\rm U})}
\hspace{0.1cm}+ \hspace{0.1cm}A_{\rm T}\cdot {\rm e}^{\hspace{0.05cm}{\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}(2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm T}\hspace{0.05cm}\cdot \hspace{0.05cm}t- \varphi_{\rm T})}
\hspace{0.1cm}+\hspace{0.1cm} A_{\rm O}\cdot {\rm e}^{\hspace{0.05cm}{\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}(2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm O}\hspace{0.05cm}\cdot \hspace{0.05cm}t- \varphi_{\rm O})}. $$

{{GraueBox|TEXT=
$\text{Example 3:}$ 
Shown the constellation arises i.e. in the [[Modulationsverfahren/Zweiseitenband-Amplitudenmodulation#AM-Signale_und_-Spektren_bei_harmonischem_Eingangssignal|double sideband amplitude modulation]] of the message signal $x_{\rm N}(t) = A_{\rm N}\cdot \cos\left(2\pi f_{\rm N}\cdot t- \varphi_{\rm N}\right)$ with the carrier signal $x_{\rm T}(t) = A_{\rm T}\cdot \cos\left(2\pi f_{\rm T}\cdot t - \varphi_{\rm T}\right)$. This is discussed frequently in the Exercises.

There are some limitations to the program parameters in this approach:
* For the frequencies, it always applies $f_{\rm O} = f_{\rm T} + f_{\rm N}$ und $f_{\rm U} = f_{\rm T} - f_{\rm N}$.

*Without distortions the amplitude of the sidebands are $A_{\rm O}= A_{\rm U}= A_{\rm N}/2$.
*The respective phase relationships can be seen in the following graphic.

[[File:Zeigerdiagramm_2_neu.png|center|frame|Spectrum $X_+(f)$ of the analytic signal for different phase constellations |class=fit]]}}

==Exercises==
[[File:Zeigerdiagramm_aufgabe_2.png|right]]
*First select the task number.
*A task description is displayed.
*Parameter values are adjusted.
*Solution after pressing &bdquo;Hide solition”.

The number &bdquo;0” will reset the program and output a text with further explanation of the applet.
 
In the following, $\rm Green$ denotes the lower sideband   ⇒   $\big (A_{\rm U}, f_{\rm U}, \varphi_{\rm U}\big )$,  
$\rm Red$ the carrier   ⇒   $\big (A_{\rm T}, f_{\rm T}, \varphi_{\rm T}\big )$ and
$\rm Blue$ the upper sideband   ⇒   $\big (A_{\rm O}, f_{\rm O}, \varphi_{\rm O}\big )$.
{{BlaueBox|TEXT=
'''(1)'''   Consider and interpret the analytic signal $x_+(t)$ for $\text{Red:} \hspace{0.15cm} A_{\rm T} = 1.5\ \text{V}, \ f_{\rm T} = 50 \ \text{kHz}, \ \varphi_{\rm T} = 0^\circ$, $A_{\rm U} = A_{\rm O} = 0$.

:Which signal values $x_+(t)$ result for $t = 0$, $t = 5 \ \rm µ s$ and $t = 20 \ \rm µ s$? What are the corresponding signal values for $x(t)$? }}

:: For a cosine signal, let $x_+(t= 0) = A_{\rm T} = 1.5\ \text{V}$. Then $x_+(t)$ rotates in a mathematically positive direction (one revolution per period $T_0 = 1/f_{\rm T}$):

:: $x_+(t= 20 \ {\rm µ s}) = x_+(t= 0) = 1.5\ \text{V}\hspace{0.3cm}\Rightarrow\hspace{0.3cm}x(t= 20 \ {\rm µ s}) = 1.5\ \text{V,}$
:: $x_+(t= 5 \ {\rm µ s}) = {\rm j} \cdot 1.5\ \text{V}\hspace{0.3cm}\Rightarrow\hspace{0.3cm}x(t= 5 \ {\rm µ s}) = {\rm Re}[x_+(t= 5 \ {\rm µ s})] = 0$.

{{BlaueBox|TEXT=
'''(2)'''   How do the ratios change for $\text{Red:} \hspace{0.15cm} A_{\rm T} = 1.0\ \text{V}, \ f_{\rm T} = 100 \ \text{kHz}, \ \varphi_{\rm T} = 90^\circ$?}}

::The signal $x(t)$ is now a sine signal with a smaller amplitude. The analytic signal now starts because of $\varphi_{\rm T} = 90^\circ$   ⇒   $\phi_{\rm T} = -90^\circ$ at $x_+(t= 0) = -{\rm j} \cdot A_{\rm T}$. After that, $x_+(t)$ rotates again in a mathematically positive direction, but twice as fast because of $T_0 = 10 \ \rm µ s$ as in $\rm (1)$.

{{BlaueBox|TEXT=
'''(3)'''   Now   $\text{Red:} \hspace{0.15cm} A_{\rm T} = 1\ \text{V}, \ f_{\rm T} = 100 \ \text{kHz}, \ \varphi_{\rm T} = 0^\circ$,   $\text{Green:} \hspace{0.15cm} A_{\rm U} = 0.4\ \text{V}, \ f_{\rm U} = 80 \ \text{kHz}, \ \varphi_{\rm U} = 0^\circ$,   $\text{Blue:} \hspace{0.15cm} A_{\rm O} = 0.4\ \text{V}, \ f_{\rm O} = 120 \ \text{kHz}, \ \varphi_{\rm O} = 0^\circ$.

:Consider and interpret the physical signal $x(t)$ and the analytic signal $x_+(t)$.}}

::The Signal $x(t)$ results in the double sideband Amplitude Modulation '''(DSB–AM)''' of the message signals $A_{\rm N}\cdot \cos\left(2\pi f_{\rm N}\cdot t\right)$ with $A_{\rm N} = 0.8\ \text{V}$, $f_{\rm N} = 20\ \text{kHz}$. The carrier $x_{\rm T}(t)$ with $f_{\rm T} = 100\ \text{kHz}$ is also cosinusoidal. The degree of modulation is $m = A_{\rm N}/A_{\rm T} = 0.8$ and the period $T_{\rm 0} = 50\ \text{µs}$.

::In the phase diagram, the (red) carrier rotates faster than the (green) lower sideband and slower than the (blue) upper sideband. The analytic signal $x_+(t)$ results as the geometric sum of the three rotating pointers. It seems that the blue pointer is leading the carrier and the green pointer is following the carrier.

{{BlaueBox|TEXT=
'''(4)'''   The settings of task '''(3)''' still apply. Which signal values are obtained at $t=0$, $t=2.5 \ \rm µ s$, $t= 5 \ \rm µ s$ and $t=10 \ \rm µ s$? }}

::At time $t=0$, all pointers are in the direction of the real axis, so that $x(t=0) = {\rm Re}\big [x+(t= 0)\big] = A_{\rm U} + A_{\rm T} + A_{\rm O} = 1.8\ \text{V}$.

::Until the time $t=2.5 \ \rm µ s$, the red carrier has rotated by $90^\circ$, the blue one by $108^\circ$ and the green one by $72^\circ$. We have $x(t=2.5 \ \rm µ s) = {\rm Re}\big [x_+(t= 2.5 \ \rm µ s)\big] = 0$, because now the pointer group points in the direction of the imaginary axis. The other sought signal values are $x(t=5 \ \rm µ s) = {\rm Re}\big [x_+(t= 5 \ \rm µ s)\big] = -1.647\ \text{V}$ and $x(t=10 \ \rm µ s) = {\rm Re}\big [x_+(t= 10 \ \rm µ s)\big] = 1.247\ \text{V}$.
::For $x_+(t)$ a spiral shape results, alternating with a smaller radius and then with a larger radius.

{{BlaueBox|TEXT=
'''(5)'''   How should the phase parameters $\varphi_{\rm T}$, $\varphi_{\rm U}$ and $\varphi_{\rm O}$ be set if both the carrier $x_{\rm T}(t)$ and the message signal $x_{\rm N}(t)$ are sinusoidal?}}

::The parameter selection $\varphi_{\rm T} = \varphi_{\rm U} = \varphi_{\rm O}=90^\circ$ describes the signals $x_{\rm T}(t) = A_{\rm T}\cdot \sin\left(2\pi f_{\rm T}\cdot t\right)$ and $x_{\rm N}(t) = A_{\rm N}\cdot \cos\left(2\pi f_{\rm N}\cdot t\right)$. If, in addition, the message $x_{\rm N}(t)$ is sinusoidal, then $\varphi_{\rm O}=\varphi_{\rm T} - 90^\circ = 0$ and $\varphi_{\rm U}=\varphi_{\rm T} + 90^\circ = 180^\circ$ must be set.

{{BlaueBox|TEXT=
'''(6)'''   The settings of task '''(3)''' apply except $A_{\rm T} = 0.6\ \text{V}$. Which modulation method is described here?

: What are the consequences of this? What changes with $A_{\rm T} = 0$? }}

::It is a '''DSB–AM with carrier''' with the modulation degree $m=0.8/0.6 = 1.333$. For $m > 1$, however, [[Modulationsverfahren/Synchrondemodulation|synchronous demodulation]] is required. [[Modulationsverfahren/Hüllkurvendemodulation|envelope detection]] no longer works.

::With $A_{\rm T} = 0$   ⇒   $m \to \infty$ results in a '''DSB–AM suppressed carrier'''. For this, one also needs coherent demodulation.

{{BlaueBox|TEXT=
'''(7)'''     Now let   $\text{Red:} \hspace{0.15cm} A_{\rm T} = 1\ \text{V}, \ f_{\rm T} = 100 \ \text{kHz}, \ \varphi_{\rm T} = 0^\circ$,   $\text{Green:} \hspace{0.15cm} A_{\rm U} = 0$,   $\text{Blue:} \hspace{0.15cm} A_{\rm O} = 0.8\ \text{V}, \ f_{\rm O} = 120 \ \text{kHz}, \ \varphi_{\rm O} = 90^\circ$.

:Which constellation is described here? What changes with $A_{\rm U} = 0.8\ \text{V}$ und $A_{\rm O} = 0$?}}

::In both cases, it is a [[Modulationsverfahren/Einseitenbandmodulation|Single sideband]] '''(ESB–AM)''' with the modulation degree $\mu = 0.8$ (in ESB we denote the degree of modulation with $\mu$ instead of $m$). The carrier signal is cosinusoidal and the message signal is sinusoidal.

:: $A_{\rm O} = 0.8\ \text{V}$, $A_{\rm U} = 0$ is an OSB modulation. The green pointer is missing and the blue pointer rotates faster compared to the red carrier.

:: $A_{\rm U} = 0.8\ \text{V}$, $A_{\rm O} = 0$ is a USB modulation. The blue pointer is missing and the green pointer rotates slower compared to the red carrier.

{{BlaueBox|TEXT=
'''(8)'''   Now let   $\text{Red:} \hspace{0.05cm} A_{\rm T} = 1\ \text{V}, \ f_{\rm T} = 100 \ \text{kHz}, \ \varphi_{\rm T} = 0^\circ$,   $\text{Green:} \hspace{0.05cm} A_{\rm U} = 0.4\ \text{V}, \ f_{\rm U} = 80 \ \text{kHz}, \ \varphi_{\rm U} = -90^\circ$,   $\text{Blue:} \hspace{0.05cm} A_{\rm O} = 0.2\ \text{V}, \ f_{\rm O} = 120 \ \text{kHz}, \ \varphi_{\rm O} = +90^\circ$.

:Which constellation could be described here? Which shape results for the equivalent lowpass–signal $x_{\rm TP}(t)$?   ⇒   &bdquo;locus”?}}

::It could be a ZSB–AM of a sinusoidal signal with cosinusoidal carrier and modulation degree $m=0.8$, in which the upper sideband is attenuated by a factor of 2. The equivalent lowpass–signal $x_{\rm TP}(t)$ has an elliptical trace in the complex plane.

==Applet Manual==
 
[[File:Zeigerdiagramm_abzug.png|right]]

* The red parameters $(A_{\rm T}, \ f_{\rm T}, \ \varphi_{\rm T})$ and the red pointer mark the '''T'''räger(Carrier).
* The green parameters $(A_{\rm U}, \ f_{\rm U} < f_{\rm T}, \ \varphi_{\rm U})$ mark the '''U'''ntere Seitenband(Lower sideband).
* The blue parameters $(A_{\rm O}, \ f_{\rm O} > f_{\rm T}, \ \varphi_{\rm O})$ mark the '''O'''bere Seitenband(upper sideband).
*All pointers rotate in a mathematically positive direction (counterclockwise).

 
Meaning of the letters in the adjacent graphic:

    '''(A)'''     Plot of the analytic signal $x_{\rm +}(t)$

    '''(B)'''     Plot of for the physical signal $x(t)$

    '''(C)'''     Parameter input via slider: amplitudes, frequencies, phase values

    '''(D)'''     Control elements:   Start – Step – Pause/Continue – Reset

    '''(E)'''     Speed of animation:   &bdquo;Speed”   ⇒   Value: 1, 2 oder 3

    '''(F)'''     &bdquo;Trace”   ⇒   On or Off, trace of complex signal values $x_{\rm +}(t)$

    '''(G)'''     Numeric output of the time $t$ and the signal values  ${\rm Re}[x_{\rm +}(t)] = x(t)$  and  ${\rm Im}[x_{\rm +}(t)]$

    '''(H)'''     Variations for the graphical representation

$\hspace{1.5cm}$Zoom–Functions &bdquo;$+$” (Enlarge), &bdquo;$-$” (Decrease) and $\rm o$ (Reset to default)

$\hspace{1.5cm}$Move with &bdquo;$\leftarrow$” (Section to the left, ordinate to the right), &bdquo;$\uparrow$” &bdquo;$\downarrow$” and &bdquo;$\rightarrow$”

    '''(I)'''     Experiment section:  Task selection and task

    '''(J)'''     Experiment section:  solution
 

==About the Authors==
This interactive calculation was designed and realized at the [http://www.lnt.ei.tum.de/startseite Lehrstuhl für Nachrichtentechnik] of the [https://www.tum.de/ Technical University of Munich] .
*The original version was created in 2005 by [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Studierende#Ji_Li_.28Bachelorarbeit_EI_2003.2C_Diplomarbeit_EI_2005.29|Ji Li]] as part of her Diploma thesis using &bdquo;FlashMX–Actionscript” (Supervisor: [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Mitarbeiter_und_Dozenten#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28am_LNT_seit_1974.29|Günter Söder]]).
*In 2018 this Applet was redesigned and updated to &bdquo;HTML5” by [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Studierende#Xiaohan_Liu_.28Bachelorarbeit_2018.29|Xiaohan Liu]] as part of her Bachelor's thesis (Supervisor: [[Biografien_und_Bibliografien/Beteiligte_der_Professur_Leitungsgebundene_%C3%9Cbertragungstechnik#Tasn.C3.A1d_Kernetzky.2C_M.Sc._.28bei_L.C3.9CT_seit_2014.29|Tasnád Kernetzky]]).

==Once again: Open Applet in new Tab==

{{LntAppletLink|physAnSignal}}

[[Category:Applets|^Verzerrungen^]]

Applets:Physical Signal & Analytic Signal

2018-08-10T14:12:19Z

Xiaohan: /* About the Authors */

{{LntAppletLink|physAnSignal}}

==Applet Description==
 
This applet shows the relationship between the physical bandpass signal $x(t)$ and the associated analytic signal $x_+(t)$. It is assumed that the bandpass signal $x(t)$ has a frequency-discrete spectrum $X(f)$:
:$$x(t) = x_{\rm U}(t) + x_{\rm T}(t) + x_{\rm O}(t) = A_{\rm U}\cdot \cos\left(2\pi f_{\rm U}\cdot t- \varphi_{\rm U}\right)+A_{\rm T}\cdot \cos\left(2\pi f_{\rm T}\cdot t- \varphi_{\rm T}\right)+A_{\rm O}\cdot \cos\left(2\pi f_{\rm O}\cdot t- \varphi_{\rm O}\right). $$
The physical signal $x(t)$ is thus composed of three [[Signaldarstellung/Harmonische_Schwingung|harmonic oscillations]], a constellation that can be found, for example, in the ''Double-sideband Amplitude Modulation''
*of the message signal $x_{\rm N}(t) = A_{\rm N}\cdot \cos\left(2\pi f_{\rm N}\cdot t- \varphi_{\rm N}\right)$   ⇒   in German:   '''N'''achrichtensignal
*with the carrier signal $x_{\rm T}(t) = A_{\rm T}\cdot \cos\left(2\pi f_{\rm T}\cdot t - \varphi_{\rm T}\right)$   ⇒   in German:   '''T'''rägersignal.

The nomenclature is also adapted to this case:
* $x_{\rm O}(t)$ denotes the &bdquo;upper sideband”   (in German:   '''O'''beres Seitenband) with the amplitude $A_{\rm O}= A_{\rm N}/2$, the frequency $f_{\rm O} = f_{\rm T} + f_{\rm N}$ and the phase $\varphi_{\rm O} = \varphi_{\rm T} + \varphi_{\rm N}$.
*Similarly, for the &bdquo;lower sideband”   (in German:   '''U'''nteres Seitenband) $x_{\rm U}(t)$ with $f_{\rm U} = f_{\rm T} + f_{\rm N}$, $A_{\rm U}= A_{\rm O}$ and $\varphi_{\rm U} = -\varphi_{\rm O}$.

The associated analytical signal is:

:$$x_+(t) = x_{\rm U+}(t) + x_{\rm T+}(t) + x_{\rm O+}(t) = A_{\rm U}\cdot {\rm e}^{\hspace{0.05cm}{\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}(2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm U}\hspace{0.05cm}\cdot \hspace{0.05cm}t- \varphi_{\rm U})}
\hspace{0.1cm}+ \hspace{0.1cm}A_{\rm T}\cdot {\rm e}^{\hspace{0.05cm}{\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}(2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm T}\hspace{0.05cm}\cdot \hspace{0.05cm}t- \varphi_{\rm T})}
\hspace{0.1cm}+\hspace{0.1cm} A_{\rm O}\cdot {\rm e}^{\hspace{0.05cm}{\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}(2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm O}\hspace{0.05cm}\cdot \hspace{0.05cm}t- \varphi_{\rm O})}. $$

[[File:Zeigerdiagramm_2a_version2.png|right|frame|Analytical signal at the time $t=0$]]
The program displays $x_+(t)$ as the vectorial sum of three rotating pointers (all with counterclockwise) as a violet dot (see figure for start time $t=0$):

*The (red) pointer of the carrier $x_{\rm T+}(t)$ with length $A_{\rm T}$ and zero phase position $\varphi_{\rm T} = 0$ rotates at constant angular velocity $2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm T}$ (one revolution in time $1/f_{\rm T})$.

*The (blue) pointer of the upper sideband $x_{\rm O+}(t)$ with length $A_{\rm O}$ and zero phase position $\varphi_{\rm O}$ rotates at the angular velocity $2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm O}$, which is slightly faster than $x_{\rm T+}(t)$.

*The (green) pointer of the lower sideband $x_{\rm U+}(t)$ with length $A_{\rm U}$ and zero phase position $\varphi_{\rm U}$ rotates at the angular velocity $2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm U}$, which is slightly faster than $x_{\rm T+}(t)$.

The time trace of $x_+(t)$ is also referred to below as '''Pointer Diagram'''. The relationship between the physical bandpass signal $x(t)$ and the associated analytic signal $x_+(t)$ is:

:$$x(t) = {\rm Re}\big [x_+(t)\big ].$$

''Note:''   In the figure $\varphi_{\rm O} = +30^\circ$. This leads to the angle with respect to the coordinate system at $t=0$:   $\phi_{\rm O}=-\varphi_{\rm O}=-30^\circ$. Similarly, the null phase angle $\varphi_{\rm U}=-30^\circ$ of the lower sideband leads to the phase angle to be considered in the complex plane:   $\phi_{\rm U}=+30^\circ$.

[[Applets:Physikalisches_Signal_%26_Analytisches_Signal|'''German description''']]

==Theoretical Background==
 
===Description of Bandpass Signals===
[[File:Zeigerdiagramm_1a.png|right|frame|Bandpass–spectrum $X(f)$ |class=fit]]
We consider '''bandpass signals''' $x(t)$ with the property that their spectra $X(f)$ are not in the range around the frequency $f=0$, but around a carrier frequency $f_{\rm T}$. In most cases it can also be assumed that the bandwidth is $B \ll f_{\rm T}$.

The figure shows such a bandpass spectrum $X(f)$. Assuming that the associated $x(t)$ is a physical signal and thus real, the spectral function $X(f)$ has a symmetry with respect to the frequency $f = 0$, if $x(t)$ is an even function   ⇒   $x(-t)=x(t)$, $X(f)$ is real and even.

Besides the physical signal $x(t)\ \circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\, \ X(f)$, one can also use the following descriptions of Bandpass signals:
*the analytic signal $x_+(t)\ \circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\, \ X_+(f)$, see next page,
*the equivalent low-pass –signal   (in German:   äquivalentes '''T'''ief'''P'''ass–Signal) $x_{\rm TP}(t)\ \circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\, \ X_{\rm TP}(f)$, see Applet [[Applets:Physical_Signal_%26_Equivalent_Low-pass_Signal|Physical Signal & Equivalent Low–pass signal]].
 

===Analytical Signal – Frequency Domain===

The '''analytical signal''' $x_+(t)$ belonging to the physical signal $x(t)$ is the time function whose spectrum fulfills the following property:
[[File:Zeigerdiagramm_3a.png|right|frame|Construction of the spectral function $X_+(f)$ |class=fit]]
:$$X_+(f)=\big[1+{\rm sign}(f)\big] \cdot X(f) = \left\{ {2 \cdot
X(f) \; \hspace{0.2cm}\rm for\hspace{0.2cm} {\it f} > 0, \atop {\,\,\,\, \rm 0 \; \hspace{0.9cm}\rm for\hspace{0.2cm} {\it f} < 0.} }\right.$$

The ''signum function'' is for positive values of $f$ equal to $+1$ and for negative $f$ values equal to $-1$.
* The (double-sided) limit returns $\sign(0)=0$.
* The index „+” should make it clear that $X_+(f)$ only has parts at positive frequencies.

From the graph you can see the calculation rule for $X_+(f)$:

The actual band pass spectrum $X(f)$ becomes
* doubled at the positive frequencies, and
* set to zero at the negative frequencies.

Due to the asymmetry of $X_+(f)$ with respect to the frequency $f=0$, it can already be said that the time function $x_+(t)$ except for a trivial special case $x_+(t)=0 \ \circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\,X_+(f)=0$ is always complex.
 

===Analytical Signal – Time Domain===
At this point, it is necessary to briefly discuss another spectral transformation.

{{BlaueBox|TEXT=
$\text{Definition:}$ 
For the '''Hilbert transformed ''' $ {\rm H}\left\{x(t)\right\}$ of a time function $x(t)$ we have::

:$$y(t) = {\rm H}\left\{x(t)\right\} = \frac{1}{ {\rm \pi} } \cdot
\hspace{0.03cm}\int_{-\infty}^{+\infty}\frac{x(\tau)}{ {t -
\tau} }\hspace{0.15cm} {\rm d}\tau.$$

This particular integral is not solvable in a simple, conventional way, but must be evaluated using the [https://de.wikipedia.org/wiki/Cauchyscher_Hauptwert Cauchy principal value theorem].

Accordingly, in the frequency domain:
:$$Y(f) = {\rm -j \cdot sign}(f) \cdot X(f) \hspace{0.05cm} .$$}}

The above result can be summarized with this definition as follows:
* The analytic signal $x_+(t)$ is obtained from the physical bandpass signal $x(t)$ by adding an imaginary part to $x(t)$ according to the Hilbert transform:

:$$x_+(t) = x(t)+{\rm j} \cdot {\rm H}\left\{x(t)\right\} .$$

*$\text{H}\{x(t)\}$ disappears only for the case $x(t) = \rm const.$. For all other signal forms, the analytic signal $x_+(t)$ is complex.

* From the analytic signal $x_+(t)$, the physical bandpass signal can be easily determined by the following operation:
:$$x(t) = {\rm Re}\big[x_+(t)\big] .$$

{{GraueBox|TEXT=
$\text{Example 1:}$  The principle of the Hilbert transformation should be further clarified by the following graphic:
*After the left representation $\rm(A)$ one gets from the physical signal $x(t)$ to the analytic signal $x_+(t)$, by adding an imaginary part ${\rm j} \cdot y(t)$.
*Here $y(t) = {\rm H}\left\{x(t)\right\}$ is a real time function that can be indicated in the spectral domain by multiplying the spectrum $X(f)$ with ${\rm - j} \cdot \sign(f)$.

[[File:P_ID2729__Sig_T_4_2_S2b_neu.png|center|frame|To clarify the Hilbert transform]]

The right representation $\rm(B)$ is equivalent to $\rm(A)$. Now $x_+(t) = x(t) + z(t)$ stand with the purely imaginary function $z(t)$. A comparison of the two pictures shows that in fact $z(t) = {\rm j} \cdot y(t)$.}}
 

===Representation of the Harmonic Oscillation as an Analytical Signal===

The spectral function $X(f)$ of a harmonic oscillation $x(t) = A\cdot\text{cos}(2\pi f_{\rm T}\cdot t - \varphi)$ is known to consist of two dirac functions in the frequencies
* $+f_{\rm T}$ with the complex weight $A/2 \cdot \text{e}^{-\text{j}\hspace{0.05cm}\varphi}$,
* $-f_{\rm T}$ with the complex weight $A/2 \cdot \text{e}^{+\text{j}\hspace{0.05cm}\varphi}$.

Thus, the spectrum of the analytic signal (that is, without the Dirac function at the frequency $f =-f_{\rm T}$, but doubling at $f =+f_{\rm T}$):

:$$X_+(f) = A \cdot {\rm e}^{-{\rm j} \hspace{0.05cm}\varphi}\cdot\delta (f - f_{\rm
T}) .$$

The associated time function is obtained by applying the [[Signaldarstellung/Gesetzmäßigkeiten_der_Fouriertransformation#Verschiebungssatz|Displacement law]]:

:$$x_+(t) = x_{\rm U+}(t) + x_{\rm T+}(t) + x_{\rm O+}(t) = A_{\rm U}\cdot {\rm e}^{\hspace{0.05cm}{\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}(2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm U}\hspace{0.05cm}\cdot \hspace{0.05cm}t- \varphi_{\rm U})}
\hspace{0.1cm}+ \hspace{0.1cm}A_{\rm T}\cdot {\rm e}^{\hspace{0.05cm}{\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}(2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm T}\hspace{0.05cm}\cdot \hspace{0.05cm}t- \varphi_{\rm T})}
\hspace{0.1cm}+\hspace{0.1cm} A_{\rm O}\cdot {\rm e}^{\hspace{0.05cm}{\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}(2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm O}\hspace{0.05cm}\cdot \hspace{0.05cm}t- \varphi_{\rm O})}. $$

This equation describes a pointer rotating at constant angular velocity $\omega_{\rm T} = 2\pi f_{\rm T}$.

{{GraueBox|TEXT=
$\text{Example 2:}$  Here the coordinate system is rotated by $90^\circ$ (real part up, imaginary part to the left) contrary to the usual representation.

[[File:P_ID712__Sig_T_4_2_S3.png|center|frame|Pointer diagram of a harmonic oscillation]]

Based on this graphic, the following statements are possible:
* At the start time $t = 0$, the pointer of length $A$ (signal amplitude) lies with the angle $-\varphi$ in the complex plane. In the example shown, $\varphi=45^\circ$.
* For times $t>0$, the constant angular velocity vector $\omega_{\rm T}$ rotates in a mathematically positive direction, that is, counterclockwise.
* The tip of the pointer is thus always on a circle with radius $A$ and needs exactly the time $T_0$, i.e. the period of the harmonic oscillation $x(t)$ for one revolution.
* The projection of the analytic signal $x_+(t)$ on the real axis, marked by red dots, gives the instantaneous values of $x(t)$.}}
 

===Analytical Signal Representation of a Sum of Three Harmonic Oscillations===

In our applet, we always assume a set of three rotating pointers. The physical signal is:
:$$x(t) = x_{\rm U}(t) + x_{\rm T}(t) + x_{\rm O}(t) = A_{\rm U}\cdot \cos\left(2\pi f_{\rm U}\cdot t- \varphi_{\rm U}\right)+A_{\rm T}\cdot \cos\left(2\pi f_{\rm T}\cdot t- \varphi_{\rm T}\right)+A_{\rm O}\cdot \cos\left(2\pi f_{\rm O}\cdot t- \varphi_{\rm O}\right). $$
* Each of the three harmonic oscillations $x_{\rm T}(t)$, $x_{\rm U}(t)$ and $x_{\rm O}(t)$ is represented by an amplitude $(A)$, a frequency $(f)$ and a phase value $(\varphi)$.
*The indices are based on the [[Modulationsverfahren/Zweiseitenband-Amplitudenmodulation|Double sideband Amplitude Modulation]] method. &bdquo;T” stands for &bdquo;carrier”, &bdquo;U” for &bdquo;lower sideband” and &bdquo;O” for &bdquo;Upper Sideband”.
*Accordingly, $f_{\rm U} < f_{\rm T}$ and $f_{\rm O} > f_{\rm T}$. There are no restrictions for the amplitudes and phases.

The associated analytical signal is:
:$$x_+(t) = x_{\rm U+}(t) + x_{\rm T+}(t) + x_{\rm O+}(t) = A_{\rm U}\cdot {\rm e}^{\hspace{0.05cm}{\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}(2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm U}\hspace{0.05cm}\cdot \hspace{0.05cm}t- \varphi_{\rm U})}
\hspace{0.1cm}+ \hspace{0.1cm}A_{\rm T}\cdot {\rm e}^{\hspace{0.05cm}{\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}(2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm T}\hspace{0.05cm}\cdot \hspace{0.05cm}t- \varphi_{\rm T})}
\hspace{0.1cm}+\hspace{0.1cm} A_{\rm O}\cdot {\rm e}^{\hspace{0.05cm}{\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}(2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm O}\hspace{0.05cm}\cdot \hspace{0.05cm}t- \varphi_{\rm O})}. $$

{{GraueBox|TEXT=
$\text{Example 3:}$ 
Shown the constellation arises i.e. in the [[Modulationsverfahren/Zweiseitenband-Amplitudenmodulation#AM-Signale_und_-Spektren_bei_harmonischem_Eingangssignal|double sideband amplitude modulation]] of the message signal $x_{\rm N}(t) = A_{\rm N}\cdot \cos\left(2\pi f_{\rm N}\cdot t- \varphi_{\rm N}\right)$ with the carrier signal $x_{\rm T}(t) = A_{\rm T}\cdot \cos\left(2\pi f_{\rm T}\cdot t - \varphi_{\rm T}\right)$. This is discussed frequently in the Exercises.

There are some limitations to the program parameters in this approach:
* For the frequencies, it always applies $f_{\rm O} = f_{\rm T} + f_{\rm N}$ und $f_{\rm U} = f_{\rm T} - f_{\rm N}$.

*Without distortions the amplitude of the sidebands are $A_{\rm O}= A_{\rm U}= A_{\rm N}/2$.
*The respective phase relationships can be seen in the following graphic.

[[File:Zeigerdiagramm_2_neu.png|center|frame|Spectrum $X_+(f)$ of the analytic signal for different phase constellations |class=fit]]}}

==Exercises==
[[File:Zeigerdiagramm_aufgabe_2.png|right]]
*First select the task number.
*A task description is displayed.
*Parameter values are adjusted.
*Solution after pressing &bdquo;Hide solition”.

The number &bdquo;0” will reset the program and output a text with further explanation of the applet.
 
In the following, $\rm Green$ denotes the lower sideband   ⇒   $\big (A_{\rm U}, f_{\rm U}, \varphi_{\rm U}\big )$,  
$\rm Red$ the carrier   ⇒   $\big (A_{\rm T}, f_{\rm T}, \varphi_{\rm T}\big )$ and
$\rm Blue$ the upper sideband   ⇒   $\big (A_{\rm O}, f_{\rm O}, \varphi_{\rm O}\big )$.
{{BlaueBox|TEXT=
'''(1)'''   Consider and interpret the analytic signal $x_+(t)$ for $\text{Red:} \hspace{0.15cm} A_{\rm T} = 1.5\ \text{V}, \ f_{\rm T} = 50 \ \text{kHz}, \ \varphi_{\rm T} = 0^\circ$, $A_{\rm U} = A_{\rm O} = 0$.

:Which signal values $x_+(t)$ result for $t = 0$, $t = 5 \ \rm µ s$ and $t = 20 \ \rm µ s$? What are the corresponding signal values for $x(t)$? }}

:: For a cosine signal, let $x_+(t= 0) = A_{\rm T} = 1.5\ \text{V}$. Then $x_+(t)$ rotates in a mathematically positive direction (one revolution per period $T_0 = 1/f_{\rm T}$):

:: $x_+(t= 20 \ {\rm µ s}) = x_+(t= 0) = 1.5\ \text{V}\hspace{0.3cm}\Rightarrow\hspace{0.3cm}x(t= 20 \ {\rm µ s}) = 1.5\ \text{V,}$
:: $x_+(t= 5 \ {\rm µ s}) = {\rm j} \cdot 1.5\ \text{V}\hspace{0.3cm}\Rightarrow\hspace{0.3cm}x(t= 5 \ {\rm µ s}) = {\rm Re}[x_+(t= 5 \ {\rm µ s})] = 0$.

{{BlaueBox|TEXT=
'''(2)'''   How do the ratios change for $\text{Red:} \hspace{0.15cm} A_{\rm T} = 1.0\ \text{V}, \ f_{\rm T} = 100 \ \text{kHz}, \ \varphi_{\rm T} = 90^\circ$?}}

::The signal $x(t)$ is now a sine signal with a smaller amplitude. The analytic signal now starts because of $\varphi_{\rm T} = 90^\circ$   ⇒   $\phi_{\rm T} = -90^\circ$ at $x_+(t= 0) = -{\rm j} \cdot A_{\rm T}$. After that, $x_+(t)$ rotates again in a mathematically positive direction, but twice as fast because of $T_0 = 10 \ \rm µ s$ as in $\rm (1)$.

{{BlaueBox|TEXT=
'''(3)'''   Now   $\text{Red:} \hspace{0.15cm} A_{\rm T} = 1\ \text{V}, \ f_{\rm T} = 100 \ \text{kHz}, \ \varphi_{\rm T} = 0^\circ$,   $\text{Green:} \hspace{0.15cm} A_{\rm U} = 0.4\ \text{V}, \ f_{\rm U} = 80 \ \text{kHz}, \ \varphi_{\rm U} = 0^\circ$,   $\text{Blue:} \hspace{0.15cm} A_{\rm O} = 0.4\ \text{V}, \ f_{\rm O} = 120 \ \text{kHz}, \ \varphi_{\rm O} = 0^\circ$.

:Consider and interpret the physical signal $x(t)$ and the analytic signal $x_+(t)$.}}

::The Signal $x(t)$ results in the double sideband Amplitude Modulation '''(DSB–AM)''' of the message signals $A_{\rm N}\cdot \cos\left(2\pi f_{\rm N}\cdot t\right)$ with $A_{\rm N} = 0.8\ \text{V}$, $f_{\rm N} = 20\ \text{kHz}$. The carrier $x_{\rm T}(t)$ with $f_{\rm T} = 100\ \text{kHz}$ is also cosinusoidal. The degree of modulation is $m = A_{\rm N}/A_{\rm T} = 0.8$ and the period $T_{\rm 0} = 50\ \text{µs}$.

::In the phase diagram, the (red) carrier rotates faster than the (green) lower sideband and slower than the (blue) upper sideband. The analytic signal $x_+(t)$ results as the geometric sum of the three rotating pointers. It seems that the blue pointer is leading the carrier and the green pointer is following the carrier.

{{BlaueBox|TEXT=
'''(4)'''   The settings of task '''(3)''' still apply. Which signal values are obtained at $t=0$, $t=2.5 \ \rm µ s$, $t= 5 \ \rm µ s$ and $t=10 \ \rm µ s$? }}

::At time $t=0$, all pointers are in the direction of the real axis, so that $x(t=0) = {\rm Re}\big [x+(t= 0)\big] = A_{\rm U} + A_{\rm T} + A_{\rm O} = 1.8\ \text{V}$.

::Until the time $t=2.5 \ \rm µ s$, the red carrier has rotated by $90^\circ$, the blue one by $108^\circ$ and the green one by $72^\circ$. We have $x(t=2.5 \ \rm µ s) = {\rm Re}\big [x_+(t= 2.5 \ \rm µ s)\big] = 0$, because now the pointer group points in the direction of the imaginary axis. The other sought signal values are $x(t=5 \ \rm µ s) = {\rm Re}\big [x_+(t= 5 \ \rm µ s)\big] = -1.647\ \text{V}$ and $x(t=10 \ \rm µ s) = {\rm Re}\big [x_+(t= 10 \ \rm µ s)\big] = 1.247\ \text{V}$.
::For $x_+(t)$ a spiral shape results, alternating with a smaller radius and then with a larger radius.

{{BlaueBox|TEXT=
'''(5)'''   How should the phase parameters $\varphi_{\rm T}$, $\varphi_{\rm U}$ and $\varphi_{\rm O}$ be set if both the carrier $x_{\rm T}(t)$ and the message signal $x_{\rm N}(t)$ are sinusoidal?}}

::The parameter selection $\varphi_{\rm T} = \varphi_{\rm U} = \varphi_{\rm O}=90^\circ$ describes the signals $x_{\rm T}(t) = A_{\rm T}\cdot \sin\left(2\pi f_{\rm T}\cdot t\right)$ and $x_{\rm N}(t) = A_{\rm N}\cdot \cos\left(2\pi f_{\rm N}\cdot t\right)$. If, in addition, the message $x_{\rm N}(t)$ is sinusoidal, then $\varphi_{\rm O}=\varphi_{\rm T} - 90^\circ = 0$ and $\varphi_{\rm U}=\varphi_{\rm T} + 90^\circ = 180^\circ$ must be set.

{{BlaueBox|TEXT=
'''(6)'''   The settings of task '''(3)''' apply except $A_{\rm T} = 0.6\ \text{V}$. Which modulation method is described here?

: What are the consequences of this? What changes with $A_{\rm T} = 0$? }}

::It is a '''DSB–AM with carrier''' with the modulation degree $m=0.8/0.6 = 1.333$. For $m > 1$, however, [[Modulationsverfahren/Synchrondemodulation|synchronous demodulation]] is required. [[Modulationsverfahren/Hüllkurvendemodulation|envelope detection]] no longer works.

::With $A_{\rm T} = 0$   ⇒   $m \to \infty$ results in a '''DSB–AM suppressed carrier'''. For this, one also needs coherent demodulation.

{{BlaueBox|TEXT=
'''(7)'''     Now let   $\text{Red:} \hspace{0.15cm} A_{\rm T} = 1\ \text{V}, \ f_{\rm T} = 100 \ \text{kHz}, \ \varphi_{\rm T} = 0^\circ$,   $\text{Green:} \hspace{0.15cm} A_{\rm U} = 0$,   $\text{Blue:} \hspace{0.15cm} A_{\rm O} = 0.8\ \text{V}, \ f_{\rm O} = 120 \ \text{kHz}, \ \varphi_{\rm O} = 90^\circ$.

:Which constellation is described here? What changes with $A_{\rm U} = 0.8\ \text{V}$ und $A_{\rm O} = 0$?}}

::In both cases, it is a [[Modulationsverfahren/Einseitenbandmodulation|Single sideband]] '''(ESB–AM)''' with the modulation degree $\mu = 0.8$ (in ESB we denote the degree of modulation with $\mu$ instead of $m$). The carrier signal is cosinusoidal and the message signal is sinusoidal.

:: $A_{\rm O} = 0.8\ \text{V}$, $A_{\rm U} = 0$ is an OSB modulation. The green pointer is missing and the blue pointer rotates faster compared to the red carrier.

:: $A_{\rm U} = 0.8\ \text{V}$, $A_{\rm O} = 0$ is a USB modulation. The blue pointer is missing and the green pointer rotates slower compared to the red carrier.

{{BlaueBox|TEXT=
'''(8)'''   Now let   $\text{Red:} \hspace{0.05cm} A_{\rm T} = 1\ \text{V}, \ f_{\rm T} = 100 \ \text{kHz}, \ \varphi_{\rm T} = 0^\circ$,   $\text{Green:} \hspace{0.05cm} A_{\rm U} = 0.4\ \text{V}, \ f_{\rm U} = 80 \ \text{kHz}, \ \varphi_{\rm U} = -90^\circ$,   $\text{Blue:} \hspace{0.05cm} A_{\rm O} = 0.2\ \text{V}, \ f_{\rm O} = 120 \ \text{kHz}, \ \varphi_{\rm O} = +90^\circ$.

:Which constellation could be described here? Which shape results for the equivalent lowpass–signal $x_{\rm TP}(t)$?   ⇒   &bdquo;locus”?}}

::It could be a ZSB–AM of a sinusoidal signal with cosinusoidal carrier and modulation degree $m=0.8$, in which the upper sideband is attenuated by a factor of 2. The equivalent lowpass–signal $x_{\rm TP}(t)$ has an elliptical trace in the complex plane.

==Applet Manual==
 
[[File:Zeigerdiagramm_abzug.png|right]]

* The red parameters $(A_{\rm T}, \ f_{\rm T}, \ \varphi_{\rm T})$ and the red pointer mark the '''T'''räger(Carrier).
* The green parameters $(A_{\rm U}, \ f_{\rm U} < f_{\rm T}, \ \varphi_{\rm U})$ mark the '''U'''ntere Seitenband(Lower sideband).
* The blue parameters $(A_{\rm O}, \ f_{\rm O} > f_{\rm T}, \ \varphi_{\rm O})$ mark the '''O'''bere Seitenband(upper sideband).
*All pointers rotate in a mathematically positive direction (counterclockwise).

 
Meaning of the letters in the adjacent graphic:

    '''(A)'''     Plot of the analytic signal $x_{\rm +}(t)$

    '''(B)'''     Plot of for the physical signal $x(t)$

    '''(C)'''     Parameter input via slider: amplitudes, frequencies, phase values

    '''(D)'''     Control elements:   Start – Step – Pause/Continue – Reset

    '''(E)'''     Speed of animation:   &bdquo;Speed”   ⇒   Value: 1, 2 oder 3

    '''(F)'''     &bdquo;Trace”   ⇒   On or Off, trace of complex signal values $x_{\rm +}(t)$

    '''(G)'''     Numeric output of the time $t$ and the signal values  ${\rm Re}[x_{\rm +}(t)] = x(t)$  and  ${\rm Im}[x_{\rm +}(t)]$

    '''(H)'''     Variations for the graphical representation

$\hspace{1.5cm}$Zoom–Functions &bdquo;$+$” (Enlarge), &bdquo;$-$” (Decrease) and $\rm o$ (Reset to default)

$\hspace{1.5cm}$Move with &bdquo;$\leftarrow$” (Section to the left, ordinate to the right), &bdquo;$\uparrow$” &bdquo;$\downarrow$” and &bdquo;$\rightarrow$”

    '''(I)'''     Experiment section:  Task selection and task

    '''(J)'''     Experiment section:  solution
 

==About the Authors==
This interactive calculation was designed and realized at the [http://www.lnt.ei.tum.de/startseite Lehrstuhl für Nachrichtentechnik] of the [https://www.tum.de/Technical University of Munich] .
*The original version was created in 2005 by [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Studierende#Ji_Li_.28Bachelorarbeit_EI_2003.2C_Diplomarbeit_EI_2005.29|Ji Li]] as part of her Diploma thesis using &bdquo;FlashMX–Actionscript” (Supervisor: [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Mitarbeiter_und_Dozenten#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28am_LNT_seit_1974.29|Günter Söder]]).
*In 2018 this Applet was redesigned and updated to &bdquo;HTML5” by [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Studierende#Xiaohan_Liu_.28Bachelorarbeit_2018.29|Xiaohan Liu]] as part of her Bachelor's thesis (Supervisor: [[Biografien_und_Bibliografien/Beteiligte_der_Professur_Leitungsgebundene_%C3%9Cbertragungstechnik#Tasn.C3.A1d_Kernetzky.2C_M.Sc._.28bei_L.C3.9CT_seit_2014.29|Tasnád Kernetzky]]).

==Once again: Open Applet in new Tab==

{{LntAppletLink|physAnSignal}}

[[Category:Applets|^Verzerrungen^]]

Applets:Physical Signal & Analytic Signal

2018-08-10T14:10:49Z

Xiaohan: /* Applet Manual */

{{LntAppletLink|physAnSignal}}

==Applet Description==
 
This applet shows the relationship between the physical bandpass signal $x(t)$ and the associated analytic signal $x_+(t)$. It is assumed that the bandpass signal $x(t)$ has a frequency-discrete spectrum $X(f)$:
:$$x(t) = x_{\rm U}(t) + x_{\rm T}(t) + x_{\rm O}(t) = A_{\rm U}\cdot \cos\left(2\pi f_{\rm U}\cdot t- \varphi_{\rm U}\right)+A_{\rm T}\cdot \cos\left(2\pi f_{\rm T}\cdot t- \varphi_{\rm T}\right)+A_{\rm O}\cdot \cos\left(2\pi f_{\rm O}\cdot t- \varphi_{\rm O}\right). $$
The physical signal $x(t)$ is thus composed of three [[Signaldarstellung/Harmonische_Schwingung|harmonic oscillations]], a constellation that can be found, for example, in the ''Double-sideband Amplitude Modulation''
*of the message signal $x_{\rm N}(t) = A_{\rm N}\cdot \cos\left(2\pi f_{\rm N}\cdot t- \varphi_{\rm N}\right)$   ⇒   in German:   '''N'''achrichtensignal
*with the carrier signal $x_{\rm T}(t) = A_{\rm T}\cdot \cos\left(2\pi f_{\rm T}\cdot t - \varphi_{\rm T}\right)$   ⇒   in German:   '''T'''rägersignal.

The nomenclature is also adapted to this case:
* $x_{\rm O}(t)$ denotes the &bdquo;upper sideband”   (in German:   '''O'''beres Seitenband) with the amplitude $A_{\rm O}= A_{\rm N}/2$, the frequency $f_{\rm O} = f_{\rm T} + f_{\rm N}$ and the phase $\varphi_{\rm O} = \varphi_{\rm T} + \varphi_{\rm N}$.
*Similarly, for the &bdquo;lower sideband”   (in German:   '''U'''nteres Seitenband) $x_{\rm U}(t)$ with $f_{\rm U} = f_{\rm T} + f_{\rm N}$, $A_{\rm U}= A_{\rm O}$ and $\varphi_{\rm U} = -\varphi_{\rm O}$.

The associated analytical signal is:

:$$x_+(t) = x_{\rm U+}(t) + x_{\rm T+}(t) + x_{\rm O+}(t) = A_{\rm U}\cdot {\rm e}^{\hspace{0.05cm}{\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}(2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm U}\hspace{0.05cm}\cdot \hspace{0.05cm}t- \varphi_{\rm U})}
\hspace{0.1cm}+ \hspace{0.1cm}A_{\rm T}\cdot {\rm e}^{\hspace{0.05cm}{\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}(2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm T}\hspace{0.05cm}\cdot \hspace{0.05cm}t- \varphi_{\rm T})}
\hspace{0.1cm}+\hspace{0.1cm} A_{\rm O}\cdot {\rm e}^{\hspace{0.05cm}{\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}(2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm O}\hspace{0.05cm}\cdot \hspace{0.05cm}t- \varphi_{\rm O})}. $$

[[File:Zeigerdiagramm_2a_version2.png|right|frame|Analytical signal at the time $t=0$]]
The program displays $x_+(t)$ as the vectorial sum of three rotating pointers (all with counterclockwise) as a violet dot (see figure for start time $t=0$):

*The (red) pointer of the carrier $x_{\rm T+}(t)$ with length $A_{\rm T}$ and zero phase position $\varphi_{\rm T} = 0$ rotates at constant angular velocity $2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm T}$ (one revolution in time $1/f_{\rm T})$.

*The (blue) pointer of the upper sideband $x_{\rm O+}(t)$ with length $A_{\rm O}$ and zero phase position $\varphi_{\rm O}$ rotates at the angular velocity $2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm O}$, which is slightly faster than $x_{\rm T+}(t)$.

*The (green) pointer of the lower sideband $x_{\rm U+}(t)$ with length $A_{\rm U}$ and zero phase position $\varphi_{\rm U}$ rotates at the angular velocity $2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm U}$, which is slightly faster than $x_{\rm T+}(t)$.

The time trace of $x_+(t)$ is also referred to below as '''Pointer Diagram'''. The relationship between the physical bandpass signal $x(t)$ and the associated analytic signal $x_+(t)$ is:

:$$x(t) = {\rm Re}\big [x_+(t)\big ].$$

''Note:''   In the figure $\varphi_{\rm O} = +30^\circ$. This leads to the angle with respect to the coordinate system at $t=0$:   $\phi_{\rm O}=-\varphi_{\rm O}=-30^\circ$. Similarly, the null phase angle $\varphi_{\rm U}=-30^\circ$ of the lower sideband leads to the phase angle to be considered in the complex plane:   $\phi_{\rm U}=+30^\circ$.

[[Applets:Physikalisches_Signal_%26_Analytisches_Signal|'''German description''']]

==Theoretical Background==
 
===Description of Bandpass Signals===
[[File:Zeigerdiagramm_1a.png|right|frame|Bandpass–spectrum $X(f)$ |class=fit]]
We consider '''bandpass signals''' $x(t)$ with the property that their spectra $X(f)$ are not in the range around the frequency $f=0$, but around a carrier frequency $f_{\rm T}$. In most cases it can also be assumed that the bandwidth is $B \ll f_{\rm T}$.

The figure shows such a bandpass spectrum $X(f)$. Assuming that the associated $x(t)$ is a physical signal and thus real, the spectral function $X(f)$ has a symmetry with respect to the frequency $f = 0$, if $x(t)$ is an even function   ⇒   $x(-t)=x(t)$, $X(f)$ is real and even.

Besides the physical signal $x(t)\ \circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\, \ X(f)$, one can also use the following descriptions of Bandpass signals:
*the analytic signal $x_+(t)\ \circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\, \ X_+(f)$, see next page,
*the equivalent low-pass –signal   (in German:   äquivalentes '''T'''ief'''P'''ass–Signal) $x_{\rm TP}(t)\ \circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\, \ X_{\rm TP}(f)$, see Applet [[Applets:Physical_Signal_%26_Equivalent_Low-pass_Signal|Physical Signal & Equivalent Low–pass signal]].
 

===Analytical Signal – Frequency Domain===

The '''analytical signal''' $x_+(t)$ belonging to the physical signal $x(t)$ is the time function whose spectrum fulfills the following property:
[[File:Zeigerdiagramm_3a.png|right|frame|Construction of the spectral function $X_+(f)$ |class=fit]]
:$$X_+(f)=\big[1+{\rm sign}(f)\big] \cdot X(f) = \left\{ {2 \cdot
X(f) \; \hspace{0.2cm}\rm for\hspace{0.2cm} {\it f} > 0, \atop {\,\,\,\, \rm 0 \; \hspace{0.9cm}\rm for\hspace{0.2cm} {\it f} < 0.} }\right.$$

The ''signum function'' is for positive values of $f$ equal to $+1$ and for negative $f$ values equal to $-1$.
* The (double-sided) limit returns $\sign(0)=0$.
* The index „+” should make it clear that $X_+(f)$ only has parts at positive frequencies.

From the graph you can see the calculation rule for $X_+(f)$:

The actual band pass spectrum $X(f)$ becomes
* doubled at the positive frequencies, and
* set to zero at the negative frequencies.

Due to the asymmetry of $X_+(f)$ with respect to the frequency $f=0$, it can already be said that the time function $x_+(t)$ except for a trivial special case $x_+(t)=0 \ \circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\,X_+(f)=0$ is always complex.
 

===Analytical Signal – Time Domain===
At this point, it is necessary to briefly discuss another spectral transformation.

{{BlaueBox|TEXT=
$\text{Definition:}$ 
For the '''Hilbert transformed ''' $ {\rm H}\left\{x(t)\right\}$ of a time function $x(t)$ we have::

:$$y(t) = {\rm H}\left\{x(t)\right\} = \frac{1}{ {\rm \pi} } \cdot
\hspace{0.03cm}\int_{-\infty}^{+\infty}\frac{x(\tau)}{ {t -
\tau} }\hspace{0.15cm} {\rm d}\tau.$$

This particular integral is not solvable in a simple, conventional way, but must be evaluated using the [https://de.wikipedia.org/wiki/Cauchyscher_Hauptwert Cauchy principal value theorem].

Accordingly, in the frequency domain:
:$$Y(f) = {\rm -j \cdot sign}(f) \cdot X(f) \hspace{0.05cm} .$$}}

The above result can be summarized with this definition as follows:
* The analytic signal $x_+(t)$ is obtained from the physical bandpass signal $x(t)$ by adding an imaginary part to $x(t)$ according to the Hilbert transform:

:$$x_+(t) = x(t)+{\rm j} \cdot {\rm H}\left\{x(t)\right\} .$$

*$\text{H}\{x(t)\}$ disappears only for the case $x(t) = \rm const.$. For all other signal forms, the analytic signal $x_+(t)$ is complex.

* From the analytic signal $x_+(t)$, the physical bandpass signal can be easily determined by the following operation:
:$$x(t) = {\rm Re}\big[x_+(t)\big] .$$

{{GraueBox|TEXT=
$\text{Example 1:}$  The principle of the Hilbert transformation should be further clarified by the following graphic:
*After the left representation $\rm(A)$ one gets from the physical signal $x(t)$ to the analytic signal $x_+(t)$, by adding an imaginary part ${\rm j} \cdot y(t)$.
*Here $y(t) = {\rm H}\left\{x(t)\right\}$ is a real time function that can be indicated in the spectral domain by multiplying the spectrum $X(f)$ with ${\rm - j} \cdot \sign(f)$.

[[File:P_ID2729__Sig_T_4_2_S2b_neu.png|center|frame|To clarify the Hilbert transform]]

The right representation $\rm(B)$ is equivalent to $\rm(A)$. Now $x_+(t) = x(t) + z(t)$ stand with the purely imaginary function $z(t)$. A comparison of the two pictures shows that in fact $z(t) = {\rm j} \cdot y(t)$.}}
 

===Representation of the Harmonic Oscillation as an Analytical Signal===

The spectral function $X(f)$ of a harmonic oscillation $x(t) = A\cdot\text{cos}(2\pi f_{\rm T}\cdot t - \varphi)$ is known to consist of two dirac functions in the frequencies
* $+f_{\rm T}$ with the complex weight $A/2 \cdot \text{e}^{-\text{j}\hspace{0.05cm}\varphi}$,
* $-f_{\rm T}$ with the complex weight $A/2 \cdot \text{e}^{+\text{j}\hspace{0.05cm}\varphi}$.

Thus, the spectrum of the analytic signal (that is, without the Dirac function at the frequency $f =-f_{\rm T}$, but doubling at $f =+f_{\rm T}$):

:$$X_+(f) = A \cdot {\rm e}^{-{\rm j} \hspace{0.05cm}\varphi}\cdot\delta (f - f_{\rm
T}) .$$

The associated time function is obtained by applying the [[Signaldarstellung/Gesetzmäßigkeiten_der_Fouriertransformation#Verschiebungssatz|Displacement law]]:

:$$x_+(t) = x_{\rm U+}(t) + x_{\rm T+}(t) + x_{\rm O+}(t) = A_{\rm U}\cdot {\rm e}^{\hspace{0.05cm}{\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}(2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm U}\hspace{0.05cm}\cdot \hspace{0.05cm}t- \varphi_{\rm U})}
\hspace{0.1cm}+ \hspace{0.1cm}A_{\rm T}\cdot {\rm e}^{\hspace{0.05cm}{\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}(2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm T}\hspace{0.05cm}\cdot \hspace{0.05cm}t- \varphi_{\rm T})}
\hspace{0.1cm}+\hspace{0.1cm} A_{\rm O}\cdot {\rm e}^{\hspace{0.05cm}{\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}(2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm O}\hspace{0.05cm}\cdot \hspace{0.05cm}t- \varphi_{\rm O})}. $$

This equation describes a pointer rotating at constant angular velocity $\omega_{\rm T} = 2\pi f_{\rm T}$.

{{GraueBox|TEXT=
$\text{Example 2:}$  Here the coordinate system is rotated by $90^\circ$ (real part up, imaginary part to the left) contrary to the usual representation.

[[File:P_ID712__Sig_T_4_2_S3.png|center|frame|Pointer diagram of a harmonic oscillation]]

Based on this graphic, the following statements are possible:
* At the start time $t = 0$, the pointer of length $A$ (signal amplitude) lies with the angle $-\varphi$ in the complex plane. In the example shown, $\varphi=45^\circ$.
* For times $t>0$, the constant angular velocity vector $\omega_{\rm T}$ rotates in a mathematically positive direction, that is, counterclockwise.
* The tip of the pointer is thus always on a circle with radius $A$ and needs exactly the time $T_0$, i.e. the period of the harmonic oscillation $x(t)$ for one revolution.
* The projection of the analytic signal $x_+(t)$ on the real axis, marked by red dots, gives the instantaneous values of $x(t)$.}}
 

===Analytical Signal Representation of a Sum of Three Harmonic Oscillations===

In our applet, we always assume a set of three rotating pointers. The physical signal is:
:$$x(t) = x_{\rm U}(t) + x_{\rm T}(t) + x_{\rm O}(t) = A_{\rm U}\cdot \cos\left(2\pi f_{\rm U}\cdot t- \varphi_{\rm U}\right)+A_{\rm T}\cdot \cos\left(2\pi f_{\rm T}\cdot t- \varphi_{\rm T}\right)+A_{\rm O}\cdot \cos\left(2\pi f_{\rm O}\cdot t- \varphi_{\rm O}\right). $$
* Each of the three harmonic oscillations $x_{\rm T}(t)$, $x_{\rm U}(t)$ and $x_{\rm O}(t)$ is represented by an amplitude $(A)$, a frequency $(f)$ and a phase value $(\varphi)$.
*The indices are based on the [[Modulationsverfahren/Zweiseitenband-Amplitudenmodulation|Double sideband Amplitude Modulation]] method. &bdquo;T” stands for &bdquo;carrier”, &bdquo;U” for &bdquo;lower sideband” and &bdquo;O” for &bdquo;Upper Sideband”.
*Accordingly, $f_{\rm U} < f_{\rm T}$ and $f_{\rm O} > f_{\rm T}$. There are no restrictions for the amplitudes and phases.

The associated analytical signal is:
:$$x_+(t) = x_{\rm U+}(t) + x_{\rm T+}(t) + x_{\rm O+}(t) = A_{\rm U}\cdot {\rm e}^{\hspace{0.05cm}{\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}(2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm U}\hspace{0.05cm}\cdot \hspace{0.05cm}t- \varphi_{\rm U})}
\hspace{0.1cm}+ \hspace{0.1cm}A_{\rm T}\cdot {\rm e}^{\hspace{0.05cm}{\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}(2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm T}\hspace{0.05cm}\cdot \hspace{0.05cm}t- \varphi_{\rm T})}
\hspace{0.1cm}+\hspace{0.1cm} A_{\rm O}\cdot {\rm e}^{\hspace{0.05cm}{\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}(2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm O}\hspace{0.05cm}\cdot \hspace{0.05cm}t- \varphi_{\rm O})}. $$

{{GraueBox|TEXT=
$\text{Example 3:}$ 
Shown the constellation arises i.e. in the [[Modulationsverfahren/Zweiseitenband-Amplitudenmodulation#AM-Signale_und_-Spektren_bei_harmonischem_Eingangssignal|double sideband amplitude modulation]] of the message signal $x_{\rm N}(t) = A_{\rm N}\cdot \cos\left(2\pi f_{\rm N}\cdot t- \varphi_{\rm N}\right)$ with the carrier signal $x_{\rm T}(t) = A_{\rm T}\cdot \cos\left(2\pi f_{\rm T}\cdot t - \varphi_{\rm T}\right)$. This is discussed frequently in the Exercises.

There are some limitations to the program parameters in this approach:
* For the frequencies, it always applies $f_{\rm O} = f_{\rm T} + f_{\rm N}$ und $f_{\rm U} = f_{\rm T} - f_{\rm N}$.

*Without distortions the amplitude of the sidebands are $A_{\rm O}= A_{\rm U}= A_{\rm N}/2$.
*The respective phase relationships can be seen in the following graphic.

[[File:Zeigerdiagramm_2_neu.png|center|frame|Spectrum $X_+(f)$ of the analytic signal for different phase constellations |class=fit]]}}

==Exercises==
[[File:Zeigerdiagramm_aufgabe_2.png|right]]
*First select the task number.
*A task description is displayed.
*Parameter values are adjusted.
*Solution after pressing &bdquo;Hide solition”.

The number &bdquo;0” will reset the program and output a text with further explanation of the applet.
 
In the following, $\rm Green$ denotes the lower sideband   ⇒   $\big (A_{\rm U}, f_{\rm U}, \varphi_{\rm U}\big )$,  
$\rm Red$ the carrier   ⇒   $\big (A_{\rm T}, f_{\rm T}, \varphi_{\rm T}\big )$ and
$\rm Blue$ the upper sideband   ⇒   $\big (A_{\rm O}, f_{\rm O}, \varphi_{\rm O}\big )$.
{{BlaueBox|TEXT=
'''(1)'''   Consider and interpret the analytic signal $x_+(t)$ for $\text{Red:} \hspace{0.15cm} A_{\rm T} = 1.5\ \text{V}, \ f_{\rm T} = 50 \ \text{kHz}, \ \varphi_{\rm T} = 0^\circ$, $A_{\rm U} = A_{\rm O} = 0$.

:Which signal values $x_+(t)$ result for $t = 0$, $t = 5 \ \rm µ s$ and $t = 20 \ \rm µ s$? What are the corresponding signal values for $x(t)$? }}

:: For a cosine signal, let $x_+(t= 0) = A_{\rm T} = 1.5\ \text{V}$. Then $x_+(t)$ rotates in a mathematically positive direction (one revolution per period $T_0 = 1/f_{\rm T}$):

:: $x_+(t= 20 \ {\rm µ s}) = x_+(t= 0) = 1.5\ \text{V}\hspace{0.3cm}\Rightarrow\hspace{0.3cm}x(t= 20 \ {\rm µ s}) = 1.5\ \text{V,}$
:: $x_+(t= 5 \ {\rm µ s}) = {\rm j} \cdot 1.5\ \text{V}\hspace{0.3cm}\Rightarrow\hspace{0.3cm}x(t= 5 \ {\rm µ s}) = {\rm Re}[x_+(t= 5 \ {\rm µ s})] = 0$.

{{BlaueBox|TEXT=
'''(2)'''   How do the ratios change for $\text{Red:} \hspace{0.15cm} A_{\rm T} = 1.0\ \text{V}, \ f_{\rm T} = 100 \ \text{kHz}, \ \varphi_{\rm T} = 90^\circ$?}}

::The signal $x(t)$ is now a sine signal with a smaller amplitude. The analytic signal now starts because of $\varphi_{\rm T} = 90^\circ$   ⇒   $\phi_{\rm T} = -90^\circ$ at $x_+(t= 0) = -{\rm j} \cdot A_{\rm T}$. After that, $x_+(t)$ rotates again in a mathematically positive direction, but twice as fast because of $T_0 = 10 \ \rm µ s$ as in $\rm (1)$.

{{BlaueBox|TEXT=
'''(3)'''   Now   $\text{Red:} \hspace{0.15cm} A_{\rm T} = 1\ \text{V}, \ f_{\rm T} = 100 \ \text{kHz}, \ \varphi_{\rm T} = 0^\circ$,   $\text{Green:} \hspace{0.15cm} A_{\rm U} = 0.4\ \text{V}, \ f_{\rm U} = 80 \ \text{kHz}, \ \varphi_{\rm U} = 0^\circ$,   $\text{Blue:} \hspace{0.15cm} A_{\rm O} = 0.4\ \text{V}, \ f_{\rm O} = 120 \ \text{kHz}, \ \varphi_{\rm O} = 0^\circ$.

:Consider and interpret the physical signal $x(t)$ and the analytic signal $x_+(t)$.}}

::The Signal $x(t)$ results in the double sideband Amplitude Modulation '''(DSB–AM)''' of the message signals $A_{\rm N}\cdot \cos\left(2\pi f_{\rm N}\cdot t\right)$ with $A_{\rm N} = 0.8\ \text{V}$, $f_{\rm N} = 20\ \text{kHz}$. The carrier $x_{\rm T}(t)$ with $f_{\rm T} = 100\ \text{kHz}$ is also cosinusoidal. The degree of modulation is $m = A_{\rm N}/A_{\rm T} = 0.8$ and the period $T_{\rm 0} = 50\ \text{µs}$.

::In the phase diagram, the (red) carrier rotates faster than the (green) lower sideband and slower than the (blue) upper sideband. The analytic signal $x_+(t)$ results as the geometric sum of the three rotating pointers. It seems that the blue pointer is leading the carrier and the green pointer is following the carrier.

{{BlaueBox|TEXT=
'''(4)'''   The settings of task '''(3)''' still apply. Which signal values are obtained at $t=0$, $t=2.5 \ \rm µ s$, $t= 5 \ \rm µ s$ and $t=10 \ \rm µ s$? }}

::At time $t=0$, all pointers are in the direction of the real axis, so that $x(t=0) = {\rm Re}\big [x+(t= 0)\big] = A_{\rm U} + A_{\rm T} + A_{\rm O} = 1.8\ \text{V}$.

::Until the time $t=2.5 \ \rm µ s$, the red carrier has rotated by $90^\circ$, the blue one by $108^\circ$ and the green one by $72^\circ$. We have $x(t=2.5 \ \rm µ s) = {\rm Re}\big [x_+(t= 2.5 \ \rm µ s)\big] = 0$, because now the pointer group points in the direction of the imaginary axis. The other sought signal values are $x(t=5 \ \rm µ s) = {\rm Re}\big [x_+(t= 5 \ \rm µ s)\big] = -1.647\ \text{V}$ and $x(t=10 \ \rm µ s) = {\rm Re}\big [x_+(t= 10 \ \rm µ s)\big] = 1.247\ \text{V}$.
::For $x_+(t)$ a spiral shape results, alternating with a smaller radius and then with a larger radius.

{{BlaueBox|TEXT=
'''(5)'''   How should the phase parameters $\varphi_{\rm T}$, $\varphi_{\rm U}$ and $\varphi_{\rm O}$ be set if both the carrier $x_{\rm T}(t)$ and the message signal $x_{\rm N}(t)$ are sinusoidal?}}

::The parameter selection $\varphi_{\rm T} = \varphi_{\rm U} = \varphi_{\rm O}=90^\circ$ describes the signals $x_{\rm T}(t) = A_{\rm T}\cdot \sin\left(2\pi f_{\rm T}\cdot t\right)$ and $x_{\rm N}(t) = A_{\rm N}\cdot \cos\left(2\pi f_{\rm N}\cdot t\right)$. If, in addition, the message $x_{\rm N}(t)$ is sinusoidal, then $\varphi_{\rm O}=\varphi_{\rm T} - 90^\circ = 0$ and $\varphi_{\rm U}=\varphi_{\rm T} + 90^\circ = 180^\circ$ must be set.

{{BlaueBox|TEXT=
'''(6)'''   The settings of task '''(3)''' apply except $A_{\rm T} = 0.6\ \text{V}$. Which modulation method is described here?

: What are the consequences of this? What changes with $A_{\rm T} = 0$? }}

::It is a '''DSB–AM with carrier''' with the modulation degree $m=0.8/0.6 = 1.333$. For $m > 1$, however, [[Modulationsverfahren/Synchrondemodulation|synchronous demodulation]] is required. [[Modulationsverfahren/Hüllkurvendemodulation|envelope detection]] no longer works.

::With $A_{\rm T} = 0$   ⇒   $m \to \infty$ results in a '''DSB–AM suppressed carrier'''. For this, one also needs coherent demodulation.

{{BlaueBox|TEXT=
'''(7)'''     Now let   $\text{Red:} \hspace{0.15cm} A_{\rm T} = 1\ \text{V}, \ f_{\rm T} = 100 \ \text{kHz}, \ \varphi_{\rm T} = 0^\circ$,   $\text{Green:} \hspace{0.15cm} A_{\rm U} = 0$,   $\text{Blue:} \hspace{0.15cm} A_{\rm O} = 0.8\ \text{V}, \ f_{\rm O} = 120 \ \text{kHz}, \ \varphi_{\rm O} = 90^\circ$.

:Which constellation is described here? What changes with $A_{\rm U} = 0.8\ \text{V}$ und $A_{\rm O} = 0$?}}

::In both cases, it is a [[Modulationsverfahren/Einseitenbandmodulation|Single sideband]] '''(ESB–AM)''' with the modulation degree $\mu = 0.8$ (in ESB we denote the degree of modulation with $\mu$ instead of $m$). The carrier signal is cosinusoidal and the message signal is sinusoidal.

:: $A_{\rm O} = 0.8\ \text{V}$, $A_{\rm U} = 0$ is an OSB modulation. The green pointer is missing and the blue pointer rotates faster compared to the red carrier.

:: $A_{\rm U} = 0.8\ \text{V}$, $A_{\rm O} = 0$ is a USB modulation. The blue pointer is missing and the green pointer rotates slower compared to the red carrier.

{{BlaueBox|TEXT=
'''(8)'''   Now let   $\text{Red:} \hspace{0.05cm} A_{\rm T} = 1\ \text{V}, \ f_{\rm T} = 100 \ \text{kHz}, \ \varphi_{\rm T} = 0^\circ$,   $\text{Green:} \hspace{0.05cm} A_{\rm U} = 0.4\ \text{V}, \ f_{\rm U} = 80 \ \text{kHz}, \ \varphi_{\rm U} = -90^\circ$,   $\text{Blue:} \hspace{0.05cm} A_{\rm O} = 0.2\ \text{V}, \ f_{\rm O} = 120 \ \text{kHz}, \ \varphi_{\rm O} = +90^\circ$.

:Which constellation could be described here? Which shape results for the equivalent lowpass–signal $x_{\rm TP}(t)$?   ⇒   &bdquo;locus”?}}

::It could be a ZSB–AM of a sinusoidal signal with cosinusoidal carrier and modulation degree $m=0.8$, in which the upper sideband is attenuated by a factor of 2. The equivalent lowpass–signal $x_{\rm TP}(t)$ has an elliptical trace in the complex plane.

==Applet Manual==
 
[[File:Zeigerdiagramm_abzug.png|right]]

* The red parameters $(A_{\rm T}, \ f_{\rm T}, \ \varphi_{\rm T})$ and the red pointer mark the '''T'''räger(Carrier).
* The green parameters $(A_{\rm U}, \ f_{\rm U} < f_{\rm T}, \ \varphi_{\rm U})$ mark the '''U'''ntere Seitenband(Lower sideband).
* The blue parameters $(A_{\rm O}, \ f_{\rm O} > f_{\rm T}, \ \varphi_{\rm O})$ mark the '''O'''bere Seitenband(upper sideband).
*All pointers rotate in a mathematically positive direction (counterclockwise).

 
Meaning of the letters in the adjacent graphic:

    '''(A)'''     Plot of the analytic signal $x_{\rm +}(t)$

    '''(B)'''     Plot of for the physical signal $x(t)$

    '''(C)'''     Parameter input via slider: amplitudes, frequencies, phase values

    '''(D)'''     Control elements:   Start – Step – Pause/Continue – Reset

    '''(E)'''     Speed of animation:   &bdquo;Speed”   ⇒   Value: 1, 2 oder 3

    '''(F)'''     &bdquo;Trace”   ⇒   On or Off, trace of complex signal values $x_{\rm +}(t)$

    '''(G)'''     Numeric output of the time $t$ and the signal values  ${\rm Re}[x_{\rm +}(t)] = x(t)$  and  ${\rm Im}[x_{\rm +}(t)]$

    '''(H)'''     Variations for the graphical representation

$\hspace{1.5cm}$Zoom–Functions &bdquo;$+$” (Enlarge), &bdquo;$-$” (Decrease) and $\rm o$ (Reset to default)

$\hspace{1.5cm}$Move with &bdquo;$\leftarrow$” (Section to the left, ordinate to the right), &bdquo;$\uparrow$” &bdquo;$\downarrow$” and &bdquo;$\rightarrow$”

    '''(I)'''     Experiment section:  Task selection and task

    '''(J)'''     Experiment section:  solution
 

==About the Authors==
This interactive calculation was designed and realized at the [http://www.lnt.ei.tum.de/startseite Lehrstuhl für Nachrichtentechnik] of the [https://www.tum.de/ Technischen Universität München] .
*The original version was created in 2005 by [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Studierende#Ji_Li_.28Bachelorarbeit_EI_2003.2C_Diplomarbeit_EI_2005.29|Ji Li]] as part of her Diploma thesis using &bdquo;FlashMX–Actionscript” (Supervisor: [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Mitarbeiter_und_Dozenten#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28am_LNT_seit_1974.29|Günter Söder]]).
*In 2018 this Applet was redesigned and updated to &bdquo;HTML5” by [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Studierende#Xiaohan_Liu_.28Bachelorarbeit_2018.29|Xiaohan Liu]] as part of her Bachelor's thesis (Supervisor: [[Biografien_und_Bibliografien/Beteiligte_der_Professur_Leitungsgebundene_%C3%9Cbertragungstechnik#Tasn.C3.A1d_Kernetzky.2C_M.Sc._.28bei_L.C3.9CT_seit_2014.29|Tasnád Kernetzky]]).

==Once again: Open Applet in new Tab==

{{LntAppletLink|physAnSignal}}

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