Signal Representation/Analytical Signal and its Spectral Function - Revision history

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	All these spectra are generally complex.		All these spectra are generally complex.

	If one considers the  [[Signal_Representation/~~Fourier_Transform_Theorems~~#Assignment_Theorem\|»Assignment Theorem«]]  of the Fourier transform,  then the following statements are possible on basis of the graph:		If one considers the  [[Signal_Representation/The_Fourier_Transform_Theorems#Assignment_Theorem\|»Assignment Theorem«]]  of the Fourier transform,  then the following statements are possible on basis of the graph:
	*The even part  $X_{\rm +g}(f)$  of  $X_{+}(f)$  leads after the Fourier transform to a real time signal,  and the odd part  $X_{\rm +u}(f)$  to an imaginary one.		*The even part  $X_{\rm +g}(f)$  of  $X_{+}(f)$  leads after the Fourier transform to a real time signal,  and the odd part  $X_{\rm +u}(f)$  to an imaginary one.

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	*If we denote the imaginary part with  $y(t)$,  the analytical signal is:		*If we denote the imaginary part with  $y(t)$,  the analytical signal is:
	:$$x_+(t)= x(t) + {\rm j} \cdot y(t) .$$		:$$x_+(t)= x(t) + {\rm j} \cdot y(t) .$$
	*According to the generally valid laws of Fourier transform corresponding to the  [[Signal_Representation/~~Fourier_Transform_Theorems~~#Assignment_Theorem\|»Assignment Theorem«]],  the following applies to the spectral function of the imaginary part:		*According to the generally valid laws of Fourier transform corresponding to the  [[Signal_Representation/The_Fourier_Transform_Theorems#Assignment_Theorem\|»Assignment Theorem«]],  the following applies to the spectral function of the imaginary part:
	:$${\rm j} \cdot Y(f) = X_{\rm +u}(f)= {\rm sign}(f) \cdot X(f)\hspace{0.3cm}\Rightarrow\hspace{0.3cm}Y(f) = \frac{{\~~rmsign~~}(f)}{ {\rm j}}\cdot X(f).$$		:$${\rm j} \cdot Y(f) = X_{\rm +u}(f)= {\rm sign}(f) \cdot X(f)\hspace{0.3cm}\Rightarrow\hspace{0.3cm}Y(f) = \frac{{\rm sign}(f)}{ {\rm j}}\cdot X(f).$$
	*After transforming this equation into the time domain,  the multiplication becomes the  [[Signal_Representation/The_Convolution_Theorem_and_Operation\|»convolution«]],  and one gets:		*After transforming this equation into the time domain,  the multiplication becomes the  [[Signal_Representation/The_Convolution_Theorem_and_Operation\|»convolution«]],  and one gets:
	:$$y(t) = \frac{1}{ {\rm \pi} t} \hspace{0.05cm}\star\hspace{0.05cm}x(t) = \frac{1}{ {\rm \pi}} \cdot\hspace{0.03cm}\int_{-\infty}^{+\infty}\frac{x(\tau)}{ {t -\tau}}\hspace{0.15cm} {\rm d}\tau.$$		:$$y(t) = \frac{1}{ {\rm \pi} t} \hspace{0.05cm}\star\hspace{0.05cm}x(t) = \frac{1}{ {\rm \pi}} \cdot\hspace{0.03cm}\int_{-\infty}^{+\infty}\frac{x(\tau)}{ {t -\tau}}\hspace{0.15cm} {\rm d}\tau.$$
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	Thus, the spectrum of the analytical signal is  $($without the Dirac delta function at the frequency  $f =-f_{\rm T})$:		Thus, the spectrum of the analytical signal is  $($without the Dirac delta function at the frequency  $f =-f_{\rm T})$:

	:$$X_+(f) = A \cdot {\rm e}^{-{\rm j} \hspace{0.05cm}\varphi}\cdot\delta (f - f_{\~~rmT~~}) .$$		:$$X_+(f) = A \cdot {\rm e}^{-{\rm j} \hspace{0.05cm}\varphi}\cdot\delta (f - f_{\rm T}) .$$

	The corresponding time function is obtained by applying the  [[Signal_Representation/~~Fourier_Transform_Theorems~~#Shifting_Theorem\|»Shifting Theorem«]]:		The corresponding time function is obtained by applying the  [[Signal_Representation/The_Fourier_Transform_Theorems#Shifting_Theorem\|»Shifting Theorem«]]:

	:$$x_+(t) = A \cdot {\rm e}^{\hspace{0.05cm} {\rm j}\hspace{0.05cm}\cdot\hspace{0.05cm}( 2 \pi f_{\rm T} t\hspace{0.05cm}-\hspace{0.05cm} \varphi)}.$$		:$$x_+(t) = A \cdot {\rm e}^{\hspace{0.05cm} {\rm j}\hspace{0.05cm}\cdot\hspace{0.05cm}( 2 \pi f_{\rm T} t\hspace{0.05cm}-\hspace{0.05cm} \varphi)}.$$
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	*At time  $t = 1\,µ\text {s}$  the tip of the resulting pointer for the given parameter values is		*At time  $t = 1\,µ\text {s}$  the tip of the resulting pointer for the given parameter values is

	:$$ \begin{align*}x_+(t = 1 {\rm \hspace{0.05cm}µ s}) & = 1 \cdot {\rm e}^{-{\rm		:$$ \begin{align}x_+(t = 1 {\rm \hspace{0.05cm}µ s}) & = 1 \cdot {\rm e}^{-{\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}60^\circ}\cdot {\rm e}^{{\rm j}\hspace{0.05cm}2 \pi \hspace{0.05cm}\cdot \hspace{0.1cm}40\hspace{0.05cm} \cdot \hspace{0.1cm} 0.001} + 2\cdot {\rm e}^{{\rm j}\hspace{0.05cm}2 \pi \hspace{0.05cm}\cdot \hspace{0.1cm}50\hspace{0.05cm} \cdot \hspace{0.1cm} 0.001}- 1\cdot {\rm e}^{{\rm j}\hspace{0.05cm}2 \pi \hspace{0.05cm}\cdot \hspace{0.1cm}60\hspace{0.05cm} \cdot \hspace{0.1cm} 0.001} = \\& = 1 \cdot {\rm e}^{-{\rm j}\hspace{0.05cm}\cdot\hspace{0.05cm}45.6^\circ} + 2\cdot {\rm e}^{{\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}18^\circ}- 1\cdot {\rm e}^{{\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}21.6^\circ} \approx1.673- {\rm j} \cdot 0.464.\end{align}$$
	j}\hspace{0.05cm}\cdot \hspace{0.05cm}60^\circ}\cdot {\rm e}^{{\rm
	j}\hspace{0.05cm}2 \pi \hspace{0.05cm}\cdot \hspace{0.1cm}40
	\hspace{0.05cm} \cdot \hspace{0.1cm} 0.001} + 2\cdot {\rm e}^{{\rm
	j}\hspace{0.05cm}2 \pi \hspace{0.05cm}\cdot \hspace{0.1cm}50
	\hspace{0.05cm} \cdot \hspace{0.1cm} 0.001}- 1\cdot {\rm e}^{{\rm
	j}\hspace{0.05cm}2 \pi \hspace{0.05cm}\cdot \hspace{0.1cm}60
	\hspace{0.05cm} \cdot \hspace{0.1cm} 0.001} = \\
	& = 1 \cdot {\rm e}^{-{\rm j}\hspace{0.05cm}\cdot
	\hspace{0.05cm}45.6^\circ} + 2\cdot {\rm e}^{{\rm
	j}\hspace{0.05cm}\cdot \hspace{0.05cm}18^\circ}- 1\cdot {\rm
	e}^{{\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}21.6^\circ} \~~approx~~
	1.673- {\rm j} \cdot 0.464.\end{align*}$$
	*The resulting pointer tip does not lie on a circle like a single oscillation, but a complicated geometric figure is created.		*The resulting pointer tip does not lie on a circle like a single oscillation, but a complicated geometric figure is created.

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	$\text{Definition:}$  The  »'''analytical signal'''«  $x_+(t)$  belonging to the physical signal  $x(t)$  is that time function, whose spectrum fulfills the following property:		$\text{Definition:}$  The  »'''analytical signal'''«  $x_+(t)$  belonging to the physical signal  $x(t)$  is that time function, whose spectrum fulfills the following property:
	[[File:EN_Sig_T_4_2_S1a.png\|right\|frame\|Analytical signal in the frequency domain]]		[[File:EN_Sig_T_4_2_S1a.png\|right\|frame\|Analytical signal in the frequency domain]]
	:$$X_+(f)=\big[1+{\rm sign}(f)\big] \cdot X(f) = \left\{ {2 \~~cdot~~		:$$X_+(f)=\big[1+{\rm sign}(f)\big] \cdot X(f) = \left\{ {2 \cdotX(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.$$
	X(f) \; \hspace{0.2cm}\rm for\hspace{0.2cm} {\it f} > 0, \atop {\,\,\,\, \rm 0 \; \hspace{0.9cm}\rm for\hspace{0.2cm} {\it f} < 0.} }\right.$$

	The  »'''sign function'''«  is equal to  $+1$  for positive $f$–values and for negative  $f$-values equal to  $-1$.		The  »'''sign function'''«  is equal to  $+1$  for positive $f$–values and for negative  $f$-values equal to  $-1$.
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	*on the left shows the  $($discrete and complex$)$  spectrum  $X(f)$  of the  "physical band-pass signal"		*on the left shows the  $($discrete and complex$)$  spectrum  $X(f)$  of the  "physical band-pass signal"

	:$$x(t) = 4\hspace{0.05cm}{\rm V}		:$$x(t) = 4\hspace{0.05cm}{\rm V}\cdot {\cos} ( 2 \pi f_{\rm u} \hspace{0.03cm}t) + 6\hspace{0.05cm}{\rm V}\cdot {\sin} ( 2 \pi f_{\rm o} \hspace{0.03cm}t),$$
	\cdot {\cos} ( 2 \pi f_{\rm u} \hspace{0.03cm}t) + 6\hspace{0.05cm}{\rm V}
	\cdot {\sin} ( 2 \pi f_{\rm o} \hspace{0.03cm}t),$$

	*on the right the  $($also discrete and complex$)$  spectrum  $X_{+}(f)$  of the corresponding  "analytical signal"  $x_{+}(t)$.}}		*on the right the  $($also discrete and complex$)$  spectrum  $X_{+}(f)$  of the corresponding  "analytical signal"  $x_{+}(t)$.}}
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	:$$x_+(t)= x(t) + {\rm j} \cdot y(t) .$$		:$$x_+(t)= x(t) + {\rm j} \cdot y(t) .$$
	*According to the generally valid laws of Fourier transform corresponding to the  [[Signal_Representation/Fourier_Transform_Theorems#Assignment_Theorem\|»Assignment Theorem«]],  the following applies to the spectral function of the imaginary part:		*According to the generally valid laws of Fourier transform corresponding to the  [[Signal_Representation/Fourier_Transform_Theorems#Assignment_Theorem\|»Assignment Theorem«]],  the following applies to the spectral function of the imaginary part:
	:$${\rm j} \cdot Y(f) = X_{\rm +u}(f)= {\rm sign}(f) \cdot X(f)		:$${\rm j} \cdot Y(f) = X_{\rm +u}(f)= {\rm sign}(f) \cdot X(f)\hspace{0.3cm}\Rightarrow\hspace{0.3cm}Y(f) = \frac{{\rmsign}(f)}{ {\rm j}}\cdot X(f).$$
	\hspace{0.3cm}\Rightarrow\hspace{0.3cm}Y(f) = \frac{{\rm
	~~sign~~}(f)}{ {\rm j}}\cdot X(f).$$
	*After transforming this equation into the time domain,  the multiplication becomes the  [[Signal_Representation/The_Convolution_Theorem_and_Operation\|»convolution«]],  and one gets:		*After transforming this equation into the time domain,  the multiplication becomes the  [[Signal_Representation/The_Convolution_Theorem_and_Operation\|»convolution«]],  and one gets:
	:$$y(t) = \frac{1}{ {\rm \pi} t} \hspace{0.05cm}\star		:$$y(t) = \frac{1}{ {\rm \pi} t} \hspace{0.05cm}\star\hspace{0.05cm}x(t) = \frac{1}{ {\rm \pi}} \cdot\hspace{0.03cm}\int_{-\infty}^{+\infty}\frac{x(\tau)}{ {t -\tau}}\hspace{0.15cm} {\rm d}\tau.$$
	\hspace{0.05cm}x(t) = \frac{1}{ {\rm \pi}} \cdot
	\hspace{0.03cm}\int_{-\infty}^{+\infty}\frac{x(\tau)}{ {t -
	\tau}}\hspace{0.15cm} {\rm d}\tau.$$

	==Representation with Hilbert transform==		==Representation with Hilbert transform==
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	$\text{Definition:}$  For the  »'''Hilbert transform'''«  $ {\rm H}\left\{x(t)\right\}$  of a time function  $x(t)$  applies:		$\text{Definition:}$  For the  »'''Hilbert transform'''«  $ {\rm H}\left\{x(t)\right\}$  of a time function  $x(t)$  applies:

	:$$y(t) = {\rm H}\left\{x(t)\right\} = \frac{1}{ {\rm \pi} } \cdot		:$$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.$$
	\hspace{0.03cm}\int_{-\infty}^{+\infty}\frac{x(\tau)}{ {t -
	\tau} }\hspace{0.15cm} {\rm d}\tau.$$

	*This particular integral cannot be solved in a simple,  conventional way,  but must be evaluated using the  [https://en.wikipedia.org/wiki/Cauchy_principal_value »Cauchy principal value«].		*This particular integral cannot be solved in a simple,  conventional way,  but must be evaluated using the  [https://en.wikipedia.org/wiki/Cauchy_principal_value »Cauchy principal value«].
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	Thus, the spectrum of the analytical signal is  $($without the Dirac delta function at the frequency  $f =-f_{\rm T})$:		Thus, the spectrum of the analytical signal is  $($without the Dirac delta function at the frequency  $f =-f_{\rm T})$:

	:$$X_+(f) = A \cdot {\rm e}^{-{\rm j} \hspace{0.05cm}\varphi}\cdot\delta (f - f_{\rm		:$$X_+(f) = A \cdot {\rm e}^{-{\rm j} \hspace{0.05cm}\varphi}\cdot\delta (f - f_{\rmT}) .$$
	T}) .$$

	The corresponding time function is obtained by applying the  [[Signal_Representation/Fourier_Transform_Theorems#Shifting_Theorem\|»Shifting Theorem«]]:		The corresponding time function is obtained by applying the  [[Signal_Representation/Fourier_Transform_Theorems#Shifting_Theorem\|»Shifting Theorem«]]:

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

	This equation describes a rotating pointer with constant angular velocity  $\omega_{\rm T} = 2\pi f_{\rm T}$.		This equation describes a rotating pointer with constant angular velocity  $\omega_{\rm T} = 2\pi f_{\rm T}$.
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	[[File:P_ID715__Sig_T_4_2_S4.png\|right\|frame\|Pointer diagram of a sum of three oscillations]]		[[File:P_ID715__Sig_T_4_2_S4.png\|right\|frame\|Pointer diagram of a sum of three oscillations]]

	:$$X_+(f) = \sum_{i=1}^{I}A_i \cdot {\rm e}^{-{\rm j}\hspace{0.05cm}\cdot\hspace{0.05cm}		:$$X_+(f) = \sum_{i=1}^{I}A_i \cdot {\rm e}^{-{\rm j}\hspace{0.05cm}\cdot\hspace{0.05cm}\varphi_i}\cdot\delta (f - f_{i}) .$$
	\varphi_i}\cdot\delta (f - f_{i}) .$$

	#The left graphic shows such a spectrum for the example  $I = 3$.		#The left graphic shows such a spectrum for the example  $I = 3$.
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	In the right graphic the corresponding time function is indicated.  This is in general:		In the right graphic the corresponding time function is indicated.  This is in general:

	:$$x_+(t) = \sum_{i=1}^{I}A_i \cdot {\rm e}^{ {\rm j}\hspace{0.05cm}\cdot\hspace{0.05cm}(\omega_i		:$$x_+(t) = \sum_{i=1}^{I}A_i \cdot {\rm e}^{ {\rm j}\hspace{0.05cm}\cdot\hspace{0.05cm}(\omega_i\hspace{0.05cm}\cdot\hspace{0.05cm} t \hspace{0.05cm}-\hspace{0.05cm} \varphi_i)}.$$
	\hspace{0.05cm}\cdot\hspace{0.05cm} t \hspace{0.05cm}-\hspace{0.05cm} \varphi_i)}.$$

	To note about this graphic:		To note about this graphic:

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	{{Display}}		{{Display}}
			[[de:Signaldarstellung/Analytisches_Signal_und_zugehörige_Spektralfunktion]]

Guenter at 15:48, 19 June 2023

2023-06-19T15:48:21Z

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	[[Aufgaben:Exercise 4.4: Vector Diagram for DSB-AM\|Exercise 4.4: Vector Diagram for DSB-AM]]		[[Aufgaben:Exercise 4.4: Vector Diagram for DSB-AM\|Exercise 4.4: Vector Diagram for DSB-AM]]

	[[Aufgaben:Exercise 4.4Z: Vector Diagram for DSB-AM\|Exercise 4.4Z: Vector Diagram for ~~DSB~~-AM]]		[[Aufgaben:Exercise 4.4Z: Vector Diagram for DSB-AM\|Exercise 4.4Z: Vector Diagram for SSB-AM]]

Guenter at 15:42, 19 June 2023

2023-06-19T15:42:31Z

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	From the graphic you can see the calculation rule for  $X_+(f)$:  The actual band-pass spectrum  $X(f)$  will		From the graphic you can see the calculation rule for  $X_+(f)$:  The actual band-pass spectrum  $X(f)$  will
	*be doubled at the positive frequencies, and		*be doubled at the positive frequencies, and

	*set to zero at the negative frequencies.}}		*set to zero at the negative frequencies.}}
	<br clear=all>		<br clear=all>

Guenter at 15:41, 19 June 2023

2023-06-19T15:41:28Z

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	==Pointer diagram representation of the harmonic oscillation==		==Pointer diagram representation of the harmonic oscillation==
	<br>		<br>
	The spectral function  $X(f)$  of a harmonic oscillation  $x(t) = A \cdot \text{cos}(2\pi f_{\rm T}t - \varphi)$  consists of two Dirac delta functions at ~~the~~ frequencies		The spectral function  $X(f)$  of a harmonic oscillation  $x(t) = A \cdot \text{cos}(2\pi f_{\rm T}t - \varphi)$  consists of two Dirac delta functions at frequencies
	* $+f_{\rm T}$  with ~~the~~ complex weight   $A/2 \cdot \text{e}^{-\text{j}\hspace{0.05cm}\varphi}$,		* $+f_{\rm T}$  with complex weight   $A/2 \cdot \text{e}^{-\text{j}\hspace{0.05cm}\varphi}$,
	* $-f_{\rm T}$  with ~~the~~ complex weight   $A/2 \cdot \text{e}^{+\text{j}\hspace{0.05cm}\varphi}$.
			* $-f_{\rm T}$  with complex weight   $A/2 \cdot \text{e}^{+\text{j}\hspace{0.05cm}\varphi}$.


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	T}) .$$		T}) .$$

	The corresponding time function is obtained by applying the  [[Signal_Representation/Fourier_Transform_Theorems#Shifting_Theorem\|~~$\text{~~Shifting Theorem}$]]:		The corresponding time function is obtained by applying the  [[Signal_Representation/Fourier_Transform_Theorems#Shifting_Theorem\|»Shifting Theorem«]]:

	:$$x_+(t) = A \cdot {\rm e}^{\hspace{0.05cm} {\rm j}\hspace{0.05cm}\cdot\hspace{0.05cm}( 2 \pi f_{\rm T} t		:$$x_+(t) = A \cdot {\rm e}^{\hspace{0.05cm} {\rm j}\hspace{0.05cm}\cdot\hspace{0.05cm}( 2 \pi f_{\rm T} t
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	This equation describes a rotating pointer with constant angular velocity  $\omega_{\rm T} = 2\pi f_{\rm T}$.		This equation describes a rotating pointer with constant angular velocity  $\omega_{\rm T} = 2\pi f_{\rm T}$.

	In the following, we will also refer to the time course of an analytical and frequency-discrete signal  $x_+(t)$  as  ~~$\text{Pointer Diagram}$~~.		In the following,  we will also refer to the time course of an analytical and frequency-discrete signal  $x_+(t)$  as  »'''pointer diagram'''«.
	{{GraueBox\|TEXT=		{{GraueBox\|TEXT=
	$\text{Example 3:}$  For illustrative reasons the coordinate system here is rotated (real part upwards, imaginary part to the left), contrary to the usual representation by  $90^\circ$.		$\text{Example 3:}$  For illustrative reasons the coordinate system here is rotated  $($real part upwards,  imaginary part to the left$)$,  contrary to the usual representation by  $90^\circ$.

	[[File:P_ID712__Sig_T_4_2_S3.png\|right\|frame\|Pointer diagram of a harmonic oscillation]]		[[File:P_ID712__Sig_T_4_2_S3.png\|right\|frame\|Pointer diagram of a harmonic oscillation]]

	On the basis of this diagram the following statements are possible:		On the basis of this diagram the following statements are possible:
	*At the start time  $t = 0$  the pointer of length  $A$  (amplitude) lies with angle  $-\varphi$  in the complex plane.  In the drawn example,  $\varphi = 45^\circ$.		*At the start time  $t = 0$  the pointer of length  $A$  $($amplitude$)$  lies with angle  $-\varphi$  in the complex plane.  In the drawn example,  $\varphi = 45^\circ$.
	*For ~~the~~ times  $t > 0$  the pointer rotates with constant angular velocity (circular frequency)  $\omega_{\rm T}$  in mathematically positive direction, i.e. counterclockwise.
	*The top of the pointer thus always lies on a circle with radius  $A$  and requires exactly the time  $T_0$, i.e. the period duration of the harmonic oscillation  $x(t)$  for one rotation.		*For times  $t > 0$  the pointer rotates with constant angular velocity  $($circular frequency$)$  $\omega_{\rm T}$  in mathematically positive direction,  i.e. counterclockwise.
	*The projection of the analytical signal  $x_+(t)$  onto the real axis, marked by red dots, provides the instantaneous values of  $x(t)$.}}
			*The top of the pointer thus always lies on a circle with radius  $A$  and requires exactly the time  $T_0$,  i.e. the  »period duration«  of the harmonic oscillation  $x(t)$  for one rotation.

			*The projection of the analytical signal  $x_+(t)$  onto the real axis,  marked by red dots,  provides the instantaneous values of  $x(t)$.}}


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	==Pointer diagram of a sum of harmonic oscillations==		==Pointer diagram of a sum of harmonic oscillations==
	<br>		<br>
	For further description we assume the following spectrum for the analytical signal:		For the further description we assume the following spectrum for the analytical signal:

	[[File:P_ID715__Sig_T_4_2_S4.png\|right\|frame\|Pointer diagram of a sum of three oscillations]]		[[File:P_ID715__Sig_T_4_2_S4.png\|right\|frame\|Pointer diagram of a sum of three oscillations]]
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	\varphi_i}\cdot\delta (f - f_{i}) .$$		\varphi_i}\cdot\delta (f - f_{i}) .$$

	The left graphic shows such a spectrum for the example  $I = 3$.  If one chooses  $I$  relatively large and the distance between adjacent spectral lines correspondingly small, then (frequency–) continuous spectral functions  $X_+(f)$  can also be approximated ~~with the above equation~~.		#The left graphic shows such a spectrum for the example  $I = 3$.
			#If one chooses  $I$  relatively large and the distance between adjacent spectral lines correspondingly small,  then with this equation frequency–continuous spectral functions  $X_+(f)$  can also be approximated.


	In the right graphic the corresponding time function is indicated.  This is in general:		In the right graphic the corresponding time function is indicated.  This is in general:
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	To note about this graphic:		To note about this graphic:
	*The sketch shows the initial position of the pointers at ~~the~~ start time  $t = 0$  corresponding to the amplitudes  $A_i$  and the phase positions  $\varphi_i$.		*The sketch shows the initial position of the pointers at start time  $t = 0$  corresponding to the amplitudes  $A_i$  and the phase positions  $\varphi_i$.

	*The tip of the resulting pointer compound is marked by the violet cross.  One obtains by vectorial addition of the three individual pointers for the time  $t = 0$:		*The tip of the resulting pointer compound is marked by the violet cross.  One obtains by vectorial addition of the three individual pointers for the time  $t = 0$:
	:$$x_+(t= 0) = \big [1 \cdot \cos(60^\circ) - 1 \cdot {\rm j} \cdot \sin(60^\circ) \big ]+ 2 \cdot \cos(0^\circ)+1 \cdot \cos(180^\circ) = 1.500 - {\rm j} \cdot 0.866.$$		:$$x_+(t= 0) = \big [1 \cdot \cos(60^\circ) - 1 \cdot {\rm j} \cdot \sin(60^\circ) \big ]+ 2 \cdot \cos(0^\circ)+1 \cdot \cos(180^\circ) = 1.500 - {\rm j} \cdot 0.866.$$
	*For times  $t > 0$  the three pointers rotate at different angular velocities  $\omega_i = 2\pi f_i$.  The red pointer rotates faster than the green one, but slower than the blue one.		*For times  $t > 0$  the three pointers rotate at different angular velocities  $\omega_i = 2\pi f_i$.  The red pointer rotates faster than the green one,  but slower than the blue one.
	*Since all pointers rotate counterclockwise, the resulting pointer  $x_+(t)$  will also tend to move in this direction.
			*Since all pointers rotate counterclockwise, the resulting pointer  $x_+(t)$  will also tend to move in this direction.

	*At time  $t = 1\,µ\text {s}$  the tip of the resulting pointer for the given parameter values is		*At time  $t = 1\,µ\text {s}$  the tip of the resulting pointer for the given parameter values is

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	The interactive applet  [[Applets:Physical_Signal_%26_Analytic_Signal\|"Physical Signal and Analytical Signal"]]  illustrates  $x_+(t)$  for the sum of three harmonic oscillations.		The interactive applet  [[Applets:Physical_Signal_%26_Analytic_Signal\|»Physical Signal and Analytical Signal«]]  illustrates  $x_+(t)$  for the sum of three harmonic oscillations.

	==Exercises for the chapter==		==Exercises for the chapter==

Guenter at 15:11, 19 June 2023

2023-06-19T15:11:57Z

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Guenter at 16:25, 12 December 2022

2022-12-12T16:25:46Z

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	{{BlaueBox\|TEXT=		{{BlaueBox\|TEXT=
	$\text{Definition:}$  The  $\text{analytical signal}$  $x_+(t)$  belonging to the physical signal  $x(t)$  is that time function, whose spectrum fulfills the following property:		$\text{Definition:}$  The  $\text{analytical signal}$  $x_+(t)$  belonging to the physical signal  $x(t)$  is that time function, whose spectrum fulfills the following property:
	[[File:~~Sig_T_4_2_S1a_Version2~~.png\|right\|frame\|Analytical signal in the frequency domain ~~'''KORREKTUR'''~~]]		[[File:EN_Sig_T_4_2_S1a.png\|right\|frame\|Analytical signal in the frequency domain]]
	:$$X_+(f)=\big[1+{\rm sign}(f)\big] \cdot X(f) = \left\{ {2 \cdot		:$$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.$$		X(f) \; \hspace{0.2cm}\rm for\hspace{0.2cm} {\it f} > 0, \atop {\,\,\,\, \rm 0 \; \hspace{0.9cm}\rm for\hspace{0.2cm} {\it f} < 0.} }\right.$$

Guenter at 17:18, 10 December 2022

2022-12-10T17:18:39Z

← Older revision		Revision as of 19:18, 10 December 2022
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	==Pointer diagram representation of the harmonic oscillation==		==Pointer diagram representation of the harmonic oscillation==
	<br>		<br>
	The spectral function  $X(f)$  of a harmonic oscillation  $x(t) = A \cdot \text{cos}(2\pi f_{\rm T}t - \varphi)$  consists of two Dirac ~~'''KORREKTUR:~~ delta~~'''~~ functions at the frequencies		The spectral function  $X(f)$  of a harmonic oscillation  $x(t) = A \cdot \text{cos}(2\pi f_{\rm T}t - \varphi)$  consists of two Dirac delta 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   $A/2 \cdot \text{e}^{-\text{j}\hspace{0.05cm}\varphi}$,
	* $-f_{\rm T}$  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 is  $($without the Dirac ~~'''KORREKTUR:~~ delta~~'''~~ function at the frequency  $f =-f_{\rm T})$:		Thus, the spectrum of the analytical signal is  $($without the Dirac delta function at the frequency  $f =-f_{\rm T})$:

	:$$X_+(f) = A \cdot {\rm e}^{-{\rm j} \hspace{0.05cm}\varphi}\cdot\delta (f - f_{\rm		:$$X_+(f) = A \cdot {\rm e}^{-{\rm j} \hspace{0.05cm}\varphi}\cdot\delta (f - f_{\rm

Hwang at 11:55, 8 December 2022

2022-12-08T11:55:32Z

← Older revision		Revision as of 13:55, 8 December 2022
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	{{Header		{{Header
	\|Untermenü=~~Bandpass~~ Signals		\|Untermenü=Band-Pass Signals
	\|Vorherige Seite=Differences and Similarities of LP and BP Signals		\|Vorherige Seite=Differences and Similarities of LP and BP Signals
	\|Nächste Seite=Equivalent Low-Pass Signal and Its Spectral Function		\|Nächste Seite=Equivalent Low-Pass Signal and Its Spectral Function