Modulation Methods/General Model of Modulation - Revision history

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2026-03-16T13:47:07Z

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	<br>		<br>
	The modulated signal   $s(t)$  is  "band-pass".  As already described in the book  "Signal Representation",  such a band-pass signal  $s(t)$  is often characterized by its associated  [[Signal_Representation/Analytical_Signal_and_Its_Spectral_Function\|$\text{analytical signal}$]]  $s_+(t)$.  It is important to note:		The modulated signal   $s(t)$  is  "band-pass".  As already described in the book  "Signal Representation",  such a band-pass signal  $s(t)$  is often characterized by its associated  [[Signal_Representation/Analytical_Signal_and_Its_Spectral_Function\|$\text{analytical signal}$]]  $s_+(t)$.  It is important to note:
	*The analytical signal $s_+(t)$  is obtained from the real physical signal  $s(t)$,  by adding to it  (as an imaginary part)  its [[Signal_Representation/~~Analytical_Signal_and_Its_Spectral_Function~~#Representation_with_Hilbert_transform\|$\text{Hilbert transform}$]]:		*The analytical signal $s_+(t)$  is obtained from the real physical signal  $s(t)$,  by adding to it  (as an imaginary part)  its [[Signal_Representation/Analytical_Signal_and_its_Spectral_Function#Representation_with_Hilbert_transform\|$\text{Hilbert transform}$]]:
	:$$s_+(t) = s(t) + {\rm j} \cdot {\rm H}\{ s(t)\}\hspace{0.05cm}.$$		:$$s_+(t) = s(t) + {\rm j} \cdot {\rm H}\{ s(t)\}\hspace{0.05cm}.$$

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	*The spectrum  $S_+(f)$  of the analytic signal is obtained from the two-sided spectrum  $S(f)$ by doubling it for positive frequencies and setting it to zero for negative frequencies:		*The spectrum  $S_+(f)$  of the analytic signal is obtained from the two-sided spectrum  $S(f)$ by doubling it for positive frequencies and setting it to zero for negative frequencies:
	:$$S_+(f) =\big[ 1 + {\rm sign}(f)\big] \cdot S(f) = \left\{ \begin{array}{c} 2 \cdot S(f) \\ 0 \\ \end{array} \right.\quad \begin{array}{{10}c} {\rm{for}} \\ {\rm{for}} \\ \end{array}\begin{array}{{20}c} f>0 \hspace{0.05cm}, \\ f<0 \hspace{0.05cm}, \\ \end{array} \hspace{1.3cm}		:$$S_+(f) =\big[ 1 + {\rm sign}(f)\big] \cdot S(f) = \left\{ \begin{array}{c} 2 \cdot S(f) \\ 0 \\ \end{array} \right.\quad \begin{array}{{10}c} {\rm{for}} \\ {\rm{for}} \\ \end{array}\begin{array}{{20}c} f>0 \hspace{0.05cm}, \\ f<0 \hspace{0.05cm}, \\ \end{array} \hspace{1.3cm}\text{with}\hspace{1.3cm}{\rm sign}(f) = \left\{ \begin{array}{c} +1 \\ -1 \\ \end{array} \right.\quad \begin{array}{{10}c} {\rm{for}} \\ {\rm{for}} \\ \end{array}\begin{array}{{20}c} f>0 \hspace{0.05cm}, \\ f<0 \hspace{0.05cm}. \\ \end{array}$$
	\text{with}\hspace{1.3cm}
	{\rm sign}(f) = \left\{ \begin{array}{c} +1 \\ -1 \\ \end{array} \right.\quad \begin{array}{{10}c} {\rm{for}} \\ {\rm{for}} \\ \end{array}\begin{array}{{20}c} f>0 \hspace{0.05cm}, \\ f<0 \hspace{0.05cm}. \\ \end{array}$$

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	*The spectrum  $S_{\rm TP}(f)$  of the equivalent low-pass signal is obtained from  $S_+(f)$  by shifting it to the left by  $f_{\rm T}$  and is thus in the frequency range around  $f =0$:		*The spectrum  $S_{\rm TP}(f)$  of the equivalent low-pass signal is obtained from  $S_+(f)$  by shifting it to the left by  $f_{\rm T}$  and is thus in the frequency range around  $f =0$:
	:$$S_{\rm TP}(f) = S_+(f + f_{\rm T}) \hspace{0.05cm}.$$		:$$S_{\rm TP}(f) = S_+(f + f_{\rm T}) \hspace{0.05cm}.$$
	*For the corresponding time function,  according to the  [[Signal_Representation/~~Fourier_Transform_Laws~~#Shifting_Theorem\|$\text{Shifting Theorem}$]]  holds:		*For the corresponding time function,  according to the  [[Signal_Representation/The_Fourier_Transform_Theorems#Shifting_Theorem\|$\text{Shifting Theorem}$]]  holds:
	:$$s_{\rm TP}(t) = s_+(t) \cdot {\rm e}^{{-\rm j}\hspace{0.08cm}		:$$s_{\rm TP}(t) = s_+(t) \cdot {\rm e}^{{-\rm j}\hspace{0.08cm}\omega_{\rm T} \hspace{0.03cm}t }\hspace{0.05cm}.$$
	\omega_{\rm T} \hspace{0.03cm}t }\hspace{0.05cm}.$$

	*The equivalent low-pass signal of an unmodulated harmonic oscillation is constant for all times.  The  "locus curve"  in this special case consists of a single point:		*The equivalent low-pass signal of an unmodulated harmonic oscillation is constant for all times.  The  "locus curve"  in this special case consists of a single point:
	:$$s(t) = A_{\rm T} \cdot \cos(\omega_{\rm T}\cdot t + \phi_{\rm T}) \hspace{0.3cm} \Leftrightarrow \hspace{0.3cm} s_+(t) = {\rm e}^{\hspace{0.03cm}{\rm j} \hspace{0.03cm}(\omega_{\rm T}\hspace{0.05cm} t \hspace{0.05cm} + \phi_{\rm T})}\hspace{0.05cm},$$		:$$s(t) = A_{\rm T} \cdot \cos(\omega_{\rm T}\cdot t + \phi_{\rm T}) \hspace{0.3cm} \Leftrightarrow \hspace{0.3cm} s_+(t) = {\rm e}^{\hspace{0.03cm}{\rm j} \hspace{0.03cm}(\omega_{\rm T}\hspace{0.05cm} t \hspace{0.05cm} + \phi_{\rm T})}\hspace{0.05cm},$$
	:$$ s_+(t) = {\rm e}^{\hspace{0.03cm}{\rm j} \hspace{0.03cm}(\omega_{\rm T}\hspace{0.05cm} t \hspace{0.05cm} + \phi_{\rm T})}\hspace{0.3cm} \Leftrightarrow \hspace{0.3cm}		:$$ s_+(t) = {\rm e}^{\hspace{0.03cm}{\rm j} \hspace{0.03cm}(\omega_{\rm T}\hspace{0.05cm} t \hspace{0.05cm} + \phi_{\rm T})}\hspace{0.3cm} \Leftrightarrow \hspace{0.3cm}s_{\rm TP}(t) = A_{\rm T} \cdot {\rm e}^{\hspace{0.03cm}{\rm j} \hspace{0.03cm} \cdot \hspace{0.05cm} \phi_{\rm T}}\hspace{0.05cm}.$$
	s_{\rm TP}(t) = A_{\rm T} \cdot {\rm e}^{\hspace{0.03cm}{\rm j} \hspace{0.03cm} \cdot \hspace{0.05cm} \phi_{\rm T}}\hspace{0.05cm}.$$
	<br clear=all>		<br clear=all>
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	$\text{Important result:}$  For an amplitude or phase modulated signal with carrier frequency  $f_{\rm T}$  it holds that:		$\text{Important result:}$  For an amplitude or phase modulated signal with carrier frequency  $f_{\rm T}$  it holds that:

	:$$s(t) = a(t) \cdot \cos(\omega_{\rm T}\cdot t + \phi(t)) \hspace{0.3cm} \Leftrightarrow \hspace{0.3cm}		:$$s(t) = a(t) \cdot \cos(\omega_{\rm T}\cdot t + \phi(t)) \hspace{0.3cm} \Leftrightarrow \hspace{0.3cm}s_{\rm TP}(t) = a(t) \cdot {\rm e}^{\hspace{0.03cm}{\rm j} \hspace{0.03cm} \cdot \hspace{0.05cm} \phi (t)}\hspace{0.05cm}.$$
	s_{\rm TP}(t) = a(t) \cdot {\rm e}^{\hspace{0.03cm}{\rm j} \hspace{0.03cm} \cdot \hspace{0.05cm} \phi (t)}\hspace{0.05cm}.$$

	The envelope  $a(t)$  and the phase response  $ϕ(t)$  of the  (physical)  band-pass signal  $s(t)$  are also preserved in the equivalent low-pass signal  $s_{\rm TP}(t)$. }}		The envelope  $a(t)$  and the phase response  $ϕ(t)$  of the  (physical)  band-pass signal  $s(t)$  are also preserved in the equivalent low-pass signal  $s_{\rm TP}(t)$. }}

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2026-03-16T12:28:09Z

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	{{Display}}		{{Display}}
			[[de:Modulationsverfahren/Allgemeines_Modell_der_Modulation]]

Guenter at 17:09, 12 January 2023

2023-01-12T17:09:11Z

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Hwang at 16:59, 2 January 2023

2023-01-02T16:59:43Z

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	$\text{Notes:}$		$\text{Notes:}$
	*In the following,  we also refer to the time-dependent course of  $s_{\rm TP}(t)$  in the complex plane as the  »'''locus curve'''«.,		*In the following,  we also refer to the time-dependent course of  $s_{\rm TP}(t)$  in the complex plane as the  »'''locus curve'''«.
	*The  »'''pointer diagram'''«  describes the course of the analytical signal  $s_+(t)$ .		*The  »'''pointer diagram'''«  describes the course of the analytical signal  $s_+(t)$ .
	*The topic presented here is illustrated in the time domain with two interactive  "HTML 5/JS"  applets:		*The topic presented here is illustrated in the time domain with two interactive  "HTML 5/JS"  applets:

Hwang at 16:50, 2 January 2023

2023-01-02T16:50:04Z

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	==Describing the physical signal using the equivalent low-pass signal==		==Describing the physical signal using the equivalent low-pass signal==
	<br>		<br>
	Some facts concerning modulation at the transmitter and demodulation at the receiver can be explained best by means of the [[Signal_Representation/~~Equivalent_Low_Pass_Signal_and_Its_Spectral_Function~~\|$\text{equivalent low-pass signal}$]]  (German:  "äquivalentes Tiefpass–Signal"   ⇒   subscript  "TP")  according to the definition given in the book  [[Signal_Representation/Equivalent_Low-Pass_Signal_and_its_Spectral_Function\|"Signal Representation"]].		Some facts concerning modulation at the transmitter and demodulation at the receiver can be explained best by means of the [[Signal_Representation/Equivalent_Low-Pass_Signal_and_its_Spectral_Function\|$\text{equivalent low-pass signal}$]]  (German:  "äquivalentes Tiefpass–Signal"   ⇒   subscript  "TP")  according to the definition given in the book  [[Signal_Representation/Equivalent_Low-Pass_Signal_and_its_Spectral_Function\|"Signal Representation"]].
	[[File:Mod_T_1_3_S5a_version2.png\|right\|frame\|The equivalent lowpass signal in the frequency domain]]		[[File:Mod_T_1_3_S5a_version2.png\|right\|frame\|The equivalent lowpass signal in the frequency domain]]

Hwang at 16:41, 2 January 2023

2023-01-02T16:41:56Z

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	In these basic methods,  two of the three oscillation parameters are thus always kept constant.		In these basic methods,  two of the three oscillation parameters are thus always kept constant.
	*Additionally,  there are also variants with more than one time dependency for amplitude,  frequency,  and phase, resp.		*Additionally,  there are also variants with more than one time dependency for amplitude,  frequency,  and phase, resp.
	*An example of this is  [[Modulation_Methods/Single-Sideband_Modulation\|$\text{~~single~~-~~sideband modulation~~}$]],  where both  $a(t)$  and  ${\it ϕ}(t)$  are affected by the source signal  $q(t)$.		*An example of this is  [[Modulation_Methods/Single-Sideband_Modulation\|$\text{Single-Sideband Modulation}$]],  where both  $a(t)$  and  ${\it ϕ}(t)$  are affected by the source signal  $q(t)$.


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	When describing   $\rm AM$,  $\rm FM$  and  $\rm PM$,  the source signal  $q(t)$  is usually assumed to be continuous in time and value.		When describing   $\rm AM$,  $\rm FM$  and  $\rm PM$,  the source signal  $q(t)$  is usually assumed to be continuous in time and value.
	*However,  the above equations can also be applied to a rectangular source signal.		*However,  the above equations can also be applied to a rectangular source signal.
	*In this case,  $q(t)$  is continuous in time but discrete in value.  Thus,  it also describes methods for  [[Modulation_Methods/Linear_Digital_Modulation\|$\text{~~linear digital modulation~~}$]].		*In this case,  $q(t)$  is continuous in time but discrete in value.  Thus,  it also describes methods for  [[Modulation_Methods/Linear_Digital_Modulation\|$\text{Linear Digital Modulation}$]].

Hwang at 16:33, 2 January 2023

2023-01-02T16:33:48Z

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Reed at 17:38, 23 March 2022

2022-03-23T17:38:29Z

← Older revision		Revision as of 19:38, 23 March 2022
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	$\text{Notes:}$		$\text{Notes:}$
	*In the following,  we also refer to the time-dependent course of  $s_{\rm TP}(t)$  in the complex plane as the  '''locus curve'''.,		*In the following,  we also refer to the time-dependent course of  $s_{\rm TP}(t)$  in the complex plane as the  '''locus curve'''.,
	*The&nbsp~~; &raquo~~;'''pointer diagram'''~~«~~  describes the course of the analytical signal  $s_+(t)$ .		*The  '''pointer diagram'''  describes the course of the analytical signal  $s_+(t)$ .
	*The topic presented here is illustrated in the time domain with two interactive  "HTML 5/JS"  applets:		*The topic presented here is illustrated in the time domain with two interactive  "HTML 5/JS"  applets:
	::(1)  [[Applets:Physikalisches_Signal_%26_Analytisches_Signal\|Physical & Analytic Signal ]],		::(1)  [[Applets:Physikalisches_Signal_%26_Analytisches_Signal\|Physical & Analytic Signal ]],

Reed at 17:36, 23 March 2022

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	$\text{Notes:}$		$\text{Notes:}$
	*In the following,  we also refer to the time-dependent course of  $s_{\rm TP}(t)$  in the complex plane as the&nbsp~~; &raquo~~;'''locus curve'''~~«~~.,		*In the following,  we also refer to the time-dependent course of  $s_{\rm TP}(t)$  in the complex plane as the  '''locus curve'''.,
	*The  »'''pointer diagram'''«  describes the course of the analytical signal  $s_+(t)$ .		*The  »'''pointer diagram'''«  describes the course of the analytical signal  $s_+(t)$ .
	*The topic presented here is illustrated in the time domain with two interactive  "HTML 5/JS"  applets:		*The topic presented here is illustrated in the time domain with two interactive  "HTML 5/JS"  applets:

Reed at 17:34, 23 March 2022

2022-03-23T17:34:58Z

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	[[File:EN_Mod_T_1_3_S5b_neu.png \|right\|frame\| Transmitted signals for amplitude and angle modulation]]		[[File:EN_Mod_T_1_3_S5b_neu.png \|right\|frame\| Transmitted signals for amplitude and angle modulation]]

	Upper ~~graphic~~   ⇒   '''Amplitude modulation'''  $\rm (AM)$:		Upper graph   ⇒   '''Amplitude modulation'''  $\rm (AM)$:
	*Here,  the source signal  $q(t)$  can be seen in the envelope  $a(t)$.		*Here,  the source signal  $q(t)$  can be seen in the envelope  $a(t)$.
	*Since the zero crossings of the carrier  $z(t)$  stay the same:  $ϕ(t) = 0$.  The equivalent low-pass signal  $s_{\rm TP}(t) = a(t)$  is real.		*Since the zero crossings of the carrier  $z(t)$  stay the same:  $ϕ(t) = 0$.  The equivalent low-pass signal  $s_{\rm TP}(t) = a(t)$  is real.
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	Bottom ~~graphic~~   ⇒   '''Angle modulation'''  $\rm (WM)$:		Bottom graph   ⇒   '''Angle modulation'''  $\rm (WM)$:
	*Here,  the envelope  $a(t)$  is constant   ⇒   the equivalent low-pass signal  $s_{\rm TP}(t) = A_{\rm T} · e^{\rm j·ϕ(t)}$  describes a circular arc.		*Here,  the envelope  $a(t)$  is constant   ⇒   the equivalent low-pass signal  $s_{\rm TP}(t) = A_{\rm T} · e^{\rm j·ϕ(t)}$  describes a circular arc.
	*The information about the source signal  $q(t)$  is found here in the location of the zero crossings  $s(t)$.		*The information about the source signal  $q(t)$  is found here in the location of the zero crossings  $s(t)$.