Modulation Methods/Phase Modulation (PM) - Revision history

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	$\rm Definition\text{:}$  An  »'''angle modulation'''«  – abbreviated as   $\rm WM$  – is present whenever the modulated signal can be represented as follows:		$\rm Definition\text{:}$  An  »'''angle modulation'''«  – abbreviated as   $\rm WM$  – is present whenever the modulated signal can be represented as follows:
	:$$s(t) = A_{\rm T} \cdot \cos\big[\psi(t)\big] = A_{\rm T} \cdot \cos\hspace{-0.1cm}\big[ω_{\rm T} · t + ϕ(t)\big]		:$$s(t) = A_{\rm T} \cdot \cos\big[\psi(t)\big] = A_{\rm T} \cdot \cos\hspace{-0.1cm}\big[ω_{\rm T} · t + ϕ(t)\big]\hspace{0.05cm}.$$
	\hspace{0.05cm}.$$
	*Here,  as in amplitude modulation,  $A_{\rm T}$  denotes the amplitude of the carrier signal  $z(t)$.		*Here,  as in amplitude modulation,  $A_{\rm T}$  denotes the amplitude of the carrier signal  $z(t)$.
	*However,  all the information about the source signal  $q(t)$  is now captured by the   "angular function"  $ψ(t)$.}}		*However,  all the information about the source signal  $q(t)$  is now captured by the   "angular function"  $ψ(t)$.}}
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	$\rm Definition\text{:}$  If the phase function $ϕ(t)$  is proportional to the applied source signal  $q(t)$,  we are dealing with  »'''phase modulation'''«  $\rm (PM)$,  and it holds that:		$\rm Definition\text{:}$  If the phase function $ϕ(t)$  is proportional to the applied source signal  $q(t)$,  we are dealing with  »'''phase modulation'''«  $\rm (PM)$,  and it holds that:
	:$$\phi(t)= K_{\rm PM} \cdot q(t)\hspace{0.05cm}\hspace{0.3cm}\Rightarrow		:$$\phi(t)= K_{\rm PM} \cdot q(t)\hspace{0.05cm}\hspace{0.3cm}\Rightarrow\hspace{0.3cm}\psi(t)= \omega_{\rm T} \cdot t +\phi(t)\hspace{0.3cm}\Rightarrow \hspace{0.3cm}s(t) = A_{\rm T}\cdot \cos \big[\psi(t)\big]\hspace{0.05cm}.$$
	\hspace{0.3cm}\psi(t)= \omega_{\rm T} \cdot t +
	\phi(t)\hspace{0.3cm}\Rightarrow \hspace{0.3cm}s(t) = A_{\rm T}
	\cdot \cos \big[\psi(t)\big]\hspace{0.05cm}.$$
	Here, $K_{\rm PM}$  denotes the modulator constant with appropriate dimensions.  If  $q(t)$  describes a voltage waveform,  this constant has the unit  $\rm 1/V$. }}		Here, $K_{\rm PM}$  denotes the modulator constant with appropriate dimensions.  If  $q(t)$  describes a voltage waveform,  this constant has the unit  $\rm 1/V$. }}

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	*These differ by the parameters  $η = 1$  and  $η = 3$, resp.:		*These differ by the parameters  $η = 1$  and  $η = 3$, resp.:
	[[File: P_ID1070__Mod_T_3_1_S2a_neu.png\|right\|frame\|Signal characteristics for phase modulation with  $η = 1$  resp.  $η = 3$]]		[[File: P_ID1070__Mod_T_3_1_S2a_neu.png\|right\|frame\|Signal characteristics for phase modulation with  $η = 1$  resp.  $η = 3$]]
	:$$s_\eta(t) = A_{\rm T} \cdot \cos \hspace{-0.1cm}\big[\omega_{\rm T} \cdot t +		:$$s_\eta(t) = A_{\rm T} \cdot \cos \hspace{-0.1cm}\big[\omega_{\rm T} \cdot t +\eta \cdot \sin (\omega_{\rm N} \cdot t) \big]\hspace{0.05cm}.$$
	\eta \cdot \sin (\omega_{\rm N} \cdot t) \big]\hspace{0.05cm}.$$
	*Dotted in gray is the cosine carrier signal  $z(t)$,  in each case based on  $f_{\rm T} = 20 \ \rm kHz$ .		*Dotted in gray is the cosine carrier signal  $z(t)$,  in each case based on  $f_{\rm T} = 20 \ \rm kHz$ .

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	Thus,  the phase modulated signal and the corresponding equivalent low-pass signal are:		Thus,  the phase modulated signal and the corresponding equivalent low-pass signal are:
	:$$s(t) = A_{\rm T} \cdot \cos \big[\omega_{\rm T} \cdot t + \eta		:$$s(t) = A_{\rm T} \cdot \cos \big[\omega_{\rm T} \cdot t + \eta\cdot \sin (\omega_{\rm N} \cdot t) \big]\hspace{0.05cm},$$
	\cdot \sin (\omega_{\rm N} \cdot t) \big]\hspace{0.05cm},$$		:$$s_{\rm TP}(t) = A_{\rm T} \cdot {\rm e}^{\hspace{0.05cm}{\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}\eta \hspace{0.05cm}\cdot\hspace{0.05cm}\sin (\omega_{\rm N} \cdot t) }\hspace{0.05cm}.$$
	:$$s_{\rm TP}(t) = A_{\rm T} \cdot {\rm e}^{\hspace{0.05cm}{\rm j}
	\hspace{0.05cm}\cdot \hspace{0.05cm}\eta \hspace{0.05cm}\cdot
	\hspace{0.05cm}\sin (\omega_{\rm N} \cdot t) }\hspace{0.05cm}.$$

	This signal is periodic and can be represented by a   [[Signal_Representation/Fourier_Series#Complex_Fourier_series\|$\text{complex Fourier series}$]].  Thus,  in general one obtains:		This signal is periodic and can be represented by a   [[Signal_Representation/Fourier_Series#Complex_Fourier_series\|$\text{complex Fourier series}$]].  Thus,  in general one obtains:
	:$$s_{\rm TP}(t) = \sum_{n = - \infty}^{+\infty}D_{n} \cdot {\rm		:$$s_{\rm TP}(t) = \sum_{n = - \infty}^{+\infty}D_{n} \cdot {\rm e}^{\hspace{0.05cm}{\rm j} \hspace{0.05cm}\cdot\hspace{0.05cm}n\hspace{0.05cm}\cdot \hspace{0.05cm}\omega_{\rm N}\hspace{0.05cm}\cdot \hspace{0.05cm} t} \hspace{0.05cm}.$$
	e}^{\hspace{0.05cm}{\rm j} \hspace{0.05cm}\cdot
	\hspace{0.05cm}n\hspace{0.05cm}\cdot \hspace{0.05cm}\omega_{\rm N}
	\hspace{0.05cm}\cdot \hspace{0.05cm} t} \hspace{0.05cm}.$$

	In the special case considered here   (sinusoidal source signal,  cosinusoidal carrier)  the typically complex Fourier coefficients  $D_n$  are all real and given by   $n$–th order  »'''Bessel functions'''«  of the first kind  ${\rm J}_n(η)$  as follows:		In the special case considered here   (sinusoidal source signal,  cosinusoidal carrier)  the typically complex Fourier coefficients  $D_n$  are all real and given by   $n$–th order  »'''Bessel functions'''«  of the first kind  ${\rm J}_n(η)$  as follows:
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	$\text{Important intermediate result:}$		$\text{Important intermediate result:}$
	Now it should be mathematically proven that in the case of phase modulation,  the equivalent low-pass signal can indeed be converted into the following function series:		Now it should be mathematically proven that in the case of phase modulation,  the equivalent low-pass signal can indeed be converted into the following function series:
	:$$s_{\rm TP}(t) = A_{\rm T} \cdot {\rm e}^{\hspace{0.05cm}{\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm}\eta \hspace{0.05cm}\cdot \hspace{0.05cm}\sin (\omega_{\rm N} \cdot t) } \hspace{0.3cm}\Rightarrow \hspace{0.3cm} s_{\rm TP}(t) = A_{\rm T} \cdot \sum_{n = - \infty}^{+\infty}{\rm J}_n (\eta) \cdot {\rm e}^{\hspace{0.05cm}{\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm}n\hspace{0.05cm}\cdot		:$$s_{\rm TP}(t) = A_{\rm T} \cdot {\rm e}^{\hspace{0.05cm}{\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm}\eta \hspace{0.05cm}\cdot \hspace{0.05cm}\sin (\omega_{\rm N} \cdot t) } \hspace{0.3cm}\Rightarrow \hspace{0.3cm} s_{\rm TP}(t) = A_{\rm T} \cdot \sum_{n = - \infty}^{+\infty}{\rm J}_n (\eta) \cdot {\rm e}^{\hspace{0.05cm}{\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm}n\hspace{0.05cm}\cdot\hspace{0.05cm}\omega_{\rm N} \hspace{0.05cm}\cdot \hspace{0.05cm} t}.$$}}
	\hspace{0.05cm}\omega_{\rm N} \hspace{0.05cm}\cdot \hspace{0.05cm} t}.$$}}


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	'''(5)'''  The Bessel functions exhibit the following symmetrical properties:		'''(5)'''  The Bessel functions exhibit the following symmetrical properties:
	:$${\rm J}_{-n} (\eta) = ( - 1)^n \cdot {\rm J}_{n} (\eta)\hspace{0.3cm}		:$${\rm J}_{-n} (\eta) = ( - 1)^n \cdot {\rm J}_{n} (\eta)\hspace{0.3cm}\Rightarrow \hspace{0.3cm} {\rm J}_{ - 1} (\eta) = - {\rm J}_{1}(\eta),\hspace{0.3cm}{\rm J}_{ - 2} (\eta) = {\rm J}_{2}(\eta),\hspace{0.3cm}{\rm J}_{ - 3} (\eta) = - {\rm J}_{3}(\eta),\hspace{0.3cm}{\rm J}_{ - 4} (\eta) = {\rm J}_{4} (\eta).$$
	\Rightarrow \hspace{0.3cm} {\rm J}_{ - 1} (\eta) = - {\rm J}_{1}
	(\eta),\hspace{0.3cm}{\rm J}_{ - 2} (\eta) = {\rm J}_{2}
	(\eta),\hspace{0.3cm}{\rm J}_{ - 3} (\eta) = - {\rm J}_{3}
	(\eta),\hspace{0.3cm}{\rm J}_{ - 4} (\eta) = {\rm J}_{4} (\eta).$$

	'''(6)'''  Considering this fact and the factor  $A_{\rm T}$ omitted so far,  we get the desired result:		'''(6)'''  Considering this fact and the factor  $A_{\rm T}$ omitted so far,  we get the desired result:
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	:$${\rm J}_n (\eta) = \frac{1}{2\pi}\cdot \int_{-\pi}^{+\pi} {{\rm e}^{\hspace{0.05cm}{\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}(\eta \hspace{0.05cm}\cdot \hspace{0.05cm}\sin(\alpha) -\hspace{0.05cm} n \hspace{0.05cm}\cdot \hspace{0.05cm}\alpha)}}\hspace{0.1cm}{\rm d}\alpha\hspace{0.05cm},$$		:$${\rm J}_n (\eta) = \frac{1}{2\pi}\cdot \int_{-\pi}^{+\pi} {{\rm e}^{\hspace{0.05cm}{\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}(\eta \hspace{0.05cm}\cdot \hspace{0.05cm}\sin(\alpha) -\hspace{0.05cm} n \hspace{0.05cm}\cdot \hspace{0.05cm}\alpha)}}\hspace{0.1cm}{\rm d}\alpha\hspace{0.05cm},$$
	and can be approximated by a series according to the next equation:		and can be approximated by a series according to the next equation:
	:$${\rm J}_n (\eta) = \sum\limits_{k=0}^{\infty}\frac{(-1)^k \cdot (\eta/2)^{n \hspace{0.05cm} + \hspace{0.05cm} 2		:$${\rm J}_n (\eta) = \sum\limits_{k=0}^{\infty}\frac{(-1)^k \cdot (\eta/2)^{n \hspace{0.05cm} + \hspace{0.05cm} 2\hspace{0.02cm}\cdot \hspace{0.05cm}k}}{k! \cdot (n+k)!} \hspace{0.05cm}.$$
	\hspace{0.02cm}\cdot \hspace{0.05cm}k}}{k! \cdot (n+k)!} \hspace{0.05cm}.$$

	The adjacent graph shows the first three summands $(k = 0,\ 1,\ 2)$  of each of the series  ${\rm J}_0(η)$, ... ,  ${\rm J}_3(η).$		The adjacent graph shows the first three summands $(k = 0,\ 1,\ 2)$  of each of the series  ${\rm J}_0(η)$, ... ,  ${\rm J}_3(η).$
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	$\text{Without proof:}$		$\text{Without proof:}$
	*Based on the equivalent low-pass signal just calculated,  we obtain for the  »'''analytical signal'''«:		*Based on the equivalent low-pass signal just calculated,  we obtain for the  »'''analytical signal'''«:
	:$$s_{\rm +}(t) = s_{\rm TP}(t) \cdot {\rm e}^{\hspace{0.05cm}{\rm j} \hspace{0.05cm}\cdot		:$$s_{\rm +}(t) = s_{\rm TP}(t) \cdot {\rm e}^{\hspace{0.05cm}{\rm j} \hspace{0.05cm}\cdot\hspace{0.05cm}\omega_{\rm T} \hspace{0.05cm}\cdot \hspace{0.05cm} t}= A_{\rm T} \cdot\sum_{n = - \infty}^{+\infty}{\rm J}_n (\eta) \cdot {\rm e}^{\hspace{0.05cm}{\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm}(\omega_{\rm T}\hspace{0.05cm}+\hspace{0.05cm} n\hspace{0.05cm}\cdot \hspace{0.05cm}\omega_{\rm N}) \hspace{0.05cm}\cdot \hspace{0.05cm} t}$$
	\hspace{0.05cm}\omega_{\rm T} \hspace{0.05cm}\cdot \hspace{0.05cm} t}= A_{\rm T} \cdot
	\sum_{n = - \infty}^{+\infty}{\rm J}_n (\eta) \cdot {\rm e}^{\hspace{0.05cm}{\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm}(\omega_{\rm T}\hspace{0.05cm}+\hspace{0.05cm} n\hspace{0.05cm}\cdot \hspace{0.05cm}\omega_{\rm N}) \hspace{0.05cm}\cdot \hspace{0.05cm} t}$$

	*By Fourier transform, we get the  »'''spectrum of the analytical signal'''«:		*By Fourier transform, we get the  »'''spectrum of the analytical signal'''«:
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	If the source signal is composed of the sum of two sinusoidal oscillations,  the signals at the output of the phase modulator are:		If the source signal is composed of the sum of two sinusoidal oscillations,  the signals at the output of the phase modulator are:
	:$$s(t) = A_{\rm T} \cdot \cos \big[\omega_{\rm T} \cdot t + \eta_1 \cdot \sin (\omega_{\rm 1} \cdot t) + \eta_2 \cdot \sin(\omega_{\rm 2} \cdot t)\big]\hspace{0.05cm},$$		:$$s(t) = A_{\rm T} \cdot \cos \big[\omega_{\rm T} \cdot t + \eta_1 \cdot \sin (\omega_{\rm 1} \cdot t) + \eta_2 \cdot \sin(\omega_{\rm 2} \cdot t)\big]\hspace{0.05cm},$$
	:$$s_{\rm TP}(t) = A_{\rm T} \cdot {\rm e}^{\hspace{0.05cm}{\rm j} \hspace{0.05cm}\cdot [\hspace{0.05cm}\eta_1 \hspace{0.05cm}\cdot \hspace{0.08cm}\sin (\omega_{\rm 1} \hspace{0.05cm}\cdot		:$$s_{\rm TP}(t) = A_{\rm T} \cdot {\rm e}^{\hspace{0.05cm}{\rm j} \hspace{0.05cm}\cdot [\hspace{0.05cm}\eta_1 \hspace{0.05cm}\cdot \hspace{0.08cm}\sin (\omega_{\rm 1} \hspace{0.05cm}\cdot\hspace{0.05cm} t) \hspace{0.05cm}+ \hspace{0.05cm}\eta_2 \hspace{0.05cm}\cdot \hspace{0.08cm}\sin (\omega_{\rm 2} \hspace{0.05cm}\cdot \hspace{0.05cm}t)]}\hspace{0.05cm}.$$
	\hspace{0.05cm} t) \hspace{0.05cm}+ \hspace{0.05cm}\eta_2 \hspace{0.05cm}\cdot \hspace{0.08cm}\sin (\omega_{\rm 2} \hspace{0.05cm}\cdot \hspace{0.05cm}t)]}\hspace{0.05cm}.$$

	*For ease of representation,  we now set  $A_{\rm T} = 1$  and get:		*For ease of representation,  we now set  $A_{\rm T} = 1$  and get:
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	*The spectral functions of the two terms are:		*The spectral functions of the two terms are:
	:$${\rm e}^{\hspace{0.05cm}{\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm}\eta_1 \hspace{0.05cm}\cdot		:$${\rm e}^{\hspace{0.05cm}{\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm}\eta_1 \hspace{0.05cm}\cdot\hspace{0.08cm}\sin (2 \pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm 1} \hspace{0.01cm}\cdot \hspace{0.05cm} t) } \hspace{0.2cm}\circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\, \hspace{0.2cm} B_1(f) = \sum_{n = - \infty}^{+\infty}{\rm J}_n (\eta_1) \cdot \delta (f - n \cdot f_{\rm 1})\hspace{0.05cm},$$
	\hspace{0.08cm}\sin (2 \pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm 1} \hspace{0.01cm}\cdot \hspace{0.05cm} t) } \hspace{0.2cm}\circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\, \hspace{0.2cm} B_1(f) = \sum_{n = - \infty}^{+\infty}{\rm J}_n (\eta_1) \cdot \delta (f - n \cdot f_{\rm 1})\hspace{0.05cm},$$		:$${\rm e}^{\hspace{0.05cm}{\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm}\eta_2 \hspace{0.05cm}\cdot\hspace{0.08cm}\sin (2 \pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm 2} \hspace{0.01cm}\cdot\hspace{0.05cm} t) } \hspace{0.2cm}\circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\, \hspace{0.2cm} B_2(f) = \sum_{n = - \infty}^{+\infty}{\rm J}_n (\eta_2) \cdot \delta (f - n \cdot f_{\rm 2})\hspace{0.05cm}.$$
	:$${\rm e}^{\hspace{0.05cm}{\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm}\eta_2 \hspace{0.05cm}\cdot
	\hspace{0.08cm}\sin (2 \pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm 2} \hspace{0.01cm}\cdot
	\hspace{0.05cm} t) } \hspace{0.2cm}\circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\, \hspace{0.2cm} B_2(f) = \sum_{n = - \infty}^{+\infty}{\rm J}_n (\eta_2) \cdot \delta (f - n \cdot f_{\rm 2})\hspace{0.05cm}.$$

	*The Bessel functions $B_1(f)$  and  $B_2(f)$  describe line spectra spaced in frequency at  $f_1$  and  $f_2$, whose weights are determined by  $η_1$  and  $η_2$.		*The Bessel functions $B_1(f)$  and  $B_2(f)$  describe line spectra spaced in frequency at  $f_1$  and  $f_2$, whose weights are determined by  $η_1$  and  $η_2$.

Maintenance script: Add German interlanguage link

2026-03-16T12:28:16Z

Add German interlanguage link

← Older revision		Revision as of 14:28, 16 March 2026
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	{{Display}}		{{Display}}
			[[de:Modulationsverfahren/Phasenmodulation_(PM)]]

Hwang at 12:43, 18 January 2023

2023-01-18T12:43:56Z

← Older revision		Revision as of 14:43, 18 January 2023
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	*The low-pass spectrum  $S_{\rm TP}(f) = B_1(f) \star B_2(f)$  consists of nine Dirac lines and is sketched in the diagram on the right.		*The low-pass spectrum  $S_{\rm TP}(f) = B_1(f) \star B_2(f)$  consists of nine Dirac delta lines and is sketched in the diagram on the right.

Guenter at 16:43, 16 January 2023

2023-01-16T16:43:04Z

← Older revision		Revision as of 18:43, 16 January 2023
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	*on the right,  the transmitted signal  $s(t)$ ⇒   blue waveforms, compared to the carrier signal  $z(t)$   ⇒   red waveforms,		*on the right,  the transmitted signal  $s(t)$ ⇒   blue waveforms, compared to the carrier signal  $z(t)$   ⇒   red waveforms,
	*on the left,  the equivalent low-pass signal  $s_{\rm TP}(t)$  in the complex plane.		*on the left,  the equivalent low-pass signal  $s_{\rm TP}(t)$  in the complex plane.
	[[File:~~EN_Mod_T_3_1_S1b_v2~~.png\|right\|frame\|Physical signal and equivalent low-pass signal for angle and amplitude modulation ~~'''KORREKTUR: nachlaufend, vorlaufend'''~~]]		[[File:EN Mod T 3 1 S1b v3.png\|right\|frame\|Physical signal and equivalent low-pass signal for angle and amplitude modulation]]

Guenter at 13:09, 13 January 2023

2023-01-13T13:09:40Z

Guenter at 13:03, 13 January 2023

2023-01-13T13:03:56Z

← Older revision		Revision as of 15:03, 13 January 2023
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	== # OVERVIEW OF THE THIRD MAIN CHAPTER # ==		== # OVERVIEW OF THE THIRD MAIN CHAPTER # ==
	<br>		<br>
	The third chapter describes  »'''angle modulation'''«   $\rm (WM)$ (from ~~the~~ German   "Winkelmodulation"). This is a generic term for		The third chapter describes  »'''angle modulation'''«   $($short: "$\rm WM)$"  from German   "Winkelmodulation"$)$.
	*  "phase modulation"  $\rm (PM)$,
	*  "frequency modulation"   $\rm (FM)$.		This is a generic term for
			*  »phase modulation«  $\rm (PM)$,

			*  »frequency modulation«   $\rm (FM)$.

Hwang at 13:33, 9 January 2023

2023-01-09T13:33:05Z

← Older revision		Revision as of 15:33, 9 January 2023
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	The lower graph corresponds to   [[Modulation_Methods/Double-Sideband_Amplitude_Modulation\|$\text{Double-Sideband Amplitude Modulation}$]]  $\rm (DSB-AM)$  as described in Chapter 2,  ~~characterised~~ by		The lower graph corresponds to   [[Modulation_Methods/Double-Sideband_Amplitude_Modulation\|$\text{Double-Sideband Amplitude Modulation}$]]  $\rm (DSB-AM)$  as described in Chapter 2,  characterized by
	*the time-dependent envelope  $a(t)$  according to the signal  $q(t)$,		*the time-dependent envelope  $a(t)$  according to the signal  $q(t)$,
	*equidistant zero crossings of  $s(t)$  according to the carrier  $z(t)$,  and		*equidistant zero crossings of  $s(t)$  according to the carrier  $z(t)$,  and

Hwang at 13:32, 9 January 2023

2023-01-09T13:32:30Z

← Older revision		Revision as of 15:32, 9 January 2023
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	The lower graph corresponds to   [[Modulation_Methods/Double-Sideband_Amplitude_Modulation\|$\text{~~double~~-~~sideband amplitude modulation~~}$]]  $\rm (DSB-AM)$  as described in Chapter 2,  characterised by		The lower graph corresponds to   [[Modulation_Methods/Double-Sideband_Amplitude_Modulation\|$\text{Double-Sideband Amplitude Modulation}$]]  $\rm (DSB-AM)$  as described in Chapter 2,  characterised by
	*the time-dependent envelope  $a(t)$  according to the signal  $q(t)$,		*the time-dependent envelope  $a(t)$  according to the signal  $q(t)$,
	*equidistant zero crossings of  $s(t)$  according to the carrier  $z(t)$,  and		*equidistant zero crossings of  $s(t)$  according to the carrier  $z(t)$,  and

Hwang at 13:31, 9 January 2023

2023-01-09T13:31:43Z

← Older revision		Revision as of 15:31, 9 January 2023
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	*on the right,  the transmitted signal  $s(t)$ ⇒   blue waveforms, compared to the carrier signal  $z(t)$   ⇒   red waveforms,		*on the right,  the transmitted signal  $s(t)$ ⇒   blue waveforms, compared to the carrier signal  $z(t)$   ⇒   red waveforms,
	*on the left,  the equivalent low-pass signal  $s_{\rm TP}(t)$  in the complex plane.		*on the left,  the equivalent low-pass signal  $s_{\rm TP}(t)$  in the complex plane.
	[[File:EN_Mod_T_3_1_S1b_v2.png\|right\|frame\|Physical signal and equivalent low-pass signal for angle and amplitude modulation]]		[[File:EN_Mod_T_3_1_S1b_v2.png\|right\|frame\|Physical signal and equivalent low-pass signal for angle and amplitude modulation '''KORREKTUR: nachlaufend, vorlaufend''']]

Hwang at 13:28, 9 January 2023

2023-01-09T13:28:01Z

← Older revision		Revision as of 15:28, 9 January 2023
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	The upper graph applies in the case of angle modulation   $\rm (WM)$:		The upper graph applies in the case of angle modulation   $\rm (WM)$:
	*The equivalent low-pass signal  $s_{\rm TP}(t) = A_{\rm T} · {\rm e}^{ \hspace{0.05cm}{\rm j}\hspace{0.05cm}· \hspace{0.05cm}ϕ(t)}$  describes an arc   ⇒   constant envelope   $a(t) = A_{\rm T}$.		*The equivalent low-pass signal  $s_{\rm TP}(t) = A_{\rm T} · {\rm e}^{ \hspace{0.05cm}{\rm j}\hspace{0.05cm}· \hspace{0.05cm}ϕ(t)}$  describes an arc   ⇒   constant envelope   $a(t) = A_{\rm T}$.
	*Thus,  the information about the source signal  $q(t)$  is exclusively found in the zero intercepts of  $s(t)$.		*Thus,  the information about the source signal  $q(t)$  is exclusively found in the zero intercepts '''KORREKTUR: crossings''' of  $s(t)$.
	*Should  $ϕ(t) < 0$  hold,  then the zero crossings of  $s(t)$  occur later than those of  $z(t)$.  Otherwise – when  $ϕ(t) > 0$,  the zero crossings of  $s(t)$  come before  $z(t)$.		*Should  $ϕ(t) < 0$  hold,  then the zero crossings of  $s(t)$  occur later than those of  $z(t)$.  Otherwise – when  $ϕ(t) > 0$,  the zero crossings of  $s(t)$  come before  $z(t)$.

← Older revision		Revision as of 15:09, 13 January 2023
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	== # OVERVIEW OF THE THIRD MAIN CHAPTER # ==		== # OVERVIEW OF THE THIRD MAIN CHAPTER # ==
	<br>		<br>
	The third chapter describes  »'''angle modulation'''«   $($short: "$\rm WM)$"  from German   "Winkelmodulation"$)$.		The third chapter describes  »'''angle modulation'''«   $($short:  "$\rm WM$"  from German   "Winkelmodulation"$)$.

	This is a generic term for		This is a generic term for
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	In detail,  the chapter covers:		In detail,  the chapter covers:
	#~~the~~  »similarities and differences«  between phase and frequency modulation,		#The  »similarities and differences«  between phase and frequency modulation,
	#the  »realization of the associated demodulators«,		#the  »realization of the associated demodulators«,
	#the  »signal characteristics and spectral functions«  of angle-modulated signals and the influence of band limiting,		#the  »signal characteristics and spectral functions«  of angle-modulated signals and the influence of band limiting,
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	*The locus curve is an   "arc"  with radius  $A_{\rm T}$.  It follows that the envelope of an angle-modulated signal is always constant:		*The locus curve is an   "arc"  with radius  $A_{\rm T}$.  It follows that the envelope of an angle-modulated signal is always constant:
	:$$a(t) = \|s_{\rm TP}(t)\|= A_{\rm T}= {\rm const.}$$		:$$a(t) = \|s_{\rm TP}(t)\|= A_{\rm T}= {\rm const.}$$
	*The equivalent low-pass signal in angle modulation is always complex and determined by a time-dependent   "phase function"  $ϕ(t)$  (in radians),  which determines the zero ~~intercepts '''KORREKTUR:~~ crossings~~'''~~ of  $s(t)$:		*The equivalent low-pass signal in angle modulation is always complex and determined by a time-dependent   "phase function"  $ϕ(t)$  (in radians),  which determines the zero crossings of  $s(t)$:
	:$$s_{\rm TP}(t)= A_{\rm T} \cdot {\rm e}^{\hspace{0.05cm}{\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}\phi(t)}\hspace{0.05cm}.$$		:$$s_{\rm TP}(t)= A_{\rm T} \cdot {\rm e}^{\hspace{0.05cm}{\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}\phi(t)}\hspace{0.05cm}.$$
	*For a symmetric source signal,   $ϕ(t)$  can take on all values between  $±ϕ_{\rm max}$,  where  $ϕ_{\rm max}$  indicates the   »'''phase deviation'''«.  The larger the phase deviation,  the more intense the modulation.		*For a symmetric source signal,   $ϕ(t)$  can take on all values between  $±ϕ_{\rm max}$,  where  $ϕ_{\rm max}$  indicates the   »'''phase deviation'''«.  The larger the phase deviation,  the more intense the modulation.
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	The upper graph applies in the case of angle modulation   $\rm (WM)$:		The upper graph applies in the case of angle modulation   $\rm (WM)$:
	*The equivalent low-pass signal  $s_{\rm TP}(t) = A_{\rm T} · {\rm e}^{ \hspace{0.05cm}{\rm j}\hspace{0.05cm}· \hspace{0.05cm}ϕ(t)}$  describes an arc   ⇒   constant envelope   $a(t) = A_{\rm T}$.		*The equivalent low-pass signal  $s_{\rm TP}(t) = A_{\rm T} · {\rm e}^{ \hspace{0.05cm}{\rm j}\hspace{0.05cm}· \hspace{0.05cm}ϕ(t)}$  describes an arc   ⇒   constant envelope   $a(t) = A_{\rm T}$.
	*Thus,  the information about the source signal  $q(t)$  is exclusively found in the zero ~~intercepts '''KORREKTUR:~~ crossings~~'''~~ of  $s(t)$.		*Thus,  the information about the source signal  $q(t)$  is exclusively found in the zero crossings of  $s(t)$.
	*Should  $ϕ(t) < 0$  hold,  then the zero crossings of  $s(t)$  occur later than those of  $z(t)$.  Otherwise – when  $ϕ(t) > 0$,  the zero crossings of  $s(t)$  come before  $z(t)$.		*Should  $ϕ(t) < 0$  hold,  then the zero crossings of  $s(t)$  occur later than those of  $z(t)$.  Otherwise – when  $ϕ(t) > 0$,  the zero crossings of  $s(t)$  come before  $z(t)$.

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	{{GraueBox\|TEXT=		{{GraueBox\|TEXT=
	[[File:EN_Mod_T_3_1_S3c.png \|right\|frame\| Example of an equivalent low-pass–signal in phase modulation ~~'''Korrektur'''~~]]		[[File:EN_Mod_T_3_1_S3c.png \|right\|frame\| Example of an equivalent low-pass–signal in phase modulation]]
	$\rm Example\ 3\text{:}$  The Bessel functions yield the following values for the modulation index $η = 1$:		$\rm Example\ 3\text{:}$  The Bessel functions yield the following values for the modulation index $η = 1$:
	:$${\rm J}_0 = 0.765,\hspace{0.3cm}{\rm J}_1 = - {\rm J}_{ - 1} = 0.440, \hspace{0.3cm}{\rm J}_2 = {\rm J}_{ - 2} = 0.115,\hspace{0.3cm}{\rm J}_3 = - {\rm J}_{ - 3} = 0.020\hspace{0.05cm}.$$		:$${\rm J}_0 = 0.765,\hspace{0.3cm}{\rm J}_1 = - {\rm J}_{ - 1} = 0.440, \hspace{0.3cm}{\rm J}_2 = {\rm J}_{ - 2} = 0.115,\hspace{0.3cm}{\rm J}_3 = - {\rm J}_{ - 3} = 0.020\hspace{0.05cm}.$$