Modulation Methods/Double-Sideband Amplitude Modulation - Revision history

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	Those familiar with the  [[Signal_Representation/Fourier_Transform_Laws\|$\text{laws of spectral transformation}$]]  and in particular with the  [[Signal_Representation/The_Convolution_Theorem_and_Operation\|$\text{Convolution Theorem}$]]  can immediately give a solution for the spectrum  $S(f)$  of the modulator output signal:		Those familiar with the  [[Signal_Representation/Fourier_Transform_Laws\|$\text{laws of spectral transformation}$]]  and in particular with the  [[Signal_Representation/The_Convolution_Theorem_and_Operation\|$\text{Convolution Theorem}$]]  can immediately give a solution for the spectrum  $S(f)$  of the modulator output signal:
	:$$S(f)= Z(f) \star Q(f) = 1/2 \cdot \delta (f + f_{\rm T})\star Q(f)+1/2 \cdot \delta (f - f_{\rm T})\star Q(f) = 1/2 \cdot Q (f + f_{\rm T})+ 1/2 \cdot Q(f - f_{\rm T})		:$$S(f)= Z(f) \star Q(f) = 1/2 \cdot \delta (f + f_{\rm T})\star Q(f)+1/2 \cdot \delta (f - f_{\rm T})\star Q(f) = 1/2 \cdot Q (f + f_{\rm T})+ 1/2 \cdot Q(f - f_{\rm T})\hspace{0.05cm}.$$
	\hspace{0.05cm}.$$

	{{BlaueBox\|TEXT=		{{BlaueBox\|TEXT=
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	*The channel bandwidth  $B_{\rm K}$  must be at least twice the signal bandwidth  $B_{\rm NF}$,  which gives the name  <br>»'''Double-Sideband Amplitude Modulation'''«  $\text{(DSB–AM)}$.		*The channel bandwidth  $B_{\rm K}$  must be at least twice the signal bandwidth  $B_{\rm NF}$,  which gives the name  <br>»'''Double-Sideband Amplitude Modulation'''«  $\text{(DSB–AM)}$.

	*It should be noted that $B_{\rm NF}$ and $B_{\rm K}$  are absolute and  [[Signal_Representation/~~Fourier_Transform_Theorems~~#Reciprocity_Theorem_of_time_duration_and_bandwidth\|$\text{non-equivalent bandwidths}$]].  The latter are defined over rectangles of equal area and are denoted in our tutorial by  $Δf_q$  and  $Δf_{\rm K}$,  resp.		*It should be noted that $B_{\rm NF}$ and $B_{\rm K}$  are absolute and  [[Signal_Representation/The_Fourier_Transform_Theorems#Reciprocity_Theorem_of_time_duration_and_bandwidth\|$\text{non-equivalent bandwidths}$]].  The latter are defined over rectangles of equal area and are denoted in our tutorial by  $Δf_q$  and  $Δf_{\rm K}$,  resp.

	*The spectral function  $S(f)$  does not include any Dirac-lines at the carrier frequency  $(\pm f_{\rm T})$.  Therefore, this method is also referred to as  "DSB-AM '''without carrier'''".		*The spectral function  $S(f)$  does not include any Dirac-lines at the carrier frequency  $(\pm f_{\rm T})$.  Therefore, this method is also referred to as  "DSB-AM '''without carrier'''".
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	:$$z_{\rm R}(t) = \frac{4}{\pi} \cdot \cos(\omega_{\rm T}\cdot t)-\frac{4}{3\pi} \cdot \cos(3\omega_{\rm T}\cdot t) +\frac{4}{5\pi} \cdot \cos(5\omega_{\rm T}\cdot t)- \text{ ...}$$		:$$z_{\rm R}(t) = \frac{4}{\pi} \cdot \cos(\omega_{\rm T}\cdot t)-\frac{4}{3\pi} \cdot \cos(3\omega_{\rm T}\cdot t) +\frac{4}{5\pi} \cdot \cos(5\omega_{\rm T}\cdot t)- \text{ ...}$$
	*The associated spectral function consists of Dirac delta lines at  $±f_{\rm T}, ±3f_{\rm T}, ±5f_{\rm T}$, etc.  Convolution with  $Q(f)$  leads to the spectrum <br>(the subscript stands for "ring modulator"):		*The associated spectral function consists of Dirac delta lines at  $±f_{\rm T}, ±3f_{\rm T}, ±5f_{\rm T}$, etc.  Convolution with  $Q(f)$  leads to the spectrum <br>(the subscript stands for "ring modulator"):
	:$$S_{\rm RM}(f) = \frac{2}{\pi} \cdot Q (f \pm f_{\rm T})-\frac{2}{3\pi} \cdot Q (f \pm 3f_{\rm		:$$S_{\rm RM}(f) = \frac{2}{\pi} \cdot Q (f \pm f_{\rm T})-\frac{2}{3\pi} \cdot Q (f \pm 3f_{\rm T})+\frac{2}{5\pi} \cdot Q (f \pm 5f_{\rm T}) -\text{ ...} \hspace{0.05cm}$$
	T})+\frac{2}{5\pi} \cdot Q (f \pm 5f_{\rm T}) -\text{ ...} \hspace{0.05cm}$$
	*From this, it can be seen that by appropriately band-limiting $($e.g. to  $±2f_{\rm T})$ and attenuating with  $π/4 ≈ 0.785$  the familiar DSB-AM spectrum can be obtained:		*From this, it can be seen that by appropriately band-limiting $($e.g. to  $±2f_{\rm T})$ and attenuating with  $π/4 ≈ 0.785$  the familiar DSB-AM spectrum can be obtained:
	:$$S(f) = {1}/{2} \cdot Q (f \pm f_{\rm T})\hspace{0.05cm}.$$		:$$S(f) = {1}/{2} \cdot Q (f \pm f_{\rm T})\hspace{0.05cm}.$$
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	*Therefore,  the equation for the modulated signal is:		*Therefore,  the equation for the modulated signal is:
	:$$s(t) = q(t) \cdot z(t) = A_{\rm N} \cdot \cos(\omega_{\rm N} t + \phi_{\rm		:$$s(t) = q(t) \cdot z(t) = A_{\rm N} \cdot \cos(\omega_{\rm N} t + \phi_{\rm N})\cdot \cos(\omega_{\rm T} t + \phi_{\rm T})\hspace{0.05cm}.$$
	N})\cdot \cos(\omega_{\rm T} t + \phi_{\rm T})\hspace{0.05cm}.$$

	This equation can be transformed using the trigonometric addition theorem:		This equation can be transformed using the trigonometric addition theorem:
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	*Using a Fourier transform we arrive at the spectral function:		*Using a Fourier transform we arrive at the spectral function:
	:$$S(f) = {A_{\rm N}}/{4} \cdot \big[\delta ( f - f_{\rm T} - f_{\rm		:$$S(f) = {A_{\rm N}}/{4} \cdot \big[\delta ( f - f_{\rm T} - f_{\rm N})+\delta ( f + f_{\rm T} + f_{\rm N})\big)] + {A_{\rm N}}/{4} \cdot \big[ \delta ( f - f_{\rm T}+ f_{\rm N})+\delta ( f+ f_{\rm T} - f_{\rm N} ) \big]\hspace{0.05cm}.$$
	N})+\delta ( f + f_{\rm T} + f_{\rm N})\big)] + {A_{\rm N}}/{4} \cdot \big[ \delta ( f - f_{\rm T}+ f_{\rm N})+\delta ( f+ f_{\rm T} - f_{\rm N} ) \big]\hspace{0.05cm}.$$

	This result,  which would also have been arrived at via convolution,  states:		This result,  which would also have been arrived at via convolution,  states:

Maintenance script: Add German interlanguage link

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	{{Display}}		{{Display}}
			[[de:Modulationsverfahren/Zweiseitenband-Amplitudenmodulation]]

Hwang at 12:42, 18 January 2023

2023-01-18T12:42:38Z

Guenter at 17:35, 12 January 2023

2023-01-12T17:35:48Z

Show changes

Hwang at 19:49, 2 January 2023

2023-01-02T19:49:27Z

← Older revision		Revision as of 21:49, 2 January 2023
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	<br>		<br>
	[[File:Mod_T_2_1_S6_version2.png\|right\|frame\|Spectrum of the analytical signal in two different viewpoints]]		[[File:Mod_T_2_1_S6_version2.png\|right\|frame\|Spectrum of the analytical signal in two different viewpoints]]
	In the further course of this chapter,  for the sake of simplifying the graphs,  the spectrum  $S_+(f)$  of the  [[Modulation_Methods/General_Model_of_Modulation#~~Beschreibung_des_physikalischen_Signals_mit_Hilfe_des_analytischen_Signals~~\|$\text{analytical signal}$]]  is usually given instead of the actual, physical spectrum  $S(f)$.		In the further course of this chapter,  for the sake of simplifying the graphs,  the spectrum  $S_+(f)$  of the  [[Modulation_Methods/General_Model_of_Modulation#Describing_the_physical_signal_using_the_analytic_signal\|$\text{analytical signal}$]]  is usually given instead of the actual, physical spectrum  $S(f)$.

	As an example,  let us consider  "DSB-AM with carrier"  and the following signals:		As an example,  let us consider  "DSB-AM with carrier"  and the following signals:

Hwang at 19:42, 2 January 2023

2023-01-02T19:42:50Z

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	*In both bracket expressions, the first Dirac delta function indicates the one for positive frequencies.		*In both bracket expressions, the first Dirac delta function indicates the one for positive frequencies.
	*The weights of all Dirac delta functions are equal and each is  $A_{\rm N}/4$.		*The weights of all Dirac delta functions are equal and each is  $A_{\rm N}/4$.
	*The sum of these weights   - that is, the integral over  $S(f)$ –  is equal to the signal value  $s(t = 0) = A_{\rm N}$.		*The sum of these weights   - that is, the integral over  $S(f)$ -   is equal to the signal value  $s(t = 0) = A_{\rm N}$.
	*The Dirac lines remain for  $ϕ_{\rm T} ≠ 0$  and/or  $ϕ_{\rm N} ≠ 0$  at the same frequencies.  However, complex rotation factors must then be added to the weights  $A_{\rm N}/4$ .		*The Dirac lines remain for  $ϕ_{\rm T} ≠ 0$  and/or  $ϕ_{\rm N} ≠ 0$  at the same frequencies.  However, complex rotation factors must then be added to the weights  $A_{\rm N}/4$ .

Hwang at 17:24, 2 January 2023

2023-01-02T17:24:13Z

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	*Due to the operation of this switch,  the harmonic oscillation  $z(t)$  can also be replaced by a periodic  (rectangular)  square wave signal with identical period duration:		*Due to the operation of this switch,  the harmonic oscillation  $z(t)$  can also be replaced by a periodic  (rectangular)  square wave signal with identical period duration:
	:$$z_{\rm R}(t) = \left\{ \begin{array}{c} +1 \\ -1 \\ \end{array} \right.\quad \begin{array}{{10}c} {\rm{for}} \\ {\rm{for}} \\ \end{array}\begin{array}{{20}c} {z(t) >0,} \\ {z(t) <0.} \\ \end{array}$$		:$$z_{\rm R}(t) = \left\{ \begin{array}{c} +1 \\ -1 \\ \end{array} \right.\quad \begin{array}{{10}c} {\rm{for}} \\ {\rm{for}} \\ \end{array}\begin{array}{{20}c} {z(t) >0,} \\ {z(t) <0.} \\ \end{array}$$
	*The modulated signal  $s(t)$  is then obtained as the product of the source signal  $q(t)$  and this square wave signal  $z_{\rm R}(t)$,  whereas in ideal DSB-AM one multiplies $q(t)$  by a cosine signal.  The carrier  $z(t)$  is not itself included in the signal $s(t)$.  Since this is supplied via the center taps of the transformers, the induced voltages cancel out   ⇒   »'''DSB-AM without carrier'''«.		*The modulated signal  $s(t)$  is then obtained as the product of the source signal  $q(t)$  and this square wave '''KORREKTUR: rectangular''' signal  $z_{\rm R}(t)$,  whereas in ideal DSB-AM one multiplies $q(t)$  by a cosine signal.  The carrier  $z(t)$  is not itself included in the signal $s(t)$.  Since this is supplied via the center taps of the transformers, the induced voltages cancel out   ⇒   »'''DSB-AM without carrier'''«.

Hwang at 17:17, 2 January 2023

2023-01-02T17:17:12Z

← Older revision		Revision as of 19:17, 2 January 2023
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	This result is still true if the restrictions made in the last section are removed:		This result is still true if the restrictions made in the last section are removed:
	#real-valued spectrum  $Q(f)$,		#real-valued spectrum  $Q(f)$,
	#carrier phase  $ϕ_{\rm T} = 0$		#carrier phase  $ϕ_{\rm T} = 0$.

	In general,  this results in a complex-valued spectrum  $S(f)$.		In general,  this results in a complex-valued spectrum  $S(f)$.

Hwang at 17:15, 2 January 2023

2023-01-02T17:15:59Z

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	*The factor  $1/2$  results from the carrier amplitude  $A_{\rm T} = 1$.  Thus,  $s(t = 0) = q(t = 0)$  and the integrals over their spectral functions  $S(f)$  and  $Q(f)$  must also be equal.		*The factor  $1/2$  results from the carrier amplitude  $A_{\rm T} = 1$.  Thus,  $s(t = 0) = q(t = 0)$  and the integrals over their spectral functions  $S(f)$  and  $Q(f)$  must also be equal.
	*The channel bandwidth  $B_{\rm K}$  must be at least twice the signal bandwidth  $B_{\rm NF}$,  which gives the name  <br>»'''Double-Sideband Amplitude Modulation'''«  $\text{(DSB–AM)}$.		*The channel bandwidth  $B_{\rm K}$  must be at least twice the signal bandwidth  $B_{\rm NF}$,  which gives the name  <br>»'''Double-Sideband Amplitude Modulation'''«  $\text{(DSB–AM)}$.
	*It should be noted that $B_{\rm NF}$ and $B_{\rm K}$  are absolute and  [[Signal_Representation/~~Fourier_Transform_Laws~~#~~Reziprozit.C3.A4tsgesetz_von_Zeitdauer_und_Bandbreite~~\|$\text{non-equivalent bandwidths}$]].  The latter are defined over rectangles of equal area and are denoted in our tutorial by  $Δf_q$  and  $Δf_{\rm K}$,  resp.		*It should be noted that $B_{\rm NF}$ and $B_{\rm K}$  are absolute and  [[Signal_Representation/Fourier_Transform_Theorems#Reciprocity_Theorem_of_time_duration_and_bandwidth\|$\text{non-equivalent bandwidths}$]].  The latter are defined over rectangles of equal area and are denoted in our tutorial by  $Δf_q$  and  $Δf_{\rm K}$,  resp.
	*The spectral function  $S(f)$  does not include any Dirac-lines at the carrier frequency  $(\pm f_{\rm T})$.  Therefore, this method is also referred to as  "DSB-AM '''without carrier'''".		*The spectral function  $S(f)$  does not include any Dirac-lines at the carrier frequency  $(\pm f_{\rm T})$.  Therefore, this method is also referred to as  "DSB-AM '''without carrier'''".
	*The frequency components		*The frequency components

Hwang at 17:09, 2 January 2023

2023-01-02T17:09:42Z

← Older revision		Revision as of 14:42, 18 January 2023
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	*The Fourier series representation of the periodic rectangular signal $z_{\rm R}(t)$  is:		*The Fourier series representation of the periodic rectangular signal $z_{\rm R}(t)$  is:
	:$$z_{\rm R}(t) = \frac{4}{\pi} \cdot \cos(\omega_{\rm T}\cdot t)-\frac{4}{3\pi} \cdot \cos(3\omega_{\rm T}\cdot t) +\frac{4}{5\pi} \cdot \cos(5\omega_{\rm T}\cdot t)- \text{ ...}$$		:$$z_{\rm R}(t) = \frac{4}{\pi} \cdot \cos(\omega_{\rm T}\cdot t)-\frac{4}{3\pi} \cdot \cos(3\omega_{\rm T}\cdot t) +\frac{4}{5\pi} \cdot \cos(5\omega_{\rm T}\cdot t)- \text{ ...}$$
	*The associated spectral function consists of Dirac lines at  $±f_{\rm T}, ±3f_{\rm T}, ±5f_{\rm T}$, etc.  Convolution with  $Q(f)$  leads to the spectrum <br>(the subscript stands for "ring modulator"):		*The associated spectral function consists of Dirac delta lines at  $±f_{\rm T}, ±3f_{\rm T}, ±5f_{\rm T}$, etc.  Convolution with  $Q(f)$  leads to the spectrum <br>(the subscript stands for "ring modulator"):
	:$$S_{\rm RM}(f) = \frac{2}{\pi} \cdot Q (f \pm f_{\rm T})-\frac{2}{3\pi} \cdot Q (f \pm 3f_{\rm		:$$S_{\rm RM}(f) = \frac{2}{\pi} \cdot Q (f \pm f_{\rm T})-\frac{2}{3\pi} \cdot Q (f \pm 3f_{\rm
	T})+\frac{2}{5\pi} \cdot Q (f \pm 5f_{\rm T}) -\text{ ...} \hspace{0.05cm}$$		T})+\frac{2}{5\pi} \cdot Q (f \pm 5f_{\rm T}) -\text{ ...} \hspace{0.05cm}$$
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	This result,  which would also have been arrived at via convolution,  states:		This result,  which would also have been arrived at via convolution,  states:
	#The spectrum consists of four Dirac lines at frequencies  $±(f_{\rm T} + f_{\rm N})$  and  $±(f_{\rm T} - f_{\rm N})$.		#The spectrum consists of four Dirac delta lines at frequencies  $±(f_{\rm T} + f_{\rm N})$  and  $±(f_{\rm T} - f_{\rm N})$.
	#In both bracket expressions, the first Dirac delta function indicates the one for positive frequencies.		#In both bracket expressions, the first Dirac delta function indicates the one for positive frequencies.
	#The weights of all Dirac delta functions are equal and each is  $A_{\rm N}/4$.		#The weights of all Dirac delta functions are equal and each is  $A_{\rm N}/4$.
	#The sum of these weights   - that is, the integral over  $S(f)$ -   is equal to the signal value  $s(t = 0) = A_{\rm N}$.		#The sum of these weights   - that is, the integral over  $S(f)$ -   is equal to the signal value  $s(t = 0) = A_{\rm N}$.
	#The Dirac lines remain for  $ϕ_{\rm T} ≠ 0$  and/or  $ϕ_{\rm N} ≠ 0$  at the same frequencies.  However, complex rotation factors must then be added to the weights  $A_{\rm N}/4$ .		#The Dirac delta lines remain for  $ϕ_{\rm T} ≠ 0$  and/or  $ϕ_{\rm N} ≠ 0$  at the same frequencies.  However, complex rotation factors must then be added to the weights  $A_{\rm N}/4$ .


	{{GraueBox\|TEXT=		{{GraueBox\|TEXT=
	$\text{Example 3:}$  The following diagram shows the spectral functions  $S(f)$  for different values of  $ϕ_{\rm T}$  and  $ϕ_{\rm N}$,  respectively.  The other parameters are assumed to be  $f_{\rm T} = 50\text{ kHz}$,  $f_{\rm N} = 10\text{ kHz}$  and  $A_{\rm N} = 4\text{ V}$.  Thus,  the magnitudes of all Dirac lines are $A_{\rm N}/4 = 1\text{ V}$.		$\text{Example 3:}$  The following diagram shows the spectral functions  $S(f)$  for different values of  $ϕ_{\rm T}$  and  $ϕ_{\rm N}$,  respectively.  The other parameters are assumed to be  $f_{\rm T} = 50\text{ kHz}$,  $f_{\rm N} = 10\text{ kHz}$  and  $A_{\rm N} = 4\text{ V}$.  Thus,  the magnitudes of all Dirac delta lines are $A_{\rm N}/4 = 1\text{ V}$.

	[[File:EN_Mod_T_2_1_S4.png\|right\|frame\|Typical spectra for DSB-AM]]		[[File:EN_Mod_T_2_1_S4.png\|right\|frame\|Typical spectra for DSB-AM]]

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	:$$Z(f) = {1}/{2} \cdot \delta (f + f_{\rm T})+{1}/{2} \cdot \delta (f - f_{\rm T})\hspace{0.05cm}.$$		:$$Z(f) = {1}/{2} \cdot \delta (f + f_{\rm T})+{1}/{2} \cdot \delta (f - f_{\rm T})\hspace{0.05cm}.$$

	Those familiar with the  [[Signal_Representation/Fourier_Transform_Laws\|laws of spectral transformation]]  and in particular with the  [[Signal_Representation/The_Convolution_Theorem_and_Operation\|Convolution Theorem]]  can immediately give a solution for the spectrum  $S(f)$  of the modulator output signal:		Those familiar with the  [[Signal_Representation/Fourier_Transform_Laws\|$\text{laws of spectral transformation}$]]  and in particular with the  [[Signal_Representation/The_Convolution_Theorem_and_Operation\|$\text{Convolution Theorem}$]]  can immediately give a solution for the spectrum  $S(f)$  of the modulator output signal:
	:$$S(f)= Z(f) \star Q(f) = 1/2 \cdot \delta (f + f_{\rm T})\star Q(f)+1/2 \cdot \delta (f - f_{\rm T})\star Q(f) = 1/2 \cdot Q (f + f_{\rm T})+ 1/2 \cdot Q(f - f_{\rm T})		:$$S(f)= Z(f) \star Q(f) = 1/2 \cdot \delta (f + f_{\rm T})\star Q(f)+1/2 \cdot \delta (f - f_{\rm T})\star Q(f) = 1/2 \cdot Q (f + f_{\rm T})+ 1/2 \cdot Q(f - f_{\rm T})
	\hspace{0.05cm}.$$		\hspace{0.05cm}.$$
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	{{BlaueBox\|TEXT=		{{BlaueBox\|TEXT=
	$\text{Please note:}$  This equation takes into account		$\text{Please note:}$  This equation takes into account
	*that the convolution of a shifted Dirac delta function  $δ(x - x_0)$  with an arbitrary function $f(x)$  yields the   '''shifted function'''  $f(x - x_0)$.}}		*that the convolution of a shifted Dirac delta function  $δ(x - x_0)$  with an arbitrary function $f(x)$  yields the   »'''shifted function'''«  $f(x - x_0)$.}}


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	*Due to the system-theoretic approach with positive and negative frequencies, $S(f)$  is composed of two parts around  $\pm f_{\rm T}$,  each of which have the same shape as  $Q(f)$.		*Due to the system-theoretic approach with positive and negative frequencies, $S(f)$  is composed of two parts around  $\pm f_{\rm T}$,  each of which have the same shape as  $Q(f)$.
	*The factor  $1/2$  results from the carrier amplitude  $A_{\rm T} = 1$.  Thus,  $s(t = 0) = q(t = 0)$  and the integrals over their spectral functions  $S(f)$  and  $Q(f)$  must also be equal.		*The factor  $1/2$  results from the carrier amplitude  $A_{\rm T} = 1$.  Thus,  $s(t = 0) = q(t = 0)$  and the integrals over their spectral functions  $S(f)$  and  $Q(f)$  must also be equal.
	*The channel bandwidth  $B_{\rm K}$  must be at least twice the signal bandwidth  $B_{\rm NF}$,  which gives the name  <br>'''Double-Sideband Amplitude Modulation'''  $\text{(DSB–AM)}$.		*The channel bandwidth  $B_{\rm K}$  must be at least twice the signal bandwidth  $B_{\rm NF}$,  which gives the name  <br>»'''Double-Sideband Amplitude Modulation'''«  $\text{(DSB–AM)}$.
	*It should be noted that $B_{\rm NF}$ and $B_{\rm K}$  are absolute and  [[Signal_Representation/Fourier_Transform_Laws#Reziprozit.C3.A4tsgesetz_von_Zeitdauer_und_Bandbreite\|non-equivalent bandwidths]].  The latter are defined over rectangles of equal area and are denoted in our tutorial by  $Δf_q$  and  $Δf_{\rm K}$,  resp.		*It should be noted that $B_{\rm NF}$ and $B_{\rm K}$  are absolute and  [[Signal_Representation/Fourier_Transform_Laws#Reziprozit.C3.A4tsgesetz_von_Zeitdauer_und_Bandbreite\|$\text{non-equivalent bandwidths}$]].  The latter are defined over rectangles of equal area and are denoted in our tutorial by  $Δf_q$  and  $Δf_{\rm K}$,  resp.
	*The spectral function  $S(f)$  does not include any Dirac-lines at the carrier frequency  $(\pm f_{\rm T})$.  Therefore, this method is also referred to as  "DSB-AM '''without carrier'''".		*The spectral function  $S(f)$  does not include any Dirac-lines at the carrier frequency  $(\pm f_{\rm T})$.  Therefore, this method is also referred to as  "DSB-AM '''without carrier'''".
	*The frequency components		*The frequency components
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	==Ring modulator==		==Ring modulator==
	<br>		<br>
	One possibility to realize  "double-sideband amplitude modulation with carrier suppression"  is offered by a so-called  '''ring modulator''',  also known as  "double push-pull diode modulator".  Below you can see the circuit on the left and a simple functional diagram on the right.		One possibility to realize  "double-sideband amplitude modulation with carrier suppression"  is offered by a so-called  »'''ring modulator'''«,  also known as  "double push-pull diode modulator".  Below you can see the circuit on the left and a simple functional diagram on the right.

	Without claiming to be complete, the principle can be stated as follows:		Without claiming to be complete, the principle can be stated as follows:
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	*Due to the operation of this switch,  the harmonic oscillation  $z(t)$  can also be replaced by a periodic  (rectangular)  square wave signal with identical period duration:		*Due to the operation of this switch,  the harmonic oscillation  $z(t)$  can also be replaced by a periodic  (rectangular)  square wave signal with identical period duration:
	:$$z_{\rm R}(t) = \left\{ \begin{array}{c} +1 \\ -1 \\ \end{array} \right.\quad \begin{array}{{10}c} {\rm{for}} \\ {\rm{for}} \\ \end{array}\begin{array}{{20}c} {z(t) >0,} \\ {z(t) <0.} \\ \end{array}$$		:$$z_{\rm R}(t) = \left\{ \begin{array}{c} +1 \\ -1 \\ \end{array} \right.\quad \begin{array}{{10}c} {\rm{for}} \\ {\rm{for}} \\ \end{array}\begin{array}{{20}c} {z(t) >0,} \\ {z(t) <0.} \\ \end{array}$$
	*The modulated signal  $s(t)$  is then obtained as the product of the source signal  $q(t)$  and this square wave signal  $z_{\rm R}(t)$,  whereas in ideal DSB-AM one multiplies $q(t)$  by a cosine signal.  The carrier  $z(t)$  is not itself included in the signal $s(t)$.  Since this is supplied via the center taps of the transformers, the induced voltages cancel out   ⇒   '''DSB-AM without carrier'''.		*The modulated signal  $s(t)$  is then obtained as the product of the source signal  $q(t)$  and this square wave signal  $z_{\rm R}(t)$,  whereas in ideal DSB-AM one multiplies $q(t)$  by a cosine signal.  The carrier  $z(t)$  is not itself included in the signal $s(t)$.  Since this is supplied via the center taps of the transformers, the induced voltages cancel out   ⇒   »'''DSB-AM without carrier'''«.


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	A comparison of these signal waveforms shows:		A comparison of these signal waveforms shows:
	*By adding the DC component  $A_{\rm T}$  the signal  $q(t)$  can now be seen in the envelope of  $s(t)$.		*By adding the DC component  $A_{\rm T}$  the signal  $q(t)$  can now be seen in the envelope of  $s(t)$.
	*Thus,  [[Modulation_Methods/Envelope_Demodulation\|envelope demodulation]]  can be applied,  which is easier and cheaper to implement than coherent  [[Modulation_Methods/Synchronous_Demodulation\|synchronous demodulation]].		*Thus,  [[Modulation_Methods/Envelope_Demodulation\|$\text{envelope demodulation}$]]  can be applied,  which is easier and cheaper to implement than coherent  [[Modulation_Methods/Synchronous_Demodulation\|$\text{synchronous demodulation}$]].
	*However,  a prerequisite for the application of the envelope demodulator is a modulation depth  $m <1$.		*However,  a prerequisite for the application of the envelope demodulator is a modulation depth  $m <1$.
	*This parameter is defined as follows:		*This parameter is defined as follows:
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	<br>		<br>
	[[File:Mod_T_2_1_S6_version2.png\|right\|frame\|Spectrum of the analytical signal in two different viewpoints]]		[[File:Mod_T_2_1_S6_version2.png\|right\|frame\|Spectrum of the analytical signal in two different viewpoints]]
	In the further course of this chapter,  for the sake of simplifying the graphs,  the spectrum  $S_+(f)$  of the  [[Modulation_Methods/General_Model_of_Modulation#Beschreibung_des_physikalischen_Signals_mit_Hilfe_des_analytischen_Signals\|analytical signal]]  is usually given instead of the actual, physical spectrum  $S(f)$.		In the further course of this chapter,  for the sake of simplifying the graphs,  the spectrum  $S_+(f)$  of the  [[Modulation_Methods/General_Model_of_Modulation#Beschreibung_des_physikalischen_Signals_mit_Hilfe_des_analytischen_Signals\|$\text{analytical signal}$]]  is usually given instead of the actual, physical spectrum  $S(f)$.

	As an example,  let us consider  "DSB-AM with carrier"  and the following signals:		As an example,  let us consider  "DSB-AM with carrier"  and the following signals:
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	==Amplitude modulation with a quadratic characteristic curve==		==Amplitude modulation with a quadratic characteristic curve==
	<br>		<br>
	Nonlinearities are usually undesirable and troublesome in Communications Engineering.  As explained in the chapter  [[Linear_and_Time_Invariant_Systems/Nonlinear_Distortions\|Nonlinear Distortions]]   of the book  "Linear and Time Invariant Systems”,  they lead to the facts that:		Nonlinearities are usually undesirable and troublesome in Communications Engineering.  As explained in the chapter  [[Linear_and_Time_Invariant_Systems/Nonlinear_Distortions\|"Nonlinear Distortions"]]   of the book  "Linear and Time Invariant Systems”,  they lead to the facts that:

	*the superposition principle is no longer applicable,		*the superposition principle is no longer applicable,