Modulation Methods/Further AM Variants - Revision history

Guenter at 09:52, 13 January 2023

2023-01-13T09:52:43Z

Hwang at 22:08, 2 January 2023

2023-01-02T22:08:21Z

Hwang at 22:02, 2 January 2023

2023-01-02T22:02:08Z

Reed at 14:21, 9 April 2022

2022-04-09T14:21:53Z

Reed at 14:21, 9 April 2022

2022-04-09T14:21:31Z

Reed at 14:21, 9 April 2022

2022-04-09T14:21:18Z

Guenter at 13:34, 18 January 2022

2022-01-18T13:34:19Z

Guenter at 13:28, 18 January 2022

2022-01-18T13:28:33Z

Guenter at 13:08, 18 January 2022

2022-01-18T13:08:03Z

Reed at 19:29, 22 December 2021

2021-12-22T19:29:34Z

@@ Line 9: / Line 9: @@
 When transmitting signals using single-sideband modulation&nbsp; $\rm (SSB–AM)$&nbsp; the following problems occur:
 *To suppress the unwanted sideband,&nbsp; a filter with a very high edge slope must be used.
 *Such steep-edged filters exhibit strong group delay distortions,&nbsp; especially at the limit of the passband.
@@ Line 17: / Line 18: @@
 The present description is based on the textbook &nbsp; [Mäu88]<ref>Mäusl, R.:&nbsp; Analoge Modulationsverfahren.&nbsp;  Heidelberg: Dr. Hüthig, 1988.</ref>.&nbsp;  According to it,&nbsp; the VSB-AM can be characterized as follows:
 *A certain frequency range of the actually suppressed sideband – in the considered example of the LSB – is additionally used with a relatively flat decreasing transfer function.
 *On the receiver side,&nbsp; a  selection curve linearly increasing in frequency with a so-called&nbsp; "Nyquist edge"&nbsp; is used in the transition range from the suppressed sideband to the transmitted sideband.
 *The demodulation performs a convolution of the sidebands around the carrier,&nbsp; so that as a result the message content of a band with the same amplitude for all frequencies is obtained.
@@ Line 26: / Line 29: @@
 [[File:EN_Mod_T_2_5_S1b.png|right|frame|Illustrating the Nyquist edge for PAL–B/G Television]]
 In this schematic representation,&nbsp; one recognizes:
 *The radiated spectrum&nbsp; (only positive frequencies are drawn)&nbsp; ranges from &nbsp;$f_{\rm T} - 1.25 \ \rm MHz$&nbsp; to &nbsp;$f_{\rm T} + 5.75 \ \rm MHz$.&nbsp; The lower vestigial sideband including Nyquist edge is &nbsp;$\approx 1.25 \ \rm MHz$&nbsp; wide.
 *The green dashed line shows the receiver passband.&nbsp; The image carrier&nbsp; '''(B)'''&nbsp; &ndash; abbreviation&nbsp; "B"&nbsp; from German "Bild" &nbsp; &rArr; &nbsp;  "image") &ndash; &nbsp; at carrier frequency  &nbsp;$f_{\rm T}$&nbsp; is centered on the Nyquist edge.
 *The luminance signal &nbsp; '''(L)'''&nbsp; goes up to about &nbsp;$5 \ \rm MHz$.&nbsp;  It contains the information for the image brightness and for the color&nbsp; "green".
 *The chrominance signal &nbsp; '''(C)'''&nbsp; is embedded in the upper part.  Two orthogonal carriers are [[Modulation_Methods/Further_AM_Variants#Quadrature_Amplitude_Modulation_.28QAM.29|$\text{QAM-modulated}$]]&nbsp; at &nbsp;$4.43 \ \rm MHz$&nbsp; for&nbsp; "red"&nbsp; and&nbsp; "blue";&nbsp; the carrier is suppressed.
 *The audio carrier &nbsp; '''(T)'''&nbsp; &ndash; abbreviation&nbsp; "T"&nbsp; from German "Ton" &nbsp; &rArr; &nbsp; "sound/audio") &ndash; &nbsp; is at &nbsp;$f_{\rm T} + 5.5 \ \rm MHz$&nbsp; and is &nbsp;$12  \ \rm  dB$&nbsp; lower than the image carrier.
 *If there is a two-channel sound&nbsp; (stereo)&nbsp; transmission &nbsp; &rArr; &nbsp; a second &nbsp; '''(T)'''&nbsp; carrier at &nbsp;$5.75 \ \rm MHz$.}}
@@ Line 43: / Line 51: @@
 *The transmitted signal is composed of two mutually orthogonal components:
 :$$s(t) = q_1(t)  \cdot \cos (\omega_{\rm T}\cdot t) - q_2(t)  \cdot \sin (\omega_{\rm T}\cdot t)\hspace{0.05cm}.$$
 *For frequency and phase synchronous demodulation,&nbsp; the signal in the upper branch before the low-pass &nbsp;$H_{\rm E1}(f)$&nbsp;  is:
 :$$b_1(t) = q_1(t)  \cdot 2 \cdot \cos^2 (\omega_{\rm T}\cdot t)  - q_2(t)  \cdot 2 \cdot
   \cos (\omega_{\rm T}\cdot t)\cdot \sin (\omega_{\rm T}\cdot t)  $$
 :$$\Rightarrow \hspace{0.3cm} b_1(t) =  q_1(t)\cdot \big[ 1 + \cos (2 \omega_{\rm T}\cdot t) \big] - q_2(t)\cdot \sin (2 \omega_{\rm T}\cdot t) \hspace{0.05cm}.$$
 *Thus,&nbsp; by limiting to frequencies &nbsp;$|f| < f_{\rm T}$,&nbsp; we obtain in the upper and lower branches,&nbsp; resp.:
 :$$v_1(t) = q_1(t),$$
 :$$v_2(t) = q_2(t)\hspace{0.05cm}.$$
 *When there is a phase offset &nbsp;$Δϕ_{\rm T}$&nbsp; between the transmitted and received carrier signals,&nbsp; in addition to attenuation of the intended participant,<br>crosstalk from the second participant occurs,&nbsp; resulting in nonlinear distortion:
 :$$v_1(t) = \alpha_{11} \cdot q_1(t)+ \alpha_{12} \cdot q_2(t) \hspace{0.05cm}, $$
@@ Line 73: / Line 84: @@
 It should be further noted regarding this arrangement:
 *The receiver-side carrier signals can have an arbitrary and also time-dependent phase offset &nbsp;$Δϕ_{\rm T}$&nbsp; with respect to the carrier signals at the transmitter, as long as the phase difference between the two branches remains exactly &nbsp;$90^\circ$&nbsp;.
 *For the signals in the upper and lower branches – after the multiplier and low-pass filtering, respectively – it holds:
 :$$b_1(t) = \cos(\Delta \phi_{\rm T}) \cdot q(t), $$
 :$$b_2(t) = -\sin(\Delta \phi_{\rm T}) \cdot q(t).$$
 *This ensures that the sink signal &nbsp;$v(t)$&nbsp; matches the source signal &nbsp;$q(t)$&nbsp; in magnitude,&nbsp; regardless of the phase offset &nbsp;$Δϕ_{\rm T}$:
 :$$v(t) = \sqrt{ b_1^2(t) + b_2^2(t)} = \sqrt{ q^2(t) } = \vert q(t) \vert \hspace{0.05cm}.$$
 *A prerequisite for operability&nbsp; &ndash;  that is,&nbsp; for the result &nbsp;$v(t) = q(t)$&nbsp; &ndash;&nbsp; is that at all times &nbsp;$q(t) ≥ 0$.&nbsp;  In an analog transmission system,&nbsp; this fact could be enforced using the&nbsp; "DSB-AM with carrier"&nbsp; modulation method,&nbsp; for example.
 *This form of non-coherent demodulation&nbsp; (or modifications of it)&nbsp; is mainly applied in some&nbsp; &raquo;'''digital modulation methods'''&laquo;,&nbsp; which are discussed in detail in the fourth chapter of&nbsp; [[Modulation_Methods|"this book"]].}}

@@ Line 25: / Line 25: @@
 [[File:EN_Mod_T_2_5_S1b.png|right|frame|Illustrating the Nyquist edge for PAL–B/G Television]]
-In this schematic representation,&nbsp; one recognises:
+In this schematic representation,&nbsp; one recognizes:
 *The radiated spectrum&nbsp; (only positive frequencies are drawn)&nbsp; ranges from &nbsp;$f_{\rm T} - 1.25 \ \rm MHz$&nbsp; to &nbsp;$f_{\rm T} + 5.75 \ \rm MHz$.&nbsp; The lower vestigial sideband including Nyquist edge is &nbsp;$\approx 1.25 \ \rm MHz$&nbsp; wide.
 *The green dashed line shows the receiver passband.&nbsp; The image carrier&nbsp; '''(B)'''&nbsp; &ndash; abbreviation&nbsp; "B"&nbsp; from German "Bild" &nbsp; &rArr; &nbsp;  "image") &ndash; &nbsp; at carrier frequency  &nbsp;$f_{\rm T}$&nbsp; is centered on the Nyquist edge.

@@ Line 22: / Line 22: @@
 {{GraueBox|TEXT=
-$\text{Example 1:}$&nbsp; The vestigial sideband method is used for&nbsp; (analog)&nbsp; color television,&nbsp; whose frequency spectrum according to the CCIR standard is shown in the graphic.&nbsp; The frequencies given refer to the &nbsp;[https://en.wikipedia.org/wiki/PAL PAL–B/G Television format]&nbsp; used in Germany.
+$\text{Example 1:}$&nbsp; The vestigial sideband method is used for&nbsp; (analog)&nbsp; color television,&nbsp; whose frequency spectrum according to the CCIR standard is shown in the graphic.&nbsp; The frequencies given refer to the &nbsp;[https://en.wikipedia.org/wiki/PAL $\text{PAL–B/G Television format}$]&nbsp; used in Germany.
 [[File:EN_Mod_T_2_5_S1b.png|right|frame|Illustrating the Nyquist edge for PAL–B/G Television]]
@@ Line 29: / Line 29: @@
 *The green dashed line shows the receiver passband.&nbsp; The image carrier&nbsp; '''(B)'''&nbsp; &ndash; abbreviation&nbsp; "B"&nbsp; from German "Bild" &nbsp; &rArr; &nbsp;  "image") &ndash; &nbsp; at carrier frequency  &nbsp;$f_{\rm T}$&nbsp; is centered on the Nyquist edge.
 *The luminance signal &nbsp; '''(L)'''&nbsp; goes up to about &nbsp;$5 \ \rm MHz$.&nbsp;  It contains the information for the image brightness and for the color&nbsp; "green".
-*The chrominance signal &nbsp; '''(C)'''&nbsp; is embedded in the upper part.  Two orthogonal carriers are [[Modulation_Methods/Further_AM_Variants#Quadrature_Amplitude_Modulation_.28QAM.29|QAM-modulated]]&nbsp; at &nbsp;$4.43 \ \rm MHz$&nbsp; for&nbsp; "red"&nbsp; and&nbsp; "blue";&nbsp; the carrier is suppressed.
+*The chrominance signal &nbsp; '''(C)'''&nbsp; is embedded in the upper part.  Two orthogonal carriers are [[Modulation_Methods/Further_AM_Variants#Quadrature_Amplitude_Modulation_.28QAM.29|$\text{QAM-modulated}$]]&nbsp; at &nbsp;$4.43 \ \rm MHz$&nbsp; for&nbsp; "red"&nbsp; and&nbsp; "blue";&nbsp; the carrier is suppressed.
 *The audio carrier &nbsp; '''(T)'''&nbsp; &ndash; abbreviation&nbsp; "T"&nbsp; from German "Ton" &nbsp; &rArr; &nbsp; "sound/audio") &ndash; &nbsp; is at &nbsp;$f_{\rm T} + 5.5 \ \rm MHz$&nbsp; and is &nbsp;$12  \ \rm  dB$&nbsp; lower than the image carrier.
 *If there is a two-channel sound&nbsp; (stereo)&nbsp; transmission &nbsp; &rArr; &nbsp; a second &nbsp; '''(T)'''&nbsp; carrier at &nbsp;$5.75 \ \rm MHz$.}}
@@ Line 59: / Line 59: @@
 {{BlaueBox|TEXT=
 $\text{Definition:}$&nbsp; Demodulators can be classified in the following way:
-*A demodulator is said to be&nbsp; '''coherent''',&nbsp; if in addition to the required frequency synchronization,&nbsp; it requires accurate information about the phase of the transmit-side carrier signal &nbsp;$z(t)$&nbsp; to reconstruct the source signal.
+*A demodulator is said to be&nbsp; &raquo;'''coherent'''&laquo;,&nbsp; if in addition to the required frequency synchronization,&nbsp; it requires accurate information about the phase of the transmit-side carrier signal &nbsp;$z(t)$&nbsp; to reconstruct the source signal.
-*If this phase information is not required,&nbsp; the demodulator is said to be an&nbsp;  '''incoherent demodulator'''&nbsp; or&nbsp;  "non-coherent demodulator".}}
+*If this phase information is not required,&nbsp; the demodulator is said to be an&nbsp; &raquo;'''incoherent demodulator'''&laquo;&nbsp; or&nbsp;  "non-coherent demodulator".}}
-An example of an incoherent demodulator is the&nbsp; [[Modulation_Methods/Envelope_Demodulation|envelope demodulator]].
+An example of an incoherent demodulator is the&nbsp; [[Modulation_Methods/Envelope_Demodulation|$\text{envelope demodulator}$]].
 {{GraueBox|TEXT=
@@ Line 79: / Line 79: @@
 :$$v(t) = \sqrt{ b_1^2(t) + b_2^2(t)} = \sqrt{ q^2(t) } = \vert q(t) \vert \hspace{0.05cm}.$$
 *A prerequisite for operability&nbsp; &ndash;  that is,&nbsp; for the result &nbsp;$v(t) = q(t)$&nbsp; &ndash;&nbsp; is that at all times &nbsp;$q(t) ≥ 0$.&nbsp;  In an analog transmission system,&nbsp; this fact could be enforced using the&nbsp; "DSB-AM with carrier"&nbsp; modulation method,&nbsp; for example.
-*This form of non-coherent demodulation&nbsp; (or modifications of it)&nbsp; is mainly applied in some&nbsp; '''digital modulation methods''',&nbsp; which are discussed in detail in the fourth chapter of&nbsp; [[Modulation_Methods|this book]].}}
+*This form of non-coherent demodulation&nbsp; (or modifications of it)&nbsp; is mainly applied in some&nbsp; &raquo;'''digital modulation methods'''&laquo;,&nbsp; which are discussed in detail in the fourth chapter of&nbsp; [[Modulation_Methods|"this book"]].}}
 ==Exercises for the chapter==

@@ Line 13: / Line 13: @@
 [[File:EN_Mod_T_2_5_S1.png|right|frame|Spectrum&nbsp; (of the analytical signal)&nbsp; for vestigial sideband amplitude modulation]]
-The problem can be greatly mitigated if instead single-sideband AM one uses &nbsp; "vestigial sideband amplitude modulation"&nbsp; $\rm (VSB–AM)$,&nbsp; as shown in the adjacent graph.
+The problem can be greatly mitigated if instead of single-sideband AM one uses &nbsp; "vestigial sideband amplitude modulation"&nbsp; $\rm (VSB–AM)$,&nbsp; as shown in the adjacent graph.
 The present description is based on the textbook &nbsp; [Mäu88]<ref>Mäusl, R.:&nbsp; Analoge Modulationsverfahren.&nbsp;  Heidelberg: Dr. Hüthig, 1988.</ref>.&nbsp;  According to it,&nbsp; the VSB-AM can be characterized as follows:

@@ Line 9: / Line 9: @@
 When transmitting signals using single-sideband modulation&nbsp; $\rm (SSB–AM)$&nbsp; the following problems occur:
 *To suppress the unwanted sideband,&nbsp; a filter with a very high edge slope must be used.
-*Such a steep-edged filters exhibit strong group delay distortions,&nbsp; especially at the limit of the passband.
+*Such steep-edged filters exhibit strong group delay distortions,&nbsp; especially at the limit of the passband.

@@ Line 2: / Line 2: @@
 {{Header
 |Untermenü=Amplitude Modulation and Demodulation
-|Vorherige Seite=Einseitenbandmodulation
+|Vorherige Seite=Single-Sideband Modulation
-|Nächste Seite=Phasenmodulation (PM)
+|Nächste Seite=Phase Modulation (PM)
 }}
 ==Vestigial sideband amplitude modulation==

← Older revision		Revision as of 13:34, 18 January 2022
Line 64:		Line 64:


−	An example of an incoherent demodulator is the   [[Modulation_Methods/Envelope_Demodulation\|envelope demodulator]].	+	An example of an incoherent demodulator is the  [[Modulation_Methods/Envelope_Demodulation\|envelope demodulator]].

	{{GraueBox\|TEXT=		{{GraueBox\|TEXT=

@@ Line 51: / Line 51: @@
 :$$v_2(t) = q_2(t)\hspace{0.05cm}.$$
 *When there is a phase offset &nbsp;$Δϕ_{\rm T}$&nbsp; between the transmitted and received carrier signals,&nbsp; in addition to attenuation of the intended participant,<br>crosstalk from the second participant occurs,&nbsp; resulting in nonlinear distortion:
-:$$v_1(t) = \alpha_{11} \cdot q_1(t)+ \alpha_{12} \cdot q_2(t) \hspace{0.05cm}, \hspace{0.3cm} v_2(t) = \alpha_{21} \cdot q_1(t)+ \alpha_{22} \cdot q_2(t)$$
+:$$v_1(t) = \alpha_{11} \cdot q_1(t)+ \alpha_{12} \cdot q_2(t) \hspace{0.05cm}, $$
 :$$\Rightarrow\hspace{0.3cm}\alpha_{11} =  \alpha_{22} = \cos(\Delta \phi_{\rm T}) \hspace{0.05cm},  \hspace{0.3cm} \alpha_{12} =  -\alpha_{21} = \sin(\Delta \phi_{\rm T}) \hspace{0.05cm}.$$
 ==Incoherent (non-coherent) Demodulation==
 <br>
 {{BlaueBox|TEXT=
-$\text{Definition:}$&nbsp; A demodulator is said to be&nbsp; '''coherent''', if in addition to the required frequency synchronization, it requires accurate information about the phase of the transmit-side carrier signal &nbsp;$z(t)$&nbsp; to reconstruct the message signal.
+$\text{Definition:}$&nbsp; Demodulators can be classified in the following way:
-If this phase information is not required, the demodulator is said to be an&nbsp;  '''incoherent demodulator'''.}}
+*If this phase information is not required,&nbsp; the demodulator is said to be an&nbsp;  '''incoherent demodulator'''&nbsp; or&nbsp;  "non-coherent demodulator".}}
-An example of an incoherent (or non-coherent) demodulator is the &nbsp; [[Modulation_Methods/Envelope_Demodulation|envelope demodulator]].
+An example of an incoherent demodulator is the &nbsp; [[Modulation_Methods/Envelope_Demodulation|envelope demodulator]].
 {{GraueBox|TEXT=
-$\text{Example 2:}$&nbsp; A second example is shown in the following block diagram.&nbsp; In contrast to quadrature amplitude modulation, here the orthogonality between cosine and sine functions is not used for the simultaneous transmission of a second source signal, but rather to simplify the receiver-side device.
+$\text{Example 2:}$&nbsp; A second example is shown in the following block diagram.&nbsp; In contrast to quadrature amplitude modulation,&nbsp;  here the orthogonality between cosine and sine functions is not used for the simultaneous transmission of a second source signal,&nbsp; but rather to simplify the receiver-side device.
-[[File:P_ID1054__Mod_T_2_5_S3_neu.png |center|frame|Incoherent demodulation with&nbsp; $\rm DSB-AM$]]
+[[File:P_ID1054__Mod_T_2_5_S3_neu.png |right|frame|Incoherent demodulation with&nbsp; $\rm DSB-AM$]]
 It should be further noted regarding this arrangement:
@@ Line 76: / Line 76: @@
 :$$b_1(t) = \cos(\Delta \phi_{\rm T}) \cdot q(t), $$
 :$$b_2(t) = -\sin(\Delta \phi_{\rm T}) \cdot q(t).$$
-*This ensures that the sink signal &nbsp;$v(t)$&nbsp; matches the source signal &nbsp;$q(t)$&nbsp;, at least in magnitude, regardless of the phase offset &nbsp;$Δϕ_{\rm T}$&nbsp;:
+*This ensures that the sink signal &nbsp;$v(t)$&nbsp; matches the source signal &nbsp;$q(t)$&nbsp; in magnitude,&nbsp; regardless of the phase offset &nbsp;$Δϕ_{\rm T}$:
 :$$v(t) = \sqrt{ b_1^2(t) + b_2^2(t)} = \sqrt{ q^2(t) } = \vert q(t) \vert \hspace{0.05cm}.$$
-*A prerequisite for operability –  that is, for the result &nbsp;$v(t) = q(t)$&nbsp; – is that at all times &nbsp;$q(t) ≥ 0$&nbsp;.  In an analog message system, this fact could be enforced using the "DSB-AM with carrier" modulation method, for example.
+*A prerequisite for operability&nbsp; &ndash;  that is,&nbsp; for the result &nbsp;$v(t) = q(t)$&nbsp; &ndash;&nbsp; is that at all times &nbsp;$q(t) ≥ 0$.&nbsp;  In an analog transmission system,&nbsp; this fact could be enforced using the&nbsp; "DSB-AM with carrier"&nbsp; modulation method,&nbsp; for example.
-*This form of non-coherent demodulation - or modifications of it - is mainly applied in some '''digital modulation methods''', which are discussed in detail in the fourth chapter of [[Modulation_Methods|this book]].}}
+*This form of non-coherent demodulation&nbsp; (or modifications of it)&nbsp; is mainly applied in some&nbsp; '''digital modulation methods''',&nbsp; which are discussed in detail in the fourth chapter of&nbsp; [[Modulation_Methods|this book]].}}
 ==Exercises for the chapter==

@@ Line 7: / Line 7: @@
 ==Vestigial sideband amplitude modulation==
 <br>
-When transmitting signals using single sideband modulation &nbsp; $\rm (SSB–AM)$&nbsp; the following problems occur:
+When transmitting signals using single-sideband modulation&nbsp; $\rm (SSB–AM)$&nbsp; the following problems occur:
-*To suppress the unwanted sideband, a filter with a very high edge slope must be used.
+*To suppress the unwanted sideband,&nbsp; a filter with a very high edge slope must be used.
-*Such a steep-edged filters exhibit strong group delay distortions, especially at the limit of the passband.
+*Such a steep-edged filters exhibit strong group delay distortions,&nbsp; especially at the limit of the passband.
-[[File:EN_Mod_T_2_5_S1.png|right|frame|Spectrum (of the analytical signal) for vestigial sideband amplitude modulation]]
+[[File:EN_Mod_T_2_5_S1.png|right|frame|Spectrum&nbsp; (of the analytical signal)&nbsp; for vestigial sideband amplitude modulation]]
-the problem can be greatly mitigated if instead single-sideband AM one uses &nbsp; ''vestigial sideband amplitude modulation''&nbsp; $\rm (VSB–AM)$&nbsp;, as shown in the adjacent graph.
+The problem can be greatly mitigated if instead single-sideband AM one uses &nbsp; "vestigial sideband amplitude modulation"&nbsp; $\rm (VSB–AM)$,&nbsp; as shown in the adjacent graph.
-The present description is based on the textbook &nbsp; [Mäu88]<ref>Mäusl, R.: ''Analoge Modulationsverfahren.''  Heidelberg: Dr. Hüthig, 1988.</ref>.&nbsp;  According to it, the VSB-AM can be characterized as follows:
+The present description is based on the textbook &nbsp; [Mäu88]<ref>Mäusl, R.:&nbsp; Analoge Modulationsverfahren.&nbsp;  Heidelberg: Dr. Hüthig, 1988.</ref>.&nbsp;  According to it,&nbsp; the VSB-AM can be characterized as follows:
 *A certain frequency range of the actually suppressed sideband – in the considered example of the LSB – is additionally used with a relatively flat decreasing transfer function.
-*On the receiver side, a  selection curve linearly increasing in frequency with a so-called "Nyquist edge" is used in the transition range from the suppressed sideband to the transmitted sideband.
+*On the receiver side,&nbsp; a  selection curve linearly increasing in frequency with a so-called&nbsp; "Nyquist edge"&nbsp; is used in the transition range from the suppressed sideband to the transmitted sideband.
-*The demodulation performs a convolution of the sidebands around the carrier, so that as a result the message content of a band with the same amplitude for all frequencies is obtained.
+*The demodulation performs a convolution of the sidebands around the carrier,&nbsp; so that as a result the message content of a band with the same amplitude for all frequencies is obtained.
 {{GraueBox|TEXT=
-$\text{Example 1:}$&nbsp; The vestigial sideband method is used for (analog) color television, whose frequency spectrum according to the CCIR standard is shown in the graphic. The frequencies given refer to the &nbsp;[https://en.wikipedia.org/wiki/PAL PAL–B/G Television format] used in Germany.
+$\text{Example 1:}$&nbsp; The vestigial sideband method is used for&nbsp; (analog)&nbsp; color television,&nbsp; whose frequency spectrum according to the CCIR standard is shown in the graphic.&nbsp; The frequencies given refer to the &nbsp;[https://en.wikipedia.org/wiki/PAL PAL–B/G Television format]&nbsp; used in Germany.
-[[File:EN_Mod_T_2_5_S1b.png|right|frame|Illustrating the Nyquist edge for PAL]]
+[[File:EN_Mod_T_2_5_S1b.png|right|frame|Illustrating the Nyquist edge for PAL–B/G Television]]
-In this schematic representation, one recognises:
+In this schematic representation,&nbsp; one recognises:
-*The radiated spectrum(only positive frequencies are drawn) ranges from &nbsp;$f_{\rm T} - 1.25 \ \rm MHz$&nbsp; to &nbsp;$f_{\rm T} + 5.75 \ \rm MHz$.&nbsp; Thus, the lower vestigial sideband including the Nyquist edge is approximately &nbsp;$1.25 \ \rm MHz$&nbsp; wide .
+*The radiated spectrum&nbsp; (only positive frequencies are drawn)&nbsp; ranges from &nbsp;$f_{\rm T} - 1.25 \ \rm MHz$&nbsp; to &nbsp;$f_{\rm T} + 5.75 \ \rm MHz$.&nbsp; The lower vestigial sideband including Nyquist edge is &nbsp;$\approx 1.25 \ \rm MHz$&nbsp; wide.
-*The green dashed line shows the receiver passband. The image carrier '''(B)'''&nbsp; at carrier frequency  &nbsp;$f_{\rm T}$&nbsp; is centered on the Nyquist edge.
+*The green dashed line shows the receiver passband.&nbsp; The image carrier&nbsp; '''(B)'''&nbsp; &ndash; abbreviation&nbsp; "B"&nbsp; from German "Bild" &nbsp; &rArr; &nbsp;  "image") &ndash; &nbsp; at carrier frequency  &nbsp;$f_{\rm T}$&nbsp; is centered on the Nyquist edge.
-*The luminance signal &nbsp; '''(L)'''&nbsp; goes up to about &nbsp;$5 \ \rm MHz$.&nbsp;.  It contains the information for the image brightness and the color "green".
+*The luminance signal &nbsp; '''(L)'''&nbsp; goes up to about &nbsp;$5 \ \rm MHz$.&nbsp;  It contains the information for the image brightness and for the color&nbsp; "green".
-*The chrominance signal &nbsp; '''(C)'''&nbsp; is embedded in the upper part.  Two orthogonal carriers are [[Modulation_Methods/Further_AM_Variants#Quadrature_Amplitude_Modulation_.28QAM.29|QAM-modulated]] at &nbsp;$4.43 \ \rm MHz$&nbsp; for "red" and "blue"; the carrier is suppressed.
+*The chrominance signal &nbsp; '''(C)'''&nbsp; is embedded in the upper part.  Two orthogonal carriers are [[Modulation_Methods/Further_AM_Variants#Quadrature_Amplitude_Modulation_.28QAM.29|QAM-modulated]]&nbsp; at &nbsp;$4.43 \ \rm MHz$&nbsp; for&nbsp; "red"&nbsp; and&nbsp; "blue";&nbsp; the carrier is suppressed.
-*The audio carrier &nbsp; '''(T)'''&nbsp; (abbreviation T from German "Ton" i.e., sound/audio) is at &nbsp;$f_{\rm T} + 5.5 \ \rm MHz$&nbsp; and is &nbsp;$12  \ \rm  dB$&nbsp; lower than the image carrier.  If there is a stereo or two-channel sound transmission, a second sound carrier follows at &nbsp;$5.75 \ \rm MHz$&nbsp;.}}
+*The audio carrier &nbsp; '''(T)'''&nbsp; &ndash; abbreviation&nbsp; "T"&nbsp; from German "Ton" &nbsp; &rArr; &nbsp; "sound/audio") &ndash; &nbsp; is at &nbsp;$f_{\rm T} + 5.5 \ \rm MHz$&nbsp; and is &nbsp;$12  \ \rm  dB$&nbsp; lower than the image carrier.
 ==Quadrature Amplitude Modulation (QAM)==
 <br>
-By exploiting the orthogonality of cosine and sine functions, a channel can be used twice for simultaneous transmission of two source signals &nbsp;$q_1(t)$&nbsp; and &nbsp;$q_2(t)$&nbsp; without mutual interference. This method is called ''quadrature amplitude modulation'' $\rm (QAM)$.
+By exploiting the orthogonality of cosine and sine functions,&nbsp; a channel can be used twice for simultaneous transmission of two source signals &nbsp;$q_1(t)$&nbsp; and &nbsp;$q_2(t)$&nbsp; without mutual interference.&nbsp; This method is called&nbsp; "quadrature amplitude modulation"&nbsp; $\rm (QAM)$.
-[[File:P_ID1053__Mod_T_2_5_S2_neu.png |center|frame|Model of quadrature amplitude modulation]]
+[[File:P_ID1053__Mod_T_2_5_S2_neu.png |right|frame|Model of quadrature amplitude modulation]]
 The QAM system has the following characteristics:
-*The transmit signal is composed of two mutually orthogonal components:
+*The transmitted signal is composed of two mutually orthogonal components:
 :$$s(t) = q_1(t)  \cdot \cos (\omega_{\rm T}\cdot t) - q_2(t)  \cdot \sin (\omega_{\rm T}\cdot t)\hspace{0.05cm}.$$
-*For frequency and phase synchronous demodulation, the signal in the upper branch before the low pass is &nbsp;$H_{\rm E1}(f)$:
+*For frequency and phase synchronous demodulation,&nbsp; the signal in the upper branch before the low-pass &nbsp;$H_{\rm E1}(f)$&nbsp;  is:
 :$$b_1(t) = q_1(t)  \cdot 2 \cdot \cos^2 (\omega_{\rm T}\cdot t)  - q_2(t)  \cdot 2 \cdot
-  \cos (\omega_{\rm T}\cdot t)\cdot \sin (\omega_{\rm T}\cdot t)=  q_1(t)\cdot \big[ 1 + \cos (2 \omega_{\rm T}\cdot t) \big] - q_2(t)\cdot \sin (2 \omega_{\rm T}\cdot t) \hspace{0.05cm}.$$
+  \cos (\omega_{\rm T}\cdot t)\cdot \sin (\omega_{\rm T}\cdot t)  $$
-*Thus, by limiting to frequencies &nbsp;$|f| < f_{\rm T}$&nbsp;, we obtain in the upper and lower branches, respectively:
+:$$\Rightarrow \hspace{0.3cm} b_1(t) =  q_1(t)\cdot \big[ 1 + \cos (2 \omega_{\rm T}\cdot t) \big] - q_2(t)\cdot \sin (2 \omega_{\rm T}\cdot t) \hspace{0.05cm}.$$
-:$$v_1(t) = q_1(t),\hspace{0.3cm} v_2(t) = q_2(t)\hspace{0.05cm}.$$
+*Thus,&nbsp; by limiting to frequencies &nbsp;$|f| < f_{\rm T}$,&nbsp; we obtain in the upper and lower branches,&nbsp; resp.:
-*When there is a phase offset &nbsp;$Δϕ_{\rm T}$&nbsp; between the transmitted and received carrier signals, in addition to attenuation of the intended participant, crosstalk from the second participant occurs, resulting in nonlinear distortion:
+:$$v_1(t) = q_1(t),$$
 :$$v_1(t) = \alpha_{11} \cdot q_1(t)+ \alpha_{12} \cdot q_2(t) \hspace{0.05cm}, \hspace{0.3cm} v_2(t) = \alpha_{21} \cdot q_1(t)+ \alpha_{22} \cdot q_2(t)$$
 :$$\Rightarrow\hspace{0.3cm}\alpha_{11} =  \alpha_{22} = \cos(\Delta \phi_{\rm T}) \hspace{0.05cm},  \hspace{0.3cm} \alpha_{12} =  -\alpha_{21} = \sin(\Delta \phi_{\rm T}) \hspace{0.05cm}.$$

@@ Line 58: / Line 58: @@
 $\text{Definition:}$&nbsp; A demodulator is said to be&nbsp; '''coherent''', if in addition to the required frequency synchronization, it requires accurate information about the phase of the transmit-side carrier signal &nbsp;$z(t)$&nbsp; to reconstruct the message signal.
-If this phase information is not required, the demodulator is said to be&nbsp;  '''incoherent  Demodulator'''.}}
+If this phase information is not required, the demodulator is said to be an&nbsp;  '''incoherent demodulator'''.}}
@@ Line 64: / Line 64: @@
 {{GraueBox|TEXT=
-$\text{Example 2:}$&nbsp; Ein zweites Beispiel zeigt das folgende Blockschaltbild.&nbsp; Im Gegensatz zur Quadratur–Amplitudenmodulation wird hier die Orthogonalität zwischen Cosinus– und Sinusfunktion nicht zur gleichzeitigen Übertragung eines zweiten Quellensignals herangezogen, sondern zur Vereinfachung der Empfangseinrichtung genutzt.
+$\text{Example 2:}$&nbsp; A second example is shown in the following block diagram.&nbsp; In contrast to quadrature amplitude modulation, here the orthogonality between cosine and sine functions is not used for the simultaneous transmission of a second source signal, but rather to simplify the receiver-side device.
-[[File:P_ID1054__Mod_T_2_5_S3_neu.png |center|frame|Inkohärente Demodulation bei&nbsp; $\rm ZSB-AM$]]
+[[File:P_ID1054__Mod_T_2_5_S3_neu.png |center|frame|Incoherent demodulation with&nbsp; $\rm DSB-AM$]]
-Zu dieser Anordnung ist weiter anzumerken:
+It should be further noted regarding this arrangement:
-*Die empfängerseitigen Trägersignale können gegenüber den Trägersignalen beim Sender einen beliebigen und auch zeitabhängigen Phasenversatz &nbsp;$Δϕ_{\rm T}$&nbsp; aufweisen, so lange die Phasendifferenz zwischen den beiden Zweigen weiterhin genau &nbsp;$90^\circ$&nbsp; beträgt.
+*The receiver-side carrier signals can have an arbitrary and also time-dependent phase offset &nbsp;$Δϕ_{\rm T}$&nbsp; with respect to the carrier signals at the transmitter, as long as the phase difference between the two branches remains exactly &nbsp;$90^\circ$&nbsp;.
-*Für die Signale im oberen und unteren Zweig – jeweils nach dem Multiplizierer und der Tiefpassfilterung – gilt nämlich:
+*For the signals in the upper and lower branches – after the multiplier and low-pass filtering, respectively – it holds:
 :$$b_1(t) = \cos(\Delta \phi_{\rm T}) \cdot q(t), $$
 :$$b_2(t) = -\sin(\Delta \phi_{\rm T}) \cdot q(t).$$
-*Damit ist gewährleistet, dass das Sinkensignal &nbsp;$v(t)$&nbsp; unabhängig vom Phasenversatz &nbsp;$Δϕ_{\rm T}$&nbsp; mit dem Quellensignal &nbsp;$q(t)$&nbsp; zumindest betragsmäßig übereinstimmt:
+*This ensures that the sink signal &nbsp;$v(t)$&nbsp; matches the source signal &nbsp;$q(t)$&nbsp;, at least in magnitude, regardless of the phase offset &nbsp;$Δϕ_{\rm T}$&nbsp;:
 :$$v(t) = \sqrt{ b_1^2(t) + b_2^2(t)} = \sqrt{ q^2(t) } = \vert q(t) \vert \hspace{0.05cm}.$$
-*Voraussetzung für die Funktionsfähigkeit – also für das Ergebnis &nbsp;$v(t) = q(t)$&nbsp; – ist, dass zu allen Zeiten &nbsp;$q(t) ≥ 0$&nbsp; ist.&nbsp; Bei einem analogen Nachrichtensystem könnte man diesen Sachverhalt beispielsweise mit dem Modulationsverfahren „ZSB–AM mit Träger” erzwingen.
+*A prerequisite for operability –  that is, for the result &nbsp;$v(t) = q(t)$&nbsp; – is that at all times &nbsp;$q(t) ≥ 0$&nbsp;.  In an analog message system, this fact could be enforced using the "DSB-AM with carrier" modulation method, for example.
-*Angewandt wird diese Form von nichtkohärenter Demodulation – oder Modifikationen hiervon – vorwiegend bei einigen&nbsp; '''Digitalen Modulationsverfahren''', die im vierten Kapitel [[Modulation_Methods|dieses Buches]] ausführlich behandelt werden.}}
+*This form of non-coherent demodulation - or modifications of it - is mainly applied in some '''digital modulation methods''', which are discussed in detail in the fourth chapter of [[Modulation_Methods|this book]].}}
-==Aufgaben zum Kapitel==
+==Exercises for the chapter==
 <br>
-[[Aufgaben:Aufgabe_2.12:_Zur_nichtkohärenten_Demodulation|Aufgabe 2.12: Zur nichtkohärenten Demodulation]]
+[[Aufgaben:Exercise_2.12:_Non-coherent_Demodulation|Exercise 2.12: Non-coherent Demodulation]]
-[[Aufgaben:Aufgabe_2.13:_Quadratur-Amplitudenmodulation_(QAM)|Aufgabe 2.13: Quadratur-Amplitudenmodulation (QAM)]]
+[[Aufgaben:Exercise_2.13:_Quadrature_Amplitude_Modulation|Exercise 2.13: Quadrature Amplitude Modulation (QAM)]]
-==Quellenverzeichnis==
+==References==
 <references/>
 {{Display}}