Aufgaben:Exercise 2.6Z: Synchronous Demodulator - Revision history

Guenter at 13:29, 6 October 2021

2021-10-06T13:29:31Z

Guenter at 13:22, 6 October 2021

2021-10-06T13:22:59Z

Guenter at 13:19, 6 October 2021

2021-10-06T13:19:48Z

Guenter at 13:13, 6 October 2021

2021-10-06T13:13:17Z

Oezer at 14:48, 20 September 2021

2021-09-20T14:48:58Z

Oezer at 14:46, 20 September 2021

2021-09-20T14:46:44Z

Oezer at 14:45, 20 September 2021

2021-09-20T14:45:19Z

Oezer at 14:44, 20 September 2021

2021-09-20T14:44:48Z

Oezer at 14:44, 20 September 2021

2021-09-20T14:44:11Z

Oezer at 14:43, 20 September 2021

2021-09-20T14:43:12Z

@@ Line 97: / Line 97: @@
 *Using the trigonometric relation&nbsp; $\cos^2(\omega_{\rm T} t)  =   {1}/{2} \cdot\big[ 1  +
-  \cos(2\omega_{\rm T} t)\big]$,&nbsp;
+  \cos(2\omega_{\rm T} t)\big]$,&nbsp; one obtains:
 :$$b(t) = {K}/{2} \cdot q(t) + {K}/{2} \cdot q(t)\cdot
-  \cos(2\omega_{\rm T} t)$$ is obtained.
+  \cos(2\omega_{\rm T} t).$$
 *The second component is located at around twice the carrier frequency&nbsp; &rArr; &nbsp; $2 f_{\rm T}$.&nbsp;
 *This is removed by the low-pass filter&nbsp; $($with the cut-off frequency&nbsp; $  f_{\rm G} = f_{\rm T})$&nbsp;.
 *Hence, the following is obtained: &nbsp; $v(t) = {K}/{2} \cdot q(t) .$
-*An ideal demodulation&nbsp; &rArr; &nbsp; $v(t) =  q(t)$ is obtained with&nbsp; $\underline {K = 2}$&nbsp;.
+*An ideal demodulation&nbsp; &rArr; &nbsp; $v(t) =  q(t)$&nbsp; is obtained with&nbsp; $\underline {K = 2}$&nbsp;.
@@ Line 112: / Line 112: @@
    \big[ \cos(\Delta \varphi)+ \cos(2\omega_{\rm T} t - \Delta \varphi) \big]$$
-and the downstream low-pass filter, which again removes the component at around twice the carrier frequency, the following is obtained here with&nbsp; $ {K = 2}$:
+and the downstream low-pass filter,&nbsp; which removes the component at around twice the carrier frequency, the following is obtained here with&nbsp; $ {K = 2}$:
 :$$v(t) = q(t) \cdot \cos(\Delta \varphi).$$
-<u>Proposed solutions 2 and 5</u> are correct:
+<u>Proposed solutions 2 and 5</u>&nbsp; are correct:
-*A phase shift&nbsp;$\Delta \varphi$&nbsp; only results in frequency-independent attenuation and not in attenuation or phase distortions.
+*A phase shift&nbsp;$\Delta \varphi$&nbsp; only results in frequency-independent attenuation and not in attenuation distortions or phase distortions.
-*A phase shift by&nbsp;$\varphi =\pm 60^\circ$&nbsp; results in halving of the signal respectively.
+*A phase shift by&nbsp;$\varphi =\pm 60^\circ$&nbsp; results in halving of the signal amplitude.
-'''(3)'''&nbsp; Here, <u>proposed solution 4</u> is correct.
+'''(3)'''&nbsp; Here, <u>proposed solution 4</u>&nbsp; is correct.
-*Exactly the same phase shift&nbsp; $\Delta \varphi$&nbsp; occurs for both summands, and phase distortions occur here:
+*Exactly the same phase shift&nbsp; $\Delta \varphi$&nbsp; occurs for both summands,&nbsp; and phase distortions occur here:
 :$$v(t)=  {2 \, \rm V}  \cdot{\rm cos}\big[ \omega_2 \cdot (t - \tau_2) \big]+
 {1 \, \rm V}  \cdot{\rm sin}\big[ \omega_5 t \cdot (t - \tau_5)\big],$$
@@ Line 129: / Line 129: @@
 \varphi}{\omega_5}.$$
-*A phase shift of&nbsp;$\varphi =60^\circ$ corresponding to&nbsp; $\pi/3$&nbsp; leads to the following delay times here:
+*A phase shift of&nbsp; $\varphi =60^\circ$ corresponding to&nbsp; $\pi/3$&nbsp; leads to the following delay times here:
 :$$\tau_2  =  \frac{\pi/3}{2 \pi \cdot 2\,\,{\rm  kHz }} \approx
 .3\,{\rm &micro; s }, \hspace{0.5cm}

@@ Line 55: / Line 55: @@
   \beta)+ \sin(\alpha + \beta)
   \big] \hspace{0.05cm}.$$
-*The signal designations result from the German original of this exercise.&nbsp; '''Here again as a listing''' &nbsp;  $q(t)$ &nbsp; &rArr; &nbsp; source signal, &nbsp; $v(t)$ &nbsp; &rArr; &nbsp; sink signal, &nbsp; $z(t)$ &nbsp; &rArr; &nbsp; transmit-site carrier signal, &nbsp; $s(t)$ &nbsp; &rArr; &nbsp; transmission signal (HF),&nbsp; $r(t)$ &nbsp; &rArr; &nbsp; received signal, &nbsp; $z_{\rm E}(t)$ &nbsp; &rArr; &nbsp; receive-site carrier signal, &nbsp; $b(t)$ &nbsp; &rArr; &nbsp; HF signal before low-pass
+*The signal designations result from the German original of this exercise.&nbsp; '''Here again as a listing''' &nbsp;  $q(t)$ &nbsp; &rArr; &nbsp; source signal, &nbsp; $v(t)$ &nbsp; &rArr; &nbsp; sink signal, &nbsp; $z(t)$ &nbsp; &rArr; &nbsp; transmit-site carrier signal, &nbsp; $s(t)$ &nbsp; &rArr; &nbsp; transmission signal (BP),&nbsp; $r(t)$ &nbsp; &rArr; &nbsp; received signal  (BP), &nbsp; $z_{\rm E}(t)$ &nbsp; &rArr; &nbsp; receive-site carrier signal, &nbsp; $b(t)$ &nbsp; &rArr; &nbsp; BP signal before low-pass.

@@ Line 62: / Line 62: @@
 <quiz display=simple>
-{What is the sink signal&nbsp;$v(t)$&nbsp; for ZSB-AM and phase-synchronous synchronous modulation&nbsp; &rArr; &nbsp; $\Delta \varphi = 0$? <br>How should&nbsp; $K$&nbsp; be chosen such that &nbsp;$v(t) = q(t)$&nbsp; holds?
+{What is the sink signal&nbsp; $v(t)$&nbsp; for DSB-AM and phase-synchronous modulation&nbsp; &rArr; &nbsp; $\Delta \varphi = 0$? <br>How should&nbsp; $K$&nbsp; be chosen such that &nbsp;$v(t) = q(t)$&nbsp; holds?
 |type="{}"}
 $K \ = \ $ { 2 3% }
-{The following holds: &nbsp;$K = 2$. Specify the sink signal&nbsp;$v(t)$&nbsp; considering a phase shift&nbsp;$\Delta \varphi$&nbsp;. <br>Which of the following statements are true?
+{The following holds: &nbsp;$K = 2$.&nbsp; Specify the sink signal&nbsp; $v(t)$&nbsp; considering a phase shift&nbsp;$\Delta \varphi$.&nbsp; Which of the following statements are true?
 |type="[]"}
-- $v(t) = q(t)$ holds independently of&nbsp;$\Delta \varphi$&nbsp;.
+- $v(t) = q(t)$&nbsp; holds independently of&nbsp; $\Delta \varphi$&nbsp;.
 + $\Delta \varphi \ne 0$&nbsp; results in frequency-independent attenuation.
-- A phase shift&nbsp;$\Delta \varphi \ne 0$&nbsp; results in attenuation distortions.
+- A phase shift&nbsp; $\Delta \varphi \ne 0$&nbsp; results in attenuation distortions.
-- A phase shift&nbsp;$\Delta \varphi \ne 0$&nbsp; results in phase distortions.
+- A phase shift&nbsp; $\Delta \varphi \ne 0$&nbsp; results in phase distortions.
-+ $v(t) = q(t)/2$ holds with&nbsp;$\Delta \varphi = \hspace{-0.05cm}-\hspace{0.05cm}60^\circ$&nbsp;.
++ $v(t) = q(t)/2$&nbsp; holds with&nbsp; $\Delta \varphi = \hspace{-0.05cm}-\hspace{0.05cm}60^\circ$.
-{Which statements hold for synchronous demodulation of the ESB&ndash;signal if a phase shift of&nbsp;$\Delta \varphi$&nbsp; is considered?
+{Which statements hold for synchronous demodulation of the SSB signal if a phase shift of&nbsp;$\Delta \varphi$&nbsp; is considered?
 |type="[]"}
-- Regardless of&nbsp;$\Delta \varphi$,&nbsp; &nbsp;$v(t) = q(t)$ holds.
+- Regardless of&nbsp; $\Delta \varphi$,&nbsp; $v(t) = q(t)$&nbsp; holds.
 - $\Delta \varphi \ne 0$&nbsp; results in frequency-independent attenuation.
-- A phase shift&nbsp;$\Delta \varphi \ne 0$&nbsp; results in attenuation distortions.
+- A phase shift&nbsp; $\Delta \varphi \ne 0$&nbsp; results in attenuation distortions.
-+ A phase shift&nbsp;$\Delta \varphi \ne 0$&nbsp; results in phase distortions.
++ A phase shift&nbsp; $\Delta \varphi \ne 0$&nbsp; results in phase distortions.
-- $v(t) = q(t)/2$ holds with&nbsp;$\Delta \varphi = \hspace{-0.05cm}-\hspace{0.05cm}60^\circ$&nbsp;.
+- $v(t) = q(t)/2$&nbsp; holds with&nbsp; $\Delta \varphi = \hspace{-0.05cm}-\hspace{0.05cm}60^\circ$&nbsp;.

@@ Line 3: / Line 3: @@
 }}
-[[File:P_ID913__LZI_Z_2_6_neu.png|right|frame|AM&ndash;modulator (top) and synchronous demodulator (bottom)]]
+[[File:P_ID913__LZI_Z_2_6_neu.png|right|frame|Amplitude modulator (top), <br>synchronous demodulator]]
 The depicted block diagram shows a transmission system
-*with&nbsp; [[Modulation_Methods/Double-Sideband_Amplitude_Modulation|Double-Sideband Amplitude Modulation]]&nbsp; (ZSB-AM)
+*with&nbsp; [[Modulation_Methods/Double-Sideband_Amplitude_Modulation|Double-Sideband Amplitude Modulation]]&nbsp; $\rm(DSB\hspace{0.03cm}&ndash;\hspace{-0.1cm}AM)$
-*and&nbsp; [[Modulation_Methods/Synchronous_Demodulation|Synchronous Demodulation]] (SD).
+*and&nbsp; [[Modulation_Methods/Synchronous_Demodulation|Synchronous Demodulation]]&nbsp; $\rm (SD)$.
@@ Line 13: / Line 13: @@
 \cdot {\rm sin}(\omega_5  t ) .$$
-*This signal is multiplied by the dimensionless carrier signal&nbsp; $z(t) = \cos(\omega_{\rm T} \cdot T)$&nbsp; of frequency&nbsp; $f_{\rm T} = 50 \ \rm kHz$. For ZSB&ndash;AM, the dashed block is irrelevant so that the following holds for the transmit signal:
+*This signal is multiplied by the dimensionless carrier signal&nbsp; $z(t) = \cos(\omega_{\rm T} \cdot T)$&nbsp; of carrier frequency&nbsp; $f_{\rm T} = 50 \ \rm kHz$. <br>For DSB&ndash;AM, the dashed block is irrelevant so that the following holds for the transmission signal:
 :$$s(t) = q(t) \cdot  {\rm cos}(\omega_{\rm T}  t ) .$$
-*In the synchronous demodulator, the receiver signal&nbsp; $r(t)$&nbsp; &ndash; identical to the transmit signal&nbsp; $s(t)$&nbsp; in an ideal channel &ndash; is multiplied by the receive-site carrier signal&nbsp; $z_{\rm E}(t)$&nbsp; where the following applies:
+*In the synchronous demodulator, the received signal&nbsp; $r(t)$&nbsp; &ndash; in an ideal channel identical to the signal&nbsp; $s(t)$&nbsp; &ndash; is multiplied by the receive-site carrier signal&nbsp; $z_{\rm E}(t)$&nbsp; where the following applies:
 :$$z_{\rm E}(t) = K \cdot  {\rm cos}(\omega_{\rm T}  t - \Delta \varphi ) .$$
-*This signal should not only be frequency-synchronous with&nbsp; $z(t)$&nbsp; but also phase-synchronous &ndash; hence the name "synchronous demodulator".
+*This signal should not only be frequency-synchronous with&nbsp; $z(t)$&nbsp; but also phase-synchronous &nbsp; &rArr; &nbsp;  hence the name "synchronous demodulator".
 *The above approach takes into account a phase shift between&nbsp; $z(t)$&nbsp; and&nbsp; $z_{\rm E}(t)$, which should ideally be&nbsp; $\Delta \varphi = 0$&nbsp; but often cannot be avoided in real systems.
-*The output signal&nbsp; $b(t)$&nbsp;  of the second multiplier includes components around twice the carrier frequency in addition to the desired NF component.
+*The output signal&nbsp; $b(t)$&nbsp;  of the second multiplier includes components around twice the carrier frequency in addition to the desired low-frequency component.
-*Using an ideal low-pass filter &ndash; for example with cut-off frequency&nbsp; $f_{\rm T}$&nbsp; &ndash; the sink signal&nbsp; $v(t)$,&nbsp; which ideally should be equal to the source signal&nbsp; $q(t)$,&nbsp; can be obtained.
+*Using an ideal low-pass filter&nbsp; $\rm (LP)$&nbsp; (e.g. with cut-off frequency&nbsp; $f_{\rm T}$)&nbsp; the sink signal&nbsp; $v(t)$,&nbsp; which ideally should be equal to the source signal&nbsp; $q(t)$,&nbsp; can be obtained.
@@ Line 44: / Line 44: @@
 Please note:
+*The exercise belongs to the chapter&nbsp; [[Linear_and_Time_Invariant_Systems/Linear_Distortions|Linear Distortions]].
+*The topic&nbsp; "amplitude modulation/synchronous demodulator"&nbsp; is discussed in detail in the book&nbsp; [[Modulation_Methods]].
 *The following trigonometric relationships are given:
 :$$\cos^2(\alpha) =  {1}/{2} \cdot \big [ 1 +
@@ Line 61: / Line 55: @@
   \beta)+ \sin(\alpha + \beta)
   \big] \hspace{0.05cm}.$$

@@ Line 129: / Line 129: @@
 :$$v(t)=  {2 \, \rm V}  \cdot{\rm cos}\big[ \omega_2 \cdot (t - \tau_2) \big]+
 {1 \, \rm V}  \cdot{\rm sin}\big[ \omega_5 t \cdot (t - \tau_5)\big],$$
-:$${\rm wobei}\hspace{0.5cm}\tau_2 = \frac{\Delta \varphi}{\omega_2}
+:$${\rm where}\hspace{0.5cm}\tau_2 = \frac{\Delta \varphi}{\omega_2}
 \hspace{0.5cm}\ne \hspace{0.5cm} \tau_5 = \frac{\Delta
 \varphi}{\omega_5}.$$

@@ Line 26: / Line 26: @@
-For the transmitter, multiplication by the carrier signal&nbsp; $z(t)$&nbsp; generally results in two sidebands. In&nbsp; [[Modulation_Methods/Single-Sideband_Modulation|Single-Sideband Modulation]]&nbsp; (ESB&ndash;AM) only one of the two bands is transmitted, for example the lower sideband (USB). Thus, the following is obtained with an ideal channel:
+For the transmitter, multiplication by the carrier signal&nbsp; $z(t)$&nbsp; generally results in two sidebands. In&nbsp; [[Modulation_Methods/Single-Sideband_Modulation|Single-Sideband Modulation]]&nbsp; (ESB&ndash;AM), only one of the two bands is transmitted, for example the lower sideband (USB). Thus, the following is obtained with an ideal channel:
 :$$r(t) = s(t)=  {1 \, \rm V} \cdot {\rm cos}\big [(\omega_{\rm T} -
 \omega_2  )\cdot t \big ] - {0.5 \, \rm V} \cdot {\rm sin}\big [(\omega_{\rm T} -

← Older revision		Revision as of 14:44, 20 September 2021
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	*The output signal  $b(t)$  of the second multiplier includes components around twice the carrier frequency in addition to the desired NF component.		*The output signal  $b(t)$  of the second multiplier includes components around twice the carrier frequency in addition to the desired NF component.
−	*Using an ideal low-pass filter – for example with cut-off frequency  $f_{\rm T}$  – the sink signal  $v(t)$  ~~can be obtained~~ which ideally should be equal to the source signal  $q(t)$ .	+	*Using an ideal low-pass filter – for example with cut-off frequency  $f_{\rm T}$  – the sink signal  $v(t)$,  which ideally should be equal to the source signal  $q(t)$,  can be obtained.

← Older revision		Revision as of 14:43, 20 September 2021
Line 23:		Line 23:

	*The output signal  $b(t)$  of the second multiplier includes components around twice the carrier frequency in addition to the desired NF component.		*The output signal  $b(t)$  of the second multiplier includes components around twice the carrier frequency in addition to the desired NF component.
−	*Using an ideal low-pass filter– for example with cut-off frequency  $f_{\rm T}$  – the sink signal  $v(t)$  can be obtained which ideally should be equal to the source signal  $q(t)$ .	+	*Using an ideal low-pass filter – for example with cut-off frequency  $f_{\rm T}$  – the sink signal  $v(t)$  can be obtained which ideally should be equal to the source signal  $q(t)$ .