Aufgaben:Exercise 2.4Z: Low-pass Influence with Synchronous Demodulation - Revision history

Reed at 15:32, 25 March 2022

2022-03-25T15:32:01Z

Guenter at 10:45, 6 December 2021

2021-12-06T10:45:39Z

Guenter at 14:09, 5 December 2021

2021-12-05T14:09:51Z

Guenter at 13:54, 5 December 2021

2021-12-05T13:54:50Z

Guenter at 13:54, 5 December 2021

2021-12-05T13:54:20Z

Reed at 16:42, 1 December 2021

2021-12-01T16:42:43Z

Reed: /* Solution */

2021-12-01T16:20:09Z

Solution

Reed at 16:08, 1 December 2021

2021-12-01T16:08:04Z

Reed at 15:55, 1 December 2021

2021-12-01T15:55:53Z

Reed at 15:55, 1 December 2021

2021-12-01T15:55:34Z

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 The graph shows the source signal &nbsp;$q(t)$&nbsp; at the top and the transmission signal &nbsp;$s(t)$&nbsp; in the middle.
-The sink signal &nbsp;$v(t)$&nbsp; is shown at the bottom (violet waveform).
+The sink signal &nbsp;$v(t)$&nbsp; is shown at the bottom (purple waveform).
 *This obviously does not match the source signal (blue dashed curve).
 *The reason for this undesired result &nbsp;$v(t) ≠ q(t)$&nbsp; could be a missing or wrongly dimensioned low-pass filter.

@@ Line 79: / Line 79: @@
-'''(2)'''&nbsp; <u>Answers 1 and 3</u> are correct:
+'''(2)'''&nbsp; <u>Answers 1 and 3</u>&nbsp; are correct:
-*A prerequisite for distortion-free demodulation is that all spectral components up to a certain frequency &nbsp; $f_1$&nbsp; are transmitted equally and as unattenuated as possible, and all components at frequencies &nbsp; $f > f_2$&nbsp; are completely suppressed.
+*A prerequisite for distortion-free demodulation is that all spectral components up to a certain frequency &nbsp; $f_1$&nbsp; are transmitted equally and as unattenuated as possible,&nbsp; and all components at frequencies &nbsp; $f > f_2$&nbsp; are completely suppressed.
 *The rectangular and trapezoidal low-pass filters satisfy this condition.
-'''(3)'''&nbsp; It must be ensured that the &nbsp; $\text{5 kHz}$ component still lies in the passband:
+'''(3)'''&nbsp; It must be ensured that the&nbsp; $\text{5 kHz}$&nbsp; component still lies in the passband:
 :$$f_{\text{1, min}}\hspace{0.15cm}\underline{ =5 \ \rm kHz}.$$
-'''(4)'''&nbsp; All spectral components in the vicinity of twice the carrier frequency – more precisely between &nbsp; $\text{95 kHz}$&nbsp; and&nbsp; $\text{ 105 kHz}$&nbsp; – must be completely suppressed:
+'''(4)'''&nbsp; All spectral components in the vicinity of twice the carrier frequency – more precisely between&nbsp; $\text{95 kHz}$&nbsp; and&nbsp; $\text{ 105 kHz}$&nbsp; – must be completely suppressed:
 :$$f_{\text{2, max}}\hspace{0.15cm}\underline{ =95 \ \rm kHz}.$$
 *Otherwise nonlinear distortion would arise.
@@ Line 96: / Line 96: @@
-'''(5)'''&nbsp; <u>Answer 2</u> is correct:
+'''(5)'''&nbsp; <u>Answer 2</u>&nbsp; is correct:
-*The cutoff frequency}&nbsp; $f_{\rm G} = \text{ 4 kHz}$&nbsp; would result in (linear) distortions, since the &nbsp; $\text{5 kHz}$ component would be cut off.
+*The cutoff frequency}&nbsp; $f_{\rm G} = \text{ 4 kHz}$&nbsp; would result in&nbsp; (linear)&nbsp; distortions,&nbsp; since the&nbsp; $\text{5 kHz}$&nbsp; component would be cut off.
-*The lowpass with cutoff frequency &nbsp; $f_{\rm G} = \text{6 kHz}$ is preferable, since with &nbsp; $f_{\rm G} = \text{10 kHz}$&nbsp;, more noise components would be superimposed on the useful signal $v(t)$.
+*The low-pass with cutoff frequency&nbsp; $f_{\rm G} = \text{6 kHz}$&nbsp; is preferable,&nbsp; since with&nbsp; $f_{\rm G} = \text{10 kHz}$&nbsp; more noise components would be superimposed on the signal&nbsp; $v(t)$.
 {{ML-Fuß}}

@@ Line 71: / Line 71: @@
 ===Solution===
 {{ML-Kopf}}
-'''(1)'''&nbsp; <u>The first statement</u> is correct:
+'''(1)'''&nbsp; <u>The first statement</u>&nbsp; is correct:
-*The sink signal shown&nbsp; $v(t)$&nbsp; exactly matches the signal &nbsp; $b(t)$&nbsp; given in the equation and thus also contains components around twice the carrier frequency.
+*The sink signal&nbsp; $v(t)$&nbsp; shown in the bottom graphic exactly matches the signal &nbsp; $b(t)$&nbsp; given in the equation and thus also contains components around twice the carrier frequency.
 *The filter &nbsp; $H_{\rm E}(f)$&nbsp; is either missing completely or its upper cutoff frequency&nbsp; $f_2$&nbsp; is too high.
-*Regarding the lower cutoff frequency &nbsp; $f_1$&nbsp;, the only statement possible is that it is smaller than the smallest frequency &nbsp; $\text{(2 kHz)}$ occurring in the signal &nbsp; $b(t)$&nbsp;.
+*Regarding the lower cutoff frequency &nbsp; $f_1$,&nbsp; the only statement possible is that it is smaller than the smallest frequency &nbsp; $\text{(2 kHz)}$ occurring in the signal&nbsp; $b(t)$.
-*Whether or not a DC component is removed by the filter is unclear, since such a component is not present in the signal &nbsp; $b(t)$&nbsp;.
+*Whether or not a DC component is removed by the filter is unclear,&nbsp; since such a component is not present in the signal &nbsp; $b(t)$.

@@ Line 26: / Line 26: @@
 Hints:
 *This exercise belongs to the chapter &nbsp; [[Modulation_Methods/Synchronous_Demodulation|Synchronous Demodulation]].
 *Particular reference is made to the page &nbsp;  [[Modulation_Methods/Synchronous_Demodulation#Block_diagram_and_time_domain_representation|Block diagram and time domain representation]].

@@ Line 6: / Line 6: @@
 Let us consider the same communication system as in  &nbsp;[[Aufgaben:Exercise_2.4:_Frequency_and_Phase_Offset|Exercise 2.4]].&nbsp; But this time, we will assume perfect frequency and phase synchronization for the synchronous demodulator $\rm (SD)$&nbsp;.
-The source signal &nbsp;$q(t)$, the transmission signal &nbsp;$s(t)$&nbsp; and the signal in the synchronous demodulator before the low-pass filter &nbsp;$b(t)$&nbsp; are given as follows:
+The source signal &nbsp;$q(t)$,&nbsp; the transmitted signal &nbsp;$s(t)$&nbsp; and the signal &nbsp;$b(t)$&nbsp; in the synchronous demodulator before the low-pass filter  are given as follows:
-:$$q(t)  =  q_1(t) + q_2(t)\hspace{0.2cm}{\rm mit }$$
+:$$q(t)  =  q_1(t) + q_2(t)\hspace{0.2cm}{\rm with }$$
 ::$$q_1(t)  =  2\,{\rm V} \cdot \cos(2 \pi \cdot 2\,{\rm kHz} \cdot t)\hspace{0.05cm},$$
 ::$$q_2(t)  =  1\,{\rm V} \cdot \sin(2 \pi \cdot 5\,{\rm kHz} \cdot t)\hspace{0.05cm},$$
@@ Line 13: / Line 13: @@
 :$$b(t)  = s(t) \cdot 2 \cdot \sin(2 \pi \cdot 50\,{\rm kHz} \cdot t)\hspace{0.05cm}.$$
-The graph shows the source signal &nbsp;$q(t)$&nbsp; at the top and the transmission signal &nbsp;$s(t)$ in the middle.
+The graph shows the source signal &nbsp;$q(t)$&nbsp; at the top and the transmission signal &nbsp;$s(t)$&nbsp; in the middle.
 The sink signal &nbsp;$v(t)$&nbsp; is shown at the bottom (violet waveform).
 *This obviously does not match the source signal (blue dashed curve).
-*The reason for this undesired result &nbsp;$v(t) ≠ q(t)$&nbsp; could be a missing or wrongly dimensioned low-pass filter, for example.
+*The reason for this undesired result &nbsp;$v(t) ≠ q(t)$&nbsp; could be a missing or wrongly dimensioned low-pass filter.
-In the subtasks&nbsp; '''(3)'''&nbsp; and&nbsp; '''(4)'''&nbsp;, a so-called &nbsp;[[Linear_and_Time_Invariant_Systems/Some_Low-Pass_Functions_in_Systems_Theory#Trapezoidal_low-pass_filters|trapezoidal low-pass filter]]&nbsp; is used, whose frequency response is as follows:
+In the subtasks&nbsp; '''(3)'''&nbsp; and&nbsp; '''(4)'''&nbsp;, a&nbsp; [[Linear_and_Time_Invariant_Systems/Some_Low-Pass_Functions_in_Systems_Theory#Trapezoidal_low-pass_filters|"trapezoidal low-pass filter"]]&nbsp; is used,&nbsp; whose frequency response is as follows:
 :$$H_{\rm E}(f) = \left\{ \begin{array}{l} \hspace{0.25cm}1 \\ \frac{f_2 -|f|}{f_2 -f_1} \\ \hspace{0.25cm} 0 \\ \end{array} \right.\quad \quad \begin{array}{*{10}c} {\rm{f\ddot{u}r}} \\ {\rm{f\ddot{u}r}} \\ {\rm{f\ddot{u}r}} \\ \end{array}\begin{array}{*{20}c} {\hspace{0.94cm}\left| \hspace{0.005cm} f\hspace{0.05cm} \right| < f_1,} \\ {f_1 \le \left| \hspace{0.005cm}f\hspace{0.05cm} \right| \le f_2,} \\ {\hspace{0.94cm}\left|\hspace{0.005cm} f \hspace{0.05cm} \right| > f_2.} \\ \end{array}$$
@@ Line 48: / Line 48: @@
 - The lower cutoff frequency is not zero.
 {With which of the low-pass functions listed below is ideal demodulation - that is, &nbsp; $v(t) = q(t)$&nbsp; – possible in principle?
-+ Right angle low-pass,
+|type="[]"}
 - Gaussian low-pass,
-+ Trapezoidal low-pass,
++ trapezoidal low-pass,
-- Slit low-pass.
+- slit low-pass.
 {What is the minimum lower corner frequency &nbsp;$f_1$&nbsp; of a trapezoidal low-pass filter one can choose to avoid distortion?
@@ Line 63: / Line 64: @@
-{Which cutoff frequency &nbsp;$f_{\rm G}$&nbsp; of an ideal rectangular low-pass would you choose if distortion is not negligible?
+{Which cutoff frequency&nbsp; (German:&nbsp; "Grenzfrequenz" &nbsp; &rArr; &nbsp; subscript&nbsp;"G") &nbsp;$f_{\rm G}$&nbsp; of an ideal rectangular low-pass filter would you choose if distortion is not negligible?
 |type="()"}
 - $f_{\rm G} = 4 \ \rm kHz$,

@@ Line 76: / Line 76: @@
 {{ML-Kopf}}
 '''(1)'''&nbsp; <u>The first statement</u> is correct:
-*The sink signal shown&nbsp; $v(t)$&nbsp; stimmt exakt mit dem als Gleichung gegebenen Signal&nbsp; $b(t)$&nbsp; überein und enthält somit auch Anteile um die doppelte Trägerfrequenz.
+*The sink signal shown&nbsp; $v(t)$&nbsp; exactly matches the signal &nbsp; $b(t)$&nbsp; given in the equation and thus also contains components around twice the carrier frequency. stimmt exakt mit dem als Gleichung gegebenen Signal überein und enthält somit auch Anteile um die doppelte Trägerfrequenz.
-*Das Filter&nbsp; $H_{\rm E}(f)$&nbsp; fehlt entweder ganz oder dessen obere Grenzfrequenz&nbsp; $f_2$&nbsp; ist zu hoch.
+*The filter &nbsp; $H_{\rm E}(f)$&nbsp; is either missing completely or its upper cutoff frequency&nbsp; $f_2$&nbsp; is too high.
 *Bezüglich der unteren Grenzfrequenz&nbsp; $f_1$&nbsp; ist nur die Aussage möglich, dass diese kleiner ist als die kleinste im Signal&nbsp; $b(t)$&nbsp; vorkommende Frequenz&nbsp; $\text{(2 kHz)}$.
 *Ob ein Gleichanteil durch das Filter entfernt wird oder nicht, ist unklar, da ein solcher im Signal&nbsp; $b(t)$&nbsp; nicht enthalten ist.

@@ Line 42: / Line 42: @@
 <quiz display=simple>
-{Welche Aussagen sind über das Filter &nbsp;$H_{\rm E}(f)$&nbsp; möglich, das zur Gewinnung des auf der Angabenseite dargestellten Sinkensignals benutzt wurde?
+{What statements can be made about the &nbsp;$H_{\rm E}(f)$&nbsp; filter used to obtain the sink signal shown on the page?
 |type="[]"}
-+ Die obere Grenzfrequenz ist zu hoch.
++ The upper cutoff frequency is too high.
-- Die obere Grenzfrequenz ist zu niedrig.
+- The upper cutoff frequency is too low.
-- Die untere Grenzfrequenz ist ungleich Null.
+- The lower cutoff frequency is not zero.
-{Mit welchen der nachfolgend aufgeführten Tiefpassfunktionen ist eine ideale Demodulation – das heißt&nbsp; $v(t) = q(t)$&nbsp; – prinzipiell möglich?
+{With which of the low-pass functions listed below is ideal demodulation - that is, &nbsp; $v(t) = q(t)$&nbsp; – possible in principle?
-|type="[]"}
++ Right angle low-pass,
-+ Rechtecktiefpass,
+- Gaussian low-pass,
-- Gaußtiefpass,
++ Trapezoidal low-pass,
-+ Trapeztiefpass,
+- Slit low-pass.
-{Wie ist die untere Eckfrequenz &nbsp;$f_1$&nbsp; eines Trapeztiefpasses mindestens zu wählen, damit keine Verzerrungen entstehen?
+{What is the minimum lower corner frequency &nbsp;$f_1$&nbsp; of a trapezoidal low-pass filter one can choose to avoid distortion?
 |type="{}"}
 $f_{\text{1, min}} \ = \ $ { 5 3% } $\ \text{kHz}$
-{Wie groß darf die obere Eckfrequenz &nbsp;$f_2$&nbsp; des Trapeztiefpasses höchstens sein, damit keine Verzerrungen entstehen?
+{What is the maximum upper corner frequency &nbsp;$f_2$&nbsp; of the trapezoidal low-pass that avoids distortion?
 |type="{}"}
 $f_{\text{2, max}} \ = \ $ { 95 3% } $\ \text{kHz}$
-{Welche Grenzfrequenz &nbsp;$f_{\rm G}$&nbsp; eines idealen, rechteckförmigen Tiefpasses würden Sie wählen, wenn Rauschstörungen nicht zu vernachlässigen sind?
+{Which cutoff frequency &nbsp;$f_{\rm G}$&nbsp; of an ideal rectangular low-pass would you choose if distortion is not negligible?
 |type="()"}
 - $f_{\rm G} = 4 \ \rm kHz$,
@@ Line 74: / Line 73: @@
 </quiz>
-===Musterlösung===
+===Solution===
 {{ML-Kopf}}
-'''(1)'''&nbsp; Richtig ist die <u>erste Aussage</u>:
+'''(1)'''&nbsp; <u>The first statement</u> is correct:
-*Das dargestellte Sinkensignal&nbsp; $v(t)$&nbsp; stimmt exakt mit dem als Gleichung gegebenen Signal&nbsp; $b(t)$&nbsp; überein und enthält somit auch Anteile um die doppelte Trägerfrequenz.
+*The sink signal shown&nbsp; $v(t)$&nbsp; stimmt exakt mit dem als Gleichung gegebenen Signal&nbsp; $b(t)$&nbsp; überein und enthält somit auch Anteile um die doppelte Trägerfrequenz.
 *Das Filter&nbsp; $H_{\rm E}(f)$&nbsp; fehlt entweder ganz oder dessen obere Grenzfrequenz&nbsp; $f_2$&nbsp; ist zu hoch.
 *Bezüglich der unteren Grenzfrequenz&nbsp; $f_1$&nbsp; ist nur die Aussage möglich, dass diese kleiner ist als die kleinste im Signal&nbsp; $b(t)$&nbsp; vorkommende Frequenz&nbsp; $\text{(2 kHz)}$.

@@ Line 33: / Line 33: @@
 ''Hints:''
 *This exercise belongs to the chapter &nbsp; [[Modulation_Methods/Synchronous_Demodulation|Synchronous Demodulation]].
-*Particular reference is made to the page &nbsp;  [[Modulation_Methods/Synchronous_Demodulation#Block_diagram_and_time_domain_representation|#Block diagram and time domain representation]].
+*Particular reference is made to the page &nbsp;  [[Modulation_Methods/Synchronous_Demodulation#Block_diagram_and_time_domain_representation|Block diagram and time domain representation]].
 *In contrast to &nbsp;[[Aufgaben:Exercise_2.4:_Frequency_and_Phase_Offset|Exercise 2.4]]&nbsp;, &nbsp;$f_1$&nbsp; and &nbsp;$f_2$&nbsp; do not describe signal frequencies, but instead relate to the low-pass filter.