Aufgaben:Exercise 2.6: Two-Way Channel - Revision history

Guenter at 09:29, 6 October 2021

2021-10-06T09:29:45Z

Guenter at 09:09, 6 October 2021

2021-10-06T09:09:33Z

Guenter at 09:08, 6 October 2021

2021-10-06T09:08:58Z

Guenter: Guenter moved page Aufgaben:Aufgabe 2.6: Zweiwegekanal to Aufgaben:Exercise 2.6: Two-Way Channel

2021-09-20T13:15:58Z

Guenter moved page Aufgabe 2.6: Zweiwegekanal to Exercise 2.6: Two-Way Channel

Guenter at 10:12, 20 September 2021

2021-09-20T10:12:32Z

Oezer at 08:11, 20 September 2021

2021-09-20T08:11:58Z

Oezer at 08:00, 20 September 2021

2021-09-20T08:00:54Z

Oezer at 08:00, 20 September 2021

2021-09-20T08:00:38Z

Oezer at 07:59, 20 September 2021

2021-09-20T07:59:30Z

Oezer at 07:59, 20 September 2021

2021-09-20T07:59:09Z

@@ Line 77: / Line 77: @@
 ===Solution===
 {{ML-Kopf}}
-'''(1)'''&nbsp; <u>Statements 1 and 2</u> are correct:
+'''(1)'''&nbsp; <u>Statements 1 and 2</u>&nbsp; are correct:
-*$h(t) = \delta(t)$&nbsp; is true with&nbsp;$z_1 = 1$, $T_1 = 0$ and $z_2 =0$&nbsp; and correspondingly&nbsp;$H(f) = 1$ so that&nbsp;$y(t) = x(t)$&nbsp; will always hold.
+*$h(t) = \delta(t)$&nbsp; is true with&nbsp; $z_1 = 1$,&nbsp; $T_1 = 0$&nbsp; and&nbsp; $z_2 =0$&nbsp; and&nbsp; correspondingly&nbsp; $H(f) = 1$&nbsp; so that&nbsp; $y(t) = x(t)$&nbsp; will always hold.
-*Each distortion-free channel impulse response&nbsp;$h(t)$&nbsp; consists of a single Dirac function, for example at&nbsp;$t = T_1$.
+*Each distortion-free channel impulse response&nbsp; $h(t)$&nbsp; consists of a single Dirac function,&nbsp; for example at&nbsp; $t = T_1$.
-*This case is accounted for in the model by&nbsp;$z_2 =0$&nbsp;. Thus, the frequency response is:
+*This case is accounted for in the model by&nbsp; $z_2 =0$.&nbsp; Thus, the frequency response is:
 :$$H(f)=  z_1\cdot {\rm e}^{-{\rm j}\cdot \hspace{0.05cm}2 \pi f T_1} \ \Rightarrow \ y(t) = z_1 \cdot x(t- T_1).$$
-*In contrast, the channel will lead to linear distortions whenever&nbsp;$z_1$&nbsp; and&nbsp;$z_2$&nbsp; are simultaneously non-zero.
+*In contrast, the channel will lead to linear distortions whenever&nbsp; $z_1$&nbsp; and&nbsp; $z_2$&nbsp; are simultaneously non-zero.
-'''(2)'''&nbsp; The Fourier transformation of the impulse response&nbsp;$h(t)$&nbsp; results in the equation:
+'''(2)'''&nbsp; The Fourier transform of the impulse response&nbsp; $h(t)$&nbsp; results in the equation:
 :$$H(f) =  z_1\cdot {\rm e}^{-{\rm j}\hspace{0.05cm} \cdot \hspace{0.05cm}2 \pi f T_1}+ z_2\cdot {\rm e}^{-{\rm j}\hspace{0.05cm} \cdot \hspace{0.05cm}2 \pi f T_2}
    .$$
-*The following is obtained with&nbsp;$z_1 = 1$, $T_1 = 0$, $z_2 =0.5$&nbsp; and&nbsp;$T_2 = 1 \ \rm ms$&nbsp;:
+*The following is obtained with&nbsp; $z_1 = 1$,&nbsp; $T_1 = 0$,&nbsp; $z_2 =0.5$&nbsp; and&nbsp; $T_2 = 1 \ \rm ms$:
 :$$H(f) =1 + 0.5 \cdot {\rm e}^{-{\rm j}\hspace{0.05cm} \cdot \hspace{0.05cm}2 \pi f T_2}.$$
-*Broken down by real and imaginary part, this yields:
+*Broken down by real and imaginary part,&nbsp; this yields:
 :$${\rm Re}\big[H(f)\big] = 1 + 0.5 \cdot \cos(2 \pi f \cdot 1\,{\rm  ms}) \ \Rightarrow \ \underline{{\rm Re}[H(f = f_1 =1 \ \rm kHz)] = 1.5}, $$
 :$${\rm Im}\big[H(f)\big] = -0.5 \cdot \sin(2 \pi f \cdot 1\,{\rm  ms}) \ \Rightarrow \ \underline{{\rm Im}\big[H(f = f_1 =1 \ \rm kHz)\big] = 0}, $$
-'''(3)'''&nbsp; The <u>first answer</u> is the only correct one:
+'''(3)'''&nbsp; The&nbsp; <u>first answer</u>&nbsp; is the only correct one:
-*From&nbsp; '''(2)'''&nbsp; it follows that the absolute value function is&nbsp;$|H(f)| = 1.5$&nbsp; and the phase function &nbsp;$b(f) \equiv 0$ for all multiples of&nbsp;$f_1 =1 \ \rm kHz$&nbsp;.
+*From&nbsp; '''(2)'''&nbsp; it follows the absolute value function is&nbsp; $|H(f)| = 1.5$&nbsp; and the phase function &nbsp; $b(f) \equiv 0$&nbsp; for all multiples of&nbsp; $f_1 =1 \ \rm kHz$ &nbsp; &rArr; &nbsp; $f= n \cdot f_1$.
 *Thus, the phase delay time is also zero in each case for these discrete frequency values.
-*But since the spectrum&nbsp;$X_1(f)$&nbsp; of the Dirac comb has spectral lines exactly at these frequencies, &nbsp;$y_1(t) = 1.5 \cdot x_1(t)$ holds.
+*But since the spectrum&nbsp; $X_1(f)$&nbsp; of the Dirac comb has spectral lines exactly at these frequencies, &nbsp; $y_1(t) = 1.5 \cdot x_1(t)$&nbsp; holds.
@@ Line 110: / Line 110: @@
    = \sqrt{1.25 +  \cos(2 \pi f \cdot T_2) }.$$
-*Thus, the following is obtained for the frequency&nbsp;$f_2 =0.25 \ \rm kHz$&nbsp;:
+*Thus, the following is obtained for the frequency&nbsp; $f_2 =0.25 \ \rm kHz$:
 :$$|H(f)| = \sqrt{1.25 +  \cos(\frac{\pi}{2} ) }= \sqrt{1.25} = 1.118.$$
@@ Line 123: / Line 123: @@
 *Thus, the phase delay time for this frequency is:
 :$$\tau_2 = \frac {b(f_2)}{2 \pi f_2}  = \frac {0.464}{2 \pi \cdot
-.25\,{\rm  kHz}} \approx 0.3\,{\rm  ms},$$
+.25\,{\rm  kHz}} \approx 0.3\,{\rm  ms}.$$
 *Hence, the following holds for the output signal:
@@ Line 134: / Line 134: @@
-'''(5)'''&nbsp; Both frequencies have the same attenuation factor&nbsp;$\alpha = 1.118$&nbsp;. Therefore, no attenuation distortions are observed.
+'''(5)'''&nbsp; Both frequencies have the same attenuation factor&nbsp; $\alpha = 1.118$&nbsp;. Therefore, no attenuation distortions are observed.
-*The following is obtained for the phase function with&nbsp;$f_3 = 1.25 \ \rm kHz$&nbsp; and &nbsp;$T_2 = 1 \ \rm ms$&nbsp;:
+*The following is obtained for the phase function with&nbsp; $f_3 = 1.25 \ \rm kHz$&nbsp; and &nbsp; $T_2 = 1 \ \rm ms$&nbsp;:
 :$$b(f = f_3)  = - {\rm arctan}\hspace{0.1cm}\frac{-0.5 \cdot \sin(
 .5 \pi)}{1+0.5 \cdot \cos(2.5 \pi)}= 0.464 = b(f = f_2),$$
-:so exactly the same value as for the frequency&nbsp;$f_2 = 0.25 \ \rm kHz$.
+:so exactly the same value as for the frequency&nbsp; $f_2 = 0.25 \ \rm kHz$.
-*Despite this, however, phase distortions now occur since for&nbsp;$f_3$&nbsp; the phase delay time is only&nbsp;$\tau  = 60 \  &micro; \rm s$&nbsp; anymore.
+*Despite this,&nbsp; however,&nbsp; phase distortions now occur since for&nbsp; $f_3$&nbsp; the phase delay time is only&nbsp; $\tau  = 60 \  &micro; \rm s$&nbsp; anymore.
-*So, the following can be formulated for the output signal:
+*So,&nbsp; the following can be formulated for the output signal:
 :$$y_3(t) = 1.118 \cdot \cos(2 \pi f_2 \cdot (t - 0.3\,{\rm  ms}) +
 .118
@@ Line 150: / Line 150: @@
 \cos(2 \pi f_3 \cdot  t - 27^\circ).$$
-Accordingly, the correct <u>answer is 3</u>:
+Accordingly, the correct&nbsp; <u>answer is 3</u>:
-*So, there are phase distortions although for both oscillations,&nbsp;$\varphi_2 = \varphi_3= 27^\circ$&nbsp; holds.
+*So, there are phase distortions although for both oscillations,&nbsp; $\varphi_2 = \varphi_3= 27^\circ$&nbsp; holds.
 *To avoid phase distortions
-**the phase delay times $\tau_2$ and $\tau_3$ would have to be equal and
+**the phase delay times&nbsp; $\tau_2$&nbsp; and&nbsp; $\tau_3$&nbsp; would have to be equal and
-**the phase values&nbsp;$\varphi_2$&nbsp; and &nbsp;$\varphi_3$&nbsp; would have to increase linearly with the corresponding frequencies.
+**the phase values&nbsp; $\varphi_2$&nbsp; and&nbsp; $\varphi_3$&nbsp; would have to increase linearly with the corresponding frequencies.
 {{ML-Fuß}}

@@ Line 10: / Line 10: @@
 *Except for a few combinations of the system parameters&nbsp; $z_1$,&nbsp; $T_1$,&nbsp; $z_2$&nbsp; and&nbsp;  $T_2$,&nbsp; this channel will result in linear distortions.
 *There is a distortion-free channel at hand only if not a single input signal is distorted by it.
-*This means: &nbsp; Even if the two-way channel is per se distorting,&nbsp;  there may be special cases where indeed&nbsp; $y(t) = \alpha \cdot x(t - \tau)$&nbsp; applies.
+*This means: &nbsp; Even if the two-way channel is per se distorting,&nbsp;  there may be special cases where indeed&nbsp; $y(t) = \alpha \cdot x(t - \tau)$.

@@ Line 4: / Line 4: @@
 [[File:P_ID912__LZI_A_2_6.png|right|frame|Impulse response of the two-way channel]]
-The so-called two-way channel is characterised by the following impulse response&nbsp; $($with&nbsp; $T_1 < T_2)$:
+The so-called&nbsp; "two-way channel"&nbsp; is characterised by the following impulse response&nbsp; $($with&nbsp; $T_1 < T_2)$:
 :$$h(t) = z_1 \cdot \delta ( t - T_1) + z_2 \cdot \delta (
 t - T_2).$$
 *Except for a few combinations of the system parameters&nbsp; $z_1$,&nbsp; $T_1$,&nbsp; $z_2$&nbsp; and&nbsp;  $T_2$,&nbsp; this channel will result in linear distortions.
-*There is a distortion-free channel at hand only if not a single input signal is distorted by it. This means: &nbsp; Even if the channel is distorting, there may be special cases where indeed&nbsp; $y(t) = \alpha \cdot x(t - \tau)$&nbsp; applies.
+*There is a distortion-free channel at hand only if not a single input signal is distorted by it.
 The test signals applied to the system input are:
-*a&nbsp; [[Signal_Representation/Time_Discrete_Signal_Representation#Dirac_comb_in_time_and_frequency_domain|Dirac comb]]&nbsp; $x_1(t)$&nbsp; at a time interval of&nbsp; $T_0 = 1 \ \rm ms$ whose spectral function&nbsp; $X_1(f)$&nbsp; is also a Dirac comb with an interval of&nbsp; $f_0 = 1/T_0 = 1 \ \rm kHz$:
+*a&nbsp; [[Signal_Representation/Time_Discrete_Signal_Representation#Dirac_comb_in_time_and_frequency_domain|Dirac comb]]&nbsp; $x_1(t)$&nbsp; at a time interval of&nbsp; $T_0 = 1 \ \rm ms$,&nbsp; whose spectral function&nbsp; $X_1(f)$&nbsp; is also a Dirac comb with an interval of&nbsp; $f_0 = 1/T_0 = 1 \ \rm kHz$:
 :$$x_1(t) = \sum_{n = - \infty}^{+\infty} \delta ( t - n \cdot T_0) ,\hspace{0.5cm} X_1(f) = T_0 \cdot \sum_{k = - \infty}^{+\infty} \delta ( f - k
 \cdot f_0) ,$$
@@ Line 33: / Line 34: @@
 *The task belongs to the chapter &nbsp; [[Linear_and_Time_Invariant_Systems/Linear_Distortions|Linear Distortions]].
-*To spare you calculations the result for the parameter set&nbsp; $\big [z_1 = 1$,&nbsp; $T_1 = 0$,&nbsp; $z_2 =0.5$,&nbsp;  $T_2 = 1 \ \rm ms\big ]$&nbsp; is given:
+*To spare you calculations&nbsp; the result for the parameter set&nbsp; $\big [z_1 = 1$,&nbsp; $T_1 = 0$,&nbsp; $z_2 =0.5$,&nbsp;  $T_2 = 1 \ \rm ms\big ]$&nbsp; is given:
 :$$|H(f = f_2)| = |H(f = f_3)| = \sqrt{1.25}   \approx 1.118, \; \; \; \;  b(f = f_2) = b(f = f_3) = \arctan (0.5)  \approx  0.464.$$
@@ Line 44: / Line 45: @@
 + The parameter set&nbsp;  $\big[z_1 = 1$, $T_1 = 0$, $z_2 =0 \big]$&nbsp; is the only possible one to describe the ideal channel.
 + Any distortion-free channel is captured by the two combinations&nbsp;  $\big[z_1 \ne 0, \; z_2 = 0 \big]$&nbsp;  or&nbsp;  $\big[z_1 = 0, \; z_2 \ne 0 \big]$&nbsp;.
-- The values&nbsp;  $\big[z_1 \ne 0\big]$&nbsp; and&nbsp;  $\big[z_2 \ne 0\big]$&nbsp; result in a distortion-free channel if&nbsp; $T_1$&nbsp; and&nbsp; $T_2$&nbsp; are optimally adjusted.
+- The values&nbsp;  $\big[z_1 \ne 0\big]$&nbsp; and&nbsp;  $\big[z_2 \ne 0\big]$&nbsp; result in a distortion-free channel&nbsp; if&nbsp; $T_1$&nbsp; and&nbsp; $T_2$&nbsp; are optimally adjusted.
@@ Line 53: / Line 54: @@
-{The Dirac comb&nbsp;$x_1(t)$&nbsp; is applied to the input of the system with the same parameters as in subtask&nbsp; '''(2)'''&nbsp;. <br>Which statements are true for the output signal&nbsp;$y_1(t)$&nbsp;?
+{The Dirac comb&nbsp; $x_1(t)$&nbsp; is applied to the input of the system with the same parameters as in subtask&nbsp; '''(2)'''&nbsp;. <br>Which statements are true for the output signal&nbsp; $y_1(t)$&nbsp;?
 |type="[]"}
 + $y_1(t)$&nbsp; is attenuated/amplified by a constant compared to&nbsp;$x_1(t)$&nbsp;.
@@ Line 60: / Line 61: @@
-{Compute the signal&nbsp;$y_2(t)$&nbsp; as the system response to the cosine signal&nbsp;$x_2(t)$. &nbsp; What is the signal value at time&nbsp;$t = 0$&nbsp;?
+{Compute the signal&nbsp; $y_2(t)$&nbsp; as the system response to the cosine signal&nbsp; $x_2(t)$. &nbsp; What is the signal value at time&nbsp; $t = 0$&nbsp;?
 |type="{}"}
 $y_2(t = 0) \ = \ $  { 0.996 3% }
-{Which statements are true regarding the signals&nbsp;$x_3(t)$&nbsp; and &nbsp;$y_3(t)$&nbsp;?
+{Which statements are true regarding the signals&nbsp; $x_3(t)$&nbsp; and &nbsp; $y_3(t)$&nbsp;?
 |type="[]"}
-- $y_3(t)$&nbsp; does not exhibit any distortions with respect to&nbsp;$x_3(t)$&nbsp;.
+- $y_3(t)$&nbsp; does not exhibit any distortions with respect to&nbsp; $x_3(t)$&nbsp;.
-- $y_3(t)$&nbsp; exhibits attenuation distortions with respect to&nbsp;$x_3(t)$&nbsp;.
+- $y_3(t)$&nbsp; exhibits attenuation distortions with respect to&nbsp; $x_3(t)$&nbsp;.
-+ $y_3(t)$&nbsp; exhibits phase distortions with respect to&nbsp;$x_3(t)$&nbsp;.
++ $y_3(t)$&nbsp; exhibits phase distortions with respect to&nbsp; $x_3(t)$&nbsp;.

← Older revision		Revision as of 10:12, 20 September 2021
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−	[[Category:Linear and Time-Invariant Systems: Exercises\|^2.3 ~~Lineare Verzerrungen~~^]]	+	[[Category:Linear and Time-Invariant Systems: Exercises\|^2.3 Linear Distortions^]]

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 </quiz>
-===Solutions===
+===Solution===
 {{ML-Kopf}}
 '''(1)'''&nbsp; <u>Statements 1 and 2</u> are correct:

@@ Line 128: / Line 128: @@
 .3\,{\rm  ms})).$$
-*The signal value at zero-time is thus:
+*The signal value at zero-time is therefore:
 :$$y_2(t=0) = 1.118 \cdot \cos(-2 \pi \cdot 0.25\,{\rm  kHz} \cdot
 .3\,{\rm  ms}) \approx 1.118 \cdot  0.891 \hspace{0.15cm}\underline{= 0.996}.$$