Aufgaben:Exercise 2.5: Distortion and Equalization - Revision history

Guenter at 14:56, 5 October 2021

2021-10-05T14:56:53Z

Guenter at 14:46, 5 October 2021

2021-10-05T14:46:13Z

Guenter at 14:42, 5 October 2021

2021-10-05T14:42:50Z

Guenter at 14:42, 5 October 2021

2021-10-05T14:42:04Z

Guenter at 14:39, 5 October 2021

2021-10-05T14:39:13Z

Guenter at 14:36, 5 October 2021

2021-10-05T14:36:44Z

Guenter: Guenter moved page Aufgaben:Exercise 2.5: Distortion and Equalisation to Aufgaben:Exercise 2.5: Distortion and Equalization

2021-10-05T12:21:24Z

Guenter moved page Exercise 2.5: Distortion and Equalisation to Exercise 2.5: Distortion and Equalization

Javier: Text replacement - "equalise" to "equalize"

2021-09-21T12:51:18Z

Text replacement - "equalise" to "equalize"

Javier: Text replacement - "equalisation" to "equalization"

2021-09-21T12:50:08Z

Text replacement - "equalisation" to "equalization"

Guenter: Guenter moved page Aufgaben:Aufgabe 2.5: Verzerrung und Entzerrung to Aufgaben:Exercise 2.5: Distortion and Equalisation

2021-09-20T13:13:37Z

Guenter moved page Aufgabe 2.5: Verzerrung und Entzerrung to Exercise 2.5: Distortion and Equalisation

@@ Line 92: / Line 92: @@
 ===Solution===
 {{ML-Kopf}}
-'''(1)'''&nbsp; <u>Proposed solutions 1 and 3</u> are correct:
+'''(1)'''&nbsp; <u>Proposed solutions 1 and 3</u>&nbsp; are correct:
 *A linear system is already implicitly assumed by specifying a frequency response so that nonlinear distortions cannot occur.
 *Since&nbsp; $H(f)$&nbsp; is purely real, phase distortions can also be ruled out.
@@ Line 104: / Line 104: @@
-'''(3)'''&nbsp; <u>Proposed solution 3</u> is correct:
+'''(3)'''&nbsp; <u>Proposed solution 3</u>&nbsp; is correct:
 *In this case, the following is obtained for the output signal:
 :$$y_1(t)=  1\,{\rm  V}\cdot \cos(2 \pi \cdot f_1 \cdot t) + {1}/{4}\cdot 1\,{\rm  V}\cdot \sin(2 \pi \cdot f_2 \cdot t).$$
@@ Line 112: / Line 112: @@
-'''(4)'''&nbsp; The output signal&nbsp; $y_2(t)$&nbsp; has the following form taking into account the fundamental frequency&nbsp; $f_0 = 3 \ \rm kHz$&nbsp;:
+'''(4)'''&nbsp; The output signal&nbsp; $y_2(t)$&nbsp; has the following form taking into account the basic frequency&nbsp; $f_0 = 3 \ \rm kHz$:
 :$$y_2(t)=  \frac{8\,{\rm  V}}{\pi^2} \left( \cos(\omega_0  t) +
   \frac{3}{8}\cdot \frac{1}{9} \cdot \cos(3\omega_0  t)\right)
 .$$
-*The factor&nbsp; $3/8$&nbsp; describes&nbsp; $H(f = 9 \ \rm kHz)$. All other spectral components at&nbsp; $15 \ \rm kHz$,&nbsp; $21 \ \rm kHz$,&nbsp; etc. are suppressed by the system.
+*The factor&nbsp; $3/8$&nbsp; describes&nbsp; $H(f = 9 \ \rm kHz)$.&nbsp; All other spectral components at&nbsp; $15 \ \rm kHz$,&nbsp; $21 \ \rm kHz$,&nbsp; etc. are suppressed by the system.
-*The strongest deviations between&nbsp; $x_2(t)$&nbsp; and&nbsp; $y_2(t)$&nbsp; will occur at the triangle peaks since the missing high frequencies have the strongest effect here. For example, for the time&nbsp; $\underline{t= 0}$ one obtains:
+*The strongest deviations between&nbsp; $x_2(t)$&nbsp; and&nbsp; $y_2(t)$&nbsp; will occur at the triangle peaks since the missing high frequencies have the strongest effect here.
 :$$y_2(t=0)=  \frac{8\,{\rm  V}}{\pi^2} \left( 1 +
   {3}/{72}\right)= 0.844\,{\rm
@@ Line 128: / Line 129: @@
-'''(5)'''&nbsp; With the fundamental frequency&nbsp; $f_0 = 2 \ \rm kHz$&nbsp; and the transmission values&nbsp;  $H(3f_0) = 0.75$,&nbsp; $H(5f_0) = 0.25$,&nbsp; $H(7f_0) = 0$&nbsp; the following is obtained:
+'''(5)'''&nbsp; With the basic frequency&nbsp; $f_0 = 2 \ \rm kHz$&nbsp; and the values&nbsp;  $H(3f_0) = 0.75$,&nbsp; $H(5f_0) = 0.25$,&nbsp; $H(7f_0) = 0$&nbsp; the following is obtained:
 :$$y_2(t=0)=  \frac{8\,{\rm  V}}{\pi^2} \left( 1 +
   \frac{3}{4}\cdot \frac{1}{9} + \frac{1}{4} \cdot\frac{1}{25}\right)= 0.886\,{\rm
@@ Line 137: / Line 138: @@
-'''(6)'''&nbsp; In the range up to&nbsp; $4  \  \rm kHz$,&nbsp; $H_{\rm E}(f) = H(f) = 1$&nbsp; is to be set. In contrast, in the range from&nbsp; $4  \  \rm kHz$&nbsp; to&nbsp; $12  \  \rm kHz$ the following holds:
+'''(6)'''&nbsp; In the range up to&nbsp; $4  \  \rm kHz$,&nbsp; $H_{\rm E}(f) = H(f) = 1$&nbsp; is to be set.&nbsp; In contrast, in the range from&nbsp; $4  \  \rm kHz$&nbsp; to&nbsp; $12  \  \rm kHz$ the following holds:
 :$$H_{\rm  E}(f)=  \frac{1}{H(f)} =
   \frac{1}{1.5 \cdot \big[1 - f/(12\,{\rm  kHz})\big]}

@@ Line 65: / Line 65: @@
-{For the test signal&nbsp; $x_2(t)$&nbsp;  with&nbsp; $\underline{f_0 = 3 \ \rm kHz}$,&nbsp; what is the maximum deviation&nbsp;$\varepsilon_{\rm max} = |y_2(t_0) - x_2(t_0)|$. <br>At what time&nbsp;$t_0$&nbsp; does&nbsp; $\varepsilon_{\rm max}$&nbsp; occur for the first time?
+{For the test signal&nbsp; $x_2(t)$&nbsp;  with&nbsp; $\underline{f_0 = 3 \ \rm kHz}$,&nbsp; what is the maximum deviation&nbsp; $\varepsilon_{\rm max} = |y_2(t_0) - x_2(t_0)|$. <br>At what time&nbsp; $t_0$&nbsp; does&nbsp; $\varepsilon_{\rm max}$&nbsp; occur for the first time?
 |type="{}"}
 $\varepsilon_\text{max} \ = \ $ { 0.156 3% } $\ \rm V$
@@ Line 71: / Line 71: @@
-{What is the maximum deviation&nbsp;$\varepsilon_{\rm max}$&nbsp; with&nbsp;$\underline{f_0 = 2 \ \rm kHz}$?
+{What is the maximum deviation&nbsp; $\varepsilon_{\rm max}$&nbsp; with&nbsp; $\underline{f_0 = 2 \ \rm kHz}$?
 |type="{}"}
 $\varepsilon_\text{max} \ = \ $ { 0.114 3% } $\ \rm V$

@@ Line 35: / Line 35: @@
 ''Please note:''
 *The task belongs to the chapter&nbsp; [[Linear_and_Time_Invariant_Systems/Linear_Distortions|Linear Distortions]].
-*In particular, reference is made to the page&nbsp;  [[Linear_and_Time_Invariant_Systems/Linear_Distortions#Equalization_method|Equalization methods]].
+*In particular, reference is made to the page&nbsp;  [[Linear_and_Time_Invariant_Systems/Linear_Distortions#Equalization_methods|Equalization methods]].
 *The term&nbsp; "overall distortion"&nbsp; used in the formulation of the questions refers to the input signal&nbsp; $x(t)$&nbsp; and the output signal&nbsp; $z(t)$.

@@ Line 35: / Line 35: @@
 ''Please note:''
 *The task belongs to the chapter&nbsp; [[Linear_and_Time_Invariant_Systems/Linear_Distortions|Linear Distortions]].
-*In particular, reference is made to the page&nbsp;  [[Linear_and_Time_Invariant_Systems/Linear_Distortions#Entzerrungsverfahren|Entzerrungsverfahren]].
+*In particular, reference is made to the page&nbsp;  [[Linear_and_Time_Invariant_Systems/Linear_Distortions#Equalization_method|Equalization methods]].
-*The term "overall distortion" used in the formulation of the questions refers to the input signal&nbsp; $x(t)$&nbsp; and the output signal&nbsp; $z(t)$.
+*The term&nbsp; "overall distortion"&nbsp; used in the formulation of the questions refers to the input signal&nbsp; $x(t)$&nbsp; and the output signal&nbsp; $z(t)$.

@@ Line 21: / Line 21: @@
   + {1}/{25} \cdot \cos(5\omega_0  t) + \hspace{0.05cm}\text{...}\big].$$
 :It should be noted that the basic frequency is&nbsp; $f_0 = 2 \ \rm kHz$&nbsp; or&nbsp; $3\ \rm kHz$.&nbsp; At time&nbsp; $t = 0$&nbsp; the signal value in both cases is&nbsp; $1 \ \rm V$.
-*A rectangular pulse&nbsp; $x_3(t)$&nbsp; with amplitude&nbsp; $A = 1 \ \rm V$&nbsp; and duration&nbsp; $T = 1 \ \rm ms$. Since its spectrum&nbsp; $X_3(f)$&nbsp; extends to infinity, &nbsp; $H(f)$&nbsp; always results in linear distortions here.
+*A rectangular pulse&nbsp; $x_3(t)$&nbsp; with amplitude&nbsp; $A = 1 \ \rm V$&nbsp; and duration&nbsp; $T = 1 \ \rm ms$. <br>Since its spectrum&nbsp; $X_3(f)$&nbsp; extends to infinity, &nbsp; $H(f)$&nbsp; always results in linear distortions here.

@@ Line 3: / Line 3: @@
 }}
-[[File:P_ID907__LZI_A_2_5.png|right|frame|Trapezoidal spectrum (top) and <br>associated impulse response (bottom)]]
+[[File:P_ID907__LZI_A_2_5.png|right|frame|Trapezoidal spectrum (top), <br>associated impulse response)]]
-A communication system with input&nbsp; $x(t)$&nbsp; and output&nbsp; $y(t)$, which is fully described by the trapezoidal frequency response&nbsp; $H(f)$&nbsp; according to the top graph, is considered. Using the roll-off factor&nbsp; $r = 0.5$&nbsp; and the equivalent bandwidth&nbsp; $\Delta f = 16 \ \rm kHz$&nbsp; the corresponding impulse response, which is computable by applying the inverse Fourier transform, is:
+A communication system with input&nbsp; $x(t)$&nbsp; and output&nbsp; $y(t)$,&nbsp; which is fully described by the trapezoidal frequency response&nbsp; $H(f)$&nbsp; according to the top graph, is considered.&nbsp; Using the roll-off factor&nbsp; $r = 0.5$&nbsp; and the equivalent bandwidth&nbsp; $\Delta f = 16 \ \rm kHz$&nbsp; the corresponding impulse response,&nbsp; which is computable by applying the inverse Fourier transform, is:
 :$$h(t) = \Delta f \cdot {\rm si}(\pi \cdot \Delta f \cdot t )\cdot
 {\rm si}(\pi \cdot r \cdot \Delta f \cdot t
-) .$$
+) = \Delta f \cdot {\rm sinc}(\Delta f \cdot t )\cdot
+{\rm sinc}(r \cdot \Delta f \cdot t
 The available input signals are:
 *The sum of two harmonic oscillations:
 :$$x_1(t) =  {1\, \rm V} \cdot \cos(\omega_1 \cdot  t) + {1\, \rm V} \cdot \sin(\omega_2 \cdot
   t).$$
-:Here, the following holds:&nbsp; $\omega_1 = 2\pi \cdot 2000 \ {\rm 1/s}$&nbsp; and&nbsp; $\omega_2 \gt \omega_1$.
+:Here, the following holds: &nbsp; $\omega_1 = 2\pi \cdot 2000 \ {\rm 1/s}$&nbsp; and&nbsp; $\omega_2 \gt \omega_1$.
 *A periodic triangular signal:
 :$$x_2(t) =  \frac{8\, \rm V}{\pi^2} \cdot \big[\cos(\omega_0  t) + {1}/{9} \cdot \cos(3\omega_0  t)
   + {1}/{25} \cdot \cos(5\omega_0  t) + \hspace{0.05cm}\text{...}\big].$$
-:It should be noted that the fundamental frequency is&nbsp; $f_0 = 2 \ \rm kHz$&nbsp; or&nbsp; $3\ \rm kHz$&nbsp;. At time&nbsp; $t = 0$&nbsp; the signal value in both cases is&nbsp; $1 \ \rm V$.
+:It should be noted that the basic frequency is&nbsp; $f_0 = 2 \ \rm kHz$&nbsp; or&nbsp; $3\ \rm kHz$.&nbsp; At time&nbsp; $t = 0$&nbsp; the signal value in both cases is&nbsp; $1 \ \rm V$.
 *A rectangular pulse&nbsp; $x_3(t)$&nbsp; with amplitude&nbsp; $A = 1 \ \rm V$&nbsp; and duration&nbsp; $T = 1 \ \rm ms$. Since its spectrum&nbsp; $X_3(f)$&nbsp; extends to infinity, &nbsp; $H(f)$&nbsp; always results in linear distortions here.

← Older revision	Revision as of 12:21, 5 October 2021
(No difference)

@@ Line 21: / Line 21: @@
-From subtask&nbsp; '''(6)'''&nbsp; onwards, it shall be attempted to eliminate the distortions possibly generated by&nbsp; $H(f)$&nbsp; by means of a downstream equaliser with
+From subtask&nbsp; '''(6)'''&nbsp; onwards, it shall be attempted to eliminate the distortions possibly generated by&nbsp; $H(f)$&nbsp; by means of a downstream equalizer with
 * frequency response&nbsp; $H_{\rm E}(f)$,
 * input signal&nbsp; $y(t)$,&nbsp; and
@@ Line 73: / Line 73: @@
-{What curve shape should the equaliser&nbsp; $H_{\rm E}(f)$&nbsp; have to compensate all distortions of&nbsp; $H(f)$&nbsp; in the best possible way? <br>What magnitude value arises as a result for&nbsp; $\underline{f = 10 \ \rm kHz}$?
+{What curve shape should the equalizer&nbsp; $H_{\rm E}(f)$&nbsp; have to compensate all distortions of&nbsp; $H(f)$&nbsp; in the best possible way? <br>What magnitude value arises as a result for&nbsp; $\underline{f = 10 \ \rm kHz}$?
 |type="{}"}
 $|H_E(f = 10 \ \rm kHz)| \ = \ $ { 4 3% }
@@ Line 147: / Line 147: @@
 '''(7)'''&nbsp; <u>Proposed solution 1</u> is correct:
 *Both&nbsp; $x_2(t)$&nbsp; and&nbsp; $x_3(t)$&nbsp; also contain spectral components at frequencies greater than&nbsp; $12  \  \rm kHz$.
-*If these have been truncated by&nbsp; $H(f)$&nbsp; &rArr; &nbsp; band limitation, they can no longer be reconstructed by the equaliser.
+*If these have been truncated by&nbsp; $H(f)$&nbsp; &rArr; &nbsp; band limitation, they can no longer be reconstructed by the equalizer.
 *This means that only the signal&nbsp; $x_1(t)$&nbsp; can be recovered by&nbsp; $H_{\rm E}(f)$&nbsp; but only if&nbsp; $f_2 < 12  \  \rm kHz$ holds:
 :$$z_1(t)=  \underline{1} \cdot 1\,{\rm  V}\cdot \cos(2 \pi \cdot f_1 \cdot t) +  \underline{4} \cdot \frac{1}{4}\cdot 1\,{\rm  V}\cdot \sin(2 \pi \cdot f_2 \cdot t).$$

@@ Line 78: / Line 78: @@
-{For which of the listed signals is complete equalisation possible? <br>$z(t) = x(t)$&nbsp; should be understood by "complete equalisation".
+{For which of the listed signals is complete equalization possible? <br>$z(t) = x(t)$&nbsp; should be understood by "complete equalization".
 |type="[]"}
 + For signal&nbsp; $x_1(t)$&nbsp; with&nbsp; $f_2 = 10 \ \rm kHz$,

← Older revision	Revision as of 13:13, 20 September 2021
(No difference)