Aufgaben:Exercise 2.4: Distortion Factor and Distortion Power - Revision history

Guenter at 12:56, 1 October 2021

2021-10-01T12:56:40Z

Guenter at 12:45, 1 October 2021

2021-10-01T12:45:09Z

Guenter at 10:25, 1 October 2021

2021-10-01T10:25:20Z

Javier: Text replacement - "analyse" to "analyze"

2021-09-22T11:47:37Z

Text replacement - "analyse" to "analyze"

Guenter at 10:10, 20 September 2021

2021-09-20T10:10:06Z

Guenter: Guenter moved page Aufgaben:Aufgabe 2.4: Klirrfaktor und Verzerrungsleistung to Aufgaben:Exercise 2.4: Distortion Factor and Distortion Power

2021-09-20T10:07:17Z

Guenter moved page Aufgabe 2.4: Klirrfaktor und Verzerrungsleistung to Exercise 2.4: Distortion Factor and Distortion Power

Oezer at 15:00, 15 September 2021

2021-09-15T15:00:34Z

Oezer at 14:59, 15 September 2021

2021-09-15T14:59:47Z

Oezer at 14:59, 15 September 2021

2021-09-15T14:59:19Z

Oezer at 14:58, 15 September 2021

2021-09-15T14:58:08Z

@@ Line 111: / Line 111: @@
 '''(3)'''&nbsp; <u>The first two proposed solutions</u> are correct:
 *Here, the nonlinear distortions cause the lower half-wave to be more peaked than the upper half-wave.
-*In addition, &nbsp; $y_{\rm max} = 1.75 \ \rm V$&nbsp; and&nbsp; $y_{\rm min} = -2.25 \ \rm V$ hold since&nbsp; $y(t)$&nbsp; does not contain any direct (DC) signals. Hence, there is no symmetry with respect to the zero line anymore.
+*In addition, &nbsp; $y_{\rm max} = 1.75 \ \rm V$&nbsp; and&nbsp; $y_{\rm min} = -2.25 \ \rm V$ hold since&nbsp; $y(t)$&nbsp; does not contain any direct (DC) signals. <br>Hence, there is no symmetry with respect to the zero line anymore.
 *For a nonlinear system, the distortion factor&nbsp; $K$&nbsp; is independent of the frequency of the cosine input signal but strongly dependent on its amplitude.
-'''(4)'''&nbsp; The root mean square (rms) value of a cosine signal is known to be&nbsp; $\sqrt{0.5}$&ndash;times the amplitude. The square of this is the "power":
+'''(4)'''&nbsp; The root mean square (rms) value of a cosine signal is known to be&nbsp; $\sqrt{0.5}$&ndash;times the amplitude.&nbsp; The square of this is the "power":
 :$$P_x = \frac{A_x^2}{2} = \frac{(2 \,{\rm V})^2}{2}\hspace{0.15cm}\underline{ = 2\,{\rm V^2}}.$$
-*Actually, the power also depends on the reference resistance&nbsp;$R$&nbsp; and has the unit "watt".
+*Actually, the power also depends on the reference resistance&nbsp;$R$&nbsp; and has the unit "Watt".
 *$P_x =  2 \ \rm W$ is obtained with&nbsp;$R = 1 \ \rm \Omega$&nbsp;, so exactly the same numerical value as in this simpler calculation.
@@ Line 125: / Line 125: @@
 '''(5)'''&nbsp; Denoting
-*the amplitude of the fundamental wave of&nbsp; $y_2(t)$ by&nbsp; $A_1$&nbsp; and
+*the amplitude of the basic wave of&nbsp; $y_2(t)$&nbsp; by&nbsp; $A_1$&nbsp; and
-*the so-called harmonics by&nbsp; $A_2$,&nbsp; $A_3$&nbsp; and&nbsp; $A_4$&nbsp;,
+*the so-called harmonics by&nbsp; $A_2$,&nbsp; $A_3$&nbsp; and&nbsp; $A_4$,&nbsp;
@@ Line 135: / Line 135: @@
 \,{\rm V})^2 \big] \hspace{0.15cm}\underline{\approx 0.031 \,{\rm V}^2}.$$
-Here,&nbsp; $A_x$&nbsp; denotes the amplitude of the input signal. The signs of the harmonics do not matter in this calculation.
+Here,&nbsp; $A_x$&nbsp; denotes the amplitude of the input signal.&nbsp; The signs of the harmonics do not matter in this calculation.
@@ Line 153: / Line 153: @@
 + \text{...} }{A_x^2}.$$
-*When calculating the distortion power&nbsp; $P_{\rm V}$&nbsp; a falsification of the amplitude of the fundamental wave&nbsp; $($this is now&nbsp; $A_1$&nbsp; instead of&nbsp; $A_x)$&nbsp; is also taken into account. Moreover, the distortion power is not in terms of&nbsp; $A_1^2$ but in terms of&nbsp; $A_x^2$&nbsp;.
+*When calculating the distortion power&nbsp; $P_{\rm V}$&nbsp; a falsification of the amplitude of the basic wave&nbsp; $($this is now&nbsp; $A_1$&nbsp; instead of&nbsp; $A_x)$&nbsp; is also taken into account.&nbsp; Moreover, the distortion power is not in terms of&nbsp; $A_1^2$&nbsp; but in terms of&nbsp; $A_x^2$&nbsp;.
 *Generally, the following relationship holds between the signal&ndash;to&ndash;distortion&ndash;power ratio and the distortion factor:
@@ Line 162: / Line 162: @@
-''Remarks:''
+Remarks:
-*A distortion factor of&nbsp; $1\%$&nbsp; in this case corresponds to the result&nbsp; $10 \cdot \lg \rho_{\rm V} = 40 \,{\rm dB}$.
+*A distortion factor of&nbsp; $1\%$&nbsp; corresponds in this case  to the result&nbsp; $10 \cdot \lg \rho_{\rm V} = 40 \,{\rm dB}$.
-*Using the distortion factor&nbsp; $K = 0.125$&nbsp; from subtask&nbsp; '''(2)'''&nbsp; &nbsp; $10 \cdot \lg \rho_{\rm V} = 18.06 \,{\rm dB}$&nbsp; would have been obtained immediately with the approximation&nbsp; $A_1 \approx A_x$&nbsp;.
+*Using the distortion factor&nbsp; $K = 0.125$&nbsp; from subtask&nbsp; '''(2)''' &nbsp; &rArr; &nbsp; $10 \cdot \lg \rho_{\rm V} = 18.06 \,{\rm dB}$&nbsp; would have been obtained immediately with the approximation&nbsp; $A_1 \approx A_x$&nbsp;.
 *The actual value &nbsp; $(18.10 \ \rm dB)$&nbsp; calculated in subtask&nbsp; '''(7)'''&nbsp; differs from this only insignificantly.

@@ Line 40: / Line 40: @@
 ''Please note:''
-*The task belongs to the chapter&nbsp; [[Linear_and_Time_Invariant_Systems/Nonlinear_Distortion|Nonlinear Distortion]].
+*The task belongs to the chapter&nbsp; [[Linear_and_Time_Invariant_Systems/Nonlinear_Distortion|Nonlinear Distortions]].
 *All powers required here refer to the resistance&nbsp;$R = 1 \ \rm \Omega$&nbsp; and thus have the unit&nbsp;${\rm V}^2$.

@@ Line 3: / Line 3: @@
 }}
-[[File:P_ID897__LZI_A_2_4.png|right|frame|On the meaning of the distortion factor]]
+[[File:P_ID897__LZI_A_2_4.png|right|frame|Given input and output signals]]
 A cosine signal
 :$$x_1(t) =  A_x  \cdot \cos(\omega_0 t)$$
-with the amplitude&nbsp; $A_x = 1 \ \rm V$&nbsp; is applied to the input of a message transmission system to test it. Then, the following signal occurs at the system output:
+with the amplitude&nbsp; $A_x = 1 \ \rm V$&nbsp; is applied to the input of a communication system to test it.
 :$$y_1(t) = {0.992 \,\rm V}  \cdot \cos(\omega_0 t) - {0.062 \,\rm
 V} \cdot \cos(2\omega_0 t)+ \hspace{0.05cm}\text{...}$$
-In the upper graph the signals&nbsp; $x_1(t)$&nbsp; and&nbsp; $y_1(t)$&nbsp; are shown. Harmonics with amplitudes smaller than&nbsp; $10 \ \rm mV$&nbsp; are not considered here.
+In the upper graph the signals&nbsp; $x_1(t)$&nbsp; and&nbsp; $y_1(t)$&nbsp; are shown. Harmonics with amplitudes&nbsp; $\lt 10 \ \rm mV$&nbsp; are not considered here.
@@ Line 20: / Line 22: @@
 \hspace{0.05cm}\text{...}$$
-It is obvious that the indices "1" and "2" respectively denote the normalised amplitude of the input signal.
+It is obvious that the indices&nbsp; "1"&nbsp; and&nbsp; "2"&nbsp; respectively denote the normalized amplitude of the input signal.
 The system is supposed to be analyzed based on the signal&ndash;to&ndash;distortion&ndash;power ratio
@@ Line 27: / Line 29: @@
 defined in the section&nbsp;[[Linear_and_Time_Invariant_Systems/Classification_of_the_Distortions#Quantitative_measure_for_the_signal_distortions| Quantitative measure for the signal distortions]]&nbsp; and the distortion factor&nbsp; $K$&nbsp;:
 * $P_x$&nbsp; denotes the power of the input signal.
-* The distortion power&nbsp; $P_{\rm V}$&nbsp; respectively represents the power&nbsp; (the root mean square)&nbsp; of the difference signal&nbsp; $\varepsilon(t) = y(t) - x(t)$&nbsp;.
+* The distortion power&nbsp; (German:&nbsp; "Verrzerrungsleistung" &nbsp; &rArr; &nbsp; "V") &nbsp;$P_{\rm V}$&nbsp; represents the power&nbsp; (the root mean square)&nbsp; of the difference signal&nbsp; $\varepsilon(t) = y(t) - x(t)$&nbsp;.
+*To determine the powers&nbsp; $P_{x}$&nbsp; and&nbsp; $P_{\rm V}$&nbsp; it is necessary to take the average of the squared signals in each case. However, it is easier to calculate the powers in the frequency domain in this task.

@@ Line 22: / Line 22: @@
 It is obvious that the indices "1" and "2" respectively denote the normalised amplitude of the input signal.
-The system is supposed to be analysed based on the signal&ndash;to&ndash;distortion&ndash;power ratio
+The system is supposed to be analyzed based on the signal&ndash;to&ndash;distortion&ndash;power ratio
 :$$\rho_{\rm V} = { P_{x}}/{P_{\rm V}} \hspace{0.3cm} \Rightarrow \hspace{0.3cm}  10 \cdot \lg \hspace{0.1cm}\rho_{\rm V} =
 \cdot \lg \hspace{0.1cm}{ P_{x}}/{P_{\rm V}}\hspace{0.3cm}  \left( {\rm in \hspace{0.15cm} dB} \right)$$

← Older revision		Revision as of 10:10, 20 September 2021
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−	[[Category:Linear and Time-Invariant Systems: Exercises\|^2.2 ~~Nichtlineare Verzerrungen~~^]]	+	[[Category:Linear and Time-Invariant Systems: Exercises\|^2.2 Nonlinear Distortions^]]

@@ Line 129: / Line 129: @@
-the distortion power is thus by computing in the frequency domain:
+the distortion power is thus by computing it in the frequency domain:
 :$$P_{\rm V} = \frac{1}{2} \cdot \big[ (A_1 - A_x)^2 + A_2^2+
 A_3^2+ A_4^2\big] =  \frac{1}{2} \cdot \big[ (-2