Aufgaben:Exercise 2.2: DC Component of Signals - Revision history

Guenter at 16:30, 17 May 2021

2021-05-17T16:30:42Z

Guenter at 14:49, 12 April 2021

2021-04-12T14:49:22Z

Guenter at 14:42, 12 April 2021

2021-04-12T14:42:22Z

Guenter at 14:40, 12 April 2021

2021-04-12T14:40:49Z

Guenter at 14:36, 12 April 2021

2021-04-12T14:36:45Z

Guenter: Guenter moved page Aufgaben:Exercise 2.2: Direct Current Component of Signals to Aufgaben:Exercise 2.2: DC Component of Signals

2021-04-12T14:33:46Z

Guenter moved page Exercise 2.2: Direct Current Component of Signals to Exercise 2.2: DC Component of Signals

Noah at 09:21, 26 March 2021

2021-03-26T09:21:24Z

Javier: Text replacement - "Category:Exercises for Signal Representation" to "Category:Signal Representation: Exercises"

2021-03-23T12:39:29Z

Text replacement - "Category:Exercises for Signal Representation" to "Category:Signal Representation: Exercises"

Noah at 01:52, 4 January 2021

2021-01-04T01:52:37Z

Noah at 01:40, 4 January 2021

2021-01-04T01:40:51Z

@@ Line 3: / Line 3: @@
 }}
-[[File:P_ID273__Sig_A_2_2.png|right|frame|Square wave signal with and without DC component]]
+[[File:P_ID273__Sig_A_2_2.png|right|frame|Square wave signal with/ without DC component]]
 The graph shows six time signals defined for all times $($from&nbsp; $-\infty$&nbsp; to&nbsp; $+\infty)$.&nbsp; For all sample signals&nbsp; $x_i(t)$&nbsp; the associated spectral function can be written as:

@@ Line 17: / Line 17: @@
 ''Hint:''
-*This exercise belongs to the chapter&nbsp; [[Signal_Representation/Direct_Current_Signal_-_Limit_Case_of_a_Periodic_Signal|Direct Current Signal - Limit Case of a Periodic Signal]].
+*This exercise belongs to the chapter&nbsp; <br>[[Signal_Representation/Direct_Current_Signal_-_Limit_Case_of_a_Periodic_Signal|Direct Current Signal - Limit Case of a Periodic Signal]].
@@ Line 66: / Line 66: @@
 '''(2)'''&nbsp; Only <u>solution 5 is correct</u>:
-*If the DC component &nbsp; $1\text{V}$ is subtracted from the signal &nbsp; $x_5(t)$&nbsp;, the residual signal&nbsp; $\Delta x_5(t) = x5(t) - 1\text{V}$&nbsp; is equal to zero.
+*If the DC component &nbsp; $1\text{V}$ is subtracted from the signal &nbsp; $x_5(t)$,&nbsp; the residual signal&nbsp; $\Delta x_5(t) = x5(t) - 1\text{V}$&nbsp; is  zero.
 *Accordignly, the spectral function is&nbsp; $\Delta X_5(f) = 0$.
-*For all other time courses&nbsp; $\Delta x_i(t)$&nbsp; is not equal to zero and thus the associated spectral function &nbsp; $\Delta X_i(f)$ is also not equal to zero.
+*For all other time courses&nbsp; $\Delta x_i(t)ßne 0$&nbsp; and thus the associated spectral function &nbsp; $\Delta X_i(f)\ne 0$,&nbsp; too.
 '''(3)'''&nbsp; Given a periodic signal, averaging over a period duration is sufficient to calculate the DC signal component&nbsp; $A_0$&nbsp;.
-*For the sample signal&nbsp;  $x_3(t)$&nbsp; the period duration is&nbsp; $T_0 = 3\,\text{ms}$. This results in the required DC component:$$A_0=\rm \frac{1}{3\,ms}\cdot \big[1\,V\cdot 1\,ms+(-1\,V)\cdot 2\,ms \big]
+*For signal&nbsp;  $x_3(t)$&nbsp; the period duration is&nbsp; $T_0 = 3\,\text{ms}$.&nbsp; This results in the required DC component:$$A_0=\rm \frac{1}{3\,ms}\cdot \big[1\,V\cdot 1\,ms+(-1\,V)\cdot 2\,ms \big]
 \hspace{0.15cm}\underline{=-0.333\,V}.$$
@@ Line 79: / Line 79: @@
 '''(4)'''&nbsp; The signal&nbsp; $x_4(t)$&nbsp; can be written as:&nbsp; $x_4(t) = 0.5 \,{\rm V} + Δx_4(t)$.
-*Here&nbsp; $Δx_4(t)$&nbsp; denotes a square wave pulse with amplitude&nbsp; $0.5 \,{\rm V} $&nbsp; and duration&nbsp; $4 \,{\rm ms} $, which due to its finite duration does not contribute to the DC signal component..
+*Here&nbsp; $Δx_4(t)$&nbsp; denotes a rectangular pulse with amplitude&nbsp; $0.5 \,{\rm V} $&nbsp; and duration&nbsp; $4 \,{\rm ms} $,
 *Therefore&nbsp; $A_0 \hspace{0.15cm}\underline{=0.5 \,{\rm V}}$ applies here.
@@ Line 92: / Line 93: @@
 :$$A_0=\lim_{T_{\rm M}\to \infty}\frac{1}{T_{\rm M}}\int _{-T_{\rm M}/2}^{0}0 {\rm V} \cdot\, {\rm d } {\it t }+\lim_{T_{\rm M}\to \infty}\frac{1}{T_{\rm M}}\int _{0}^{+T_{\rm M}/2}1 \rm V \ {\rm d }{\it t }.$$
-*Only the second term makes a contribution. From this follows again :&nbsp; $A_0 \hspace{0.15cm}\underline{=0.5 \,{\rm V}}$.
+*Only the second term makes a contribution.&nbsp; From this follows again :&nbsp; $A_0 \hspace{0.15cm}\underline{=0.5 \,{\rm V}}$.
 {{ML-Fuß}}
 __NOEDITSECTION__
 [[Category:Signal Representation: Exercises|^2.2 Direct Current Signal^]]

@@ Line 11: / Line 11: @@
 Here:
 *$A_0$&nbsp; is the DC component of the signal.
-*$\Delta X_i(f)$&nbsp; is the spectrum of the residual signal reduced by the DC component&nbsp;
+*$\Delta X_i(f)$&nbsp; is the spectrum of the residual signal reduced by the DC component:&nbsp;
 :$$\Delta x_i(t) = x_i(t) - A_0.$$
@@ Line 23: / Line 20: @@

@@ Line 5: / Line 5: @@
 [[File:P_ID273__Sig_A_2_2.png|right|frame|Square wave signal with and without DC component]]
-The graph shows some time signals defined for all times $($from&nbsp; $-\infty$&nbsp; to&nbsp; $+\infty)$&nbsp;. For all six sample signals&nbsp; $x_i(t)$&nbsp; the associated spectral function can be written as:
+The graph shows six time signals defined for all times $($from&nbsp; $-\infty$&nbsp; to&nbsp; $+\infty)$.&nbsp; For all sample signals&nbsp; $x_i(t)$&nbsp; the associated spectral function can be written as:
 :$$X_i(f)=A_0\cdot{\rm \delta}(f)+\Delta X_i(f).$$
-Here
+Here:
-*$A_0$&nbsp; is the DC component, and
+*$A_0$&nbsp; is the DC component of the signal.
-*$\Delta X_i(f)$&nbsp; is the spectrum of the residual signal reduced by the DC component&nbsp; $\Delta x_i(t) = x_i(t) - A_0$.
+*$\Delta X_i(f)$&nbsp; is the spectrum of the residual signal reduced by the DC component&nbsp;
+:$$\Delta x_i(t) = x_i(t) - A_0.$$

← Older revision		Revision as of 14:36, 12 April 2021
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−	[[Category:Signal Representation: Exercises\|^2.2 Direct Current ~~Signal - Limit Case of a Periodic~~ Signal^]]	+	[[Category:Signal Representation: Exercises\|^2.2 Direct Current Signal^]]

← Older revision		Revision as of 12:39, 23 March 2021
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−	[[Category:~~Exercises for~~ Signal Representation\|^2.2 Direct Current Signal - Limit Case of a Periodic Signal^]]	+	[[Category:Signal Representation: Exercises\|^2.2 Direct Current Signal - Limit Case of a Periodic Signal^]]

@@ Line 78: / Line 78: @@
-'''(3)'''&nbsp; With a periodic signal, averaging over a period duration is sufficient to calculate the DC signal component&nbsp; $A_0$&nbsp;.
+'''(3)'''&nbsp; Given a periodic signal, averaging over a period duration is sufficient to calculate the DC signal component&nbsp; $A_0$&nbsp;.
-*Beim Beispielsignal&nbsp;  $x_3(t)$&nbsp; ist diese&nbsp; $T_0 = 3\,\text{ms}$. Damit ergibt sich der gesuchte Gleichanteil zu
+*For the sample signal&nbsp;  $x_3(t)$&nbsp; the period duration is&nbsp; $T_0 = 3\,\text{ms}$. This results in the required DC component:$$A_0=\rm \frac{1}{3\,ms}\cdot \big[1\,V\cdot 1\,ms+(-1\,V)\cdot 2\,ms \big]
 \hspace{0.15cm}\underline{=-0.333\,V}.$$
-'''(4)'''&nbsp; Für das Signal&nbsp; $x_4(t)$&nbsp; kann geschrieben werden:&nbsp; $x_4(t) = 0.5 \,{\rm V} + Δx_4(t)$.
+'''(4)'''&nbsp; The signal&nbsp; $x_4(t)$&nbsp; can be written as:&nbsp; $x_4(t) = 0.5 \,{\rm V} + Δx_4(t)$.
-*Hierbei bezeichnet&nbsp; $Δx_4(t)$&nbsp; einen Rechteckimpuls mit Amplitude&nbsp; $0.5 \,{\rm V} $&nbsp; und Dauer&nbsp; $4 \,{\rm ms} $, der wegen seiner endlichen Dauer nicht zum Gleichsignalanteil beiträgt.
+*Here&nbsp; $Δx_4(t)$&nbsp; denotes a square wave pulse with amplitude&nbsp; $0.5 \,{\rm V} $&nbsp; and duration&nbsp; $4 \,{\rm ms} $, which due to its finite duration does not contribute to the DC signal component..
-*Deshalb gilt hier&nbsp; $A_0 \hspace{0.15cm}\underline{=0.5 \,{\rm V}}$.
+*Therefore&nbsp; $A_0 \hspace{0.15cm}\underline{=0.5 \,{\rm V}}$ applies here.
-'''(5)'''&nbsp; Die allgemeine Gleichung zur Berechnung des Gleichsignalanteils lautet:
+'''(5)'''&nbsp; The general equation for calculating the DC signal component is:
 :$$A_0=\lim_{T_{\rm M}\to \infty}\frac{1}{T_{\rm M}}\int_{-T_{\rm M}/2}^{+T_{\rm M}/2}x(t)\, {\rm d }t.$$
-*Spaltet man dieses Integral in zwei Teilintegrale auf, so erhält man:
+*If one splits this integral into two partial integrals, one obtains:
 :$$A_0=\lim_{T_{\rm M}\to \infty}\frac{1}{T_{\rm M}}\int _{-T_{\rm M}/2}^{0}0 {\rm V} \cdot\, {\rm d } {\it t }+\lim_{T_{\rm M}\to \infty}\frac{1}{T_{\rm M}}\int _{0}^{+T_{\rm M}/2}1 \rm V \ {\rm d }{\it t }.$$
-*Nur der zweite Term liefert einen Beitrag. Daraus folgt wiederum&nbsp; $A_0 \hspace{0.15cm}\underline{=0.5 \,{\rm V}}$.
+*Only the second term makes a contribution. From this follows again :&nbsp; $A_0 \hspace{0.15cm}\underline{=0.5 \,{\rm V}}$.
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@@ Line 31: / Line 31: @@
 <quiz display=simple>
-{Welche der Signale beinhalten einen Gleichanteil, das heißt, bei welchen Signalen ist&nbsp; $A_0 \neq 0$?
+{Which of the signals contains a DC component, i.e. for which signals is &nbsp; $A_0 \neq 0$?
 |type="[]"}
 + Signal&nbsp; $x_1(t),$
-- Signal&nbsp; $x_2(t),$
+- signal&nbsp; $x_2(t),$
-+ Signal&nbsp; $x_3(t),$
++ signal&nbsp; $x_3(t),$
-+ Signal&nbsp; $x_4(t),$
++ signal&nbsp; $x_4(t),$
-+ Signal&nbsp; $x_5(t),$
++ signal&nbsp; $x_5(t),$
-+ Signal&nbsp; $x_6(t).$
++ signal&nbsp; $x_6(t).$
-{Bei welchen der Signale gilt für das „Restspektrum”&nbsp; $\Delta X_i(f) =0$?
+{For which of the signals is the „residual spectrum”&nbsp; $\Delta X_i(f) =0$?
 |type="[]"}
 - Signal&nbsp; $x_1(t),$
-- Signal&nbsp; $x_2(t),$
+- signal&nbsp; $x_2(t),$
-- Signal&nbsp; $x_3(t),$
+- signal&nbsp; $x_3(t),$
-- Signal&nbsp; $x_4(t),$
+- signal&nbsp; $x_4(t),$
-+ Signal&nbsp; $x_5(t),$
++ signal&nbsp; $x_5(t),$
-- Signal&nbsp; $x_6(t).$
+- signal&nbsp; $x_6(t).$
-{Wie groß ist der Gleichanteil des Signals&nbsp; $x_3(t)$?
+{What is the DC component of the signal&nbsp; $x_3(t)$?
 |type="{}"}
 $x_3(t)\hspace{-0.1cm}:\,\,A_0 \ = \ $  { -0.35--0.31 } &nbsp; ${\rm V}$
-{Wie groß ist der Gleichanteil des Signals&nbsp; $x_4(t)$?
+{What is the DC component of the signal&nbsp; $x_4(t)$?
 |type="{}"}
 $x_4(t)\hspace{-0.1cm}:\,\,A_0\ = \ $ { 0.5 3% } &nbsp; ${\rm V}$
-{Wie groß ist der Gleichanteil des Signals&nbsp; $x_6(t)$?
+{What is the DC component of the signal&nbsp; $x_6(t)$?
 |type="{}"}
 $x_6(t)\hspace{-0.1cm}:\,\,A_0\ = \ $ { 0.5 3% } &nbsp; ${\rm V}$
@@ Line 65: / Line 65: @@
 </quiz>
-===Musterlösung===
+===Solution===
 {{ML-Kopf}}
-'''(1)'''&nbsp; Richtig sind die <u>Antworten 1, 3, 4, 5 und 6</u>.
+'''(1)'''&nbsp; The correct <u>answers are 1, 3, 4, 5 and 6</u>.
-*Alle Signale mit Ausnahme von&nbsp; $x_2(t)$&nbsp; beinhalten einen Gleichsignalanteil.
+*All signals except&nbsp; $x_2(t)$&nbsp; contain a DC signal component.
+'''(3)'''&nbsp; With a periodic signal, averaging over a period duration is sufficient to calculate the DC signal component&nbsp; $A_0$&nbsp;.
 *Beim Beispielsignal&nbsp;  $x_3(t)$&nbsp; ist diese&nbsp; $T_0 = 3\,\text{ms}$. Damit ergibt sich der gesuchte Gleichanteil zu