Aufgaben:Exercise 4.2: Rectangular Spectra - Revision history

Guenter at 13:49, 5 May 2021

2021-05-05T13:49:48Z

Guenter at 13:34, 5 May 2021

2021-05-05T13:34:51Z

Guenter at 13:27, 5 May 2021

2021-05-05T13:27:29Z

Guenter at 12:20, 5 May 2021

2021-05-05T12:20:19Z

Guenter: Guenter moved page Aufgaben:Exercise 4.2: Rectangular-Shaped Spectra to Aufgaben:Exercise 4.2: Rectangular Spectra

2021-05-05T09:31:16Z

Guenter moved page Exercise 4.2: Rectangular-Shaped Spectra to Exercise 4.2: Rectangular Spectra

Javier: Text replacement - "bandpass" to "band-pass"

2021-05-03T15:13:21Z

Text replacement - "bandpass" to "band-pass"

Guenter at 08:50, 23 April 2021

2021-04-23T08:50:58Z

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

2021-03-23T12:39:18Z

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

Noah at 19:27, 4 February 2021

2021-02-04T19:27:27Z

Noah at 18:46, 4 February 2021

2021-02-04T18:46:04Z

@@ Line 28: / Line 28: @@
 ''Hints:''
 *This exercise belongs to the chapter&nbsp; [[Signal_Representation/Differences_and_Similarities_of_Low-Pass_and_Band-Pass_Signals|Differences and Similarities of Low-Pass and Band-Pass Signals]].
-*In this task, the function $\rm si(x) = \rm sin(x)/x = \rm sinc(x/π)$ is used.
+*In this task, the function&nbsp; $\rm si(x) = \rm sin(x)/x = \rm sinc(x/π)$&nbsp; is used.
 *Consider the following trigonometric relationship in the solution:
@@ Line 62: / Line 62: @@
 {{ML-Kopf}}
-'''(1)'''&nbsp;  The time&nbsp; $T_u$ &nbsp; &rArr; &nbsp; first zero of the LP signal&nbsp; $u(t)$&nbsp; &ndash; is equal to the reciprocal of the width of the rectangular spectrum, i.e. &nbsp; $1/(2\, \text{kHz} ) \hspace{0.15 cm}\underline{= 0.5 \, \text{ms}}$.
+'''(1)'''&nbsp;  The time&nbsp; $T_u$ &nbsp; &rArr; &nbsp; first zero of the low-pass signal&nbsp; $u(t)$&nbsp; &ndash; is equal to the reciprocal of the width of the rectangular spectrum, i.e. &nbsp; $1/(2\, \text{kHz} ) \hspace{0.15 cm}\underline{= 0.5 \, \text{ms}}$.
-*The pulse amplitude is equal to the rectangular area as shown in the sample solution for&nbsp; [[Aufgaben:Aufgabe_4.1:_Tiefpass-_und_Bandpass-Signale|task 4.1]]&nbsp;. From this follows&nbsp; $u_0\hspace{0.15 cm}\underline{= 2 \, \text{V}}$.
+*The pulse amplitude is equal to the rectangular area as shown in the sample solution for&nbsp; [[Aufgaben:Aufgabe_4.1:_Tiefpass-_und_Bandpass-Signale|Exercise 4.1]]&nbsp;.&nbsp; From this follows&nbsp; $u_0\hspace{0.15 cm}\underline{= 2 \, \text{V}}$.
-[[File:P_ID704__Sig_A_4_2_b_neu.png|250px|right|frame|Multiplication with cosine]]
+[[File:P_ID704__Sig_A_4_2_b_neu.png|250px|right|frame|Multiplication with a cosine function]]
 '''(2)'''&nbsp;  The band-pass spectrum can be represented with&nbsp; $f_{\rm T} = 4\, \text{kHz}$&nbsp; as follows:
@@ Line 73: / Line 73: @@
 \delta(f- f_{\rm T})+ \delta(f+ f_{\rm T})\right].$$
-According to the&nbsp; [[Signal_Representation/Fourier_Transform_Laws#Verschiebungssatz|Verschiebungssatz]]&nbsp;, the following then applies to the associated time signal:
+According to the&nbsp; [[Signal_Representation/Fourier_Transform_Theorems#Shifting_Theorem|Shifting Theorem]],&nbsp; the following then applies to the associated time signal:
 :$$w(t) = 2 \cdot u(t) \cdot {\cos} ( 2 \pi f_{\rm T} t) =   2 u_0
@@ Line 80: / Line 80: @@
 The graph shows
 *above the low&ndash;pass signal $u(t)$,
-*then the oscillation $c(t) = 2 · \cos(2 \pi fTt$ ),
+*then the oscillation $c(t) = 2 · \cos(2 \pi f_{\rm T}t$ ),
 *below the band-pass signal&nbsp; $w(t) = u(t) \cdot c(t)$.
@@ Line 100: / Line 100: @@
 '''(3)'''&nbsp;  Proposed <u>solution 1 is correct</u>:
-*If we compare the spectral function&nbsp; $W(f)$&nbsp; of this task with the spectrum&nbsp; $D(f)$&nbsp; in the sample solution to&nbsp;  [[Aufgaben:4.1_TP-_und_BP-Signale|task 4.1]], we see that&nbsp; $w(t)$&nbsp; and&nbsp; $d(t)$&nbsp; are identical signals.
+*If we compare the spectral function&nbsp; $W(f)$&nbsp; of this task with the spectrum&nbsp; $D(f)$&nbsp; in the sample solution to&nbsp;  [[Aufgaben:Exercise_4.1:_Low-Pass_and_Band-Pass_Signals|Exercise 4.1]], we see that&nbsp; $w(t)$&nbsp; and&nbsp; $d(t)$&nbsp; are identical.
-*Etwas aufwändiger ist dieser Beweis im Zeitbereich. Mit&nbsp; $f_2 = 2 \,\text{kHz}$&nbsp; kann für das hier betrachtete Signal geschrieben werden:
+*This proof is somewhat more complex in the time domain.&nbsp; With&nbsp; $f_2 = 2 \,\text{kHz}$&nbsp; can be written for the signal considered here:
 :$$w(t )  =  4\hspace{0.05cm}{\rm V}
@@ Line 108: / Line 108: @@
 ({4\hspace{0.05cm}{\rm V}})/({\pi f_2 t})\cdot \sin (\pi f_2 t) \cdot \cos ( 4 \pi f_2 t) .$$
-*Wegen der trigonometrischen Beziehung
+*Because of the trigonometric relationship
 :$$\sin (\alpha) \cdot \cos (\beta)  =   {1}/{2} \cdot \big[ \sin

@@ Line 18: / Line 18: @@
 - 6 \hspace{0.05cm}{\rm V} \cdot {\rm si} ( 3 \pi f_2\hspace{0.05cm} t)$$
-whose spectrum was determined in &nbsp;[[Aufgaben:Aufgabe_4.1:_Tiefpass-_und_Bandpass-Signale|task 4.1]]&nbsp;. Let&nbsp; $f_2 = 2 \ \rm kHz.$
+whose spectrum was determined in &nbsp;[[Aufgaben:Exercise_4.1Z:_High-Pass_System|Exercise 4.1Z]]&nbsp;. Let&nbsp; $f_2 = 2 \ \rm kHz.$
@@ Line 28: / Line 28: @@
 ''Hints:''
 *This exercise belongs to the chapter&nbsp; [[Signal_Representation/Differences_and_Similarities_of_Low-Pass_and_Band-Pass_Signals|Differences and Similarities of Low-Pass and Band-Pass Signals]].
+*In this task, the function $\rm si(x) = \rm sin(x)/x = \rm sinc(x/π)$ is used.
 *Consider the following trigonometric relationship in the solution:
@@ Line 38: / Line 39: @@
 <quiz display=simple>
-{What are the values of the parameters&nbsp; $u_0$&nbsp; and&nbsp; $T_u$&nbsp; of the low-pass signal?
+{What are the parameter values&nbsp; $u_0$&nbsp; and&nbsp; $T_u$&nbsp; of the low-pass signal?
 |type="{}"}
 $u_0\ = \ $  { 2 3% }  &nbsp;$\text{V}$
 $T_u\ = \ $  { 0.5 3% } &nbsp;$\text{ms}$
-{Calculate the band-pass signal&nbsp; $w(t)$. What are the signal values at&nbsp; $t = 0$&nbsp; and&nbsp; $t = 62.5 \, {\rm &micro;}\text{s}$?
+{Calculate the band-pass signal&nbsp; $w(t)$.&nbsp; What are the signal values at&nbsp; $t = 0$&nbsp; and&nbsp; $t = 62.5 \, {\rm &micro;}\text{s}$?
 |type="{}"}
 $w(t=0)\ = \ $ { 4 3% } &nbsp;$\text{V}$
 $w(t=62.5 \,{\rm &micro;}  \text{s})\ = \ $ { 0. } &nbsp;$\text{V}$
-{Which statements are true regarding the band-pass signals&nbsp; $d(t)$&nbsp; and&nbsp; $w(t)$&nbsp;? Justify your result in the time domain.
+{Which statements are true regarding the band-pass signals&nbsp; $d(t)$&nbsp; and&nbsp; $w(t)$&nbsp;?&nbsp; Justify your result in the time domain.
-|type="[]"}
+|type="()"}
 + The signals&nbsp; $d(t)$&nbsp; and&nbsp; $w(t)$&nbsp; are identical.
 - $d(t)$&nbsp; and&nbsp; $w(t)$&nbsp; differ by a constant factor.

@@ Line 1: / Line 1: @@
-{{quiz-Header|Buchseite=Signal Representation/Differences and Similarities of LP and BP Signals
+{{quiz-Header|Buchseite=Signal_Representation/Differences_and_Similarities_of_Low-Pass_and_Band-Pass_Signals
 }}
-[[File:P_ID695__Sig_A_4_2_neu.png|250px|right|frame|Rectangular low&ndash;pass and band-pass spectra]]
+[[File:P_ID695__Sig_A_4_2_neu.png|250px|right|frame|Given low&ndash;pass and band-pass spectra]]
 We consider two signals&nbsp; $u(t)$&nbsp; and&nbsp; $w(t)$&nbsp; with rectangular spectra&nbsp; $U(f)$&nbsp; and&nbsp; $W(f)$ respectively.
 *It is obvious that
@@ Line 27: / Line 27: @@
 ''Hints:''
-*The task belongs to the chapter&nbsp; [[Signal_Representation/Differences_and_Similarities_of_LP_and_BP_Signals|Differences and Similarities of LP and BP Signals]].
+*This exercise belongs to the chapter&nbsp; [[Signal_Representation/Differences_and_Similarities_of_Low-Pass_and_Band-Pass_Signals|Differences and Similarities of Low-Pass and Band-Pass Signals]].
 *Consider the following trigonometric relationship in the solution:

← Older revision		Revision as of 12:20, 5 May 2021
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	__NOEDITSECTION__		__NOEDITSECTION__
−	[[Category:Signal Representation: Exercises\|^4.1 Differences~~/Similarities of LP~~ and ~~BP Signals~~^]]	+	[[Category:Signal Representation: Exercises\|^4.1 Differences between Low-Pass and Band-Pass^]]

@@ Line 3: / Line 3: @@
 }}
-[[File:P_ID695__Sig_A_4_2_neu.png|250px|right|frame|Rectangular low&ndash;pass and bandpass spectra]]
+[[File:P_ID695__Sig_A_4_2_neu.png|250px|right|frame|Rectangular low&ndash;pass and band-pass spectra]]
 We consider two signals&nbsp; $u(t)$&nbsp; and&nbsp; $w(t)$&nbsp; with rectangular spectra&nbsp; $U(f)$&nbsp; and&nbsp; $W(f)$ respectively.
 *It is obvious that
@@ Line 10: / Line 10: @@
 :is a low-pass signal whose two parameters&nbsp; $u_0$&nbsp; and&nbsp; $T_u$&nbsp; are to be determined in subtask&nbsp; '''(1)'''&nbsp;.
-*In contrast, the spectrum&nbsp; $W(f)$ shows that&nbsp; $w(t)$&nbsp; describes a bandpass signal.
+*In contrast, the spectrum&nbsp; $W(f)$ shows that&nbsp; $w(t)$&nbsp; describes a band-pass signal.
-This task also refers to the bandpass signal
+This task also refers to the band-pass signal
 :$$d(t)  =   10 \hspace{0.05cm}{\rm V} \cdot {\rm si} ( 5 \pi f_2 \hspace{0.05cm}t)
@@ Line 43: / Line 43: @@
 $T_u\ = \ $  { 0.5 3% } &nbsp;$\text{ms}$
-{Calculate the bandpass signal&nbsp; $w(t)$. What are the signal values at&nbsp; $t = 0$&nbsp; and&nbsp; $t = 62.5 \, {\rm &micro;}\text{s}$?
+{Calculate the band-pass signal&nbsp; $w(t)$. What are the signal values at&nbsp; $t = 0$&nbsp; and&nbsp; $t = 62.5 \, {\rm &micro;}\text{s}$?
 |type="{}"}
 $w(t=0)\ = \ $ { 4 3% } &nbsp;$\text{V}$
 $w(t=62.5 \,{\rm &micro;}  \text{s})\ = \ $ { 0. } &nbsp;$\text{V}$
-{Which statements are true regarding the bandpass signals&nbsp; $d(t)$&nbsp; and&nbsp; $w(t)$&nbsp;? Justify your result in the time domain.
+{Which statements are true regarding the band-pass signals&nbsp; $d(t)$&nbsp; and&nbsp; $w(t)$&nbsp;? Justify your result in the time domain.
 |type="[]"}
 + The signals&nbsp; $d(t)$&nbsp; and&nbsp; $w(t)$&nbsp; are identical.
@@ Line 67: / Line 67: @@
 [[File:P_ID704__Sig_A_4_2_b_neu.png|250px|right|frame|Multiplication with cosine]]
-'''(2)'''&nbsp;  The bandpass spectrum can be represented with&nbsp; $f_{\rm T} = 4\, \text{kHz}$&nbsp; as follows:
+'''(2)'''&nbsp;  The band-pass spectrum can be represented with&nbsp; $f_{\rm T} = 4\, \text{kHz}$&nbsp; as follows:
 :$$ W(f)   = U(f- f_{\rm T}) + U(f+ f_{\rm T}) =   U(f)\star \left[
@@ Line 80: / Line 80: @@
 *above the low&ndash;pass signal $u(t)$,
 *then the oscillation $c(t) = 2 · \cos(2 \pi fTt$ ),
-*below the bandpass signal&nbsp; $w(t) = u(t) \cdot c(t)$.
+*below the band-pass signal&nbsp; $w(t) = u(t) \cdot c(t)$.

← Older revision	Revision as of 09:31, 5 May 2021
(No difference)

← Older revision		Revision as of 08:50, 23 April 2021
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	__NOEDITSECTION__		__NOEDITSECTION__
−	[[Category:Signal Representation: Exercises\|^4.1 Differences ~~and~~ Similarities of LP and BP Signals^]]	+	[[Category:Signal Representation: Exercises\|^4.1 Differences/Similarities of LP and BP Signals^]]

← Older revision		Revision as of 12:39, 23 March 2021
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	__NOEDITSECTION__		__NOEDITSECTION__
−	[[Category:~~Exercises for~~ Signal Representation\|^4.1 Differences and Similarities of LP and BP Signals^]]	+	[[Category:Signal Representation: Exercises\|^4.1 Differences and Similarities of LP and BP Signals^]]

@@ Line 3: / Line 3: @@
 }}
-[[File:P_ID695__Sig_A_4_2_neu.png|250px|right|frame|Rechteckförmige Tiefpass&ndash; und Bandpass&ndash;Spektren]]
+[[File:P_ID695__Sig_A_4_2_neu.png|250px|right|frame|Rectangular low&ndash;pass and bandpass spectra]]
 We consider two signals&nbsp; $u(t)$&nbsp; and&nbsp; $w(t)$&nbsp; with rectangular spectra&nbsp; $U(f)$&nbsp; and&nbsp; $W(f)$ respectively.
 *It is obvious that
@@ Line 66: / Line 66: @@
-[[File:P_ID704__Sig_A_4_2_b_neu.png|250px|right|frame|Multiplikation mit Cosinus]]
+[[File:P_ID704__Sig_A_4_2_b_neu.png|250px|right|frame|Multiplication with cosine]]
-'''(2)'''&nbsp;  Das Bandpass&ndash;Spektrum kann mit&nbsp; $f_{\rm T} = 4\, \text{kHz}$&nbsp;  wie folgt dargestellt werden:
+'''(2)'''&nbsp;  The bandpass spectrum can be represented with&nbsp; $f_{\rm T} = 4\, \text{kHz}$&nbsp; as follows:
 :$$ W(f)   = U(f- f_{\rm T}) + U(f+ f_{\rm T}) =   U(f)\star \left[
 \delta(f- f_{\rm T})+ \delta(f+ f_{\rm T})\right].$$
-Entsprechend dem&nbsp; [[Signal_Representation/Fourier_Transform_Laws#Verschiebungssatz|Verschiebungssatz]]&nbsp; gilt dann für das dazugehörige Zeitsignal:
+According to the&nbsp; [[Signal_Representation/Fourier_Transform_Laws#Verschiebungssatz|Verschiebungssatz]]&nbsp;, the following then applies to the associated time signal:
 :$$w(t) = 2 \cdot u(t) \cdot {\cos} ( 2 \pi f_{\rm T} t) =   2 u_0
   \cdot {\rm si} ( \pi {t}/{T_{\rm u}})\cdot {\cos} ( 2 \pi f_{\rm T} t). $$
-Die Grafik zeigt
+The graph shows
-*oben das Tiefpass&ndash;Signal $u(t)$,
+*above the low&ndash;pass signal $u(t)$,
-*dann die Schwingung $c(t) = 2 · \cos(2 \pi fTt$ ),
+*then the oscillation $c(t) = 2 · \cos(2 \pi fTt$ ),
-*unten das Bandpass&ndash;Signal&nbsp; $w(t) = u(t) \cdot c(t)$.
+*below the bandpass signal&nbsp; $w(t) = u(t) \cdot c(t)$.
-Insbesondere erhält man zum Zeitpunkt&nbsp; $t = 0$:
+In particular, at time&nbsp; $t = 0$ one obtains:
 :$$w(t = 0)  =   2 \cdot u_0 \hspace{0.15 cm}\underline{= 4 \hspace{0.05cm}{\rm V}}.$$
-Der Zeitpunkt&nbsp; $t=62.5 \,{\rm &micro;} \text{s}$&nbsp; entspricht genau einer viertel Periodendauer des Signals&nbsp; $c(t)$:
+The time&nbsp; $t=62.5 \,{\rm &micro;} \text{s}$&nbsp; corresponds exactly to a quarter of the period of the signal&nbsp; $c(t)$:
 :$$ w(t = 62.5 \hspace{0.05cm}{\rm &micro; s})  =  2 u_0 \cdot  {\rm si} ( \pi \cdot \frac{62.5 \hspace{0.05cm}{\rm &micro;  s}}
@@ Line 98: / Line 98: @@
-'''(3)'''&nbsp;  Richtig ist der  <u>Lösungsvorschlag 1</u>:
+'''(3)'''&nbsp;  Proposed <u>solution 1 is correct</u>:
-*Vergleicht man die Spektralfunktion&nbsp; $W(f)$&nbsp; dieser Aufgabe mit dem Spektrum&nbsp; $D(f)$&nbsp; in der Musterlösung zu&nbsp;  [[Aufgaben:4.1_TP-_und_BP-Signale|Aufgabe 4.1]], so erkennt man, dass&nbsp; $w(t)$&nbsp; und&nbsp; $d(t)$&nbsp; identische Signale sind.
+*If we compare the spectral function&nbsp; $W(f)$&nbsp; of this task with the spectrum&nbsp; $D(f)$&nbsp; in the sample solution to&nbsp;  [[Aufgaben:4.1_TP-_und_BP-Signale|task 4.1]], we see that&nbsp; $w(t)$&nbsp; and&nbsp; $d(t)$&nbsp; are identical signals.
 *Etwas aufwändiger ist dieser Beweis im Zeitbereich. Mit&nbsp; $f_2 = 2 \,\text{kHz}$&nbsp; kann für das hier betrachtete Signal geschrieben werden:
@@ Line 112: / Line 112: @@
 (\alpha + \beta)+ \sin (\alpha - \beta)\big]$$
-:kann obige Gleichung umgeformt werden:
+:the above equation can be transformed:
 :$$w(t )  =
@@ Line 119: / Line 119: @@
 \hspace{0.05cm}{\rm V} \cdot \frac{\sin (3\pi f_2 t)}{3\pi f_2 t}.$$
-*Damit ist gezeigt, dass beide Signale tatsächlich identisch sind  &nbsp; ⇒  &nbsp; Lösungsvorschlag 1:
+*This shows that both signals are actually identical  &nbsp; ⇒  &nbsp; Proposed solution 1:
 :$$w(t)  =   10 \hspace{0.05cm}{\rm V} \cdot {\rm si} ( 5 \pi f_2 t)