Aufgaben:Exercise 3.3Z: Rectangular Pulse and Dirac Delta - Revision history

Guenter: Guenter moved page Aufgaben:Exercise 3.3Z: Rectangular Pulse and Dirac Impulse to Aufgaben:Exercise 3.3Z: Rectangular Pulse and Dirac Delta

2021-05-25T16:42:42Z

Guenter moved page Exercise 3.3Z: Rectangular Pulse and Dirac Impulse to Exercise 3.3Z: Rectangular Pulse and Dirac Delta

Guenter at 13:15, 24 May 2021

2021-05-24T13:15:59Z

Guenter at 12:32, 23 April 2021

2021-04-23T12:32:15Z

Guenter at 12:31, 23 April 2021

2021-04-23T12:31:24Z

Guenter at 12:30, 23 April 2021

2021-04-23T12:30:44Z

Guenter at 12:28, 23 April 2021

2021-04-23T12:28:40Z

Guenter at 12:09, 23 April 2021

2021-04-23T12:09:25Z

Guenter at 12:06, 23 April 2021

2021-04-23T12:06:57Z

Guenter at 12:03, 23 April 2021

2021-04-23T12:03:55Z

Guenter: Guenter moved page Aufgaben:Exercise 3.3Z: Rectangular- and Dirac Pulse to Aufgaben:Exercise 3.3Z: Rectangular Pulse and Dirac Impulse

2021-04-23T11:55:53Z

Guenter moved page Exercise 3.3Z: Rectangular- and Dirac Pulse to Exercise 3.3Z: Rectangular Pulse and Dirac Impulse

@@ Line 3: / Line 3: @@
 }}
-[[File:P_ID507__Sig_Z_3_3.png|right|frame|Verschiedene Rechteckimpulse]]
+[[File:P_ID507__Sig_Z_3_3.png|right|frame|Various rectangular pulses]]
 We consider here a multitude of symmetrical rectangular functions&nbsp; $x_k(t)$.&nbsp; The individual rectangles differ in amplitudes (heights)
 :$$A_k  = k \cdot A$$

@@ Line 83: / Line 83: @@
 &nbsp;
-'''(4)'''&nbsp;  In the limiting case&nbsp; $k \rightarrow \infty$&nbsp; the then infinitely high and infinitely narrow&nbsp; [[Signal_Representation/Special_Cases_of_Pulses#Rectangular_pulse|Rectangular pulse]]&nbsp; changes into the&nbsp; [[Signal_Representation/Special_Cases_of_Impulse_Signals#Dirac_Delta_Impulse|Dirac delta impulse]].
+'''(4)'''&nbsp;  In the limiting case&nbsp; $k \rightarrow \infty$&nbsp; the then infinitely high and infinitely narrow&nbsp; [[Signal_Representation/Special_Cases_of_Pulses#Rectangular_pulse|Rectangular pulse]]&nbsp; changes into the&nbsp; [[Signal_Representation/Special_Cases_of_Pulses#Dirac_.28delta.29_impulse|Dirac delta impulse]].
 *Its spectrum is constant for all frequencies.
 *Thus the spectral value&nbsp; $X_{\infty}(f = 2 \,\text{kHz})\hspace{0.15 cm}\underline{=1 \text{ mV/Hz}}$&nbsp; also applies at the frequency &nbsp; $f = 2 \,\text{kHz}$&nbsp;.

@@ Line 83: / Line 83: @@
 &nbsp;
-'''(4)'''&nbsp;  In the limiting case&nbsp; $k \rightarrow \infty$&nbsp; the then infinitely high and infinitely narrow&nbsp; [[Signal_Representation/Special_Cases_of_Pulses#Rectangular_pulse|Rectangular pulse]]&nbsp; changes into the&nbsp; [[Signal_Representation/Special_Cases_of_Impulse_Signals#Dirac_Delta_Impulse|Dirac Delta impulse]].
+'''(4)'''&nbsp;  In the limiting case&nbsp; $k \rightarrow \infty$&nbsp; the then infinitely high and infinitely narrow&nbsp; [[Signal_Representation/Special_Cases_of_Pulses#Rectangular_pulse|Rectangular pulse]]&nbsp; changes into the&nbsp; [[Signal_Representation/Special_Cases_of_Impulse_Signals#Dirac_Delta_Impulse|Dirac delta impulse]].
 *Its spectrum is constant for all frequencies.
 *Thus the spectral value&nbsp; $X_{\infty}(f = 2 \,\text{kHz})\hspace{0.15 cm}\underline{=1 \text{ mV/Hz}}$&nbsp; also applies at the frequency &nbsp; $f = 2 \,\text{kHz}$&nbsp;.

@@ Line 83: / Line 83: @@
 &nbsp;
-'''(4)'''&nbsp;  In the limiting case&nbsp; $k \rightarrow \infty$&nbsp; the then infinitely high and infinitely narrow&nbsp; [[Signal_Representation/Special_Cases_of_Impulse_Signals#Rectangular_Impulse|Rectangular pulse]]&nbsp; changes into the&nbsp; [[Signal_Representation/Special_Cases_of_Impulse_Signals#Dirac_Delta_Impulse|Dirac Delta impulse]].
+'''(4)'''&nbsp;  In the limiting case&nbsp; $k \rightarrow \infty$&nbsp; the then infinitely high and infinitely narrow&nbsp; [[Signal_Representation/Special_Cases_of_Pulses#Rectangular_pulse|Rectangular pulse]]&nbsp; changes into the&nbsp; [[Signal_Representation/Special_Cases_of_Impulse_Signals#Dirac_Delta_Impulse|Dirac Delta impulse]].
 *Its spectrum is constant for all frequencies.
 *Thus the spectral value&nbsp; $X_{\infty}(f = 2 \,\text{kHz})\hspace{0.15 cm}\underline{=1 \text{ mV/Hz}}$&nbsp; also applies at the frequency &nbsp; $f = 2 \,\text{kHz}$&nbsp;.

@@ Line 25: / Line 25: @@
 *You can check your results using the two interactive applets&nbsp;
-::[[Applets:Pulses_and_Spectra|Pulses and Spectra]],
+:[[Applets:Pulses_and_Spectra|Pulses and Spectra]],
-::[[Applets:Frequency_%26_Impulse_Responses|Frequency & Impulse Responses]].
+:[[Applets:Frequency_%26_Impulse_Responses|Frequency & Impulse Responses]].
@@ Line 64: / Line 64: @@
 ===Solutions===
 {{ML-Kopf}}
-'''(1)'''&nbsp;  Proposed <u>solutions 1 and 2</u> are correct:
+'''(1)'''&nbsp;  The <u>proposed solutions 1 and 2</u> are correct:
 *The spectral value at frequency&nbsp; $f = 0$&nbsp; is always equal to the area under the time function according to&nbsp; [[Signal_Representation/Fourier_Transform_and_Its_Inverse#The_First_Fourier_Integral|the first Fourier integral]]&nbsp;:
 :$$X( f ) = \int_{ - \infty }^{ + \infty } {x( t )}  \cdot {\rm{e}}^{ - {\rm{j2\pi }}ft} \hspace{0.1cm} {\rm d}t \hspace{0.5cm} \Rightarrow \hspace{0.5cm} \;X( {f = 0} ) = \int_{ - \infty }^{ + \infty } {x( t )}\hspace{0.1cm}  {\rm d}t.$$
@@ Line 71: / Line 71: @@
-'''(2)'''&nbsp;  Proposed <u>solutions 1 and 3</u> are correct:
+'''(2)'''&nbsp;  The <u>proposed solutions 1 and 3</u> are correct:
 *Due to equal pulse areas, the spectral value is not changed at the frequency&nbsp; $f = 0$&nbsp;.
-*The equidistant zero crossings now occur at the interval&nbsp; $f_2 = 1/T_2 = 4 \,\text{kHz}$&nbsp; auf.
+*The equidistant zero crossings now occur at the interval&nbsp; $f_2 = 1/T_2 = 4 \,\text{kHz}$.
@@ Line 83: / Line 83: @@
 &nbsp;
-'''(4)'''&nbsp;  In the limiting case&nbsp; $k \rightarrow \infty$&nbsp; the then infinitely high and infinitely narrow&nbsp; [[Signal_Representation/Special_Cases_of_Impulse_Signals#Rectangular_Impulse|Rectangular impulse]]&nbsp; changes into the&nbsp; [[Signal_Representation/Special_Cases_of_Impulse_Signals#Dirac_Delta_Impulse|Dirac Delta impulse]]&nbsp;.
+'''(4)'''&nbsp;  In the limiting case&nbsp; $k \rightarrow \infty$&nbsp; the then infinitely high and infinitely narrow&nbsp; [[Signal_Representation/Special_Cases_of_Impulse_Signals#Rectangular_Impulse|Rectangular pulse]]&nbsp; changes into the&nbsp; [[Signal_Representation/Special_Cases_of_Impulse_Signals#Dirac_Delta_Impulse|Dirac Delta impulse]].
 *Its spectrum is constant for all frequencies.
-*Thus the spectral value&nbsp; $X_{\infty}(f = 2 \,\text{kHz})\hspace{0.15 cm}\underline{=1 \text{ mV/Hz}}$ also applies at the frequency &nbsp; $f = 2 \,\text{kHz}$&nbsp;.
+*Thus the spectral value&nbsp; $X_{\infty}(f = 2 \,\text{kHz})\hspace{0.15 cm}\underline{=1 \text{ mV/Hz}}$&nbsp; also applies at the frequency &nbsp; $f = 2 \,\text{kHz}$&nbsp;.
 {{ML-Fuß}}

@@ Line 25: / Line 25: @@
 *You can check your results using the two interactive applets&nbsp;
-::[[Applets:Impulse_und_Spektren|Impulses and Spectra]],
+::[[Applets:Applets:Pulses_and_Spectra|Impulses and Spectra]],
 ::[[Applets:Frequenzgang_und_Impulsantwort|Frequenzgang und Impulsantwort]].

@@ Line 4: / Line 4: @@
 [[File:P_ID507__Sig_Z_3_3.png|right|frame|Verschiedene Rechteckimpulse]]
-We consider here a multitude of symmetrical rectangular functions&nbsp; $x_k(t)$. The individual rectangles differ in amplitudes (heights)
+We consider here a multitude of symmetrical rectangular functions&nbsp; $x_k(t)$.&nbsp; The individual rectangles differ in amplitudes (heights)
 :$$A_k  = k \cdot A$$
 and different pulse durations (widths)
@@ Line 21: / Line 21: @@
 ''Hints:''
-*This exercise belongs to the chapter&nbsp; [[Signal_Representation/Special_Cases_of_Impulse_Signals|Special Cases of Impulse Signals]].
+*This task belongs to the chapter&nbsp;  [[Signal_Representation/Special_Cases_of_Pulses|Special Cases of Pulses]].
-*You can check your results using the two interactive applets&nbsp; [[Applets:Impulse_und_Spektren|Impulses and Spectra]]&nbsp; sowie&nbsp;  [[Applets:Frequenzgang_und_Impulsantwort|Frequenzgang und Impulsantwort]]&nbsp;.
+*You can check your results using the two interactive applets&nbsp;
@@ Line 31: / Line 34: @@
 <quiz display=simple>
-{Which of the following statements are true regarding the spectrum&nbsp; $X_1(f)$&nbsp;?
+{Which of the following statements are true regarding the spectrum&nbsp; $X_1(f)$?
 |type="[]"}
 + The spectral value&nbsp; $X_1(f = 0)$&nbsp; is equal to&nbsp; $10^{–3} \,\text{V/Hz}$.
@@ Line 38: / Line 41: @@
-{Which of the following statements are true regarding the spectrum&nbsp; $X_2(f)$&nbsp; zu?
+{Which of the following statements are true regarding the spectrum&nbsp; $X_2(f)$?
 |type="[]"}
-+ The spectral value&nbsp; $X_2(f = 0)$&nbsp; ist gleich&nbsp; $10^{–3} \,\text{V/Hz}$.
++ The spectral value is&nbsp; $X_2(f = 0)$&nbsp; is equal to&nbsp; $10^{–3} \,\text{V/Hz}$.
 - $X_2(f)$&nbsp; has zeros at the interval of&nbsp; $2\, \text{kHz}$.
 + $X_2(f)$&nbsp; has zeros at the interval of&nbsp; $4 \,\text{kHz}$.
-{Let&nbsp; $k = 10$. Calculate the frequency&nbsp; $f_{10}$&nbsp; of the first zero and the spectral value at&nbsp; $f = 2 \,\text{kHz}$.
+{Let&nbsp; $k = 10$.&nbsp; Calculate the frequency&nbsp; $f_{10}$&nbsp; of the first zero and the spectral value at&nbsp; $f = 2 \,\text{kHz}$.
 |type="{}"}
 $f_{10} \ = \ ${ 20 3% } &nbsp;$\text{kHz}$
@@ Line 51: / Line 54: @@
-{What is the spectral value at&nbsp; bei $f = 2 \,\text{kHz}$&nbsp; in the limiting case&nbsp; $k \rightarrow \infty$? Interpret the result.
+{What is the spectral value at&nbsp; $f = 2 \,\text{kHz}$&nbsp; in the limiting case&nbsp; $k \rightarrow \infty$?&nbsp; Interpret the result.
 |type="{}"}
 $X_{\infty}(f = 2 \,\text{kHz})\ = \ $ { 1 3% } &nbsp;$\text{mV/Hz}$