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

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	'''(1)'''  The <u>proposed solutions 1 and 2</u> are correct:		'''(1)'''  The <u>proposed solutions 1 and 2</u> are correct:
	*The spectral value at frequency  $f = 0$  is always equal to the area under the time function according to  [[Signal_Representation/~~Fourier_Transform_and_Its_Inverse~~#~~The_First_Fourier_Integral~~\|the first Fourier integral]] :		*The spectral value at frequency  $f = 0$  is always equal to the area under the time function according to  [[Signal_Representation/The_Fourier_Transform_and_its_Inverse#The_first_Fourier_integral\|the first Fourier integral]] :
	:$$X( f ) = \int_{ - \infty }^{+ \infty} {x( t )} \cdot {\rm{e}}^{- {\rm{j2\pi}}ft} \hspace{0.1cm} {\rm d}t \hspace{0.5cm} \Rightarrow \hspace{0.5cm} \;X( {f = 0} ) = \int_{ - \infty }^{+ \infty} {x( t )}\hspace{0.1cm} {\rm d}t.$$		:$$X( f ) = \int_{ - \infty }^{+ \infty} {x( t )} \cdot {\rm{e}}^{- {\rm{j2\pi}}ft} \hspace{0.1cm} {\rm d}t \hspace{0.5cm} \Rightarrow \hspace{0.5cm} \;X( {f = 0} ) = \int_{ - \infty }^{+ \infty} {x( t )}\hspace{0.1cm} {\rm d}t.$$
	*In the present case, the pulse area is always  $A \cdot T = 10^{–3} \,\text{Vs} = 1\, \text{mV/Hz}$.		*In the present case, the pulse area is always  $A \cdot T = 10^{–3} \,\text{Vs} = 1\, \text{mV/Hz}$.
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	[[Category:Signal Representation: Exercises\|^3.2 Special Cases of Pulses^]]		[[Category:Signal Representation: Exercises\|^3.2 Special Cases of Pulses^]]
	[[de:Aufgaben:3.~~3Z_Rechteck~~-~~_und_Diracimpuls~~]]		[[de:Aufgaben:Aufgabe 3.3Z: Rechteck- und Diracimpuls]]

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	'''(1)'''  The <u>proposed solutions 1 and 2</u> are correct:		'''(1)'''  The <u>proposed solutions 1 and 2</u> are correct:
	*The spectral value at frequency  $f = 0$  is always equal to the area under the time function according to  [[Signal_Representation/Fourier_Transform_and_Its_Inverse#The_First_Fourier_Integral\|the first Fourier integral]] :		*The spectral value at frequency  $f = 0$  is always equal to the area under the time function according to  [[Signal_Representation/Fourier_Transform_and_Its_Inverse#The_First_Fourier_Integral\|the first Fourier integral]] :
	:$$X( f ) = \int_{ - \infty }^{ + \infty } {x( t )} \cdot {\rm{e}}^{ - {\rm{j2\pi }}ft} \hspace{0.1cm} {\rm d}t \hspace{0.5cm} \Rightarrow \hspace{0.5cm} \;X( {f = 0} ) = \int_{ - \infty }^{ + \infty } {x( t )}\hspace{0.1cm} {\rm d}t.$$		:$$X( f ) = \int_{ - \infty }^{+ \infty} {x( t )} \cdot {\rm{e}}^{- {\rm{j2\pi}}ft} \hspace{0.1cm} {\rm d}t \hspace{0.5cm} \Rightarrow \hspace{0.5cm} \;X( {f = 0} ) = \int_{ - \infty }^{+ \infty} {x( t )}\hspace{0.1cm} {\rm d}t.$$
	*In the present case, the pulse area is always  $A \cdot T = 10^{–3} \,\text{Vs} = 1\, \text{mV/Hz}$.		*In the present case, the pulse area is always  $A \cdot T = 10^{–3} \,\text{Vs} = 1\, \text{mV/Hz}$.
	*Because of  $T_1 = 500 \,µ\text{s}$  the spectrum  $X_1(f)$  has zero crossings at the interval  $f_1 = 1/T_1 = 2 \,\text{kHz}$ .		*Because of  $T_1 = 500 \,µ\text{s}$  the spectrum  $X_1(f)$  has zero crossings at the interval  $f_1 = 1/T_1 = 2 \,\text{kHz}$ .

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	[[Category:Signal Representation: Exercises\|^3.2 Special Cases of Pulses^]]		[[Category:Signal Representation: Exercises\|^3.2 Special Cases of Pulses^]]
			[[de:Aufgaben:3.3Z_Rechteck-_und_Diracimpuls]]

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

← Older revision	Revision as of 18:42, 25 May 2021
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Guenter at 13:15, 24 May 2021

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	[[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  $x_k(t)$.  The individual rectangles differ in amplitudes (heights)		We consider here a multitude of symmetrical rectangular functions  $x_k(t)$.  The individual rectangles differ in amplitudes (heights)
	:$$A_k = k \cdot A$$		:$$A_k = k \cdot A$$

Guenter at 12:32, 23 April 2021

2021-04-23T12:32:15Z

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

Guenter at 12:31, 23 April 2021

2021-04-23T12:31:24Z

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

Guenter at 12:30, 23 April 2021

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

Guenter at 12:28, 23 April 2021

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	*You can check your results using the two interactive applets		*You can check your results using the two interactive applets
	::[[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]].


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	===Solutions===		===Solutions===
	{{ML-Kopf}}		{{ML-Kopf}}
	'''(1)'''  ~~Proposed~~ <u>solutions 1 and 2</u> are correct:		'''(1)'''  The <u>proposed solutions 1 and 2</u> are correct:
	*The spectral value at frequency  $f = 0$  is always equal to the area under the time function according to  [[Signal_Representation/Fourier_Transform_and_Its_Inverse#The_First_Fourier_Integral\|the first Fourier integral]] :		*The spectral value at frequency  $f = 0$  is always equal to the area under the time function according to  [[Signal_Representation/Fourier_Transform_and_Its_Inverse#The_First_Fourier_Integral\|the first Fourier integral]] :
	:$$X( f ) = \int_{ - \infty }^{ + \infty } {x( t )} \cdot {\rm{e}}^{ - {\rm{j2\pi }}ft} \hspace{0.1cm} {\rm d}t \hspace{0.5cm} \Rightarrow \hspace{0.5cm} \;X( {f = 0} ) = \int_{ - \infty }^{ + \infty } {x( t )}\hspace{0.1cm} {\rm d}t.$$		:$$X( f ) = \int_{ - \infty }^{ + \infty } {x( t )} \cdot {\rm{e}}^{ - {\rm{j2\pi }}ft} \hspace{0.1cm} {\rm d}t \hspace{0.5cm} \Rightarrow \hspace{0.5cm} \;X( {f = 0} ) = \int_{ - \infty }^{ + \infty } {x( t )}\hspace{0.1cm} {\rm d}t.$$
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	'''(2)'''  ~~Proposed~~ <u>solutions 1 and 3</u> are correct:		'''(2)'''  The <u>proposed solutions 1 and 3</u> are correct:
	*Due to equal pulse areas, the spectral value is not changed at the frequency  $f = 0$ .		*Due to equal pulse areas, the spectral value is not changed at the frequency  $f = 0$ .
	*The equidistant zero crossings now occur at the interval  $f_2 = 1/T_2 = 4 \,\text{kHz}$~~  auf~~.		*The equidistant zero crossings now occur at the interval  $f_2 = 1/T_2 = 4 \,\text{kHz}$.


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

Guenter at 12:09, 23 April 2021

2021-04-23T12:09:25Z

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	*You can check your results using the two interactive applets		*You can check your results using the two interactive applets
	::[[~~Applets:~~Applets:Pulses_and_Spectra\|~~Impulses~~ and Spectra]],		::[[Applets:Pulses_and_Spectra\|Pulses and Spectra]],
	::[[Applets:~~Frequenzgang_und_Impulsantwort~~\|~~Frequenzgang und Impulsantwort~~]].		::[[Applets:Frequency_%26_Impulse_Responses\|Frequency & Impulse Responses]].