Aufgaben:Exercise 3.5Z: Integration of Dirac Functions - Revision history

Guenter at 13:21, 24 May 2021

2021-05-24T13:21:21Z

Guenter at 12:48, 27 April 2021

2021-04-27T12:48:02Z

Guenter at 12:40, 27 April 2021

2021-04-27T12:40:24Z

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

2021-03-23T12:39:19Z

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

Noah at 17:04, 27 January 2021

2021-01-27T17:04:37Z

Noah at 17:02, 27 January 2021

2021-01-27T17:02:44Z

Oezdemir: Oezdemir moved page Aufgaben:Aufgabe 3.5Z: Integration von Diracfunktionen to Aufgaben:Exercise 3.5Z: Integration of Dirac Functions

2020-11-02T15:29:11Z

Oezdemir moved page Aufgabe 3.5Z: Integration von Diracfunktionen to Exercise 3.5Z: Integration of Dirac Functions

Oezdemir at 15:24, 2 November 2020

2020-11-02T15:24:24Z

Javier: Text replacement - "Signaldarstellung/Gesetzmäßigkeiten der Fouriertransformation" to "Signal_Representation/Fourier_Transform_Laws"

2020-09-01T09:49:42Z

Text replacement - "Signaldarstellung/Gesetzmäßigkeiten der Fouriertransformation" to "Signal_Representation/Fourier_Transform_Laws"

Javier: Text replacement - "Signal_Representation/Gesetzmäßigkeiten_der_Fouriertransformation" to "Signal_Representation/Fourier_Transform_Laws"

2020-09-01T09:49:19Z

Text replacement - "Signal_Representation/Gesetzmäßigkeiten_der_Fouriertransformation" to "Signal_Representation/Fourier_Transform_Laws"

@@ Line 3: / Line 3: @@
 }}
-[[File:P_ID515__Sig_Z_3_5_neu.png|right|frame|Integration von Diracfunktionen ]]
+[[File:P_ID515__Sig_Z_3_5_neu.png|right|frame|Integration of Dirac functions ]]
 Like in&nbsp; [[Aufgaben:3.5_Differentiation_eines_Dreicksignals|Exercise 3.5]]&nbsp; the spectrum&nbsp; ${Y(f)}$&nbsp; of the signal
 :$$y( t ) = \left\{ \begin{array}{c} A \\   - A \\  0 \\  \end{array} \right.\quad \begin{array}{*{20}c}   {{\rm{for}}}  \\   {{\rm{for}}}  \\   {\rm{else.}}  \\ \end{array}\;\begin{array}{*{20}c}   { - T \le t < 0,}  \\   {0 < t \le T,}  \\   {}  \\\end{array}$$

@@ Line 48: / Line 48: @@
 ===Solution===
 {{ML-Kopf}}
-'''(1)'''&nbsp;  In the description of the task you will find the Fourier correspondence between&nbsp; ${u(t)}$&nbsp; and&nbsp; ${U(f)}$.
+'''(1)'''&nbsp;  In the task description you will find the Fourier correspondence between&nbsp; ${u(t)}$&nbsp; and&nbsp; ${U(f)}$.
-*Since both the time functions&nbsp; ${u(t)}$&nbsp; and&nbsp; ${x(t)}$&nbsp; and the corresponding spectra&nbsp; ${U(f)}$&nbsp; and&nbsp; ${X(f)}$&nbsp; are even and real,&nbsp; ${X(f)}$&nbsp; can be easily calculated by applying the "Vertauschungssatz":
+*Since both the time functions&nbsp; ${u(t)}$&nbsp; and&nbsp; ${x(t)}$&nbsp; and the corresponding spectra&nbsp; ${U(f)}$&nbsp; and&nbsp; ${X(f)}$&nbsp; are even and real,&nbsp; ${X(f)}$&nbsp; can be easily calculated by applying the "Duality Theorem":
 :$$X( f ) =  - 2 \cdot A \cdot T  + 2 \cdot A \cdot T \cdot \cos \left( {{\rm{2\pi }}fT} \right).$$
 *Because of the relation&nbsp; $\sin^{2}(\alpha) = (1 – \cos(\alpha))/2$&nbsp; it can also be written for this:
 :$$X( f ) =  - 4 \cdot A \cdot T \cdot \sin ^2 ( {{\rm{\pi }}fT} ).$$
-:*At frequency&nbsp; $f = 0$&nbsp;&nbsp; ${x(t)}$&nbsp; has no spectral components&nbsp; &rArr; &nbsp;  ${X(f = 0)} \;\underline{= 0}$.
+:*At frequency&nbsp; $f = 0$&nbsp; the signal&nbsp; ${x(t)}$&nbsp; has no spectral components&nbsp; &rArr; &nbsp;  ${X(f = 0)} \;\underline{= 0}$.
-:*For&nbsp; $f = 1 \,\text{kHz}$&nbsp; &ndash; also&nbsp; $f \cdot T = 0.5$&nbsp; &ndash; &nbsp; on the other hand:
+:*For&nbsp; $f = 1 \,\text{kHz}$&nbsp; &ndash; and also&nbsp; $f \cdot T = 0.5$&nbsp; &ndash; &nbsp; on the other hand:
 :$$X( f   =  1\;{\rm{kHz}} )  =    - 4 \cdot A \cdot T = -2 \cdot 10^{ - 3} \;{\rm{V/Hz}}\;  \Rightarrow \;
 |X( {f = 1\;{\rm{kHz}}} )|   \hspace{0.15 cm}\underline{=  2  \;{\rm{mV/Hz}}}{\rm{.}}$$
@@ Line 60: / Line 60: @@
-'''(2)'''&nbsp; The spectrum&nbsp; ${Y(f)}$&nbsp; can be determined from&nbsp; ${X(f)}$&nbsp; by applying the integration theorem.
+'''(2)'''&nbsp; The spectrum&nbsp; ${Y(f)}$&nbsp; can be determined from&nbsp; ${X(f)}$&nbsp; by applying the&nbsp; "Integration Theorem".
 *Because of&nbsp; ${X(f = 0)} = 0$&nbsp; the Dirac function does not have to be taken into account at the frequency&nbsp; $f = 0$&nbsp; and one obtains:
 :$$Y( f ) = \frac{X( f )}{{{\rm{j}} \cdot 2{\rm{\pi }}fT}} = \frac{{ - 4 \cdot A \cdot T \cdot \sin ^2 ( {{\rm{\pi }}fT} )}}{{{\rm{j}}\cdot 2{\rm{\pi }}fT}} = 2{\rm{j}} \cdot A \cdot T \cdot \frac{{\sin ^2 ( {{\rm{\pi }}fT} )}}{{{\rm{\pi }}fT}}.$$
-*Of course, the result is the same as in&nbsp; [[Aufgaben:3.5_Differentiation_eines_Dreicksignals|Task 3.5]]:
+*Of course, the result is the same as in&nbsp; [[Aufgaben:3.5_Differentiation_eines_Dreicksignals|Exercise 3.5]]:
-:*At frequency&nbsp; $f = 0$&nbsp;&nbsp;  ${y(t)}$&nbsp; also has no spectral components&nbsp; &rArr; &nbsp;  ${Y(f = 0)} \;\underline{= 0}$.
+:*At frequency&nbsp; $f = 0$&nbsp; The signal&nbsp;  ${y(t)}$&nbsp; also has no spectral components&nbsp; &rArr; &nbsp;  ${Y(f = 0)} \;\underline{= 0}$.
-:*For&nbsp; $f = 1\,\text{kHz} \ \ (f \cdot T = 0.5)$&nbsp; one obtains a value smaller by a factor&nbsp; $\pi$&nbsp; compared to&nbsp; $X(f)$&nbsp;:
+:*For&nbsp; $f = 1\,\text{kHz} \ \ (f \cdot T = 0.5)$&nbsp; one obtains a value smaller by a factor&nbsp; $\pi$&nbsp; compared to&nbsp; $X(f)$:
 :$$|Y( {f = 1\;{\rm{kHz}}} )| =  \frac{4 \cdot A \cdot T}{\rm{\pi }} \hspace{0.15 cm}\underline{=  {\rm{0}}{\rm{.636}}  \;{\rm{mV/Hz}}}{\rm{.}}$$
 {{ML-Fuß}}

@@ Line 4: / Line 4: @@
 [[File:P_ID515__Sig_Z_3_5_neu.png|right|frame|Integration von Diracfunktionen ]]
-Like in&nbsp; [[Aufgaben:3.5_Differentiation_eines_Dreicksignals|Task 3.5]]&nbsp; the spectrum&nbsp; ${Y(f)}$&nbsp; of the signal
+Like in&nbsp; [[Aufgaben:3.5_Differentiation_eines_Dreicksignals|Exercise 3.5]]&nbsp; the spectrum&nbsp; ${Y(f)}$&nbsp; of the signal
-:$$y( t ) = \left\{ \begin{array}{c} A \\   - A \\  0 \\  \end{array} \right.\quad \begin{array}{*{20}c}   {{\rm{f \ddot{u}r}}}  \\   {{\rm{f\ddot{u} r}}}  \\   {\rm{else.}}  \\ \end{array}\;\begin{array}{*{20}c}   { - T \le t < 0,}  \\   {0 < t \le T,}  \\   {}  \\\end{array}$$
+:$$y( t ) = \left\{ \begin{array}{c} A \\   - A \\  0 \\  \end{array} \right.\quad \begin{array}{*{20}c}   {{\rm{for}}}  \\   {{\rm{for}}}  \\   {\rm{else.}}  \\ \end{array}\;\begin{array}{*{20}c}   { - T \le t < 0,}  \\   {0 < t \le T,}  \\   {}  \\\end{array}$$
-can be determined. Again,&nbsp; $A = 1 \,\text{V}$&nbsp; and&nbsp; $T = 0.5 \,\text{ms}$ apply.
+can be determined.&nbsp; Again,&nbsp; $A = 1 \,\text{V}$&nbsp; and&nbsp; $T = 0.5 \,\text{ms}$&nbsp; apply.
 Assume the time signal&nbsp; ${x(t)}$&nbsp; according to the middle sketch, which is composed of three Dirac pulses at&nbsp; $–T$,&nbsp; $0$&nbsp; and&nbsp; $+T$&nbsp; with the pulse weights&nbsp; ${AT}$,&nbsp; $-2{AT}$&nbsp; und&nbsp; ${AT}$&nbsp;.
-The spectral function&nbsp; ${X(f)}$&nbsp; can be given directly by applying the&nbsp; [[Signal_Representation/Fourier_Transform_Laws#Vertauschungssatz|Vertauschungssatz]]&nbsp; if one takes into account that the time function belonging to&nbsp; ${U(f)}$&nbsp; is as follows (see lower sketch):
+The spectral function&nbsp; ${X(f)}$&nbsp; can be given directly by applying the&nbsp; [[Signal_Representation/Fourier_Transform_Theorems#Duality_Theorem|Duality Theorem]]&nbsp; if one takes into account that the time function belonging to&nbsp; ${U(f)}$&nbsp; is as follows (see lower sketch):
 :$$u( t ) =  - 2A + 2A \cdot \cos ( {2{\rm{\pi }}f_0 t} ).$$
@@ Line 21: / Line 19: @@
 ''Hints:''
-*This exercise belongs to the chapter&nbsp; [[Signal_Representation/Fourier_Transform_Laws|Fourier Transform Laws]].
+*This exercise belongs to the chapter&nbsp; [[Signal_Representation/Fourier_Transform_Theorems|Fourier Transform Theorems]].
-*All of these laws are illustrated with examples in the learning video&nbsp; [[Gesetzmäßigkeiten_der_Fouriertransformation_(Lernvideo)|Gesetzmäßigkeiten der Fouriertransformation]]&nbsp;.
+*All the laws presented here  are illustrated with examples in the (German language) learning video<br> &nbsp; &nbsp; &nbsp;[[Gesetzmäßigkeiten_der_Fouriertransformation_(Lernvideo)|Gesetzmäßigkeiten der Fouriertransformation]] &nbsp; &rArr; &nbsp;  "Regularities to the Fourier transform".
-The following relationship exists between&nbsp; ${x(t)}$&nbsp; and&nbsp; ${y(t)}$&nbsp;:
+*The following relationship exists between&nbsp; ${x(t)}$&nbsp; and&nbsp; ${y(t)}$&nbsp;:
 :$$y( t ) = \frac{1}{T} \cdot \hspace{-0.1cm} \int_{ - \infty }^{\hspace{0.05cm}t} {x( \tau  )}\, {\rm d}\tau .$$
-*The&nbsp; [[Signal_Representation/Fourier_Transform_Laws#Integration_Theorem|Integration_Theorem]]&nbsp; reads in a correspondingly adapted form:
+*The&nbsp; [[Signal_Representation/Fourier_Transform_Laws#Integration_Theorem|Integration Theorem]]&nbsp; reads in a correspondingly adapted form:
 :$$\frac{1}{T}\cdot \hspace{-0.1cm} \int_{ - \infty }^{\hspace{0.05cm}t} {x( \tau  )}\,\, {\rm d}\tau\ \ \circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\, \ \ X( f ) \cdot \left( {\frac{1}{{{\rm{j}}\cdot 2{\rm{\pi }\cdot }fT}} + \frac{1}{2T}\cdot {\rm \delta} ( f )} \right).$$
@@ Line 33: / Line 31: @@
 <quiz display=simple>
-{Calculate the spectral function&nbsp; ${X(f)}$. What is its magnitude at the frequencies&nbsp; $f = 0$&nbsp; and&nbsp; $f = 1\, \text{kHz}$?
+{Calculate the spectral function&nbsp; ${X(f)}$.&nbsp; What is its magnitude at the frequencies&nbsp; $f = 0$&nbsp; and&nbsp; $f = 1\, \text{kHz}$?
 |type="{}"}
 $|{X(f = 0)}| \ = \ $ { 0. }  &nbsp;$\text{mV/Hz}$
@@ Line 39: / Line 37: @@
-{Calculate the spectral function&nbsp; ${Y(f)}$. What values result at the frequencies&nbsp; $f = 0$&nbsp; and&nbsp; $f = 1\, \text{kHz}$?
+{Calculate the spectral function&nbsp; ${Y(f)}$.&nbsp; What values result at the frequencies&nbsp; $f = 0$&nbsp; and&nbsp; $f = 1\, \text{kHz}$?
 |type="{}"}
 $|{Y(f = 0)}|\ = \ $ { 0. }  &nbsp;$\text{mV/Hz}$

← Older revision		Revision as of 12:39, 23 March 2021
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	__NOEDITSECTION__		__NOEDITSECTION__
−	[[Category:~~Exercises for~~ Signal Representation\|^3.3 Fourier Transform Theorems^]]	+	[[Category:Signal Representation: Exercises\|^3.3 Fourier Transform Theorems^]]

@@ Line 8: / Line 8: @@
 can be determined. Again,&nbsp; $A = 1 \,\text{V}$&nbsp; and&nbsp; $T = 0.5 \,\text{ms}$ apply.
-Assum ethe time signal&nbsp; ${x(t)}$&nbsp; according to the middle sketch, which is composed of three Dirac pulses at&nbsp; $–T$,&nbsp; $0$&nbsp; and&nbsp; $+T$&nbsp; with the pulse weights&nbsp; ${AT}$,&nbsp; $-2{AT}$&nbsp; und&nbsp; ${AT}$&nbsp;.
+Assume the time signal&nbsp; ${x(t)}$&nbsp; according to the middle sketch, which is composed of three Dirac pulses at&nbsp; $–T$,&nbsp; $0$&nbsp; and&nbsp; $+T$&nbsp; with the pulse weights&nbsp; ${AT}$,&nbsp; $-2{AT}$&nbsp; und&nbsp; ${AT}$&nbsp;.
 The spectral function&nbsp; ${X(f)}$&nbsp; can be given directly by applying the&nbsp; [[Signal_Representation/Fourier_Transform_Laws#Vertauschungssatz|Vertauschungssatz]]&nbsp; if one takes into account that the time function belonging to&nbsp; ${U(f)}$&nbsp; is as follows (see lower sketch):
@@ Line 21: / Line 21: @@
 ''Hints:''
-*This exercise belongs to the chpater&nbsp; [[Signal_Representation/Fourier_Transform_Laws|Fourier Transform Laws]].
+*This exercise belongs to the chapter&nbsp; [[Signal_Representation/Fourier_Transform_Laws|Fourier Transform Laws]].
 *All of these laws are illustrated with examples in the learning video&nbsp; [[Gesetzmäßigkeiten_der_Fouriertransformation_(Lernvideo)|Gesetzmäßigkeiten der Fouriertransformation]]&nbsp;.
 The following relationship exists between&nbsp; ${x(t)}$&nbsp; and&nbsp; ${y(t)}$&nbsp;:

← Older revision	Revision as of 15:29, 2 November 2020
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← Older revision		Revision as of 15:24, 2 November 2020
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−	[[Category:Exercises for Signal Representation\|^3. ~~Aperiodische Signale - Impulse~~^]]	+	[[Category:Exercises for Signal Representation\|^3.3 Fourier Transform Theorems^]]

← Older revision		Revision as of 09:49, 1 September 2020
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−	{{quiz-Header\|Buchseite=~~Signaldarstellung~~/~~Gesetzmäßigkeiten der Fouriertransformation~~	+	{{quiz-Header\|Buchseite=Signal_Representation/Fourier_Transform_Laws
	}}		}}

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 Ausgegangen wird vom Zeitsignal&nbsp; ${x(t)}$&nbsp; gemäß der mittleren Skizze, das sich aus drei Diracimpulsen bei&nbsp; $–T$,&nbsp; $0$&nbsp; und&nbsp; $+T$&nbsp; mit den Impulsgewichte&nbsp; ${AT}$,&nbsp; $-2{AT}$&nbsp; und&nbsp; ${AT}$&nbsp; zusammensetzt.
-Die Spektralfunktion&nbsp; ${X(f)}$&nbsp; kann durch Anwendung des&nbsp; [[Signal_Representation/Gesetzmäßigkeiten_der_Fouriertransformation#Vertauschungssatz|Vertauschungssatzes]]&nbsp; direkt angegeben werden, wenn man berücksichtigt, dass die zu&nbsp; ${U(f)}$&nbsp; gehörige Zeitfunktion wie folgt lautet (siehe untere Skizze):
+Die Spektralfunktion&nbsp; ${X(f)}$&nbsp; kann durch Anwendung des&nbsp; [[Signal_Representation/Fourier_Transform_Laws#Vertauschungssatz|Vertauschungssatzes]]&nbsp; direkt angegeben werden, wenn man berücksichtigt, dass die zu&nbsp; ${U(f)}$&nbsp; gehörige Zeitfunktion wie folgt lautet (siehe untere Skizze):
 :$$u( t ) =  - 2A + 2A \cdot \cos ( {2{\rm{\pi }}f_0 t} ).$$
@@ Line 21: / Line 21: @@
 ''Hinweise:''
-*Die Aufgabe gehört zum  Kapitel&nbsp; [[Signal_Representation/Gesetzmäßigkeiten_der_Fouriertransformation|Gesetzmäßigkeiten der Fouriertransformation]].
+*Die Aufgabe gehört zum  Kapitel&nbsp; [[Signal_Representation/Fourier_Transform_Laws|Gesetzmäßigkeiten der Fouriertransformation]].
 *Alle diese Gesetzmäßigkeiten werden im Lernvideo&nbsp; [[Gesetzmäßigkeiten_der_Fouriertransformation_(Lernvideo)|Gesetzmäßigkeiten der Fouriertransformation]]&nbsp; an Beispielen verdeutlicht.
 *Zwischen&nbsp; ${x(t)}$&nbsp; und&nbsp; ${y(t)}$&nbsp; besteht folgender Zusammenhang:
 :$$y( t ) = \frac{1}{T} \cdot \hspace{-0.1cm} \int_{ - \infty }^{\hspace{0.05cm}t} {x( \tau  )}\, {\rm d}\tau .$$
-*Der&nbsp; [[Signal_Representation/Gesetzmäßigkeiten_der_Fouriertransformation#Integrationssatz|Integrationssatz]]&nbsp; lautet in entsprechend angepasster Form:
+*Der&nbsp; [[Signal_Representation/Fourier_Transform_Laws#Integrationssatz|Integrationssatz]]&nbsp; lautet in entsprechend angepasster Form:
 :$$\frac{1}{T}\cdot \hspace{-0.1cm} \int_{ - \infty }^{\hspace{0.05cm}t} {x( \tau  )}\,\, {\rm d}\tau\ \ \circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\, \ \ X( f ) \cdot \left( {\frac{1}{{{\rm{j}}\cdot 2{\rm{\pi }\cdot }fT}} + \frac{1}{2T}\cdot {\rm \delta} ( f )} \right).$$