Aufgaben:Exercise 3.5: Differentiation of a Triangular Pulse - Revision history

Guenter at 13:19, 24 May 2021

2021-05-24T13:19:53Z

Guenter at 12:23, 27 April 2021

2021-04-27T12:23:18Z

Guenter at 12:13, 27 April 2021

2021-04-27T12:13:23Z

Guenter at 16:08, 26 April 2021

2021-04-26T16:08:53Z

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 16:30, 27 January 2021

2021-01-27T16:30:15Z

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Oezdemir: Oezdemir moved page Aufgaben:Aufgabe 3.5: Differentiation eines Dreicksignals to Aufgaben:Exercise 3.5: Differentiation of a Triangular Pulse

2020-11-02T15:28:23Z

Oezdemir moved page Aufgabe 3.5: Differentiation eines Dreicksignals to Exercise 3.5: Differentiation of a Triangular Pulse

Oezdemir at 15:24, 2 November 2020

2020-11-02T15:24:16Z

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"

2020-09-01T09:49:31Z

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

@@ Line 3: / Line 3: @@
 }}
-[[File:P_ID514__Sig_A_3_5.png|250px|right|frame|Dreiecksignal und <br>differenziertes Dreiecksignal]]
+[[File:P_ID514__Sig_A_3_5.png|250px|right|frame|Triangular signal and <br>differentiated triangular signal]]
 We are looking for the spectrum&nbsp; $Y(f)$&nbsp; of the signal

@@ Line 48: / Line 48: @@
 |type="[]"}
 + The zeros of&nbsp; $X(f)$&nbsp; also remain in&nbsp; $Y(f)$&nbsp;.
-- For&nbsp; $f \rightarrow \infty$&nbsp; has&nbsp; $Y(f)$&nbsp; the same course as&nbsp; $X(f)$.
+- For large&nbsp; $f$&ndash;values,&nbsp; $Y(f)$&nbsp; satisfies the same bound as&nbsp; $X(f)$.
-+ For&nbsp; $f \rightarrow \infty$&nbsp; ist&nbsp; $Y(f)$&nbsp; is twice as large as the spectrum of a rectangular pulse of duration&nbsp; $T$.
++ For large&nbsp; $f$&ndash;values,&nbsp; $|X(f)|$&nbsp; is smaller than the magnitude spectrum of a rectangular pulse of duration&nbsp; $T$.
 {Calculate&nbsp; $Y(f)$&nbsp; starting from the rectangular pulse by applying the displacement theorem. Which statement is true here?
@@ Line 82: / Line 82: @@
 *For the frequencies at which the sine function delivers the values&nbsp; $\pm 1$&nbsp; ,&nbsp; $|Y_{\text{max}}(f)|$&nbsp; and&nbsp; $|Y(f)|$&nbsp; are identical.
-*For the square-wave pulse of amplitude&nbsp; $A$&nbsp;  the corresponding bound is&nbsp; $A/(\pi \cdot |f|)$.
+*For the rectangular pulse of same amplitude&nbsp; $A$&nbsp;  the corresponding bound is&nbsp; $A/(\pi \cdot |f|)$.
 *In contrast, the spectrum&nbsp; $X(f)$&nbsp; of the triangular pulse falls asymptotically faster:
@@ Line 94: / Line 94: @@
 :$$y(t) = r( {t + T/2} ) - r( {t - T/2} ).$$
-*By applying the displacement theorem twice, one obtains:
+*By applying the shifting theorem twice, one obtains:
 :$$Y( f ) = R( f ) \cdot {\rm{e}}^{{\rm{j\pi }}fT}  - R( f ) \cdot {\rm{e}}^{ - {\rm{j\pi }}fT} .$$

@@ Line 30: / Line 30: @@
 ''Hints:''
 *This task belongs to the chapter&nbsp; [[Signal_Representation/Fourier_Transform_Theorems|Fourier Transform Theorems]].
-*All the laws presented here - including the&nbsp; [[Signal_Representation/Fourier_Transform_Laws#Shifting_Theorem|Shifting Theorem]]&nbsp; and the&nbsp; [[Signal_Representation/Fourier_Transform_Laws#Integration_Theorem|Integration Theorem]]&nbsp; – 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".
+*All the laws presented here - including the&nbsp; [[Signal_Representation/Fourier_Transform_Laws#Shifting_Theorem|Shifting Theorem]]&nbsp; and the&nbsp; [[Signal_Representation/Fourier_Transform_Theorems#Differentiation_Theorem|Differentiation Theorem]]&nbsp; – 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".
 *In subtask&nbsp; '''(3)'''&nbsp; the spectrum&nbsp; $Y(f)$&nbsp; is to be calculated starting from a symmetrical rectangular pulse&nbsp; $r(t)$&nbsp; with amplitude&nbsp; $A$&nbsp; and duration&nbsp; $T$&nbsp; and its spectrum&nbsp; $R(f) = A \cdot T \cdot \text{si}(\pi fT)$&nbsp;.&nbsp; This is achieved by applying the&nbsp; [[Signal_Representation/Fourier_Transform_Laws#Shifting_Theorem|Shifting Theorem]].
-*In&nbsp; [[Aufgaben:3.5Z_Integration_von_Diracfunktionen|Exercise 3.5Z]]&nbsp; the spectrum&nbsp; $Y(f)$&nbsp; is calculated starting from a signal consisting of three Dirac functions by applying the integration theorem.
+*In&nbsp; [[Aufgaben:3.5Z_Integration_von_Diracfunktionen|Exercise 3.5Z]]&nbsp; the spectrum&nbsp; $Y(f)$&nbsp; is calculated starting from a signal consisting of three Dirac functions by applying the&nbsp; [[Signal_Representation/Fourier_Transform_Theorems#Integration_Theorem|Integration Theorem]].

@@ Line 7: / Line 7: @@
 We are looking for the spectrum&nbsp; $Y(f)$&nbsp; of the signal
-:$$y\left( t \right) = \left\{ \begin{array}{c} A \\   - A \\  0 \\  \end{array} \right.\quad \begin{array}{*{20}c}   {{\rm{f \ddot{u}r}}}  \\   {{\rm{f\ddot{u} r}}}  \\   {{\rm{f\ddot{u}r}}}  \\ \end{array}\;\begin{array}{*{20}c}   { - T \le t < 0,}  \\   {0 < t \le T,}  \\   {{\rm{else}}{\rm{.}}}  \\\end{array}$$
+:$$y\left( t \right) = \left\{ \begin{array}{c} A \\   - A \\  0 \\  \end{array} \right.\quad \begin{array}{*{20}c}   {{\rm{f or}}}  \\   {{\rm{for}}}  \\      \\ \end{array}\;\begin{array}{*{20}c}   { - T \le t < 0,}  \\   {0 < t \le T,}  \\   {{\rm{else}}{\rm{.}}}  \\\end{array}$$
 Let&nbsp; $A = 1\,{\rm V}$&nbsp;  and&nbsp; $T = 0.5\,{\rm ms}$ apply.
@@ Line 15: / Line 15: @@
 :$$X( f ) = A \cdot T \cdot {\mathop{\rm si}\nolimits} ^2 ( {{\rm{\pi }}fT} ),$$
-where&nbsp; $\text{si}(x) = \text{sin}(x)/x$&nbsp;.
+where&nbsp; $\text{si}(x) = \text{sin}(x)/x$.
-A comparison of the two time signals shows that the following relationship exists between the functions&nbsp; $x(t)$&nbsp; and&nbsp; $y(t)$&nbsp;:
 :$$y(t) = T \cdot \frac{{{\rm d}x(t)}}{{{\rm d}t}}.$$
@@ Line 28: / Line 29: @@
 ''Hints:''
-*This exercise belongs to the chapter&nbsp; [[Signal_Representation/Fourier_Transform_Laws|Fourier Transform Laws]].
+*This task belongs to the chapter&nbsp; [[Signal_Representation/Fourier_Transform_Theorems|Fourier Transform Theorems]].
-*All the laws presented there - including the&nbsp; [[Signal_Representation/Fourier_Transform_Laws#Shifting_Theorem|Shifting Theorem]]&nbsp; and the&nbsp; [[Signal_Representation/Fourier_Transform_Laws#Integration_Theorem|Integration Theorem]]&nbsp; – are illustrated with examples in the learning video&nbsp; [[Gesetzmäßigkeiten_der_Fouriertransformation_(Lernvideo)|Gesetzmäßigkeiten der Fouriertransformation]]&nbsp;.
+*All the laws presented here - including the&nbsp; [[Signal_Representation/Fourier_Transform_Laws#Shifting_Theorem|Shifting Theorem]]&nbsp; and the&nbsp; [[Signal_Representation/Fourier_Transform_Laws#Integration_Theorem|Integration Theorem]]&nbsp; – 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".
-*In subtask&nbsp; '''(3)'''&nbsp; the spectrum&nbsp; $Y(f)$&nbsp; is to be calculated starting from a symmetrical rectangular pulse&nbsp; $r(t)$&nbsp; with amplitude&nbsp; $A$&nbsp; and duration&nbsp; $T$&nbsp; and its spectrum&nbsp; $R(f) = A \cdot T \cdot \text{si}(\pi fT)$&nbsp;. This is achieved by applying the&nbsp; [[Signal_Representation/Fourier_Transform_Laws#Shifting_Theorem|Shifting Theorem]].
+*In subtask&nbsp; '''(3)'''&nbsp; the spectrum&nbsp; $Y(f)$&nbsp; is to be calculated starting from a symmetrical rectangular pulse&nbsp; $r(t)$&nbsp; with amplitude&nbsp; $A$&nbsp; and duration&nbsp; $T$&nbsp; and its spectrum&nbsp; $R(f) = A \cdot T \cdot \text{si}(\pi fT)$&nbsp;.&nbsp; This is achieved by applying the&nbsp; [[Signal_Representation/Fourier_Transform_Laws#Shifting_Theorem|Shifting Theorem]].
-*In&nbsp; [[Aufgaben:3.5Z_Integration_von_Diracfunktionen|Task 3.5Z]]&nbsp; the spectrum&nbsp; $Y(f)$&nbsp; is calculated starting from a signal consisting of three Dirac functions by applying the integration theorem.
+*In&nbsp; [[Aufgaben:3.5Z_Integration_von_Diracfunktionen|Exercise 3.5Z]]&nbsp; the spectrum&nbsp; $Y(f)$&nbsp; is calculated starting from a signal consisting of three Dirac functions by applying the integration theorem.

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

← 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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 ''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 dort dargelegten Gesetzmäßigkeiten – unter Anderem auch der&nbsp; [[Signal_Representation/Gesetzmäßigkeiten_der_Fouriertransformation#Verschiebungssatz|Verschiebungssatz]]&nbsp; und der&nbsp; [[Signal_Representation/Gesetzmäßigkeiten_der_Fouriertransformation#Integrationssatz|Integrationssatz]]&nbsp; – werden im Lernvideo&nbsp; [[Gesetzmäßigkeiten_der_Fouriertransformation_(Lernvideo)|Gesetzmäßigkeiten der Fouriertransformation]]&nbsp; an Beispielen verdeutlicht.
+*Alle dort dargelegten Gesetzmäßigkeiten – unter Anderem auch der&nbsp; [[Signal_Representation/Fourier_Transform_Laws#Verschiebungssatz|Verschiebungssatz]]&nbsp; und der&nbsp; [[Signal_Representation/Fourier_Transform_Laws#Integrationssatz|Integrationssatz]]&nbsp; – werden im Lernvideo&nbsp; [[Gesetzmäßigkeiten_der_Fouriertransformation_(Lernvideo)|Gesetzmäßigkeiten der Fouriertransformation]]&nbsp; an Beispielen verdeutlicht.
-*In der Teilaufgabe&nbsp; '''(3)'''&nbsp; soll das Spektrum&nbsp; $Y(f)$&nbsp; ausgehend von einem symmetrischen Rechteckimpuls&nbsp; $r(t)$&nbsp; mit Amplitude&nbsp; $A$&nbsp; und Dauer&nbsp; $T$&nbsp; sowie dessen Spektrum&nbsp; $R(f) = A \cdot T \cdot \text{si}(\pi fT)$&nbsp; berechnet werden. Dies erreicht man durch zweimalige Anwendung des&nbsp; [[Signal_Representation/Gesetzmäßigkeiten_der_Fouriertransformation#Verschiebungssatz|Verschiebungssatzes]].
+*In der Teilaufgabe&nbsp; '''(3)'''&nbsp; soll das Spektrum&nbsp; $Y(f)$&nbsp; ausgehend von einem symmetrischen Rechteckimpuls&nbsp; $r(t)$&nbsp; mit Amplitude&nbsp; $A$&nbsp; und Dauer&nbsp; $T$&nbsp; sowie dessen Spektrum&nbsp; $R(f) = A \cdot T \cdot \text{si}(\pi fT)$&nbsp; berechnet werden. Dies erreicht man durch zweimalige Anwendung des&nbsp; [[Signal_Representation/Fourier_Transform_Laws#Verschiebungssatz|Verschiebungssatzes]].
 *In&nbsp; [[Aufgaben:3.5Z_Integration_von_Diracfunktionen|Aufgabe 3.5Z]]&nbsp; wird das Spektrum&nbsp; $Y(f)$&nbsp; ausgehend von einem aus drei Diracfunktionen bestehenden Signal durch Anwendung des Integrationssatzes berechnet.