Aufgaben:Exercise 5.2: Inverse Discrete Fourier Transform - Revision history

Guenter at 14:38, 16 May 2021

2021-05-16T14:38:34Z

Guenter at 14:17, 16 May 2021

2021-05-16T14:17:10Z

Guenter at 14:15, 16 May 2021

2021-05-16T14:15:56Z

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

2021-03-23T12:39:51Z

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

Noah at 19:01, 20 March 2021

2021-03-20T19:01:43Z

Show changes

Oezdemir: Oezdemir moved page Aufgaben:Aufgabe 5.2: Inverse Diskrete Fouriertransformation to Aufgaben:Exercise 5.2: Inverse Discrete Fourier Transform

2020-11-29T18:50:47Z

Oezdemir moved page Aufgabe 5.2: Inverse Diskrete Fouriertransformation to Exercise 5.2: Inverse Discrete Fourier Transform

Oezdemir at 18:49, 29 November 2020

2020-11-29T18:49:54Z

Javier: Text replacement - "Signaldarstellung/Diskrete Fouriertransformation (DFT)" to "Signal_Representation/Discrete_Fourier_Transform_(DFT)"

2020-09-01T13:29:45Z

Text replacement - "Signaldarstellung/Diskrete Fouriertransformation (DFT)" to "Signal_Representation/Discrete_Fourier_Transform_(DFT)"

Javier: Text replacement - "Signal_Representation/Diskrete_Fouriertransformation_(DFT)" to "Signal_Representation/Discrete_Fourier_Transform_(DFT)"

2020-09-01T13:29:33Z

Text replacement - "Signal_Representation/Diskrete_Fouriertransformation_(DFT)" to "Signal_Representation/Discrete_Fourier_Transform_(DFT)"

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

2020-08-24T08:27:51Z

Text replacement - "Category:Aufgaben zu Signaldarstellung" to "Category:Exercises for Signal Representation"

@@ Line 76: / Line 76: @@
 ===Solution===
 {{ML-Kopf}}
-'''(1)'''&nbsp;  From the IDFT equation, with&nbsp; $D(\mu) = 0$&nbsp; for&nbsp; $\mu \ne 0$:
+'''(1)'''&nbsp;  From the IDFT equation,&nbsp; with&nbsp; $D(\mu) = 0$&nbsp; for&nbsp; $\mu \ne 0$:
 :$$d(\nu) = D(0) \cdot w^0 = D(0) =1\hspace{0.5cm}(0 \le \nu \le 7)\ \hspace{0.5cm} \Rightarrow\hspace{0.5cm}\hspace{0.15 cm}\underline{d(0) = d(1) = 1}.$$
-*This set of parameters describes the discrete form of the Fourier correspondence of the DC signal:
+*This parameter set describes the discrete form of the Fourier correspondence of the DC signal:
 :$$x(t) = 1 \hspace{0.2cm}\circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\, \hspace{0.2cm}
@@ Line 86: / Line 86: @@
-'''(2)'''&nbsp;  All spectral coefficients are zero except&nbsp; $D_1 = D_7 = 0.5$. It follows for&nbsp; $0 ≤ ν ≤ 7$:
+'''(2)'''&nbsp;  All spectral coefficients are zero except&nbsp; $D_1 = D_7 = 0.5$.&nbsp; It follows for&nbsp; $0 ≤ ν ≤ 7$:
 :$$d(\nu) = 0.5 \cdot {\rm e}^{-{\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm} (\pi /4) \hspace{0.05cm}\cdot \hspace{0.05cm} \nu} + 0.5 \cdot {\rm e}^{-{\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm} (\pi /4) \hspace{0.05cm}\cdot \hspace{0.05cm} \nu}
@@ Line 104: / Line 104: @@
-'''(3)'''&nbsp; Compared to subtask&nbsp; '''(2)'''&nbsp;, the oscillation frequency is now twice as large, namely&nbsp; $2 f_{\rm A}$&nbsp; instead of&nbsp; $f_{\rm A}$:
+'''(3)'''&nbsp; Compared to subtask&nbsp; '''(2)''',&nbsp; the oscillation frequency is now twice as large, namely&nbsp; $2 f_{\rm A}$&nbsp; instead of&nbsp; $f_{\rm A}$:
 :$$x(t) = \cos(2 \pi \cdot (2f_{\rm A}) \cdot t) \hspace{0.2cm}\circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\, \hspace{0.2cm}
@@ Line 132: / Line 132: @@
 '''(5)'''&nbsp; The Discrete Fourier Transform is also linear. Therefore, the superposition principle is still applicable:
 *The coefficients&nbsp; $D(\mu )$&nbsp; from column&nbsp; $\rm E$&nbsp; result as the sums of columns&nbsp; $\rm A$&nbsp; and&nbsp; $\rm D$.
-*Therefore, the alternating sequence&nbsp;  $\langle \hspace{0.1cm}d(ν) \hspace{0.1cm}\rangle $&nbsp;  becomes the sequence shifted up by&nbsp; $1$&nbsp; according to subtask&nbsp; '''(4)'''&nbsp;:
+*Therefore, the alternating sequence&nbsp;  $\langle \hspace{0.1cm}d(ν) \hspace{0.1cm}\rangle $&nbsp;  becomes the sequence shifted up by&nbsp; $1$&nbsp; according to subtask&nbsp; '''(4)''':
 :$$ \hspace{0.15 cm}\underline{d(0) =d(2) =d(4) =d(6)= 2}, \hspace{0.2cm}\hspace{0.15 cm}\underline{d(1) =d(3) =d(5) =d(7)  = 0}

← Older revision		Revision as of 14:17, 16 May 2021
Line 39:		Line 39:
	*This task belongs to the chapter  [[Signal_Representation/Discrete_Fourier_Transform_(DFT)\|Discrete Fourier Transformation (DFT)]].		*This task belongs to the chapter  [[Signal_Representation/Discrete_Fourier_Transform_(DFT)\|Discrete Fourier Transformation (DFT)]].

−	*The topic dealt with here is also dealt with in the interactive applet  [[Applets:~~Diskrete_Fouriertransformation_und_Inverse~~\|Discrete Fourier Transform and Inverse]].	+	*The topic dealt with here is also dealt with in the interactive applet  [[Applets:Discrete_Fouriertransform_and_Inverse\|Discrete Fourier Transform and Inverse]].

@@ Line 5: / Line 5: @@
 [[File:P_ID1138__Sig_A_5_2.png|250px|right|frame|Five different sets for the spectral coefficients&nbsp; $D(\mu)$]]
-With the ''Discrete Fourier Transform''&nbsp; (DFT), the following are obtained
+With the&nbsp; '''Discrete Fourier Transform'''&nbsp; $\rm (DFT)$,
-*from the&nbsp; $N$&nbsp; time coefficients&nbsp; $d(\nu)$ &nbsp; &rArr;   &nbsp; samples of the continuous-time signal&nbsp; $x(t)$ –
+*the&nbsp; $N$&nbsp; spectral range coefficients&nbsp; $D(\mu)$&nbsp;  are calculated
-are calculated. With&nbsp; $\nu = 0$, ... , $N – 1$&nbsp; and&nbsp; $\mu = 0$, ... , $N – 1$&nbsp; holds:
 :$$D(\mu) = \frac{1}{N} \cdot \sum_{\nu = 0 }^{N-1}
@@ Line 21: / Line 22: @@
   \hspace{0.05cm}.$$
-For the ''Inverse Discrete Fourier Transform''&nbsp; (IDFT)  &nbsp; ⇒  &nbsp; „inverse function” of the DFT, the following applies accordingly:
+For the&nbsp; '''Inverse Discrete Fourier Transform'''&nbsp; $\rm (DFT)$  &nbsp; ⇒  &nbsp; "inverse function" of the DFT, the following applies accordingly:
 :$$d(\nu) =  \sum_{\mu = 0 }^{N-1}
   D(\mu) \cdot  {w}^{-\nu \hspace{0.05cm} \cdot \hspace{0.05cm}\mu} \hspace{0.05cm}.$$
-In this task, the time coefficients&nbsp; $d(\nu)$&nbsp;  are to be determined for various example sequences (which are labelled&nbsp; $\rm A$, ... ,&nbsp; $\rm E$ in the table above)&nbsp; $D(\mu)$&nbsp; ermittelt werden. Thus,&nbsp; $N = 8$ always applies.
+In this task, the time coefficients&nbsp; $d(\nu)$&nbsp;  are to be determined for various sequences&nbsp; $D(\mu)$&nbsp;  (which are labelled&nbsp; $\rm A$, ... ,&nbsp; $\rm E$&nbsp; in the table above).&nbsp; Thus,&nbsp; $N = 8$ always applies.
@@ Line 46: / Line 46: @@
 <quiz display=simple>
-{What are the time coefficients&nbsp; $d(\nu)$&nbsp; for the&nbsp; $D(\mu)$–values of column&nbsp; $\rm A$?
+{What are the time coefficients&nbsp; $d(\nu)$&nbsp; for the&nbsp; $D(\mu)$&nbsp; values of column&nbsp; $\rm A$?
 |type="{}"}
 $d(0)\ = \ $  { 1 3% }
 $d(1)\ = \ $ { 1 3% }
-{What are the time coefficients&nbsp; $d(ν)$&nbsp; for the&nbsp; $D(\mu)$–values of column&nbsp; $\rm B$?
+{What are the time coefficients&nbsp; $d(ν)$&nbsp; for the&nbsp; $D(\mu)$&nbsp; values of column&nbsp; $\rm B$?
 |type="{}"}
 $d(0)\ = \ $ { 1 3% }
 $d(1)\ = \ $ { 0.707 3% }
-{What are the time coefficients&nbsp; $d(ν)$&nbsp; for the&nbsp; $D(\mu)$–values of column&nbsp; $\rm C$?
+{What are the time coefficients&nbsp; $d(ν)$&nbsp; for the&nbsp; $D(\mu)$&nbsp; values of column&nbsp; $\rm C$?
 |type="{}"}
 $d(0)\ = \ $ { 1 3% }
 $d(1)\ = \ $ { 0. }
-{What are the time coefficients&nbsp; $d(ν)$&nbsp; for the&nbsp; $D(\mu)$–values of column&nbsp; $\rm D$?
+{What are the time coefficients&nbsp; $d(ν)$&nbsp; for the&nbsp; $D(\mu)$&nbsp; values of column&nbsp; $\rm D$?
 |type="{}"}
 $d(0)\ = \ ${ 1 3% }
 $d(1)\ = \ $ { -1.03--0.97 }
-{What are the time coefficients&nbsp; $d(ν)$&nbsp; for the&nbsp; $D(\mu)$–values of column&nbsp; $\rm E$?
+{What are the time coefficients&nbsp; $d(ν)$&nbsp; for the&nbsp; $D(\mu)$&nbsp; values of column&nbsp; $\rm E$?
 |type="{}"}
 $d(0)\ = \ $ { 2 3% }

← Older revision		Revision as of 12:39, 23 March 2021
Line 139:		Line 139:

	__NOEDITSECTION__		__NOEDITSECTION__
−	[[Category:~~Exercises for~~ Signal Representation\|^5.2 Discrete Fourier Transform^]]	+	[[Category:Signal Representation: Exercises\|^5.2 Discrete Fourier Transform^]]

← Older revision		Revision as of 18:49, 29 November 2020
Line 138:		Line 138:

	__NOEDITSECTION__		__NOEDITSECTION__
−	[[Category:Exercises for Signal Representation\|^5. ~~Zeit- und frequenzdiskrete Signaldarstellung~~^]]	+	[[Category:Exercises for Signal Representation\|^5.2 Discrete Fourier Transform^]]

← Older revision	Revision as of 18:50, 29 November 2020
(No difference)

← Older revision		Revision as of 13:29, 1 September 2020
Line 1:		Line 1:

−	{{quiz-Header\|Buchseite=~~Signaldarstellung~~/~~Diskrete Fouriertransformation~~ (DFT)	+	{{quiz-Header\|Buchseite=Signal_Representation/Discrete_Fourier_Transform_(DFT)
	}}		}}

@@ Line 36: / Line 36: @@
 ''Hinweise:''
-*Die Aufgabe gehört zum  Kapitel&nbsp; [[Signal_Representation/Diskrete_Fouriertransformation_(DFT)|Diskrete Fouriertransformation (DFT)]].
+*Die Aufgabe gehört zum  Kapitel&nbsp; [[Signal_Representation/Discrete_Fourier_Transform_(DFT)|Diskrete Fouriertransformation (DFT)]].
 *Die hier behandelte Thematik wird auch im interaktiven Applet&nbsp; [[Applets:Diskrete_Fouriertransformation_und_Inverse|Diskrete Fouriertransformation und Inverse]] behandelt.

← Older revision		Revision as of 08:27, 24 August 2020
Line 138:		Line 138:

	__NOEDITSECTION__		__NOEDITSECTION__
−	[[Category:~~Aufgaben zu Signaldarstellung~~\|^5. Zeit- und frequenzdiskrete Signaldarstellung^]]	+	[[Category:Exercises for Signal Representation\|^5. Zeit- und frequenzdiskrete Signaldarstellung^]]