Aufgaben:Exercise 5.4: Comparison of Rectangular and Hanning Window - Revision history

Hwang at 13:24, 18 January 2023

2023-01-18T13:24:10Z

Javier: Text replacement - "analyse" to "analyze"

2021-09-22T11:47:37Z

Text replacement - "analyse" to "analyze"

Guenter at 15:16, 19 May 2021

2021-05-19T15:16:33Z

Guenter: Guenter moved page Aufgaben:Exercise 5.4: Comparison of rectangular and Hanning window to Aufgaben:Exercise 5.4: Comparison of Rectangular and Hanning Window

2021-05-19T12:37:34Z

Guenter moved page Exercise 5.4: Comparison of rectangular and Hanning window to Exercise 5.4: Comparison of Rectangular and Hanning Window

Guenter: Guenter moved page Aufgaben:Exercise 5.4: Comparison of Rectangular And Hanning Window to Aufgaben:Exercise 5.4: Comparison of rectangular and Hanning window

2021-05-19T12:36:00Z

Guenter moved page Exercise 5.4: Comparison of Rectangular And Hanning Window to Exercise 5.4: Comparison of rectangular and Hanning window

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

2021-03-23T12:39:40Z

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

Noah at 19:07, 22 March 2021

2021-03-22T19:07:52Z

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Oezdemir: Oezdemir moved page Aufgaben:Aufgabe 5.4: Vergleich von Rechteck- und Hanningfenster to Aufgaben:Exercise 5.4: Comparison of Rectangular And Hanning Window

2020-11-29T20:57:53Z

Oezdemir moved page Aufgabe 5.4: Vergleich von Rechteck- und Hanningfenster to Exercise 5.4: Comparison of Rectangular And Hanning Window

Oezdemir at 20:56, 29 November 2020

2020-11-29T20:56:53Z

Javier: Text replacement - "Signaldarstellung/Spektralanalyse" to "Signal_Representation/Spectrum_Analysis"

2020-09-01T13:33:55Z

Text replacement - "Signaldarstellung/Spektralanalyse" to "Signal_Representation/Spectrum_Analysis"

@@ Line 80: / Line 80: @@
 + The DFT spectrum&nbsp; $Y_{\rm A}(f)$&nbsp; is identical to the actual spectrum&nbsp; $X(f)$.
-{Using the Hanning window and&nbsp;  $T_{\rm P} = 8 \ \text{ms} $, what is&nbsp; $Y(f)$&nbsp; when the input spectrum&nbsp; $X(f) = Y_{\rm A}(f)$&nbsp; is applied? <br>Give the weights of the Dirac lines at&nbsp; $f_1= 1\ \text{kHz}$ &nbsp; &rArr; &nbsp; $G(f_1)$&nbsp; and at &nbsp; $f_2 = 1.125\ \text{kHz}$&nbsp; &rArr; &nbsp; $G(f_2)$.
+{Using the Hanning window and&nbsp;  $T_{\rm P} = 8 \ \text{ms} $, what is&nbsp; $Y(f)$&nbsp; when the input spectrum&nbsp; $X(f) = Y_{\rm A}(f)$&nbsp; is applied? <br>Give the weights of the Dirac delta lines at&nbsp; $f_1= 1\ \text{kHz}$ &nbsp; &rArr; &nbsp; $G(f_1)$&nbsp; and at &nbsp; $f_2 = 1.125\ \text{kHz}$&nbsp; &rArr; &nbsp; $G(f_2)$.
 |type="{}"}
 $G(f_1 = 1.000 \ \text{kHz})\ = \ $ { 0.625 3% } &nbsp;$\text{V}$
@@ Line 96: / Line 96: @@
 '''(1)'''&nbsp;  <u>Solutions 1 and 4</u> are correct:
 *Using the Hanning window, three Dirac functions should be recognisable even if&nbsp; $x(t)$&nbsp; contains only one frequency &nbsp; ⇒ &nbsp; the rectangular window was used.
-*With&nbsp; $T_{\rm P} = 4 \ \text{ms}$&nbsp;, the frequency resolution is&nbsp; $f_{\rm A}= 1/T_{\rm P} = 0.25 \ \text{kHz}$.&nbsp; Thus the frequency&nbsp; $f_2$&nbsp; does not lie in the given grid and&nbsp; $Y(f)$&nbsp; would be composed of very many Dirac lines.&nbsp; This means: &nbsp; the third statement is wrong.
+*With&nbsp; $T_{\rm P} = 4 \ \text{ms}$&nbsp;, the frequency resolution is&nbsp; $f_{\rm A}= 1/T_{\rm P} = 0.25 \ \text{kHz}$.&nbsp; Thus the frequency&nbsp; $f_2$&nbsp; does not lie in the given grid and&nbsp; $Y(f)$&nbsp; would be composed of very many Dirac delta lines.&nbsp; This means: &nbsp; the third statement is wrong.
 [[File:P_ID1167__Sig_A_5_4a.png|right|frame|Output signal&nbsp; $y(t)$&nbsp; with the rectangular window]]

@@ Line 11: / Line 11: @@
 Unknown and thus to be estimated are its parameters&nbsp; $A_1$,&nbsp; $f_1$,&nbsp; $A_2$&nbsp; and&nbsp; $f_2$.
-After weighting the signal with the window function&nbsp; $w(t)$&nbsp;, the product&nbsp; $y(t) = x(t) \cdot w(t)$&nbsp; is subjected to a&nbsp; [[Signal_Representation/Discrete_Fourier_Transform_(DFT)|Discrete Fourier Transform]]&nbsp; (DFT) with the parameters&nbsp; $N = 512$&nbsp; and&nbsp; $T_{\rm P}$.&nbsp; The time&nbsp; $T_{\rm P}$&nbsp; of the signal section to be analysed can be set by the user as desired.
+After weighting the signal with the window function&nbsp; $w(t)$&nbsp;, the product&nbsp; $y(t) = x(t) \cdot w(t)$&nbsp; is subjected to a&nbsp; [[Signal_Representation/Discrete_Fourier_Transform_(DFT)|Discrete Fourier Transform]]&nbsp; (DFT) with the parameters&nbsp; $N = 512$&nbsp; and&nbsp; $T_{\rm P}$.&nbsp; The time&nbsp; $T_{\rm P}$&nbsp; of the signal section to be analyzed can be set by the user as desired.
 Two functions are available for windowing, each of which is zero for&nbsp; $|t| > T_{\rm P}/2$:
@@ Line 50: / Line 50: @@
   \hspace{0.05cm}.$$
-In the graph, two further spectral functions&nbsp; $Y_{\rm B}(f)$&nbsp; and&nbsp; $Y_{\rm C}(f)$&nbsp; are shown, which result when a&nbsp; $1 \ \text{kHz}$&nbsp; signal is analysed by DFT and the DFT parameter&nbsp; $T_{\rm P} = 8.5 \ \text{ms}$&nbsp; is chosen unfavourably.
+In the graph, two further spectral functions&nbsp; $Y_{\rm B}(f)$&nbsp; and&nbsp; $Y_{\rm C}(f)$&nbsp; are shown, which result when a&nbsp; $1 \ \text{kHz}$&nbsp; signal is analyzed by DFT and the DFT parameter&nbsp; $T_{\rm P} = 8.5 \ \text{ms}$&nbsp; is chosen unfavourably.
 *For one of the images the rectangular window is used, for the other the Hanning window.

@@ Line 11: / Line 11: @@
 Unknown and thus to be estimated are its parameters&nbsp; $A_1$,&nbsp; $f_1$,&nbsp; $A_2$&nbsp; and&nbsp; $f_2$.
-After weighting the signal with the window function&nbsp; $w(t)$&nbsp;, the product&nbsp; $y(t) = x(t) \cdot w(t)$&nbsp; is subjected to a&nbsp; [[Signal_Representation/Discrete_Fourier_Transform_(DFT)|Discrete Fourier Transform]]&nbsp; (DFT) with the parameters&nbsp; $N = 512$&nbsp; and&nbsp; $T_{\rm P}$&nbsp;. The time period&nbsp; $T_{\rm P}$&nbsp; of the signal section to be analysed can be set by the user as desired.
+After weighting the signal with the window function&nbsp; $w(t)$&nbsp;, the product&nbsp; $y(t) = x(t) \cdot w(t)$&nbsp; is subjected to a&nbsp; [[Signal_Representation/Discrete_Fourier_Transform_(DFT)|Discrete Fourier Transform]]&nbsp; (DFT) with the parameters&nbsp; $N = 512$&nbsp; and&nbsp; $T_{\rm P}$.&nbsp; The time&nbsp; $T_{\rm P}$&nbsp; of the signal section to be analysed can be set by the user as desired.
-Two functions are available for windowing, each of which is zero for&nbsp; $|t| > T_{\rm P}/2$&nbsp;:
+Two functions are available for windowing, each of which is zero for&nbsp; $|t| > T_{\rm P}/2$:
 *The&nbsp; '''rectangular window''':
 :$${w} (\nu)   = \left\{ \begin{array}{c} 1 \\
 \\  \end{array} \right.\quad
-\begin{array}{*{10}c}    {\rm{f\ddot{u}r}}
+\begin{array}{*{10}c}    {\rm{for}}
 \\    \\ \end{array}\begin{array}{*{20}c}
 -N/2 \le \nu < N/2 \hspace{0.05cm}, \\
-{\rm sonst} \hspace{0.05cm}, \\
+{\rm else} \hspace{0.05cm}, \\
 \end{array}$$
@@ Line 31: / Line 31: @@
 :$${w} (\nu)   = \left\{ \begin{array}{c} 0.5 + 0.5 \cdot \cos (2 \pi \cdot {\nu}/{N}) \\
 \\  \end{array} \right.\quad
-\begin{array}{*{10}c}    {\rm{f\ddot{u}r}}
+\begin{array}{*{10}c}    {\rm{for}}
 \\    \\ \end{array}\begin{array}{*{20}c}
 -N/2 \le \nu < N/2 \hspace{0.05cm}, \\
-{\rm sonst} \hspace{0.05cm}, \\
+{\rm else} \hspace{0.05cm}, \\
 \end{array}$$
@@ Line 50: / Line 50: @@
   \hspace{0.05cm}.$$
-In the graph above, two further spectral functions&nbsp; $Y_{\rm B}(f)$&nbsp; and&nbsp; $Y_{\rm C}(f)$&nbsp; are shown, which result when a&nbsp; $1 \ \text{kHz}$–Signal&nbsp; signal is analysed by DFT and the DFT parameter&nbsp; $T_{\rm P} = 8.5 \ \text{ms}$&nbsp; is chosen unfavourably.
+In the graph, two further spectral functions&nbsp; $Y_{\rm B}(f)$&nbsp; and&nbsp; $Y_{\rm C}(f)$&nbsp; are shown, which result when a&nbsp; $1 \ \text{kHz}$&nbsp; signal is analysed by DFT and the DFT parameter&nbsp; $T_{\rm P} = 8.5 \ \text{ms}$&nbsp; is chosen unfavourably.
 *For one of the images the rectangular window is used, for the other the Hanning window.
@@ Line 64: / Line 64: @@
 ''Hints:''
 *This task belongs to the chapter&nbsp; [[Signal_Representation/Spectrum_Analysis|Spectrum Analysis]].
-*Note that the frequency resolution&nbsp; $f_{\rm A}$&nbsp; is equal to the reciprocal of the adjustable parameter&nbsp; $T_{\rm P}$&nbsp;.
+*Note that the frequency resolution&nbsp; $f_{\rm A}$&nbsp; is equal to the reciprocal of the adjustable parameter&nbsp; $T_{\rm P}$.
@@ Line 77: / Line 78: @@
 - The Hanning window was used for weighting.
 - The DFT parameter&nbsp; $T_{\rm P} = 4\ \text{ms}$&nbsp; was used.
-+ The DFT spectrum $Y_{\rm A}(f)$ is identical to the actual spectrum $X(f)$.
++ The DFT spectrum&nbsp; $Y_{\rm A}(f)$&nbsp; is identical to the actual spectrum&nbsp; $X(f)$.
-{Using the Hanning window and&nbsp;  $T_{\rm P} = 8 \ \text{ms} $, what is, &nbsp; $Y(f)$&nbsp; when the input spectrum&nbsp; $X(f) = Y_{\rm A}(f)$&nbsp; is applied? <br>Give the weights of the diraclines at&nbsp; $f_1= 1\ \text{kHz}$&nbsp; and&nbsp; $f_2 = 1.125\ \text{kHz}$&nbsp;.
+{Using the Hanning window and&nbsp;  $T_{\rm P} = 8 \ \text{ms} $, what is&nbsp; $Y(f)$&nbsp; when the input spectrum&nbsp; $X(f) = Y_{\rm A}(f)$&nbsp; is applied? <br>Give the weights of the Dirac lines at&nbsp; $f_1= 1\ \text{kHz}$ &nbsp; &rArr; &nbsp; $G(f_1)$&nbsp; and at &nbsp; $f_2 = 1.125\ \text{kHz}$&nbsp; &rArr; &nbsp; $G(f_2)$.
 |type="{}"}
 $G(f_1 = 1.000 \ \text{kHz})\ = \ $ { 0.625 3% } &nbsp;$\text{V}$
 $G(f_2 = 1.125 \ \text{kHz})\ = \ $ { 0.5 3% } &nbsp;$\text{V}$
-{We consider the&nbsp; $1\ \text{kHz}$ cosine signal&nbsp; $x(t)$. Which spectrum -&nbsp; $Y_{\rm B}(f)$&nbsp; or&nbsp; $Y_{\rm C}(f)$&nbsp; – results with the rectangular or the Hanning window, respectively, if the DFT parameter&nbsp;  $T_{\rm P} = 8.5 \ \text{ms}$&nbsp; is chosen unfavourably?
+{We consider the&nbsp; $1\ \text{kHz}$ cosine signal&nbsp; $x(t)$.&nbsp; Which spectrum -&nbsp; $Y_{\rm B}(f)$&nbsp; or&nbsp; $Y_{\rm C}(f)$&nbsp; – results with the rectangular or the Hanning window, respectively, if the DFT parameter&nbsp;  $T_{\rm P} = 8.5 \ \text{ms}$&nbsp; is chosen unfavourably?
 |type="()"}
-- $Y_{\rm B}(f)$ results with rectangular windowing.
+- $Y_{\rm B}(f)$&nbsp; results with rectangular windowing.
-+ $Y_{\rm B}(f)$ results with the Hanning window.
++ $Y_{\rm B}(f)$&nbsp; results with the Hanning window.
 </quiz>
@@ Line 93: / Line 94: @@
 ===Solution===
 {{ML-Kopf}}
-'''(1)'''&nbsp;  <u>Proposed solutions 1 and 4</u> are correct:
+'''(1)'''&nbsp;  <u>Solutions 1 and 4</u> are correct:
-*Using the Hanning window, three Dirac functions should be recognisable even if&nbsp; $x(t)$&nbsp; contains only one frequency &nbsp; ⇒ &nbsp; rectangular window was used.
+*Using the Hanning window, three Dirac functions should be recognisable even if&nbsp; $x(t)$&nbsp; contains only one frequency &nbsp; ⇒ &nbsp; the rectangular window was used.
-*With&nbsp; $T_{\rm P} = 4 \ \text{ms}$&nbsp;, the frequency resolution is&nbsp; $f_{\rm A}= 1/T_{\rm P} = 0.25 \ \text{kHz}$. Thus the frequency&nbsp; $f_2$&nbsp; does not lie in the given grid and&nbsp; $Y(f)$&nbsp; would be composed of very many diraclines. This means: &nbsp; die third statement is wrong.
+*With&nbsp; $T_{\rm P} = 4 \ \text{ms}$&nbsp;, the frequency resolution is&nbsp; $f_{\rm A}= 1/T_{\rm P} = 0.25 \ \text{kHz}$.&nbsp; Thus the frequency&nbsp; $f_2$&nbsp; does not lie in the given grid and&nbsp; $Y(f)$&nbsp; would be composed of very many Dirac lines.&nbsp; This means: &nbsp; the third statement is wrong.
-[[File:P_ID1167__Sig_A_5_4a.png|right|frame|$\text{Example signal 1}$&nbsp; for spectral analysis]]
+[[File:P_ID1167__Sig_A_5_4a.png|right|frame|Output signal&nbsp; $y(t)$&nbsp; with the rectangular window]]
-*As can be seen from the graph,&nbsp; $x(t)$&nbsp; has the period duration&nbsp; $T_{\rm 0} = 8 \ \text{ms}$. If one chooses the DFT parameter equal to&nbsp; $T_{\rm P} = 4 \ \text{ms}$&nbsp; (or an integer multiple thereof), the periodic continuation&nbsp; ${\rm P}\{ x(t)\} $&nbsp;  in the interval&nbsp; $|t| \leq T_{\rm P}/2$&nbsp; coincides with&nbsp; $x(t)$&nbsp;, so that the weighting function&nbsp; $w(t)$&nbsp; has no disturbing effect: &nbsp;
+*As can be seen from the graph,&nbsp; $x(t)$&nbsp; has the period duration&nbsp; $T_{\rm 0} = 8 \ \text{ms}$.
 *The DFT spectrum&nbsp; $Y(f)$&nbsp; thus agrees with the actual spectrum.
-'''(2)'''&nbsp; Because of$T_{\rm 0} = 8 \ \text{ms}$&nbsp;, the Hanning spectrum&nbsp; $W(f)$&nbsp;
+'''(2)'''&nbsp; Because of&nbsp; $T_{\rm 0} = 8 \ \text{ms}$&nbsp;, the Hanning spectrum&nbsp; $W(f)$&nbsp;
 *consists of three Dirac functions at positive frequencies
-*and three axisymmetrical diracs at negative frequencies
+*and three axisymmetrical Diracs at negative frequencies
 are composed. For the positive frequencies, the spectral function is:
 :$$W(f) =0.5\cdot {\rm \delta}(f) + 0.25\cdot {\rm \delta}(f-f_{\rm A})+ 0.25\cdot {\rm \delta}(f+f_{\rm A})\hspace{0.05cm}.$$
-The output spectrum results from the convolution between&nbsp; $X(f)$&nbsp; and&nbsp; $W(f)$. At positive frequencies, there are now four diracs with the following weights:
+The output spectrum results from the convolution between&nbsp; $X(f)$&nbsp; and&nbsp; $W(f)$.&nbsp; At positive frequencies, there are now four Diracs with the following weights:
 :$$\begin{align*} G(f = 0.875\,{\rm kHz}) & = 1\, {\rm V}\cdot 0.25 = 0.250\, {\rm
@@ Line 128: / Line 130: @@
-'''(3)'''&nbsp; <u> The second solution suggestion</u> is correct::
+'''(3)'''&nbsp; <u> Solution 2</u> is correct:
-*The rectangular window then delivers a very strongly distorted result if the window width&nbsp; $T_{\rm P}$&nbsp; (as here) is not adapted to the frequency of the cosine signal.
+*The rectangular window delivers a very strongly distorted result if the window width&nbsp; $T_{\rm P}$&nbsp; (as here) is not adapted to the cosine frequency.
-*In this case, the Hanning window is more suitable.
+*In this case, the Hanning window is more suitable.&nbsp; Then the measured spectrum&nbsp; $Y_{\rm B}(f)$ results.
 {{ML-Fuß}}
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 [[Category:Signal Representation: Exercises|^5.4 Spectrum Analysis^]]

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