(New page: =Discrete-time Fourier transform of a window function= Used in ECE438. ---- Consider the perfect discrete-time window function <math>w[n]= \left\{ \begin{array}{ll} 1,&\text{ if }0 \...)
 
 
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[[Category:discrete-time Fourier transform]]
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[[Category:ECE438Fall2010Boutin]]
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=Discrete-time Fourier transform of a window function=
 
=Discrete-time Fourier transform of a window function=
 
Used in [[ECE438]].
 
Used in [[ECE438]].
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This page can be used to study the frequency-domain behavior of a discrete-time window function, as its length increases.
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Consider the perfect discrete-time window function
 
Consider the perfect discrete-time window function
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0, & \text{ else}.
 
0, & \text{ else}.
 
\end{array}
 
\end{array}
  \right.</math>
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  \right.,</math>
  
The DTFT of that window function is
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for any integer values of n. Note that N represents the length of the window. The DTFT of this window function is
  
 
<math>W(\omega) =\frac{e^{\frac{-j \omega (N-1)}{2}} \sin\left( \frac{\omega N}{2}\right)}{\sin \left( \frac{\omega}{2} \right)}</math>.
 
<math>W(\omega) =\frac{e^{\frac{-j \omega (N-1)}{2}} \sin\left( \frac{\omega N}{2}\right)}{\sin \left( \frac{\omega}{2} \right)}</math>.
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[[Image:W_of_omega_N_equal_10000.png|500px]]
 
[[Image:W_of_omega_N_equal_10000.png|500px]]
  
Observe the close resemblance of this graph to that of the magnitude of the Fourier transform of the corresponding continuous-time window.  
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Observe the close resemblance of this graph to that of the magnitude of the Fourier transform of the signal <math>x[n]=1</math>, for any n integer (in other words, an "infinite-length" window).
 
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[[ECE438|Back to ECE438]]
 
[[ECE438|Back to ECE438]]
  
 
[[2010_Fall_ECE_438_Boutin|Back to ECE438 Fall 2010]]
 
[[2010_Fall_ECE_438_Boutin|Back to ECE438 Fall 2010]]

Latest revision as of 08:25, 29 December 2010


Discrete-time Fourier transform of a window function

Used in ECE438.

This page can be used to study the frequency-domain behavior of a discrete-time window function, as its length increases.


Consider the perfect discrete-time window function

$ w[n]= \left\{ \begin{array}{ll} 1,&\text{ if }0 \leq n < N \\ 0, & \text{ else}. \end{array} \right., $

for any integer values of n. Note that N represents the length of the window. The DTFT of this window function is

$ W(\omega) =\frac{e^{\frac{-j \omega (N-1)}{2}} \sin\left( \frac{\omega N}{2}\right)}{\sin \left( \frac{\omega}{2} \right)} $.


Below is the graph of the magniture of $ W(\omega) $ for $ N=15 $.

W of omega N equal 15.png


Below is the graph of the magniture of $ W(\omega) $ for $ N=100 $. Observe that the ripples are "thinner" and more numerous than in the previous case of $ N=10 $.

W of omega N equal 100.png


Below is the graph of the magniture of $ W(\omega) $ for $ N=10000 $.

W of omega N equal 10000.png

Observe the close resemblance of this graph to that of the magnitude of the Fourier transform of the signal $ x[n]=1 $, for any n integer (in other words, an "infinite-length" window).


Back to ECE438

Back to ECE438 Fall 2010

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