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| <math> X[k] \  </math>
 
| <math> X[k] \  </math>
 
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|-
| align="right" style="padding-right: 1em;" | name
+
| align="right" style="padding-right: 1em;" |
| <math>type signal here\ </math>  
+
| <math>\ \delta[n] </math>  
 
|  
 
|  
| <math> type transform here \! \ </math>
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| <math>\ 1 </math>
 
|-
 
|-
| align="right" style="padding-right: 1em;" | name
+
| align="right" style="padding-right: 1em;" |  
| <math>type signal here \ </math>  
+
| <math>\ 1 </math>  
 
|  
 
|  
| <math>type transform here</math>
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| <math>\ N\delta[k] </math>
 +
|-
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| align="right" style="padding-right: 1em;" |
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| <math>\ e^{j2\pi k_0 n} </math>
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|
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| <math>\ N\delta[((k - k_0))_N] </math>
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|-
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| align="right" style="padding-right: 1em;" |
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| <math>\ \cos(\frac{2\pi}{N}k_0n) </math>
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|
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| <math>\ \frac{N}{2}(\delta[((k - k_0))_N] + \delta[((k+k_0))_N]) </math>
 
|}
 
|}
  

Revision as of 11:11, 27 November 2011

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Discrete Fourier Transform

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Discrete Fourier Transform Pairs and Properties (info)
Definition CT Fourier Transform and its Inverse
Discrete Fourier Transform $ X [k] = \sum_{n=0}^{N-1} x[n]e^{-j 2\pi \frac{k n}{N}} \, $
Inverse Discrete Fourier Transform $ \,x [n] = (1/N) \sum_{k=0}^{N-1} X[k] e^{j 2\pi\frac{kn}{N}} \, $
Discrete Fourier Transform Pairs (info)
$ x[n] \ $ $ \longrightarrow $ $ X[k] \ $
$ \ \delta[n] $ $ \ 1 $
$ \ 1 $ $ \ N\delta[k] $
$ \ e^{j2\pi k_0 n} $ $ \ N\delta[((k - k_0))_N] $
$ \ \cos(\frac{2\pi}{N}k_0n) $ $ \ \frac{N}{2}(\delta[((k - k_0))_N] + \delta[((k+k_0))_N]) $
Discrete Fourier Transform Properties
$ x[n] \ $ $ \longrightarrow $ $ X[k] \ $
Linearity $ ax[n]+by[n] \ $ $ aX[k]+bY[k] \ $
Circular Shift $ x[((n-m))_N] \ $ $ X[k]e^{(-j\frac{2 \pi}{N}km)} \ $
Duality $ X[n] \ $ $ NX[((-k))_N] \ $
Multiplication $ x[n]y[n] \ $ $ \frac{1}{N} X[k]\circledast Y[k], \ \circledast \text{ denotes the circular convolution} $
Convolution $ x(t) \circledast y(t) \ $ $ X[k]Y[k] \ $
time reversal $ \ x(-t) $ $ \ X(-f) $
Other Discrete Fourier Transform Properties
Parseval's Theorem $ \sum_{n=0}^{N-1}|x[n]|^2 = \frac{1}{N} \sum_{k=0}^{N-1}|X[k]|^2 $

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