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| <math> NX[((-k))_N] \  </math>
 
| <math> NX[((-k))_N] \  </math>
 
|-
 
|-
| align="right" style="padding-right: 1em;" | multiplication property
+
| align="right" style="padding-right: 1em;" | Multiplication
| <math>x[n]y[n] \ </math>  
+
| <math> x[n]y[n] \ </math>  
 
|  
 
|  
| <math> write DFT here</math>
+
| <math> \frac{1}{N} X[k]\circledast Y[k], \  \circledast \text{ denotes the circular convolution} </math>
 
|-
 
|-
| align="right" style="padding-right: 1em;" | convolution property
+
| align="right" style="padding-right: 1em;" | Convolution
| <math>x(t)*y(t) \!</math>  
+
| <math>x(t) \circledast y(t) \ </math>  
 
|  
 
|  
| <math> X(f)Y(f) \!</math>
+
| <math> X[k]Y[k] \ </math>
 
|-
 
|-
 
| align="right" style="padding-right: 1em;" | time reversal  
 
| align="right" style="padding-right: 1em;" | time reversal  
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! style="background: none repeat scroll 0% 0% rgb(238, 238, 238);" colspan="2" | Other Discrete Fourier Transform Properties
 
! style="background: none repeat scroll 0% 0% rgb(238, 238, 238);" colspan="2" | Other Discrete Fourier Transform Properties
 
|-
 
|-
| align="right" style="padding-right: 1em;" | property
+
| align="right" style="padding-right: 1em;" | Parseval's Theorem
| <math>type math here</math>
+
| <math> \sum_{n=0}^{N-1}|x[n]|^2  = \frac{1}{N} \sum_{k=0}^{N-1}|X[k]|^2 </math>
 
|}
 
|}
 
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Revision as of 12:33, 25 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] \ $
name $ type signal here\ $ $ type transform here \! \ $
name $ type signal here \ $ $ type transform here $
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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Alumni Liaison

Ph.D. on Applied Mathematics in Aug 2007. Involved on applications of image super-resolution to electron microscopy

Francisco Blanco-Silva