Line 57: Line 57:
 
| <math>\longrightarrow</math>
 
| <math>\longrightarrow</math>
 
| <math> X[k] \  </math>
 
| <math> X[k] \  </math>
 +
|-
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| align="right" style="padding-right: 1em;" | Linearity
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| <math> ax[n]+by[n] \  </math>
 +
|
 +
| <math> aX[k]+bY[k] \  </math>
 +
|-
 +
| align="right" style="padding-right: 1em;" | Circular Shift
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| <math> x[((n-m))_N] \  </math>
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|
 +
| <math> X[k]e^{(-j\frac{2 \pi}{N}km)} \  </math>
 +
|-
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| align="right" style="padding-right: 1em;" | Duality
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| <math> X[n] \  </math>
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|
 +
| <math> NX[((-k))_N] \  </math>
 
|-
 
|-
 
| align="right" style="padding-right: 1em;" | multiplication property
 
| align="right" style="padding-right: 1em;" | multiplication property

Revision as of 11:37, 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 property $ x[n]y[n] \ $ $ write DFT here $
convolution property $ x(t)*y(t) \! $ $ X(f)Y(f) \! $
time reversal $ \ x(-t) $ $ \ X(-f) $
Other Discrete Fourier Transform Properties
property $ type math here $

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Ph.D. 2007, working on developing cool imaging technologies for digital cameras, camera phones, and video surveillance cameras.

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