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= Discrete Fourier Transform =
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'''[[Collective_Table_of_Formulas|Collective Table of Formulas]]'''
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Definition: let x[n] be a DT signal with Period N.
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Discrete Fourier transforms (DFT) Pairs and Properties
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<math> X [k] = \sum_{k=0}^{N-1} x[n].e^{-J.2pi.kn/N}</math>
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click [[Collective_Table_of_Formulas|here]] for [[Collective_Table_of_Formulas|more formulas]]
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</center>
  
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<math> x [n] = (1/N) \sum_{k=0}^{N-1} X[k].e^{J.2pi.kn/N}</math>
 
 
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{|
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|-
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! style="background: none repeat scroll 0% 0% rgb(228, 188, 126); font-size: 110%;" colspan="2" | Discrete Fourier Transform Pairs and Properties  [[Discrete Fourier Transform|(info)]]
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|-
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! style="background: none repeat scroll 0% 0% rgb(238, 238, 238);" colspan="2" | Definition Discrete Fourier Transform and its Inverse
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|-
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| Let x[n] be a periodic DT signal, with period N.
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|-
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| align="right" style="padding-right: 1em;" |  N-point [[Discrete Fourier Transform|Discrete Fourier Transform]]
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| <math>X [k] = \sum_{n=0}^{N-1} x[n]e^{-j 2\pi \frac{k n}{N}} \, </math>
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|-
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| align="right" style="padding-right: 1em;" | Inverse Discrete Fourier Transform
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| <math>\,x [n] = (1/N) \sum_{k=0}^{N-1} X[k] e^{j 2\pi\frac{kn}{N}} \,</math>
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|}
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{|
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|-
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! style="background: none repeat scroll 0% 0% rgb(238, 238, 238);" colspan="4" | Discrete Fourier Transform Pairs [[Discrete Fourier Transform| (info)]]
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|-
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| align="right" style="padding-right: 1em;" |
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| <math> x[n] \  \text{ (period } N) </math>
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| <math>\longrightarrow </math>
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| <math> X_N[k] \  \  (N \text{ point DFT)}</math>
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|-
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| align="right" style="padding-right: 1em;" | 
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| <math>\ \sum_{k=-\infty}^\infty \delta[n+Nk] = \left\{ \begin{array}{ll} 1, & \text{ if } n=0, \pm N, \pm 2N , \ldots\\ 0, & \text{ else.} \end{array}\right.</math>
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|
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| <math>\ 1 \text{ (period } N) </math>
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|-
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| align="right" style="padding-right: 1em;" |
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| <math>\ 1 \text{ (period } N) </math>
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|
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| <math>\ N\sum_{m=-\infty}^\infty \delta[k+Nm] = \left\{ \begin{array}{ll} N, & \text{ if } n=0, \pm N, \pm 2N , \ldots\\ 0, & \text{ else.} \end{array}\right.</math>
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|-
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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>
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|}
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{|
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|-
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! style="background: none repeat scroll 0% 0% rgb(238, 238, 238);" colspan="4" | Discrete Fourier Transform Properties
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|-
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| align="right" style="padding-right: 1em;" |
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| <math> x[n] \  </math>
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| <math>\longrightarrow</math>
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| <math> X[k] \  </math>
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|-
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| align="right" style="padding-right: 1em;" | Linearity
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| <math> ax[n]+by[n] \  </math>
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|
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| <math> aX[k]+bY[k] \  </math>
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|-
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| align="right" style="padding-right: 1em;" | Circular Shift
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| <math> x[((n-m))_N] \  </math>
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|
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| <math> X[k]e^{(-j\frac{2 \pi}{N}km)} \  </math>
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|-
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| align="right" style="padding-right: 1em;" | Duality
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| <math> X[n] \  </math>
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|
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| <math> NX[((-k))_N] \  </math>
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|-
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| align="right" style="padding-right: 1em;" | Multiplication
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| <math> x[n]y[n] \ </math>
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|
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| <math> \frac{1}{N} X[k]\circledast Y[k], \  \circledast \text{ denotes the circular convolution} </math>
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|-
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| align="right" style="padding-right: 1em;" | Convolution
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| <math>x(t) \circledast y(t) \ </math>
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|
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| <math> X[k]Y[k] \ </math>
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|-
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| align="right" style="padding-right: 1em;" |
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| <math>\ x^*[n] </math>
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|
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| <math>\ X^*[((-k))_N] </math>
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|-
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| align="right" style="padding-right: 1em;" |
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| <math>\ x^*[((-n))_N] </math>
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|
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| <math>\ X^*[k] </math>
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|-
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| align="right" style="padding-right: 1em;" |
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| <math>\ \Re\{x[n]\} </math>
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|
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| <math>\ X_{ep}[k] = \frac{1}{2}\{X[((k))_N] + X^*[((-k))_N]\} </math>
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|-
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| align="right" style="padding-right: 1em;" |
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| <math>\ j\Im\{x[n]\} </math>
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|
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| <math>\ X_{op}[k] = \frac{1}{2}\{X[((k))_N] - X^*[((-k))_N]\} </math>
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|-
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| align="right" style="padding-right: 1em;" |
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| <math>\ x_{ep}[n] = \frac{1}{2}\{x[n] + x^*[((-n))_N]\} </math>
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|
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| <math>\ \Re\{X[k]\} </math>
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|-
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| align="right" style="padding-right: 1em;" |
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| <math>\ x_{op}[n] = \frac{1}{2}\{x[n] - x^*[((-n))_N]\} </math>
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|
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| <math>\ j\Im\{X[k]\} </math>
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|}
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{|
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|-
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! style="background: none repeat scroll 0% 0% rgb(238, 238, 238);" colspan="2" | Other Discrete Fourier Transform Properties
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|-
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| align="right" style="padding-right: 1em;" | Parseval's Theorem
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| <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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|}
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----
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[[ECE438|Go to Relevant Course Page: ECE 438]]
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[[ECE538|Go to Relevant Course Page: ECE 538]]
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[[Collective_Table_of_Formulas|Back to Collective Table]]
 
[[Collective_Table_of_Formulas|Back to Collective Table]]
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[[Category:Formulas]]
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[[Category:discrete Fourier transform]]
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[[Category:Fourier transform]]
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[[Category:ECE438]]

Latest revision as of 15:28, 23 April 2013

Collective Table of Formulas

Discrete Fourier transforms (DFT) Pairs and Properties

click here for more formulas

Discrete Fourier Transform Pairs and Properties (info)
Definition Discrete Fourier Transform and its Inverse
Let x[n] be a periodic DT signal, with period N.
N-point 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] \ \text{ (period } N) $ $ \longrightarrow $ $ X_N[k] \ \ (N \text{ point DFT)} $
$ \ \sum_{k=-\infty}^\infty \delta[n+Nk] = \left\{ \begin{array}{ll} 1, & \text{ if } n=0, \pm N, \pm 2N , \ldots\\ 0, & \text{ else.} \end{array}\right. $ $ \ 1 \text{ (period } N) $
$ \ 1 \text{ (period } N) $ $ \ N\sum_{m=-\infty}^\infty \delta[k+Nm] = \left\{ \begin{array}{ll} N, & \text{ if } n=0, \pm N, \pm 2N , \ldots\\ 0, & \text{ else.} \end{array}\right. $
$ \ 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] \ $
$ \ x^*[n] $ $ \ X^*[((-k))_N] $
$ \ x^*[((-n))_N] $ $ \ X^*[k] $
$ \ \Re\{x[n]\} $ $ \ X_{ep}[k] = \frac{1}{2}\{X[((k))_N] + X^*[((-k))_N]\} $
$ \ j\Im\{x[n]\} $ $ \ X_{op}[k] = \frac{1}{2}\{X[((k))_N] - X^*[((-k))_N]\} $
$ \ x_{ep}[n] = \frac{1}{2}\{x[n] + x^*[((-n))_N]\} $ $ \ \Re\{X[k]\} $
$ \ x_{op}[n] = \frac{1}{2}\{x[n] - x^*[((-n))_N]\} $ $ \ j\Im\{X[k]\} $
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 $

Go to Relevant Course Page: ECE 438

Go to Relevant Course Page: ECE 538

Back to Collective Table

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

Francisco Blanco-Silva