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= Discrete Fourier Transform =
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Please help building this page!
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'''[[Collective_Table_of_Formulas|Collective Table of Formulas]]'''
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*Let's try to follow the same table syntax as for [[CT_Fourier_Transform_(frequency_in_hertz)|this table]]
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</font size>
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*You can copy and paste the formulas from these pages:
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**[[Student_summary_Discrete_Fourier_transform_ECE438F09]]
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**[[Discrete_Time_Fourier_Transform_Properties_(DTFT)_-_Mohammed_Almathami]]
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Discrete Fourier transforms (DFT) Pairs and Properties
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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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----
 
{|
 
{|
 
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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  [[More on CT Fourier transform|(info)]]
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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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! style="background: none repeat scroll 0% 0% rgb(238, 238, 238);" colspan="2" | Definition CT Fourier Transform and its Inverse
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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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| align="right" style="padding-right: 1em;" |  [[Discrete Fourier Transform|Discrete Fourier Transform]]
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| Let x[n] be a periodic DT signal, with period N.
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| <math>X [k] = \sum_{k=0}^{N-1} x[n]e^{-j 2\pi \frac{k n}{N}} \, </math>
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| align="right" style="padding-right: 1em;" | Inverse Discrete Fourier Transform  
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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
 
| <math>\,x [n] = (1/N) \sum_{k=0}^{N-1} X[k] e^{j 2\pi\frac{kn}{N}} \,</math>
 
| <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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| align="right" style="padding-right: 1em;" |  
 
| align="right" style="padding-right: 1em;" |  
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| <span class="texhtml">''x''[''n'']</span>  
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| <math> x[n] \  \text{ (period } N) </math>
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| <math>\longrightarrow</math>
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| <math>\longrightarrow </math>
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| <math> X[k] </math>
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| <math> X_N[k] \  \  (N \text{ point DFT)}</math>
 
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| align="right" style="padding-right: 1em;" | name
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| align="right" style="padding-right: 1em;" |
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| <math>type signal here\ </math>  
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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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| <math> type transform here \! \ </math>
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| <math>\ 1 \text{ (period } N) </math>
 
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|-
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| align="right" style="padding-right: 1em;" | name
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| align="right" style="padding-right: 1em;" |  
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| <math>type signal here \ </math>  
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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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| align="right" style="padding-right: 1em;" |
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| <math>\ e^{j2\pi k_0 n} </math>
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| <math>\ N\delta[((k - k_0))_N] </math>
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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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| <math>type transform here</math>
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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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| align="right" style="padding-right: 1em;" |  
 
| align="right" style="padding-right: 1em;" |  
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| <span class="texhtml">''x''[''n'']</span>  
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| <math> x[n] \  </math>
 
| <math>\longrightarrow</math>
 
| <math>\longrightarrow</math>
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| <math> X[k] </math>
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| <math> X[k] \  </math>
 
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| align="right" style="padding-right: 1em;" | multiplication property
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| align="right" style="padding-right: 1em;" | Linearity
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| <math>x[n]y[n] \ </math>  
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| <math> ax[n]+by[n] \  </math>
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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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| <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>
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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> write DFT here</math>
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| <math> \frac{1}{N} X[k]\circledast Y[k], \  \circledast \text{ denotes the circular convolution} </math>
 
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| align="right" style="padding-right: 1em;" | convolution property
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| align="right" style="padding-right: 1em;" | Convolution
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| <math>x(t)*y(t) \!</math>  
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| <math>x(t) \circledast y(t) \ </math>  
 
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|  
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| <math> X(f)Y(f) \!</math>
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| <math> X[k]Y[k] \ </math>
 
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|-
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| align="right" style="padding-right: 1em;" | time reversal
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| align="right" style="padding-right: 1em;" |
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| <math>\ x(-t) </math>  
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| <math>\ x^*[n] </math>  
 
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|  
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| <math>\ X(-f)</math>
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| <math>\ X^*[((-k))_N] </math>
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| align="right" style="padding-right: 1em;" |
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| <math>\ x^*[((-n))_N] </math>
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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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| <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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| <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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| <math>\ j\Im\{X[k]\} </math>
 
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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
 
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| align="right" style="padding-right: 1em;" | property
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| align="right" style="padding-right: 1em;" | Parseval's Theorem
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| <math>type math here</math>
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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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[[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]]
  
 
[[Category:Formulas]]
 
[[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

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

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