(14 intermediate revisions by 4 users not shown)
Line 1: Line 1:
−
<div style="font-family: Verdana, sans-serif; font-size: 14px; text-align: justify; width: 80%; margin: auto; border: 1px solid #aaa; padding: 1em; text-align:right;">
+
<center><font size= 4>
−
{|
+
'''[[Collective_Table_of_Formulas|Collective Table of Formulas]]'''
−
|-
+
</font size>
−
|'''If you enjoy using this [[Collective_Table_of_Formulas|collective table of formulas]], please consider  [https://donate.purdue.edu/DesignateGift.aspx?allocation=017637&appealCode=11213&amount=25&allocationDescription=RheaProjectMimiBoutin donating to Project Rhea] or [[Donations | becoming a sponsor]].'''
+
−
| [[Image:DonateNow.png]]
+
−
|-
+
−
|}
+
−
</div>
+
  
 +
Discrete Fourier transforms (DFT) Pairs and Properties
  
−
= Discrete Fourier Transform =
+
click [[Collective_Table_of_Formulas|here]] for [[Collective_Table_of_Formulas|more formulas]]
−
Please help building this page!
+
−
*Let's try to follow the same table syntax as for [[CT_Fourier_Transform_(frequency_in_hertz)|this table]]
+
−
*You can copy and paste the formulas from these pages:
+
−
**[[Student_summary_Discrete_Fourier_transform_ECE438F09]]
+
−
**[[Discrete_Time_Fourier_Transform_Properties_(DTFT)_-_Mohammed_Almathami]]
+
  
 +
</center>
 +
 +
----
 
{|
 
{|
 
|-
 
|-
−
! 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)]]
+
! 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)]]
 
|-
 
|-
−
! style="background: none repeat scroll 0% 0% rgb(238, 238, 238);" colspan="2" | Definition CT Fourier Transform and its Inverse
+
! style="background: none repeat scroll 0% 0% rgb(238, 238, 238);" colspan="2" | Definition Discrete Fourier Transform and its Inverse
 
|-
 
|-
−
| align="right" style="padding-right: 1em;" |  [[Discrete Fourier Transform|Discrete Fourier Transform]]  
+
| Let x[n] be a periodic DT signal, with period N.
 +
|-
 +
| align="right" style="padding-right: 1em;" |  N-point [[Discrete Fourier Transform|Discrete Fourier Transform]]  
 
| <math>X [k] = \sum_{n=0}^{N-1} x[n]e^{-j 2\pi \frac{k n}{N}} \, </math>
 
| <math>X [k] = \sum_{n=0}^{N-1} x[n]e^{-j 2\pi \frac{k n}{N}} \, </math>
 
|-
 
|-
−
| align="right" style="padding-right: 1em;" | Inverse Discrete Fourier Transform  
+
| 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>
 
|}
 
|}
Line 34: Line 30:
 
|-
 
|-
 
| align="right" style="padding-right: 1em;" |  
 
| align="right" style="padding-right: 1em;" |  
−
| <math> x[n] \  </math>
+
| <math> x[n] \  \text{ (period } N) </math>
−
| <math>\longrightarrow</math>
+
| <math>\longrightarrow </math>
−
| <math> X[k] \  </math>
+
| <math> X_N[k] \  \  (N \text{ point DFT)}</math>
 
|-
 
|-
−
| align="right" style="padding-right: 1em;" | name
+
| align="right" style="padding-right: 1em;" |
−
| <math>type signal here\ </math>  
+
| <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>  
 
|  
 
|  
−
| <math> type transform here \! \ </math>
+
| <math>\ 1 \text{ (period } N) </math>
 
|-
 
|-
−
| align="right" style="padding-right: 1em;" | name
+
| align="right" style="padding-right: 1em;" |  
−
| <math>type signal here \ </math>  
+
| <math>\ 1 \text{ (period } N) </math>
 +
|
 +
| <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>
 +
|-
 +
| align="right" style="padding-right: 1em;" |
 +
| <math>\ e^{j2\pi k_0 n} </math>
 +
|
 +
| <math>\ N\delta[((k - k_0))_N] </math>
 +
|-
 +
| align="right" style="padding-right: 1em;" |
 +
| <math>\ \cos(\frac{2\pi}{N}k_0n) </math>  
 
|  
 
|  
−
| <math>type transform here</math>
+
| <math>\ \frac{N}{2}(\delta[((k - k_0))_N] + \delta[((k+k_0))_N]) </math>
 
|}
 
|}
  
Line 83: Line 89:
 
| <math> X[k]Y[k] \ </math>
 
| <math> X[k]Y[k] \ </math>
 
|-
 
|-
−
| align="right" style="padding-right: 1em;" | time reversal
+
| align="right" style="padding-right: 1em;" |
−
| <math>\ x(-t) </math>  
+
| <math>\ x^*[n] </math>  
 
|  
 
|  
−
| <math>\ X(-f)</math>
+
| <math>\ X^*[((-k))_N] </math>
 +
|-
 +
| align="right" style="padding-right: 1em;" |
 +
| <math>\ x^*[((-n))_N] </math>
 +
|
 +
| <math>\ X^*[k] </math>
 +
|-
 +
| align="right" style="padding-right: 1em;" |
 +
| <math>\ \Re\{x[n]\} </math>
 +
|
 +
| <math>\ X_{ep}[k] = \frac{1}{2}\{X[((k))_N] + X^*[((-k))_N]\} </math>
 +
|-
 +
| align="right" style="padding-right: 1em;" |
 +
| <math>\ j\Im\{x[n]\} </math>
 +
|
 +
| <math>\ X_{op}[k] = \frac{1}{2}\{X[((k))_N] - X^*[((-k))_N]\} </math>
 +
|-
 +
| align="right" style="padding-right: 1em;" |
 +
| <math>\ x_{ep}[n] = \frac{1}{2}\{x[n] + x^*[((-n))_N]\} </math>
 +
|
 +
| <math>\ \Re\{X[k]\} </math>
 +
|-
 +
| align="right" style="padding-right: 1em;" |
 +
| <math>\ x_{op}[n] = \frac{1}{2}\{x[n] - x^*[((-n))_N]\} </math>
 +
|
 +
| <math>\ j\Im\{X[k]\} </math>
 
|}
 
|}
  
Line 97: Line 128:
 
|}
 
|}
 
----
 
----
 +
[[ECE438|Go to Relevant Course Page: ECE 438]]
 +
 +
[[ECE538|Go to Relevant Course Page: ECE 538]]
 +
 
[[Collective_Table_of_Formulas|Back to Collective Table]]
 
[[Collective_Table_of_Formulas|Back to Collective Table]]
  
 
[[Category:Formulas]]
 
[[Category:Formulas]]
 +
[[Category:discrete Fourier transform]]
 +
[[Category:Fourier transform]]
 +
[[Category:ECE438]]

Latest revision as of 14:28, 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

Retrieved from "https://www.projectrhea.org/rhea/index.php?title=Discrete_Fourier_Transform_table&oldid=55666"
Shortcuts
Help Main Wiki Page Random Page Special Pages Log in
Search

Alumni Liaison

Correspondence Chess Grandmaster and Purdue Alumni

Prof. Dan Fleetwood