(6 intermediate revisions by 2 users not shown)
Line 1: Line 1:
<div style="font-family: Verdana, sans-serif; font-size: 14px; text-align: center; width: 70%; margin: auto; border: 1px solid #aaa; padding: 1em; text-align:center;">
+
<center><font size= 4>
{|
+
'''[[Collective_Table_of_Formulas|Collective Table of Formulas]]'''
|-
+
</font size>
|
+
'''This [[Collective Table of Formulas|Collective table of formulas]] is proudly sponsored'''<br> '''by the [http://www.facebook.com/hkn.beta Nice Guys of Eta Kappa Nu].''' <br><br> Visit us at the HKN Lounge in EE24 for hot coffee and fresh bagels only $1 each!
+
  
| &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;[[Image:HKNlogo.jpg]]
+
Discrete Fourier transforms (DFT) Pairs and Properties
|}
+
</div>
+
  
 +
click [[Collective_Table_of_Formulas|here]] for [[Collective_Table_of_Formulas|more formulas]]
  
= Discrete Fourier Transform =
+
</center>
Please help building this page!
+
*You can copy and paste the formulas from these pages:
+
**[[Student_summary_Discrete_Fourier_transform_ECE438F09]]
+
**[[Discrete_Time_Fourier_Transform_Properties_(DTFT)_-_Mohammed_Almathami]]
+
  
 +
----
 
{|
 
{|
 
|-
 
|-
! 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 Discrete 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
Line 142: Line 136:
 
[[Category:Formulas]]
 
[[Category:Formulas]]
 
[[Category:discrete Fourier transform]]
 
[[Category:discrete Fourier transform]]
 +
[[Category:Fourier transform]]
 +
[[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

Alumni Liaison

Have a piece of advice for Purdue students? Share it through Rhea!

Alumni Liaison