Revision as of 11:11, 27 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	$\,x [n] = (1/N) \sum_{k=0}^{N-1} X[k] e^{j 2\pi\frac{kn}{N}} \,$


$x[n] \$	$\longrightarrow$	$X[k] \$
$\ \delta[n]$		$\ 1$
$\ 1$		$\ N\delta[k]$
$\ e^{j2\pi k_0 n}$		$\ N\delta[((k - k_0))_N]$
$\ \cos(\frac{2\pi}{N}k_0n)$		$\ \frac{N}{2}(\delta[((k - k_0))_N] + \delta[((k+k_0))_N])$

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] \$
time reversal	$\ x(-t)$		$\ X(-f)$

Other Discrete Fourier Transform Properties
Parseval's Theorem	$\sum_{n=0}^{N-1}\|x[n]\|^2 = \frac{1}{N} \sum_{k=0}^{N-1}\|X[k]\|^2$

@@ Line 38: / Line 38: @@
 | <math> X[k] \  </math>
 |-
-| align="right" style="padding-right: 1em;" | name
+| align="right" style="padding-right: 1em;" |
-| <math>type signal here\ </math>
+| <math>\ \delta[n] </math>
 |
-| <math> type transform here \! \ </math>
+| <math>\ 1 </math>
 |-
-| align="right" style="padding-right: 1em;" | name
+| align="right" style="padding-right: 1em;" |
-| <math>type signal here \ </math>
+| <math>\ 1 </math>
 |
-| <math>type transform here</math>
+| <math>\ N\delta[k] </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>\ \frac{N}{2}(\delta[((k - k_0))_N] + \delta[((k+k_0))_N]) </math>
 |}