Revision as of 13:48, 27 November 2011

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Discrete Fourier Transform

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


$x[n] \ \text{ (period } N)$	$\longrightarrow$	$X_N[k] \ \ (N \text{ point DFT)$
$\ \sum_{k=-\infty}^\infty \delta[n+Nk]$		$\ 1 \text{ (period } N)$
$\ 1 \text{ (period } N)$		$\ N\delta[k]$
$\ e^{j2\pi k_0 n}$		$\ N\delta[((k - k_0))_N]$
$\ \cos(\frac{2\pi}{N}k_0n)$		$\ \frac{N}{2}(\delta[((k - k_0))_N] + \delta[((k+k_0))_N])$

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 19: / Line 19: @@
 ! 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(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>
 |-
 | 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>
 |}
@@ Line 35: / Line 37: @@
 | <math> x[n] \  \text{ (period } N) </math>
 | <math>\longrightarrow</math>
-| <math> X[k] \  \  \text{ (period } N) </math>
+| <math> X_N[k] \  \  (N \text{ point DFT) </math>
 |-
 | align="right" style="padding-right: 1em;" |