Revision as of 17:32, 4 October 2011

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

Discrete Fourier Transform Pairs (info)
	$x [n]$	$\longrightarrow$	$$ X[k] $$
name	$type signal here\$		$type transform here \! \$
name	$type signal here \$		$$ type transform here $$

Discrete Fourier Transform Properties
	$x [n]$	$\longrightarrow$	$$ X[k] $$
multiplication property	$x[n]y[n] \$		$$ write DFT here $$
convolution property	$x(t)*y(t) \!$		$X(f)Y(f) \!$
time reversal	$\ x(-t)$		$\ X(-f)$

Other Discrete Fourier Transform Properties
property	$$ type math here $$

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 | align="right" style="padding-right: 1em;" |  [[Discrete Fourier Transform|Discrete Fourier Transform]]
-| <math>X [k] = \sum_{k=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>
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 | align="right" style="padding-right: 1em;" | Inverse Discrete Fourier Transform