Revision as of 11:37, 25 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	$\,x [n] = (1/N) \sum_{k=0}^{N-1} X[k] e^{j 2\pi\frac{kn}{N}} \,$

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] \$
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 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	$$ type math here $$

@@ Line 57: / Line 57: @@
 | <math>\longrightarrow</math>
 | <math> X[k] \  </math>
+|-
+| align="right" style="padding-right: 1em;" | Linearity
+| <math> ax[n]+by[n] \  </math>
+|
+| <math> aX[k]+bY[k] \  </math>
+|-
+| align="right" style="padding-right: 1em;" | Circular Shift
+| <math> x[((n-m))_N] \  </math>
+|
+| <math> X[k]e^{(-j\frac{2 \pi}{N}km)} \  </math>
+|-
+| align="right" style="padding-right: 1em;" | Duality
+| <math> X[n] \  </math>
+|
+| <math> NX[((-k))_N] \  </math>
 |-
 | align="right" style="padding-right: 1em;" | multiplication property