Difference between revisions of "Table DT Fourier Transforms" - Rhea

Revision as of 06:51, 8 March 2011

Discrete-time Fourier Transform Pairs and Properties
DT Fourier transform and its Inverse
DT Fourier Transform	$\,\mathcal{X}(\omega)=\mathcal{F}(x[n])=\sum_{n=-\infty}^{\infty}x[n]e^{-j\omega n}\,$
Inverse DT Fourier Transform	$\,x[n]=\mathcal{F}^{-1}(\mathcal{X}(\omega))=\frac{1}{2\pi} \int_{0}^{2\pi}\mathcal{X}(\omega)e^{j\omega n} d \omega\,$

DT Fourier Transform Pairs
	$x[n] \$	$\longrightarrow$	$\mathcal{X}(\omega) \$
DTFT of a complex exponential	$e^{jw_0n} \$		$\pi\sum_{l=-\infty}^{+\infty}\delta(w-w_0-2\pi l) \$
(info) DTFT of a rectangular window	$w[n]= \$		add formula here
	$a^{n} u[n], \|a\|<1 \$		$\frac{1}{1-ae^{-j\omega}} \$
	$(n-1)a^{n} u[n], \|a\|<1 \$		$\frac{1}{(1-ae^{-j\omega})^2} \$
	$\sin\left(\omega _0 n\right) u[n] \$		$\frac{1}{2j}\left( \frac{1}{1-e^{-j(\omega -\omega _0)}}-\frac{1}{1-e^{-j(\omega +\omega _0)}}\right)$

DT Fourier Transform Properties
	$x[n] \$	$\longrightarrow$	$\mathcal{X}(\omega) \$
multiplication property	$x[n]y[n] \$		$\frac{1}{2\pi} \int_{2\pi} X(\theta)Y(\omega-\theta)d\theta$
convolution property	$x[n]*y[n] \!$		$X(\omega)Y(\omega) \!$
time reversal	$\ x[-n]$		$\ X(-\omega)$

Other DT Fourier Transform Properties
Parseval's relation	$\frac {1}{N} \sum_{n=-\infty}^{\infty}\left\| x[n] \right\|^2 =$

@@ Line 19: / Line 19: @@
 |-
 | align="right" style="padding-right: 1em;" |  || <math>a^{n} u[n],  |a|<1 \ </math> || ||<math>\frac{1}{1-ae^{-j\omega}} \ </math>
+|-
+| align="right" style="padding-right: 1em;" |  || <math>(n-1)a^{n} u[n],  |a|<1 \ </math> || ||<math>\frac{1}{(1-ae^{-j\omega})^2} \ </math>
 |-
 | align="right" style="padding-right: 1em;" |  || <math>\sin\left(\omega _0 n\right) u[n] \ </math>  || ||<math>\frac{1}{2j}\left( \frac{1}{1-e^{-j(\omega -\omega _0)}}-\frac{1}{1-e^{-j(\omega +\omega _0)}}\right)</math>