Revision as of 07:50, 28 October 2009

Discrete-time Fourier Transform Pairs and Properties

Please feel free to add onto this table! And if you see a mistake, please correct it.

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) \$
	$a^{n} u[n], \|a\|<1 \$		$\frac{1}{1-ae^{-j\omega}} \$
	$\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	$\ x[-n]$		$\ X(-\omega)$

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 =Discrete-time Fourier Transform Pairs and Properties=
-Please feel free to add onto this table!
+Please feel free to add onto this table! And if you see a mistake, please correct it.
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 |-
 | 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>
+|-
+|}
+{|
+|-
+! colspan="4" style="background: #eee;" | DT Fourier Transform Properties
+|-
+| align="right" style="padding-right: 1em;" |   || <math>x[n]</math> || <math>\longrightarrow</math>|| <math> \mathcal{X}(\omega) </math>
+|-
+| align="right" style="padding-right: 1em;" | multiplication property|| <math>x[n]y[n] \ </math> || || <math>\frac{1}{2\pi} \int_{2\pi} X(\theta)Y(\omega-\theta)d\theta</math>
+|-
+| align="right" style="padding-right: 1em;" |  convolution property || <math>x[n]*y[n] \!</math> || ||<math> X(\omega)Y(\omega) \!</math>
+|-
+| align="right" style="padding-right: 1em;" | time reversal ||<math>\ x[-n] </math> || ||<math>\ X(-\omega)</math>
 |-
 |}
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