Revision as of 06:38, 27 October 2009

Discrete-time Fourier Transform Pairs and Properties

Please feel free to add onto this table!

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}} \$

@@ Line 9: / Line 9: @@
 |-
 | align="right" style="padding-right: 1em;" | Inverse DT Fourier Transform || <math>\,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\,</math>
+|}
+{|
 |-
-! colspan="2" style="background: #eee;" | DT Fourier Transform Pairs
+! colspan="4" style="background: #eee;" | DT Fourier Transform Pairs
+|-
+| align="right" style="padding-right: 1em;" |   || <math>x[n]</math> || <math>\longrightarrow</math>|| <math> \mathcal{X}(\omega) </math>
 |-
-| align="right" style="padding-right: 1em;" | DTFT of a complex exponential || <math>e^{jw_0n} \longrightarrow 2\pi\sum_{l=-\infty}^{+\infty}\delta(w-w_0-2\pi l) \ </math>
+| align="right" style="padding-right: 1em;" | DTFT of a complex exponential || <math>e^{jw_0n}</math> || || <math>\pi\sum_{l=-\infty}^{+\infty}\delta(w-w_0-2\pi l) \ </math>
 |-
 |-
-| align="right" style="padding-right: 1em;" | [[:DT Fourier an_ECE301Fall2008mboutin]] || {{:DT Fourier an_ECE301Fall2008mboutin}}
+| align="right" style="padding-right: 1em;" |  || <math>a^{n} u[n],  |a|<1 \ </math> || ||<math>\frac{1}{1-ae^{-j\omega}} \ </math>
 |-
 |}
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