Revision as of 05:25, 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
DT Fourier Transform Pair_ECE301Fall2008mboutin	$e^{jw_0n} \longrightarrow 2\pi\sum_{l=-\infty}^{+\infty}\delta(w-w_0-2\pi l) \$
DT Fourier an_ECE301Fall2008mboutin	$a^{n} u[n], \|a\|<1 \longrightarrow \frac{1}{1-ae^{-j\omega}} \$

@@ Line 8: / Line 8: @@
 | align="right" style="padding-right: 1em;" | DT Fourier Transform || <math>\,\mathcal{X}(\omega)=\mathcal{F}(x[n])=\sum_{n=-\infty}^{\infty}x[n]e^{-j\omega n}\,</math>
 |-
-| align="right" style="padding-right: 1em;" | [[DT Inverse Fourier Transform_ECE301Fall2008mboutin]] || {{:DT Inverse Fourier Transform_ECE301Fall2008mboutin}}
+| 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