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| 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;" | DTFT of a complex exponential || <math>e^{jw_0n} \ </math> || || <math>\pi\sum_{l=-\infty}^{+\infty}\delta(w-w_0-2\pi l) \ </math>
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| align="right" style="padding-right: 1em;" | ([[DTFT_rectangular_window|info]]) DTFT of a rectangular window || <math>w[n]= \ </math> || || add formula here
 
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| 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>a^{n} u[n],  |a|<1 \ </math> || ||<math>\frac{1}{1-ae^{-j\omega}} \ </math>

Latest revision as of 12:52, 22 October 2010

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}} \ $
$ \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 = $

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Alumni Liaison

Correspondence Chess Grandmaster and Purdue Alumni

Prof. Dan Fleetwood