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[[Category: Defintions]]
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[[Category: Fourier]]
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[[Category: Fourier Transform]]
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{| class="wikitable" style="margin: 1em auto 1em auto" border="1" cellpadding="5" cellspacing="0"
 
{| class="wikitable" style="margin: 1em auto 1em auto" border="1" cellpadding="5" cellspacing="0"
 
! Time Domain !! Fourier Domain
 
! Time Domain !! Fourier Domain

Latest revision as of 11:28, 10 December 2008


Time Domain Fourier Domain
$ x(t)=\frac{1}{2 \pi} \int_{-\infty}^{\infty} X(j \omega)e^{j \omega t}d \omega $ $ X(j \omega)=\int_{-\infty}^\infty x(t) e^{-j \omega t}d t $
$ 1\ $ $ 2 \pi \delta (\omega)\ $
$ -0.5+u(t)\ $ $ \frac{1}{j \omega}\ $
$ \delta (t)\ $ $ 1\ $
$ \delta (t-c)\ $ $ e^{-j \omega c}\ $
$ u(t)\ $ $ \pi \delta(\omega)+\frac{1}{j \omega} $
$ e^{-bt}u(t)\ $ $ \frac{1}{j \omega + b} $
$ cos \omega_0 t\ $ $ \pi [\delta ( \omega + \omega_0 ) + \delta ( \omega - \omega_0 ) ]\ $
$ cos ( \omega_0 t + \theta )\ $ $ \pi [ e^{-j \theta} \delta ( \omega + \omega_0 ) + e^{j \theta} \delta ( \omega - \omega_0 )]\ $
$ sin \omega_0 t\ $ $ j \pi [ \delta ( \omega + \omega_0 ) - \delta ( \omega - \omega_0 ) ]\ $
$ sin ( \omega_0 t + \theta )\ $ $ j \pi [ e^{-j \theta} \delta ( \omega + \omega_0 ) - e^{j \theta} \delta ( \omega - \omega_0 ) ]\ $
$ rect \left ( \frac{t}{\tau} \right ) $ $ \tau sinc \frac{\tau \omega}{2 \pi} $
$ \tau sinc \frac{\tau t}{2 \pi} $ $ 2 \pi p_\tau\ ( \omega ) $
$ \left ( 1-\frac{2 |t|}{\tau} \right ) p_\tau (t) $ $ \frac{\tau}{2} sinc^2 \frac{\tau \omega}{4 \pi} $
$ \frac{\tau}{2} sinc^2 \left ( \frac{\tau t}{4 \pi} \right ) $ $ 2 \pi \left ( 1-\frac{2|\omega|}{\tau} \right ) p_\tau (\omega) $

Notes:

$ sinc(x) = \frac {sin(x)}{x} $
$ p_\tau (t)\ $ is the rectangular pulse function of width $ \tau\ $


Source courtesy Wikibooks.org

Alumni Liaison

Recent Math PhD now doing a post-doctorate at UC Riverside.

Kuei-Nuan Lin