(New page: {| ! Time Domain !! Fourier Domain |- | <math> x(t)=\frac{1}{2 \pi} \int_{-\infty}^{\infty} X(j \omega)e^{j \omega t}d \omega </math> | <math> X(j \omega)=\int_{-\infty}^\infty x(t) e^{-j...)
 
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{| class="wikitable" style="margin: 1em auto 1em auto" border="1" cellpadding="5" cellspacing="0"
{|
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! Time Domain !! Fourier Domain
 
! Time Domain !! Fourier Domain
 
|-
 
|-
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| <math> X(j \omega)=\int_{-\infty}^\infty x(t) e^{-j \omega t}d t </math>
 
| <math> X(j \omega)=\int_{-\infty}^\infty x(t) e^{-j \omega t}d t </math>
 
|-
 
|-
| <math> 1\ </math> || <math> 2 \pi \delta (\omega) </math>
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| <math> 1\ </math> || <math> 2 \pi \delta (\omega)\ </math>
 
|-
 
|-
| <math> 0.5 + u(t)\ </math> || <math> \frac{1}{j \omega}\ </math>
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| <math> -0.5+u(t)\ </math> || <math> \frac{1}{j \omega}\ </math>
 
|-
 
|-
| <math> \delta (t) \ </math> || <math> 1\ </math>
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| <math> \delta (t)\ </math> || <math> 1\ </math>
 
|-
 
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| <math> \delta (t-c)\ </math> || <math> e j \omega c </math>
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| <math> \delta (t-c)\ </math> || <math> e^{-j \omega c}\ </math>
 
|-
 
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| <math> u(t) </math> || <math> \pi \delta(\omega)+\frac{1}{j \omega} </math>
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| <math> u(t)\ </math> || <math> \pi \delta(\omega)+\frac{1}{j \omega} </math>
 
|-
 
|-
| <math> e ^{bt}u(t) </math> || <math> \frac{1}{j \omega + b} </math>
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| <math> e^{-bt}u(t)\ </math> || <math> \frac{1}{j \omega + b} </math>
 
|-
 
|-
| <math> cos \omega_0 t </math> || <math> \pi [\delta ( \omega + \omega_0 ) + \delta ( \omega - \omega_0 )] </math>
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| <math> cos \omega_0 t\ </math> || <math> \pi [\delta ( \omega + \omega_0 ) + \delta ( \omega - \omega_0 ) ]\ </math>
 
|-
 
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|cos(ω0t + θ) π[e − jθδ(ω + ω0) + ejθδ(ω − ω0)]?
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| <math> cos ( \omega_0 t + \theta )\ </math>
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| <math> \pi [ e^{-j \theta} \delta ( \omega + \omega_0 ) + e^{j \theta} \delta ( \omega - \omega_0 )]\ </math>
 
|-
 
|-
|sinω0t jπ[δ(ω + ω0) − δ(ω − ω0)]?
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| <math> sin \omega_0 t\ </math> || <math> j \pi [ \delta ( \omega + \omega_0 ) - \delta ( \omega - \omega_0 ) ]\ </math>
 
|-
 
|-
|sin(ω0t + θ) [e − jθδ(ω + ω0) − ejθδ(ω − ω0)]?
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| <math> sin ( \omega_0 t + \theta )\ </math>
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| <math> j \pi [ e^{-j \theta} \delta ( \omega + \omega_0 ) - e^{j \theta} \delta ( \omega - \omega_0 ) ]\ </math>
 
|-
 
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|rect(\frac{t}{\tau}) \tau sinc \frac{\tau \omega}{2 \pi}
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| <math> rect \left ( \frac{t}{\tau} \right ) </math> || <math> \tau sinc \frac{\tau \omega}{2 \pi} </math>
 
|-
 
|-
|\tau sinc \frac{\tau t}{2 \pi} 2πpτ(ω)
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| <math> \tau sinc \frac{\tau t}{2 \pi} </math> || <math> 2 \pi p_\tau\ ( \omega ) </math>
 
|-
 
|-
(1-\frac{2 |t|}{\tau})p_\tau (t) \frac{\tau}{2} sinc^2 \frac{\tau \omega}{4 \pi}
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| <math> \left ( 1-\frac{2 |t|}{\tau} \right ) p_\tau (t) </math> || <math> \frac{\tau}{2} sinc^2 \frac{\tau \omega}{4 \pi} </math>
 
|-
 
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\frac{\tau}{2} sinc^2 ( \frac{\tau t}{4 \pi} ) 2 \pi (1-\frac{2|\omega|}{\tau})p_\tau (\omega)
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| <math> \frac{\tau}{2} sinc^2 \left ( \frac{\tau t}{4 \pi} \right ) </math>
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| <math> 2 \pi \left ( 1-\frac{2|\omega|}{\tau} \right ) p_\tau (\omega) </math>
 
|}
 
|}
    Note: sinc(x) = sin(x) / x ; pτ(t) is the rectangular pulse function of width τ
 
  
Note: Source courtesy Wikibooks.org
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Notes:
 +
: <math> sinc(x) = \frac {sin(x)}{x} </math> <br>
 +
: <math> p_\tau (t)\ </math> is the rectangular pulse function of width <math> \tau\ </math>
 +
 
 +
 
 +
Source courtesy Wikibooks.org

Revision as of 10:27, 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

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Basic linear algebra uncovers and clarifies very important geometry and algebra.

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