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! colspan="2" style="background:  #e4bc7e; font-size: 110%;" | CT Fourier Transform Pairs and Properties (frequency <math>\omega</math> in radians per time unit) [[more_on_CT_Fourier_transform|(info)]]
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! colspan="2" style="background: #eee;" | Definition CT Fourier Transform and its Inverse
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! style="background: none repeat scroll 0% 0% rgb(228, 188, 126); font-size: 110%;" colspan="2" | CT Fourier Transform Pairs and Properties (frequency <span class="texhtml">ω</span> in radians per time unit) [[More on CT Fourier transform|(info)]]
 
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| align="right" style="padding-right: 1em;" | [[more_on_CT_Fourier_transform|(info)]] CT Fourier Transform || <math>\mathcal{X}(\omega)=\mathcal{F}(x(t))=\int_{-\infty}^{\infty} x(t) e^{-j\omega t} dt</math>
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! style="background: none repeat scroll 0% 0% rgb(238, 238, 238);" colspan="2" | Definition CT Fourier Transform and its Inverse
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| align="right" style="padding-right: 1em;" | [[more_on_CT_Fourier_transform|(info)]] Inverse DT Fourier Transform || <math>\, x(t)=\mathcal{F}^{-1}(\mathcal{X}(\omega))=\frac{1}{2\pi} \int_{-\infty}^{\infty}\mathcal{X}(\omega)e^{j\omega t} d \omega\,</math>
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| align="right" style="padding-right: 1em;" | [[More on CT Fourier transform|(info)]] CT Fourier Transform  
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| <math>\mathcal{X}(\omega)=\mathcal{F}(x(t))=\int_{-\infty}^{\infty} x(t) e^{-i\omega t} dt</math>
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|-
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| align="right" style="padding-right: 1em;" | [[More on CT Fourier transform|(info)]] Inverse DT Fourier Transform  
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| <math>\, x(t)=\mathcal{F}^{-1}(\mathcal{X}(\omega))=\frac{1}{2\pi} \int_{-\infty}^{\infty}\mathcal{X}(\omega)e^{i\omega t} d \omega\,</math>
 
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! colspan="4" style="background: #eee;" | CT Fourier Transform Pairs
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! style="background: none repeat scroll 0% 0% rgb(238, 238, 238);" colspan="4" | CT Fourier Transform Pairs
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| align="right" style="padding-right: 1em;" |  || <math>x(t)</math> || <math>\longrightarrow</math>|| <math> \mathcal{X}(\omega) </math>
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| align="right" style="padding-right: 1em;" | CTFT of a unit impulse || <math>\delta (t)\ </math> || || <math> 1 \! \ </math>
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| align="right" style="padding-right: 1em;" | CTFT of a shifted unit impulse || <math>\delta (t-t_0)\ </math> || || <math> e^{jwt_0} \ \ </math>
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|-
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| align="right" style="padding-right: 1em;" | CTFT of a complex exponential || <math>e^{jw_0t}\ </math> || || <math> 2\pi \delta (\omega - \omega_0) \ </math>
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||
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| align="right" style="padding-right: 1em;" | || <math>e^{-at}u(t)\ </math>, where <math>a\in {\mathbb R}, a>0 </math> || || <math>\frac{1}{a+j\omega}  \ </math>
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||
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| align="right" style="padding-right: 1em;" |  || <math>te^{-at}u(t)\ </math>, where <math>a\in {\mathbb R}, a>0 </math> || || <math>\left( \frac{1}{a+j\omega}\right)^2 \ </math>
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||
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| align="right" style="padding-right: 1em;" | CTFT of a cosine || <math>\cos(\omega_0 t) \ </math> || || <math> \pi \left[\delta (\omega - \omega_0) + \delta (\omega + \omega_0)\right] \ </math>
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| align="right" style="padding-right: 1em;" |  
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||
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| <span class="texhtml">''x''(''t'')</span>
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| <math>\longrightarrow</math>
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| <math> \mathcal{X}(\omega) </math>
 
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| align="right" style="padding-right: 1em;" | CTFT of a sine || <math>sin(\omega_0 t) \ </math> || || <math> \frac{\pi}{j} \left[\delta (\omega - \omega_0) - \delta (\omega + \omega_0)\right]  \ </math>
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| align="right" style="padding-right: 1em;" | CTFT of a unit impulse
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||
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| <math>\delta (t)\ </math>  
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|  
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| <math> 1 \! \ </math>
 
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|-
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| align="right" style="padding-right: 1em;" | CTFT of a rect || <math>\left\{\begin{array}{ll}1, &  \text{ if }|t|<T,\\ 0, & \text{else.}\end{array} \right. \ </math> || || <math> \frac{2 \sin \left( T \omega \right)}{\omega}  \ </math>
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| align="right" style="padding-right: 1em;" | CTFT of a shifted unit impulse
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||
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| <math>\delta (t-t_0)\ </math>  
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|  
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| <math>e^{iwt_0}</math>
 
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|-
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| align="right" style="padding-right: 1em;" | CTFT of a sinc || <math>\frac{2 \sin \left( W t  \right)}{\pi t }  \ </math> || || <math>\left\{\begin{array}{ll}1, &  \text{ if }|\omega| <W,\\ 0, & \text{else.}\end{array} \right.  \ </math>
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| align="right" style="padding-right: 1em;" | CTFT of a complex exponential
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||  
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| <math>e^{iw_0t}</math>  
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|  
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| <math> 2\pi \delta (\omega - \omega_0) \ </math>  
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|  
 
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|-
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| align="right" style="padding-right: 1em;" | CTFT of a periodic function || <math>\sum^{\infty}_{k=-\infty} a_{k}e^{jkw_{0}t} \ </math> || || <math>2\pi\sum^{\infty}_{k=-\infty}a_{k}\delta(w-kw_{0}) \ </math>
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| align="right" style="padding-right: 1em;" |  
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||
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| <math>e^{-at}u(t)\ </math>, where <math>a\in {\mathbb R}, a>0 </math>  
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|  
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| <math>\frac{1}{a+i\omega}</math>  
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|  
 
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| align="right" style="padding-right: 1em;" | CTFT of an impulse train || <math>\sum^{\infty}_{n=-\infty} \delta(t-nT) \ </math> || || <math>\frac{2\pi}{T}\sum^{\infty}_{k=-\infty}\delta(w-\frac{2\pi k}{T}) \ </math>
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| align="right" style="padding-right: 1em;" |  
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||
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| <math>te^{-at}u(t)\ </math>, where <math>a\in {\mathbb R}, a>0 </math>
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|
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| <math>\left( \frac{1}{a+i\omega}\right)^2</math>  
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|  
 
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| align="right" style="padding-right: 1em;" | CTFT of a cosine
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| <math>\cos(\omega_0 t) \ </math>
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|
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| <math> \pi \left[\delta (\omega - \omega_0) + \delta (\omega + \omega_0)\right] \ </math>
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| align="right" style="padding-right: 1em;" | CTFT of a sine
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| <math>sin(\omega_0 t)  \ </math>
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| <math>\frac{\pi}{i} \left[\delta (\omega - \omega_0) - \delta (\omega + \omega_0)\right]</math>
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| align="right" style="padding-right: 1em;" | CTFT of a rect
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| <math>\left\{\begin{array}{ll}1, &  \text{ if }|t|<T,\\ 0, & \text{else.}\end{array} \right. \ </math>
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|
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| <math> \frac{2 \sin \left( T \omega \right)}{\omega}  \ </math>
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|-
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| align="right" style="padding-right: 1em;" | CTFT of a sinc
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| <math>\frac{2 \sin \left( W t  \right)}{\pi t }  \ </math>
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|
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| <math>\left\{\begin{array}{ll}1, &  \text{ if }|\omega| <W,\\ 0, & \text{else.}\end{array} \right.  \ </math>
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|
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| align="right" style="padding-right: 1em;" | CTFT of a periodic function
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| <math>\sum^{\infty}_{k=-\infty} a_{k}e^{ikw_{0}t}</math>
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|
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| <math>2\pi\sum^{\infty}_{k=-\infty}a_{k}\delta(w-kw_{0}) \ </math>
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| align="right" style="padding-right: 1em;" | CTFT of an impulse train
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| <math>\sum^{\infty}_{n=-\infty} \delta(t-nT)  \ </math>
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|
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| <math>\frac{2\pi}{T}\sum^{\infty}_{k=-\infty}\delta(w-\frac{2\pi k}{T}) \ </math>
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|
 
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{|
 
{|
 
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! colspan="4" style="background: #eee;" | CT Fourier Transform Properties
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! style="background: none repeat scroll 0% 0% rgb(238, 238, 238);" colspan="4" | CT Fourier Transform Properties
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| align="right" style="padding-right: 1em;" |  || <math>x(t)</math> || <math>\longrightarrow</math>|| <math> \mathcal{X}(\omega) </math>
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| align="right" style="padding-right: 1em;" | multiplication property|| <math>x(t)y(t) \ </math> || || <math>\frac{1}{2\pi} X(\omega)*Y(\omega) =\frac{1}{2\pi} \int_{-\infty}^{\infty} X(\theta)Y(\omega-\theta)d\theta</math>
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| align="right" style="padding-right: 1em;" | convolution property || <math>x(t)*y(t) \!</math> || ||<math> X(\omega)Y(\omega) \!</math>
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| align="right" style="padding-right: 1em;" | time reversal ||<math>\ x(-t) </math> || ||<math>\ X(-\omega)</math>
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| align="right" style="padding-right: 1em;" |  
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| <span class="texhtml">''x''(''t'')</span>  
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| <math>\longrightarrow</math>
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| <math> \mathcal{X}(\omega) </math>
 
|-
 
|-
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| align="right" style="padding-right: 1em;" | multiplication property
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| <math>x(t)y(t) \ </math>
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|
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| <math>\frac{1}{2\pi} X(\omega)*Y(\omega) =\frac{1}{2\pi} \int_{-\infty}^{\infty} X(\theta)Y(\omega-\theta)d\theta</math>
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|-
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| align="right" style="padding-right: 1em;" | convolution property
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| <math>x(t)*y(t) \!</math>
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|
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| <math> X(\omega)Y(\omega) \!</math>
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|-
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| align="right" style="padding-right: 1em;" | time reversal
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| <math>\ x(-t) </math>
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|
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| <math>\ X(-\omega)</math>
 
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! colspan="2" style="background: #eee;" | Other CT Fourier Transform Properties
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! style="background: none repeat scroll 0% 0% rgb(238, 238, 238);" colspan="2" | Other CT Fourier Transform Properties
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| align="right" style="padding-right: 1em;" | Parseval's relation || <math>\int_{-\infty}^{\infty} |x(t)|^2 dt = \frac{1}{2\pi} \int_{-\infty}^{\infty} |\mathcal{X}(w)|^2 dw</math>
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| align="right" style="padding-right: 1em;" | Parseval's relation  
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| <math>\int_{-\infty}^{\infty} |x(t)|^2 dt = \frac{1}{2\pi} \int_{-\infty}^{\infty} |\mathcal{X}(w)|^2 dw</math>
 
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[[ MegaCollectiveTableTrial1|Back to Collective Table]]
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[[MegaCollectiveTableTrial1|Back to Collective Table]]  
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[[Category:Formulas]]
 
[[Category:Formulas]]

Revision as of 04:57, 5 April 2010

CT Fourier Transform Pairs and Properties (frequency ω in radians per time unit) (info)
Definition CT Fourier Transform and its Inverse
(info) CT Fourier Transform $ \mathcal{X}(\omega)=\mathcal{F}(x(t))=\int_{-\infty}^{\infty} x(t) e^{-i\omega t} dt $
(info) Inverse DT Fourier Transform $ \, x(t)=\mathcal{F}^{-1}(\mathcal{X}(\omega))=\frac{1}{2\pi} \int_{-\infty}^{\infty}\mathcal{X}(\omega)e^{i\omega t} d \omega\, $
CT Fourier Transform Pairs
x(t) $ \longrightarrow $ $ \mathcal{X}(\omega) $
CTFT of a unit impulse $ \delta (t)\ $ $ 1 \! \ $
CTFT of a shifted unit impulse $ \delta (t-t_0)\ $ $ e^{iwt_0} $
CTFT of a complex exponential $ e^{iw_0t} $ $ 2\pi \delta (\omega - \omega_0) \ $
$ e^{-at}u(t)\ $, where $ a\in {\mathbb R}, a>0 $ $ \frac{1}{a+i\omega} $
$ te^{-at}u(t)\ $, where $ a\in {\mathbb R}, a>0 $ $ \left( \frac{1}{a+i\omega}\right)^2 $
CTFT of a cosine $ \cos(\omega_0 t) \ $ $ \pi \left[\delta (\omega - \omega_0) + \delta (\omega + \omega_0)\right] \ $
CTFT of a sine $ sin(\omega_0 t) \ $ $ \frac{\pi}{i} \left[\delta (\omega - \omega_0) - \delta (\omega + \omega_0)\right] $
CTFT of a rect $ \left\{\begin{array}{ll}1, & \text{ if }|t|<T,\\ 0, & \text{else.}\end{array} \right. \ $ $ \frac{2 \sin \left( T \omega \right)}{\omega} \ $
CTFT of a sinc $ \frac{2 \sin \left( W t \right)}{\pi t } \ $ $ \left\{\begin{array}{ll}1, & \text{ if }|\omega| <W,\\ 0, & \text{else.}\end{array} \right. \ $
CTFT of a periodic function $ \sum^{\infty}_{k=-\infty} a_{k}e^{ikw_{0}t} $ $ 2\pi\sum^{\infty}_{k=-\infty}a_{k}\delta(w-kw_{0}) \ $
CTFT of an impulse train $ \sum^{\infty}_{n=-\infty} \delta(t-nT) \ $ $ \frac{2\pi}{T}\sum^{\infty}_{k=-\infty}\delta(w-\frac{2\pi k}{T}) \ $
CT Fourier Transform Properties
x(t) $ \longrightarrow $ $ \mathcal{X}(\omega) $
multiplication property $ x(t)y(t) \ $ $ \frac{1}{2\pi} X(\omega)*Y(\omega) =\frac{1}{2\pi} \int_{-\infty}^{\infty} X(\theta)Y(\omega-\theta)d\theta $
convolution property $ x(t)*y(t) \! $ $ X(\omega)Y(\omega) \! $
time reversal $ \ x(-t) $ $ \ X(-\omega) $
Other CT Fourier Transform Properties
Parseval's relation $ \int_{-\infty}^{\infty} |x(t)|^2 dt = \frac{1}{2\pi} \int_{-\infty}^{\infty} |\mathcal{X}(w)|^2 dw $

Back to Collective Table

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

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