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{|
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[[Category:Formulas]]
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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)
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[[Category:Fourier transform]]
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[[Category:ECE301]]
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[[Category:ECE438]]
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<center><font size= 4>
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'''[[Collective_Table_of_Formulas|Collective Table of Formulas]]'''
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</font size>
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Table of Continuous-time (CT)  Fourier Transform Pairs and Properties
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 +
as a function of <math>\omega</math> in radians per time unit
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 +
(used in [[ECE301]])
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</center>
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 +
----
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{|
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|-
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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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|-
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! colspan="2" style="background: #eee;" | Definition CT Fourier Transform and its Inverse
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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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| align="right" style="padding-right: 1em;" | 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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| align="right" style="padding-right: 1em;" | [[More on CT Fourier transform|(info)]] Inverse CT Fourier Transform  
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|-
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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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| align="right" style="padding-right: 1em;" | 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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|}
 
|}
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{|
 
{|
 
|-
 
|-
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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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|-
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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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|-
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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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|-
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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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−
|-
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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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|-
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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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|-
 
|-
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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;" | <br>
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||  
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| align="right" style="padding-right: 1em;" |
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| signal (function of t)
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| <math>\longrightarrow</math>  
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| Fourier transform (function of <math>\omega</math>)
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|  
 
|-
 
|-
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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;" | 1
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||  
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| align="right" style="padding-right: 1em;" | CTFT of a unit impulse
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| <math>\delta (t)\ </math>  
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|  
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| <math> 1 \ </math>  
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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;" | 2
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||  
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| align="right" style="padding-right: 1em;" | CTFT of a shifted unit impulse
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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;" | 3
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||  
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| align="right" style="padding-right: 1em;" | CTFT of a complex exponential
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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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|  
 
|-
 
|-
−
| 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;" | 4
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||
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| align="right" style="padding-right: 1em;" |
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| <math>e^{-at}u(t),\ </math> <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;" | 5
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||
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| align="right" style="padding-right: 1em;" |
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| <math>te^{-at}u(t),\ </math> <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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|  
 
|-
 
|-
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| align="right" style="padding-right: 1em;" | 6
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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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|
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|
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|-
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| align="right" style="padding-right: 1em;" | 7
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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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|
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| <math>\frac{\pi}{i} \left[\delta (\omega - \omega_0) - \delta (\omega + \omega_0)\right]</math>
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|
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|
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|-
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| align="right" style="padding-right: 1em;" | 8
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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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|
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|-
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| align="right" style="padding-right: 1em;" | 9
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| align="right" style="padding-right: 1em;" | CTFT of a sinc
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| <math>\frac{\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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|
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|-
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| align="right" style="padding-right: 1em;" | 10
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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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|
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|
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|-
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| align="right" style="padding-right: 1em;" | 11
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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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|-
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| align="right" style="padding-right: 1em;" | 12
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| align="right" style="padding-right: 1em;" |
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| <math> 1 \  </math>
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|
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| <math>2\pi \delta (\omega) \ </math>
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|
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|
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|-
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| align="right" style="padding-right: 1em;" | 13
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| align="right" style="padding-right: 1em;" | CTFT of a Periodic Square Wave
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| <math>x(t+T)=x(t)=\left\{\begin{array}{ll}1, &  |t|\leq T_1,\\ 0, & T_1<|t|\leq T/2 \end{array} \right.</math>
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|
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| <math>\sum^{\infty}_{k=-\infty}\frac{2 \sin(k\frac{2\pi}{T}T_1)}{k}\delta(\omega-k\frac{2\pi}{T})</math>
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|
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|
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|-
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| align="right" style="padding-right: 1em;" | 14
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| align="right" style="padding-right: 1em;" | CTFT of a Step Function
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| <math> u(t) \  </math>
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|
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| <math>\frac{1}{j\omega}+\pi\delta(\omega)</math>
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|
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|
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|-
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| align="right" style="padding-right: 1em;" | 15
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| align="right" style="padding-right: 1em;" |
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| <math> e^{-\alpha |t|}  \ </math>
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|
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| <math>\frac{2\alpha}{\alpha^{2}+\omega^{2}}</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" |  
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|-
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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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|-
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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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|-
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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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|-
 
|-
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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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| align="right" style="padding-right: 1em;" |
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| <math> x(t) \ </math>  
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| <math>\longrightarrow</math>
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| <math> \mathcal{X}(\omega) </math>
 
|-
 
|-
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|}
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| align="right" style="padding-right: 1em;" | 16
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| align="right" style="padding-right: 1em;" | [[Info multiplication property FT w|(info)]] 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} \mathcal{X}(\omega)*\mathcal{Y}(\omega) =\frac{1}{2\pi} \int_{-\infty}^{\infty} \mathcal{X}(\theta)\mathcal{Y}(\omega-\theta)d\theta</math>
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|-
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| align="right" style="padding-right: 1em;" | 17
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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> \mathcal{X}(\omega)\mathcal{Y}(\omega) \!</math>
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|-
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| align="right" style="padding-right: 1em;" | 18
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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>\ \mathcal{X}(-\omega)</math>
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|-
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| align="right" style="padding-right: 1em;" | 19
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| align="right" style="padding-right: 1em;" | Frequency Shifting
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| <math>e^{j\omega_0 t}x(t)</math>
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|
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| <math> \mathcal{X} (\omega - \omega_0) </math>
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|-
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| align="right" style="padding-right: 1em;" | 20
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| align="right" style="padding-right: 1em;"| Conjugation
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| <math> x^{*}(t) \  </math>
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|
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| <math> \mathcal{X}^{*} (-\omega)</math>
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|-
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| align="right" style="padding-right: 1em;" | 21
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| align="right" style="padding-right: 1em;" | Time and Frequency Scaling
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| <math> x(at) \  </math>
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|
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| <math>\frac{1}{|a|} \mathcal{X} (\frac{\omega}{a})</math>
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|-
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| align="right" style="padding-right: 1em;" | 23
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| align="right" style="padding-right: 1em;" | Differentiation in Frequency
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| <math> tx(t) \  </math>
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|
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| <math>j\frac{d}{d\omega} \mathcal{X} (\omega)</math>
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|-
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| align="right" style="padding-right: 1em;" | 24
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| align="right" style="padding-right: 1em;" | Symmetry
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| <math> x(t)\ \text{ real and even}</math>
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|
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| <math> \mathcal{X} (\omega) \ \text{ real and even} </math>
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|-
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| align="right" style="padding-right: 1em;" | 25
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| align="right" style="padding-right: 1em;" |
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| <math> x(t) \ \text{ real and odd}</math>
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|
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| <math> \mathcal{X} (\omega) \ \text{ purely imaginary and odd} </math>
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|-
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| align="right" style="padding-right: 1em;" | 26
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| align="right" style="padding-right: 1em;" | Duality
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| <math> \mathcal{X} (-t) </math>
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|
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| <math> 2 \pi x (\omega) \  </math>
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|-
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| align="right" style="padding-right: 1em;" | 27
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| align="right" style="padding-right: 1em;" | Differentiation
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| <math>\frac{d^{n}x(t)}{dt^{n}}</math>
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|
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| <math> (j \omega)^{n} \mathcal{X} (\omega) </math>
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|-
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| align="right" style="padding-right: 1em;" | 28
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| align="right" style="padding-right: 1em;" | Linearity
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| <math> ax(t) + by(t) \  </math>
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|
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| <math> a \mathcal{X}(\omega) + b \mathcal{Y} (\omega) </math>
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|-
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| align="right" style="padding-right: 1em;" | 29
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| align="right" style="padding-right: 1em;" | Time Shifting
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| <math> x(t-t_0) \  </math>
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|
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| <math>e^{-j\omega t_0}X(\omega)</math>
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|}  
  
 
{|
 
{|
 
|-
 
|-
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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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|-  
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|-
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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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<br>
 
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[[ MegaCollectiveTableTrial1|Back to Collective Table]]
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[[Collective Table of Formulas|Back to Collective Table]]

Latest revision as of 11:45, 24 August 2016


Collective Table of Formulas

Table of Continuous-time (CT) Fourier Transform Pairs and Properties

as a function of $ \omega $ in radians per time unit

(used in ECE301)

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 CT 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

signal (function of t) $ \longrightarrow $ Fourier transform (function of $ \omega $)
1 CTFT of a unit impulse $ \delta (t)\ $ $ 1 \ $
2 CTFT of a shifted unit impulse $ \delta (t-t_0)\ $ $ e^{-iwt_0} $
3 CTFT of a complex exponential $ e^{iw_0t} $ $ 2\pi \delta (\omega - \omega_0) \ $
4 $ e^{-at}u(t),\ $ $ a\in {\mathbb R}, a>0 $ $ \frac{1}{a+i\omega} $
5 $ te^{-at}u(t),\ $ $ a\in {\mathbb R}, a>0 $ $ \left( \frac{1}{a+i\omega}\right)^2 $
6 CTFT of a cosine $ \cos(\omega_0 t) \ $ $ \pi \left[\delta (\omega - \omega_0) + \delta (\omega + \omega_0)\right] \ $
7 CTFT of a sine $ sin(\omega_0 t) \ $ $ \frac{\pi}{i} \left[\delta (\omega - \omega_0) - \delta (\omega + \omega_0)\right] $
8 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} \ $
9 CTFT of a sinc $ \frac{\sin \left( W t \right)}{\pi t } \ $ $ \left\{\begin{array}{ll}1, & \text{ if }|\omega| <W,\\ 0, & \text{else.}\end{array} \right. \ $
10 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}) \ $
11 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}) $
12 $ 1 \ $ $ 2\pi \delta (\omega) \ $
13 CTFT of a Periodic Square Wave $ x(t+T)=x(t)=\left\{\begin{array}{ll}1, & |t|\leq T_1,\\ 0, & T_1<|t|\leq T/2 \end{array} \right. $ $ \sum^{\infty}_{k=-\infty}\frac{2 \sin(k\frac{2\pi}{T}T_1)}{k}\delta(\omega-k\frac{2\pi}{T}) $
14 CTFT of a Step Function $ u(t) \ $ $ \frac{1}{j\omega}+\pi\delta(\omega) $
15 $ e^{-\alpha |t|} \ $ $ \frac{2\alpha}{\alpha^{2}+\omega^{2}} $
CT Fourier Transform Properties
$ x(t) \ $ $ \longrightarrow $ $ \mathcal{X}(\omega) $
16 (info) multiplication property $ x(t)y(t) \ $ $ \frac{1}{2\pi} \mathcal{X}(\omega)*\mathcal{Y}(\omega) =\frac{1}{2\pi} \int_{-\infty}^{\infty} \mathcal{X}(\theta)\mathcal{Y}(\omega-\theta)d\theta $
17 convolution property $ x(t)*y(t) \ $ $ \mathcal{X}(\omega)\mathcal{Y}(\omega) \! $
18 time reversal $ \ x(-t) $ $ \ \mathcal{X}(-\omega) $
19 Frequency Shifting $ e^{j\omega_0 t}x(t) $ $ \mathcal{X} (\omega - \omega_0) $
20 Conjugation $ x^{*}(t) \ $ $ \mathcal{X}^{*} (-\omega) $
21 Time and Frequency Scaling $ x(at) \ $ $ \frac{1}{|a|} \mathcal{X} (\frac{\omega}{a}) $
23 Differentiation in Frequency $ tx(t) \ $ $ j\frac{d}{d\omega} \mathcal{X} (\omega) $
24 Symmetry $ x(t)\ \text{ real and even} $ $ \mathcal{X} (\omega) \ \text{ real and even} $
25 $ x(t) \ \text{ real and odd} $ $ \mathcal{X} (\omega) \ \text{ purely imaginary and odd} $
26 Duality $ \mathcal{X} (-t) $ $ 2 \pi x (\omega) \ $
27 Differentiation $ \frac{d^{n}x(t)}{dt^{n}} $ $ (j \omega)^{n} \mathcal{X} (\omega) $
28 Linearity $ ax(t) + by(t) \ $ $ a \mathcal{X}(\omega) + b \mathcal{Y} (\omega) $
29 Time Shifting $ x(t-t_0) \ $ $ e^{-j\omega t_0}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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