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[[Category:Formulas]]
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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
 +
 +
as a function of <math>\omega</math> in radians per time unit
 +
 +
(used in [[ECE301]])
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</center>
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 +
----
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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)]]
 
 
|-
 
|-
 
! style="background: none repeat scroll 0% 0% rgb(238, 238, 238);" colspan="2" | Definition CT Fourier Transform and its Inverse
 
! style="background: none repeat scroll 0% 0% rgb(238, 238, 238);" colspan="2" | Definition CT Fourier Transform and its Inverse
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| <math>\mathcal{X}(\omega)=\mathcal{F}(x(t))=\int_{-\infty}^{\infty} x(t) e^{-i\omega t} dt</math>
 
| <math>\mathcal{X}(\omega)=\mathcal{F}(x(t))=\int_{-\infty}^{\infty} x(t) e^{-i\omega t} dt</math>
 
|-
 
|-
−
| align="right" style="padding-right: 1em;" | [[More on CT Fourier transform|(info)]] Inverse DT Fourier Transform  
+
| align="right" style="padding-right: 1em;" | [[More on CT Fourier transform|(info)]] Inverse CT Fourier Transform  
 
| <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>
 
| <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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{|
 
{|
 
|-
 
|-
−
! style="background: none repeat scroll 0% 0% rgb(238, 238, 238);" colspan="4" |
 
 
! style="background: none repeat scroll 0% 0% rgb(238, 238, 238);" colspan="4" | CT Fourier Transform Pairs
 
! style="background: none repeat scroll 0% 0% rgb(238, 238, 238);" colspan="4" | CT Fourier Transform Pairs
 
|-
 
|-
 
| align="right" style="padding-right: 1em;" | <br>  
 
| align="right" style="padding-right: 1em;" | <br>  
 
| align="right" style="padding-right: 1em;" |  
 
| align="right" style="padding-right: 1em;" |  
−
| <span class="texhtml">''x''(''t'')</span>
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| signal (function of t)
 
| <math>\longrightarrow</math>  
 
| <math>\longrightarrow</math>  
−
| <math> \mathcal{X}(\omega) </math>  
+
| Fourier transform (function of <math>\omega</math>)
 
|  
 
|  
 
|-
 
|-
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| <math>\delta (t)\ </math>  
 
| <math>\delta (t)\ </math>  
 
|  
 
|  
−
| '''<span class="texhtml">1</span><br>'''
+
| <math> 1 \ </math>  
 
|  
 
|  
 
|-
 
|-
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| align="right" style="padding-right: 1em;" | 9  
 
| align="right" style="padding-right: 1em;" | 9  
 
| align="right" style="padding-right: 1em;" | CTFT of a sinc  
 
| align="right" style="padding-right: 1em;" | CTFT of a sinc  
−
| <math>\frac{2 \sin \left( W t  \right)}{\pi t }  \ </math>  
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| <math>\frac{\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>  
 
| <math>\left\{\begin{array}{ll}1, &  \text{ if }|\omega| <W,\\ 0, & \text{else.}\end{array} \right.  \ </math>  
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| <math>\sum^{\infty}_{n=-\infty} \delta(t-nT)  \ </math>  
 
| <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><br>'''
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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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| align="right" style="padding-right: 1em;" | 12  
 
| align="right" style="padding-right: 1em;" | 12  
 
| align="right" style="padding-right: 1em;" |  
 
| align="right" style="padding-right: 1em;" |  
−
| '''1<br> '''
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| <math> 1 \  </math>  
 
|  
 
|  
−
| <span class="texhtml">2πδ(ω)
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| <math>2\pi \delta (\omega) \ </math>
−
</span>
+
 
|  
 
|  
 
|  
 
|  
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| align="right" style="padding-right: 1em;" | 13  
 
| align="right" style="padding-right: 1em;" | 13  
 
| align="right" style="padding-right: 1em;" | CTFT of a Periodic Square Wave  
 
| 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>
 
|  
 
|  
−
<math>\left\{\begin{array}{ll}1, &  |t|<T_1,\\ 0, & T_1<|t|<=\frac{T}{2}\end{array} \right.</math>
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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>  
−
 
+
−
and
+
−
 
+
−
<span class="texhtml">''x''(''t'' + ''T'') = ''x''(''t'')</span>
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−
 
+
−
|
+
−
| <math>\sum^{\infty}_{k=-\infty}\frac{2 \sin(k\omega_0T_1}{k}\delta(\omega-k\omega_0)</math>  
+
 
|  
 
|  
 
|  
 
|  
 
|-
 
|-
−
| align="right" style="padding-right: 1em;" |  
+
| align="right" style="padding-right: 1em;" | 14  
−
14  
+
−
 
+
 
| align="right" style="padding-right: 1em;" | CTFT of a Step Function  
 
| align="right" style="padding-right: 1em;" | CTFT of a Step Function  
−
| <span class="texhtml">''u''(''t'')</span>  
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| <math> u(t) \  </math>
 
|  
 
|  
 
| <math>\frac{1}{j\omega}+\pi\delta(\omega)</math>  
 
| <math>\frac{1}{j\omega}+\pi\delta(\omega)</math>  
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| align="right" style="padding-right: 1em;" | 15  
 
| align="right" style="padding-right: 1em;" | 15  
 
| align="right" style="padding-right: 1em;" |  
 
| align="right" style="padding-right: 1em;" |  
−
| <span class="texhtml">''e''<sup> − α &#124; ''t'' &#124; </sup></span>  
+
| <math> e^{-\alpha |t|}  \ </math>  
 
|  
 
|  
 
| <math>\frac{2\alpha}{\alpha^{2}+\omega^{2}}</math>  
 
| <math>\frac{2\alpha}{\alpha^{2}+\omega^{2}}</math>  
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|-
 
|-
 
| align="right" style="padding-right: 1em;" |  
 
| align="right" style="padding-right: 1em;" |  
−
| align="right" style="padding-right: 1em;" |  
+
| align="right" style="padding-right: 1em;" |
−
| <span class="texhtml">''x''(''t'')</span>  
+
| <math> x(t) \ </math>  
 
| <math>\longrightarrow</math>  
 
| <math>\longrightarrow</math>  
 
| <math> \mathcal{X}(\omega) </math>
 
| <math> \mathcal{X}(\omega) </math>
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| align="right" style="padding-right: 1em;" | 17  
 
| align="right" style="padding-right: 1em;" | 17  
 
| align="right" style="padding-right: 1em;" | convolution property  
 
| align="right" style="padding-right: 1em;" | convolution property  
−
| <math>x(t)*y(t) \!</math>  
+
| <math>x(t)*y(t) \ </math>  
 
|  
 
|  
 
| <math> \mathcal{X}(\omega)\mathcal{Y}(\omega) \!</math>
 
| <math> \mathcal{X}(\omega)\mathcal{Y}(\omega) \!</math>
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| <math>e^{j\omega_0 t}x(t)</math>  
 
| <math>e^{j\omega_0 t}x(t)</math>  
 
|  
 
|  
−
| <span class="texhtml">''X''(ω − ω<sub>0</sub>)</span>
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| <math> \mathcal{X} (\omega - \omega_0) </math>
 
|-
 
|-
 
| align="right" style="padding-right: 1em;" | 20  
 
| align="right" style="padding-right: 1em;" | 20  
−
| align="right" style="padding-right: 1em;" | Conjugation  
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| align="right" style="padding-right: 1em;"| Conjugation
−
| <span class="texhtml">''x''<sup> * </sup>(''t'')</span>  
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| <math> x^{*}(t) \  </math>  
 
|  
 
|  
−
| <span class="texhtml">''X''<sup> * </sup>( − ω)</span>
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| <math> \mathcal{X}^{*} (-\omega)</math>
 
|-
 
|-
 
| align="right" style="padding-right: 1em;" | 21  
 
| 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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| align="right" style="padding-right: 1em;" | Time and Frequency Scaling
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| <span class="texhtml">''x''(''a''''t'''''<b>)</b></span>  
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| <math> x(at) \  </math>  
 
|  
 
|  
−
| <math>\frac{1}{|a|}X(\frac{\omega}{a})</math>
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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;" | 22
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−
| align="right" style="padding-right: 1em;" | Multiplication
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−
| <span class="texhtml">''x''(''t'')''y''(''t'')</span>
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−
|
+
−
| <math>\frac{1}{2\pi}X(\omega)*Y(\omega)</math><br>
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|-
 
|-
 
| align="right" style="padding-right: 1em;" | 23  
 
| align="right" style="padding-right: 1em;" | 23  
 
| align="right" style="padding-right: 1em;" | Differentiation in Frequency  
 
| align="right" style="padding-right: 1em;" | Differentiation in Frequency  
−
| <span class="texhtml">''t''''x'''''<b>(''t'')</b></span>  
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| <math> tx(t) \  </math>
 
|  
 
|  
−
| <math>j\frac{d}{d\omega}X(\omega)</math>
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| <math>j\frac{d}{d\omega} \mathcal{X} (\omega)</math>
 
|-
 
|-
 
| align="right" style="padding-right: 1em;" | 24  
 
| align="right" style="padding-right: 1em;" | 24  
−
| align="right" style="padding-right: 1em;" | Symmetry  
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| align="right" style="padding-right: 1em;" | Symmetry
−
| <span class="texhtml">''x''(''t'') &nbsp; &nbsp; real and even</span>  
+
| <math> x(t)\ \text{ real and even}</math>  
 
|  
 
|  
−
| <span class="texhtml">''X''(ω) &nbsp; &nbsp; real and even</span>
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| <math> \mathcal{X} (\omega) \ \text{ real and even} </math>
 
|-
 
|-
 
| align="right" style="padding-right: 1em;" | 25  
 
| align="right" style="padding-right: 1em;" | 25  
 
| align="right" style="padding-right: 1em;" |  
 
| align="right" style="padding-right: 1em;" |  
−
| <span class="texhtml">''x''(''t'') &nbsp; &nbsp; real and odd</span>  
+
| <math> x(t) \ \text{ real and odd}</math>  
 
|  
 
|  
−
| <span class="texhtml">''X''(ω) &nbsp; &nbsp; perely imaginary and odd</span>
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| <math> \mathcal{X} (\omega) \ \text{ purely imaginary and odd} </math>
 
|-
 
|-
 
| align="right" style="padding-right: 1em;" | 26  
 
| align="right" style="padding-right: 1em;" | 26  
 
| align="right" style="padding-right: 1em;" | Duality  
 
| align="right" style="padding-right: 1em;" | Duality  
−
| x(-t)  
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| <math> \mathcal{X} (-t) </math>
 
|  
 
|  
−
| '''<span class="texhtml">2π''X''(ω)</span>'''
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| <math> 2 \pi x (\omega) \  </math>
 
|-
 
|-
 
| align="right" style="padding-right: 1em;" | 27  
 
| align="right" style="padding-right: 1em;" | 27  
 
| align="right" style="padding-right: 1em;" | Differentiation  
 
| align="right" style="padding-right: 1em;" | Differentiation  
−
| '''<math>\frac{d^{n}x(t)}{dt^{n}}</math>'''
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| <math>\frac{d^{n}x(t)}{dt^{n}}</math>  
−
|
+
−
| '''<span class="texhtml">(''j''ω)<sup>''n''</sup>''X''(ω)</span>'''
+
−
|-
+
−
| align="right" style="padding-right: 1em;" |
+
−
| align="right" style="padding-right: 1em;" | <br>
+
−
|
+
−
|
+
 
|  
 
|  
 +
| <math> (j \omega)^{n} \mathcal{X} (\omega) </math>
 
|-
 
|-
 
| align="right" style="padding-right: 1em;" | 28  
 
| align="right" style="padding-right: 1em;" | 28  
 
| align="right" style="padding-right: 1em;" | Linearity  
 
| align="right" style="padding-right: 1em;" | Linearity  
−
| <span class="texhtml">''a''''x'''''<b>(</b>'''''t'') + ''b''''''''y''(''t'')''</span>  
+
| <math> ax(t) + by(t) \  </math>  
 
|  
 
|  
−
| <span class="texhtml">''a''''X'''''<b>(ω) + </b>'''''b''''''''Y''(ω)''</span>
+
| <math> a \mathcal{X}(\omega) + b \mathcal{Y} (\omega) </math>
 
|-
 
|-
 
| align="right" style="padding-right: 1em;" | 29  
 
| align="right" style="padding-right: 1em;" | 29  
 
| align="right" style="padding-right: 1em;" | Time Shifting  
 
| align="right" style="padding-right: 1em;" | Time Shifting  
−
| <span class="texhtml">''x''(''t'' − ''t''<sub>0</sub>)</span>  
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| <math> x(t-t_0) \  </math>
 
|  
 
|  
−
|  
+
| <math>e^{-j\omega t_0}X(\omega)</math>  
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<math>e^{-j\omega t_0}X(\omega)</math>  
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|}  
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+
−
|}
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−
 
+
−
<br>
+
  
 
{|
 
{|
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|}
  
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<br> <br>
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<br>  
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Sources:
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Class Text Book
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−
 
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http://www1.na.infn.it/~cavalier/Download/SICSI_LAES/Lucidi_DSP/FourierTransformPairs.pdf
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−
 
+
 
----
 
----
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[[Collective Table of Formulas|Back to Collective Table]]
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[[Collective Table of Formulas|Back to Collective Table]]
+
−
 
+
−
[[Category:Formulas]]
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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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