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[[Category:Fourier transform]]
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[[Category:ECE301]]
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|'''If you enjoy using this [[Collective_Table_of_Formulas|collective table of formulas]], please consider  [https://donate.purdue.edu/DesignateGift.aspx?allocation=017637&appealCode=11213&amount=25&allocationDescription=RheaProjectMimiBoutin donating to Project Rhea] or [[Donations | becoming a sponsor]].'''
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[[Category:ECE438]]
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'''[[Collective_Table_of_Formulas|Collective Table of Formulas]]'''
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Continuous-time Fourier Transform Pairs and Properties
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as a function of frequency f in hertz
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(used in [[ECE438]])
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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">f</span> in hertz) [[More on CT Fourier transform|(info)]]
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! style="background: none repeat scroll 0% 0% rgb(238, 238, 238);" colspan="2" | CT Fourier Transform and its Inverse
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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;" |  CT Fourier Transform  
 
| align="right" style="padding-right: 1em;" |  CT Fourier Transform  
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! style="background: none repeat scroll 0% 0% rgb(238, 238, 238);" colspan="4" | CT Fourier Transform Pairs [[More_on_CTFTs| (info)]]
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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;" |  
 
| align="right" style="padding-right: 1em;" |  
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| <span class="texhtml">''x''(''t'')</span>
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| signal (function of t)  
 
| <math>\longrightarrow</math>
 
| <math>\longrightarrow</math>
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| <math> X(f) </math>
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| Fourier transform (function of f)
 
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| align="right" style="padding-right: 1em;" | CTFT of a unit impulse  
 
| align="right" style="padding-right: 1em;" | CTFT of a unit impulse  
 
| <math>\delta (t)\ </math>  
 
| <math>\delta (t)\ </math>  
 
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| <math> 1 \! \ </math>
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| <math> 1 \ </math>
 
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| align="right" style="padding-right: 1em;" | CTFT of a shifted unit impulse  
 
| align="right" style="padding-right: 1em;" | CTFT of a shifted unit impulse  
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| <math>e^{-at}u(t)\ </math>, where <math>a\in {\mathbb R}, a>0 </math>  
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| <math>e^{-at}u(t), \  \text{ where } a\in {\mathbb R}, a>0 </math>  
 
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| <math>\frac{1}{a+i2\pi f}</math>  
 
| <math>\frac{1}{a+i2\pi f}</math>  
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| <math>te^{-at}u(t)\ </math>, where <math>a\in {\mathbb R}, a>0 </math>  
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| <math>te^{-at}u(t), \  \text{ where } a\in {\mathbb R}, a>0 </math>  
 
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| <math>\left( \frac{1}{a+i2\pi f}\right)^2</math>  
 
| <math>\left( \frac{1}{a+i2\pi f}\right)^2</math>  
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| <span class="texhtml">''x''(''t'')</span>  
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| <math> x(t) \  </math>  
 
| <math>\longrightarrow</math>
 
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| <math> X(f) </math>
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| <math> X(f) \  </math>
 
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| align="right" style="padding-right: 1em;" | multiplication property
 
| align="right" style="padding-right: 1em;" | multiplication property
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| <math> X(f)*Y(f) =\int_{-\infty}^{\infty} X(\theta)Y(f-\theta)d\theta</math>
 
| <math> X(f)*Y(f) =\int_{-\infty}^{\infty} X(\theta)Y(f-\theta)d\theta</math>
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| align="right" style="padding-right: 1em;" | time shifting property
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| <math>x(t-t_0) \ </math>
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| <math> X(f)e^{-j 2 \pi f t_0}  \ </math>
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| align="right" style="padding-right: 1em;" | frequency shifting (also called "modulation") property
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| <math>x(t) e^{j 2 \pi f_0 t}  \ </math>
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| <math> X(f-f_0)  \  </math>
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| align="right" style="padding-right: 1em;" | scaling and shifting property
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| <math>x\left( \frac{ t- t_0}{a} \right) \ </math>
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| <math> |a| X(af) e^{-j 2 \pi f t_0}    \  </math>
 
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| align="right" style="padding-right: 1em;" | convolution property  
 
| align="right" style="padding-right: 1em;" | convolution property  
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| <math>x(t)*y(t) \ </math>  
 
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| <math> X(f)Y(f) \ </math>
 
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| align="right" style="padding-right: 1em;" | time reversal  
 
| align="right" style="padding-right: 1em;" | time reversal  
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[[Collective_Table_of_Formulas|Back to Collective Table]]
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[[Collective_Table_of_Formulas|Back to Collective Table]]
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[[Category:Formulas]]
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Latest revision as of 20:01, 4 March 2015


Collective Table of Formulas

Continuous-time Fourier Transform Pairs and Properties

as a function of frequency f in hertz

(used in ECE438)


CT Fourier Transform and its Inverse
CT Fourier Transform $ X(f)=\mathcal{F}(x(t))=\int_{-\infty}^{\infty} x(t) e^{-i2\pi ft} dt $
Inverse DT Fourier Transform $ \, x(t)=\mathcal{F}^{-1}(X(f))=\int_{-\infty}^{\infty}X(f)e^{i2\pi ft} df \, $
CT Fourier Transform Pairs
signal (function of t) $ \longrightarrow $ Fourier transform (function of f)
CTFT of a unit impulse $ \delta (t)\ $ $ 1 \ $
CTFT of a shifted unit impulse $ \delta (t-t_0)\ $ $ e^{-i2\pi ft_0} $
CTFT of a complex exponential $ e^{iw_0t} $ $ \delta (f - \frac{\omega_0}{2\pi}) \ $
$ e^{-at}u(t), \ \text{ where } a\in {\mathbb R}, a>0 $ $ \frac{1}{a+i2\pi f} $
$ te^{-at}u(t), \ \text{ where } a\in {\mathbb R}, a>0 $ $ \left( \frac{1}{a+i2\pi f}\right)^2 $
CTFT of a cosine $ \cos(\omega_0 t) \ $ $ \frac{1}{2} \left[\delta (f - \frac{\omega_0}{2\pi}) + \delta (f + \frac{\omega_0}{2\pi})\right] \ $
CTFT of a sine $ sin(\omega_0 t) \ $ $ \frac{1}{2i} \left[\delta (f - \frac{\omega_0}{2\pi}) - \delta (f + \frac{\omega_0}{2\pi})\right] $
CTFT of a rect $ \left\{\begin{array}{ll}1, & \text{ if }|t|<T,\\ 0, & \text{else.}\end{array} \right. \ $ $ \frac{\sin \left(2\pi Tf \right)}{\pi f} \ $
CTFT of a sinc $ \frac{ \sin \left( W t \right)}{\pi t } \ $ $ \left\{\begin{array}{ll}1, & \text{ if }|f| <\frac{W}{2\pi},\\ 0, & \text{else.}\end{array} \right. \ $
CTFT of a periodic function $ \sum^{\infty}_{k=-\infty} a_{k}e^{ikw_{0}t} $ $ \sum^{\infty}_{k=-\infty}a_{k}\delta(f-\frac{kw_{0}}{2\pi}) \ $
CTFT of an impulse train $ \sum^{\infty}_{n=-\infty} \delta(t-nT) \ $ $ \frac{1}{T}\sum^{\infty}_{k=-\infty}\delta(f-\frac{k}{T}) \ $
CT Fourier Transform Properties
$ x(t) \ $ $ \longrightarrow $ $ X(f) \ $
multiplication property $ x(t)y(t) \ $ $ X(f)*Y(f) =\int_{-\infty}^{\infty} X(\theta)Y(f-\theta)d\theta $
time shifting property $ x(t-t_0) \ $ $ X(f)e^{-j 2 \pi f t_0} \ $
frequency shifting (also called "modulation") property $ x(t) e^{j 2 \pi f_0 t} \ $ $ X(f-f_0) \ $
scaling and shifting property $ x\left( \frac{ t- t_0}{a} \right) \ $ $ |a| X(af) e^{-j 2 \pi f t_0} \ $
convolution property $ x(t)*y(t) \ $ $ X(f)Y(f) \ $
time reversal $ \ x(-t) $ $ \ X(-f) $
Other CT Fourier Transform Properties
Parseval's relation $ \int_{-\infty}^{\infty} |x(t)|^2 dt = \int_{-\infty}^{\infty} |X(f)|^2 df $

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Abstract algebra continues the conceptual developments of linear algebra, on an even grander scale.

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