(New page: {| |- ! 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> ...)
 
 
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*<span style="color:green"> I suggest putting a link next to each formula explaining how to obtain it from the formula in terms of <math>\omega</math>. Also, I would not use the "mathcal" (curly) font for the transform variable, just a capital letter instead. </span> --[[User:Mboutin|Mboutin]] 08:52, 3 September 2010 (UTC).
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**<span style="color:green"> Fixed the X(f) notation </span> -[[User:sbiddand|Sbiddand]]
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*<span style="color:green">The explanation for each formula still needs to be added! In particular, some students said it was not clear how to get the convolution property in terms of f. So this needs to be explained clearly.</span> --[[User:Mboutin|Mboutin]] 09:04, 7 September 2010 (UTC)
 +
**<span style="color:green"> Provided explanation for each formula. </span> -[[User:zhao148|Zhao]]
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***<span style="color:green"> Modified explanation for each formula. </span> -[[User:zhao148|Zhao]] 17:20, 15 September 2010 (UTC).
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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 per time unit) [[More on CT Fourier transform|(info)]]
 
! 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 per time unit) [[More on CT Fourier transform|(info)]]
 
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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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! style="background: none repeat scroll 0% 0% rgb(238, 188, 126);" colspan="2" | (Click title to see explanation on how to obtain the formula in terms of f in hertz)
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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  
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| align="right" style="padding-right: 1em;" |  [[Explain_CTFT|CT Fourier Transform]]
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| <math>\mathcal{X}(f)=\mathcal{F}(x(t))=\int_{-\infty}^{\infty} x(t) e^{-i2\pi ft} dt</math>
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| <math>X(f)=\mathcal{F}(x(t))=\int_{-\infty}^{\infty} x(t) e^{-i2\pi ft} dt</math>
 
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| align="right" style="padding-right: 1em;" | Inverse DT Fourier Transform  
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| align="right" style="padding-right: 1em;" | [[Explain_InverseCTFT|Inverse CT Fourier Transform]]
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| <math>\, x(t)=\mathcal{F}^{-1}(\mathcal{X}(f))=\int_{-\infty}^{\infty}\mathcal{X}(f)e^{i2\pi ft} df \,</math>
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| <math>\, x(t)=\mathcal{F}^{-1}(X(f))=\int_{-\infty}^{\infty}X(f)e^{i2\pi ft} df \,</math>
 
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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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! 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;" |  
 
| <span class="texhtml">''x''(''t'')</span>  
 
| <span class="texhtml">''x''(''t'')</span>  
 
| <math>\longrightarrow</math>
 
| <math>\longrightarrow</math>
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| <math> \mathcal{X}(f) </math>
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| <math> X(f) </math>
 
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| align="right" style="padding-right: 1em;" | CTFT of a unit impulse  
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| align="right" style="padding-right: 1em;" | [[Explain_unitimpulse|CTFT of a unit impulse]]
 
| <math>\delta (t)\ </math>  
 
| <math>\delta (t)\ </math>  
 
|  
 
|  
 
| <math> 1 \! \ </math>
 
| <math> 1 \! \ </math>
 
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| align="right" style="padding-right: 1em;" | CTFT of a shifted unit impulse  
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| align="right" style="padding-right: 1em;" | [[Explain_CTFT_shifted_unitimpulse|CTFT of a shifted unit impulse]]
 
| <math>\delta (t-t_0)\ </math>  
 
| <math>\delta (t-t_0)\ </math>  
 
|  
 
|  
 
| <math>e^{-i2\pi ft_0}</math>
 
| <math>e^{-i2\pi ft_0}</math>
 
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|-
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| align="right" style="padding-right: 1em;" | CTFT of a complex exponential  
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| align="right" style="padding-right: 1em;" | [[Explain_CTFT_cpxexp|CTFT of a complex exponential]]
 
| <math>e^{iw_0t}</math>  
 
| <math>e^{iw_0t}</math>  
 
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| align="right" style="padding-right: 1em;" | CTFT of a cosine  
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| align="right" style="padding-right: 1em;" | [[Explain_CTFT_cos|CTFT of a cosine]]
 
| <math>\cos(\omega_0 t) \ </math>  
 
| <math>\cos(\omega_0 t) \ </math>  
 
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| align="right" style="padding-right: 1em;" | CTFT of a sine  
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| align="right" style="padding-right: 1em;" | [[Explain_CTFT_sin|CTFT of a sine]]
 
| <math>sin(\omega_0 t)  \ </math>  
 
| <math>sin(\omega_0 t)  \ </math>  
 
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| align="right" style="padding-right: 1em;" | CTFT of a rect  
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| align="right" style="padding-right: 1em;" | [[Explain_CTFT_rect|CTFT of a rect]]
 
| <math>\left\{\begin{array}{ll}1, &  \text{ if }|t|<T,\\ 0, & \text{else.}\end{array} \right. \ </math>  
 
| <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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| align="right" style="padding-right: 1em;" | CTFT of a sinc  
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| align="right" style="padding-right: 1em;" | [[Explain_CTFT_sinc|CTFT of a sinc]]
 
| <math>\frac{2 \sin \left( W t  \right)}{\pi t }  \ </math>  
 
| <math>\frac{2 \sin \left( W t  \right)}{\pi t }  \ </math>  
 
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| align="right" style="padding-right: 1em;" | CTFT of a periodic function  
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| align="right" style="padding-right: 1em;" | [[Explain_CTFT_periofunc|CTFT of a periodic function]]
 
| <math>\sum^{\infty}_{k=-\infty} a_{k}e^{ikw_{0}t}</math>  
 
| <math>\sum^{\infty}_{k=-\infty} a_{k}e^{ikw_{0}t}</math>  
 
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| align="right" style="padding-right: 1em;" | CTFT of an impulse train  
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| align="right" style="padding-right: 1em;" | [[Explain_CTFT_impulsetrain|CTFT of an impulse train]]
 
| <math>\sum^{\infty}_{n=-\infty} \delta(t-nT)  \ </math>  
 
| <math>\sum^{\infty}_{n=-\infty} \delta(t-nT)  \ </math>  
 
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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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! style="background: none repeat scroll 0% 0% rgb(238, 238, 238);" colspan="4" | CT Fourier Transform Properties  
 
|-
 
|-
 
| align="right" style="padding-right: 1em;" |  
 
| align="right" style="padding-right: 1em;" |  
 
| <span class="texhtml">''x''(''t'')</span>  
 
| <span class="texhtml">''x''(''t'')</span>  
 
| <math>\longrightarrow</math>
 
| <math>\longrightarrow</math>
−
| <math> \mathcal{X}(f) </math>
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| <math> X(f) </math>
 
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|-
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| align="right" style="padding-right: 1em;" | multiplication property
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| align="right" style="padding-right: 1em;" | [[Explain_CTFT_multiprop|multiplication property]]
 
| <math>x(t)y(t) \ </math>  
 
| <math>x(t)y(t) \ </math>  
 
|  
 
|  
 
| <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;" | convolution property  
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| align="right" style="padding-right: 1em;" | [[Explain_CTFT_convprop|convolution property]]
 
| <math>x(t)*y(t) \!</math>  
 
| <math>x(t)*y(t) \!</math>  
 
|  
 
|  
 
| <math> X(f)Y(f) \!</math>
 
| <math> X(f)Y(f) \!</math>
 
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| align="right" style="padding-right: 1em;" | time reversal  
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| align="right" style="padding-right: 1em;" | [[Explain_CTFT_timerev|time reversal]]
 
| <math>\ x(-t) </math>  
 
| <math>\ x(-t) </math>  
 
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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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! 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  
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| align="right" style="padding-right: 1em;" | [[Explain_CTFT_Parseval|Parseval's relation]]
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| <math>\int_{-\infty}^{\infty} |x(t)|^2 dt = \int_{-\infty}^{\infty} |\mathcal{X}(f)|^2 df</math>
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| <math>\int_{-\infty}^{\infty} |x(t)|^2 dt = \int_{-\infty}^{\infty} |X(f)|^2 df</math>
 
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----
 
----
  
−
[[MegaCollectiveTableTrial1|Back to Collective Table]]  
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[[MegaCollectiveTableTrial1|Back to Collective Table]] | [[2010_Fall_ECE_438_Boutin|Back to 438 main page]]
  
 
[[Category:Formulas]]
 
[[Category:Formulas]]

Latest revision as of 12:26, 15 September 2010

  • I suggest putting a link next to each formula explaining how to obtain it from the formula in terms of $ \omega $. Also, I would not use the "mathcal" (curly) font for the transform variable, just a capital letter instead. --Mboutin 08:52, 3 September 2010 (UTC).
  • The explanation for each formula still needs to be added! In particular, some students said it was not clear how to get the convolution property in terms of f. So this needs to be explained clearly. --Mboutin 09:04, 7 September 2010 (UTC)
    • Provided explanation for each formula. -Zhao
      • Modified explanation for each formula. -Zhao 17:20, 15 September 2010 (UTC).
CT Fourier Transform Pairs and Properties (frequency f in hertz per time unit) (info)
(Click title to see explanation on how to obtain the formula in terms of f in hertz)
Definition 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 CT Fourier Transform $ \, x(t)=\mathcal{F}^{-1}(X(f))=\int_{-\infty}^{\infty}X(f)e^{i2\pi ft} df \, $
CT Fourier Transform Pairs
x(t) $ \longrightarrow $ $ X(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)\ $, where $ a\in {\mathbb R}, a>0 $ $ \frac{1}{a+i2\pi f} $
$ te^{-at}u(t)\ $, 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{2 \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 $
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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