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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)]]
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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;" |  [[Explain_CTFT|CT Fourier Transform]]
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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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|-
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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}(X(f))=\int_{-\infty}^{\infty}X(f)e^{i2\pi ft} df \,</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 Pairs
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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> X(f) </math>
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| align="right" style="padding-right: 1em;" | [[Explain_unitimpulse|CTFT of a unit impulse]]
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| <math>\delta (t)\ </math>
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| <math> 1 \! \ </math>
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| align="right" style="padding-right: 1em;" | [[Explain_CTFT_shifted_unitimpulse|CTFT of a shifted unit impulse]]
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| <math>\delta (t-t_0)\ </math>
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| <math>e^{-i2\pi ft_0}</math>
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| align="right" style="padding-right: 1em;" | [[Explain_CTFT_cpxexp|CTFT of a complex exponential]]
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| <math>e^{iw_0t}</math>
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| <math> \delta (f - \frac{\omega_0}{2\pi}) \ </math>
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| align="right" style="padding-right: 1em;" |
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| <math>e^{-at}u(t)\ </math>, where <math>a\in {\mathbb R}, a>0 </math>
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| <math>\frac{1}{a+i2\pi f}</math>
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| align="right" style="padding-right: 1em;" |
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| <math>te^{-at}u(t)\ </math>, where <math>a\in {\mathbb R}, a>0 </math>
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| <math>\left( \frac{1}{a+i2\pi f}\right)^2</math>
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| align="right" style="padding-right: 1em;" | [[Explain_CTFT_cos|CTFT of a cosine]]
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| <math>\cos(\omega_0 t) \ </math>
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|
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| <math> \frac{1}{2} \left[\delta (f - \frac{\omega_0}{2\pi}) + \delta (f + \frac{\omega_0}{2\pi})\right] \ </math>
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|
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| align="right" style="padding-right: 1em;" | [[Explain_CTFT_sin|CTFT of a sine]]
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| <math>sin(\omega_0 t)  \ </math>
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|
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| <math>\frac{1}{2i} \left[\delta (f - \frac{\omega_0}{2\pi}) - \delta (f + \frac{\omega_0}{2\pi})\right]</math>
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| align="right" style="padding-right: 1em;" | [[Explain_CTFT_rect|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{\sin \left(2\pi Tf \right)}{\pi f}  \ </math>
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|-
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| align="right" style="padding-right: 1em;" | [[Explain_CTFT_sinc|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 }|f| <\frac{W}{2\pi},\\ 0, & \text{else.}\end{array} \right.  \ </math>
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|
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|-
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| align="right" style="padding-right: 1em;" | [[Explain_CTFT_periofunc|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>\sum^{\infty}_{k=-\infty}a_{k}\delta(f-\frac{kw_{0}}{2\pi}) \ </math>
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|-
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| align="right" style="padding-right: 1em;" | [[Explain_CTFT_impulsetrain|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{1}{T}\sum^{\infty}_{k=-\infty}\delta(f-\frac{k}{T}) \ </math>
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|
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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="4" | CT Fourier Transform Properties
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|-
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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> X(f) </math>
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|-
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| align="right" style="padding-right: 1em;" | [[Explain_CTFT_multiprop|multiplication property]]
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| <math>x(t)y(t) \ </math>
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|
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| <math> X(f)*Y(f) =\int_{-\infty}^{\infty} X(\theta)Y(f-\theta)d\theta</math>
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|-
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| align="right" style="padding-right: 1em;" | [[Explain_CTFT_convprop|convolution property]]
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| <math>x(t)*y(t) \!</math>
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|
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| <math> X(f)Y(f) \!</math>
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|-
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| align="right" style="padding-right: 1em;" | [[Explain_CTFT_timerev|time reversal]]
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| <math>\ x(-t) </math>
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|
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| <math>\ X(-f)</math>
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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" | Other CT Fourier Transform Properties
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|-
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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} |X(f)|^2 df</math>
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|}
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----
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[[MegaCollectiveTableTrial1|Back to Collective Table]] | [[2010_Fall_ECE_438_Boutin|Back to 438 main page]]
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[[Category:Formulas]]
  
  

Revision as of 03:39, 4 September 2013


HW1_Solution_ECE438F13

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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