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| <math>ax(t)+by(t)</math>  
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| <math>e^{j\omega_0 t}x(t)</math>
 
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| <math>x(t-t_0)</math>
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| align="right" style="padding-right: 1em;" | Time and Frequency Scaling
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| align="right" style="padding-right: 1em;" | Linearity
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| <span class="texhtml">''a''''x''(''t'') + ''b''''y''(''t'')</span>
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| <span class="texhtml">''a''''X''(ω) + ''b''''Y''(ω)</span>
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| <span class="texhtml">''x''(''t'' − ''t''<sub>0</sub>)</span>
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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>
 
| <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>  
 
<br>  

Revision as of 12:29, 10 April 2011

CT Fourier Transform Pairs and Properties (frequency ω in radians per time unit) (info)
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 DT 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

x(t) $ \longrightarrow $ $ \mathcal{X}(\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{2 \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
 		2πδ(ω)

13 CTFT of a Periodic Square Wave

$ \left\{\begin{array}{ll}1, & |t|<T_1,\\ 0, & T_1<|t|<=\frac{T}{2}\end{array} \right. $

and

x(t + T) = x(t)

$ \sum^{\infty}_{k=-\infty}\frac{2 \sin(k\omega_0T_1}{k}\delta(\omega-k\omega_0) $

14

CTFT of a Step Function u(t) $ \frac{1}{j\omega}+\pi\delta(\omega) $
CT Fourier Transform Properties
x(t) $ \longrightarrow $ $ \mathcal{X}(\omega) $
(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 $
convolution property $ x(t)*y(t) \! $ $ \mathcal{X}(\omega)\mathcal{Y}(\omega) \! $
time reversal $ \ x(-t) $ $ \ \mathcal{X}(-\omega) $
Frequency Shifting $ e^{j\omega_0 t}x(t) $ $ X(\omega-\omega_0) $
Conjugation $ x^{*}(t) $ $ X^{*}(-\omega) $
Time and Frequency Scaling $ x(at) $ $ \frac{1}{|a|}X(\frac{\omega}{a}) $
Linearity a'x(t) + b'y(t) a'X(ω) + b'Y(ω)
Time Shifting x(t − t0)

$ 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 $



Sources:


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Basic linear algebra uncovers and clarifies very important geometry and algebra.

Dr. Paul Garrett