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+ | *<span style="color:green"> Fixed mistake of shifted unit impulse CTFT. Change X to <math>\mathcal{X}</math> of CTFT properties. </span> -[[User:zhao148|Zhao]] | ||
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| <math>\delta (t-t_0)\ </math> | | <math>\delta (t-t_0)\ </math> | ||
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− | | <math>e^{iwt_0}</math> | + | | <math>e^{-iwt_0}</math> |
|- | |- | ||
| align="right" style="padding-right: 1em;" | CTFT of a complex exponential | | align="right" style="padding-right: 1em;" | CTFT of a complex exponential | ||
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| <math>x(t)y(t) \ </math> | | <math>x(t)y(t) \ </math> | ||
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− | | <math>\frac{1}{2\pi} X(\omega)*Y(\omega) =\frac{1}{2\pi} \int_{-\infty}^{\infty} X(\theta)Y(\omega-\theta)d\theta</math> | + | | <math>\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</math> |
|- | |- | ||
| 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> X(\omega)Y(\omega) \!</math> | + | | <math> \mathcal{X}(\omega)\mathcal{Y}(\omega) \!</math> |
|- | |- | ||
| align="right" style="padding-right: 1em;" | time reversal | | align="right" style="padding-right: 1em;" | time reversal | ||
| <math>\ x(-t) </math> | | <math>\ x(-t) </math> | ||
| | | | ||
− | | <math>\ X(-\omega)</math> | + | | <math>\ \mathcal{X}(-\omega)</math> |
|} | |} | ||
Revision as of 20:43, 9 September 2010
- Fixed mistake of shifted unit impulse CTFT. Change X to $ \mathcal{X} $ of CTFT properties. -Zhao
CT Fourier Transform Pairs and Properties (frequency ω in radians per time unit) (info) | |
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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) $ | ||
CTFT of a unit impulse | $ \delta (t)\ $ | $ 1 \! \ $ | ||
CTFT of a shifted unit impulse | $ \delta (t-t_0)\ $ | $ e^{-iwt_0} $ | ||
CTFT of a complex exponential | $ e^{iw_0t} $ | $ 2\pi \delta (\omega - \omega_0) \ $ | ||
$ e^{-at}u(t)\ $, where $ a\in {\mathbb R}, a>0 $ | $ \frac{1}{a+i\omega} $ | |||
$ te^{-at}u(t)\ $, where $ a\in {\mathbb R}, a>0 $ | $ \left( \frac{1}{a+i\omega}\right)^2 $ | |||
CTFT of a cosine | $ \cos(\omega_0 t) \ $ | $ \pi \left[\delta (\omega - \omega_0) + \delta (\omega + \omega_0)\right] \ $ | ||
CTFT of a sine | $ sin(\omega_0 t) \ $ | $ \frac{\pi}{i} \left[\delta (\omega - \omega_0) - \delta (\omega + \omega_0)\right] $ | ||
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} \ $ | ||
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. \ $ | ||
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}) \ $ | ||
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}) \ $ |
CT Fourier Transform Properties | |||
---|---|---|---|
x(t) | $ \longrightarrow $ | $ \mathcal{X}(\omega) $ | |
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) $ |
Other CT Fourier Transform Properties | |
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Parseval's relation | $ \int_{-\infty}^{\infty} |x(t)|^2 dt = \frac{1}{2\pi} \int_{-\infty}^{\infty} |\mathcal{X}(w)|^2 dw $ |