| Line 17: | Line 17: | ||
! style="background: none repeat scroll 0% 0% rgb(238, 238, 238);" colspan="4" | CT Fourier Transform Pairs | ! style="background: none repeat scroll 0% 0% rgb(238, 238, 238);" colspan="4" | CT Fourier Transform Pairs | ||
|- | |- | ||
| − | | align="right" style="padding-right: 1em;" | <br> | + | | align="right" style="padding-right: 1em;" | <br> |
| align="right" style="padding-right: 1em;" | | | align="right" style="padding-right: 1em;" | | ||
| <span class="texhtml">''1x''(''t'')</span> | | <span class="texhtml">''1x''(''t'')</span> | ||
| Line 24: | Line 24: | ||
| | | | ||
|- | |- | ||
| − | | align="right" style="padding-right: 1em;" | 1 | + | | align="right" style="padding-right: 1em;" | 1 |
| 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> | ||
| | | | ||
| − | | <math> | + | | <font class="Apple-style-span" color="#FF0000" size="3">'''<math>e^{-iwt_0}</math><br>'''</font> |
| | | | ||
|- | |- | ||
| − | | align="right" style="padding-right: 1em;" | 2 | + | | align="right" style="padding-right: 1em;" | 2 |
| align="right" style="padding-right: 1em;" | CTFT of a shifted unit impulse | | align="right" style="padding-right: 1em;" | CTFT of a shifted unit impulse | ||
| <math>\delta (t-t_0)\ </math> | | <math>\delta (t-t_0)\ </math> | ||
| Line 38: | Line 38: | ||
| | | | ||
|- | |- | ||
| − | | align="right" style="padding-right: 1em;" | 3 | + | | align="right" style="padding-right: 1em;" | 3 |
| align="right" style="padding-right: 1em;" | CTFT of a complex exponential | | align="right" style="padding-right: 1em;" | CTFT of a complex exponential | ||
| <math>e^{iw_0t}</math> | | <math>e^{iw_0t}</math> | ||
| Line 46: | Line 46: | ||
| | | | ||
|- | |- | ||
| − | | align="right" style="padding-right: 1em;" | 4 | + | | align="right" style="padding-right: 1em;" | 4 |
| align="right" style="padding-right: 1em;" | | | align="right" style="padding-right: 1em;" | | ||
| <math>e^{-at}u(t),\ </math> <math>a\in {\mathbb R}, a>0 </math> | | <math>e^{-at}u(t),\ </math> <math>a\in {\mathbb R}, a>0 </math> | ||
| Line 54: | Line 54: | ||
| | | | ||
|- | |- | ||
| − | | align="right" style="padding-right: 1em;" | 5 | + | | align="right" style="padding-right: 1em;" | 5 |
| align="right" style="padding-right: 1em;" | | | align="right" style="padding-right: 1em;" | | ||
| <math>te^{-at}u(t),\ </math> <math>a\in {\mathbb R}, a>0 </math> | | <math>te^{-at}u(t),\ </math> <math>a\in {\mathbb R}, a>0 </math> | ||
| Line 62: | Line 62: | ||
| | | | ||
|- | |- | ||
| − | | align="right" style="padding-right: 1em;" | 6 | + | | align="right" style="padding-right: 1em;" | 6 |
| align="right" style="padding-right: 1em;" | CTFT of a cosine | | align="right" style="padding-right: 1em;" | CTFT of a cosine | ||
| <math>\cos(\omega_0 t) \ </math> | | <math>\cos(\omega_0 t) \ </math> | ||
| Line 70: | Line 70: | ||
| | | | ||
|- | |- | ||
| − | | align="right" style="padding-right: 1em;" | 7 | + | | align="right" style="padding-right: 1em;" | 7 |
| align="right" style="padding-right: 1em;" | CTFT of a sine | | align="right" style="padding-right: 1em;" | CTFT of a sine | ||
| <math>sin(\omega_0 t) \ </math> | | <math>sin(\omega_0 t) \ </math> | ||
| Line 78: | Line 78: | ||
| | | | ||
|- | |- | ||
| − | | align="right" style="padding-right: 1em;" | 8 | + | | align="right" style="padding-right: 1em;" | 8 |
| align="right" style="padding-right: 1em;" | CTFT of a rect | | align="right" style="padding-right: 1em;" | 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> | ||
| Line 86: | Line 86: | ||
| | | | ||
|- | |- | ||
| − | | align="right" style="padding-right: 1em;" | 9 | + | | align="right" style="padding-right: 1em;" | 9 |
| align="right" style="padding-right: 1em;" | CTFT of a sinc | | align="right" style="padding-right: 1em;" | 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> | ||
| Line 94: | Line 94: | ||
| | | | ||
|- | |- | ||
| − | | align="right" style="padding-right: 1em;" | 10 | + | | align="right" style="padding-right: 1em;" | 10 |
| align="right" style="padding-right: 1em;" | CTFT of a periodic function | | align="right" style="padding-right: 1em;" | 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> | ||
| Line 102: | Line 102: | ||
| | | | ||
|- | |- | ||
| − | | align="right" style="padding-right: 1em;" | 11 | + | | align="right" style="padding-right: 1em;" | 11 |
| align="right" style="padding-right: 1em;" | CTFT of an impulse train | | align="right" style="padding-right: 1em;" | CTFT of an impulse train | ||
| <math>\sum^{\infty}_{n=-\infty} \delta(t-nT) \ </math> | | <math>\sum^{\infty}_{n=-\infty} \delta(t-nT) \ </math> | ||
| Line 110: | Line 110: | ||
| | | | ||
|- | |- | ||
| − | | align="right" style="padding-right: 1em;" | 12 | + | | align="right" style="padding-right: 1em;" | 12 |
| align="right" style="padding-right: 1em;" | | | align="right" style="padding-right: 1em;" | | ||
| − | | | + | | '''Failed to parse (lexing error): 1 \! \''' |
| + | <br> | ||
| + | |||
| | | | ||
| | | | ||
| Line 149: | Line 151: | ||
|- | |- | ||
| align="right" style="padding-right: 1em;" | | | align="right" style="padding-right: 1em;" | | ||
| − | | align="right" style="padding-right: 1em;" | Lin | + | | align="right" style="padding-right: 1em;" | Lin |
| | | | ||
| | | | ||
| Line 163: | Line 165: | ||
|} | |} | ||
| + | <br> | ||
| + | Sources: | ||
| − | + | <br> | |
| − | + | ||
| − | + | ||
---- | ---- | ||
Revision as of 11:49, 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 | |||||||
|---|---|---|---|---|---|---|---|
| |
1x(t) | $ \longrightarrow $ | $ \mathcal{X}(\omega) $ | ||||
| 1 | CTFT of a unit impulse | $ \delta (t)\ $ | $ e^{-iwt_0} $ |
||||
| 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 | Failed to parse (lexing error): 1 \! \
|
||||||
| 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) $ | |||||
| Lin | |||||||
| 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:
