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| <math>e^{-i2\pi ft_0}</math> | | <math>e^{-i2\pi ft_0}</math> | ||
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| − | | align="right" style="padding-right: 1em;" | CTFT of a complex exponential | + | | 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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| Line 53: | Line 53: | ||
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| − | | align="right" style="padding-right: 1em;" | CTFT of a cosine | + | | 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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Revision as of 16:20, 9 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).
- Fixed the X(f) notation -Sbiddand
- 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)
| CT Fourier Transform Pairs and Properties (frequency f in hertz per time unit) (info) | |
|---|---|
| Definition CT Fourier Transform and its Inverse (Click title to see explanation) | |
| 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 $ |
