| 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)\ $ | 1 |
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| 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}) $ |
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| 12 | 1 |
2πδ(ω)
|
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| 13 | 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) $ | ||||
| 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 $ |
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