| 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 |
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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 | 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) $ | ||||
| 15 | e − α | t | | $ \frac{2\alpha}{\alpha^{2}+\omega^{2}} $ | |||||
| 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(ω − ω0) | |||||
| Conjugation | x * (t) | X * ( − ω) | |||||
| Time and Frequency Scaling | x(a't) | $ \frac{1}{|a|}X(\frac{\omega}{a}) $ | |||||
| Multiplication | x(t)y(t) | $ \frac{1}{2\pi}X(\omega)*Y(\omega) $ | |||||
| |
Differentiation in Frequency | t'x(t) | $ j\frac{d}{d\omega}X(\omega) $ | ||||
| Symmetry | x(t) real and even | X(ω) real and even | |||||
| x(t) real and odd | X(ω) perely imaginary and odd | ||||||
| Duality | x(-t) | $ 2\pi X(\omega) $ | |||||
| Differentiation | $ \frac{d^{n}x(t)}{dt^{n}} $ | $ (j\omega)^{n}X(\omega) $ | |||||
| |
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| 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:
Class Text Book
http://www1.na.infn.it/~cavalier/Download/SICSI_LAES/Lucidi_DSP/FourierTransformPairs.pdf
