Table of Continuous-time (CT) Fourier Transform Pairs and Properties
as a function of $ \omega $ in radians per time unit
(used in ECE301)
| 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 CT 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 | ||||||
|---|---|---|---|---|---|---|
| |
signal (function of t) | $ \longrightarrow $ | Fourier transform (function of $ \omega $) | |||
| 1 | CTFT of a unit impulse | $ \delta (t)\ $ | $ 1 \ $ | |||
| 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{\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 | $ 1 \ $ | $ 2\pi \delta (\omega) \ $ | ||||
| 13 | CTFT of a Periodic Square Wave | $ x(t+T)=x(t)=\left\{\begin{array}{ll}1, & |t|<T_1,\\ 0, & T_1<|t|\leq T/2 \end{array} \right. $ | $ \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^{-\alpha |t|} \ $ | $ \frac{2\alpha}{\alpha^{2}+\omega^{2}} $ | ||||
| CT Fourier Transform Properties | |||||||
|---|---|---|---|---|---|---|---|
| $ x(t) \ $ | $ \longrightarrow $ | $ \mathcal{X}(\omega) $ | |||||
| 16 | (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 $ | ||||
| 17 | convolution property | $ x(t)*y(t) \ $ | $ \mathcal{X}(\omega)\mathcal{Y}(\omega) \! $ | ||||
| 18 | time reversal | $ \ x(-t) $ | $ \ \mathcal{X}(-\omega) $ | ||||
| 19 | Frequency Shifting | $ e^{j\omega_0 t}x(t) $ | $ \mathcal{X} (\omega - \omega_0) $ | ||||
| 20 | Conjugation | $ x^{*}(t) \ $ | $ \mathcal{X}^{*} (-\omega) $ | ||||
| 21 | Time and Frequency Scaling | $ x(at) \ $ | $ \frac{1}{|a|} \mathcal{X} (\frac{\omega}{a}) $ | ||||
| 23 | Differentiation in Frequency | $ tx(t) \ $ | $ j\frac{d}{d\omega} \mathcal{X} (\omega) $ | ||||
| 24 | Symmetry | $ x(t)\ \text{ real and even} $ | $ \mathcal{X} (\omega) \ \text{ real and even} $ | ||||
| 25 | $ x(t) \ \text{ real and odd} $ | $ \mathcal{X} (\omega) \ \text{ purely imaginary and odd} $ | |||||
| 26 | Duality | $ \mathcal{X} (-t) $ | $ 2 \pi x (\omega) \ $ | ||||
| 27 | Differentiation | $ \frac{d^{n}x(t)}{dt^{n}} $ | $ (j \omega)^{n} \mathcal{X} (\omega) $ | ||||
| 28 | Linearity | $ ax(t) + by(t) \ $ | $ a \mathcal{X}(\omega) + b \mathcal{Y} (\omega) $ | ||||
| 29 | Time Shifting | $ x(t-t_0) \ $ | $ 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 $ |
