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 $x(t+T)=x(t)=\left\{\begin{array}{ll}1, & |t|\leq T_1,\\ 0, & T_1<|t|\leq T/2 \end{array} \right.$ $\sum^{\infty}_{k=-\infty}\frac{2 \sin(k\frac{2\pi}{T}T_1)}{k}\delta(\omega-k\frac{2\pi}{T})$
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 $\int_{-\infty}^{\infty} |x(t)|^2 dt = \frac{1}{2\pi} \int_{-\infty}^{\infty} |\mathcal{X}(w)|^2 dw$