CT Fourier Transform (frequency in radians per time unit) - Rhea

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	$\int_{-\infty}^{\infty} \|x(t)\|^2 dt = \frac{1}{2\pi} \int_{-\infty}^{\infty} \|\mathcal{X}(w)\|^2 dw$

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CT Fourier Transform (frequency in radians per time unit) - Rhea

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