Signals and Systems/Table of Fourier Transforms

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Signals and Systems

[edit] Fourier Transform

$F(j\omega) = \mathcal{F} \left\{f(t) \right\} = \int_{-\infty}^\infty f(t) e^{-j\omega t}dt$

[edit] Inverse Fourier Transform

$\mathcal{F}^{-1}\left\{F(j\omega) \right\} = f(t) = \frac{1}{2\pi}\int_{-\infty}^\infty F(j\omega) e^{j\omega t} d\omega$

[edit] Table of Fourier Tranforms

This table contains some of the most commonly encountered Fourier transforms.

	Time Domain	Frequency Domain
	$x(t) = \mathcal{F}^{-1}\left\{ X(\omega) \right\}$	$X(\omega) = \mathcal{F} \left\{ x(t) \right\}$
1	$x(t)=\frac{1}{2 \pi} \int_{-\infty}^{\infty} X(j \omega)e^{j \omega t}d \omega$	$X(j \omega)=\int_{-\infty}^\infty x(t) e^{-j \omega t}d t$
2	$1 \,$	$2\pi\delta(\omega) \,$
3	$-0.5 + u(t) \,$	$\frac{1}{j \omega} \,$
4	$\delta (t) \,$	$1 \,$
5	$\delta (t-c) \,$	$e^{-j \omega c} \,$
6	$u(t) \,$	$\pi \delta(\omega)+\frac{1}{j \omega} \,$
7	$e^{-bt}u(t) \,$	$\frac{1}{j \omega+b} \,$
8	$\cos \omega_0 t \,$	$\pi \left[ \delta(\omega+\omega_0)+\delta(\omega-\omega_0) \right] \,$
9	$\cos (\omega_0 t + \theta) \,$	$\pi \left[ e^{-j \theta}\delta(\omega+\omega_0)+e^{j \theta}\delta(\omega-\omega_0) \right] \,$
10	$\sin \omega_0 t \,$	$j \pi \left[ \delta(\omega +\omega_0)-\delta(\omega-\omega_0) \right] \,$
11	$\sin (\omega_0 t + \theta) \,$	$j \pi \left[ e^{-j \theta}\delta(\omega +\omega_0)-e^{j \theta}\delta(\omega-\omega_0) \right] \,$
12	$\mbox{rect} \left( \frac{t}{\tau} \right) \,$	$\tau \mbox{sinc} \left( \frac{\tau \omega}{2 \pi} \right) \,$
13	$\tau \mbox{sinc} \left( \frac{\tau t}{2 \pi} \right) \,$	$2 \pi p_\tau(\omega)\,$
14	$\left( 1-\frac{2 \|t\|}{\tau} \right) p_\tau (t) \,$	$\frac{\tau}{2} \mbox{sinc}^2 \left( \frac{\tau \omega}{4 \pi} \right) \,$
15	$\frac{\tau}{2} \mbox{sinc}^2 \left( \frac{\tau t}{4 \pi} \right) \,$	$2 \pi \left( 1-\frac{2\|\omega\|}{\tau} \right) p_\tau (\omega) \,$

Notes:	$sinc(x) = sin(x) / x$ $p τ (t)$ is the rectangular pulse function of width $τ$
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