Signals and Systems/Table of Fourier Transforms

Fourier Transform

$F(j\omega )=ℱ\{f(t)\}=\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}f(t)e^{-j\omega t}dt$

Inverse Fourier Transform

${ℱ}^{-1}\{F(j\omega )\}=f(t)=\frac{1}{2\pi }\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}F(j\omega )e^{j\omega t}d\omega$

Table of Fourier Tranforms

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

Time Domain	Fourier Domain
$x(t)=\frac{1}{2\pi }\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}X(j\omega )e^{j\omega t}d\omega$	$X(j\omega )=\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}x(t)e^{-j\omega t}dt$
$1$	$2\pi \delta (\omega )$
$-0.5+u(t)$	$\frac{1}{j\omega }$
$\delta (t)$	$1$
$\delta (t-c)$	$e^{-j\omega c}$
$u(t)$	$\pi \delta (\omega )+\frac{1}{j\omega }$
$e^{-bt}u(t)$	$\frac{1}{j\omega +b}$
$\cos {\omega }_{0}t$	$\pi [\delta (\omega +{\omega }_0)+\delta (\omega -{\omega }_0)]$
$\cos ({\omega }_{0}t+\theta )$	$\pi [e^{-j\theta }\delta (\omega +{\omega }_0)+e^{j\theta }\delta (\omega -{\omega }_0)]$
$\sin {\omega }_{0}t$	$j\pi [\delta (\omega +{\omega }_0)-\delta (\omega -{\omega }_0)]$
$\sin ({\omega }_{0}t+\theta )$	$j\pi [e^{-j\theta }\delta (\omega +{\omega }_0)-e^{j\theta }\delta (\omega -{\omega }_0)]$
$rect(\frac{t}{\tau })$	$\tau sinc\frac{\tau \omega }{2\pi }$
$\tau sinc\frac{\tau t}{2\pi }$	$2\pi p_{\tau }(\omega )$
$(1-\frac{2|t|}{\tau })p_{\tau }(t)$	$\frac{\tau }{2}sin{c}^2\frac{\tau \omega }{4\pi }$
$\frac{\tau }{2}sin{c}^2(\frac{\tau t}{4\pi })$	$2\pi (1-\frac{2|\omega |}{\tau })p_{\tau }(\omega )$

Note: $sinc(x)=\sin (x)/x$ ; $p_{\tau }(t)$ is the rectangular pulse function of width $\tau$