Signals and Systems/Table of Fourier Transforms

Fourier Transform

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

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

Table of Fourier Transforms

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	$X(j\omega )=\int _{{-\infty }}^{\infty }x(t)e^{{-j\omega t}}dt$	$x(t)={\frac {1}{2\pi }}\int _{{-\infty }}^{{\infty }}X(j\omega )e^{{j\omega t}}d\omega$
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	$e^{{-bt}}u(t)\,(b>0)$	${\frac {1}{j\omega +b}}\,$
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	$\tau {\mbox{sinc}}\left({\frac {\tau t}{2\pi }}\right)\,$	$2\pi {\mbox{rect}}\left({\frac {\omega }{\tau }}\right)\,$
14	$\left(1-{\frac {2\|t\|}{\tau }}\right){\mbox{rect}}\left({\frac {t}{\tau }}\right)\,$	${\frac {\tau }{2}}{\mbox{sinc}}^{2}\left({\frac {\tau \omega }{4\pi }}\right)\,$
15	${\frac {\tau }{2}}{\mbox{sinc}}^{2}\left({\frac {\tau t}{4\pi }}\right)\,$	$2\pi \left(1-{\frac {2\|\omega \|}{\tau }}\right){\mbox{rect}}\left({\frac {\omega }{\tau }}\right)\,$
16	$e^{{-a\|t\|}},\Re \{a\}>0\,$	${\frac {2a}{a^{2}+\omega ^{2}}}\,$

Notes:	${\mbox{sinc}}(x)=\sin(x)/x$ ${\mbox{rect}}\left({\frac {t}{\tau }}\right)$ is the rectangular pulse function of width $\tau$ $u(t)$ is the Heaviside step function $\delta (t)$ is the Dirac delta function

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