Fourier Transform

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

Inverse Fourier Transform

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

Table of Fourier Transforms

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

	Time Domain	Frequency Domain
	${\displaystyle x(t)={\mathcal {F}}^{-1}\left\{X(\omega )\right\}}$	${\displaystyle X(\omega )={\mathcal {F}}\left\{x(t)\right\}}$
1	${\displaystyle X(j\omega )=\int _{-\infty }^{\infty }x(t)e^{-j\omega t}dt}$	${\displaystyle x(t)={\frac {1}{2\pi }}\int _{-\infty }^{\infty }X(\omega )e^{j\omega t}d\omega }$
2	${\displaystyle 1\,}$	${\displaystyle 2\pi \delta (\omega )\,}$
3	${\displaystyle -0.5+u(t)\,}$	${\displaystyle {\frac {1}{j\omega }}\,}$
4	${\displaystyle \delta (t)\,}$	${\displaystyle 1\,}$
5	${\displaystyle \delta (t-c)\,}$	${\displaystyle e^{-j\omega c}\,}$
6	${\displaystyle u(t)\,}$	${\displaystyle \pi \delta (\omega )+{\frac {1}{j\omega }}\,}$
7	${\displaystyle e^{-bt}u(t)\,(b>0)}$	${\displaystyle {\frac {1}{j\omega +b}}\,}$
8	${\displaystyle \cos \omega _{0}t\,}$	${\displaystyle \pi \left[\delta (\omega +\omega _{0})+\delta (\omega -\omega _{0})\right]\,}$
9	${\displaystyle \cos(\omega _{0}t+\theta )\,}$	${\displaystyle \pi \left[e^{-j\theta }\delta (\omega +\omega _{0})+e^{j\theta }\delta (\omega -\omega _{0})\right]\,}$
10	${\displaystyle \sin \omega _{0}t\,}$	${\displaystyle j\pi \left[\delta (\omega +\omega _{0})-\delta (\omega -\omega _{0})\right]\,}$
11	${\displaystyle \sin(\omega _{0}t+\theta )\,}$	${\displaystyle j\pi \left[e^{-j\theta }\delta (\omega +\omega _{0})-e^{j\theta }\delta (\omega -\omega _{0})\right]\,}$
12	${\displaystyle {\mbox{rect}}\left({\frac {t}{\tau }}\right)\,}$	${\displaystyle \tau {\mbox{sinc}}\left({\frac {\tau \omega }{2\pi }}\right)\,}$
13	${\displaystyle \tau {\mbox{sinc}}\left({\frac {\tau t}{2\pi }}\right)\,}$	${\displaystyle 2\pi {\mbox{rect}}\left({\frac {\omega }{\tau }}\right)\,}$
14	${\displaystyle \left(1-{\frac {2|t|}{\tau }}\right){\mbox{rect}}\left({\frac {t}{\tau }}\right)\,}$	${\displaystyle {\frac {\tau }{2}}{\mbox{sinc}}^{2}\left({\frac {\tau \omega }{4\pi }}\right)\,}$
15	${\displaystyle {\frac {\tau }{2}}{\mbox{sinc}}^{2}\left({\frac {\tau t}{4\pi }}\right)\,}$	${\displaystyle 2\pi \left(1-{\frac {2|\omega |}{\tau }}\right){\mbox{rect}}\left({\frac {\omega }{\tau }}\right)\,}$
16	${\displaystyle e^{-a|t|},\Re \{a\}>0\,}$	${\displaystyle {\frac {2a}{a^{2}+\omega ^{2}}}\,}$
Notes:	${\displaystyle {\mbox{sinc}}(x)=\sin(\pi x)/(\pi x)}$ ${\displaystyle {\mbox{rect}}\left({\frac {t}{\tau }}\right)}$ is the rectangular pulse function of width ${\displaystyle \tau }$ ${\displaystyle u(t)}$ is the Heaviside step function ${\displaystyle \delta (t)}$ is the Dirac delta function
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