Engineering Handbook/Mathematics/Fourier Transformation

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	$X(j\omega )=\int _{-\infty }^{\infty }x(t)e^{-j\omega t}dt$	$x(t)={\frac {1}{2\pi }}\int _{-\infty }^{\infty }X(\omega )e^{j\omega t}d\omega$
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:	${\mbox{sinc}}(x)=\sin(\pi x)/(\pi x)$ ${\mbox{rect}}\left({\frac {t}{\tau }}\right)$ is the rectangular pulse function of width $\tau$ $u(t)$ is the Heaviside step function $\delta (t)$ is the Dirac delta function