# Signals and Systems/Table of Fourier Transforms

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