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 Tranforms

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