Signals and Systems â€“ Table of Fourier Transform Pairs

Fourier Transform

Fourier transform is a transformation technique that transforms signals from the continuous-time domain to the corresponding frequency domain and vice-versa.

The Fourier transform of a continuous-time function $x(t)$ is defined as,

$$\mathrm{X(\omega)=\int_{-\infty}^{\infty} x(t)e^{-j\omega t}dt… (1)}$$

Inverse Fourier Transform

The inverse Fourier transform of a continuous-time function is defined as,

$$\mathrm{x(t)=\frac{1}{2\pi}\int_{-\infty}^{\infty}X(\omega)\:e^{j\omega t}d\omega… (2)}$$

Equations (1) and (2) for $X(\omega)$ and $x(t)$ are known as Fourier transform pair and can be represented as −

$$\mathrm{X(\omega)=F[x(t)]}$$

And

$$\mathrm{x(t)=F^{-1}[X(\omega)]}$$

Table of Fourier Transform Pairs

Function,x(t)Fourier Transform, X(ω)
$\delta(t)$
1
$\delta(t-t_{0})$
$e^{-j \omega t_{0}}$
1$2\pi \delta(\omega)$
u(t)$\pi\delta(\omega)+\frac{1}{j\omega}$
$\sum_{n=−\infty}^{\infty}\delta(t-nT)$
$\omega_{0}\sum_{n=−\infty}^{\infty}\delta(\omega-n\omega_{0});\:\:\left(\omega_{0}=\frac{2\pi}{T} \right)$
sgn(t)
$\frac{2}{j\omega}$
$e^{j\omega_{0}t}$
$2\pi\delta(\omega-\omega_{0})$
$cos\:\omega_{0}t$
$\pi[\delta(\omega-\omega_{0})+\delta(\omega+\omega_{0})]$
$sin\:\omega_{0}t$
$-j\pi[\delta(\omega-\omega_{0})-\delta(\omega+\omega_{0})]$
$e^{-at}u(t);\:\:\:a >0$
$\frac{1}{a+j\omega}$
$t\:e^{at}u(t);\:\:\:a >0$
$\frac{1}{(a+j\omega)^{2}}$
$e^{-|at|};\:\:a >0$
$\frac{2a}{a^{2}+\omega^{2}}$
$e^{-|t|}$
$\frac{2}{1+\omega^{2}}$
$\frac{1}{\pi t}$
$-j\:sgn(\omega)$
$\frac{1}{a^{2}+t^{2}}$
$\frac{\pi}{a}e^{-a|\omega|}$
$\Pi (\frac{t}{τ})$
$τ\:sin c(\frac{\omega τ}{2})$
$\Delta(\frac{t}{τ})$
$\frac{τ}{2}sin C^{2}(\frac{\omega τ}{4})$
$\frac{sin\:at}{\pi t}$
$P_{a}(\omega)=\begin{cases}1 & for\:|\omega|\:< a\0 & for\:|\omega|\: > a \end{cases}$
$cos\:\omega_{0}t\:u(t)$
$\frac{\pi}{2}[\delta(\omega-\omega_{0})+\delta(\omega+\omega_{0})]+\left [ \frac{j\omega}{(j\omega)^{2}+\omega_{0}^{2}} \right ]$
$sin\:\omega_{0}t\:u(t)$
$-j\frac{\pi}{2}[\delta(\omega-\omega_{0})-\delta(\omega+\omega_{0})]+\left [\frac{\omega_{0}}{(j\omega)^{2}+\omega_{0}^{2}} \right ]$