Time Domain Fourier Domain
$x(t)=\frac{1}{2 \pi} \int_{-\infty}^{\infty} X(j \omega)e^{j \omega t}d \omega$ $X(j \omega)=\int_{-\infty}^\infty x(t) e^{-j \omega t}d t$
$1\$ $2 \pi \delta (\omega)\$
$-0.5+u(t)\$ $\frac{1}{j \omega}\$
$\delta (t)\$ $1\$
$\delta (t-c)\$ $e^{-j \omega c}\$
$u(t)\$ $\pi \delta(\omega)+\frac{1}{j \omega}$
$e^{-bt}u(t)\$ $\frac{1}{j \omega + b}$
$cos \omega_0 t\$ $\pi [\delta ( \omega + \omega_0 ) + \delta ( \omega - \omega_0 ) ]\$
$cos ( \omega_0 t + \theta )\$ $\pi [ e^{-j \theta} \delta ( \omega + \omega_0 ) + e^{j \theta} \delta ( \omega - \omega_0 )]\$
$sin \omega_0 t\$ $j \pi [ \delta ( \omega + \omega_0 ) - \delta ( \omega - \omega_0 ) ]\$
$sin ( \omega_0 t + \theta )\$ $j \pi [ e^{-j \theta} \delta ( \omega + \omega_0 ) - e^{j \theta} \delta ( \omega - \omega_0 ) ]\$
$rect \left ( \frac{t}{\tau} \right )$ $\tau sinc \frac{\tau \omega}{2 \pi}$
$\tau sinc \frac{\tau t}{2 \pi}$ $2 \pi p_\tau\ ( \omega )$
$\left ( 1-\frac{2 |t|}{\tau} \right ) p_\tau (t)$ $\frac{\tau}{2} sinc^2 \frac{\tau \omega}{4 \pi}$
$\frac{\tau}{2} sinc^2 \left ( \frac{\tau t}{4 \pi} \right )$ $2 \pi \left ( 1-\frac{2|\omega|}{\tau} \right ) p_\tau (\omega)$

Notes:

$sinc(x) = \frac {sin(x)}{x}$
$p_\tau (t)\$ is the rectangular pulse function of width $\tau\$

Source courtesy Wikibooks.org

## Alumni Liaison

Ph.D. on Applied Mathematics in Aug 2007. Involved on applications of image super-resolution to electron microscopy

Francisco Blanco-Silva