| $f(t)$ |
$F(\omega)$ |
| $\delta(t)$ |
1 |
| $1$ |
$2\pi\delta(\omega)$ |
| $\delta(t-t_0)$ |
$e^{\large{-j\omega t_0}}$ |
| $e^{\large{j2\pi f_0 t}}$ |
$2\pi\delta(\omega-\omega_0)$ |
| $\cos(2\pi f_0 t)$ |
$\pi [\delta(\omega-\omega_0) + \delta(\omega+\omega_0)]$ |
| $\sin(2\pi f_0 t)$ |
$-j\pi [\delta(\omega-\omega_0) - \delta(\omega+\omega_0)]$ |
| $e^{-a t}u(t), a > 0$ |
$\dfrac{1}{a + j\omega}$ |
| $te^{\large{-a t}}u(t), a > 0$ |
$\dfrac{1}{(a + j\omega)^2}$ |
| $e^{\large{-a |t|}}$ |
$\dfrac{2a}{a^2 + \omega^2}$ |
| $\dfrac{2a}{a^2 + t^2}$ |
$2 \pi e^{\large{-2 \pi a |f|}}$ |
| $e^{\large{-\pi t^2}}$ |
$e^{\large{-\pi f^2}}$ |
| $u(t)$ |
$\pi\delta(\omega) + \dfrac{1}{j\omega}$ |
| $\textrm{sgn}(t)$ |
$\dfrac{2}{j\omega}$ |
| $\dfrac{d}{dt} \delta(t)$ |
$j\omega$ |
| $\sum\limits_{n=-\infty}^\infty \delta(t-nT_0)$ |
$ \dfrac{1}{T_0} ~ \sum\limits_{n=-\infty}^\infty \delta \left(\omega-\dfrac{2\pi n}{T_0} \right)$ |