| 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
