| 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(ω0t + θ) π[e − jθδ(ω + ω0) + ejθδ(ω − ω0)]? | |
| sinω0t jπ[δ(ω + ω0) − δ(ω − ω0)]? | |
| sin(ω0t + θ) jπ[e − jθδ(ω + ω0) − ejθδ(ω − ω0)]? | |
| rect(\frac{t}{\tau}) \tau sinc \frac{\tau \omega}{2 \pi} | |
| \tau sinc \frac{\tau t}{2 \pi} 2πpτ(ω) |
Note: sinc(x) = sin(x) / x ; pτ(t) is the rectangular pulse function of width τ
Note: Source courtesy Wikibooks.org
