Table of Fourier transform
| S.No | F(t) | F(ω) |
|---|---|---|
| 1 | \( e^{-at}u(t) \) | \( \frac {1}{a+j\omega}, \ a\gt 0 \) |
| 2 | \( e^{at}u(-t) \) | \( \frac {1}{a-j\omega}, \ a\gt 0 \) |
| 3 | \( e^{-a|t|} \) | \( \frac {2a}{a^2+\omega^2}, \ a\gt 0 \) |
| 4 | \( te^{-at}u(t) \) | \( \frac {1}{(a+j\omega)^2}, \ a\gt 0 \) |
| 5 | \( t^ne^{-at}u(t) \) | \( \frac {n!}{(a+j\omega)^{n+1}}, \ a\gt 0 \) |
| 6 | \( \delta (t) \) | \( 1 \) |
| 7 | \( 1 \) | \( 2\pi \delta(\omega) \) |
| 8 | \( e^{j\omega_0t} \) | \( 2\pi \delta(\omega-\omega_0) \) |
| 9 | \( \cos \omega_0t \) | \( \pi[ \delta(\omega-\omega_0) + \delta(\omega+\omega_0)] \) |
| 10 | \( \sin \omega_0t \) | \( j\pi[ \delta(\omega+\omega_0) - \delta(\omega-\omega_0)] \) |
| 11 | \( u(t) \) | \( \pi \delta(\omega) + \frac {1}{j\omega} \) |
| 12 | \( \text{sgn} \ t \) | \( \frac {2}{j\omega} \) |
| 13 | \( \cos \omega_0t \ u(t)\) | \( \frac {\pi}{2}[ \delta(\omega-\omega_0) + \delta(\omega+\omega_0)] + \frac {j\omega}{\omega_0^2-\omega^2} \) |
| 14 | \( \sin \omega_0t \ u(t)\) | \( \frac {\pi}{2j}[ \delta(\omega-\omega_0) - \delta(\omega+\omega_0)] + \frac {\omega_0}{\omega_0^2-\omega^2} \) |
| 15 | \( e^{-at} \sin \omega_0t \ u(t) \) | \( \frac {\omega_0}{(a+j\omega)^2+\omega_0^2} \) |
| 16 | \( e^{-at} \cos \omega_0t \ u(t) \) | \( \frac {a+j\omega}{(a+j\omega)^2+\omega_0^2} \) |
| 17 | \( \text{rect} \ (\frac {t}{\tau}) \) | \( \tau \ \text{sinc} (\frac {\omega\tau}{2})\) |
| 18 | \( \frac {W}{\pi} \text{sinc} (Wt) \) | \( \text{rect} \ (\frac {\omega}{2W}) \) |
| 19 | \( \Delta (\frac {t}{\tau}) \) | \( \frac {\tau}{2} \ \text{sinc}^2 (\frac {\omega\tau}{4}) \) |
| 20 | \( \frac {W}{2\pi} \text{sinc}^2 (\frac {Wt}{2}) \) | \( \Delta (\frac {\omega}{2W}) \) |
| 21 | \( \sum_{n=-\infty}^{\infty} \delta(t-nT) \) | \( \omega_0 \sum_{n=-\infty}^{\infty} \delta(\omega-n\omega_0) \) |
| 22 | \( e^{-t^2/2\sigma^2} \) | \( \sigma \sqrt {2\pi} e^{\sigma^2\omega^2/2} \) |
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