Table of Fourier transforms [Jaeyoung Wiki]

This page is read only. You can view the source, but not change it. Ask your administrator if you think this is wrong.
====== Table of Fourier transforms ======

===== CTFT =====

$$ x(t) \leftrightarrow X(j\omega) $$

^ $x(t)$ (CT signal) ^ $X(j\omega)$ (CTFT) ^
| $\delta(t)$ | $1$ |
| $\delta(t - t_0)$ | $e^{-j\omega t_0}$ |
| $1$ | $2\pi \delta(\omega)$ |
| $e^{j\omega_0 t}$ | $2\pi \delta(\omega - \omega_0)$ |
| $e^{-at}u(t), \mathrm{Re}\{a\} > 0$ | $\frac{1}{\alpha + j\omega}$ |
| $u(t)$ | $\frac{1}{j\omega} + \pi \delta(\omega)$ |
| $\frac{\sin \omega_c t}{\pi t}$ | $\left\{ \begin{array}{ll} 1, & -\omega_c < \omega < \omega_c \\ 0, & \mathrm{otherwise} \end{array}\right.$ |
| $\left. \begin{array}{ll} 1, & -M \leq n \leq M \\ 0, & \mathrm{otherwise} \end{array} \right\}$ | $\frac{\sin \omega M}{\omega / 2}$ |

===== DTFT =====

$$ x[n] \leftrightarrow X(j\Omega) $$

^ $x[n]$ (DT signal) ^ $X(j\Omega), -\pi \lt \Omega \leq \pi $ (DTFT)^
| $\delta[n]$ | $1$ |
| $\delta[n - n_0]$ | $e^{-j\Omega n_0}$ |
| $1$ | $2\pi \delta(\Omega)$ |
| $e^{j\Omega_0 n} (-\pi \lt \Omega_0 \leq \pi)$ | $2\pi \delta(\Omega - \Omega_0)$ |
| $a^n u[n], |a| \lt 1$ | $\frac{1}{1 - ae^{-j\Omega}}$ |
| $u[n]$ | $\frac{1}{1-e^{-j\Omega}} + \pi\delta(\Omega)$ |
| $\frac{\sin \Omega_c n}{\pi n}$ | $\left\{ \begin{array}{ll} 1, & -\Omega_c \lt \Omega \lt \Omega_c \\ 0, & \mathrm{otherwise}\end{array}\right.$ |
| $\left. \begin{array}{ll} 1, & -M \leq n \leq M \\ 0, & \mathrm{otherwise} \end{array} \right\}$ | $\frac{\sin [\Omega(2M+1)/2]}{\sin(\Omega/2)}$ 

===== References =====

  * //Oppenheim, A. and Verghese, G., 2016. Signals, systems and inference. 1st ed.//