Fourier Transform Table

Table: Properties of FT

Property	Aperiodic signal	Fourier transform
	$x(t)$	$X(j\omega)$
	$y(t)$	$Y(j\omega)$
Linearity	$ax(t) + by(t)$	$aX(j\omega) + bY(j\omega)$
Time Shifting	$x(t - t_0)$	$e^{-j\omega t_0} X(j\omega)$
Frequency Shifting	$e^{j\omega_0 t} x(t)$	$X(j(\omega - \omega_0))$
Conjugation	$x^*(t)$	$X^*(-j\omega)$
Time Reversal	$x(-t)$	$X(-j\omega)$
Time and Frequency Scaling	$x(at)$	$\frac{1}{ \lvert a \lvert } X\left(\frac{j\omega}{a}\right)$
Convolution	$x(t) * y(t)$	$X(j\omega)Y(j\omega)$
Multiplication	$x(t)y(t)$	$\frac{1}{2\pi} \int_{-\infty}^{\infty} X(j\theta)Y(j(\omega - \theta)) d\theta$
Differentiation in Time	$\frac{d}{dt} x(t)$	$j\omega X(j\omega)$
Integration	$\int_{-\infty}^{t} x(\tau) d\tau$	$\frac{1}{j\omega} X(j\omega) + \pi X(0) \delta(\omega)$
Differentiation in Frequency	$t x(t)$	$j \frac{d}{d\omega} X(j\omega)$
Conjugate Symmetry for Real Signals	$x(t)$ real	$X(j\omega) = X^*(-j\omega)$ $\mathfrak{Re}\{X(j\omega)\} = \mathfrak{Re}\{X(-j\omega)\}$ $\mathfrak{Im}\{X(j\omega)\} = -\mathfrak{Im}\{X(-j\omega)\}$ $\lvert X(j\omega) \rvert = \lvert X(-j\omega) \rvert$ $\angle X(j\omega) = -\angle X(-j\omega)$
Symmetry for Real and Even Signals	$x(t)$ real and even	$X(j\omega)$ real and even
Symmetry for Real and Odd Signals	$x(t)$ real and odd	$X(j\omega)$ purely imaginary and odd
Even-Odd Decomposition for Real Signals	$x_e(t) = \mathcal{Ev}\{x(t)\}$ $x_o(t) = \mathcal{Od}\{x(t)\}$	$\mathfrak{Re}\{X(j\omega)\}$ $j\mathfrak{Im}\{X(j\omega)\}$
Parseval’s Relation for Aperiodic Signals	$\int_{-\infty}^{\infty} \lvert x(t) \rvert ^2 dt = \frac{1}{2\pi} \int_{-\infty}^{\infty} \lvert X(j\omega) \rvert ^2 d\omega$

Table: Commonly Used FT

Signal	Fourier Transform	Fourier Series Coefficients (if periodic)
$\sum_{k=-\infty}^{+\infty} a_k e^{jk\omega_0 t}$	$2\pi \sum_{k=-\infty}^{+\infty} a_k \delta(\omega - k\omega_0)$	$a_k$
$e^{j\omega_0 t}$	$2\pi \delta(\omega - \omega_0)$	$a_1 = 1$, $a_k = 0$, otherwise
$\cos \omega_0 t$	$\pi[\delta(\omega - \omega_0) + \delta(\omega + \omega_0)]$	$a_1 = \frac{1}{2}$, $a_k = 0$, otherwise
$\sin \omega_0 t$	$\frac{\pi}{j}[\delta(\omega - \omega_0) - \delta(\omega + \omega_0)]$	$a_1 = -\frac{1}{2j}$, $a_k = 0$, otherwise
$x(t) = 1$	$2\pi \delta(\omega)$	$a_0 = 1$, $a_k = 0$, $k \neq 0$
Periodic square wave $x(t) = \begin{cases} 1, & \lvert t \rvert < T_1 \\ 0, & T_1 < \lvert t\lvert \leq \frac{\tau}{2} \end{cases}$ $x(t + T) = x(t)$	$\sum_{k=-\infty}^{+\infty} \frac{2\sin k\omega_0 T_1}{k} \delta(\omega - k\omega_0)$	$\frac{\omega_0 T_1}{\pi} \text{sinc}\left(\frac{k\omega_0 T_1}{\pi}\right) = \frac{\sin k\omega_0 T_1}{k\pi}$
$\sum_{n=-\infty}^{\infty} \delta(t - nT)$	$\frac{2\pi}{T} \sum_{k=-\infty}^{\infty} \delta(\omega - \frac{2\pi k}{T})$	$a_k = \frac{1}{T}$ for all $k$
$x(t) = \begin{cases} 1, & \lvert t \lvert < T_1 \\ 0, & \lvert t\lvert > T_1 \end{cases}$	$\frac{2\sin\omega T_1}{\omega}$	-
$\frac{\sin Wt}{\pi t}$	$X(j\omega) = \begin{cases} 1, & \lvert \omega \lvert < W \\ 0, & \lvert \omega \lvert > W \end{cases}$	-
$\delta(t)$	$1$	-
$u(t)$	$\frac{1}{j\omega} + \pi \delta(\omega)$	-
$\delta(t - t_0)$	$e^{-j\omega t_0}$	-
$e^{-a t}u(t), \Re(a) > 0$	$\frac{1}{a + j\omega}$	-
$t e^{-a t}u(t), \Re(a) > 0$	$\frac{1}{(a + j\omega)^2}$	-
$\frac{t^n}{n!}e^{-a t}u(t), \Re(a) > 0$	$\frac{1}{(a + j\omega)^{n+1}}$	-