附录三 常用的傅里叶变换对

序号	连续时间信号	傅里叶变换
1	$\delta(t)$	$1$
2	$\delta(t-t_0)$	$e^{-j\omega t_0}$
3	$e^{-a t}u(t), \text{Re}(a)>0$	$\frac{1}{j\omega + a}$
4	$e^{-\vert a\vert t}, \text{Re}(a)>0$	$\frac{2a}{\omega^2 + a^2}$
5	$\delta^{(n)}(t)$	$(j\omega)^n$
6	$1$	$2\pi\delta(\omega)$
7	$e^{j\omega_0 t}$	$2\pi\delta(\omega-\omega_0)$
8	$g_{\tau}(t)=\begin{cases}1, & \vert t\vert <\tau/2 \\ 0, & \vert t\vert >\tau/2 \end{cases}$	$\tau \text{Sa}(\frac{\omega\tau}{2})$
9	$\cos(\omega_0 t)$	$\pi[\delta(\omega-\omega_0) + \delta(\omega+\omega_0)]$
10	$\sin(\omega_0 t)$	$\frac{\pi}{j}[\delta(\omega-\omega_0) - \delta(\omega+\omega_0)]$
11	$\frac{sin(Wt)}{\pi t}$	$g_{2W}(\omega) = \begin{cases} 1, & \vert \omega\vert <W \\ 0, & \vert \omega\vert >W \end{cases}$
12	$u(t)$	$\frac{1}{j\omega} + \pi\delta(\omega)$
13	$\text{sgn}(t)$	$\frac{2}{j\omega}$
14	$\frac{1}{\pi t}$	$-jsgn(\omega)$
15	$te^{-at}u(t), \quad \text{Re}\{a\}>0$	$\frac{1}{(a+j\omega)^2}$
16	$e^{at}u(-t), \quad \text{Re}\{a\}>0$	$\frac{1}{a-j\omega}$
17	周期信号$\sum_{k=-\infty}^{+\infty} a_k e^{jk\omega_0 t}$	$2\pi\sum_{k=-\infty}^{+\infty} a_k \delta(\omega-k\omega_0)$