常用傅里叶变换对

矩形脉冲函数：

$$ G_{\tau}(t) \stackrel{F S}{\longleftrightarrow} \tau S_{a}\left(\frac{\tau}{2} \omega\right) $$

脉冲宽度为T，以x=0为对称轴

单边指数函数:

$$ e^{-a t} u(t) \stackrel{F S}{\longleftrightarrow}\frac{1}{a+j \omega} $$

单位冲击函数:

$$ \delta(t) \stackrel{FS}{\longleftrightarrow}1 $$

单位阶跃函数:

$$ u(t) \stackrel{F S}{\longleftrightarrow} \frac{1}{j w}+\pi \delta(w) $$

常数函数:

$$ 1 \stackrel{FS}{\longleftrightarrow}2\pi\delta(\omega) $$

抽样函数：

$$ S_{a}\left(\omega_{c} t\right) \stackrel{F S}{\longleftrightarrow} \frac{\pi}{\omega_{c}} G_{2} \omega_{c}(\omega) $$

$Sa(x) = \frac{\mathrm{sin}(x)}{x}$

三角脉冲函数：

$$ \Lambda_{2 \tau}(t) \stackrel{F S}{\longleftrightarrow} \tau S_{a}^{2}\left(\frac{\tau}{s} \omega\right) $$

脉冲宽度为$2\tau$.

$\Lambda_{2\tau(t)}=\frac{1}{\tau}G_\tau(t) * G_{\tau}(t)$, 使用傅里叶变换的积分性质可以证明.

虚指数函数:

$$ \begin{aligned} e^{j w_0 t} \stackrel{F S}{\longleftrightarrow}& 2 \pi \delta\left(\omega-\omega_0\right) \\ e^{-j w_2 t} \stackrel{F S}{\longleftrightarrow}& 2 \pi \delta\left(\omega+\omega_0\right) \\ \end{aligned} $$

其它与指数有关的函数:

$$ t e^{-a t} u(t) \stackrel{F S}{\longleftrightarrow} \frac{1}{(a+j w)^{2}} $$

$$ \begin{aligned}e^{-a t} \sin \left(\omega_{0} t\right) u(t) \stackrel{F S}{\longleftrightarrow} \frac{\omega_{0}}{(a+j \omega)^{2}+\omega_{0}^{2}} \\ e^{-a t} \cos \left(\omega_{0} t\right) u(t) \stackrel{F S}{\longleftrightarrow} \frac{a+j \omega}{(a+j \omega)^{2}+\omega_{0}^{2}}\end{aligned} $$

三角函数:

$$ \begin{aligned} \cos \left(\omega_b t\right)&=\frac{1}{2}\left(e^{j \omega_0 t}+e^{-j \omega_0 t}\right) \\ & \stackrel{ F S}{\longleftrightarrow} \pi\left[\delta\left(\omega-\omega_0\right)+\delta\left(\omega+\omega_0\right)\right] \end{aligned} $$

$$ \begin{aligned} \sin \left(\omega_{0} t\right) & =\frac{1}{2 j}\left(e^{j \omega_{0} t}-e^{-j \omega_{0} t}\right) \\ \qquad & \stackrel{F S}{\longleftrightarrow}\frac{\pi}{j}\left[\delta\left(j \omega-j \omega_{0}\right)-\delta\left(j \omega +j \omega_{0}\right)\right]\end{aligned} $$

转化为指数函数再进行运算.

调频信号:

$$ \begin{aligned}f(t) \cos \left(\omega_{0} t\right)\stackrel{FS}{ \longleftrightarrow }\frac{1}{2}\left[F\left(j \omega-j \omega_{0}\right)+F\left(j \omega_{+j} \omega_{0}\right)\right] \end{aligned} $$

$$ \begin{aligned}f(t) \sin \left(\omega_{0} t\right) \stackrel{F S}{\longleftrightarrow} \frac{1}{2 j}\left[F\left(j \omega-j \omega_{0}\right)-F\left(j \omega+j \omega_{0}\right)\right] \end{aligned} $$