傅里叶变换与卷积

傅里叶变换:

$$F(w)={\int }_{-\infty }^{+\infty }f(t)e^{-iwt}dt$$

傅里叶变换逆变换：

$$f(t)=\frac{1}{2\pi }{\int }_{-\infty }^{+\infty }F(w)e^{iwt}dw$$
忽略前面常数项：$\frac{1}{2\pi }$
$$F^{-1}(t)={\int }_{-\infty }^{+\infty }F(w)e^{iwt}dw$$

卷积

$$x(t)\ast g(t)={\int }_0^{\tau }x(\tau )g(t-\tau )d\tau$$

时域的卷积等于频域的乘积

证：$F(x(t)\ast g(t))=F(x(t))F(g(t))$

$$\begin{array}{rl}F(x(t)\ast g(t))& ={\int }_{-\infty }^{+\infty }(x(t)\ast g(t))e^{-iwt}dt\\ & ={\int }_{-\infty }^{+\infty }{\int }_0^{\tau }x(\tau )g(t-\tau )d\tau e^{-iwt}dt\\ & ={\int }_{-\infty }^{+\infty }{\int }_{-\infty +\tau }^{+\infty }x(\tau )g(t-\tau )e^{-iwt}d\tau dt\\ & 令:t-\tau =ut=u+\tau dt=du+d\tau =du\\ & t\in (-\infty +\tau ,\infty )\Rightarrow u=t-\tau \in (-\infty ,\infty )\\ & ={\int }_{-\infty }^{+\infty }{\int }_{-\infty }^{+\infty }x(\tau )g(u)e^{-iw(u+\tau )}d\tau du\\ & ={\int }_{-\infty }^{+\infty }x(\tau )e^{-iw\tau }d\tau {\int }_{-\infty }^{+\infty }g(u)e^{-iwu}du\\ & =F(x(t))F(g(t))\end{array}$$

进一步推导：
x ∗ g = F − 1 { F { x } F { g } } x*g=F^{-1}\lbrace F\lbrace x\rbrace F \lbrace g \rbrace \rbrace x∗g=F−1{F{x}F{g}}

参考：
https://zhuanlan.zhihu.com/p/54505069
https://blog.youkuaiyun.com/visionlabs/article/details/50485483