信号与系统-从傅里叶级数到傅里叶变换

傅里叶级数复数表达式为

$$\{\begin{array}{c}x(t)={\sum }_{k=-\infty }^{+\infty }{a}_k{e}^{jk{\omega }_{0}t}\\ a_k=\frac{1}{T_0}{\int }_0^{T_0}x(t)e^{-jk{\omega }_{0}t}dt\end{array}$$

其中$x(t)$是以$T_0$为周期，${\omega }_0=\frac{2\pi }{T_0}$

定义：

$$X(j\omega )={\int }_{T_0}x(t)e^{-j\omega t}dt={\int }_{-\infty }^{\infty }x(t)e^{-j\omega t}dt$$

则有：

$$\begin{array}{rl}{a}_k& =\frac{1}{T_0}{\int }_{T_0}{e}^{-jk{\omega }_{0}t}dt\\ & =\frac{1}{T_0}X(jk{\omega }_0)\end{array}$$

$$\begin{array}{rl}x(t)& =\sum _{k=-\infty }^{+\infty }\frac{1}{T_0}X(jk{\omega }_0)e^{jk{\omega }_{0}t}\\ & =\frac{1}{{\omega }_0{T}_0}\sum _{k=-\infty }^{+\infty }X(jk{\omega }_0)e^{jk{\omega }_{0}t}{\omega }_0\end{array}$$

非周期$x(t)$可以推出$T_0=+\infty$进而可以推出${\omega }_0=\frac{2\pi }{{\omega }_0}$趋于0

$$x(t)=\frac{1}{2\pi }{\int }_{-\infty }^{+\infty }X(j\omega )e^{j\omega t}d\omega$$

非周期函数$x(t)$的傅里叶变换对

$$\{\begin{array}{c}X(j\omega )={\int }_{-\infty }^{+\infty }x(t)e^{-j\omega t}\\ x(t)=\frac{1}{2\pi }{\int }_{-\infty }^{+\infty }X(j\omega )e^{j\omega t}d\omega \end{array}$$

$X(j\omega )$和$x(t)$互为傅里叶变换对

参考文献

https://www.bilibili.com/video/BV1g94y1Q76G?p=20&vd_source=b3a15f5d8887594220b3104779be9fc3