The Zero-State Response of a Linear Time Invariant System Equals to the Convolution of the Input Signal and the Unit-pulse Response

Feb. 11, 2022

以离散系统为例。在离散的线性时不变(linear time invariant, LTI)系统中，输入为单位脉冲函数$\delta(k)$时产生单位脉冲响应$h(k)$，则有：

image-20220710133729178

根据卷积和的定义，对于任意输入信号$f(k)$：

\[\begin{aligned} 输入&=\sum^{\infty}_{i=-\infty}f(i)\delta(k-i)=f(k)*\delta(k)=f(k)\\响应&=\sum^{\infty}_{i=-\infty}f(i)h(k-i)=f(k)*h(k)\end{aligned}\]

即：

image-20220710133840501