信号与系统-2-卷积的运算性质

卷积的运算性质

• 卷积代数
• 卷积的微积分
• 与冲激函数或阶跃函数的卷积
• 其他结论

卷积代数

卷积代数的运算性质与代数运算类似

• 交换律：两函数在卷积积分中的次序可交换$$f_1(t)\ast f_2(t)=f_2(t)\ast f_1(t)$$
• 分配律：并联系统的冲激响应等于组成并联系统的各子系统冲激响应之和$$f_1(t)\ast [f_2(t)+f_3(t)]=f_1(t)\ast f_2(t)+f_1(t)\ast f_3(t)$$
• 结合律：串联系统的冲激响应等于组成串联系统的各子系统冲激响应的卷积$$[f_1(t)\ast f_2(t)]\ast f_3(t)=f_1(t)\ast [f_2(t)\ast f_3(t)]$$

卷积的微积分

卷积的微积分与代数运算大不相同

• 两个函数卷积后的导数等于其中一个函数的导数与另一函数的卷积$$\begin{array}{rl}\frac{d}{dt}[f_1(t)\ast f_2(t)]& =f_1(t)\ast \frac{d{f}_2(t)}{dt}\\ & =\frac{d{f}_1(t)}{dt}\ast f_2(t)\end{array}$$
• 两函数卷积后的积分等于其中一个函数之积分与另一函数之卷积$$\begin{array}{rl}{\int }_{-\infty }^t[f_1(\lambda )\ast f_2(\lambda )]d\lambda & =f_1(t)\ast {\int }_{-\infty }^t{f}_2(\lambda )d\lambda \\ & =f_2(t)\ast {\int }_{-\infty }^t{f}_1(\lambda )d\lambda \end{array}$$

与冲激函数或阶跃函数的卷积

• 函数$f(t)$与单位冲激函数$\delta (t)$卷积的结果仍然是函数$f(t)$本身$$f(t)\ast \delta (t)=f(t)$$ $proof:$ $$f(t)\ast \delta (t)={\int }_{-\infty }^{+\infty }\delta (\tau )\cdot f(t-\tau )d\tau =f(t)$$

• 函数$f(t)$与延迟的单位冲激函数$\delta (t-t_0)$卷积的结果，相当于把函数$f(t)$延时$t_0$ $$f(t)\ast \delta (t-t_0)=f(t_0)$$

其他结论

• $$f(t)\ast {\delta }^{\prime }(t)=f^{\prime }(t)$$
  $proof:$ $$\begin{array}{rl}f(t)\ast {\delta }^{\prime }(t)& ={\int }_{-\infty }^{+\infty }{\delta }^{\prime }(t)f(t-\tau )d\tau \\ & ={\int }_{-\infty }^{+\infty }f(t-\tau )d\delta (\tau )\\ & =0-{\int }_{-\infty }^{+\infty }\delta (\tau )d[f(t-\tau )]\\ & ={\int }_{-\infty }^{+\infty }\delta (\tau )f^{\prime }(t-\tau )d\tau \\ & =f^{\prime }(t)\end{array}$$
• $$f(t)\ast u(t)={\int }_{-\infty }^{t}f(\tau )d\tau$$
  $proof:$ $$\begin{array}{rl}f(t)\ast u(t)& ={\int }_{-\infty }^{+\infty }f(\tau )\cdot u(t-\tau )d\tau \\ & ={\int }_{-\infty }^{t}f(\tau )d\tau \end{array}$$