任意激励下的时域分析(单位冲击函数和单位冲击响应)

单位脉冲函数(Unit Pulse Function)

使用单位阶跃函数定义单位脉冲函数：

$$\begin{array}{rl}p(t)& =\frac{1}{\mathrm{\Delta }}ϵ(t)-\frac{1}{\mathrm{\Delta }}ϵ(t-\mathrm{\Delta })\\ S& =\frac{1}{\mathrm{\Delta }}\ast \mathrm{\Delta }={\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}p(t)dt=1& \text{面积为1}\end{array}$$

单位冲击函数(Unit Impulse Function)

使用单位脉冲函数定义单位冲击函数：

$$\begin{array}{rl}\delta (t)& =\underset{\mathrm{\Delta }\to 0}{lim}p(t)\end{array}$$

函数特点：

$$\begin{array}{rl}\delta (t)& =\{\begin{array}{ll}0& t<0\\ \to +\mathrm{\infty }& t\to 0\\ 0& t>0\end{array}\\ S& ={\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}\delta (t)dt={\int }_{0^{-}}^{0^{+}}\delta (t)dt=1& \text{面积为1}\end{array}$$

单位冲击函数的延时（平移性质）

$$\begin{array}{r}f(t)=\delta (t-t_0)\end{array}$$

函数特点：

$$\begin{array}{rlr}f(t)& =0& t\ne t_0\\ S& ={\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}f(t)dt={\int }_{t_0^{-}}^{t_0^{+}}f(t)dt=1& \text{面积为1}\end{array}$$

单位冲击函数和其他函数的乘积

$$\begin{array}{r}f(t)\delta (t)=f(0)\delta (t)\end{array}$$

$$\begin{array}{rl}S& ={\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}f(t)\delta (t)dt\\ & ={\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}f(0)\delta (t)dt\\ & =f(0){\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}\delta (t)dt\\ & =f(0)\ast 1\\ & =f(0)\end{array}$$

延时单位冲击函数和其他函数的乘积

$$\begin{array}{rl}& f(t)\delta (t-t_0)\\ & =f(t_0)\delta (t-t_0)\end{array}$$

$$\begin{array}{rl}& {\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}f(t)\delta (t-t_0)dt\\ & ={\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}f(t_0)\delta (t-t_0)dt\\ & =f(t_0){\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}\delta (t-t_0)dt\\ & =f(t_0)\ast 1\\ & =f(t_0)\end{array}$$

单位阶跃函数和单位冲击函数的关系

$$\begin{array}{rl}{\int }_{-\mathrm{\infty }}^t\delta (t)dt& =\{\begin{array}{ll}{\int }_{-\mathrm{\infty }}^t\delta (t)dt=0& t<0^{-}\\ {\int }_{-\mathrm{\infty }}^{0^{+}}\delta (t)dt=1& t=0^{+}\\ {\int }_{-\mathrm{\infty }}^t\delta (t)dt=1& t>0^{+}\end{array}\\ & =ϵ(t)\\ \frac{d}{dt}ϵ(t)& =\delta (t)\end{array}$$

• 单位冲击函数$\delta (t)$的积分是单位阶跃函数$ϵ(t)$，
• 即: 单位阶跃函数$ϵ(t)$的导数是单位冲击函数$\delta (t)$