单位冲激偶信号 ${\delta }^{\prime }(t)$ 的基本性质

1. ${\delta }^{\prime }(t)$的面积为零：${\int }_{-\infty }^{\infty }{\delta }^{\prime }(t)dt=0$

2. 筛选特性：$x(t){\delta }^{\prime }(t-t_0)=x(t_0){\delta }^{\prime }(t-t_0)-x^{\prime }(t_0)\delta (t-t_0)$

   推导过程：
   $$\begin{array}{rl}[x(t)\delta (t-t_0){]}^{\prime }& =x(t_0){\delta }^{\prime }(t-t_0)\\ & =x^{\prime }(t)\delta (t-t_0)+x(t){\delta }^{\prime }(t-t_0)\\ & =x^{\prime }(t_0)\delta (t-t_0)+x(t){\delta }^{\prime }(t-t_0)\\ x(t){\delta }^{\prime }(t-t_0)& =x(t_0){\delta }^{\prime }(t-t_0)-x^{\prime }(t_0)\delta (t-t_0)\end{array}$$

3. 取样特性：${\int }_{-\infty }^{+\infty }x(t){\delta }^{\prime }(t-t_0)dt=-x^{\prime }(t_0)$

   注意积分区间是否包含冲激点。

4. 微分器：$x(t)\ast {\delta }^{\prime }(t)=x^{\prime }(t)$，$x(t)\ast {\delta }^{\prime }(t-t_0)=x^{\prime }(t-t_0)$

   推导过程：
   $$\begin{array}{rl}x(t)\ast {\delta }^{\prime }(t-t_0)& ={\int }_{-\infty }^{+\infty }x(k){\delta }^{\prime }\left((t-k)-t_0\right)dk\\ & ={\int }_{-\infty }^{+\infty }-x(k){\delta }^{\prime }\left(k-(t-t_0)\right)dk\\ & =-{\int }_{-\infty }^{+\infty }x(t-t_0){\delta }^{\prime }(k-(t-t_0))-x^{\prime }(t-t_0)\delta (k-(t-t_0))dk\\ & =x^{\prime }(t-t_0)\end{array}$$
   注意：卷积运算 $f(t)\ast g(t)={\int }_{-\infty }^{\infty }f(k)g(t-k)$，若 $g(t)=h(t-t_0)$，则$g(t-k)=h(t-k-t_0)$，所以 $f(t)\ast h(t-t_0)={\int }_{-\infty }^{\infty }f(k)h((t-k)-t_0)$。

5. 展缩特性：${\delta }^{\prime }(at+b)=\frac{1}{a|a|}{\delta }^{\prime }(t+\frac{b}{a})$

   推导过程可见：https://blog.csdn.net/weixin_44252933/article/details/123654783

6. 这是一个奇函数