奇异函数 - 冲激信号

单位冲激函数δ的基本特性

$\int_{-\infty}^{\infty} x(t) \delta\left(t-t_{0}\right) \mathrm{d} t=\int_{-\infty}^{\infty} x\left(t+t_{0}\right) \delta(t) \mathrm{d} t=x\left(t_{0}\right)$
$\delta(a t)=\frac{1}{|a|} \delta(t)$
$x(t) \delta\left(t-t_{0}\right)=x\left(t_{0}\right) \delta\left(t-t_{0}\right)$
$x(t) * \delta\left(t-t_{0}\right)=x\left(t-t_{0}\right)$
$\delta\left(t-t_{1}\right) * \delta\left(t-t_{2}\right)=\delta\left(t-t_{1}-t_{2}\right)$
$\frac{\mathrm{d} u(t)}{\mathrm{d} t}=\delta(t)$