# 冲激不变法

## 讨论

${\displaystyle T}$ 为采样周期对连续时间系统的冲激响应 ${\displaystyle h_{c}(t)}$ 采样得到了离散时间系统的冲激响应 ${\displaystyle h[n]}$

${\displaystyle h[n]=Th_{c}(nT)\,}$

${\displaystyle H(e^{j\omega })=\sum _{k=-\infty }^{\infty }{H_{c}\left(j{\frac {\omega }{T}}+j{\frac {2{\pi }}{T}}k\right)}\,}$

${\displaystyle H(e^{j\omega })=H_{c}(j\omega /T)\,}$ for ${\displaystyle |\omega |\leq \pi \,}$

### 系统函数中极点的效应

${\displaystyle H_{c}(s)=\sum _{k=1}^{N}{\frac {A_{k}}{s-s_{k}}}\,}$

${\displaystyle h_{c}(t)={\begin{cases}\sum _{k=1}^{N}{A_{k}e^{s_{k}t}},&t\geq 0\\0,&{\mbox{otherwise}}\end{cases}}}$

${\displaystyle h[n]=Th_{c}(nT)\,}$
${\displaystyle h[n]=T\sum _{k=1}^{N}{A_{k}e^{s_{k}nT}u[n]}\,}$

${\displaystyle H(z)=T\sum _{k=1}^{N}{\frac {A_{k}}{1-e^{s_{k}T}z^{-1}}}\,}$

### 修正公式

${\displaystyle h[n]=T\left(h_{c}(nT)-{\frac {1}{2}}h_{c}(0)\delta [n]\right)\,}$
${\displaystyle h[n]=T\sum _{k=1}^{N}{A_{k}e^{s_{k}nT}}\left(u[n]-{\frac {1}{2}}\delta [n]\right)\,}$

${\displaystyle H(z)=T\sum _{k=1}^{N}{{\frac {A_{k}}{1-e^{s_{k}T}z^{-1}}}-{\frac {T}{2}}\sum _{k=1}^{N}A_{k}}.}$

## 参考文献

1. ^ Jackson, L.B. A correction to impulse invariance. IEEE Signal Processing Letters. 2000-10-01, 7 (10): 273–275. ISSN 1070-9908. doi:10.1109/97.870677.

### 其他来源

• Oppenheim, Alan V. and Schafer, Ronald W. with Buck, John R. Discrete-Time Signal Processing. Second Edition. Upper Saddle River, New Jersey: Prentice-Hall, 1999.
• Sahai, Anant. Course Lecture. Electrical Engineering 123: Digital Signal Processing. University of California, Berkeley. 5 April 2007.