冲激不变法

冲激不变法是利用连续时间滤波器来设计离散时间无限冲激响应（IIR）滤波器的一种方法，这种方法中对连续时间系统的冲激响应进行采样以产生离散时间系统的冲激响应。离散时间系统的频率响应就会是连续时间系统的频率响应的位移后的拷贝之和；如果连续时间系统的频带大致限制在小于采样的奈奎斯特频率的范围内，则离散时间系统的频率响应会大致与连续系统的频带相同，低于奈奎斯特频率。

讨论

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

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

因此，该系统的频率响应为

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

如果连续时间滤波器是大致是带限的（即当 $|\Omega |\geq \pi /T$ 时，$H_{c}(j\Omega )<\delta$），则每次采样的频率低于 π（即奈奎斯特频率低于 1/(2T) Hz）的离散时间系统的频率响应就会大致为连续时间系统的频率响应：

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

与双线性变换比较

注意到会出现混叠，包含有奈奎斯特频率以下与超过该频率连续时间滤波器的非零的响应混叠。雙線性轉換使用不同的映射方法，将连续系统能够达到无限频率的频率响应，映射到在离散时间系统中至多能达到奈奎斯特频率的范围，避免了冲激不变法线性地对频率映射产生的循环混叠，从而可以替代冲激不变法。

系统函数中极点的效应

If the continuous poles at $s=s_{k}$, the system function can be written in partial fraction expansion as

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

Thus, using the inverse Laplace transform, the impulse response is

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

The corresponding discrete-time system's impulse response is then defined as the following

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

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

Performing a z-transform on the discrete-time impulse response produces the following discrete-time system function

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

Thus the poles from the continuous-time system function are translated to poles at z = e^s_kT. The zeros, if any, are not so simply mapped.

零极点

If the system function has zeros as well as poles, they can be mapped the same way, but the result is no longer an impulse invariance result: the discrete-time impulse response is not equal simply to samples of the continuous-time impulse response. This method is known as the matched Z-transform method, or pole–zero mapping. In the case of all-pole filters, the methods are equivalent.

稳定性和因果性

Since poles in the continuous-time system at s = s_k transform to poles in the discrete-time system at z = exp(s_kT), poles in the left half of the s-plane map to inside the unit circle in the z-plane; so if the continuous-time filter is causal and stable, then the discrete-time filter will be causal and stable as well.

修正公式

When a causal continuous-time impulse response has a discontinuity at $t=0$, the expressions above are not consistent. This is because $h_{c}(0)$ should really only contribute half its value to $h[0]$.

Making this correction gives

$h[n]=T\left(h_{c}(nT)-{\frac {1}{2}}h_{c}(0)\delta [n]\right)\,$

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

Performing a z-transform on the discrete-time impulse response produces the following discrete-time system function

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

参见

• 無限脈衝響應濾波器
• 雙線性轉換
• 匹配Z变换方法
• 连续时间滤波器：

  切比雪夫滤波器

  巴特沃斯滤波器

  椭圆函数滤波器

参考文献

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.

外部链接

• Impulse Invariant Transform at CircuitDesign.info Brief explanation, an example, and application to Continuous Time Sigma Delta ADC's.