2.3 Impulse-invariant design

Pre-classical , adhoc-but-easy method of converting an analog prototype filter to a digital IIRfilter. Does not preserve any optimality.

Impulse invariance means that digital filter impulse response exactly equals samples of the analog prototype impulseresponse: $$\forall n\colon h(n)=h_a(nT)$$ How is this done?

The impulse response of a causal, stable analog filter is simply a sum of decaying exponentials: $$H_a(s)=\frac{b_0+b_{1}s+b_{2}s^2+\mathrm{...}+b_{p}s^p}{1+a_{1}s+a_{2}s^2+\mathrm{...}+a_{p}s^p}=\frac{A_1}{s-s_1}+\frac{A_2}{s-s_2}+\mathrm{...}+\frac{A_p}{s-s_p}$$ which implies $$h_a(t)=(A_{1}e^{s_{1}t}+A_{2}e^{s_{2}t}+\mathrm{...}+A_{p}e^{s_{p}t})u(t)$$ For impulse invariance, we desire $$h(n)=h_a(nT)=(A_{1}e^{s_{1}nT}+A_{2}e^{s_{2}nT}+\mathrm{...}+A_{p}e^{s_{p}nT})u(n)$$ Since $$A_{k}e^{s_{k}Tn}u(n)\equiv \frac{A_{k}z}{z-e^{s_{k}T}}$$ where $\left|z\right|> \left|e^{s_{k}T}\right|$ , and $$H(z)=\sum_{k=1}^p A_k\frac{z}{z-e^{s_{k}T}}$$ where $\left|z\right|> \max\{k , \left|e^{s_{k}T}\right|\}$ .

This technique is used occasionally in digital simulations of analog filters.