It is not possible to obtain a causal and stable inverse system (a perfect compensator)...

It is not possible to obtain a causal and stable inverse system (a perfect compensator) for a nonminimum-phase system. In this problem, we study an approach to compensating for only the magnitude of the frequency response of a nonminimum-phase system.

Suppose that a stable nonminimum-phase LTI discrete-time system with a rational system function H(z) is cascaded with a compensating system H_c(z) as shown in Figure P5.70.

(a) How should H_c(z) be chosen so that it is stable and causal and so that the magnitude of the overall effective frequency response is unity? (Recall that H(z) can always be represented as H(z) = H_ap(z)H_min(z).)

(b) What are the corresponding system functions Hc(z) and G(z)?

(c) Assume that

Determine H_min(z),Hap(z),H_c(z), and G(z) for this case, and construct the pole–zero plots for each system function.