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 linear time-invariant discrete-time system with a rational system function H(z) is cascaded with a compensating system Hc( z) as shown in Figure PS.64-1.

 

 

                

 

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L                                                      I                   Figure PS.64-1

(a)    How should Hc( 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) = Hap(z)Hmin(z).)

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

(c)  Assume that

H(z) = (1 - 0.8ei03 ,,. z- 1)( 1 - 0.Be-1° 3,,. z-1)(1 - l.2ei0·7,,. z-1)(1 -1.Ze-1° 7,,. z-1).

Find Hmin(Z), Hap(Z), Hc( Z), and G(z) for this case, and construct the pole-zero plots for each system function.