Consider the class of sequences that are real and stable and whose z-transforms are of t...

Consider the class of sequences that are real and stable and whose z-transforms are of the form

where |a_k|, |b_k |, |c_k |, |d_k | < 1. Let denote the complex cepstrum of x[n].

(a) Let y[n] = x[−n]. Determine in terms of .

(b) If x[n] is causal, is it also minimum phase? Explain.

(c) Suppose that x[n] is a finite-duration sequence such that

with |a_k| < 1 and |b_k | < 1. The function X(z) has zeros inside and outside the unit circle. Suppose that we wish to determine y[n] such that |Y(e^jω)| = |X(e^jω)| and Y(z) has no zeros outside the unit circle. One approach that achieves this objective is depicted in Figure P13.14. Determine the required sequence . A possible application of the system in Figure P13.14 is to stabilize an unstable system by applying the transformation of Figure P13.14 to the sequence of coefficients of the denominator of the system function.