Let x[n] be a sequence with z-transform X(z) and complex cepstrum xˆ [n]. The magnitude...

Let x[n] be a sequence with z-transform X(z) and complex cepstrum xˆ [n]. The magnitude squared function for X(z) is

Since V (e^jω) = |X(e^jω)|2 ≥ 0, the complex cepstrum corresponding to V (z) can be computed without phase unwrapping.

(a) Obtain a relationship between the complex cepstrum and the complex cepstrum

(b) Express the complex cepstrum in terms of the cepstrum cx [n].

(c) Determine the sequence such that

is the complex cepstrum of a minimum-phase sequence x min [n] for which

(d) Suppose that X(z) is as given by Eq. (13.32).Usethe result of part (c) and Eqs. (13.36a), (13.36b), and (13.36c) to find the complex cepstrum of the minimum-phase sequence, and work backward to find X min(z).

The technique employed in part (d) may be used in general to obtain a minimum-phase factorization of a magnitude-squared function.