An interesting use of exponential weighting is in computing the complex cepstrum without...

An interesting use of exponential weighting is in computing the complex cepstrum without phase unwrapping. Assume that X(z) has no poles and zeros on the unit circle. Then it is possible to find an exponential weighting factor α in the product w[n] = αⁿx[n], such that none of the poles or zeros of X(z) are shifted across the unit circle in forming

(a) Assuming that no poles or zeros of X(z) move across the unit circle, show that

(b) Now suppose that instead of the complex cepstrum, we compute c_x [n] and c_w[n]. Use the result of part (a) to obtain expressions for both c_x [n] and c_w[n] in terms of

(c) Now show that

Since c_x [n] and c_w[n] can be computed from log |X(e^jω)| and log |W(e^jω)|, respectively, Eq. (P13.32-2) is the basis for computing the complex cepstrum without computing the phase of X(e^jω). Discuss some potential problems that might arise with this approach.