Let x[n] be a causal stable sequence with z-transform X(z). The complex cepstrum x[n] is defined as

Question:

Let x[n] be a causal stable sequence with z-transform X(z). The complex cepstrum x[n] is defined as the inverse transform of the logarithm of X(z); i.e.,

where the ROC of X(z) includes the unit circle. (Strictly speaking, taking the logarithm of a complex number requires some careful considerations. Furthermore, the logarithm of a calid z-transform may not be a valid z-transform. For now, we assume that this operation is valid.) Determine the complex cepstrum for the sequence