In Section 12.3, we defined a sequence xˆ [n] referred to as the complex cepstrum of a s...

In Section 12.3, we defined a sequence xˆ [n] referred to as the complex cepstrum of a sequence x[n], and indicated that a causal complex cepstrum ˆx [n] is equivalent to the minimum-phase condition of Section 5.4 on x[n]. The sequence ˆx[n] is the inverse Fourier transform of Xˆ (ejω) as defined in Eq. (12.53). Note that because X(ejω) and Xˆ (ejω) are defined, the ROC of both X(z) and Xˆ (z) must include the unit circle.

(a) Justify the statement that the singularities (poles) of Xˆ (z) will occur wherever X(z) has either poles or zeros. Use this fact to prove that if ˆx[n] is causal, x[n] is minimum phase.

(b) Justify the statement that if x[n] is minimum phase the constraints of the ROC require ˆx [n] to be causal. We can examine this property for the case whenx[n] can be written as a superposition of complex exponentials. Specifically, consider a sequence x[n] whose z-transform is

Where A > 0 and ak, bk, ck and dk all have magnitude less than one.

(c) Write an expression for Xˆ (z) = logX(z).

(d) Solve for ˆx[n] by taking the inverse z-transform of your answer in part (c).

(e) Based on part (d) and the expression for X(z), argue that for sequences x[n] of this form, a causal complex cepstrum is equivalent to having minimum phase.