Oneof the interesting and important properties of minimum-phase sequences is the minimum...

Oneof the interesting and important properties of minimum-phase sequences is the minimum energy delay property; i.e., of all the causal sequences having the same Fourier transform magnitude function |H(e^jω)|, the quantity

is maximum for all n ≥ 0 when h[n] is the minimum-phase sequence. This result is proved as follows: Let h_min[n] be a minimum-phase sequence with z-transform Hmin(z). Furthermore, let z_k be a zero of H_min(z) so that we can express H_min(z) as H_min(z) = Q(z)(1 – z_kz ⁻¹), |zk | < 1,

where Q(z) is again minimum phase. Now consider another sequence h[n] with z-transform H(z) such that

|H(e^jω)| = |Hmin(e^jω)|

and such that H(z) has a zero at z = 1/z∗k instead of at z_k.

(a) Express H(z) in terms of Q(z).

(b) Express h[n] and h_min[n] in terms of the minimum-phase sequence q[n] that has z transform Q(z).

(c) To compare the distribution of energy of the two sequences, show that

(d) Using the result of part (c), argue that