LR Rabiner and RW Schafer, June 3

DRAFT: L. R. Rabiner and R. W. Schafer, June 3, 2009
BIBLIOGRAPHY 493
Assume that all the poles of H(z) are inside the unit circle. Use.
Eq. (8.85) to obtain a recursion relation between the complex cepstrum,
ˆ h[n], and the coefficients {αk}. (Hint: How is the complex
cepstrum of 1/H(z) related to ˆ h[n]?)
8.5 Consider a finite length minimum phase sequence x[n] with complex
cepstrum ˆx[n], and a sequence
with complex cepstrum ˆy[n].
y[n] = α n x[n]
(a) If 0 < α < 1, how will ˆy[n] be related to ˆx[n]?
(b) How should α be chosen so that y[n] would no longer be minimum
phase?
(c) How should α be chosen so that y[n] is maximum phase?
8.6 Show that if x[n] is minimum phase, then x[−n] is maximum phase.
8.7 Consider a sequence, x[n], with complex cepstrum ˆx[n]. The ztransform
of ˆx[n] is
ˆX(z) = log[X(z)] =
∞
m=−∞
ˆx[m]z −m
where X(z) is the z-transform of x[n]. The z-transform ˆ X(z) is
sampled at N equally spaced points on the unit circle, to obtain
ˆXp[k] = ˆ 2π j
X(e N k ) 0 ≤ k ≤ N − 1
Using the inverse DFT, we compute
ˆxp[n] = 1
N−1
2π
ˆXp[k]e
j N
N
k=0
kn
0 ≤ n ≤ N − 1
which serves as an approximation to the complex cepstrum.
(a) Express ˆ Xp[k] in terms of the true complex cepstrum, ˆx[m].
(b) Substitute the expression obtained in (a) into the inverse DFT
expression for ˆxp[n] and show that
ˆxp[n] =
∞
ˆx[n + rN]
r=−∞