LR Rabiner and RW Schafer, June 3

DRAFT: L. R. Rabiner and R. W. Schafer, June 3, 2009
BIBLIOGRAPHY 495
xn[m] = s[m]w[n − m]
as the input to the homomorphic processing system.
(a) State the condition under which we can approximate xn[m] as
where
x[m] = pn[m] ∗ hv[m]
pn[m] = p[m]w[n − m]
(b) For the special case n = 0, find the z-transform of p0[m] in terms
of the z-transform of w[m].
(c) Express the complex cepstrum, ˆp0[m], in terms of ˆw[m].
8.11 The z-transform of a signal x[n] is defined as
N−1
X(z) = x[n]z −n
n=0
We evaluate X(z) at a set of points
zk = AW −k
k = 0, 1, . . . , M − 1
where A and W are arbitrary complex numbers. If we make the
simple substitution
nk = [n2 + k 2 − (k − n) 2 ]
2
then X(zk) can be written in the form
N−1
X(zk) = P [k] y[n]g[k − n]
n=0
i.e., X(zk) is a convolution of y[n] and g[n].
(a) Determine P [k], y[n] and g[n] in terms of x[n], A, and W .
(b) Sketch the points zk in the z-plane.
(c) Can you suggest how the FFT can be used to evaluate the above
expression for X(zk)?
8.12 (MATLAB Exercise) Write a MATLAB program to compute the
cepstrum of the signal:
in 3 ways, namely:
x1[n] = a n u[n] |a| < 1