LR Rabiner and RW Schafer, June 3

DRAFT: L. R. Rabiner and R. W. Schafer, June 3, 2009
492 BIBLIOGRAPHY
PROBLEMS
8.1 The complex cepstrum, ˆx[n], of a sequence x[n] is the inverse Fourier
transform of the complex log spectrum
ˆX(e jω ) = log |X(e jω )| + j arg[X(e jω )]
Show that the (real) cepstrum, c[n], defined as the inverse Fourier
transform of the log magnitude, is the even part of ˆx[n]; i.e., show
that
c[n] =
ˆx[n] + ˆx[−n]
2
8.2 A linear time-invariant system has the transfer function:
⎡
⎤
⎢ 1 − 4z−1
H(z) = 8 ⎣
1 − 1
6 z−1
⎥
⎦
(a) Find the complex cepstral coefficients, ˆ h[n], for all n.
(b) Plot ˆ h[n] versus n for the range −10 ≤ n ≤ 10.
(c) Find the (real) cepstrum coefficients, c[n], for all n.
8.3 Consider an all-pole model of the vocal tract transfer function of the
form
where
V (z) =
1
q
(1 − ckz −1 )(1 − c ∗ kz −1 )
k=1
ck = rke jθk .
Show that the corresponding cepstrum is
ˆv[n] = 2
q (rk) n
n cos(θkn)
k=1
8.4 Consider an all-pole model for the combined vocal tract, glottal
pulse, and radiation system of the form
H(z) =
1 −
G
p
αkz −k
k=1