LR Rabiner and RW Schafer, June 3

DRAFT: L. R. Rabiner and R. W. Schafer, June 3, 2009
484CHAPTER 8. THE CEPSTRUM AND HOMOMORPHIC SPEECH PROCESSING
where ⇐⇒ denotes the unique relationship between a sequence and its DTFT.
An interesting result can be obtained if we represent the complex cepstrum as
ˆh[n] = c[n] + d[n], (8.112)
where c[n] = Ev{ ˆ h[n]} is the even part and d[n] = Odd{ ˆ h[n]} is the odd part of
the complex cepstrum. Recalling that the DTFT of the complex cepstrum is,
by definition, ˆ H(e jω ) = log |H(e jω )| + j arg{H(e jω )}, it can be shown that the
following DTFT relations hold:
and
nc[n] ⇐⇒ j d log |H(ejω )|
, (8.113a)
dω
nd[n] ⇐⇒ − d arg{H(ejω )}
. (8.113b)
dω
The DTFT expression on the right in (8.113b) is the group delay function [15]
for H(ejω ); i.e.,
grd{H(e jω )} = − d arg{H(ejω )}
. (8.114)
dω
Now if h[n] is assumed to be obtained by all-pole modeling as discussed in
Section 8.6, the complex cepstrum satisfies ˆ h[n] = 0 for n < 0. This means that
ˆh[n] = 2c[n] = 2d[n] for n > 0. If we define l[n] = n, then the liftered cepstrum
distance
D =
∞
m=−∞
|l[m]c[m] − l[m]¯c[m]| =
is equivalent to either
D = 1
π
d log |H(e
2π
jω )|
dω
or
−π
∞
m=−∞
l[m]d[m] − l[m] ¯ d[m] (8.115a)
− d log | ¯ H(e jω )|
dω
dω. (8.115b)
D = 1
π
grd{H(e
2π −π
jω )} − grd{ ¯ H(e jω )} dω, (8.115c)
The result of (8.115b) was also given by Tohkura [26].
Instead of l[n] = n for all n, or the lifter of (8.110), Itakura proposed the
lifter
l[n] = n s e −n2 /2τ 2
. (8.116)
This lifter has great flexibility. For example, if s = 0 we have simply low
quefrency liftering of the cepstrum. If s = 1 and τ is large, we have essentially
l[n] = n for small n with high quefrency tapering. The effect of liftering with
Eq. (8.116) is illustrated in Figure 8.40, which shows in (a) the short-time Fourier
transform of a segment of voiced speech along with a linear predictive analysis
spectrum with p = 12. In (b) is shown the liftered group delay spectrum for
s = 1 and τ ranging from 5 to 30. Observe that as τ increases, the formant
frequencies are increasingly emphasized. If larger values of s are used, even
greater enhancement of the resonance structure is observed.