LR Rabiner and RW Schafer, June 3

DRAFT: L. R. Rabiner and R. W. Schafer, June 3, 2009
8.4. COMPUTING THE SHORT-TIME CEPSTRUM AND COMPLEX CEPSTRUM OF SPEECH461
zeros without first computing the complex cepstrum. To see how this can be
done, we write the discrete-time Fourier transform in terms of the numerical
values of the zeros as
X(e jω ) = Ae −jωM0
Mi
(1 − ame −jω Mo
) (1 − bme jω ) (8.68)
m=1
Assuming A > 0, the phase of X(e jω ) is
arg[X(e jω )] = −jωM0 +
Mi
m=1
m=1
ARG{(1 − ame −jω )} +
Mo
m=1
ARG{(1 − bme jω )}
(8.69)
Our notation in Eq. (8.69) implies that the sum of the terms on the right is the
continuous phase function arg{X(e jω )} that we require in the computation of
the complex cepstrum even though the right hand side terms are principal value
phases. This is true because of the following relations for the principal value
phases of the individual polynomial factors:
−π/2 < ARG{(1 − ae −jω )}] < π/2 if |a| < 1 (8.70a)
−π/2 < ARG{(1 − be jω )} < π/2 if |b| < 1, (8.70b)
which hold for −π < ω ≤ π. Furthermore, it can be shown that
ARG{(1 − ae −jω )} = 0 for ω = 0 and π (8.71a)
ARG{(1 − be jω )} = 0 for ω = 0 and π. (8.71b)
Since |am| < 1 for m = 1, . . . , Mi and |bm| < 1 for m = 1, . . . , Mo in Eq. (8.69),
it follows that the principal values all of the terms in that equation can be
computed without discontinuities, and therefore the sum will have no discontinuities.
Also note that the term −jωMo can be included or not as desired.
Generally it would be omitted since its inverse DTFT can dominate the complex
cepstrum values.
To compute the sampled values of the continuous phase at frequencies 2πk/N,
we simply need to compute N-point DFT’s of the individual terms and then
compute and add the principal-value phases of those terms. Specifically, the
continuous phase of the DFT X[k] of the finite-length sequence x[n] is
arg{X[k]} =
+
Mo
m=1
Mi
m=1
ARG{(1 − ame −j2πk/N )}
ARG{(1 − bme −j2π(N−k)/N )} 0 ≤ k ≤ N − 1. (8.72)
The contributions due to the zeros that are inside the unit circle (am’s) can
be computed by using an N-point FFT algorithm to compute the DFTs of
the sequences δ[n] − amδ[n − 1]. Similarly the contributions due to the zeros
that are outside the unit circle are obtained by computing the DFTs of the
sequences δ[n] − bmδ[n − N + 1]. The computation can be reduced significantly