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 SPEECH457
transform of the logarithm of the magnitude of the discrete Fourier transform
of the finite-length input sequence; i.e.,
˜c[n] = 1
N−1
N
k=0
2π j
log |X[k]|e N kn , 0 ≤ n ≤ N − 1. (8.60)
As before, the cepstrum computed using the discrete Fourier transform is related
to the true cepstrum computed by Eq. (8.59) by the quefrency-aliasing formula
˜c[n] =
∞
r=−∞
c[n + rN] 0 ≤ n ≤ N − 1. (8.61)
Furthermore, just as c[n] is the even part of ˆx[n], ˜c[n] is the N-periodic even
part of ˜ˆx[n], i.e.,
˜c[n] = ˜ˆx[n] + ˜ˆx[N − n]
. (8.62)
2
Figure 8.23 shows how the computations leading to Eq. (8.61) are implemented
using the DFT and the inverse DFT. In this figure, the operation of computing
the cepstrum using the DFT as in Eq. (8.60) is denoted ˜ C{·}.
C { }
~
x[ n ] X [k]
log X [ k]
DFT
Log
Magnitude
IDFT
Figure 8.23: Computation of the cepstrum using the DFT.
c~
[ n]
The effect of quefrency aliasing can be significant for high-quefrency components
such as those corresponding to voiced excitation. A simple example will
illustrate this point. Consider a finite-length input x[n] = δ[n] + αδ[n − Np]. Its
discrete-time Fourier transform is
X(e jω ) = 1 + αe −jωNp . (8.63)
Using the power series expansion of Eq. (8.22), the complex logarithm can be
expressed as
ˆX(e jω ) = log{1 + αe −jωNp ∞
m+1 m
(−1) α
} =
e
m
−jωmNp , (8.64)
m=1
from which it follows that the complex cepstrum is
∞
m+1 m
(−1) α
ˆx[n] =
δ[n − mNp]. (8.65)
m
m=1
If, instead of this exact analysis, we compute the complex cepstrum as the
inverse N-point DFT of the complex logarithm of the N-point DFT of x[n] (i.e.,