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 SPEECH455
discrete-time Fourier transform of the complex logarithm of the Fourier transform
of the input.
As we have already observed, there are questions of uniqueness of this set of
equations. In order to clearly define the complex cepstrum with Eqs. (8.56a)-
(8.56c), we must provide a unique definition of the complex logarithm of the
Fourier transform. To do this, it is helpful to impose the constraint that the
complex cepstrum of a real input sequence be also a real sequence. Recall that
for a real sequence the real part of the Fourier transform is an even function
and the imaginary part is odd. Therefore, if the complex cepstrum is to be a
real sequence, we must define the log magnitude function to be an even function
of ω and the phase must be defined to be an odd function of ω. As we have
already asserted, a further sufficient condition for the complex logarithm to
be unique is that the phase be computed so that it is a continuous periodic
function of ω with period of 2π [13, 20]. Algorithms for the computation of
an appropriate phase function typically start with the principal value phase
sampled at the DFT frequencies as a basis for search for discontinuities of size
2π. Due to the sampling, care must be taken to locate the frequencies at which
the discontinuities occur. A simple approach, that generally works well if the
phase is densely sampled, is to search for jumps (either positive or negative)
of size greater than some prescribed tolerance. 9 Once the frequencies where
the principal value “wraps around” are found, the appropriate multiples of 2π
radians can be added or subtracted to produce the “unwrapped phase” [20, 27].
Another method of computing the phase is discussed in Section 8.4.2.
Although Eqs. (8.56a)-(8.56c) can be useful for theoretical analysis,the are
not in a form that is useful for computation, since Eq. (8.56c) requires the evaluation
of an integral. However, we can approximate Eq. (8.56c) by using the
discrete Fourier transform. The discrete Fourier transform (DFT) of a finite
length sequence is identical to a sampled version of the discrete-time Fourier
transform (DTFT) of that same sequence [15]; i.e., X[k] = X(e j2πk/N ). Furthermore,
the discrete Fourier transform can be efficiently computed by a fast
Fourier transform algorithm [15]. Thus, the approach that is suggested for computing
the complex cepstrum is to replace all of the DTFT operations in Figure
8.4 by corresponding DFT operations. The resulting implementation of the
characteristic system is depicted in Figure 8.21 and defined by the equations
D ∗{
}
~
x[ n] X [k]
Complex X ˆ [ k]
DFT
Log
IDFT
~
xˆ
[ n]
Figure 8.21: Computation of the complex cepstrum using the DFT (implementation
of the approximate inverse system for convolution ˜ D∗{·}).
9 The unwrap( ) function in Matlab uses a default tolerance of π, which is reasonable,
since jumps of close to π radians can occur because of zeros that are very close to the unit
circle.