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 SPEECH459
Therefore, in this example, only the two circled samples at n = Np = 75 and
n = N − Np = 256 − 75 = 181 are in what could be considered their “correct”
locations (assuming samples at 128 < n < 256 to be “negative quefrency” samples).
Again, by increasing N, we can mitigate the aliasing effects since the
cepstrum approaches zero for large n.
8.4.2 Computation Based on the z-Transform
An approach to computing the complex cepstrum of finite-length sequences that
does not require phase unwrapping is suggested by the example of Section 8.3.
In that example, the z-transforms of all the convolved components of the model
had closed-form expressions as rational functions. By factoring the numerator
and denominator polynomials, it was possible to compute the complex cepstrum
exactly. Short-time analysis of natural speech signals is based upon finite-length
(windowed) segments of the speech waveform, and if a sequence x[n] has finite
length, then its z-transform is a polynomial in z −1 of the form
X(z) =
M
x[n]z −n . (8.66a)
n=0
Such an M th -order polynomial in z −1 can be represented in terms of its roots
as
Mi
X(z) = x[0] (1 − amz −1 Mo
) (1 − b −1
m=1
m=1
m z −1 ), (8.66b)
where the quantities am are the (complex) zeros that lie inside the unit circle
(minimum-phase part) and the quantities b −1
m are the zeros that are outside the
unit circle (maximum-phase part); i.e., |am| < 1 and |bm| < 1. We assume that
no zeros lie precisely on the unit circle. 11 If we factor a term −b −1
m z −1 out of
each factor of the product at far right in Eq. (8.66b), then Eq. (8.66b) can be
expressed as
where
X(z) = Az −Mo
Mi
(1 − amz −1 Mo
) (1 − bmz). (8.66c)
m=1
A = x[0](−1) Mo
Mo
m=1
m=1
b −1
m . (8.66d)
This representation of a windowed frame of speech can be obtained by using a
polynomial rooting algorithm to find the zeros am and b −1
m that lie inside and
outside the unit circle, respectively for the polynomial whose coefficients are the
sequence x[n].
11 Perhaps not surprisingly, it is rare that a computed root of a polynomial is precisely on
the unit circle; however, as previously mentioned, most of the zeros lie close to the unit circle
for high-order polynomials.