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 SPEECH463
right hand side of the equation, but in the case of minimum- and maximumphase
sequences, Eq. (8.76) can be specialized to a form that permits recursive
computation.
Specifically, consider a minimum-phase signal xmnp[n] and its corresponding
complex cepstrum ˆxmnp[n], which, by definition, have the properties xmnp[n] = 0
and ˆxmnp[n] = 0 for n < 0. If we impose these conditions, Eq. (8.76) becomes
nxmnp[n] =
n
kˆxmnp[k]xmnp[n − k]. (8.77)
k=0
If we separate the k = n term from the sum on the right we obtain
n−1
nxmnp[n] = nˆxmnp[n]xmnp[0] + kˆxmnp[k]xmnp[n − k]. (8.78)
Finally, dividing both sides by n and solving for ˆxmnp[n] in Eq. (8.78) we have
the result that we are seeking; i.e.,
ˆxmnp[n] = xmnp[n]
xmnp[0] −
n−1
k=0
k=0
k
ˆxmnp[k]
n
xmnp[n − k]
xmnp[0]
n > 0, (8.79)
where Eq. (8.79) holds only for n > since we cannot divide by 0. Of course, by
definition, both xmnp[n] and ˆxmnp[n] are zero for n < 0.
Observe that Eq. (8.79) can be used to compute ˆxmnp[n] for n > 0 if we
know ˆxmnp[0]. Since
∞
ˆXmnp(z) = ˆxmnp[n]z −n
∞
= log xmnp[n]z −n
, (8.80)
it follows that
lim
n→∞
n=0
n=0
ˆXmnp(z) = ˆxmnp[0] = log{xmnp[0]}. (8.81)
Therefore, putting Eq. (8.81) together with Eq. (8.79) we can finally write the
recusion
⎧
0
⎪⎨ log{xmnp[0]}
ˆxmnp[n] =
xmnp[n]
⎪⎩ xmnp[0]
n < 0
n = 0
−
n−1
k
ˆxmnp[k]
n
xmnp[n − k]
xmnp[0]
n > 0,
(8.82)
k=0
which is a recursive relationship that can be used to implement the characteristic
system for convolution D∗{·} if it is known that the input is a minimum-phase
signal. The inverse characteristic system can be implemented recursively by
simply rearranging Eq. (8.82) to obtain
⎧
0
⎪⎨ exp{ˆxmnp[0]}
xmnp[n] =
n−1
k
⎪⎩
ˆxmnp[n]xmnp[0] + ˆxmnp[k]xmnp[n − k]
n
n < 0
n = 0
n > 0.
(8.83)
k=0