LR Rabiner and RW Schafer, June 3

DRAFT: L. R. Rabiner and R. W. Schafer, June 3, 2009
8.5. HOMOMORPHIC FILTERING OF NATURAL SPEECH 471
cepstrum by l[n] corresponds to convolving its DTFT L(ejω ) with the complex
logarithm ˆ X(ejω ) as in
ˆY (e jω π
) = ˆX(e jθ )L(e j(ω−θ) )dθ. (8.95)
−π
This operation, which is simply linear filtering of the complex logarithm of the
DTFT, was also called “liftering” by Bogert et al. [1], and therefore l[n] is often
called a “lifter”. The resulting windowed complex cepstrum is processed by the
inverse characteristic system to recover the desired component.
This is illustrated by the thick lines in Figures. 8.29a and 8.29b which show
the log magnitude and phase obtained in the process of implementing the inverse
characteristic system (i.e., ˆ Y (ejω )) when l[n] is of the form
⎧
⎪⎨ 1, |n| < nco
llp[n] = 0.5 |n| = nco
(8.96)
⎪⎩
0, |n| > nco,
where, in general, nco is chosen to be less than the pitch period, Np, and in this
example, nco = 50 as shown in Figure 8.30a. 18
When using the DFT implementation, the lifter in Eq. (8.96) must conform
to the sample ordering of the DFT, i.e., the negative quefrencies fall in the
interval N/2 < n ≤ N −1 for an N-point DFT. Thus, for DFT implementations,
the lowpass lifter has the form,
⎧
1, 0 ≤ n < nco
⎪⎨ 0.5 n = nco
˜llp[n] = 0, nco < n < N − nco
(8.97)
0.5 n = N − nco
⎪⎩
1 N − nco < n ≤ N − 1.
For simplicity, we shall henceforth define lifters in DTFT form as in Eq. (8.96),
recognizing that the DFT form is always obtained by the process that yielded
Eq. (8.97).
The thick lines that are superimposed on the plots of log |X(e jω )| and
arg{X(e jω )} in Figures 8.29a and 8.29b show the real and imaginary parts
of ˆ Y (e jω ) corresponding to the liftered complex cepstrum ˆy[n] = llp[n]ˆx[n].
By comparing these plots to the corresponding plots with thin lines in Figures
8.29b and 8.29c respectively, it can be seen that ˆ Y (e jω ) is a lowpass filtered
(smoothed) version of ˆ X(e jω ). The result of the lowpass liftering is to remove
the effect of the excitation in the short-time Fourier transform. That is, retaining
only the low quefrency components of the complex cepstrum is a way of
estimating ˆ HV (e jω ) = log |HV (e jω )| + j arg{HV (e jω )}, the complex logarithm
of the frequency response of the vocal tract system. We see that the smoothed
log magnitude function in Figure 8.29a clearly displays formant resonances at
about 500, 1500, 2250, and 3100 Hz. Also note that if the lifter llp[n] is applied
18 A one-sample transition is included in Eq. (8.96). Expanding or omitting this transition
completely usually has little effect.