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 473
Log Magnitude
Phase (Radians)
1
0
−1
−2
−3
2
0
−2
(a) Log Magnitude of Excitation Component
0 500 1000 1500 2000 2500 3000 3500 4000
(b) Phase of Excitation Component
0 500 1000 1500 2000 2500 3000 3500 4000
Frequency (Hz)
Figure 8.33: Homomorphic filtering of voiced speech; (a) and (b) estimate of
log magnitude and phase of Ew(e jω ).
ˆy[n] = lhp[n]ˆx[n], which is shown in Figure 8.32b approximates an impulse train
with spacing equal to the pitch period and amplitudes retaining the shape of
the Hamming window used to weight the input signal. Thus, with the highpass
lifter, y[n] serves as an estimate of ew[n].
If the same value of nco is used for both the lowpass and highpass lifters,
then llp[n] + lhp[n] = 1 for all n. Thus, the choice of the lowpass and highpass
lifters defines ew[n] and hV [n] so that hV [n] ∗ ew[n] = x[n]; i.e., convolution of
the waveforms in Figures 8.32a and 8.32b will result in the original windowed
speech signal shown in Figure 8.32c. In terms of the corresponding discrete-time
Fourier transform, adding the curves in Figures 8.33a and 8.33b to the smooth
curves plotted with thick lines in Figures 8.29a and 8.29b respectively results in
the rapidly varying curves in Figures 8.29a and 8.29b.
8.5.4 Minimum-Phase Analysis
Since the cepstrum is the inverse Fourier transform of the logarithm of the
magnitude of the Fourier transform of the windowed speech segment, it is also
the even part of the complex cepstrum. If the input signal is known to have the
minimum-phase property, we also know that the complex cepstrum is zero for
n < 0 and therefore, it can be obtained from the cepstrum by the operation in
Eq. (8.35), which can be seen to be equivalent to multiplying the cepstrum by
a lifter; i.e., ˆxmnp[n] = lmnp[n]c[n] where
⎧
⎪⎨ 0 n < 0
lmnp[n] = 1
⎪⎩
2
n = 0
0 < n.
(8.99a)