LR Rabiner and RW Schafer, June 3

DRAFT: L. R. Rabiner and R. W. Schafer, June 3, 2009
470CHAPTER 8. THE CEPSTRUM AND HOMOMORPHIC SPEECH PROCESSING
4
2
0
−2
(a) Complex Cepstrum
−4
−150 −100 −50 0 50 100 150
0.5
0
−0.5
(b) Cepstrum
−150 −100 −50 0
Quefrency (Samples)
50 100 150
Figure 8.30: Homomorphic analysis of voiced speech; (a) Complex cepstrum
˜ˆx[n]; (b) Cepstrum ˜c[n]. (For comparison to Figure 8.27, the samples of ˜ ˆx[n]
and ˜c[n] were reordered by placing the “negative quefrency” samples from the
interval N/2 < n ≤ N − 1 before the samples in the range 0 ≤ n < N/2.)
The plots in Figure 8.29a and 8.29b and the cepstrum plots in Figure 8.30
suggest how homomorphic filtering can be used to separate the excitation and
vocal tract components. First, note that the impulses in the complex cepstrum
due to the periodic excitation tend to be separated from the low quefrency components.
This suggests that the appropriate system for short-time homomorphic
filtering of speech is as depicted in Figure 8.31, which shows a segment of speech
s[ n]
X X
wn [ ]
D { }
-1{
}
∗
xn [ ] xn ˆ[ ]
ln [ ]
D
yn ˆ[ ] ∗ yn [ ]
Figure 8.31: Implementation of a system for short-time homomorphic filtering
of speech.
selected by the window, w[n], with the complex cepstrum computed as discussed
in Section 8.4. 17 The desired component of the input is selected by what might
be termed a “cepstrum window”, denoted l[n]. This type of filtering is appropriately
called “frequency-invariant linear filtering” since multiplying the complex
17 −1
For theoretical analysis, the operators D∗{·} and D∗ {·} would be represented in terms
of the DTFT, but in practice, we would use the operators ˜ D∗{·} and ˜ D −1
∗ {·} implemented
using the DFT with N large enough to avoid aliasing in the cepstrum.