LR Rabiner and RW Schafer, June 3

DRAFT: L. R. Rabiner and R. W. Schafer, June 3, 2009
8.3. HOMOMORPHIC ANALYSIS OF THE SPEECH MODEL 451
log e | S(e j2π FT ) |
arg [ S(e j2π FT ) ]
ARG [ S(e j2π FT ) ]
6
4
2
0
−2
(a) Log Magnitude Spectrum of Synthetic Voiced Speech
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0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000
10
5
0
(b) Continuous Phase of Synthetic Voiced Speech
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0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000
4
2
0
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(c) Principal Value Phase of Synthetic Voiced Speech
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frequency in Hz
Figure 8.19: Frequency-domain representation of the complex cepstrum. (a)
Log magnitude log |S(e j2πF T )| (real part of ˆ S(e j2πF T )) (b) Continuous phase
arg{S(e j2πF T )} (imaginary part of ˆ S(e j2πF T )), (c) Principal value phase
ARG{S(e j2πF T )}. The heavy lines in (a) and (b) represent ˆ HV (e j2πF T ) =
log |HV (e j2πF T )| + j arg{HV (e j2πF T )} .
since no z-transform representation exists directly for the random noise input
signal itself. However, if we employ the autocorrelation and power spectrum
representation for the model for unvoiced speech production, we can obtain
similar results to those for voiced speech.
Recall that for unvoiced speech, we have no glottal pulse excitation so the
model output is s[n] = hU [n] ∗ u[n] = v[n] ∗ r[n] ∗ (AU u[n]), where u[n] is a unitvariance
white noise sequence. The autocorrelation representation of unvoiced
speech is therefore
φss[n] = φvv[n] ∗ φrr[n] ∗ (A 2 U δ[n]) = A 2 U φvv[n] ∗ φrr[n], (8.47)
where φvv[n] and φrr[n] are the deterministic autocorrelation functions of the
vocal tract and radiation systems respectively. These are combined by convolution.
The z-transform of φss[n] exists and is given by
where
Φss(z) = A 2 U Φvv(z)Φrr(z), (8.48)
Φvv(z) = V (z)V (z −1 ) (8.49a)
Φrr(z) = R(z)R(z −1 ) (8.49b)