LR Rabiner and RW Schafer, June 3

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DRAFT: L. R. Rabiner and R. W. Schafer, June 3, 2009 444CHAPTER 8. THE CEPSTRUM AND HOMOMORPHIC SPEECH PROCESSING speech” even though it is only partially determined by the vocal tract configuration. Similarly, for unvoiced sections of speech, we can represent the pressure signal at the lips as s[n] = u[n] ∗ hU [n], (8.37c) where u[n] is the (unit-variance) random unvoiced excitation signal and hU [n] is the convolution of the vocal tract response, v[n], with the radiation load response, r[n], including scaling by the unvoiced section gain control AU , i.e., hU [n] = AU · v[n] ∗ r[n]. (8.37d) Again, it it is generally convenient to refer to hU [n] as “the vocal tract impulse response for unvoiced speech” even though it is only partially determined by the vocal tract configuration. 8.3.1 Homomorphic Analysis of the Model for Voiced Speech To illustrate how homomorphic systems and cepstrum analysis can be applied to speech, we apply the results of Section 8.2 to a discrete-time model for a sustained /AE/ vowel with fundamental frequency f0 = 125 Hz. As a model for the glottal pulse, we use the model proposed by Rosenberg [19], ⎧ ⎨ 0.5[1 − cos(π(n + 1)/N1)] 0 ≤ n ≤ N1 − 1 g[n] = cos(0.5π(n + 1 − N1)/N2) N1 ≤ n ≤ N1 + N2 − 2 (8.38) ⎩ 0 otherwise. This glottal pulse model is illustrated in Figure 8.13(a) where g[n] in Eq. (8.38) is plotted for N1 = 25 and N2 = 10. This finite-length sequence has length 34 samples, so its z-transform G(z) is a polynomial of the form G(z) = z −33 33 (−b −1 k ) 33 (1 − bkz) (8.39) k=1 with 33 zeros, which are plotted in Figure 8.13(a). Note that all 33 roots of G(z) are outside the unit circle for this example; i.e., this glottal pulse is a maximum-phase sequence. The vocal tract system is specified in terms of its formant frequencies and bandwidths in V (z) given as a product of second-order sections of the form V (z) = k=1 1 5 (1 − 2e −2πσkT cos(2πFkT )z −1 + e −4πσkT . (8.40) −2 z ) k=1 The model for an /AE/ vowel is given in Table 8.2 in terms of the set of analog center frequencies and bandwidths of the first 5 formants. The sampling rate in Eq. (8.40) is assumed to be Fs = 1/T = 10000 Hz thus providing adequate
DRAFT: L. R. Rabiner and R. W. Schafer, June 3, 2009 8.3. HOMOMORPHIC ANALYSIS OF THE SPEECH MODEL 445 g [ n ] r [ n ] 1.2 1 0.8 0.6 0.4 0.2 (a) Glottal Pulse 0 0 1 2 time nT in ms 3 4 1 0.5 0 −0.5 −1 (c) Radiation Load 0 1 2 time nT in ms 3 4 v [ n ] p [ n ] 1 0.5 0 −0.5 (b) Vocal Tract Impulse Response −1 0 5 10 15 20 25 time nT in ms 1 0.8 0.6 0.4 0.2 (d) Voiced Excitation 0 0 5 10 15 20 25 time nT in ms Figure 8.13: Time-domain representation of speech model: (a) Glottal pulse g[n] as given by Eq. (8.38). (b) Vocal tract impulse response, v[n]. (c) Radiation load impulse response r[n]. (d) Periodic excitation p[n]. k Fk (Hz) 2σk (Hz) 1 660 60 2 1720 100 3 2410 120 4 3500 175 5 4500 250 Table 8.2: Formant frequencies (in Hz) and bandwidths (in Hz) for /AE/ vowel model. bandwidth for five formant frequencies. The first 251 samples of v[n] and the 10 poles of V (z) are shown in Figures 8.13(b) and 8.14(b) respectively. (Note that the individual signal samples are shown connected by straight lines in this case.) The model for the radiation load is the simple first difference system R(z) = 1 − γz −1 . (8.41) The corresponding impulse response r[n] = δ[n] − γδ[n − 1] is shown in Figure 8.13(c) and the single zero is shown in Figure 8.14(c) for the specific value of γ = 0.96.
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LR Rabiner and RW Schafer, June 3

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