LR Rabiner and RW Schafer, June 3

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DRAFT: L. R. Rabiner and R. W. Schafer, June 3, 2009 452CHAPTER 8. THE CEPSTRUM AND HOMOMORPHIC SPEECH PROCESSING are deterministic z-transforms representing the shaping of the power spectrum by the vocal tract and radiation respectively. The power spectrum of the synthetic speech output would therefore be Φss(e jω ) = A 2 U |V (e jω )| 2 |R(e jω )| 2 . (8.50) From Eq. (8.48) or (8.50) it becomes clear that if we apply the characteristic system for convolution to the convolution of autocorrelation functions in Eq. (8.47), we do not need the complex cepstrum since Φss(e jω ) is real and non-negative so the cepstrum and complex cepstrum are identical. The cepstrum of an autocorrelation function would be the inverse discrete-time Fourier transform of the logarithm of the Fourier transform of the autocorrelation function; i.e., the cepstrum of the autocorrelation function is the inverse DTFT of the logarithm of the power spectrum. Thus, the real logarithm of the power spectrum is ˆΦss(e jω ) = 2(log AU ) + 2 log |V (e jω )| + 2 log |R(e jω )|, (8.51) so that the cepstrum (complex cepstrum) of the autocorrelation function is ˆφss[n] = 2(log AU )δ[n] + 2(ˆv[n] + ˆv[−n])/2 + 2(ˆr[n] + ˆr[−n])/2. (8.52) Therefore, the cepstrum of the autocorrelation function is an even function of quefrency n, and, because it is based on the power spectrum (which involves squared-magnitudes) it is twice the size of the cepstra of the deterministic components due to the vocal tract and radiation systems. By our assumption, both v[n] and r[n] represent minimum phase systems, so ˆhU [n] = ˆφss[n]/2 n ≥ 0 0 otherwise. (8.53) However, note that since phase is not included in the power spectrum representation, we cannot distinguish between minimum phase and maximum phase components if they were to exist in a more detailed model. As a simple example, if the vocal tract and radiation systems of the previous section were excited by a white noise input instead of the periodic glottal pulse input, the components of the cepstrum due to g[n] and p[n] would be omitted. The synthetic sound produced would be like a hoarse whisper instead of the voiced vowel /AE/. 7 Thus, the theoretical log power spectrum would be given by Eq. (8.51) and the cepstrum of the autocorrelation function would be given by Eq. (8.52). Using the same values for the parameters of the vocal tract and radiation systems, we obtain the plots in Figure 8.20. In this case, there is no periodic component in the log spectrum and therefore no isolated peak in the cepstrum. As discussed above, the absence of such a peak in the expected quefrency range of the pitch period serves as an indicator of unvoiced excitation. Algorithms for pitch detection based on the cepstrum are discussed in Chapter 10. 7 A more realistic model of an unvoiced fricative would differ in the locations and bandwidths of poles and might also include zeros.
DRAFT: L. R. Rabiner and R. W. Schafer, June 3, 2009 8.4. COMPUTING THE SHORT-TIME CEPSTRUM AND COMPLEX CEPSTRUM OF SPEECH453 log e | S(e j2π FT ) | c [ n ] 4 2 0 −2 −4 −6 −8 −10 (a) Log Power Spectrum of Synthetic Unvoiced Speech −12 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 frequency in Hz 0.2 0 −0.2 −0.4 −0.6 −0.8 −1 (b) Cepstrum of Autocorrelation of Synthetic Unvoiced Speech −25 −20 −15 −10 −5 0 5 10 15 20 25 quefrency nT in ms Figure 8.20: Homomorphic analysis of unvoiced speech: (a) Log magnitude log{Φss(e j2πF T )} (b) Cepstrum of autocorrelation function ˆ φss[n] 8.4 Computing the Short-Time Cepstrum and Complex Cepstrum of Speech In the previous section, we computed exact expressions for the complex cepstrum of the output of a discrete-time model for speech production. This was possible because the synthetic speech signal was created with known systems and known excitations for which z-transform representations could be determined. This model is implicitly assumed in most speech analysis techniques; however a major difference in practice is that we base our analysis on short segments of a given natural speech signal. We simply assume that the short segment of a natural speech signal could have been a short segment of the model output. Since the speech signal changes continuously with time, we employ a sequence of analyses to track those changes. That is the approach that we shall now develop for homomorphic speech analysis leading to short-time versions of the cepstrum and complex cepstrum.
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LR Rabiner and RW Schafer, June 3

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