LR Rabiner and RW Schafer, June 3

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DRAFT: L. R. Rabiner and R. W. Schafer, June 3, 2009 478CHAPTER 8. THE CEPSTRUM AND HOMOMORPHIC SPEECH PROCESSING 0.5 0 −0.5 Section of Speech Wave and Window for Short-time Cepstrum Analysis 1 −1 0 50 100 150 200 250 time nT (in msec) Figure 8.37: 2001 samples of speech waveform at the transition from the fricative /SH/ to the diphthong /EY/ for an utterance of the word /SH/ /EY/ /D/. The short-time homomorphic analysis of this segment is plotted in Figure 8.38. window on the left in Figure 8.37 corresponds to the unvoiced example in Section 8.5.5 and to frame 1 in Figure 8.38. The other window, positioned at 1000 samples (12.5 ms) after the beginning of the plot in Figure 8.37, corresponds to the voiced example considered in detail in Section 8.5.3 and to frame 10 in Figure 8.38. The set of plots in Figure 8.38 confirm all the assertions that we have made about the properties of the cepstrum and about how homomorphic filtering can achieve a degree of separation between the excitation and vocal tract system impulse response. For the given window length (401 samples) and effective spacing between frames (100 samples), the first 5 frames include only the fricative /SH/. This is evident in the cepstrum, the short-time Fourier transform, and the liftered spectrum in each of the first 5 frames. Frame 6 begins at 75 ms and ends at 125 ms; thus, it straddles the transition from unvoiced to voiced speech. This is manifested in the cepstrum as a lack of a strong peak because the rapid variations in the short-time spectrum are more random than periodic. However, the formant structure of the voiced part of the window, being of larger amplitude, tends to give the smoothed log spectrum the formant resonance pattern that is characteristic of the later voiced frames. Frames 7–15 contain only voiced speech of the diphthong /EY/, and the corresponding cepstrums show a clear peak at quefrency corresponding to the pitch period. The strong peak in the cepstrum corresponds to the clearly defined periodic structure in the short-time Fourier transform. Subsequent frames contain only voiced speech, and the cepstrum peak moves slightly to the right with increasing frame number, implying a slight drop in pitch frequency. The smoothed log spectra in frames 7–15 show how the formant frequencies change with time. In Chapter 10 we discuss how the short-time cepstrum can be the basis for algorithms for pitch and voiced/unvoiced detection and also for estimating the time variation of formant frequencies. In Chapter 11, we will also discuss the role of the cepstrum in speech coding.
DRAFT: L. R. Rabiner and R. W. Schafer, June 3, 2009 8.6. CEPSTRUM ANALYSIS OF ALL-POLE MODELS 479 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 (a) Short-Time Log Spectra F1 F2 F3 0 1 2 frequency F (in kHz) 3 4 window number → 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 (b) Short-Time Cepstra 0 5 10 15 20 25 quefrency nT (in msec) Figure 8.38: Short-time homomorphic analysis of the waveform in Figure 8.37 at 15 window positions separated by 100 samples (12.5 ms). 8.6 Cepstrum Analysis of All-Pole Models In Section 8.4 we showed several ways of computing the short-time cepstrum and complex cepstrum, and in Section 8.5 we illustrated how these methods can be applied to natural speech signals. Another, somewhat indirect, method of cepstrum analysis is based on all-pole modeling of speech using the methods of linear prediction that are discussed in detail in Chapter 9. Although it may be somewhat awkward to introduce linear predictive models before discussing how they are derived, this should not cause much difficulty, since our focus is simply on the all-pole model that results. Linear predictive analysis is simply a means to compute the model directly from short-segments of the speech signal. For our purposes at this point, it is sufficient to assert that short-time analysis techniques can be applied to estimate a vocal tract system model of the
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LR Rabiner and RW Schafer, June 3

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