LR Rabiner and RW Schafer, June 3

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DRAFT: L. R. Rabiner and R. W. Schafer, June 3, 2009 476CHAPTER 8. THE CEPSTRUM AND HOMOMORPHIC SPEECH PROCESSING Amplitude 0.05 0 −0.05 log magnitude −2 −4 −6 (a) Speech Segment with Hamming Window 0 50 100 150 200 250 300 350 400 Time (Samples) (b) Log Magnitude 0 −1 −2 0 500 1000 1500 2000 2500 3000 3500 4000 frequency in Hz (c) Cepstrum 1 0 DFT of Windowed Signal Estimate of Vocal Tract Spectrum −150 −100 −50 0 Quefrency (Samples) 50 100 150 Figure 8.35: Homomorphic filtering of unvoiced speech; (a) Windowed unvoiced speech segment x[n], (b) log magnitude of short-time Fourier transform log |X(e jω )| (Heavy line shows estimate of vocal tract spectrum log |HU (e jω )|), (c) corresponding cepstrum. nco = 20 as shown. As in the case of voiced speech, we can compute zero-phase, minimum-phase, or maximum-phase impulse responses as the output of the inverse characteristic system D−1 ∗ {·} with the liftered cepstrum as input. Figure 8.36a shows the zerophase impulse response corresponding to the smoothed log magnitude in Figure 8.35b obtained using the lifter in Eq. (8.97) with nco = 20, (shown superimposed on the cepstrum in Figure 8.36c). Figure 8.36b shows the corresponding minimum-phase impulse response obtained using the lifter of Eq. (8.99b) with nco = 20. Not shown in Figure 8.36 is the maximum-phase impulse response, which would simply be a time-reversed version of the impulse response in Figure 8.36b. Observe that the impulse response for the unvoiced example varies quite rapidly in time compared to the impulse responses derived for voiced speech (plotted in Figures 8.32a and 8.34). This is because the log spectrum for the fricative peaks at about 2700 Hz, while the peak of the voiced spectrum is at low frequencies.
DRAFT: L. R. Rabiner and R. W. Schafer, June 3, 2009 8.5. HOMOMORPHIC FILTERING OF NATURAL SPEECH 477 Amplitude 0.2 0.1 0 −0.05 (a) Zero−phase Impulse Response Estimate −200 −150 −100 −50 0 50 100 150 200 0.15 0.1 0.05 0 (b) Minimum−Phase Impulse Response Estimate −200 −150 −100 −50 0 Time (Samples) 50 100 150 200 Figure 8.36: Homomorphic analysis of unvoiced speech; (a) Zero-phase estimate of vocal tract impulse response hU [n], (b) Zero-phase estimate of vocal tract impulse response hU [n]. 8.5.6 Summary Illustration of Short-Time Cepstrum Analysis The previous discussion and examples show that it is possible to obtain approximations to some of the basic components of the speech waveform by homomorphic filtering. While the examples show that it is possible to separate a windowed speech segment into plausible excitation and vocal tract impulse response components, complete deconvolution of the speech waveform is often not necessary. Rather, we may only require estimates of basic parameters such as pitch period and formant frequencies. For this purpose, the cepstrum is entirely sufficient. Thus, in many speech analysis applications, it is not necessary to compute the unwrapped phase. Notice, for example, by comparing Figures 8.30b to Figure 8.35c, that the cepstrum provides a basis for distinguishing between voiced and unvoiced speech, and furthermore, the pitch period of voiced speech is placed clearly in evidence in the cepstrum. Also note that the formant frequencies show up clearly in the smoothed log magnitude estimate of the vocal tract frequency response, which can be obtained by applying the lifter of Eq. (8.97) or (8.99b) to the cepstrum. As an illustration of the effectiveness of the short-time cepstrum in highlighting the time-varying properties of speech, consider the 2001-point (250 ms) segment of speech shown in Figure 8.37. This figure shows two Hamming windows of length 401 samples (50 ms at 8 kHz sampling rate). Short-time cepstrum analysis and homomorphic smoothing was performed on this waveform every 100 samples (12.5 ms) to obtain a cepstrum and smoothed log magnitude function for each frame. Figure 8.38 shows 15 frames of this processing with the first frame positioned at the beginning of the waveform in Figure 8.37. The
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LR Rabiner and RW Schafer, June 3

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