LR Rabiner and RW Schafer, June 3

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DRAFT: L. R. Rabiner and R. W. Schafer, June 3, 2009 488CHAPTER 8. THE CEPSTRUM AND HOMOMORPHIC SPEECH PROCESSING resulting parameter sets the delta cepstrum (for the estimate of the first derivative) and the delta-delta cepstrum (for the estimate of the second derivative). Conceptually, the simplest way of computing the delta cepstrum parameters would be as a first difference of cepstral vectors, of the form: ∆mfccm[n] = mfccm[n] − mfccm−1[n]. (8.119) This simple difference is a poor approximation to the first derivative of the mel-cepstral coefficients and is not generally used in practice. Instead the first derivative is often implemented as a least-squares approximation to the local slope (over a region around the current time sample), thereby providing a locally smoothed estimate of the first derivative of the form: ∆mfccm[n] = M k=−M k(mfccm+k[n]) M k=−M k 2 . (8.120) It is readily shown that the computation of the delta cepstrum values via Eq. (8.120) is equivalent to fitting the cepstral values of neighboring frames (M frames before frame m and M frames after frame m) by a straight line of slope 1. (It is a simple matter to extend the fitting procedure to a second order polynomial [18].) One of the key advantages of using differential parameters such as the delta cepstrum or the delta-delta cepstrum is that the differencing operation removes the effect of simple linear filtering on the parameter values as discussed in Section 8.7.1, thereby making them less sensitive to channel shaping effects that might occur in a speech communication system. This simple cepstral normalization technique has been called cepstral mean subtraction (CMS) or cepstral mean normalization (CMN) and has been widely used in speech recognition systems [6]. 8.8 Summary In this chapter we have presented the basic ideas of homomorphic signal processing as applied to speech. The main idea of homomorphic speech processing is the separation or deconvolution of a segment of speech into a component representing the vocal tract impulse response, and a component representing the excitation source. The way in which such separation is achieved is through linear filtering of the inverse Fourier transform of the log spectrum of the signal, i.e., the cepstrum of the signal. Computational considerations in implementing a homomorphic speech processing system were described. Finally, some typical methods for estimating speech parameters based on the homomorphic model of speech were discussed.
DRAFT: L. R. Rabiner and R. W. Schafer, June 3, 2009 Bibliography [1] B. Bogert, M. Healy and J. Tukey, The Quefrency Alanysis of Time Series for Echoes, Proc. Symp. on Time Series Analysis, M. Rosenblatt, Editor, Ch. 15, pp. 209-243 J. Wiley, New York 1963. [2] S. B. Davis and P. Mermelstein, Comparison of Parametric Representations for Monosyllabic Word Recognition, IEEE Trans. on Acoustics, Speech and Signal Proc., Vol. ASSP-28, No. 4, pp. 357-366, Aug. 1980. [3] S. Furui, Cepstral analysis technique for automatic speaker verification, IEEE Trans. Acoust., Speech, and Sig. Proc., vol. ASSP-29, no. 2, pp. 254-272, April 1981. [4] S. Furui, “Speaker Independent Isolated Word Recognizer Using Dynamic Features of Speech Spectra,” IEEE Trans. on Acoustics, Speech and Signal Proc., Vol. ASSP-34, pp. 52-59, 1986. [5] C. P. Hughes and A. Nikeghbali, “The Zeros of Random Polynomials Cluster Near the Unit Circle,” arXiv:math/0406376v3 [math.CV], http://arxiv.org/PS cache/math/pdf/0406/0406376v3.pdf. [6] X. Huang, A. Acero and H.-W. Hon, Spoken Language Processing, Prentice- Hall Inc., Englewood Cliffs, NJ, 2001. [7] F. Itakura and T. Umezaki, “Distance measure for speech recognition based on the smoothed group delay spectrum,” in Proc. IEEE ICASSP, vol. 12, April 1987, pp. 1257-1260. [8] B.-H. Juang, L. R. Rabiner, and J. G. Wilpon, “On the use of bandpass liftering in speech recognition,” IEEE Trans. Acoust., Speech, and Sig. Proc., vol. ASSP-35, no. 7, pp. 947-954, July 1987. [9] J. D. Markel and A. H. Gray “Cep[stral Distance and the Frequency Domain,” J. Acoust. Soc. Am., vol. 58, S97, 1975. [10] A. M. Noll, “Cepstrum pitch determination,” J. Acoust. Soc. Am., vol. 41, no. 2, pp. 293-309, February 1967. [11] A. V. Oppenheim, Superposition in a Class of Nonlinear Systems, Tech. Report No. 432, Research Lab of Electronics, MIT, Cambridge, Mass., March 1965. 489
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LR Rabiner and RW Schafer, June 3

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