l' ;O~S$I'r:!J 112 5. Periodogram and Ulackman-Tukey t.ome~rt. AkI:N s'I""Q 8. Mo. L. Y.. and Cobbold. R. S. C.. Speckle in C. W. Doppler ultrasound spectra: a CH APTER 6 (J simulation study, IEEE Trails. Ultra.HIII.Ferroelec. Freq. Control UFFC-33:747-753. 1986. 9. Rascom. P. A. 1.. Cohbold. R. S. C. and Roelofs. B. H. Moo Innuence of spectral hruadeninJlon C. W. Doppler ultrasound speclra: a Iteolllciric appmach. Ultra.wlI/III Af('d. mol. 12:JK7-:W5. IIJK6. Cepstrum Analysis 10. Reddy. S. Mooand Kirlin. R. L.. Spectral analysis of auditory evoked potentials with pseudorandom noise excitalion. IE££ Troll.'. 8itl/II('d. EII1(.BI\tF.-2ti(1I):479-4K7.1979. II. Hampton. R. L.. Experiments using pseudorandom noise. Sill/lIlllliall 4:246-295. 1965. 12. Goldstein. R. F.. "Eleclroencephalic Audiometry in Modern Developments in Audiol- ogy" (J. Berger. Ed.). Academic Press. New York. 1973. 13. Akselrod. SooGordon. DooUbel. F. A.. Shannon. D. c.. Barger. A. Cooand Cohen. R. J.. Power spectrum analysis of heart rate nucluation: a quantitative probe of beat to beat cardiovascular conlrol. Sd('IIC'('213:22()-222. 1981. 14. DeBoer. R. W.. Karemaker. J. Mooand Strackee. J.. Comparing spectra of a series of point events particularly for heart rate variability spectra. IEEE TrllllJ. Bio/lled. EIII(. BI\1E-31:3114-387. 19114. 15. Eckberg. D. L.. Human sinus arrhythmia as an index of vagal cardiac outnow. J. Appl. 1'1I)'siol.54(4):961-966. 1983. 16. Craelius. W.. Akay. M.. and Tangella. M.. Heart rate variability as an index of auto- nomic imbalance in patients with recent myocardial infarction. Med. Bioi. COII/p. 30:385-388. 1992. 17. Myers G. A.. Martin. G. J.. Magid. N. M.. Barnell. P. SooSchaad. J. W.. Weiss. J. 5.. In this chapter, a tutorial review of the cepstrum method is provided. Lesch. M.. and Singer. D. H.. Power spectral analysis of heart rate variability in sudden While details of the power and complex cepstra are discussed. extensive . J cardiac deaths. IEEE Trail... 8io/llcd. £111(.BME.33:1149-1158. 1986. derivations of the formulae are given elsewhere [1-7]. In addition to cepstrum signal processing, topics for review include biomedical applica- tions in the areas of electrocardiogram (ECG) and heart sound signal analyses [1-7] and speech signal processing [2]. Previolls studies suggest that cepstrum analysis is well suited to data which consist of wavelets [I]. This is true even if the shapes of the wave- lets are not known prior to analysis. For instance, the power cepstrum was successfully applied in radar analysis, where the arrival time of the main wavelet was determined by reducing interference [4]. and in marine exploration, where source depth was determined and the ocean bottom was mapped [4]. Considerable emphasis is given in this chapter to cep- strum applications in medicine, i,ncJuding diastolic heart sound analysis for the detection of coronary artery disease. ECG pattern classification [ 10,13], and speech signal decomposition for theoretical as well as band- width compression application purposes [2]. The cepstrum method serves as an alternative approach to linear pre- diction in that it does not make any assumption regarding the characteris- tics of the data sequence. Bogert el al. [I] developed the cepstrum ap- proach to find echo arrival times in a composite signal by decomposing the nonadditive constituents. The term cepstrum represents the power spectrum; it is defined as a function of pseudo-time, I, the spectral ripple frequency or quefrency. The cepstrum terms defined by Bogert et al. are 113 114 6. Cepstrum Analysis 6.2 The Power Cepstrum 115 summarized below (I.4J: This equation can then be written as the multiplication of the Fourier transform of the two sequences. frequency quefrency spectrum cepstrum IX(z)/z= !Y(zW '!V(z)!2. «(j.4) phase saphe Upon taking the logarithm of both sides of the equation. we obtain amplitude gamnitmle filtering liftering harmonic rahmonic ~ .l~gIX(zW= logl Y(zW + log!V(z)!2. (6.5) To further elaborate on the power spectrum analysis, let us assume that period repiod the excitation function (signal) is given as Throughout the chapter. both terms are used so as not to confuse readers. venT) = S(IIT) + cS(IIT - noT). (6.6) 6.1 The Cepstra where S(n) denotes the unit impulse function in a sampled data sequence. On the basis of this equation. Eq. (6.4) can be further written as Cepstrum analysis is concerned with the deconvolution of two signal (6.7) types: the fundamental (basic) wavelet and a train of impulses (excitation function) [1.4J.The composite signal can be represented in terms of By taking the logarithm of both sides of this equation and substituting power. complex, or phase cepstra. In this chapter. emphasis is placed on z = ejw,we expand Eq. (6.5) as the power and complex cepstra. Readers interested in phase cepstra are 10glX(ejldW = logl Y(ejwW + 10g(1 + c2 + 2c cos(wnoT» (6.8) referred to [1.3.4J. 2c = log!Y(ejo,W + log(l + c2) + log (1 + ~ cos(CJJf/oT)). 6.2 The Power Cepstrum (6.9) The power cepstrum was first described and used by Bogert C'tal. [I] in The details of these derivations are described elsewhere [3]. It is obvious 1963. The purpose of the study was to determine echo arrival times in a from Eq. (6.9) that the logarithm of the magnitude squared of the z- composite signal since the delayed echoes appear as ripples in the loga- transform of the data sequence xC,,) will have sinusoidal components rithmic spectrum of the input data sequence X(II). In practice. the power (ripples). The amplitudes and frequencies of these ripples correspond to cepstrum is an effective tool provided that the frequencies of the basic the amplitude c of the excitation function and the time delay, 110T. wavelet and excitation function do not overlap. By taking the inverse z-transform of Eq. (6.9). the data sequence x(nT) The power cepstrum of the signal is defined as the square of inverse z- can now be expressed in terms of its components. 11is assumed that the transform of the logarithm of the magnitude squared of the z-transform of power cepstra of these components are additive, each corresponding to the data sequence, which can be written as different frequency bands, Xl'c(lIT)= (z.I{log!X(ZW})2 (6.1) (6.10) where YPCis the power cepstrum of the basic wavelet. and Vpcis the power cepstrum of the excitation signal. Xpc(lIT) = C~j £ log!X(z)j2z,,-1dZr. (6.2) ;1; Note that in the above equation the cross-product term was neglected where X(,) represents the ,-transform of the data sequence X(IIT). Let us Ifttie data sequences yPCand Vpchave different frequency ranges, they can assume that the data sequence consists of two convoluted sequences be easily obtained by filtering in the pseudo-frequency domain. Y(IIT) and v(nT). which represent the basic wavelet and excitation func- In summary, after taking the inverse z-transform and obtaining the tion. respectively. The data sequence X(IIT) can be written as power cepstrum, the peaks produced by the excitation function can be x(nT) )'(nT) * V(IIT). (6.3) identified at the quefrencies (delays) of nT. Assuming that vpc(nT) is an = '" 116 6. Ccpstrum Analysis 6.3 The Complex Cepstrum 117 impulse function, the peaks of the power cepstrum can be detected if the form loglY(z)i2 is quefrency limited to less than 110T and the ripples of the loglY(Z)12havea period(repoid)less than (110T)-I. V(IIT) = 8nT + e8(IIT - noT). (6.16) While power cepstrum methods have been successfully applied to By taking the z-transformand substitutingz = ejw,we have biomedical signals including the ECG and diastolic heart sounds, the methods are limited by their failure to maintain the phase information- V(z) = V(ejw7)= I + ce-~"oT (6.17) required for precise recovery of analyzed signals. and X(ejwT) = Y(ejwT)(I + ee-jw"oT). (6.18) 6.3 The Complex Cepstrum Taking the logarithm of both sides of Eq. (6.18), The complex cepstrum is an outgrowth of homomorphicsystem theory log X(ejwT)= log Y(eiwT)+ log(l + ce-jwnoT). (6.19) developed by Oppenheim [2]. Although the power cepstrum can be used for detecting echoes, it cannot be used for wavelet recovery since the Where c < I, the wavelet component dominates and the data sequence phase information is lost [2-5]. The complex cepstrum of a data sequence exhibits minimum phase characteristics. This is most evident when Eq. can be defined as the inverse z-transform of the complex logarithm of the (6.19) is expanded to the form z-transform of the data sequence as follows, 2 log X(eiwT) = log Y(eiwT)+ ce-jwnoT- ~ e-v.-""T. (6.20) X(IIT) = _1. 10g(X(z»zn-1 dz, (6.11) 27TJ f.c Finally, the complex cepstrum of the data sequence x(n) is obtained by where .i(IIT) represents the complex cepstrum and X(z) representsthe z- taking the inverse z-transform of Eq. (6.20), transform of the data sequence x(nT). c2 Let us assume that the input sequence is the convolution of two se- x(nT) = y(nT) + c8(nT - noT)- 2" 8(nT - 2noT). (6.21) quences as follows, X(IIT) = Y(IIT) '" V(IIT). (6.12) It is evident in Eq. (6.21) that the complex cepstrum includes the com- plex cepstrum of the wavelet as well as ripples of the excitation function where Y(IIT) represents the basic wavelet and V(IIT) represents the excita- at the positive frequencies (noT).