572 Chapter 13.Fourier and Spectral Applications There are many variant procedures that all fall under the rubric of LPC If the spectral character of the data is time-variable,then it is best not to use a single set of LP coefficients for the whole data set,but rather to partition the data into segments,computing and storing different LP coefficients for each segment. If the data are really well characterized by their LP coefficients,and you can tolerate some small amount oferror,then don't bother storing all of the residuals.Just do linear prediction until you are outside of tolerances,then reinitialize(using M sequential stored residuals)and continue predicting. In some applications,most notably speech synthesis,one cares only about the spectral content of the reconstructed signal,not the relative phases. In this case,one need not store any starting values at all,but only the LP coefficients for each segment of the data.The output is reconstructed by driving these coefficients with initial conditions consisting of all zeros except for one nonzero spike.A speech synthesizer chip may have of order 10 LP coefficients,which change perhaps 20 to 50 times per second. Some people believe that it is interesting to analyze a signal by LPC,even when the residuals zi are not small.The xi's are then interpreted as the underlying"input signal"which,when filtered through the all-poles filter defined by the LP coefficients(see $13.7),produces the observed "output University Press. signal."LPC reveals simultaneously,it is said,the nature of the filter and the particular input that is driving it.We are skeptical of these applications, the literature,however,is full of extravagant claims. Programs CITED REFERENCES AND FURTHER READING: OF SCIENTIFIC Childers,D.G.(ed.)1978,Modem Spectrum Analysis(New York:IEEE Press),especially the 、学 paper by J.Makhoul (reprinted from Proceedings of the IEEE,vol.63,p.561,1975). Burg,J.P.1968.reprinted in Childers,1978.[1] Anderson,N.1974,reprinted in Childers,1978.[2] Cressie.N.1991,in Spatia/Statistics and Digital Image Analysis(Washington:National Academy Press).[3] Press,W.H.,and Rybicki,G.B.1992,Astrophysica/Journal,vol.398,pp.169-176.[4] Numerical Recipes 10621 43106 13.7 Power Spectrum Estimation by the (outside Maximum Entropy (All Poles)Method Software. The FFT is not the only way to estimate the power spectrum of a process,nor is it necessarily the best way for all purposes.To see how one might devise another method, let us enlarge our view for a moment,so that it includes not only real frequencies in the Nyquist interval-fe<f<fe,but also the entire complex frequency plane.From that vantage point,let us transform the complex f-plane to a new plane,called the z-transform plane or z-plane,by the relation 2三e2mfA (13.7.1) where A is,as usual,the sampling interval in the time domain.Notice that the Nyquist interval on the real axis of the f-plane maps one-to-one onto the unit circle in the complex z-plane.572 Chapter 13. Fourier and Spectral Applications Permission is granted for internet users to make one paper copy for their own personal use. Further reproduction, or any copyin Copyright (C) 1988-1992 by Cambridge University Press. Programs Copyright (C) 1988-1992 by Numerical Recipes Software. Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5) g of machine￾readable files (including this one) to any server computer, is strictly prohibited. To order Numerical Recipes books or CDROMs, visit website http://www.nr.com or call 1-800-872-7423 (North America only), or send email to directcustserv@cambridge.org (outside North America). There are many variant procedures that all fall under the rubric of LPC. • If the spectral character of the data is time-variable, then it is best not to use a single set of LP coefficients for the whole data set, but rather to partition the data into segments, computing and storing different LP coefficients for each segment. • If the data are really well characterized by their LP coefficients, and you can tolerate some small amount of error, then don’t bother storing all of the residuals. Just do linear prediction until you are outside of tolerances, then reinitialize (using M sequential stored residuals) and continue predicting. • In some applications, most notably speech synthesis, one cares only about the spectral content of the reconstructed signal, not the relative phases. In this case, one need not store any starting values at all, but only the LP coefficients for each segment of the data. The output is reconstructed by driving these coefficients with initial conditions consisting of all zeros except for one nonzero spike. A speech synthesizer chip may have of order 10 LP coefficients, which change perhaps 20 to 50 times per second. • Some people believe that it is interesting to analyze a signal by LPC, even when the residuals xi are not small. The xi’s are then interpreted as the underlying “input signal” which, when filtered through the all-poles filter defined by the LP coefficients (see §13.7), produces the observed “output signal.” LPC reveals simultaneously, it is said, the nature of the filter and the particular input that is driving it. We are skeptical of these applications; the literature, however, is full of extravagant claims. CITED REFERENCES AND FURTHER READING: Childers, D.G. (ed.) 1978, Modern Spectrum Analysis (New York: IEEE Press), especially the paper by J. Makhoul (reprinted from Proceedings of the IEEE, vol. 63, p. 561, 1975). Burg, J.P. 1968, reprinted in Childers, 1978. [1] Anderson, N. 1974, reprinted in Childers, 1978. [2] Cressie, N. 1991, in Spatial Statistics and Digital Image Analysis (Washington: National Academy Press). [3] Press, W.H., and Rybicki, G.B. 1992, Astrophysical Journal, vol. 398, pp. 169–176. [4] 13.7 Power Spectrum Estimation by the Maximum Entropy (All Poles) Method The FFT is not the only way to estimate the power spectrum of a process, nor is it necessarily the best way for all purposes. To see how one might devise another method, let us enlarge our view for a moment, so that it includes not only real frequencies in the Nyquist interval −fc <f<fc, but also the entire complex frequency plane. From that vantage point, let us transform the complex f-plane to a new plane, called the z-transform plane or z-plane, by the relation z ≡ e2πif∆ (13.7.1) where ∆ is, as usual, the sampling interval in the time domain. Notice that the Nyquist interval on the real axis of the f-plane maps one-to-one onto the unit circle in the complex z-plane