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Additional info for Adaptive Digital Filters, 2nd Edition (Signal Processing and Communications)
B. Moore, ‘‘Identiﬁcation of ARMA Parameters of Time Series,’’ IEEE Transactions AC-20, 104–107 (February 1975). J. Lamperti, Stochstic Processes, Springer, New York, 1977. R. G. Jacquot, Modern Digital Control Systems, Marcel Dekker, New York, 1981. A. Papoulis, ‘‘Predictable Processes and Wold’s Decomposition: A Review,’’ IEEE Transactions ASSP-33, 933–938 (August 1985). S. M. Kay and S. L. Marple, ‘‘Spectrum Analysis: A Modern Perspective,’’ Proc. IEEE 69, 1380–1419 (November 1981). V. F. Pisarenko, ‘‘The Retrieval of Harmonics from a Covariance Function,’’ Geophysical J.
The rational function decomposition of HðzÞHðzÀ1 Þ yields, after simpliﬁcation, rðpÞ ¼ N=q X i jPi jn cos½n Argðpi Þ þ i ð2:74Þ i¼1 where the real parameters i and i are the parameters of the decomposition and hence are related to the poles Pi . 32 Chapter 2 It is worth pointing out that the same expression is obtained for the generating ﬁlter of the type FIR/IIR, but then the parameters i and i are no longer related to the poles: they are independent. A limitation of AR spectra is that they do not take on zero values, whereas MA spectra do.
Give the corresponding signal power spectrum. Find the eigenvalues of the matrix 2 3 1:00 0:70 0:08 R3 ¼ 4 0:70 1:00 0:70 5 0:08 0:70 1:00 and the coefﬁcients of the minimum eigenvalue ﬁlter. Locate the zeros of that ﬁlter and give the harmonic spectrum. Compare with the prediction spectrum obtained in the previous exercise. 56 Chapter 2 REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. T. W. Anderson, The Statistical Analysis of Time Series, Wiley, New York, 1971. G. E. P. Box and G.