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By Paulo Sergio Ramirez DINIZ (auth.)

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Nawab, Signals and Systems, Prentice Hall, Englewood Cliffs, NJ, 2nd edition, 1997. 4. A. V. Oppenheim and R. W. Schaffer, Discrete-Time Signal Processing, Prentice Hall, Englewood Cliffs, NJ, 1989. 5. A. Antoniou, Digital Filters: Analysis, Design, and Applications, McGraw HilI, New York, NY, 2nd edition, 1993. 6. L. B. Jackson, Digital Filters and Signal Processing, Kluwer Academic Publishers, Norwell, MA, 3rd edition, 1996. 7. R. A. Roberts and C. T. Mullis, Digital Signal Processing, Addison-Wesley, Reading, MA, 1987.

For the special case where ai = 0 for i = 1,2, ... ,N the process is identified as a moving average (MA) process. This terminology indicates that the process depends on a linear combination of the present and past sampies of the input signal. In summary, an ARMA process can be generated by applying a white noise to the input of a digital filter with poles and zeros, whereas for the AR and MA cases the digital filters are all-pole and all-zero filters, respectively. M arkov Process A stochastic process is called a Markov process if its past has no influence in the future if the present is specified [13], [15].

66) 6. The sum of the eigenvalues of R is equal to the trace of R, and the product of the eigenvalues of R is equal to the determinant of R. Proof: tr[Q-1RQ] = tr[A] where, tr[A] = 2:[:0 aii· Since tr[A' A] = tr[AA'], we have N tr[Q-l RQ] = tr[RQQ-l] = tr[RI] = tr[R] = LAi i=O Also det[Q-l RQJ = det[RJ det[QJ det[Q-l] = det[R] = det[A] = n[:o Ai D 34 CHAPTER 2 7. 68) where the minimum and maximum values are reached when the vector w is chosen to be the eigenvector corresponding to the minimum and maximum eigenvalues, respectively.

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