By Paulo Sergio Ramirez DINIZ (auth.)
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Extra resources for Adaptive Filtering: Algorithms and Practical Implementation
Therefore, for Hermitian matrices with repeated eigenvalues it is always possible to find a complete set of ort ho normal eigenvectors. 63) i=O that is known as spectral decomposition. 64) In addition, since qi = AiR-1qi, the eigenvectors of a matrix and of its inverse coincide, whereas the eigenvalues are reciprocals of each other. 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.
As an illustration, suppose a white noise is applied as input to a lowpass filter with impulse response h(k) and sharp cutoff at a given frequency Wl. The autocorrelation function ofthe output signal y(k) will not be a single impulse, it will be h(k) * h( -k). Therefore, the signal y(k) will look like a band-limited random signal, in this case, a slow-varying noise. Some properties of the function R x (e iw ) of a discrete-time and stationary stochastic process are worth mentioning. The power spectrum density is a periodic function of w, with period 271", as can be verified from its definition.
13. P. Strobach, Linear Prediction Theory, Springer Verlag, New York, NY, 1990. 14. B. Widrow and S. D. Stearns, Adaptive Signal Processing, Prentice Hall, Englewood Cliffs, NJ, 1985. 15. J. R. Treichler, C. R. , and M. G. Larimore, Theory and Design 0/ Adaptive Filters, John Wiley & Sons, New York, NY, 1987. 16. B. Farhang-Boroujeny, Adaptive Filters: Theory and Applications, John Wiley & Sons, New York, NY, 1998. 17. S. Haykin, Adaptive Filter Theory, Prentice Hall, Englewood Cliffs, NJ, 4th edition, 2002.
Adaptive Filtering: Algorithms and Practical Implementation by Paulo Sergio Ramirez DINIZ (auth.)