By Paulo Sergio Ramirez DINIZ (auth.)
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Additional info for Adaptive Filtering: Algorithms and Practical Implementation
Multiplying the equation above by R in one instance and by AN-1 on the other instance, and subtracting the results, it yields aO(AO - AN)(AO - AN-t)qo + a1(A1 - AN)(A1 - AN-t)q1 + ... + aN-2(AN-2 - AN-1)qN-2 = 0 By repeating the same steps above several times, we end up with Since we assumed AO nonzero, then ao = O. i- Al, AO i- A2, ... AO i- AN, and Qo was assumed The same line of thought can be used to show that ao = a1 = a2 = ... 59). Therefore, the eigenvectors corresponding to different eigenvalues are linearly independent.
Two processes are considered jointly WSS if and only if any linear combination of them is also WSS. 25) must be WSS, for any constants k l and k2 , if Xl (k) and X2 (k) are jointly WSS. This property implies that both Xl (k) and X2 (k) have shift-invariant means and autocorrelations, and that their cross-correlation is also shift invariant. 26) where Pxr(k),Xi (k) (Yr> Yi) is the joint probability density function (pdf) of xr(k) and xi(k). 27) where * denotes complex conjugate, since we assurne for now that we are dealing with complex signals, and Pxr(k),xi(k),xr(l),xi(l) (Yr> Yi, Zr> Zi) is the joint probability density function of the random variables x(k) and x(l).
As a consequence, it is important to examine the main properties of the matrix R. Some properties of the correlation matrix come from the statistical nature of the adaptive filtering problem, whereas other properties derive from the linear algebra theory. 52) where x H (k) is the Hermitian transposition of x(k), that means transposition followed by complex conjugation or vice versa. The main properties of the R matrix are listed below: 1. The matrix R is positive semidefinite. Proof: Given an arbitrary complex weight vector w, we can form a signal given by The magnitude squared of y(k) is The mean-square (MS) value of y(k) is then given by MS[y(k)) = E[ly(kW) = wH E[x(k)x H (k))w = wHRw ~ 0 28 CHAPTER 2 Therefore, the matrix R is positive semidefinite.