By Paulo Sergio Ramirez DINIZ (auth.)
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Extra info for Adaptive Filtering: Algorithms and Practical Implementation
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). 29) i=l where x(k) is a white noise, is called autoregressive moving average (ARMA) process. The coefficients ai and bj are the parameters of the ARMA process. , r(l) "10 for some l "I O.
The linear constraints are satisfied if C T w condition to be satisfied becomes = f. 96) Therefore, for the GSe structure shown in Fig. 97) Minimization of the output energy is achieved with a proper choice of Wl. 99) Given that Wl,o is the solution to an unconstrained minimization problem of transformed quantities, any unconstrained adaptive filter can be used to estimate recursively this optimal solution.
From now on in this section, we discuss so me properties of the correlation matrix related to its eigenvalues and eigenvectors. 56) Rq=Aq where I is the (N + 1) by (N + 1) identity matrix. 55) is called characteristic equation of R, and has (N + 1) solutions for A. We denote the (N + 1) eigenvalues of R by Ao, Al, ... , AN. 56), however we consider only those particular values of A that are linked to a nonzero eigenvector q. Some important properties related to the eigenvalues and eigenvectors of R, that will be useful in the following chapters, are listed below.
Adaptive Filtering: Algorithms and Practical Implementation by Paulo Sergio Ramirez DINIZ (auth.)