 By Paulo Sergio Ramirez Diniz (auth.)

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Extra resources for Adaptive Filtering: Algorithms and Practical Implementation

Example text

Its performance, however, serves as a comparison pattern for gradient-based algorithms. gw(k) It is worth noting that several alternative gradient-based algorithms available replace gw(k) by an estimate gw(k), and they differ in the way the gradient vector is estimated. 79) and, as can be noted, it depends on the vector p and the matrix R , that are usually not available. 96) Now some of the main properties related to the convergence behavior of the steepest-descent algorithm in stationary environment are described.

Identically, the maximum value for 'R occurs when Wi = 0 for i :j; I, where A/ is the largest eigenvalue. o There are several ways to define the norm of a matrix. In this book the norm of a matrix R, denoted by IIRII. 69) Note that the norm of R is a measure of how a vector w grows in magnitude, when it is multiplied by R. 68). 70) Fundamentals of Adaptive Filtering where Amax 35 is the maximum eigenvalue of R. 71) In case there is an error in the vector p , originated by quantization or estimation, how does it affect the solution of the system of linear equations?

N. 57) o 30 CHAPTER 2 2. 58) 0 0 >'N 0 Proof: RQ R[qo ql . · qN] = [>'Oqo >'lql . ·>'NqNl >'0 0 o >'1 Q 0 o 0 =QA 0 Therefore , since Q is invertible because the q;'s are linearly independent , we can show that o 3. The nonzero eigenvectors qo, ql ' . . qN that correspond to different eigenvalues are linearly independent. , multiplying the equation above by R in one instance and by )\N -1 on the other instance, and subtracting the results, it yields ao(>'o - >'N)(>'0 - >'N- t}qo + a1 (>'1 - >'N)(>'1 - >'N- t}q1 + .. 