Adaptive Filters

CHAPTER 38 Regularized Prediction Problems

The derivation of the fast array algorithm in Chapter 37 was based on the realization that, by proper choice of the regularization matrix Π as in (37.13), the successive differences δP_i–1 in (37.15) will have rank 2 with a constant signature matrix, S = (1 ⊕ –1), and with two-column factors . In this chapter, we provide an interpretation for the columns of . In the process of doing so, we shall arrive at other efficient implementations of RLS. These implementations will not be in array form, but in terms of explicit sets of equations; they are known as the fast Kalman filter, the fast a posteriori error sequential technique (FAEST), and the fast transversal filter (FTF).

First, however, to facilitate the presentation, we need to adopt a more explicit notation in order to indicate the fact that the {Pi, P_i–1} are M × M matrices. For this reason, we shall write PM,I and PM,_I–1, instead of P_i and P_i–1. We shall also write U_M,_I instead of u_i ΠM instead of Π, W_M,I instead of w_i γM (i) instead of γ(i), and GM,I instead of g_i. The subscript M in all these variables is used to indicate the order of the underlying estimation problem, i.e., the dimension of the regression vectors{u_M,i}

Thus, consider again the regularized least-squares problem