Adaptive Filters

CHAPTER 35 QR and Inverse QR Algorithms

In preparation for our discussions of RLS array methods, we now recall briefly the defining equations of RLS for ease of reference.

Consider a collection of data , where the {u_J} are 1 × M and the {d(j)} are scalars, in addition to an M × 1 column vector , an M × M positive-definite matrix Π, and a scalar λ satisfying 0 << λ ≤ 1. Then the solution, w_N, of the regularized least-squares problem

(35.1)

and the corresponding minimum cost, ξ(N), can be obtained recursively as follows (recall Alg. 30.2). Start with , and repeat for i ≥ 0:

(35.2)

Moreover, the following relations also hold at each iteration i:

(35.3)

We further remarked following Alg. 30.2 that the {w_i} also satisfy the following construction. Start with , and repeat for i ≥ 0:

(35.4) ...