Adaptive Filters

CHAPTER 40 Three Basic Estimation Problems

The recursive least-squares algorithms described so far in Parts VIII {Least-Squares Methods) and IX {Fast RLS Algorithms), including array variants and fast least-squares variants, are usually qualified as fixed-order algorithms. The qualification “fixed-order” means that, from one iteration to another, these implementations propagate quantities that relate to estimation problems of fixed-order.

In this part, we shall study RLS algorithms that are order-recursive in nature, as opposed to fixed-order. They are widely known as lattice filters and have several desirable properties such as improved numerical behavior, stability, modularity, in addition to computational efficiency. In these implementations, least-squares problems of increasing orders are solved successively so that, in addition to time-updates, the lattice filters rely heavily on order-updates for various quantities.

Our treatment of least-squares lattice filters has at least three features:

1. First, all relevant order-recursive relations are derived without assuming any struc ture in the regression vectors.
2. Second, and because of the above, the derivation is able to show that it is possible to design efficient lattice filters even for cases where the regressors do not possess shift structure. This generalization is achieved by pinpointing the variable whose update is affected by data structure, and by showing what kind of structure enables an efficient order-recursive ...