Adaptive Filters

CHAPTER 32 Order and Time-Update Relations

This chapter can be skipped on a first reading. Its results will only be needed in later chapters when we study fast fixed-order and order-recursive least-squares filters in Parts IX (Fast RLS Algorithms) and X (Lattice Filters).

However, there is a reason why we choose to present the material at this location, and not in later chapters. The reason is that the results we present here are often derived in the literature in the absence of regularization and under the assumption of some structure in the data matrix H (such as requiring its successive rows to be shifted versions of one another, as explained later in the introductory remarks to Chapter 37). In comparison, the derivation given here indicates that the results hold irrespective of data structure, i.e., for any H. In addition, the arguments incorporate regularization and clarify its role in order-update relations. In so doing, the results will allow us to provide later in Chapter 40 a treatment of lattice filters in the presence of regularization. The results also allow the class of lattice filters to be extended to more general data structures, other than the classical tapped-delay-line structure, as was shown in detail in Chapter 16 of Sayed (2003).

For now, it suffices to treat the material in this chapter as an application of the concepts and geometric constructions of the earlier sections.

32.1 BACKWARD ORDER-UPDATE RELATIONS

Consider a weighted regularized least-squares ...