Fast Recursive Least-Squares Ladder Algorithms

Abstract

Chapters 7 and 8 have illuminated the classes of ladder algorithms that are based on a pure order recursive construction of the ladder form. In this type of algorithm, the central problem appeared to be the order recursive updating of the covariance C_m(t) according to

$$ {{{\text{C}}}_{{\text{m}}}}{\text{(t)}}{\mkern 1mu} {\mkern 1mu} {\text{ = }}{\mkern 1mu} {{{\text{C}}}_{{{\text{m - 1}}}}}{\text{(t)}}{\mkern 1mu} {\text{ + }}{\mkern 1mu} \ldots {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\text{.}} $$

(9.1)

Two solutions to the problem (9.1) have been presented. The first approach, discussed in Chap. 7, was based on the incorporation of transversal predictor parameters as intermediate recursion variables in Levinson-type algorithms. Alternatively, Chap. 8 introduced the algorithms of the PORLA type where the objective was to replace the Levinson-type recursion by inner product recursions, where inner products, also termed “generalized residual energies”, were used as intermediate recursion variables, hence avoiding the explicit computation of transversal predictor parameters.