Analysis of a recursive least squares hyperbolic rotation algorithm for signal processing

Abstract

The application of hyperbolic plane rotations to the least squares downdating problem arising in windowed recursive least squares signal processing is studied. A forward error analysis is given to show that this algorithm can be expected to perform well in the presence of rounding errors, provided that the problem is not too ill conditioned. The hyperbolic rotation algorithm is shown to be forward (weakly) stable and, in fact, comparable to an orthogonal downdating method shown to be backward stable by Stewart. It is shown in detail how the method's accuracy depends upon the conditioning. Numerical comparisons are made with the usual method based upon orthogonal rotations as implemented in LINPACK. Both methods have the important advantage over the classical normal equations appraoch that they can be effectively implemented on special purpose signal processing devices requiring shorter word-lengths. However, the hyperbolic rotation requires n 2 fewer multiplications and additions for each downdating step than the orthogonal method, where n is the number of least squares filter coefficients. In addition, it is more amenable to implementation on a variety of vector and parallel machines. In many signal processing applications n is not large, and if n processors are available, then the downdating process can be accomplished in 2 n time steps by the hyperbolic rotation method.