Fast recursive algorithms for a class of linear equations

Abstract

In many signal processing applications, one often seeks the solution of a linear system of equations by means of fast algorithms. The special form of the matrix associated with the linear system may permit the development of algorithms requiring 0 (p <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> ) or fewer operations. Hankel and Toeplitz matrices provide well known examples and various fast schemes have been developed in the literature to cover these cases. These techniques have common characteristics so that they may be generalized to cover a wider class of linear systems. The purpose of this paper is to develop fast algorithms that cover this wider set of systems. An important feature of the general scheme introduced here is that it leads to the definition of two broad classes of matrices, called diagonal innovation matrices (DIM) and peripheral innovation matrices (PIM), for which fast schemes can be developed. The class of PIM matrices includes many structures appearing in signal processing applications. Most of them are extensively studied in this paper and Fortran coding is provided. Finally, ARMA modeling is considered and within the general framework already introduced, fast methods for the determination of the autoregressive (AR) portion of the ARMA model are presented.