Least squares ARMA modeling of linear time-varying systems: Lattice filter structures and fast RLS algorithms

The problem of modeling linear time-varying systems is one of considerable importance in many varied and diverse applications. This paper looks at AR, MA, and ARMA modeling of linear time-varying systems with lattice filters. There are two parts to this paper. The first part develops the lattice filter structures. In particular, a new ARMA(N, M) lattice filter structure is presented (N: AR order, M: MA order) which is fully consistent with the geometric characteristics of the AR and MA lattice filter structures in that it evaluates all optimal ARMA (i, j) filters of order lower than (N, M), and each such filter is realized in terms of a fully orthogonal set of realization vectors. For uncorrelated input, this ARMA lattice filter simplifies greatly, resulting in the white input ARMA lattice filter. The second part of the paper develops new fast RLS algorithms for the evaluation of the lattice filter coefficients. There are two classes of algorithms presented. One class is based on projection error sample averaging, while the other uses input-output sample averaging and some orthogonality relations. This latter approach is equivalent to the exact RLS formulation. The algorithms of the former class are the most efficient modeling algorithms known to the authors, and the algorithms of the latter class are of comparable or better efficiency than other algorithms in the same class.