ARMA Modeling of time varying systems with lattice filters

This paper looks at least squares ARMA modeling of linear time varying systems with lattice filters. The modeling problem is formulated in a Hilbert space as it is an enlightening approach, that provides very powerful orthogonality relations to work with. There are two parts to this paper. The first part presents a new ARMA lattice filter ARMA(N,M) which is fully consistent with the geometrical characteristics of the AR and MA lattice filters in that it is realized in terms of a fully orthogonal lattice basis and it evaluates all optimal ARMA(i,j) filters of lower order. It therefore goes further in the basis orthogonalization than the ARMA lattice of Lee, Friedlander and Morf and it does not require that N=M. The second part of the paper presents a new fast RLS algorithm for the evaluation of the lattice filter coefficients. The algorithm is based on an inner product factorization and differs from other RLS lattice algorithms in that the projection of the so called pinning vector does not appear in any of the time updates. The algorithm is formulated as a sliding window algorithm, but it embeds a growing window (prewindowed) algorithm which is realized simply by dropping terms from the sliding window algorithm.