Adaptive Cumulant-Based Estimation of ARMA Parameters

Time-recursive lattice algorithms for the estimation of the AR parameters of an ARMA process, using a 1-D slice of the k-th order cumulant are presented. The cumulant matrix can be viewed as the cross-correlation of the observed process, y(n), and an associated process, z(n), and, hence, is not generally Hermitian. Cumulant-based AR modeling is shown to be equivalent to linear prediction with non-conventional orthogonality conditions. Our algorithm leads to a pair of lattices, one excited by y(n) and the other by z(n); the lattices being coupled through order- and time-update equations. Parameter estimates are shown to be asymptoticaly unbiased and to converge in mean-square; the lattice algorithm has zero mis-adjustment noise. Lattice algorithms for cumulant-based joint process estimation are also given. Algorithms for time-adaptive estimation of MA. and hence of ARMA, processes are also presented. One version of our lattice algorithm is shown to solve the over-determined problem, provided the (1,2,4)-generalized inverse is used. These algorithms may be used to estimate the parameters of non-minimum phase ARMA models, which may have inherent all-pass factors, from outputs corrupted with colored Gaussian noise.