On the Existence and Construction of Solutions to the Partial Lossless Inverse Scattering Problem with Applications to Estimation Theory

The partial lossless inverse scattering (PLIS) problem plays a fundamental role in the solution of -stationary estimation problems, as exemplified in the construction of fast algorithms for the inversion of α-stationary matrices. We treat some open problems connected with this theory. First we derive necessary and sufficient conditions for a Hermitian matrix to be positive (say, a covariance matrix) in terms of its α-stationary wave representation. Next we propose a generalization of the recursive construction of PLIS solutions which was initiated by Kailath, Lev-Ari, and their co-workers. We show that each invariant subspace of the backward shift operator has a corresponding solution of the PLIS problem, provided the operator that generalizes the classical Pick matrix is positive definite. We also show that, under fairly general conditions, all solutions of H[formula omitted] type are essentially obtained in this way. Furthermore, we give an important characterization of the residual “wave” quantities generated by the scatterer. Finally, we put the results in the context of the theory of stochastic processes and derive stochastic models forward and backward order-recursive innovation filters.