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, which plays a fundamental role in the solution of a α-stationary estimation problems, is considered. Some open problems connected with this theory are treated. Necessary and sufficient conditions for a Hermitian matrix to be positive (say, a covariance matrix) are derived in terms of its α-stationary wave representation. A generalization of the recursive construction of PLIS solutions is proposed. It is shown that each invariant subspace of the backward shift operator has a corresponding solution of the PLIS problem, provided the operator that generalized the classical Pick matrix is positive definite. It is also shown that, under fairly general conditions, all solutions of the H ^∞ type are essentially obtained in this way, and a characterization of the residual wave quantities generated by the scatterer is given. Stochastic models and forward and backward order-recursive innovation filters are derived