TY - JOUR

T1 - On the Existence and Construction of Solutions to the Partial Lossless Inverse Scattering Problem with Applications to Estimation Theory

AU - Alpay, Daniel

AU - Dewilde, Patrick

AU - Dym, Harry

N1 - Funding Information:
Manuscript received January 20, 1988; revised September 6, 1988. This work was supported in part by ZWO, in part by the Weizmann Institute under a Meyerhoff Fellowship, in part by the Delft University of Technology, and in part by STW under grants for the DEL 77.1260 and MEGA projects. The material in this paper was partially presented at MTNS, Phoenix, AZ, 1977. D. Alpay and H. Dym are with the Department of Mathematics, the Weizmann Institute, Rehovot. Israel. P. Dewilde is with the Department of Electrical Engineering, Delft University of Technology, P.O. Box 5031, 2600 GA, Delft, The Netherlands. IEEE Log Number 8931191.

PY - 1989/1/1

Y1 - 1989/1/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=0024772281&partnerID=8YFLogxK

U2 - 10.1109/18.45275

DO - 10.1109/18.45275

M3 - Article

AN - SCOPUS:0024772281

SN - 0018-9448

VL - 35

SP - 1184

EP - 1205

JO - IEEE Transactions on Information Theory

JF - IEEE Transactions on Information Theory

IS - 6

ER -