Lossless chain scattering matrices and optimum linear prediction: The vector case

In this paper we give a systematic treatment of the exact and approximate realization of a positive real matrix‐valued function on the open unit disc by means of a lossless circuit connected to a passive load. We discuss the mathematical properties of the chain scattering matrix which describes the lossless circuit and rederive a form of the classical Darlington synthesis theorem generalized to roomy matrix‐valued transmission functions. We then develop a matrix version of an algorithm due to Schur for the construction of approximate realizations which produces (minimal degree) Nevanlinna–Pick approximants to the original positive real matrix. We further identify the normalized inverse of one of the outer factors of the approximant to the positive real matrix as the orthogonal projection of the identity onto a suitably defined subspace, give its interpretation as a reproducing kernel, and establish strong convergence under mild conditions on the growth of the order of the approximation. Finally we interpret and apply the mathematical theory developed in the body of the paper to the theory of prediction for vector‐valued second order stationary stochastic sequences and briefly discuss connections with the theory of maximum entropy extensions and of inverse scattering.