A Schur-based approach to the four-block problem

We descibe an alternative solution to the four-block problem using the method of (generalized) Schur analysis. We first reduce the general problem to a simpler one by invoking an inner-outer factorization with a block-diagonal inner matrix. Then using small-sized spectral factorizations we are able to parametrize an unknown entry in terms of a Schurtype matrix function that satisfies a finite number of interpolation conditions of the Hermite-Féjer type. We describe a simple recursive solution that determines the Schur function in terms of a transmission-line cascade of elementary J-lossless sections. A state-space realization for each section is given, as well as a parametrization of all solutions to the four-block problem in terms of a linear fractional transformation. Formulas for a global solution are also given; though they are computationally less effective.