A recursive Schur-based solution of the four-block problem

We describe a new 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 a coprime factorization with a block-diagonal inner matrix. Then, using convenient spectral factorizations, we are able to parameterize the unknown entry in terms of a Schur-type matrix function, which is shown to satisfy a finite number of interpolation conditions of the Hermite-Fejer type. All possible interpolating functions are then determined via a simple recursive procedure that constructs a transmission-line (or lattice) cascade of elementary J-lossless sections. This also leads to a parameterization of all solutions of the four-block problem in terms of a linear fractional transformation.