Surface Acoustic Wave Filters, Unitary Extensions and Schur Analysis

We study the concept of a mixed matrix in connection with linear fractional transformations of lossless and passive matrix-valued rational functions, and show that they can be parametrized by sequences of elementary chain matrices. These notions are exemplified on a model of a Surface Acoustic Wave filter for which a state-space realization is carried out in detail. The issue of optimal design of such filters –as yet unsolved– naturally raises a Darlington synthesis problem with both symmetry and interpolation constraints whith control on the McMillan degree. As a partial answer, we give necessary and sufficient conditions for symmetric Darlington synthesis to be possible without increasing the McMillan degree for a symmetric rational contractive matrix which is strictly contractive in at least one point of the unit circle .