Matrix representations and parameterizations of lossless and reciprocal multiports

Abstract

Stimulated by recent work on the synthesis of a certain class of surface acoustic wave (SAW) filters, we discuss the exceedingly classical subject of physical realizability criteria for lossless, and reciprocal linear multiports. We give an essentially coordinate-free formulation of the realizability conditions for passive, lossless and reciprocal multiports. For this sake we describe a linear multiport by its external behavior (the linear subspace generated by all admissible signal pairs at the ports) and analyze the properties of this space in terms of Gramians of its bases with respect to physically relevant metrics. The standard matrix representations (impedance matrix, scattering matrix, …) arise by the introduction of various affine coordinates in the external behavior. Instead of discussing their well-known coordinate-dependent properties we focus on the so called P-matrix as an attractive non-standard example and study its properties and relations to other matrix representations. As a major result we show that the parameterization of lossless multiports as induced by the P-matrix is largely equivalent to the Arov-Dewilde-Dym parameterization of J-inner functions.