Minimal symmetric Darlington synthesis

TL;DR

This paper addresses the minimal symmetric Darlington synthesis problem for rational symmetric Schur functions, providing a characterization of minimal extensions and their degrees, motivated by applications in Surface Acoustic Wave filters.

Contribution

It introduces a method to determine the minimal McMillan degree of symmetric extensions under specific constraints, with a constructive approach based on symmetric realizations and spectral factorization.

Findings

Minimal McMillan degree is given by zeros of odd multiplicity of I-SS*

Provides a constructive characterization of all minimal extensions

Applicable to Surface Acoustic Wave filter design

Abstract

We consider the symmetric Darlington synthesis of a p x p rational symmetric Schur function S with the constraint that the extension is of size 2p x 2p. Under the assumption that S is strictly contractive in at least one point of the imaginary axis, we determine the minimal McMillan degree of the extension. In particular, we show that it is generically given by the number of zeros of odd multiplicity of I-SS*. A constructive characterization of all such extensions is provided in terms of a symmetric realization of S and of the outer spectral factor of I-SS*. The authors's motivation for the problem stems from Surface Acoustic Wave filters where physical constraints on the electro-acoustic scattering matrix naturally raise this mathematical issue.