Analytical Derivation of Scattering and Admittance Rational Functions from Coupling Matrix

The coupling matrix provides an immediate tool to describe a network, widely used in the context of microwave filters. While several techniques exist to obtain and manipulate the coupling matrix for a given response, the opposite is traditionally achieved by numerically evaluating the response of the network at each frequency of interest and fitting the response to obtain the rational expressions embedding poles and zeros. The present manuscript instead proposes an analytic way to retrieve directly the admittance and scattering functions, as ratios of polynomials, from the coupling matrix. Applications include commercial software packages and more robust microwave filter optimizations during design.