Microwave Filter Synthesis and Tuning: What Circuit and Approximation Theory Can Do for You

Whereas microwave filters are by construction distributed structures, they are infinite dimensional linear dynamical systems. When roughly tuned, they admit a remarkable finite dimensional approximation based on lumped elements, namely their circuital approximation. This explains the success of the synthesis approach developed by G. Matthaei and pursued by authors like R.J. Cameron who refined and further developed the model known now as the coupling matrix approach. 

While tuning a filter, it is therefore natural to try to fit a circuit to measured data in order to infer pertinent dimensional correction from it. The first step in this direction is to perform a rational approximation of the 2x2 filter's response at given MacMillan degree: a task belonging to approximation theory. In this talk, we will review methods developed in the last decade to do so, in particular bounded analytical extensions techniques followed by Hankel norm approximation approaches. Some details will be given on how to solve the phase loading problem which is inevitable when dealing with filter measurements.  Eventually the realization step consisting in constructing a circuit with a particular topology representing the previously obtained rational matrix will be discussed. Compatibility conditions will be given for the solvability of this polynomial, multivariate nonlinear problem. Eventually techniques from computer algebra will be presented in order to obtain an effective approach.