Coupling matrix synthesis problem: an approach based on continuation techniques and the "cheater’s homotopy"

Microwave filters are usually realised by means of an inter-connected network of resonators. The coupling matrix framework introduced in 1974 [1] has become through the years one of the central tools to address synthesis questions in this area. A possible reason for this success is the ability to capture within a simple circuit model the interaction between the topology of the coupling network and the related class of 2×2 achievable scattering responses. In this direction the work of C. Bell [2] is the first to introduce canonical forms, i.e., that is coupling topologies with a minimal number of coupling elements able to accommodate all pole responses (i.e. responses with at most N-2 TZs here). Note that the concept of minimality is here loosely defined as the property that the number of non-zero couplings appearing in these coupling forms "cannot be further reduced". In the same vein of fundamental work S. Amari linked the length of the shortest path in the coupling graph to the maximal number of transmission zeros a topology can accommodate [3]. The work of R.J. Cameron [4]–[6] introducing sequences of rotations for a vast variety of practical coupling topologies and extending the (N + 2) × (N + 2) coupling matrix to the access ports firmly installed the coupled resonator model as an essential tool in microwave filter synthesis. The developments due to G. Macchiarella and S. Tamiazzo [7] were also decisive here in that they allowed for computation of topologies made of cascaded triplets and quadruplets within the coupling matrix framework without the need of Darlington-like extractions techniques. At the same time optimisation techniques were designed [8], [9] to tackle the situation of topologies for which no explicit synthesis scheme was known.At that time some practitioners noticed that in some cases these optimization-based methods, when initialised with different starting points, yielded distinct coupling matrices whereas corresponding to the same response. The papers [10], [11] and professorial thesis [12] are the first to clarify the situation by introducing, for a coupling topology, the notions of non-redundancy, admissible responses, and reduced order, that is number of distinct complex coupling matrices solving the CM synthesis problem for a given topology. The later is an invariant of the topology and do not depend of the response to be realised. The practical computations of these matrices is here achieved by solving a system of polynomial equations by means of a Gröbner basis approach. For fast computations the tool Dedale-HF [13] based on continuation techniques is developed and allows the user to solve exhaustively the coupling matrix synthesis problem for some listed coupling topologies. The practical use of such computations, allowing, for example, network simplifications, is illustrated in [14].In recent years the subject of computation of multiple solutions has regained accrued interest. In particular [15] details how interval arithmetic can be used to compute exhaustively all real solutions for a particular topology. The paper [16] presents a heuristic based on Levenberg-Marquardt’s method to again compute all real solutions to a CM synthesis problem and applies it in [17] to find for a particular response all possible realistic topologies, and therewith all possible real CM solutions. In the latter particular attention is paid to some compatibility conditions between the studied topology and its associated folded form.In this talk we will first recall the notions of non-redundancy and admissible responses which allow to derive a precise notion of minimality echoing the tentative definition by C. Bell of 1984. In particular we will show that minimality is always associated with a set of admissible responses, a disregarded fact that often leads to apparent contradictions: the cul-de-sac form has far less non-zero couplings than its counterparts, the arrow and folded forms. In this context how can the aforementioned canonical forms be minimal?In a second part of the talk we will detail the continuation mechanism used in the software package Dedale-HF which is based on an efficient parametrisation of similarity transforms via the matrix exponential map. This mechanism is primarily used within Dedale-HF to move pre-computed fibers of solutions for a randomly chosen response toward a fiber of equivalent coupling matrices realizing the response prescribed by the user. Here the exhaustive nature of the pre-computed fiber, containing real as well as complex elements is essential: during the continuation process some of the real solutions may become complex and vice versa. We will also show how this technique can be used to build a heuristic, called by mathematicians the "cheater’s homotopy", that iteratively constructs solution sets for a particular topology. The latter appears therefore as an alternative to the Gröbner basis approach, or interval arithmetic, when the problem size appears to be intractable. The heuristic is shown to perform astonishingly well on test problems where the number of solutions is currently known as well as on open problems such as the extended box topology of degree 14 with a postulated reduced order of 46080.