**Analytical methods for the conception and tuning of microwave devices**

Room Euler Violet – February 6, 2019 at 02:00 PM.

In this presentation I will expose part of the work I’ve been pursuing since my appointment at Inria in 2001. Passive microwave devices such as filters, multiplexers, power-dividers are ruled by Maxwell equations that describes the wave scattering phenomenon at the origin of their functioning. The central mathematical object is here the scattering matrix, that describes the linear dependency between the Fourier transforms of the incoming and outcoming modal power waves propagating in the transmission lines (or wave-guides) used to feed this type of hardware. Designing or conceiving this devices amounts to determine their physical dimensions, the size of their electromagnetic cavities and coupling irises, such as to meet certain frequency specifications formulated on their scattering matrix. Tuning procedures tackle the problem of correcting these dimensional parameters at hand of harmonic measurements performed on the system.

We will present here a de-embedding procedure for filters that aim to furnish a stable rational model of given MacMillan degree, when starting from narrow band incomplete harmonic measurements of these devices. We will show why the partial nature of the frequency measurement induces an ill-posed stable rational approximation problem and naturally brings us to state an analytic completion problem, best formulated in the framework of the classical Hardy spaces of holomorphic functions. The main question we will try to answer here is: how much additional information to the harmonic measurements is needed in order to perform a completion procedure in a satisfactory manner ? Without spoiling the suspense around this question, we can roughly answer it this way: as opposed to what the analytic continuation principle may suggest, quite a lot. After introducing elementary bounded extremal problems, we will detail a mixed norm version of these, where the modulus of the transfer needs to be specified for frequencies outside the measurement band. We will explain how this led us to consider a completion problem where additional information is provided, by means of a finite dimensional description of possible extensions of the data. The latter is at heart of the dedicated software toolbox Presto-HF that was transferred to academic, as well as industrial, practitioners of the filter community.

Working in close connection with filter specialists led us to consider a preliminary stage in the manufacturing process of filters: the synthesis of an ideal frequency response. We will describe here a procedure for the computation of multiband responses, with a guaranteed optimal selectivity at fixed degree and number of transmission zeros. A specific alternation property is presented in order to characterize optimal solutions of a set of signed quasi-convex sub-problems, in terms of alternating sequences of extremal points. As compared with Achieser’s result on uniform real valued rational approximation on an interval, our approach consists in an adaptation of the latter to the multi-band case and to the solving of a Zolotariov problem of the third kind.

Deembedding techniques as well as frequency response synthesis procedures have in common that they all end up with a rational 2×2 scattering matrix that needs to be realized as a circuit to proceed further in the filter’s tuning or synthesis process. The circuit used here is the low-pass prototype, which consists of ideally coupled resonators. The coupling topology, that is the way resonators are coupled, or not, one to another is crucial here. We present here a result stating the existence of a canonical circuital realisation called the arrow form. For circuits with non-canonical coupling topologies we develop an algebraic approach for the corresponding structured realisation problem, together with an abstract framework clarifying the compatibility conditions between coupling topology and class of frequency responses. Some formal results at the crossing of circuit theory and algebraic geometry will be discussed. Software implementations relying on the Groebner bases engine FGb and the dedicated toolbox Dedale-HF will be discussed.

We will eventually discuss contributions to the classical problem of broadband matching in electronics, encountered when conceiving energy efficient multiplexers and antennas. An approach based on Nevanlinna-Pick interpolation will be presented.