## Section: New Results

### Synthesis of compact multiplexers and the polynomial structure of n×n inner matrices

Participants : Martine Olivi, Fabien Seyfert, Stéphane Bila [ Xlim ] , Hussein Ezzedin [ Xlim ] , Amine Rouini.

The objective of our work in the ANR Filipix is the derivation of efficient algorithms for the synthesis of microwave multiplexers. In our opinion, an efficient frequency design process calls for the understanding of the structure of n×n loss-less reciprocal rational functions for n>2 . Whereas the case n = 2 is completely understood and a keystone of filter synthesis, very little seems to be known about the polynomial structure of such matrices when they involve more than 2 ports.

We therefore started with the analysis of the 3×3 case typically of practical use in the manufacturing of diplexers. Based on recent results obtained on minimal degree reciprocal Darlington synthesis [42] we derived a polynomial model for 3×3 reciprocal inner rational matrices with given MacMillan degree. The latter writes as follow:

where we define

^{*}= p

_{1}p

_{1}

^{*}+ p

_{2}p

_{2}

^{*}

and the following divisibility conditions must hold

If the polynomials p and r of degree kn-1 are given together with a condition at infinity on S , then one can show that there exist 2^{k} (3×3 ) inner extensions (of MacMillan degree n ) of S_{1, 1} . Their computation involves mainly linear algebra. During his internship, Amine Rouini designed and programmed a procedure which makes effective this extension process. From a practical synthesis point of view the extension process that starts with the polynomials p_{1} and p_{2} is more relevant but remains technically problematic: some issues concerning the stability of the derived polynomial q are still unsolved for the moment. The study of particular forms of the polynomial model in connexion with some special circuit topologies used for the implementation of the diplexer are also currently under investigation.