## Section: New Results

### Synthesis and Tuning of broad band microwave filters

Participants : Smain Amari, Magued Bekheit [ RMC, Kingston, Canada ] , Fabien Seyfert.

Some important results have been obtained in order to handle tuning and synthesis of broad band
filters. One of the major problems when dealing with wide band filters is the break down
of the classical low pass model which relies on a narrow band assumption.
We showed however that
there exists a unifying “low pass formalism” which is valid in the narrow band as well as in the wide band situations. The
latter relies on the following remark. Let S be any inner, real, symmetric
(S^{t} = S ), rational matrix, which is identity at infinity
and has MacMillan degree n . Then the rational matrix S_{r} defined by:

is again an inner, complex, symmetric matrix, which is identity at infinity,
and has MacMillan degree n . It
can be shown that S is entirely characterised by the knowledge of its
reduced “complex” version S_{r} . Measurements of S on two conjugate
frequency bands are mapped to measurements of S_{r} on a single band,
which up
to the use of a linear frequency transformation can be cast to the normalized
band [-1, 1] . Usual techniques used to recover rational models from low pass
responses measured on a single frequency interval can therefore be used to
recover high pass responses via the use of the generalized reduced response
S_{r} . Implementation attempts of the latter in the Presto-Hf software were
started and encouraging results where obtained for the tuning
of an ultra-wide band filter realized with suspended strip lines. Figure
11 shows data and their rational approximation of this
10^{th} order filter (reduced order 5) with a bandwidth ratio of
approximately 10% (in collaboration with the university of Ulm, Germany).

Concerning the synthesis of the response of such filters we had already shown
that the latter amounts to a Zolotarev problem with a non-polynomial weight
(with a square root singularity). For fixed transmission zeros we were however
able to derive explicit formulas for the optimal (in the Chebychev sense)
filter function F_{n} :

where is a suitable parabolic frequency transformation and the
z_{k}^{'}s are prescribed transmission zeros. Recurrence formulas for the practical
computation of F_{N} have been derived and implemented as part of the
Dedale-HF software package. As for the realization of such responses, first results were obtained with resonant coupling elements
6.8

In collaboration with the RMC and possibly with XLim and ST-Microelectronics (Tours) our goal is now to test the validity of our unified approach on real examples. Data collection campaigns obtained during tuning phases are scheduled. Joint publications about the topic are also in progress.