## Section: New Results

### Frequency approximation and OMUX design

Participants : Laurent Baratchart, Jean-Paul Marmorat, Fabien Seyfert, Damien Pacaud [ Thales Alenia Space, Toulouse ] .

An OMUX (Output MUltipleXor) can be modeled in the frequency domain through scattering matrices of filters, like those described in section 4.3 , connected in parallel onto a common guide. The problem of designing an OMUX with specified performance in a given frequency range naturally translates into a set of constraints on the values of the scattering matrices and of the phase shift introduced by the guide in the considered bandwidth.

An OMUX simulator on a Matlab platform was designed last year and checked against a number of designs proposed by Thales Alenia Space (a.k.a. TAS). Under the terms of a contract with TAS (2007), it has been used to design a dedicated software to optimize OMUXes whose second release to TAS has taken place this year.

The software proceeds by adding channels recursively, applying to the new channel the above short-circuit and reflection-in-the bandwidth rules. This yields an initial guess for the global “optimizer” which seems to regularly outperform those currently used by TAS. More extensive tests are being conducted. A natural sequel should consist of the study of the so-called “manifold-peaks” that may impede a design based on ideal assumptions of losslesness.

A new problem was brought in by Damien Pacaud (TAS) concerning a de-embedding problem one encounters while tuning T-junction diplexers. Let S be the measured scattering matrix of a diplexer composed of a junction with response T and two filtering devices with response A and B as plotted on figure 12 . The de-embedding question is the following: given S and T , is it possible to derive A and B ? Although the question may appear classical very little seems to be known about it in the literature and among measurements specialists.

Using algebraic elimination techniques we derived the following interesting relation:

_{1, 1}-S

_{1, 2}*S

_{1, 3}/S

_{2, 3}= T

_{1, 1}-T

_{1, 2}*T

_{1, 3}/T

_{2, 3}

which shows that the reflexion term S_{1, 1} can be deduced, from the
remaining measurements (independently from A and B ). As a
consequence of this redundancy we showed that de-embedding problem, in
its current statement, is ill posed. Even if additional loss-less
conditions are made, the following statement holds: for every phase parameter
of the transmission A_{1, 2} there exists a unique set of
measurements A and B that are compatible with S and
T . Moreover closed form expressions exists for the derivation A
and B .

In order to overcome the ill posed character of the de-embedding problem we currently study approaches where several measuring campaigns are made, for example while varying the short circuit position of the junction T . Generalisation of our preliminary results to general multiplexers are also of great interest.