## Section: New Results

### The Zolotarev problem and multi-band filter design

Participants : Vincent Lunot [ Thales Alenia Space, Cannes ] , Fabien Seyfert.

The theoretical developments took place over the last two years, while deepenings of the numerical aspects were carried out in 2007. This study was conducted under contract with the CNES and Thales-Alenia-Space (Toulouse), and was part of V. Lunot's doctoral work [80] . The problem goes as follows. On introducing the ratio of the transmission and reflexion entries of a scattering matrix, the design of a multi-band filter response (see section 4.3 ) reduces to the following optimization problem of Zolotarev type [86] :

where (resp. ) is a finite union of compact
intervals I_{i} of the real line corresponding to the pass-bands
(resp. stop-bands), and P_{m}(K) stands for the set of polynomials of degree
less than m with coefficients in the field K . Depending on the physical
symmetry of the filter, it is interesting to solve problem
(1 ) either for (“real” problem) or
(“mixed” problem), or else
(“complex” problem).
The “real” Zolotarev problem can be decomposed into a sequence of
concave maximization problems, whose solution we were able to characterize
in terms of an alternation property. Based on this,
a Remez-like algorithm has been
derived in the polynomial case
(i.e., when the denominator q of the scattering matrix is fixed),
which allows for the computation of a dual-band response
(see Figure 10 )
according to the frequency specifications
(see Figure 9 for an example
from the spacecraft SPOT5 (CNES)).
We have designed an algorithm in the rational
case which, unlike linear programming, avoids sampling over
all frequencies. This raises the issue of
the “generic normality” (*i.e.* the maximum degree) of the approximant
with respect to the geometry of
the intervals. This question remains open.
The design of efficient procedures
to tackle the “mixed” and “complex”
cases remains a challenge.
The software *easyFF* was registered at the APP under the number IDDN.FR.001.150004.000.S.P.2009.000.3150. A license agreement is beeing worked for a permanent distribution of this software to our academic partners: Xlim and the Royal
Military College of Canada. Applications of the Remez algorithm
to filter synthesis are described in [52] , [82] .
An article on the general approach based on linear programming has
been published [81] .