Team apics

Overall Objectives
Scientific Foundations
Application Domains
New Results
Contracts and Grants with Industry
Other Grants and Activities

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] :

Im21 ${~\mtext letting:~E_{n,m}{(K,K^')}={{p\#8712 P_m{(K)},q\#8712 P_n{(K^')}~\mtext such~\mtext that~\#8704 x\#8712 I~,~\mfenced o=| c=| \mfrac {p(x)}{q(x)}\#8804 1}},}$

Im22 ${~\mtext solve:~\munder max{{(p,q)}\#8712 E_{m,n}{(K,K^')}}\munder min{x\#8712 J}\mfenced o=| c=| \mfrac {p(x)}{q(x)}}$(1)

where Im23 ${I=\#8899 I_i}$ (resp. Im24 ${J=\#8899 J_i}$ ) is a finite union of compact intervals Ii of the real line corresponding to the pass-bands (resp. stop-bands), and Pm(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 Im25 ${K=K^'=\#119825 }$ (“real” problem) or Im26 ${K=\#119810 ,K^'=\#119825 }$ (“mixed” problem), or else Im27 ${K=K^'=\#119810 }$ (“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] .

Figure 9. SPOT5 specifications
Figure 10. 7th order dual-band response and its critical points


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