**Delay extraction and determination of high order rational model**

The delay extraction as well as the high order rational model derivation are both based on the resolution of some tailored extremal problems in the Hardy space of analytic functions H^{2}. The idea is here, starting from measurement of the S-parameters on a frequency band and from a qualitative description of the system’s behaviour outside the measurement band, to derive a high order rational model as well as an estimated value of the delay components due to the access feeds. Detailed information about this procedure can be found in following papers [1], [2].

[1] “Extraction of coupling parameters for microwave filters: determination of a stable rational model from scattering data”, Seyfert, F.; Baratchart, L.; Marmorat, J.-P.; Bila, S.; Sombrin, J., *Microwave Symposium Digest, 2003 IEEE MTT-S International* , vol.1, no., pp.25,28 vol.1, 8-13 June 2003

[2] “Identification of microwave filters by analytic and rational H2 approximation“, M. Olivi, F. Seyfert and J.P. Marmorat, *Automatica 49(2013) 317-325 *, doi: 10.1016/j.automatica.2012.10.005.

**Derivation of a rational model of prescribed MacMillan degree**

The knowledge of a high order, stable, rational model allows to cast the task of computing a rational approximant of prescribed MacMillan degree, to a model order reduction problem. The MacMillan degree of the system is known, since Ho and Kalman’s work, to be the rank of the Hankel matrix of the system: a well known model order reduction method consists therefore in “approaching” high rank Hankel matrix, by a Hankel matrix of prescribed rank. An interesting characteristic of this approach, is that it amounts in practice to compute a singular value decomposition (SVD) of the Hankel matrix of the system (of high order), in order to get its best approximant of given rank in terms of Hankel norm [3]. No local optimization is involved here and the SVD, which is a non trivial algebraic operation on matrices for which efficient methods have been developed, delivers here the global optimum to the reduction problem. The draw back of this approach is however the interpretation in the frequency domain of the Hankel norm: it is a quotient norm of L^{∞} type, and therefore not always adapted to noisy measurements data. We therefore initialize, with the obtained reduced system, a descent algorithm for rational approximation that minimizes the L^{2} norm to the measurements. The optimization is performed among all rational matrices of prescribed MacMillan degree. The RARL2 matlab package is used for this: it offers an efficient parametrization of rational functions in the matricial case (the filter’s S-parameters are 2×2) well adapted for optimization purposes. For details and references about its functioning, see RARL2‘s page and [2].

[3] “All Optimal Hankel Norm Approximations of Linear Multivariable Systems and their L^{∞} Error Bounds”, K. Glover, International Journal of control, Vol. 39, pp. 1115-1193, 1984

**Circuital realization**

A circuital realization of the obtained rational scattering matrix of the filter is then computed by classical techniques: see Dedale-HF’s tutorial “From S TO M“, on this matter as well as Cameron’s paper [4]. This yields a coupling matrix in canonical form, which can be further transformed to match other coupling topologies. See Dedale-HF page for this, and [5].

[4] R. J. Cameron. General coupling matrix synthesis methods for chebyshev filtering functions. IEEE Transaction on Microwave Theory and Techniques, 47(4):433–442, 1999

[5] R.J. Cameron, J.C. Faugere, F. Seyfert, “An exhaustive Approach to the Coupling Matrix Synthesis Problem, Application to the Design of High Degree Asymmetric Filters”, International Journal of RF and Computer-Aided Engeneering, 17:1(4–12), 2007