Constrained L2L^2-approximation by polynomials on subsets of the circle

We study best approximation to a given function, in the least square sense on a subset of the unit circle, by polynomials of given degree which are pointwise bounded on the complementary subset. We show that the solution to this problem, as the degree goes large, converges to the solution of a bounded extremal problem for analytic functions which is instrumental in system identification. We provide a numerical example on real data from a hyperfrequency filter

Greedy Algorithms and Rational Approximation in One and Several Variables

International audienceWe will review the recent development of rational approximation in one and several real and complex variables. The concept rational approximation is closely related to greedy algorithms, based on a dictionary consisting of Szeg˝ o kernels in the present context

System identification of microwave filters from multiplexers by rational interpolation

International audienceMicrowave multiplexers are multi-port structures composed of several two-port filters connected to a common junction. This paper addresses the de-embedding problem, in which the goal is to determine the filtering components given the measured scattering parameters of the overall multiplexer at several frequencies. Due to structural properties, the transmission zeros of the filters play a crucial role in this problem, and, consequently, in our approach. We propose a system identification algorithm for deriving a rational model of the filters' scattering matrix. The approach is based on rational interpolation with derivative constraints, with the interpolation conditions being located precisely at the filters' transmission zeros

Electromagnetic Optimization of Microwave Filters using Adjoint Sensitivities

submitted to Transactions on Microwave Theory and TechniquesThis paper introduces a novel computer-aided tuning (CAT) method for coupled-resonator microwave bandpass filters. The method is based on the estimation of the Jacobian of the function that relates the physical filter design parameters to the extracted coupling parameters. Lately commercial full-wave electromagnetic (EM) simulators provide the adjoint sensitivities of the S-parameters with respect to the geometrical parameters. This information leads to an efficient estimation of the Jacobian since it no longer requires finite difference based evaluation. The tuning method first extracts the physically implemented coupling matrix and estimates the corresponding Jacobian. Next it compares the extracted coupling matrix to the target coupling matrix (golden goal). Using the difference between the coupling matrices and the pseudo-inverse of the estimated Jacobian, a correction that brings the design parameters closer to the golden goal is obtained. This process is repeated iteratively until the correction becomes sufficiently small with respect to a user specified goal. In the case of coupling structures with multiple solutions, the Jacobian is calculated for each admissible solution. This paper presents a criterion to identify the physical solution among the different solutions. The CAT method is applied to the design of a cascaded triplet (CT) filter implemented in a microstrip technology. This filter is a well-known examples of a non-canonical coupling structure

Electromagnetic Optimization of Microwave Filters using Adjoint Sensitivities

submitted to Transactions on Microwave Theory and TechniquesThis paper introduces a novel computer-aided tuning (CAT) method for coupled-resonator microwave bandpass filters. The method is based on the estimation of the Jacobian of the function that relates the physical filter design parameters to the extracted coupling parameters. Lately commercial full-wave electromagnetic (EM) simulators provide the adjoint sensitivities of the S-parameters with respect to the geometrical parameters. This information leads to an efficient estimation of the Jacobian since it no longer requires finite difference based evaluation. The tuning method first extracts the physically implemented coupling matrix and estimates the corresponding Jacobian. Next it compares the extracted coupling matrix to the target coupling matrix (golden goal). Using the difference between the coupling matrices and the pseudo-inverse of the estimated Jacobian, a correction that brings the design parameters closer to the golden goal is obtained. This process is repeated iteratively until the correction becomes sufficiently small with respect to a user specified goal. In the case of coupling structures with multiple solutions, the Jacobian is calculated for each admissible solution. This paper presents a criterion to identify the physical solution among the different solutions. The CAT method is applied to the design of a cascaded triplet (CT) filter implemented in a microstrip technology. This filter is a well-known examples of a non-canonical coupling structure

Parametric Modeling of the Coupling Parameters of Planar Coupled-Resonator Microwave Filters

International audience—The design of planar coupled-resonator microwave filters is widely based on coupling matrix theory. In this framework a coupling matrix is first obtained during the synthesis step. Next this coupling matrix is physically implemented by correctly dimensioning the geometrical parameters of the filter. The implementation step is carried out using simplified empirical design curves relating the coupling coefficients and geometrical parameters. The curves typically only provide initial values and EM optimization is often needed such that the filter response meets the specifications. This paper proposes to extract paramet-ric models that relate the filters design parameters directly to the coupling parameters. The advantage of such models is that they allow to tune the filter in a numerically cheap way and that they provide physical insight in the filters behavior. This paper explains how such models can be extracted from EM simulations and illustrates the technique for the design of an 8 pole cascaded quadruplet filter in a microstrip technology

Linearized Active Circuits: Transfer Functions and Stability

International audienceWe study the properties of electronic circuits after linearization around a fixed operating point in the context of closed-loop stability analysis. When distributed elements, like transmission lines, are present in the circuit it is known that unstable circuits can be created without poles in the complex right half-plane. This undermines existing closed-loop stability analysis techniques that determine stability by looking for right half-plane poles. We observed that the problematic circuits rely on unrealistic elements with an infinite bandwidth. In this paper, we therefore define a class of realistic linearized components and show that a circuit composed of realistic elements is only unstable with poles in the complex right half-plane. Furthermore, we show that the amount of right half-plane poles in a realistic circuit is finite, even when distributed elements are present. In the second part of the paper, we provide examples of component models that are realistic and show that the class includes many existing models, including ones for passive devices, active devices and transmission lines

Minimax principle and lower bounds in H2^{2}-rational approximation

In Press. This is the corrected proof as pubished online by the journalInternational audienceWe derive some lower bounds in rational approximation of given degree to functions in the Hardy space $H^2$ of the disk. We apply these to asymptotic errors rates in approximation to Blaschke products and to Cauchy integrals on geodesic arcs.We also explain how to compute such bounds, either using Adamjan-Arov-Krein theory or linearized errors, and we present a couple of numerical experiments on several types of functions. We dwell on the Adamjan-Arov-Krein theory and a maximin principle developed in the article "An L^p analog of AAK theory for p >= 2", by L. Baratchart and F. Seyfert, in the Journal of Functional Analysis, 191 (1), pp. 52-122, 2012