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

Constrained extremal problems in the Hardy space H2 and Carleman's formulas

We study some approximation problems on a strict subset of the circle by analytic functions of the Hardy space H2 of the unit disk (in C), whose modulus satisfy a pointwise constraint on the complentary part of the circle. Existence and uniqueness results, as well as pointwise saturation of the constraint, are established. We also derive a critical point equation which gives rise to a dual formulation of the problem. We further compute directional derivatives for this functional as a computational means to approach the issue. We then consider a finite-dimensional polynomial version of the bounded extremal problem

Extended Path Filter Configurations

International audienceIn this work we introduce a method for increasing the maximum number of transmission zeros in the response of path filters. This recently introduced inline filter configuration allows for up to four transmission zeros on the imaginary axis. The solution proposed in this work, while maintaining the inline configuration, increases the number of transmission zeros up to N-1 (with N order of the filter). The novel concept allowing the additional zeros introduction is verified by means of the design of a waveguide filter of order 8 with 6 transmission zeros

A unified approach to Nevanlinna-Pick interpolation problems

International audienceThis work deals with Complex-valued interpolation by a Schur rational function of given degree at a set of nodes located in the closed lower half-plane, with prescribed maximum points for the modulus (i.e. points where it is equal to 1) on the real axis. The motivation comes from broadband matching, for which the technique we develop offers a new tool

Boundary Nevanlinna-Pick interpolation with prescribed peak points. Application to impedance matching

International audienceWe study a generalized Nevanlinna Pick interpolation problem on the half-plane for rational functions of prescribed degree, where peak points are imposed and interpolation conditions may lie on the real axis. This generalizes previous work by T. Giorgiou, C. Byrnes, A Lindquist and A. Megretski. The problem is motivated by the issue of broadband matching in electronics and microwave system design. We prove existence and uniqueness of a solution by differential-topological techniques. The approach is put to work numerically on a real example, using a continuation method

Estimating unstable poles in simulations of microwave circuits

International audienceThe impedance of a microwave circuit has an infinite number of poles due to the distributed elements. This complicates locating those poles with a rational approximation. In this paper, we propose an algorithm to locate the unstable poles of a circuit with distributed elements. The proposed method exploits the fact that a realistic circuit can only have a finite number of unstable poles. We first determine the unstable part whose poles coincide with the unstable poles of the circuit. A rational approximation of the unstable part is used to estimate the unstable poles. Having the ability to trace a circuit's poles as a function of the circuit parameters is a useful design tool. Pole tracking techniques have been used for the design of oscillators [1], [2], to stabilise power amplifiers [3]-[5] and during the design of frequency dividers [1], [6]. The core of a pole tracking tool is a robust automatic algorithm to estimate the poles of a circuit. To determine the poles of a circuit, a two-step procedure is followed. First, an impedance Z(jω) of the circuit is determined at discrete frequency points between 0 and f max with an AC simulation (Fig. 1). Then, the poles of Z(jω) are determined in a post-processing step. In lumped circuits, Z(jω) is a rational function, so a rational approximation can be used to determine the circuit poles. The impedance presented by a circuit with distributed elements, like transmission lines, is non-rational but still meromorphic 1 [7]. This makes estimating the poles of a microwave circuit more difficult. A good fit of Z(jω) can be obtained with a high-order rational function, but the rational approximation will contain spurious poles that do not correspond to poles of the underlying function [8]. To circumvent this problem, another approach proposed in [9], is to compute low-order rational approximants of the circuit's response restricted to small frequency intervals. This local version of the rational approximation scheme, yields precise estimates of poles when these are close enough to the imaginary axis. In this paper, we propose an algorithm that can estimate the unstable poles of a circuit without performing a rational approximation of a non-rational function. The proposed technique exploits the fact that the equilibrium solution of a realistic circuit 2 can only have a finite amount of poles in the right half-plane [10]

Constrained extremal problems in H2 and Carleman's formulas

International audienceWe consider the extremal problem of best approximation to some function $f$ in $L^2(I)$, with $I$ a subset of the circle, by the trace of a Hardy function whose modulus is bounded pointwise by some gauge function on the complementary subset

Polynomial structure of 3 x 3 reciprocal inner matrices

International audienceThe objective of our work is the derivation of efficient algorithms for the synthesis of microwave multiplexers. In our opinion, an efficient frequency design process calls for the understanding of the structure of n x n inner (or lossless) reciprocal rational functions for n > 2. Whereas the case n = 2 is completely understood and a keystone of filter synthesis very little seems to be known about the polynomial structure of such matrices when they involve more than 2 ports. We therefore start with the analysis of the 3 x 3 case typically of practical use in the manufacturing of diplexers. Based on recent results obtained on minimal degree reciprocal Darlington synthesis, we derive a polynomial model for 3 x 3 reciprocal inner rational matrices with given McMillan degree

Generalized Nevanlinna-Pick interpolation on the boundary. Application to impedance matching

This work deal with interpolation of complex values by a Schur rational functions of prescribed degree at a set of nodes located in the closed disk, with prescribed maximum points for the modulus (i.e. where it is equal to 1) on the circle. The motivation comes from broadband matching, for which the technique we develop offers a new tool

Path Filters: A Class Of True In-Line Topologies With Transmission Zeros

International audienceIn this paper we present a comprehensive discussion of a new class of inline microwave filters with transmission zeros in the response, namely the path filters. The main features of this filters class are highlighted, and an original (synthesis-based) design approach is presented, relying on the derivation of suitable characteristic polynomials. In addition to the classical Generalized Chebyshev characteristic, two new characteristics are introduced (namely the Bounded Chebyshev and the Reduced Chebyshev), that allow improving the flexibility in the requirements assignment of path filters. A new method for the synthesis of the lowpass prototype is also introduced, that overcome the limitation in the classical synthesis based on the manipulation of the transversal prototype (whose synthesis may fail in case of path filters). Finally, the proposed approach for designing the class of considered filters has been validated by several examples that include the evaluation of the characteristic polynomials, the prototype synthesis and the dimensioning of the physical structures in waveguide technology