Some Remarks on the Vector Fitting Iteration

Abstract

Summary. Vector Fitting (VF) is an iterative technique to construct rational approximations based on multiple frequency domain samples, introduced by Gustavsen and Semlyen [1,3]. VF is nowadays widely investigated and used in the Power Systems and Microwave Engineering communities. Numerical experiments show that VF has favorable convergence properties. However, so far, no theoretical proof for its convergence, or conditions to guarantee convergence, have been published. This paper gives a description of a general iterative Least-Squares framework for rational approximation and shows that VF fits into this framework.

Macromodeling of Multiport Systems Using a Fast Implementation of the Vector Fitting Method

“…It is known that the iterative procedure of VF is equivalent to the Sanathanan-Koerner iteration [7], [8] using an implicit weighting scheme [5]. This iteration is known to have good convergence properties if the signal-to-noise ratios are sufficiently high.…”

Rational Modeling Algorithm for Passive Microwave Structures and Systems

“…The trivial null solution of (1) is avoided by setting one coefficientc t 0 = 1 or by adding an additional relaxation constraint to (1), as in [9]. It was shown in a previous report that this process is related to the Sanathanan-Koerner iteration [16].…”

Robust Parametric Macromodeling Using Multivariate Orthonormal Vector Fitting

“…By repeating this process iteratively , updated values of the coefficients and can be derived by minimizing the following cost function [16]: (4) The trivial null solution can typically be avoided by fixing one coefficient (e.g., the constant term of the denominator) to unity. This can be done without loss of generality since both the numerator and denominator can be divided by the same constant value.…”

“…The starting poles of the frequency-dependent basis functions and are chosen as stable complex conjugate pairs , which have small real parts , and their imaginary parts linearly spaced over the frequency range of interest such that (15) (16) It was shown in [10] that this distribution of the poles reduces the probability that poles must be relocated over long distances and, therefore, avoids that the Sanathanan-Koerner weighting exhibits a large dynamic variation, which breaks down the numerical conditioning of the system equations [20].…”

“…1. Based on a dense set of -parameter data samples, a multivariate macromodel is calculated as a function of a varying iris height with -mm over the frequency range (12)(13)(14)(15)(16)(17)(18).…”