Complex-curve fitting

TLDR

The mathematical analysis of linear dynamic systems often requires fitting the frequency response with a ratio of two frequency‑dependent polynomials. The paper presents a method for evaluating polynomial coefficients in frequency‑response fitting. The method minimizes the weighted sum of squared errors between the actual magnitude and the polynomial ratio at selected frequencies, reducing coefficient determination to solving determinants whose elements depend on amplitude ratio and phase shift, and is limited to systems without poles on the imaginary axis. This solution is particularly adaptable to digital computing methods due to its simplicity in required programming.

Abstract

The mathematical analysis of linear dynamic systems, based on experimental test results, often requires that the frequency response of the system be fitted by an algebraic expression. The form in which this expression is usually desired is that of a ratio of two frequency-dependent polynomials. In this paper, a method of evaluation of the polynomial coefficients is presented. It is based on the minimization of the weighted sum of the squares of the errors between the absolute magnitudes of the actual function and the polynomial ratio, taken at various values of frequency (the independent variable). The problem of the evaluation of the unknown coefficients is reduced to that of the numerical solution of certain determinants. The elements of these determinants are functions of the amplitude ratio and phase shift, taken at various values of frequency. This form of solution is particularly adaptable to digital computing methods, be-because of the simplicity in the required programming. The treatment is restricted to systems which have no poles on the imaginary axis; i.e., to systems having a finite, steady-state (zero frequency) magnitude.