Perturbation theory-based performance analysis of TLS-Prony method for natural frequency extraction

Abstract

The total least squares Prony method is used for estimating s-plane natural frequencies from noisy late time response. It consists of three steps: In the first step, the minimum eigenvector of a matrix, whose entries are defined from the noisy late time response, should be computed. In the second step, the roots of a polynomial, whose coefficients are the entries of the minimum eigenvector defined in the first step, are estimated. The roots in the second step are defined as z-plane natural frequencies. In the third step, s-plane natural frequencies are obtained from z-plane natural frequencies. In this paper, a rigorous derivation of the mean square error of the estimator in each step due to an additive noise in the late time response is presented: In the first step, it is shown how the minimum eigenvector of a matrix is perturbed due to an additive noise in the late time response. In the second step, it is derived how the roots of a polynomial are perturbed due to perturbation in the coefficients of the polynomial. In the third step, it is derived how the s-plane natural frequencies are perturbed due to perturbation in z-plane natural frequencies. The mean square error for the first step, the second step and the third step are presented in (26), (36), and (44), respectively, and the validity of these expressions is illustrated in the results using simulated response and experimentally measured data. In conclusion, the mean square errors of the minimum eigenvector, the z plane natural frequencies, and the s plane natural frequencies for the total least squares Prony method can be obtained analytically from (26), (36) and (44) without a computationally intensive Monte-Carlo simulation.