Analytical investigation into error propagation of power spectral density transmissibility (PSDT) based on coherence function

Abstract

Power spectral density transmissibility (PSDT) has attracted widespread interests because it is insensitive to the nature of excitation and has no demand for measuring the input. The classical spectral estimation procedure is usually subject to measurement and spectral leakage errors, which might cause distortion in the estimation of PSDT. This study investigates how the error of PSD estimation propagates into PSDT estimation based on analytical approximation formulas. Based on perturbation theory in tandem with statistical moments of the ratio of random variables, both first- and second-order asymptotic expressions of the mean and variance of PSDT estimation are derived in terms of coherence functions, PSDT measurements, and the number of averages. Given these theoretical findings, it is further revealed that the variance of PSDT estimation around the system poles approaches zero, and the variances of the magnitude and phase of the mode shape decrease to local minima. The performance of the analytical error propagation formulas is validated through numerical and field test data. Parametric studies into the effects of recording durations, window types, and reference outputs on the statistics of PSDT estimation and mode shape estimation are also conducted.