The statistical errors in the estimated correlation function matrix for operational modal analysis

Abstract

Given the random vibration of a linear and time-invariant system, the correlation function matrix is equivalent to free decays when the system is excited by Gaussian white noise. Correlation-driven Operational Modal Analysis utilises these properties to identify modal parameters from systems in operation based on the response only. Due to the finite length of the system response, the correlation function matrix must be estimated and this introduces statistical errors. This article focuses on the statistical errors due to this estimation process and the effect it has on the envelope and zero crossings of the estimated correlation function matrix. It is proven that the estimated correlation function matrix is a Gaussian stochastic process. Furthermore, it is proven that the envelope of the modal correlation function matrix is Rice distributed. This causes the tail region of the correlation function to become erroneous - called the noise tail. The zero crossings are unbiassed, but the random error related to the crossings increases fast in the noise tail. The theory is tested on a simulated case and there is a high agreement between theory and simulation. A new expression for the minimal time length is introduced based on the bias error on the envelope.