Estimation of impulse response by using a nonwhite input—analysis of estimation errors by geometrical approach

Abstract

When the ordinary least-square method is applied in the estimation of the impulse response of the linear system using a nonwhite input, the result is affected greatly by the observation noise and the error in the numerical calculation, and the accuracy of the estimation is greatly deteriorated. This paper presents a mathematical analysis based on a geometrical approach for the error generation mechanism and provides a clear physical description from the standpoint of the system identification. It is shown that the nonwhite input has the following problem. The excitation turns out to be weak for the linear system in which the impulse response vector corresponds to the eigenvector for a small eigenvalue of the autocorrelation matrix. This weakness of the excitation is the major cause of the deterioration of the estimation accuracy in the ordinary least-square method. Based on those results, a new physical interpretation is given to the eigenvalue truncation method and the regularization method, which are already proposed as improvements of the ordinary least-square method. The proposed error analysis can also be considered as the basic theory which can be applied not only to the estimation of the impulse response, but also to the parameter estimation of the transfer function model and the design of the optimal input for the identification.