Estimation of a system pulse transfer function in the presence of noise

Statistical estimation theory is applied to derive effective techniques for measurement of the pulse transfer function of a linear system from normal operating records obscured by additive noise. It is shown that the problem is equivalent to that of fitting a hyperplane to a set of observed points with random errors in certain coordinates. A method of Koopmans is applied to obtain generalized least squares estimates which are also maximum likelihood estimates when the noise is white and Gaussian. The estimates of the coefficients are obtained as the components of the eigenvector corresponding to the smallest eigenvalue of a matrix equation involving the sample auto- and cross-correlation functions of the input and output records and the covariance matrix of the corresponding noise components. Expressions for the sampling variances and biases are given. The properties of the simpler standard least squares estimates are also considered. The appropriate modifications for nonwhite noise are described.