Monte Carlo evaluation of methods for pulse transfer function estimation

This paper describes the results of a Monte Carlo evaluation made of the methods proposed in current literature for the estimation of the pulse transfer function of a linear, time-invariant dynamic system with feedback. Considered are two basic methods for estimating the coefficients of a pulse transfer function, given only the normal operating input and output of the system obscured by noise and over a limited period of time. The most commonly proposed method is a linear method in which a set of simultaneous linear equations is formed from the sampled data and the coefficients obtained by a matrix inversion. The other method is an eigenvector method proposed by Levin which uses the eigenvector associated with the smallest eigenvalue of a matrix formed from the sampled data. This paper presents a set of examples designed to compare linear and eigenvector estimation methods and to verify experimentally the theoretical results and approximations given by Levin. The comparison shows that the eigenvector method generally gives estimates with equal or smaller rms error
\sqrt{Variance+(Bias)^2}
than the linear method. The eigenvector estimates had bias magnitudes which were consistently less than their standard deviations; the linear estimates did not, and thus had rms errors which often consisted largely of the bias. The approximate covariance matrix given by Levin for the coefficients estimated with the eigenvector method is found to be reasonably accurate.