On the variance error in system identification

Abstract

AbstractWe investigate how the noise and input power spectral densities influence the system identification performance. Consider the problem of identifying a linear system using the prediction error method, where Φ(eiω) is the ratio of the input power spectral density to the noise power spectral density. It is shown that the asymptotic estimation variance of the transfer function estimate is determined entirely by the non-causal part of Φ(eiω) and its first few derivatives evaluated at the poles of the system. As a result, the set of all input power spectral densities for which the asymptotic variance error remains constant, span an infinite dimensional linear manifold, which can be characterized by the solutions of a Nevanlinna-Pick interpolation problem. Since the solutions of the degree constrained Nevanlinna-Pick interpolation problem can be freely parameterized by the spectral zeros, it is possible to freely choose the zeros of Φ(eiω) without disturbing the asymptotic variance error. This provide some interesting insights on the small sample performance of the prediction error method.