Variance error, interpolation and experiment design

We investigate how the variance error associated with the prediction error identification is related to the power spectral densities of the input and the additive noise at the output. Let Φ(eiω) be the ratio of the input power spectral density (PSD) to the output-noise PSD. We characterize the set of all functions ΦΦ for which the variance error remains constant. This analysis results a minimal, finite-dimensional, affine parameterization of the variance error. This parameterization connects our analysis with the theory of Nevanlinna–Pick interpolation. It is shown that the set of all ΦΦ for which the variance error remains constant can be characterized by the solutions of a Nevanlinna–Pick interpolation problem. This insight has interesting consequences in optimal input design, where it is possible to use some recent tools in analytic interpolation theory to tune shape of the input PSD to suit certain needs.