Cramer-Rao bounds for deterministic modal analysis

How accurately can deterministic modes be identified from a finite record of noisy data? In this paper we answer this question by computing the Cramer-Rao bound on the error covariance matrix of any unbiased estimator of mode parameters. The bound is computed for many of the standard parametric descriptions of a mode, including autoregressive and moving average parameters, poles and residues, and poles and zeros. Asymptotic, frequency domain versions of the CramerRao bound bring insight into the role played by poles and zeros. Application of the bound to secondand fourth-order systems illustrates the coupling between estimator errors and illuminates the influence of mode locations on our ability to identify them. Application of the bound to the estimation of an energy spectrum illuminates the accuracy of estimators that presume to resolve spectral peaks.