Cramer-Rao bounds on eigenvalue estimates from impulse response data: The multi-observation case

We examine the effect of having multiple observations in the estimation of non-random modes of linear dynamical systems from noisy impulse response data. Specifically, for this estimation problem, we develop an explicit algebraic characterization of the Fisher information matrix and hence Cramer-Rao bound in terms of the eigenvalues and residues of the transfer function, and so develop some simple bounds on the minimum possible error variance for eigenvalue estimates in terms of the eigenvalues' locations. We focus especially on developing a relationship between the Cramer-Rao bound on pole estimates for the multi-observation case, and those when each single observation is used separately for estimation.