New Results on Deterministic Cramér–Rao Bounds for Real and Complex Parameters

The Cramér-Rao bounds (CRB) is a lower bound of great interest for system analysis and design in the asymptotic region [high signal-to-noise ratio (SNR) and/or large number of snapshots], as it is simple to calculate and it is usually possible to obtain closed form expressions. The first part of the paper is a generalization to complex parameters of the Barankin rationale for deriving MSE lower bounds, that is the minimization of a norm under a set of linear constraints. With the norm minimization approach the study of Fisher information matrix (FIM) singularity, constrained CRB and regularity conditions become straightforward corollaries of the derivation. The second part provides new results useful for system analysis and design: a general reparameterization inequality, the equivalence between reparameterization and equality constraints, and an explicit relationship between parameters unidentifiability and FIM singularity.