The Cramer-Rao bound on frequency estimates of signals closely spaced in frequency (unconditional case)

The paper analyzes the Cramer-Rao (CR) bound on frequency estimation covariance for the unconditional (or stochastic) signal model. It addresses the problem of n signals closely spaced in (temporal or spatial) frequency. The main result is that for this regime, the CR bound decomposes into a product of simple scalar factors that individually reflect frequency separation, signal powers and covariances, data sampling grid, and sample size. The factored expression provides useful insight into the behavior of the bound for closely spaced frequencies. The result also leads to a new formula for the signal-to-noise (SNR) threshold at which an unbiased frequency estimator can resolve signals closely spaced in frequency. Interestingly, with a simple modification, the formulae are identical to those recently obtained for the conditional (or deterministic) signal model