THE CRAMER-RAO BOUND ON FREQUENCY ESTIMATES OF SIGNALS CLOSELY SPACED IN FREQUENCY

This paper examines the Cramer-Rao (CR) lower bound on the variance of frequency estimates for the problem of n signals closely spaced in frequency. The dependence of the bound upon maximum frequency separation (delta-omega), the signal-to-noise ratio (SNR), and the number N of data vectors (or snapshots), bears importantly upon the performance of high-resolution techniques for spectrum analysis. Previously these dependences have been explored via simulation due to the complexity of the Fisher information matrix. The main results presented herein are simple analytic expressions for the CR bound in terms of delta-omega, SNR and N valid for small delta-omega. The results are applicable to the conditional (deterministic) signal model. The results show that the CR bound on frequency estimates is proportional to (delta-omega)-2(n-1)/N.SNR. Therefore, the bound increases rapidly as the signal separation is reduced. Examples indicate that the expressions closely approximate the exact CR bounds whenever the signal separation is smaller than one resolution cell (e.g., reciprocal record duration, or beamwidth). Based upon the results, it is argued that the threshold SNR at which an unbiased estimator can resolve n closely spaced signals is at least proportional to (delta-omega)-2n/N. The results are quite general, and apply to many different types of temporal and spatial sampling grids.