Lower bounds on bearing estimation errors of any number of narrowband far-field sources

The authors present a closed-form, simple expression for a lower bound on the variance of an unbiased estimator of the bearing of a narrowband source in the presence of any number of interfering sources received by an arbitrary M-sensor array. The sources radiate narrowband, statistically independent Gaussian signals. The spectral levels of all signals are unknown a priori. The bound is a good approximation of the Cramer-Rao lower bound (CRLB) in large signal-to-noise-ratio (SNR) and/or observation time conditions. Since the CRLB is used as a measure of efficiency for source location estimation algorithms, the bound gives some insight into the asymptotic properties of an optimal estimator. It shows that the estimation of the bearing of the ith source is inversely proportional to the SNR of this source and is not a function of the SNRs of the other sources. The incremental error due to the presence of other sources is uniquely determined by the beam pattern of the conventional array and its first derivative at the different source spatial separations.