Simple approximations to the Cramer-Rao lower bound on direction of arrival for closely spaced sources

It is demonstrated that the Cramer-Rao lower bound (CRLB) on the directions of arrival (DOAs) of two closely spaced uncorrelated plane wave sources impinging on a uniform line array can be approximated by simple algebraic formulas which provide insight into the behavior of the bound with respect to signal-to-noise ratios (SNRs), source separation, etc. These formulas are sufficiently accurate for general use with the number of sensors between about 6 and 60, for source separations down to about 0.1 of a classical beamwidth, and for SNRs from -30 to +30 dB