An approximate expression for the Cramer-Rao bound on DOA estimates of closely spaced sources in broadband line-array beamforming

It is demonstrated that the Cramer-Rao lower bound on DOA estimation with two uncorrelated but closely-spaced wideband sources is well approximated by a straightforward modification of a previously published, extremely simple, non-matrix, narrowband formula. The approximation is useful, in the context of a uniform line array with constant SNR across the band, for a number of sensors from about 6 to 60, SNRs from -30 dB to +30 dB, source separations down to about 0.1 beamwidths and bandwidths approaching an octave. This covers many of the situations in the array processing literature. Variants of the basic formula are also discussed