A generalized Cramer–Rao lower bound for line arrays

Abstract

The Cramer–Rao lower bound (CRLB) on the variance of an estimate is a consequence of the underlying likelihood function. That is to say, the more realistic the likelihood function, the more realistic the bound. It is shown here that by including the forward motion of a line array in the likelihood function for the bearing of a continuous broadband signal, the CRLB on the bearing estimate is lower than that found for the case of the stationary array. This is a consequence of the fact that there is bearing information contained in the Doppler that is not exploited by conventional beamformers. Further, it is shown that this improvement in performance requires that a nuisance parameter in the form of a source frequency be jointly estimated along with the bearing. An example is given which uses a recursive processor and a synthetic broadband signal from a nonmoving source. It is shown that the performance improvement relative to a stationary array of equal physical aperture is a function of the array speed, the integration time, the bearing angle and the direction of motion of the array, i.e., whether the motion incurs ‘‘up’’ or ‘‘down’’ Doppler.