Nonuniform linear antenna arrays minimising Cramer-Rao bounds for joint estimation of single source range and direction of arrival

Different antenna placements in nonuniform linear arrays for the joint estimation of the range and direction-of-arrival (DOA) of one source are considered. It is shown that the Cramer-Rao bound (CRB) for the range and DOA estimates can be expressed as a function of the angles at which the source sees the antennas. This new formulation of the CRB easily reveals the geometrical properties that the array should have in order to minimise the CRB. Based on the new representation of the CRB, a result found in the literature (which was concerned with the special case of a number of antennas that is divisible by four) is generalised to include the case of an arbitrary number of antennas. Furthermore, it is shown that symmetric arrays are close to uniformly optimal for different DOAs when the range is much larger than the aperture of the arrays.