Cramer-Rao bounds for antenna array design

We study the impact of the geometry of the (planar) antenna array on the accuracy of the estimated direction(s) of arrivals of an emitting source. We develop explicit Cramer-Rao bounds (CRBs) of the azimuth and elevation angles that show a simple structure. In particular, for a fixed elevation angle, the CRBs are cosine functions of the source azimuth, and so, regardless of the array geometry. The amplitude and extremes of these functions depend on the array geometry. Hence, the array configuration can be chosen in order to ensure a desired (an)isotropic behavior. To do so, we propose a pragmatic methodology that also takes into account the array ambiguity problem. The array design problem is simplified by limiting the array search within a family of V-shaped arrays that are advantageously characterized by a single parameter, the angle between the two branches. A performance measure is proposed, then analytically expressed, to assess the array directivity and gain with respect to the more standardly used uniform circular array.