A Generalized theorem on the average array directivity factor

The beampattern of an array consisting of $N$ elements is determined by the beampatterns of the individual elements, their placement, and the weights assigned to them. For each look direction, it is possible to design weights that maximize the array directivity factor (DF). For the case of an array of omnidirectional elements using optimal weights, it has been shown that the average DF over all look directions equals the number of elements. The validity of this theorem is not dependent on array geometry. We generalize this theorem by means of an alternative proof. The chief contributions of this letter are a) a compact and direct proof, b) generalization to arrays containing directional elements (such as cardioids and dipoles), and c) generalization to arbitrary wave propagation models. A discussion of the theorem's ramifications on array processing is provided.