Optimum design of directive antenna arrays subject to random variations

This paper discusses the optimum design of discrete, directive antenna arrays of arbitrary geometrical configuration in space, when the excitations and spatial positions of the elements vary in a random fashion about their nominal values. Under certain assumptions the expected power pattern of an array turns out to be the power pattern of the nominal array, plus a “background” power level which has the same dependence on direction as the pattern of a single element. A set of excitations which maximizes the theoretical directivity of an array may correspond to a superdirective design, in which the background power level will completely swamp the desired pattern unless the excitations and positions of the elements are controlled with extraordinary precision. A method is given for maximizing the gain of the array while holding the expected background power level constant, when the precision with which the excitations and positions can be controlled is known. The method is illustrated with numerical examples.