Constrained and unconstrained synthesis of array factor for circular arrays

We consider arrays of N isotropic elements equally spaced on a circular ring. Our purpose is to determine the N element excitations which minimize the mean square distance between the corresponding array factor and a given radiation pattern. At first we examine the unconstrained synthesis problem, obtaining a closed form solution. Then we consider the case where each excitation is constrained to belong to a prescribed set of complex numbers. To solve this problem, we propose an iterative technique which produces a sequence of excitation vectors. Each point of the sequence is derived from the preceding one by modifying one of its N components within the corresponding constraint set, in such a way as to minimize the mean square distance. We show that if the constraint sets are closed and convex, then the sequence converges to the unique point of global minimum of the mean square distance for any starting point of the sequence. If the constraint sets are non-convex, the sequence admits a convergent subsequence, and the limit of the latter is a good approximation of the solution to the problem, in some cases of interest. Some examples illustrate the effectiveness of the method.