ON THE DESIGN OF MAXIMALLY SPARSE BEAMFORMING ARRAYS

The problem of designing nonuniformly spaced arrays is formulated as one of constrained optimization in which the cost function is chosen to select the array with the minimum number of elements. The response of the array is controlled by a set of inequality point response constraints. It is shown that a suitable cost function for this problem is the l(p) quasi-norm for 0 < p < 1 and that there exists a number p1 such that for all 0 < p < p1 the resulting array is maximally sparse. Furthermore it is shown that a solution to the problem lies at an extreme point of the simplex formed by the point constaints. A simplex search algorithm is described which will converge to a local minimum of the cost function on this simplex. The algorithm is illustrated in application to the design of sparse linear and planar arrays.