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 constraints. A simplex search algorithm is described which will converge to a local minimum of the cost function on this simplex. The algorithm is applied to the design of sparse linear and planar arrays