Application of Convex Relaxation to Array Synthesis Problems

A general procedure to solve efficiently non convex array synthesis problems is presented. It is based on the SemiDefinite Relaxation (SDR) technique. The way to properly relax the constraints in order to formulate the synthesis of shaped beams, phase-only arrays and reconfigurable arrays as semidefinite programming problems is detailed. These so-approximated array synthesis problems are then convex, easy to implement and can be efficiently solved using off-the-shelf numerical routines. The conditions under which the relaxed problems provide the optimal solution to the original non convex synthesis problems are specified. Various representative numerical comparisons with arrays designed by other approaches show the validity of the proposed method and illustrate its potentialities.