Phase-only beam synthesis by iterative semidefinite relaxations with rank refinement

In phased array antennas, by varying the complex element weights beam patterns with desired shapes can be synthesized and/or steered to desired directions. These complex weights can be implemented by using amplitude controllers and phase shifters at the system level. Since controlling the phase of an RF signal is much easier than controlling its power, many systems do not have an individual amplitude controller for each element. Hence, beamshaping and steering are to be achieved by varying only the element phases. In this work, a new approach is proposed for phase-only beam synthesis problem. In this approach, the phase-only beam synthesis is formulated as a non-convex quadratically constrained quadratic problem (QCQP). Then, it is relaxed to a convex semidefinite problem (SDP), which generally provides an undesired high rank solution. An iterative technique is developed to obtain a rank-1 solution to the relaxed convex SDP. Conducted experiments show that, proposedmethod can successfully synthesize beam shapes with desired characteristics and steering directions by using only the element phases.