A Semidefinite Relaxation Method for Directivity Maximization and Sparse Array Synthesis of a Flat-Top Beam Array

A Semidefinite Relaxation (SDR) method combined with Successive Convex Approximation (SCA) is proposed to maximize the directivity and sparsity of the flat-top beam array. First, the complex excitations of the antenna elements are defined as optimization variables, and the problem of directivity maximization with flat-top beam is then reformulated as an SDR problem. Next, the non-convex constraint in the SDR is transformed into the optimization objective according to the equivalence between the trace of the matrix and itsl2-norm. Finally, based on the SCA method, a lower bound function of the first-order Taylor expansion of the matrixl2-norm is used to approximate the non-convex terms. The new contribution of this proposed method lies in three points. 1) It achieves high optimization accuracy because the rank-one constraint in the SDR is equivalently transformed and solved within the SCA method. 2) The the optimization results are less affected by the initial excitation, which increases the computation efficiency. 3) It exhibits strong applicability, which can achieve sparse array synthesis for flat-top beams based on directivity maximization. The proposed method is validated by optimizing several linear and planar arrays. On the condition of different initial excitations, the optimized directivity of a 41-element linear array are identical, and those of a 14×14 planar array have the difference less than 0.19 dB. The sparsification rate of 44% is achieved for the linear array and 35.2% for the planar array.