Optimal Non-Parametric Window-Based Linear-Phase Weighting in Uniform Linear Array Beamforming

This paper formulates beamforming synthesis as a multi-objective, multi-variable optimization problem to minimize the number of array elements while meeting constraints on half-power beamwidth, sidelobe level, and radiation attenuation in undesired directions. The problem is cast into a weighted single-objective formulation by unifying normalized beamwidth and sidelobe metrics. Twelve non-parametric window functions—parameterized solely by array length—are investigated. To identify the optimal window and minimum array size, two optimization algorithms are developed. While both guarantee the identical global optimum, the second algorithm significantly lowers computational complexity through progressive search-space pruning. A rigorous complexity analysis formalizes these efficiency gains. Simulations across five distinct constraint scenarios demonstrate that the framework reliably identifies the optimal window function and the minimum feasible element count while satisfying all radiation requirements.