Discrete Variable Beamforming for Intelligent Reflecting Surfaces

In this paper, we present a novel algorithm called Optimal Partitioning Algorithm (OPA) to solve the discrete beamforming problem where the weight of each unit cell in an IRS is constrained to a binary set. OPA is based on the idea that the optimal partition of the weights must arise due to a separating line in the complex plane. Unlike the conventional quantization methods and evolutionary algorithms that are heuristic in nature, our proposed algorithm comes with proof of optimality while also being very fast due to its linear time complexity, making it vastly superior to its counterparts. Using our algorithm to obtain optimal weights, we analyze the existence of grating lobes and explore various methods to mitigate them, such as the use of triangular lattices and prephasing. We show that our algorithm provides a rigorous framework for the execution of the above strategies.