The problem of optimizing discrete phases in a reconfigurable intelligent surface (RIS) to maximize the received power at a user equipment is addressed. Necessary and sufficient conditions to achieve this maximization are given. These conditions are employed in an algorithm to achieve the maximization. New versions of the algorithm are given that are proven to achieve convergence in N or fewer steps whether the direct link is completely blocked or not, where N is the number of the RIS elements, whereas previously published results achieve this in KN or 2N number of steps where K is the number of discrete phases. Thus, for a discrete-phase RIS, the techniques presented in this paper achieve the optimum received power in the smallest number of steps published in the literature. In addition, in each of those N steps, the techniques presented in this paper determine only one or a small number of phase shifts with a simple elementwise update rule, which result in a substantial reduction of computation time, as compared to the algorithms in the literature. As a secondary result, we define the uniform polar quantization (UPQ) algorithm which is an intuitive quantization algorithm that can approximate the continuous solution with an approximation ratio of sinc^2(1/K) and achieve low time-complexity, given perfect knowledge of the channel.

翻译：本文研究了在可重构智能表面（RIS）中优化离散相位以最大化用户设备接收功率的问题。给出了实现该最大化的充要条件，并利用这些条件设计了一种算法以实现最大化。本文提出了该算法的新版本，证明无论是直视链路完全被阻断还是未被阻断，该算法均能在N步或更少步数内收敛，其中N为RIS单元数，而此前发表的结果需KN或2N步（K为离散相位数）。因此，对于离散相位RIS，本文提出的技术以文献中发布的最少步数实现了最优接收功率。此外，在这N步的每一步中，本文技术仅通过简单的逐元素更新规则确定一个或少量相移，相比文献中的算法显著降低了计算时间。作为辅助结果，我们定义了均匀极性量化（UPQ）算法，这是一种直观的量化算法，可在完美信道信息条件下以sinc^2(1/K)的近似比逼近连续解，并实现低时间复杂度。