Received Power Maximization Using Nonuniform Discrete Phase Shifts for RISs With a Limited Phase Range

Abstract: To maximize the received power at a user equipment, the problem of optimizing a reconfigurable intelligent surface (RIS) with a limited phase range R < 2{\pi} and nonuniform discrete phase shifts with adjustable gains is addressed. Necessary and sufficient conditions to achieve this maximization are given. These conditions are employed in two algorithms to achieve the global optimum in linear time for R {\ge} {\pi} and R < {\pi}, where R is the limited RIS phase range. With a total number of N(2K + 1) complex vector additions, it is shown for R {\ge} {\pi} and R < {\pi} that the global optimality is achieved in NK or fewer and N(K + 1) or fewer steps, respectively, where N is the number of RIS elements and K is the number of discrete phase shifts which may be placed nonuniformly over the limited phase range R. In addition, we define two quantization algorithms that we call nonuniform polar quantization (NPQ) algorithm and extended nonuniform polar quantization (ENPQ) algorithm, where the latter is a novel quantization algorithm for RISs with a significant phase range restriction, i.e., R < {\pi}. With NPQ, we provide a closed-form solution for the approximation ratio with which an arbitrary set of nonuniform discrete phase shifts can approximate the continuous solution. We also show that with a phase range limitation, equal separation among the nonuniform discrete phase shifts maximizes the normalized performance. Furthermore, we show that the gain of using K {\ge} 3 with R < {\pi}/2 and K {\ge} 4 with R < {\pi} is only marginal. Finally, we prove that when R < 2{\pi}/3, ON/OFF selection for the RIS elements brings significant performance compared to the case when the RIS elements are strictly ON.