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

TL;DR

This work addresses maximizing received power in RIS-assisted links when the RIS phase range is limited ($R<2\pi$) and the available phase shifts are nonuniform and discrete. It derives necessary and sufficient optimality conditions and proposes two linear-time algorithms that achieve global optimum for $R\ge\pi$ and $R<\pi$, respectively, along with a practical nonuniform quantization method (NPQ) and an enhanced ENPQ variant that exploits ON/OFF RIS gains for small $R$. The authors show that equally spaced nonuniform phase shifts over the RIS range maximize the average performance, and provide closed-form approximation ratios for NPQ and ENPQ, illustrating how close these schemes come to the continuous optimum as $N$ grows. They also analyze the impact of phase-range limitation on performance and demonstrate substantial gains from gain flexibility when $R<\pi$, including significant improvements from ENPQ over NPQ in such regimes. Overall, the paper offers scalable, theory-backed strategies for RIS design under realistic phase-range constraints, with practical quantization schemes and performance guarantees.

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 < 2π 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} π and R < π, 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} π and R < π 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 < π. 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 < π/2 and K {\ge} 4 with R < π is only marginal. Finally, we prove that when R < 2π/3, ON/OFF selection for the RIS elements brings significant performance compared to the case when the RIS elements are strictly ON.

Figures (19)

• Figure 1: Nonuniform phase placement for $R \in [0,2\pi]$.
• Figure 2: PDF of $\delta_n$, i.e., the quantization error.
• Figure 3: $E(R,K)$ vs $R$ for $K \in \{2,3,4,6,8\}$
• Figure 4: An illustration for the optimality of $\theta_n^*=\phi_k$ given $\mu \in {\rm arc} (s_{nk}:s_{n,k+1})$.
• Figure 5: CDF plots for SNR Boost with PA24, PA24, Algorithm 1, and nonuniform polar quantization (NPQ), for $K=4$.