Optimal Beamforming of RIS-Aided Wireless Communications: An Alternating Inner Product Maximization Approach

TL;DR

This work tackles discrete uni-modular phase optimization for RIS-assisted beamforming by formulating it as general $\ell_p$-norm maximization and introducing a concise alternating inner product maximization framework. A key innovation is the divide-and-sort (DaS) search that delivers a polynomial-sized candidate set for discrete inner products, enabling efficient and scalable solutions, along with a post-rounding lifting mechanism that mitigates performance loss from quantization. The method achieves monotonic convergence, demonstrates notable SNR gains in simulations, and is validated through prototyping and field trials, including 4-bit quantization performing almost as well as continuous phase configurations. The practical impact is a scalable, robust RIS optimization approach suitable for large-scale deployments with moderate quantization, reducing hardware costs while preserving performance.

Abstract

This paper investigates a general discrete $\ell_p$-norm maximization problem, with the power enhancement at steering directions through reconfigurable intelligent surfaces (RISs) as an instance. We propose a mathematically concise iterative framework composed of alternating inner product maximizations, well-suited for addressing $\ell_1$- and $\ell_2$-norm maximizations with either discrete or continuous uni-modular variable constraints. The iteration is proven to be monotonically non-decreasing. Moreover, this framework exhibits a distinctive capability to mitigate performance degradation due to discrete quantization, establishing it as the first post-rounding lifting approach applicable to any algorithm intended for the continuous solution. Additionally, as an integral component of the alternating iterations framework, we present a divide-and-sort (DaS) method to tackle the discrete inner product maximization problem. In the realm of $\ell_\infty$-norm maximization with discrete uni-modular constraints, the DaS ensures the identification of the global optimum with polynomial search complexity. We validate the effectiveness of the alternating inner product maximization framework in beamforming through RISs using both numerical experiments and field trials on prototypes. The results demonstrate that the proposed approach achieves higher power enhancement and outperforms other competitors. Finally, we show that discrete phase configurations with moderate quantization bits (e.g., 4-bit) exhibit comparable performance to continuous configurations in terms of power gains.

Figures (17)

• Figure 1: RIS-aided point-to-point MISO communication. (a) line-of-sight (LoS) scenario. (b) non-line-of-sight (NLoS) scenario.
• Figure 2: The approach to find a suitable solution to \ref{['E:Subproblem']}. (a) When $\psi$ is given and held fixed, the step to determine the best phase configuration $\Omega_{i, \text{opt}} (\psi)$ is to check which region $\psi$ belongs to. (b) The piecewise constant nature of the function $\Omega_{i, \text{opt}} (\psi)$.
• Figure 3: Illustration of region rearrangement. There are two groups of partitions ($n=2$), represented by red (on the left) and blue arcs (in the middle), each containing $2^B$ regions. After the rearrangement, these two groups combine to form a total of $2 \times 2^B$ non-overlapping regions (on the right).
• Figure 4: Convergence behavior and lifting performance. We set $\mathbf{A} \in \mathbb{C}^{10 \times 100}$ and $B = 2$. Both discrete and continuous alternating inner product maximizations converge rapidly, requiring only a small number of iterations. The proposed lifting approach recovers the hard rounding loss. (a) and (b) illustrate the value of cost function for solving $(Q_1)$ and $(Q_2)$, respectively.
• Figure 5: Lift the solution obtained using Manopt. We set $\mathbf{A} \in \mathbb{C}^{10 \times 100}$ and $B = 1$, conducting 20,000 trials for $p=1, 2$. The distribution of the relative lifting gain is visualized, with the red line marking the median. (a) and (b) illustrate the results for solving $(Q_1)$ and $(Q_2)$, respectively.

Theorems & Definitions (6)

• Theorem 1
• proof
• Remark 1
• Remark 2
• Proposition 2
• Remark 3