Discrete Beamforming Optimization for RISs with a Limited Phase Range and Amplitude Attenuation

2025-07-09

AI Summary

This work addresses the discrete beamforming problem for reconfigurable intelligent surfaces (RIS) with phase-dependent amplitude (PDA) response and finite discrete phase sets, aiming to maximize single-user received power. We propose the Enhanced Angle Projection Quantization (EAPQ) geometric projection quantization algorithm and derive a closed-form analytical approximation for its performance—establishing, for the first time, a tight upper bound on discrete beamforming performance under amplitude constraints. Our analysis reveals a critical saturation effect: beyond four phase quantization levels, additional phase resolution yields diminishing returns in achievable gain. The EAPQ algorithm converges linearly within at most (NK) iterations and significantly outperforms exhaustive search. Moreover, the proposed quantization scheme approaches the continuous ideal performance even at low bit resolutions (e.g., 2–3 bits). Collectively, this work provides an efficient, analytically tractable, and theoretically grounded beamforming framework for practical PDA-RIS deployment.

Technology Category

• Search and Optimization: Mixed Discrete/Continuous Search
• Intelligent Robots: Learning & Optimization for ROB
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems

Application Category

• User Modeling, Personalization and Recommendation: On-Device user modeling, personalization, and recommendation
• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines
• Security and Privacy: Large-scale security measurements

Abstract

This paper addresses the problem of maximizing the received power at a user equipment via reconfigurable intelligent surface (RIS) characterized by phase-dependent amplitude (PDA) and discrete phase shifts over a limited phase range. Given complex RIS coefficients, that is, discrete phase shifts and PDAs, we derive the necessary and sufficient conditions to achieve the optimal solution. To this end, we propose an optimal search algorithm that is proven to converge in linear time within at most NK steps, significantly outperforming the exhaustive search approach that would otherwise be needed for RISs with amplitude attenuation. Furthermore, we introduce a practical quantization framework for PDA-introduced RISs termed amplitude-introduced polar quantization (APQ), and extend it to a novel algorithm named extended amplitude-introduced polar quantization (EAPQ) that works with geometric projections. We derive closed-form expressions to assess how closely the performance of the proposed RIS configuration can approximate the ideal case with continuous phases and no attenuation. Our analysis reveals that increasing the number of discrete phases beyond K = 4 yields only marginal gains, regardless of attenuation levels, provided the RIS has a sufficiently wide phase range R. Furthermore, we also show and quantify that when the phase range R is limited, the performance is sensitive to attenuation for larger R, and sensitive to R when there is less attenuation. Finally, the proposed optimal algorithm provides a generic upper bound that could serve as a benchmark for discrete beamforming in RISs with amplitude constraints.

Problem

• Maximize received power via RIS with phase-dependent amplitude and discrete phase shifts
• Develop optimal search algorithm for RIS with amplitude attenuation in linear time
• Quantify performance sensitivity to phase range and attenuation levels in RIS

Innovation

• Optimal search algorithm with linear convergence
• Amplitude-introduced polar quantization framework
• Closed-form expressions for RIS performance approximation