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

arXiv: https://arxiv.org/abs/2507.07342 · HTML: https://mlsanity.com/paper/2507.07342/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 8 of its 14 references that are in the map.
- First posted on arXiv 9 Jul 2025.

## Builds on (most cited first, 8 of 8)

- [Intelligent Reflecting Surface: Practical Phase Shift Model and Beamforming Optimization](https://mlsanity.com/paper/1907.06002/) (2019)
- [RIS-Aided Wireless Communications: Prototyping, Adaptive Beamforming, and Indoor/Outdoor Field Trials](https://mlsanity.com/paper/2103.00534/) (2021)
- [A Linear Time Algorithm for the Optimal Discrete IRS Beamforming](https://mlsanity.com/paper/2211.04704/) (2022)
- [Practical Modeling and Beamforming for Intelligent Reflecting Surface Aided Wideband Systems](https://mlsanity.com/paper/2006.01465/) (2020)
- [Intelligent reflecting surface enhanced wideband MIMO-OFDM communications: From practical model to reflection optimization](https://mlsanity.com/paper/2007.14243/) (2020)
- [RIS-aided Near-Field Localization under Phase-Dependent Amplitude Variations](https://mlsanity.com/paper/2204.12783/) (2022)
- [RIS with insufficient phase shifting capability: Modeling, beamforming, and experimental validations](https://mlsanity.com/paper/2307.02297/) (2023)
- [Received Power Maximization Using Nonuniform Discrete Phase Shifts for RISs With a Limited Phase Range](https://mlsanity.com/paper/2406.16210/) (2024)

## Abstract (arXiv)

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.