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Beamforming Design for Beyond Diagonal RIS-Aided Cell-Free Massive MIMO Systems

Yizhuo Li, Jiakang Zheng, Bokai Xu, Yiyang Zhu, Jiayi Zhang, Dusit Niyato, Bo Ai

TL;DR

The paper tackles sum-SE maximization in BD-RIS–aided CF mMIMO by formulating a unified beamforming framework that accommodates hybrid transmitting/reflecting BD-RIS architectures. It develops an alternating optimization method that decouples active and passive beamforming, employing Lagrangian dual and quadratic transforms for tractability. To handle unitary constraints on BD-RIS, it introduces a Riemannian L-BFGS algorithm on the Stiefel manifold, achieving near full-second-order performance with reduced complexity. Simulations demonstrate superior performance of BD-RIS over traditional RIS, with fast convergence and robustness to CSI errors, highlighting practical potential for scalable 6G deployments.

Abstract

Reconfigurable intelligent surface (RIS)-aided cell-free (CF) massive multiple-input multiple-output (mMIMO) is a promising technology for further improving spectral efficiency (SE) with low cost and power consumption. However, conventional RIS has inevitable limitations due to its diagonal scattering matrix. In contrast, beyond-diagonal RIS (BD-RIS) has gained great attention. This correspondence focuses on integrating a hybrid transmitting and reflecting BD-RIS into CF mMIMO systems to enhance coverage and spatial multiplexing. This requires completing the beamforming design under the transmit power constraints and unitary constraints of the BD-RIS, by optimizing active and passive beamformer simultaneously. To tackle this issue, we introduce an alternating optimization algorithm that decomposes it using fractional programming and solves the subproblems alternatively. Moreover, to address the challenge introduced by the unitary constraint on the beamforming matrix of the BD-RIS, we propose a Riemannian limited-memory Broyden-Fletcher-Goldfarb-Shanno (R-L-BFGS) algorithm to solve the problem optimally. Simulation results show that our algorithm achieves faster convergence and finds higher-quality solutions compared with baselines, while also demonstrating a favorable performance-complexity trade-off.

Beamforming Design for Beyond Diagonal RIS-Aided Cell-Free Massive MIMO Systems

TL;DR

The paper tackles sum-SE maximization in BD-RIS–aided CF mMIMO by formulating a unified beamforming framework that accommodates hybrid transmitting/reflecting BD-RIS architectures. It develops an alternating optimization method that decouples active and passive beamforming, employing Lagrangian dual and quadratic transforms for tractability. To handle unitary constraints on BD-RIS, it introduces a Riemannian L-BFGS algorithm on the Stiefel manifold, achieving near full-second-order performance with reduced complexity. Simulations demonstrate superior performance of BD-RIS over traditional RIS, with fast convergence and robustness to CSI errors, highlighting practical potential for scalable 6G deployments.

Abstract

Reconfigurable intelligent surface (RIS)-aided cell-free (CF) massive multiple-input multiple-output (mMIMO) is a promising technology for further improving spectral efficiency (SE) with low cost and power consumption. However, conventional RIS has inevitable limitations due to its diagonal scattering matrix. In contrast, beyond-diagonal RIS (BD-RIS) has gained great attention. This correspondence focuses on integrating a hybrid transmitting and reflecting BD-RIS into CF mMIMO systems to enhance coverage and spatial multiplexing. This requires completing the beamforming design under the transmit power constraints and unitary constraints of the BD-RIS, by optimizing active and passive beamformer simultaneously. To tackle this issue, we introduce an alternating optimization algorithm that decomposes it using fractional programming and solves the subproblems alternatively. Moreover, to address the challenge introduced by the unitary constraint on the beamforming matrix of the BD-RIS, we propose a Riemannian limited-memory Broyden-Fletcher-Goldfarb-Shanno (R-L-BFGS) algorithm to solve the problem optimally. Simulation results show that our algorithm achieves faster convergence and finds higher-quality solutions compared with baselines, while also demonstrating a favorable performance-complexity trade-off.

Paper Structure

This paper contains 10 sections, 21 equations, 1 figure, 3 algorithms.

Table of Contents

  1. Introduction
  2. SYSTEM MODEL
  3. BEAMFORMING DESIGN
  4. Overview of the Beamforming Design Framework
  5. Active Beamforming: Solve $\mathbf{w}$
  6. Passive Beamforming: Solve $\{$$\mathbf{\Theta}_\mathrm{t}, \mathbf{\Theta}_\mathrm{r}$$\}$
  7. Riemannian BFGS Method
  8. Proposed Riemannian L-BFGS
  9. Simulation Results
  10. Conclusion

Figures (1)

  • Figure 1: (a): Sum-SE against the number of $\text{iterations } I_{\mathrm{o}}$ with $L=3$, $K=4$, $M=16$, $N=2$, ${P}_{l,\mathrm{max}}=0.001$, $G=2$. (b): Sum-SE against the transmit power per AP with $L=3$, $K=4$, $M=32$, $N=2$, $G=2$. (c): Sum-SE against the CSI error parameter $\delta$ with $L=3$, $K=4$, $M=32$, $N=2$, ${P}_{l,\mathrm{max}}=0.003$, $G=2$. (d): Computational complexity comparison: (1) average CPU time per iteration and (2) average number of iterations to convergence, versus the number of BD-RIS cells $M$. The experimental platform is built on a Windows 11 system with 13th Gen Intel(R) Core(TM) i7-13650HX.