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Reciprocal Beyond-Diagonal Reconfigurable Intelligent Surface (BD-RIS): Scattering Matrix Design via Manifold Optimization

Marko Fidanovski, Iván Alexander Morales Sandoval, Hyeon Seok Rou, Giuseppe Thadeu Freitas de Abreu, Emil Björnson

Abstract

Beyond-diagonal reconfigurable intelligent surfaces (BD-RISs) are emerging as a transformative technology in wireless communications, enabling enhanced performance and quality of service (QoS) of wireless systems in harsh urban environments due to their relatively low cost and advanced signal processing capabilities. Generally, BD-RIS systems are employed to improve robustness, increase achievable rates, and enhance energy efficiency of wireless systems in both direct and indirect ways. The direct way is to produce a favorable propagation environment via the design of optimized scattering matrices, while the indirect way is to reap additional improvements via the design of multiple-input multiple-output (MIMO) beamformers that further exploit the latter "engineered" medium. In this article, the problem of sum-rate maximization via BD-RIS is examined, with a focus on feasibility, namely low-complexity physical implementation, by enforcing reciprocity in the BD-RIS design in a manner that adheres to the geometry of the manifold of symmetric matrices. To that end, the sum-rate objective is transformed into a quadratic function via fractional programming (FP), augmented via the also quadratic reciprocity constraint in the form of a regularization term, while the unitary constraint is dealt with via a manifold optimization framework. Simulation results demonstrate the effectiveness of the proposed method in outperforming current state-of-the-art (SotA) approaches in terms of sum-rate maximization.

Reciprocal Beyond-Diagonal Reconfigurable Intelligent Surface (BD-RIS): Scattering Matrix Design via Manifold Optimization

Abstract

Beyond-diagonal reconfigurable intelligent surfaces (BD-RISs) are emerging as a transformative technology in wireless communications, enabling enhanced performance and quality of service (QoS) of wireless systems in harsh urban environments due to their relatively low cost and advanced signal processing capabilities. Generally, BD-RIS systems are employed to improve robustness, increase achievable rates, and enhance energy efficiency of wireless systems in both direct and indirect ways. The direct way is to produce a favorable propagation environment via the design of optimized scattering matrices, while the indirect way is to reap additional improvements via the design of multiple-input multiple-output (MIMO) beamformers that further exploit the latter "engineered" medium. In this article, the problem of sum-rate maximization via BD-RIS is examined, with a focus on feasibility, namely low-complexity physical implementation, by enforcing reciprocity in the BD-RIS design in a manner that adheres to the geometry of the manifold of symmetric matrices. To that end, the sum-rate objective is transformed into a quadratic function via fractional programming (FP), augmented via the also quadratic reciprocity constraint in the form of a regularization term, while the unitary constraint is dealt with via a manifold optimization framework. Simulation results demonstrate the effectiveness of the proposed method in outperforming current state-of-the-art (SotA) approaches in terms of sum-rate maximization.

Paper Structure

This paper contains 11 sections, 2 theorems, 56 equations, 7 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. System Model
  3. Scattering Matrix Definition
  4. Problem Formulation
  5. A Note on the General MU-MIMO Case
  6. Stage 1: Scattering Matrix Design
  7. Solution for the Group-Connected Architecture
  8. Simulation Results
  9. Conclusion
  10. Proof of Lemma \ref{['lem:gradAconjA']}
  11. Proof of Lemma \ref{['lem:lem2']}

Key Result

Lemma 1

Consider $\mathbf{A} \in \mathbb{C}^{N \times N}$ with entries defined as Then where with $a_{i,k}^{\star}$ and $a_{k,i} \in \mathbb{C}$, such that they can be written in the form $x=\Re{x}+j\Im{x}$.

Figures (7)

  • Figure 1: Illustration of the system model, where a BS with $N$TX antennas serves $K$ single-antenna users through an $R$-Element RBD-RIS, without a LoS link$^2$ between the BS and the users.
  • Figure 2: Beampattern illustrating the responses of a BD-RIS with different levels of connectivity. The plots are shown in terms of the auxiliary quantities $u\triangleq \sin(\phi)\cos(\theta)$ and $v \triangleq \sin(\phi)\sin(\theta)$, where $\phi$ and $\theta$ are the azimuth and elevation angles, swept in the ranges $[-\pi,\pi]$ and $[-\pi/2,\pi/2]$, respectively. The blue lines correspond to the directions of two users, located at $(\phi_1, \theta_1) = (-40^\circ,10^\circ)$ and $(\phi_2, \theta_2) = (-20^\circ,10^\circ)$.
  • Figure 3: Convergence of Algorithm \ref{['alg:cga']} vs. number of iterations $I$ for different connectivity structures, with $P_{\mathrm{max}} = 20\,\mathrm{dBm}$, $K = 2$, $N = 2$, $R = 32$, and $K$ factor $K_f = -\infty$ dB.
  • Figure 4: Comparison of sum-rate performance of the proposed vs. SotA YahyaOJCS2024ZheyuArxiv2024 scattering matrix design with uniform power allocation, considering the single-connected "SC", group-connected "GC", with group sizes of 2 and 4, "GC(2)" and "GC(4)", and the fully-connected "FC" architecture.
  • Figure 5: CDF of sum-rate performance of the proposed vs. SotA YahyaOJCS2024ZheyuArxiv2024 scattering matrix design with uniform power allocation, considering the fully-connected "FC", group-connected "GC", with group sizes of 2 and 4, "GC(2)" and "GC(4)", and the single-connected "SC" architecture.
  • ...and 2 more figures

Theorems & Definitions (3)

  • Lemma 1
  • Lemma 2
  • proof