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STAR Beyond Diagonal RISs with Amplification: Modeling and Optimization

Chandan Kumar Sheemar, Giovanni Iacovelli, Wali Ullah Khan, George C. Alexandropoulos, Stefano Tomasin, Symeon Chatzinotas

Abstract

This paper develops a physically consistent signal model with hardware constraints for a simultaneous transmitting and reflecting beyond-diagonal RIS (STAR BD-RIS) endowed with per-element amplification and lossless power splitting. We explicitly decouple (i) amplification via a diagonal gain matrix, (ii) element-wise reflection/transmission splitting, and (iii) passive beyond-diagonal coupling on each branch, while enforcing practical feasibility through per-element emission caps and an aggregate RIS power budget under the operating covariance. Building on this model, we cast downlink sum-rate maximization as an equivalent weighted minimum mean-square error (WMMSE) problem and propose an alternating optimization framework with provable monotonic descent. The method admits closed-form updates for MMSE combiners and weights, waterfilling-like beamformer updates via a single dual variable, a per-element amplification update that satisfies emission constraints, and a STAR power-splitting update based on cyclic coordinate descent with a global acceptance test. For the beyond-diagonal coupling matrices, we derive Riemannian gradient steps on the complex Stiefel manifold with QR/polar retraction method, preserving passivity at every iterate. Furthermore, the proposed approach decouples the optimization of the reflective and transmissive responses of the BD-RIS, enabling efficient distributed implementation. Numerical results demonstrate substantial sum-rate gains compared to the conventional passive BD-RIS.

STAR Beyond Diagonal RISs with Amplification: Modeling and Optimization

Abstract

This paper develops a physically consistent signal model with hardware constraints for a simultaneous transmitting and reflecting beyond-diagonal RIS (STAR BD-RIS) endowed with per-element amplification and lossless power splitting. We explicitly decouple (i) amplification via a diagonal gain matrix, (ii) element-wise reflection/transmission splitting, and (iii) passive beyond-diagonal coupling on each branch, while enforcing practical feasibility through per-element emission caps and an aggregate RIS power budget under the operating covariance. Building on this model, we cast downlink sum-rate maximization as an equivalent weighted minimum mean-square error (WMMSE) problem and propose an alternating optimization framework with provable monotonic descent. The method admits closed-form updates for MMSE combiners and weights, waterfilling-like beamformer updates via a single dual variable, a per-element amplification update that satisfies emission constraints, and a STAR power-splitting update based on cyclic coordinate descent with a global acceptance test. For the beyond-diagonal coupling matrices, we derive Riemannian gradient steps on the complex Stiefel manifold with QR/polar retraction method, preserving passivity at every iterate. Furthermore, the proposed approach decouples the optimization of the reflective and transmissive responses of the BD-RIS, enabling efficient distributed implementation. Numerical results demonstrate substantial sum-rate gains compared to the conventional passive BD-RIS.
Paper Structure (22 sections, 82 equations, 7 figures, 3 algorithms)

This paper contains 22 sections, 82 equations, 7 figures, 3 algorithms.

Table of Contents

  1. Introduction
  2. State-of-the-Art and Motivation
  3. On the STAR D-RISs
  4. On the STAR BD-RISs
  5. Main Contributions
  6. System Model
  7. On the STAR BD-RIS Operation with Amplification
  8. Signal Model
  9. Problem Formulation
  10. Proposed Solution
  11. WMMSE Transformation of the Sum-Rate Maximization Problem
  12. Optimal MMSE Combiners and MSE Weights
  13. WMMSE-Based Digital Beamforming
  14. WMMSE-Based Amplification Matrix Optimization
  15. WMMSE-Driven Optimal Power Splitting
  16. ...and 7 more sections

Figures (7)

  • Figure 1: Multi-Functional BD-RIS-assisted communications system with simultaneous transmission, reception, and amplification.
  • Figure 2: Convergence of the proposed methods at $P_{BS}=20$dBm.
  • Figure 3: Sum-rate as a function of $P_{\mathrm{BS}}$ with $1$ transmissive and $1$ reflective user.
  • Figure 4: Sum-rate as a function of $P_{\mathrm{BS}}$ with $1$ transmissive and $1$ reflective users.
  • Figure 5: Sum-rate as a function of BD-RIS size with $1$ user per zone.
  • ...and 2 more figures