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MIMO Capacity Maximization with Beyond-Diagonal RIS

Ignacio Santamaria, Mohammad Soleymani, Eduard Jorswieck, Jesús Gutiérrez

TL;DR

The paper tackles maximizing the capacity of a MIMO link aided by a beyond-diagonal reconfigurable intelligent surface (BD-RIS) with a fully connected, unitary and symmetric scattering matrix. It proposes an alternating optimization framework: for a fixed BD-RIS matrix, the transmit covariance is optimally allocated via eigen decomposition and waterfilling; for a fixed transmit strategy, the BD-RIS is optimized on the unitary manifold using a concave lower bound and Takagi factorization to enforce the unitary-symmetric constraints. The resulting algorithm consistently achieves higher capacity than a diagonal RIS, with gains amplified by more streams, more BD-RIS elements, and higher transmit power. The work shows BD-RIS can substantially improve link conditioning and throughput in MIMO systems and lays groundwork for extensions to group-connected architectures and multi-user networks, offering practical impact for next-generation wireless design.

Abstract

This paper addresses the problem of maximizing the capacity of a multiple-input multiple-output (MIMO) link assisted by a beyond-diagonal reconfigurable intelligent surface (BD-RIS). We maximize the capacity by alternately optimizing the transmit covariance matrix, and the BD-RIS scattering matrix, which, according to network theory, should be unitary and symmetric. These constraints make the optimization of BD-RIS more challenging than that of diagonal RIS. To find a stationary point of the capacity we maximize a sequence of quadratic problems in the manifold of unitary matrices. This leads to an efficient algorithm that always improves the capacity obtained by a diagonal RIS. Through simulation examples, we study the capacity improvement provided by a passive BD-RIS architecture over the conventional RIS model in which the phase shift matrix is diagonal.

MIMO Capacity Maximization with Beyond-Diagonal RIS

TL;DR

The paper tackles maximizing the capacity of a MIMO link aided by a beyond-diagonal reconfigurable intelligent surface (BD-RIS) with a fully connected, unitary and symmetric scattering matrix. It proposes an alternating optimization framework: for a fixed BD-RIS matrix, the transmit covariance is optimally allocated via eigen decomposition and waterfilling; for a fixed transmit strategy, the BD-RIS is optimized on the unitary manifold using a concave lower bound and Takagi factorization to enforce the unitary-symmetric constraints. The resulting algorithm consistently achieves higher capacity than a diagonal RIS, with gains amplified by more streams, more BD-RIS elements, and higher transmit power. The work shows BD-RIS can substantially improve link conditioning and throughput in MIMO systems and lays groundwork for extensions to group-connected architectures and multi-user networks, offering practical impact for next-generation wireless design.

Abstract

This paper addresses the problem of maximizing the capacity of a multiple-input multiple-output (MIMO) link assisted by a beyond-diagonal reconfigurable intelligent surface (BD-RIS). We maximize the capacity by alternately optimizing the transmit covariance matrix, and the BD-RIS scattering matrix, which, according to network theory, should be unitary and symmetric. These constraints make the optimization of BD-RIS more challenging than that of diagonal RIS. To find a stationary point of the capacity we maximize a sequence of quadratic problems in the manifold of unitary matrices. This leads to an efficient algorithm that always improves the capacity obtained by a diagonal RIS. Through simulation examples, we study the capacity improvement provided by a passive BD-RIS architecture over the conventional RIS model in which the phase shift matrix is diagonal.
Paper Structure (14 sections, 1 theorem, 13 equations, 4 figures, 1 algorithm)

This paper contains 14 sections, 1 theorem, 13 equations, 4 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. System Model and Problem Formulation
  3. System Model
  4. Problem Formulation
  5. Capacity maximization with passive BD-RIS
  6. Optimizing ${\bf R}_{xx}$ for a given $\bm{\Theta}$
  7. Optimizing $\bm{\Theta}$ for a given ${\bf R}_{xx}$
  8. Simulation Results
  9. Scenario description
  10. Results varying the BD-RIS position
  11. Results varying the number of BD-RIS elements
  12. Results varying the Tx power
  13. Conclusions
  14. Acknowledgment

Key Result

Lemma 1

Let us denote given a feasible solution at iteration $t$, $\bm{\Theta}_t$, the following concave-lower bound of the capacity exists. where $C_t = C(\bm{\Theta}_t)$, ${\bf H}_t = \overline{{\bf H}}_{eq}(\bm{\Theta}_t)$, and ${\bf R}_t = \frac{1}{\sigma_n^2} {\bf I}_{N_R} - \left(\sigma_n^2 {\bf I}_{N_R} + {\bf H}_t {\bf H}_t^H\right)^{-1}$.

Figures (4)

  • Figure 1: Achievable rate for a $4\times 4$ MIMO link assisted by an BD-RIS with $M=100$ elements.
  • Figure 2: Achievable rates for an increasing number of BD-RIS elements $M$ for $2\times 2$ and $4\times 4$ MIMO channels.
  • Figure 3: Achievable rate vs. transmit power for BD-RIS and RIS.
  • Figure 4: Number of active streams vs. transmit power for BD-RIS and RIS architectures.

Theorems & Definitions (1)

  • Lemma 1: TCOMbounds16