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Capacity Maximization for MIMO Channels Assisted by Beyond-Diagonal RIS

Emil Björnson, Özlem Tuğfe Demir

TL;DR

This work addresses capacity maximization for MIMO channels aided by a beyond-diagonal RIS (BD-RIS), where the end-to-end channel is ${\bf H} = {\bf F} {\bm{\Theta}} {\bf G}^{\mathrm{H}}$ with a unitary reflection ${\bm{\Theta}}$ and jointly optimized transmit covariance ${\bf Q}$. It derives a closed-form optimal BD-RIS configuration, ${\bm{\Theta}} = {\bf V}_{F} {\bf V}_{G}^{\mathrm{H}} {\bf Q} = {\bf U}_{G} \mathrm{diag}(q_1,...,q_{N_t}) {\bf U}_{G}^{\mathrm{H}}$, achieving capacity $C = \sum_{i=1}^{K} \log_2\left(1 + \dfrac{q_i \sigma_i^2({\bf F}) \sigma_i^2({\bf G}^{\mathrm{H}})}{N_0}\right)$ with $q_i$ from waterfilling and $K = \min(N_t,N_r,M)$. The paper provides a geometric interpretation: the optimal BD-RIS pairs the strongest singular directions of ${\bf F}$ and ${\bf G}$, and at high SNR a broader set of configurations becomes optimal through permutations of these directions. It also analyzes the equal-singular-value (semi-unitary) case, showing a simplified capacity expression and a special case where a conventional RIS suffices. Numerical results corroborate the theory, showing BD-RIS gains over conventional RIS that grow with the number of elements, while certain regimes (e.g., $M \le \min(N_t,N_r)$ with semi-unitary channels) recover RIS performance. Overall, the work clarifies when BD-RIS yields gains and provides a tractable, closed-form design for capacity optimization, including non-reciprocal implementations that approach the theoretical optimum.

Abstract

Reconfigurable intelligent surfaces (RISs) can improve the capacity of wireless communication links by passively beamforming the impinging signals in desired directions. This feature has been demonstrated both analytically and experimentally for conventional RISs, consisting of independently reflecting elements. To further enhance reconfigurability, a new architecture called beyond-diagonal RIS (BD-RIS) has been proposed. It allows for controllable signal flows between RIS elements, resulting in a non-diagonal reflection matrix, unlike the conventional RIS architecture. Previous studies on BD-RIS-assisted communications have predominantly considered single-antenna transmitters/receivers. One recent work provides an iterative capacity-improving algorithm for multiple-input multiple-output (MIMO) setups but without providing geometrical insights. In this paper, we derive the first closed-form capacity-maximizing BD-RIS reflection matrix for a MIMO channel. We describe how this solution pairs together propagation paths, how it behaves when the signal-to-noise ratio is high, and what capacity is achievable with ideal semi-unitary channel matrices. The analytical results are corroborated numerically.

Capacity Maximization for MIMO Channels Assisted by Beyond-Diagonal RIS

TL;DR

This work addresses capacity maximization for MIMO channels aided by a beyond-diagonal RIS (BD-RIS), where the end-to-end channel is with a unitary reflection and jointly optimized transmit covariance . It derives a closed-form optimal BD-RIS configuration, , achieving capacity with from waterfilling and . The paper provides a geometric interpretation: the optimal BD-RIS pairs the strongest singular directions of and , and at high SNR a broader set of configurations becomes optimal through permutations of these directions. It also analyzes the equal-singular-value (semi-unitary) case, showing a simplified capacity expression and a special case where a conventional RIS suffices. Numerical results corroborate the theory, showing BD-RIS gains over conventional RIS that grow with the number of elements, while certain regimes (e.g., with semi-unitary channels) recover RIS performance. Overall, the work clarifies when BD-RIS yields gains and provides a tractable, closed-form design for capacity optimization, including non-reciprocal implementations that approach the theoretical optimum.

Abstract

Reconfigurable intelligent surfaces (RISs) can improve the capacity of wireless communication links by passively beamforming the impinging signals in desired directions. This feature has been demonstrated both analytically and experimentally for conventional RISs, consisting of independently reflecting elements. To further enhance reconfigurability, a new architecture called beyond-diagonal RIS (BD-RIS) has been proposed. It allows for controllable signal flows between RIS elements, resulting in a non-diagonal reflection matrix, unlike the conventional RIS architecture. Previous studies on BD-RIS-assisted communications have predominantly considered single-antenna transmitters/receivers. One recent work provides an iterative capacity-improving algorithm for multiple-input multiple-output (MIMO) setups but without providing geometrical insights. In this paper, we derive the first closed-form capacity-maximizing BD-RIS reflection matrix for a MIMO channel. We describe how this solution pairs together propagation paths, how it behaves when the signal-to-noise ratio is high, and what capacity is achievable with ideal semi-unitary channel matrices. The analytical results are corroborated numerically.

Paper Structure

This paper contains 9 sections, 4 theorems, 17 equations, 4 figures.

Table of Contents

  1. Introduction
  2. System Model and Problem Formulation
  3. Capacity Maximizing BD-RIS Configuration
  4. On the capacity of channels with equal singular values
  5. Numerical Results
  6. Conclusions
  7. Proof of Theorem \ref{['th:capacity']}
  8. Proof of Corollary \ref{['corollary']}
  9. Proof of Corollary \ref{['th:orthogonal']}

Key Result

Theorem 1

The optimization problem eq:Capacity is solved byA (block-)diagonal matrix with $a_1,\ldots,a_N$ is denoted by $\mathrm{diag}(a_1,\ldots,a_N)$. and the maximum capacity value is where the transmit powers $q_1,\ldots,q_{K}$ are computed using the waterfilling algorithm: $q_i = \mu - N_0 / ( \sigma_i^2 ({\bf F}) \sigma_i^2 ({\bf G}^{\hbox{\tiny $\mathrm{H}$}}) )$, where $\mu$ is selected so that $

Figures (4)

  • Figure 1: Theorem \ref{['th:capacity']} proves that the optimal BD-RIS configuration will reflect one transmitter-RIS path along one RIS-receiver. The pairing is made in the order of strength.
  • Figure 2: The capacity achieved with different RIS architectures and configurations versus the number of elements ($M$).
  • Figure 3: The capacity achieved versus the SNR for $M=64$.
  • Figure 4: The capacity achieved versus the number of elements ($M$) when having semi-unitary channels.

Theorems & Definitions (4)

  • Theorem 1
  • Corollary 1
  • Corollary 2
  • Lemma 1