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Capacity Analysis of Cascaded BD-RIS Assisted MIMO Systems

M. S. S. Manasa, Praful D. Mankar, Sundaram Vanka

TL;DR

This work analyzes capacity in cascaded BD-RIS aided MIMO systems, addressing both transmitter and RIS design under unitary phase constraints. It derives closed-form optimal BD-RIS phase shifts for cascaded configurations and shows that uniform power allocation (UPA) can achieve performance close to SVD-WF, reducing transmitter complexity. The paper also develops approximate ergodic-capacity expressions, including a high-SNR scaling law that reveals how capacity grows with the cascaded RIS dimensions, and provides a practical N_req relation to meet target rates. The results demonstrate significant capacity gains for cascaded BD-RIS over single RIS setups and offer design guidelines for RIS size and deployment in high-SNR regimes, with extensions to diagonal passive RIS discussed as well.

Abstract

This paper examines the cascaded deployment of beyond diagonal (BD) reconfigurable intelligent surfaces (RISs) and explores its potential to enhance the performance of MIMO systems. We first derive the jointly optimal closed form solutions for the RISs in cascade with SVD water filling (SVD WF) and uniform power allocation (UPA) precoding strategies. The optimally configured cascaded-RIS with UPA is shown to achieve performance comparable to that with the SVD WF approach, suggesting that cascaded-RISs can also aid in reducing transmitter complexity. Furthermore, the approximate ergodic capacity for UPA is derived, along with its high SNR approximation which provides multiple useful insights into the dimension and deployment of cascaded RISs. The analytical results establish a clear tradeoff among transmit power, RIS size, and achievable capacity, providing insights for practical deployment in high SNR cascaded RIS MIMO systems.

Capacity Analysis of Cascaded BD-RIS Assisted MIMO Systems

TL;DR

This work analyzes capacity in cascaded BD-RIS aided MIMO systems, addressing both transmitter and RIS design under unitary phase constraints. It derives closed-form optimal BD-RIS phase shifts for cascaded configurations and shows that uniform power allocation (UPA) can achieve performance close to SVD-WF, reducing transmitter complexity. The paper also develops approximate ergodic-capacity expressions, including a high-SNR scaling law that reveals how capacity grows with the cascaded RIS dimensions, and provides a practical N_req relation to meet target rates. The results demonstrate significant capacity gains for cascaded BD-RIS over single RIS setups and offer design guidelines for RIS size and deployment in high-SNR regimes, with extensions to diagonal passive RIS discussed as well.

Abstract

This paper examines the cascaded deployment of beyond diagonal (BD) reconfigurable intelligent surfaces (RISs) and explores its potential to enhance the performance of MIMO systems. We first derive the jointly optimal closed form solutions for the RISs in cascade with SVD water filling (SVD WF) and uniform power allocation (UPA) precoding strategies. The optimally configured cascaded-RIS with UPA is shown to achieve performance comparable to that with the SVD WF approach, suggesting that cascaded-RISs can also aid in reducing transmitter complexity. Furthermore, the approximate ergodic capacity for UPA is derived, along with its high SNR approximation which provides multiple useful insights into the dimension and deployment of cascaded RISs. The analytical results establish a clear tradeoff among transmit power, RIS size, and achievable capacity, providing insights for practical deployment in high SNR cascaded RIS MIMO systems.

Paper Structure

This paper contains 13 sections, 4 theorems, 37 equations, 2 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. RIS Architectures --- Diagonal and Beyond-Diagonal
  3. RIS Deployment -- Multi-RIS Systems
  4. Optimal Configuration for RIS systems
  5. System Model
  6. Optimal RIS Phase shift matrix
  7. Optimal Precoding
  8. Optimal RIS phase shift
  9. Special Case 1: Single RIS System
  10. Special Case 2: Diagonal Passive RIS
  11. Ergodic Capacity for cascaded RIS
  12. Numerical Results
  13. Conclusion

Key Result

Theorem 1

For UPA precoding, i.e. $\mathbf{Q} = \frac{P_t}{M}\mathbf{I}_M$, the optimal phase shift matrices for cascaded RISs are given by and the achievable capacity is given by where $R=\min(R_0,R_1,\ldots,R_L)$, $R_l$ and $\lambda_{lk}$ are the rank and the $k$-th eigenvalue of $\mathbf{H}_l$.

Figures (2)

  • Figure 1: (a) EC vs. SNR for single D-RIS and BD-RIS, dashed lines represent Monte Carlo result for BD-RIS with SVD-WF (b) EC vs. SNR for cascaded BD-RISs, (c) EC vs. $N$ for cascaded BD-RISs.
  • Figure 2: Number of RIS elements reequired to achieve a target capacity. The solid and dashed curves represent the analytical and simulation results, respectively.

Theorems & Definitions (10)

  • Theorem 1
  • proof
  • Remark 1
  • Theorem 2
  • proof
  • Theorem 3
  • proof
  • Corollary 3.1
  • proof
  • Remark 2