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Low-Complexity Planar Beyond-Diagonal RIS Architecture Design Using Graph Theory

Matteo Nerini, Zheyu Wu, Shanpu Shen, Bruno Clerckx

TL;DR

This work shows how graph theory can identify BD-RIS architectures that can be realized on a double-layer PCB, introducing planar-connected RIS and maximal-planar-connected RIS as the most flexible yet practically implementable designs under planar constraints. It provides necessary planarity conditions, analyzes existing BD-RIS architectures for planarity, and presents three maximal-planar-connected examples achieving high degrees of freedom with 4N−6 interconnections, while preserving planarity. Through joint precoding and RIS optimization, the paper demonstrates a trade-off between sum-rate performance and circuit complexity, revealing that maximal-planar-connected RIS offers a favorable balance for practical deployment. The results offer concrete design guidelines for low-complexity BD-RIS architectures capable of nearing fully-connected performance in multi-user scenarios.

Abstract

Reconfigurable intelligent surfaces (RISs) enable programmable control of the wireless propagation environment and are key enablers for future networks. Beyond-diagonal RIS (BD-RIS) architectures enhance conventional RIS by interconnecting elements through tunable impedance components, offering greater flexibility with higher circuit complexity. However, excessive interconnections between BD-RIS elements require multi-layer printed circuit board (PCB) designs, increasing fabrication difficulty. In this letter, we use graph theory to characterize the BD-RIS architectures that can be realized on double-layer PCBs, denoted as planar-connected RISs. Among the possible planar-connected RISs, we identify the ones with the most degrees of freedom, expected to achieve the best performance under practical constraints.

Low-Complexity Planar Beyond-Diagonal RIS Architecture Design Using Graph Theory

TL;DR

This work shows how graph theory can identify BD-RIS architectures that can be realized on a double-layer PCB, introducing planar-connected RIS and maximal-planar-connected RIS as the most flexible yet practically implementable designs under planar constraints. It provides necessary planarity conditions, analyzes existing BD-RIS architectures for planarity, and presents three maximal-planar-connected examples achieving high degrees of freedom with 4N−6 interconnections, while preserving planarity. Through joint precoding and RIS optimization, the paper demonstrates a trade-off between sum-rate performance and circuit complexity, revealing that maximal-planar-connected RIS offers a favorable balance for practical deployment. The results offer concrete design guidelines for low-complexity BD-RIS architectures capable of nearing fully-connected performance in multi-user scenarios.

Abstract

Reconfigurable intelligent surfaces (RISs) enable programmable control of the wireless propagation environment and are key enablers for future networks. Beyond-diagonal RIS (BD-RIS) architectures enhance conventional RIS by interconnecting elements through tunable impedance components, offering greater flexibility with higher circuit complexity. However, excessive interconnections between BD-RIS elements require multi-layer printed circuit board (PCB) designs, increasing fabrication difficulty. In this letter, we use graph theory to characterize the BD-RIS architectures that can be realized on double-layer PCBs, denoted as planar-connected RISs. Among the possible planar-connected RISs, we identify the ones with the most degrees of freedom, expected to achieve the best performance under practical constraints.
Paper Structure (11 sections, 5 theorems, 5 equations, 6 figures, 1 table)

This paper contains 11 sections, 5 theorems, 5 equations, 6 figures, 1 table.

Table of Contents

  1. Introduction
  2. BD-RIS Graph Theoretical Model
  3. Planar-Connected RIS
  4. Maximal-Planar-Connected RIS
  5. Example 1 ($3$-band-connected RIS)
  6. Example 2
  7. Example 3
  8. Performance and Complexity Evaluation
  9. Sum Rate Maximization
  10. Numerical Results
  11. Conclusion

Key Result

Proposition 1

Any forest-connected RIS, including the single-connected RIS and the tree-connected RIS as special cases, is planar-connected.

Figures (6)

  • Figure 1: (a) The complete graph on five vertices $K_5$ (non-planar), and (b) the complete graph on four vertices $K_4$ (planar).
  • Figure 2: Recursive construction of a planar drawing of the graph of a $3$-band-connected RIS with $N$ elements.
  • Figure 3: Admittance matrix $\mathbf{Y}$ of the three examples of maximal-planar-connected RIS, with tunable entries in black and zero entries in white.
  • Figure 4: Example 2 of maximal-planar-connected RIS, with $N=8$ elements.
  • Figure 5: Example 3 of maximal-planar-connected RIS, with $N=8$ elements.
  • ...and 1 more figures

Theorems & Definitions (12)

  • Definition 1
  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Proposition 3
  • proof
  • Proposition 4
  • proof
  • Proposition 5
  • ...and 2 more