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Beyond Diagonal Reconfigurable Intelligent Surfaces Utilizing Graph Theory: Modeling, Architecture Design, and Optimization

Matteo Nerini, Shanpu Shen, Hongyu Li, Bruno Clerckx

TL;DR

Tree-connected RIS, whose corresponding graph is a tree, is proven to be the least complex BD-RIS architecture able to achieve the performance upper bound in multiple-input singleoutput (MISO) systems and to optimize tree-connected RIS is optimized through a low-complexity iterative algorithm.

Abstract

Recently, beyond diagonal reconfigurable intelligent surface (BD-RIS) has been proposed to generalize conventional RIS. BD-RIS has a scattering matrix that is not restricted to being diagonal and thus brings a performance improvement over conventional RIS. While different BD-RIS architectures have been proposed, it still remains an open problem to develop a systematic approach to design BD-RIS architectures achieving the optimal trade-off between performance and circuit complexity. In this work, we propose novel modeling, architecture design, and optimization for BD-RIS based on graph theory. This graph theoretical modeling allows us to develop two new efficient BD-RIS architectures, denoted as tree-connected and forest-connected RIS. Tree-connected RIS, whose corresponding graph is a tree, is proven to be the least complex BD-RIS architecture able to achieve the performance upper bound in multiple-input single-output (MISO) systems. Besides, forest-connected RIS allows us to strike a balance between performance and complexity, further decreasing the complexity over tree-connected RIS. To optimize tree-connected RIS, we derive a closed-form global optimal solution, while forest-connected RIS is optimized through a low-complexity iterative algorithm. Numerical results confirm that tree-connected (resp. forest-connected) RIS achieves the same performance as fully-connected (resp. group-connected) RIS, while reducing the complexity by up to 16.4 times.

Beyond Diagonal Reconfigurable Intelligent Surfaces Utilizing Graph Theory: Modeling, Architecture Design, and Optimization

TL;DR

Tree-connected RIS, whose corresponding graph is a tree, is proven to be the least complex BD-RIS architecture able to achieve the performance upper bound in multiple-input singleoutput (MISO) systems and to optimize tree-connected RIS is optimized through a low-complexity iterative algorithm.

Abstract

Recently, beyond diagonal reconfigurable intelligent surface (BD-RIS) has been proposed to generalize conventional RIS. BD-RIS has a scattering matrix that is not restricted to being diagonal and thus brings a performance improvement over conventional RIS. While different BD-RIS architectures have been proposed, it still remains an open problem to develop a systematic approach to design BD-RIS architectures achieving the optimal trade-off between performance and circuit complexity. In this work, we propose novel modeling, architecture design, and optimization for BD-RIS based on graph theory. This graph theoretical modeling allows us to develop two new efficient BD-RIS architectures, denoted as tree-connected and forest-connected RIS. Tree-connected RIS, whose corresponding graph is a tree, is proven to be the least complex BD-RIS architecture able to achieve the performance upper bound in multiple-input single-output (MISO) systems. Besides, forest-connected RIS allows us to strike a balance between performance and complexity, further decreasing the complexity over tree-connected RIS. To optimize tree-connected RIS, we derive a closed-form global optimal solution, while forest-connected RIS is optimized through a low-complexity iterative algorithm. Numerical results confirm that tree-connected (resp. forest-connected) RIS achieves the same performance as fully-connected (resp. group-connected) RIS, while reducing the complexity by up to 16.4 times.
Paper Structure (25 sections, 4 theorems, 61 equations, 10 figures, 2 algorithms)

This paper contains 25 sections, 4 theorems, 61 equations, 10 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. BD-RIS Modeling Utilizing Graph Theory
  3. Diagonal RIS
  4. Beyond Diagonal RIS
  5. Graph Theoretical Modeling
  6. BD-RIS Architecture Design Utilizing Graph Theory
  7. Tree-Connected RIS
  8. Two Examples of Tree-Connected RIS
  9. Tridiagonal RIS
  10. Arrowhead RIS
  11. Forest-Connected RIS
  12. Optimization of Tree- and Forest-Connected RIS for MISO Systems
  13. Tree-Connected RIS Optimization
  14. Forest-Connected RIS Optimization
  15. Optimizing $\mathbf{B}$ with Fixed $\mathbf{w}$
  16. ...and 10 more sections

Key Result

Lemma 1

A BD-RIS with associated graph $\mathcal{G}$ is MISO optimal if and only if $\mathcal{G}$ is a connected graph.

Figures (10)

  • Figure 1: RIS classification tree.
  • Figure 2: Examples of $4$-port BD-RIS architectures (left) and their corresponding graphs $\mathcal{G}$ (right), where $\mathcal{G}$ is (a) empty, (b) disconnected and acyclic, (c) connected and acyclic, i.e., a tree, (d) connected and cyclic, (e) complete.
  • Figure 3: Tridiagonal RIS with $N=8$.
  • Figure 4: Arrowhead RIS with $N=5$ where port 1 is the central vertex.
  • Figure 5: Two-dimensional coordinate system for the RIS-aided MISO system.
  • ...and 5 more figures

Theorems & Definitions (8)

  • Lemma 1
  • proof
  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Proposition 3
  • proof