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Global Optimal Closed-Form Solutions for Intelligent Surfaces With Mutual Coupling: Is Mutual Coupling Detrimental or Beneficial?

Matteo Nerini, Hongyu Li, Bruno Clerckx

TL;DR

The paper addresses optimizing BD-RIS architectures under mutual coupling to maximize the RIS-aided channel gain. It develops a global optimal closed-form approach for fully-connected and tree-connected RISs by reformulating the problem with auxiliary variables that diagonalize mutual coupling, leading to a unitary symmetric matrix $ar{oldsymbol{ heta}}$ that achieves a tight upper bound on $|h|^2$. Closed-form expressions for the channel-gain upper bounds are derived in terms of the mutual-coupling matrices $ ext{Z}_{II}$ or $ ext{Y}_{II}$ and the direct-link components. The authors also derive scaling laws for the average channel gain under Rayleigh fading and prove that mutual coupling increases average performance when optimized accordingly, with numerical results validating the theory and showing BD-RIS can match FC performance at lower circuit complexity. Overall, the work provides actionable, globally optimal designs for BD-RIS under mutual coupling and clarifies the beneficial role of coupling in practical RIS deployments.

Abstract

Reconfigurable Intelligent Surface (RIS) is a breakthrough technology enabling the dynamic control of the propagation environment in wireless communications through programmable surfaces. To improve the flexibility of conventional diagonal RIS (D-RIS), beyond diagonal RIS (BD-RIS) has emerged as a family of more general RIS architectures. However, D-RIS and BD-RIS have been commonly explored neglecting mutual coupling effects, while the global optimization of RIS with mutual coupling, its performance limits, and scaling laws remain unexplored. This study addresses these gaps by deriving global optimal closed-form solutions for BD-RIS with mutual coupling to maximize the channel gain, specifically fully- and tree-connected RISs. Besides, we provide the expression of the maximum channel gain achievable in the presence of mutual coupling and its scaling law in closed form. By using the derived scaling laws, we analytically prove that mutual coupling increases the channel gain on average under Rayleigh fading channels. Our theoretical analysis, confirmed by numerical simulations, shows that both fully- and tree-connected RISs with mutual coupling achieve the same channel gain upper bound when optimized with the proposed global optimal solutions. Furthermore, we observe that a mutual coupling-unaware optimization of RIS can cause a channel gain degradation of up to 5 dB.

Global Optimal Closed-Form Solutions for Intelligent Surfaces With Mutual Coupling: Is Mutual Coupling Detrimental or Beneficial?

TL;DR

The paper addresses optimizing BD-RIS architectures under mutual coupling to maximize the RIS-aided channel gain. It develops a global optimal closed-form approach for fully-connected and tree-connected RISs by reformulating the problem with auxiliary variables that diagonalize mutual coupling, leading to a unitary symmetric matrix that achieves a tight upper bound on . Closed-form expressions for the channel-gain upper bounds are derived in terms of the mutual-coupling matrices or and the direct-link components. The authors also derive scaling laws for the average channel gain under Rayleigh fading and prove that mutual coupling increases average performance when optimized accordingly, with numerical results validating the theory and showing BD-RIS can match FC performance at lower circuit complexity. Overall, the work provides actionable, globally optimal designs for BD-RIS under mutual coupling and clarifies the beneficial role of coupling in practical RIS deployments.

Abstract

Reconfigurable Intelligent Surface (RIS) is a breakthrough technology enabling the dynamic control of the propagation environment in wireless communications through programmable surfaces. To improve the flexibility of conventional diagonal RIS (D-RIS), beyond diagonal RIS (BD-RIS) has emerged as a family of more general RIS architectures. However, D-RIS and BD-RIS have been commonly explored neglecting mutual coupling effects, while the global optimization of RIS with mutual coupling, its performance limits, and scaling laws remain unexplored. This study addresses these gaps by deriving global optimal closed-form solutions for BD-RIS with mutual coupling to maximize the channel gain, specifically fully- and tree-connected RISs. Besides, we provide the expression of the maximum channel gain achievable in the presence of mutual coupling and its scaling law in closed form. By using the derived scaling laws, we analytically prove that mutual coupling increases the channel gain on average under Rayleigh fading channels. Our theoretical analysis, confirmed by numerical simulations, shows that both fully- and tree-connected RISs with mutual coupling achieve the same channel gain upper bound when optimized with the proposed global optimal solutions. Furthermore, we observe that a mutual coupling-unaware optimization of RIS can cause a channel gain degradation of up to 5 dB.

Paper Structure

This paper contains 21 sections, 4 theorems, 76 equations, 6 figures, 1 table.

Table of Contents

  1. Introduction
  2. RIS-Aided Channel Model With Mutual Coupling
  3. Optimizing Fully-Connected RIS With Mutual Coupling
  4. Problem Formulation and Global Optimal Solution
  5. Channel Gain Upper Bound
  6. Optimizing Tree-Connected RIS With Mutual Coupling
  7. Problem Formulation and Global Optimal Solution
  8. Channel Gain Upper Bound
  9. Channel Gain Scaling Laws
  10. Scaling Law with Mutual Coupling
  11. Scaling Law with No Mutual Coupling
  12. Impact of Mutual Coupling on the Average Channel Gain
  13. Numerical Results
  14. Optimizing Fully- and Tree-Connected RISs
  15. Scaling Laws
  16. ...and 6 more sections

Key Result

Proposition 1

For a reciprocal and lossy $N$-port network with impedance matrix $\mathbf{Z}\in\mathbb{C}^{N\times N}$ and admittance matrix $\mathbf{Y}=\mathbf{Z}^{-1}$, the matrices $\Re\{\mathbf{Z}\}$ and $\Re\{\mathbf{Y}\}$ are positive semi-definite.

Figures (6)

  • Figure 1: RIS-aided system modeled with multiport network theory.
  • Figure 2: Circuit topology of a (a) fully-connected RIS and (b) tree-connected RIS, with $N_I=4$ elements.
  • Figure 3: Channel gain versus the number of RIS elements for different values of inter-element distance $d$.
  • Figure 4: Simulated mean values of the terms appearing in the channel gain expression and corresponding theoretical closed-form.
  • Figure 5: Channel gain upper bounds and their scaling laws versus the number of RIS elements for different values of inter-element distance $d$.
  • ...and 1 more figures

Theorems & Definitions (8)

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