A physics-compliant diagonal representation for wireless channels parametrized by beyond-diagonal reconfigurable intelligent surfaces

TL;DR

This paper introduces a physics-compliant diagonal representation for BD-RIS-parametrized wireless channels by decomposing the RIS load into static parts and tunable loads, yielding a chain cascade RE-SLC-IL whose end-to-end channel can be written in a form analogous to the conventional D-RIS model. The key insight is that the cascade of RE and SLC (denoted K) terminated by a diagonal IL provides a structure that allows existing physics-based D-RIS algorithms to be applied to BD-RIS scenarios without developing new BD-RIS-specific methods. The authors connect this diagonal scattering-parameter framework to the PhysFad coupled-dipole formalism, establishing operational equivalence with D-RIS and extending PhysFad to BD-RIS, including analytical, numerical, and experimental perspectives. An experimentally grounded case study demonstrates end-to-end BD-RIS channel estimation and RSSI optimization using D-RIS-style algorithms, highlighting ambiguities that do not hinder predictive performance and showing practical viability of the approach. Overall, the work enables a paradigm shift where BD-RIS system-level optimization can leverage established physics-compliant D-RIS tools, informing comparisons that focus on tunable-element counts rather than element counts alone.

Abstract

The parametrization of wireless channels by so-called "beyond-diagonal reconfigurable intelligent surfaces" (BD-RIS) is mathematically characterized by a matrix whose off-diagonal entries are partially or fully populated. Physically, this corresponds to tunable coupling mechanisms between the RIS elements that originate from the RIS control circuit. Here, we derive a physics-compliant diagonal representation for BD-RIS-parametrized channels. We recognize that any RIS control circuit can always be separated into its static parts (SLC) and a set of tunable individual loads (IL). Therefore, a BD-RIS-parametrized channel results from the chain cascade of three systems: i) radio environment (RE), ii) SLC, and iii) IL. RE and SLC are static non-diagonal systems whose cascade K is terminated by the tunable diagonal system IL. This physics-compliant representation in terms of K and IL is directly analogous to that for conventional ("diagonal") RIS (D-RIS). Therefore, scenarios with BD-RIS can also readily be captured by the physics-compliant coupled-dipole model PhysFad, as we show. In addition, physics-compliant algorithms for system-level optimization with D-RIS can be directly applied to scenarios with BD-RIS. We demonstrate this important implication of our conceptual finding in a case study on end-to-end channel estimation and optimization in a BD-RIS-parametrized rich-scattering environment. Our case study is the first experimentally grounded system-level optimization for BD-RIS: We obtain the characteristics of RE and IL from experimental measurements and a commercial PIN diode, respectively. Altogether, our physics-compliant diagonal representation for BD-RIS enables a paradigm shift in how practitioners in wireless communications and signal processing implement system-level optimizations for BD-RIS because it enables them to directly apply existing physics-compliant D-RIS algorithms.

Figures (6)

• Figure 1: Generic multi-port chain cascade analysis of BD-RIS-parametrized wireless channels. The radio environment (RE) is an $N_\mathrm{RE}$-port system characterized by the scattering matrix $\mathbf{S}^\mathrm{RE}$, where $N_\mathrm{RE}$ is the sum of $N_\mathrm{T}$ (number of transmitting antennas), $N_\mathrm{R}$ (number of receiving antennas) and $N_\mathrm{S}$ (number of RIS elements). $\mathbf{S}^\mathrm{RE}$ lumps together all scattering effects in the RE, including scattering originating from objects and walls as well as the structural scattering by the antennas and RIS elements. The $N_\mathrm{S}$ auxiliary ports associated with the RIS elements are terminated by a load circuit (L). L can be separated into its static parts (SLC) and individual tunable lumped elements (IL), characterized by the scattering matrices $\mathbf{S}^\mathrm{SLC}$ and $\mathbf{S}^\mathrm{IL}$, respectively. SLC and IL have $N_\mathrm{SLC}=N_\mathrm{S}+N_\mathrm{C}$ and $N_\mathrm{C}$ ports, respectively. The $N_\mathrm{C}$ ports of SLC associated with tunable lumped elements are terminated by IL. (In the case of a conventional D-RIS, each RIS element auxiliary port is directly and uniquely connected to one individual tunable load, implying $N_\mathrm{C}=N_\mathrm{S}$ and $\mathbf{S}^\mathrm{SLC} = \left[ \mathbf{0}_{N_\mathrm{S}} \ \ \mathbf{I}_{N_\mathrm{S}}; \mathbf{I}_{N_\mathrm{S}} \ \ \mathbf{0}_{N_\mathrm{S}}\right]$.) The conventional perspective on this chain cascade (albeit never explicited so far) consists in first evaluating the cascade of SLC and IL, referred to as L, which yields a generally "beyond-diagonal" scattering matrix $\mathbf{S}^\mathrm{L}$ that terminates the RIS element auxiliary ports of the RE. The alternative perspective emphasized in this paper is to first evaluate the cascade of RE and SLC, referred to as K; the auxiliary ports of K associated with tunable lumped elements are then terminated by $\mathbf{S}^\mathrm{IL}$ which is always diagonal.
• Figure 2: (A) Ideal 2-port T network. (B) The three impedances in (A) are replaced by auxiliary ports, yielding a 5-port network whose scattering matrix can be determined analytically, see Appendix \ref{['AppendixA']} and Eq. (\ref{['eq_T']}). (C) Ideal 2-port $\pi$ network. (D) The three impedances in (C) are replaced by auxiliary ports, yielding a 5-port network whose scattering matrix can be determined analytically, see Eq. (\ref{['eq_pi']}).
• Figure 3: Experimentally grounded study of a SISO link inside a rich-scattering environment parametrized by a 6-element RIS with group-connected reconfigurable impedance network having three groups and group size two.
• Figure 4: End-to-end BD-RIS-parametrized channel estimation in the rich-scattering environment from Fig. \ref{['Fig3']}A using a physics-compliant D-RIS algorithm. The parameters to be estimated are $r_\mathrm{A}$ and $r_\mathrm{B}$, as well as the blocks $\mathcal{RT}$, $\mathcal{AC}$ and $\mathcal{CC}$ of $\mathbf{S}^\mathrm{K}$. Given these parameters, for any given $\mathbf{b}$ we can evaluate $\mathbf{S}^\mathrm{IL}$ using Eq. (\ref{['eq_b']}) and then $h$ using Eq. (\ref{['eq22_new']}). (A) Amplitude and phase of the experimentally grounded ground truth (blue) and two separate estimates (red and green) of the estimated parameters. For ease of visualization, we display the matrix-valued blocks $\mathcal{AC}$ and $\mathcal{CC}$ of $\mathbf{S}^\mathrm{K}$ in vectorized form. (B) Amplitude and phase of ground-truth (blue) and predicted (red) end-to-end wireless SISO channel $h$ for 100 random unseen configurations of the BD-RIS in the rich-scattering environment from Fig. \ref{['Fig3']}A.
• Figure 5: Probability density function of the RSSI for the considered SISO link, as well as predicted and ground-truth RSSI for the configuration $\mathbf{b}_\mathrm{opt}$ obtained with Algorithm \ref{['algo1']} based on the end-to-end channel estimate from Sec. \ref{['subsec_chanest']}.