Lossy Beyond Diagonal Reconfigurable Intelligent Surfaces: Modeling and Optimization

TL;DR

This work addresses the gap between idealized lossless BD-RIS models and practical realizations by introducing a lossy BD-RIS model based on admittance parameters. It develops a circuit-grounded formulation in which each tunable admittance traces a circle in the complex plane, capturing the coupling between real and imaginary parts due to losses, and maps this to the BD-RIS scattering matrix across group- and forest-connected architectures. For SISO systems, the authors propose a double-loop MM-ADMM algorithm and a lower-complexity ADMM-based alternative to maximize the received power, while for MU-MISO systems they employ fractional programming and a BCD framework to jointly optimize the BS precoder and the BD-RIS phase/amplitude matrix; ADMM is again used to handle lossy constraints. Simulations show all BD-RIS architectures outperform D-RIS under losses, with group-connected BD-RIS often preferred in lossy scenarios, and reveal that optimal architectures in the lossless case do not necessarily remain optimal when losses are present, underscoring the need for loss-aware design and optimization in RIS-enabled networks.

Abstract

Beyond diagonal reconfigurable intelligent surface (BD-RIS) has emerged as an advancement and generalization of the conventional diagonal RIS (D-RIS) by introducing tunable interconnections between RIS elements, enabling smarter wave manipulation and enlarged coverage. While BD-RIS has demonstrated advantages over D-RIS in various aspects, most existing works rely on the assumption of a lossless model, leaving practical considerations unaddressed. This paper thus proposes a lossy BD-RIS model and develops corresponding optimization algorithms for various BD-RIS-aided communication systems. First, by leveraging admittance parameter analysis, we model each tunable admittance based on a lumped circuit with losses and derive an expression of a circle characterizing the real and imaginary parts of each tunable admittance. We then consider the received signal power maximization in single-user single-input single-output (SISO) systems with the proposed lossy BD-RIS model. To solve the optimization problem, we design an effective algorithm by carefully exploiting the problem structure. Specifically, an alternating direction method of multipliers (ADMM) framework is custom-designed to deal with the complicated constraints associated with lossy BD-RIS. Furthermore, we extend the proposed algorithmic framework to more general multiuser multiple-input single-output (MU-MISO) systems, where the transmit precoder and BD-RIS scattering matrix are jointly designed to maximize the sum-rate of the system. Finally, simulation results demonstrate that all BD-RIS architectures still outperform D-RIS in the presence of losses, but the optimal BD-RIS architectures in the lossless case are not necessarily optimal in the lossy case, e.g. group-connected BD-RIS can outperform fully- and tree-connected BD-RISs in SISO systems with relatively high losses, whereas the opposite always holds true in the lossless case.

Figures (12)

• Figure 1: Examples of a 36-element BD-RIS with (a) a group-connected reconfigurable admittance network, (b) a forest-connected reconfigurable admittance network in tridiagonal form, and (c) a forest-connected reconfigurable admittance network in arrowhead form, each with a group size of 3, along with the corresponding circuit model for each admittance component.
• Figure 2: Imaginary part of each $Y_{m_g,n_g}$ as a function of its real part at a signal frequency of $f = 2.4$ GHz with $L_1 = 6$ nH. Circles represent all possible values of the admittance component for varying $R$. The practical range corresponds to $C_{m_g,n_g} \in [0.35, 3.20]$ pF and $L_2 = 0.7$ nH.
• Figure 3: (a): Average rate versus $L_1$ for SISO systems ($R=2.5\ \Omega$, $M=32$, $\bar{M}=\{1,4,32\}$, $P=20$ dBm); (b): Sum-rate versus $L_1$ for MU-MISO systems ($R=3\ \Omega$, $N=K=4$, $M=32$, $\bar{M}=\{1,4,32\}$, $P=20$ dBm).
• Figure 4: (a) Convergence of the proposed MM-ADMM algorithm; (b) Convergence of the ADMM algorithm in the inner loop of MM-ADMM algorithm ($R=1 \ \Omega$, $M=30$, $\bar{M}=\{1,3,30\}$, $P=20$ dBm).
• Figure 5: Convergence of the low-complexity algorithm ($R=1 \ \Omega$, $M=30$, $\bar{M}=\{1,3,30\}$, $P=20$ dBm).