A Tutorial on Beyond-Diagonal Reconfigurable Intelligent Surfaces: Modeling, Architectures, System Design and Optimization, and Applications

Written by its inventors, this first tutorial on Beyond-Diagonal Reconfigurable Intelligent Surfaces (BD-RISs) provides the readers with the basics and fundamental tools necessary to appreciate, understand, and contribute to this emerging and disruptive technology. Conventional (Diagonal) RISs (D-RISs) are characterized by a diagonal scattering matrix $\mathbfΘ$ such that the wave manipulation flexibility of D-RIS is extremely limited. In contrast, BD-RIS refers to a novel and general framework for RIS where its scattering matrix is not limited to be diagonal (hence, the ``beyond-diagonal’’ terminology) and consequently, all entries of $\mathbfΘ$ can potentially help shaping waves for much higher manipulation flexibility. This physically means that BD-RIS can artificially engineer and reconfigure coupling across elements of the surface thanks to inter-element reconfigurable components which allow waves absorbed by one element to flow through other elements. Consequently, BD-RIS opens the door to more general and versatile intelligent surfaces that subsumes existing RIS architectures as special cases. In this tutorial, we share all the secret sauce to model, design, and optimize BD-RIS and make BD-RIS transformative in many different applications. Topics discussed include physics-consistent and multi-port network-aided modeling; transmitting, reflecting, hybrid, and multi-sector mode analysis; reciprocal and non-reciprocal architecture designs and optimal performance-complexity Pareto frontier of BD-RIS; signal processing, optimization, and channel estimation for BD-RIS; hardware impairments (discrete-value impedance and admittance, lossy interconnections and components, wideband effects, mutual coupling) of BD-RIS; benefits and applications of BD-RIS in communications, sensing, power transfer.

💡 Research Summary

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This paper presents the first comprehensive tutorial on Beyond‑Diagonal Reconfigurable Intelligent Surfaces (BD‑RIS), a new class of metasurfaces that overcomes the fundamental limitations of conventional diagonal RIS (D‑RIS). While D‑RIS can only tune the phase of each element independently, resulting in a diagonal scattering matrix Θ, BD‑RIS introduces inter‑element reconfigurable connections (e.g., tunable capacitors, inductors, non‑reciprocal phase shifters) that allow the scattering matrix to become a full, possibly non‑symmetric, unitary matrix. This extra degree of freedom enables waves absorbed by one element to flow through others, dramatically expanding wave‑manipulation capabilities.

The tutorial is organized into several parts. First, a toy SISO example illustrates how BD‑RIS can improve received power compared with D‑RIS, providing intuition for the subsequent sections. Next, a physics‑consistent multi‑port network model is derived: each RIS element is treated as a port with an associated impedance/admittance matrix, and the overall scattering matrix is obtained via standard Z‑to‑S parameter conversion. This model naturally incorporates loss, mutual coupling, and frequency‑dependent behavior, making it suitable for realistic hardware design.

The authors then classify BD‑RIS operation into four fundamental modes—transmission, reflection, hybrid, and multi‑sector—each corresponding to a different way of connecting the ports. They discuss a wide range of possible architectures: block‑diagonal, fully connected, and permuted matrices, as well as reciprocal (symmetric Θ) versus non‑reciprocal (asymmetric Θ) designs. By mapping each architecture to its corresponding Θ structure, they derive a performance‑complexity Pareto frontier, showing that modestly connected (block‑diagonal) designs can achieve near‑optimal performance with far lower hardware overhead than fully connected implementations.

Optimization and signal‑processing techniques are surveyed in depth. For the SISO case, the tutorial formulates joint transmit‑beamforming and BD‑RIS configuration as a non‑convex problem with unitary constraints on Θ. Solution methods include alternating optimization, matrix factorization (QR/SVD) with unitary projection, semidefinite programming (SDP) relaxations, and meta‑heuristics such as genetic algorithms and deep reinforcement learning. Channel estimation is addressed by exploiting the multi‑port nature of the surface: least‑squares, compressed‑sensing, and novel protocols for estimating asymmetric channels in non‑reciprocal configurations are presented, together with a discussion of the challenges posed by the enlarged parameter space.

Hardware impairments specific to BD‑RIS are modeled and quantified. Four key non‑idealities are considered: (i) discrete‑valued impedance/admittance due to digital control, (ii) lossy interconnections and components that break the unitary property, (iii) wideband effects where the impedance matrix varies with frequency, and (iv) mutual coupling among closely spaced elements. The authors provide analytical expressions for performance degradation under each impairment and suggest design guidelines (e.g., optimal quantization levels, low‑loss materials, compensation networks) to mitigate them.

The tutorial then showcases the benefits of BD‑RIS across several 6G‑relevant applications. In communications, BD‑RIS can extend indoor‑outdoor coverage, assist cell‑edge users, and serve as an auxiliary antenna array for massive MIMO or stacked‑surface systems. In sensing, the additional degrees of freedom improve radar detection in non‑line‑of‑sight environments. In wireless power transfer, BD‑RIS can act as an efficient power relay, and in integrated scenarios (SWIPT, ISAC) it can simultaneously deliver information, power, and sensing data. Numerical results demonstrate that non‑reciprocal hybrid modes achieve higher spectral efficiency and lower bit‑error rates than traditional reflective RIS, while block‑diagonal architectures attain performance close to the fully connected optimum with far reduced circuit complexity.

Finally, the authors identify open research challenges: real‑time large‑scale optimization, fabrication of low‑loss reconfigurable components, broadband multi‑frequency operation, security and privacy in programmable environments, and standardization for network‑level integration. They suggest future directions such as machine‑learning‑driven online configuration, metasurface‑based terahertz implementations, and cross‑layer design that jointly considers PHY‑layer RIS control and higher‑layer network protocols.

In summary, this tutorial provides a unified, physics‑grounded framework for modeling, designing, optimizing, and deploying BD‑RIS, positioning it as a cornerstone technology for the next generation of intelligent wireless environments.