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Beyond-Diagonal Reconfigurable Intelligent Surfaces

Updated 17 November 2025

BD-RIS are intelligent metasurfaces that use non-diagonal scattering matrices to enable controlled inter-port electromagnetic coupling and advanced wave manipulation.
They offer flexible architectures—ranging from single- to fully-connected designs—that balance hardware complexity with performance enhancements like full-space coverage and joint transmission/reflection.
BD-RIS leverage network theoretic models and circuit realizations with robust optimization techniques to achieve near-ideal channel shaping and improved integrated sensing and communication.

Beyond-Diagonal Reconfigurable Intelligent Surface (BD-RIS) is a generalized class of intelligent metasurfaces wherein the scattering matrix is no longer restricted to be diagonal, thereby enabling controlled electromagnetic (EM) coupling among ports. This architectural departure from conventional diagonal RIS (D-RIS) unlocks advanced degrees of freedom for manipulating wave propagation, supporting functionalities such as full-space coverage, joint transmission and reflection, amplitude and phase engineering, and integrated sensing and communication. BD-RIS architectures, characterized by network-theoretic models and circuit realizations, support a spectrum of hardware and optimization trade-offs—from minimal-complexity designs compatible with IoT applications to fully-connected networks delivering near-ideal channel shaping in 6G systems.

1. General Principles and Physical Modeling

In contrast to D-RIS, whose scattering matrix $\Theta$ is strictly diagonal ( $\Theta = \mathrm{diag}(e^{j\theta_1}, \ldots, e^{j\theta_M})$ ), a BD-RIS allows $\Theta \in \mathbb{C}^{M\times M}$ to be non-diagonal, subject only to passivity ( $\Theta^H\Theta\preceq I$ ), losslessness ( $\Theta^H\Theta=I$ ), and reciprocity ( $\Theta = \Theta^T$ for reciprocal architectures) (Li et al., 22 May 2025). This is physically realized through a multiport passive network embedding tunable admittances between ports. The scattering matrix is related to the physical admittance by

$\Theta = (Y_0 I + Y)^{-1}(Y_0 I - Y),$

where $Y$ is the $M\times M$ network admittance, $Y_0$ is the reference port admittance, and $Y$ may have a general block or full structure (Li et al., 22 May 2025, Chen et al., 19 Apr 2024, Wu et al., 14 Oct 2025).

From a circuit perspective, each off-diagonal $Y_{mn}$ is rendered by a tunable resonant network (e.g., varactor-inductor modules), supporting both amplitude and phase adjustment and inter-element power splitting (Peng et al., 28 Apr 2025, Ming et al., 13 Apr 2025). The physical implementation of this general inter-port coupling is foundational for BD-RIS’s enhanced flexibility.

2. BD-RIS Architectures and Circuit Complexity

BD-RIS enables a continuous range of architectures by enforcing structural constraints on the admittance or susceptance matrix $Y$ (or $B$ ), directly determining the sparsity pattern and, by extension, the circuit complexity (Zhou et al., 22 Sep 2025, Zhou et al., 27 Nov 2024):

Architecture	Admittance Pattern	Circuit Complexity
Single-connected	$Y$ diagonal	$M$ tunable loads
Tree-connected	Arrowhead/tridiagonal $Y$	$2M-1$
Group-connected	Block-diagonal (size $\bar M$ )	$M + G\bar M(\bar M-1)/2$
Fully-connected	$Y$ full symmetric	$M(M+1)/2$
Stem/Cluster-connected	$Y$ stem/cluster block	$O(MQ)$ (for $Q$ stems)
Band-connected	$\|j-n\| \leq q$ (q-bandwidth)	$O(Mq)$

A key result is that proper selection of group, stem, or band parameters allows designers to match the channel shaping power of a fully-connected architecture at significantly reduced hardware cost—specifically, “stem-connected” or “band-connected” architectures with $Q=2M-1$ interconnections achieve the same channel gain as a fully-connected topology of $M\times M$ links (Zhou et al., 22 Sep 2025, Wu et al., 14 Oct 2025).

3. Principal Operating Modes and Functionalities

BD-RIS designs generalize operating modes beyond phase-only reflection:

Reflecting Mode: Standard, half-space operation. $\Theta$ is unrestricted apart from structural constraints.
Transmitting Mode: Transmission to the opposite half-space (via back-to-back antenna topology) (Li et al., 2022).
Hybrid Mode: Simultaneous transmission and reflection with independent phase and variable power splitting (enabled by two-port power splitters) (Ming et al., 13 Apr 2025, Wang et al., 2023).
Multi-sector Mode: Surface is subdivided into $L$ directional sectors, each forming a unitary subnetwork, yielding highly directive full-space coverage (Li et al., 2022).

The hybrid and multi-sector modes are uniquely enabled by BD-RIS and cannot be realized by classical D-RIS—off-diagonal admittances couple energy, steer beams through arbitrary “scattering pathways,” and support independent, simultaneous multi-directional wavefront shaping.

4. System-Level Modeling, Problem Formulations, and Algorithms

Communication, Sensing, and Joint ISAC:

System models uniformly link the end-to-end channel as $H_{\text{eff}} = H_B \Theta H_F$ (for MIMO) or $h = h_{RI}^H \Theta h_{IT}$ (for SISO), extended to include direct links and multiuser MIMO (Björnson et al., 27 Nov 2024, Zhao et al., 21 Jul 2024, Chen et al., 30 Sep 2025).
Optimization problems include sum-rate maximization, min-CRB (Cramér-Rao bound) for sensing, and dual-function trade-off metrics:

$\max_{\Theta, W} ~ \rho\cdot \frac{\text{Sum-Rate}}{V_c} + (1-\rho) \cdot \frac{1}{V_s}(-\operatorname{tr}(F^{-1}))$

for ISAC (Chen et al., 30 Sep 2025, Chen et al., 19 Apr 2024, Wang et al., 2023).

Constraints on $\Theta$ depend on architecture: unitary/symmetric (for fully/group), diagonal unit-modulus (single), block-unitary (group), or structurally sparse for stem/band (Zhou et al., 22 Sep 2025).
Solution methodologies: Joint optimization is non-convex; tractable updates are obtained via alternating optimization (AO), block coordinate descent, projected gradient or conjugate gradient on the complex Stiefel manifold (for unitary blocks), or structure-oriented projection algorithms (Zhou et al., 22 Sep 2025, Zhou et al., 27 Nov 2024, Fidanovski et al., 24 Sep 2025). For joint active/passive design, closed-form projections onto the power sphere or symmetry/unitarity constraints are used (Chen et al., 30 Sep 2025).

Remarkably, the capacity-optimal $\Theta$ for MIMO channels is achieved by matching the right singular vectors of $H_F$ to the left singular vectors of $H_B$ ( $\Theta^*=V_B U_F^H$ ), thus enabling optimal parallelization of end-to-end eigenchannels not attainable by diagonal RIS (Björnson et al., 27 Nov 2024, Zhao et al., 21 Jul 2024).

5. Channel Estimation and Hardware Non-idealities

Channel Estimation:

For BD-RIS, estimation of the cascaded channel (RIS-augmented channel) requires tailored pilot and RIS pattern design. For group/fully-connected blocks, the minimum-variance unbiased estimator’s MSE is $\sigma^2 / (P_u N \bar{M})$ , where $\bar{M}$ is the group size; thus, richer coupling increases both expressive power and the training overhead (Li et al., 26 Mar 2024).
Fast channel estimation becomes critical as the number of unknowns scales quadratically with $M$ in fully-connected systems (Khan et al., 5 Feb 2025).

Hardware Impairments and Losses:

Physical constraints—mutual coupling, impedance mismatch, admittance quantization, frequency dependence, and component losses—are captured in physics-consistent models via multiport network theory (Wu et al., 14 Oct 2025, Hougne, 30 Sep 2024, Peng et al., 28 Apr 2025).
In practical settings, moderate/intermediate group-connected architectures can outperform fully-connected ones as loss increases, owing to a better trade-off between beneficial coupling and dissipative loss (Peng et al., 28 Apr 2025).
Frequency-dependent modeling (wideband/OFDM) reveals that the performance gains of BD-RIS are more pronounced as the complexity of element-wise coupling increases, but so is the sensitivity to accurate circuit modeling (Li et al., 12 May 2024).

6. Performance–Complexity Trade-offs and Design Guidelines

Power gain and capacity: In SISO Rayleigh/LoS channels, fully-connected BD-RIS achieves an average gain ~62% higher than single-connected (D-RIS) (Li et al., 22 May 2025, Nerini et al., 20 Dec 2024).
Multiplexing and sum-rate: In multiuser MIMO, stem/cluster/band-connected architectures can match fully-connected performance when stem or band width equals twice the number of data streams minus one ( $Q=2M-1$ ), with only $O(M)$ hardware complexity rather than $O(M^2)$ (Zhou et al., 22 Sep 2025, Zhou et al., 27 Nov 2024).
Dynamic grouping: Adapting the BD-RIS interconnection topology to channel state information and traffic can yield 12–21% higher sum-rate than any fixed architecture at moderate hardware cost, with the benefit rising as system size increases (Li et al., 2022).
Design guidelines: For high spectral efficiency under hardware constraints, designers are encouraged to employ stem- or dynamically group-connected BD-RIS with $Q \sim 2M-1$ , or exploit multi-sector or hybrid modes for directional full-space coverage or simultaneous multi-beam operation (Li et al., 2022, Ming et al., 13 Apr 2025).

7. Applications, Prototyping, and Future Directions

Integrated Sensing and Communications (ISAC):

Transmitter-side BD-RIS allows substantial dual-functional gains: up to 30% sum-rate increase or 50% CRB reduction compared to diagonal RIS for the same hardware size, and can accommodate more sensing targets without radar degradation (Chen et al., 30 Sep 2025, Chen et al., 19 Apr 2024).

Physical Layer Security, IoT, and Wideband Systems:

Fine-grain interference nulling, dynamic spectrum enhancement in vehicular and dense IoT deployments, and multi-carrier power transfer are all achievable via optimized BD-RIS topologies (Khan et al., 5 Feb 2025, Li et al., 22 May 2025).
Prototyped 4x4-cell BD-RIS arrays with hybrid control realize independent dual-beam steering and tunable transmission/reflectivity, experimentally validating the analytic Thevenin equivalent model (Ming et al., 13 Apr 2025).

Physics-Compliant Modeling and Legacy Compatibility:

Through a three-block chain-cascade abstraction, all existing D-RIS optimization tools can be directly applied to BD-RIS once the static coupling network is characterized, mitigating the complexity of deployment and management (Hougne, 30 Sep 2024).

Open Challenges:

Hardware scalability, fast calibration, low-overhead channel estimation, and robust optimization under strong coupling and EM non-idealities remain active research areas for realizing the full potential of BD-RIS in ultra-dense, programmable EM environments for 6G and beyond (Li et al., 22 May 2025, Wu et al., 14 Oct 2025).

References:

(Li et al., 22 May 2025) “A Tutorial on Beyond-Diagonal Reconfigurable Intelligent Surfaces: Modeling, Architectures, System Design and Optimization, and Applications”
(Chen et al., 30 Sep 2025) “Transmitter-Side Beyond-Diagonal RIS-Enabled Integrated Sensing and Communications”
(Chen et al., 19 Apr 2024) “Transmitter Side Beyond-Diagonal RIS for mmWave Integrated Sensing and Communications”
(Zhou et al., 22 Sep 2025) “Generalized Beyond-Diagonal RIS Architectures: Theory and Design via Structure-oriented Symmetric Unitary Projection”
(Wu et al., 14 Oct 2025) “Beyond-Diagonal RIS Architecture Design and Optimization under Physics-Consistent Models”
(Zhou et al., 27 Nov 2024) “A Novel Q-stem Connected Architecture for Beyond-Diagonal Reconfigurable Intelligent Surfaces”
(Ming et al., 13 Apr 2025) “A Hybrid Transmitting and Reflecting Beyond Diagonal Reconfigurable Intelligent Surface with Independent Beam Control and Power Splitting”
(Hougne, 30 Sep 2024) “A physics-compliant diagonal representation for wireless channels parametrized by beyond-diagonal reconfigurable intelligent surfaces”
(Peng et al., 28 Apr 2025) “Lossy Beyond Diagonal Reconfigurable Intelligent Surfaces: Modeling and Optimization”
(Li et al., 2022) “A Dynamic Grouping Strategy for Beyond Diagonal Reconfigurable Intelligent Surfaces with Hybrid Transmitting and Reflecting Mode”
(Nerini et al., 20 Dec 2024) “Dual-Polarized Beyond Diagonal RIS”
(Wang et al., 2023) “A Dual-Function Radar-Communication System Empowered by Beyond Diagonal Reconfigurable Intelligent Surface”
(Zhao et al., 21 Jul 2024) “MIMO Channel Shaping and Rate Maximization Using Beyond Diagonal RIS”
(Björnson et al., 27 Nov 2024) “Capacity Maximization for MIMO Channels Assisted by Beyond-Diagonal RIS”
(Khan et al., 5 Feb 2025) “Beyond Diagonal RIS: A New Frontier for 6G Internet of Things Networks”
(Li et al., 26 Mar 2024) “Channel Estimation and Beamforming for Beyond Diagonal Reconfigurable Intelligent Surfaces”
(Li et al., 12 May 2024) “Beyond Diagonal Reconfigurable Intelligent Surfaces in Wideband OFDM Communications: Circuit-Based Modeling and Optimization”
(Li et al., 2022) “Beyond Diagonal Reconfigurable Intelligent Surfaces: A Multi-Sector Mode Enabling Highly Directional Full-Space Wireless Coverage”
(Li et al., 2022) “Beyond Diagonal Reconfigurable Intelligent Surfaces: From Transmitting and Reflecting Modes to Single-, Group-, and Fully-Connected Architectures”
(Fidanovski et al., 24 Sep 2025) “Reciprocal Beyond-Diagonal Reconfigurable Intelligent Surface (BD-RIS): Scattering Matrix Design via Manifold Optimization”