Beyond Diagonal IRS: Advanced Wave Control

• Beyond Diagonal Intelligent Reflecting Surface (BD-IRS) is a reconfigurable platform that enables coupled amplitude-phase control through inter-element connections, extending traditional independent phase shifting.
• It leverages multi-port circuit models and graph-theoretic approaches to optimize signal reflection and transmission across wideband and multi-user systems.
• Sparse architectures such as tree-connected and group-connected BD-IRS yield near-optimal performance with reduced circuit complexity compared to fully-connected configurations.

Searching arXiv for recent BD-RIS/BD-IRS papers to ground the article in current literature. arXiv_search(query="Beyond diagonal RIS OR BD-RIS OR BD-IRS", max_results=10) Beyond diagonal intelligent reflecting surfaces, also termed beyond-diagonal reconfigurable intelligent surfaces in much of the literature, generalize conventional RIS by removing the restriction that the surface scattering matrix be diagonal. In the diagonal model, each element independently applies a phase shift, typically written as $$\Theta=\operatorname{diag}(e^{j\theta_1},\dots,e^{j\theta_M})$$. In BD-IRS, inter-element reconfigurable connections make the scattering matrix non-diagonal, so incident energy on one port can be rerouted and re-radiated by another. Under ideal lossless modeling, the feasible set is commonly unitary, and under reciprocal network realizations it is additionally symmetric; under more general passive modeling, the constraint relaxes to $\Theta^H\Theta\preceq I_M$. This shift enlarges the wave-manipulation degrees of freedom from element-wise phase control to coupled amplitude-phase processing and unifies reflective, transmissive, hybrid, and multi-sector intelligent surfaces within one framework (Li et al., 22 May 2025, Li et al., 2022).

1. Conceptual foundation and formal definition

The defining feature of BD-IRS is controlled inter-element coupling. In microwave-network language, an $N$-port surface relates incoming and outgoing waves through $\mathbf b=\mathbf S\mathbf a$, with lossless operation imposing $\mathbf S^H\mathbf S=I_N$. Conventional RIS constrains $\mathbf S$ or $\Theta$ to be diagonal, whereas BD-IRS permits off-diagonal entries that arise from reconfigurable impedance or admittance networks among ports. This means that a wave absorbed by one element can flow through other elements before re-radiation, rather than being processed locally and independently (Li et al., 22 May 2025).

A canonical reciprocal lossless model uses a real symmetric susceptance matrix $B$ and reference impedance $Z_0$, yielding

$\Theta=(I+jZ_0B)^{-1}(I-jZ_0B),$

with $\Theta^H\Theta\preceq I_M$0 and $\Theta^H\Theta\preceq I_M$1. In this formulation, a nonzero off-diagonal entry of $\Theta^H\Theta\preceq I_M$2 corresponds to a tunable admittance between two ports. The diagonal RIS appears as the special case in which $\Theta^H\Theta\preceq I_M$3 is diagonal; BD-IRS corresponds to sparse, block-sparse, or fully populated non-diagonal $\Theta^H\Theta\preceq I_M$4 (Nerini et al., 2023).

A common misconception is that BD-IRS means an arbitrary dense complex matrix. The physically consistent models in the literature are more constrained. Depending on whether the surface is passive or active, reciprocal or non-reciprocal, narrowband or wideband, the admissible matrices are unitary, symmetric-unitary, block-diagonal unitary, asymmetric unitary, or frequency-dependent matrices induced by circuit parameters rather than chosen independently entry by entry (Li et al., 22 May 2025).

2. Multi-port circuit modeling and frequency dependence

The modern BD-IRS literature is built on multi-port network theory. In the tutorial treatment, impedance, admittance, and scattering descriptions are linked through

$\Theta^H\Theta\preceq I_M$5

with reciprocity implying $\Theta^H\Theta\preceq I_M$6, $\Theta^H\Theta\preceq I_M$7, and $\Theta^H\Theta\preceq I_M$8, while losslessness implies $\Theta^H\Theta\preceq I_M$9, $N$0, and hence $N$1. The same framework is also used to include mutual coupling through the embedded array network and to derive end-to-end channels of the form

$N$2

which makes clear that the RIS action is not, in general, separable from mutual electromagnetic interaction (Li et al., 22 May 2025).

Practical broadband modeling sharpens this picture further. In the OFDM system of “Beyond Diagonal IRS Aided OFDM: Rate Maximization under Frequency-Dependent Reflection,” the interconnection between element $N$3 and element $N$4 is realized by an $N$5–$N$6–$N$7–$N$8 chain, with all tunable capacitors collected into a real nonnegative matrix $N$9. The resulting admittance matrix $\mathbf b=\mathbf S\mathbf a$0 depends jointly on $\mathbf b=\mathbf S\mathbf a$1 and the operating frequency, and the reflection matrix becomes

$\mathbf b=\mathbf S\mathbf a$2

At OFDM subcarrier $\mathbf b=\mathbf S\mathbf a$3, $\mathbf b=\mathbf S\mathbf a$4 with $\mathbf b=\mathbf S\mathbf a$5. The key consequence is that the reflection matrices over different subcarriers are coupled through the same capacitance matrix, so the standard narrowband practice of optimizing each subcarrier independently is no longer physically consistent (Yuan et al., 8 Sep 2025).

This frequency dependence is not a minor implementation detail. It changes the design variable itself: the object of optimization is no longer a free collection $\mathbf b=\mathbf S\mathbf a$6 but a common circuit realization that induces all $\mathbf b=\mathbf S\mathbf a$7 simultaneously. This suggests that wideband BD-IRS design is fundamentally a circuit-system co-design problem rather than a straightforward extension of narrowband passive beamforming (Yuan et al., 8 Sep 2025).

3. Architectures and the performance-complexity trade-off

The literature distinguishes several BD-IRS architectures by interconnection topology. The baseline classes are single-connected, group-connected, and fully-connected. Single-connected RIS is diagonal. Group-connected RIS partitions the ports into groups and fully interconnects only within each group, giving a block-diagonal scattering matrix. Fully-connected RIS allows all ports to interconnect and yields the largest feasible set of scattering matrices under the relevant physical constraints (Li et al., 2022).

Graph-theoretic modeling makes this taxonomy precise. In the formulation of “Beyond Diagonal Reconfigurable Intelligent Surfaces Utilizing Graph Theory,” the RIS is represented by a simple graph $\mathbf b=\mathbf S\mathbf a$8, where each edge corresponds to a tunable inter-port admittance and each vertex also has a ground admittance. The total number of tunable elements is therefore $\mathbf b=\mathbf S\mathbf a$9. Two particularly important architectures are tree-connected and forest-connected RIS. Tree-connected RIS uses a connected acyclic graph with $\mathbf S^H\mathbf S=I_N$0, so its circuit complexity is $\mathbf S^H\mathbf S=I_N$1, which is much smaller than the $\mathbf S^H\mathbf S=I_N$2 tunable admittances of a fully-connected realization. The same work proves that tree-connected RIS is the least complex BD-RIS architecture able to achieve the performance upper bound in MISO systems, while forest-connected RIS provides an intermediate performance-complexity trade-off and can reduce complexity by up to $\mathbf S^H\mathbf S=I_N$3 times relative to the corresponding fully-connected benchmark in the reported example (Nerini et al., 2023).

Other architectural refinements broaden this trade-off. Dynamic grouping permits a permuted block-diagonal structure adapted to CSI rather than fixed grouping, and the reported simulations show that the dynamically group-connected architecture outperforms fixed group-connected architectures (Li et al., 2022). Multi-sector BD-RIS arranges antennas into $\mathbf S^H\mathbf S=I_N$4 sectors on a polygon prism, providing full-space coverage with highly directional sector beams and a distinct scaling law in the number of sectors (Li et al., 2022). Hybrid transmitting-and-reflecting BD-RIS splits the scattering into reflection and transmission submatrices constrained by

$\mathbf S^H\mathbf S=I_N$5

thereby unifying reflective, transmissive, and simultaneous transmitting/reflecting operation in a single network model (Li et al., 2022, Mahmood et al., 2024).

Dual polarization adds a further dimension to the architectural question. Nerini and Clerckx show that, in dual-polarized systems, group-connected RIS with group size $\mathbf S^H\mathbf S=I_N$6 provides remarkable gains over conventional RIS in both Rayleigh and line-of-sight channels while maintaining reduced circuit complexity, and in the worst-case LoS plus opposite-polarization regime the group size-$\mathbf S^H\mathbf S=I_N$7 architecture achieves the same performance as the fully-connected design with complexity $\mathbf S^H\mathbf S=I_N$8 rather than $\mathbf S^H\mathbf S=I_N$9 (Nerini et al., 2024). A plausible implication is that, in several practically relevant regimes, the most meaningful design choice is not “diagonal versus fully-connected” but rather which sparse topology sits on the Pareto frontier for the channel and hardware regime of interest.

4. Optimization frameworks and algorithmic methods

The optimization problems induced by BD-IRS are typically non-convex because the scattering matrix is constrained by unitary, symmetric, block, or circuit-induced structure and is coupled with transmit precoding, receive filtering, or power allocation. Several algorithmic families recur across the literature: closed-form synthesis when the structure is sufficiently rigid, block-coordinate descent, fractional programming, manifold optimization on Stiefel or unitary manifolds, alternating optimization, successive convex approximation, semidefinite relaxations, ADMM, and penalty dual decomposition (Li et al., 22 May 2025).

Closed-form global optimization is available in several important cases. “Closed-Form Global Optimization of Beyond Diagonal Reconfigurable Intelligent Surfaces” proves that, for reflective or transmissive SISO systems and extensions to single-user MIMO and multi-user MISO, the theoretical performance upper bounds can be exactly achieved for any channel realization. The fully-connected solution is obtained with cubic complexity in the number of RIS elements, while group-connected architectures scale linearly in the number of RIS elements when the group size is fixed (Nerini et al., 2022). In the graph-theoretic setting, tree-connected RIS admits a unique closed-form global optimum obtained by solving a real-valued linear system, whereas forest-connected RIS is optimized by a provably convergent low-complexity alternating algorithm with per-iteration complexity $\mathbf S$0 (Nerini et al., 2023).

For sum-rate maximization in multi-user systems, the standard design pattern is to alternate between active beamforming and BD-IRS updates. In the unifying 2022 framework, fractional programming with the Lagrangian dual and quadratic transforms rewrites the log-SINR objective using auxiliary variables, and the group- or fully-connected BD-RIS update is then solved by Riemannian conjugate-gradient optimization on the complex Stiefel manifold. The single-connected case reduces to element-wise amplitude and phase updates under a per-cell power-splitting constraint (Li et al., 2022).

Wideband OFDM introduces an additional layer. In (Yuan et al., 8 Sep 2025), the exact circuit expression is first relaxed using the passive constraint $\mathbf S$1, producing a tractable problem over surrogate subcarrier reflection matrices $\mathbf S$2 and subcarrier powers $\mathbf S$3. With fixed $\mathbf S$4, power allocation reduces to standard water-filling. With fixed powers, the $\mathbf S$5 update is handled by successive convex approximation after linearizing the quadratic magnitude terms. Once the alternating optimization converges, a common capacitance matrix is recovered by solving

$\mathbf S$6

followed by recomputation of the actual subcarrier responses and water-filling power allocation (Yuan et al., 8 Sep 2025).

Sensing-oriented BD-IRS optimization often replaces rate objectives by information-theoretic or estimation-theoretic criteria. In the CRB-based sensing letter, the design variable is a unitary BD-RIS scattering matrix, and the objective is to maximize

$\mathbf S$7

subject to $\mathbf S$8, which is solved by an adaptive Riemannian steepest ascent algorithm on the complex Stiefel manifold (Zhang et al., 15 Aug 2025). In integrated sensing and communication, the reflection optimization is further coupled to posterior Cramér–Rao bound constraints and multi-user rate fairness, leading to a penalty dual decomposition method and a separate TDMA-based alternative that removes sensing-communication mutual interference (Zheng et al., 22 May 2025).

5. Communication, sensing, and computation applications

The primary motivation for BD-IRS is system-level performance enhancement under realistic propagation and hardware constraints. In communication systems, gains have been reported across narrowband MISO/MIMO, THz links, OFDM, transmitter-side RIS-assisted massive MIMO, and hybrid indoor/outdoor deployments. In the low-resolution THz downlink problem, joint design of a $\mathbf S$9-bit digital precoder and a discrete-phase fully-connected BD-IRS yields a spectral-efficiency improvement over the conventional diagonal IRS, with the reported gain remaining roughly $\Theta$0–$\Theta$1 as the BS antenna count grows and the design converging within $\Theta$2–$\Theta$3 outer iterations in the cited configuration (Khan et al., 2024). In transmitter-side massive MIMO, optimizing a fully-connected unitary BD-RIS from statistical CSI gives a per-user SE gain that grows from $\Theta$4 at $\Theta$5 to $\Theta$6 at $\Theta$7 over the diagonal benchmark (Mishra et al., 2023).

Hybrid transmitting-and-reflecting BD-IRS is especially relevant for full-space coverage. In THz indoor/outdoor service with reflective and transmissive users, the fully-connected hybrid BD-IRS with jointly optimized beamforming improves the system sum rate by approximately $\Theta$8 over TDMA, $\Theta$9 over FDMA, and $B$0 over STAR-IRS in the reported study (Mahmood et al., 2024). A complementary hardware-oriented design based on two phase-reconfigurable antenna arrays and tunable two-port power splitters provides independent beam steering control of reflected and transmitted waves together with tunable power splitting in the same aperture, frequency band, and polarization (Ming et al., 13 Apr 2025).

Sensing and integrated sensing-and-communication form a second major application cluster. In pure sensing, the CRB-optimization framework for AOA estimation under a unitary BD-RIS constraint shows that a fully-connected BD-RIS with $B$1 can match or outperform a diagonal RIS with $B$2, and that the proposed Riemannian algorithm converges in fewer than $B$3 iterations (Zhang et al., 15 Aug 2025). In uplink ISAC with a device-based sensing model, the BD-IRS reflection matrix is optimized to maximize the minimum expected achievable rate subject to a posterior CRB constraint, with reported design insights that fully- and group-connected BD-IRS significantly outperform single-connected IRS, that group-connected designs offer a good complexity-performance trade-off, and that TDMA can outperform SDMA under severe sensing-communication interference or loose sensing requirements (Zheng et al., 22 May 2025). In dual-function radar-communication, a hybrid reflecting/transmitting BD-RIS supports full-space coverage and fair target detection via max-min SCNR optimization, and the reported numerical results show that the cell-wise fully- and group-connected designs outperform STAR-RIS by up to $B$4 dB when the QoS threshold is large (Wang et al., 2023).

BD-IRS has also been integrated with mobile edge computing and UAV deployment. In a UAV-BD-IRS-enabled MEC network with partial offloading, joint optimization of UAV placement, computational resource allocation, communication resources, and the BD-IRS matrix yields a $B$5 increase over traditional diagonal IRS, a $B$6 improvement over IRS on buildings, and a $B$7 enhancement in worst-case latency compared to binary offloading schemes (Mahmood et al., 2023). These results indicate that the added analog-domain degrees of freedom of BD-IRS can translate into network-level gains beyond link-level rate or SNR metrics.

6. Hardware realization, nonidealities, and active/hybrid extensions

BD-IRS is inseparable from circuit realization. The tutorial literature emphasizes discrete-value impedance and admittance tuning, lossy interconnections and components, wideband effects, and mutual coupling as first-order design issues rather than secondary implementation losses (Li et al., 22 May 2025). This is reflected in hardware prototypes. In the hybrid transmitting-and-reflecting prototype of (Ming et al., 13 Apr 2025), each cell contains two $B$8-bit phase-reconfigurable antennas and a tunable two-port power splitter built upon a varactor in parallel with a bias inductor. The splitter controls the power ratio of $B$9 over $Z_0$0 from $Z_0$1 dB to $Z_0$2 dB, each antenna provides $Z_0$3 MHz bandwidth at $Z_0$4 GHz, and a $Z_0$5 prototype experimentally verifies beam steering in reflection mode, transmission mode, and hybrid mode with independent reflected and transmitted beams (Ming et al., 13 Apr 2025).

A common misconception is that passive BD-IRS alone resolves the multiplicative path-loss bottleneck. The active BD-RIS literature addresses this directly by introducing reflection-type amplifiers into beyond-diagonal architectures. In the multiport-network model of active BD-RIS, the end-to-end channel becomes

$Z_0$6

so amplification improves the useful signal but also injects dynamic noise through $Z_0$7. The corresponding power constraint includes both amplified signal and amplified noise terms. In the reported results, active BD-RIS can achieve higher spectral efficiency than active/passive diagonal RIS and passive BD-RIS, and to achieve the same spectral efficiency the number of elements required by active BD-RIS is less than half of that required by active diagonal RIS (Shen et al., 14 Mar 2026).

Hybrid active/passive and hybrid transmitting/reflecting extensions expand the design space further. “Active Beyond-Diagonal Reconfigurable Intelligent Surface with Hybrid Transmitting and Reflecting Mode” derives a physics-compliant model for an active BD-RIS with full-space coverage and proposes reciprocal and non-reciprocal implementations with cell-wise single, group, and fully connections. Under the same total power budget, the proposed active hybrid-mode BD-RIS substantially outperforms active and passive simultaneous transmitting and reflecting RISs as well as passive BD-RISs with hybrid mode (Liu et al., 15 Apr 2026). Ntougias and Krikidis introduce a family of hybrid BD-RIS architectures in which two reflecting subsurfaces are active or passive group-connected BD-RISs; their SISO analysis shows that these architectures attain the same or higher receive SNR than diagonal counterparts while using significantly fewer reflect-type amplifiers (Ntougias et al., 7 May 2026).

The resulting picture is nuanced. Fully-connected BD-IRS offers the largest feasible set of scattering responses, but the literature repeatedly shows that specific sparse architectures can achieve the same upper bound in specific regimes, while active extensions mitigate multiplicative fading at the cost of amplifier noise, extra power consumption, and additional RF complexity. This suggests that the central design question for BD-IRS is not whether inter-element coupling is useful—the evidence across communications, sensing, and networking indicates that it is—but rather how much coupling, which physical topology, and which level of hardware sophistication are justified for a given deployment scenario (Nerini et al., 2023, Li et al., 22 May 2025).