BD-RIS: Beyond Diagonal Surfaces

• BD-RIS are reconfigurable intelligent surfaces defined by non-diagonal scattering matrices that enable multiport wave manipulation beyond conventional RIS.
• They support diverse architectures—from single-connected to fully-connected modes—offering richer beam pattern design and improved sidelobe shaping.
• Advanced optimization and estimation techniques are essential to overcome challenges in channel estimation, hardware losses, and wideband operation.

Beyond Diagonal Reconfigurable Intelligent Surface (BD-RIS) denotes a class of reconfigurable intelligent surfaces whose scattering matrix is not restricted to be diagonal, so incident power at one port can be reradiated through other ports via reconfigurable inter-element connections. In the passive and lossless idealization, the scattering matrix is typically constrained by $\Theta^H\Theta=I$, and the framework unifies conventional single-connected RIS, group-connected and fully-connected networks, as well as reflective, transmissive, hybrid, and multi-sector operating modes (Li et al., 2022, Li et al., 22 May 2025). Relative to diagonal RIS, BD-RIS enlarges the controllable design space from element-wise phase shifts to multi-port wave manipulation, but this added flexibility makes channel estimation, topology design, and hardware realization substantially more involved (Zhao et al., 5 Dec 2025).

1. Fundamental model and physical interpretation

A BD-RIS is naturally described as a passive multi-port network. If $\mathbf{a}\in\mathbb{C}^N$ and $\mathbf{b}\in\mathbb{C}^N$ denote the incident and outgoing wave vectors, the surface implements $\mathbf{b}=\mathbf{S}\mathbf{a}$, with scattering matrix $\mathbf{S}\in\mathbb{C}^{N\times N}$; equivalent impedance and admittance descriptions are related through

$$\Theta=(Z+Z_0I)^{-1}(Z-Z_0I)=(Y_0I+Y)^{-1}(Y_0I-Y),$$

with $Y_0=1/Z_0$ (Li et al., 22 May 2025). Passivity implies $\Theta^H\Theta\preceq I$, while losslessness implies $\Theta^H\Theta=I$ (Li et al., 22 May 2025).

The distinction from conventional RIS is structural rather than cosmetic. A diagonal RIS imposes

$$\Theta_{\rm D}=\mathrm{diag}(e^{j\theta_1},\dots,e^{j\theta_N}),$$

whereas BD-RIS allows a full or block-structured matrix with nonzero off-diagonal terms (Li et al., 2022). In a fully-connected BD-RIS, the scattering matrix has $\mathbf{a}\in\mathbb{C}^N$0 real parameters subject to the unitary constraint $\mathbf{a}\in\mathbb{C}^N$1, whereas a diagonal RIS has only $\mathbf{a}\in\mathbb{C}^N$2 (Zhao et al., 5 Dec 2025). This additional freedom enables simultaneous amplitude and phase reconfiguration, richer beam patterns, and more flexible sidelobe shaping (Zhang et al., 15 Aug 2025).

A recurrent conceptual point in the literature is that the off-diagonal structure should not be conflated with uncontrolled electromagnetic mutual coupling. A physics-compliant interpretation separates the radio environment, the static load circuit, and the individually tunable loads; within that decomposition, the off-diagonal behavior of a BD-RIS-parametrized channel originates from the RIS control circuit, while the tunable part can still be represented by a strictly diagonal matrix of individual loads (Hougne, 2024). This observation is central because it connects BD-RIS to existing circuit-theoretic and physics-compliant RIS models rather than placing it outside them.

2. Architectural taxonomy and operating modes

BD-RIS architectures are usually classified by how ports are interconnected. The canonical forms are summarized below.

Architecture	Matrix structure	Typical note
Single-connected	Diagonal $\mathbf{a}\in\mathbb{C}^N$3	Conventional RIS
Group-connected	$\mathbf{a}\in\mathbb{C}^N$4	DoF/complexity trade-off
Fully-connected	Arbitrary unitary $\mathbf{a}\in\mathbb{C}^N$5	Maximum flexibility
Dynamic group-connected	$\mathbf{a}\in\mathbb{C}^N$6	CSI-adaptive grouping
Tree/forest-connected	Sparse off-diagonal structure	Reduced hardware complexity

In the group-connected model, the $\mathbf{a}\in\mathbb{C}^N$7 ports are partitioned into disjoint groups, each internally interconnected and constrained by block-wise unitarity; the single-connected and fully-connected cases arise as the extremes $\mathbf{a}\in\mathbb{C}^N$8 and $\mathbf{a}\in\mathbb{C}^N$9, respectively (Zhang et al., 15 Aug 2025). Dynamic grouping generalizes this by introducing a permutation matrix $\mathbf{b}\in\mathbb{C}^N$0 so that grouping itself becomes a CSI-dependent optimization variable (Li et al., 2022).

A second classification concerns radiation mode. Reflective and transmissive modes correspond to single-sided operation, while hybrid operation jointly supports reflection and transmission under

$\mathbf{b}\in\mathbb{C}^N$1

for the appropriate block or cell structure (Li et al., 2022). Multi-sector BD-RIS extends this idea beyond two half-spaces: the surface is arranged on the faces of an $\mathbf{b}\in\mathbb{C}^N$2-sided polygon prism, with each sector covering $\mathbf{b}\in\mathbb{C}^N$3 of the azimuth and the global scattering matrix partitioned into subblocks $\mathbf{b}\in\mathbb{C}^N$4 that can reradiate energy across sectors (Li et al., 2022). This makes BD-RIS broader than STAR-RIS or IOS formulations tied to two-sector full-space coverage.

The taxonomy has been pushed further along several axes. Dual-polarized BD-RIS incorporates polarization-dependent channel factors and shows that performance-complexity trade-offs can differ materially from the uni-polarized case (Nerini et al., 2024). Hybrid transmitting-and-reflecting BD-RIS hardware has also been demonstrated with independent beam steering of the reflected and transmitted waves and tunable power splitting inside the same aperture (Ming et al., 13 Apr 2025). Collectively, these variations indicate that “BD-RIS” is not a single architecture but a family of multi-port surfaces defined by their circuit graph, reciprocity properties, and radiation geometry (Li et al., 22 May 2025).

3. Beamforming and optimization methodologies

The central design problem is typically to optimize $\mathbf{b}\in\mathbb{C}^N$5 jointly with active beamforming or power allocation. In communication settings this often takes the form of sum-rate maximization; in integrated sensing and communication (ISAC), rate is coupled to sensing metrics such as a Cramér–Rao bound (CRB) (Zhang et al., 15 Aug 2025). Representative formulations include

$\mathbf{b}\in\mathbb{C}^N$6

and related weighted-sum SNR or normalized rate-CRB objectives (Zhang et al., 15 Aug 2025, Esmaeilbeig et al., 2024).

The nonconvexity is driven by bilinear channel coupling and manifold constraints. Consequently, the literature centers on alternating optimization, fractional programming, successive convex approximation, and Riemannian or manifold methods. In the multi-user MISO downlink, one widely used pattern is to update auxiliary variables and BS precoders in closed form, then update BD-RIS blocks on a complex Stiefel manifold by Riemannian conjugate-gradient or related methods (Li et al., 2022). In ISAC, the unitary block constraints define a product of Stiefel manifolds, so Euclidean gradient steps are infeasible without tangent-space projection and retraction (Zhang et al., 15 Aug 2025). Survey-style case studies also compare alternating optimization, manifold optimization, successive convex approximation, and gradient-based projection, with per-iteration complexities reported as $\mathbf{b}\in\mathbb{C}^N$7, $\mathbf{b}\in\mathbb{C}^N$8, roughly $\mathbf{b}\in\mathbb{C}^N$9, and $\mathbf{b}=\mathbf{S}\mathbf{a}$0, respectively (Khan et al., 29 Dec 2025).

A distinct line of work studies what the surface can achieve passively before digital precoding is added. For passive maximum-ratio transmission, the scattering matrix is matched to the cascaded channel and then projected onto the admissible BD-RIS set; for passive interference nulling, $\mathbf{b}=\mathbf{S}\mathbf{a}$1 is designed so that the equivalent channel becomes diagonal (Yahya et al., 2024). The required degrees of freedom differ sharply across architectures: for exact nulling, the diagonal RIS condition is $\mathbf{b}=\mathbf{S}\mathbf{a}$2, while the fully-connected BD-RIS condition is $\mathbf{b}=\mathbf{S}\mathbf{a}$3 (Yahya et al., 2024). This clarifies why off-diagonal couplings are especially valuable in interference-limited multiuser settings.

Reciprocity introduces an additional design layer. Reciprocal BD-RIS imposes $\mathbf{b}=\mathbf{S}\mathbf{a}$4 alongside unitarity, leading to symmetric-unitary manifolds or penalty-based relaxations followed by reciprocity-enforcing projection (Fidanovski et al., 24 Sep 2025). The significance is not purely mathematical: symmetry is directly tied to feasible low-complexity physical implementation.

4. Channel estimation and training overhead

Channel estimation is one of the principal bottlenecks in BD-RIS systems because the enlarged scattering matrix induces many more effective unknowns than in diagonal RIS. In an LS-based uplink formulation, vectorization of BD-RIS blocks leads to a linear model

$\mathbf{b}=\mathbf{S}\mathbf{a}$5

with LS estimate

$\mathbf{b}=\mathbf{S}\mathbf{a}$6

and mean-square error

$\mathbf{b}=\mathbf{S}\mathbf{a}$7

For the pilot-pattern design problem under BD-RIS unitary constraints, the minimum is attained if and only if the training matrix is scaled-unitary, and the resulting minimum MSE is

$\mathbf{b}=\mathbf{S}\mathbf{a}$8

(Li et al., 2024). That work also quantifies a central trade-off: more interelement connections improve the communication performance while increasing the training overhead for channel estimation (Li et al., 2024).

More recent work addresses the estimation problem by abandoning a purely cascaded view. The two-stage individual channel estimation framework separates the slowly varying BS–RIS channel $\mathbf{b}=\mathbf{S}\mathbf{a}$9 from the fast RIS–user channels $\mathbf{S}\in\mathbb{C}^{N\times N}$0 (Zhao et al., 5 Dec 2025). The BS–RIS link is estimated at a bistatic full-duplex base station by exploiting a sparse Saleh–Valenzuela representation and solving a compressive-sensing problem such as

$\mathbf{S}\in\mathbb{C}^{N\times N}$1

followed by coarse/fine angle refinement; the RIS–user links are then estimated by LS using the reconstructed $\mathbf{S}\in\mathbb{C}^{N\times N}$2 (Zhao et al., 5 Dec 2025). This directly targets the mismatch between BD-RIS channel timescales and the pilot burden of naively estimating a high-dimensional cascaded channel.

The pilot-overhead comparison is explicit. Conventional cascaded estimation needs at least $\mathbf{S}\in\mathbb{C}^{N\times N}$3 slots per user, for a total of $\mathbf{S}\in\mathbb{C}^{N\times N}$4. The two-timescale individual scheme requires approximately

$\mathbf{S}\in\mathbb{C}^{N\times N}$5

which is reported as much smaller than $\mathbf{S}\in\mathbb{C}^{N\times N}$6 for large $\mathbf{S}\in\mathbb{C}^{N\times N}$7 (Zhao et al., 5 Dec 2025). In the reported numerical study, the proposed method achieves an average NMSE reduction of approximately $\mathbf{S}\in\mathbb{C}^{N\times N}$8 dB, and for $\mathbf{S}\in\mathbb{C}^{N\times N}$9 the overhead reduction exceeds $$\Theta=(Z+Z_0I)^{-1}(Z-Z_0I)=(Y_0I+Y)^{-1}(Y_0I-Y),$$0 (Zhao et al., 5 Dec 2025). This suggests that in BD-RIS, structural modeling and timescale separation are not optional refinements but prerequisites for scalable CSI acquisition.

5. Hardware realism, circuit models, and physics-compliant representations

A substantial part of the BD-RIS literature is devoted to moving beyond the lossless narrowband idealization. In a lossy lumped-circuit model, each tunable inter-port admittance is implemented by a varactor-based parallel resonant circuit, and eliminating the capacitance yields a circle constraint in the complex-admittance plane:

$$\Theta=(Z+Z_0I)^{-1}(Z-Z_0I)=(Y_0I+Y)^{-1}(Y_0I-Y),$$1

This model leads to custom MM–ADMM and ADMM-based solvers for SISO power maximization and MU-MISO sum-rate maximization (Peng et al., 28 Apr 2025). A notable consequence is architectural reordering under losses: all BD-RIS architectures still outperform diagonal RIS, but the optimal architecture in the lossless case is not necessarily optimal in the lossy case; specifically, group-connected BD-RIS can outperform fully- and tree-connected BD-RISs in SISO systems with relatively high losses, whereas the opposite always holds in the lossless case (Peng et al., 28 Apr 2025).

Wideband operation introduces another non-ideality. In OFDM systems, the BD-RIS scattering matrix becomes frequency dependent through the admittance matrix, and the susceptance of each tunable element can be approximated around a center frequency by a linear function of frequency and its center-frequency value (Li et al., 2024). Simulation results show that BD-RIS outperforms conventional RIS in wideband OFDM, but ignoring wideband variation causes up to $$\Theta=(Z+Z_0I)^{-1}(Z-Z_0I)=(Y_0I+Y)^{-1}(Y_0I-Y),$$2 average-rate loss for complex circuits such as group-connected BD-RIS with $$\Theta=(Z+Z_0I)^{-1}(Z-Z_0I)=(Y_0I+Y)^{-1}(Y_0I-Y),$$3, while the loss is negligible, below $$\Theta=(Z+Z_0I)^{-1}(Z-Z_0I)=(Y_0I+Y)^{-1}(Y_0I-Y),$$4, for simpler architectures or low-resolution cases (Li et al., 2024). This makes circuit-aware frequency dependence especially important precisely where BD-RIS complexity is highest.

Physics-compliant modeling provides an important counterpoint to abstract matrix formulations. A BD-RIS control circuit can always be decomposed into a static load circuit and a set of individually tunable loads, yielding a three-block chain cascade: radio environment, static load circuit, and diagonal tunable loads (Hougne, 2024). The end-to-end channel can then be written as

$$\Theta=(Z+Z_0I)^{-1}(Z-Z_0I)=(Y_0I+Y)^{-1}(Y_0I-Y),$$5

where the only tunable object is the strictly diagonal matrix $$\Theta=(Z+Z_0I)^{-1}(Z-Z_0I)=(Y_0I+Y)^{-1}(Y_0I-Y),$$6 (Hougne, 2024). The practical implication is that physics-compliant D-RIS algorithms for estimation and optimization can be transferred directly to BD-RIS-parametrized channels.

Hardware prototyping has also begun to validate hybrid BD-RIS operation. A $$\Theta=(Z+Z_0I)^{-1}(Z-Z_0I)=(Y_0I+Y)^{-1}(Y_0I-Y),$$7 prototype built from two phase-reconfigurable antenna arrays interconnected by tunable two-port power splitters achieved a tunable power ratio of $$\Theta=(Z+Z_0I)^{-1}(Z-Z_0I)=(Y_0I+Y)^{-1}(Y_0I-Y),$$8 over $$\Theta=(Z+Z_0I)^{-1}(Z-Z_0I)=(Y_0I+Y)^{-1}(Y_0I-Y),$$9 from $Y_0=1/Z_0$0 dB to $Y_0=1/Z_0$1 dB, $Y_0=1/Z_0$2-bit phase reconfiguration with $Y_0=1/Z_0$3 MHz bandwidth at $Y_0=1/Z_0$4 GHz, and measured beam steering in reflection, transmission, and hybrid modes (Ming et al., 13 Apr 2025). Reported efficiencies are approximately $Y_0=1/Z_0$5 in reflection mode and $Y_0=1/Z_0$6 in transmission mode, with insertion loss below $Y_0=1/Z_0$7 dB in pure modes (Ming et al., 13 Apr 2025). Such demonstrations narrow the gap between circuit abstraction and electromagnetic implementation.

6. Performance regimes, applications, and open problems

BD-RIS has been studied across communication, sensing, and joint communication-sensing scenarios. In BD-RIS-enabled ISAC with unitary scattering matrices, fully-connected BD-RIS achieves the highest sum rate, with gains of up to $Y_0=1/Z_0$8–$Y_0=1/Z_0$9 and the lowest CRB relative to single-connected or group-connected modes; in the same setting, a log-barrier Riemannian algorithm improves rate by $\Theta^H\Theta\preceq I$0–$\Theta^H\Theta\preceq I$1 over a fixed-step conjugate-gradient benchmark (Zhang et al., 15 Aug 2025). In transmitter-side mmWave ISAC, a fully-connected BD-RIS provides approximately $\Theta^H\Theta\preceq I$2 bps/Hz higher sum rate at the same $\Theta^H\Theta\preceq I$3 for $\Theta^H\Theta\preceq I$4, and more than $\Theta^H\Theta\preceq I$5 bps/Hz as $\Theta^H\Theta\preceq I$6 grows to $\Theta^H\Theta\preceq I$7 (Chen et al., 2024). These results are consistent with the broader claim that off-diagonal coupling is especially valuable when a single surface must satisfy multiple spatial objectives simultaneously.

Coverage-oriented architectures reveal a different benefit. In multi-sector BD-RIS, increasing the number of sectors improves directional control over the full $\Theta^H\Theta\preceq I$8 azimuth, and an $\Theta^H\Theta\preceq I$9 design yields up to $\Theta^H\Theta=I$0 sum-rate improvement at $\Theta^H\Theta=I$1 compared with STAR-RIS and group-connected two-sector baselines under the idealized pattern model (Li et al., 2022). Dynamic grouping yields further gains by adapting the circuit partition itself to CSI: for $\Theta^H\Theta=I$2 and $\Theta^H\Theta=I$3 or $\Theta^H\Theta=I$4, dynamic grouping provides $\Theta^H\Theta=I$5–$\Theta^H\Theta=I$6 sum-rate gain over any fixed grouping, and the gain widens to approximately $\Theta^H\Theta=I$7 for $\Theta^H\Theta=I$8, $\Theta^H\Theta=I$9 (Li et al., 2022).

Performance-complexity trade-offs become especially sharp in dual-polarized settings. Under opposite polarizations and line-of-sight, the Pareto frontier of received power versus tunable-load count is achieved by forming $$\Theta_{\rm D}=\mathrm{diag}(e^{j\theta_1},\dots,e^{j\theta_N}),$$0 groups of size two and $$\Theta_{\rm D}=\mathrm{diag}(e^{j\theta_1},\dots,e^{j\theta_N}),$$1 singleton groups, and at $$\Theta_{\rm D}=\mathrm{diag}(e^{j\theta_1},\dots,e^{j\theta_N}),$$2 the group-size-two design with complexity $$\Theta_{\rm D}=\mathrm{diag}(e^{j\theta_1},\dots,e^{j\theta_N}),$$3 attains the same received power as a fully-connected BD-RIS (Nerini et al., 2024). This result is often interpreted as evidence that a small amount of structured coupling can recover the full BD-RIS advantage without quadratic circuit complexity.

Several common misconceptions are explicitly corrected by the literature. First, fully-connected BD-RIS is not universally optimal once circuit loss is modeled; SISO rankings can invert in favor of group-connected architectures (Peng et al., 28 Apr 2025). Second, richer coupling does not come for free at the signaling layer; training overhead increases with connectivity unless structure such as sparsity, grouping, or multi-timescale stationarity is exploited (Li et al., 2024, Zhao et al., 5 Dec 2025). Third, BD-RIS does not invalidate diagonal physics-based models; it can often be recast into a diagonal tunable-load representation after absorbing static circuitry into the surrounding network (Hougne, 2024).

Open problems remain concentrated in hardware and system integration. Reported challenges include hardware losses and impedance-component mismatches, fine-grained control of dense tunable networks, frequency dependence in wideband operation, accurate channel estimation with minimal overhead, optimal topology design under hardware constraints, and coordination across multiple BD-RIS surfaces (Khan et al., 29 Dec 2025, Li et al., 22 May 2025). The current literature therefore presents BD-RIS not as a single settled architecture, but as an active design space in which circuit topology, physics-compliant modeling, estimation strategy, and optimization algorithm are tightly coupled.