Movable Beyond-Diagonal RIS Architectures

• MA-BD-RIS is a unified metasurface architecture that enables controlled spatial repositioning and dynamic inter-element electromagnetic interconnections.
• It employs a joint optimization framework integrating fractional programming, closed-form beamforming, PPADMM for BD configuration, and SCA for element placement.
• Simulation results show robust performance gains and favorable trade-offs between spatial movability and circuit connectivity in multi-user MISO systems.

Movable Beyond-Diagonal Reconfigurable Intelligent Surfaces (MA-BD-RIS) describe a unified metasurface architecture enabling both controlled spatial repositioning (movability) and dynamic inter-element electromagnetic interconnections (beyond-diagonal connectivity) for wireless channel manipulation. By integrating mechanical relocation and programmable circuit-level scattering, MA-BD-RIS act as channel extenders and enablers of joint analog and physical-domain beamforming. This paradigm generalizes classical fixed “diagonal” phase-shift RIS, group- or fully-connected RIS, and mechanically steerable reflectarrays, delivering robust performance gains under a broad range of channel and deployment regimes (Xu et al., 11 Jan 2026, Li et al., 2022).

1. System and Channel Modeling

MA-BD-RIS settings consider a downlink multi-user, multiple-input single-output (MU-MISO) system. The base station (BS) is equipped with $N_t$ fixed antennas serving $K$ single-antenna users via an RIS of $M$ passive elements, arranged into $N_G$ movable groups of $N_E = M/N_G$ elements per group.

The BS–user direct path is presumed blocked. The transmit signal is

$$\mathbf{x} = \mathbf{W}\mathbf{s} = \sum_{k=1}^K \mathbf{w}_k s_k,\quad \mathbb{E}[\mathbf{s}\mathbf{s}^H] = \mathbf{I}_K,$$

where $\mathbf{W} = [\mathbf{w}_1,\ldots,\mathbf{w}_K]$ are precoders. The received signal at user $k$ is given by

$$y_k(\mathbf{c}) = \mathbf{h}_k^H(\mathbf{c})\,\boldsymbol{\Theta}\,\mathbf{H}(\mathbf{c})\,\mathbf{W}\,\mathbf{s} + n_k, \quad n_k \sim \mathcal{CN}(0, \sigma^2),$$

with user SINR

$$\gamma_k = \frac{|\mathbf{h}_k^H(\mathbf{c})\,\boldsymbol{\Theta}\,\mathbf{H}(\mathbf{c})\,\mathbf{w}_k|^2} {\sum_{i\neq k}|\mathbf{h}_k^H(\mathbf{c})\,\boldsymbol{\Theta}\,\mathbf{H}(\mathbf{c})\,\mathbf{w}_i|^2 + \sigma^2}.$$

The channel $\mathbf{H}(\mathbf{c})$ is composed by concatenating the response of each group at position $\mathbf{c}_g$, modeled via field-response vectors and geometric far-field propagation; movability is captured via $$\mathbf{t}_{g,m} = \mathbf{c}_g + \boldsymbol{\delta}_{g,m}$$ for the group and intra-group offset.

2. MA-BD-RIS Architecture, Scattering Matrix, and Variables

The principal distinguishing feature is the use of a beyond-diagonal (BD) scattering matrix

$$\boldsymbol{\Theta} = \mathrm{blkdiag}(\boldsymbol{\Theta}_1, \ldots, \boldsymbol{\Theta}_{N_G}) \in \mathbb{C}^{M\times M}$$

that models arbitrary linear interconnections within each group. For BD-RIS, this matrix is block-diagonal, with each block satisfying symmetry ($\boldsymbol{\Theta}^T=\boldsymbol{\Theta}$), losslessness ($$\boldsymbol{\Theta}^H\boldsymbol{\Theta} = \mathbf{I}$$), and parameterized by imaginary admittance

$$\mathbf{Y} = j\mathbf{B}, \quad \mathbf{B}^T = \mathbf{B}, \quad \mathbf{B} = \mathrm{blkdiag}(\mathbf{B}_1,\ldots,\mathbf{B}_{N_G}),$$

with

$$\boldsymbol{\Theta} = (\mathbf{I} + Z_0j\mathbf{B})^{-1}(\mathbf{I} - Z_0j\mathbf{B}),\quad Z_0=50\Omega.$$

Movability variables are the set of reference group positions $\{\mathbf{c}_g\} \subset \mathbb{R}^2$. Placement constraints are imposed as feasible region ($\mathbf{c}_g \in \mathcal{C}_R$) and minimum inter-group distance ($\|\mathbf{c}_g - \mathbf{c}_{g'}\| \ge D$), ensuring non-colliding motion and practical deployment.

A typical table of variables for MA-BD-RIS is:

Symbol/Term	Meaning	Constraint/Domain
$\mathbf{W}$	Transmit precoding matrix (BS)	$\mathrm{Tr}(\mathbf{W}^H\mathbf{W}) \le P$
$\mathbf{B}$	BD-RIS imaginary admittance	Block-diagonal, Hermitian
$\mathbf{c}_g$	Reference group position	$\mathbf{c}_g \in \mathcal{C}_R$

3. Joint Optimization Formulation

MA-BD-RIS operation is governed by the following joint sum-rate maximization problem: $$\max_{\mathbf{W},\,\mathbf{B},\,\{\mathbf{c}_g\}} \sum_{k=1}^{K} \log_2(1 + \gamma_k)$$ subject to:

• Transmit power: $\mathrm{Tr}(\mathbf{W}^H\mathbf{W}) \le P$.
• Placement: $\mathbf{c}_g \in \mathcal{C}_R$, $\|\mathbf{c}_g-\mathbf{c}_{g'}\|\geq D$.
• BD-RIS structure: $\mathbf{B}$ block-diagonal, Hermitian; $\boldsymbol{\Theta}$ as above.

The problem is non-convex, entangling digital beamforming, analog BD-RIS configuration, and real-world element placement. This encapsulates the hardware degrees-of-freedom spanning circuit DoF (from BD interconnection), and spatial DoF (from group mobility).

4. Solution Algorithm: Block Optimization and Subproblem Solvers

A sequential alternating optimization scheme is employed with three primary blocks (Xu et al., 11 Jan 2026):

• Fractional Programming Initialization: The sum-rate is re-expressed via auxiliary variables ($\rho_k, \psi_k$) into a tractable quadratic-linear surrogate.
• Beamforming Updates: For fixed RIS and placement, closed-form updates yield

$$\mathbf{w}_k^{\rm opt} = (\mathbf{Q}+\lambda \mathbf{I})^{-1}\mathbf{q}_k,$$

with $\lambda$ enforcing the power constraint via bisection.

• BD-RIS Configuration (PPADMM): ADMM targets the augmented Lagrangian in $\mathbf{B},\ \mathbf{U}$, with each subproblem reduced to small-dimensional linear systems; distinct upper-triangular entries are vectorized and updated with proximal regularization for robust convergence.
• Element Placement via SCA: Each group’s position $\mathbf{c}_g$ is updated independently using a second-order Taylor expansion to approximate the non-convex objective by a concave quadratic surrogate in local coordinates, subject to linearized inter-group distance constraints; results in small QCQPs solvable by standard convex solvers.

The outer iterations cyclically update the beamforming, BD-RIS, and placement, converging to a stationary point in a few tens of iterations under standard assumptions.

5. Computational Complexity and Algorithmic Properties

The modular, block-wise structure enables tractable computational scaling:

• Beamforming: Single $N_t \times N_t$ matrix inversion per update ($\mathcal{O}(N_t^3)$).
• BD-RIS (PPADMM): Main cost is solving for $\mathbf{B}$ in a dimension equal to the nonzero upper-triangular entries of the block-diagonal admittance (typically $\frac{1}{2}N_G N_E (N_E+1)$); $\mathbf{U}$-subproblem involves $M \times M$ inversions per user ($\mathcal{O}(KM^3)$ naive, reduced by caching).
• Placement: $N_G$ parallel low-dimensional (2-variable) convex QCQPs.
• Convergence: Each block solved to stationarity; overall alternating method guaranteed to converge to a stationary point; the PPADMM component is globally convergent for appropriate proximal weighting (Xu et al., 11 Jan 2026).

6. Performance Analysis and Trade-offs

Simulations demonstrate regime-specific performance trends (Xu et al., 11 Jan 2026):

• For small $M$ (e.g., $M \leq 64$) or rich-scattering (large $L$), movable-only RIS architectures (minimal connectivity, high spatial reconfiguration) outperform highly connected designs due to the spatial SNR boost from exploiting favorable channel “hotspots.”
• For large $M$ ($M \geq 128$) or massive $N_t$, highly connected BD-RIS (full inter-element connectivity) surpass movable-only architectures as circuit-based beamforming becomes more effective.
• Limited movability ($l_s \approx 1.1$, near half-wavelength) already produces substantial rate gains over fixed, purely diagonal RIS.
• There exists a group size $N_E$ that balances the tradeoff: intra-group connectivity vs. inter-group mobility, maximizing spatial DoF within connectivity/hardware budget.

Numerical results include:

• With $M=32,\,L=8$, movable-only RIS exceeds fixed phased-array RIS sum-rate by 20%.
• For $M \geq 128$, fully-connected BD-RIS achieves 10–15% gain over movable-only (Xu et al., 11 Jan 2026).

7. Design Guidelines and Practical Deployment

A unified MA-BD-RIS framework underscores a fundamental tradeoff between spatial movability and circuit connectivity. Key design principles are (Xu et al., 11 Jan 2026):

• Small-Scale or SNR-Limited Scenarios: Favor maximal movability with minimal circuit interconnection; low-complexity hardware suffices.
• Large-Scale or Massive BS Array: Adopt higher inter-element connectivity to leverage circuit DoF for beam shaping; group- or fully-connected architectures recommended.
• Hybrid Regimes: Moderate group size ($N_G$) enables a tunable mix between movability and connectivity.
• Practical region sizing: Movability region should modestly exceed half-wavelength ($l_s \approx 1.1-1.3$), and $N_G$ be dimensioned according to performance goals and hardware constraints.

In summary, MA-BD-RIS architectures represent a flexible and generalizable class of programmable metasurfaces, offering unified spatial and circuit-level adaptation for robust channel shaping, with optimization and implementation dictated by scenario requirements and system resource constraints (Xu et al., 11 Jan 2026, Li et al., 2022).