---
title: Movable Beyond-Diagonal RIS Architectures
url: https://www.emergentmind.com/topics/movable-beyond-diagonal-reconfigurable-intelligent-surfaces-ma-bd-ris
type: topic
---

# Movable Beyond-Diagonal RIS Architectures

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 [2601.06837][2203.08184].

## 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 [2601.06837]:

- **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 [2601.06837].

## 6. Performance Analysis and Trade-offs

Simulations demonstrate regime-specific performance trends [2601.06837]:
- 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 [2601.06837].

## 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 [2601.06837]:
- **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 [2601.06837][2203.08184].

Source: https://www.emergentmind.com/topics/movable-beyond-diagonal-reconfigurable-intelligent-surfaces-ma-bd-ris