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Beyond Diagonal Reconfigurable Intelligent Surfaces: From Transmitting and Reflecting Modes to Single-, Group-, and Fully-Connected Architectures

Hongyu Li, Shanpu Shen, Bruno Clerckx

arXiv:2205.02866v2cs.ITeess.SP

TL;DR

Existing RIS research largely centered on single-connected reflective surfaces with diagonal phase-shift matrices, leaving modes and architectures insufficiently unified. The paper proposes BD-RIS, a general non-diagonal model, and jointly designs the transmit precoder and RIS matrix for sum-rate maximization. Simulations compare nine cases and report substantial gains for more connected hybrid architectures.

  • Problem

    Most existing research focuses on single-connected reflective RISs with diagonal phase-shift matrices, while a comprehensive model unifying RIS modes and architectures is lacking.

  • Method

    The paper models BD-RIS through a multi-port reconfigurable impedance network and jointly designs the transmit precoder and BD-RIS matrix, using manifold-based optimization for the unitary constraint.

  • Results

    Under the stated settings, CW-FC and CW-GC hybrid BD-RISs achieve around 75% and 37% higher sum-rate than CW-SC hybrid BD-RISs, while CW-FC hybrid gains around 20% over CW-FC reflective/transmissive designs.

  • Takeaways & Limitations

    BD-RIS provides a unified nine-case framework in which more connected architectures and hybrid operation can deliver higher simulated sum-rate than corresponding single-connected or reflective/transmissive designs.

Abstract

from arXiv · show

Reconfigurable intelligent surfaces (RISs) are envisioned as a promising technology for future wireless communications. With various hardware realizations, RISs can work under different modes (reflective/transmissive/hybrid) or have different architectures (single/group/fully-connected). However, most existing research focused on single-connected reflective RISs, mathematically characterized by diagonal phase shift matrices, while there is a lack of a comprehensive study for RISs unifying different modes/architectures. In this paper, we solve this issue by analyzing and proposing a general RIS-aided communication model. Specifically, we establish an RIS model not limited to diagonal phase shift matrices, a novel branch referred to as beyond diagonal RIS (BD-RIS), unifying modes and architectures. With the proposed model, we develop efficient algorithms to jointly design transmit precoder and BDRIS matrix to maximize the sum-rate for RIS-aided systems. We also provide simulation results to compare the performance of BD-RISs with different modes/architectures. Simulation results show that under the same mode, fully- and group-connected RIS can effectively increase the sum-rate performance compared with single-connected RIS, and that hybrid RIS outperforms reflective/transmissive RIS with the same architecture.

I. INTRODUCTION

The paper introduces BD-RIS as a unified framework for RIS modes and architectures beyond conventional diagonal, reflective, single-connected designs. It develops a general communication model and joint design methods, then evaluates nine mode/architecture cases.

  • Motivation: RISs use tunable elements to modify wireless propagation through controllable phase shifts and amplitudes while consuming little power.Each element can independently switch PIN diodes between ON and OFF states.
  • Research gap: Most prior RIS research focused on conventional single-connected reflective RISs represented by diagonal phase shift matrices.Beyond-diagonal models instead permit scattering matrices that also control signal magnitudes.
  • Unified framework: BD-RIS classifies RISs by whether their scattering matrix is diagonal and unifies three modes with three architectures, producing nine cases.The modes are reflective, transmissive, and hybrid; architectures are cell-wise single-, group-, and fully-connected.
  • Model: The proposed model represents the RIS as antennas connected through a group-connected multi-port reconfigurable impedance network, where port connections enable flexible modes.STAR-RIS is established as a special case of block-diagonal, cell-wise single-connected BD-RIS.
  • Optimization: The paper jointly designs the transmit precoder and BD-RIS matrix to maximize sum-rate, providing general and lower-complexity solutions together with initialization, convergence, and complexity analyses.The efficient solution targets the cell-wise single-connected case and has similar performance to the general solution.
  • Evaluation: 75% and 37% higher sum-rate are achieved by fully- and group-connected hybrid BD-RISs than cell-wise single-connected hybrid ones under Rayleigh fading; fully-connected hybrid gains around 20% over reflective/transmissive counterparts under Rician fading.These comparisons use the paper’s specific parameter settings and channel conditions.

III. ARCHITECTURE/MODE ANALYSIS AND DESIGN

The architecture analysis distinguishes cell-wise single-, group-, and fully-connected BD-RISs by their circuit connectivity. The single-connected case reduces to diagonal scattering matrices and corresponds to STAR-RIS.

  • Cell-wise single-connected: In cell-wise single-connected BD-RIS, cells are not connected to one another.A two-cell example is given for this architecture.
  • Cell-wise single-connected: Cell-wise single-connected BD-RIS uses diagonal Φr and Φt matrices whose entries represent per-cell transmissive and reflective coefficients.The entries are subject to the architecture’s stated constraints.
  • Relation to STAR-RIS: The cell-wise single-connected BD-RIS architecture is essentially the STAR-RIS.This identifies STAR-RIS with the single-connected branch of the BD-RIS architecture classification.

2) Cell-Wise Fully-Connected (CW-FC) Architecture:

CW-FC BD-RIS connects all cells through reconfigurable impedance components, producing full scattering matrices and the most general constraint among the architectures described. The architecture increases circuit complexity as the number of cells grows, motivating group-connected alternatives.

  • CW-FC connects all BD-RIS cells through reconfigurable impedance components.
  • Its reflection and transmission matrices are full matrices satisfying the architecture’s general constraint.
  • CW-FC can achieve the best performance because it has the most general constraint among the considered architectures.
  • The two antennas within each cell are connected through a 2-port fully-connected reconfigurable impedance network.
  • 3) Cell-Wise Group-Connected (CW-GC) Architecture:: CW-GC divides cells into equal-sized groups, applies CW-FC within each group, and yields block-diagonal reflection and transmission matrices.
  • 3) Cell-Wise Group-Connected (CW-GC) Architecture:: CW-GC trades RIS performance against circuit complexity, while grouping strategies beyond equal-sized groups remain future work.

IV. JOINT TRANSMIT PRECODER AND BD-RIS MATRIX DESIGN

The paper formulates joint transmit-precoder and BD-RIS design as sum-rate maximization under transmit-power and BD-RIS constraints. It transforms the nonconvex problem into block updates solved iteratively by coordinate descent.

  • The system assumes blocked direct BS-user links and exact instantaneous CSI at the BS; the design provides an upper bound under these assumptions.
  • The design jointly optimizes the transmit precoder and BD-RIS matrices to maximize MU-MISO sum-rate under power and BD-RIS constraints.
  • Fractional programming transforms the original objective into a more tractable multivariable block-optimization problem using auxiliary variables.
  • Algorithm 1 initializes the BD-RIS coefficients and precoder, then updates ι, τ, and W before updating the BD-RIS matrices.
  • Block coordinate descent iteratively updates the auxiliary vectors, transmit precoder, and BD-RIS beamformer until convergence.

C. Auxiliary Vectors: Blocks ι and τ

The auxiliary-vector subproblems isolate fractional terms introduced during objective transformation. With the other blocks fixed, the auxiliary updates are unconstrained convex optimizations, while the precoder subproblem is convex and solved using a Lagrange multiplier.

  • Fixing W, Φt, Φr, and one auxiliary vector makes the subproblem for the other auxiliary vector unconstrained and convex.
  • With Φt, Φr, τ, and ι fixed, the transmit-precoder subproblem is convex.
  • The precoder solution uses a Lagrange multiplier for the power constraint, with the optimal multiplier obtained by bisection search.

E. BD-RIS Matrix: Block {Φt, Φr}

The BD-RIS matrix subproblem is decomposed by architecture into group-level designs, with a general CW-GC solution based on manifold optimization and specialized reductions for CW-FC and CW-SC.

  • Architecture specialization: CW-SC and CW-FC architectures are special cases of CW-GC, obtained by setting G = M and G = 1, respectively.The general solution therefore applies to both architectures.
  • General solution for CW-GC BD-RIS: The CW-GC formulation designs each group’s transmissive and reflective matrices while fixing the remaining group pairs.Quadratic terms couple different groups, motivating a group-by-group block update.
  • Manifold optimization: The CW-GC subproblem has a unitary constraint and is solved on a complex Stiefel manifold using a Riemannian conjugate-gradient method.The manifold approach avoids relaxation-induced performance loss and addresses the unitary constraint directly.
  • Manifold optimization: The Riemannian conjugate-gradient procedure computes Euclidean and projected Riemannian gradients, chooses descent directions, and performs retraction steps until convergence.The algorithm uses a Riemannian Polak–Ribiere update and backtracking for the step size.
  • General solution for CW-GC BD-RIS: After solving all groups, the optimal reflective and transmissive matrices are split from the resulting group matrices.The procedure is summarized in Algorithm 2.
  • Efficient CW-SC specialization: For CW-SC, diagonal matrices enable an efficient algorithm with similar performance but lower computational complexity than the general solution.CW-FC can instead be solved directly using the general algorithm with G = 1.

2) An Efficient Solution for CW-SC BD-RIS:

The efficient CW-SC solution separates phase and amplitude optimization for each transmissive-reflective coefficient pair, using direct phase selection and one-dimensional amplitude search.

  • Per-element optimization: The CW-SC subproblem updates one pair of coefficients, φt,m and φr,m, while fixing all other pairs.The diagonal structure permits element-wise updates.
  • Algorithm: Algorithm 3 repeatedly updates all coefficient pairs until convergence and returns the optimized CW-SC BD-RIS matrix.Its input includes channel quantities, auxiliary variables, the precoder, and current transmissive and reflective matrices.
  • Phase shift: Optimal phase shifts are obtained directly by setting cos(∠χi,m − θi,m) = −1.This separates phase selection from the subsequent amplitude optimization.
  • Amplitude: With optimal phases, the amplitude problem becomes a real-valued convex optimization over αt,m ∈ (0, 1).The corresponding reflective amplitude is constrained through the transmissive-reflective amplitude relation.
  • Amplitude: Because no closed-form amplitude solution is available, golden-section search finds the minimum of the one-dimensional objective.The search is applied within the iterative CW-SC design procedure.

F. Initialization

Initialization uses diagonal CW-SC BD-RIS coefficients with random phases, followed by an MMSE precoder and power normalization.

  • BD-RIS initialization: Initial transmissive and reflective BD-RIS matrices are chosen as diagonal matrices corresponding to CW-SC cases.This provides initial coefficients across different modes and connections.
  • BD-RIS initialization: Each nonzero initial coefficient has constant amplitude 1/√2 and a random phase in [0, 2π).The initialization is applied to both transmissive and reflective matrices.
  • Transmit precoder initialization: An MMSE precoder initializes W after the BD-RIS coefficients are set.The precoder is computed using the initialized effective channel quantities.
  • Transmit precoder initialization: The initialized precoder is additionally normalized to satisfy the transmit-power constraint.This normalization follows the MMSE initialization.

G. Convergence Analysis

The proposed algorithms are not accompanied by a strict convergence proof, but simulations report convergence within limited iterations and similar performance for the efficient CW-SC method.

  • Convergence limitation: Algorithm 1 lacks a strict convergence proof because the {Φt, Φr} update is not guaranteed to reach a global optimum.The remaining block updates are monotonic, and the authors state that the induced loss is negligible.
  • Simulation assessment: Simulations show that the proposed solutions converge within limited iterations under Rayleigh and Rician fading channel realizations.The reported convergence behavior is used to demonstrate algorithm robustness.
  • Simulation assessment: The efficient CW-SC Algorithm 3 achieves similar performance to CW-SC Algorithm 2 with lower complexity.The comparison is based on the convergence simulations and the subsequent complexity discussion.

H. Complexity Analysis

The proposed algorithms have architecture-dependent complexity because BD-RIS circuit connectivity changes the number of optimized matrix elements. Greater connectivity improves performance but increases both circuit and optimization complexity.

  • BD-RIS optimization: CW-GC BD-RIS optimization divides each iteration into G sub-problems solved by a manifold conjugate-gradient method with complexity O{Icg M^3}.CW-FC and CW-SC are special cases of CW-GC, with G=1 and G=M, respectively.
  • Architecture-specific complexity: Joint transmit-beamformer and CW-SC BD-RIS design has complexity O{I(K^2M^2 + IbsKN^3 + IscM)}.
  • Complexity–performance trade-off: Group- and fully-connected architectures require more reconfigurable impedance components, increasing topology and optimization complexity alongside their performance improvement.Their matrix nonzero elements scale differently with M, producing distinct computational costs.

V. PERFORMANCE EVALUATION

Simulations compare nine BD-RIS mode–architecture combinations in an RIS-aided MU-MISO system. Hybrid operation and greater connectivity generally deliver higher sum-rate, while the proposed framework also identifies open modeling and hardware-realization directions.

  • Setup: Simulations evaluate BD-RIS sum-rate across nine modes and architectures using channels with small-scale and large-scale fading.The BS–RIS and RIS–user channels follow distance-dependent pathloss and fading models.
  • Transmit-power evaluation: 20% higher sum-rate is achieved by hybrid CW-FC than transmissive/reflective CW-FC under the reported Rician-fading setting.The paper attributes this to hybrid BD-RIS serving users from two sides and exploiting multiuser diversity.
  • Cell-count evaluation: All schemes’ sum-rates increase with M, while CW-FC/GC curves have larger slopes than CW-SC because their nonzero matrix elements grow quadratically rather than linearly.
  • User-count evaluation: Hybrid schemes outperform reflective/transmissive schemes at matched architecture and user counts, although the gap narrows as the number of users grows.The hybrid scheme remains better when the total simultaneously served user count is fixed.
  • Conclusions and future work: The unified model covers reflective, transmissive, and hybrid modes with CW-SC, CW-GC, and CW-FC architectures, while wideband extension and physical realization remain future work.The paper also proposes joint precoder and BD-RIS matrix design and identifies integration with RSMA, WPT, SWIPT, and ISAC as future applications.
  • Transmit-power evaluation: 75% and 37% higher sum-rate are reported for hybrid CW-FC and CW-GC, respectively, than hybrid CW-SC under the reported Rayleigh-fading setting.

APPENDIX PROOF OF CONVEXITY OF OBJECTIVE (42A)

The appendix proves that objective (42a) is convex over αt,m ∈ (0,1), establishing a unique minimum point for the associated one-dimensional optimization.

  • Objective representation: The proof represents the objective as the real-valued quadratic function z̄(x)=ax^2+bx+c with coefficients determined by υm, χt,m, and χr,m.
  • Derivative analysis: The analysis examines first-, second-, and third-order derivatives across cases a≥0 and a<0 to characterize the objective’s monotonicity.
  • Conclusion: The objective is convex on x∈(0,1) and has only one minimum point.
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