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

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.Source: §I · p. 1

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.Source: §I · pp. 1–2

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.Source: §I · p. 2

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.Source: §I · p. 2

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.Source: §I · p. 2

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.Source: §I · p. 2

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.Source: 1) Cell-Wise Single-Connected (CW-SC)… · p. 4
• 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.Source: 1) Cell-Wise Single-Connected (CW-SC)… · p. 4

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.Source: 1) Cell-Wise Single-Connected (CW-SC)… · p. 4

• CW-FC connects all BD-RIS cells through reconfigurable impedance components.Source: 2) Cell-Wise Fully-Connected (CW-FC)… · p. 4
• Its reflection and transmission matrices are full matrices satisfying the architecture’s general constraint.Source: 2) Cell-Wise Fully-Connected (CW-FC)… · p. 4
• CW-FC can achieve the best performance because it has the most general constraint among the considered architectures.Source: 2) Cell-Wise Fully-Connected (CW-FC)… · p. 4
• The two antennas within each cell are connected through a 2-port fully-connected reconfigurable impedance network.Source: 2) Cell-Wise Fully-Connected (CW-FC)… · p. 4

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.Source: 3) Cell-Wise Group-Connected (CW-GC)… · p. 5
• CW-GC trades RIS performance against circuit complexity, while grouping strategies beyond equal-sized groups remain future work.Source: 3) Cell-Wise Group-Connected (CW-GC)… · p. 5

• The system assumes blocked direct BS-user links and exact instantaneous CSI at the BS; the design provides an upper bound under these assumptions.Source: A. Problem Formulation · p. 6
• The design jointly optimizes the transmit precoder and BD-RIS matrices to maximize MU-MISO sum-rate under power and BD-RIS constraints.Source: A. Problem Formulation · p. 6
• Fractional programming transforms the original objective into a more tractable multivariable block-optimization problem using auxiliary variables.Source: A. Problem Formulation · p. 6, B. Overview of the Joint Design… · p. 6
• Algorithm 1 initializes the BD-RIS coefficients and precoder, then updates ι, τ, and W before updating the BD-RIS matrices.Source: B. Overview of the Joint Design… · p. 7
• Block coordinate descent iteratively updates the auxiliary vectors, transmit precoder, and BD-RIS beamformer until convergence.Source: B. Overview of the Joint Design… · pp. 6–7

• Fixing W, Φt, Φr, and one auxiliary vector makes the subproblem for the other auxiliary vector unconstrained and convex.Source: C. Auxiliary Vectors: Blocks ι and τ · p. 7
• With Φt, Φr, τ, and ι fixed, the transmit-precoder subproblem is convex.Source: D. Transmit Precoder: Block W · p. 7
• The precoder solution uses a Lagrange multiplier for the power constraint, with the optimal multiplier obtained by bisection search.Source: D. Transmit Precoder: Block W · p. 7

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.Source: E. BD-RIS Matrix: Block {Φt, Φr} · p. 7, 1) A General Solution for CW-GC… · p. 8

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.Source: 1) A General Solution for CW-GC… · pp. 7–8

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.Source: 1) A General Solution for CW-GC… · p. 8
• 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.Source: 1) A General Solution for CW-GC… · p. 8

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.Source: 1) A General Solution for CW-GC… · pp. 8–9

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.Source: 1) A General Solution for CW-GC… · p. 8

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.Source: 2) An Efficient Solution for CW-SC… · p. 9

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.Source: 2) An Efficient Solution for CW-SC… · p. 9

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.Source: 2) An Efficient Solution for CW-SC… · p. 9

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.Source: 2) An Efficient Solution for CW-SC… · p. 9
• 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.Source: 2) An Efficient Solution for CW-SC… · p. 9

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.Source: F. Initialization · p. 10
• 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.Source: F. Initialization · p. 10

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.Source: F. Initialization · p. 10
• The initialized precoder is additionally normalized to satisfy the transmit-power constraint.This normalization follows the MMSE initialization.Source: F. Initialization · p. 10

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.Source: G. Convergence Analysis · p. 10

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.Source: G. Convergence Analysis · p. 10
• 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.Source: G. Convergence Analysis · p. 10

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.Source: 1) The General Solution for CW-GC BD-RIS · p. 10

Architecture-specific complexity

• Joint transmit-beamformer and CW-SC BD-RIS design has complexity O{I(K^2M^2 + IbsKN^3 + IscM)}.Source: 2) The Efficient Solution for CW-SC… · p. 10

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.Source: 2) The Efficient Solution for CW-SC… · pp. 10–11

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.Source: §V · p. 11

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.Source: §V · p. 11

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.Source: §V · p. 12

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.Source: §V · p. 12

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.Source: §VI · pp. 12–13, 1) Wideband Modeling · p. 13, 2) Interplay with Other Techniques · p. 13

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.Source: §VI · pp. 12–13

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.Source: APPENDIX PROOF OF CONVEXITY OF… · p. 13

Derivative analysis

• The analysis examines first-, second-, and third-order derivatives across cases a≥0 and a<0 to characterize the objective’s monotonicity.Source: APPENDIX PROOF OF CONVEXITY OF… · p. 13

Conclusion

• The objective is convex on x∈(0,1) and has only one minimum point.Source: APPENDIX PROOF OF CONVEXITY OF… · p. 13