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Beyond Diagonal Reconfigurable Intelligent Surfaces: A Multi-Sector Mode Enabling Highly Directional Full-Space Wireless Coverage

Hongyu Li, Shanpu Shen, Bruno Clerckx

arXiv:2209.00301v2eess.SPcs.ITmath.IT

TL;DR

Prior RIS models – reflect signals to only one side, while the prior hybrid BD-RIS approach achieves better performance than STAR-RIS. The paper models antennas in L sectors connected through a group-connected impedance network, derives continuous and discrete constraints and a gain-aware channel model, and designs continuous and discrete MU-MISO sum-rate algorithms. Maximum received power is positively correlated with sector count, while increasing sectors can reduce antenna count and codebook resolution at the same sum-rate requirement.

  • Problem

    Prior RIS models – reflect signals to only one side, while the prior hybrid BD-RIS approach achieves better performance than STAR-RIS.

    Source: §I · p. 2

  • Method

    The paper models antennas in L sectors connected through a group-connected impedance network, derives continuous and discrete constraints and a gain-aware channel model, and designs continuous and discrete MU-MISO sum-rate algorithms.

    Source: §I · p. 2

  • Results

    Maximum received power is positively correlated with sector count, while increasing sectors can reduce antenna count and codebook resolution at the same sum-rate requirement.

    Source: §VII · p. 13

  • Takeaways & Limitations

    Increasing sector count can lower finite-codebook resolution, which the authors identify as beneficial for controlling practical RIS realization costs.

    Source: §I · p. 2

Abstract

from arXiv · show

Reconfigurable intelligent surface (RIS) has gained much traction due to its potential to manipulate the propagation environment via nearly-passive reconfigurable elements. In our previous work, we have analyzed and proposed a beyond diagonal RIS (BD-RIS) model, which is not limited to traditional diagonal phase shift matrices, to unify different RIS modes/architectures. In this paper, we create a new branch of BD-RIS supporting a multi-sector mode. A multi-sector BD-RIS is modeled as multiple antennas connected to a multi-port group-connected reconfigurable impedance network. More specifically, antennas are divided into $L$ ($L \ge 2$) sectors and arranged as a polygon prism with each sector covering $1/L$ space. Different from the recently introduced concept of intelligent omni-surface (or simultaneously transmitting and reflecting RIS), the multi-sector BD-RIS not only achieves a full-space coverage, but also has significant performance gains thanks to the highly directional beam of each sector.We derive the constraint of the multi-sector BD-RIS and the corresponding channel model taking into account the relationship between antenna beamwidth and gain. With the proposed model, we first derive the scaling law of the received signal power for a multi-sector BD-RIS-assisted single-user system. We then propose efficient beamforming design algorithms to maximize the sum-rate of the multi-sector BD-RIS-assisted multiuser system. Simulation results verify the effectiveness of the proposed design and demonstrate the performance enhancement of the proposed multi-sector BD-RIS.

I. INTRODUCTION

The introduction motivates a multi-sector BD-RIS as a full-space alternative to prior RIS models and introduces its model, channel characterization, and optimization contributions. The model partitions coverage into sectors and connects each cell’s antennas through a reconfigurable impedance network.

I. INTRODUCTION

  • Multi-sector BD-RIS extends full-space coverage with directional sector beams, while prior models were limited to one-sided reflection or incurred greater circuit complexity.The proposed model includes STAR-RIS/IOS as a special instance.Source: §I · p. 2
  • The paper derives continuous- and discrete-value scattering constraints and a channel model accounting for antenna beamwidth and gain.The model also derives received-power scaling with sector count for an SU-SISO system.Source: §I · p. 2
  • For MU-MISO systems, the proposed sum-rate design derives a closed-form continuous solution and an efficient algorithm for discrete values.Discrete values are selected from finite codebooks.Source: §I · p. 2
  • For the same sum-rate requirement, increasing the number of sectors can reduce the total RIS antenna count and the resolution of finite codebooks.The reduced codebook resolution is described as beneficial for controlling practical RIS realization costs.Source: §I · p. 2

II. MULTI-SECTOR BD-RIS MODEL

  • The model’s scattering matrix characterizes the reconfigurable impedance network separately from antenna-array characteristics, and its lossless-network constraint is specified.The channel model is developed by specifying the RIS antenna arrangement and architecture.Source: §II · p. 3
  • An M-cell multi-sector BD-RIS has L antennas per cell, arranged across L sectors, with each cell’s antennas connected to an L-port fully connected impedance network.Each sector covers 1/L of space; the RIS model is passive.Source: §II · p. 3

A. Cell-Wise Single-Connected Multi-Sector BD-RIS

The section focuses on a cell-wise single-connected multi-sector BD-RIS, whose cells are not connected to one another. Its antenna arrangement uses the flexibility of group-connected impedance networks to support multiple sectors, with increasing sector count trading higher performance for greater circuit complexity and cost.

A. Cell-Wise Single-Connected Multi-Sector BD-RIS

  • The cell-wise single-connected architecture keeps different RIS cells electrically separate, while each cell can use different inter-cell topologies in other architecture variants.The cell-wise group- and fully-connected variants are described as offering higher performance gains at the cost of greater circuit-topology complexity; the paper focuses on the single-connected case.Source: A. Cell-Wise Single-Connected… · p. 4
  • A 2-cell, 4-sector design uses the same impedance-network topology as an 8-element group-connected RIS with group size 4, but places antennas along polygon sides rather than in one direction.The group-connected network supports multi-sector operation through the antenna arrangement; the cited group-connected RIS uses that network flexibility for more flexible reflection.Source: A. Cell-Wise Single-Connected… · p. 5, Figure 5 · p. 5
  • Increasing the number of sectors increases performance but requires more circuit complexity and cost.The described M-cell, L-sector single-connected design requires ML antennas and M fully connected L-port reconfigurable impedance networks, each with 2 impedance components.Source: A. Cell-Wise Single-Connected… · p. 5
  • The section distinguishes continuous amplitude and phase control from discrete values selected from finite codebooks.These are the two element-value cases considered for the multi-sector BD-RIS.Source: A. Cell-Wise Single-Connected… · p. 5

1) Continuous Case: 

The section gives separate element constraints for continuous and discrete control. In the discrete case, amplitude-square and phase values are uniformly spaced and selected using codebooks controlled by bit counts.

1) Continuous Case: 

  • In the continuous case, each element has an amplitude β_i between 0 and 1, its phase ϑ_i is in [0, 2π), and the amplitudes in each group sum to 1.Source: 1) Continuous Case:  · p. 5

2) Discrete Case:

  • In the discrete case, amplitude-square values are uniformly spaced over [0, 1], with the codebook controlled by A bits.Source: 2) Discrete Case · p. 5
  • Discrete phase values are uniformly spaced around the unit circle, with the codebook controlled by B bits.The discrete case also transforms the element constraint to use values selected from finite codebooks.Source: 2) Discrete Case · p. 5

B. Channel Model

The channel model builds path loss from transmitter-to-surface and surface-to-user propagation while accounting for antenna gains. It considers idealized radiation patterns and assumes far-field distances for the transmitter, surface, and users.

B. Channel Model

  • The power captured by a sector antenna from the transmitter is modeled as a generalization of the free-space Friis transmission equation.Source: B. Channel Model · p. 5
  • The far-field assumption treats distances between the transmitter/users and the multi-sector BD-RIS as much larger than the sizes of those devices, ignoring antenna spacing and distances among sectors.Source: B. Channel Model · p. 5
  • The path-loss derivation separates transmitter and user effects from the BD-RIS contribution by assuming constant transmitter and user antenna gains, while accounting for RIS antenna gain.The derivation combines power captured from the transmitter, power scattered by RIS antennas, and power received by users to form the cascaded path loss.Source: B. Channel Model · p. 6

1) Idealized Radiation Pattern:

2) Practical Radiation Pattern:

The practical radiation-pattern model links sector coverage to antenna beamwidth and gain, and represents each sector as a uniform planar array. The resulting channel model combines large-scale path loss with small-scale fading.

2) Practical Radiation Pattern:

  • The channel combines large-scale fading represented by path-loss components with small-scale fading, including Rician fading with line-of-sight and Rayleigh non-line-of-sight components.Source: 2) Practical Radiation Pattern · p. 6
  • Adjacent sectors are slightly overlapped to guarantee full-space coverage.Source: 2) Practical Radiation Pattern · p. 6
  • Each sector is modeled as an M-antenna uniform planar array, and its steering vector is used to form the line-of-sight transmitter-RIS and RIS-user channel components.Source: 2) Practical Radiation Pattern · p. 7
  • STAR-RIS/IOS is a special instance of the multi-sector BD-RIS with L = 2; the model also accounts for how sector beamwidth and antenna gain affect fading and system performance.The paper connects the multi-sector model to a group-connected reconfigurable impedance network.Source: 2) Practical Radiation Pattern · p. 7
  • The established communication model combines the channel, path-loss, fading, and continuous- or discrete-value BD-RIS models for performance evaluation.Source: 2) Practical Radiation Pattern · p. 7

III. POTENTIAL APPLICATIONS OF MULTI-SECTOR BD-RIS

The section identifies millimeter-wave/terahertz systems and cell-free networks as potential applications for multi-sector BD-RIS, then examines received-power scaling with sector count.

III. POTENTIAL APPLICATIONS OF MULTI-SECTOR BD-RIS

A. Millimeter Wave/Terahertz Communications and Sensing

  • For millimeter-wave/terahertz communications and sensing, increasing sector count raises antenna gain and beam directionality, while full-space coverage can extend coverage around obstacles.The passage also identifies high spreading loss, sparse multipath, and blockage as relevant challenges.Source: A. Millimeter Wave/Terahertz… · p. 7

B. Cell-Free Networks

  • In cell-free networks, full-space coverage and flexible placement may simplify deployment, and higher performance and wider coverage may reduce the number of required surfaces.For L > 2, the structure is described as more adaptable than STAR-RIS, which usually needs attachment to building surfaces.Source: B. Cell-Free Networks · p. 7

IV. SCALING LAW

  • The scaling-law analysis focuses on how received power varies with sector count L, a factor distinct from the RIS-element-count focus of prior scaling work.It considers both idealized and practical radiation patterns; the stated maximum power depends on cell count M and path loss.Source: §IV · pp. 7–8

1) Idealized Radiation Pattern:

The scaling analysis relates received power to sector count under idealized and practical radiation patterns, with outcomes depending on whether sector size or total antenna count is fixed.

1) Idealized Radiation Pattern:

  • Under the idealized pattern, maximum received power is proportional to L when other terms are fixed; fixing total antennas ML instead makes M decrease as L grows, creating a trade-off.The idealized-pattern result assumes LoS path-loss components of 2.Source: 1) Idealized Radiation Pattern · p. 8

2) Practical Radiation Pattern:

  • With M = 32 antennas per sector, received power increases as L grows; with total antennas ML = 180 fixed, it still increases significantly, as antenna gain dominates shrinking sector dimension.The figure plots average received power versus L; practical-pattern power can approach the idealized upper bound.Source: Figure 7 · p. 8, 2) Practical Radiation Pattern · p. 9
  • Under the practical pattern, received power is proportional to L when M is fixed, but its dependence on L is less straightforward when total antennas ML are fixed.The analysis relates sector count to antenna gain and sector dimension.Source: 2) Practical Radiation Pattern · p. 8

V. BEAMFORMING DESIGN FOR MULTI-SECTOR BD-RIS-ASSISTED MU-MISO SYSTEM

The paper formulates sum-rate maximization for a multi-sector BD-RIS-assisted MU-MISO system and proposes algorithms for continuous and discrete surface designs.

V. BEAMFORMING DESIGN FOR MULTI-SECTOR BD-RIS-ASSISTED MU-MISO SYSTEM

  • The proposed design framework targets sum-rate maximization in multi-sector BD-RIS-assisted MU-MISO systems, covering both continuous and discrete cases.The design alternates across optimization blocks, including the precoder and the sector-specific surface variables.Source: §V · p. 9, A. Problem Formulation and… · p. 9

A. Problem Formulation and Optimization Framework

  • The sum-rate problem is treated as a sum-of-functions-of-ratio optimization and transformed into a four-block optimization using fractional programming.Solutions for auxiliary-variable and precoder blocks follow the prior framework.Source: A. Problem Formulation and… · p. 9

B. Solution to {φl}∀l∈L

  • For the surface-design block, the method successively optimizes each cell while fixing the other cells, for both continuous and discrete designs.This cell-wise procedure is applied to sector vectors φ_l.Source: B. Solution to {φl}∀l∈L · p. 9

1) Continuous Case with Constraint (9):

The continuous design derives cell-wise phase and amplitude solutions, while the discrete design quantizes continuous solutions and adjusts them to satisfy the discrete constraint.

1) Continuous Case with Constraint (9):

  • For continuous amplitudes and phase shifts, phase choices minimize the cell subproblem, while amplitude design reduces to a real-valued problem with a multiplier found by bisection.The multiplier is selected within bounds so that g(µ⋆) = 1.Source: 1) Continuous Case with Constraint (9) · pp. 9–10

2) Discrete Case with Constraint (10):

  • For discrete values, the method quantizes continuous phase and amplitude solutions, then adjusts the quantized result to satisfy the constraint.The adjustment uses phases that increment or decrement the amplitude-square levels.Source: 2) Discrete Case with Constraint (10) · p. 10

3) Summary:

The design procedure separates continuous and discrete cases, iterating their respective updates until convergence.

3) Summary:

  • The continuous and discrete procedures are organized as Steps 1–4 for multi-sector BD-RIS design.The supplied procedure directs the design to the continuous or discrete branch after solving sub-problem (40).Source: 3) Summary · p. 10
  • For the continuous case, calculate χ_i and obtain θ⋆_i for each i in L_m.The procedure repeats the continuous update until the vectors φ⋆_l converge.Source: 3) Summary · pp. 10–11
  • For the discrete case, calculate χ_i and obtain θ⋆_i for each i in L_m.The discrete procedure then quantizes the continuous solution β⋆_m.Source: 3) Summary · pp. 10–11

VI. PERFORMANCE EVALUATION

The evaluation compares multi-sector BD-RIS designs across sector counts, radiation patterns, architectures, and codebook resolutions. The conclusion links added sectors to antenna and resolution reductions, while identifying dimensioning and alternative design metrics as open considerations.

VI. PERFORMANCE EVALUATION

  • With idealized radiation patterns, multi-sector BD-RIS outperforms STAR and CW-GC/FC; with practical patterns, it outperforms STAR and CW-GC and nearly matches CW-FC.The L = 6 design also achieves higher sum-rate than CW-GC at the same circuit complexity.Source: Figure 9 · p. 11, §VI · p. 12
  • More sectors can reduce total antenna count at similar sum-rate, while idealized radiation patterns outperform practical ones at fixed sector count.For practical patterns, ML = 120 with L = 3 performs similarly to ML = 96 with L = 6.Source: §VI · p. 11
  • Higher codebook resolutions improve sum-rate, but more sectors reduce the performance gap between resolutions and can support lower-resolution designs.With practical-pattern modeling, A = 2 and B = 2 are sufficient for satisfactory performance; with idealized-pattern modeling, A = 3 and B = 3 approach the continuous case.Source: Figure 10 · p. 12, §VI · p. 12

VII. CONCLUSION AND FUTURE WORK

  • The model includes STAR-RIS/IOS as a special instance, derives an antenna-gain-aware channel model and received-power scaling law, and reports positive correlation between maximum received power and sector count.For MU-MISO, more sectors can also reduce antennas and codebook resolutions at the same sum-rate requirement.Source: §VII · p. 13

1) The Dimensioning of Multi-Sector BD-RIS:

  • Antenna size and sector count depend jointly on operating frequency, performance requirements, and circuit complexity or cost.The paper identifies dimensioning as a balance between performance and hardware complexity.Source: 1) The Dimensioning of Multi-Sector… · p. 13

2) Different Multi-Sector BD-RIS Design Approaches:

APPENDIX A PROOF OF LEMMA 1

Appendix A applies a Lagrange method and checks a positive-semidefinite condition; Appendix B presents an inequality in its proof of Lemma 2.

APPENDIX A PROOF OF LEMMA 1

  • Applying the Lagrange method gives (Ψ_m + μI_L)β_m = χ_m, subject to β_m^Tβ_m = 1.The proof then turns to showing Ψ_m + μI_L is positive semidefinite.Source: Appendix A · p. 13
  • Rearranging (49) completes the proof of Lemma 1.Source: Appendix A · p. 13

APPENDIX B PROOF OF LEMMA 2

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