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A Tutorial on Beyond-Diagonal Reconfigurable Intelligent Surfaces: Modeling, Architectures, System Design and Optimization, and Applications

Hongyu Li, Matteo Nerini, Shanpu Shen, Bruno Clerckx

arXiv:2505.16504v2eess.SPcs.ITmath.IT

TL;DR

BD-RIS research still faces unexplored architecture design and practical channel-estimation constraints. This tutorial surveys BD-RIS modeling and design, reporting gains over D-RIS and STAR-RIS in example systems.

  • Problem

    The BD-RIS architecture design space remains largely unexplored, while passive channel-estimation overhead grows with element count and circuit complexity.

    Source: 2) Related Works and Discussions · p. 22, 2) Future Research Direction · p. 36

  • Method

    The tutorial develops and surveys BD-RIS modeling, modes and architectures, optimization, channel estimation, hardware impairments, benefits, applications, and challenges.

    Source: §IX · pp. 36–37

  • Results

    At M = 128, a 4-sector BD-RIS improves sum-rate by 150% over STAR-RIS, while SISO BD-RIS achieves approximately 62% higher channel gain than D-RIS asymptotically.

    Source: §II · p. 6, C. Enabling Flexible Modes with… · p. 23

  • Takeaways & Limitations

    BD-RIS offers potential for higher performance, larger coverage, and denser connectivity, with hardware and signal-processing challenges remaining research directions.

    Source: §IX · pp. 36–37

    Current passive channel-estimation training overhead grows with element count and circuit complexity, and perfect-hardware methods may not apply to impaired BD-RIS.

    Source: 2) Related Works and Discussions · p. 22

Abstract

from arXiv · show

Written by its inventors, this first tutorial on Beyond-Diagonal Reconfigurable Intelligent Surfaces (BD-RISs) provides the readers with the basics and fundamental tools necessary to appreciate, understand, and contribute to this emerging and disruptive technology. Conventional (Diagonal) RISs (D-RISs) are characterized by a diagonal scattering matrix $\mathbf{\Theta}$ such that the wave manipulation flexibility of D-RIS is extremely limited. In contrast, BD-RIS refers to a novel and general framework for RIS where its scattering matrix is not limited to be diagonal (hence, the ``beyond-diagonal'' terminology) and consequently, all entries of $\mathbf{\Theta}$ can potentially help shaping waves for much higher manipulation flexibility. This physically means that BD-RIS can artificially engineer and reconfigure coupling across elements of the surface thanks to inter-element reconfigurable components which allow waves absorbed by one element to flow through other elements. Consequently, BD-RIS opens the door to more general and versatile intelligent surfaces that subsumes existing RIS architectures as special cases. In this tutorial, we share all the secret sauce to model, design, and optimize BD-RIS and make BD-RIS transformative in many different applications. Topics discussed include physics-consistent and multi-port network-aided modeling; transmitting, reflecting, hybrid, and multi-sector mode analysis; reciprocal and non-reciprocal architecture designs and optimal performance-complexity Pareto frontier of BD-RIS; signal processing, optimization, and channel estimation for BD-RIS; hardware impairments (discrete-value impedance and admittance, lossy interconnections and components, wideband effects, mutual coupling) of BD-RIS; benefits and applications of BD-RIS in communications, sensing, power transfer.

I. INTRODUCTION

The tutorial presents BD-RIS as a generalization of diagonal RIS that adds tunable inter-element coupling, then surveys its modeling, architectures, optimization, hardware impairments, benefits, applications, and challenges.

I. INTRODUCTION

  • RIS is presented as a 6G candidate for reconfiguring wireless propagation, alongside approaches aimed at spectral efficiency, resource use, sensing, and power transfer.The tutorial also describes RIS as a planar surface of tunable elements that can be deployed to bypass obstacles and redirect waves.Source: §I · p. 1, A. Background and Motivation · p. 2

A. Background and Motivation

  • D-RIS controls only diagonal scattering entries, while BD-RIS interconnects elements so off-diagonal entries can be tuned and waves can flow between elements.The tutorial motivates this additional flexibility as a response to D-RIS’s limited wave manipulation.Source: A. Background and Motivation · p. 2, B. BD-RIS · p. 2

B. BD-RIS

  • BD-RIS supports varied circuit topologies and element arrangements, including reciprocal or non-reciprocal designs and layouts that can enable wider coverage and denser connectivity.The described scattering matrices include full, block-diagonal, and permuted structures; flexible inter-element connections permit signals to flow between directions.Source: B. BD-RIS · pp. 2–3
  • Envisioned uses include indoor-outdoor coverage, integrated access and backhaul, wireless power transfer, sensing, and combined communication, sensing, and power-transfer systems.The tutorial also describes BD-RIS as applicable to D-RIS-enabled scenarios, with more flexible deployment and wave manipulation.Source: B. BD-RIS · p. 3

C. Contributions and Organization

  • The tutorial covers physics-consistent modeling, modes and architectures, optimization and channel estimation, four hardware impairment classes, applications, and future research challenges.It presents itself as the first comprehensive tutorial on BD-RIS and includes a SISO example to explain the rationale and benefits.Source: C. Contributions and Organization · pp. 3–4

II. BD-RIS-AIDED SISO: A TOY EXAMPLE

  • In the SISO asymptotic regime, BD-RIS achieves approximately a 62% higher channel gain than D-RIS under i.i.d. Rayleigh fading.The reported comparison is illustrated in Fig. 4.Source: §II · p. 6, Figure 4 · p. 6

III. MODELING, MODE ANALYSIS, AND ARCHITECTURE DESIGN OF BD-RIS

The section introduces BD-RIS modeling, mode analysis, and architecture design, beginning with multi-port network analysis as a foundation. It explains why this framework is suitable for representing RIS elements and wireless-system behavior.

III. MODELING, MODE ANALYSIS, AND ARCHITECTURE DESIGN OF BD-RIS

  • The section presents modeling, mode analysis, and architecture design as the core topics for understanding BD-RIS.It includes technical details, graphical illustrations, summaries, and discussions.Source: §III · p. 6

A. Preliminaries on Multi-Port Network Analysis

  • Multi-port network analysis provides a simplified alternative to field analysis using Maxwell’s equations for studying wireless systems.The method can explicitly capture hardware effects such as mutual coupling and antenna mismatching.Source: A. Preliminaries on Multi-Port Network… · p. 6
  • RIS suits multi-port modeling because each RIS element can be treated as a port in a network.The subsection introduces impedance, admittance, and scattering parameters as foundations for BD-RIS modeling.Source: A. Preliminaries on Multi-Port Network… · p. 6

1) Impedance, Admittance, and Scattering Parameters:

A passive RIS is modeled as a multi-port reconfigurable impedance network described by impedance, admittance, or scattering matrices. Its mathematical properties distinguish diagonal from beyond-diagonal RIS and reciprocal from non-reciprocal BD-RIS, with the latter supporting varied modes and architectures.

1) Impedance, Admittance, and Scattering Parameters:

  • The network’s voltage, current, incident-wave, and reflected-wave vectors are related through impedance, admittance, and scattering matrices.The scattering matrix relates incident waves to reflected waves; impedance and admittance relate voltage and current.Source: 1) Impedance, Admittance, and… · p. 6

2) Reciprocal and Lossless Networks:

  • Reciprocal networks have symmetric impedance, admittance, and scattering matrices, while reciprocal lossless networks deliver zero net real power.The lossless condition is associated with purely imaginary impedance and admittance entries and a unitary scattering matrix.Source: 2) Reciprocal and Lossless Networks · p. 7

1) Impedance, Admittance, and Scattering Parameters:

  • An M-element passive RIS is modeled as an M-antenna array connected to an M-port reconfigurable impedance network, characterized by ZI, YI, or Θ.These are its impedance, admittance, and scattering matrices, respectively.Source: B. Modeling and Classification · p. 7

B. Modeling and Classification

  • A passive reconfigurable impedance network does not reflect more power than the incident waves, and its lossless scattering matrix is unitary.The unitary property is the stated lossless-network condition.Source: B. Modeling and Classification · p. 7

1) D-RIS and BD-RIS:

  • D-RIS has diagonal matrices, whereas BD-RIS is not restricted to diagonal matrices; even lossless BD-RIS can tune entry amplitudes and phases.For the four-element example, D-RIS supports phase control, while BD-RIS entries can have magnitudes up to one under unit-norm column constraints.Source: 1) D-RIS and BD-RIS · p. 7

2) Reciprocal and Non-Reciprocal BD-RIS:

  • BD-RIS may be reciprocal or non-reciprocal; non-reciprocity is an additional degree of freedom, and the designs support varied modes and architectures.The classification depends on the reconfigurable impedance network, element arrangement, and circuit topology.Source: 2) Reciprocal and Non-Reciprocal BD-RIS · p. 7

C. Mode Analysis

BD-RIS modes and reciprocal circuit architectures are classified by their supported scattering behavior and inter-element connections. Interconnections enable hybrid and multi-sector operation, while reciprocal architectures range from single-connected to fully- and group-connected designs.

C. Mode Analysis

  • RIS classification distinguishes reciprocal from non-reciprocal designs, supported modes, and inter-element architectures.Source: C. Mode Analysis · p. 8

2) Hybrid Transmitting and Reflecting Mode:

  • Hybrid mode connects back-to-back elements so incident signals can be partly reflected within a sector and partly scattered into another for full-space coverage.Pure transmission is an extreme case in which signals are scattered entirely toward the other sector.Source: 2) Hybrid Transmitting and Reflecting… · p. 8

3) Multi-Sector Mode:

  • Multi-sector mode connects L elements around an L-sided polygon, scattering signals within and across sectors; its uni-directional sector coverage can yield higher gains than hybrid mode.Source: 3) Multi-Sector Mode · p. 8

D. Architecture Design: Reciprocal Architectures

2) Fully-Connected:

  • Fully-connected BD-RIS uses interconnections to exploit off-diagonal scattering entries, increasing wave-manipulation flexibility, and has a symmetric unitary scattering matrix when lossless.Source: 2) Fully-Connected · p. 9

3) Group-Connected:

  • Group-connected architecture partitions ports into fully connected groups, trading circuit complexity against flexibility; single-connected and fully-connected designs are its G = M and G = 1 special cases.Source: 3) Group-Connected · pp. 9–10

4) Tree-Connected:

Tree, forest, band, and stem architectures use structured interconnections to reduce circuit complexity or match fully-connected performance under specified settings. Their circuit constraints are represented by the admittance matrix, whose sparsity reflects the topology.

4) Tree-Connected:

  • Tree-connected architectures can match fully-connected performance in single-user MISO systems using 2M − 1 admittance components.Tridiagonal and arrowhead designs are two tree topologies; their constraints are represented by sparse admittance matrices even though the resulting impedance and scattering matrices are full.Source: 4) Tree-Connected · p. 10

5) Forest-Connected:

  • Forest-connected designs divide ports into G tree-connected groups, using G(2M̄ − 1) components and block-diagonal admittance matrices.Setting G = M gives single-connected architecture, while G = 1 gives tree-connected architecture.Source: 5) Forest-Connected · p. 10

6) Band- and Stem-Connected:

  • Band- and stem-connected architectures constrain interconnections to support optimal designs for multi-user MIMO, with the number of links theoretically limited by system degrees of freedom.Band width q controls adjacent links; stem width q controls the number of ports linked to all others, and stem-connected architecture includes arrowhead tree-connected design when q = 1.Source: 6) Band- and Stem-Connected · pp. 10–11

7) Dynamically Connected:

Dynamically connected architectures reconfigure both admittances and switch states, while non-reciprocal circuits relax matrix symmetry. The paper also shows that element arrangement and circuit topology jointly determine supported modes.

7) Dynamically Connected:

  • Dynamically connected architecture jointly controls admittance values and switch states, and can realize fixed architectures or varied groupings by selecting interconnection links.Fully-connected and single-connected designs correspond to all switches on and all switches off, respectively.Source: 7) Dynamically Connected · pp. 11–12

E. Architecture Design: Non-Reciprocal Architectures

  • Non-reciprocal circuits such as isolators, gyrators, and circulators yield asymmetric unitary scattering blocks in lossless grouped designs.Reciprocal designs offer simpler, potentially more cost-effective circuits and topology constraints reflected in matrix sparsity; non-reciprocal designs offer greater wave-manipulation flexibility.Source: E. Architecture Design: Non-Reciprocal… · pp. 12–13

F. Unified Modes and Architectures

IV. SIGNAL PROCESSING FOR BD-RIS

The tutorial surveys optimization approaches for BD-RIS, including methods for unitary or symmetric-unitary scattering matrices and for architecture-constrained admittance matrices. It describes a closed-form global optimum for tree- and forest-connected designs in a simple SISO case.

IV. SIGNAL PROCESSING FOR BD-RIS

A. Optimizing BD-RIS: A Simple SISO Case

  • For symmetric-unitary Θ, the tutorial describes decomposition, projection, and auxiliary-variable strategies to separate the symmetry and unitary constraints.Source: 2) Solutions for Unitary · p. 15
  • The decomposition Θ = UD U^T interprets BD-RIS as a power divider controlled by U followed by a phase-shifter network D; D-RIS fixes U = IM.In BD-RIS, inter-element connections let waves partially flow to other elements before reflection.Source: 2) Solutions for Unitary · p. 16
  • Tree- and forest-connected SISO designs admit a closed-form global optimum obtained by solving linear equations over nonzero entries of YI, with O(M^3) complexity.Source: 3) Solutions for YI · p. 16

B. Survey on BD-RIS Optimization and Performance Analysis

The section surveys BD-RIS optimization across wireless scenarios and expresses joint precoder and surface design through architecture-dependent admittance- or scattering-matrix constraints. It describes iterative designs and scenario-specific solutions for reciprocal and non-reciprocal surfaces.

B. Survey on BD-RIS Optimization and Performance Analysis

  • The section frames its coverage as optimization and performance analysis across scenarios, including multi-user MIMO under specified channel-modeling assumptions.The channel model assumes perfect matching, no mutual coupling or specular reflection, and unilateral approximation at all devices.Source: B. Survey on BD-RIS Optimization and… · p. 16
  • Joint BD-RIS and precoder optimization can be formulated in admittance-matrix or scattering-matrix form, with constraints adapted to the architecture.The scattering-matrix formulation covers single-, group-, and fully connected architectures as well as non-reciprocal architectures.Source: B. Survey on BD-RIS Optimization and… · p. 17
  • For MISO, alternating surface and precoder design uses maximum-ratio transmission for a fixed surface and SISO solutions for a fixed precoder.The procedure iterates until the objective converges.Source: B. Survey on BD-RIS Optimization and… · p. 17
  • For non-reciprocal BD-RIS, surveyed solutions include SISO closed forms and upper bounds, MISO grouping strategies, and multi-sector beamforming for specific users.The coordinated group-connected design uses scattering matrices with more than M non-zero entries.Source: B. Survey on BD-RIS Optimization and… · p. 18

2) Capacity Maximization for MIMO:

For MISO and SIMO received-power maximization, the surveyed approaches alternate BD-RIS design with transmit-covariance design, with additional closed-form solutions for specified channel and architecture settings.

2) Capacity Maximization for MIMO:

  • The MISO/SIMO formulation uses a transmit covariance matrix in place of the precoder and includes receiver noise power.The setup has one user with Nr receive antennas.Source: 2) Capacity Maximization for MIMO · p. 18
  • For reciprocal, fully connected BD-RIS, a common approach iteratively designs the surface and transmit covariance matrix.The surface design can decouple Θ as ΨᴴΨ with unitary Ψ; rank-1 LoS MIMO channels also permit a closed-form Θ solution.Source: 2) Capacity Maximization for MIMO · p. 18
  • For non-reciprocal BD-RIS, manifold and matrix-decomposition methods address the design, with closed forms for near- and far-field cases reported as globally optimal.The manifold method applies to a specified non-reciprocal constraint set.Source: 2) Capacity Maximization for MIMO · p. 18

3) Sum-Rate Maximization for Multi-User MISO:

For multi-user MISO sum-rate design, the section surveys iterative joint optimization and architecture-specific reciprocal and non-reciprocal methods. It also covers multi-user MISO energy-efficiency maximization and power minimization.

3) Sum-Rate Maximization for Multi-User MISO:

  • Multi-user MISO sum-rate optimization is a multi-variable problem handled by separate surface and precoder design or block coordinate descent.The setup uses one data stream per single-antenna user.Source: 3) Sum-Rate Maximization for… · p. 18
  • Reciprocal-architecture sum-rate methods include low-complexity two-stage designs, joint designs, quasi-Newton optimization with heuristic scheduling, and multi-sector closed-form approaches.The multi-sector approach also supports achievable sum-rate expressions and extensions to energy efficiency, error probability, and outage probability.Source: 3) Sum-Rate Maximization for… · p. 18
  • For non-reciprocal multi-user MISO sum-rate design, surveyed methods include learning-based optimization and a hybrid-mode constraint described as a Stiefel manifold.The learning-based methods support larger-dimensional optimizations.Source: 3) Sum-Rate Maximization for… · p. 18

4) Energy Efficiency Maximization and Power Minimization for Multi-User MISO:

  • For energy-efficiency maximization and power minimization, surveyed reciprocal designs include constraint-decoupling optimization, STARS meta-learning, and a universal admittance-matrix framework.The universal framework introduces auxiliary variables to eliminate a matrix inverse and applies to arbitrary admittance constraints, including several connected architectures.Source: 4) Energy Efficiency Maximization and… · p. 19

C. Channel Estimation

BD-RIS channel estimation depends on its scattering-matrix structure, so passive D-RIS patterns do not directly carry over. The least-squares discussion contrasts the resulting cascaded-channel structures and their pattern-design constraints.

C. Channel Estimation

  • Passive D-RIS estimation methods do not directly work for BD-RIS because the cascaded-channel structure and feasible pattern constraints differ.The cascaded-channel structure is mathematically determined by the RIS scattering matrix.Source: C. Channel Estimation · p. 19, 1) Least-Squares Estimation · p. 20
  • Semi-passive estimation separates base-station–RIS and RIS–user channels with relatively low training overhead, but performance depends on RF-chain count and entails power consumption.Passive estimation reduces the RF-chain requirement but generally estimates the cascaded channel instead of the separate links.Source: C. Channel Estimation · p. 19

1) Least-Squares Estimation:

  • The least-squares setup considers uplink estimation of the BD-RIS-aided channel with a blocked direct link and derives an estimator for the cascaded channel.If a direct link exists, it can first be estimated with the BD-RIS turned off, then removed before estimating the cascaded channel.Source: 1) Least-Squares Estimation · pp. 19–20
  • Under the stated least-squares setup, estimation performance depends on the BD-RIS pattern design.The passage gives the MSE in terms of the training-pattern matrix.Source: 1) Least-Squares Estimation · p. 20
  • D-RIS can use orthogonal patterns such as discrete Fourier or Hadamard matrices, but those solutions are not feasible for the BD-RIS pattern-design problem.The comparison highlights the distinct constraints on the two RIS pattern designs.Source: 1) Least-Squares Estimation · p. 21

2) Related Works and Discussions:

BD-RIS research reports performance gains from added scattering flexibility, while also identifying estimation challenges tied to architecture, training overhead, and hardware. Its benefits include low-complexity designs that approach fully connected performance, flexible coverage modes, and large gains in distributed deployments.

2) Related Works and Discussions:

  • Existing BD-RIS channel-estimation methods are early-stage, architecture-dependent, and can incur high training overhead and estimation error that increases with group size.Methods based on non-reciprocal architectures may not readily extend to other architectures or practical hardware impairments.Source: 2) Related Works and Discussions · pp. 21–22

A. Boosting Received Power and Rates

  • BD-RIS increases received power by up to 62% for SISO and sum-rate by up to 43% for MU-MISO over D-RIS.The cited comparison attributes these gains to BD-RIS flexibility in manipulating amplitude and phase of diagonal and off-diagonal scattering-matrix entries.Source: A. Boosting Received Power and Rates · p. 22

B. Enabling Low-Complexity Architectures with High Performance

  • For MISO, the least circuit complexity for an optimal BD-RIS is characterized by a tree-connected condition.The review also gives a multi-user MIMO minimum-complexity condition and identifies band- and stem-connected designs as examples that attain it.Source: B. Enabling Low-Complexity… · pp. 22–23
  • Tree-connected BD-RIS can match fully connected performance in SISO at the lowest possible circuit complexity, with arrowhead and tridiagonal designs as examples.For multi-user MIMO, band- and stem-connected designs with proper width q can also match fully connected performance with much less circuit complexity.Source: B. Enabling Low-Complexity… · pp. 22–23, Figure 17 · p. 22, Figure 18 · p. 22

C. Enabling Flexible Modes with Highly-Directional Wireless Coverage

  • At M = 128, reflecting BD-RIS improves sum-rate by up to 75% over single-connected D-RIS, while a four-sector fully connected BD-RIS improves it by 150% over STAR-RIS.The reported four-sector gain is attributed to the higher antenna gains of more directional antennas.Source: C. Enabling Flexible Modes with… · p. 23, Figure 19 · p. 23

D. Providing Orders of Magnitude Gains in Distributed Deployments

  • Distributed-over-localized BD-RIS gain grows exponentially with path-loss exponent and reaches several orders of magnitude under the stated sufficient and necessary distance-product conditions.The receiver-location benefit is greatest when the receiver is far from localized BD-RIS and close to distributed BD-RIS.Source: D. Providing Orders of Magnitude Gains… · pp. 23–24, Figure 21 · p. 24

E. Enabling Simultaneously Optimal Transmissions for Uplink and Downlink with Non-Reciprocal Architectures

Non-reciprocal BD-RIS can support optimal uplink and downlink transmissions for users at different locations, while dual-polarized analyses find gains over D-RIS. The hardware discussion describes discrete-value design approaches and notes unresolved design problems for non-reciprocal architectures.

E. Enabling Simultaneously Optimal Transmissions for Uplink and Downlink with Non-Reciprocal Architectures

  • When uplink and downlink users are not aligned, non-reciprocal BD-RIS can point both beams toward their signal directions, whereas reciprocal BD-RIS fails to point to the downlink user.This supports simultaneous optimal communication with a full-duplex base station; the benefit has also been shown in multi-user multi-antenna scenarios [97].Source: E. Enabling Simultaneously Optimal… · p. 24, Figure 22 · p. 24

F. Providing Enhanced Gains in Dual-Polarized Systems

  • For same-polarization links, dual-polarized fully connected BD-RIS can increase SISO received power by up to 62% over dual-polarized D-RIS.For opposite polarizations, the gain varies with channel fading; the analysis reports gains above one for both Rayleigh and LoS channels, with greater benefits at smaller χ.Source: F. Providing Enhanced Gains in… · p. 25, Figure 24 · p. 25

VI. BD-RIS WITH HARDWARE IMPAIRMENTS

  • The hardware discussion treats continuous-value, lossless impedance networks without frequency dependence or mutual coupling as idealizations that are not fully achievable in real implementations.The described impairment analysis varies one hardware factor at a time while assuming other factors are perfect; joint analysis is left as a future direction.Source: §VI · p. 25

A. Discrete-Value Impedance and Admittance

  • For reciprocal BD-RIS, discrete-value reactance codebooks can be learned offline and then used for online design; one resolution bit is reported as sufficient for satisfactory fully connected performance close to the continuous-value case.For non-reciprocal architectures, discrete-value expression and design remain unexplored, and the exact codebook problem has no existing solution.Source: A. Discrete-Value Impedance and… · pp. 25–26

B. Lossy Interconnections and Admittance Components

BD-RIS hardware losses arise both in interconnections and in reconfigurable admittance components, and each source constrains feasible circuit values. The cited analyses find negligible system impact from interconnection loss at small spacing, but significant losses can destroy the optimum of fully- and tree-connected architectures.

B. Lossy Interconnections and Admittance Components

1) Lossy Interconnections:

  • Lossy interconnections constrain off-diagonal admittance values to a circle in the complex plane, while their system-performance impact is negligible for small inter-element spacing.With lossless components and half-wavelength-multiple line lengths, the simplified constraint couples the real and imaginary parts; lossless interconnections allow arbitrary imaginary values.Source: 1) Lossy Interconnections · p. 27, Figure 27 · p. 27, Figure 28 · p. 27

2) Lossy Admittance Components:

  • Parasitic resistance constrains each component’s admittance to a circle, and significant component losses can destroy the optimum of fully- and tree-connected SISO architectures.The component is modeled using inductors, tunable capacitance, and an equivalent resistor representing varactor parasitic resistance.Source: 2) Lossy Admittance Components · pp. 27–28, Figure 31 · p. 28, Figure 30 · p. 28

C. Wideband Effect

The section addresses frequency-dependent circuit responses and physical mutual coupling, which complicate BD-RIS modeling. It reviews reciprocal wideband modeling and mutual-coupling-aware analyses, including reported performance effects and remaining scope boundaries.

C. Wideband Effect

  • Wideband BD-RIS modeling must capture frequency-dependent, coupled scattering-matrix entries; ignoring frequency variation creates a larger performance gap as circuit complexity increases.The cited reciprocal model links admittance values across frequencies through a common capacitor, and the reported susceptance is approximately linear over some practical bandwidth.Source: C. Wideband Effect · pp. 28–29, Figure 33 · p. 29

D. Mutual Coupling Effect

  • Mutual coupling becomes stronger as element spacing shrinks, challenging the common assumption of sufficiently large spacing; physics-consistent channel models represent the RIS through scattering, impedance, or admittance matrices.The impedance matrix’s off-diagonal entries characterize physical mutual coupling, which depends on inter-element spacing.Source: D. Mutual Coupling Effect · pp. 29–30, II ZIT · p. 31

II ZIT

  • The combined impedance ZI + ZII captures artificial and physical coupling, which BD-RIS can reconfigure to compensate for or exploit; reported gains over D-RIS grow as spacing decreases.Under Rayleigh fading, fully- and tree-connected BD-RIS can also increase average channel gain with mutual coupling, while inaccurate modeling can cause non-negligible sum-rate loss.Source: II ZIT · pp. 30–31

VII. APPLICATIONS OF BD-RIS

The tutorial surveys BD-RIS applications across channel shaping, security, communications, sensing, energy transfer, and computing. It also identifies practical design and implementation challenges as directions for future research.

1) BD-RIS for Channel Shaping:

  • BD-RIS can reach a larger dynamic range of channel singular values than D-RIS, supporting increased power gain and achievable rates, and can improve interference nulling in MIMO interference channels.The cited work attributes improved interference nulling or minimization to BD-RIS’s potential to reduce degrees of freedom.Source: 1) BD-RIS for Channel Shaping · pp. 31–32

2) BD-RIS Aiding Channel Reciprocity Attack and Physical Layer Security:

  • Non-reciprocal BD-RIS can break uplink-downlink channel reciprocity, with cited applications in physical-layer security and one-way secure communications, as well as potential malicious degradation of downlink performance.The cited security uses include supporting simultaneous optimal uplink and downlink transmission when legitimate users are not aligned.Source: 2) BD-RIS Aiding Channel Reciprocity… · p. 32

3) BD-RIS in Terahertz Communications and Massive MIMO:

  • For Terahertz links, BD-RIS can help bypass blockages and increase coverage and rates; integrated radome BD-RIS can reconfigure highly directional beams from a small-dimensional active antenna array.The cited massive-MIMO works also report performance benefits in a cell-free network.Source: 3) BD-RIS in Terahertz Communications… · p. 32

B. BD-RIS for Sensing and ISAC

  • A multi-sector self-sensing BD-RIS can scatter signals across sectors for full-space coverage, while BD-RIS-aided ISAC works report gains in coverage, SNR, power, throughput, tradeoff, or outage performance.BD-RIS is also reported to bridge active and passive localization and increase localization precision.Source: B. BD-RIS for Sensing and ISAC · pp. 32–33

C. BD-RIS for WPT and SWIPT

  • BD-RIS can flexibly shape wireless channels for WPT, with cited work reporting improved harvested energy for energy users and spectral efficiency for information users in SWIPT.Source: C. BD-RIS for WPT and SWIPT · p. 33

1) BD-RIS and Stacked Intelligent Metasurface (SIM); 2) BD-RIS with Advanced Multiple Access:

3) BD-RIS Aiding Unmanned Aerial Vehicle (UAV) and Non-Terrestrial Networks (NTN); 4) BD-RIS Aiding Mobile Edge Computing (MEC):

VIII. CHALLENGES AND FUTURE RESEARCH DIRECTIONS OF BD-RIS

  • BD-RIS remains at an early study stage, with challenges identified in implementation, circuit design, optimization, and transmission protocols.Source: §VIII · pp. 33–34

A. BD-RIS Implementation

The tutorial surveys BD-RIS implementation modes and architectures, then identifies practical design and evaluation gaps spanning hardware, beamforming, channel acquisition, and complexity.

A. BD-RIS Implementation

  • Existing STAR-RIS, hybrid-mode, and tridiagonal reflective prototypes show that BD-RIS can be implemented with existing RF techniques, though comprehensive implementation remains underway.Reflective, hybrid, and multi-sector modes require different antenna-array arrangements, while network components vary with architecture and may include varactors, PIN diodes, switches, or non-reciprocal devices.Source: A. BD-RIS Implementation · p. 34

1) Challenge:

  • More reconfigurable admittance components increase circuit complexity, cost, power losses, and drive-circuit consumption, potentially weakening BD-RIS performance benefits; theoretical methods may also delay effective transmission.These burdens grow especially as the number of elements or groups increases.Source: 1) Challenge · pp. 34–35

2) Future Research Direction:

  • Practical prototypes are needed to determine which architectures best balance performance, hardware feasibility, cost, power consumption, and computational complexity.Existing tree/forest and stem/band performance-complexity findings remain analytical, and a fair power-consumption model still needs practical validation.Source: 2) Future Research Direction · p. 35

B. Active BD-RIS

  • Active BD-RIS could address passive RIS performance limits, but requires modeling the antenna array, reconfigurable impedance network, and reflection power amplifier as a series connection.Research directions include power-scaling laws, performance-complexity or performance-power trade-offs, reduced channel-estimation overhead, and amplifier impairments.Source: B. Active BD-RIS · p. 35, 1) Challenge · p. 35, 2) Future Research Direction · p. 35

C. Artificial Intelligence (AI)-Driven Beamforming Solutions

D. CSI-Free Protocols

E. New BD-RIS Architectures

IX. CONCLUSION

  • The tutorial consolidates BD-RIS modeling, modes, architectures, signal processing, impairment analysis, and applications in communications, sensing, and power transfer.It also identifies implementation and signal-processing challenges relevant to possible commercialization.Source: §IX · pp. 36–37
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