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Modeling and Architecture Design of Reconfigurable Intelligent Surfaces Using Scattering Parameter Network Analysis

Shanpu Shen, Bruno Clerckx, Ross Murch

arXiv:2011.11362v2eess.SPcs.IT

TL;DR

RIS research has emphasized system-level optimization, but tractable electromagnetic-compliant communication models remain an open issue, alongside limited received signal power. This paper uses scattering-parameter network analysis to derive such a model and proposes group and fully connected impedance networks. Compared with the single connected case, the proposed architectures report up to 62% higher received power, 21% fewer elements at equal power, and further gains under distance-dependent pathloss and Rician fading.

  • Problem

    Tractable RIS communication models that satisfy electromagnetic equations remain an open problem, while received RIS signal power is limited.

  • Method

    The paper derives an electromagnetic RIS communication model through scattering-parameter network analysis and develops group and fully connected reconfigurable impedance networks.

  • Results

    Compared with the single connected case, fully and group connected networks increase received signal power by up to 62%, reduce required RIS elements by up to 21% at equal power, and achieve up to 48% and 34% gains under Rician fading with distance-dependent pathloss.

  • Takeaways & Limitations

    Adjusting both the phases and magnitudes of impinging waves provides more general and efficient RIS architectures than phase-only single connected networks.

Abstract

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Reconfigurable intelligent surfaces (RISs) are an emerging technology for future wireless communication. The vast majority of recent research on RIS has focused on system level optimizations. However, developing straightforward and tractable electromagnetic models that are suitable for RIS aided communication modeling remains an open issue. In this paper, we address this issue and derive communication models by using rigorous scattering parameter network analysis. We also propose new RIS architectures based on group and fully connected reconfigurable impedance networks that can adjust not only the phases but also the magnitudes of the impinging waves, which are more general and more efficient than conventional single connected reconfigurable impedance network that only adjusts the phases of the impinging waves. In addition, the scaling law of the received signal power of an RIS aided system with reconfigurable impedance networks is also derived. Compared with the single connected reconfigurable impedance network, our group and fully connected reconfigurable impedance network can increase the received signal power by up to 62%, or maintain the same received signal power with a number of RIS elements reduced by up to 21%. We also investigate the proposed architecture in deployments with distance-dependent pathloss and Rician fading channel, and show that the proposed group and fully connected reconfigurable impedance networks outperform the single connected case by up to 34% and 48%, respectively.

I. INTRODUCTION

The paper addresses the open problem of developing tractable, electromagnetic-compliant RIS communication models while improving the limited received signal power of RIS systems. It derives a scattering-parameter model and proposes more flexible impedance-network architectures with reported power and element-count benefits.

  • Motivation: RIS research has largely focused on system-level optimization, while tractable models satisfying electromagnetic equations remain an open problem.Prior work includes physical-optics, free-space pathloss, and practical phase-shift models, but the modeling challenge remains.
  • Motivation: Limited received signal power and low composite channel gain motivate developing more efficient RIS architectures.The paper notes that RIS may require many elements to compete with massive MIMO and decode-and-forward relays.
  • Contributions: The paper derives a physical and electromagnetic-compliant RIS communication model using rigorous scattering-parameter network analysis.The general model accounts for impedance mismatch and mutual coupling, while a perfect-matching, no-coupling case becomes tractable.
  • Contributions: Fully connected and group connected networks adjust both phases and magnitudes of impinging waves, unlike single connected networks that adjust only phases.Their scattering matrices are respectively full or block diagonal rather than diagonal with unit-modulus entries.
  • Results: 62% higher received signal power and 21% fewer RIS elements are reported for fully and group connected networks versus the single connected case.The element reduction applies when maintaining the same received signal power.
  • Results: 48% and 34% increases in received signal power are reported for fully connected and group connected networks, respectively, under distance-dependent pathloss and Rician fading.The paper also reports that small group sizes approach fully connected performance while maintaining low complexity.

A. Transmitter and Receiver

The RIS communication network connects transmitter, receiver, and RIS ports through source, load, and reconfigurable impedance relationships. The RIS network is represented by a reciprocal scattering matrix derived from its impedance matrix.

  • A. Transmitter and Receiver: Transmit antennas connect in series with voltage sources and source impedances, which relate the transmitter wave vectors.The source reflection coefficients form a diagonal matrix, and the source vector contains the transmit voltage sources.
  • A. Transmitter and Receiver: Receive antennas connect in series with load impedances, which relate the receiver wave vectors through diagonal load reflection coefficients.Each diagonal entry represents the reflection coefficient of one receive-antenna load impedance.
  • B. Reconfigurable Intelligent Surface: The RIS ports connect to a reconfigurable impedance network whose scattering matrix Θ is expressed through its impedance matrix ZI.The network is passive and reciprocal, giving symmetric impedance and scattering matrices.
  • B. Reconfigurable Intelligent Surface: Single connected networks use separate ground-connected impedances and produce diagonal impedance and scattering matrices.The four-element example is shown in Fig. 2(a).
  • B. Reconfigurable Intelligent Surface: Fully connected networks connect each RIS port to other ports to provide a more general architecture for improving received signal power.The four-element example is shown in Fig. 2(b).

1) Single Connected Reconfigurable Impedance Network:

The paper contrasts single, fully connected, and group connected reconfigurable impedance networks. Group connectivity interpolates between the simpler single-connected architecture and the more general fully connected architecture, trading performance enhancement against circuit complexity.

  • Single Connected Reconfigurable Impedance Network: Single-connected networks use separate ground-connected reconfigurable impedances, producing a diagonal impedance matrix with one component per RIS element.Their phase shifts have unit modulus because the impedances are purely reactive.
  • Fully Connected Reconfigurable Impedance Network: Fully connected networks connect every port to ground and to the other ports, requiring NI(NI + 1) /2 reconfigurable impedance components.Their impedance matrix is full and symmetric.
  • Fully Connected Reconfigurable Impedance Network: The fully connected scattering matrix is complex symmetric unitary, and the single-connected network is a special case of it.This makes the fully connected architecture more general than the conventional architecture.
  • Group Connected Reconfigurable Impedance Network: For an 8-element RIS, the illustrated group configurations use either 4 groups of size 2 or 2 groups of size 4.The group-connected design is proposed as a tradeoff between performance enhancement and the quadratic complexity of full connectivity.
  • Group Connected Reconfigurable Impedance Network: Group-connected networks partition an NI-element RIS into G groups, each using an NG-port fully connected network and yielding a block-diagonal scattering matrix.They require NI(NG + 1) /2 components; NG = 1 gives single connectivity, while NG = NI gives full connectivity.

III. RIS AIDED COMMUNICATION MODEL

The general RIS communication model relates transmit and receive voltages through scattering-network matrices and represents the resulting channel as a function of the RIS scattering matrix. The RIS configuration can then be optimized to control the channel and enhance system performance, although the general expression is difficult to evaluate directly.

  • III. RIS AIDED COMMUNICATION MODEL: The model derives the receive voltage from the transmit voltage through vR = (ΓR + I) TRT (I + ΓT TT T + TT T )^-1 vT.This follows from the relationships between incident and reflected waves and the voltage vectors.
  • III. RIS AIDED COMMUNICATION MODEL: Defining transmit and receive voltages as x and y gives the channel matrix H = (ΓR + I) TRT (I + ΓT TT T + TT T )^-1.The additive white Gaussian noise is ignored in the stated input-output relation.
  • III. RIS AIDED COMMUNICATION MODEL: The submatrices TRT and TT T depend on the RIS configuration Θ, so the channel is represented as H(Θ).This makes the RIS scattering matrix the design variable controlling the communication channel.
  • III. RIS AIDED COMMUNICATION MODEL: Optimizing Θ can intelligently control H(Θ) to enhance wireless-system performance.
  • III. RIS AIDED COMMUNICATION MODEL: The general model includes impedance mismatch and mutual coupling at the transmitter, RIS, and receiver.The associated matrix inversions make explicit expressions for H(Θ) difficult to obtain and complicate optimization.

B. RIS Aided Communication Model with Perfect Matching and No Mutual Coupling

Under perfect matching and no mutual coupling, the scattering-network model simplifies into channel matrices linking the transmitter, RIS, and receiver. The resulting formulation supports single-, fully-, and group-connected RIS architectures, with the latter two able to adjust both wave phases and magnitudes.

  • RIS Aided Communication Model with Perfect Matching and No Mutual Coupling: Perfect matching and no mutual coupling set the relevant antenna scattering matrices to zero and simplify the network representation.The assumption can be approximately achieved by matching antennas to Z0 and spacing them farther than half a wavelength.
  • RIS Aided Communication Model with Perfect Matching and No Mutual Coupling: The simplified channel is H = (SRT + SRIΘSIT)(I + STIΘSIT)^-1.The inverse term represents second-order reflections between the transmitter and RIS and back to the transmitter.
  • RIS Aided Communication Model with Perfect Matching and No Mutual Coupling: Because the second reflection STIΘSIT is extremely small and proportional to the squared transmitter-to-RIS pathloss, the inverse term can often be approximated by I.
  • RIS Aided Communication Model with Perfect Matching and No Mutual Coupling: The transmission scattering matrices SRT, SIT, and SRI are equivalently the transmitter-to-receiver, transmitter-to-RIS, and RIS-to-receiver channel matrices.
  • RIS Aided Communication Model with Perfect Matching and No Mutual Coupling: Single-connected, fully connected, and group-connected networks impose diagonal, full symmetric-unitary, and block-diagonal symmetric-unitary scattering structures, respectively.The proposed group and fully connected architectures can adjust both phases and magnitudes, unlike the conventional single-connected architecture.

IV. SCALING LAW

The scaling-law analysis quantifies received power versus RIS size in a SISO system with perfect matching and no mutual coupling. It simplifies the received power to the squared magnitude of the RIS-mediated channel under normalized transmit power and omitted direct propagation.

  • IV. SCALING LAW: The analysis studies received signal-power scaling with the number of RIS elements NI in a SISO system.It assumes NT = 1, NR = 1, perfect matching, and no mutual coupling.
  • IV. SCALING LAW: The received signal is modeled as y = hRT x + hRIΘhIT x + n before the direct channel is omitted for the scaling analysis.The terms represent the direct path, RIS-mediated path, and additive white Gaussian noise.
  • IV. SCALING LAW: With transmit power PT = 1 and the direct channel omitted, the received signal power is PR = |hRIΘhIT|^2.Here hIT and hRI denote the transmitter-to-RIS and RIS-to-receiver channels.

A. Single Connected Reconfigurable Impedance Network

The single connected network optimizes received power by adjusting only element-wise phases, whereas fully connected networks can also adjust magnitudes and may perform better.

  • A. Single Connected Reconfigurable Impedance Network: The single connected network optimizes the received signal power through a diagonal unit-modulus matrix.Its ports are not connected to one another, so only phases of the impinging waves can be adjusted.
  • A. Single Connected Reconfigurable Impedance Network: A closed-form optimizer for the fully connected network is difficult to derive, so the paper uses quasi-Newton optimization to approach a tight upper bound.Monte Carlo results confirm that the upper bound is tight.
  • A. Single Connected Reconfigurable Impedance Network: The fully connected network achieves higher received power than the single connected case when the channel-gain moduli are linearly independent.Equality is achieved only under the stated proportionality condition.
  • A. Single Connected Reconfigurable Impedance Network: Fully connected networks can adjust both phases and magnitudes of impinging waves, unlike the single connected architecture.The single connected case is analogous to equal-gain combining, whereas the fully connected case is analogous to maximum-ratio combining.

C. Group Connected Reconfigurable Impedance Network

The group connected architecture uses block-diagonal complex symmetric unitary networks, providing an intermediate design between single and fully connected RIS architectures.

  • C. Group Connected Reconfigurable Impedance Network: The group connected network partitions the RIS into groups whose blocks are complex symmetric unitary matrices.The optimization uses one such block for each group.
  • C. Group Connected Reconfigurable Impedance Network: A closed-form solution for the optimal group matrices is difficult, so each block is optimized numerically with a quasi-Newton method.Monte Carlo results confirm that the resulting upper bound is tight.
  • C. Group Connected Reconfigurable Impedance Network: The group connected architecture provides a tradeoff between the single connected and fully connected cases.The single and fully connected architectures correspond to group sizes N_G = 1 and N_G = N_I, respectively.
  • C. Group Connected Reconfigurable Impedance Network: In LoS channels, the single, group, and fully connected architectures have the same performance.Their optimal configurations coincide under the stated LoS assumption.

E. Rayleigh Fading Channel

Under i.i.d. Rayleigh fading, larger connected groups improve average received power over the single connected architecture, with fully connected networks providing the largest gain.

  • E. Rayleigh Fading Channel: Under i.i.d. Rayleigh fading, group connected architectures achieve higher average received power than the single connected architecture, and fully connected achieves the highest power.The single and fully connected cases are the group-size extremes N_G = 1 and N_G = N_I.
  • E. Rayleigh Fading Channel: The gains of group and fully connected architectures arise when the channel-gain moduli are linearly independent.Small group sizes offer satisfactory gains while maintaining lower complexity.
  • E. Rayleigh Fading Channel: 1.26, 1.37, 1.43, 1.49, and 1.52 are the group-connected power gains for N_G = 2, 3, 4, 6, and 8, respectively.The gain increases with group size but does not grow without limit.
  • E. Rayleigh Fading Channel: 1.62 is the limiting power gain of the fully connected architecture over the single connected architecture.This is the maximum-group-size case and agrees with the derived asymptotic limit.
  • E. Rayleigh Fading Channel: 11%, 14%, 16%, 18%, and 19% are the RIS-element reductions for N_G = 2, 3, 4, 6, and 8 at equal average received power.The fully connected case reduces the required number of elements by 21%.

V. PERFORMANCE EVALUATION

The evaluation formulates RIS SISO power maximization and tests the three architectures in a distance-dependent pathloss model with mixed Rayleigh and Rician fading.

  • V. PERFORMANCE EVALUATION: The received signal power is modeled as P_R = P_T ||h_RT + h_RI Θ h_IT||^2 and optimized over the RIS network matrix.The direct and RIS-assisted paths are combined coherently in the optimization.
  • V. PERFORMANCE EVALUATION: The group-connected optimization is constrained by a block-diagonal Θ with complex symmetric unitary blocks.The paper converts the constrained problem into an unconstrained optimization over reactance-matrix entries.
  • V. PERFORMANCE EVALUATION: The evaluation uses a 2D SISO deployment with the transmitter at (0, 0), receiver at (52, 0), and the RIS centered at (50, 2).The RIS is a half-wavelength-spaced uniform linear array.
  • V. PERFORMANCE EVALUATION: The channel model combines distance-dependent pathloss with Rayleigh transmitter-receiver and RIS-receiver channels and a Rician transmitter-RIS channel.The specified pathloss exponents are α_RT = 3.5, α_IT = 2, and α_RI = 2.8.
  • V. PERFORMANCE EVALUATION: The group connected architecture outperforms the single connected architecture, while the fully connected architecture achieves the highest received power.The fully connected received power does not change with the Rician factor, whereas the single connected power increases with it.
  • V. PERFORMANCE EVALUATION: Power gain increases with more reconfigurable components and optimization computations, creating a performance-complexity tradeoff.The study compares circuit-topology and optimization-computational complexity across the three architectures.

VI. CONCLUSIONS AND FUTURE WORK

The paper presents a scattering-parameter RIS model and more general connected architectures that improve received power over the single-connected design. It also identifies future work in channel estimation, broader deployments, and discrete optimization.

  • The proposed model accounts for impedance mismatch and mutual coupling and reduces to the conventional RIS model under special conditions.
  • Fully and group connected architectures use full or block-diagonal scattering matrices to adjust both impinging-wave phases and magnitudes.
  • 62% higher received signal power and 21% fewer RIS elements at equal power are reported for fully and group connected networks versus the single-connected case.
  • 48% and 34% higher received signal power are reported for fully and group connected networks, respectively, under distance-dependent pathloss and Rician fading.
  • Future work: Future work includes more efficient channel estimation, extensions to multi-user and multi-cell scenarios, and optimization with discrete values of Θ.

APPENDIX

The appendix introduces scattering parameters for arbitrary one-port and N-port networks. It relates incident waves to reflected waves through the scattering matrix, with reflection coefficient as the one-port special case.

  • A one-port network produces a reflected wave from an incident wave, with port voltage equal to the sum of the two waves.
  • The reflection coefficient completely characterizes a one-port input impedance, and passive impedances satisfy |Γ| ≤ 1.
  • For a pure reactive impedance, Γ = e^jθ and |Γ| = 1, supporting RIS phase shifting with unit-modulus reflection.
  • For an N-port network, the scattering relation b = Sa maps incident waves to reflected waves and characterizes the network at its ports.
  • The one-port scattering matrix is a scalar reflection coefficient, while a SISO wireless link can be represented as a two-port network.
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