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Reconfigurable Intelligent Surface assisted Two-Way Communications: Performance Analysis and Optimization

Saman Atapattu, Rongfei Fan, Prathapasinghe Dharmawansa, Gongpu Wang, Jamie Evans, Theodoros A. Tsiftsis

arXiv:2001.07907v2eess.SPcs.IT

TL;DR

The paper studies RIS-assisted two-way communications and the limited communication-theoretic understanding of single and moderate numbers of reflective elements under multipath fading. It derives exact single-element results, develops gamma-based multiple-element approximations, and formulates RIS phase optimization for non-reciprocal channels, showing outage decreases at (log(ρ)/ρ)^L while spectral efficiency increases with L.

  • Problem

    Communication-theoretic foundations for RIS systems with single and moderate numbers of reflective elements under multipath fading remain insufficiently understood for network design.

  • Method

    The paper derives exact single-element performance expressions, approximates multiple-element products of Rayleigh variables with gamma distributions, and optimizes RIS phases for minimum user SINR in non-reciprocal channels.

  • Results

    User outage probability decreases at (log(ρ)/ρ)^L, spectral efficiency increases with L, and Scheme 2 achieves lower outage probability than Scheme 1.

  • Takeaways & Limitations

    The analysis provides performance approximations and RIS phase-optimization formulations for two-way systems with reciprocal and non-reciprocal channels.

Abstract

from arXiv · show

In this paper, we investigate the two-way communication between two users assisted by a re-configurable intelligent surface (RIS). The scheme that two users communicate simultaneously over Rayleigh fading channels is considered. The channels between the two users and RIS can either be reciprocal or non-reciprocal. For reciprocal channels, we determine the optimal phases at the RIS to maximize the signal-to-interference-plus-noise ratio (SINR). We then derive exact closed-form expressions for the outage probability and spectral efficiency for single-element RIS. By capitalizing the insights obtained from the single-element analysis, we introduce a gamma approximation to model the product of Rayleigh random variables which is useful for the evaluation of the performance metrics in multiple-element RIS. Asymptotic analysis shows that the outage decreases at $\left(\log(ρ)/ρ\right)^L$ rate where $L$ is the number of elements, whereas the spectral efficiency increases at $\log(ρ)$ rate at large average SINR $ρ$. For non-reciprocal channels, the minimum user SINR is targeted to be maximized. For single-element RIS, closed-form solution is derived whereas for multiple-element RIS the problem turns out to be non-convex. The latter one is solved through semidefinite programming relaxation and a proposed greedy-iterative method, which can achieve higher performance and lower computational complexity, respectively.

I. INTRODUCTION

The paper introduces RIS-assisted two-way communications to address the limited analytical understanding of RIS systems beyond one-way links. It studies reciprocal and non-reciprocal channels, comparing one-slot simultaneous transmission with a two-slot interference-free alternative.

  • Motivation: RIS uses passive, tunable reflectors that redirect existing radio waves with negligible energy consumption.The concept is positioned as an alternative to power-hungry active components and can integrate into infrastructure such as building walls.
  • Research gap: Prior RIS research mainly considered one-way communications, leaving RIS-assisted two-way performance limits largely unquantified.The paper identifies this gap as its novelty and motivation.
  • Transmission schemes: Scheme 1 exchanges both users’ information in one time-slot but requires transmission and reception antennas and incurs self- and loop-interference.The analytical framework focuses on Scheme 1 because Scheme 2 can be deduced from it.
  • Transmission schemes: Scheme 2 uses two orthogonal time-slots, allowing single-antenna users and eliminating self- and loop-interference.It is treated as twice one-way communications, trading one additional time-slot for interference-free transmission.
  • Contributions: For reciprocal channels, the paper derives exact single-element results and Gamma-based multiple-element approximations for outage probability and spectral efficiency.The multiple-element approximation models each product of two Rayleigh random variables with a Gamma random variable.

B. Non-Reciprocal Channels

For non-reciprocal channels, forward and backward RIS links differ because of antenna placement or hardware. The model assumes full CSI and uses phase information to formulate the received signals and SINRs.

  • Channel model: Non-reciprocity occurs when transmit and receive antennas are far apart or use non-reciprocal hardware.Consequently, forward and backward channels between each user and the RIS may differ.
  • Channel model: Each user’s non-reciprocal channel is represented by separate transmit- and receive-side amplitudes and phases.The model uses ht,ℓ, hr,ℓ, gt,ℓ, and gr,ℓ for the two users’ directional links.
  • Channel knowledge: The users are assumed to know all channel vectors, while each RIS element knows the phases of its associated directional channels.This CSI supports RIS reflection configuration and subsequent SINR optimization.
  • Signal processing: The received signal and SINR expressions are formulated for both users under the non-reciprocal model.The paper derives U1’s signal first and obtains U2’s SINR through analogous processing.
  • Channel estimation: Channel estimation uses Scheme 2 twice, enabling one-way estimation methods before configuring the RIS for Scheme 1.The users provide the required phase information to a RIS micro-controller through a low-latency backhaul.

A. Optimum Phase Design at RIS

The paper aligns RIS phases to maximize instantaneous SINR in reciprocal channels, then derives outage expressions for one element and Gamma approximations for multiple elements. The approximation enables tractable performance analysis when exact characterization is difficult.

  • Phase optimization: The optimal RIS phase for each element is selected from the channel phase structure to maximize each user’s instantaneous SINR.The resulting maximum SINRs are obtained by substituting the optimal phases into the SINR expression.
  • Outage probability: The single-element outage probability follows from the SINR CDF evaluated at the SINR threshold.Outage is defined as the probability that SINR does not exceed γth.
  • Reciprocal channels: For L = 1, the SINR depends on the square of a product of two i.i.d. Rayleigh random variables.Its distribution is derived using the Rayleigh product CDF and a variable transformation.
  • Multiple-element analysis: For L ≥2, the SINR contains a sum of products of Rayleigh random variables, making exact CDF characterization difficult.The paper therefore seeks an approximate PDF and CDF for each product term.
  • Gamma approximation: A product of two i.i.d. Rayleigh random variables is approximated by a Gamma distribution using matched mean and variance.The Gamma model has shape parameter k and scale parameter θ, with mean kθ and variance kθ^2.
  • Approximation accuracy: The Gamma approximation closely matches simulation and the exact CCDF across very small, moderate, and large variance values.The reported KL divergence is approximately 2.3 × 10^-4, with little dependence on σ.

C. Spectral Efficiency

Spectral efficiency is defined as log2(1 + SINR) and averaged over the SINR distribution. The paper derives separate expressions for single-element and multiple-element reciprocal RIS cases.

  • Definition: Spectral efficiency is measured as log2(1 + SINR) in bits/sec/Hz.Average spectral efficiency is obtained by integrating this quantity over the SINR PDF.
  • General expression: Integration by parts provides an alternative average spectral-efficiency expression involving the SINR complementary distribution.The resulting expression is used for subsequent reciprocal-channel analysis.
  • Single-element RIS: For L = 1, the paper evaluates average spectral efficiency using the exact single-element SINR characterization.The resulting closed-form expression uses Meijer G functions.
  • Multiple-element RIS: For L ≥2, average spectral efficiency is evaluated using the Gamma-based SINR approximation.The expression involves generalized hypergeometric and logarithmic Gamma functions.

D. Asymptotic Analysis

At high SINR, RIS-assisted two-way networks exhibit logarithmic outage-rate scaling with the number of elements and logarithmic spectral-efficiency growth. Interference scaling determines whether transmit-power increases continue improving performance.

  • High SINR: (log(ρ)/ρ)^L outage decay holds for an L-element RIS at high SINR over Rayleigh fading channels.This contrasts with the (1/ρ)^L rate observed for traditional multiple-relay networks.
  • High SINR: log(ρ) spectral-efficiency growth holds for L-element RIS-assisted two-way networks at high SINR.The result applies over Rayleigh fading channels.
  • Interference scaling: Interference independent of transmit power permits asymptotic results obtained by replacing ρ with 1/ω in the outage expressions.In this regime, loop-interference variance dominates outage probability.
  • Interference scaling: Interference proportional to transmit power makes ρ converge to a constant proportional to 1/ω, limiting high-power improvements.The corresponding asymptotic results follow from the single- and multiple-element expressions after replacing ρ by 1/ω.
  • Open issue: The precise array-gain expression G(L, γ_th, ω, σ) remains unresolved.The paper leaves this expression for future work.
  • Element-count trade-offs: Rates of spectral-efficiency improvement and power saving decrease as L increases.Very large RISs may therefore be less effective relative to the overhead of additional channel estimations and phase adjustments.

2) For Large L (or LIS):

For large RISs, the aggregate channel is modeled through a central-limit approximation, but this approach does not readily yield closed-form average spectral efficiency. Non-reciprocal channels also require joint minimum-SINR phase optimization.

  • Large L approximation: The sum of product-Rayleigh terms is approximated by a Gaussian random variable for sufficiently large L.Its mean is Lπσ^2/4 and its variance is L(16−π^2)σ^4/16.
  • Large L approximation: The Gaussian approximation produces an error-function CDF for the aggregate channel and transfers to the SINR CDF.The SINR CDF is obtained by evaluating the aggregate-channel CDF at the corresponding threshold transformation.
  • Large L approximation: The CLT approximation may not provide closed-form or built-in-special-function expressions for average spectral efficiency.This is identified as a disadvantage of the approach.
  • Non-reciprocal channels: For non-reciprocal channels, maximizing each user’s instantaneous SINR is not straightforward because the optimal RIS phases depend on all channel phases.The paper instead formulates maximization of the minimum user SINR.

A. For L = 1

For multiple elements, non-reciprocal RIS phase design becomes a non-convex max–min SINR problem. The paper develops an SDP-relaxation route and a lower-complexity greedy-iterative alternative.

  • Problem formulation: The multiple-element objective maximizes min(γ_1, γ_2) over all RIS phases.This formulation targets balanced user performance under non-reciprocal channels.
  • Problem formulation: The max–min SINR problem is non-convex because both user SINRs are non-convex functions of the phase matrix.The rank-one constraint is the source of non-convexity after matrix reformulation.
  • SDP-relax method: Dropping the rank-one constraint converts the reformulated problem into a semidefinite relaxation.For fixed t, the relaxed problem becomes an SDP feasibility problem.
  • SDP-relax method: Bisection search finds the maximal achievable t, with fixed-t feasibility complexity scaling at O((2L)^2).The searched bounds are updated until the required tolerance is reached.
  • SDP-relax method: Gaussian randomization recovers a rank-one solution from the relaxed matrix; larger K can improve the solution but increases computational complexity.The paper calls the complete procedure the SDP-relax method.
  • Greedy-iterative method: The greedy-iterative method optimizes one RIS phase at a time using a discretized search and stops below a predefined utility-improvement threshold.Its total complexity is O(N_rK), allowing a performance–complexity trade-off through K.

V. FURTHER DISCUSSION

Scheme 2 avoids self- and loop-interference by separating the two directions into successive one-way transmissions. Its outage and spectral-efficiency behavior can therefore be analyzed for both reciprocal and non-reciprocal channels, with a power boundary identifying when Scheme 1 is preferable.

  • Transmission schemes: Scheme 2 transmits in two successive time-slots, eliminating self-interference and loop-interference and reducing the system to two one-way communications.Each user transmits in a separate slot, so the maximum instantaneous SNR can be used for analysis.
  • Outage comparison: Scheme 2 always achieves lower outage probability than Scheme 1 when loop interference is non-zero.The comparison uses ρ = P/σ2_w for the transmit-power-based average SINR.
  • Asymptotic behavior: Scheme 2 has an outage decrease at log(ρ)/ρ rate for the single-element asymptotic case.This rate is stated for the user outage probability in the large-SINR regime.
  • Spectral efficiency: Scheme 2 incurs a factor 1/2 in average spectral efficiency because only one user transmits in each resource block.The expressions are obtained by multiplying the one-way spectral-efficiency expressions by 1/2 and substituting ρ = P/σ2_w.
  • Scheme selection: The transmit-power boundary at which Scheme 1 outperforms Scheme 2 is approximated for multiple-element RIS using the preceding analytical expressions.The boundary is derived using equations (25), (26), and (42).
  • Channel reciprocity: The analytical expressions for Scheme 2 remain valid for non-reciprocal channels because the users do not transmit simultaneously.The absence of simultaneous transmission removes the relevant interference terms.

B. With Phase Adjustment Errors or Uncertainties

The paper models RIS phase-adjustment errors with uniform or von Mises distributions and derives exact or approximate SINR distributions accordingly. Exact evaluation is available in a worst-case uniform setting, while more general cases may require gamma-based approximations and remain analytically difficult.

  • Error models: Phase-adjustment errors at reciprocal-channel RIS elements are modeled as random variables caused by channel-estimation or phase-discretization errors.The paper considers independent errors with uniform or von Mises distributions.
  • Uniform-error analysis: For uniformly distributed errors over (−π, π), the paper gives an SINR CDF for i.i.d. Rayleigh cascaded channels.The derivation uses the CDF of a cascade channel and a linear random-variable transformation.
  • Performance metrics: The exact outage probability is obtained by evaluating the SINR CDF at the threshold, Pout = Fγ(γth).The corresponding exact average spectral efficiency is then evaluated using the outage-analysis expressions.
  • Analytical limitation: For general error cases, deriving the SINR CDF may be difficult, so the paper suggests gamma approximations while leaving the bivariate-gamma derivation for future work.Each sum can be approximated by a gamma random variable before applying a bivariate gamma model.
  • Non-reciprocal channels: For non-reciprocal channels, uniform and von Mises error models may be mismatched but can still be used with appropriately selected parameters.The parameters are δ for the uniform model and μ and κ for the von Mises model.

VI. SIMULATION RESULTS

The simulations validate the analytical results and show how loop-interference scaling and RIS size affect outage, spectral efficiency, power savings, and the crossover between transmission schemes.

  • Analytical outage and spectral-efficiency results exactly match simulations for L = 1, confirming the accuracy of the analysis.
  • At ν = 0, outage decreases as log(P)/P and spectral efficiency increases as log(P); at ν = 1, both exhibit interference-induced floors.
  • Scheme 2 always outperforms Scheme 1 in outage probability whenever loop interference is non-zero.
  • For L ≥ 2, outage decreases as [log(P)/P]^L, while larger L improves outage but yields diminishing power savings.
  • Scheme 1 outperforms Scheme 2 at high P when loop interference is power-independent, but Scheme 2 outperforms it when interference grows with P.
  • When L increases from 2 to 16, transmit power falls by around 19 dBm, whereas increasing L from 16 to 64 saves only 12 dBm.
  • When loop-interference power is below noise power, ω < 10^-10, noise dominates and produces a power floor; for L = 16, it is around -120 dBm.

C. For Phase Adjustment Errors

Phase adjustment errors substantially affect outage performance but not the asymptotic spectral-efficiency growth rate. Smaller error ranges recover nearly the no-error performance and reduce required transmit power.

  • For Phase Adjustment Errors: As δ decreases from π to 0, outage diversity changes from log(P)/P to [log(P)/P]^L.
  • For Phase Adjustment Errors: For L = 16 and 10^-4 outage probability, required power is approximately 27 dBm at δ = π, -9 dBm at δ = π/2, and -14 dBm at δ ≤ π/4.
  • For Phase Adjustment Errors: Compared with δ = π, power savings reach 99.97% at δ = π/2 and 99.99% when δ ≤ π/4.
  • For Phase Adjustment Errors: Spectral efficiency increases as log(P) for every δ, with gains over δ = π of 22.4%, 27.8%, 29.1%, and 29.5% for δ = π/2, π/4, π/8, and 0.
  • For Phase Adjustment Errors: The analytical results for δ = π exactly match simulations, and the performance gap from the no-error case is negligible when δ ≤ π/8.
  • For Non-reciprocal Channels: For non-reciprocal channels, fairness algorithms give both users nearly equal spectral efficiency, whereas user-specific phase adjustment favors U1 and harms U2.

APPENDIX A PROOF OF THEOREM 1

The appendix proves the high-SINR asymptotic behavior of outage probability and spectral efficiency using expansions and bounds for the analytical expressions.

  • For L = 1, the proof derives a high-SINR outage approximation by retaining dominant terms in the closed-form expression.
  • For L ≥ 2, the proof establishes an outage bound and rewrites it in terms of outage probabilities to prove the theorem.
  • The spectral-efficiency proof uses Mellin-transform evaluation and residue summation for the relevant special-function expression.
  • At high SINR, hypergeometric-function terms have negligible effect for L ≥ 2, yielding spectral-efficiency expressions that grow as log(ρ).
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