PaperlayerView exact revision on arXiv

Source-linked AI summary

Passive Beamforming and Information Transfer Design for Reconfigurable Intelligent Surfaces Aided Multiuser MIMO Systems

Wenjing Yan, Xiaojun Yuan, Zhen-Qing He, Xiaoyan Kuai

arXiv:1912.10209v2cs.ITeess.SP

TL;DR

The paper studies how RIS-aided multiuser MIMO systems can simultaneously support primary communication and passive RIS information transfer. It develops stochastic and simplified beamforming methods, a turbo message passing receiver, and extensions to multi-RIS systems, with simulations demonstrating the proposed designs' performance.

  • Problem

    Multiuser MIMO PBIT requires new beamforming and receiver designs because multiplexing invalidates single-SNR and rank-one approaches, while RIS randomness creates stochastic and bilinear estimation problems.

  • Method

    The paper formulates beamforming as a two-step stochastic program with SAA and deterministic alternating-optimization algorithms, and uses iterative TMP modules to estimate user signals and RIS states.

  • Results

    Simulations demonstrate the advantages of the proposed passive beamforming and receiver designs, with TMP tightly approaching lower bounds and optimized designs improving BER by about 4–5 dB in reported settings.

  • Takeaways & Limitations

    Single-RIS designs extend to multi-RIS systems with minor modifications to the prior distribution of RIS-element on-off states.

Abstract

from arXiv · show

This paper investigates the passive beamforming and information transfer (PBIT) technique for the multiuser multiple-input multiple-output (MIMO) systems with the aid of a reconfigurable intelligent surface (RIS), where the RIS enhances the primary communication via passive beamforming and at the same time delivers additional information by the spatial modulation (which adjusts the on-off states of the reflecting elements). For the passive beamforming design, we propose to maximize the sum channel capacity of the RIS-aided multiuser MIMO channel and formulate the problem as a two-step stochastic program. A sample average approximation (SAA) based iterative algorithm is developed for the efficient passive beamforming design of the considered scheme. To strike a balance between complexity and performance, we then propose a simplified beamforming algorithm by approximating the stochastic program as a deterministic alternating optimization problem. For the receiver design, the signal detection at the receiver is a bilinear estimation problem since the RIS information is multiplicatively modulated onto the reflected signals of the reflecting elements. To solve this bilinear estimation problem, we develop a turbo message passing (TMP) algorithm in which the factor graph associated with the problem is divided into two modules: one for the estimation of the user signals and the other for the estimation of the RIS's on-off states. The two modules are executed iteratively to yield a near-optimal low-complexity solution. Furthermore, we extend the design of the multiuser MIMO PBIT scheme from single-RIS to multi-RIS, by leveraging the similarity between the single-RIS and multi-RIS system models. Extensive simulation results are provided to demonstrate the advantages of our passive beamforming and receiver designs.

I. INTRODUCTION

RISs can enhance wireless communication through reconfigurable passive reflection while also conveying information through reflecting-element states. This paper addresses beamforming and receiver design for multiuser MIMO PBIT systems, including single- and multi-RIS settings.

  • RIS background: RISs use low-cost, nearly passive reflecting elements to adjust incident-wave phases and improve received signal energy, coverage, and interference conditions.They can be installed on building facades, ceilings, or walls and controlled adaptively.
  • PBIT motivation: PBIT simultaneously enhances primary communication through passive beamforming and delivers additional RIS information through spatial modulation of reflecting-element indices.Potential RIS data sources include environmental sensing, control-link acknowledgements, and CSI uploaded for resource allocation.
  • Beamforming design: In multiuser MIMO, the paper maximizes conditional mutual information as an approximation to sum channel capacity and formulates passive beamforming as a two-step stochastic program.The approximation uses the observation that RIS information rates are typically lower than user information rates.
  • Receiver design: The receiver must jointly recover user signals and RIS states from a bilinear model, while multiuser multiplexing makes the received signal matrix non-rank-one.These properties make rank-1 matrix-factorization techniques developed for earlier PBIT systems inapplicable.
  • Receiver design: A turbo message passing algorithm separates user-signal and RIS-state estimation into two iteratively executed factor-graph modules.The design targets near-optimal, low-complexity inference where PBiGAMP is computationally expensive and performs poorly for the problem's low-rank measurement matrices.
  • Multi-RIS extension: The paper extends passive beamforming and receiver designs from single-RIS to multi-RIS systems by exploiting model similarity and modifying the on-off-state prior distribution.Multiple RISs cooperatively enhance user–BS communication and RIS–BS information transfer.

B. Problem Description

The beamforming design maximizes conditional mutual information for the RIS-aided PBIT uplink, using stochastic optimization and sample-average approximation under unit-modulus phase constraints.

  • Problem formulation: The receiver must recover both user information X and RIS information s under perfect CSI, while phase shifts are optimized for recovery performance.The design criterion is the sum channel capacity, expressed through mutual information.
  • Problem formulation: The passive beamforming problem is converted from maximizing I(x, s; y) to maximizing conditional mutual information I(x; y|s).The conversion uses the chain rule and the lower data rate of RIS sensors or IoT devices.
  • Stochastic beamforming design: The conditional-mutual-information design is formulated as a two-stage stochastic program over the RIS phase matrix Θ and auxiliary variables.The stochasticity arises from the RIS on-off state vector s.
  • Stochastic beamforming design: SAA replaces enumeration of all 2^N RIS-state scenarios with ℓs independently generated replications, whose solution approaches the original stochastic-program solution.The approximation converges exponentially fast with increasing ℓs, making a moderate sample size sufficient for relatively accurate design.
  • Stochastic beamforming design: The resulting alternating optimization first updates Θ for fixed per-sample objectives and then solves those objectives over Φ and Σ for fixed Θ.The auxiliary matrix Φ is a combining matrix, while Σ is positive semidefinite.
  • SDP relaxation: The non-convex QCQP is relaxed to an SDP and converted to a feasible phase vector through Gaussian randomization.The SDP can be solved with convex optimization software, but its relaxed solution is not guaranteed to be rank one.

B. Solve {Q(Θ, s[i])} for Given Θ

For fixed Θ, the algorithm optimizes the auxiliary combining and covariance variables for each sampled RIS state.

  • Per-sample optimization: For each sampled state s[i], the algorithm minimizes J[i](Φ, Σ) over the combining matrix Φ and positive semidefinite matrix Σ.The conditional covariance matrices Cxy|s[i] and Cyy|s[i] enter the per-sample objective.
  • Per-sample optimization: Setting the derivative with respect to Φ to zero yields the optimal Φ for a given RIS state.The optimization is performed separately for each sampled state.
  • Per-sample optimization: Setting the derivative with respect to Σ to zero yields the optimal Σ for a given RIS state.Σ is constrained to be positive semidefinite in the formulation.

C. Overall Iterative Algorithm

The simplified beamforming method exchanges minimization and expectation to obtain a more computationally friendly deterministic objective, then solves it by alternating optimization.

  • Simplified beamforming method: The simplified method exchanges minimization over (Φ, Σ) with expectation over s to modify the stochastic beamforming objective.This produces the deterministic problem in (26).
  • Simplified beamforming method: The modified objective can be evaluated explicitly, making it more computationally friendly than the sample-average formulation.The resulting problem remains non-convex and is solved suboptimally.
  • Alternating optimization: For fixed (Φ, Σ), the method reduces the update to an optimization problem over the RIS phase matrix Θ.The phase design is subsequently reformulated as a QCQP and approximated through an SDP.
  • Alternating optimization: Algorithm 2 iteratively computes (Φ, Σ), solves for Δ, applies Gaussian randomization, and updates θ until the objective reduction or iteration limit criterion is met.The algorithm is presented as the simplified beamforming procedure.

C. Overall Iterative Algorithm

The receiver jointly estimates user signals and RIS states in a bilinear model, motivating turbo message passing that alternates between two detector modules.

  • Receiver formulation: The BS receiver seeks to recover both user information X and RIS on-off states s using a maximum a posteriori formulation.The joint recovery is difficult because X and s appear bilinearly in the observation model.
  • Receiver formulation: Although PBiGAMP applies to the rewritten bilinear model, its i.i.d.-Gaussian measurement assumption is violated by the rank-1 matrices Hn.The PBiGAMP complexity is O(MNKT^2), which is unaffordable for large T.
  • Turbo message passing: Turbo message passing divides the factor graph into an X-detector and an s-detector to avoid the two PBiGAMP issues.The modules exchange messages iteratively until convergence or a maximum iteration count.
  • Turbo message passing: The factor graph representation models the joint probability for M = N = K = 3 and T = 2, while the turbo graph depicts the two-module message-passing structure.The figures use variable nodes and factor nodes to represent the factorized posterior and its decomposition.
  • Turbo message passing: The X-detector estimates user signals from Y and s-detector messages, while the s-detector estimates RIS states from Y and X-detector messages.Each module outputs means and variances that are passed to the other module.

B. Design of X-Detector

The X-detector converts the observation into a linear estimation problem by treating residual user-signal interference as equivalent noise, then recovers X with GGAMP-SBL.

  • Linear model: The X-detector models Y as a known linear transform of X plus equivalent noise from residual s and additive noise.The transform uses the mean of s, while the residual contribution is assumed independent and circularly symmetric complex Gaussian.
  • GGAMP-SBL estimation: GGAMP-SBL recovers X from Y using an arbitrary measurement matrix and Gaussian priors for each x_kt.The algorithm updates marginal posterior estimates through generalized approximate message passing within expectation-maximization iterations.
  • GGAMP-SBL estimation: The E-step estimates marginal posteriors of x_kt, while the M-step updates their prior variances during each EM iteration.Damping is applied to selected updates to support GAMP convergence, and tolerance parameters terminate the inner and outer iterations.
  • Outputs: The algorithm outputs the marginal posterior means of X from the final iteration.Its inputs include Y, the effective channel matrix, equivalent-noise variances, and Gaussian prior parameters.

C. Design of s-Detector

The s-detector similarly isolates RIS-state estimation by treating residual uncertainty in X as equivalent noise and applying GGAMP-SBL to the resulting model.

  • Linear model: The s-detector represents Y as a linear transformation of s plus equivalent noise caused by residual X and additive noise.The transformation is constructed from the estimated user-signal mean and the channel matrices H_n.
  • Linear model: The effective measurement matrix A is formed by vectorizing the channel-weighted mean contributions from the reflecting elements.A is assembled from vec(A_n), where A_n = H_n times the estimated mean X.
  • Noise modeling: The s-detector assumes independent circularly symmetric complex Gaussian equivalent-noise elements and uses their variances in detection.The variance expression is derived separately for the s-detector model.
  • GGAMP-SBL detection: GGAMP-SBL detects s after replacing the X-detector’s measurement matrix, likelihood, and prior with their s-detector counterparts.The prior changes from Gaussian parameters for x_kt to Gaussian parameters for the RIS states s_n.

D. Algorithm Summary

The TMP receiver alternates between GGAMP-SBL estimation of user signals X and RIS states s, passing updated means and variances between the two detectors until termination.

  • Turbo iteration: The two detector stages exchange posterior means and variances across turbo iterations.The algorithm reuses the previous iteration’s estimates to initialize each subsequent detector.
  • X-detector stage: TMP first updates the effective channel and equivalent-noise variances before initializing and estimating X with GGAMP-SBL.The X estimate is then mapped to the nearest constellation point.
  • s-detector stage: TMP next updates the effective matrix and noise variances, initializes s, and estimates RIS states with GGAMP-SBL.A hard decision is made on each estimated s_n after the s-detector stage.
  • Termination and output: TMP terminates using a tolerance condition or a maximum turbo-iteration count and outputs the final marginal posterior means.The algorithm summary identifies ε_td as the turbo-iteration tolerance parameter and τ_max as the iteration limit.
  • Complexity: The TMP computational complexity is dominated by steps 2 and 8 and scales as O(MNKT).These steps update the equivalent-noise variances for the X- and s-detectors, respectively.

VI. EXTENSION ON MULTI-RIS

The multi-RIS extension uses multiple independently controlled RIS modules whose phase shifts are jointly designed, while retaining the single-RIS system-model form and adapting state distributions and detector parameters.

  • System model: The multi-RIS system contains L RIS modules, each with its own controller and sensors, while jointly designing all reflecting-element phase shifts.The joint phase-shift design targets user-BS communication enhancement.
  • System model: Each RIS has N_l reflecting elements, a diagonal state matrix S_l carrying sensor information, and a diagonal phase-shift matrix Θ_l.The state vector contains the on-off states of the passive elements in RIS_l.
  • System model: The multi-RIS received-signal model retains the same expression as the single-RIS model, with concatenated user-RIS and RIS-BS channels.X and W retain their single-RIS definitions in the resulting observation model.
  • Design modifications: The main difference from the single-RIS case is the probability distribution of the combined RIS-state vector s.This distribution reflects the states across all RIS modules.
  • Design modifications: The SAA beamforming, simplified beamforming, and TMP receiver designs are adapted by using the multi-RIS state distribution and corresponding parameter changes.The resulting modifications produce the multi-RIS beamforming and receiver designs.

VII. NUMERICAL RESULTS

The simulations model direct links with Rayleigh fading and RIS reflect links with Rician fading, using fixed array dimensions and randomized propagation parameters.

  • Channel generation: The direct user-BS link H0 is modeled with independently distributed Rayleigh small-scale fading.
  • Channel generation: RIS reflect links use Rician fading to capture both line-of-sight and non-line-of-sight effects.The RIS is assumed to provide line-of-sight links from users to the RIS and from the RIS to the base station.
  • Simulation assumptions: Rician factors are set to κ1 = 3 dB and κ2 = 10 dB, while angles and phases are randomly drawn from uniform distributions.
  • Simulation assumptions: The simulations use 16 × N RIS arrays, 16 × N_l arrays for each multi-RIS, and an 8 × M base-station array.
  • Simulation assumptions: The large-scale fading factors are independently sampled with f0,k from [−10, 0] dB and f1,k from [−13, 0] dB.

B. Simulations for Passive Beamforming Design

Simulations evaluate passive beamforming and detector designs in single- and multi-RIS PBIT systems, showing rate–complexity tradeoffs and BER improvements from optimization.

  • Simulation setup: The experiments average results over 100 random realizations for rate evaluation and 500 random realizations for detector evaluation.The detector simulations use QPSK-modulated user symbols and set N1 = ... = NL = 32 in multi-RIS cases.
  • Passive beamforming: 0.5 to 2 bits: SAA beamforming exceeds simplified beamforming for N = 32 to 128 at iteration number 10^4.The simplified algorithm converges near iteration 200 and is at least two orders of magnitude faster.
  • Passive beamforming: 2 to 6 bits: optimizing θ increases the single-RIS achievable-rate gain as N grows from 32 to 256.The direct-link contribution decreases from 5 bit to 1 bit over the same range.
  • Passive beamforming: 2 to 6 bits: optimizing θ increases the multi-RIS rate gain as L grows from 1 to 8 under both sparsity settings.Random sparsity outperforms uniform sparsity by 2 bits throughout the considered range of L.
  • Detector design: About 4 dB: optimized θ improves the BER of X across the considered SNR range in the single-RIS system.TMP significantly outperforms PBiGAMP and tightly approaches the lower bounds, while θ optimization has little impact on detecting s.
  • Detector design: About 4 dB and 5 dB: optimization improves performance at average BER = 10^-4 under uniform and random sparsity, respectively.TMP tightly approaches the lower bounds for both multi-RIS sparsity settings.

APPENDIX A

The appendix derives a conditional-mutual-information expression for the PBIT channel by introducing an auxiliary variable and optimizing over Gaussian conditional distributions.

  • Mutual-information derivation: The derivation starts from the Blahut–Arimoto representation of I(x; y|s), with expectations over the joint distribution of y, x, and s.
  • Channel representation: An auxiliary variable z = (G2ΘSG1 + H0)x separates the effective noiseless signal from additive white Gaussian noise.
  • Input model: The appendix adopts Gaussian x because I(x; y|s) is maximized under Gaussian input, with diagonal covariance entries given by the users’ powers pk.
  • Auxiliary distribution: The optimal auxiliary conditional distribution is Gaussian, parameterized by coefficient matrix Φ, conditional mean Φy, and conditional variance Σ.
  • Final expression: Substituting the channel model, the distribution of s, and the Gaussian auxiliary distribution yields the target expression used in the paper’s optimization.
  • Bilinear representation: The appendix also writes each received element as a bilinear term (s̃)^T B_m x_t plus noise and derives its variance using independence assumptions.
Loading 1912.10209v2…