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Joint Reflecting and Precoding Designs for SER Minimization in Reconfigurable Intelligent Surfaces Assisted MIMO Systems

Jia Ye, Shuaishuai Guo, Mohamed-Slim Alouini

arXiv:1906.11466v2eess.SP

TL;DR

The paper proposes joint reflecting and precoding schemes for RIS-assisted MIMO communications. Simulations report improved SER, including advantages over relaying systems, robustness with imperfect CSI, and benefits from additional reflecting elements.

  • Problem

    Hardware complexity and system update cost remain practical issues for these communication systems.

  • Method

    The paper proposes joint reflecting and precoding schemes, including an MMED-Precoding design based on the desired squared minimum distance.

  • Results

    Simulation results show that the proposed algorithms significantly improve SER performance across different scheme combinations.

  • Takeaways & Limitations

    The RIS-assisted system with proper reflecting and precoding design can outperform relaying systems, remain robust with imperfect CSI, and benefit from more reflecting elements.

Abstract

from arXiv · show

This paper investigates the use of a reconfigurable intelligent surface (RIS) to aid point-to-point multi-data-stream multiple-input multiple-output (MIMO) wireless communications. With practical finite alphabet input, the reflecting elements at the RIS and the precoder at the transmitter are alternatively optimized to minimize the symbol error rate (MSER). In the reflecting optimization with a fixed precoder, two reflecting design methods are developed, referred as eMSER-Reflecting and vMSER-Reflecting. In the optimization of the precoding matrix with a fixed reflecting pattern, the matrix optimization is transformed to be a vector optimization problem and two methods are proposed to solve it, which are referred as MSER-Precoding and MMED-Precoding. The superiority of the proposed designs is investigated by simulations. Simulation results demonstrate that the proposed reflecting and precoding designs can offer a lower SER than existing designs with the assumption of complex Gaussian input. Moreover, we compare RIS with a full-duplex Amplify-and-Forward (AF) relay system in terms of SER to show the advantage of RIS.

I. INTRODUCTION

RIS is presented as a flexible, low-energy technology for improving wireless communication, but practical implementation remains challenging. Prior work largely optimizes objectives such as rate, efficiency, power, SINR, or secrecy rather than jointly minimizing SER for finite-alphabet RIS-aided MIMO.

  • Motivation: Hardware complexity and system update cost remain obstacles to practical implementation despite advances in Massive MIMO, mmWave communications, and ultra-dense small-cell deployments.RIS was proposed as a promising solution for meeting communication-quality demands in a green and effective way.
  • RIS technology: RIS uses reconfigurable reflecting elements to alter electromagnetic-wave phases and adapt reflecting surfaces to time-varying propagation.The surface can use components such as PIN diodes and smart controllers, while MEMS and metamaterials enable real-time reconfiguration.
  • RIS technology: RIS offers passive operation without dedicated energy for decoding, channel estimation, or transmission, while its reflected path carries the same useful information as the direct path.The RIS reflects ambient RF signals without a transmitter module or additional information of its own.
  • RIS technology: RIS can improve propagation conditions, multipath richness, coverage, received signal power, interference management, security, and energy consumption.Its lightweight, conformal geometry also supports installation on building facades, ceilings, and walls.
  • Prior work: Existing RIS research has optimized outage probability, signal power, SINR, data rate, spectral efficiency, secrecy rate, and related objectives.The reviewed literature includes designs for phase shifts, power allocation, beamforming, interference management, and security.
  • Research gap: Prior work had not jointly optimized reflecting and precoding for RIS-assisted MIMO communications using SER minimization with finite-alphabet input.Earlier studies evaluated error performance without optimizing both phase shifts and the precoder.

B. Contributions

The paper formulates finite-alphabet SER minimization for RIS-assisted point-to-point multi-stream MIMO and alternates reflecting and precoding optimization. It proposes multiple reflecting and precoding schemes, then evaluates their performance, complexity, robustness, and comparison with AF relaying.

  • Reflecting design: vMSER-Reflecting reduces reflecting-design complexity by transforming a non-convex constraint into a convex one and optimizing the reflecting vector with vector gradient descent.The method jointly solves for the optimal reflecting vector.
  • Problem formulation: The paper formulates joint SER minimization for a RIS-enhanced point-to-point multi-data-stream MIMO system with a multi-antenna transmitter and receiver.Because precoding and reflecting are coupled in the objective but physically separate, the paper uses an alternating strategy.
  • Reflecting design: eMSER-Reflecting optimizes reflecting angles element by element through coordinate descent, but has high computational complexity.The reflecting optimization is reformulated to minimize SER with the precoder fixed.
  • Precoding design: MSER-Precoding transforms matrix optimization into vector optimization and solves it with projected gradient descent.The reformulation uses new forms for the channel matrix, precoder, and transmitted symbol vectors.
  • Precoding design: At high SNR, SER minimization is equivalent to maximizing minimum Euclidean distance, motivating the QCQP-based MMED-Precoding method.MMED-Precoding achieves comparable performance with substantially reduced computational complexity.
  • Evaluation: Simulations show that the proposed designs improve SER over SNR-maximizing designs for finite-alphabet input and can achieve the same or better performance than full-duplex AF relaying.The evaluation also considers exhaustive search, computational complexity, CPU running time, imperfect CSI, and different numbers of reflecting elements.

C. Organization

The paper describes an RIS-assisted MIMO system, formulates SER minimization under power and unit-modulus phase constraints, and organizes the study around alternating reflecting and precoding designs, complexity analysis, and numerical comparisons.

  • Organization: The paper covers system modeling, reflecting and precoding designs, computational complexity, numerical comparisons, and conclusions.
  • System model: The system uses an RIS-assisted MIMO link in which the RIS acts as a passive relay between transmitter and receiver.
  • System model: The received signal combines the RIS-reflected channel H2ΦH1 with the direct channel Hd before precoding, transmission, and additive noise.
  • Problem formulation: The design objective is to minimize SER for finite-alphabet data while respecting maximum transmit power and unit-modulus RIS phase constraints.
  • Problem formulation: The joint problem is non-convex because the reflecting pattern and precoder are coupled, so the paper alternately optimizes them to obtain sub-optimal solutions.

III. JOINT REFLECTING AND PRECODING DESIGN

The joint design decouples the coupled RIS phase and precoder variables through alternating optimization. Reflecting phases are updated with coordinate-based procedures while maintaining unit-modulus feasibility.

  • Alternating optimization: The RIS phase matrix is optimized with the precoder fixed, then the precoder is updated with the reflecting pattern fixed.
  • Alternating optimization: The alternating process repeats until the objective or solutions reach an acceptable convergence level, yielding sub-optimal designs.
  • Reflecting designs: For reflecting optimization, each phase is updated while the remaining phase shifts are fixed, and the coordinate process is repeated until a stopping criterion is met.
  • Reflecting designs: The eMSER-Reflecting search uses θ_n directly, so φ_n = exp(jθ_n) always satisfies the unit-modulus constraint.

2) vMMSR-Reflecting Scheme:

The vMSER-Reflecting scheme relaxes the unit-modulus reflecting constraint into a convex ℓ∞-constrained problem, approximates it with increasing ℓp penalties and a barrier function, and iteratively updates projected gradient directions.

  • Constraint relaxation: The method relaxes the non-convex unit-modulus constraint into a convex ℓ∞-constrained optimization problem.
  • Constraint relaxation: The ℓ∞ constraint is approximated with an ℓp formulation using a gradually increased large p.
  • Barrier formulation: A logarithmic barrier function incorporates the non-negative constraint and penalizes violations of the approximated ℓp constraint.
  • Feasibility: The relaxation may produce infeasible unit-modulus elements, so element-wise normalization restores feasibility while retaining good performance under the stated sensitivity condition.
  • Iterative optimization: Algorithm 2 computes a gradient search direction, projects it onto the tangent plane, and iteratively updates the reflecting solution.

B. Precoding Designs

The precoding design transforms matrix optimization into vector optimization and solves the resulting problems with iterative gradient-based methods. The MSER-Precoding algorithm monotonically decreases its objective while producing feasible critical-point solutions.

  • Feasibility: The precoding procedure maintains feasibility for all generated solutions under the transmit-power constraint.
  • Vector reformulation: The received-signal model is reformulated by rebuilding the channel, precoder, and data stream so the precoding matrix can be represented as a vector.
  • MSER-Precoding: The resulting precoding problem is solved iteratively using gradient directions, tangent-plane projection, line search, and a stopping criterion.
  • MSER-Precoding: In Algorithm 3, PS(f_k+1) ≤ PS(f_k), so the objective is non-increasing at every iteration.
  • MSER-Precoding: Because the reformulated problem is non-convex, the algorithm generates a critical point rather than guaranteeing a global optimum.

2) MMED-Precoding Scheme:

MMED-Precoding reformulates the SER-related precoder optimization as a maximum-minimum Euclidean-distance problem under a transmit-power constraint. The resulting non-convex QCQP is solved using an existing method, then scaled to satisfy feasibility.

  • In the high-SNR regime, minimizing SER can be formulated as maximizing the minimum Euclidean distance between noise-free received signal vectors.
  • MMED-Precoding maximizes the minimum quadratic distance subject to tr(Q) ≤ Pmax.
  • Introducing an auxiliary variable r yields an equivalent epigraph formulation for the precoding problem.
  • The formulation is a large-scale non-convex QCQP solved using the method from.
  • After optimization with the power constraint released, the solution is scaled to satisfy the power constraint and ensure feasibility.

C. Alternating Optimization

Alternating optimization alternates reflecting-element and precoder updates from a feasible initialization until the SER objective stabilizes. Its objective sequence is monotone decreasing and bounded below, establishing convergence, but non-convexity limits the result to a sub-optimal solution.

  • Algorithm 4 initializes feasible reflecting and precoding variables, alternately optimizes them, and stops when the SER change meets εT.
  • The SER objective PS(φk, fk) forms a monotone decreasing sequence during alternating optimization.
  • Because the sequence is bounded below by zero, the proposed alternating optimization algorithms are always converged.
  • The approach gradually reduces SER but guarantees only a sub-optimal solution because the optimization problem is non-convex.
  • The alternating-optimization solution remains feasible because the proposed reflecting and precoding schemes are always feasible.
  • The section analyzes computational complexity, including gradient calculations, angle searches, and matrix computations for the proposed algorithms.

2) Complexity of vMSER-Reflecting:

The complexity study compares reflecting–precoding combinations, runtime, and convergence behavior against exhaustive search and SNR-oriented baselines. vMSER combinations approach eMSER performance with lower complexity, while exhaustive search scales exponentially.

  • Precoding complexity: MMED-Precoding does not need SNR information and generally has much lower complexity than MSER-Precoding because it requires fewer BFGS iterations.
  • Evaluation setup: The proposed alternating-optimization combinations are evaluated against exhaustive search, existing SNR-oriented designs, varying surface sizes, CSI quality, and full-duplex AF relaying.
  • CPU running time and SER: Exhaustive-search complexity increases exponentially with 2NtNs + N, restricting it to systems with small Nt, Ns, and N.
  • CPU running time and SER: All combinations achieve almost optimal performance compared with the exhaustive-search solution, validating the proposed designs’ effectiveness.
  • CPU running time and SER: vMSER combinations achieve performance close to eMSER combinations with much lower computation complexity, making them more suitable for practical implementation.
  • Convergence behavior: The average number of alternating-optimization iterations increases as N and Nt increase, reflecting more variables and constraints.

B. Comparison with Existing Algorithms

The proposed reflecting and precoding designs target SER directly for finite-alphabet inputs, unlike baselines that optimize approximations, received power, SNR, or interference suppression. Simulations report better SER, with complexity–performance trade-offs among the proposed methods.

  • Reflecting schemes: In a high-SNR comparison, SDP-Reflecting is about 2−3 dB worse than eMSER-Reflecting and vMSER-Reflecting.The reported gap is attributed to the exponential Q-function approximation being accurate mainly at low SNR.
  • Reflecting schemes: The proposed reflecting schemes provide favorable SER compared with random reflecting, showing that phase-shift design strongly influences system SER.
  • Precoding schemes: All proposed precoding schemes provide lower SER than random precoding, while MSER-Precoding performs slightly better than MMED-Precoding.
  • Precoding schemes: MMED-Precoding has lower complexity than MSER-Precoding and is therefore described as more appealing for realistic systems.
  • Precoding schemes: Eigen-Precoding is much faster but about 4-5 dB worse than MSER-Precoding and MMER-Precoding in the given setup.
  • Finite-alphabet inputs: The Gaussian-input optimization assumption should be reconsidered for finite-alphabet systems because continuous-input designs cannot replace optimization for limited discrete data.
  • Precoding schemes: The proposed precoders minimize SER directly, whereas ZF precoding targets interference nulling, producing significant SER gains for the proposed schemes.

C. Impact of the Number of Reflecting Elements

As the number of RIS reflecting elements increases, careful phase-shift design becomes increasingly important for reducing SER. Joint reflecting and precoding optimization provides the strongest reported improvement, while additional elements alone do not guarantee gains.

  • Reflecting design becomes more important than precoding design for affecting SER as the number of reflecting elements increases.
  • A larger RIS provides no obvious SER gain when its reflecting elements are not carefully designed.
  • Joint reflecting and precoding design achieves optimal performance at the cost of more iterations.
  • The system SER can be significantly decreased by adding RIS reflecting elements and using the proposed reflecting design.
  • With imperfect CSI, proposed schemes still optimize SER effectively versus random joint design, although performance degrades and reaches a floor as SNR grows.
  • For σ2_e = 0.3, vMSER-MMED achieves about 10-15 dB SER gain over random design with perfect CSI and about 6-7 dB over SDP-Eigen with perfect CSI.

E. Comparison with AF Relay Cooperating System

The comparison with full-duplex AF relaying shows that AF can achieve lower SER under noise-free relay conditions, but RIS performance approaches or surpasses AF as relay noise increases or more reflecting elements are added.

  • AF relaying achieves much lower SER than an equally sized RIS-assisted system when the relay has no noise.
  • The AF advantage arises because the relay consumes additional power to forward signals, whereas RIS provides no power gain.
  • The SER gap between RIS and AF decreases as relay-noise variance increases, because AF forwarding introduces more noise power.
  • RIS outperforms AF relaying when more reflecting elements are equipped.
  • Adding 10 reflecting elements provides more than 7-8 dB SER gain and nearly matches the noise-free AF system.
  • The proposed reflecting and precoding schemes improve SER performance across scheme combinations, with reflecting schemes playing a more critical role than precoding schemes.
  • The RIS-assisted system remains robust with imperfect CSI and can surpass relaying performance without consuming additional power.
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