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

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.Source: §I · p. 2

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.Source: §I · p. 2
• 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.Source: §I · p. 2
• 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.Source: §I · pp. 2–3

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.Source: A. Prior Work · pp. 3–4

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.Source: A. Prior Work · p. 5

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.Source: B. Contributions · pp. 5–6

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.Source: B. Contributions · p. 5

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.Source: B. Contributions · p. 5

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.Source: B. Contributions · p. 6
• 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.Source: B. Contributions · p. 6

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.Source: B. Contributions · p. 6

Organization

• The paper covers system modeling, reflecting and precoding designs, computational complexity, numerical comparisons, and conclusions.Source: C. Organization · p. 6

System model

• The system uses an RIS-assisted MIMO link in which the RIS acts as a passive relay between transmitter and receiver.Source: §II · p. 7
• The received signal combines the RIS-reflected channel H2ΦH1 with the direct channel Hd before precoding, transmission, and additive noise.Source: §II · p. 8

Problem formulation

• The design objective is to minimize SER for finite-alphabet data while respecting maximum transmit power and unit-modulus RIS phase constraints.Source: §II · pp. 8–9
• 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.Source: §II · p. 9

Alternating optimization

• The RIS phase matrix is optimized with the precoder fixed, then the precoder is updated with the reflecting pattern fixed.Source: §III · p. 9
• The alternating process repeats until the objective or solutions reach an acceptable convergence level, yielding sub-optimal designs.Source: §III · p. 9

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.Source: 1) eMSER-Refelcting Scheme · p. 10
• The eMSER-Reflecting search uses θ_n directly, so φ_n = exp(jθ_n) always satisfies the unit-modulus constraint.Source: 1) eMSER-Refelcting Scheme · p. 10

Constraint relaxation

• The method relaxes the non-convex unit-modulus constraint into a convex ℓ∞-constrained optimization problem.Source: 2) vMMSR-Reflecting Scheme · pp. 10–11
• The ℓ∞ constraint is approximated with an ℓp formulation using a gradually increased large p.Source: 2) vMMSR-Reflecting Scheme · p. 11

Barrier formulation

• A logarithmic barrier function incorporates the non-negative constraint and penalizes violations of the approximated ℓp constraint.Source: 2) vMMSR-Reflecting Scheme · p. 11

Feasibility

• The relaxation may produce infeasible unit-modulus elements, so element-wise normalization restores feasibility while retaining good performance under the stated sensitivity condition.Source: 2) vMMSR-Reflecting Scheme · p. 12

Iterative optimization

• Algorithm 2 computes a gradient search direction, projects it onto the tangent plane, and iteratively updates the reflecting solution.Source: 2) vMMSR-Reflecting Scheme · p. 13

Feasibility

• The precoding procedure maintains feasibility for all generated solutions under the transmit-power constraint.Source: B. Precoding Designs · p. 13, 1) MMSE-Precoding Scheme · p. 15

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.Source: B. Precoding Designs · pp. 13–14

MSER-Precoding

• The resulting precoding problem is solved iteratively using gradient directions, tangent-plane projection, line search, and a stopping criterion.Source: 1) MMSE-Precoding Scheme · p. 15
• In Algorithm 3, PS(f_k+1) ≤ PS(f_k), so the objective is non-increasing at every iteration.Source: 1) MMSE-Precoding Scheme · p. 15
• Because the reformulated problem is non-convex, the algorithm generates a critical point rather than guaranteeing a global optimum.Source: 1) MMSE-Precoding Scheme · p. 15

• In the high-SNR regime, minimizing SER can be formulated as maximizing the minimum Euclidean distance between noise-free received signal vectors.Source: 2) MMED-Precoding Scheme · p. 16
• MMED-Precoding maximizes the minimum quadratic distance subject to tr(Q) ≤ Pmax.Source: 2) MMED-Precoding Scheme · p. 16
• Introducing an auxiliary variable r yields an equivalent epigraph formulation for the precoding problem.Source: 2) MMED-Precoding Scheme · p. 17
• The formulation is a large-scale non-convex QCQP solved using the method from.Source: 2) MMED-Precoding Scheme · p. 17
• After optimization with the power constraint released, the solution is scaled to satisfy the power constraint and ensure feasibility.Source: 2) MMED-Precoding Scheme · p. 17

• Algorithm 4 initializes feasible reflecting and precoding variables, alternately optimizes them, and stops when the SER change meets εT.Source: C. Alternating Optimization · p. 18
• The SER objective PS(φk, fk) forms a monotone decreasing sequence during alternating optimization.Source: C. Alternating Optimization · pp. 17–18
• Because the sequence is bounded below by zero, the proposed alternating optimization algorithms are always converged.Source: C. Alternating Optimization · p. 18
• The approach gradually reduces SER but guarantees only a sub-optimal solution because the optimization problem is non-convex.Source: C. Alternating Optimization · p. 18
• The alternating-optimization solution remains feasible because the proposed reflecting and precoding schemes are always feasible.Source: C. Alternating Optimization · p. 18
• The section analyzes computational complexity, including gradient calculations, angle searches, and matrix computations for the proposed algorithms.Source: §IV · p. 18, 1) Complexity Order of eMSER-Reflecting · pp. 18–19

Precoding complexity

• MMED-Precoding does not need SNR information and generally has much lower complexity than MSER-Precoding because it requires fewer BFGS iterations.Source: 2) Complexity of MMED-Precoding Scheme · p. 19

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.Source: §V · p. 20

CPU running time and SER

• Exhaustive-search complexity increases exponentially with 2NtNs + N, restricting it to systems with small Nt, Ns, and N.Source: A. CPU Running Time Comparison and… · p. 20
• All combinations achieve almost optimal performance compared with the exhaustive-search solution, validating the proposed designs’ effectiveness.Source: A. CPU Running Time Comparison and… · p. 20
• vMSER combinations achieve performance close to eMSER combinations with much lower computation complexity, making them more suitable for practical implementation.Source: A. CPU Running Time Comparison and… · p. 21

Convergence behavior

• The average number of alternating-optimization iterations increases as N and Nt increase, reflecting more variables and constraints.Source: A. CPU Running Time Comparison and… · p. 21

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.Source: B. Comparison with Existing Algorithms · p. 23
• The proposed reflecting schemes provide favorable SER compared with random reflecting, showing that phase-shift design strongly influences system SER.Source: B. Comparison with Existing Algorithms · p. 23

Precoding schemes

• All proposed precoding schemes provide lower SER than random precoding, while MSER-Precoding performs slightly better than MMED-Precoding.Source: B. Comparison with Existing Algorithms · p. 24
• MMED-Precoding has lower complexity than MSER-Precoding and is therefore described as more appealing for realistic systems.Source: B. Comparison with Existing Algorithms · p. 24
• Eigen-Precoding is much faster but about 4-5 dB worse than MSER-Precoding and MMER-Precoding in the given setup.Source: B. Comparison with Existing Algorithms · pp. 24–25

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.Source: B. Comparison with Existing Algorithms · pp. 25–26

Precoding schemes

• The proposed precoders minimize SER directly, whereas ZF precoding targets interference nulling, producing significant SER gains for the proposed schemes.Source: B. Comparison with Existing Algorithms · p. 25

• Reflecting design becomes more important than precoding design for affecting SER as the number of reflecting elements increases.Source: C. Impact of the Number of Reflecting… · p. 27
• A larger RIS provides no obvious SER gain when its reflecting elements are not carefully designed.Source: C. Impact of the Number of Reflecting… · p. 27
• Joint reflecting and precoding design achieves optimal performance at the cost of more iterations.Source: C. Impact of the Number of Reflecting… · p. 27
• The system SER can be significantly decreased by adding RIS reflecting elements and using the proposed reflecting design.Source: C. Impact of the Number of Reflecting… · p. 27
• With imperfect CSI, proposed schemes still optimize SER effectively versus random joint design, although performance degrades and reaches a floor as SNR grows.Source: D. Superiority in Presence of CSI… · p. 28
• 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.Source: D. Superiority in Presence of CSI… · p. 28

• AF relaying achieves much lower SER than an equally sized RIS-assisted system when the relay has no noise.Source: E. Comparison with AF Relay… · p. 28
• The AF advantage arises because the relay consumes additional power to forward signals, whereas RIS provides no power gain.Source: E. Comparison with AF Relay… · pp. 28–29
• The SER gap between RIS and AF decreases as relay-noise variance increases, because AF forwarding introduces more noise power.Source: E. Comparison with AF Relay… · p. 29
• RIS outperforms AF relaying when more reflecting elements are equipped.Source: E. Comparison with AF Relay… · p. 29
• Adding 10 reflecting elements provides more than 7-8 dB SER gain and nearly matches the noise-free AF system.Source: E. Comparison with AF Relay… · p. 29
• The proposed reflecting and precoding schemes improve SER performance across scheme combinations, with reflecting schemes playing a more critical role than precoding schemes.Source: §VI · p. 29
• The RIS-assisted system remains robust with imperfect CSI and can surpass relaying performance without consuming additional power.Source: §VI · pp. 29–30