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Asymptotic Max-Min SINR Analysis of Reconfigurable Intelligent Surface Assisted MISO Systems

Qurrat-Ul-Ain Nadeem, Abla Kammoun, Anas Chaaban, Merouane Debbah, Mohamed-Slim Alouini

arXiv:1903.08127v3cs.IT

TL;DR

The paper asks how RIS phases and linear precoding can maximize the minimum SINR in a multi-user MISO system with a BS-to-RIS LoS channel. It analyzes rank-one and full-rank cases, deriving closed-form or deterministic OLP characterizations and using them for RIS phase design. Rank-one operation becomes ineffective as the number of users grows, while the full-rank analysis enables phase optimization; the rank-one case is therefore limited for serving more than one user.

  • Problem

    The paper addresses the previously unstudied problem of RIS parameter design for maximizing minimum SINR in multi-user MISO systems, where obtaining global CSI creates prohibitive signal-exchange overhead.

  • Method

    The paper analyzes rank-one and full-rank BS-to-RIS LoS channels, characterizes the OLP, derives deterministic approximations using RMT, and optimizes RIS phases using projected gradient ascent.

  • Results

    For rank-one LoS, OLP reduces to MRT and the minimum SINR is bounded by a quantity that goes to zero with K; full-rank analysis supports RIS phase optimization.

  • Takeaways & Limitations

    The rank-one BS-to-RIS scenario is limited in efficiently serving more than one user, whereas the proposed full-rank framework enables RIS design using channel large-scale statistics.

  • Takeaways & Limitations

    The rank-one BS-to-RIS LoS scenario is limited in efficiently serving more than one user.

Abstract

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This work focuses on the downlink of a single-cell multi-user system in which a base station (BS) equipped with $M$ antennas communicates with $K$ single-antenna users through a reconfigurable intelligent surface (RIS) installed in the line-of-sight (LoS) of the BS. RIS is envisioned to offer unprecedented spectral efficiency gains by utilizing $N$ passive reflecting elements that induce phase shifts on the impinging electromagnetic waves to smartly reconfigure the signal propagation environment. We study the minimum signal-to-interference-plus-noise ratio (SINR) achieved by the optimal linear precoder (OLP), that maximizes the minimum SINR subject to a given power constraint for any given RIS phase matrix, for the cases where the LoS channel matrix between the BS and the RIS is of rank-one and of full-rank. In the former scenario, the minimum SINR achieved by the RIS-assisted link is bounded by a quantity that goes to zero with $K$. For the high-rank scenario, we develop accurate deterministic approximations for the parameters of the asymptotically OLP, which are then utilized to optimize the RIS phase matrix. Simulation results show that RISs can outperform half-duplex relays with a small number of passive reflecting elements while large RISs are needed to outperform full-duplex relays.

I. INTRODUCTION

RISs are passive, reconfigurable surfaces proposed to improve propagation conditions and spectral efficiency with low energy use. This paper studies RIS-assisted multi-user MISO design under realistic channel rank and correlation conditions, focusing on max-min SINR and reduced CSI requirements.

  • RIS motivation: RISs address hardware-cost, power-consumption, poor-scattering, and blockage challenges associated with massive MIMO and mmWave communication.The introduction links these challenges to RF-chain costs, poor propagation environments, and high path and penetration losses.
  • RIS motivation: RISs use low-cost reflecting elements whose independently controlled phase shifts reconfigure propagation while consuming very low energy.Their passive operation avoids active transmit modules such as power amplifiers.
  • Related technologies: Unlike active relays, RISs reflect signals without power amplifiers or active retransmission, and they can operate full-duplex without self-interference.LISs instead use electromagnetically active surfaces capable of transmitting.
  • Related work: Prior RIS designs optimize objectives such as received power, energy, spectral efficiency, outage, or rates, but generally require global CSI for all users and the BS.The paper identifies the resulting signal-exchange overhead as prohibitively high.
  • Paper contributions: This paper studies max-min SINR RIS design for multi-user MISO with a BS-to-RIS LoS channel and correlated Rayleigh RIS-to-user channels.It accounts for both channel rank and correlation structure, extending prior optimization criteria considered in RIS parameter design.
  • Paper contributions: For rank-one LoS, OLP reduces to MRT and the minimum SINR is bounded by a quantity tending to zero with K; for full-rank channels, RMT approximations support phase optimization.The proposed projected-gradient algorithm requires only channel large-scale statistics rather than global CSI.

II. SYSTEM MODEL

The paper models a RIS-assisted downlink MISO system with passive phase-adjusting elements, quasi-static fading, and a blocked direct BS-user link. It highlights substantial CSI, training, synchronization, and real-time overhead challenges, motivating phase optimization from slowly varying statistics.

  • System configuration: The system comprises an M-antenna BS, K single-antenna users, and an RIS with N passive reflecting elements.The RIS is installed near a high-rise building to assist communication from the BS.
  • Channel assumptions: The model assumes quasi-static flat-fading channels, normalized noise variance, and blocked direct BS-user links.The blocked-link assumption is associated with indoor and mmWave or sub-mmWave scenarios.
  • RIS and channel model: The RIS response is modeled by a diagonal phase-shift matrix with fixed amplitude reflection coefficient α and element phases θ_n.Each phase satisfies θ_n ∈ [0, 2π], while α ∈ (0, 1].
  • CSI acquisition: Obtaining CSI is a critical challenge because nearly passive RIS elements lack receive RF chains for channel estimation.Element-by-element training creates large overhead when N is large.
  • Practical overhead: Codebook-based beam training has codebook size on the order of N, creating substantial complexity and real-time overhead.Updating RIS phases with fast fading and synchronizing the BS and RIS can also increase latency.
  • Statistical optimization: For large M, N, and K, the optimized RIS phase matrix can use users’ spatial correlation matrices and other slowly varying large-scale statistics.The BS shares the optimized phase matrix with the RIS controller after several coherence intervals, reducing signal exchange overhead and synchronization delays.
  • Scope boundary: The precoder is computed assuming perfect CSI at the BS, so the reported results serve as performance upper bounds when CSI errors are present.Perfect CSI is especially challenging in RIS-assisted systems because many links are involved.

B. Optimal Linear Precoder

For a fixed RIS phase matrix, the optimal linear precoder solves a max-min SINR problem under a power constraint. The analysis shows that rank-one BS-to-RIS LoS channels fundamentally limit multi-user performance, whereas additional rank can provide multiple degrees of freedom.

  • Max-min OLP: For any RIS phase matrix, the OLP jointly selects precoding vectors and user powers to maximize the minimum SINR under the BS power constraint.The OLP and optimal powers are characterized through fixed-point equations.
  • Rank-one channel: With a rank-one BS-to-RIS LoS channel, the OLP is maximum-ratio transmission and the minimum SINR has a closed-form expression.The RIS-assisted channel then offers only one degree of freedom.
  • Rank-one channel: As K increases, the rank-one RIS-assisted link’s minimum SINR is upper bounded by a quantity that goes to zero, regardless of M and N.The pinhole first communication leg limits the channel degrees of freedom and the number of efficiently served users.
  • High-rank alternatives: Supporting simultaneous transmission of multiple users requires sufficient rank in H1, with rank(H1) ≥ K; multiple RISs can introduce such rank.With L RISs having orthogonal rank-one BS channels, the model has up to L degrees of freedom and can serve up to L users.
  • Full-rank channel: When H1 is full-rank, the paper applies random matrix theory to analyze the OLP and derives asymptotic deterministic quantities.The broader multi-RIS extension is left outside the work because it makes the optimization problems harder.

B. Optimization of Φ for a Single-User Setting

The single-user RIS phase-design problem is formulated under unit-modulus phase constraints. The paper derives an optimal phase alignment for the rank-one setting and motivates higher-rank designs for multi-user operation.

  • Single-user formulation: For a single user, RIS phase optimization maximizes the minimum SINR subject to unit-modulus constraints on every RIS element.The formulation assumes infinitely resolvable phase shifters.
  • Rank-one design: The rank-one phase-design problem is transformed into maximizing the magnitude of a weighted sum of RIS element contributions.The transformed variables retain unit magnitude.
  • Rank-one design: The optimal solution aligns each RIS phase with the corresponding effective channel phase.The maximum is attained when arg(¯g_n) = arg(v_n) for every element.
  • High-rank extension: For multi-user operation, the paper states that rank(H1) must exceed K and discusses multiple RISs, scattering, spacing, or spherical-wave propagation as ways to increase rank.The paper’s high-rank analysis assumes a full-rank LoS matrix and uses random matrix theory.
  • High-rank extension: The high-rank analysis derives deterministic approximations for OLP SINR, fixed-point parameters, and allocated powers to design RIS parameters.The considered common-correlation-matrix assumption simplifies the analysis; differing user correlation matrices are left for future work.

A. Large System Analysis

The large-system analysis replaces fast-fading-dependent OLP quantities with deterministic equivalents based on large-scale channel statistics. These equivalents remain accurate at moderate dimensions and substantially reduce computation and update frequency.

  • Computational motivation: The original fixed-point computation is demanding because matrix inversions must be repeatedly recomputed for large M and K.The quantities depend directly on channel vectors that vary with small-scale fading and must otherwise be updated every realization.
  • Implementation consequence: Using deterministic equivalents substantially reduces computational complexity because fast-fading-rate solutions of the OLP fixed-point and power equations are no longer required.The reduction also enables RIS phase optimization using large-scale channel information.
  • Deterministic equivalents: Random matrix theory is used to show that OLP parameters and the minimum SINR approach explicit deterministic quantities in the large-system regime.The SINR deterministic equivalent is specified through a unique positive fixed-point solution.
  • Deterministic equivalents: The deterministic equivalents are tight even for moderate system dimensions.This supports using the asymptotic quantities beyond only the limiting regime.
  • Parameter structure: The deterministic user parameters require only channel attenuation coefficients β_ks and P_max, while asymptotic power allocation favors users with weaker channel conditions.The parameter q_k acts as a downlink user-priority parameter.
  • Asymptotic OLP: The asymptotically optimal precoding vectors and powers approximate the corresponding OLP quantities.The paper provides deterministic equivalents for both q_k-related parameters and optimal transmit powers.
  • Implementation consequence: The deterministic quantities depend only on large-scale statistics, including slowly varying spatial correlations and channel attenuation coefficients.They need not be recomputed at every channel realization.

B. Optimization of Φ

The RIS phase matrix is optimized through projected gradient ascent on a deterministic equivalent of the minimum SINR, subject to unit-modulus phase constraints. The method converges to a stationary objective value, but global optimality is unavailable because the problem is nonconvex; phase updates can be performed only when channel statistics change.

  • Optimization formulation: The large-system problem replaces τ* with its deterministic equivalent and optimizes the RIS phase matrix under |φ_n| = 1.The objective and constraint are formulated for the large-system setting.
  • Projected gradient ascent: The phase-gradient method uses the derivative of the deterministic equivalent with respect to each φ_n and backtracking line search to select the step size.The projection solves a closest-feasible-point problem satisfying the unit-modulus constraint.
  • Projected gradient ascent: Projected gradient ascent increases the deterministic equivalent of the minimum SINR at each iteration and converges in objective value.The ascent direction uses derivatives with respect to the RIS phases, and each iterate is projected onto the feasible set.
  • Optimization limitations: No global optimality claim is possible because the RIS optimization problem is nonconvex with respect to Φ.The algorithm is presented as a preliminary starting point for max-min SINR optimization in RIS-assisted multi-user MISO systems.
  • Statistical phase updates: Optimized phases need not be recomputed at every channel realization; updating them over several coherence intervals reduces BS–RIS signaling overhead and real-time delays.The BS and RIS need synchronization when the correlation matrices change.
  • Results with Direct Channel - Special Case: With direct channels, the general RMT analysis does not readily provide deterministic equivalents for per-user quantities, whereas a special equal-average-SNR case yields a common correlation matrix and Corollary 3.In that special case, the RIS phases can be designed using the derivative expression with R replaced by R̃.

V. SIMULATION RESULTS AND DISCUSSIONS

Simulations compare RIS-assisted MISO links with HD-AF and FD-AF relays under single-user layouts, channel models, and varying RIS sizes and distances. RISs can outperform HD-AF relays with moderate element counts, while matching FD-AF performance requires substantially larger surfaces.

  • Distance effects: RIS performance decreases faster with increasing user distance than relay performance, requiring larger surfaces for comparable rates.RISs with moderate N are competitive in small-cell scenarios, whereas large RISs are needed to outperform relays for cell-edge users in larger cells.
  • Relay benchmarks: FD-AF relays outperform HD-AF relays in the simulations because the assumed FD self-interference is very low, although practical systems may experience higher interference.The FD-AF comparison therefore uses an optimistic self-interference assumption.

2) Multiple Users:

The multi-user analysis examines RIS-assisted MISO systems under rank-one BS-to-RIS channels and evaluates how the minimum SINR changes with users, antennas, and multiple RISs. A single rank-one RIS cannot efficiently support many users, whereas multiple surfaces can provide sufficient channel rank.

  • Single RIS: The minimum user SINR under OLP is evaluated for a multi-user MISO system and compared with its closed-form expression and an asymptotic bound.The plotted quantities are scaled with K while M and K grow at the same rate.
  • Single RIS: Kτ*u converges to 1 as K increases regardless of the number of RIS elements or BS antennas, validating the asymptotic bound.The exact scaled minimum SINR remains below this bound.
  • Single RIS: The rank-one link is therefore limited in efficiently serving more than one user.The analysis states that the system cannot support multiple users under a rank-one link.
  • Multiple RISs: For L = 8 rank-one RISs and K = 5 users, the minimum SINR increases as more RISs are introduced compared with a single-RIS system.The surfaces are deployed at specified xRIS positions with yRIS = 30m and M = 8.
  • Multiple RISs: For L ≤ 4, the channel lacks enough rank to support five users, whereas for L ≥ 5 increasing M improves performance through the massive MIMO effect.The results support using L ≥ K rank-one surfaces to serve K users efficiently.

B. Full Rank H1

For full-rank BS-to-RIS LoS channels, the paper develops deterministic equivalents for OLP parameters and uses them to optimize RIS phase shifts. Simulations validate the approximation and show that optimized phase design outperforms CoM beamforming, while reflection loss increases the required RIS size.

  • Validation: The deterministic equivalent closely matches Monte Carlo minimum-SINR simulations even for moderate system dimensions.The validation compares τ̄ with simulated τ* against Pmax.
  • Phase optimization: The proposed phase design performs significantly better than the Center of Means reflect-beamforming scheme.The comparison evaluates minimum user rate through log2(1+τ̄).
  • Relay comparison: With α = 1, N = 15 passive elements outperform HD-AF and N = 180 outperform FD-AF; with α = .8, the counts rise to N = 22 and N = 270.The proposed algorithm is sub-optimal, so more efficient reflect beamforming could yield higher gains.
  • Practical implications: RIS deployment is motivated by lower hardware complexity and energy consumption than active relays, which require dedicated power and active electronic components.FD relays additionally require self-interference cancellation circuits.
  • Analysis and design: The full-rank analysis uses random-matrix-theory tools to develop deterministic approximations for OLP parameters and optimize the RIS phase matrix.The design maximizes the minimum SINR using projected gradient ascent.
  • Scope and extensions: The study leaves channel estimation, imperfect CSI, discrete phase levels, multiple-RIS analysis, and joint deployment optimization as open directions.These issues are identified as challenges or extensions for RIS-assisted systems.

APPENDIX A

The appendix derives closed-form or bounded expressions for OLP behavior and establishes asymptotic convergence results. These derivations support the rank-one SINR characterization and the large-system deterministic equivalent.

  • Rank-one derivation: The rank-one OLP proof applies the Woodbury formula to simplify the precoding expression and complete the closed-form result.The derivation uses aHa = M and obtains a normalized channel direction.
  • Multiple-RIS bound: For multiple rank-one RISs, the appendix uses bounds rather than exact closed forms and assumes orthogonal LoS channel vectors across RISs.The resulting bound is converted into the minimum-SINR result of Corollary 2.
  • Asymptotic convergence: The appendix establishes deterministic-equivalent behavior under boundedness conditions ensured by the stated assumptions.The derivation explicitly invokes lim sup d < ∞.
  • Asymptotic convergence: The asymptotic analysis studies Kτ*u − 1 through convergence results and vanishing probability terms.The argument uses distributional convergence and properties of proper-distribution cumulative distribution functions.

APPENDIX C

The appendix proves deterministic equivalents for the OLP parameters by replacing random coefficients with a common quantity, establishing its convergence, and then deriving convergence of the remaining parameters.

  • Analytical caveat: Direct application of standard random-matrix tools to the quadratic form is not analytically correct because the coefficients and τ* depend on the channel vectors.The proof instead exploits weak dependence heuristically and then supplies a rigorous argument.
  • Deterministic-equivalent construction: The proof first approximates all d_ks by a common quantity d̃ defined as the unique solution of a fixed-point equation.The approximation uses the expected weak dependence of d_i, i≠k, on h̃_k and results from Lemmas 4 and 5.
  • Convergence proof: The convergence proof establishes lim sup e_K ≤ 1 and lim inf e_1 ≥ 1, implying the required convergence of the coefficient sequence.The upper bound follows by contradiction using monotonicity of the relevant right-hand side and the constraint μ ≥ 1.
  • Convergence of τ*: Although d̃ remains random through τ*, the appendix derives a deterministic equivalent τ̄ by resolvent-identity calculations showing |τ*−τ̄| → 0.The vanishing terms are discarded only after the convergence is justified through the resolvent identity.
  • OLP parameter convergence: Combining the convergence of τ* and the d_ks yields convergence of the OLP parameters q*_ks to their deterministic equivalents.The appendix concludes the proof of Theorem 3 after putting these convergence results together.

APPENDIX D PROOF OF LEMMA 2

The appendix derives the phase-gradient expression for the deterministic equivalent τ̄ using the implicit function theorem and resolvent-based random-matrix identities.

  • Implicit differentiation: The derivative of τ̄ with respect to each RIS phase φ_n is computed by applying the implicit function theorem to g(τ̄,Φ).The function is defined through τ̄ and a resolvent involving the RIS phase matrix Φ.
  • Derivative evaluation: The calculation of ∂g/∂φ_n uses intermediate trace terms T1 and T2, which are combined to obtain the required derivative expression.The resulting derivative is substituted into the implicit-function relation together with the defining equation for τ̄.
  • Channel and resolvent setup: The channel matrix is modeled as H=[h_1,…,h_K], with h_k=R^1/2z_k and z_k distributed as CN(0,I_M).The resolvent is defined for HBH^H with diagonal B containing the coefficients b_k.
  • Random-matrix tools: The derivation relies on rank-one perturbation, quadratic-form convergence, and a fixed-point result for the resolvent matrix.These classical identities and convergence results provide the random-matrix tools used in the proof.
  • Asymptotic conditions: Under the stated asymptotic regime and bounded-interval conditions, the relevant resolvent quantities satisfy the claimed convergence.The appendix states the convergence result after specifying the asymptotic assumptions.
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