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Weighted Sum-Rate Maximization for Reconfigurable Intelligent Surface Aided Wireless Networks

Huayan Guo, Ying-Chang Liang, Jie Chen, Erik G. Larsson

arXiv:1912.11999v1eess.SP

TL;DR

The paper addresses weighted sum-rate maximization in RIS-aided multiuser MISO downlink systems by jointly designing AP beamforming and RIS phases under perfect and imperfect CSI. It proposes a low-complexity perfect-CSI algorithm and extends it with stochastic SCA for imperfect CSI. Simulations show significant gains over benchmarks and very small degradation when channel uncertainty is below 10%.

  • Problem

    Joint RIS phase and AP beamforming optimization is challenging because the variables are deeply coupled, while accurate RIS-related CSI is difficult to obtain.

  • Method

    The paper uses a low-complexity BCD and fractional-programming approach for perfect CSI, extending it to imperfect CSI with stochastic SCA.

  • Results

    The proposed joint design schemes achieve significant performance gains over benchmarks, with very small degradation when channel estimation uncertainty is below 10%.

  • Takeaways & Limitations

    The proposed methods provide stationary joint-design solutions for RIS-aided WSR maximization in both perfect- and imperfect-CSI settings.

  • Takeaways & Limitations

    The reported performance remains especially favorable only when channel uncertainty is smaller than 10%.

Abstract

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Reconfigurable intelligent surfaces (RIS) is a promising solution to build a programmable wireless environment via steering the incident signal in fully customizable ways with reconfigurable passive elements. In this paper, we consider a RIS-aided multiuser multiple-input single-output (MISO) downlink communication system. Our objective is to maximize the weighted sum-rate (WSR) of all users by joint designing the beamforming at the access point (AP) and the phase vector of the RIS elements, while both the perfect channel state information (CSI) setup and the imperfect CSI setup are investigated. For perfect CSI setup, a low-complexity algorithm is proposed to obtain the stationary solution for the joint design problem by utilizing the fractional programming technique. Then, we resort to the stochastic successive convex approximation technique and extend the proposed algorithm to the scenario wherein the CSI is imperfect. The validity of the proposed methods is confirmed by numerical results. In particular, the proposed algorithm performs quite well when the channel uncertainty is smaller than 10%.

I. INTRODUCTION

The paper studies RIS-aided multiuser MISO downlink transmission, jointly optimizing AP beamforming and RIS phases to maximize weighted sum-rate under perfect and imperfect CSI. It develops lower-complexity optimization methods and extends them to uncertain channel information.

  • Motivation: RIS elements passively adjust incident electromagnetic-wave reflections, offering low power consumption and nearly no additional thermal noise.RIS can improve wireless propagation by creating virtual links when direct AP-user links suffer deep fading or shadowing.
  • Imperfect CSI: High-accuracy CSI for RIS-related channels is difficult to obtain because the RIS is passive and the channel dimensions grow with the number of elements.The paper considers partial channel knowledge for RIS phase optimization and channel-estimation errors under imperfect CSI.
  • Problem formulation: The optimization jointly designs AP beamforming and RIS phases to maximize the users' weighted sum-rate under an AP transmit-power constraint.Each user treats signals intended for other users as interference when decoding its own signal.
  • Challenges: Joint design is difficult because RIS phases cannot directly suppress inter-user interference, leaving beamforming and phase optimization deeply coupled.This coupling makes conventional alternating optimization slow and potentially expensive when both subproblems require iterative solutions.
  • Contributions: For perfect CSI, the paper proposes a lower-complexity BCD-based joint design, then extends the approach to imperfect CSI using stochastic SCA.The imperfect-CSI formulation maximizes average WSR and is designed to be independent of specific channel-model assumptions.
  • System model: The system comprises a multi-antenna AP, one RIS, and multiple single-antenna users communicating over quasi-static flat-fading channels.The RIS phase-shift matrix is diagonal, with each element applying a unit-modulus phase shift to reflected signals.

B. Discussion on Channel Estimation

High-accuracy CSI for RIS-related channels is difficult because training overhead and estimation errors grow with the RIS configuration. The paper therefore formulates imperfect-CSI design around average WSR while retaining conventional operation after RIS configuration.

  • Estimating the AP–RIS and RIS–user channels accurately is challenging because the channel dimension grows linearly with the number of RIS elements.
  • Brute-force estimation sequentially measures each RIS element, creating training overhead proportional to N.Grouping adjacent elements can reduce this overhead.
  • Compressive sensing reduces training overhead by exploiting the low-rank property of the RIS-aided link.
  • Semi-passive RIS designs integrate active elements with channel-estimation capability and may make training overhead negligible using deep learning and compressive sensing.
  • Under imperfect CSI, channel estimates and estimation-error distributions define random channel realizations, and RIS phases maximize expected achievable WSR.The AP may know the combined channel for beamforming while RIS-related channels remain partially known.

III. ALTERNATING OPTIMIZATION FOR THE PERFECT CSI SETUP

For perfect CSI, the paper alternates between AP beamforming and RIS phase optimization. The beamforming block uses WMMSE, while the phase block is solved on a complex circle manifold using RCG.

  • Alternating optimization decomposes the joint variables into blocks and updates each while fixing the others at their latest values.
  • With fixed RIS phases, AP beamforming becomes conventional multiuser MISO WSR maximization, for which WMMSE obtains a stationary solution.
  • With fixed beamforming, RIS phase optimization uses effective direct and RIS-link channels under unit-modulus constraints.
  • Because the phase objective is differentiable on a complex circle manifold, the RCG algorithm can obtain a stationary solution.Each RCG iteration comprises three key steps.

1) Compute Riemannian Gradient:

The section develops the optimization context for joint beamforming and RIS phase design, emphasizing the limitations of alternating optimization and introducing fractional-programming-based block updates.

  • The RIS phase and AP beamforming variables are deeply coupled because RIS cannot directly suppress inter-user interference or allocate transmit power.
  • Alternating optimization can converge slowly and become computationally unacceptable when both subproblems require iterative solvers.
  • High phase-update precision may require large IR values to avoid stopping at an uninteresting point.
  • Independent solution of the two subproblems is difficult to extend to imperfect CSI, which requires coordinated W and θ design.
  • The proposed method applies closed-form fractional programming, decomposes the transformed problem into four blocks, and uses BCD updates for α, β, W, and θ.

1) Lagrangian Dual Transform:

The transformed optimization introduces auxiliary variables and uses block coordinate descent, with a prox-linear update reducing the beamforming-update cost and eliminating repeated searches.

  • Lagrangian Dual Transform: The closed-form fractional-programming transformation introduces auxiliary variables α and β to reformulate the sum-of-logarithms-of-ratios objective.
  • Lagrangian Dual Transform: BCD cyclically updates α, β, W, and θ to obtain a stationary solution of the transformed problem.
  • Lagrangian Dual Transform: The conventional W update requires an expensive matrix inverse and one-dimensional search for the transmit-power dual variable λ.
  • Lagrangian Dual Transform: The prox-linear BCD rule eliminates the λ search and expensive matrix inverse from the W update.
  • Lagrangian Dual Transform: The W-update complexity is reduced to O(KM^2), with no iteration required.
  • Lagrangian Dual Transform: The prox-linear update converges because the relevant objective is strongly convex and satisfies the KL property.

D. Successive Convex Approximation for Updating θ

The RIS phase update converts unit-modulus complex variables into real phases and uses a convex surrogate to handle the resulting non-convex optimization.

  • D. Successive Convex Approximation for Updating θ: The RIS phase constraint imposes unit modulus on every element, |θ_n| = 1.
  • D. Successive Convex Approximation for Updating θ: The phase variables are reparameterized as θ_n = e^jϕ_n with real ϕ_n.
  • D. Successive Convex Approximation for Updating θ: Because the phase objective is non-convex, the method solves a surrogate problem using successive convex approximation.
  • D. Successive Convex Approximation for Updating θ: The surrogate is constructed using a second-order Taylor expansion and the phase vector is updated by minimizing it.

E. Algorithm Development

The algorithm develops a BCD procedure with Armijo-controlled phase updates and establishes stationary-solution behavior while reducing complexity relative to iterative alternating optimization.

  • E. Algorithm Development: The BCD block order cyclically updates α, β, ϕ, β, W, and α until convergence of the objective.
  • E. Algorithm Development: The proposed method updates α, β, and W with complexities O(KNM), O(KNM), and O(KM^2), respectively.
  • E. Algorithm Development: The phase-update complexity is dominated by U and is O(K^2N^2), giving total complexity O(IO(2KNM + KM^2 + K^2N^2)).
  • E. Algorithm Development: Armijo search selects the phase step size κ to accelerate convergence when phase updates yield small objective improvements.
  • E. Algorithm Development: Every stationary solution of the phase subproblem is a critical point of the BCD method and a stationary solution of the original problem.
  • E. Algorithm Development: The proposed algorithm reduces complexity with respect to N by IR times because the θ block requires no iterative updating.

V. EXTEND THE NON-CONVEX BCD FOR THE IMPERFECT CSI SETUP

The imperfect-CSI problem is transformed and solved through a non-convex BCD framework that updates auxiliary variables, beamforming, and RIS phases. The resulting inner-layer solution provides a stationary solution for the transformed problem.

  • The imperfect-CSI problem P(B) is equivalently transformed using the closed-form fractional-programming approach.
  • The RIS phase vector is represented by ϕ, with unit-modulus constraints |θ_n| = 1 for every RIS element.The phase relation is θ_n = e^jϕ_n.
  • The algorithm obtains a stationary solution by iteratively updating auxiliary variables α and β, beamforming W, and RIS phases.The updates continue until the transformed objective fA1 converges.
  • Because the inner-layer subproblem reaches a stationary solution, the solution of P(B2) is also stationary for P(B1).

B. Stochastic Successive Convex Approximation for P(B2)

Stochastic SCA addresses the non-convex phase-update problem under random channel realizations by constructing recursive surrogate functions and applying gradient-projection updates.

  • Each new channel realization is handled by updating the RIS phase using a sample-average approximation.The realization ξ_r is introduced at iteration r.
  • Because the per-realization objective remains non-convex, stochastic SCA replaces it with an approximation around the previous iterate.The surrogate ˆh_i(ϕ, ϕ_i−1; ξ_i) approximates ˆg(ϕ; ξ_i).
  • Convergence requires the surrogate to be continuously differentiable, strongly convex, and uniformly bounded in its second-order derivatives.
  • The surrogate designed for the perfect-CSI setup satisfies the stochastic SCA requirements.
  • The recursive approximation updates the surrogate with δ_r = r^-0.501 and computes the new phase vector through gradient projection.The step size κ_r is selected by the Armijo rule.

C. Algorithm Development

The stochastic SCA algorithm processes successive channel realizations, updates recursive surrogates, and uses Armijo search to refine RIS phases. Its complexity is characterized for the simulated RIS-aided multiuser MISO setting and evaluated against three baselines.

  • Algorithm Development: For each new channel realization, the algorithm computes a gradient, updates the recursive surrogate h_r, and obtains ϕ_r using Armijo search.
  • Algorithm Development: The proposed stochastic SCA algorithm has complexity involving the iteration counts I_A and I_S.
  • Algorithm Development: The complexity expression includes the system dimensions K, N, and M, with terms involving K2N2 and KNM2.
  • Algorithm Development: Compared with Algorithm 2, Algorithm 4 costs I_S times the complexity with respect to M for each outer-loop iteration, while its N-dependent complexity remains approximately unchanged.
  • Simulation Setup: The simulations use one 4-antenna AP, four single-antenna users, and a RIS-aided link with LOS components in the AP–RIS and RIS–user channels.
  • Simulation Setup: The channel model combines Rayleigh fading for direct links with Rician fading for RIS-aided channels and sets the Rician factor to ε = 10.
  • Simulation Setup: At the reference point, the RIS-aided path-loss for N = 1 is 154.32 dB, compared with about 117.23 dB for the direct link.The passage states that the direct link therefore cannot be ignored.
  • Baselines: Performance is evaluated against no-RIS, random-phase, and multi-start upper-bound baselines.

B. Weighted Sum Rate Analyses

The simulations show that jointly optimizing AP beamforming and RIS phases substantially improves WSR, while performance depends on channel uncertainty, RIS size, and deployment location. The proposed schemes remain effective across user locations, with an optimal RIS position observed in the tested deployment range.

  • Transmit-power analysis: About 4 dB gain is achieved by joint beamforming and phase optimization when N = 100, while unoptimized RIS phases provide negligible improvement.Under imperfect CSI, the proposed algorithm still achieves about 3 dB gain when the channel uncertainty parameter is at most 0.5.
  • Convergence analysis: The proposed algorithm has much lower per-iteration complexity than alternating optimization, although its convergence is slightly slower in the perfect CSI setup.As channel uncertainty increases, the proposed algorithm requires more convergence steps.
  • RIS-size analysis: All schemes with optimized RIS phases gain substantially as N increases, but the RIS phase design does not achieve a squared gain in this setting.At N = 200, the proposed algorithm performs similarly to the case with PT = 8 dBm, and the authors attribute the absence of squared gain to the relatively small RIS aperture gain.
  • RIS deployment and user locations: The average rate is maximized at RIS horizontal coordinate DI = 195 m when PT = 5 dBm and N = 100.Moving the RIS from 200 m toward 205 m decreases the average rate, while moving it toward 170 m first increases and then decreases the rate because the RIS-link path loss is the product of two path losses.
  • RIS deployment and user locations: With the RIS deployed at (195m, 0m), the proposed schemes show stable performance gains over CDF curves for random user locations.The reported gains remain consistent with the deployment-location results in Fig. 6.
  • Overall findings: Extensive simulations show significant gains over benchmarks with 100 passive RIS elements, while performance degradation remains very small below 10% channel-estimation uncertainty.The conclusion summarizes the proposed joint-design schemes’ benchmark gains and their robustness to sufficiently small uncertainty.
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