Sum Rate Maximization for Multiuser MISO Downlink with Intelligent Reflecting Surface

1. INTRODUCTION

• IRSs deploy many passive reflecting elements whose independently controlled phase shifts can constructively align desired signals and destructively align interference.A smart controller coordinates phase configuration with other network components via a dedicated control link,.Source: §1 · p. 1
• Prior IRS studies considered power minimization, sum-rate maximization, symbol-level precoding, physical-layer security, energy transfer, and OFDM applications,,,,,,,,.Source: §1 · p. 1
• Continuous phase shifters are costly for large IRSs, prompting studies of finite-resolution designs for transmit-power minimization and average sum-rate maximization,.Continuous phase shifts vary from 0 to 2π, whereas discrete designs use finite phase sets.Source: §1 · p. 1
• The paper jointly optimizes BS transmit beamformers and continuous or discrete IRS phases for multiuser MISO sum-rate maximization.It derives a reformulation with structured convex constraints and a smooth objective, then proposes a block-coordinated accelerated projected gradient algorithm.Source: §1 · p. 1
• Simulations show the proposed algorithm achieves higher sum rate than a state-of-the-art design across different phase-shifter resolutions.Source: §1 · p. 1

2. SYSTEM MODEL AND PROBLEM STATEMENT

• The BS with M antennas unicasts independent data to K single-antenna users, while an IRS with N reflecting elements assists the downlink.The transmitted data symbol for user k has unit average power and is paired with beamformer v_k.Source: §2 · p. 1
• The IRS phase-shift matrix is Θ = √ηDiag(θ), with θ_n = e^jφ_n and η representing the reflection coefficient.The BS-to-IRS channel is G, and θ collects the element-wise reflection coefficients.Source: §2 · p. 1
• Continuous phases use the unit-modulus set FCP, whereas discrete phases use FDP = {1, ω, ..., ω^(L−1)} with ω = e^j2π/L.L is the number of realizable phase angles, and the discrete values are equally spaced.Source: §2 · p. 2
• The design jointly chooses BS beamformers and IRS phases to maximize the sum of users’ rates under a total BS transmit-power budget.The feasible phase set F is either FCP or FDP.Source: §2 · p. 2
• The problem is hard because beamformers and phase shifts are tightly coupled, while phase constraints are nonconvex and discrete phases yield an essentially NP-hard MINP.The paper therefore develops an efficient approximate approach based on reformulation and block-coordinated APG.Source: §2 · p. 2

3. A TRACTABLE APPROACH TO PROBLEM (3)

• The reformulation makes all constraints convex, including the convex hull of the continuous- or discrete-phase feasible set, while retaining a tightly coupled nonconvex objective.A gradually increased penalty avoids an ill-posed problem during the divide-and-conquer procedure.Source: §3 · p. 2, §3.1 · p. 2

3.1. Solution to problem (7)

• With fixed IRS phases, the transmission-variable subproblem is exactly a WMMSE problem solved by cyclic block-coordinate updates of u, w, and V.The transmit-power multiplier is selected by bisection to satisfy the complementary condition.Source: §3.1 · p. 2
• For continuous phases, the convex hull is the unit-modulus disk, whereas for discrete phases it is a regular polygon whose vertices are the allowed phase values.These convex hulls provide the feasible sets used by the subsequent projected updates.Source: §3.1 · p. 2, §3 · p. 2

3.2. Solution to problem (8)

• Using only one APG iteration per inexact MM update reduces computation and enables more frequent phase updates, with convergence to a KKT solution under technical conditions.The complete procedure is summarized as the GEMM method.Source: §3.2 · p. 3
• The IRS subproblem is a nonconvex difference-of-convex optimization handled by GEMM, which majorizes the objective by linearizing the concave penalty term.The surrogate majorizes fλ and matches it at the current iterate.Source: §3.2 · p. 3
• APG accelerates each GEMM update through extrapolation, backtracking-selected step sizes, and projection onto the convexified phase set.The continuous-phase projection has a closed form, and the discrete-phase projection also admits a closed form.Source: §3.2 · p. 3

• The evaluation averages results over 300 Rayleigh-fading channel realizations using eight users, eight BS antennas, and the stated path-loss and stopping configurations.Source: §4 · p. 3
• The proposed algorithm converges quickly within a few iterations for different phase-shifter resolutions at N = 100 and Pmax = 5dBm.Source: §4 · p. 3, Figure 1 · p. 4
• For continuous phase shifts, the proposed design achieves better sum rate performance than the algorithms in [5].Source: §4 · p. 4, Figure 2 · p. 4
• For 1-bit and 2-bit IRSs, the proposed design outperforms quantized AO and BCD, showing that explicit discrete-phase optimization avoids serious degradation from naive quantization.Source: §4 · p. 4
• All IRS-aided schemes attain much higher sum rate than WMMSE without IRS, while the proposed design outperforms the other schemes across the tested numbers of reflecting elements.Source: §4 · p. 4, Figure 3 · p. 4

• The proposed block-coordinated APG algorithm jointly designs transmit beamforming and IRS phase shifts for MU-MISO downlink sum-rate maximization.Source: §5 · p. 4
• Numerical results show that including an IRS effectively improves spectral efficiency over conventional wireless systems.Source: §5 · p. 4
• The proposed design attains higher sum rates than state-of-the-art designs, especially under discrete-phase constraints.Source: §5 · p. 4

• Applying the rate-MMSE relationship makes the original sum-rate problem equivalent to an optimization over receiver variables, weights, beamformers, and IRS phases.Source: §6 · p. 4
• The identity log(1 + SINR_k) = max over u_k and w_k converts each user rate into a weighted MMSE expression.Source: §6 · p. 4
• For a µ-Lipschitz objective, subtracting λ||x||^2 over the convex hull preserves global optima when λ exceeds µ for continuous phases or µ/sin(π/L) for discrete phases.Source: §6 · p. 4