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Weighted Sum-Rate Optimization for Intelligent Reflecting Surface Enhanced Wireless Networks

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

arXiv:1905.07920v2eess.SP

TL;DR

The paper addresses weighted sum-rate maximization in an IRS-aided multiuser MISO downlink with jointly optimized BS and IRS beamforming, including discrete phase shifts. It decouples the non-convex problem and alternates closed-form active and passive updates; simulations show significant gains over benchmarks, with small degradation for 2-bit phase shifts.

  • Problem

    The paper targets the difficult joint WSR optimization of BS active and IRS passive beamforming under practical discrete phase-shift constraints and large IRS sizes.

  • Method

    Lagrangian and fractional-programming transforms decouple the problem for alternating active/passive optimization, while three closed-form low-complexity algorithms solve the passive-beamforming subproblem across reflection assumptions.

  • Results

    The proposed schemes achieve significant capacity gains over benchmark systems; continuous phase shifting nearly matches ideal performance, and 2-bit phase shifting incurs only small degradation.

  • Takeaways & Limitations

    Low-complexity joint beamforming can support IRS-aided multiuser MISO optimization under both continuous and discrete phase-shift models.

Abstract

from arXiv · show

Intelligent reflecting surface (IRS) is a promising solution to build a programmable wireless environment for future communication systems. In practice, an IRS consists of massive low-cost elements, which can steer the incident signal in fully customizable ways by passive beamforming. In this paper, we consider an IRS-aided multiuser multiple-input single-output (MISO) downlink communication system. In particular, the weighted sum-rate of all users is maximized by joint optimizing the active beamforming at the base-station (BS) and the passive beamforming at the IRS. In addition, we consider a practical IRS assumption, in which the passive elements can only shift the incident signal to discrete phase levels. This non-convex problem is firstly decoupled via Lagrangian dual transform, and then the active and passive beamforming can be optimized alternatingly. The active beamforming at BS is optimized based on the fractional programming method. Then, three efficient algorithms with closed-form expressions are proposed for the passive beamforming at IRS. Simulation results have verified the effectiveness of the proposed algorithms as compared to different benchmark schemes.

I. INTRODUCTION

The paper studies IRS-aided multiuser MISO downlink WSR maximization with jointly optimized BS active and IRS passive beamforming, including practical discrete phase shifts. It develops alternating low-complexity algorithms and evaluates their performance against benchmarks.

  • Motivation: IRS passively reflects incident RF waves toward specified directions with low power consumption and nearly no additional thermal noise.The IRS is positioned to provide virtual links when direct BS-user links experience deep fading or shadowing.
  • Challenges: Existing IRS studies commonly assume continuous phase shifters, whereas practical hardware may restrict reflection coefficients to discrete values.Heuristic continuous-solution quantization can have unpredictable performance loss, while large-scale SDR methods are not scalable.
  • Approach: The proposed framework decouples active and passive beamforming, alternately optimizes them, and uses closed-form methods for the resulting subproblems.The active beamforming uses fractional-programming transforms, while passive beamforming is treated through QCQP formulations and related algorithms.
  • Contributions: Three low-complexity passive-beamforming algorithms support ideal, continuous, and discrete phase-shift assumptions.The convex ideal-reflection-coefficient problem is solved optimally, and the resulting procedures are extended to non-convex phase-shift cases.

B. Received Signal at User k

The system model defines transmitted user symbols and BS beamforming vectors, then characterizes received signals, interference, SINR, and the BS power constraint. These quantities support the joint WSR optimization under IRS reflection-coefficient feasibility constraints.

  • Signal Model: Independent zero-mean unit-variance symbols are transmitted from the BS using user-specific beamforming vectors w_k.The transmit beamforming matrix is W = [w_1, w_2, · · ·, w_K].
  • Received Signal: Each user receives a superposition of desired and interfering signals plus additive white Gaussian noise.Signals intended for other users are treated as interference when decoding user k's symbol.
  • Optimization Objective: The optimization maximizes users' weighted sum-rate by jointly designing BS transmit beamforming and the IRS reflection-coefficient matrix.The weights represent user priorities, while the BS transmit power is constrained and each IRS coefficient must belong to the feasible set.
  • Challenges: The resulting problem is generally non-convex because of its objective and feasible sets, so the paper seeks a low-complexity suboptimal solution.The stated challenges are decoupling the optimization variables and scaling reflection-coefficient adjustment to large N.

III. WSR MAXIMIZATION FOR DOWNLINK TRANSMISSION

The WSR problem is decoupled using a Lagrangian dual transform, then solved by alternating updates of auxiliary SINR variables, BS beamforming, and IRS reflection coefficients. The BS subproblem uses quadratic-transform fractional programming and alternating optimization, without a guarantee of global optimality.

  • A. Lagrangian Dual Transform: The logarithmic WSR objective is equivalently transformed using auxiliary variables α for users’ decoding SINRs.For fixed W and Θ, each optimal α_k is updated before optimizing the beamforming variables.
  • A. Lagrangian Dual Transform: With α fixed, optimizing W and Θ becomes a sum of multiple-ratio fractional-programming subproblems.The ratio-induced non-convexity is addressed using fractional-programming techniques, followed by alternating optimization of W and Θ.
  • B. Transmit Beamforming: For fixed Θ, the BS beamforming problem is reformulated through the quadratic transform with auxiliary variables β.The resulting problem is optimized over W and β under the BS transmit-power constraint.
  • B. Transmit Beamforming: Alternately updating W and β solves convex subproblems but does not guarantee global optimality.The resulting formulation is biconvex, so fixing one variable while optimizing the other provides a practical iterative method.
  • B. Transmit Beamforming: The beamforming update uses a dual variable for the power constraint and obtains W and β by stationarity conditions.The dual variable λ0 is optimally determined for the transmit-power constraint.

C. Optimizing Reflection Response Matrix Θ

With α and W fixed, the IRS reflection response is optimized by transforming the objective into a function of θ and an auxiliary vector ε. The resulting passive-beamforming subproblem is a QCQP whose only non-convexity comes from discrete reflection constraints.

  • C. Optimizing Reflection Response Matrix Θ: For fixed α and W, optimizing Θ is translated into optimizing its reflection-coefficient vector θ.The coefficients satisfy the feasible-set constraints for each IRS element.
  • C. Optimizing Reflection Response Matrix Θ: The IRS objective is reformulated with the quadratic transform using auxiliary variables ε.The vector ε contains one auxiliary variable ε_k for each user, and θ and ε are updated alternately.
  • C. Optimizing Reflection Response Matrix Θ: The transformed objective includes the quadratic form f4(θ) = f3a(θ, ε◦) for fixed optimal auxiliary variables.This connects the auxiliary-variable update to the final reflection-coefficient optimization.
  • C. Optimizing Reflection Response Matrix Θ: The passive-beamforming subproblem is a QCQP with a quadratic concave objective.Its non-convexity is introduced only by the discrete feasible-set constraint on each reflection coefficient.

D. Algorithm Development

Algorithm 1 alternates nominal-SINR, transmit-beamforming, and IRS reflection-coefficient updates from feasible initial points. Under a stated condition on the IRS update, the objective is monotonically nondecreasing and the algorithm converges.

  • D. Algorithm Development: Algorithm 1 initializes feasible W^(0) and Θ^(0), then repeatedly updates α, W, and IRS reflection coefficients.The process stops when the objective function f1a converges.
  • D. Algorithm Development: The nominal SINR α is updated first in each iteration, followed by transmit beamforming and IRS reflection-coefficient updates.The W and Θ updates use the fractional-programming subproblems developed earlier.
  • D. Algorithm Development: Algorithm 1 is guaranteed to converge when the IRS vector θ satisfies the condition specified in (29).Under that condition, the objective function is monotonically nondecreasing after every iteration.

IV. REFLECTION COEFFICIENTS ADJUSTMENT FOR (P4)

The paper develops low-complexity methods for the IRS reflection-coefficient QCQP, motivated by the difficulty and poor scalability of existing approaches. The nearest point projection method solves a convex relaxation first and projects its solution into discrete feasible sets.

  • IV. Reflection Coefficients Adjustment for (P4): The reflection-coefficient subproblem (P4) is the only step of Algorithm 1 without a closed-form solution.This section therefore focuses on solving (P4) efficiently.
  • IV. Reflection Coefficients Adjustment for (P4): When θ_n belongs to F1, (P4a) is convex; for F2 or F3, it is non-convex and finding the optimum is challenging.The distinction follows from whether the element constraints are convex in the corresponding feasible set.
  • IV. Reflection Coefficients Adjustment for (P4): SDR has computational complexity O(N^6), making it unsuitable for large-scale IRS settings.The paper motivates low-complexity and scalable alternatives because IRSs may contain massive numbers of passive elements.
  • A. Nearest Point Projection: The nearest point projection method first solves the convex F1 problem optimally, then projects θ to the nearest feasible point in F2 or F3.The projected solution is suboptimal for the non-convex problem, while the first-stage solution is optimal for the convex formulation.
  • IV. Reflection Coefficients Adjustment for (P4): For the convex case, the problem is equivalently transformed into a dual problem using Lagrange dual decomposition.The dual variables correspond to the element-wise constraints, and the duality gap is zero under Slater’s condition.

2) Projection Step:

The projection step obtains feasible reflection coefficients by projecting optimal relaxed solutions onto non-convex feasible sets. Because projection need not produce a local optimum, updates are accepted only under a convergence condition, while the LDD implementation has high complexity.

  • The optimal reflection coefficient θ• is projected onto the nearest feasible point in F2 or F3 to obtain a suboptimal non-convex solution.
  • The projected θ• is not necessarily a local optimum, so it is updated only when constraint (29) is satisfied to guarantee Algorithm 1 convergence.
  • The LDD method has O(N), O(N^3), and O(N^2) costs for summation, matrix inversion, and final multiplication, respectively.
  • The resulting LDD complexity is O(N^6), motivating lower-complexity replacements for the NPP method.

B. Iterative Reflection Coefficient Updating

The ICU algorithm updates IRS reflection coefficients one at a time using closed-form coordinate solutions. It converges to a local optimum, reaches the global optimum under the convex F1 setting, and reduces NPP complexity to O(N^2).

  • B. Iterative Reflection Coefficient Updating: ICU extends single-user alternating optimization to multiuser systems by optimizing one of the N reflection coefficients while fixing the others.
  • B. Iterative Reflection Coefficient Updating: For each coefficient, the resulting subproblem is a concave quadratic function with closed-form solutions across the convex and non-convex feasible cases.
  • B. Iterative Reflection Coefficient Updating: ICU repeatedly optimizes all reflection coefficients in order from n = 1 to n = N.
  • B. Iterative Reflection Coefficient Updating: Because each coordinate update is optimal with the others fixed, ICU converges to a local optimum of (P4a), and the F1 case is globally optimal.
  • B. Iterative Reflection Coefficient Updating: The ICU complexity is O(N^2), and replacing LDD with ICU reduces NPP complexity to O(N^2).

C. Alternating Direction Method of Multipliers

ADMM introduces an auxiliary vector and penalty formulation to update IRS coefficients in parallel. It offers lower complexity than LDD and SDR, but non-convex feasible sets retain duality and optimality limitations.

  • C. Alternating Direction Method of Multipliers: ADMM is proposed because ICU updates coefficients serially and may take a long time when N is large, whereas parallel updates are desired.
  • C. Alternating Direction Method of Multipliers: The method introduces an auxiliary vector q and penalty term enforcing q ≠ θ through equality constraints handled by Lagrange variables.
  • C. Alternating Direction Method of Multipliers: For F2 or F3, the dual objective provides only an upper bound because the primal problem is non-convex and has a duality gap.
  • C. Alternating Direction Method of Multipliers: ADMM alternates θ and q updates, applies projection operations, and then updates the Lagrange variables.
  • C. Alternating Direction Method of Multipliers: ADMM convergence for F2 or F3 requires a suitable penalty parameter, but convergence need not yield a global or local optimum.

3) Disucssion:

The simulations use a four-antenna BS, a four-user IRS-aided femtocell, and averaged Rayleigh-fading realizations. Under the stated geometry, IRS reflection can materially improve received power and joint beamforming yields significant sum-rate gains over random passive beamforming.

  • A. Simulation Scenario: The simulated femtocell places the BS at (0, 0), the IRS at (LI, 50 m), and four users within a 10 m-radius circle centered at (200 m, 0).
  • A. Simulation Scenario: The BS has M = 4 antennas, IRS reflection efficiency is η = 0.8, bandwidth is 200 kHz, and receiver noise power is −117 dBm.
  • A. Simulation Scenario: The direct and IRS-aided links use path-loss exponents D = 3.5 and I = 2, respectively, with the latter modeled as free-space propagation.
  • 3) Disucssion:: With ξ = 10 dB and N = 10, the direct-link path loss is about −111 dB, while the IRS-aided link is about −122 dB and may double average received power.
  • 3) Disucssion:: Results average over 10^4 channel realizations generated from 100 user-location snapshots and 100 independent small-scale-fading realizations per snapshot.

B. Benchmarks and Initialization

The proposed IRS-aided joint beamforming schemes substantially improve sum rate over conventional and random-passive-beamforming baselines. Performance depends on IRS size, reflection gain, quantization, and deployment location, with clear trade-offs among these factors.

  • Benchmarks: Joint active and passive beamforming achieves about 3 dB gain over the no-IRS baseline at N = 10 and ξ = 10 dB.All three proposed algorithms have almost the same performance, while random passive beamforming provides very small gain.
  • Benchmarks: The proposed schemes maintain stable performance gains across CDF curves, indicating good performance with high probability across user locations.
  • IRS Size and Material: R = 20 bps/Hz requires increasing N from 10 to 40, whereas increasing PT from 0 dBm to 5 dBm achieves the same rate with 5 dB.Increasing N strengthens only the IRS-assisted link, while increasing BS transmit power benefits both direct and IRS-assisted links.
  • IRS Size and Material: Increasing ξ from 10 dB to 15 dB enables about R = 20 bps/Hz, making reflection gain more effective than increasing PT or N for improving R.The reflection gain is counted twice during reflection; at large ξ, even random passive beamforming can obtain significant rate gain.
  • Deployment Location: IRS-aided performance improves when the IRS is closer to the BS or user cluster, while the center location LI = 100 m is the worst case.Very close placement can worsen propagation conditions, creating a trade-off between propagation quality and double-fading effects.
  • Conclusion: The conclusion reports significant capacity gains over no-IRS and random-passive-beamforming systems, while a 2-bit quantizer incurs only small performance degradation.All three low-complexity passive-beamforming algorithms apply to continuous and discrete phase shifts.

APPENDIX A PROOF OF LEMMA 3

The appendix establishes an equivalent coordinate-ascent form for the ADMM iteration under a positive-definiteness condition. It then uses this equivalence to support convergence of the ADMM algorithm.

  • APPENDIX A PROOF OF LEMMA 3: Substituting the preceding relationships transforms the original functions and replaces the auxiliary variable ¯λ with a function of q.
  • APPENDIX A PROOF OF LEMMA 3: When µ 2 I_N − U ≻ 0, the ADMM iteration is equivalent to coordinate ascent on V(q, θ).
  • APPENDIX A PROOF OF LEMMA 3: The appendix concludes that the ADMM algorithm converges.
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