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Intelligent Reflecting Surface Enhanced Wireless Network via Joint Active and Passive Beamforming

Qingqing Wu, Rui Zhang

arXiv:1810.03961v4cs.ITmath.OC

TL;DR

The paper addresses non-convex IRS-aided multiuser MISO beamforming, jointly optimizing AP transmission and IRS reflections to minimize transmit power under SINR constraints, with simulations showing improved network performance over conventional systems.

  • Problem

    Joint AP transmit and IRS reflection beamforming is difficult to optimize because the SINR constraints produce a non-convex problem.

  • Method

    The paper jointly optimizes AP transmit beamforming and IRS phase shifts, whose elements independently control reflected-signal phases.

  • Results

    Simulations show that deploying an IRS and jointly optimizing its reflection with AP transmission significantly improves energy saving, coverage extension, and achievable rate over conventional systems.

  • Takeaways & Limitations

    Within the evaluated practical setups, IRS-assisted joint optimization improves wireless-network performance across energy saving, coverage extension, and achievable rate.

Abstract

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Intelligent reflecting surface (IRS) is envisioned to be a new and revolutionizing technology for achieving spectrum and energy efficient wireless communication networks cost-effectively in the future. Specifically, an IRS consists of a large number of low-cost passive elements each reflecting the incident signal with a certain phase shift to collaboratively achieve beamforming and/or interference suppression at designated receivers. In this paper, we study an IRS-aided multiuser multiple-input single-output (MISO) wireless system where one IRS is deployed to assist in the communication from a multi-antenna access point (AP) to multiple single-antenna users. As such, each user receives the superposed signals from the AP as well as the IRS via its reflection. We aim to minimize the total transmit power at the AP by jointly optimizing the transmit beamforming by active antenna array at the AP and reflect beamforming by passive phase shifters at the IRS, subject to users' individual signal-to-interference-plus-noise ratio (SINR) constraints. However, the formulated problem is non-convex and difficult to be solved optimally.

I. INTRODUCTION

The introduction motivates IRS as a low-cost, energy- and spectrum-efficient technology whose reconfigurable passive elements can enhance desired signals and suppress interference. It formulates joint AP transmit and IRS reflect beamforming under SINR constraints, then outlines solution methods and power-scaling results.

  • Motivation and IRS concept: IRS addresses wireless networks’ energy, hardware-cost, and interference challenges through a planar array of individually controlled passive elements.The elements reconfigure reflected-signal propagation without active transmit modules.
  • Motivation and IRS concept: Adjusting IRS phase shifts coherently boosts desired received power and destructively suppresses interference at unintended receivers.This reconfigurability also supports enhanced security and privacy.
  • Implementation and feasibility: IRS offers flexible, lightweight deployment and can integrate transparently into existing cellular or WiFi networks, especially for dense indoor applications.An experimental two-user testbed reported greatly improved spectral efficiency with IRS.
  • Problem formulation: The paper jointly optimizes AP active transmit beamforming and IRS passive reflect beamforming to minimize total AP transmit power under users’ SINR constraints.The resulting problem is generally difficult because its SINR constraints are non-convex.
  • Contributions and results: For sufficiently large N, AP transmit power for a single user near the IRS decreases in the order of N2, while IRS reduces required SINR-target power in single-user and multiuser setups.The authors develop semidefinite-relaxation and alternating-optimization-based methods, extending the single-user designs to multiuser suboptimal algorithms.

II. SYSTEM MODEL AND PROBLEM FORMULATION · A. System Model · B. Problem Formulation

The paper models an IRS-aided single-cell downlink from a multi-antenna AP to K single-antenna users and formulates joint AP transmit and IRS reflect beamforming as AP-power minimization under individual SINR constraints. The resulting problem is non-convex because beamforming and phase shifts are coupled, while a rank condition provides a sufficient feasibility criterion.

  • A. System Model: An IRS assists downlink communication from a multi-antenna AP to K single-antenna users in a single-cell network.The AP has M transmit antennas, and the IRS has N reflecting units.
  • A. System Model: The IRS controller switches between receiving mode for channel estimation and reflecting mode for data transmission.The model assumes perfect CSI at the AP, quasi-static flat fading, TDD, and channel reciprocity for downlink CSI acquisition.
  • A. System Model: The IRS reflection-coefficients matrix is Θ = diag(β1e^jθ1, ···, βNe^jθN), with θn ∈ [0, 2π) and βn ∈ [0, 1].The composite AP-IRS-user channel concatenates the AP-IRS link, IRS reflection, and IRS-user link.
  • A. System Model: Linear transmit precoding assigns each user one dedicated AP beamforming vector, and each user receives direct and IRS-reflected signals plus noise.The transmitted user data are independent, zero-mean, unit-variance random variables.
  • B. Problem Formulation: The optimization minimizes total AP transmit power by jointly selecting AP transmit beamforming and IRS reflect beamforming under individual SINR constraints for all users.The formulation uses W = [w1, ···, wK], Hr, and Hd to represent the beamformers and channel matrices.
  • B. Problem Formulation: Continuous IRS phase shifts are used to characterize fundamental performance limits, while discrete phase-shift design is left for future work; the reflection amplitudes are set to βn = 1.The practical implementation may select phase shifts from finitely many values.
  • B. Problem Formulation: The problem is difficult because its objective and some constraints are convex, but the SINR constraints are non-convex due to coupled transmit beamforming and phase shifts.The paper applies SDR and alternating optimization approximately, first for the single-user case and then for the multiuser case.
  • B. Problem Formulation: Problem (P1) is feasible for any finite user SINR targets γk’s if rank(G^H H_r + H_d) = K.The additional AP-IRS-user link makes this rank condition easier to satisfy than the no-IRS condition rank(Hd) = K, because reflected links can prevent channel alignment.

III. SINGLE-USER SYSTEM · A. SDR

For the single-user case, the paper removes inter-user interference but retains a non-convex joint beamforming problem. The SDR approach converts the phase-design subproblem into a convex SDP, provides a bound for evaluating alternatives, and requires rank-one recovery with a 4-approximation guarantee.

  • III. SINGLE-USER SYSTEM: The single-user setup sets K = 1, so inter-user interference is absent and the original problem simplifies after dropping the user index.
  • III. SINGLE-USER SYSTEM: The resulting problem remains non-convex because its constraint LHS is not jointly concave in w and θ.
  • III. SINGLE-USER SYSTEM: The paper solves the single-user problem using SDR and alternating optimization, with both techniques later extended to the multiuser system.
  • A. SDR: SDR is first applied to obtain a lower bound on the optimal value and evaluate performance gaps from suboptimal solutions.
  • A. SDR: For any fixed phase shift θ, maximum-ratio transmission is the optimal transmit beamforming solution, and the optimal transmit power satisfies P ∗= γσ2.
  • A. SDR: The phase optimization becomes a non-convex QCQP with unit-modulus constraints, which is reformulated as a homogeneous QCQP before lifting.
  • A. SDR: Relaxing the rank-one constraint produces a convex SDP solvable by CVX, but the relaxed solution may have rank(V) ≠ 1 and only upper-bounds the original objective.
  • A. SDR: 4-approximation of the optimal objective value is guaranteed by SDR followed by a sufficiently large number of randomizations.

B. Alternating Optimization

The proposed alternating-optimization algorithm reduces complexity relative to the SDR-based solution by iteratively optimizing AP transmit beamforming and IRS phase shifts. Both updates have closed-form solutions, and the algorithm converges because the objective is non-increasing and bounded below.

  • B. Alternating Optimization: The algorithm alternates between optimizing the AP transmit beamforming direction and power and optimizing the IRS phase shifts.The iterations continue until convergence.
  • B. Alternating Optimization: For fixed transmit beamforming direction, the joint transmit-power and phase-shift problem has a closed-form solution despite being non-convex.The derivation exploits the special structure of the objective function.
  • B. Alternating Optimization: Each IRS phase shift aligns the reflected AP–IRS–user signal with the direct AP–user signal to achieve coherent signal combining.The optimal phase is independent of the amplitude of the corresponding reflected channel component.
  • B. Alternating Optimization: The method is practically appealing because transmit beamforming and IRS phase shifts are obtained in closed form without invoking an SDP solver.Its convergence follows from exact subproblem solutions, a non-increasing objective, and a lower bound imposed by the SNR constraint.

C. Power Scaling Law with Infinitely Large Surface

With optimal IRS phase design, the received SNR gains a squared scaling with the number of reflecting elements, enabling AP transmit power reduction by 1/N2 without compromising user SNR. Unit and random phase shifts yield linear received-power gains, while AF relays achieve only linear SNR scaling because relay noise increases with N.

  • IRS power scaling: 1/N2 AP transmit-power reduction preserves the user received SNR under optimal IRS phase design.The squared gain results from combining an order-N transmit beamforming gain with an order-N aperture gain.
  • IRS power scaling: Order N received-power gains are achieved with both unit and random IRS phase shifts.This demonstrates practical usefulness without channel knowledge for optimally setting the phase shifts.
  • IRS power scaling: The IRS noise power remains constant as N increases, so user receive SNR inherits the received-signal power’s squared gain.The constant noise power contrasts with the relay case, where effective noise grows with N.
  • AF relay comparison: Only linear SNR scaling with N is obtained by a full-duplex AF relay, even with perfect self-interference cancellation.The relay result is an upper bound for imperfect self-interference cancellation.
  • AF relay comparison: Relay signal power scales as N2, but effective receiver noise also scales linearly with N, producing a lower SNR scaling order than the IRS.Half-duplex AF relays have the same receive-SNR scaling order as full-duplex AF relays.

IV. MULTIUSER SYSTEM · A. Alternating Optimization Algorithm

For the general multiuser setup, the paper proposes two efficient suboptimal algorithms by extending the single-user approaches. The alternating algorithm jointly updates AP transmit beamforming and IRS phase shifts, using MMSE-based beamforming and SDR-based phase optimization.

  • IV. MULTIUSER SYSTEM: The multiuser section proposes two efficient algorithms to solve (P1) suboptimally by generalizing the two single-user approaches.
  • A. Alternating Optimization Algorithm: The alternating method applies alternating optimization, designing AP transmit beamforming with MMSE to handle multiuser interference instead of single-user MRT.
  • A. Alternating Optimization Algorithm: For fixed IRS phases, (P1) becomes the conventional multiuser MISO downlink power-minimization problem, solvable by SOCP, SDP, or uplink-downlink-duality fixed-point iteration.
  • A. Alternating Optimization Algorithm: For fixed transmit beamforming, the phase-shift problem has non-convex unit-modulus constraints and is approximately solved by semidefinite relaxation after converting the constraints to quadratic form.
  • A. Alternating Optimization Algorithm: The objective value of (P3) is non-increasing across iterations, while the SDR-derived feasible solution need not satisfy every SINR constraint with equality.
  • A. Alternating Optimization Algorithm: The algorithm alternately solves (P3) and (P4) or (P4’), initializing each iteration with the previous solution and using Gaussian randomization when extracting phase shifts.
  • A. Alternating Optimization Algorithm: The iteration starts with (P3) because it is feasible for arbitrary phase shifts under the stated rank condition, whereas (P4) may be infeasible for arbitrary beamforming.
  • A. Alternating Optimization Algorithm: Problem (P4’) introduces SINR-residual slack variables while preserving (P4)’s feasible V set and generally producing more efficient converged phase-shift solutions.

B. Two-Stage Algorithm

The two-stage algorithm decouples joint active and passive beamforming into phase-shift optimization followed by MMSE-based transmit beamforming. It reduces computational complexity by solving each subproblem once, but may incur performance loss.

  • Algorithm overview: The method decouples the joint beamforming design into separate phase-shift and transmit-beamforming subproblems, yielding lower complexity than alternating optimization.The phase shifts and transmit beamforming are optimized in separate stages.
  • Stage 1: Phase-shift optimization: The first stage optimizes IRS phase shifts through weighted effective channel gain maximization to align user-channel phases and enhance IRS beamforming gain.The gain is especially targeted toward users near the IRS.
  • Stage 2: Transmit-beamforming optimization: The second stage solves problem (P3) with fixed phase shifts to obtain the optimal MMSE-based transmit beamforming.This transmit-beamforming subproblem is solved after the phase shifts are obtained.
  • Stage 1: Phase-shift optimization: The non-convex phase-shift problem is reformulated as a homogeneous QCQP and solved using semidefinite relaxation and Gaussian randomization.The phase shifts satisfy unit-modulus constraints.
  • Stage 1: Phase-shift optimization: Because one common phase-shift set serves users with different channels, their combined channel power gains generally cannot be maximized simultaneously and must be balanced.This multiuser coupling makes the phase-shift problem non-convex.
  • Complexity and performance: The two-stage algorithm solves (P5) and (P3) once each, reducing computational complexity relative to alternating optimization but potentially causing performance loss.The performance tradeoff is evaluated in the next section.

V. SIMULATION RESULTS · A. Single-User System · 1) AP Transmit Power versus AP-User Distance:

The simulations evaluate an IRS-aided single-user system with joint AP transmit and IRS reflection beamforming under a 10 dB SNR target. The proposed designs achieve near-optimal transmit power and extend coverage by exploiting both direct and reflected links as user distance changes.

  • V. SIMULATION RESULTS: The simulation uses a 3D geometry with a ULA at the AP, a URA at the IRS, half-wavelength spacing, and IRS center-to-AP distance d0.The AP and IRS reference antennas are at (0, 0, 0) and (0, d0, 0), respectively, with d0 > 0.
  • A. Single-User System: The comparison includes the SDP lower bound, SDR with Gaussian randomization, alternating optimization, AP-user MRT, AP-IRS MRT, random phase shift, and a benchmark without IRS.The AP-user and AP-IRS MRT schemes respectively align the AP beamformer with the direct channel and the AP-IRS rank-one channel.
  • 1) AP Transmit Power versus AP-User Distance:: The two proposed schemes achieve near-optimal transmit power relative to the lower bound and significantly outperform the benchmark schemes.This comparison is made as transmit power is plotted against the AP-user horizontal distance d.
  • 1) AP Transmit Power versus AP-User Distance:: At about 13 dBm, network coverage increases from about 33 m without IRS to beyond 50 m with the proposed joint beamforming designs.The results indicate that passive IRS deployment can extend coverage without installing an additional AP or active relay.
  • 1) AP Transmit Power versus AP-User Distance:: AP-user MRT is effective near the AP but costly near the IRS, whereas AP-IRS MRT behaves oppositely; improper beamforming can make IRS use worse than no IRS.For example, AP-IRS MRT performs worse than the no-IRS case for d ≤35 m, while joint designs balance direct and reflected transmission.

2) AP Transmit Power versus Number of Reflecting Elements: · 3) Comparison with AF Relay:

Increasing IRS reflecting elements can sharply reduce AP transmit power when the user is near the IRS, but the gain weakens with location. Under a fixed power budget, the IRS can match AF-relay rates at smaller N and is expected to outperform them at sufficiently large N.

  • 2) AP Transmit Power versus Number of Reflecting Elements:: For d = 50 m, the AP-IRS MRT scheme achieves near-optimal transmit power because the IRS-reflected signal is much stronger than the direct AP signal.The user is very close to the IRS in this case.
  • 2) AP Transmit Power versus Number of Reflecting Elements:: The proposed schemes’ required transmit power scales approximately in the order of N^2 as the number of reflecting elements increases.This scaling is observed even when the AP-IRS channel is LoS rather than Rayleigh fading.
  • 2) AP Transmit Power versus Number of Reflecting Elements:: The number of reflecting elements can be selected according to the IRS location and the target user SNR or AP coverage.This follows from the location-dependent transmit-power gains.
  • 3) Comparison with AF Relay:: The AF-relay comparison ignores the AP-user direct link, assumes perfect SIC for FD AF relaying, and gives both systems the same total transmit-power budget P = 5 mW.All IRS transmit power is assigned to the AP, whereas FD AF relay power is optimally allocated between the AP and relay by exhaustive search; HD AF relay is also a benchmark.
  • 3) Comparison with AF Relay:: When N is small, the IRS-aided system matches FD/HD AF relay rates using more reflecting elements, while passive elements require no transmit RF chains and cost less than active relay antennas.The comparison evaluates achievable rate in bps/Hz versus N.
  • 3) Comparison with AF Relay:: Doubling N from 400 to 800 increases IRS rate about 2 bps/Hz versus about 1 bps/Hz for FD AF relay, so IRS is expected to eventually outperform FD/HD AF relaying.The difference follows from IRS and AF-relay SNR gains scaling as N^2 and N, respectively.

B. Multiuser System · 1) AP Transmit Power versus User SINR Target:

The multiuser simulations show that the proposed IRS-assisted joint beamforming designs reduce AP transmit power and converge effectively. Their gains arise from IRS-assisted desired-signal enhancement and interference suppression, with performance depending on user distance and channel scattering.

  • B. Multiuser System: The eight-user setup places four users near the IRS and four cell-edge users near the AP, with d2 = 3 m and d1 = 20 m.The IRS and AP channel parameters are set to αAI = αIu = 2.8 and βAI = βIu = 3 dB.
  • B. Multiuser System: The proposed Alternating optimization w/ IRS and Two-stage algorithm w/ IRS are compared with MMSE and zero-forcing beamforming without the IRS.A random phase-shift IRS with MMSE AP beamforming is also used as a benchmark.
  • B. Multiuser System: The proposed algorithm’s required transmit power decreases quickly with iterations, and solving (P4’) achieves lower converged power than solving (P4).The convergence test uses M = 4, active users Uk, k = 1, 2, 3, 4, γ = 20 dB, and two-stage phase initialization.
  • 1) AP Transmit Power versus User SINR Target:: For the near-IRS user, IRS reflection remarkably increases desired-signal power and oppositely aligns reflected interference to suppress direct-link interference.As the SINR target increases, direct-link interference rises monotonically while combined-channel interference decreases.
  • 1) AP Transmit Power versus User SINR Target:: Beam directions adapt to user distance: the far user remains aligned with its direct channel, whereas the near user uses the combined channel at low SINR and converges to ZF at high SINR.IRS-assisted interference cancellation suppresses interference at the near user, motivating joint optimization of AP beamforming and IRS phase shifts.
  • 1) AP Transmit Power versus User SINR Target:: When βAI ≥−10 dB, both proposed algorithms require increasing transmit power as the AP-IRS Rician factor rises because stronger LoS increases channel correlation and interference.The results favor deploying the IRS in relatively rich scattering environments, which provide spatial degrees of freedom for serving multiple users.
  • 1) AP Transmit Power versus User SINR Target:: Higher Rician factors make near-IRS reflected signals dominate and reduce spatial multiplexing through more severe multiuser interference.The implication is that relatively rich scattering avoids a strong LoS, low-rank G and supplies sufficient spatial degrees of freedom from the AP to the IRS.

3) Comparison with Massive MIMO:

The comparison evaluates an IRS-aided system against massive MIMO under a joint transmission protocol and accounts for coordination delay. A hybrid AP–IRS deployment can match or outperform AP-only systems with more active antennas, even with moderate delay.

  • Transmission protocol: The IRS uses receive RF chains for channel estimation, exchanges estimated AP-user channels and optimized beamforming with the AP, then transmits collaboratively.The protocol assumes the AP–IRS channel is quasi-static, changing much more slowly than the AP-user and IRS-user channels.
  • Delay model: The IRS-aided system incurs additional coordination and computation delay τ, with delay ratio ρ = τ/Tc and the assumption τ < Tc.Users are served by the AP during coordination and by both the AP and IRS during the remaining coherence time.
  • Ideal-delay comparison: An AP with M = 20 active antennas and an IRS with N = 80 passive reflecting elements achieves nearly the same performance as an AP-only deployment with M = 50.This result is reported for the ideal negligible-delay case, ρ = 0.

VI. CONCLUSIONS

The paper concludes that jointly optimizing AP transmit beamforming and IRS passive reflection can improve IRS-aided multiuser wireless performance while reducing transmit power and implementation cost. It also identifies channel-estimation limitations and future directions for feedback-based beamforming and relay comparisons.

  • Main contributions: The proposed approach jointly optimizes active AP transmit beamforming and passive IRS reflect beamforming under user SINR constraints to minimize transmit power.Semidefinite relaxation and alternating optimization provide algorithms trading system performance against computational complexity.
  • Performance findings: For a single user, receive SNR increases quadratically with the IRS reflecting-element count, outperforming conventional massive MIMO or multi-antenna AF relay efficiency.For multiuser systems, jointly designed IRS interference suppression and AP beamforming improve all users, including those far from the IRS.
  • Performance findings: Extensive simulations show that deploying an IRS and jointly optimizing reflection with AP transmission improves energy saving, coverage extension, and achievable rate over conventional setups without an IRS.The study also provides insights into optimal IRS deployment and the delay-performance trade-off for practical design.
  • Practical considerations: When the IRS lacks receive RF chains, it cannot directly estimate associated AP/user channels; feedback from APs or users is proposed as a future solution.With receive RF chains, pilot-assisted channel-estimation methods can be similarly applied to the IRS.
  • Future work: Future work should compare IRS energy efficiency with full-duplex AF relaying, whose higher spectrum efficiency requires effective SIC and additional energy consumption.A parallel work is noted as showing greater IRS energy efficiency than half-duplex multi-antenna AF relaying.
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