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Beamforming Optimization for Wireless Network Aided by Intelligent Reflecting Surface with Discrete Phase Shifts

Qingqing Wu, Rui Zhang

arXiv:1906.03165v4cs.ITcs.ETeess.SP

TL;DR

The paper addresses practical IRS-assisted multiuser transmission with finite phase shifts, optimizing AP precoding and IRS phases under SINR constraints. It develops exact and approximate designs and shows that discrete phase shifts preserve the asymptotic squared power gain, with a constant phase-level-dependent loss.

  • Problem

    Prior IRS studies mainly assume continuous phase shifts that are difficult to implement, leaving the performance and squared power gain of finite-level phase shifts to be investigated.

  • Method

    The paper jointly optimizes continuous AP transmit precoding and discrete IRS phase shifts, using optimal and successive-refinement algorithms for single-user and multiuser cases.

  • Results

    Discrete phase shifts retain the same asymptotic squared power gain as continuous shifts, while incurring only a constant power loss in dB dependent on the number of phase-shift levels.

  • Takeaways & Limitations

    Even 1-bit IRS phase shifters can preserve the asymptotic squared power gain of continuous phase shifts.

Abstract

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Intelligent reflecting surface (IRS) is a cost-effective solution for achieving high spectrum and energy efficiency in future wireless networks by leveraging massive low-cost passive elements that are able to reflect the signals with adjustable phase shifts. Prior works on IRS mainly consider continuous phase shifts at reflecting elements, which are practically difficult to implement due to the hardware limitation. In contrast, we study in this paper an IRS-aided wireless network, where an IRS with only a finite number of phase shifts at each element is deployed to assist in the communication from a multi-antenna access point (AP) to multiple single-antenna users. We aim to minimize the transmit power at the AP by jointly optimizing the continuous transmit precoding at the AP and the discrete reflect phase shifts at the IRS, subject to a given set of minimum signal-to-interference-plus-noise ratio (SINR) constraints at the user receivers. The considered problem is shown to be a mixed-integer non-linear program (MINLP) and thus is difficult to solve in general. To tackle this problem, we first study the single-user case with one user assisted by the IRS and propose both optimal and suboptimal algorithms for solving it. Besides, we analytically show that as compared to the ideal case with continuous phase shifts, the IRS with discrete phase shifts achieves the same squared power gain in terms of asymptotically large number of reflecting elements, while a constant proportional power loss is incurred that depends only on the number of phase-shift levels. The proposed designs for the single-user case are also extended to the general setup with multiple users among which some are aided by the IRS. Simulation results verify our performance analysis as well as the effectiveness of our proposed designs as compared to various benchmark schemes.

I. INTRODUCTION

IRSs use low-cost passive reflecting elements to improve wireless spectrum and energy efficiency, but practical finite phase shifts introduce alignment loss that this paper analyzes and addresses.

  • Motivation: IRSs are proposed as a cost-effective approach to improving spectrum and energy efficiency through low-cost passive reflecting elements with adjustable phase shifts.Their passive architecture can provide favorable performance scaling with fewer hardware and energy costs than active technologies.
  • Multiuser operation: IRS beamforming can enhance desired signals and suppress interference, improving SINR flexibility for users at different locations.Users near the IRS can tolerate more AP interference, allowing more flexible AP precoding for other users.
  • Practical phase shifts: Finite phase shifts are more practical than continuous control but can misalign reflected and non-reflected signals, causing performance degradation.The paper motivates investigating this loss for low-resolution IRS implementations.
  • Paper scope: The paper studies a multiuser system in which a multi-antenna AP jointly optimizes active precoding and finite discrete IRS phase shifts.The considered phase-shift design contrasts with prior work based mainly on continuous phase shifts.
  • Algorithms: The resulting optimization is an NP-hard MINLP, motivating globally optimal and low-complexity successive-refinement algorithms.The single-user problem can be transformed into an ILP, while the refinement method targets close-to-optimal performance.
  • Main findings: As N →∞, discrete phase shifts retain the IRS O(N^2) squared power gain while incurring a constant dB loss dependent only on phase-shift levels.Simulations also compare the proposed algorithms with quantization- and codebook-based schemes and the no-IRS case.
  • Main findings: Finite-level IRSs can achieve the same multiuser SINR performance as conventional massive MIMO while using fewer active AP antennas and reducing energy consumption.The paper also reports that the proposed joint designs outperform quantization-based and codebook-based IRS schemes.

A. System Model

The system is a downlink multiuser MISO network assisted by an IRS, modeled with ideal single-reflection channels, perfect CSI, continuous AP precoding, and discrete IRS phase shifts.

  • System architecture: The considered network has an AP with M antennas, K single-antenna users, and an IRS with N reflecting elements.The paper focuses on downlink communication and notes extension to uplink through channel reciprocity.
  • Channel assumptions: The model keeps only first-order IRS reflections and assumes perfect channel state information is known at the AP during each coherence time.Signals reflected by the IRS two or more times are ignored because of substantial path loss.
  • IRS model: Each IRS element applies a complex reflection coefficient with amplitude β_n and phase θ_n, represented collectively by a diagonal matrix Θ.The reflected vector is modeled as ŷ = Θx̂, with Θ containing the element-wise reflection coefficients.
  • IRS model: The IRS maps incident signals to reflected signals through an N × N diagonal phase-shift matrix, while practical operation fixes β_n = 1.Independent amplitude and phase control is considered costly, so elements are designed to maximize signal reflection.
  • Discrete control: Each element uses one of L = 2^b uniformly quantized phase levels over [0, 2π), with spacing Δθ = 2π/L.The phase-shift set is F = {0, Δθ, · · ·, (L −1)Δθ}.
  • Transmission model: The AP uses conventional continuous linear precoding, transmits independent zero-mean unit-variance symbols, and combines direct and IRS-reflected signals at each receiver.User noise is modeled as independent additive white Gaussian noise, and the received SINR follows from the combined signal model.
  • User reception: A user far from the IRS may effectively be served through AP precoding alone, although Θ can indirectly affect that user through other users’ precoders and interference.This gives the SINR expression applicability to users at arbitrary locations.
  • Control: The IRS controller coordinates with the AP over a separate wireless link to exchange channel information and control element phases in real time.The paper notes that its downlink results are extendable to uplink using channel reciprocity.

B. Problem Formulation

The paper formulates AP power minimization under user SINR and discrete IRS constraints, then exploits single-user structure to derive exact and low-complexity solution methods.

  • B. Problem Formulation: The design target is minimizing AP transmit power by jointly optimizing AP precoders W and IRS phase shifts θ under user SINR requirements.Each phase shift is restricted to the finite set F.
  • B. Problem Formulation: Coupling between W and θ makes the SINR constraints non-convex, while discrete phase restrictions make the overall problem an NP-hard MINLP.The paper states that no standard method efficiently obtains a globally optimal solution in general.
  • B. Problem Formulation: Relaxing discrete phases to continuous values does not remove non-convexity, and direct nearest-value quantization can be ineffective for low-resolution multiuser systems with severe interference.The quantization approach is nevertheless used to characterize asymptotically large-N performance.
  • B. Problem Formulation: For asymptotically large N, direct quantization is used to characterize discrete-phase performance relative to continuous phase shifts.This comparison concerns the regime of an asymptotically large number of reflecting elements.
  • III. SINGLE-USER SYSTEM: The single-user case assumes K = 1 and can correspond to orthogonal multiple access, eliminating multiuser interference and simplifying the formulation.For fixed θ, maximum-ratio transmission is optimal, and minimizing power becomes maximizing the combined user-channel gain.
  • III. SINGLE-USER SYSTEM: In the single-user problem, the optimal transmit power is determined from the required SINR, noise power, and optimized combined channel gain.The channel expression is rewritten using the IRS phase vector and an associated channel-dependent matrix.
  • III. SINGLE-USER SYSTEM: A successive-refinement algorithm reduces computational complexity by selecting IRS phase shifts iteratively and can achieve close-to-optimal performance.The method is motivated by the need for an efficient alternative to the exact solution.

A. Optimal Solution

The optimal single-user phase-shift problem is reformulated as an integer linear program using SOS1 variables, enabling globally optimal solution by branch-and-bound.

  • The single-user problem (P2) can be reformulated as an ILP whose globally optimal solution is obtainable.
  • SOS1 vectors encode discrete phase-shift choices with one active entry among otherwise zero entries.They express optimization variables constrained to discrete sets.
  • Binary variables represent element phase shifts and phase-shift differences, including an ambiguity variable for differences modulo 2π.
  • The reformulated problem is an ILP and can be solved optimally with branch-and-bound.

B. Suboptimal Solution

The suboptimal approach successively refines IRS phase shifts, providing convergence with lower complexity while characterizing the asymptotic performance and power-loss tradeoff of discrete phase shifts.

  • The proposed successive refinement algorithm alternately optimizes each IRS phase shift while fixing the remaining N−1 shifts.It is designed to reduce the exponential worst-case complexity of the optimal method.
  • The objective is non-decreasing across iterations and converges to a locally optimal solution because it is upper-bounded by a finite value.
  • As N →∞, the discrete-phase IRS retains the asymptotic squared power gain O(N^2), while its power ratio depends only on the number of phase-shift levels.
  • The reported losses are η(1) = −3.9 dB, η(2) = −0.9 dB, and η(3) = −0.2 dB for 1-, 2-, and 3-bit phase shifters, respectively.Using 2 or 3 bits is described as sufficient for close-to-optimal performance with approximately 0.9 dB or 0.2 dB loss.
  • The tradeoff between element count N and phase-shifter resolution 2^b allows comparable received power at different hardware-cost configurations.

IV. MULTIUSER SYSTEM

For multiple users, the paper develops optimal and suboptimal phase-shift designs that jointly address AP precoding and IRS configuration under SINR constraints.

  • The general setup allows multiple users to share a time-frequency dimension, with only some users aided by a nearby IRS.
  • The paper proposes optimal and suboptimal algorithms for the multiuser problem (P1).
  • For fixed phase shifts, the AP precoder is obtained using the MMSE-based linear precoder and fixed-point iteration based on uplink-downlink duality.
  • The globally optimal phase shifts require exhaustive search over all possibilities, solving the corresponding precoding problem for each case.
  • The suboptimal design uses a ZF precoder and successive phase-shift refinement to eliminate multiuser interference with lower complexity.
  • The ZF-based successive refinement algorithm is guaranteed to converge because its objective is lower-bounded by a finite value.

V. NUMERICAL RESULTS

The numerical section specifies a three-dimensional IRS-assisted simulation model using path loss, Rician fading, and fixed system geometry and SINR settings.

  • The simulations use a three-dimensional coordinate setup with a ULA at the AP and a URA at the IRS.
  • The IRS has N = N_yN_z reflecting elements, with N_y fixed at 4 while N_z increases linearly with N.
  • The AP-user and IRS-user channels are generated with distance-dependent path loss and scenario-specific path-loss exponents and Rician factors.
  • All channels use Rician fading, combining deterministic line-of-sight and Rayleigh fading components.
  • The AP-user link is modeled with α_Au = 3.5 and β_Au = 0 because the IRS serves users suffering severe AP-user attenuation.

A. Single-User System

The single-user evaluation compares discrete-phase IRS designs with continuous-phase and no-IRS benchmarks across distance and reflecting-element count. Channel-aware successive refinement and quantization perform near optimally, while discrete phase shifts retain practical gains but incur a bounded loss versus continuous shifts.

  • Performance Comparison with Benchmark Schemes: The study evaluates AP transmit power against AP-user distance using M = 4, N = 16, and γ = 25 dB.The compared schemes include continuous-phase lower bound, optimal and suboptimal discrete-phase algorithms, quantization, codebook, and no-IRS benchmarks.
  • Performance Comparison with Benchmark Schemes: 1-bit phase shifters require significantly less transmit power than no IRS when the user is near the IRS.This demonstrates a signal-hot-spot effect even with coarse, low-cost phase shifters.
  • Performance Comparison with Benchmark Schemes: Discrete phase shifts lose performance relative to the continuous-phase lower bound because reflected and direct multipath signals cannot be perfectly phase-aligned.The resulting mismatch produces performance loss at the receiver.
  • Performance Comparison with Benchmark Schemes: The proposed successive refinement and quantization schemes both achieve near-optimal single-user performance and outperform the codebook scheme.The comparison favors optimizing phase shifts from actual channels over selecting vectors from a predefined codebook.

2) Impact of Discrete Phase Shifts:

As the number of reflecting elements grows, discrete-phase performance gaps approach phase-level-dependent constants, while successive refinement provides efficient multiuser optimization. In multiuser simulations, IRS-assisted methods reduce AP power, with ZF-based refinement near optimal and increasingly advantageous at larger system sizes.

  • Impact of Discrete Phase Shifts: As N increases, the quantization-to-continuous-phase performance gap first grows and then approaches a constant determined by η(b).The reported constants are η(1) = −3.9 dB and η(2) = −0.9 dB.
  • Impact of Discrete Phase Shifts: When IRS-reflected power dominates received power, phase-quantization loss converges to the asymptotic value predicted by Proposition 1.The successive-refinement advantage over quantization is more evident for b = 1 than for b = 2.
  • Impact of Discrete Phase Shifts: The quantization scheme has higher complexity because it first obtains continuous phase shifts using an SDP solver.Successive refinement avoids this initial SDP step.
  • Performance Comparison with Benchmark Schemes: ZF-based and MMSE-based successive refinement perform almost identically across a wide range of SINR targets and considered setups.The MMSE-based method generally serves as a lower bound for the ZF-based method.
  • Performance Comparison with Benchmark Schemes: IRS-assisted algorithms achieve significant AP transmit-power reductions versus no IRS, while ZF-based refinement is near optimal and outperforms quantization and codebook schemes.Its gain over benchmark schemes becomes more pronounced as system size increases.
  • Performance Comparison with Benchmark Schemes: Quantization performs worse than the codebook scheme at high SINR because coarse phase errors affect both desired-signal combining and interference cancellation.The quantization loss is smaller in the low-SINR regime than in the high-SINR regime.

2) AP Transmit Power versus Number of Users:

With more users, IRS-assisted joint active and passive beamforming limits AP transmit-power growth more effectively than no IRS. Users near the IRS require less additional power, while controllable reflected paths improve channel conditioning and spatial multiplexing.

  • AP Transmit Power versus Number of Users: Adding a user near the IRS requires less additional AP transmit power than adding a user far from the IRS.The comparison attributes this difference to passive beamforming gain.
  • AP Transmit Power versus Number of Users: When the number of users approaches the number of AP antennas, no-IRS transmit power increases much faster than IRS-assisted transmit power.The setup uses M = 8, N = 48, and b = 1.
  • AP Transmit Power versus Number of Users: Joint active and passive beamforming more effectively suppresses multiuser interference in the IRS-assisted system.The IRS adds controllable multipath signals that can transform a poorly conditioned MIMO channel into a better conditioned one.
  • AP Transmit Power versus Number of Users: Adding controllable IRS paths can increase channel rank and provide spatial multiplexing gain compared with highly correlated AP-user channels without IRS.The paper illustrates this mechanism for K = M = 2.

3) IRS-aided Small MIMO versus Large MIMO without IRS:

IRS-assisted small-MIMO designs can reduce active-antenna requirements while maintaining transmit-power and user-SINR targets, offering a cost-effective alternative to large MIMO. The paper also reports practical discrete-phase-shift performance and identifies several deployment and hardware limitations for future work.

  • IRS-aided Small MIMO versus Large MIMO without IRS: IRS deployment can significantly reduce RF power consumption and active-antenna hardware cost while preserving the considered large-MIMO performance gain.The paper presents this as a cost-effective alternative to large MIMO without IRS.
  • IRS-aided Small MIMO versus Large MIMO without IRS: The IRS-aided system provides flexibility to trade off AP active antennas, IRS passive elements, and phase-shifter resolution against system performance and cost.The design variables include the number of AP antennas M, IRS elements N, and phase-shifter levels 2^b.
  • IRS-aided Small MIMO versus Large MIMO without IRS: Discrete-phase IRSs retain the asymptotic squared power gain O(N^2) of continuous phase shifts, with only a constant power loss in dB.The paper specifically reports this asymptotic behavior even with 1-bit phase shifters.
  • IRS-aided Small MIMO versus Large MIMO without IRS: Directly quantizing optimized continuous phase shifts is near-optimal for one user but incurs non-negligible degradation in multiuser settings because of severe co-channel interference.The comparison highlights that single-user conclusions do not transfer directly to multiuser operation.
  • IRS-aided Small MIMO versus Large MIMO without IRS: The scope is limited to a single AP/IRS design with phase-shift optimization; joint amplitude-phase control and multi-AP/multi-IRS coordination remain more challenging open problems.The paper also leaves multi-cell deployment and joint AP/IRS placement for future work.
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