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Intelligent Reflecting Surface: Practical Phase Shift Model and Beamforming Optimization

Samith Abeywickrama, Rui Zhang, Qingqing Wu, Chau Yuen

arXiv:2002.10112v1cs.ITeess.SP

TL;DR

The paper addresses the practical gap between ideal IRS phase-shift assumptions and phase-dependent reflection amplitudes. It proposes a practical model and jointly optimizes AP and IRS beamforming with iterative algorithms, finding substantial gains over ideal-model-based optimization.

  • Problem

    Ideal IRS models assume full reflection regardless of phase shift, although this is practically difficult and can make phase-alignment-based designs non-optimal.

  • Method

    The paper develops a phase-dependent amplitude model, formulates joint AP transmit and IRS reflect beamforming optimization under users’ SINR constraints, and solves it with iterative AO and penalty-based techniques.

  • Results

    Simulation results show substantial performance gains for beamforming optimized with the practical phase shift model compared with the conventional ideal model.

  • Takeaways & Limitations

    Accounting for phase-dependent reflection amplitude is important when designing IRS beamforming for practical wireless systems.

Abstract

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Intelligent reflecting surface (IRS) that enables the control of wireless propagation environment has recently emerged as a promising cost-effective technology for boosting the spectrum and energy efficiency in future wireless communication systems. Prior works on IRS are mainly based on the ideal phase shift model assuming the full signal reflection by each of the elements regardless of its phase shift, which, however, is practically difficult to realize. In contrast, we propose in this paper the practical phase shift model that captures the phase-dependent amplitude variation in the element-wise reflection coefficient. Based on the proposed model and considering an IRS-aided multiuser system with an IRS deployed to assist in the downlink communications from a multi-antenna access point (AP) to multiple single-antenna users, we formulate an optimization problem to minimize the total transmit power at the AP by jointly designing the AP transmit beamforming and the IRS reflect beamforming, subject to the users' individual signal-to-interference-plus-noise ratio (SINR) constraints. Iterative algorithms are proposed to find suboptimal solutions to this problem efficiently by utilizing the alternating optimization (AO) or penalty-based optimization technique. Moreover, we analyze the asymptotic performance loss of the IRS-aided system that employs practical phase shifters but assumes the ideal phase shift model for beamforming optimization, as the number of IRS elements goes to infinity. Simulation results unveil substantial performance gains achieved by the proposed beamforming optimization based on the practical phase shift model as compared to the conventional ideal model.

I. INTRODUCTION

The paper replaces the ideal IRS reflection assumption with a practical phase-dependent amplitude model and develops joint beamforming optimization for IRS-aided multiuser systems. It proposes efficient iterative algorithms and reports substantial gains over ideal-model-based optimization.

  • Motivation: Prior IRS studies commonly assume unity reflection amplitude regardless of phase shift, although hardware limitations make full reflection practically difficult.The practical element response is phase dependent, with losses from semiconductor devices, metals, and dielectric substrates contributing to reduced reflection amplitude.
  • Motivation: Phase alignment alone is generally non-optimal under phase-dependent amplitude, so IRS phase shifts must balance reflected-signal amplitude against alignment.The paper motivates an analytical phase shift model that captures this amplitude–phase relationship.
  • Contributions: For IRS-aided multiuser downlink transmission, the authors minimize AP transmit power by jointly optimizing AP transmit and IRS reflect beamforming under individual user SINR constraints.The resulting problem is non-convex, motivating iterative suboptimal solution methods.
  • Contributions: The paper proposes an analytical phase-shifter model applicable to various IRS semiconductor devices and validates its accuracy against experimental results from the literature.This model characterizes the fundamental relationship between reflection amplitude and phase shift.
  • Contributions: Two iterative approaches use alternating optimization and penalty-based optimization to solve the single-user problem, then extend these approaches to the general multiuser case.The multiuser algorithms provide different complexity–performance tradeoffs, with the penalty-based method offering better performance at higher complexity.
  • Contributions: Simulation results show substantial performance gains from practical-model beamforming optimization compared with the conventional ideal model.The paper also analyzes asymptotic performance loss when practical phase shifters are optimized using the ideal phase shift model.

II. SYSTEM MODEL

The system models IRS-assisted downlink communication from a multi-antenna AP to multiple single-antenna users, including practical reflecting-element hardware and phase-dependent reflection behavior.

  • The considered MISO system uses an AP with M antennas, an IRS with N reflecting elements, and K single-antenna users.
  • The IRS elements are programmable through a smart controller that communicates with the AP over a separate wireless link.
  • The model assumes negligible multiply reflected signals, quasi-static flat fading over each transmission block, and channel knowledge at the AP.
  • Each IRS reflection coefficient v_n has amplitude |v_n| in [0,1] and phase arg(v_n) in [−π,π), unlike the ideal model's unity amplitude.
  • A. Equivalent Circuit Model: The equivalent reflecting-element circuit includes inductive, capacitive, and resistive parameters, with resistance representing power dissipation from semiconductor, metal, and dielectric losses.
  • A. Equivalent Circuit Model: The element reflection amplitude varies with phase: it is lowest near zero phase and approaches unity near phase shifts of π or −π.
  • A. Equivalent Circuit Model: The circuit model agrees with reported experimental results and is used to motivate practical phase-shift-aware beamforming design.

B. Proposed Phase Shift Model

The paper proposes an analytical phase-shift model in which each IRS element's reflection amplitude depends on its phase, capturing practical hardware behavior for beamforming design.

  • The proposed model represents each reflection coefficient as v_n = β_n(θ_n)e^jθ_n, with phase θ_n and phase-dependent amplitude β_n(θ_n).
  • The amplitude function uses β_min, φ, and α, which respectively characterize the minimum amplitude, horizontal displacement, and curve steepness.
  • Setting β_min = 1 or α = 0 reduces the practical model to the ideal unity-amplitude phase-shift model.
  • The model parameters can be obtained by standard curve fitting because the element circuits are fixed after fabrication.
  • The proposed model closely matches practical-element simulation results and is adopted for subsequent IRS beamforming design.

IV. SINGLE-USER BEAMFORMING OPTIMIZATION

The single-user problem minimizes AP transmit power by jointly designing AP and IRS beamforming under the practical phase-shift model. The analysis shows that ideal-model beamforming can incur substantial power loss, although practical phase shifters retain O(N^2) power scaling asymptotically.

  • Problem formulation: The single-user formulation jointly optimizes AP transmit beamforming and IRS reflect beamforming to minimize AP transmit power under a minimum SNR requirement.For K = 1, inter-user interference is absent, and MRT is optimal for the AP beamforming subproblem.
  • Problem formulation: The resulting effective-channel power-gain problem remains non-convex because the practical reflection constraints are non-convex.The non-convexity makes the problem difficult to solve optimally in general.
  • Asymptotic power loss: As N →∞, the practical-to-ideal power ratio depends on βmin and α but not on N, while the O(N^2) power scaling order remains.This preserves the squared power-scaling order established under the ideal phase-shift model.
  • Asymptotic power loss: 5.5 dB power loss occurs for βmin = 0.2 and α = 1.6 under the ideal IRS assumption, while η(0.8, 1.6) = −1.1 dB.The reported numerical values indicate greater sensitivity to βmin than to α.
  • Implications: The results motivate incorporating hardware imperfections into beamforming design for practical IRS-aided systems.The paper evaluates practical-model beamforming using two optimization techniques.

C. AO-based Algorithm

The AO-based algorithm alternately optimizes one IRS phase shift while fixing the others. It uses a trust region and either one-dimensional search or a closed-form quadratic approximation to update each phase.

  • AO procedure: The AO method iteratively optimizes one reflecting-element phase shift while keeping the remaining phase shifts fixed.The iterations continue until the objective value converges.
  • Trust-region design: Because practical amplitude depends on phase, the ideal-model phase solution is no longer optimal for a practical IRS.The phase must balance the phase alignment term against the phase-dependent reflection amplitude.
  • Trust-region design: The trust region encloses the candidate optimal phase between arg(ϕn) and (−1)^λπ, with λ determined by the sign of arg(ϕn).The region accounts for the phase-dependent amplitude behavior.
  • Phase update: A one-dimensional search can obtain a high-quality approximate phase solution, but it may be computationally inefficient.The search interval is [arg(ϕn), (−1)^λπ].
  • Phase update: A closed-form approximation fits a quadratic through three sampled trust-region points and selects the resulting approximate optimizer.The three points are θA, θB, and θC, with corresponding objective values f1, f2, and f3.

D. Penalty-based Algorithm

The penalty-based method replaces difficult equality constraints with penalty terms and uses nested iterations. Block coordinate descent and convex approximations efficiently update the auxiliary variables and IRS phases.

  • Penalty framework: The method penalizes equality-constraint violations, then increases the penalty coefficient in an outer loop until convergence.The inner loop solves the penalized problem while the outer loop updates μ.
  • Inner optimization: For fixed phase variables, the auxiliary vector is updated using block coordinate descent and a convex-concave approximation.The concave-convex procedure linearizes the relevant term with a first-order Taylor expansion.
  • Inner optimization: The auxiliary-vector subproblem becomes an unconstrained convex optimization problem with a closed-form update.The update is obtained by setting the first-order derivative of the objective to zero.
  • Phase optimization: For fixed auxiliary variables, the phase variables are separable and can be optimized through N independent subproblems in parallel.Each subproblem uses the phase of the corresponding auxiliary variable and a trust region.
  • Phase optimization: The phase subproblem is handled with a trust region around its optimizer and a closed-form approximation obtained by quadratic fitting.The overall procedure is summarized in Algorithm 2 and terminates when constraint violation falls below a threshold.

V. MULTIUSER BEAMFORMING OPTIMIZATION

The multiuser extension minimizes AP transmit power under individual SINR constraints by jointly optimizing AP and IRS beamforming. Auxiliary variables and nested penalty-based block updates make the coupled non-convex problem tractable.

  • Problem formulation: The multiuser formulation minimizes total AP transmit power while jointly optimizing AP transmit and IRS reflect beamforming under individual user SINR constraints.The minimum SINR requirement of user k is γk > 0.
  • Problem formulation: The problem is non-convex because AP beamformers and the IRS vector are coupled and the SINR constraints are non-convex.The reformulated problem retains coupling through newly added equality constraints.
  • Penalty-based solution: Auxiliary variables decouple the AP beamformers from the IRS vector, after which equality-constraint violations are penalized.The resulting penalized problem is solved with a two-layer iterative algorithm.
  • Penalty-based solution: The inner loop partitions variables into four blocks and alternately optimizes each block with the others fixed.The AP beamformers, IRS vector, phase variables, and auxiliary variables are updated through corresponding subproblems.
  • Computational structure: User-specific variables separate across the objective and constraints, allowing K independent subproblems to be solved in parallel.Several block updates use closed-form expressions, while the SINR-related update uses bisection search.
  • Computational structure: The overall Algorithm 3 complexity is O(IinnIout(N^3 + M^3 + K^2(N^2 + NM + M^2) + K^2 log2(1/ε3))).Iinn and Iout denote the inner and outer iteration numbers required for convergence.

C. Two-Stage Algorithm

The two-stage algorithm first optimizes IRS phase shifts for weighted effective channel gain, then obtains AP transmit beamforming with those phase shifts. It offers lower complexity than the extended penalty-based algorithm while addressing practical phase-shift constraints.

  • Phase-shift optimization: The first stage optimizes IRS phase shifts by maximizing weighted effective channel power gain under the practical reflection-coefficient constraint.The phase shifts align different user channels to maximize active and passive beamforming gains.
  • Algorithm implementation: The phase-shift subproblem can be solved using the penalty-based technique, while the beamforming subproblem uses the resulting phase shifts.This provides an alternative implementation path for the first-stage optimization.
  • Transmit-beamforming optimization: The second stage solves the conventional multiuser MISO downlink power-minimization problem to obtain optimal AP transmit beamforming.Its solution can use an MMSE-based linear precoder computed through fixed-point iteration and uplink-downlink duality.

VI. SIMULATION RESULTS

The simulations evaluate practical IRS beamforming in a 3D multiuser wireless setup with Rayleigh fading, specified path-loss exponents, and a rectangular IRS array. The study uses a 2.4 GHz carrier and practical phase-shift parameters.

  • Simulation setup: The simulation system uses a 2.4 GHz carrier frequency and a three-dimensional coordinate system with a ULA at the AP and a URA at the IRS.The reference-distance signal attenuation is about 40 dB.
  • Simulation setup: The IRS contains N = NyNz reflecting elements arranged along the y- and z-axes with half-wavelength spacing.Ny and Nz denote the numbers of elements along the two axes.
  • Channel and user model: Channels are modeled with Rayleigh fading, and the AP-IRS, IRS-user, and AP-user path-loss exponents are 2.2, 2.8, and 3.8, respectively.Users are uniformly and randomly distributed within a cluster centered at (dx, d, 0) with radius r.
  • Deployment assumptions: The IRS is positioned for users with weak AP-user channels and locations that can avoid severe blockage with the AP.This deployment context motivates the reflected link in the simulation scenarios.

A. Single-User Case

In the single-user simulations, practical IRS reflection causes an asymptotic power loss that becomes predictable for large N, while practical-model optimization substantially outperforms ideal-model phase design. Discrete phase resolution further increases the importance of modeling hardware imperfections.

  • Asymptotic performance: As N increases, the ideal and practical cases first separate and then approach a constant determined by η(βmin, α).The practical cases use βmin = 0.8, 0.5, and 0.2, while the ideal case uses βmin = 1.
  • Asymptotic performance: When N is moderate, IRS hardware loss becomes more pronounced because the AP-user and IRS-user signal powers are comparable.For sufficiently large N, the reflected IRS signal dominates and the loss converges to the asymptotic result in Proposition 1.
  • Continuous phase shifts: The gap between practical-model optimization and ideal-model phase design increases as the user moves closer to the IRS.Closer IRS placement strengthens the reflected channel, making practical reflection design more important; moving toward the AP reduces the gap as the direct channel dominates.
  • Discrete phase shifts: With discrete phase shifts, one-bit designs have nearly identical performance under practical and ideal models, but their performance gap increases with b.The discrete levels are equally spaced, with U = 2^b possible phase values.

B. Multiuser Case

In the multiuser case, higher SINR targets expose the limits of simplified two-stage designs because interference becomes a bottleneck. Adding users raises required AP power, and practical IRS reflection may fail to preserve full spatial multiplexing gains.

  • SINR-target comparison: As the SINR target γ increases, the extended penalty-based and two-stage methods diverge, with the two-stage method performing worse than scheme 4) at high SINR.The two-stage method has only a small loss in the low-SINR regime compared with the extended penalty-based method.
  • SINR-target comparison: High SINR targets make multiuser interference the performance bottleneck, increasing the importance of jointly optimizing AP transmit and IRS reflect beamforming.The simulations indicate that both the practical phase-shift model and more sophisticated optimization are needed in this regime.
  • Number of users: Required AP transmit power increases as users are added from K = 4 to K = 7, while performance gaps between schemes remain almost constant.The experiment uses equal user SINR targets and successively adds users to the cluster.
  • Number of users: At K = M = 8, no-IRS transmit power increases more sharply than ideal-IRS power because the no-IRS MIMO channel becomes poorly conditioned.The ideal IRS adds strong multipath components that make the effective MIMO channel well-conditioned.
  • Number of users: With βmin = 0.2, IRS reflection is insufficient to recover the full spatial multiplexing gain, causing considerable power loss.This contrasts with the ideal IRS case, whose stronger reflected paths improve effective-channel conditioning.
  • Conclusion: The paper concludes that practical phase-shift beamforming optimization outperforms optimization based on the conventional ideal model in both single-user and multiuser setups.The conclusion reports significant performance loss when the ideal model is used for beamforming optimization.
  • Conclusion: Future work should examine the performance difference in more general IRS-aided systems, including OFDM, NOMA, physical-layer security, and SWIPT systems.These systems define the stated scope boundary for the current study.

APPENDIX A: PROOF OF PROPOSITION 1

The appendix proves Proposition 1 by comparing practical and ideal IRS models in the large-N regime and by establishing local optimization steps for phase variables. It also constructs a quadratic surrogate from three sampled points to locate a stationary point.

  • Asymptotic comparison: For large N, the IRS-reflected signal dominates the direct AP-user link, allowing the latter to be ignored in the practical-power analysis.The proof then derives Ppractical under this asymptotic simplification.
  • Asymptotic comparison: Under the ideal phase shift model, each element has unit amplitude, βn(θn) = 1, and the corresponding power is denoted Pideal.The comparison uses the same phase solution θ for the ideal model.
  • Phase optimization: The proof uses inequalities involving βn and cosine terms to show that selected phase perturbations can produce function values no smaller than the current value.These steps cover cases based on arg(ϕn), λ, and the neighboring phase offsets δ or ∆.
  • Phase optimization: Three phase points and their function values determine quadratic coefficients a0, a1, and a2, whose stationary point supplies the candidate phase update.The coefficients are obtained by evaluating the quadratic at θA, θB, and θC, followed by substitution into the stationary-point expression.
  • Phase optimization: The argument extends the neighboring-phase inequality procedure to λ = 1 and concludes the proof of the stated proposition.The existence of a suitable ∆ depends on the values of vn and ψn.
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