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Intelligent Reflecting Surface: Practical Phase Shift Model and Beamforming Optimization

Samith Abeywickrama, Rui Zhang, Chau Yuen

arXiv:1907.06002v4eess.SPcs.IT

TL;DR

Ideal IRS models assume full reflection regardless of phase, although practical elements exhibit phase-dependent amplitude variation. The paper proposes a practical model, jointly optimizes transmit and IRS beamforming for achievable rate, and solves the non-convex problem with alternating optimization; simulations show substantial gains over the ideal model.

  • Problem

    Prior IRS work commonly assumes unity reflection amplitude regardless of phase shift, although hardware losses make this assumption difficult to realize.

  • Method

    The paper models phase-dependent reflection amplitude and jointly optimizes transmit and IRS reflect beamforming using a low-complexity alternating-optimization algorithm.

  • Results

    Simulation results show substantial performance gains from joint beamforming optimization under the practical phase shift model compared with the conventional ideal model.

  • Takeaways & Limitations

    The conventional ideal phase shift model can produce significant performance loss relative to the proposed practical model in IRS beamforming optimization.

Abstract

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Intelligent reflecting surface (IRS) that enables the control of the wireless propagation environment has been looked upon as a promising 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 a practical phase shift model that captures the phase-dependent amplitude variation in the element-wise reflection coefficient. Applying this new model to an IRS-aided wireless system, we formulate a problem to maximize its achievable rate by jointly optimizing the transmit beamforming and the IRS reflect beamforming. The formulated problem is non-convex and difficult to be optimally solved in general, for which we propose a low-complexity suboptimal solution based on the alternating optimization (AO) technique. Simulation results unveil a substantial performance gain achieved by the joint beamforming optimization based on the proposed phase shift model as compared to the conventional ideal model.

I. INTRODUCTION

IRSs can improve wireless efficiency by controlling propagation, but ideal full-reflection assumptions conflict with practical phase-dependent reflection losses. This paper introduces a practical model and jointly optimizes transmit and IRS beamforming using a low-complexity alternating-optimization approach.

  • IRSs enhance spectrum and energy efficiency by controlling wireless propagation through reconfigurable passive reflecting elements.
  • Ideal IRS models assume unity reflection amplitude at every element regardless of phase shift, an assumption that is difficult to realize because of hardware limitations.
  • Practical reflecting elements experience phase-dependent amplitude variation, with losses in semiconductor devices, metals, and dielectric substrates reducing reflection amplitude.
  • With phase-dependent amplitude, phase shifts must balance reflection amplitude and phase alignment, so ideal-model reflection design can degrade performance.
  • The paper proposes a practical phase shift model, jointly optimizes transmit and IRS reflect beamforming for achievable rate, and uses alternating optimization to obtain a low-complexity suboptimal solution.

A. Equivalent Circuit Model

The paper models each IRS reflecting element with an equivalent circuit whose impedance determines a phase-dependent reflection coefficient. The model captures practical energy dissipation and the resulting non-uniform relationship between reflection amplitude and phase shift.

  • Equivalent circuit: Each reflecting element is represented by a parallel resonant circuit, with metallic parts modeled as inductors and circuit parameters describing its impedance.The equivalent parameters include bottom-layer inductance L1, top-layer inductance L2, effective capacitance Cn, effective resistance Rn, and incident-signal angular frequency ω.
  • Reflection coefficient: The reflection coefficient vn describes the fraction of the electromagnetic wave reflected at the impedance discontinuity between free-space impedance Z0 and element impedance Zn(Cn, Rn).Changing the circuit parameters changes vn, allowing the reflected waves to be controlled and programmed.
  • Element response: A practical reflecting element can achieve almost 2π phase tuning, while its amplitude and phase generally vary together as Cn and Rn change.The circuit behavior is evaluated using Cn from 0.47 pF to 2.35 pF at the stated component values and 2.4 GHz operating frequency.
  • Element response: Reflection amplitude is lowest near zero phase shift because enhanced currents increase dielectric, metallic, and ohmic losses, whereas it approaches unity near phase shifts of π or −π.The reported numerical behavior agrees with experimental results in the literature, supporting the circuit model’s practical accuracy.
  • Practicality: Ideal unit-amplitude reflection would require zero energy dissipation, but practical semiconductor, metallic, and dielectric losses make that condition unattainable.Consequently, beamforming design should incorporate a practical phase shift model rather than assume unit amplitude for every phase.

B. Proposed Phase Shift Model

The proposed phase shift model explicitly links each IRS element’s reflection amplitude to its phase shift through a circuit-dependent analytical function. It closely matches practical-element simulations and is then used for beamforming design.

  • Model formulation: The model writes the reflection coefficient as vn = βn(θn)e^jθn, with phase shift θn and corresponding amplitude βn(θn).It is intended to characterize the amplitude–phase relationship for a variety of semiconductor devices used in IRS implementations.
  • Model parameters: The amplitude function uses βmin, φ, and k to represent the minimum amplitude, horizontal displacement, and curve steepness associated with a specific circuit implementation.These parameters can be obtained by standard curve fitting after the IRS circuit is fabricated.
  • Model parameters: When k = 0, the proposed expression reduces to the ideal phase shift model with unit amplitude for every element and phase shift.For practical circuits, the parameters are selected to capture the non-uniform amplitude response.
  • Validation and use: The proposed model closely matches the practical reflecting-element simulation results and is adopted for subsequent IRS beamforming design.The paper assumes identical reflecting-element circuits, so the same βmin, φ, and k apply to all elements.

A. Problem Formulation

The paper formulates joint transmit and IRS reflect beamforming to maximize achievable rate under the practical phase-shift model. After fixing the IRS reflection, transmit beamforming has a closed-form optimum, but the resulting IRS problem remains non-convex.

  • The optimization jointly chooses the AP beamformer w and IRS reflection vector v to maximize achievable rate under the AP transmit-power constraint.
  • For any fixed IRS reflection v, maximum-ratio transmission gives the optimal transmit beamforming solution.
  • Substituting the optimal transmit beamformer reduces the problem to optimizing the IRS reflection coefficients.
  • Each reflection coefficient must satisfy v_n = β_n(θ_n)e^{jθ_n}, linking its amplitude to its phase shift.
  • The reduced problem is non-convex and generally difficult to solve optimally, motivating an alternating-optimization solution.

B. Proposed AO Algorithm

The proposed AO algorithm optimizes one IRS element's phase shift at a time while holding the others fixed. Each resulting scalar subproblem is non-convex, so the method uses an efficiently computable approximate solution.

  • AO iteratively optimizes the phase shift of one reflecting element while fixing all other element phase shifts.
  • The per-element optimization is formulated as a single-variable non-convex problem after isolating the selected element's amplitude and phase terms.
  • The method repeatedly cycles through all N elements until the objective converges, using a closed-form approximate solution for each scalar subproblem.

C. An Approximate Solution to (P2)

The paper restricts each element's search to a trust region between phase alignment and the phase of maximum reflection amplitude. It then approximates the scalar objective with a quadratic model evaluated at three points.

  • Under the practical model, phase alignment alone may be suboptimal because zero phase shift can produce the lowest reflection amplitude.
  • The selected phase should balance amplitude and phase alignment, deviating toward π or −π when the aligned phase is near zero.
  • A high-quality numerical solution searches over the trust region from arg(ϕ_n) to (−1)^λπ.
  • The closed-form approximation fits a quadratic through three equally sampled trust-region points, including θ_A and θ_B.
  • The proposition provides the approximate scalar solution, and the resulting AO procedure iterates these updates until convergence.

V. SIMULATION RESULTS

Simulations evaluate achievable rate in an IRS-assisted MISO system under continuous and discrete phase shifts. The practical-model AO designs outperform ideal-model beamforming when IRS-reflected links are influential, with the gap increasing as the IRS becomes stronger.

  • The simulated system uses a two-antenna AP, one single-antenna user, and an IRS whose AP and IRS are separated by 500 m.
  • Fig. 5 averages achievable rate over 1000 channel realizations while varying the AP-user horizontal distance.
  • As the user moves toward the IRS, the practical-model optimized design increasingly outperforms ideal-model beamforming evaluated with the practical model.
  • At d = 498 m, the performance gap between the practical optimized and ideal-model schemes increases with N as the IRS reflecting channel strengthens.
  • With discrete phase shifts, achievable-rate performance increases with the number of representation bits b; b = 2 under the practical model exceeds the ideal model with b →∞.

VI. CONCLUSION

The paper proposes a practical IRS phase shift model and jointly optimizes transmit and reflect beamforming to maximize achievable rate. Simulations show that using the conventional ideal model can cause significant performance loss relative to the practical model.

  • The practical phase shift model captures IRS behavior more realistically than the conventional ideal model.
  • The authors formulate and solve a joint transmit and reflect beamforming optimization problem for an IRS-aided MISO system.The optimization maximizes achievable rate using alternating optimization.
  • Beamforming optimization based on the conventional ideal phase shift model may lead to significant performance loss compared with the proposed practical model.
  • Future work should examine the performance difference in broader IRS-assisted systems, including multi-user, OFDM, physical-layer security, and SWIPT settings.

APPENDIX A BEAMFORMING OPTIMIZATION

The appendix constructs a quadratic approximation using three selected phase points and their function values, then derives the stationary point used in the optimization result.

  • Three phase points θA, θB, and θC, together with their function values, determine the quadratic approximation.
  • The constants a0, a1, and a2 are obtained by substituting the three phase points into the approximation conditions.
  • The stationary point of g(θn) is substituted into the derived expression to obtain (16), completing the proof.
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