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Enabling Panoramic Full-Angle Reflection via Aerial Intelligent Reflecting Surface

Haiquan Lu, Yong Zeng, Shi Jin, Rui Zhang

arXiv:2001.07339v1cs.ITmath.IT

TL;DR

The paper addresses limited deployment flexibility, half-space reflection, and repeated-reflection losses in terrestrial IRS systems. It proposes AIRS-based 3D networking and jointly optimizes beamforming, placement, and phase shifts for worst-case area SNR, using a closed-form single-location solution and an efficient area-coverage design. Numerical results show significant improvement over heuristic AIRS deployment schemes.

  • Problem

    Terrestrial IRS deployment is constrained, typically covers only half-space, and may require multiple lossy reflections, motivating flexible panoramic aerial reflection.

    Source: §I · pp. 1–2

  • Method

    The paper jointly optimizes transmit beamforming, AIRS placement, and phase shifts, deriving a closed-form single-location solution and an efficient area-coverage suboptimal design.

    Source: §I · p. 2, B. Area Coverage Enhancement · p. 4, §V · p. 6

  • Results

    Numerical results show that the proposed design significantly improves performance over heuristic AIRS deployment schemes.

    Source: §V · p. 6, §IV · p. 6

  • Takeaways & Limitations

    AIRS provides a 3D networking architecture for efficient sky-based signal coverage with panoramic reflection and fewer required reflections than terrestrial IRS.

    Source: §I · pp. 1–2, §V · p. 6

Abstract

from arXiv · show

This paper proposes a new three dimensional (3D) networking architecture enabled by aerial intelligent reflecting surface (AIRS) to achieve panoramic signal reflection from the sky. Compared to the conventional terrestrial IRS, AIRS not only enjoys higher deployment flexibility, but also is able to achieve 360$^\circ$ panoramic full-angle reflection and requires fewer reflections in general due to its higher likelihood of having line of sight (LoS) links with the ground nodes. We focus on the problem to maximize the worst-case signal-to-noise ratio (SNR) in a given coverage area by jointly optimizing the transmit beamforming, AIRS placement and phase shifts. The formulated problem is non-convex and the optimization variables are coupled with each other in an intricate manner. To tackle this problem, we first consider the special case of single-location SNR maximization to gain useful insights, for which the optimal solution is obtained in closed-form. Then for the general case of area coverage, an efficient suboptimal solution is proposed by exploiting the similarity between phase shifts optimization for IRS and analog beamforming for the conventional phase array. Numerical results show that the proposed design can achieve significant performance gain than heuristic AIRS deployment schemes.

I. INTRODUCTION

Terrestrial IRS deployments face installation constraints, half-space coverage, and attenuation from multiple reflections, motivating AIRS for flexible 3D, panoramic reflection. The paper formulates and addresses worst-case coverage optimization using AIRS placement, phase shifts, and transmit beamforming.

Motivation

  • Terrestrial IRS installation can require difficult site selection, rent, urban-landscape considerations, and property-owner approval.Source: §I · p. 1
  • Terrestrial IRS generally serves only terminals on one side, whereas AIRS enables 360° panoramic full-angle reflection between ground nodes.Source: §I · pp. 1–2, Figure 1 · p. 1
  • Multiple terrestrial-IRS reflections in complex environments cause signal attenuation through scattering toward undesired directions.Source: §I · p. 1, Figure 2 · p. 1

AIRS architecture

  • AIRS uses aerial platforms such as balloons or UAVs to provide intelligent reflection from the sky with flexible placement or trajectory optimization.Source: §I · pp. 1–2, Figure 3 · p. 2

Problem and approach

  • The paper maximizes minimum target-area SNR by jointly optimizing source beamforming, AIRS placement, and phase shifts despite a coupled non-convex formulation.Source: §I · p. 2
  • Single-location optimization yields a closed-form solution, while area coverage uses an efficient design exploiting similarity between IRS phase shifts and analog beamforming.Source: §I · p. 2

II. SYSTEM MODEL AND PROBLEM FORMULATION

The system model places a passive reflecting array on an aerial platform above a rectangular target area and models source–AIRS and AIRS–area links under high-probability LoS conditions. The formulation maximizes the area’s minimum received SNR over AIRS placement, reflection phases, and source transmit beamforming.

System model

  • The modeled system uses a source with M antennas and an AIRS ULA of N passive elements separated by d < λ above a rectangular coverage area.The direct source-to-area link is assumed negligible because of severe blockage.Source: §II · p. 2
  • The AIRS is placed at altitude H, with horizontal reference coordinate q, while source-to-AIRS and AIRS-to-location distances determine the modeled links.Source: §II · p. 2

Channel model

  • The model assumes aerial-to-ground links are LoS with high probability and therefore adopts a free-space path-loss model for channel power gains.Source: §II · p. 2

Signal model

  • The AIRS uses element-wise phase shifts represented by a diagonal phase-shift matrix, while the source applies a unit-norm transmit beamforming vector.Source: §II · pp. 2–3

Channel model

  • AIRS placement affects both source-to-AIRS path loss and the source-link angles of departure and arrival.Source: §II · p. 2

Problem formulation

  • The optimization maximizes the minimum SNR over the rectangular area by jointly selecting AIRS placement q, phase vector θ, and beamforming vector v.The problem is difficult because the objective is an area-wide minimum and q, θ, and v are intricately coupled in a highly non-convex objective.Source: §II · p. 3

III. PROPOSED SOLUTIONS

The proposed solution first derives the optimal transmit beamforming for the AIRS-assisted link, showing that it is independent of the destination location.

III. PROPOSED SOLUTIONS

  • The optimal transmit beamforming vector is the source array response toward the AIRS, normalized to unit norm.This follows because the concatenated channel produces a rank-one matrix whose dominant eigenvector is the source-to-AIRS steering vector.Source: §III · p. 3
  • For fixed placement, phase shifts, and destination location, maximum-ratio transmission is obtained as the dominant eigenvector of the concatenated channel matrix.The channel outer product is rank one, simplifying the beamforming optimization.Source: §III · p. 3
  • The optimal beamforming remains valid for every destination location because it is independent of the AIRS-to-destination position.Thus, the same source beamforming choice is optimal regardless of w.Source: §III · p. 3

A. Single-Location SNR Maximization

For a single destination location, the paper obtains a closed-form AIRS placement characterized by the height-to-distance ratio, with symmetric optimal horizontal locations.

A. Single-Location SNR Maximization

  • The single-location problem removes the inner area minimization because the destination position is known.The resulting optimization focuses directly on maximizing SNR at one specified location.Source: A. Single-Location SNR Maximization · p. 3
  • At the optimum, AIRS elements coherently add their reflected rays through coordinated phase shifts.The common phase component does not alter the coherent-combining principle.Source: A. Single-Location SNR Maximization · p. 3
  • The closed-form optimal horizontal placement depends only on ρ, the AIRS height-to-source-distance ratio.This removes dependence on transmit SNR in the single-location placement result.Source: A. Single-Location SNR Maximization · pp. 3–4
  • For ρ ≥ 1/2, the AIRS should be placed above the midpoint between the source and destination nodes.The result identifies the midpoint as the optimal horizontal position in this regime.Source: A. Single-Location SNR Maximization · p. 4
  • For ρ < 1/2, two optimal horizontal AIRS locations exist symmetrically about the source-destination midpoint.The two-location symmetry is illustrated by the deployment-coefficient curve in Fig. 4.Source: A. Single-Location SNR Maximization · p. 4, Figure 4 · p. 4

B. Area Coverage Enhancement

For area coverage, the paper decouples phase-shift design from AIRS placement by approximating the worst-case objective through array gain before optimizing placement.

B. Area Coverage Enhancement

  • The proposed area-coverage method uses a two-step suboptimal design that separates phase-shift optimization from AIRS placement.The decomposition exploits the similarity between IRS phase control and analog beamforming in conventional phase arrays.Source: B. Area Coverage Enhancement · p. 4
  • The objective separates into an AIRS passive-beamforming array-gain term and a concatenated path-loss term.The array-gain term depends on phase shifts and location, whereas the path-loss term accounts for the concatenated propagation loss.Source: B. Area Coverage Enhancement · p. 4
  • The first step maximizes worst-case AIRS array gain for a fixed placement while ignoring concatenated path loss.This approximation is motivated by array gain being more sensitive to location variation than path loss under the stated distance condition.Source: B. Area Coverage Enhancement · p. 4
  • The second step substitutes the phase-shift solution into the worst-case SNR objective and optimizes the AIRS placement.Thus, placement is selected using the resulting area-wide SNR rather than array gain alone.Source: B. Area Coverage Enhancement · p. 4

1) Phase Shifts Optimization:

The phase-shift design treats AIRS reflection like analog beamforming, relating array gain to spatial-frequency deviation and partitioning elements when full-array coverage is insufficient.

Phase alignment

  • To maximize coherent reception at a target location, each reflecting element receives a phase shift that compensates its spatial-frequency-dependent phase.The common phase offset is arbitrary, while element-dependent shifts align the reflected signals at the selected location.Source: 1) Phase Shifts Optimization · p. 4

Array-gain behavior

  • Array gain decreases as spatial-frequency deviation increases, with nulls and beamwidth determined by the effective sub-array aperture.The beamwidth is inversely proportional to the array aperture, while the 3-dB beamwidth provides the coverage criterion.Source: 1) Phase Shifts Optimization · p. 4

Full-array coverage

  • The full-array architecture covers the target area when its 3-dB beamwidth exceeds the maximum spatial-frequency deviation across that area.The phase shifts are then designed for the center of the rectangular area.Source: 1) Phase Shifts Optimization · p. 5

Sub-array coverage

  • When full-array coverage fails, the AIRS partitions its N elements and spatial-frequency range into L sub-arrays, trading reduced peak gain for wider coverage.Each sub-array uses N/L elements, reducing its spatial-frequency span and increasing its beamwidth by L times.Source: 1) Phase Shifts Optimization · p. 5, Figure 6 · p. 5

Worst-case design

  • The resulting worst-case SNR for a fixed placement occurs at a boundary location after the phase shifts are selected for full-array or sub-array coverage.The placement-dependent architecture is chosen according to the relationship between maximum spatial-frequency deviation and 3-dB beamwidth.Source: 2) AIRS Placement Optimization · p. 5

2) AIRS Placement Optimization:

AIRS placement optimization reduces to a one-dimensional search because the optimal placement lies on the x-axis, yielding an efficient suboptimal solution for the area-coverage problem.

2) AIRS Placement Optimization:

  • Because the placement objective lacks a closed-form expression, the AIRS horizontal coordinate is optimized by one-dimensional search with q = [x, 0]^T.The placement-dependent phase-shift solution is first obtained, then evaluated during the search.Source: 2) AIRS Placement Optimization · p. 5

IV. NUMERICAL RESULTS

Numerical results show that optimizing AIRS placement substantially improves single-location and area-coverage SNR compared with heuristic placements.

Single-location SNR

  • A target SNR of 5 dB requires about 360 elements with benchmark placement but only 140 with optimal placement.For both placements, SNR increases with the number of reflecting elements and coherent phase shifts are applied.Source: §IV · p. 6, Figure 7 · p. 6

Area coverage

  • For area coverage, worst-case SNR generally decreases as the AIRS moves from the source toward the target area, making placement above the source optimal.Near the target, larger spatial-frequency deviation requires sub-arrays and sacrifices peak gain; near the source, concatenated path loss is smaller.Source: §IV · p. 6, Figure 8 · p. 6

Transmit-power evaluation

  • With N = 200, optimal placement above the source significantly outperforms placement above the rectangular area's center as transmit power varies.The comparison supports jointly optimizing AIRS deployment and active/passive beamforming.Source: §IV · p. 6, Figure 9 · p. 6

V. CONCLUSION

The paper develops AIRS-based 3D networking for sky-based coverage, combining closed-form single-location optimization with suboptimal sub-array design for general area coverage.

V. CONCLUSION

  • The proposed design jointly optimizes transmit beamforming, AIRS placement, and phase shifts to maximize worst-case SNR in a target area.The single-location problem has a closed-form optimal solution, while area coverage uses an efficient sub-array-based suboptimal solution.Source: §V · p. 6
  • Numerical results show significant performance improvement over heuristic AIRS deployment schemes.Source: §V · p. 6

APPENDIX A PROOF OF PROPOSITION 2

The proof reduces the placement problem to a scalar ratio ξ by projecting the reference point onto the source–target line. It then analyzes the resulting cubic derivative across discriminant cases to establish the minimizing value ξ = 1.

Reduction

  • The placement problem is reduced to q = ξ ŵ, where ξ is a ratio coefficient and the perpendicular component is set to zero.This simultaneously minimizes the relevant distances in the projection argument.Source: Appendix A · p. 6

Cubic analysis

  • The derivative is transformed into the depressed cubic f′(ζ) = ζ^3 + aζ + b, whose roots are classified using the discriminant.The proof considers the cases of one, repeated, or three real solutions.Source: Appendix A · pp. 6–7

Case 1

  • When the derivative is monotonically increasing, the monotonicity of f(ξ) is characterized through its stationary point, as illustrated in Fig. 10.The cited case has one real solution for the derivative equation.Source: Appendix A · p. 7, Figure 10 · p. 7

Conclusion

  • Across the analyzed cases, f(ξ) reaches its minimum at ξ = 1.The proof states this conclusion after the monotonicity analysis for the relevant cases.Source: Appendix A · p. 7

Case 3

  • In the three-real-solution case, the proof examines derivative intervals and plots the resulting monotonicity behavior in Fig. 11.The case is identified by a negative discriminant.Source: Appendix A · p. 7, Figure 11 · p. 7

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