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Aerial Intelligent Reflecting Surface: Joint Placement and Passive Beamforming Design with 3D Beam Flattening

Haiquan Lu, Yong Zeng, Shi Jin, Rui Zhang

arXiv:2007.13295v2eess.SP

TL;DR

The paper addresses worst-case SNR coverage with an aerial intelligent reflecting surface by jointly designing source beamforming, AIRS placement, and 3D passive beamforming. It derives closed-form single-location solutions and proposes decoupled beamforming-placement optimization with adjustable flattened beams for area coverage. The proposed designs achieve significant performance gains over benchmark schemes, including about 25 dB in one reported placement comparison.

  • Problem

    The paper studies how to extend ground-source communication coverage over a target area using AIRS while maximizing the minimum SNR across all locations despite a difficult joint non-convex design problem.

  • Method

    The paper jointly designs source MRT, AIRS placement, and 3D passive beamforming, using closed-form single-location analysis and sub-array-based beam broadening and flattening for area coverage.

  • Results

    About 25 dB SNR gain is reported for optimized AIRS placement over a benchmark placement in the rectangular-target-area comparison.

  • Takeaways & Limitations

    AIRS placement and passive beamforming can be efficiently coordinated by balancing target-area angular span with cascaded channel path loss.

Abstract

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This paper proposes a new three-dimensional (3D) wireless passive relaying system enabled by aerial IRS (AIRS). Compared to the conventional terrestrial IRS, AIRS enjoys more deployment flexibility as well as wider-range signal reflection, thanks to its high altitude and thus more likelihood of establishing line-of-sight (LoS) links with ground source/destination nodes. Specifically, we aim to maximize the worst-case signal-to-noise ratio (SNR) over all locations in a target area by jointly optimizing the transmit beamforming for the source node and the placement as well as 3D passive beamforming for the AIRS. The formulated problem is non-convex and thus difficult to solve. To gain useful insights, we first consider the special case of maximizing the SNR at a given target location, for which the optimal solution is obtained in closed-form. The result shows that the optimal horizontal AIRS placement only depends on the ratio between the source-destination distance and the AIRS altitude. Then for the general case of AIRS-enabled area coverage, we propose an efficient solution by decoupling the AIRS passive beamforming design to maximize the worst-case array gain, from its placement optimization by balancing the resulting angular span and the cascaded channel path loss. Our proposed solution is based on a novel 3D beam broadening and flattening technique, where the passive array of the AIRS is divided into sub-arrays of appropriate size, and their phase shifts are designed to form a flattened beam pattern with adjustable beamwidth catering to the size of the coverage area. Both the uniform linear array (ULA)-based and uniform planar array (UPA)-based AIRSs are considered in our design, which enable two-dimensional (2D) and 3D passive beamforming, respectively. Numerical results show that the proposed designs achieve significant performance gains over the benchmark schemes.

I. INTRODUCTION

The paper introduces AIRS as a flexible 3D passive-relaying architecture and formulates joint source beamforming, AIRS placement, and 3D passive beamforming for worst-case SNR coverage. It develops closed-form single-location insights and an efficient area-coverage design using beam broadening and flattening.

  • Motivation: Terrestrial IRS deployment is constrained by installation difficulties and typically supports terminals in only half of the space.Building-wall or facade deployment also causes signal attenuation when multiple reflections scatter power toward undesired directions.
  • Motivation: AIRS mounts passive IRS elements on aerial platforms to provide elevated, flexible 3D wireless relaying.Its elevated position supports more likely LoS links with ground nodes, while aerial placement adds a network-design degree of freedom.
  • Motivation: AIRS can provide panoramic or full-angle reflection and often achieve desired signal manipulation with one reflection in complex urban environments.This reduces signal power loss associated with multiple terrestrial-IRS reflections.
  • Challenges: AIRS introduces practical challenges involving aerial-platform endurance, stability, controllability, flight control, safety, drift, vibration, and channel estimation.The paper identifies these issues as requiring further investigation.
  • Problem Formulation: The paper maximizes worst-case SNR across a target area by jointly optimizing source transmit beamforming, AIRS placement, and 3D passive beamforming.The problem is difficult because beamforming must serve all target-area locations and placement affects both cascaded path loss and coverage geometry.
  • Solution Structure: The source's optimal transmit beamforming is maximum ratio transmission (MRT), reducing the remaining problem to AIRS placement and 3D passive beamforming.The placement optimization balances angular span against cascaded channel path loss after beamforming decoupling.
  • Single-Location Design: For a single target location, the optimal AIRS placement and passive phase shifts are derived in closed form, with horizontal placement governed by ρ, the source-destination distance-to-altitude ratio.For 0 ≤ ρ ≤ 2, placement is above the midpoint; for ρ > 2, two symmetric optimal horizontal locations exist.
  • Area-Coverage Design: For area coverage, the proposed two-step method decouples passive beamforming from placement optimization using flattened beams whose width matches the target area.The array is partitioned into sub-arrays; ULA-based AIRS supports one-dimensional flattening, while UPA-based AIRS supports 3D flattening over two spatial-frequency dimensions.

II. SYSTEM MODEL AND PROBLEM FORMULATION

The system models AIRS-assisted coverage extension from a ground source to a rectangular target area, optimizing source transmission, AIRS placement, and 3D phase shifts for worst-case SNR. MRT reduces the problem to placement and passive beamforming, but the resulting min-SNR optimization remains highly non-convex.

  • System model: An AIRS assists a ground source in extending communication coverage to a rectangular terrestrial area A.The source is placed at the origin, while the target-area center lies on the x-axis.
  • System model: The AIRS uses a sub-wavelength UPA with N = NxNy reflecting elements and fixed altitude H > 0.Element counts are specified along the x- and y-axes, with a conventional source UPA containing M = MyMz antennas.
  • Channel model: LoS-dominated aerial-ground links are modeled with free-space path loss, while placement changes both channel power gain and link angles.The AIRS-to-ground and source-to-AIRS distances determine the cascaded channel behavior.
  • Problem formulation: The objective maximizes the minimum SNR across A by jointly optimizing source beamforming, AIRS placement q, and element phase shifts θ.Each reflecting element has a phase shift constrained to [0, 2π).
  • Problem reduction: MRT is optimal for the source and is independent of the target location, reducing the problem to joint AIRS placement and 3D passive beamforming.The rank-one effective channel yields the source beamforming vector aligned with the transmit array response.
  • Problem reduction: The reduced min-SNR problem is difficult because the two-dimensional worst-case objective is implicit and placement and phase shifts are intricately coupled.The optimization is explicitly characterized as highly non-convex.

III. OPTIMIZATION FOR SNR MAXIMIZATION AT SINGLE TARGET LOCATION

For a single target location, the AIRS phase shifts coherently combine reflected rays, and the optimal horizontal placement has a closed form governed by the distance-to-altitude ratio. The altitude choice also creates a practical performance trade-off under the LoS assumption.

  • Single-location formulation: For a single target location w1, the inner worst-case minimization disappears and the problem reduces to maximizing SNR at that point.The AIRS phase shifts remain constrained by 0 ≤ θn < 2π.
  • Optimal passive beamforming: The optimal phase shifts make the reflected rays add coherently at w1 for any fixed AIRS placement.A common phase offset may be shared by all reflecting elements.
  • Optimal placement: The optimal 2D AIRS placement is available in closed form.The placement is expressed through the deployment coefficient ξ∗(ρ).
  • Optimal placement: For 0 ≤ ρ ≤ 2, the AIRS is placed above the midpoint; for ρ > 2, two symmetric optimal placements exist around that midpoint.Here ρ is the ratio between source-destination distance and AIRS altitude.
  • Optimal placement: The optimal horizontal placement depends only on ρ, unlike conventional active-relay placement, which generally depends on transmit power and other factors.The comparison is stated specifically for the single-location solution.
  • Deployment trade-off: Increasing altitude H decreases optimal SNR, while sufficiently large H may be required to obtain LoS links with both source and target.This creates an altitude-selection trade-off for practical AIRS deployment.

IV. OPTIMIZATION FOR MIN-SNR MAXIMIZATION IN AREA COVERAGE

Area coverage is addressed first with ULA-based AIRS beam steering and then extended to UPA-based AIRS for 3D passive beamforming over both spatial-frequency dimensions.

  • ULA-based AIRS: The area-coverage design begins with ULA-based AIRS, where passive beamforming operates only along the x-dimension.The ULA case sets Ny = 1 and N = Nx.
  • UPA-based AIRS: The proposed solution is extended to UPA-based AIRS, enabling 3D passive beamforming over both spatial-frequency dimensions.The two dimensions are denoted by Φ̄ and Ω̄.

A. The Special Case of ULA-Based AIRS

For ULA-based AIRS, the design decouples phase-shift optimization from placement optimization: the first maximizes worst-case array gain, while the second accounts for path loss and coverage geometry. The resulting area-coverage SNR is a lower bound on the optimum.

  • ULA formulation: The ULA model sets Ny = 1 and represents each phase shift as θn = θnx.The resulting SNR is given by the reduced ULA expression.
  • Two-step optimization: The proposed two-step method decouples AIRS phase-shift design from placement optimization.The phase shifts target worst-case array gain, while placement balances angular span against cascaded path loss.
  • Two-step optimization: For fixed placement, the phase shifts maximize the minimum array gain across all target-area locations.This reframes passive beamforming as a worst-case array-design problem.
  • Approximation: The phase-shift approximation ignores location-dependent cascaded path loss during the first step, which is reasonable when array gain varies more sensitively.The stated conditions are especially small target areas or H ≫ Dx and Dy.
  • Approximation: The resulting max-min SNR after the two-step procedure is a lower bound on the optimal value of the original area-coverage problem.The bound follows from solving the approximated phase-design problem before placement optimization.

1) Beam Broadening and Flattening for Passive Beamforming:

The proposed beam broadening and flattening method partitions the AIRS array into sub-arrays and steers them to form a flattened beam whose width matches the target area. This creates a trade-off between wider coverage and lower worst-case array gain.

  • Beam flattening: The N-element array is partitioned into L sub-arrays, whose phase shifts are optimized to produce an approximately equal array gain across the target spatial-frequency interval.The sub-array size is Ns = N/L, and the design targets a flattened beam over [∆min(q), ∆max(q)].
  • Beam steering: Each sub-array steers toward a distinct spatial frequency, with adjacent directions separated by the sub-array spatial-frequency resolution 1/(Ns d̄x).The resulting shifted sub-array beams are combined to broaden the overall coverage beam.
  • Phase design: The sub-array common phases are selected to maximize array gain at intersections of adjacent sub-array patterns, reducing fluctuations between neighboring beams.For sufficiently large Ns, the intersection-point gain is comparable to the gain at individual sub-array beam directions.
  • Design trade-off: The coverage beamwidth is broadened by a factor of L^2, while the worst-case array gain undergoes the same proportional reduction for fixed N.Thus, selecting L requires balancing coverage width against worst-case gain.
  • Scaling behavior: For relatively wide target areas or large N, the worst-case array gain increases linearly with N because elements are partitioned into sub-arrays.This differs from the quadratic increase associated with full-array single-location beamforming.

2) AIRS Placement Optimization:

AIRS placement is optimized after passive beamforming by balancing the target area's angular span against cascaded path loss. For a symmetric target area, the placement reduces to a one-dimensional search along the x-axis.

  • Decoupled optimization: The passive beamforming design is decoupled from placement, allowing the placement problem to use the resulting worst-case array-gain characterization.Substituting the beamforming result simplifies the placement optimization problem.
  • Placement criterion: The optimal placement balances minimizing the target area's spatial-frequency span with minimizing cascaded path loss to the worst-case destination.These competing objectives determine a non-trivial placement rather than simply moving the AIRS toward the target area.
  • Worst-case location: For a fixed AIRS placement, the worst-case SNR occurs at the target-area location farthest from the AIRS, which lies on the area's boundary.The relevant boundary point is arg max_{w∈A} ∥q − w∥^2.
  • Geometric simplification: Because the target area is symmetric about the x-axis, the optimal placement lies on that axis as q = [qx, 0]^T.The remaining placement variable qx is optimized in one dimension.
  • Algorithm: A one-dimensional search selects qx by minimizing the placement cost over a prescribed search interval and returns the corresponding passive beamforming design.The algorithm evaluates distance, spatial-frequency span, and the resulting cost for candidate placements.

B. The General Case of UPA-Based AIRS

For UPA-based AIRS, the design extends beam broadening and flattening to both spatial-frequency dimensions. A separable phase-shift structure enables efficient 3D passive beamforming and placement optimization.

  • 3D beamforming: UPA-based AIRS supports passive beam steering over both spatial-frequency dimensions Φ̄ and Ω̄, unlike the single-axis ULA formulation.Each element's phase shift generally affects both dimensions, making the unrestricted optimization difficult.
  • Separable phase design: The proposed simplification restricts each phase shift to θ_{nx,ny} = θ_{nx} + θ_{ny}, reducing independent variables from NxNy to Nx + Ny.This separable design is sub-optimal in general but enables decoupled steering along the two axes.
  • Sub-array sizing: The target area's spatial-frequency spans are computed separately along the x- and y-axes, then used to determine the required numbers of sub-arrays Lx and Ly.The two broadened beams are tuned so their coverage widths match the target area.
  • 3D beam construction: Applying the ULA phase design independently along x and y produces a 3D broadened and flattened beam pattern matched to the target coverage area.Sub-array directions and common phase terms are then assigned for both axes.
  • Placement optimization: UPA placement optimization balances angular spans along both axes with cascaded path loss, and the optimized x-coordinate can be found by one-dimensional search.The resulting joint placement and 3D beamforming procedure provides an efficient solution for the general UPA case.

V. NUMERICAL RESULTS

Numerical results show that AIRS placement and passive beamforming strongly affect worst-case coverage SNR, with performance gains from optimized deployment and 3D beam broadening and flattening.

  • Single-location SNR maximization: For single-location maximization, the optimal placement significantly outperforms midpoint placement, while SNR increases quadratically with the number of AIRS elements.The optimal deployment places the AIRS close to the source or target for the considered setup.
  • ULA-based area coverage: As the ULA-based AIRS moves toward the coverage center, required subarrays increase stepwise because the target area's angular span widens.Broader beams require more subarray partitions and reduce worst-case array gain.
  • ULA-based area coverage: The optimal ULA placement depends on coverage-area size and balances angular span against cascaded path loss, sometimes favoring a location left of the source.For one considered setup, higher array gain dominates the increased concatenated path loss.
  • ULA-based area coverage: 25 dB SNR gain is achieved by optimized ULA placement over placement above the target-area center in the transmit-power comparison.The result demonstrates the importance of joint AIRS deployment and beamforming design.
  • UPA-based area coverage: 30 dB SNR gain is achieved by proposed 3D passive beamforming over benchmark 1D beamforming for UPA-based coverage.The proposed design maintains approximately equal array gain across target locations, whereas the benchmark cannot steer over one angular dimension and may yield zero SNR at specific locations.

VI. CONCLUSION

The paper develops passive AIRS relaying for sky-based coverage extension by jointly optimizing source transmission, AIRS placement, and 3D passive beamforming. Closed-form single-location design and beam-broadening-based area-coverage design produce significant gains over benchmark schemes.

  • Conclusion: The proposed AIRS relaying system maximizes worst-case SNR over a target area through joint source beamforming, AIRS placement, and 3D passive beamforming.The system is intended for coverage extension over the sky.
  • Conclusion: Single-location SNR maximization yields a closed-form optimal solution whose horizontal AIRS placement depends only on the source-destination distance to AIRS-altitude ratio.This provides an analytical design for the special case.
  • Conclusion: Area coverage is addressed by decoupling worst-case array-gain beamforming from placement optimization through a balance between angular span and cascaded path loss.The proposed beam broadening and flattening technique forms an adjustable flattened beam suited to the coverage-area size.
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