PaperlayerView exact revision on arXiv

Source-linked AI summary

Intelligent Reflecting Surface Deployment for Low-Altitude Coverage: Illumination Geometry, Directional Characteristics, and Optimization

Guoying Zhang, Qingqing Wu, Ailing Zheng, Xingxiang Peng, Wen Chen, Wei Feng

arXiv:2608.30586v1cs.IT

TL;DR

Fixed-downtilt terrestrial BSs provide uneven low-altitude 3D coverage, motivating passive enhancement without changing existing BS configurations. The paper models radiation-pattern-aware quasi-static rooftop IRS deployment, jointly optimizing sites, orientations, and phases for worst-case SNR. Analytical characterizations and simulations support the design, with the proposed scheme outperforming considered benchmarks across tested deployment budgets.

  • Problem

    Fixed-downtilt BSs can provide weak and spatially varying low-altitude coverage despite LoS propagation, while transmit-side remedies trade aerial performance against terrestrial service or require additional infrastructure.

  • Method

    The paper builds a radiation-pattern-aware 3D channel model and solves a budget-constrained quasi-static rooftop IRS max-min SNR problem using mixed-integer alternating optimization.

  • Results

    Higher worst-case SNR than every considered benchmark is achieved across all tested deployment budgets, while simulations validate the analytical characterizations.

  • Takeaways & Limitations

    Fixed rooftop IRS sites with optimized orientations and phase shifts can enhance low-altitude coverage while preserving the existing fixed-downtilt BS configuration.

Abstract

from arXiv · show

Terrestrial base stations (BSs) are typically configured with fixed downtilt to serve ground users, resulting in weak illumination of low-altitude airspace even under line-of-sight (LoS) propagation. In this paper, we establish a channel model that incorporates BS and intelligent reflecting surface (IRS) radiation patterns for three-dimensional (3D) low-altitude coverage while preserving the existing BS configuration. We formulate a budget-constrained IRS deployment problem that jointly determines candidate-site selection, IRS orientations, and phase shifts to maximize the worst-case signal-to-noise ratio (SNR) over the 3D low-altitude airspace. The selected sites and optimized IRS parameters remain fixed after deployment, yielding a quasi-static IRS configuration. We characterize the illumination geometry between the fixed-downtilt BS and rooftop candidates by deriving the nonnegative installation-height range satisfying the BS main-lobe condition. The separation between the mapped main-lobe height boundaries grows linearly with horizontal BS-to-site distance and decreases inversely with the number of BS antennas. We further derive an analytical lower bound on the regional worst-case normalized array gain achievable through IRS phase design over served directions with different direction spans. The resulting sufficient direction span decreases inversely with the square root of the number of IRS elements when the same worst-case normalized gain guarantee is maintained. We develop a mixed-integer alternating optimization (AO) algorithm to solve the resulting problem. Simulation results validate the analytical characterizations and show that the proposed scheme achieves higher worst-case SNR than benchmarks across different deployment budgets.

I. INTRODUCTION

The paper addresses weak and spatially varying low-altitude coverage from fixed-downtilt terrestrial BSs by deploying quasi-static rooftop IRSs without changing the BS configuration. It jointly optimizes sites, orientations, and phase shifts for worst-case 3D-airspace SNR, supported by analytical characterizations and a mixed-integer AO algorithm.

  • Motivation: Fixed-downtilt BSs can leave low-altitude locations weakly served despite LoS propagation, especially near sidelobes or radiation nulls.The weakest locations can limit service quality across the region.
  • Motivation: Changing BS downtilt improves aerial gain but reduces ground-user gain and increases inter-cell interference, while densification and active relays require additional power and RF hardware.These trade-offs motivate passive coverage enhancement.
  • Related work: IRSs provide configurable passive signal redirection through adjustable element phase shifts without RF chains, reducing power consumption and hardware cost relative to active relays.Prior aerial designs often adapt reflection coefficients to terminal locations or trajectories.
  • Contributions: The paper studies quasi-static rooftop IRS deployment under fixed BS downtilt, keeping selected sites, orientations, and phase shifts fixed across terminal locations while maximizing worst-case SNR.The formulation jointly handles candidate-site selection, orientations, and phase shifts under a deployment budget.
  • Contributions: The analysis shows main-lobe height-boundary separation grows linearly with BS-to-site distance and decreases inversely with BS antenna count, while sufficient IRS direction span scales inversely with the square root of IRS elements.These characterize illumination geometry and directional behavior under different array sizes.
  • Contributions: The proposed mixed-integer AO method validates the analytical results and achieves higher worst-case SNR than every considered benchmark across tested deployment budgets.Comparisons also show the importance of optimizing IRS orientation and phase shifts while accounting for element radiation patterns.

B. Propagation Model

The propagation model represents direct and IRS-assisted links as LoS-dominant far-field channels while incorporating BS array radiation and IRS element radiation patterns. IRS reflection phases enter the cascaded channel and the resulting received-signal SNR.

  • Channel assumptions: The direct BS-to-location, BS-to-IRS, and IRS-to-location links are modeled as frequency-flat LoS channels in the far field.The model uses propagation distances and large-scale attenuation for each link.
  • Radiation patterns: The BS antenna gain combines a 3GPP element pattern with an array factor that depends on elevation angle, downtilt, and the number of BS antennas.The element gain includes boresight gain, half-power beamwidth, and sidelobe attenuation.
  • Radiation patterns: The IRS element radiation pattern is a front-side cosine-power directional gain with directivity exponent pI ≥ 2 and zero gain over the back side.The incident angle is measured relative to the IRS normal.
  • IRS-assisted channel: The cascaded BS–IRS–location channel combines BS-to-IRS and IRS-to-location propagation, BS and IRS radiation gains, array responses, and the diagonal phase-shift matrix.Each reflecting element applies a unit-amplitude phase shift φm,n ∈ [0, 2π).
  • Received signal and SNR: The received signal includes the direct link and selected IRS-assisted links, with SNR γu(s, τ, φ) determined by site selection, IRS orientations, phase shifts, transmit power, and noise.The binary selection variables determine which candidate IRS paths contribute.
  • Channel assumptions: The far-field model requires BS-to-IRS and IRS-to-location distances to exceed each IRS aperture’s Fraunhofer distance dF,m = 2D^2_ap,m/λ.This is an explicit modeling assumption for the cascaded links.

C. Problem Formulation

Because the target airspace is continuous, the paper approximates its minimum SNR using uniform samples augmented near predicted BS array-factor null heights. It then formulates a budget-constrained mixed-integer nonconvex max-min SNR problem.

  • Finite-region approximation: The finite optimization set combines uniform airspace samples with locations at predicted weak direct-link heights near fixed-downtilt BS array-factor nulls.The null heights are computed from valid null elevation angles at each sampled horizontal distance.
  • Finite-region approximation: The resulting location set is Q = Quni ∪ Qnull ⊂ V, and the objective maximizes the minimum SNR over Q.The augmented samples target locations likely to constrain worst-case direct-link performance.
  • Max-min formulation: An auxiliary variable Γ imposes a common SNR lower bound over all sampled locations while representing deployment budget, binary site selection, orientations, and phase shifts.The formulation jointly controls the rooftop deployment and IRS configuration.
  • Problem structure: Binary site variables and nonlinear orientation- and phase-dependent cascaded channels make the resulting problem mixed-integer and nonconvex.This structure prevents direct convex optimization of the complete formulation.
  • Approximation trade-off: The sampling interval Δv controls spatial resolution and constraint count; uniform samples scale as O(Δv^-3) for a fixed BS array.Reducing Δv improves both spatial and null-surface approximations but increases samples, with the uniform grid dominating growth.

III. BS AND IRS ARRAY-SIZE ANALYSIS

The section characterizes how a fixed-downtilt BS illuminates rooftop IRS candidates, deriving installation-height ranges and array-size scaling laws for the main-lobe boundaries.

  • BS illumination geometry: The BS main-lobe interval is defined by the connected elevation-angle region containing the downtilt direction and satisfying the prescribed array-factor loss threshold.For half-wavelength antenna spacing, the interval is expressed through a unique positive boundary in the array-factor variable.
  • BS illumination geometry: The nonnegative installation-height range is obtained by mapping the elevation boundaries to panel-center heights and subtracting each candidate building height.Candidates with an empty range are removed from the retained set used for subsequent optimization.
  • Main-lobe height scaling: WηB(ρm) = ρm(tan θhigh − tan θlow), so the height-boundary separation is linear in horizontal BS-to-site distance and independent of building height.Building height shifts both installation-height boundaries equally, affecting feasibility but not their separation.
  • Main-lobe height scaling: WηB(ρm) = Θ(ρm/Nt), showing that the separation decreases at an inverse-Nt rate as the BS array grows.The two boundaries converge toward the downtilt-aligned panel-center height as Nt increases.
  • Numerical validation: With ηB = 3 dB and θtilt = 8°, doubling Nt approximately halves the slope of WηB(ρm), while relative slope error falls from 5.60% at Nt = 4 to 0.34% at Nt = 16.The comparison uses Nt ∈ {4, 8, 16} and validates the large-array approximation.

B. Directional Characteristics of a Candidate IRS

The section analyzes the worst-case directional gain of a fixed-orientation IRS using one common phase vector over the outgoing directions viewed from a candidate site.

  • Directional gain formulation: The normalized worst-case array gain minimizes the squared normalized response over all outgoing directions in the considered direction set.For the full target region, the direction set contains the outgoing directions toward all sampled locations.
  • Analytical lower bound: Proposition 2 gives an analytical lower bound on the maximum achievable worst-case normalized array gain in terms of IRS element count Nm and direction span Dm.The bound depends jointly on (Nm − 1) sin2(Dm/4) under a fixed IRS orientation.
  • Array-size scaling: The sufficient direction span scales as Θ(Nm^-1/2) for fixed worst-case gain, so quadrupling Nm approximately halves the allowed span.The equivalent condition is (Nm−1)sin2(Dm/4) ≤ 6(1−√κ̄)/π2.
  • Numerical validation: The constructive phase design's evaluated worst-case gains remain above the analytical lower bounds for the tested array sizes and direction spans.Figure 4 compares the evaluated curves and lower bounds for Nm = 36, 100, and 196.

IV. ALTERNATING OPTIMIZATION ALGORITHM

The proposed mixed-integer alternating optimization decomposes deployment, IRS orientation, and reflection-vector variables into alternating update blocks.

  • Algorithm structure: The algorithm optimizes over the retained candidate set, with each installation height fixed according to the derived height-selection rule.The reflection vector uses unit-modulus entries and defines the IRS phase-shift matrix.
  • Algorithm structure: At each outer iteration, the method alternates updates of the deployment vector, IRS orientations, and reflection vectors.These variables form the three blocks used in the alternating updates.

A. Block Updates of the Proposed AO Algorithm

The AO algorithm alternates among deployment, IRS-orientation, and phase-shift updates to improve the sampled worst-case SNR under fixed blocks.

  • Deployment update: The deployment update reformulates binary site-selection products with auxiliary variables, yielding an exact MILP for fixed IRS orientations and reflection coefficients.Branch-and-bound can return a globally optimal deployment update for the current fixed blocks.
  • Orientation update: The orientation update applies successive convex approximation to construct a concave quadratic local SNR model around the current orientations.The resulting QCQP is convex and solvable using CVX.
  • Candidate acceptance: Backtracking evaluates candidate orientations and phase shifts against the true SNR, accepting them only when the prescribed conditions hold.The orientation candidate preserves or improves the current worst-case SNR when the local model does not exceed the true SNR at every location; phase candidates are likewise accepted conditionally.
  • Phase-shift update: The phase-shift update parameterizes unit-modulus coefficients by element phases and replaces the nonconcave SNR constraints with quadratic local models.The resulting phase increment is obtained by optimizing the minimum quadratic-model value across sampled locations.

B. Overall Procedure, Convergence, and Complexity

The complete AO procedure performs deployment, orientation, and phase updates whose accepted candidates make the sampled worst-case SNR sequence monotonically non-decreasing and convergent.

  • Overall procedure: Each AO block update preserves or increases the worst-case SNR, so the merit sequence is monotonically non-decreasing.Deployment feasibility, orientation acceptance, and phase acceptance provide the blockwise guarantees.
  • Convergence: The merit sequence converges because the sampled SNR minimum is bounded above at finite transmit power.Each sampled-location SNR is a finite sum of bounded terms.
  • Complexity: The deployment MILP has exponential branch-and-bound cost in the number of candidates, while orientation and phase blocks scale polynomially.The deployment MILP therefore dominates per-iteration cost for large candidate sets.
  • Complexity: With at most Kmax outer iterations, the overall complexity combines the three block-update costs.The supplied complexity expression is stated in terms of the deployment, orientation, and phase-block costs.

A. Simulation Setup and Baselines

The simulations use a 300 m × 300 m urban area with rooftop IRS candidates, sampled 3D target airspace, and deployment budgets varied across defined baselines.

  • Scenario and sampling: The urban scenario contains a 4 × 4 building grid, one rooftop-center IRS candidate per building, and a target airspace spanning [−70, 70] m × [−70, 70] m and altitudes [50, 110] m.The basic sampling grid contains 14 × 14 × 6 = 1176 locations before augmentation with predicted direct-link null heights.
  • Simulation parameters: The main setup uses fc = 3.5 GHz, B = 20 MHz, HB = 35 m, Nt = 8, θtilt = 8°, P0 = 37 dBm, and σ2 = −92 dBm.Additional parameters include λ/2 BS spacing, 8 dBi maximum BS gain, and rooftop heights uniformly drawn from m.
  • Evaluation settings: The study evaluates IRS element counts Nm ∈ {36, 100, 196}, orientation bounds, and deployment budgets Mmax ∈ {1, 2, 3, 4, 6, 8, 10}.The retained candidate set is determined by analytical main-lobe screening and projected installation heights.
  • Baselines: The baselines include No-IRS, Random-Site, Max-BI-Power, Centroid Phase, ERP-Agnostic, and Fixed-Tilt.They respectively remove IRSs, randomize sites, rank incident power, use centroid co-phasing, ignore element radiation patterns during design, or fix orientations.

B. Validation of Analytical Results and Algorithm

The simulations validate AO convergence and the analytical height characterization, while showing that larger deployment budgets improve worst-case SNR.

  • AO convergence: The worst-case SNR increases monotonically without oscillation and approaches a stable value within about ten to twelve outer iterations for Mmax = 2, 4, and 6.The Mmax = 6 case remains above the Mmax = 4 and 2 cases near convergence.
  • Mast-height validation: The favorable IRS mast-height range shifts downward as BS downtilt increases, with the SNR peak moving from hinst = 10 m at θtilt = 4° to hinst = 4 m at θtilt = 12°.The predicted 6.05 m height separation is consistent with the observed 6 m shift between the corresponding SNR peaks.

C. Coverage Performance and Design Insights

The proposed quasi-static IRS deployment improves worst-case 3D coverage by jointly optimizing site selection, orientations, and phase shifts. Results also show that IRS directivity and orientation freedom strongly shape coverage gains, especially near fixed-BS array-factor nulls.

  • Deployment budget: 5.84 dB at Mmax = 1 rises to 17.47 dB at Mmax = 6 and 19.66 dB at Mmax = 10, outperforming all five baselines.The gain is 11.63 dB from Mmax = 1 to Mmax = 6 but only 2.19 dB from Mmax = 6 to Mmax = 10.
  • IRS element directivity: For pI = 2, 4, and 6, the 3 dB half-width narrows from 45.0° to 32.8° and 27.0°, while the 10 dB half-width decreases from 71.6° to 55.8° and 47.2°.Increasing pI raises broadside gain but narrows the useful angular range.
  • Optimized deployment: The optimized Mmax = 6 deployment distributes IRSs on different sides of the BS with different normals, while dominant reflected-power providers vary across the target airspace.The interleaved spatial pattern indicates that coverage is shared across selected IRSs rather than dominated by one panel everywhere.
  • Coverage heatmaps: At z = 95 m, the proposed deployment raises minimum SNR by about 60.3 dB, compared with about 20.0 dB at z = 55 m.The stronger improvement occurs where the altitude slice intersects a severe direct-link array-factor null.
  • Orientation freedom: The worst-case SNR increases with orientation freedom δ because larger feasible cones allow selected IRSs to direct responses toward complementary airspace regions.At δ = 0°, all IRS normals are vertical and share the same mounting orientation.
Loading 2608.30586v1…