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Performance Evaluation of HAPS-enabled Coverage Enhancement in Hard-to-Reach Areas

Hao Lin, Mustafa A. Kishk, Mohamed-Slim Alouini

arXiv:2609.05067v1cs.NI

TL;DR

The paper studies coverage holes in hard-to-reach areas where terrestrial infrastructure is limited to the perimeter. It develops a stochastic-geometry model for directional HAPSs coexisting with terrestrial networks, derives DL and UL coverage, and reports deployment guidance based on HAPS count and beamwidth.

  • Problem

    Coverage holes in hard-to-reach areas are typically overlooked, despite terrestrial deployment being constrained by geography and limited largely to transportation networks or coastlines.

  • Method

    The paper builds a stochastic-geometry framework for directional HAPSs deployed inside a coverage hole alongside terrestrial base stations around its perimeter.

  • Results

    The analysis shows that DL and UL coverage depend on HAPS number, 3 dB beamwidth, HAPS altitude, and UE location, with narrower beams improving reported UL coverage.

  • Takeaways & Limitations

    HAPS number and 3 dB beamwidth should be jointly designed by considering DL performance, UL performance, system complexity, and operational expense.

Abstract

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High altitude platform stations (HAPSs) are becoming a key component of future non-terrestrial networks (NTNs). HAPSs can serve a larger area than uncrewed aerial vehicles (UAVs) and offer lower propagation latency, maintenance expense, and energy costs than satellites. A major application of HAPSs is to serve the areas where terrestrial network (TN) deployment is infeasible, especially in hard-to-reach areas and post-disaster areas. For instance, in the Amazon rainforest, the Mediterranean region, and deserts, TN deployment is severely constrained by geographical and environmental conditions. Only areas close to transportation networks or coastlines can be covered, while large areas remain uncovered. Such coverage holes in hard-to-reach areas are typically overlooked in existing literature. Motivated by these realistic cases, in this paper, we use tools from stochastic geometry to mathematically model hard-to-reach areas where cellular terrestrial infrastructure only exists at their perimeter. We propose to deploy a HAPS constellation over this hard-to-reach area to enhance connectivity. For that setup, we derive the downlink (DL) and uplink (UL) coverage performance of the considered user equipment (UE) as a function of the location of the UE inside the coverage hole. Our results show how the number of HAPSs, beamwidth, and HAPS altitude affect the DL and UL coverage probabilities. Finally, we provide multiple useful guidelines for future HAPS deployment.

I. Introduction

The paper addresses coverage holes in hard-to-reach areas where terrestrial base stations remain confined to boundaries, proposing a stochastic-geometry HAPS framework with directional beamforming. It models HAPS–terrestrial coexistence and derives DL/UL coverage results to guide joint deployment design.

  • Motivation: Hard-to-reach regions such as deserts, rainforests, islands, and remote mountains often have terrestrial stations only near transport routes or coastlines.These areas still contain potential users in energy, agriculture, disaster-warning, fishing, and extraction applications.
  • Approach: The proposed framework combines stochastic geometry, directional HAPS antennas, and coexistence with terrain-constrained terrestrial networks.The model places terrestrial base stations in an annular perimeter region and HAPSs inside the circular coverage hole.
  • Research gap: Existing work motivates HAPS deployment for coverage enhancement, but inland and nearshore holes with perimeter-only terrestrial infrastructure remain insufficiently characterized.The paper distinguishes these settings from open-ocean coverage scenarios and studies their association and interference conditions.
  • Analysis: The paper analyzes both downlink and uplink coverage and quantifies how HAPS count, 3 dB beamwidth, and altitude affect performance.It uses a HAPS deployment inside the hole and terrestrial infrastructure around its boundary to represent geographically constrained scenarios.
  • Design implications: The resulting design guidance requires jointly selecting HAPS number and beamwidth while considering DL performance, UL performance, system complexity, and operational expense.Directional beamforming is treated as a central design variable rather than an incidental antenna feature.

A. Downlink Transmission

The downlink model represents HAPS and TBS transmissions with path loss, fading, antenna gains, and interference, while associating the UE with the access point offering the strongest average power.

  • A. Downlink Transmission: The model assigns transmit and receive gains, carrier-frequency parameters, wavelength, and environmental attenuation to HAPS and TBS links.The HAPS unit-distance received-power model includes atmospheric absorption and environmental attenuation; a corresponding TBS model is also defined.
  • A. Downlink Transmission: HAPS and TBS channel gains follow Gamma distributions derived from Nakagami-m fading, with separate path-loss exponents for the two tiers.The analysis models TBS antennas as omnidirectional and compares omnidirectional and directional HAPS antennas for tractability.
  • A. Downlink Transmission: Interfering HAPS beam off-boresight angles are modeled uniformly over [0, π], and their antenna gains are measured relative to the maximum main-lobe gain.The interfering-link geometry and antenna pattern enter the HAPS interference model.
  • A. Downlink Transmission: The downlink SINR is formulated separately for HAPS- and TBS-associated UEs, accounting for desired signal, tier-specific interference, and noise.The resulting downlink coverage probability combines the two association cases for the UE.
  • A. Downlink Transmission: The UE associates with the nearest HAPS or TBS because beam alignment gives the serving HAPS its main-lobe gain and propagation distance mainly determines average signal strength.The association rule is based on the access point with the strongest average received power.

B. Uplink Transmission

The uplink model studies a scheduled UE served by its closest HAPS over a separate frequency band, using aligned HAPS beams and grant-free same-resource-block interferers.

  • B. Uplink Transmission: The uplink analysis focuses on direct ground-to-air transmission from a scheduled UE associated with its closest HAPS.The setup covers applications such as IoT gateways collecting neighboring-device information and mobile users transmitting directly to HAPSs.
  • B. Uplink Transmission: The uplink signal model includes UE transmit gain, HAPS receive gain, channel fading, and HAPS path loss.The channel gains follow a Gamma distribution with parameter mH and unit expectation under Nakagami-m fading.
  • B. Uplink Transmission: The serving HAPS beam is aligned with the typical UE, while the ITU HAPS antenna pattern models directional beamforming.The beam direction toward the typical UE is explicitly defined in the uplink model.
  • B. Uplink Transmission: Uplink services use a different frequency band from downlink services, and interference comes from grant-free UEs transmitting on the same resource block.The interfering-UE geometry is represented through vectors and off-boresight angles relative to the serving HAPS beam.
  • B. Uplink Transmission: Uplink coverage is defined as the probability that the serving-HAPS SINR exceeds the threshold when the UE is associated with the HAPS tier.The coverage expression is obtained after deriving the required distance distributions.

III. Distance Distributions

The paper derives distance distributions from a typical UE to its closest HAPS and TBS, using deployment-region geometry and independent spatial locations to support subsequent coverage analysis.

  • III. Distance Distributions: The UE is placed inside the coverage hole, with HAPSs deployed over an inner circular region and TBSs deployed in a surrounding annular region.The HAPS and TBS deployment areas are represented through circular-area intersections and annular geometry.
  • III. Distance Distributions: The closest-HAPS distance distribution is obtained from the area of the HAPS deployment region lying within distance d_H of the UE.The paper introduces both the CCDF and PDF of D_H.
  • III. Distance Distributions: The CCDF of D_H uses the probability that all HAPSs lie farther than d_H, with independence converting this probability into a product of individual probabilities.The PDF follows by differentiating one minus the CCDF with respect to d_H.
  • III. Distance Distributions: The closest-TBS distance distribution is needed both for TBS association and for TBS interference when the UE connects to a HAPS.The TBS distance model uses the intersection between the annular deployment area and a circle centered at the UE projection.
  • III. Distance Distributions: The paper introduces the CCDF and PDF of D_T from the TBS-region intersection area and then uses both distance distributions for downlink and uplink analysis.The TBS distance distribution includes a Dirac point mass at +∞ in its formulation.

A. Downlink Analysis

The downlink analysis derives coverage probabilities conditioned on serving distance for TBS and HAPS association, then combines them using the distance distributions and interference transforms.

  • A. Downlink Analysis: For TBS association, conditional downlink coverage is derived given d_T while accounting for thermal noise, other-TBS interference, and interference from all HAPSs.The TBS-associated expression uses the HAPS interference contribution through its Laplace transform.
  • A. Downlink Analysis: The interference calculations use Laplace transforms of noise and interference, with directional HAPS antenna effects represented in the HAPS-interference term.For omnidirectional HAPS antennas, the interference transform simplifies by replacing the antenna-pattern factor with constant 1.
  • A. Downlink Analysis: For HAPS association, conditional downlink coverage is derived given d_H with interference from the remaining HAPSs and TBSs outside the association-dependent distance threshold.The closest HAPS is serving, so the HAPS interference count is reduced by one.
  • A. Downlink Analysis: The overall downlink coverage probability combines the TBS- and HAPS-associated conditional probabilities using the distributions of d_T and d_H.The distance distributions are supplied by Lemmas 2 and 1, respectively.
  • A. Downlink Analysis: Conditional downlink coverage expressions can be approximated using an upper bound for the upper incomplete Gamma function to reduce computational complexity.The approximation is stated for the conditional coverage probabilities in Theorems 1 and 2.

B. Uplink Analysis

The uplink analysis derives conditional and average UL coverage probabilities for UEs in the hard-to-reach area, accounting for serving-HAPS location and interference from grant-free UEs.

  • Conditional UL coverage: The analysis first evaluates UL coverage for a UE at z_U conditioned on the location of its closest HAPS.This conditional formulation also characterizes UL performance at different UE locations when HAPS positions are fixed.
  • Conditional UL coverage: Theorem 4 gives the UL coverage probability conditioned on the closest HAPS location.The formulation includes interference from grant-free UEs through the Laplace transform of noise plus interference.
  • Serving-HAPS distribution: The serving-HAPS location is modeled through its probability density, obtained using order statistics and the exclusion region for other HAPSs.For a candidate serving-HAPS point, one HAPS occupies that point while the remaining HAPSs lie outside the relevant circular area.
  • Average UL coverage: Theorem 5 averages UL coverage over the unknown serving-HAPS location using the conditional coverage probability and its density.This provides average UL performance when HAPS locations are random.

V. Simulation Results and Discussion

The simulations evaluate a circular hard-to-reach area with uniformly deployed HAPSs and terrestrial base stations confined to an surrounding annulus, comparing analytical and Monte Carlo results.

  • Simulation setup: 20,000 Monte Carlo simulations evaluate DL and UL performance in a circular hard-to-reach area of radius 200 km.HAPSs are uniformly deployed inside the area, while TBSs occupy an annulus with inner radius 200 km and outer radius 300 km.
  • Simulation setup: The DL setup uses TBS transmit power pTx_T = 43 dBm and TBS density λT = 0.1 TBSs/km2.The carrier frequency is 2.4 GHz, bandwidth is 10 MHz, and HAPS altitude is 20 km.
  • Simulation setup: The UL setup uses UE transmit power pTx_U = 23 dBm and grant-free interfering-UE density λU = 0.02 UEs/km2.The attenuation factor sA = 0 dB represents a favorable environment with negligible additional atmospheric and environmental attenuation.
  • Validation: Solid lines denote analytical results, while markers denote simulation results.

A. Downlink Performance

The downlink analysis studies coverage versus UE location, HAPS count, antenna beamwidth, and altitude for omnidirectional and directional HAPS antennas. Directional beamforming improves coverage, while altitude increases reduce it under the considered setting.

  • Coverage versus UE location: Directional beamforming improves DL coverage most as the HAPS count increases, reaching 8.41% and 9.82% improvement for NH = 8 and NH = 16, respectively.With NH = 4 and rU = 0 km, the improvement is 5.89%.
  • Coverage versus HAPS count and beamwidth: At the hard-to-reach-area center, achieving DL coverage probability 0.9 requires 18 HAPSs with ψb = 80° but only 8 with ψb = 2.5° when τDL = −10 dB.When τDL = 0 dB, coverage probability 0.6 requires at least 32 HAPSs for ψb = 5° and 26 for ψb = 2.5°.
  • Coverage versus HAPS count and beamwidth: With τDL = 0 dB and ψb > 20°, DL coverage probability remains below 0.5 with fewer than 32 HAPSs.The results indicate that HAPS count and antenna pattern should be jointly selected for the target performance and expense.
  • Coverage versus altitude: Increasing HAPS altitude lowers DL coverage probability for NH = 16 because larger-scale path loss reduces received signal strength.At the area center, interference mainly comes from side-lobe signals from other HAPSs because the UE is far from TBSs.
  • Approximation comparison: The random footprint association policy and uniform off-boresight angle approximation differ by less than 0.02.

B. Uplink Performance

Uplink coverage improves with narrower HAPS beams and more HAPSs, while altitude and user location shape performance. Directional antennas are needed to control interference, but deployment must balance coverage against complexity and operational constraints.

  • Coverage maps: At ψb = 10°, only UEs near HAPSs exceed 0.8 coverage probability; ψb = 5° expands high-coverage regions, while ψb = 2.5° covers most areas successfully.The analysis assumes NH = 16 HAPSs and τUL = −10 dB.
  • UE location: For rU < 140 km, UL coverage is 0.2 at ψb = 10°, 0.55 at ψb = 5°, and 0.85 at ψb = 2.5°.Beyond 140 km, more UEs associate with perimeter TBSs, reducing the HAPS-associated UL coverage probability.
  • HAPS number and beamwidth: With directional antennas, achieving UL coverage probability 0.7 requires NH = 26 at ψb = 5° and NH = 10 at ψb = 2.5°; omnidirectional antennas yield coverage close to 0.Increasing NH and narrowing the 3 dB beamwidth both improve UL coverage, but practical design must also consider DL performance, system complexity, and operational expense.
  • HAPS altitude: At NH = 16 and hH = 20 km, UL coverage is 0.20 for ψb = 10° and 0.54 for ψb = 5°; for ψb = 2.5°, it peaks at 0.86 when hH = 32 km.For ψb = 2.5°, coverage remains above 0.8 over 20 km < hH < 50 km.
  • UE transmit power: At ψb = 2.5°, pTx_U = 16 dBm is required to exceed coverage probability 0.7, whereas ψb = 5° and ψb = 10° remain below 0.68 and 0.3.These latter values are below the stated target requirement.
  • Model scope: The circular hard-to-reach-area model is tractable and can use an effective radius, but irregular geometries require modified distance distributions and the association rule omits load, backhaul, and handover effects.Future scheduling is also needed to reduce mutual interference and improve spectral efficiency.

VI. Conclusion

The paper develops and validates a stochastic-geometry framework for DL and UL coverage in hard-to-reach areas where terrestrial base stations lie around the perimeter. Its results support narrower HAPS beams and sufficient HAPS density, jointly designed with deployment complexity and expense.

  • Framework: The framework models coexistence between a HAPS network over the coverage hole and a perimeter-only TBS network, deriving conditional and average DL and UL coverage.Monte Carlo simulations validate the analytical results.
  • Coverage enhancement: Sufficient HAPS deployment can fill hard-to-reach-area coverage holes and avoid connection gaps caused by handover between HAPSs and TBSs.The conclusion reports this outcome for the proposed HAPS-based solution.
  • Design guidelines: Narrower HAPS beams improve coverage probability and reduce the HAPS number required to meet a target, but beamwidth and HAPS count should be jointly designed.The design should consider DL performance, UL performance, system complexity, and operational expense.

Appendix A Proof of Theorem 1

The proof derives the downlink SINR coverage probability by combining interference transforms from terrestrial base stations and HAPSs, while approximating directional beam-angle statistics to retain tractability.

  • The downlink SINR coverage analysis separates thermal noise, terrestrial interference, and HAPS interference for a UE associated with its closest TBS.The total interference transform factors across independent channel gains and independent HAPS and TBS point processes.
  • The HAPS interference transform conditions interfering HAPS distances on the minimum distance determined by the closest-HAPS association geometry.The conditional distance distribution is used to calculate the HAPS interference Laplace transform.
  • The proof models off-boresight angles of interfering HAPS beams as independent random variables with a common marginal distribution.An isotropic three-dimensional beam model would yield fψ(ψ) = 1/2 sin ψ, but the considered beams are steered toward scheduled ground UEs.
  • The downlink coverage probability is approximated using a tight upper bound on the upper incomplete Gamma function under Nakagami-m fading.This avoids evaluating high-order derivatives of the Laplace transform.

Appendix B Proof of Theorem 4

The proof obtains uplink coverage conditioned on the closest HAPS and represents the receiver disturbance as grant-free-user interference plus thermal noise.

  • The uplink coverage probability is conditioned on the location of the closest HAPS serving the UE.
  • The uplink interference-plus-noise term combines grant-free interfering-UE interference with thermal noise.The proof denotes this aggregate quantity as UU = IU + N0 and derives its transform components.
  • The conditional uplink coverage probability is evaluated using the aggregate interference-plus-noise transform and the thermal-noise Laplace transform.The thermal-noise transform is LN0(sU) = exp(−sUN0).
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