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Interference-Aware Downlink Power Control for High-Altitude Platform Stations with QoS and Max-Min Fairness

Rajan Shrestha, Hayder Al-Hraishawi

arXiv:2610.12380v1eess.SP

TL;DR

HAPS downlink power control must account for geometry-dependent interference, while its implications for QoS feasibility, minimum required power, and fairness have received limited attention. This paper models that coupling and proposes QoS-minimum-power and max–min schemes; the schemes approach LP benchmarks with much lower computation time.

  • Problem

    The impact of geometry-dependent inter-user interference on HAPS downlink QoS feasibility, minimum required power, and fairness has received limited attention.

    Source: §I · p. 1

  • Method

    The paper derives a statistical-CSI SINR model for geometry-dependent coupling and develops QoS-constrained power minimization and max–min SINR fairness schemes.

    Source: §I · p. 1

  • Results

    At Rreq = 2.5 bits/s/Hz, feasibility is approximately 0.95 versus less than 0.4 for all three baselines; at K = 20, computation time falls approximately 99.97%, with performance comparable to LP benchmarks.

    Source: §IV · pp. 5–6, §V · p. 6

  • Takeaways & Limitations

    Avoiding LP-based optimization supports power control on computationally constrained HAPS platforms.

    Source: §IV · p. 6

Abstract

from arXiv · show

High-altitude platform stations (HAPS) are a promising component of non-terrestrial networks (NTNs) for wide-area connectivity. However, serving multiple users over large user footprints requires efficient power-control mechanisms that account for the unique propagation characteristics and strict power constraints of HAPS. This paper develops a tractable analytical and optimization framework for downlink power allocation in HAPS systems under statistical channel state information, 3GPP NTN path loss, and Rician fading. A closed-form expression for the signal-to-interference-plus-noise ratio (SINR) with mean-channel maximal-ratio transmission (MRT) precoding is derived, capturing geometry-dependent inter-user interference coupling. Based on this, two power-control schemes are proposed: quality-of-service (QoS)-constrained power minimization and max-min SINR fairness. The QoS solution is obtained in closed form or through a low-complexity fixed-point iteration, while the max-min solution is obtained through bisection. Numerical results demonstrate up to 6 dB reduction in required transmit power at moderate-to-high target rates and improved worst-user SINR compared to the uniform power, inverse path-loss, and geometry-aware heuristic allocations. The proposed QoS-optimal scheme also achieves a QoS feasibility probability of approximately 0.95 at R_req = 2.5 bits/s/Hz and P_max = 35 dBm, while the uniform power, inverse path-loss, and geometry-aware heuristic allocations achieve less than 0.4. Moreover, the proposed schemes achieve performance comparable to the linear programming (LP)-based benchmarks at a fraction of the computational cost, reducing the average computation time by approximately 99.97% when K = 20.

I. INTRODUCTION

HAPS massive MIMO needs power control that accounts for geometry-dependent interference under tight platform power budgets. The paper develops statistical-CSI schemes for QoS-constrained power minimization and max–min fairness, alongside a model of the users’ geometry.

I. INTRODUCTION

  • Strong LoS propagation makes inter-user interference depend on user locations, angular separation, and array responses, while its impact on QoS feasibility, required power, and fairness has received limited attention.HAPS also face constraints from nighttime battery capacity, payload weight, amplifier output power, and thermal dissipation.Source: §I · p. 1
  • The framework derives a geometry-aware SINR expression and proposes QoS power minimization and max–min fairness schemes under statistical CSI, Rician fading, and 3GPP NTN channel models.The QoS scheme uses a closed-form solution or fixed-point iteration; max–min fairness is solved by bisection.Source: §I · p. 1

II. SYSTEM MODEL

  • The system model considers a HAPS serving K randomly distributed single-antenna users, with user geometry represented relative to the platform’s ground projection.Fig. 1 schematically depicts the downlink system and user geometry.Source: §II · p. 2, Figure 1 · p. 2

A. Channel Model

The channel model combines Rician fading and spatial correlation with large-scale gains derived from 3GPP NTN path-loss models. Total path loss includes basic path loss, atmospheric gas attenuation, and tropospheric scintillation.

A. Channel Model

  • The downlink channel is modeled as Rician fading, with large-scale gain β_k, Rician K-factor κ_k, and a line-of-sight steering vector indexed by broadside and azimuth departure angles.The scattered component’s correlation matrix is normalized so tr(R_k) = M, preserving the interpretations of β_k as average large-scale gain and κ_k as the LoS-to-scattered power ratio.Source: A. Channel Model · p. 2
  • The correlation matrix R_k captures spatial correlation of the scattered component, and the large-scale gain β_k is obtained from 3GPP TR 38.811 NTN models.Source: A. Channel Model · p. 2
  • Total path loss is the sum of basic path loss, atmospheric gas attenuation, and tropospheric scintillation; the basic path-loss model includes free-space loss, shadow fading, and clutter loss.Source: A. Channel Model · p. 2

B. Channel State Information (CSI)

The system uses statistical CSI for precoding because it varies more slowly and is more reliably estimated than instantaneous CSI. The signal model adopts normalized mean-channel MRT and uses the UATF bound to express user rates and geometry-dependent interference.

B. Channel State Information (CSI)

  • Statistical CSI evolves much more slowly than instantaneous channels, making it reliably estimable with low pilot overhead and robust to feedback latency.The paper attributes its slower variation to user mobility and shadow-fading coherence rather than fast fading.Source: B. Channel State Information (CSI) · p. 2

C. Signal Model and Rate Analysis

  • Normalized mean-channel MRT is adopted under statistical CSI, with scalar factors in the mean channel omitted because they do not affect the normalized beam direction.The downlink signal allocates power p_k to user k and uses precoder w_k.Source: C. Signal Model and Rate Analysis · p. 3
  • The UATF-based SINR expression separates non-coherent scattered contributions from coherent LoS interference caused by nonorthogonal user steering directions.The non-coherent term includes desired-signal variance and scattered multiuser interference, both treated as effective noise.Source: C. Signal Model and Rate Analysis · p. 3

III. PROPOSED DOWNLINK POWER CONTROL

The section rewrites the closed-form SINR into a power-control representation that makes explicit how channel statistics and HAPS-user geometry jointly shape user coupling. This representation underpins the paper’s two power-control strategies.

III. PROPOSED DOWNLINK POWER CONTROL

  • The reformulation defines an effective desired-signal gain, a self-variance coefficient, and inter-user coupling coefficients for the SINR model.These quantities reorganize the closed-form SINR into a compact power-control form.Source: §III · p. 3
  • The SINR representation makes power control depend on both large-scale channel gains and geometry-induced spatial coupling.The resulting user-coupling model is governed jointly by channel statistics and HAPS-user geometry.Source: §III · p. 3
  • The compact SINR form supports two power-control strategies tailored to different operational objectives.The section introduces these strategies after expressing the SINR in terms of the power allocation vector.Source: §III · p. 3

A. QoS-Constrained Minimum-Power Allocation

The QoS-constrained problem minimizes total downlink power while meeting per-user SINR targets. Its feasibility has both user-specific and system-wide conditions, and its minimum-power solution can be obtained in closed form or by fixed-point iteration.

A. QoS-Constrained Minimum-Power Allocation

  • The QoS allocation minimizes total downlink transmit power while guaranteeing each user’s SINR target corresponding to the required rate.The target is defined per user from the required rate.Source: A. QoS-Constrained Minimum-Power… · p. 3
  • A user-specific feasibility condition is needed: a large self-variance coefficient relative to mean-channel gain can prevent that user from meeting its target even without multiuser interference.This limit is stated under statistical CSI.Source: A. QoS-Constrained Minimum-Power… · p. 3
  • The QoS problem is feasible if and only if ρ(DZ) < 1, in which case its unique minimum-power solution is p⋆ = (I − DZ)^−1b.The solution is componentwise minimal; fixed-point iteration also converges to it when ρ(DZ) < 1.Source: A. QoS-Constrained Minimum-Power… · p. 4

B. Max–Min Fairness Power Allocation

The max–min formulation maximizes the lowest user SINR under a total transmit-power budget. It finds the largest feasible common SINR target by checking the corresponding minimum-power vector and using bisection.

B. Max–Min Fairness Power Allocation

C. Benchmark Schemes

The section introduces baseline strategies used to compare against the proposed power-control schemes.

C. Benchmark Schemes

1) Uniform Power Allocation:

Uniform allocation gives every user equal power without accounting for path-loss disparity or spatial interference coupling.

1) Uniform Power Allocation:

2) Inverse Path-Loss Allocation:

The section compares inverse path-loss, geometry-aware, and LP-based benchmarks with the proposed schemes, reporting power, fairness, feasibility, and computation results.

2) Inverse Path-Loss Allocation:

  • Inverse path-loss allocation favors users with weaker large-scale channel gains to compensate for the near-far effect, but ignores inter-user interference.Source: 2) Inverse Path-Loss Allocation · p. 5

3) Geometry-Aware Heuristic:

IV. NUMERICAL RESULTS

  • The QoS-optimal scheme requires less power than the three allocation baselines across target rates and closely approaches Direct LP; max–min fairness similarly approaches Bisection-LP and outperforms the baselines.Source: §IV · p. 5, Figure 2 · p. 5, Figure 3 · p. 5
  • At Rreq = 2.5 bits/s/Hz, QoS feasibility is approximately 0.95 for the proposed scheme versus less than 0.4 for each baseline.Feasibility counts realizations where all users meet their rate targets within the total power budget.Source: §IV · p. 5, Figure 4 · p. 6
  • Feasibility decreases as K increases and improves as M increases, with the proposed scheme remaining comparable to Direct LP.Source: Figure 5 · p. 6
  • At K = 20, both proposed schemes reduce computation time by approximately 99.97% relative to LP-based benchmarks.The comparison averages 100 independent realizations.Source: §IV · p. 6

V. CONCLUSION

The paper presents a statistical-CSI downlink power-control framework for HAPS massive MIMO and reports lower transmit power, improved minimum-user SINR, and higher QoS feasibility than conventional allocations. Its schemes achieve performance comparable to LP-based benchmarks with approximately 99.97% lower computation time at K = 20.

V. CONCLUSION

  • Approximately 99.97% lower computation time at K = 20 accompanies performance comparable to LP-based benchmarks, alongside lower transmit power, improved minimum-user SINR, and higher QoS feasibility than conventional allocations.Fig. 6 compares computation time versus number of users at Rreq = 1.5 bit/s/Hz and Pmax = 35 dBm.Source: §V · p. 6, Figure 6 · p. 6
  • A closed-form SINR expression under mean-channel MRT captures scattered and geometry-dependent coherent LoS interference, enabling tractable QoS-constrained minimum-power and max–min fairness solutions.Source: §V · p. 6
  • The framework addresses HAPS massive MIMO downlink power control under statistical CSI and 3GPP NTN channel models.Source: §V · p. 6
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