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Multiple Antenna Aided NOMA in UAV Networks: A Stochastic Geometry Approach

Tianwei Hou, Yuanwei Liu, Zhengyu Song, Xin Sun, Yue Chen

arXiv:1805.04985v1cs.IT

TL;DR

The paper addresses the limited investigation of MIMO-NOMA assisted UAV networks. It uses stochastic geometry to model a three-dimensional UAV framework with randomly roaming users and interference, derives outage and ergodic-rate expressions, and reports that outage depends strongly on target rates and power allocation, with error floors when interference power scales with UAV power.

  • Problem

    The paper investigates MIMO-NOMA assisted UAV networks, whose network performance had not been intelligently investigated in existing work.

  • Method

    The paper develops a three-dimensional stochastic-geometry UAV framework modeling randomly located NOMA users and interference sources, then derives outage-probability, ergodic-rate, asymptotic, diversity-order, and high-SNR-slope results.

  • Results

    Outage probability is strongly affected by target transmission rates and NOMA power-allocation factors, while interference power proportional to UAV power produces outage error floors.

  • Takeaways & Limitations

    The framework characterizes MIMO-NOMA UAV performance across outage, ergodic rate, diversity, and high-SNR behavior under modeled interference conditions.

Abstract

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This article investigates the multiple-input multiple-output (MIMO) non-orthogonal multiple access (NOMA) assisted unmanned aerial vehicles (UAVs) networks. By utilizing a stochastic geometry model, a new 3-Dimension UAV framework for providing wireless service to randomly roaming NOMA users has been proposed. In an effort to evaluate the performance of the proposed framework, we derive analytical expressions for the outage probability and the ergodic rate of MIMO-NOMA enhanced UAV networks. We examine tractable upper bounds for the whole proposed framework, with deriving asymptotic results for scenarios that transmit power of interference sources being proportional or being fixed to the UAV. For obtaining more insights for the proposed framework, we investigate the diversity order and high signal-to-noise (SNR) slope of MIMO-NOMA assisted UAV networks. Our results confirm that: i) The outage probability of NOMA enhanced UAV networks is affected to a large extent by the targeted transmission rates and power allocation factors of NOMA users; and ii) For the case that the interference power is proportional to the UAV power, there are error floors for the outage probabilities.

I. INTRODUCTION

The introduction motivates combining MIMO-NOMA with UAV networks, where UAVs can provide connectivity and NOMA improves spectrum efficiency. It identifies the limited study of MIMO-NOMA in UAV settings and reviews related MIMO-NOMA, stochastic-geometry, and UAV-network research.

  • UAV communications: UAVs can provide connectivity to multiple users as wireless relays or aerial base stations, including for temporary events and after disasters.Their line-of-sight connections can also enhance spectrum efficiency.
  • NOMA: NOMA improves spectrum efficiency by serving multiple users in the same frequency, time, and code block through superposition coding and successive interference cancellation.Users with better channel conditions can remove intra-channel interference at the receiver.
  • Related NOMA research: Prior NOMA studies examined fixed, cognitive-radio-inspired, and dynamic power allocation, with reported effects on system performance, fairness, and quality of service.One cited result states that only the user with higher channel gain influences system performance under the examined settings.
  • MIMO-NOMA research: Related MIMO-NOMA work studied transmit power control, detection, capacity, beamforming, and signal alignment, while also reporting challenges such as high computational complexity and restrictive receiver-antenna assumptions.Some models separated MIMO-NOMA into independent SISO-NOMA arrangements or addressed inter-cell interference with coordinated beamforming.
  • Stochastic-geometry research: Stochastic geometry has been used to capture network-topology randomness and model randomly located NOMA users, including cooperative, security-oriented, and randomly deployed-user settings.One cited analysis found that grouping users with more distinctive channel gains can improve NOMA diversity order.
  • Research gap: UAV-NOMA studies have considered environment-dependent LoS probability, single-antenna systems, and F-NOMA under LoS and NLoS transmission scenarios.The introduction states that how MIMO-NOMA assists UAV networks remains unknown and motivates further investigation.
  • Research gap: The paper addresses this gap by developing a general MIMO-NOMA assisted UAV-network framework with interference and stochastic-geometry modeling of user and interference-source locations.The stated motivation is the absence of work investigating the network performance of MIMO-NOMA assisted UAV networks.

B. Contributions

The paper proposes a general MIMO-NOMA aided UAV framework that models user and interference-source locations stochastically and analyzes LoS and NLoS propagation. It derives outage and ergodic-rate results, including bounds, asymptotics, diversity orders, and high-SNR slopes.

  • Framework: A general MIMO-NOMA aided UAV framework incorporates interference and stochastic-geometry models for user and interference-source locations.Both LoS and NLoS links are considered.
  • Outage analysis: Closed-form outage-probability expressions and tractable upper bounds are derived for paired NOMA users in LoS and NLoS scenarios.The analysis also obtains diversity orders from the developed outage results.
  • Asymptotic results: Zero diversity orders occur when interference power is proportional to transmit power, producing outage-performance error floors.This result is explicitly reported for paired NOMA users.
  • Rate analysis: Exact ergodic-rate expressions, analytical lower bounds, and special-case closed forms are developed for the proposed network.The special case uses path loss exponent three.
  • Rate and SNR behavior: Far users have zero high-SNR slopes in both LoS and NLoS scenarios, while near users share the same rate ceiling across those scenarios.The results also state that near-user ergodic rates are not affected by LoS transmission.

B. Channel Model

The channel model represents a UAV serving paired NOMA users through MIMO beamforming over composite fading channels, with stochastic user and interferer locations. It specifies the propagation, precoding, detection, NOMA decoding, and interference assumptions used for SINR analysis.

  • Channel model: The composite channel model combines large-scale fading from path loss and shadowing with small-scale Nakagami-m fading.The small-scale channel from the UAV to user k is an N × K matrix.
  • Propagation geometry: The UAV-to-user distance uses UAV height h and horizontal distance rk, with path loss exponent α defining large-scale fading.The UAV cell is modeled as a disc with coverage radius Rd.
  • Precoding and detection: Precoding and detection vectors produce the effective shared channel and the two users’ SINRs under paired NOMA decoding.The higher-gain user applies SIC, while the lower-gain user treats the higher-gain user’s signal as noise.

III. PERFORMANCE EVALUATIONS

The section evaluates downlink MIMO-NOMA UAV networks under fixed UAV power allocation, deriving channel statistics and analytical outage results. It also examines asymptotic behavior, diversity order, and interference effects.

  • Fixed power allocation is employed at the UAV to evaluate downlink MIMO-NOMA assisted UAV networks.
  • A. New Channel Statistics: New channel statistics are derived for MIMO-NOMA assisted UAV networks, including Nakagami-m fading parameters for paired users.The paired users share the same small-scale fading channel coefficient.
  • B. Outage Probabilities: Constraint 1 requires the far user's target rate to satisfy the stated condition; otherwise, the outage probabilities of paired NOMA users are one.The near user must first decode the far user's signal using SIC before decoding its own signal.
  • B. Outage Probabilities: The framework develops analytical upper bounds for the outage probabilities of far and near NOMA users under fixed power allocation.The lower bound is not estimated because it is considered meaningless when the minimum detection vector approaches zero.
  • B. Outage Probabilities: Increasing the Nakagami-m fading parameter or decreasing the far user's target rate can decrease the far-user outage probability.
  • B. Outage Probabilities: Both paired NOMA users have diversity order one when interference power increases proportionally with transmit SNR, and proportional interference strongly influences outage performance.For limited interference, an exact near-user expression is also derived.

C. Ergodic Rates

The section derives ergodic-rate expressions for far and near NOMA users, including high-SNR approximations and a fixed-path-loss special case. It shows that far-user rates are controlled by power allocation at high SNR, whereas near-user rates depend mainly on transmit SNR.

  • Theorem 3 provides a closed expression for the ergodic rate of the k′-th far user.
  • The far user’s ergodic rate is entirely dependent on power allocation factors in the high SNR regime.
  • The near user’s ergodic rate is treated through a high-SNR approximation and an exact expression for fixed path loss exponent.
  • For path loss exponent α = 3, an exact near-user ergodic-rate expression is obtained using a Meijer-G function.
  • The near user’s ergodic rate is mainly dependent on transmit SNR.
  • The high SNR slope is formulated to characterize ergodic-rate behavior in the high-SNR regime.

IV. NUMERICAL STUDIES

The numerical-studies section evaluates the MIMO-NOMA assisted UAV framework using Monte Carlo simulations and analytical results under specified simulation parameters. The cited material introduces the evaluation setup and summarizes parameters in Table II.

  • Monte Carlo simulations are conducted to illustrate the correctness of the analytical results.
  • The considered network assumes a power-allocation setting, with simulation parameters summarized in Table II.
  • Fig. 2 evaluates NOMA-user outage probability versus transmit SNR at target rates Rk′ = Rk = 1.5 BPCU.

A. Outage Probabilities

The outage analysis examines how interference, fading, target rates, UAV geometry, and propagation conditions affect paired NOMA users. Fixed interference yields diversity order one, whereas interference proportional to UAV power produces error floors and diversity order zero.

  • Higher interference power increases outage probability for both NOMA users by reducing received SINR.
  • Interference power proportional to UAV power produces error floors and diversity order zero for NOMA users in both LoS and NLoS scenarios.The dynamic interference becomes much greater than noise at high SNR, while the asymptotic result does not exist when interference is too large.
  • Increasing the small-scale fading parameter m decreases outage probability for different interference power levels in LoS transmission.The LoS link provides higher received power than the comparison NLoS case.
  • Fixed interference power gives diversity order one in both LoS and NLoS scenarios.
  • When the target rate exceeds the stated threshold, an outage ceiling remains even as UAV height approaches zero.
  • Increasing UAV-user distance raises near-user outage faster than far-user outage, while remote UAVs can give far users better outage probability through higher received power.
  • Increasing disc radius increases near-user outage probability, and at zero UAV height the framework degenerates to a traditional BS setting where far-user outage is more frequent.
  • Lowering the paired users’ target rates dramatically decreases their outage probabilities.

B. Ergodic rates

The ergodic-rate results compare near and far users across SNR, interference, and LoS/NLoS conditions, and evaluate high-SNR slopes. Far-user rates approach a ceiling governed by power allocation, while near-user rates continue increasing with slope one.

  • B. Ergodic rates: As transmit SNR increases, near-user ergodic rate increases because the received signal increases.
  • B. Ergodic rates: For α = 3, the near-user approximation and exact ergodic-rate results can be considered the same.
  • B. Ergodic rates: Fixed interference becomes relatively smaller as SNR increases, causing far-user rates with and without interference to approach one another.
  • B. Ergodic rates: Far users in LoS and NLoS scenarios have the same ergodic-rate ceiling because their high-SNR rate depends only on power-allocation coefficients.
  • B. Ergodic rates: LoS propagation gives near users higher ergodic rates than NLoS propagation by increasing received power.
  • B. Ergodic rates: The high SNR slope approaches zero for far users and one for near users.The far-user slope reflects a high-SNR rate ceiling, whereas the near-user slope increases monotonically toward one.
  • B. Ergodic rates: LoS links accelerate the increasing rate of paired users and the decreasing rate ceiling for far users, while both scenarios have the same slopes.
  • V. CONCLUSIONS: The framework derives analytical expressions for outage probability and ergodic rate, along with diversity orders and high SNR slopes.

APPENDIX A: PROOF OF LEMMA 1

The appendix derives Lemma 1 by transforming Gaussian channel representations and constructing outage-probability bounds under a stochastic-geometry interference model. It also specializes parts of the derivation to K = N = 1 and imposes spatial and ordering assumptions for the NOMA users.

  • Special case: For K = N = 1, the analysis assumes real-valued channel coefficients and explicitly writes the corresponding matrices and base vector.The general case is described as challenging to estimate before this specialization.
  • Channel representation: The channel construction uses Box–Muller transformations and independent uniformly distributed variables to obtain the channel-vector distribution.The resulting Hk is a K × 1 complex Gaussian vector with zero mean and m variance.
  • Outage-probability derivation: The derivation models interference sources under a homogeneous Poisson point process with density λI and evaluates their aggregate effect using Laplace functionals.Interferers are treated as stationary, with reception analyzed at the far user's location.
  • Interference geometry: An exclusion disc for interference sources prevents infinite received interference power in the outage calculation.The resulting expression is transformed into an upper bound after changing to polar coordinates.
  • User geometry: The outage bound incorporates uniformly distributed NOMA users in the coverage disc, with the poorer-channel user generally assigned the larger horizontal distance.The derivation then applies polar-coordinate and Riemann–Stieltjes transformations.

APPENDIX C: PROOF OF COROLLARY 1

The appendix derives a high-SNR asymptotic outage result and an approximate ergodic-rate expression for the far NOMA user. The derivation relies on user-location distributions and approximations because the exact CDF contains a lower incomplete Gamma function.

  • Asymptotic outage probability: The far-user asymptotic analysis takes transmit SNR to ρ →∞ and uses a power-series approximation for the lower incomplete Gamma function.The resulting approximation targets the upper bound of outage probability at high transmit SNR.
  • Ergodic-rate formulation: The far user's ergodic rate is expressed through an expectation involving log2(1 + SINRk′).The derivation rewrites this expectation using the far-user CDF and integral transformations.
  • User distribution: NOMA users are modeled as independently and identically distributed points in the coverage area, yielding PDFs and a CDF for the far user.These distributions support the subsequent expectation-rate calculation.
  • Approximation: The appendix estimates only an approximated ergodic rate because the exact CDF contains a lower incomplete Gamma function that is challenging to calculate.This is an explicit scope limitation of the rate derivation.

APPENDIX F: PROOF OF THEOREM 4

The appendix derives the near-user ergodic rate by transforming its integral representation and expanding terms involving the lower incomplete Gamma function. A high-SNR simplification removes the J3l − J4l contribution, while a special α = 3 case permits further transformation.

  • Near-user rate: The near-user ergodic rate begins as an integral representation and is transformed through algebraic manipulation into terms denoted J1, J2, J3, and J4.The derivation applies the same integral-based approach used for the far-user rate.
  • Integral evaluation: Polynomial expansions and standard integral identities are used to evaluate J1 and J2.The appendix cites [33, eq. (3.352.4)] for the transformation.
  • Gamma-function terms: J3 and J4 are difficult to calculate because they contain the lower incomplete Gamma function.The appendix therefore evaluates that function using an exponential-series expansion.
  • High-SNR simplification: In the high-SNR regime, the difference J3l − J4l can be eliminated from the derivation.This completes the high-SNR form of the theorem.
  • Special case: For α = 3, the lower incomplete Gamma function admits a further transformation involving a Meijer-G function.The appendix then obtains specialized expressions for J3 and J4.
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