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Downlink Coverage Analysis for a Finite 3D Wireless Network of Unmanned Aerial Vehicles
Vishnu Vardhan Chetlur, Harpreet S. Dhillon
TL;DR
The paper develops realistic system-level analysis techniques for finite three-dimensional UAV networks. It derives exact and approximate downlink coverage results, including a no-fading treatment, and identifies performance trends related to UAV height and receiver location.
Problem
Realistic system-level analysis techniques are needed for UAV networks.
Method
The paper characterizes serving and interfering-node distances and derives coverage expressions using derivatives of the Laplace transform under independent Nakagami-m fading, alongside a dominant-interferer approximation for no fading.
Results
The no-fading coverage probability cannot be obtained explicitly as the limiting case m →∞ of Nakagami-m fading, while the derived results match simulations exactly in Fig. 4.
Takeaways & Limitations
The analysis provides useful performance trends concerning UAV heights and the reference receiver’s ground location, supporting discussion of the proposed analytic approaches’ applicability.
Abstract
from arXiv · showhide
In this paper, we consider a finite network of unmanned aerial vehicles (UAVs) serving a given region. Modeling this network as a uniform binomial point process (BPP), we derive the downlink coverage probability of a reference receiver located at an arbitrary position on the ground assuming Nakagami-$m$ fading for all wireless links. The reference receiver is assumed to connect to its closest transmitting node as is usually the case in cellular systems. After deriving the distribution of distances from the reference receiver to the serving and interfering nodes, we derive an exact expression for downlink coverage probability in terms of the derivative of Laplace transform of interference power distribution. In the downlink of this system, it is not unusual to encounter scenarios in which the line-of-sight (LOS) component is significantly stronger than the reflected multipath components. To emulate such scenarios, we also derive the coverage probability in the absence of fading from the results of Nakagami-$m$ fading by taking the limit $m \to \infty$. Using asymptotic expansion of incomplete gamma function, we concretely show that this limit reduces to a redundant condition. Consequently, we derive an accurate coverage probability approximation for this case using dominant interferer-based approach in which the effect of dominant interferer is exactly captured and the residual interference from other interferers is carefully approximated. We then derive the bounds of the approximate coverage probability using Berry-Esseen theorem. Our analyses reveal several useful trends in coverage probability as a function of height of the transmitting nodes and the location of reference receiver on the ground.
I. INTRODUCTION
The paper addresses limited understanding of coverage and capacity in finite UAV networks and develops stochastic-geometry tools for system-level downlink analysis. It models deployments realistically for arbitrary ground receivers and finite regions, extending beyond common infinite or center-receiver assumptions.
- Motivation: Temporary UAV deployment can provide short-term connectivity faster and more cost-effectively than temporary conventional base stations.The motivation includes civilian events and other situations requiring temporary network resources.
- Motivation: Finite UAV networks serving given regions remain insufficiently understood in terms of terrestrial-user coverage and capacity.The paper emphasizes realistic deployments with a fixed number of UAVs.
- Motivation: Existing UAV system-level performance studies have mostly relied on field tests and simulations, which become difficult to scale with many parameters.Stochastic geometry offers tractable expressions for key performance metrics by modeling node locations probabilistically.
- Contribution: The paper develops a comprehensive downlink analysis for a finite multi-UAV network using stochastic geometry.The framework targets an arbitrarily located ground user served by a finite UAV deployment.
- Modeling choice: A homogeneous BPP is used instead of an infinite PPP because the scenario has a given, potentially small number of UAVs covering a finite region.The receiver may lie anywhere in the region and connects to the closest transmitter drawn from the BPP.
B. Contributions
The paper formulates a finite three-dimensional UAV network and derives coverage results under fading and no-fading conditions. It also develops an approximation and bounds for no-fading coverage, then studies how coverage varies with system parameters.
- Modeling of finite three-dimensional network: The framework models UAVs as a finite three-dimensional network and analyzes downlink coverage for an arbitrary ground receiver.UAVs are uniformly distributed in a finite area at a common altitude, and the receiver connects to the closest UAV.
- Modeling of finite three-dimensional network: The analysis derives serving and interfering distance distributions for the finite deployment.The ordered distances identify the serving node and closest interferer, supporting subsequent interference analysis.
- Coverage probability: For Nakagami-m fading, the paper derives an exact coverage-probability expression in terms of the Laplace transform of interference power.The model permits different fading parameters for serving and interfering links while restricting the serving parameter to integer values for tractability.
- Coverage probability: Taking m →∞ does not explicitly produce the no-fading coverage probability because the limiting condition becomes redundant.The paper uses asymptotic incomplete-gamma expansions to establish this behavior.
- Coverage probability: An accurate no-fading approximation captures the dominant interferer exactly, approximates residual interference, and is bounded using the Berry-Esseen theorem.This approach addresses the absence of an explicit expression from the Nakagami-m limit.
- Performance analysis: Coverage probability decreases with UAV altitude when the transmitter-scattering area is fixed and increases with the channel path-loss exponent.The paper presents these trends as useful system-design guidelines.
III. COVERAGE PROBABILITY
The paper characterizes serving and interfering distance distributions in finite three-dimensional UAV networks, including joint distributions needed for dominant-interferer analysis.
- Contribution: The dominant-interferer-based distance results are presented as unique to this paper and as its first application to finite cellular-network analysis.The paper also states that the results have general interest for finite wireless networks.
- Distance distributions: Distance distributions are derived for transmitters, the serving node, interferers, and the dominant interferer at arbitrary receiver locations.The analysis also provides simplified origin-receiver corollaries.
- Distance distributions: Conditioned on the serving distance, interferer distances are independent and identically distributed.A further conditioning on the dominant-interferer distance yields an i.i.d. model for the remaining interferers.
- Dominant interferer: The dominant interferer is the second closest transmitter in the absence of fading.The joint serving-distance and dominant-interferer-distance distribution supports the later approximation.
B. Coverage Probability under Nakagami-m Fading Channels
For Nakagami-m fading, the paper derives downlink coverage probability for a receiver connected to its closest transmitter using interference-distance distributions and Laplace-transform derivatives.
- Model and definition: Coverage probability is defined by the probability that the receiver’s SIR exceeds a predetermined threshold for successful communication.The SIR uses path-loss exponent α and Nakagami fading gains on desired and interfering links.
- Coverage derivation: The conditional coverage probability is expressed through derivatives of the conditional Laplace transform of interference power.The result is then deconditioned over the serving distance.
- Coverage derivation: Theorem 1 gives the coverage probability for the receiver under a Nakagami-m fading channel.The derivation uses the complementary cumulative distribution function of the desired gamma-distributed fading gain.
C. Limiting Case of No-fading
The paper studies the no-fading limit of the Nakagami-m coverage expression and finds that asymptotic analysis yields only the ordinary SIR condition rather than an explicit probability expression.
- Limit evaluation: The no-fading case is obtained by taking m →∞ and m0 →∞ in the Nakagami-m coverage expression.Evaluating the desired-link limit is more challenging than applying the limit only to interfering links.
- Asymptotic analysis: An asymptotic expansion of the incomplete gamma function shows that the limit converges to three values depending on z.The expansion is required because both the summation limit and summand approach infinity.
- Result: The coverage probability converges to 1 when z < 1, which is exactly the no-fading condition SIR > β.Thus, the limiting result supplies a redundant coverage condition rather than an explicit no-fading coverage expression.
- Contribution: The paper reports this redundant-condition insight as previously unreported for the limiting case of Nakagami-m fading.It therefore motivates an alternative approximation for coverage without fading.
D. Dominant Interferer Approach
To approximate no-fading coverage, the paper captures the dominant interferer exactly and models the aggregate interference from the remaining interferers with a Gaussian approximation.
- Approximation method: The dominant-interferer approach captures the dominant interferer exactly while approximating the aggregate interference from the remaining interferers.The remaining interference is represented by a normal random variable using its conditional mean and variance.
- Residual interference: The conditional mean and variance of the residual interference are derived from the i.i.d. distances of the remaining interferers.These quantities are given in Lemmas 8 and 9.
- Special case: For receivers at the origin, the residual-interference expressions can be simplified to closed form; otherwise, the integrals can be evaluated numerically.The formulation uses the Q-function for the Gaussian approximation.
- Coverage approximation: Theorem 3 provides an approximation to the receiver’s no-fading coverage probability using the dominant-interferer construction.The approximation combines the residual-interference Gaussian probability with the joint distribution of serving and dominant-interferer distances.
E. Bounds of Coverage Probability Approximation
The paper applies the Berry–Esseen theorem (BET) to bound the coverage-probability approximation using interference moments. The approximation converges with network size and is reported to remain accurate even for few nodes.
- Berry–Esseen analysis: BET bounds the deviation between the normal approximation and the true interference distribution using distribution moments.The analysis uses the first, second, and third moments of the centered interference variables.
- Berry–Esseen analysis: Lemma 10 provides the third absolute moment required for the coverage-probability bounds.This moment is conditioned on the relevant geometric variables and is evaluated numerically when no simple closed form is available.
- Coverage bounds: Theorem 4 gives coverage-probability bounds for a receiver at distance x0 from the origin.The bound uses the standard normal CDF, variance σ2, third moment ρ, and constant C = 0.4748.
- Accuracy and convergence: For small N, the bounds can be loose, but the approximation error decreases at rate (N −2)−1/2 as the number of transmitters grows.Thus, the approximate coverage probability converges to the actual value for large networks.
- Accuracy and convergence: The approximation is reported to be surprisingly accurate even for a small number of network nodes.This complements the asymptotic convergence result by indicating useful accuracy outside the large-N regime.
IV. RESULTS AND DISCUSSION
The results validate the analytical coverage expressions against simulations and examine fading, path-loss exponent, UAV height, and receiver location. Coverage worsens with lower path-loss exponent, greater UAV height, and changing receiver position, while SIR variance falls toward the no-fading case.
- Validation: The theoretical coverage results match the simulations exactly in the validation experiments.The study uses simulations of finite UAV networks to assess the analytical results and obtain design insights.
- Impact of fading: SIR variance decreases as m increases from Rayleigh fading (m = 1) toward no fading (m →∞).The SIR therefore becomes more concentrated as fading diminishes.
- Impact of path-loss exponent: Coverage probability degrades when the path-loss exponent α decreases because the resulting interference increase outweighs the desired-signal power increase.The comparison varies α while fixing h = 10 km, m = 1, and x0 = 4 km.
- Impact of height: Coverage probability deteriorates as UAV height h increases because serving and interfering nodes become more similar in separation from the receiver.The reported height comparison uses h values of 2, 4, 6, and 8 km.
- Impact of receiver distance: Coverage probability varies significantly with receiver location x0, highlighting the importance of modeling arbitrarily located receivers.The analysis plots coverage against x0 for different UAV heights.
- Urban applicability: The urban extension models independently blocked LOS paths and fixed attenuation η for hidden UAVs, while retaining analytical tractability.Conditioning on the number of visible UAVs yields visible and blocked BPPs; the independent-blocking assumption is numerically compared with simulations.
V. CONCLUSION
The paper presents a comprehensive downlink coverage analysis for a finite three-dimensional UAV network modeled as a BPP. It derives exact fading-based results, a no-fading approximation with bounds, and coverage trends relevant to system design.
- The network is modeled as a BPP containing N UAVs, enabling downlink analysis for a finite three-dimensional wireless network.
- The analysis characterizes distances from the reference receiver to serving and interfering nodes and derives exact coverage probability under Nakagami-m fading.The exact expression is written using derivatives of the Laplace transform of the interference power distribution.
- No-fading case: Taking m →∞ does not explicitly yield the no-fading coverage probability because the resulting limiting condition is redundant.This conclusion follows from an asymptotic expansion of the incomplete gamma function.
- No-fading case: The proposed no-fading approximation models the dominant interferer exactly and approximates residual interference from other interferers by a normal distribution using the CLT.
- No-fading case: Berry-Esseen theorem provides bounds for the approximate coverage probability and quantifies the convergence rate of the normal approximation to the true distribution.
- Performance and extensions: Coverage probability exhibits useful trends with UAV height and channel propagation characteristics, while the framework can be extended to urban shadowing and other finite-network metrics.The suggested additional metrics include throughput and energy efficiency.
APPENDIX
The appendix derives distance distributions and interference transforms needed for finite-network coverage analysis. It uses projected two-dimensional BPP geometry, order statistics, conditional independence, and asymptotic expansions.
- Distance distributions: The CDF of a transmitter’s projected distance is obtained from disk-intersection areas in the ground-plane BPP, with separate contained-disk and partial-overlap cases.
- Distance distributions: Because the receiver connects to the closest transmitter, the serving distance is R = min{Wi}, and its PDF follows by differentiating the conditional CDF.
- Interferer distances: Conditioned on the serving distance, the ordered interfering distances are derived from order-statistics densities and the unordered distances are i.i.d.
- Interference transform: The interference Laplace transform factors over interferers using independent channel gains and conditionally i.i.d. interferer distances.The gamma-channel moment generating function supplies the individual channel-gain transform.
- Asymptotic analysis: The appendix develops asymptotic estimates for incomplete gamma functions through substitutions, limiting cases, Taylor expansions, and multinomial expansions.
G. Proof of Lemma 10
The proof derives a conditional moment expression for an interference-related variable and obtains its final form using the conditional distribution of the dominant interferer distance.
- The conditional PDF of Xi is used to form an integral involving |xi|^3 fXi(xi|r, u1, x0).
- The integral is split according to the sign of µVi, with limits changed accordingly.
- Substituting the resulting expression and the conditional distribution of Ui yields the final expression.
H. Proof of Theorem 4
The proof expresses coverage probability through a conditional CDF and then applies the Berry-Esseen theorem to bound it using the standard normal CDF.
- Coverage probability is first written using the conditional CDF of MN−2.
- The Berry-Esseen theorem relates the integrand’s CDF term to the CDF of the standard normal distribution.
- The resulting inequality gives bounds on the approximate coverage probability.