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Downlink Coverage and Rate Analysis of Low Earth Orbit Satellite Constellations Using Stochastic Geometry
Niloofar Okati, Taneli Riihonen, Dani Korpi, Ilari Angervuori, Risto Wichman
TL;DR
The paper tackles the limited general theoretical understanding of massive LEO-network performance beyond constellation-specific simulations and models generic networks with stochastic geometry. It derives analytical coverage and rate expressions, then uses an effective satellite count to align the model with deterministic constellations and studies design parameters.
Problem
General performance analysis of massive LEO networks is limited because prior deterministic and location-based models typically support simulations or specific constellation geometries.
Method
The paper models satellites as a binomial point process on a sphere and derives downlink coverage-probability and average-data-rate expressions using stochastic geometry.
Results
Theoretical performance for random and real constellations matches almost perfectly, with uneven latitude distributions compensated through an effective number of satellites.
Takeaways & Limitations
The framework supports analysis and design of dense satellite networks across frequency-channel and altitude choices and can extend to merged constellations.
Abstract
from arXiv · showhide
As low Earth orbit (LEO) satellite communication systems are gaining increasing popularity, new theoretical methodologies are required to investigate such networks' performance at large. This is because deterministic and location-based models that have previously been applied to analyze satellite systems are typically restricted to support simulations only. In this paper, we derive analytical expressions for the downlink coverage probability and average data rate of generic LEO networks, regardless of the actual satellites' locality and their service area geometry. Our solution stems from stochastic geometry, which abstracts the generic networks into uniform binomial point processes. Applying the proposed model, we then study the performance of the networks as a function of key constellation design parameters. Finally, to fit the theoretical modeling more precisely to real deterministic constellations, we introduce the effective number of satellites as a parameter to compensate for the practical uneven distribution of satellites on different latitudes. In addition to deriving exact network performance metrics, the study reveals several guidelines for selecting the design parameters for future massive LEO constellations, e.g., the number of frequency channels and altitude.
I. INTRODUCTION
The paper addresses the lack of general theoretical tools for analyzing massive LEO networks by modeling satellite locations stochastically and deriving tractable performance expressions. It further compares generic and deterministic constellations and examines key design parameters.
- Motivation: General theoretical understanding of massive LEO communication-constellation performance remains missing despite extensive simulation-based planning.
- Related Works: Existing approaches are tied to specific constellation parameters, regular coverage footprints, or simulations and lack a generic analytical model of interference.
- Approach: The paper models a finite set of satellites as a binomial point process on a sphere and applies stochastic geometry to generic downlink LEO networks.
- Contributions: Exact coverage-probability and average-achievable-data-rate expressions are derived in terms of the Laplace transform of interference power.
- Contributions: The effective number of satellites compensates for uneven satellite density across latitudes, enabling accurate theoretical analysis of large deterministic constellations.
- Design Insights: Increasing frequency bands improves coverage probability, whereas data rate has an optimal channel count depending on path loss exponent; performance declines with altitude in the practical LEO range.
III. COVERAGE PROBABILITY
The section defines downlink coverage probability for a user at an arbitrary location on Earth using the SINR from its nearest satellite.
- Coverage probability is evaluated for a user in an arbitrary location on Earth.
- A user is considered covered when the SINR from its nearest satellite exceeds the threshold T.
A. Coverage Probability under Rayleigh Fading Channel
The Rayleigh-fading analysis derives coverage probability by separating noise-limited and interference-present cases, using Laplace transforms for interference.
- The serving channel uses normalized Rayleigh fading, with channel gain G0 distributed exponentially with unit mean.
- Coverage probability is split into no-interference and at-least-one-interferer cases.
- Pc(T) = P0 P(SNR > T) + (1 − P0) P(SINR > T|NI > 0).
- With K = N orthogonal channels, the system becomes noise-limited and coverage reduces to its first term.
- For Rayleigh-fading interfering channels and particular path loss exponents, the interference expression reduces to elementary functions.
B. Coverage Probability under Non-fading Channels
The non-fading analysis models the serving channel without small-scale fading and retains the same two-case coverage decomposition.
- The non-fading model applies when many satellites make multiple line-of-sight links likely and multipath components weak.
- For a non-fading serving channel, the gain is set to G0 = 1.
- Coverage probability is again separated into no-interference and at-least-one-interferer terms.
- The interference Laplace function for general fading is obtained using the preceding lemma, with a corollary covering non-fading interference.
IV. AVERAGE ACHIEVABLE RATE
The section defines average achievable rate as the ergodic capacity of the fading link normalized to bandwidth 1/K Hz.
- Average achievable rate is measured in bit/s/Hz.
- The rate represents ergodic capacity over a fading link normalized to bandwidth 1/K Hz.
A. Average Achievable Rate under Rayleigh Fading Channel
The paper derives the average downlink rate for Rayleigh-fading serving channels by averaging over both the spatial BPP and fading distribution, with simplifications for selected path-loss exponents and orthogonal channel allocation.
- The average rate under Rayleigh fading averages over both the spatial BPP and the fading distribution.
- Theorem 3 gives the average downlink rate in bits/s/Hz for an arbitrarily located user with G0 ∼ Exp(1).
- For α = 2 and α = 4, substituting the corresponding interference transforms produces more simplified expressions.
- Allocating K = N orthogonal channels eliminates the interference term and makes network performance noise-limited.
B. Average Achievable Rate under Non-fading Channels
The paper derives the average downlink rate for non-fading serving channels while allowing arbitrary fading statistics for interference, separating zero- and non-zero-interference cases.
- The average-rate derivation assumes a non-fading serving channel and arbitrary fading statistics for interfering channels.
- The rate expression is divided into separate terms for zero and non-zero interference conditions.
- Theorem 4 specifies the downlink average rate for an arbitrarily located mobile user and its serving satellite under a non-fading channel.
- For non-fading interfering channels, the relevant Laplace transform can be calculated using Corollary 3.
V. NUMERICAL RESULTS
Monte Carlo simulations validate the analytical expressions and examine coverage and data rate across constellation design parameters. The results show that more channels improve coverage but data rate can peak and then decline.
- Numerical verification: The effective number of satellites Neff compensates for uneven satellite distributions across latitudes when comparing random and practical constellations.
- Numerical verification: The analytical coverage and data-rate expressions are validated against Monte Carlo simulations under static or Rayleigh serving channels and general interference fading.
- Coverage probability: The theoretical coverage results match simulations for N = 720 satellites, with α = 2 and α = 4.
- Average achievable rate: For α = 2, data rate increases with K below 45 but decreases for K > 45 because reduced bandwidth eventually outweighs interference mitigation.
- Average achievable rate: For α = 4, the largest average data rate occurs at the lowest number of frequency channels because distant interferers are less effective.
- Frequency reuse and design trade-offs: Coverage probability increases with frequency bands, while data rate has an optimum channel count that depends on performance priorities.
B. Effective Number of Satellites
The effective number of satellites, Neff(φi, φu), compensates for uneven latitude-dependent satellite density when modeling practical constellations as random ones. Fitting Neff to coverage or rate substantially improves matching across latitudes and system parameters.
- Latitude dependence: Walker and random constellations differ mainly because practical constellations have non-uniform satellite density across latitudes.For polar constellations, coverage declines from the equator toward the poles as visible-satellite density and interference increase; inclined constellations show latitude-dependent alternation.
- Definition: Neff(φi, φu) represents the number of satellites in a uniform random constellation producing the same coverage probability as a practical constellation.It depends on constellation inclination and user latitude, and is estimated by minimizing mean absolute error across selected threshold values.
- Coverage matching: Applying Neff in Theorem 1 compensates for uneven latitude distributions and yields coverage matching Walker constellations at the corresponding user latitude.The paper reports that approximation error is rather minimal when varying satellite density across latitudes is included.
- Rate matching: Neff values refined for Theorem 3 improve agreement between random and practical constellations for average data rate.The rate-fitted values differ slightly from coverage-fitted values because they are optimized for the target performance metric.
- Altitude: For practical LEO altitudes, coverage probability declines as altitude increases when total satellites vary so Neff remains constant.With Neff available, Theorems 3 or 4 can analyze data rate for given satellite constellations under the same altitude-related assumption.
- Implications: The framework models LEO networks with exact coverage and rate expressions, while frequency reuse makes spectral efficiency suitable for commercial operation.Its scalable results apply to individual constellations and massive networks combining several constellations.
APPENDIX
The appendix derives the satellite-distance distribution from spherical-cap geometry and differentiates the resulting CDF to obtain the corresponding PDF.
- Distance distribution: The CDF of the shaded spherical-cap area is obtained from the cap surface area relative to the enclosing sphere.The derivation then relates cap area to satellite distance R.
- Distance distribution: The distance CDF is deduced by combining the spherical-cap area relation with the geometric relationship between Acap and R.
- Distance distribution: The corresponding PDF is derived by differentiating the distance CDF with respect to r.
- Rate analysis: Theorem 3’s average-rate expression is evaluated using the Laplace-transform framework for interference and the conditional term log2(1 + SINR) | NI > 0.The supplied appendix passages also identify averaging over fading, interfering distance, and the binomial interferer count as steps in the Laplace derivation.
C. Proof of Theorem 3
The proof of Theorem 3 uses the conditional logarithmic SINR expression and an expectation identity for positive random variables to derive average data rate.
- Proof strategy: The proof begins from the conditional rate term log2(1 + SINR) | NI > 0.
- Proof strategy: For a positive random variable X, the expectation is rewritten as an integral of its tail probability over t > 0.This identity provides the bridge from the SINR distribution to the average-rate expression.
D. Proof of Theorem 4
The supplied proof passages obtain Theorem 4 by following the same derivation as Theorem 3 and setting interference I to zero.
- Proof strategy: Theorem 4 follows the same principles as the preceding derivation, with I = 0 substituted into the rate analysis.