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Performance Analysis of Satellite Communication System Under the Shadowed-Rician Fading: A Stochastic Geometry Approach
Dong-Hyun Jung, Joon-Gyu Ryu, Woo-Jin Byun, Junil Choi
TL;DR
The paper studies finite LEO satellite networks under shadowed-Rician fading, where exact stochastic-geometry performance analysis and throughput optimization are needed. It models satellites with a homogeneous BPP, derives exact and Poisson-limit approximated outage and throughput expressions, and proposes an iterative constrained optimizer. Exact analyses match simulations, approximations are close, and the optimizer performs very close to two-dimensional exhaustive search.
Problem
Exact outage probability and throughput for BPP-modeled LEO systems under shadowed-Rician fading had not been analyzed in prior cited work.
Method
The paper derives serving-satellite distance and distribution-case expressions, obtains exact and Poisson-limit approximated performance formulas, and optimizes throughput under visibility and outage constraints.
Results
Exact outage and throughput analyses match Monte Carlo simulations, approximations are fairly close, and the proposed optimizer is very close to two-dimensional exhaustive search.
Takeaways & Limitations
The derived expressions support throughput optimization for finite LEO satellite systems while retaining close agreement between exact, approximated, simulated, and optimized results.
Abstract
from arXiv · showhide
In this paper, we consider downlink low Earth orbit (LEO) satellite communication systems where multiple LEO satellites are uniformly distributed over a sphere at a certain altitude according to a homogeneous binomial point process (BPP). Based on the characteristics of the BPP, we analyze the distance distributions and the distribution cases for the serving satellite. We analytically derive the exact outage probability, and its approximated expression is obtained using the Poisson limit theorem. With these derived expressions, the system throughput maximization problem is formulated under the satellite-visibility and outage constraints. To solve this problem, we reformulate it with bounded feasible sets and propose an iterative algorithm to obtain near-optimal solutions. Simulation results perfectly match the derived exact expressions for the outage probability and system throughput. The analytical results of the approximated expressions are fairly close to those of the exact ones. It is also shown that the proposed algorithm for the throughput maximization is very close to the optimal performance obtained by a two-dimensional exhaustive search.
I. INTRODUCTION
The paper addresses finite-satellite LEO communication analysis under shadowed-Rician fading using a homogeneous BPP. It derives performance expressions and an iterative throughput-optimization method, supported by simulation results.
- Motivation: LEO satellite integration can provide global coverage without deploying costly terrestrial base stations, but high altitude, timing differences, and Doppler create integration challenges.LEO satellites operate at approximately 300–2,000 km altitude, producing propagation-delay and synchronization concerns.
- Related works: Finite LEO constellations are modeled with a BPP because their satellite count is finite and their positions can appear random across a spherical surface.The BPP is used instead of the commonly adopted PPP for finite satellite networks.
- Related works: Prior BPP-based LEO studies did not analyze exact outage probability and throughput under shadowed-Rician fading.This gap motivates the paper’s system-level stochastic-geometry analysis.
- Contributions: The paper derives distance distributions for nearest, main-lobe-serving, and side-lobe-serving satellites, then analyzes three serving-satellite distribution cases.The case probabilities use the BPP void probability and account for beam-pattern effects.
- Contributions: Closed-form exact outage and throughput expressions are complemented by Poisson-limit approximations, followed by constrained throughput maximization with an iterative algorithm.The optimization includes satellite-visibility and outage constraints and uses bounded feasible sets.
- Results: The exact expressions match Monte Carlo simulations, approximations remain fairly close, and the proposed optimizer performs close to optimal solutions.The paper also compares computational complexity with exhaustive search.
III. SURFACE AREAS AND DISTANCE DISTRIBUTIONS
The analysis partitions the visible-satellite region into main- and side-lobe areas and uses spherical-cap geometry to derive the areas and distance distributions needed for serving-satellite analysis.
- Areas of interest: The visible region is partitioned into main-lobe and side-lobe regions according to the threshold polar angle ψ_th.The regions are A_ml_vis for 0 ≤ ψ ≤ ψ_th and A_sl_vis for ψ_th ≤ ψ ≤ ψ_max.
- Distance distributions: A(x) is defined as the spherical cap containing points whose terminal distance is less than x, enabling distance probabilities through area ratios.Its area grows from zero at x = a to the full sphere area at x = 2r_e + a.
- Surface areas: The complete satellite region is a sphere, while the visible and main-lobe regions are spherical caps whose areas are computed from their cap heights.The side-lobe area is obtained as the difference between the visible and main-lobe cap areas.
- Beamwidth effect: Increasing the beamwidth increases ψ_th, enlarging the angular boundary associated with the main lobe.The derivative of ψ_th with respect to ω_th is positive for positive altitude.
- Beamwidth effect: The main-lobe area shrinks as the main-lobe beamwidth increases, and these surface areas determine serving-satellite distribution probabilities.The paper uses the resulting areas in the subsequent serving-satellite analysis.
B. Distribution of Distance to Nearest Satellite
The paper models terminal-to-nearest-satellite distance through spherical-cap geometry, deriving its CDF and PDF from homogeneous BPP probabilities.
- A(x) is a spherical cap containing points whose terminal distance is less than x, with x ranging from a to 2r_e+a.Its height is q(x) = (x^2 − a^2)/(2r_e).
- The cap area grows from zero at x = a to the full-sphere area 4π(r_e+a)^2 at x = 2r_e+a.Thus, the distance event is equivalent to satellite placement in A(x), whose probability is the cap-area ratio.
- The nearest-satellite CDF is 0 for x≤a, 1−(1−κ(x))^S for a<x≤2r_e+a, and 1 beyond 2r_e+a.
- Differentiating the CDF yields the corresponding nearest-satellite PDF over a<x≤2r_e+a, with zero density elsewhere.
C. Distribution of Distance to Serving Satellite
The serving satellite is classified by whether its main or side lobe covers the terminal, or whether no satellite is visible; conditional distance distributions support the later performance analysis.
- Conditional distances: Y denotes terminal distance to a serving satellite whose main lobe is directed toward the terminal, while Z denotes the corresponding side-lobe distance.
- Conditional distances: The main-lobe boundary distance is determined by ψ_th through the spherical-distance relation given in the model.
- The derived distance distributions are used to obtain the exact system performance in the following analysis.
- Serving-satellite cases: The serving satellite has three cases: main-lobe association, side-lobe association, or no serving satellite because all satellites are invisible.The case probabilities are obtained using the BPP void probability.
- Beamwidth effect: Increasing satellite beamwidth makes association with a satellite whose main lobe points toward the terminal more likely.The paper attributes this to an increase in ψ_th with beamwidth.
B. Outage Probability
Conditioned on at least one visible satellite, the paper derives exact outage probability expressions by combining serving-case probabilities with integrals over main- and side-lobe distance domains.
- An outage occurs when the instantaneous serving-link rate falls below the required transmission rate R, assuming at least one satellite is visible.
- The exact system outage probability combines the main-lobe and side-lobe outage probabilities with the three serving-satellite distribution cases.
- Integral derivation: The derivation partitions the integration domain into D1 and D2 and sums the corresponding integrals.The D1 integral factors into x- and t-integrals, while the D2 integral is obtained after changing variable order.
- The final closed-form outage expression is exact but complicated for obtaining direct insight into system performance.A simpler tight approximation is developed later.
V. PERFORMANCE APPROXIMATION
The paper approximates finite-satellite BPP expressions with a PPP using the Poisson limit theorem, obtaining simpler outage formulas and asymptotic results while analyzing convergence.
- The Poisson limit theorem provides approximations for the three serving-satellite cases and the outage probability, with asymptotic analysis for S→∞.The asymptotic regime is intended for ultra-dense LEO satellite scenarios.
- PPP approximation: For sufficiently low satellite altitude, the bounded-area satellite process is asymptotically modeled by a PPP with density λ_s = S/(4π(r_e+a)^2).
- Distance distributions: The approximated nearest-distance CDF and PDF become closer to the exact results as x decreases, with near agreement in the satellite-visible region.As S increases, the nearest satellite is more likely to be close because of denser satellite distribution.
- Distance distributions: The approximated PDFs of Y and Z are fairly close to the exact PDFs, and serving-satellite distances become closer as S increases.
- Outage approximation: The approximated outage expression is simpler because the Poisson limit theorem removes the summation over the number of satellites.This is especially useful for evaluating systems with thousands of satellites.
- Asymptotic behavior: As S→∞, the nearest satellite is located in the main-lobe region with probability approaching one, and its distance approaches the minimum distance a.The asymptotic outage probability is therefore the outage probability for deterministic serving distance a.
- Convergence: The truncated approximated outage probability converges as the summation limit N→∞.
VI. THROUGHPUT MAXIMIZATION AND COMPLEXITY ANALYSES
The paper formulates throughput maximization over transmission rate and minimum elevation angle under visibility and outage constraints, then proposes bounded-set optimization because the original problem is non-convex.
- The paper reformulates the problem with bounded feasible sets and proposes an iterative algorithm for throughput maximization.
- Throughput is optimized over transmission rate R and minimum elevation angle θmin under satellite-visibility and outage constraints.The throughput is the successfully transferred data rate without outage.
- Low R reduces the data rate, whereas high R increases outage probability and can reduce system throughput.
- Low θmin can produce long serving-satellite distances, while high θmin increases satellite-invisible probability.
- The visibility constraint targets availability, the outage constraint targets reliability, and non-convexity makes direct optimization difficult.A two-dimensional exhaustive search can obtain optimal solutions but may require high computational complexity.
A. Problem Transformation and Iterative Algorithm
The optimization is decomposed into two bounded subproblems whose feasible limits follow from outage monotonicity, and an iterative algorithm alternately searches transmission rate and elevation angle.
- Problem Transformation and Iterative Algorithm: The derivative of θmin with respect to dmax is always negative, so θmin decreases monotonically with dmax.
- Problem Transformation and Iterative Algorithm: The outage constraint bounds feasible transmission rate because outage probability increases with R.For fixed θmin, the maximum feasible rate occurs where the outage constraint is tight.
- Problem Transformation and Iterative Algorithm: For fixed optimal R*, the outage constraint supplies a lower bound θ0(R*) because outage probability decreases with θmin.
- Problem Transformation and Iterative Algorithm: Algorithm 1 initializes θmin at an upper bound μ, searches R within its feasible interval, then searches θmin within its outage-feasible interval.The procedure updates throughput and repeats the alternating searches until convergence criteria are met.
- Problem Transformation and Iterative Algorithm: The proposed method outputs optimized R* and θ*min after iteratively solving the two subproblems.
B. Complexity Analyses
The approximated outage expression and iterative algorithm reduce computational burden relative to exact probability evaluation and two-dimensional exhaustive search.
- Complexity Analyses: Exact serving-satellite probabilities require O(S), whereas approximated probabilities require O(1) because they replace S-th-power terms with exponentials.
- Complexity Analyses: Exact outage probability has complexity O(S^2N^2τ), while the approximation has complexity O(N^2τ) when α=2.The approximated expression is therefore significantly less complex for large S because its complexity is independent of S.
- Complexity Analyses: Two-dimensional exhaustive search has complexity O(cout⌊90°/Δθ⌋⌊R̂/ΔR⌋).
- Complexity Analyses: Algorithm 1 has complexity O(coutL(⌊Rmax(μ)/ΔR⌋+⌊μ/Δθ⌋)), making the approximate expression and iterative algorithm easier to use.
VII. NUMERICAL RESULTS
Numerical results validate the exact and approximate analyses across visibility, outage, and throughput experiments, while showing how satellite count, elevation angle, fading, and frequency band affect performance.
- Visibility and serving-satellite distributions: As S increases, the serving satellite becomes more likely to lie in Avis, while the probability of satellite invisibility decreases.
- Visibility and serving-satellite distributions: For small S, the serving satellite is more likely in Aslvis because its area is 1.14 × 10^7 km2 versus 1.82 × 10^5 km2 for Amlvis.
- Visibility and serving-satellite distributions: Satellite-visible probability decreases as θmin increases or altitude decreases, and exact results match simulations while approximations remain close.
- Visibility and serving-satellite distributions: 90% satellite-visible probability with S=100 requires θmin≤{7.7,20.7}° at altitudes {600,1200} km, but is unattainable at 300 km.
- Outage and throughput: Exact outage probabilities match simulations and approximations closely; outage increases with transmission rate and more severe shadowing.
- Outage and throughput: Exact throughput curves match simulations and approximations are nearly identical; handheld terminals have higher bps/Hz throughput, while VSATs can provide higher bps with wider Ka-band bandwidth.
- Outage and throughput: As S increases, outage probability first decreases and then becomes constant as serving-satellite distance approaches the minimum distance a.
- Outage and throughput: Outage decreases as θmin increases, and handheld terminals outperform VSATs because S-band has lower path loss and fewer EIRP-density limitations.
VIII. CONCLUSIONS
The paper analyzes downlink LEO satellite systems with uniformly distributed satellites, deriving exact and approximated outage expressions and optimizing throughput. Simulations verify the analyses and show the proposed algorithm performs close to exhaustive-search optimum.
- The study models multiple LEO satellites uniformly distributed at a fixed altitude and analyzes serving-satellite distance distributions and distribution-case probabilities.
- It derives the exact outage probability and an approximation based on the Poisson limit theorem.
- The proposed iterative algorithm jointly optimizes transmission rate and minimum elevation angle to maximize system throughput.
- Simulation results verify the exact and approximated analyses and show the proposed algorithm performs close to the optimum.The comparison uses exhaustive search as the optimal reference.
- The approximated expressions are expected to provide high accuracy with lower complexity for systems containing more than thousands of satellites.
APPENDIX A PROOF OF LEMMA 1
The appendix derives serving-satellite distribution cases and outage expressions by applying void probabilities, geometric integration domains, and previously established lemmas. The derivation separates the integration into domains and obtains final outage expressions from the resulting integrals.
- The void probability gives the probability that a region contains no satellite and supports the distribution-case calculations.
- The three serving-satellite cases distinguish satellites in the minimum visible region, satellites only in a larger visible region, and complete satellite invisibility.
- The probabilities of the three cases are obtained by applying the void-probability expression to surface areas derived earlier.
- The approximated outage probability is derived similarly to the exact outage probability using the approximated distribution functions and outage terms.
- A variable transformation rewrites the integral, whose domain is divided into D3 and D4 for separate evaluation.
- Combining the evaluated integrals with the preceding equations yields the final outage expression, while the other outage term follows by similar steps.
- Some intermediate outage derivation details are omitted because of space limitations.