🤖 AI Summary
This study addresses the severe degradation in downlink SINR coverage probability and spectral efficiency caused by densification in the Walker low Earth orbit (LEO) satellite constellation under full frequency reuse. By leveraging geospatial constraints of the Earth and a stochastic channel model, the work uncovers a fundamental tension between linearly growing interference and the boundedness of the desired signal. The authors introduce the “deterministic visible ring-block lemma” to derive, for the first time, an explicit finite-N performance upper bound applicable to arbitrarily large constellations and phase configurations, and establish corresponding scaling laws under frequency reuse. The analysis rigorously shows that under full frequency reuse, both coverage probability and ergodic spectral efficiency decay as O(1/N). In contrast, under independent sparse reuse with qN = O(1), the performance upper bound scales as O(1/(qN)).
📝 Abstract
We establish densification converses for Walker LEO constellations under nearest-visible association in the full-frequency-reuse setting. Performance is evaluated under the invariant (stationary) measure induced by the constellation/Earth dynamics on the user--constellation ``phase state.'' A key Walker-specific feature, absent from unbounded planar models, is that association is restricted to a bounded visible cap determined by Earth geometry. Under power-law path-loss, a two-level antenna-gain model, i.i.d.\ nonnegative fading with unit mean and finite second moment, and nonzero noise, we prove that increasing the total satellite count $N=N_oN_s$ forces the aggregate interference to grow at least linearly in $N$, while the useful signal remains uniformly bounded above. Consequently, the downlink SINR coverage probability at any fixed threshold and the ergodic spectral efficiency both vanish as $N\to\infty$. The key technical ingredient is a deterministic visibility-annulus block lemma, uniform over all sufficiently large constellations and all "phase states", showing that a fixed fraction of visible satellites lies in a distance annulus strictly inside the horizon; this yields explicit finite-$N$ collapse bounds. In particular, we derive nonasymptotic $O(1/N)$ upper bounds on both coverage and ergodic spectral efficiency. Finally, in the case of frequency reuse through independent thinning, with activity probability $q$, we show that avoiding densification collapse necessarily requires $qN=O(1)$, equivalently a reuse factor $Ω(N)$, and we obtain a corresponding explicit $O(1/(qN))$ upper bound.