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The Skyline Process: Quantifying Sky Visibility in 3D Urban Environments

Junse Lee, François Baccelli

Abstract

Non-terrestrial networks (NTNs) are considered a promising technology for seamless, universal communication in the 6G era. However, signals from NTN elements to ground users are often blocked by high-rise buildings in dense urban environments. To quantify this blocking effect, in this paper, we propose a novel analytical framework by modeling the location of buildings as a 3D skyline process based on stochastic geometry and we derive closed-form expressions for the distribution of the blockage elevation angles. Furthermore, we extend our analysis to include spatially correlated blockage effects and analyze the spectral properties of the Skyline process using power spectral density (PSD) and autocorrelation function (ACF). Based on these theoretical findings, we present numerical results that provide insights into the design of LEO satellite networks by computing the mean number of visible satellites and the outage probability. We provide a decorrelation angle that serves as a useful threshold for obtaining the satellite diversity gain. These findings provide first analytical steps toward designing and user connection strategies to NTNs in urban environments.

The Skyline Process: Quantifying Sky Visibility in 3D Urban Environments

Abstract

Non-terrestrial networks (NTNs) are considered a promising technology for seamless, universal communication in the 6G era. However, signals from NTN elements to ground users are often blocked by high-rise buildings in dense urban environments. To quantify this blocking effect, in this paper, we propose a novel analytical framework by modeling the location of buildings as a 3D skyline process based on stochastic geometry and we derive closed-form expressions for the distribution of the blockage elevation angles. Furthermore, we extend our analysis to include spatially correlated blockage effects and analyze the spectral properties of the Skyline process using power spectral density (PSD) and autocorrelation function (ACF). Based on these theoretical findings, we present numerical results that provide insights into the design of LEO satellite networks by computing the mean number of visible satellites and the outage probability. We provide a decorrelation angle that serves as a useful threshold for obtaining the satellite diversity gain. These findings provide first analytical steps toward designing and user connection strategies to NTNs in urban environments.

Paper Structure

This paper contains 27 sections, 10 theorems, 64 equations, 11 figures.

Table of Contents

  1. Introduction
  2. Related Works
  3. Network Analysis using Stochastic Geometry
  4. Urban Blockage Modeling
  5. Sky Visibility and Proposed Extension
  6. Contributions
  7. System Model and Geometric Formulation
  8. Modeling of 3D Urban Environment
  9. Skyline Process: LoS Sky Visibility from the User
  10. Statistical Properties of the Skyline Process
  11. First-Order Statistics: Blockage Probability
  12. Skyline Process in a Given Direction
  13. Skyline Process over All Directions
  14. Characterization of the Dominant Obstacle
  15. Joint Spatial Statistics for Angular Separation
  16. ...and 12 more sections

Key Result

Theorem 1

The number of buildings intersecting the user's viewing direction $\psi$ and within a distance $R$ from the user follows a Poisson distribution with parameter $\lambda l\left(R - \frac{l}{4\pi}\right)$ for $R > \frac{l}{2\pi}$ and $\lambda \pi R^2$ otherwise. CDF of $\boldsymbol{\omega_1(\psi)}$: Here, $\omega_1(\psi)$ denotes the maximum elevation blockage angle seen by the user in the directio

Figures (11)

  • Figure 1: Illustration of the urban network model.
  • Figure 2: Illustrations of Skyline processes.
  • Figure 3: Illustration of $\mathcal{A}(\psi)$ where $\psi = \frac{\pi}{8}, \frac{\pi}{4}$ and $l=1$.
  • Figure 4: CDF of $\omega_1(\psi)$ with $\lambda = 1$ and $l = 1$.
  • Figure 5: The probability $\mathbb{P}[\omega_2(\psi)=0]$ as a function of various urban parameters.
  • ...and 6 more figures

Theorems & Definitions (18)

  • Theorem 1
  • Theorem 2
  • Example 1
  • Example 2
  • Lemma 1
  • Example 3
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Example 4
  • ...and 8 more