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On the Impact of Dynamic Beamforming on EMF Exposure and Network Coverage: A Stochastic Geometry Perspective

Quentin Gontier, Charles Wiame, Joe Wiart, François Horlin, Christo Tsigros, Claude Oestges, Philippe De Doncker

Abstract

This paper introduces a new mathematical framework for dynamic beamforming-based cellular networks, grounded in stochastic geometry. The framework is used to study the electromagnetic field exposure (EMFE) of active and idle users as a function of the distance between them. A novel multi-cosine antenna pattern is introduced, offering more accurate modeling by incorporating both main and side lobes. Results show that the cumulative distribution functions of EMFE and coverage obtained with the multi-cosine pattern align closely with theoretical models, reducing error to less than 2\%, compared to a minimum of 8\% for other models. The marginal distribution of EMFE for each user type is mathematically derived. A unique contribution is the introduction of the SCAIU (\underline{S}patial \underline{C}DF for \underline{A}ctive and \underline{I}dle \underline{U}sers), a metric that ensures coverage for active users while limiting EMFE for idle users. Network performance is analyzed using these metrics across varying distances and antenna elements. The analysis reveals that, for the chosen network parameters, with 64 antenna elements, the impact on idle user EMFE becomes negligible beyond 60~m. However, to maintain active user SINR above 10 dB and idle user EMFE below -50~dBm at 2~m, more than 256 elements are required.

On the Impact of Dynamic Beamforming on EMF Exposure and Network Coverage: A Stochastic Geometry Perspective

Abstract

This paper introduces a new mathematical framework for dynamic beamforming-based cellular networks, grounded in stochastic geometry. The framework is used to study the electromagnetic field exposure (EMFE) of active and idle users as a function of the distance between them. A novel multi-cosine antenna pattern is introduced, offering more accurate modeling by incorporating both main and side lobes. Results show that the cumulative distribution functions of EMFE and coverage obtained with the multi-cosine pattern align closely with theoretical models, reducing error to less than 2\%, compared to a minimum of 8\% for other models. The marginal distribution of EMFE for each user type is mathematically derived. A unique contribution is the introduction of the SCAIU (\underline{S}patial \underline{C}DF for \underline{A}ctive and \underline{I}dle \underline{U}sers), a metric that ensures coverage for active users while limiting EMFE for idle users. Network performance is analyzed using these metrics across varying distances and antenna elements. The analysis reveals that, for the chosen network parameters, with 64 antenna elements, the impact on idle user EMFE becomes negligible beyond 60~m. However, to maintain active user SINR above 10 dB and idle user EMFE below -50~dBm at 2~m, more than 256 elements are required.
Paper Structure (27 sections, 8 theorems, 51 equations, 8 figures, 1 table)

This paper contains 27 sections, 8 theorems, 51 equations, 8 figures, 1 table.

Table of Contents

  1. Introduction
  2. Related Works
  3. Exploring EMFE
  4. DBF models in SG
  5. EMFE assessment using SG
  6. Contributions
  7. System Model
  8. Topology
  9. Propagation Model
  10. Antenna Pattern Models
  11. Mathematical Framework
  12. Preliminaries
  13. EMFE
  14. Coverage
  15. SCAIU
  16. ...and 12 more sections

Key Result

Proposition 1

The $k$th moments ($k>0$) of the gain models are given by where $\operatorname{erf}(\cdot)$ is the error function and $\chi_{i_{\textrm{\normalfont max}}}^\dagger = \sum\limits_{i = 1}^{i_{\textrm{\normalfont max}}} \chi_i$.

Figures (8)

  • Figure 1: Scheme of the network with an AU at the origin and an IU at a distance $d$. The antenna pattern of a typical BS is shown in the lower right corner ($N \! = \! 16$).
  • Figure 2: Antenna patterns for positive angles and zoom ($N \! = \! 16$)
  • Figure 3: CDF of IU's EMFE for different antenna patterns. $N \! = \! 64$, $d \! = \! \numprint[m]{10}$.
  • Figure 4: CDF of signal and interference power contributing to the total IU's EMFE. $N \! = \! 64$, $d \! = \! \numprint[m]{10}$.
  • Figure 5: CCDF of SINR for different antenna patterns and zoom on the CCDF. $N \! = \! 64$, $d \! = \! \numprint[m]{10}$.
  • ...and 3 more figures

Theorems & Definitions (14)

  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Proposition 3
  • proof
  • Theorem 1
  • proof
  • Lemma 1
  • Theorem 2
  • ...and 4 more