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Stochastic Analysis of Homogeneous Wireless Networks Assisted by Intelligent Reflecting Surfaces

Ali H. Abdollahi Bafghi, Mahtab Mirmohseni, Masoumeh Nasiri-Kenari, Behrouz Maham, Umberto Spagnolini

TL;DR

This work develops a stochastic-geometry framework for a homogeneous wireless network assisted by Intelligent Reflecting Surfaces, with BSs, users, and IRSs distributed as independent PPPs. It derives analytical upper and lower bounds on the second moments of the desired signal and interference, and provides tail bounds for the corresponding power random variables to bound outage probability. The results reveal how the bounds scale with the number of IRS elements and IRS density, showing, for large $Q$, a dominant signal gain scaling as $O(Q^2 λ_{IRS}^4)$ and an interference gain that grows more slowly, leading to improved outage performance as IRS density increases. Numerical results corroborate the bounds and illustrate the trade-offs between signal enhancement and interference in IRS-rich deployments, offering insights for network design and IRS deployment strategies.

Abstract

In this paper, we study the impact of the existence of multiple IRSs in a homogeneous wireless network, in which all BSs, users (U), and IRSs are spatially distributed by an independent homogeneous PPP, with density $λ_{\rm BS}\rm{[BS/m^2]}$, $λ_{\rm U}\rm{[U/m^2]}$, and $λ_{\rm IRS}\rm{[IRS/m^2]}$, respectively. We utilize a uniformly random serving strategy for BS and IRS to create stochastic symmetry in the network. We analyze the performance of the network and study the effect of the existence of the IRS on the network performance. To this end, for a typical user in the system, we derive analytical upper and lower bounds on the expectation of the power (second statistical moment) of the desired signal and the interference caused by BSs and other users. After that, we obtain analytical upper bounds on the decay of the probability of the power of the desired signal and the interference for the typical user (which results in a lower bound for the cumulative distribution function (CDF)). Moreover, we derive upper bounds on the decay of the probability of the capacity of one typical user, which results in a lower bound for the outage probability. In the numerical results, we observe that the numerical calculation of the power of the desired signal and the interference is near the derived lower bounds and we show that the increment of the parameter ${(λ_{\rm IRS})}$ causes increment in powers of both the desired and interference signals. We also observe that the increment of the parameter ${λ_{\rm IRS}}$ causes the decrement of outage probability.

Stochastic Analysis of Homogeneous Wireless Networks Assisted by Intelligent Reflecting Surfaces

TL;DR

This work develops a stochastic-geometry framework for a homogeneous wireless network assisted by Intelligent Reflecting Surfaces, with BSs, users, and IRSs distributed as independent PPPs. It derives analytical upper and lower bounds on the second moments of the desired signal and interference, and provides tail bounds for the corresponding power random variables to bound outage probability. The results reveal how the bounds scale with the number of IRS elements and IRS density, showing, for large , a dominant signal gain scaling as and an interference gain that grows more slowly, leading to improved outage performance as IRS density increases. Numerical results corroborate the bounds and illustrate the trade-offs between signal enhancement and interference in IRS-rich deployments, offering insights for network design and IRS deployment strategies.

Abstract

In this paper, we study the impact of the existence of multiple IRSs in a homogeneous wireless network, in which all BSs, users (U), and IRSs are spatially distributed by an independent homogeneous PPP, with density , , and , respectively. We utilize a uniformly random serving strategy for BS and IRS to create stochastic symmetry in the network. We analyze the performance of the network and study the effect of the existence of the IRS on the network performance. To this end, for a typical user in the system, we derive analytical upper and lower bounds on the expectation of the power (second statistical moment) of the desired signal and the interference caused by BSs and other users. After that, we obtain analytical upper bounds on the decay of the probability of the power of the desired signal and the interference for the typical user (which results in a lower bound for the cumulative distribution function (CDF)). Moreover, we derive upper bounds on the decay of the probability of the capacity of one typical user, which results in a lower bound for the outage probability. In the numerical results, we observe that the numerical calculation of the power of the desired signal and the interference is near the derived lower bounds and we show that the increment of the parameter causes increment in powers of both the desired and interference signals. We also observe that the increment of the parameter causes the decrement of outage probability.
Paper Structure (21 sections, 9 theorems, 621 equations, 7 figures)

This paper contains 21 sections, 9 theorems, 621 equations, 7 figures.

Table of Contents

  1. Introduction
  2. System Model and Preliminaries
  3. System Model
  4. BSs and IRSs selection and serving strategy:
  5. Upper and Lower Bounds on $E\left\{ {{{\left| {{\mathbb{S} }({{\mathbf{x}}_{\rm{U}}})} \right|}^2}} \right\}$ and $E\left\{ {{{\left| {\mathbb{I} ({{\mathbf{x}}_{\rm{U}}})} \right|}^2}} \right\}$
  6. Bounds on $E\left\{ {{{\left| {{\mathbb{S} }({{\mathbf{x}}_{\rm{U}}})} \right|}^2}} \right\}$
  7. Bounds on $E\left\{ {{{\left| {\mathbb{I} ({{\mathbf{x}}_{\rm{U}}})} \right|}^2}} \right\}$
  8. Bounds on the CDF of the Power of Signal and Interference
  9. CDF of $\mathbb{P}\left( {\mathbb{S}\left( {{{\mathbf{x}}_{\rm{U}}}} \right)} \right)$ and $\mathbb{P}\left( {\mathbb{I}\left( {{{\mathbf{x}}_{\rm{U}}}} \right)} \right)$
  10. Outage probability
  11. Numerical Results
  12. Conclusion
  13. Proof of Theorem 1
  14. Proof of Theorem 2
  15. Proof of Theorem 3
  16. ...and 6 more sections

Key Result

Theorem 1

$E\left\{ {{{\left| {\mathbb{S} ({{\mathbf{x}}_{\rm{U}}})} \right|}^2}} \right\}$ is upper bounded as follows: where $P^{\mathbb{S} }_{\max}({\rm BS})$ is caused from the BS and $P^{\mathbb{S} }_{\max}({\rm IRS})$ is caused from the IRSs, which are given as follows:

Figures (7)

  • Figure 1: IRS assisted network where BS, U, and IRS are located in $\mathbf{x}_{\rm BS},\mathbf{x}_{\rm U},\mathbf{x}_{\rm IRS}$, and $\mathbb{I}({\mathbf{x}}_{{\rm {U}}})$ is the ensemble interference to U (dashed lines).
  • Figure 2: PPP processes in ${\cal C}({{\bf {x}}_{{0}}},{R_{{\rm {co}}}})$ and ${\cal C}({{\bf {x}}_{{0}}},{2R_{{\rm {co}}}})$ for $\ell=1$ the BSs, $\ell=2$ the Us, and $\ell=3$ the IRSs.
  • Figure 3: Illustration of the system with users, BSs, and IRSs spatially distributed with PPP. In this figure, green circles, red circles, and blue rectangles denote users, BSs, and IRSs, respectively.
  • Figure 4: Comparison of $P^{\mathbb{S}}_{\min},P^{\mathbb{I}}_{\min}$, and numerical powers of the desired signal and interference for different values of $\lambda_{\rm IRS} \rm{[IRS/m^2]}$.
  • Figure 5: $E\{{\mathbb{C}}({\bf x}_{\rm U})\}$ [bits] for different values of $\lambda_{\rm IRS} \rm{[IRS/m^2]}$.
  • ...and 2 more figures

Theorems & Definitions (9)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Theorem 6
  • Theorem 7
  • Theorem 8
  • Theorem 9