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Closed-form analysis of Multi-RIS Reflected Signals in RIS-Aided Networks Using Stochastic Geometry

Guodong Sun, Francois Baccelli

TL;DR

This paper tackles the challenge of analytically evaluating multi-RIS reflections in RIS-aided networks using stochastic geometry. It develops a framework that models RIS locations as point processes conditioned on BS placements and derives closed-form Laplace transforms for the aggregated RIS-reflected power, accommodating Rayleigh and Nakagami-$m$ fading. Key contributions include a closed-form expression for the RIS-reflection Laplace transform under PPP deployment, extensions to BPP and Nakagami fading, and a practical proximity-deployment guideline showing near-BS or near-UE RIS placement yields the best gains. The results demonstrate substantial computational efficiency over Monte Carlo simulations and provide actionable insights for RIS deployment and modeling in realistic cellular networks.

Abstract

Reconfigurable intelligent surfaces (RISs) enhance wireless communication by creating engineered signal reflection paths in addition to direct links. This work presents a stochastic geometry framework using point processes (PPs) to model multiple randomly deployed RISs conditioned on their associated base station (BS) locations. By characterizing aggregated reflections from multiple RISs using the Laplace transform, we analytically assess the performance impact of RIS-reflected signals by integrating this characterization into well-established stochastic geometry frameworks. Specifically, we derive closed-form expressions for the Laplace transform of the reflected signal power in several deployment scenarios. These analytical results facilitate performance evaluation of RIS-enabled enhancements. Numerical simulations validate that optimal RIS placement favors proximity to BSs or user equipment (UEs), and further quantify the impact of reflected interference, various fading assumptions, and diverse spatial deployment strategies. Importantly, our analytical approach shows superior computational efficiency compared to Monte Carlo simulations.

Closed-form analysis of Multi-RIS Reflected Signals in RIS-Aided Networks Using Stochastic Geometry

TL;DR

This paper tackles the challenge of analytically evaluating multi-RIS reflections in RIS-aided networks using stochastic geometry. It develops a framework that models RIS locations as point processes conditioned on BS placements and derives closed-form Laplace transforms for the aggregated RIS-reflected power, accommodating Rayleigh and Nakagami- fading. Key contributions include a closed-form expression for the RIS-reflection Laplace transform under PPP deployment, extensions to BPP and Nakagami fading, and a practical proximity-deployment guideline showing near-BS or near-UE RIS placement yields the best gains. The results demonstrate substantial computational efficiency over Monte Carlo simulations and provide actionable insights for RIS deployment and modeling in realistic cellular networks.

Abstract

Reconfigurable intelligent surfaces (RISs) enhance wireless communication by creating engineered signal reflection paths in addition to direct links. This work presents a stochastic geometry framework using point processes (PPs) to model multiple randomly deployed RISs conditioned on their associated base station (BS) locations. By characterizing aggregated reflections from multiple RISs using the Laplace transform, we analytically assess the performance impact of RIS-reflected signals by integrating this characterization into well-established stochastic geometry frameworks. Specifically, we derive closed-form expressions for the Laplace transform of the reflected signal power in several deployment scenarios. These analytical results facilitate performance evaluation of RIS-enabled enhancements. Numerical simulations validate that optimal RIS placement favors proximity to BSs or user equipment (UEs), and further quantify the impact of reflected interference, various fading assumptions, and diverse spatial deployment strategies. Importantly, our analytical approach shows superior computational efficiency compared to Monte Carlo simulations.

Paper Structure

This paper contains 17 sections, 2 theorems, 23 equations, 8 figures, 1 table.

Table of Contents

  1. Introduction
  2. System Model
  3. General Framework
  4. Classical stochastic geometry analysis
  5. Laplace Transform of reflected signals
  6. Framework adaption for RIS deployment
  7. Special Case with Closed-Form Expressions
  8. PPP and Rayleigh modeling
  9. Extension to BPP modeling and Nakagami-$m$ fading
  10. Numerical Analysis
  11. Proximity deployment guideline
  12. System level RIS reflected interference
  13. Investigation of fading models
  14. RIS placement and selection
  15. Performance evaluation efficiency
  16. ...and 2 more sections

Key Result

Lemma 1

When RISs are configured as beamformers, the coverage probability for the communication coverage threshold $T$ is given by where $\Upsilon^+ \triangleq \max\{0, \Upsilon\}$.

Figures (8)

  • Figure 1: Randomly located RISs reflect signals to assist BS-UE communication, with inter-cell BSs and RISs generating interference.
  • Figure 2: A stochastic geometry-based performance analysis framework for cellular networks (uncolored), including RIS enhancement (colored).
  • Figure 3: The RISs are located at a constant distance $R$ from the BS, with their angular distribution being random along a circle.
  • Figure 4: The square area where multiple RISs are randomly deployed is varied in proximity from the UE to the BS.
  • Figure 5: The relationship between the RIS-enhanced ergodic rate and the distance of the RIS deployment area to the BS.
  • ...and 3 more figures

Theorems & Definitions (4)

  • Lemma 1
  • proof
  • Theorem 1
  • proof