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Stochastic differential equations for performance analysis of wireless communication systems

Eya Ben Amar, Nadhir Ben Rached, Raul Tempone, Mohamed-Slim Alouini

TL;DR

Time-varying wireless channels exhibit fading dynamics that static models fail to capture. The authors develop stochastic differential equation (SDE) based channel models using Markovian projection to obtain a low-dimensional envelope SDE for the squared envelope $R=I^2+Q^2$, covering asymptotic Rayleigh, Rice, and Hoyt cases, and study fade-duration statistics through Monte Carlo (MC) simulations. They introduce an importance sampling (IS) framework for efficient estimation of the right tail of the fade-duration CCDF, deriving the optimal control via a PDE/Kolmogorov backward equation and demonstrating substantial variance reduction with a projected SDE. The framework is validated across Rayleigh, Rice, and Hoyt fading, showing how channel parameters influence tail behavior and demonstrating practical gains for reliable performance analysis in dynamic wireless environments. Overall, the work provides a principled, low-dimensional stochastic modeling approach and efficient rare-event estimation tools for time-varying fading channels.

Abstract

This paper addresses the difficulty of characterizing the time-varying nature of fading channels. The current time-invariant models often fall short of capturing and tracking these dynamic characteristics. To overcome this limitation, we explore using of stochastic differential equations (SDEs) and Markovian projection to model signal envelope variations, considering scenarios involving Rayleigh, Rice, and Hoyt distributions. Furthermore, it is of practical interest to study the performance of channels modeled by SDEs. In this work, we investigate the fade duration metric, representing the time during which the signal remains below a specified threshold within a fixed time interval. We estimate the complementary cumulative distribution function (CCDF) of the fade duration using Monte Carlo simulations, and analyze the influence of system parameters on its behavior. Finally, we leverage importance sampling, a known variance-reduction technique, to estimate the tail of the CCDF efficiently.

Stochastic differential equations for performance analysis of wireless communication systems

TL;DR

Time-varying wireless channels exhibit fading dynamics that static models fail to capture. The authors develop stochastic differential equation (SDE) based channel models using Markovian projection to obtain a low-dimensional envelope SDE for the squared envelope , covering asymptotic Rayleigh, Rice, and Hoyt cases, and study fade-duration statistics through Monte Carlo (MC) simulations. They introduce an importance sampling (IS) framework for efficient estimation of the right tail of the fade-duration CCDF, deriving the optimal control via a PDE/Kolmogorov backward equation and demonstrating substantial variance reduction with a projected SDE. The framework is validated across Rayleigh, Rice, and Hoyt fading, showing how channel parameters influence tail behavior and demonstrating practical gains for reliable performance analysis in dynamic wireless environments. Overall, the work provides a principled, low-dimensional stochastic modeling approach and efficient rare-event estimation tools for time-varying fading channels.

Abstract

This paper addresses the difficulty of characterizing the time-varying nature of fading channels. The current time-invariant models often fall short of capturing and tracking these dynamic characteristics. To overcome this limitation, we explore using of stochastic differential equations (SDEs) and Markovian projection to model signal envelope variations, considering scenarios involving Rayleigh, Rice, and Hoyt distributions. Furthermore, it is of practical interest to study the performance of channels modeled by SDEs. In this work, we investigate the fade duration metric, representing the time during which the signal remains below a specified threshold within a fixed time interval. We estimate the complementary cumulative distribution function (CCDF) of the fade duration using Monte Carlo simulations, and analyze the influence of system parameters on its behavior. Finally, we leverage importance sampling, a known variance-reduction technique, to estimate the tail of the CCDF efficiently.
Paper Structure (23 sections, 4 theorems, 107 equations, 13 figures, 1 table)

This paper contains 23 sections, 4 theorems, 107 equations, 13 figures, 1 table.

Table of Contents

  1. Introduction
  2. Motivation
  3. Literature Review
  4. Contributions
  5. Stochastic Differential Equations for Modeling Wireless Channels
  6. Problem Setting
  7. Markovian Projection
  8. Rayleigh Case
  9. Rice Fading
  10. Hoyt Case
  11. Fading Duration
  12. Problem Formulation
  13. Monte Carlo Estimator
  14. Importance Sampling for Right-Tail Estimation
  15. Motivation
  16. ...and 8 more sections

Key Result

Lemma 1

Markovian projection gyongy1986mimicking We let $X \in \mathbbm{R}^d$ solve and we consider the non-Markovian process $S=P X$, where $S \in \mathbbm{R}^{\bar{d}}$ and $P^T=\left(P_1^T, \cdots, P_{\bar{d}}^T\right)^T$ represents the projection matrix onto a dimension $1 \leq \bar{d}<d$. We let $\bar{S}^{\left(x_0\right)} \in \mathbbm{R}^{\bar{d}}$ solve for $t \in[0, T]$ where $\bar{W}$ denotes a

Figures (13)

  • Figure 1: Rayleigh Fading: Histogram of $\bar{R} (T)$ and $I(T)^2+Q(T)^2$ using $10^6$ samples with the following parameters: $T=4$, $N=100$, $I_0=0$, $Q_0=0$, $B=1$, and $\sigma=1$.
  • Figure 2: Fifty samples of $I(s)^2+Q(s)^2$ with the following parameters: $N=100$, $I_0=0$, $Q_0=0$, $k=1$, $\theta=1$, and $\beta=1$.
  • Figure 3: Comparison of the exact and approximated drift with the following parameters: $N=100$, $I_0=0$, $Q_0=0$, $k=1$, $\theta=1$, and $\beta=1$.
  • Figure 4: Rice Fading: Histogram of $\bar{R} (T)$ and $I(T)^2+Q(T)^2$ using $10^6$ samples with the following parameters: $N=100$, $I_0=0$, $Q_0=0$, $k=1$, $\theta=1$, and $\beta=1$.
  • Figure 5: Histogram of $\bar{R} (T)$ and $I(T)^2+Q(T)^2$ using $10^6$ samples with the following parameters: $N=100$, $I_0=0$, $Q_0=0$, $k_1=0.1$, $k_2=0.5$, $\beta_1=\beta_2=1$.
  • ...and 8 more figures

Theorems & Definitions (6)

  • Lemma 1
  • Proposition 1
  • Proposition 2
  • Theorem 1
  • Remark 1
  • Remark 2