From Gaussian Fading to Gilbert-Elliott: Bridging Physical and Link-Layer Channel Models in Closed Form

2026-04-03

AI Summary

This work addresses the long-standing challenge of establishing an exact closed-form mapping between physical-layer Gaussian fading and the two-state Gilbert–Elliott (GE) Markov model at the link layer, a task typically reliant on simulation or approximation. By thresholding a stationary Gaussian process into binary states over discrete time slots, the authors derive closed-form expressions for the GE state transition probabilities using Owen’s T function, requiring only the first-order correlation coefficient ρ. The study makes three key contributions: it establishes, for the first time, an exact analytical bridge from Gaussian fading to the GE model; reveals how the smoothness of the covariance kernel governs the scaling law of link persistence times; and unifies two distinct Markov approximation diagnostic criteria. Monte Carlo simulations validate the theoretical results, which reduce to the classical arcsine identity when the threshold equals the process mean.

Abstract

Dynamic fading channels are modeled at two fundamentally different levels of abstraction. At the physical layer, the standard representation is a correlated Gaussian process, such as the dB-domain signal power in log-normal shadow fading. At the link layer, the dominant abstraction is the Gilbert-Elliott (GE) two-state Markov chain, which compresses the channel into a binary ``decodable or not'' sequence with temporal memory. Both models are ubiquitous, yet practitioners who need GE parameters from an underlying Gaussian fading model must typically simulate the mapping or invoke continuous-time level-crossing approximations that do not yield discrete-slot transition probabilities in closed form. This paper provides an exact, closed-form bridge. By thresholding the Gaussian process at discrete slot boundaries, we derive the GE transition probabilities via Owen's $T$-function for any threshold, reducing to an elementary arcsine identity when the threshold equals the mean. The formulas depend on the covariance kernel only through the one-step correlation coefficient $ρ= K(D)/K(0)$, making them applicable to any stationary Gaussian fading model. The bridge reveals how kernel smoothness governs the resulting link-layer dynamics: the GE persistence time grows linearly in the correlation length $T_c$ for a smooth (squared-exponential) kernel but only as $\sqrt{T_c}$ for a rough (exponential/Ornstein--Uhlenbeck) kernel. We further quantify when the first-order GE chain is a faithful approximation of the full binary process and when it is not, reconciling two diagnostics, the one-step Markov gap and the run-length total-variation distance, that can trend in opposite directions. Monte Carlo simulations validate all theoretical predictions.

Problem

• Gaussian fading
• Gilbert-Elliott model
• channel modeling
• closed-form
• Markov chain

Innovation

• closed-form bridge
• Gaussian fading
• Gilbert-Elliott model
• Owen's T-function
• stationary Gaussian process