AI summaryⓘ
The authors study two common ways to model wireless signal fading: a detailed Gaussian model at the physical layer and a simpler two-state Markov model (Gilbert-Elliott) at the link layer. They provide an exact formula to convert between these models by analyzing how often the Gaussian signal crosses a threshold, using a specific mathematical function. This formula depends only on a simple correlation value, making it usable for many types of signal fading patterns. They also explain how the smoothness of the fading influences how long the channel stays in one state and show when the simpler Markov model accurately represents the more complex process. Their findings are supported by simulations confirming their theory.
Dynamic fading channelsGaussian processLog-normal shadow fadingGilbert-Elliott modelMarkov chainThresholdingOwen's T-functionCorrelation kernelRun-length distributionMonte Carlo simulation
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.