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.

翻译：动态衰落信道在两种根本不同层次的抽象上进行建模。在物理层，标准表示是相关高斯过程，例如对数正态阴影衰落的dB域信号功率。在链路层，主导抽象是吉尔伯特-埃利奥特（GE）两状态马尔可夫链，它将信道压缩为具有时间记忆的二进制“可解码或不可解码”序列。这两种模型随处可见，然而，当从业者需要从底层高斯衰落模型中获取GE参数时，通常必须模拟映射过程或采用连续时间电平穿越近似方法，而这些方法无法以闭式形式给出离散时隙转移概率。本文提供了一种精确的闭式桥接方法。通过在离散时隙边界对高斯过程进行阈值化处理，我们利用Owen的$T$函数推导出任意阈值下的GE转移概率，当阈值等于均值时，该表达式可简化为基本反正弦恒等式。这些公式仅通过一步相关系数$ρ= K(D)/K(0)$依赖于协方差核，因此适用于任何平稳高斯衰落模型。该桥接揭示了核平滑性如何决定链路层动力学：对于平滑（平方指数）核，GE持续时长与相关长度$T_c$呈线性增长；而对于粗糙（指数/奥恩斯坦-乌伦贝克）核，则仅与$\sqrt{T_c}$成正比。我们进一步量化了一阶GE链何时能忠实逼近完整二进制过程、何时不能，并调和了可能呈现相反趋势的两个诊断指标——一步马尔可夫间隙和游程总变差距离。蒙特卡洛仿真验证了所有理论预测。