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arxiv: 2604.03160 · v1 · submitted 2026-04-03 · 💻 cs.IT · eess.SP· math.IT

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

Bhaskar Krishnamachari , Victor Gutierrez This is my paper

Pith reviewed 2026-05-13 18:36 UTC · model grok-4.3

classification 💻 cs.IT eess.SPmath.IT
keywords Gaussian fadingGilbert-Elliott channelclosed-form mappingOwen's T-functionMarkov approximationcorrelation coefficientlink-layer modelingthresholding
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The pith

Thresholding a correlated Gaussian fading process at discrete slots produces exact Gilbert-Elliott transition probabilities in closed form via Owen's T-function.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper establishes an exact mapping from a continuous correlated Gaussian fading process to the parameters of a discrete two-state Gilbert-Elliott Markov chain. The transition probabilities are obtained by hard-thresholding the Gaussian values at each time slot and evaluating the resulting joint probabilities with Owen's T-function; the formulas depend on the underlying covariance only through the single scalar one-step correlation coefficient. When the threshold sits at the mean, the expressions simplify to an elementary arcsine relation. The same formulas also show how kernel smoothness controls link-layer persistence: linear growth with correlation length for squared-exponential kernels versus square-root growth for exponential kernels. The work further reconciles two diagnostics for the accuracy of the first-order Markov approximation.

Core claim

By thresholding the Gaussian process at discrete slot boundaries, the GE transition probabilities are derived 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 rho = K(D)/K(0), making them applicable to any stationary Gaussian fading model.

What carries the argument

Hard thresholding of the continuous Gaussian fading process to a binary decodable/not sequence at fixed slot times, with transition probabilities expressed through Owen's T-function evaluated at the one-step correlation rho.

If this is right

  • The GE persistence time scales linearly with correlation length for smooth kernels and as the square root for rough kernels.
  • The mapping applies directly to any stationary Gaussian model once its one-step correlation is known.
  • The first-order Markov approximation can be checked by comparing one-step Markov gap against run-length total-variation distance.
  • Closed-form GE parameters can be substituted into link-layer analyses without requiring Monte Carlo sampling of the physical channel.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Designers could pre-compute GE tables for common kernels and feed them into cross-layer schedulers or error-control codes.
  • The same thresholding-plus-Owen-T approach may extend to other continuous-to-binary channel abstractions such as those arising in optical or molecular communication.
  • Measuring empirical one-step correlation from field data would immediately yield usable GE parameters without needing the full covariance function.

Load-bearing premise

The binary sequence produced by hard thresholding the Gaussian process at each discrete slot is adequately described by a first-order Markov chain.

What would settle it

A Monte Carlo simulation of thresholded samples from a Gaussian process with known one-step correlation rho that produces state-transition frequencies measurably different from the Owen's T-function expressions.

Figures

Figures reproduced from arXiv: 2604.03160 by Bhaskar Krishnamachari, Victor Gutierrez.

Figure 1
Figure 1. Figure 1: Two diagnostics for the matched first-order GE chain ( [PITH_FULL_IMAGE:figures/full_fig_p012_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Sample paths of the stationary Gaussian fading process for three values of [PITH_FULL_IMAGE:figures/full_fig_p014_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Matched GE transition probability p01 = p10 versus Tc for S/σ = 0, both kernels. Solid and dashed curves are the arcsin formula (18) with ρ from (7); markers are Monte Carlo estimates with 95% CIs. Note that the two kernels map the same Tc to different values of ρ, so this figure compares GE parameters at matched Tc, not matched ρ. 2 4 6 8 10 12 14 Coherence parameter Tc 5 10 15 20 25 30 35 [TGE] Squared-e… view at source ↗
Figure 4
Figure 4. Figure 4: Expected persistence time E[TGE] versus Tc for S/σ = 0. Left: squared-exponential kernel with exact theory (16) (solid), linear asymptote (33) (dashed), and Monte Carlo (circles). Right: exponential kernel with exact theory (solid), √ Tc asymptote (34) (dashed), and Monte Carlo (circles). The contrasting growth rates (linear vs. √ Tc) are clearly visible. 15 [PITH_FULL_IMAGE:figures/full_fig_p015_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Effect of the threshold level on E[TGE], both kernels. Left: squared-exponential. Right: exponential. Solid lines are the closed-form theory (16); markers show Monte Carlo means with 95% CIs. Only nonnegative thresholds are shown (the mapping is symmetric under S/σ 7→ −S/σ). The square-root scaling of the exponential kernel is evident in the sublinear curvature of the right panel [PITH_FULL_IMAGE:figures/… view at source ↗
Figure 6
Figure 6. Figure 6: Matched GE transition probabilities (left column) and mean dwell times (right column) [PITH_FULL_IMAGE:figures/full_fig_p017_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Run-length distributions for state-1 runs ( [PITH_FULL_IMAGE:figures/full_fig_p018_7.png] view at source ↗
read the original 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 $\rho = K(D)/K(0)$, 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 0 minor

Summary. The manuscript derives exact closed-form expressions for the transition probabilities of the Gilbert-Elliott two-state Markov chain from an underlying stationary Gaussian process that models physical-layer fading. By hard-thresholding the Gaussian samples at discrete slot boundaries, the probabilities are obtained from the bivariate normal CDF via Owen's T-function for arbitrary thresholds; the expressions depend on the covariance kernel only through the one-step correlation coefficient ρ = K(D)/K(0) and reduce to the elementary arcsine law when the threshold equals the mean. The work further shows that GE persistence time scales linearly with correlation length Tc for a squared-exponential kernel but only as √Tc for an exponential kernel, and it quantifies the fidelity of the first-order Markov approximation by reconciling the one-step Markov gap with run-length total-variation distance, all corroborated by Monte Carlo simulations.

Significance. If the derivations hold, the paper supplies a parameter-light, exact bridge between ubiquitous physical-layer Gaussian fading models and link-layer GE chains that applies to any stationary kernel through ρ alone. This removes the need for simulation or continuous-time level-crossing approximations when discrete-slot transition probabilities are required. The explicit dependence on kernel smoothness yields concrete design insight (linear versus square-root scaling of persistence), while the dual-diagnostic error analysis for the Markov assumption is a useful practical contribution. Strengths include the closed-form expressions, direct reduction to the known arcsine identity, and Monte Carlo validation that matches theory across kernels.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive review and the recommendation to accept the manuscript. The summary correctly identifies the core contributions: the exact closed-form GE transition probabilities via Owen's T-function, their dependence solely on the one-step correlation ρ, the kernel-dependent scaling of persistence time, and the dual-diagnostic assessment of the first-order Markov approximation.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The central derivation obtains GE transition probabilities directly from the bivariate Gaussian CDF (equivalently Owen's T-function) applied to thresholded samples separated by lag D. The only input is the externally supplied one-step correlation rho = K(D)/K(0) for any stationary kernel; no parameters are fitted to the target quantities, no self-referential definitions appear, and no load-bearing self-citations are invoked. The first-order Markov approximation error is quantified separately via explicit diagnostics (one-step gap and run-length TV distance) with Monte Carlo validation, keeping the closed-form bridge independent of its own outputs.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The derivation rests on the standard assumption that the physical-layer fading is a stationary zero-mean Gaussian process whose one-step correlation is known, together with the modeling choice that the link-layer state is produced by hard thresholding at each discrete slot. No new entities are introduced and no parameters are fitted to data.

free parameters (2)
  • threshold
    User-selected level that defines the boundary between good and bad states in the binary process; it is a modeling choice rather than a fitted constant.
  • rho
    One-step correlation coefficient extracted from the covariance kernel evaluated at the slot duration; it is supplied by the physical-layer model rather than fitted inside the derivation.
axioms (2)
  • domain assumption The underlying fading is a stationary Gaussian random process.
    Standard physical-layer model for log-normal shadowing and similar effects; invoked when the bivariate normal distribution is used to obtain the joint probability of consecutive states.
  • domain assumption The link-layer binary sequence is produced by hard thresholding the Gaussian process at each discrete time slot.
    This step converts the continuous process into the decodable/not sequence whose transition probabilities are then derived.

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