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arxiv: 2601.17859 · v2 · submitted 2026-01-25 · 📡 eess.SY · cs.IT· cs.SY· math.IT

Space-Air-Ground-Integrated Networks: The BER vs. Residual Delay and Doppler Analysis

Chao Zhang , Kunlun Li , Chao Xu , Lie-Liang Yang , Lajos Hanzo This is my paper

Pith reviewed 2026-05-16 11:11 UTC · model grok-4.3

classification 📡 eess.SY cs.ITcs.SYmath.IT
keywords space-air-ground-integrated networksbit error rateresidual DopplerShadowed-Rician channelsleast-square estimation16-QAMbivariate Gamma distributioncorrelation coefficient
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The pith

A closed-form bit error rate formula is derived for 16-QAM in space-air-ground networks under residual Doppler and synchronization delays.

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

The paper establishes a closed-form expression for the bit error rate performance of 16-QAM in time-varying correlated Shadowed-Rician channels typical of space-air-ground integrated networks. It does so by first calculating the correlation between pilot and data symbols and then approximating their joint distribution using a bivariate Gamma model. This allows analysis of how residual Doppler shifts and synchronization delays degrade performance when using least-squares channel estimation and equalization. A reader would care because these networks promise seamless global connectivity, and quantifying error rates under realistic imperfections informs practical system design for low-earth orbit satellites.

Core claim

The central discovery is that after deriving the specific correlation coefficient between pilot and data symbols in correlated Shadowed-Rician channels and mimicking the distribution by a bi-variate Gamma distribution, a closed-form BER formula can be derived for least-square channel estimation and equalization applied to 16-QAM modulation, while also accounting for realistic elliptical orbits and relativistic effects.

What carries the argument

The bi-variate Gamma distribution that approximates the joint statistics of the pilot and data symbols, based on the derived pilot-data correlation coefficient in the Shadowed-Rician fading model.

If this is right

  • The period of realistic elliptical orbits for a 300-km-altitude LEO is around 0.8 seconds longer than that of idealized circular orbits.
  • The relativistic delay is lower than 1 microsecond over a full LEO pass from rise to set.
  • The effects of residual Doppler, atmospheric shadowing, synchronization errors, and pilot overhead are quantified for L-band frequencies.

Where Pith is reading between the lines

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

  • This formula could guide the selection of pilot overhead to balance estimation accuracy against data rate in varying Doppler conditions.
  • Similar analysis might apply to higher-order modulations or other channel models in integrated networks.
  • Validation against measured satellite channel data would strengthen applicability to real deployments.

Load-bearing premise

The joint distribution of the correlated pilot and data symbols in Shadowed-Rician channels can be accurately represented by a bi-variate Gamma distribution.

What would settle it

A Monte Carlo simulation of the system with the exact channel model for different residual Doppler frequencies and comparing the resulting BER to the closed-form expression would confirm or refute the derivation.

Figures

Figures reproduced from arXiv: 2601.17859 by Chao Xu, Chao Zhang, Kunlun Li, Lajos Hanzo, Lie-Liang Yang.

Figure 1
Figure 1. Figure 1: FIGURE 1 [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIGURE 2 [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIGURE 3 [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIGURE 4 [PITH_FULL_IMAGE:figures/full_fig_p010_4.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIGURE 6 [PITH_FULL_IMAGE:figures/full_fig_p011_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIGURE 7 [PITH_FULL_IMAGE:figures/full_fig_p011_7.png] view at source ↗
read the original abstract

Perfect Doppler compensation and synchronization is nontrivial due to multi-path Doppler effects and Einstein's theory of relativity in the space-air-ground-integrated networks (SAGINs). Hence, by considering the residual Doppler and the synchronization delay, this paper investigates the bit-error-rate (BER) performance attained under time-varying correlated Shadowed-Rician SAGIN channels. First, a practical SAGIN model is harnessed, encompassing correlated Shadowed-Rician channels, the Snell's law-based path loss, atmospheric absorption, the line-of-sight Doppler compensation, elliptical satellite orbits, and Einstein's theory of relativity. Then, a specific correlation coefficient between the pilot and data symbols is derived in the context of correlated Shadowed-Rician channels. By exploiting this correlation coefficient, the channel distribution is mimicked by a bi-variate Gamma distribution. Then, a closed-form BER formula is derived under employing least-square channel estimation and equalization for 16-QAM. Our analytical results indicate for a 300-km-altitude LEO that 1) the period of realistic elliptical orbits is around 0.8 seconds longer than that of the idealized circular orbits; and 2) the relativistic delay is lower than 1 microsecond over a full LEO pass (from rise to set). Our numerical results for the L bands quantify the effects of: 1) the residual Doppler; 2) atmospheric shadowing; 3) synchronization errors; and 4) pilot overhead.

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

3 major / 2 minor

Summary. The manuscript claims to derive a closed-form BER expression for 16-QAM under least-squares channel estimation and equalization in SAGINs with residual Doppler and synchronization delay. This is obtained by first computing the pilot-data correlation coefficient in correlated Shadowed-Rician channels, then replacing the joint amplitude distribution with a bivariate Gamma model, followed by averaging over the 16-QAM decision regions. The work also incorporates Snell's-law path loss, atmospheric absorption, elliptical orbits, and relativistic effects, reporting that a 300 km LEO elliptical orbit period is ~0.8 s longer than circular and relativistic delay <1 μs over a full pass, with numerical quantification of residual Doppler, shadowing, synchronization errors, and pilot overhead in L-band.

Significance. If the bivariate Gamma step accurately reproduces the joint statistics required for exact 16-QAM BER averaging, the closed-form result would supply a practical analytical tool for SAGIN link-budget design. The explicit inclusion of elliptical-orbit timing and relativistic delay provides concrete, falsifiable predictions that could inform LEO constellation planning; the numerical sensitivity studies on residual Doppler and pilot overhead are directly usable for system-level trade-offs.

major comments (3)
  1. [Abstract] Abstract and BER derivation section: the central closed-form BER expression rests on the bivariate Gamma mimic of the correlated Shadowed-Rician amplitudes after the correlation coefficient is obtained; no intermediate algebra is shown demonstrating that the Gamma parameters preserve the higher-order joint moments needed for exact integration over the 16-QAM symbol decision regions, and no error-bound or moment-matching verification is supplied.
  2. [Numerical results] Numerical results and model sections: the reported 0.8 s orbital-period difference and sub-microsecond relativistic delay are presented as analytical outcomes, yet the manuscript provides neither the explicit orbit-equation derivation nor a comparison against the idealized circular-orbit baseline that would confirm these quantities are load-bearing for the BER claim.
  3. [Numerical results] Validation: the manuscript contains no Monte-Carlo simulation curves comparing the closed-form BER against the exact (non-Gamma) Shadowed-Rician model under LS estimation and residual Doppler; without such verification the accuracy of the final expression cannot be assessed.
minor comments (2)
  1. Define all acronyms at first use (SAGIN, BER, LS, etc.) and ensure consistent notation for the correlation coefficient throughout.
  2. [Abstract] The abstract states that the bivariate Gamma is obtained 'by exploiting this correlation coefficient'; the corresponding equation or paragraph in the main text should be explicitly cross-referenced.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the constructive feedback. We address each major comment below and will incorporate revisions to strengthen the manuscript.

read point-by-point responses

  1. Referee: [Abstract] Abstract and BER derivation section: the central closed-form BER expression rests on the bivariate Gamma mimic of the correlated Shadowed-Rician amplitudes after the correlation coefficient is obtained; no intermediate algebra is shown demonstrating that the Gamma parameters preserve the higher-order joint moments needed for exact integration over the 16-QAM symbol decision regions, and no error-bound or moment-matching verification is supplied.

    Authors: We agree that the derivation would benefit from additional transparency. The bivariate Gamma parameters are obtained by matching the first- and second-order moments of the joint amplitude distribution derived from the pilot-data correlation coefficient in the correlated Shadowed-Rician model; these moments are the ones required for the subsequent exact integration over the 16-QAM decision regions. In the revision we will insert the intermediate algebra for the moment-matching step and include a brief numerical check confirming that higher-order moments remain sufficiently close for the BER accuracy reported. revision: yes

  2. Referee: [Numerical results] Numerical results and model sections: the reported 0.8 s orbital-period difference and sub-microsecond relativistic delay are presented as analytical outcomes, yet the manuscript provides neither the explicit orbit-equation derivation nor a comparison against the idealized circular-orbit baseline that would confirm these quantities are load-bearing for the BER claim.

    Authors: The 0.8 s difference follows directly from integrating the elliptical Keplerian orbit equations (semi-major axis and eccentricity corresponding to 300 km perigee altitude) versus the circular case via Kepler’s third law; the resulting timing offset enters the residual Doppler term used in the BER expression. We will add the explicit orbit-equation steps and a short comparison table against the circular baseline in the model section to make this dependence explicit. revision: yes

  3. Referee: [Numerical results] Validation: the manuscript contains no Monte-Carlo simulation curves comparing the closed-form BER against the exact (non-Gamma) Shadowed-Rician model under LS estimation and residual Doppler; without such verification the accuracy of the final expression cannot be assessed.

    Authors: We accept that direct Monte-Carlo validation against the exact Shadowed-Rician distribution is necessary. In the revised numerical results we will include Monte-Carlo curves generated under the precise Shadowed-Rician fading, least-squares estimation, and residual Doppler conditions, plotted alongside the closed-form expression to quantify the approximation error. revision: yes

Circularity Check

0 steps flagged

Derivation chain is self-contained with explicit approximation and no circular reductions

full rationale

The paper first derives the pilot-data correlation coefficient directly from the correlated Shadowed-Rician model, then adopts a bivariate Gamma approximation whose parameters are set by that coefficient, and finally averages the 16-QAM decision regions under LS estimation to obtain the closed-form BER. This is a standard sequence of model-based derivation followed by moment-matched approximation; the resulting BER expression is not identical to the input statistics by construction, nor does any step rename a fitted quantity as a prediction or rely on self-citation for uniqueness. The approximation choice affects accuracy but does not create a tautological loop.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 0 invented entities

The model rests on established physical laws and standard estimation techniques; no new free parameters or invented entities are introduced in the abstract.

axioms (3)
  • domain assumption Snell's law governs path loss in the SAGIN model
    Invoked to construct the practical channel model encompassing atmospheric absorption and geometry
  • domain assumption Einstein's theory of relativity determines synchronization delay
    Used to compute relativistic delay over a full LEO pass
  • standard math Least-squares estimation and equalization are applied to the received symbols
    Standard technique stated for obtaining the BER expression

pith-pipeline@v0.9.0 · 5582 in / 1434 out tokens · 31387 ms · 2026-05-16T11:11:30.986705+00:00 · methodology

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