On the Distribution Function of the Generalized Beckmann Random Variable and Its Applications in Communications
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Bibliographic record
Abstract
The Beckmann distribution has a wide range of applications in radio-frequency communications, free-space optical (FSO) communications, and underwater wireless optical communications (UWOC). However, the cumulative distribution function (cdf) of the Beckmann random variable (RV) does not have a closed-form expression, which makes it challenging to derive analytical solutions for the outage probability of systems involving Beckmann RVs. In this paper, we study the generalized Beckmann distribution, which includes the Beckmann, Rayleigh, Rician, Nakagami-m, Hoyt, κ-μ, η-μ, single-sided Gaussian, and the Beaulieu-Xie distributions as special cases. Three approaches are proposed to estimate the cdf of the generalized Beckmann distribution, including closed-form upper and lower cdf bounds, single-fold integration based on the closed-form characteristic function, and a left-tail cdf approximation. We compare the three approaches in terms of the ranges of applications and the computation time complexity. Based on the new cdf estimation techniques, one can efficiently evaluate the outage probabilities of pointing-error-limited FSO systems, UWOC systems, and maximum-ratio combining over arbitrarily correlated generalized Beckmann channels.
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Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.003 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.005 | 0.000 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
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Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from it