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arxiv: 1907.01818 · v1 · pith:URFULUPFnew · submitted 2019-07-03 · 💻 cs.IT · eess.SP· math.IT

A Simple Evaluation for the Secrecy Outage Probability Over Generalized-K Fading Channels

Hui Zhao , Yuanwei Liu , Ahemd Sultan-Salem , Mohamed-Slim Alouini This is my paper

Pith reviewed 2026-05-25 10:06 UTC · model grok-4.3

classification 💻 cs.IT eess.SPmath.IT
keywords secrecy outage probabilitygeneralized-K fadingasymptotic analysisdiversity orderwiretap channelapproximation
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The pith

A simple approximation for secrecy outage probability over generalized-K fading channels tightens as wiretap SNR decreases and reveals the secrecy diversity order.

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

This paper develops a simple approximation for the secrecy outage probability over generalized-K fading channels. The approximation simplifies the expression and becomes tighter when the average SNR of the wiretap channel is low. An asymptotic form is then derived for high SNR on the main channel, which directly exposes the secrecy diversity order in general cases. Numerical checks confirm the approximation stays accurate across the tested regimes. A reader would care because the result replaces a complex integral with an elementary expression while still capturing the key scaling behavior of secrecy performance.

Core claim

A simple approximation for the secrecy outage probability (SOP) over generalized-K fading channels is developed. This approximation becomes tighter as the average SNR of the wiretap channel decreases. Based on this simple expression, the asymptotic SOP in the high SNR region of the main channel is analyzed, revealing the secrecy diversity order in a general case.

What carries the argument

The asymptotic SOP approximation obtained by exploiting the series expansion properties of the generalized-K distribution under the independence assumption between main and wiretap channels.

If this is right

  • SOP evaluation reduces from a double integral to a single elementary expression.
  • The secrecy diversity order is obtained in closed form for arbitrary generalized-K parameters.
  • The approximation error vanishes in the regime of practical interest for physical-layer security.

Where Pith is reading between the lines

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

  • The same expansion technique may apply to other composite fading models that admit similar series representations.
  • System designers could use the diversity-order formula to set target SNR margins without Monte-Carlo simulation.

Load-bearing premise

The main and wiretap channels are statistically independent and each follows a generalized-K distribution whose parameters allow the series expansion to be truncated without losing the leading term.

What would settle it

Direct numerical comparison of the proposed closed-form SOP against the exact integral expression at successively lower wiretap SNR values, or measurement of the slope of log(SOP) versus log(SNR_main) to check the predicted diversity order.

Figures

Figures reproduced from arXiv: 1907.01818 by Ahemd Sultan-Salem, Hui Zhao, Mohamed-Slim Alouini, Yuanwei Liu.

Figure 3
Figure 3. Figure 3: Psop versus γd for me = ke = 2, γe = 5 dB, and Rs = 1. that the ASOP for kd = md is not a linear function with respect to γd in dB, despite the slowly changing slope in high SNRs. REFERENCES [1] P. S. Bithas, N. C. Sagias, P. T. Mathiopoulos, G. K. Karagiannidis, and A. A. Rontogiannis, “On the performance analysis of digital communications over generalized-K fading channels,” IEEE Commun. Lett., vol. 10, … view at source ↗
Figure 1
Figure 1. Figure 1: Psop versus γd for md = me = 2.5, kd = ke = 2, and Rs = 1. From [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Psop versus γd for kd = me = ke = 1.5, γe = 0 dB, and Rs = 1. VI. CONCLUSION In this letter, a simple SOP expression was derived with a high accuracy for a small average SNR of the wiretap channel. Although the matching becomes worse for a large average SNR of the wiretap channel, the approximate SOP converges to the exact SOP with increasing the average SNR of the main channel. To obtain the secrecy diver… view at source ↗
read the original abstract

A simple approximation for the secrecy outage probability (SOP) over generalized-K fading channels is developed. This approximation becomes tighter as the average signal-to-noise ratio (SNR) of the wiretap channel decreases. Based on this simple expression, we also analyze the asymptotic SOP in the high SNR region of the main channel. Besides simplifying the SOP expression significantly, this asymptotic SOP expression reveals the secrecy diversity order in a general case. Numerical results demonstrate the high accuracy of our proposed approximation results.

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

2 major / 3 minor

Summary. The manuscript develops a simple closed-form approximation to the secrecy outage probability (SOP) integral over generalized-K fading channels for the main and wiretap links. The approximation is shown to tighten as the average SNR of the wiretap channel decreases; an asymptotic high-SNR expansion of the main-channel link is then used to extract the secrecy diversity order in closed form as a function of the generalized-K parameters. Monte-Carlo simulations are provided to illustrate accuracy.

Significance. If the derivation and error bounds hold, the work supplies a practical simplification of the SOP expression together with an explicit secrecy-diversity-order formula that is parameter-free once the fading parameters are fixed. This is useful for physical-layer security analysis in composite fading environments and is strengthened by the explicit derivation steps and numerical validation supplied in the manuscript.

major comments (2)
  1. [§III, Eq. (14)] §III, Eq. (14): the series truncation used to obtain the simple SOP approximation is presented without an explicit remainder bound; because the claim that the approximation becomes arbitrarily tight for low wiretap SNR rests on this step, a uniform error estimate (or at least the truncation order) should be stated for the full range of generalized-K parameters (k,m).
  2. [§IV, Eq. (22)] §IV, Eq. (22): the asymptotic SOP expression that yields the secrecy diversity order assumes independence of the main and wiretap channels; the manuscript should confirm that the diversity-order result continues to hold (or state the modification) when the two links experience correlated shadowing, as this is a common practical case not covered by the independence assumption.
minor comments (3)
  1. [Notation] The notation for the generalized-K parameters (shape and scale) should include a brief reference to the standard definition in the literature (e.g., Shankar 2004) to avoid ambiguity.
  2. [Figure 2] Figure 2 caption: the curves labeled “approx.” and “asymp.” should explicitly state the SNR regime (wiretap vs. main) to which each corresponds.
  3. [§IV] A short table summarizing the diversity-order expressions for the special cases (K, Nakagami-m, Rayleigh) would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments and positive evaluation. We address each major comment below and indicate the revisions we will make.

read point-by-point responses

  1. Referee: [§III, Eq. (14)] §III, Eq. (14): the series truncation used to obtain the simple SOP approximation is presented without an explicit remainder bound; because the claim that the approximation becomes arbitrarily tight for low wiretap SNR rests on this step, a uniform error estimate (or at least the truncation order) should be stated for the full range of generalized-K parameters (k,m).

    Authors: We agree that an explicit remainder bound strengthens the presentation. In the revised manuscript we will add a short derivation of the truncation error for the series in Eq. (14), showing that the remainder is bounded uniformly for all k,m>0 and vanishes as the wiretap SNR tends to zero. This bound follows directly from the alternating-series test applied to the generalized-K PDF expansion and confirms the claimed tightness without altering the main result. revision: yes

  2. Referee: [§IV, Eq. (22)] §IV, Eq. (22): the asymptotic SOP expression that yields the secrecy diversity order assumes independence of the main and wiretap channels; the manuscript should confirm that the diversity-order result continues to hold (or state the modification) when the two links experience correlated shadowing, as this is a common practical case not covered by the independence assumption.

    Authors: The secrecy-diversity-order derivation in §IV explicitly uses the independence assumption stated in the system model. Under correlated shadowing the joint distribution changes and the diversity order would in general be modified. Because the manuscript does not claim the result for the correlated case, we will insert a brief remark noting this scope limitation while leaving the independent-channel analysis unchanged. revision: partial

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained from channel statistics

full rationale

The paper derives the SOP approximation and asymptotic diversity order directly from the generalized-K PDF (Meijer-G form), joint statistics under channel independence, and standard high-SNR series expansion. No equation reduces by construction to a fitted parameter or self-citation; Monte-Carlo curves serve as external validation. The result is therefore independent of its own outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract supplies no explicit free parameters, axioms, or invented entities; all modeling assumptions remain implicit in the generalized-K channel description.

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