On the Sum of Independent Fox's $H$-Function Variates With Applications

Rahama, Yousuf Abo; Ismail, Mahmoud H.; Hassan, Mohamed S.

doi:10.1109/tvt.2018.2827180

Cited by 37 publications

(26 citation statements)

References 15 publications

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“…Asymptotic PNZ The asymptotic PNZ at high or lowγ B regime, is computed by evaluating the residues of analytical PNZ, given in (16). According to [34], Fox's H-function can be further simplified by choosing the dominate term of the Mellin-Barnes type integral. As such, we can evaluate the residue of the PNZ at lowγ B regime, at the point…”

Section: A Asymptotic Sopmentioning

confidence: 99%

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On Physical Layer Security Over Fox's $H$-Function Wiretap Fading Channels

Kong

Kaddoum

Chergui

2019

IEEE Trans. Veh. Technol.

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Most of the well-known fading distributions, if not all of them, could be encompassed by the Fox's H-function fading. Consequently, we investigate the physical layer security (PLS) over Fox's H-function fading wiretap channels, in the presence of non-colluding and colluding eavesdroppers. In particular, for the non-colluding scenario, closed-form expressions are derived for the secrecy outage probability (SOP), the probability of nonzero secrecy capacity (PNZ), and the average secrecy capacity (ASC). These expressions are given in terms of either univariate or bivariate Fox's H-function. In order to show the effectiveness of our derivations, three metrics are respectively listed over the following frequently used fading channels, including Rayleigh, Weibull, Nakagami-m, α − µ, Fisher-Snedecor (F-S) F , and extended generalized-K (EGK). Our tractable results are not only straightforward and general, but also feasible and applicable, especially the SOP, which is usually limited to the lower bound in the literature due to the difficulty of deriving closed-from analytical expressions. For the colluding scenario, a super eavesdropper equipped with maximal ratio combining (MRC) or selection-combining (SC) schemes is characterized. The lower bound of SOP and exact PNZ are thereafter derived with closed-form expressions in terms of the multivariate Fox's H-function. In order to validate the accuracy of our analytical results, Monte-Carlo simulations are subsequently conducted for the aforementioned fading channels. One can observe that for the former non-colluding scenario, we have perfect agreement between the exact analytical and simulation results, and highly accurate approximations between the exact and asymptotic analytical results. On the contrary, the SOP and PNZ of colluding eavesdropper is greatly degraded with the increase of the number of eavesdroppers. Also, the so-called super eavesdropper with MRC is much powerful to wiretap the main channel than the one with SC.Index Terms-Physical layer security, Fox's H-function wiretap fading channels, Mellin transform, secrecy outage probability, probability of non-zero secrecy capacity, average secrecy capacity.

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Section: A Asymptotic Sopmentioning

confidence: 99%

“…where step (a) is developed by interchanging the order of two integrals. The inner integral in (42) can be further expressed as Subsequently, substituting (34) and (41), yields the following result…”

Section: Appendix a Proof Of The Theoremmentioning

confidence: 99%

On Physical Layer Security Over Fox's $H$-Function Wiretap Fading Channels

Kong

Kaddoum

Chergui

2019

IEEE Trans. Veh. Technol.

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“…Moreover, the sum of -distributed random variates (RVs) has been investigated in [40], where a new -distributed RV is used to highly approximate the sum. Similarly, the sums of other general fading models, e.g., -, -, Fisher-Snedecor F , fluctuating two-ray (FTR), and Fox's -function distributed RVs are also characterized in [41]- [47], where their application to the maximal-ratio-combining (MRC) diversity receiver are correspondingly investigated with the assistance of the obtained statistical results [42]- [47].…”

Section: B Motivationsmentioning

confidence: 99%

“…Applying the result given in [59], each single Mellin-Barnes integral is then further simplified and expanded at the single residue of the corresponding integrands at the pole closest to the contour [47], namely, the minimum pole on the right of the contour for small Fox's arguments, and subsequently applying the residue theorem, we can arrive at (20).…”

Section: Appendix B Proof For Remarkmentioning

confidence: 99%

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Effective Rate Evaluation of RIS-Assisted Communications Using the Sums of Cascaded α-μ Random Variates

Kong

Chatzinotas

et al. 2021

IEEE Access

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This paper begins with the characterization of the sum of versions of cascaded -(equivalently, * ( -)) distributed random variates (RVs), and later on, explores the feasibility and applicability of the obtained results in the recent proposed reconfigurable intelligent surface (RIS)-assisted communications. More specifically, the sum of cascaded -distributed RVs is exactly characterized by the probability density function (PDF) expressed in terms of the programmable multivariate Fox'sfunction, and approximated with the mixture of Gaussian (MoG) distribution. The approximation accuracy is further validated by the Kolmogorov-Smirnov (KS) goodness of fit test. Based on the obtained statistical results, a promising application, i.e., RIS-assisted communication, is considered and evaluated from the effective rate perspective. The exact, asymptotic, and approximated effective rate are derived with closedform expressions. Finally, our analytical results are further validated using the Monte-Carlo simulation, their accuracy is confirmed with the analytical error indicator. One can conclude that (i) the exact effective rate expression is complex in the manner of mathematical representation; (ii) the MoG-based effective rate expression is simple and provides a highly tight approximation to the exact expression, besides it indeed acts as a highly tight upper bound for the exact effective rate expression; and (iii) the MoG-based solution outperforms the multivariate Fox's -function based approach with reducing complexity.

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