I Introduction
The physical layer security of the classic Wyner’s wiretap model has been widely analysed over multipath fading channels in the recent works [1]. For example, in [2] and references therein, both the main and the wiretap channels which are the Alice/Bob and Alice/Eve channels are represented by using various models of fading scenarios such as Rayleigh, Nakagami-, and Rician.
In a wireless communication, in addition to multipath fading, the channels may subject to the shadowing effect. Therefore, several efforts have been dedicated to study the physical layer security under composite multipath/shadowing fading scenario [2]. For instance, in [3], the average security capacity (ASC), the secure outage probability (SOP), and the strictly positive secure capacity (SPSC) over generalised- (
) fading model which is composite of Nakagami-m/Gamma distributions are derived in terms of the extended generalized bivariate Meijer G-function (EGBMGF). This is because the statistical properties, namely, the probability density function (PDF), cumulative distribution function (CDF), and the moment generating function (MGF), are derived in terms of the modified Bessel functions. Therefore, to obtain simple mathematical expressions of the performance metrics over generalised-
fading channel, a mixture gamma distribution is used as an approximate framework in [4]. However, the fading parameters are assumed to be integer values.More recent, the Fisher-Snedecor fading channel has been proposed as a composite of Nakagami-/Nakagami- [5]. In contrast to the generalised- fading channel, the statistics of the Fisher-Snedecor fading channel are derived in simple closed-form expressions. Furthermore, the Fisher-Snedecor fading channel includes Nakagami-, Rayleigh, and one-sided Gaussian as special cases. Therefore, it can be employed for both line-of-sight (LoS) and non-LoS (NLoS) communications scenarios with better fitting to the empirical measurements than the generalised- () fading model. However, it has been utilised by one work in the open technical literature [6].
Motivated by there is no work has been devoted to analyse the physical layer security over Fisher-Snedecor fading channel, this paper investigates the aforementioned analysis. In particular, the ASC, the SOP, the lower bound of SOP (SOP), and the SPSC are derived when both the main and the wiretap channels subject to the Fisher-Snedecor fading channel. To this effect and the best of the authors’ knowledge, novel analytic results of the performance metrics are obtained in exact closed-form mathematically tractable expressions.
Ii Fisher-Snedecor Fading Model
The PDF of the received instantaneous SNR, , using Fisher-Snedecor distribution is expressed as [5, (5)]
(1) |
where , , , , and are the multipath index, the shape parameter, the average SNR and the beta function defined in [7, (8.380.1)], respectively.
The CDF of using Fisher-Snedecor distribution is given as [5, (4)]
(2) |
where is the hypergeometric function defined in [7, (9.14.1)].
Iii Average Secrecy Capacity
The ASC can be calculated by [4, (6)] where , , and are given as
(3) |
(4) |
(6) |
(7) |
(13) |
(5) |
Accordingly, and over Fisher-Snedecor fading scenarios are given in (6) and (7) at the top of this page. In addition, is expressed as
(8) |
where and are Meijer G-function and EGBMGF, respectively.
Proof:
Substituting (1) and (2) in (3), we have
(9) |
Invoking the identities [8, (11)], [8, (10)], and [8, (17)] with some mathematical manipulations, (9) can be rewritten as
(10) |
Using [9, (9)] to compute the integral in (10) and doing some mathematical simplifications, (6) is yielded which completes the proof of .
Following the same steps that are employed to derive , can be deduced in closed-from expression as given in (7).
To obtain , we substitute (1) in (5) and recall the identity [8, (11)]. Thus, this yields
(11) |
Employing [10, (2.24.2.4)], (8) is yielded which completes the proof of .
(18) |
Iv Secure Outage Probability
The SOP can be evaluated by [2, (14)]
(12) |
where with is the target secrecy threshold.
The SOP can be expressed in exact closed-form as given in (13) at the top of the this page.
Proof:
Inserting (1) and (2) in (12), the result is
SOP | ||||
(14) |
Assuming and and performing some mathematical simplifications, (14) becomes as follows
SOP | ||||
(15) |
Utilising the identities [8, (10)] and [8, (17)], (15) is expressed as
SOP | ||||
(16) |
Making use of [9, (9)], the derived result in (13) is yielded
V Lower Bound of the Secure Outage Probability
The SOP can be computed by [2, (17)]
(17) |
The SOP over Fisher-Snedecor fading scenarios can be derived as given in (18) at the top of this page.
Proof:
Plugging (1) and (2) in (17) and doing some mathematical manipulations, we have
(19) |
With the help of [8, (17)], (19) can be rewritten as
(20) |
Utilising [10, (2.23.2.4)] to compute the integral in (20), the result in (18) is deduced and this completes the proof.
Vi Strictly Positive Secure Capacity
The SPSC is expressed as [2, (20)]
(21) |
Consequently, the SPSC over Fisher-Snedecor fading channels can be obtained by using (13) and and inserting the result in (21).
Vii Analytical and Simulation Results
In this section, to validate our derived expressions of the physical layer security over Fisher-Snedecor fading channels, the Monte Carlo simulations that are obtained via generating realizations are compared with the analytical results. In all figures, the simulations and the numerical results of the performance metrics that are plotted versus for and (moderate shadowing) are represented by the solid lines and the stars, respectively. Moreover, two different scenarios of the shadowing impact at the eavesdropper which are light and heavy shadowing are studied by using and , respectively. In all results, a MATHEMATICA code that is provided in [9] has been used to calculate the EGBMGF. This is because it is not available as a built in function in MATLAB and MATHEMATICA software packages.
Figs. 1-5 show the ASC, the SOP, the SOP, and the SPSC over Fisher-Snedecor fading channels for dB and different values of the fading parameters and . In these figures, it can be observed that the performance becomes better, when increases. This is because small and large values of correspond to light and heavy shadowing, respectively. For instance, in Fig. 1, when and (fixed), the ASC for is approximately higher than . In the same context, when increases, the ASC decreases. This refers to less impact of the multipath on the Eve which would lead to reduce the total ASC.
In Figs. 2 and 4 that are plotted for bit/s/Hz, one can see that the values of SOP are greater than or equal to the SOP which confirms our derived expressions. Furthermore, another confirmation that proves the validation of our analysis is the perfect matching between the numerical results and their Monte Carlo simulation counterparts in all provided figures.
Viii Conclusions
In this letter, the secrecy performance of physical layer over Fisher-Snedecor fading channels is analysed. Specifically, the ASC, the SOP, the SOP, and the SPSC are derived in exact mathematically tractable closed-form expressions. The results of this work provide a good insight about the security of the physical layer over composite multipath/shadowing fading channels when the wireless channels subject to heavy, moderate, or light shadowing. Moreover, the analysis of the physical layer security over different scenarios can be deduced from the derived expressions by setting and for specific values such as the Nakagami- fading condition is obtained by inserting and where m is the Nakagami- multipath index.
References
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