I Introduction
Shannon’s informationtheoretic notion of perfect secrecy has been developed by Wyner via suggesting the wiretap channel. In the notion of this channel, an eavesdropper is presented when a legitimate user, namely, Alice, communicates with the intended receiver which is called Bob [1]. Consequently, the performance of the physical layer security over different fading channel models has been widely analysed in the open literature. For instance, the probability of strictly positive secrecy capacity (SPSC), the secure outage probability (SOP), and the average secrecy capacity (ASC) when the wireless channels subject to the additive white Gaussian noise (AWGN) and Rayleigh fading channel are given in [2] and [3], respectively. In [4], the SPSC when both the main and eavesdropper channels undergo Rician fading channel is derived. The SOP and the SPSC of the physical layer using Rician and Nakagami fading conditions for the Bob and the eavesdropper wireless channels are given in [5]. The Weibull fading channel model is used in [6] and [7] to study the SPSC and ASC, respectively.
Recently, many works have been implemented using various generalized fading distributions that unify most of the wellknown channel models. In addition, they provide results closer to the practical data than the conventional distributions, namely, Rayleigh, Nakagami, and Nakagami. In [8], the ASC over fading channel that is used to model the lineofsight (LoS) communication environment is derived. The performance of the physical layer security in nonlinear communication scenario is analysed in [9] and [10] via utilising the fading condition. Moreover, the ASC, the SOP, the SOP, and the SPSC of the physical layer over fading using the Fox’s function channel model which is a unified framework for a variety of distributions are presented in [11]. The ASC using the / and / fading scenarios for the main/eavesdropper channels is given in [12]. The more generalised fading channels and are used in [13] to derive the SOP and its asymptotic value. These fading distributions are provided in a single model which is that is also used to represent both the main channel and the eavesdropper’s channel of the classic Wyner’s wiretap model in [14].
The wireless may also affected by the multipath and shadowing simultaneously. Accordingly, the performance metrics of the physical layer security over composite fading channels have been also derived by several efforts in the open literature. For example, the analysis in [15][17] are investigated over generalised () fading channel which is a composite of Nakagami
/gamma distributions using different methods. In [11] and [18], the FisherSnedecor
distribution that is proposed as an alternative approach for the fading condition via employing the inverse Nakagami distribution instead of gamma model is used to derive the expression of the ASC, the SOP, the SOP, and the SPSC of the physical layer. The ASC and the SOP overshadowed fading are given in [19] for integer fading parameters as well as in terms of the derivative of the incomplete moment generating function (IMGF) framework that is included a bivariate confluent hypergeometric function
. This channel model is also utilised in [20] to analyse the SOPand SPSC using the exact probability density function (PDF) and the Gamma distribution as an approximate approach. However, the results in both efforts are either included double infinite series or approximated. Therefore, the authors in [21] have extensively analysed the performance of the physical layer security over
shadowed fading channel using exact closedform analytic expressions for both scenarios of the values of the fading parameters.More recent, the socalled Fluctuating Beckmann (FB) fading channel has been proposed as an extended model of the shadowed and the classical Beckmann distributions [22]. Thus, it includes the onesided Gaussian, Rayleigh, Nakagami, Rician, , , , Beckmann, Rician shadowed and the shadowed distributions as special cases. Hence, the FB fading channel is more generalised than the shadowed fading. In addition to [22], the aforementioned fading channel model has been utilised by only one previous work via studying the effective rate of communication system [23].
Motivated by there is no work has been achieved to analyse the secrecy performance of the physical layer over FB fading channels, this paper is dedicated to investigate this analysis. Our main contributions are summarised as follows:

Analysing the performance of the physical layer security when both the main and wiretap channels are subjected to FB fading channel models. In particular, novel exact closedform mathematically tractable expressions of the ASC, SOP, SOP, and SPSC are derived.

When the fading parameters, namely, and which represent the real extension of multipath clusters and shadowing severity index, respectively are arbitrary numbers, the secrecy performance metrics are obtained in terms of the extended generalised bivariate Fox’s function (EGBFHF). Although, this function is not available in the popular mathematical software packages such as MATLAB and MATHEMATICA, it has been implemented by several works such as [10] and [24].

To earn more insights into the behaviour of the physical layer security as well as the impact of the parameters of the FB fading model via using simple exact closedfrom analytic expressions of the aforementioned performance metrics, and are assumed to even and integer values. Consequently, the derived results are obtained in simple mathematical functions that are presented in all software packages.

From the provided literature in this work, the ASC, the SOP, the SOP
, and the SPSC for some special cases of the FB fading model such as Beckmann have not been yet introduced due to the complexity of their PDF and cumulative distribution function (CDF). However, these expressions can be deduced from our derived expressions because the FB fading model is a versatile representation of many distributions such as
shadowed and Beckmann.
Organization: Section II is divided into two subsections. In the first subsection, the system model that is used in this work is described whereas the general and limited formats of the probability density function (PDF) and the cumulative distribution function (CDF) of FB fading channel are given in the second subsection. The ASC, the SOP, the SOP, and the SPSC for two cases of the values of and shadowing parameters are derived in Sections III, IV, V, and VI, respectively. Section VII explains the performance of the physical layer security over some special cases of FB fading channel. In Section VIII, the Monte Carlo simulations and numerical results are presented. Finally, some conclusions are highlighted in Section IX.
Ii System and Channel Models
Iia System Model
Wyner’s wiretap channel model has been proposed three different nodes with two wireless communication links [1]. The first link is between the transmitter and the legitimate receiver which are called Alice and Bob, respectively via main channel. Thus, Bob’s channel state information (CSI) can be known by Alice. On the other side, the second wireless communication link describes the wiretap channel between the Alice and an external receiver which is named the eavesdropper (Eve). Accordingly, perfect knowledge of Eve’s channel CSI can not be assumed and hence informationtheoretic security can not be introduced. This case is happened when Eve is unknown eavesdropper and that makes Alice unable to access it’s CSI.
In this paper, the main and wiretap channels are supposed to be independent and they subject to quasistatic FB fading channels. In addition, Alice, Bob, and Alice are equipped with a single antenna and perfect knowledge of the CSI of both Bob and Eve are assumed. When Alice transmits the signal , the received signals at both Bob and Eve is given as [20]
(1) 
where , , and stand for Bob, and eavesdropper, respectively. Moreover, and
are the FB fading channel and the additive white Gaussian noise that has zero mean and fixed variance, respectively.
IiB The PDF and CDF of Fluctuating Beckmann Fading Channel Model
Case_1: The PDF of the instantaneous SNR , , for the destination (Bob), , and the eavesdropper, , channels using FB fading channel model is given by [22, eq. (5)]
(2) 
where , , , is the average SNR, is the shadowing severity parameter, is the Gamma function and is the multivariate confluent hypergeometric function defined in [25, eq. (1.7.10)]. Furthermore, are the roots of with
(3) 
where , , , , , and are real numbers for th cluster and and mutually independent Gaussian random processes.
The CDF of the FB fading channel condition is expressed as [22, eq. (6)]
(4) 
Case_2: When is integer number and is even number, the PDF and the CDF are, respectively, given by [22, eqs. (10) and (14)]
(5) 
and
(6) 
where , , , u(.) is the unit step function, and and are calculated by [22, eq.(51)] and [22, eq. (52)], respectively.
Iii Average Secrecy Capacity
The normalised ASC that is defined as the difference between the capacity of the main and wiretap channels over instantaneous SNR, , can be calculated by [15, eq. (6)] where , , and are respectively expressed as
(7) 
(8) 
(9) 
Theorem 1
The exact closedform expressions for , , and using the PDF and the CDF of Case_1 are given in (10), (11), and (12), respectively, at the bottom of this page where denotes the EGBFHF that is defined in [26, A.1]. To the best of the authors knowledge, (10), (11), and (12) are novel.
Proof:
See Appendix A.
(10) 
(11) 
(12) 
Corollary 1
For Case 2, , , and are respectively derived in simple analytic exact closedform expressions as shown in (13), (14), and (15), at the top this page. Substituting these expressions in and performing some straightforward manipulations, novel result is obtained as given in (16) at the top of this page.
Proof:
See Appendix B.
(13) 
(14) 
(15) 
(16) 
Iv Secure Outage Probability
The SOP is defined as the probability of falling the instantaneous secrecy capacity, , of the system below the target secrecy threshold, , i.e., where stands for the probability symbol. Mathematically, the SOP can be evaluated by [20, eq. (20)]
(17) 
where with denotes the target secrecy threshold.
Theorem 2
The SOP for Case_1 and Case_2 of the PDF and the CDF are expressed in exact closedform as shown in (18) and (19), respectively. In (19), stands for the binomial coefficient [25, eq. (1.1.16)] and () is the Pochhammer symbol [25, eq. (1.1.15)]. Additionally, (19) is simpler than (18) and hence better insights can be obtained for the SOP. To the best authors’ knowledge, (18) and (19) are novel and mathematically tractable.
Proof:
See Appendix C.
SOP  
(18) 
(19) 
V Lower Bound of SoP
According to [6], the SOP can be obtained from (17) when tends to . Consequently, the SOP can be computed by
(20) 
Theorem 3
The when and are arbitrary numbers, namely, Case_1, is presented in (21) whereas for Case_2, the is given in (22) where . One can see that (21) and (22) are obtained in simple exact closedform expressions. To the best of our knowledge, (21) and (22) are also new.
Proof:
See Appendix D.
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