On the Physical Layer Security Peculiarities of Wireless Communications in the Presence of the Beaulieu-Xie Shadowed Fading

Abstract

The article presents an analysis of the physical layer security of a wireless communication system functioning in the presence of multipath fading and a wiretap. Under the assumption of the equal propagation conditions (both for the legitimate receiver and the eavesdropper) described by the shadowed Beaulieu–Xie model, a closed-form expression for the secrecy outage probability was derived. The correctness of the obtained expression was numerically verified via comparison with the direct numerical integration. The truncated version of the obtained expression was analyzed for various channel parameters to establish the requirements for numerically efficient implementation (in terms of the number of summands delivering the desired precision). An in-depth study of the secrecy outage probability dependence from all the possible channel parameters for different fading scenarios was performed, including heavy fading and light fading, with and without strong dominant and multipath components. The performed research demonstrated the existence of the secrecy outage probability non-uniqueness with the respect to the average signal-to-noise ratio in the main channel and the relative distance between the legitimate and wiretap receivers.

1. Introduction

With the ever-increasing number of wireless communication technologies, the problem of personal data security is of great importance. Classically, this has been solved via coding and ciphering usually employed at the upper layer of the protocol stack [1], but starting from the mid-2000s, physical layer security (PLS) algorithms [2,3] have gained specific ground. Not opposing the classical approach, but rather assisting it, they help to exploit the specific traits of the physical wireless propagation phenomena to increase the security of the communication link.

Dating back to the pioneering work [4], the communication link between the transmitter and the legitimate receiver is assumed to be accompanied by the passive wiretap link. From a practical perspective, in the case of wireless links, one of the most critical situations occurs when the wiretap device is subtly present nearby the receiver. In this case, the propagation conditions can be assumed equal for both of the devices and the channels exhibit physical and statistical symmetry. Later, we will resort to this case as “the symmetric communication system with a wiretap”. Since we are mainly after legitimate link security maximization, the general goal of PLS algorithms can be regarded as breaking this symmetry in favor of the main link (leading to asymmetry in the results). At this state, it is evident that the PLS heavily relies upon the statistical description of the channel used. Thus, the more profound the model, the better its predictive capabilities, and the higher the adequacy of the expected security.

Nowadays, alongside the classical channel models, such as Rayleigh [5], Rice [6], Nakagami-m [6], log-normal [7], Hoyt [8], Weibull [9], which are commonly limited in applicability, the so-called generalized models have gained wide acceptance: generalized gamma [10], generalized K [11], $\alpha -\mu$ [12], $\kappa -\mu$ [13], $\kappa -\mu$ shadowed [14], fluctuating Beckmann [15], Beaulieu–Xie [16], and the combination of those channels [17]. At the same time, the increasing complexities of the novel models have led to the loss of simplicity in mathematical derivations. Recently, a new model (shadowed Beaulieu–Xie [18]) was proposed; it generalizes a wide range of simplified ones (allowing to derive closed-form representations).

The shadowed Beaulieu–Xie fading model was initially proposed in [18], where the statistical and physical models were presented, and several statistical functions were derived: an envelope probability density and cumulative distribution function, characteristic and moment-generating functions, and arbitrary-order moments. By relating the envelope probability functions with that of the instantaneous power, the expressions for the outage probability and the bit error probability for coherent/non-coherent binary frequency shift keying modulations were derived. Later on, in [16], the capacity analysis was carried out and the error rate analysis for various modulations was performed in [19]. In [20], an alternative expression for the moment-generating function of the instantaneous signal-to-noise ratio was proposed and applied to the derived average probability of the energy-based detection, receiver operating characteristic, and area under its curve. To date, the physical layer security analysis is absent. Although such an analysis was performed for the classical Beaulieu–Xie channel model (see [21]), the way the shadowing was introduced in [18] prevents a direct derivation of secrecy metrics (for instance, the secrecy outage probability (SOP)) for a shadowed channel from the results obtained in [21]. A simple attempt was performed by the authors in [22], where a numerical analysis was executed for only a single fading scenario corresponding to the experimental results given in [18], which demonstrate some existing peculiarities of the shadowed Beaulieu–Xie model SOP, but no profound analysis is present.

Thus, motivated by the problem stated above, the present research performs the closed-form analysis of the secrecy outage probability in the presence of multipath fading represented by the shadowed Beaulieu–Xie model. The major contributions of this work can be summarized as follows:

• A closed-form expression was derived for the secrecy outage probability of a symmetric wireless wiretap system functioning in the presence of multipath fading subjected to the shadowed Beaulieu–Xie model.
• A numerical simulation of the obtained expression was performed: (a) to demonstrate the correctness of the analytical work and (b) to establish the specifications required for numerical implementation (i.e., the number of summands in the truncated version of the derived solution) for various channel parameters.
• A thorough analysis of the secrecy outage probability dependence from all possible channel parameters for different fading scenarios: heavy fading and light fading, with and without strong dominant and multipath components.
• A discovered non-uniqueness of the secrecy outage probability with respect to the average signal-to-noise ratio and relative distance between the legitimate and wiretap receivers from the transmitter.

The remainder of the paper is organized as follows: Section 2 provides some preliminary results of the assumed: (a) symmetric wireless wiretap system and its description in terms of the average and instantaneous signal-to-noise ratios (SNR) for the main and the wiretap links, (b) secrecy outage probability as the metric used to quantify the security of the communication process, and (c) statistical channel models for both links; Section 3 derives the closed-form expression of the SOP and studies the aspects of its numerical implementations; Section 4 presents a thorough numerical analysis of the derived expression depending on various channel parameter values; the conclusions are drawn in Section 5.

2. The Assumed Model Description

2.1. System Model

In accordance with the classical Wyner model [4], the assumed SISO (single input single output) communication system is comprised of the main (further denoted by the index “M”) wireless transmission channel between the transmitter (Alice) and the legitimate recipient (Bob), and the wiretap channel (further denoted by the index “W”), formed between the transmitter and eavesdropper (Eve). The elements of the message $x\left(j\right)$, $j=1,\dots ,n$, (n—the length of the code word) are transmitted through a multipath fading channel and sensed simultaneously by Bob ($y_M\left(j\right)$) and Eve ($y_W\left(j\right)$). In this case, the mathematical model of the system for the $jth$ symbol being transmitted is described by the expressions:

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}\hfill & {\displaystyle y_M\left(j\right)=h_M\left(j\right)x\left(j\right)+n_M\left(j\right),}\hfill \\ \hfill & {\displaystyle y_W\left(j\right)=h_W\left(j\right)x\left(j\right)+n_W\left(j\right).}\hfill \end{array}\right.\end{array}$$

(1)

Here, it is assumed that both legal and wiretap receptions are conducted by the presence of the zero-mean complex Gaussian noise ($n_M\left(j\right)$ and $n_W\left(j\right)$, respectively), and the transmission coefficients ($h_M\left(j\right)$ and $h_W\left(j\right)$) are stochastic with arbitrary distributions in a general case, and will be defined specifically for the problem under analysis in Section 2.3.

With the fixed transmission/reception methods and signal processing algorithms, the values of the transmission coefficients $h_M\left(j\right)$ and $h_W\left(j\right)$ are determined by the total combination of physical propagation phenomena: the transmitter’s angle spectrum, attenuation, and diffraction on various obstacles along various paths, various re-reflections from the underlying surface and nearby objects, and energy losses during propagation. Since most of the factors are random, the expected communication link secrecy will fluctuate.

In the current study, we will adopt the following assumptions about the channels’ properties:

• The main and the wiretap channels are considered statistically independent.
• Both channels are described by the same generalized model with the same sets of model parameters.
• The security analysis of the considered communication systems is performed for the critically important case when the legitimate and wiretap receivers are located close to each other. This corresponds to the similarity of the signal propagation conditions in both channels and, consequently, the proximity of the values of the model parameters.
• The transmission coefficients are considered constant over the duration interval of the message, i.e., $h_M\left(j\right)\approx h_M$ and $h_W\left(j\right)\approx h_W$ [5], which corresponds to the typical case of slow fading or, equivalently, channels that are quasi-static over the time interval under consideration.

The signal receptions in the presence of noise in the main channel and the wiretap channel are characterized by instantaneous signal-to-noise ratios:

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}\hfill & {\displaystyle {\gamma }_M\left(j\right)=\frac{P{\left|h_M\left(j\right)\right|}^2}{{\sigma }_M^2}=\frac{P{\left|h_M\right|}^2}{{\sigma }_M^2}={\gamma }_M,}\hfill \\ \hfill & {\displaystyle {\gamma }_W\left(j\right)=\frac{P{\left|h_W\left(j\right)\right|}^2}{{\sigma }_W^2}=\frac{P{\left|h_W\right|}^2}{{\sigma }_W^2}={\gamma }_W,}\hfill \end{array}\right.\end{array}$$

(2)

where ${\sigma }_M^2 and {\sigma }_W^2$ represent the noise power in each channel, P is the average power of the transmitter, and the second equality takes into account the quasi-static nature of the channels in accordance with the fourth of the above assumptions.

Due to the stochastic nature of (2), the instantaneous signal-to-noise ratios (2) are mainly described in terms of their average values (for the main channel and the wiretap channel), and for further analysis, will act as independent parameters of the system model:

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}\hfill & {\displaystyle {\overline{\gamma }}_M\left(j\right)=\frac{P\mathbb{E}\left\{{\left|h_M\left(j\right)\right|}^2\right\}}{{\sigma }_M^2}=\frac{P\mathbb{E}\left\{{\left|h_M\right|}^2\right\}}{{\sigma }_M^2}={\overline{\gamma }}_M,}\hfill \\ \hfill & {\displaystyle {\overline{\gamma }}_W\left(j\right)=\frac{P\mathbb{E}\left\{{\left|h_W\left(j\right)\right|}^2\right\}}{{\sigma }_W^2}=\frac{P\mathbb{E}\left\{{\left|h_W\right|}^2\right\}}{{\sigma }_W^2}={\overline{\gamma }}_W,}\hfill \end{array}\right.\end{array}$$

(3)

here $\mathbb{E}\left\{·\right\}$ is the expectation operator.

Due to the propagation attenuation of the signal (as a result of energy dissipation during the expansion of the wavefront with the distance from the source, as well as at scattering or absorption inside obstacles) the communication system must also be characterized by the distances from the transmitter to the legitimate receiver ($d_M$) and the wiretap receiver ($d_W$).

Further on, the propagation medium is characterized with the help of the so-called path loss exponent [23]. Since it is a random variable, its effective value $\alpha$ is used (for example, $\alpha$ = 2 for propagation in free space without fading).

Assuming that the propagation conditions in both channels are similar, the average values of the SNR can be considered inversely proportional to the distances from the transmitter to the receiver with the same power $\alpha$: ${\overline{\gamma }}_M\sim 1/\phantom{1d_M^{\alpha }}{d}_M^{\alpha }$, ${\overline{\gamma }}_W\sim 1/\phantom{1d_W^{\alpha }}{d}_W^{\alpha }$ [5]:

$$\frac{{\overline{\gamma }}_M}{{\overline{\gamma }}_W}={\left(\frac{d_W}{d_M}\right)}^{\alpha }.$$

(4)

Therefore, under the assumed conditions, we will consider four parameters of the communication system model: the SNR in the main channel ${\overline{\gamma }}_M$ (which can be controlled, for example, by changing the processing algorithm in a legitimate receiver), the SNR in the wiretap channel ${\overline{\gamma }}_W$ (which participants in a legitimate link session cannot control), the average value of the path loss exponent $\alpha$ and the ratio of the distances from the transmitter to the wiretap and legitimate receivers $d_W/\phantom{d_W{d}_M}{d}_M$.

2.2. Secrecy Metric

For further analysis, it is important to define the metric that quantifies the level of the communication link secrecy. Among the plethora of such metrics (see, for instance, [24,25,26,27,28]) one of the most prominent roles (in various applications) is given to the secrecy outage probability [29,30,31,32,33]. Classically it is defined as the probability that the instantaneous capacity C falls below some pre-specified threshold value $C_{th}$:

$$\begin{array}{ccc}\hfill P_{out}\left(C_{th}\right)& =& \mathbb{P}\left(C<C_{th}\right)=\mathbb{P}\left({\gamma }_M<\left(1+{\gamma }_W\right)2^{C_{th}}-1\right)=\hfill \\ \hfill & =& {\int }_0^{\infty }{\int }_0^{\left(1+z_W\right)2^{C_{th}}-1}{f}_{{\gamma }_M,{\gamma }_W}\left(z_M,z_W\right)\mathrm{d}{z}_M\mathrm{d}{z}_W,\hfill \end{array}$$

(5)

where the second equality follows from the definition of the secrecy capacity [5].

2.3. Fading Channel Model

In this state, it is evident that for a fading channel, SOP heavily relies on the channel model. Thus, as mentioned earlier, it is important that such a model be complex enough to incorporate most of the physically meaningful effects, and at the same time can yield closed-form results. Within the research, we will adopt the shadowed Beaulieu–Xie channel model [18,34], which defines the probability density function of the instantaneous signal-to-noise ratio in the following form:

$$\begin{array}{c}\hfill f_{{\gamma }_i}\left(z_i\right)=\frac{e^{-\frac{m_X\left({\mathsf{\Omega }}_X+{\mathsf{\Omega }}_Y\right)}{{\overline{\gamma }}_i{\mathsf{\Omega }}_X}{z}_i}}{\Gamma \left(m_X\right)\sqrt{\frac{z_i{\overline{\gamma }}_i}{{\mathsf{\Omega }}_X+{\mathsf{\Omega }}_Y}}}{\left(\frac{m_X}{{\mathsf{\Omega }}_X}\right)}^{m_X}{\left(\frac{z_i\left({\mathsf{\Omega }}_X+{\mathsf{\Omega }}_Y\right)}{{\overline{\gamma }}_i}\right)}^{m_X-\frac{1}{2}}{\left(\frac{m_Y{\mathsf{\Omega }}_X}{m_Y{\mathsf{\Omega }}_X+m_X{\mathsf{\Omega }}_Y}\right)}^{m_Y}\times \\ \hfill \times {}_1{F}_1\left(m_Y;m_X;\frac{m_X^2\left({\mathsf{\Omega }}_X+{\mathsf{\Omega }}_Y\right){\mathsf{\Omega }}_Y}{{\overline{\gamma }}_i{\mathsf{\Omega }}_X\left(m_Y{\mathsf{\Omega }}_X+m_X{\mathsf{\Omega }}_Y\right)}{z}_i\right),\end{array}$$

(6)

here, the index $i=\{M;W\}$ enumerates the channel, with index M denoting the main and W the wiretap channels, respectively; ${\overline{\gamma }}_i$ stands for the average signal-to-noise ratio in each channel, ${}_1{F}_1\left(·\right)$ is the confluent hypergeometric function [35], and $\Gamma \left(·\right)$ is the Euler gamma function [35]. The model is parameterized with four characteristics: $m_X$ and $m_Y$, which define the shadowing coefficients of the overall and dominant components; ${\mathsf{\Omega }}_X$ and ${\mathsf{\Omega }}_Y$, which define the energy of the non-line-of-sight (NLoS) and line-of-sight (LoS) components respectively.

Further on, we will assume only the case of the symmetric wiretap, when the main and the wiretap channels have the same parameters, thus $m_X$, $m_Y$, ${\mathsf{\Omega }}_X$, and ${\mathsf{\Omega }}_Y$ will not be indexed with $i=\{M;W\}$, contrary to the signal-to-noise ratios (instantaneous and average). This choice corresponds to the practical situation when the eavesdropper is located nearby the legitimate receiver, which is by far one of the most important for indoor communications with a crowded environment.

Thus, it is required to derive the closed-form representation of the assumed metric (i.e., SOP) (5) within the framework of the adopted symmetric communication system (1) with a wiretap for the case of shadowed Beaulieu–Xie fading channel models (6).

3. Derived Closed-Form Results

3.1. Secrecy Outage Probability Derivation

To evaluate the closed-form expression for the SOP, it can be noticed that due to the independence of the main and the wiretap channels, their joint probability density function can be factorized, i.e., $f_{{\gamma }_M,{\gamma }_W}\left(z_1,z_2\right)=f_{{\gamma }_M}\left(z_1\right)f_{{\gamma }_W}\left(z_2\right)$.

Thus, combining SOP (5) with the assumed channel models (6) yields:

$$\begin{array}{c}\hfill P_{out}\left(C_{th}\right)={\int }_0^{\infty }{\int }_0^{\left(1+z_W\right)2^{C_{th}}-1}{f}_{{\gamma }_M}\left(z_M\right)f_{{\gamma }_W}\left(z_W\right)\mathrm{d}{z}_M\mathrm{d}{z}_W=\frac{{\left(\frac{m_Y{\mathsf{\Omega }}_X}{m_Y{\mathsf{\Omega }}_X+m_X{\mathsf{\Omega }}_Y}\right)}^{2m_Y}{\left(\frac{m_X}{{\mathsf{\Omega }}_X}\left({\mathsf{\Omega }}_X+{\mathsf{\Omega }}_Y\right)\right)}^{2m_X}}{{\Gamma }^2\left(m_X\right){\overline{\gamma }}_M^{m_X}{\overline{\gamma }}_W^{m_X}}\times \\ \hfill \times {\int }_0^{\infty }{\int }_0^{\left(1+z_W\right)2^{C_{th}}-1}{z}_W^{m_X-1}{z}_M^{m_X-1}{e}^{-\frac{m_X\left({\mathsf{\Omega }}_X+{\mathsf{\Omega }}_Y\right)}{{\overline{\gamma }}_W{\mathsf{\Omega }}_X}{z}_W}{}_1{F}_1\left(m_Y;m_X;\frac{m_X^2\left({\mathsf{\Omega }}_X+{\mathsf{\Omega }}_Y\right){\mathsf{\Omega }}_Y}{{\overline{\gamma }}_M{\mathsf{\Omega }}_X\left(m_Y{\mathsf{\Omega }}_X+m_X{\mathsf{\Omega }}_Y\right)}{z}_M\right)\times \\ \hfill \times e^{-\frac{m_X\left({\mathsf{\Omega }}_X+{\mathsf{\Omega }}_Y\right)}{{\overline{\gamma }}_M{\mathsf{\Omega }}_X}{z}_M}{}_1{F}_1\left(m_Y;m_X;\frac{m_X^2\left({\mathsf{\Omega }}_X+{\mathsf{\Omega }}_Y\right){\mathsf{\Omega }}_Y}{{\overline{\gamma }}_W{\mathsf{\Omega }}_X\left(m_Y{\mathsf{\Omega }}_X+m_X{\mathsf{\Omega }}_Y\right)}{z}_W\right)\mathrm{d}{z}_M\mathrm{d}{z}_W.\end{array}$$

(7)

At this point, one can use the classical series expansion of the confluent hypergeometric function ${}_1{F}_1(·)$ (see Equation (13.2.2) in [35]):

$${}_1{F}_1\left(a,b,z\right)=\sum _{s=0}^{\infty }\frac{{\left(a\right)}_s}{{\left(b\right)}_{s}s!}{z}^s,$$

(8)

where ${(·)}_s$ is the Pochhammer symbol [35].

Rearranging the terms and switching the order of summation and integration operation delivers:

$$\begin{array}{c}\hfill P_{out}\left(C_{th}\right)=\frac{{\left(\frac{m_Y{\mathsf{\Omega }}_X}{m_Y{\mathsf{\Omega }}_X+m_X{\mathsf{\Omega }}_Y}\right)}^{2m_Y}{\left(\frac{m_X}{{\mathsf{\Omega }}_X}\left({\mathsf{\Omega }}_X+{\mathsf{\Omega }}_Y\right)\right)}^{2m_X}}{{\Gamma }^2\left(m_X\right){\overline{\gamma }}_M^{m_X}{\overline{\gamma }}_W^{m_X}}\sum _{p=0}^{\infty }\sum _{q=0}^{\infty }\frac{{\left(m_Y\right)}_q}{{\left(m_X\right)}_q}\frac{{\left(m_Y\right)}_p}{{\left(m_X\right)}_p}\frac{{\left(\frac{m_X^2\left({\mathsf{\Omega }}_X+{\mathsf{\Omega }}_Y\right){\mathsf{\Omega }}_Y}{{\mathsf{\Omega }}_X\left(m_Y{\mathsf{\Omega }}_X+m_X{\mathsf{\Omega }}_Y\right)}\right)}^{p+q}}{q!p!{\overline{\gamma }}_W^p{\overline{\gamma }}_M^q}\times \\ \hfill \times {\int }_0^{\infty }{z}_W^{m_X+p-1}{e}^{-\frac{m_X\left({\mathsf{\Omega }}_X+{\mathsf{\Omega }}_Y\right)}{{\overline{\gamma }}_W{\mathsf{\Omega }}_X}{z}_W}\left\{{\int }_0^{\left(1+z_W\right)2^{C_{th}}-1}{z}_M^{m_X+q-1}{e}^{-\frac{m_X\left({\mathsf{\Omega }}_X+{\mathsf{\Omega }}_Y\right)}{{\overline{\gamma }}_M{\mathsf{\Omega }}_X}{z}_M}\mathrm{d}{z}_M\right\}\mathrm{d}{z}_W.\end{array}$$

(9)

To evaluate the inner integral, one can apply the definition of the lower incomplete gamma function (see Equation (8.2.1) in [35]):

$$\begin{array}{ccc}\hfill \gamma \left(n+1,z\right)& =& {\int }_0^z{t}^n{e}^{-t}\mathrm{d}t=\hfill \end{array}$$

(10)

$$\begin{array}{ccc}\hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} & =& n!\left(1-e^{-z}\sum _{k=0}^n\frac{z^k}{k!}\right),\hfill \end{array}$$

(11)

where the second equality follows from the series expansion of $\gamma \left(a,z\right)$ (see Equations (8.4.7) and (8.4.11) in [35]).

Let us combine (9) with (11) and simplify the obtained expression with the help of the following identities:

$$\frac{1}{{\left(m_X\right)}_p}\Gamma \left(m_X+p\right)=\Gamma \left(m_X\right),$$

(12)

$$\begin{array}{ccc}\hfill \sum _{p=0}^{\infty }\frac{{\left(m_Y\right)}_p}{p!}{\left(\frac{m_X{\mathsf{\Omega }}_Y}{\left(m_Y{\mathsf{\Omega }}_X+m_X{\mathsf{\Omega }}_Y\right)}\right)}^p& =& {}_2{F}_1\left(m_Y,b;b;\frac{m_X{\mathsf{\Omega }}_Y}{\left(m_Y{\mathsf{\Omega }}_X+m_X{\mathsf{\Omega }}_Y\right)}\right)=\hfill \\ \hfill & =& {\left(1-\frac{m_X{\mathsf{\Omega }}_Y}{\left(m_Y{\mathsf{\Omega }}_X+m_X{\mathsf{\Omega }}_Y\right)}\right)}^{-m_Y},\hfill \end{array}$$

(13)

where (12) follows from the definition of the Pochhammer function [35], and (13) from the definition of the Gauss hypergeometric function (see Equation (15.4.6) in [35]).

To resolve the remaining integral over $z_W$ one can make use of the Tricomi confluent hypergeometric function definition (see Equation (13.4.4) in [35]):

$$\begin{array}{c}\hfill U\left(a,b,z\right)=\frac{1}{\Gamma \left(a\right)}{\int }_0^{\infty }{e}^{-zt}{t}^{a-1}{(1+t)}^{b-a-1}\mathrm{d}t,\end{array}$$

(14)

which is valid when $\Re a>0$ and $|argz|<\raisebox{1ex}{$\pi $}\!\left/ \!\raisebox{-1ex}{$2$}\right.$ (it can be verified that those conditions are satisfied), thus delivering the following form of the secrecy outage capacity:

$$\begin{array}{c}\hfill P_{out}\left(C_{th}\right)=1-\frac{{\left(\frac{2^{C_{th}}-1}{2^{C_{th}}}\right)}^{m_X}{\left(\frac{m_Y{\mathsf{\Omega }}_X}{m_Y{\mathsf{\Omega }}_X+m_X{\mathsf{\Omega }}_Y}\right)}^{2m_Y}}{{\left(\frac{m_X}{{\overline{\gamma }}_W{\mathsf{\Omega }}_X}\left({\mathsf{\Omega }}_X+{\mathsf{\Omega }}_Y\right)\right)}^{-m_X}}{e}^{-\frac{m_X}{{\overline{\gamma }}_M{\mathsf{\Omega }}_X}\left({\mathsf{\Omega }}_X+{\mathsf{\Omega }}_Y\right)\left(2^{C_{th}}-1\right)}\times \\ \hfill \times \sum _{p=0}^{\infty }\sum _{q=0}^{\infty }\frac{{\left(m_Y\right)}_q{\left(m_Y\right)}_p}{q!p!{\overline{\gamma }}_W^p{\overline{\gamma }}_M^q}\frac{{\left(\frac{m_X^2\left({\mathsf{\Omega }}_X+{\mathsf{\Omega }}_Y\right){\mathsf{\Omega }}_Y}{{\mathsf{\Omega }}_X\left(m_Y{\mathsf{\Omega }}_X+m_X{\mathsf{\Omega }}_Y\right)}\right)}^p}{{\left(\frac{2^{C_{th}}-1}{2^{C_{th}}}\right)}^{-p}}{\left(\frac{m_X{\mathsf{\Omega }}_Y}{m_Y{\mathsf{\Omega }}_X+m_X{\mathsf{\Omega }}_Y}\right)}^q\sum _{k=0}^{m_X+q-1}\frac{{\left[\left(\frac{m_X\left({\mathsf{\Omega }}_X+{\mathsf{\Omega }}_Y\right)}{{\overline{\gamma }}_M{\mathsf{\Omega }}_X}\right)\left(2^{C_{th}}-1\right)\right]}^k}{k!}\times \\ \hfill \times U\left(m_X+p;m_X+p+1+k;\left(\frac{2^{C_{th}}-1}{2^{C_{th}}}\right)\left(\frac{m_X\left({\mathsf{\Omega }}_X+{\mathsf{\Omega }}_Y\right)}{{\mathsf{\Omega }}_X}\left(\frac{1}{{\overline{\gamma }}_W}+2^{C_{th}}\frac{1}{{\overline{\gamma }}_M}\right)\right)\right).\end{array}$$

(15)

It is worth noting that for the obtained specific arguments of the Tricomi hypergeometric function, it can be represented in terms of the finite sum (see Equation (13.2.8) in [35]):

$$\begin{array}{c}\hfill U\left(m_X+p;m_X+p+1+k;\left(\frac{2^{C_{th}}-1}{2^{C_{th}}}\right)\left(\frac{m_X\left({\mathsf{\Omega }}_X+{\mathsf{\Omega }}_Y\right)}{{\mathsf{\Omega }}_X}\left(\frac{1}{{\overline{\gamma }}_W}+2^{C_{th}}\frac{1}{{\overline{\gamma }}_M}\right)\right)\right)=\\ \hfill {\displaystyle =\frac{{\displaystyle \sum _{s=0}^k\left(\genfrac{}{}{0pt}{}{k}{s}\right){(m_X+p)}_s{\left[\left(\frac{2^{C_{th}}-1}{2^{C_{th}}}\right)\left(\frac{m_X\left({\mathsf{\Omega }}_X+{\mathsf{\Omega }}_Y\right)}{{\mathsf{\Omega }}_X}\left(\frac{1}{{\overline{\gamma }}_W}+2^{C_{th}}\frac{1}{{\overline{\gamma }}_M}\right)\right)\right]}^{-s}}}{{\displaystyle {\left[\left(\frac{2^{C_{th}}-1}{2^{C_{th}}}\right)\left(\frac{m_X\left({\mathsf{\Omega }}_X+{\mathsf{\Omega }}_Y\right)}{{\mathsf{\Omega }}_X}\left(\frac{1}{{\overline{\gamma }}_W}+2^{C_{th}}\frac{1}{{\overline{\gamma }}_M}\right)\right)\right]}^{m_X+p}}},}\end{array}$$

(16)

thus (15) can be rewritten as

$$\begin{array}{c}\hfill P_{out}\left(C_{th}\right)=1-\frac{{\left(\frac{2^{C_{th}}-1}{2^{C_{th}}}\right)}^{m_X}{\left(\frac{m_Y{\mathsf{\Omega }}_X}{m_Y{\mathsf{\Omega }}_X+m_X{\mathsf{\Omega }}_Y}\right)}^{2m_Y}}{{\left(\frac{m_X}{{\overline{\gamma }}_W{\mathsf{\Omega }}_X}\left({\mathsf{\Omega }}_X+{\mathsf{\Omega }}_Y\right)\right)}^{-m_X}}{e}^{-\frac{m_X}{{\overline{\gamma }}_M{\mathsf{\Omega }}_X}\left({\mathsf{\Omega }}_X+{\mathsf{\Omega }}_Y\right)\left(2^{C_{th}}-1\right)}\times \\ \hfill \times \sum _{p=0}^{\infty }\sum _{q=0}^{\infty }\frac{{\left(m_Y\right)}_q{\left(m_Y\right)}_p}{q!p!{\overline{\gamma }}_W^p{\overline{\gamma }}_M^q}\frac{{\left(\frac{m_X^2\left({\mathsf{\Omega }}_X+{\mathsf{\Omega }}_Y\right){\mathsf{\Omega }}_Y}{{\mathsf{\Omega }}_X\left(m_Y{\mathsf{\Omega }}_X+m_X{\mathsf{\Omega }}_Y\right)}\right)}^p}{{\left(\frac{2^{C_{th}}-1}{2^{C_{th}}}\right)}^{-p}}{\left(\frac{m_X{\mathsf{\Omega }}_Y}{m_Y{\mathsf{\Omega }}_X+m_X{\mathsf{\Omega }}_Y}\right)}^q\sum _{k=0}^{m_X+q-1}\frac{{\left[\left(\frac{m_X\left({\mathsf{\Omega }}_X+{\mathsf{\Omega }}_Y\right)}{{\overline{\gamma }}_M{\mathsf{\Omega }}_X}\right)\left(2^{C_{th}}-1\right)\right]}^k}{k!}\times \\ \hfill {\displaystyle \times \frac{{\displaystyle \sum _{s=0}^k\left(\genfrac{}{}{0pt}{}{k}{s}\right){(m_X+p)}_s{\left[\left(\frac{2^{C_{th}}-1}{2^{C_{th}}}\right)\left(\frac{m_X\left({\mathsf{\Omega }}_X+{\mathsf{\Omega }}_Y\right)}{{\mathsf{\Omega }}_X}\left(\frac{1}{{\overline{\gamma }}_W}+2^{C_{th}}\frac{1}{{\overline{\gamma }}_M}\right)\right)\right]}^{-s}}}{{\displaystyle {\left[\left(\frac{2^{C_{th}}-1}{2^{C_{th}}}\right)\left(\frac{m_X\left({\mathsf{\Omega }}_X+{\mathsf{\Omega }}_Y\right)}{{\mathsf{\Omega }}_X}\left(\frac{1}{{\overline{\gamma }}_W}+2^{C_{th}}\frac{1}{{\overline{\gamma }}_M}\right)\right)\right]}^{m_X+p}}}.}\end{array}$$

(17)

Reorganizing the terms, and performing the summation over p, i.e.,

$$\begin{array}{ccc}\hfill & & \sum _{p=0}^{\infty }\frac{{\left(m_Y\right)}_p{(m_X+p)}_s}{p!}{\left[\frac{m_X{\mathsf{\Omega }}_Y}{(m_Y{\mathsf{\Omega }}_X+m_X{\mathsf{\Omega }}_Y)\left(1+2^{C_{th}}\frac{\overline{{\gamma }_W}}{\overline{{\gamma }_W}}\right)}\right]}^p=\hfill \\ & & ={\left(m_X\right)}_s{}_2{F}_1\left(m_Y,m_X+s;m_X;\frac{m_X{\mathsf{\Omega }}_Y}{(m_Y{\mathsf{\Omega }}_X+m_X{\mathsf{\Omega }}_Y)\left(1+2^{C_{th}}\frac{\overline{{\gamma }_W}}{\overline{{\gamma }_W}}\right)}\right),\hfill \end{array}$$

(18)

yields the expression for the secrecy outage probability:

$$\begin{array}{ccc}\hfill & & P_{out}\left(C_{th}\right)=1-e^{-\frac{m_X\left({\mathsf{\Omega }}_X+{\mathsf{\Omega }}_Y\right)\left(2^{C_{th}}-1\right)}{{\overline{\gamma }}_M{\mathsf{\Omega }}_X}}\sum _{q=0}^{\infty }\sum _{k=0}^{m_X+q-1}\sum _{s=0}^k\frac{{\left(m_Y\right)}_q}{q!}{\left(\frac{m_Y{\mathsf{\Omega }}_X}{m_Y{\mathsf{\Omega }}_X+m_X{\mathsf{\Omega }}_Y}\right)}^{q+2m_Y}\times \hfill \\ & & \times \left(\genfrac{}{}{0pt}{}{k}{s}\right){\left[\frac{\left(2^{C_{th}}-1\right)m_X\left({\mathsf{\Omega }}_X+{\mathsf{\Omega }}_Y\right)}{{\mathsf{\Omega }}_X}\right]}^{k-s}{\left(1+2^{C_{th}}\frac{\overline{{\gamma }_W}}{\overline{{\gamma }_W}}\right)}^{-s-m_X}\frac{{\left(2^{C_{th}}\overline{{\gamma }_W}\right)}^s{\left(m_X\right)}_s}{{\overline{{\gamma }_M}}^{k}k!}\times \hfill \\ & & \times {}_2{F}_1\left(m_Y,m_X+s;m_X;\frac{m_X{\mathsf{\Omega }}_Y}{(m_Y{\mathsf{\Omega }}_X+m_X{\mathsf{\Omega }}_Y)\left(1+2^{C_{th}}\frac{\overline{{\gamma }_W}}{\overline{{\gamma }_W}}\right)}\right).\hfill \end{array}$$

(19)

It is worth noting that (19) is given in terms of a single infinite series (compared to (15)), which can be successfully truncated with the desired accuracy. The derived expression (19) is novel, and has not been presented in the technical scientific literature.

3.2. Derived Expression Numerical Analysis and Discussion

Although the derived expression is closed-form, from a practical point of view, it is clear that the series should be truncated. Thus, it is important to understand the number of summands required to reach the desired accuracy.

To achieve that goal and demonstrate the correctness of the derived solution, a numerical simulation was performed, and the results are illustrated in Figure 1 and Figure 2 and

Table 1. A comparison between the numerical and closed-form solutions for $C_{th}=0.3$, ${\overline{\gamma }}_W=10$ dB, $m_X=m_Y=1,{\mathsf{\Omega }}_X={\mathsf{\Omega }}_Y=0$ dB with various numbers of terms N.

${\overline{\mathit{\gamma }}}_{\mathit{M}}$, dB	${\mathit{P}}_{\mathit{out}}^{\mathit{num}}$	$\mathit{N}=10$	$\mathit{N}=20$	$\mathit{N}=30$
		${\mathit{P}}_{\mathit{out}}^{\mathit{cf}}$	${\mathit{P}}_{\mathit{out}}^{\mathit{cf}}$	${\mathit{P}}_{\mathit{out}}^{\mathit{cf}}$
0	0.94038	0.940572	0.940381	0.94038
5	0.855632	0.855978	0.855632	0.855632
10	0.705207	0.705664	0.705208	0.705207
15	0.503409	0.503896	0.50341	0.503409
20	0.306428	0.306916	0.306428	0.306428
25	0.163436	0.163924	0.163437	0.163436
30	0.0800201	0.0805084	0.0800206	0.0800201
35	0.0373851	0.0378734	0.0373856	0.0373851
40	0.0170628	0.0175511	0.0170633	0.0170628

and

Table 2. A comparison between numerical and closed-form solutions for $N=20$, ${\overline{\gamma }}_M={\overline{\gamma }}_W=10$ dB, ${\mathsf{\Omega }}_X={\mathsf{\Omega }}_Y=0$ dB, with various $C_{th}$ and shadowing coefficients.

${\mathit{C}}_{\mathit{th}}$	${\mathit{m}}_{\mathit{X}}={\mathit{m}}_{\mathit{Y}}=1$		${\mathit{m}}_{\mathit{X}}={\mathit{m}}_{\mathit{Y}}=2$		${\mathit{m}}_{\mathit{X}}={\mathit{m}}_{\mathit{Y}}=3$
	${\mathit{P}}_{\mathit{out}}^{\mathit{num}}$	${\mathit{P}}_{\mathit{out}}^{\mathit{cf}}$	${\mathit{P}}_{\mathit{out}}^{\mathit{num}}$	${\mathit{P}}_{\mathit{out}}^{\mathit{cf}}$	${\mathit{P}}_{\mathit{out}}^{\mathit{num}}$	${\mathit{P}}_{\mathit{out}}^{\mathit{cf}}$
0.1	0.571434	0.571434	0.600792	0.600797	0.623793	0.623825
0.2	0.640469	0.640469	0.695150	0.695155	0.735766	0.735798
0.3	0.705207	0.705208	0.777915	0.777920	0.827489	0.827519
0.4	0.764140	0.764141	0.846028	0.846033	0.895670	0.895697
0.5	0.816242	0.816243	0.898699	0.898704	0.941763	0.941786
0.6	0.860981	0.860982	0.937004	0.937008	0.970142	0.970159
0.7	0.898269	0.898270	0.963189	0.963192	0.986044	0.986055
0.8	0.928373	0.928374	0.979962	0.979964	0.994121	0.994128
0.9	0.951825	0.951826	0.989965	0.989967	0.997806	0.997811

.

First, the obtained closed-form solution (19) was truncated by N terms (denoted as $P_{out}^{cf}$), and the calculated results for various systems and channel parameters were compared with the results obtained via brute-force numeric integration of (7) (denoted as $P_{out}^{num}$). The relative error $err\left(N\right)=\left|\frac{P_{out}^{num}-P_{out}^{cf}}{P_{out}^{num}}\right|$ as a function of the number of terms is presented in Figure 1 and the resultant computational speedup ${ϵ}_t\left(N\right)$ (defined as the ratio between the times required to compute $P_{out}^{cf}$ and $P_{out}^{num}$) is presented in Figure 2.

It can be observed that the increase in the path loss and the threshold capacity improved the residual error (see Figure 1), whereas the improvement of the shadowing conditions (greater m) impaired it. If the $1\%$ error is tolerable then, for example, for the heavy path loss scenario with light shadowing, even the 0-term truncation gives the result. For most of the cases, 10–15 terms are enough to yield the sub-percent error.

On the other hand, the computational speedup (see Figure 2) that can be obtained with the help of the proposed solution is at least 10 times for $N\ge 14$. Moreover, it can be seen that even though ${ϵ}_t\left(N\right)$ decreases as a function of N, $P_{out}^{cf}$ is always superior over $P_{out}^{num}$ (i.e., ${ϵ}_t\left(N\right)>1$). It should be noted that, in practice, in the sphere of wireless communications, the relative error of ${10}^{-8}$ is way below the necessity. Thus, it can be seen that depending on the desired accuracy and computational time, one can choose the number of terms, respectively.

Table 1 presents the comparison for 10, 20, and 30 terms as functions of the average SNR in the main channel. It can be observed that although the precision drops with the increase of ${\overline{\gamma }}_M$, even $N=10$ can provide three-digit precision (for reasonably high SNR $\simeq 40$ dB), and $N=30$ delivers seven-digit precision, which is excessive for practical applications.

Limiting ourselves to $N=20$ (yielding in most cases five-digit precision) an analysis of the fading conditions impart was performed (see Table 2).

It was observed that the improvement in shadowing conditions (greater $m_X$ and $m_Y$) impairs the precision (up to four digits), but it must be stressed that the overall values of SOP are high and such precision is more than enough. On the other hand, increasing the requirements implied on the $C_{th}$ does not affect the quality of the obtained result.

It should be noted that in the case of a fixed number of terms, extra precision is sought, one can successfully apply well-established convergence-increasing algorithms, for instance, Shanks’ transformation (see [36]).

As the result of the numerical implementation of the derived expression for various values of the channel and the system parameters, it was decided to truncate the series in (15) for further calculations by twenty terms, delivering at least five-digit precision.

4. Simulation and Results

4.1. Simulation Setup

To analyze the dependence of the secrecy outage probability on the values of the channel and system parameters, the numeric simulation was carried out.

The assumed ranges of the parameter values are presented in

Table 3. System and channel parameters assumed for simulation.

Parameter	Parameter Value
Shadowing coefficients of the total and LoS components ($m_X,m_Y$)	$1\dots 10$
Energy of NLoS and LoS components (${\mathsf{\Omega }}_X,{\mathsf{\Omega }}_Y$), dB	$-10\dots 10$
Average SNR for the main and the wiretap channels (${\overline{\gamma }}_M,{\overline{\gamma }}_W$), dB	$0\dots 50$
The relative distance between the receiver, the eavesdropper, and the transmitter ($\raisebox{1ex}{$d_W$}\!\left/ \!\raisebox{-1ex}{$d_M$}\right.$)	$0.1\dots 10$
Path loss exponent ($\alpha$)	$1.5\dots 5$
Normalized threshold capacity ($C_{th}$)	$0.01\dots 0.99$

.

Several notes should be pointed out:

• The range of path loss exponent was chosen in accordance with the existent real-life measurements performed for various scenarios [37,38].
• The threshold channel capacity corresponding to the secrecy outage was normalized by the capacity of the non-fading Gaussian channel with Gaussian input (see [5]).
• The $\raisebox{1ex}{$d_W$}\!\left/ \!\raisebox{-1ex}{$d_M$}\right.$ range was chosen in a way to cover the possible cases when the eavesdropper was further and closer to the transmitter compared to the main receiver.
• In accordance with [18], $2m_Y$ equals the number of LoS components; thus, the analysis was performed for the scenarios with possible multiple LoS components.
• The average SNR range was set so to account for the requirements of the shadowed Beaulieu–Xie channel modulation techniques (see [18]).

It is worth mentioning that the problem under analysis was a multiparametric one; thus, a single-argument cut of a multi-argument function (as widely used) cannot give a full description. So, for further analysis, we will resort to the contour plots that give descriptions in terms of several joint parameters.

4.2. Simulation Results and Analysis

Figure 3 and Figure 4 demonstrate the influence of the system model parameters (the ratio of the distances $d_W/\phantom{d_W{d}_M}{d}_M$ and the average SNR in the main channel ${\overline{\gamma }}_M$) on the secrecy outage probability $P_{out}$, depending on the channel model parameters: the average power of multipath ${\mathsf{\Omega }}_X$, line-of-sight components ${\mathsf{\Omega }}_Y$, shadowing coefficients of the overall $m_X$, and the line-of-sight components $m_Y$. Figure 3 was obtained with significant shadowing ($m_X=m_Y=1$), and Figure 4—with weak shadowing ($m_X=m_Y=5$). Both figures show four situations covering the most indicative combinations of the average powers of LoS and NLoS components: in the case of a, both components are weak ( ${\mathsf{\Omega }}_X={\mathsf{\Omega }}_Y=-10$ dB), in the case of b, multipath components predominate in the signal (${\mathsf{\Omega }}_X\gg {\mathsf{\Omega }}_Y$), in the case of c, the line-of-sight components are strong, and multipath components are weakened (${\mathsf{\Omega }}_X\ll {\mathsf{\Omega }}_Y$), and finally, in the case of d, both components are strong (${\mathsf{\Omega }}_X={\mathsf{\Omega }}_Y=10$ dB). With the color scheme used, low values $P_{out}$ (and, therefore, a greater degree of link secrecy) correspond to dark blue; high values correspond to yellow.

A comparison of Figure 3 and Figure 4 shows that the amount of shadowing $m_X=m_Y$ has a more pronounced effect than the combination of the signal component powers ${\mathsf{\Omega }}_X,{\mathsf{\Omega }}_Y$. One of the main features of $P_{out}$ is related to the size and shape of the desirable values region of the secrecy outage probability ($P_{out}<0.2$), shown in Figure 3 and Figure 4 by the dark blue. It contains the existing global extremum of the $P_{out}$ with respect to the variables $d_W/\phantom{d_W{d}_M}{d}_M$ and ${\overline{\gamma }}_M$. This area corresponds to the most favorable transmission in terms of security. In Figure 3 it is observed for $d_W/\phantom{d_W{d}_M}{d}_M>8$ and $13<{\overline{\gamma }}_M<26$, and in Figure 4 it occupies a fairly significant part of the plot area: $d_W/\phantom{d_W{d}_M}{d}_M>2.5$ and ${\overline{\gamma }}_M<35$. Thus, the range of system parameters at which the secrecy outage probability will be small is narrower at lower m, i.e., in a situation of larger shadowing. This suggests that in a situation when the shadowing is large, care is needed in choosing the employed transmitting/receiving strategy, optimizing the average SNR. In addition, at small $m_X=m_Y$ a communication link becomes secure only when the wiretap receiver is removed from the legitimate one by a large distance, while at large, $m_X=m_Y$, a secure scenario, can be achieved even for the situation in which both receivers are at almost the same distance from the transmitter.

In Figure 3 for small $m_X=m_Y$, the region with $P_{out}<0.2$ is such that for all combinations of ${\mathsf{\Omega }}_X,{\mathsf{\Omega }}_Y$ for a fixed $d_W/\phantom{d_W{d}_M}{d}_M$ there are two values of ${\overline{\gamma }}_M$ (which practically do not depend on ${\mathsf{\Omega }}_X,{\mathsf{\Omega }}_Y$), which means that the choice of system parameters is non-unique. For large $m_X=m_Y$, as illustrated in Figure 4, the shape of such an area changes significantly and is also dependent on ${\mathsf{\Omega }}_X,{\mathsf{\Omega }}_Y$. With the equal average powers of multipath components and line-of-sight components ${\mathsf{\Omega }}_X={\mathsf{\Omega }}_Y$ (Figure 4a,d), there are $3<d_W/\phantom{d_W{d}_M}{d}_M<3.5$, which correspond to the ambiguity of the choice of ${\overline{\gamma }}_M$. With large differences between the power of the components, regardless of the sign, i.e., at ${\mathsf{\Omega }}_X\gg {\mathsf{\Omega }}_Y$ (Figure 4b) or ${\mathsf{\Omega }}_X\ll {\mathsf{\Omega }}_Y$ (Figure 4c), such ambiguity is not observed.

Analyzing contour plots with one of the arguments being constant, one should keep in mind the relationship (4) between the parameters of the system model. With a constant ratio $d_W/\phantom{d_W{d}_M}{d}_M=const$ we have $\sqrt[\alpha ]{{\overline{\gamma }}_M/\phantom{{\overline{\gamma }}_M{\overline{\gamma }}_W}{\overline{\gamma }}_W}=const$, this means that the increase in ${\overline{\gamma }}_M$ corresponds to either a proportional increase of ${\overline{\gamma }}_W$ (i.e., facilitating the reception for the wiretap receiver), or an increase in the path loss exponent $\alpha$ (i.e., increasing attenuation and worsening the overall signal propagation conditions). Thus, while maintaining $d_W/\phantom{d_W{d}_M}{d}_M<3$ and $d_W/\phantom{d_W{d}_M}{d}_M$ fixed, the increase of ${\overline{\gamma }}_M$ leads to the increase of $P_{out}$, which corresponds to the impairment of the predicted security. For the constant average main channel SNR ${\overline{\gamma }}_M=const$, (4) evaluates to ${\overline{\gamma }}_W{\left(d_W/\phantom{d_W{d}_M}{d}_M\right)}^{\alpha }=const$, i.e., the increase of $d_W/\phantom{d_W{d}_M}{d}_M$ (the withdrawal of the eavesdropper from the transmitter with a legitimate receiver being fixed) corresponds either to the reduction of ${\overline{\gamma }}_W$ (i.e., hardening the reception for the wiretap), or to the decrease of $\alpha$ (i.e., approaching the propagation conditions to the free space). Thus, while maintaining fixed ${\overline{\gamma }}_M$, the increase of $d_W/\phantom{d_W{d}_M}{d}_M$ yields the reduction of $P_{out}$, which corresponds to the improvement of the predicted security.

The influence of the path loss exponent $\alpha$ on the secrecy outage probability has the following features at different relative distances from the transmitter to the wiretap and legitimate receivers (see Figure 5):

1.

If the location of the eavesdropper is too close to the transmitter compared to the legitimate receiver (for example, $d_W/\phantom{d_W{d}_M}{d}_M=0.1$ in Figure 5a), the higher the path loss exponent $\alpha$, the greater the secrecy outage probability $P_{out}$, since in this case ${\overline{\gamma }}_M<{\overline{\gamma }}_W$ and the reception conditions for the legitimate receiver compared to the wiretap are worse, as follows from (4). In the analyzed range of parameters $\alpha$ and $C_{th}$, $P_{out}$ does not fall below 0.96.

2.

At the unit relative distances (see $d_W/\phantom{d_W{d}_M}{d}_M=1$ in Figure 5b) the signal propagation conditions in the main and the wiretap channels are equal, as $m_X=m_Y,{\mathsf{\Omega }}_X={\mathsf{\Omega }}_Y$ and ${\overline{\gamma }}_M={\overline{\gamma }}_W$ from (4). Hence, the secrecy outage probability is determined primarily by the required threshold capacity $C_{th}$: the higher $C_{th}$, the more likely the loss of link secrecy.

3.

Finally, in cases when the eavesdropper is further from the transmitter than the legitimate receiver (see, for instance, $d_W/\phantom{d_W{d}_M}{d}_M=10$ in Figure 5c), with the growth of the path loss exponent $\alpha$ there is a confrontation between two multidirectional factors. On the one hand, the reception conditions for a legitimate receiver are improving (the inequality ${\overline{\gamma }}_M>{\overline{\gamma }}_W$ becomes stronger), on the other hand, the conditions for signal propagation are worsening due to the increase of $\alpha$ similarly for both receivers. This leads to the fact that the equiprobability lines run along the horizontal axis in the area $\alpha >3$ (Figure 5c); in this region, the values of $P_{out}$ are determined by the value of $C_{th}$, as in Figure 5b. Thus, the existence and the shape of the area of $\alpha$ and $C_{th}$, which provides a low secrecy outage probability, depends on the relative distance between the receivers and the transmitter.

There is no acceptable threshold capacity $C_{th}$ (for instance, greater than 0.1) for either $d_W/\phantom{d_W{d}_M}{d}_M\approx 1$ or even $d_W/\phantom{d_W{d}_M}{d}_M<1$ that would guarantee the required probability of $P_{out}<0.2$. If the relative distance satisfied the condition $d_W/\phantom{d_W{d}_M}{d}_M>8$, then, by decreasing $C_{th}$, it is possible to ensure a safe link with a given probability of outage, and better propagation conditions (with smaller $\alpha$)—with a greater decrease in $C_{th}$.

Considering the influence of the channel parameters (i.e., average powers of NLoS ${\mathsf{\Omega }}_X$ and LoS ${\mathsf{\Omega }}_Y$ components for various combinations of shadowing conditions $m_X,m_Y$), demonstrated in Figure 6, it is evident that the contour lines of constant $P_{out}$ can be approximated on a logarithmic scale as ${\mathsf{\Omega }}_Y={\mathsf{\Omega }}_X+b$, where b is a constant term representing the LoS–NLoS component power imbalance. As b increases, $P_{out}$ also increases, and the link becomes insecure. The area corresponding to the least secrecy is located in the lower right corner of the plot at any $m_X,m_Y$. This corresponds to scenarios ${\mathsf{\Omega }}_Y<{\mathsf{\Omega }}_X+b$ and even ${\mathsf{\Omega }}_Y<<{\mathsf{\Omega }}_X+b$, which practically means that secure communication is easier to provide with strong NLoS, rather than LoS components. The specific value of the admissible power imbalance depends on the shadowing conditions of the components. The specific area of $P_{out}<0.2$ is maximized for the case of $m_X=1,m_Y=5$ (see Figure 6b), which constitutes to the situation when security is guaranteed for the greater range of power imbalances, and minimized for the case of $m_X=5,m_Y=1$ (see Figure 6c). From the point of view of the overall area of $P_{out}<0.2$ the situations depicted in Figure 6a,d are in between. The gradient of $P_{out}$ is the greatest in the cases depicted in Figure 6b,d. Thus, even a small change in ${\mathsf{\Omega }}_X,{\mathsf{\Omega }}_Y$ can cause a serious increase of $P_{out}$ making the link not secure.

An analysis of the results obtained for varying $m_X,m_Y$, and distinct combinations of ${\mathsf{\Omega }}_X,{\mathsf{\Omega }}_Y$ (see Figure 7) made it possible to conclude that in the cases of equal powers of LoS and NLoS components (i.e., ${\mathsf{\Omega }}_X={\mathsf{\Omega }}_Y=-5$ dB in Figure 7a, or ${\mathsf{\Omega }}_X={\mathsf{\Omega }}_Y=5$ dB in Figure 7c) the difference of $P_{out}$ as a function of $m_X,m_Y$ is almost intractable. In those cases, the area with $P_{out}<0.1$ (where the secrecy constraints are satisfied) is concentrated where $m_X,m_Y$ are large, which corresponds to the small overall and LoS component shadowing, and the equiprobability contours are almost parallel to the $m_Y$ axis (thus the values $m_Y$ do not impact) for $m_X<4,m_Y>6$ in case of $P_{out}=0.2$ and for $m_X<2,m_Y>3$ in case of $P_{out}=0.3$. On the other hand, for large values of $m_X$ and small $m_Y$ the lines of $P_{out}=const$ are almost parallel to $m_X$, thus $m_Y$ does not have any impact on the link security.

For the cases of unequal LoS/NLoS powers (see Figure 7b,c), the greater impact has the part of the channel spatial structure that exhibits stronger shadowing. From the results, presented in Figure 7b, which correspond to the case with strong NLoS (${\mathsf{\Omega }}_X=5$ dB) and weak LoS (${\mathsf{\Omega }}_Y=-5$ dB) components, it is evident that $P_{out}=const$ contours are almost parallel to the $m_Y$ axis for arbitrary $m_X$, which means that it does not have any impact in those conditions. Strictly the opposite is the case with strong LoS (${\mathsf{\Omega }}_Y=5$ dB) and weak NLoS (${\mathsf{\Omega }}_X=-5$ dB) components (see Figure 7c), when $P_{out}=const$ lines, starting from some threshold $m_Y$, are parallel to the $m_X$ axis. The increase of $P_{out}$ shifts this threshold to the region with smaller $m_Y$, e.g., $m_Y\approx 4$ for $P_{out}=0.2$, and $m_Y\approx 2$ for $P_{out}=0.3$.

4.3. Discussion and Analysis Summary

The performed analysis makes it possible to find specific propagation conditions (i.e., channel parameters), which deliver the desired level of link security. On the other hand, recently, a novel and very promising technology—reconfigurable intelligent surfaces [39]—demonstrated the ability of communication assistance [40,41]. Such surfaces are designed in such a way to change and control the propagation channel properties. A classic example is the controlled variation of its reflection coefficient [42], thus creating new clusters of multipath waves or suppressing the existing ones [43]. This means that the effective value of the path loss exponent, shadowing coefficients and LoS/NLoS power imbalance can be possibly controlled or manipulated. The possible further deployment of the reconfigurable intelligent surfaces (or other technologies that can help to control channel parameters) allows us to reformulate the performed descriptive analysis into specific recommendations for communication system design.

Summarizing all of the above, several general conclusions can be drawn and recommendations can be given:

• First, the recommended values of channel parameters $m_X,m_Y$ should constitute to the light shadowing, and the values of ${\mathsf{\Omega }}_X$, greater than that of ${\mathsf{\Omega }}_Y$ (i.e., favoring the multipath components of a fading channel, rather than line-of-sight components). Second, the following system parameters are recommended: $d_W/\phantom{d_W{d}_M>1}{d}_M>1$ or even $d_W/\phantom{d_W{d}_M>>1}{d}_M>>1$ (i.e., exploiting the situation when the legitimate receiver is closer than the eavesdropper), the values of the path loss exponent $\alpha$ and average signal-to-noise ratio in the main channel should correspond to the ranges, depicted in Figure 3, Figure 4 and Figure 5, the threshold capacities $C_{th}$ should be lowered down up to the lowest values that are still admissible for information transfer.
• The overall parameter values for which there is no way to achieve communication link secrecy corresponds to the following propagation scenarios: the eavesdropper is closer to the transmitter than the legitimate receiver (i.e., $d_W/\phantom{d_W{d}_M\le 0.1}{d}_M\le 0.1$, see Figure 3a), the overall signal propagation conditions are too severe (large path loss exponent $\alpha$ values), large values of the average SNR in the wiretap channel ${\overline{\gamma }}_W$ (the wiretap receiver has an advantage over the legitimate one).
• Contour lines of the constant secrecy outage probability exhibit a specific minimum in case of heavy shadowing, i.e., small $m_X,m_Y$, and eavesdropper displaced at least twice further than the legitimate receiver, i.e., $d_W/\phantom{d_W{d}_M>2}{d}_M>2$, see Figure 3. This results in non-uniqueness of the possible guidelines in the choice of main channel average SNR ${\overline{\gamma }}_M$.
• Although from the physical perspective, the meanings of several parameters are identical ($m_X,m_Y$ and ${\mathsf{\Omega }}_X,{\mathsf{\Omega }}_Y$), their impacts on the $P_{out}$ greatly differ. To reach the smaller values of $P_{out}$, it is desirable to have ${\mathsf{\Omega }}_Y<{\mathsf{\Omega }}_X$, and the overall shadowing $m_X$ has a stronger impact, than the LoS shadowing $m_Y$; moreover, the cases with $m_X>6$ are preferable, and recommendations about the choice of $m_Y$ greatly depend on the required level of secrecy.

5. Conclusions

The presented research studies the problem of the physical layer security of a wireless communication system functioning in the presence of a multipath fading channel and a wiretap. The analysis was performed under the following assumptions: a the communication system was symmetric, meaning that the physical propagation conditions are equal for the legitimate receiver and the eavesdropper; b both channels are assumed to be statistically independent; c both channels are described by the shadowed Beaulieu–Xie fading model. The link secrecy was characterized by the secrecy outage probability defined for a fixed target secrecy capacity level. For the model under consideration, the closed-form expression of the SOP is presented, and its correctness is numerically verified. It is demonstrated that the proposed expression delivers practically reasonable results (in terms of the desired precision) from all the possible channel parameters. An in-depth study of the secrecy performance was carried out for all practically meaningful fading scenarios, including heavy and light fading, with and without strong dominant and multipath components. The performed research demonstrated the existence of the secrecy outage probability non-uniqueness with respect to the average signal-to-noise ratio in the main channel and the relative distance between the legitimate and wiretap receivers.