# Physical Layer Security for CRNs over Beaulieu-Xie Fading Channels

Authors: Yanyang Zeng; Yang Hua; Yiwei Fang; Xiaohong Wang
Journal: Wireless Communications and Mobile Computing
 (2022)

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## Abstract

In recent years, the access of massive communication devices leads to the insufficient spectrum resources of wireless networks. One of the practical means to resolve the problem is to build cognitive radio networks (CRNs), which can realize the sharing of spectrum resources between primary and secondary users, thereby improving the utilization rate of wireless spectrum resources. To this end, the CRNs are utilized to establish the Wyner’s eavesdropping model over the Beaulieu-Xie fading channels. We mainly deduce the accurate expressions of secrecy outage probability and strictly positive secrecy capacity to explore the performance of physical layer security. Moreover, the better overlap between the statistical simulation and the theoretical results indicates the correctness of the theoretical analysis equation. The interesting results are that both increasing thePmax and decreasing the Cth can improve the security performance. This work is a good reference and guidance for modeling CRNs (Internet of Things, fifth generation, cell phone networks, etc.) and security performance evaluation.


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## Body

## 1. Introduction

With the wide application of wireless transmission, the useful information will face the risk of disclosure in the process of transmission, so it is particularly important to find measures to avoid and reduce the eavesdropping of signals. Recently, the issue has become a research hotspot in academia. Different from traditional encryption and decryption methods, physical layer security (PLS) utilizes the characteristics of the transmission channels and noise to ensure the secure transmission of information. It was first proposed in [1] and applied to many practical communication scenarios, such as Beyond-5G low-latency communication networks [2], multihop amplify-and-forward (AF) relay systems [3], and cognitive radio networks (CRNs) [4]. Furthermore, the authors of [5] proposed a scheme combining deep learning with transmit antenna selection to enhance the PLS of cellular networks. In [6], the authors outlined some low complexity PLS schemes appropriate for the Internet of Things (IoTs). A cooperative beamforming scheme was designed in [7] to enhance the PLS in CRNs. For exploring the PLS properties of κ−μ shadowed fading, Sun et al. in [8] deduced closed-form expressions for the lower bounds of strictly positive secrecy capacity (SPSC) and secure outage probability (SOP).Radio waves are utilized to transmit information in wireless communication networks (WCNs), but the actual communication environment is complex and variable so that there is a difference between the received and the transmitted signal, which lead to the distorted transmission of information. To better evaluate the transmission performance of WCNs, it is necessary to build and analyze system models of different fading channels [9–16]. Zhu et al. in [9] studied the performance of a full-duplex decode-and-forward (DF) system over the Rician distribution by analyzing the exact analytical expression of the outage probability (OP). The authors in [10] proposed the η−μ fading model and derived precise expressions of level crossing rate (LCR) and average fading duration (AFD). Badarneh et al. in [11] obtained precise expressions for the average bit error rate (ABER) and average channel capacity (ACC) of dual α−μ distribution to explored the performance of selective combining receivers. In [12], the outage performance of relay-assisted vehicular communication under double-Rayleigh fading was investigated. The exact expressions of probability density function (PDF), LCR, and AFD of double Hoyt channel were derived in [13]. Ata in [14] analyzed the PLS of cascaded Rayleigh fading channels by solving the exact precise expressions of SOP. The authors of [15] studied the security performance of the DF relay system under the κ−μ shadowed fading by deducing the exact accurate expressions of SOP and SPSC. In [16], the security performance of single-input multiple-output (SIMO) system under composite distribution was studied by deriving the closed expression of ACC.The access of a large number of wireless devices will lead to a sharp shortage of spectrum resources, and the emergence of CRNs can effectively solve the problem. The CRNs improve the utilization of wireless spectrum resources mainly by using spectrum sensing technology [17] [18]. The combination of CRNs and other technologies has been extensively studied by scholars [19–27]. Goldsmith et al. in [19] reviewed the basic capacity constraints of different wireless network design patterns based on cognitive radios. In [20], the performance of CRNs was discussed when there are multiple groups of cognitive radio relays. The authors of [21] proposed a new scheme for power allocation in CRNs with limited cross-layer interference. Zhang et al. in [22] investigated the effect of the relay selection scheme on the outage performance of CRNs by studying the exact expression of OP. Tashman and Hamouda in [23] analyzed the security performance of SIMO cognitive networks over κ−μ distribution by deriving the closed expression of SOP. In [24], the transmission performance of SIMO underlay CRNs over generalized K channels was studied. The authors in [25] explored the PLS of CRNs under cascaded Rayleigh channels. By analyzing the accurate expression of SOP, the authors of [26] studied the security performance when there are multiple master control units and eavesdroppers in the network. The optimal PLS scheme for users in underlay multiple-input multiple-output (MIMO) CRN was studied in [27].Exact modeling based on wireless fading channels is important for simulating specific practical scenarios and analyzing their security performance. Different fading channels have their respective characteristics, so the channels can simulate different scenarios. For instance, the Nakagami-m distribution can vary m to control the number of clusters [28]. And the Rician distribution can simulate scenarios when line-of-sight (LOS) or deterministic components are present in the received signal [29]. Combining the advantages of the above two distributions, Beaulieu and Xie proposed a new Beaulieu-Xie (BX) fading channels [30], which can represent both specular and diffuse scattering components, that is, it can simulate both LOS and non-LOS (NLOS) scenarios [31]. Therefore, the BX fading is very suitable for modeling and researching femtocells [32] and high-speed trains [33]. The accurate formulas of the LCR and AFD for BX distribution were deduced in [34]. In [35], the closed expressions for the asymptotic upper and lower bounds of the OP and the error probability of the BX fading were deduced. The theoretical expression of the OP for two BX random variables was investigated in [36]. The authors of [37] explored precise and progressive expressions for the effective rate of MIMO networks over BX distribution. Singh and Joshi in [38] derived exact expressions for the ABER of SIMO and MIMO orthogonal frequency division multiplexing systems under BX fading.As far as the author’s knowledge, there are no relevant literatures to study the security analysis based on CRNs under BX fading channels in the published database. Therefore, this work focuses on deriving the exact closed expressions of SOP and SPSC based on CRNs over the BX fading channels. On this basis, the PLS of CRN is analyzed emphatically. Moreover, the influences of the channel parameters and the maximum interference power on the security of the considering model are both studied. Finally, the correctness of the derivations in this paper is verified by simulation analysis.The organization of this paper is as follows: In this section, we mainly introduced the research background, current situation and contributions. In Section2, the system model based on the underlay CRN and the accurate expression of the PDF and cumulative distribution function (CDF) of the signal-to-noise ratio (SNR) of the BX distribution were described. Section 6 obtained the precise expression of SOP. The exact expression of SPSC was derived in Section 4. Section 5 analyzed the numerical simulation results. Section 6 summarized the paper.


## 2. System Model and Channel Characteristics

### 2.1. System Model

As shown in Figure1. The underlay CRN model includes the primary network and the secondary network, which can share frequency band resources, but the premise is that the normal communication of the primary network is not disturbed. We assume that the primary network has one transmission source (P) and N users. The secondary network consists of a signal transmitting source (S), a target receiver (D), and an eavesdropper (E), and all nodes are equipped with an antenna. The channel coefficients of S⟶P, S⟶D, and S⟶E are represented as hSP, hD, and hE, respectively. S can transmit information to P, D, and E, and the transmitted signal at S is x; then the information received at P, D, and E can be written as
(1)ySP=PShspx+nSP,(2)ySD=PShDx+nD,(3)ySE=PShEx+nE,where PS represents the transmit power of S. ySP, ySD, and ySE are the signals received at P, D, and E, respectively. ni∼CN0,σ2i=S,P,D,E are represented as additive white Gaussian noise.Figure 1
The system model.


### 2.2. Statistical Properties of the BX Distribution

In this section, the expressions of the PDF and CDF for the BX distribution are given, and to make the calculation easier and more convenient, we have performed some mathematical operations on them. In [38] (eq. (3)), the PDF of the SNR for the BX channels is written as(4)fγ=2λm−1mΩm+1γm/2−1/2e−λ2/2e−mγ/ΩIm−12mγΩλ,where m and λ reflect the shape and extension, respectively, and Ω is the average SNR. Ix⋅ is the Bessel function [39] (eq. (8.401)).According to the formula in [39] (eq. (8.445)) of(5)Iαz=∑j=0∞1j!Γα+j+1z2α+2j,(4) can be rewritten as
(6)fγ=2λm−1mΩm+1γm/2−1/2e−λ2/2e−mγ/Ω∑k=0∞1k!Γm+k2/m/Ω1/2γ1/2λ2m+2k−1=2λm−1mΩm+1e−λ2/2∑k=0∞1k!Γm+k2/m/Ω1/2λ2m+2k−1γm+k−1e−mγ/Ω=e−λ2/2∑k=0∞2−kmm+kΩ−m+kλ2kk!Γm+kγm+k−1e−m/Ωγ.Utilizing [39] (eq. (3.351.3)), the CDF of SNR is expressed as(7)Fγ=∫0γfxdx=e−λ2/2∑l=0∞λ/22ll!Γl+ml+m−1!1−e−mγ/Ω∑l1=0l+m−1m/Ωl1γl1l1!.


## 2.1. System Model

As shown in Figure1. The underlay CRN model includes the primary network and the secondary network, which can share frequency band resources, but the premise is that the normal communication of the primary network is not disturbed. We assume that the primary network has one transmission source (P) and N users. The secondary network consists of a signal transmitting source (S), a target receiver (D), and an eavesdropper (E), and all nodes are equipped with an antenna. The channel coefficients of S⟶P, S⟶D, and S⟶E are represented as hSP, hD, and hE, respectively. S can transmit information to P, D, and E, and the transmitted signal at S is x; then the information received at P, D, and E can be written as
(1)ySP=PShspx+nSP,(2)ySD=PShDx+nD,(3)ySE=PShEx+nE,where PS represents the transmit power of S. ySP, ySD, and ySE are the signals received at P, D, and E, respectively. ni∼CN0,σ2i=S,P,D,E are represented as additive white Gaussian noise.Figure 1
The system model.


## 2.2. Statistical Properties of the BX Distribution

In this section, the expressions of the PDF and CDF for the BX distribution are given, and to make the calculation easier and more convenient, we have performed some mathematical operations on them. In [38] (eq. (3)), the PDF of the SNR for the BX channels is written as(4)fγ=2λm−1mΩm+1γm/2−1/2e−λ2/2e−mγ/ΩIm−12mγΩλ,where m and λ reflect the shape and extension, respectively, and Ω is the average SNR. Ix⋅ is the Bessel function [39] (eq. (8.401)).According to the formula in [39] (eq. (8.445)) of(5)Iαz=∑j=0∞1j!Γα+j+1z2α+2j,(4) can be rewritten as
(6)fγ=2λm−1mΩm+1γm/2−1/2e−λ2/2e−mγ/Ω∑k=0∞1k!Γm+k2/m/Ω1/2γ1/2λ2m+2k−1=2λm−1mΩm+1e−λ2/2∑k=0∞1k!Γm+k2/m/Ω1/2λ2m+2k−1γm+k−1e−mγ/Ω=e−λ2/2∑k=0∞2−kmm+kΩ−m+kλ2kk!Γm+kγm+k−1e−m/Ωγ.Utilizing [39] (eq. (3.351.3)), the CDF of SNR is expressed as(7)Fγ=∫0γfxdx=e−λ2/2∑l=0∞λ/22ll!Γl+ml+m−1!1−e−mγ/Ω∑l1=0l+m−1m/Ωl1γl1l1!.


## 3. SOP Analysis

We utilize SOP as a performance evaluation metric to explore the security. SOP is defined as the probability that the value of secrecy capacity is less than the specified threshold value [40], which can be expressed as
(8)SOP=PrCS≤Cth,where CS=CD−CE, CD, and CE denote the instantaneous channel capacity of link (S⟶D) and the link (S⟶E), respectively, and Cth represents the specified threshold. The primary and the secondary network in the underlay CRNs must satisfy the following conditions [41]: (i) The maximum transmission power (Pmax) cannot be less than the transmission power of S. (ii) The maximum interference power (Ip) allowed by the main network cannot be less than the power from S to P. Therefore, the SOP in underlay CRNs is written as
(9)SOP=PrCs≤Cth=PrCs≤Cth,X≤IpPmax+PrCs≤Cth,X≥IpPmax=Φ1+Φ2,where X represents the instantaneous SNR of S⟶Ρ. In the next subsection, we will evaluate the closed-form expressions of Φ1 and Φ2 in (9).


### 3.1. Representation ofΦ1

According to the above analysis,Φ1 can be converted to
(10)Φ1=PrCs≤Cth,X≤IpPmax=PγD≤θγE+θ−1υPX≤IpPmax=I1I2,where I1=PγD≤θγE+θ−1/υ, I2=PX≤Ip/Pmax, θ=eCth, and υ=Pmax/σ2.Using (6) and (7), I1 can be obtained as
(11)I1=∫0∞FDθγE+θ−1υfEγEdγE=∫0∞e−λD2/2∑l=0∞λD/22ll!Γl+mDl+mD−1!1−e−mDθγE+θ−1/υ/ΩD∑l1=0l+mD−1mD/ΩDl1θγE+θ−1/υl1l1!×e−λE2/2∑k=0∞2−kmEmE+kΩE−mE+kλE2kk!ΓmE+kγEmE+k−1e−mE/ΩEγEdγE.Utilizing an equation in [39] (eq. (1.111)), the exact analytical solution forI1 can be represented as
(12)I1=e−λE2/2e−λD2/2∑k=0∞2−kmEmE+kΩE−mE+kλE2kk!ΓmE+k∑l=0∞λD/22ll!Γl+mDl+mD−1!×∫0∞γEmE+k−1e−mE/ΩEγEdγE−e−mDθ−1/υ/ΩD∑l1=0l+mD−1mD/ΩDl1l1!×∑t1=0l1t1l1θt1θ−1υl1−t1∫0∞e−mDθ/ΩD+mE/ΩEγEγEmE+k+t1−1dγE=e−λE2/2e−λD2/2∑k=0∞2−kmEmE+kΩE−mE+kλE2kk!ΓmE+k∑l=0∞λD/22ll!Γl+mDl+mD−1!×ΓmE+kmE/ΩEmE+k−e−mDθ−1/υ/ΩD∑l1=0l+mD−1mD/ΩDl1∑t1=0l1t1l1θt1θ−1/υl1−t1l1!ΓmE+k+t1mDθ/ΩD+mE/ΩEmE+k+t1.Using the similar method as solvingI1, I2 is obtained as
(13)I2=PrX≤IpPmax=FpIpPmax=e−λp2/2∑l=0∞λp/22ll!Γl+mpl+mp−1!1−e−mpIp/Pmax/Ωp∑l1=0l+mp−1mp/Ωpl1Ip/Pmaxl1l1!.


### 3.2. Representation ofΦ2

According to (9) and [42] (eq. (13)), the expressions for Φ2 can be rewritten as
(14)Φ2=PrCs≤Cth,X≥IpPmax=∫Ip/Pmax∞Gxfsxdx,where Gx=∫0∞FDθy+θ−1x/φfEydy, φ=Ip/σ2.Using (6) and (7) and an equation in [39] (eq. (3.381.4)), the exact Gx is expressed as
(15)Gx=∫0∞FDθy+θ−1xφfEydy=∫0∞e−λD2/2∑l=0∞λD/22ll!Γl+mDl+mD−1!×1−e−mDθ−1x/φ/ΩD∑l1=0l+mD−1mD/ΩDl1∑t1=0l1t1l1θt1θ−1/φl1−t1xl1−t1l1!yt1e−θmDy/ΩD×e−λE2/2∑k=0∞2−kmEmE+kΩE−mE+kλE2kk!ΓmE+kymE+k−1e−mE/ΩEydy=e−λD2/2∑l=0∞λD/22ll!Γl+mDl+mD−1!e−λE2/2∑k=0∞2−kmEmE+kΩE−mE+kλE2kk!ΓmE+k×∫0∞ymE+k−1e−mE/ΩEy−e−mDθ−1x/φ/ΩD∑l1=0l+mD−1mD/ΩDl1l1!×∑t1=0l1t1l1θt1θ−1φl1−t1xl1−t1ymE+k+t1−1e−mE/ΩE+θmD/ΩDydy=B1B2B3−B4xl1−t1e−mDθ−1x/φ/ΩD,where(16)B1=e−λD2/2∑l=0∞λD/22ll!Γl+mDl+mD−1!,(17)B2=e−λE2/2∑k=0∞2−kmEmE+kΩE−mE+kλE2kk!ΓmE+k,(18)B3=ΓmE+kmE/ΩEmE+k,(19)B4=∑l1=0l+mD−1mD/ΩDl1∑t1=0l1t1l1θt1θ−1/φl1−t1l1!ΓmE+k+t1mE/ΩE+θmD/ΩDmE+k+t1.Substituting (15) into (14) and using [39] (eq. (3.351.2)), Φ2 can be represented as
(20)Φ2=∫Ip/Pmax∞Gxfsxdx=∫Ip/Pmax∞B1B2B3B5xmE+k−1e−mE/ΩEx−B4B5xl1−t1+mE+k−1e−mDθ−1/φΩD+mEΩExdx=B1B2B3B5mEΩE−mE+kΓmE+k,IpPmaxmEΩE−B4B5mDθ−1/φΩD+mEΩE−l1−t1+mE+kΓl1−t1+mE+k,mDθ−1/φΩD+mEΩEIpPmax,where(21)B5=e−λE2/2∑k=0∞2−kmEmE+kΩE−mE+kλE2kk!ΓmE+k.


## 3.1. Representation ofΦ1

According to the above analysis,Φ1 can be converted to
(10)Φ1=PrCs≤Cth,X≤IpPmax=PγD≤θγE+θ−1υPX≤IpPmax=I1I2,where I1=PγD≤θγE+θ−1/υ, I2=PX≤Ip/Pmax, θ=eCth, and υ=Pmax/σ2.Using (6) and (7), I1 can be obtained as
(11)I1=∫0∞FDθγE+θ−1υfEγEdγE=∫0∞e−λD2/2∑l=0∞λD/22ll!Γl+mDl+mD−1!1−e−mDθγE+θ−1/υ/ΩD∑l1=0l+mD−1mD/ΩDl1θγE+θ−1/υl1l1!×e−λE2/2∑k=0∞2−kmEmE+kΩE−mE+kλE2kk!ΓmE+kγEmE+k−1e−mE/ΩEγEdγE.Utilizing an equation in [39] (eq. (1.111)), the exact analytical solution forI1 can be represented as
(12)I1=e−λE2/2e−λD2/2∑k=0∞2−kmEmE+kΩE−mE+kλE2kk!ΓmE+k∑l=0∞λD/22ll!Γl+mDl+mD−1!×∫0∞γEmE+k−1e−mE/ΩEγEdγE−e−mDθ−1/υ/ΩD∑l1=0l+mD−1mD/ΩDl1l1!×∑t1=0l1t1l1θt1θ−1υl1−t1∫0∞e−mDθ/ΩD+mE/ΩEγEγEmE+k+t1−1dγE=e−λE2/2e−λD2/2∑k=0∞2−kmEmE+kΩE−mE+kλE2kk!ΓmE+k∑l=0∞λD/22ll!Γl+mDl+mD−1!×ΓmE+kmE/ΩEmE+k−e−mDθ−1/υ/ΩD∑l1=0l+mD−1mD/ΩDl1∑t1=0l1t1l1θt1θ−1/υl1−t1l1!ΓmE+k+t1mDθ/ΩD+mE/ΩEmE+k+t1.Using the similar method as solvingI1, I2 is obtained as
(13)I2=PrX≤IpPmax=FpIpPmax=e−λp2/2∑l=0∞λp/22ll!Γl+mpl+mp−1!1−e−mpIp/Pmax/Ωp∑l1=0l+mp−1mp/Ωpl1Ip/Pmaxl1l1!.


## 3.2. Representation ofΦ2

According to (9) and [42] (eq. (13)), the expressions for Φ2 can be rewritten as
(14)Φ2=PrCs≤Cth,X≥IpPmax=∫Ip/Pmax∞Gxfsxdx,where Gx=∫0∞FDθy+θ−1x/φfEydy, φ=Ip/σ2.Using (6) and (7) and an equation in [39] (eq. (3.381.4)), the exact Gx is expressed as
(15)Gx=∫0∞FDθy+θ−1xφfEydy=∫0∞e−λD2/2∑l=0∞λD/22ll!Γl+mDl+mD−1!×1−e−mDθ−1x/φ/ΩD∑l1=0l+mD−1mD/ΩDl1∑t1=0l1t1l1θt1θ−1/φl1−t1xl1−t1l1!yt1e−θmDy/ΩD×e−λE2/2∑k=0∞2−kmEmE+kΩE−mE+kλE2kk!ΓmE+kymE+k−1e−mE/ΩEydy=e−λD2/2∑l=0∞λD/22ll!Γl+mDl+mD−1!e−λE2/2∑k=0∞2−kmEmE+kΩE−mE+kλE2kk!ΓmE+k×∫0∞ymE+k−1e−mE/ΩEy−e−mDθ−1x/φ/ΩD∑l1=0l+mD−1mD/ΩDl1l1!×∑t1=0l1t1l1θt1θ−1φl1−t1xl1−t1ymE+k+t1−1e−mE/ΩE+θmD/ΩDydy=B1B2B3−B4xl1−t1e−mDθ−1x/φ/ΩD,where(16)B1=e−λD2/2∑l=0∞λD/22ll!Γl+mDl+mD−1!,(17)B2=e−λE2/2∑k=0∞2−kmEmE+kΩE−mE+kλE2kk!ΓmE+k,(18)B3=ΓmE+kmE/ΩEmE+k,(19)B4=∑l1=0l+mD−1mD/ΩDl1∑t1=0l1t1l1θt1θ−1/φl1−t1l1!ΓmE+k+t1mE/ΩE+θmD/ΩDmE+k+t1.Substituting (15) into (14) and using [39] (eq. (3.351.2)), Φ2 can be represented as
(20)Φ2=∫Ip/Pmax∞Gxfsxdx=∫Ip/Pmax∞B1B2B3B5xmE+k−1e−mE/ΩEx−B4B5xl1−t1+mE+k−1e−mDθ−1/φΩD+mEΩExdx=B1B2B3B5mEΩE−mE+kΓmE+k,IpPmaxmEΩE−B4B5mDθ−1/φΩD+mEΩE−l1−t1+mE+kΓl1−t1+mE+k,mDθ−1/φΩD+mEΩEIpPmax,where(21)B5=e−λE2/2∑k=0∞2−kmEmE+kΩE−mE+kλE2kk!ΓmE+k.


## 4. SPSC Analysis

For the communication network security, SPSC is also one of the important evaluation indicators. SPSC is described as the probability that the secrecy capacity is greater than zero, which can be expressed as [40]
(22)SPSC=PCD−CE>0=1−∫0∞FDγEfEγEdγE=1−Ψ.Substituting (6) and (7) into Equation (22), we can get the expression of Ψ as
(23)Ψ=PγD≤γE=∫0∞FDγEfEγEdγE=∫0∞e−λD2/2∑l=0∞λD/22lΓl+1Γl+mDl+mD−1!1−e−mDγE/ΩD∑l1=0l+mD−1mD/ΩDl1γEl1l1!×e−λE2/2∑k=0∞2−kmEmE+kΩE−mE+kλE2kk!ΓmE+kγEmE+k−1e−mE/ΩEγEdγE=e−λE2/2∑k=0∞2−kmEmE+kΩE−mE+kλE2kk!ΓmE+ke−λD2/2∑l=0∞λD/22lΓl+1Γl+mDl+mD−1!×ΓmE+kmE/ΩEmE+k−∑l1=0l+mD−1mD/ΩDl1l1!ΓmE+k+l1mE/ΩE+mD/ΩDmE+k+l1.Combined with (22) and (23), the exact accurate analytical formula of SPSC can be obtained. In (23), with the increase of λ, λ2k and λ/22l will increase and lead to the improvement of SPSC; however, e−λ2/2 will increase and lead to the decline of SPSC.


## 5. Numerical Analysis

In this section, the simulation results of the inferences are given, and their correctness will be verified by statistical simulations. Furthermore, we analyze the performance through the simulation curves. We implement the random number generator of BX distribution by using the acceptance rejection method. It is worth noting that the SOP and SPSC contain infinite series; however, when the number of loops reaches more than 40 times, the derived formulas for SOP and SPSC will converge to a fixed value.Figure2 presents the variation of SOP under the conditions of different ΩΩD=ΩE=4,2,1. In all simulations, we set λD=λE=1, mD=mE=1, Ip=10dB, and Cth=0.1dB. The theoretical curves overlap well with the Monte Carlo simulations. ΩD is the average SNR of the main channel, the larger ΩD means that the better transmission quality of the signal. In this situation, the SOP should be smaller. In addition, the value of SOP decreases as Pmax rises, which shows that increasing the value of Pmax can enhance the confidentiality. And by increasing the ΩD, the SOP decreases and the security is improved.Figure 2
SOP versusPmax for different ΩD.Figure3 illustrates the change of SOP with the average SNR under different Cth. The parameter settings are as follows: λD=λE=1, mD=mE=1, Ip=10dB, Pmax=1dB, and ΩE=1dB. Figure 3 shows that as ΩD rises, SOP gradually decreases, indicating that increasing the average SNR of the main channel when the SNR of the eavesdropping channel is constant can enhance security. And with the rise of the set threshold Cth, the security will be weakened.Figure 3
SOP versusΩD for different Cth.Figure4 describes the variation of SOP with the average SNR and λD=λE=3,2,1. Its parameters are set to mD=mE=1, Ip=5dB, Pmax=1dB, ΩE=1dB, and Cth=0.1dB. This figure depicts the effect of the parameter λ of the BX fading on the security, Figure 4 is based on the Equations (9), (10), (12), (13), and (20). We can see that the value of λ has a large transformation on the system performance at ΩD>5dB. In other words, the rise of λ will greatly enhance the security under the condition of high SNR.Figure 4
SOP versusΩD for different λ.Figure5 plots the change of SPSC with ΩD and λ. The parameters are mD=mE=2, Cth=0.1dB, Ip=0.1dB, Pmax=1dB, and ΩE=2dB. It can be seen that the change of λ has different effects on security under high and low SNR. When ΩD<2dB, the value of SPSC reduces as λ increases. That is, at low SNR, the increase of λ weakens the security. Analysis of Figures 4 and 5 shows that under high SNR, with the rise of λ, the value of SOP decreases rapidly, and the value of SPSC gradually increases. These conclusions collectively demonstrate that the increase of λ will enhance confidentiality.Figure 5
SPSC versusΩD for different λ.Figure6 depicts the variation of SOP with m and ΩD. The remaining parameters are set as follows: λD=λE=1, ΩE=1dB, Ip=5dB, Pmax=1dB, and Cth=0.1dB. From Figure 6, the value of SOP reduces rapidly as the m rises for ΩD>6dB, which indicates that increasing m can greatly enhance the security, and when ΩD<6dB, the change of m has little effect on the confidentiality.Figure 6
SOP versusΩD for different m.Figure7 shows the variation of SPSC under different m with the average SNR. In Figure 7 we set λD=λE=2, Cth=0.1dB, ΩE=2dB, Ip=0.1dB, and Pmax=1dB. The simulation curves are drawn based on (22) and (23). Furthermore, it can be observed that at low SNR, as m increases, the value of SPSC reduces, which proved that the confidentiality is weakened. When ΩE>3dB, as m adds, the security of the system is improved. This is consistent with the results obtained in Figure 6.Figure 7
SPSC versusΩD for different m.


## 6. Conclusion

In this paper, we investigate the PLS of the CRNs over BX fading channels. The accurate expressions of SOP and SPSC are obtained. Then the impact of the parameters for the BX fading channels which inherits the advantages of both Nakagami-m and Rician fading channels on the security of the CRN is analyzed. Finally, we compared the theoretical derivation with Monte Carlo simulation; the coincidence of simulation curves proves the correctness of the theoretical formulas. And interestingly, rising Pmax, increasing ΩD, decreasing Cth, and increasing the values of λ and m under high SNR all promote the confidentiality.



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*Source: 6785754-2022-08-08.xml*