# Security Performance Analysis of Relay Networks Based onκ-μ Shadowed Channels with RHIs and CEEs

Authors: Jiangfeng Sun; Xiaohong Wang; Yiwei Fang; Xinji Tian; Mingfu Zhu; Jiangtao Ou; Chengyuan Fan
Journal: Wireless Communications and Mobile Computing
 (2022)

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## Abstract

This paper mainly describes the transmission reliability and physical layer security of multiantenna decode-and-forward relay networks over theκ-μ shadowed fading distribution. We consider the existence of residual hardware impairments (RHIs) and channel estimation errors (CEEs) at the same time. Firstly, we give the network model. On this basis, we analyze the reliability and security by deriving the exact expressions of outage probability (OP) and intercept probability (IP). To further verify and investigate the theoretical derivation, the approximate analysis of OP is obtained under the high signal-to-noise ratio. In addition, we also use an optimized antenna selection technology at the transmitter to improve the security performance. Finally, through the comparative study of theoretical simulation and Monte Carlo simulation results, the factors affecting network transmission and security are acquired. The interesting conclusion is that both RHIs and CEEs can increase OP and decrease IP; that is to say, the existence of RHIs and CEEs can reduce reliability and enhance security.


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## Body

## 1. Introduction

With the widespread application of wireless communication technology, physical layer security (PLS) technology is highly concerned by scholars because it can take advantage of channel fading to improve antieavesdropping performance. The PLS was first proposed in [1]. Then, Yang et al. in [2] described that one of the dominating troubles faced by the 5G network is how to ensure the security of wireless data transmission. PLS have been generally studied in different system models, such as cooperative nonorthogonal multiple access (NOMA) systems [3], green communication systems based on relay selection [4], and multihop relay systems [5]. Moreover, Zhu and Yao in [6] also designed a cooperative beamforming scheme to enhance the PLS of cognitive radio networks. The authors of [7] investigated the reliability of quasistatic Rayleigh fading by deriving its outage probability (OP). The authors in [8] proposed a PLS project based on channel precompensation to improve the anti-interference of the polarization shift keying modulation system. Fan et al. in [9] used the relay selection to improve PLS when there are multiple users and multiple eavesdroppers.In the actual communication process, interference from obstacles or other factors will be encountered inevitably, and the phenomenon that causes the power of the received signals to change with time and position is called channel fading. On account of the rapid development of communication networks, the researches on the transmission and security performances of various models under fading channels have been more and more favored by scholars. The authors in [10] studied the security of the fading channels when there was an eavesdropper in the system model. The authors of [11] explored the channel capacity of multiple-input multiple-output (MIMO) systems under Rayleigh fading. Zhu et al. in [12] considered the full-duplex decode-and-forward (DF) system in the Rayleigh fading and gave the upper and lower bound expressions of its OP. New closed-form expressions of the strictly positive secrecy capacity (SPSC) and secure outage probability (SOP) over κ-μ fading channels were given in [13]. Badarneh et al. deduced the precise expression of the nth moment of the dual α-μ fading in [14]. And the performance was researched in terms of channel quality estimation index and average channel capacity (ACC). By deriving the precise expressions of average duration of fades (ADF) and probability density function (PDF) for dual Hoyt fading in [15], the authors investigated the statistical characteristics of the fading model of the narrow-band amplify and forward (AF) in this fading environment. In [16], the OP of double shadowed κ-μ fading was analyzed, and the effect of double shadowed on wireless fading channels was discussed. The authors of [17] derived the accurate expressions of PDF and cumulative distribution function (CDF) of amplitude and signal-to-noise ratio (SNR) of multicascade κ-μ fading channels and obtained the closed-form expressions of OP, average symbol error probability (ASEP), and ACC.In wireless communication networks, cooperative relay technology can improve system reliability, energy efficiency, and system coverage, which has been studied extensively [18–24]. The authors in [18] researched the performance of the double-hop relay system. The application of a full-duplex AF relay system in vehicle-to-vehicle (V2V) was investigated in [19]. Majhi et al. of [20] deduced an accurate expression of the OP of orthogonal frequency division multiplexing (OFDM) relay system. The authors in [21] considered the selection of relay antenna in the V2V communication model using both physical-layer network coding and AF schemes in dual Nakagami-m channels. The OP of the two-way relay DF system under the packet fading channels was investigated in [22]. The authors in [23] studied the influence of the relay selection scheme and the number of relays on the outage performance of cognitive radio networks (CRN) by analyzing the OP. The authors of [24] analyzed the security of relay systems using AF under generalized-K fading.Due to the time-varying characteristics of the communication state, radio frequency devices will be affected by many factors, such as phase noise and power amplifier nonlinearity. These irreparable hardware impairments (HIs) will distort the signals in transmission, so it is necessary to consider the HIs in a communication system. In addition to the above-mentioned HIs that will threaten the accurate transmission of the signal, the existence of CEEs is also one of the reasons for the mismatch between the expected signal and the actual transmission signal. To make the research more practical, many researchers have considered the influence of transceiver HIs and CEEs on the system during modeling and analysis [25–31], such as cognitive amplification and forwarding multirelay networks [25], beyond fifth-generation Internet of Things collaborative NOMA relay system [26], and ambient backscatter NOMA systems [27]. The authors of [28] described the performance of multiple relay systems when there were both transceiver HIs and CEEs. The authors in [29] studied the influence of RHIs and CEEs on cooperative NOMA systems by analyzing OP and the energy efficiency. The influences of the MIMO system with CEEs, RHIs, and imperfect successive interference cancellation effects on Nakagami-m fading were investigated in [30]. Li et al. of [31] researched the reliability and security performance of cooperative multirelay systems with RHIs and CEEs by deriving and analyzing the accurate expressions of OP and IP.Through studying the existing fading channels, we found that the newly emergingκ-μ shadowed fading can characterize many fading channels by changing its parameters, such as Rice, Nakagami-m, Rayleigh, and κ-μ. The κ-μ shadowed fading was first proposed in [32]. Due to its composite characteristics, the channels can be applied to many scenarios. As shown in [33], a secure multicast scheme under κ-μ shadowed fading with multiple users and multiple eavesdroppers was studied. Sun et al. in [34] investigated the security performance of the single-input multiple-output relay communication architecture over the κ-μ shadowed distribution by deriving the precise and approximate expressions of SOP and SPSC. The closed-form expressions of the OP and the ASEP of the two-hop multiantenna wireless transmission system based on the AF relay node of the κ-μ shadowed fading system were deduced in [35]. Illi et al. in [36] researched the security of a dual-hop underwater communication system based on the κ-μ shadowed architecture. The authors in [37] studied the performance of the dual-hop autofocus communication network over the κ-μ shadowed fading.References [10–17] mainly described the research of scholars on wireless fading channel in recent years, [18–24] were aimed at elaborating a widely used cooperative relay technology, [25–31] presented the performance of the system when scholars consider the existence of CEEs and HIs in different models in recent years, and [32–37] described the relevant researches on κ-μ shadowed fading channel in recent years. Synthesizing the above researches, to the author’s knowledge, no scholars have studied the effects of RHIs and CEEs with DF under κ-μ shadowed fading. Therefore, we develop this work. First of all, we consider an eavesdropping model with DF relay and study the influence of CEEs and RHIs on the transmission performance and anti-interference ability of the network over κ-μ shadowed fading channels. In addition, to further verify and analyze the theoretical derivation, the high SNR approximate analysis of OP is obtained. Moreover, we also compared the ideal and nonideal situationsThe rest of this article is organized as follows. In Section2, the statistical characteristics of the system model and the κ-μ shadowed fading are given, and the expressions of each parameter are analyzed when the RHIs and CEEs are considered simultaneously. In Section 3, the antenna selection scheme at the transmitter is given first; then, the precise expressions of OP and IP under ideal and nonideal situations are described. In Section 4, the approximate expressions of high SNR of OP under fixed value and varying function of channel estimation variance are derived. In Section 5, the numerical simulation results are discussed in detail. Section 6 summarizes the article, and the next section is the appendix.


## 2. System Model and Statistical Features

### 2.1. System Model

In Figure1, we consider the classic Wyner wiretapping model. The model includes a transmitter (B), a relay (H), a receiver user (D), and an eavesdropper (E). The B is equipped with L antennas B1,B1,⋯,Bl,⋯,BL, H and legitimate D have an antenna, and E has Q antennas E1,E1,⋯,Eq,⋯,EQ. On account of the existence of the shadowed fading, all signals from the B can only be transmitted to H and cannot be directly sent to D or E. In addition, the information transmission mode at the relay is assumed to be DF.Figure 1
The system model.As can be seen, the communication system of this model is divided into two time slots: (i)B sends the original signals to H; (ii) the H decodes the signals and forwards them to D. To improve the antieavesdropping ability, we have added artificial noise to the model. Since it is difficult to acquire the channel state information (CSI) of all nodes, the effective way to solve this problem is channel estimation which can be divided into three types, including reference signal-based estimation [38], blind estimation [39], and semiblind estimation [40]. The method used in this paper belongs to the first category that uses training sequences to estimate the CSI. According to the method in reference [41], the actual communication channel is expressed as
(1)hXY=h^XY+eXY.In (1), XY∈BlH;HD;HEq,1≤l≤L,1≤q≤Q, hXY is the channel coefficient, and h^XY denotes the channel estimation of hXY. eXY~CN0,σeXY2 represents the CEEs and σeXY2 denotes the estimated variance. We assume two typical conditions: (i) σeXY2=M, where M is a nonnegative value; (ii) σeXY2=Ω/1+δρΩ, where Ω, δ>0, and ρ represent the variance of the channel gain, the quality parameter of the CEEs, and the variance of the average SNR, respectively.


### 2.2. The First Transfer Process

During this course, the signals fromB to H are represented as xBlH, and ExBlH2=1. We considered both RHIs and defective CSI, so the information at the H can be described as
(2)yBlH=h^BlH+eBlHPBxBlH+ηt,BlH+ηr,BlH+νBlH,where h^BlH represents the channel from the antenna chosen by the B to H. The total transmit power is P, and τ is employed to represent the selected antenna power allocation factor, and 1−τ indicates the power allocation factor of the unchecked antenna. Here, we employ PB to indicate the power of the B, then PB=τP. xBlH is the valid signal from B to H, ExBlH2=1, and νBlH~CN0,σBLH2 is the additive white Gaussian noise (AWGN). ηt,BlH and ηr,BlH are, respectively, expressed as the distortion noise caused by RHIs at the transmitting and the receiving terminal.


### 2.3. The Second Transfer Process

AtH, the relay forwards the decoding of the signal to D and E. To improve the antieavesdropping ability, when H transmits information to D and E, the influence of artificial noise is also considered. In the actual communication system, due to the existence of CEEs at D, there will be some artificial noise and residual interference [42–44]. In summary, the information at D and Eq can be written as
(3)yHD=h^HD+eHDPHxHD+ηt,HD+ηr,HD+νHD,(4)yHEq=h^HEq+eHEqPHxHEq+ηt,HEq+ηr,HEq+νHEq,where xHD and xHEq represent the actual signals transmitted to D and Eq with ExHD2=ExHEq2=1. xJHD and xJHEq are, respectively, denoted as the signals sent to D and Eq with artificial noise. And ExJHD2=ExJHEq2=1. ξ1∈0,1 and ξ2∈0,1 separately describe the factitious noise quantization coefficients at D and Eq. νHD and νHEq, respectively, represent the composite AWGN with mean value 0 and variance σHD2 and σHEq2. PS and PJ represent the transmit power of relay and factitious noise and PB=PH=Pg. ηt,HD and ηr,HD are, respectively, expressed as the distortion noise caused by RHIs during H to D communication. ηt,HEq and ηr,HEq are, respectively, expressed as the distortion noise caused by RHIs during H to Eq communication. As mentioned in [45], the distortion noise can be written as
(5)ηt~CN0,ft2Pg,ηr~CN0,fr2Pgh2.The distortion noise was regarded as the product ofηt, ηr, and h. For an achievable channel h, the composite distortion power at the receiving terminal can be described as
(6)Eηt,ηr,hηt+ηr2=Pgh2ft2+fr2=Pgh∧+e2ft2+fr2.From (6), we can get that power only rests with the pg and the h∧XY2. Where fXY=ft,XY2+fr,XY2, the messages at H, D, and Eq were described as
(7)yBlH=h^BlH+eBlHPBxBlH+ηBlH+νBlH,(8)yHD=h^HD+eHDPHxHD+ξ1PJxJHD+ηHD+νHD,(9)yHEq=h^HEq+eHEqPHxHEq+ξ2PJxJHEq+ηHEq+νHEq,where the comprehensive distortion noises from RHIs at Bl⟶H, H⟶D, and H⟶Eq are, respectively, expressed as ηBlH∼CN0,fBlH2PB, ηHD∼CN0,fHD2PH, and ηHEq∼CN0,fHEq2PH. Besides, fBlH=ft,BlH2+fr,BlH2, fHD=ft,HD2+fr,HD2, and fHEq=ft,HEq2+fr,HEq2.The signal-to-interference-plus-noise ratios (SINRs) ofBl⟶H, H⟶D, and H⟶Eq are given as
(10)γBlH=ρBlHh∧BlH2ρBlHσeBlH2+h∧BlH2fBlH2+σeBlH2fBlH2+1,(11)γHD=ρHDh∧HD2ρHDσeHD2+ξ1PJ/σHD2+ρHDfHD2h∧HD2+σeHD2+1,(12)γHEq=ρHEqh∧HEq2ρHEqσeHEq2+ξ2PJ/σHEq2+ρHEqfHEq2h∧HEq2+σeHEq2+1,where ρBlH=PB/σBlH2, ρHD=PH/σHD2, and ρHEq=PH/σHEq2.


### 2.4. Statistical Features

In the model, we consider the widely appliedκ-μ shadowed fading, and the PDF of the channel gain has been given in [46] as
(13)fh∧XY2x=mXYmXY1+κXYμXYμXYμXYΓμXYmXY+κXYμXYmXYψXYexp−x1+κXYμXYψXY×xψXYμXY−1F11mXY;μXY;μXY2κXY1+κXYxψXYmXY+κXYμXY.After some mathematical operations, the CDF expression of the channel gain can be obtained as(14)Fh∧XY2x=mXY+κXYμXYmXY−mXY1ΓμXY∑i=1∞mXYiμXYii!mXYκXYμXYmXY+κXYμXYmXYi×μXY+i−1!1−∑s=0μXY+i−1xss!1+κXYμXYψXYsexp−1+κXYμXYψXYx.The meanings ofκ, m, and μ in (13) and (14) are all channel parameters, where m is the shaping parameter of Nakagami-m fading, and μ>0 is related to the number of received signal groups. κ=d2/2μσ2 is a positive number, d2 represents the total power of the dominating constituent, and 2μσ2 is the total power of the scattered wave. ψ=Εh∧XY2, and h^XY indicates the amplitude of the channel. The capacity formula of the Shannon theorem is defined as
(15)C=12log21+γ.According to (10) and (11), we can get the expression of the instantaneous channel capacity of B⟶H and H⟶D as
(16)CBH=12log21+ρBHh∧BH2ρBHσeBH2+h∧BH2fBH2+σeBH2fBH2+1,(17)CHD=12log21+ρHDh∧HD2ρHDσeHD2+ξ1PJ/σHD2+ρHDfHD2h∧HD2+σeHD2+1.The eavesdropper employs multiple relay combining (MRC) algorithm to intercept legal information; the SINR atH⟶E is
(18)γHE=∑q=1QγHEq.The SINR ofH⟶E is expressed as
(19)γHE=QγHEq.So we can get the instantaneous capacity ofH⟶E as
(20)CHE=12log21+QρHEqh∧HEq2ρHEqσeHEq2+ξ2PJ/σHEq2+ρHEqfHEq2h∧HEq2+σeHEq2+1.


## 2.1. System Model

In Figure1, we consider the classic Wyner wiretapping model. The model includes a transmitter (B), a relay (H), a receiver user (D), and an eavesdropper (E). The B is equipped with L antennas B1,B1,⋯,Bl,⋯,BL, H and legitimate D have an antenna, and E has Q antennas E1,E1,⋯,Eq,⋯,EQ. On account of the existence of the shadowed fading, all signals from the B can only be transmitted to H and cannot be directly sent to D or E. In addition, the information transmission mode at the relay is assumed to be DF.Figure 1
The system model.As can be seen, the communication system of this model is divided into two time slots: (i)B sends the original signals to H; (ii) the H decodes the signals and forwards them to D. To improve the antieavesdropping ability, we have added artificial noise to the model. Since it is difficult to acquire the channel state information (CSI) of all nodes, the effective way to solve this problem is channel estimation which can be divided into three types, including reference signal-based estimation [38], blind estimation [39], and semiblind estimation [40]. The method used in this paper belongs to the first category that uses training sequences to estimate the CSI. According to the method in reference [41], the actual communication channel is expressed as
(1)hXY=h^XY+eXY.In (1), XY∈BlH;HD;HEq,1≤l≤L,1≤q≤Q, hXY is the channel coefficient, and h^XY denotes the channel estimation of hXY. eXY~CN0,σeXY2 represents the CEEs and σeXY2 denotes the estimated variance. We assume two typical conditions: (i) σeXY2=M, where M is a nonnegative value; (ii) σeXY2=Ω/1+δρΩ, where Ω, δ>0, and ρ represent the variance of the channel gain, the quality parameter of the CEEs, and the variance of the average SNR, respectively.


## 2.2. The First Transfer Process

During this course, the signals fromB to H are represented as xBlH, and ExBlH2=1. We considered both RHIs and defective CSI, so the information at the H can be described as
(2)yBlH=h^BlH+eBlHPBxBlH+ηt,BlH+ηr,BlH+νBlH,where h^BlH represents the channel from the antenna chosen by the B to H. The total transmit power is P, and τ is employed to represent the selected antenna power allocation factor, and 1−τ indicates the power allocation factor of the unchecked antenna. Here, we employ PB to indicate the power of the B, then PB=τP. xBlH is the valid signal from B to H, ExBlH2=1, and νBlH~CN0,σBLH2 is the additive white Gaussian noise (AWGN). ηt,BlH and ηr,BlH are, respectively, expressed as the distortion noise caused by RHIs at the transmitting and the receiving terminal.


## 2.3. The Second Transfer Process

AtH, the relay forwards the decoding of the signal to D and E. To improve the antieavesdropping ability, when H transmits information to D and E, the influence of artificial noise is also considered. In the actual communication system, due to the existence of CEEs at D, there will be some artificial noise and residual interference [42–44]. In summary, the information at D and Eq can be written as
(3)yHD=h^HD+eHDPHxHD+ηt,HD+ηr,HD+νHD,(4)yHEq=h^HEq+eHEqPHxHEq+ηt,HEq+ηr,HEq+νHEq,where xHD and xHEq represent the actual signals transmitted to D and Eq with ExHD2=ExHEq2=1. xJHD and xJHEq are, respectively, denoted as the signals sent to D and Eq with artificial noise. And ExJHD2=ExJHEq2=1. ξ1∈0,1 and ξ2∈0,1 separately describe the factitious noise quantization coefficients at D and Eq. νHD and νHEq, respectively, represent the composite AWGN with mean value 0 and variance σHD2 and σHEq2. PS and PJ represent the transmit power of relay and factitious noise and PB=PH=Pg. ηt,HD and ηr,HD are, respectively, expressed as the distortion noise caused by RHIs during H to D communication. ηt,HEq and ηr,HEq are, respectively, expressed as the distortion noise caused by RHIs during H to Eq communication. As mentioned in [45], the distortion noise can be written as
(5)ηt~CN0,ft2Pg,ηr~CN0,fr2Pgh2.The distortion noise was regarded as the product ofηt, ηr, and h. For an achievable channel h, the composite distortion power at the receiving terminal can be described as
(6)Eηt,ηr,hηt+ηr2=Pgh2ft2+fr2=Pgh∧+e2ft2+fr2.From (6), we can get that power only rests with the pg and the h∧XY2. Where fXY=ft,XY2+fr,XY2, the messages at H, D, and Eq were described as
(7)yBlH=h^BlH+eBlHPBxBlH+ηBlH+νBlH,(8)yHD=h^HD+eHDPHxHD+ξ1PJxJHD+ηHD+νHD,(9)yHEq=h^HEq+eHEqPHxHEq+ξ2PJxJHEq+ηHEq+νHEq,where the comprehensive distortion noises from RHIs at Bl⟶H, H⟶D, and H⟶Eq are, respectively, expressed as ηBlH∼CN0,fBlH2PB, ηHD∼CN0,fHD2PH, and ηHEq∼CN0,fHEq2PH. Besides, fBlH=ft,BlH2+fr,BlH2, fHD=ft,HD2+fr,HD2, and fHEq=ft,HEq2+fr,HEq2.The signal-to-interference-plus-noise ratios (SINRs) ofBl⟶H, H⟶D, and H⟶Eq are given as
(10)γBlH=ρBlHh∧BlH2ρBlHσeBlH2+h∧BlH2fBlH2+σeBlH2fBlH2+1,(11)γHD=ρHDh∧HD2ρHDσeHD2+ξ1PJ/σHD2+ρHDfHD2h∧HD2+σeHD2+1,(12)γHEq=ρHEqh∧HEq2ρHEqσeHEq2+ξ2PJ/σHEq2+ρHEqfHEq2h∧HEq2+σeHEq2+1,where ρBlH=PB/σBlH2, ρHD=PH/σHD2, and ρHEq=PH/σHEq2.


## 2.4. Statistical Features

In the model, we consider the widely appliedκ-μ shadowed fading, and the PDF of the channel gain has been given in [46] as
(13)fh∧XY2x=mXYmXY1+κXYμXYμXYμXYΓμXYmXY+κXYμXYmXYψXYexp−x1+κXYμXYψXY×xψXYμXY−1F11mXY;μXY;μXY2κXY1+κXYxψXYmXY+κXYμXY.After some mathematical operations, the CDF expression of the channel gain can be obtained as(14)Fh∧XY2x=mXY+κXYμXYmXY−mXY1ΓμXY∑i=1∞mXYiμXYii!mXYκXYμXYmXY+κXYμXYmXYi×μXY+i−1!1−∑s=0μXY+i−1xss!1+κXYμXYψXYsexp−1+κXYμXYψXYx.The meanings ofκ, m, and μ in (13) and (14) are all channel parameters, where m is the shaping parameter of Nakagami-m fading, and μ>0 is related to the number of received signal groups. κ=d2/2μσ2 is a positive number, d2 represents the total power of the dominating constituent, and 2μσ2 is the total power of the scattered wave. ψ=Εh∧XY2, and h^XY indicates the amplitude of the channel. The capacity formula of the Shannon theorem is defined as
(15)C=12log21+γ.According to (10) and (11), we can get the expression of the instantaneous channel capacity of B⟶H and H⟶D as
(16)CBH=12log21+ρBHh∧BH2ρBHσeBH2+h∧BH2fBH2+σeBH2fBH2+1,(17)CHD=12log21+ρHDh∧HD2ρHDσeHD2+ξ1PJ/σHD2+ρHDfHD2h∧HD2+σeHD2+1.The eavesdropper employs multiple relay combining (MRC) algorithm to intercept legal information; the SINR atH⟶E is
(18)γHE=∑q=1QγHEq.The SINR ofH⟶E is expressed as
(19)γHE=QγHEq.So we can get the instantaneous capacity ofH⟶E as
(20)CHE=12log21+QρHEqh∧HEq2ρHEqσeHEq2+ξ2PJ/σHEq2+ρHEqfHEq2h∧HEq2+σeHEq2+1.


## 3. Performance Analysis of Outage Probability (OP) and Intercept Probability (IP)

To improve the performance of the communication systems, we use the optimal transmit antenna selection (OTAS) strategy at the transmitter with multiple antennas and deduce the precise expressions of OP and IP to analyze its reliability and security.


### 3.1. OP Analysis

In this section, we discuss the transmission performance of multiantenna cooperative communication systems in the existence of RHIs and CEEs from the perspective of OP.


### 3.2. Outage Probability

The OP of the system model can be indicated as(21)Pout=PrCXY<Rs,where the target threshold is Rs.Under the premise of CEEs and RHIs, using the OTAS protocol, the channel gain containing the estimation error can be obtained as(22)h^BH=max1≤l≤Lh^BlH.The CDF of the channel gain duringBl⟶H transmission is written as
(23)Fh∧XY2x=mXY+κXYμXYmXY−mXY1ΓμXY∑i=1∞mXYiμXYii!×μXY+i−1!×mXYκXYμXYmXY+κXYμXYmXYi1−∑s=0μXY+i−1xss!1+κXYμXYψXYsexp−1+κXYμXYψXYxL.With the DF algorithm, the channel capacity in the information transmission ofBl⟶H and H⟶D is expressed as
(24)CH=minCBH,CHD.Through the above analysis, in the following theorem, we will research the exact expression of the OP of theκ-μ shadowed fading with RHIs and CEEs.Theorem 1.
Forκ-μ shadowed fading, the accurate expressions of OP for two scenes are expressed as follows:
(1)
Nonideal condition (with RHIs and CEEs)(25)Poutni=I1+I2−I1I2.
To ensure that OP is not zero, we set1−ΘfBH2>0 and ρHD−Θξ1PJ/σHD2+fHD2ρHD>0. I1 and I2 correspond to equations (35) and (36) in the appendix, respectively.
(2)
Ideal condition (fBH=fHD=0 and σeBH2=κeHD2=0)(26)Poutid=mBH+κBHμBHmBH−mBH1ΓμBH∑i=1∞mBHiμBHii!μBH+i−1!mBHκBHμBHmBH+κBHμBHmBHi×1−∑s=0μBH+i−1φ3ss!1+κBHμBHψBHsexp−1+κBHμBHψBHφ3L+mHD+κHDμHDmHD−mHD1ΓμHD∑i=1∞mHDiμHDii!μHD+i−1!mHDκHDμHDmHD+κHDμHDmHDi×1−∑s=0μHD+i−1φ4ss!1+κHDμHDψHDsexp−1+κHDμHDψHDφ4−mBH+κBHμBHmBH−mBH1ΓμBH∑i=1∞mBHiμBHii!μBH+i−1!mBHκBHμBHmBH+κBHμBHmBHi×1−∑s=0μBH+i−1φ3ss!1+κBHμBHψBHsexp−1+κBHμBHψBHφ3L×mHD+κHDμHDmHD−mHD1ΓμHD∑i=1∞mHDiμHDii!μHD+i−1!mHDκHDμHDmHD+κHDμHDmHDi×1−∑s=0μHD+i−1φ4ss!1+κHDμHDψHDsexp−1+κHDμHDψHDφ4,where φ3=Θ/ρBH and φ4=Θ/ρBH−Θξ1PJ/σHD2.Proof.
See Appendix A.


### 3.3. IP Analysis

IP means the probability that the capacity of the eavesdropping channel is greater thanRS, which is written as
(27)Pint=PrCHE>Rs.For the precise expressions of IP under theκ-μ shadowed fading, the two scenes will be given below.


### 3.4. Nonideal Condition

Substituting (20) into (27), we obtain
(28)Pint.i.i.dni=PrQρHEqh∧HEq2ρHEqσeHEq2+ξ2PJ/σHEq2+ρHEqfHEq2h∧HEq2+σeHEq2+1>Θ=1−Fh∧HEq2φ5.where φ5=ΞσeHEq2ρHEq+ξ2PJ/σHEq2+ρHEqfHEq2+1/ρHEq−Ξξ2PJ/σHEq2+ρHEqfHEq2, Θ=22RS−1, and Ξ=Θ/Q. Substituting (14) into (28), we can get the exact expression of OP as
(29)Pint.i.i.dni=1−mHEq+κHEqμHEqmHEq−mHEq1ΓμHEq∑i=1∞mHEqiμHEqii!mHEqκHEqμHEqmHEq+κHEqμHEqmHEqi×μHEq+i−1!1−∑s=0μHEq+i−1φ5ss!1+κHEqμHEqψHEqsexp−1+κHEqμHEqψHEqφ5.where Ξ<ρHEqσHEq2/ξ2PJ.


### 3.5. Ideal Condition

Similar to the derivation of (29), we obtain
(30)Pintid=1−mHEq+κHEqμHEqmHEq−mHEq1ΓμHEq∑i=1∞mHEqiμHEqii!mHEqκHEqμHEqmHEq+κHEqμHEqmHEqi×μHEq+i−1!1−∑s=0μHEq+i−1φ6ss!1+κHEqμHEqψHEqsexp−1+κHEqμHEqψHEqφ6,where φ6=Ξ/ρHEq−Ξξ2PJ/σHEq2, Θ=22RS−1, and Ξ=Θ/Q and we set Ξ<ρHEqσHEq2/ξ2PJ.


## 3.1. OP Analysis

In this section, we discuss the transmission performance of multiantenna cooperative communication systems in the existence of RHIs and CEEs from the perspective of OP.


## 3.2. Outage Probability

The OP of the system model can be indicated as(21)Pout=PrCXY<Rs,where the target threshold is Rs.Under the premise of CEEs and RHIs, using the OTAS protocol, the channel gain containing the estimation error can be obtained as(22)h^BH=max1≤l≤Lh^BlH.The CDF of the channel gain duringBl⟶H transmission is written as
(23)Fh∧XY2x=mXY+κXYμXYmXY−mXY1ΓμXY∑i=1∞mXYiμXYii!×μXY+i−1!×mXYκXYμXYmXY+κXYμXYmXYi1−∑s=0μXY+i−1xss!1+κXYμXYψXYsexp−1+κXYμXYψXYxL.With the DF algorithm, the channel capacity in the information transmission ofBl⟶H and H⟶D is expressed as
(24)CH=minCBH,CHD.Through the above analysis, in the following theorem, we will research the exact expression of the OP of theκ-μ shadowed fading with RHIs and CEEs.Theorem 1.
Forκ-μ shadowed fading, the accurate expressions of OP for two scenes are expressed as follows:
(1)
Nonideal condition (with RHIs and CEEs)(25)Poutni=I1+I2−I1I2.
To ensure that OP is not zero, we set1−ΘfBH2>0 and ρHD−Θξ1PJ/σHD2+fHD2ρHD>0. I1 and I2 correspond to equations (35) and (36) in the appendix, respectively.
(2)
Ideal condition (fBH=fHD=0 and σeBH2=κeHD2=0)(26)Poutid=mBH+κBHμBHmBH−mBH1ΓμBH∑i=1∞mBHiμBHii!μBH+i−1!mBHκBHμBHmBH+κBHμBHmBHi×1−∑s=0μBH+i−1φ3ss!1+κBHμBHψBHsexp−1+κBHμBHψBHφ3L+mHD+κHDμHDmHD−mHD1ΓμHD∑i=1∞mHDiμHDii!μHD+i−1!mHDκHDμHDmHD+κHDμHDmHDi×1−∑s=0μHD+i−1φ4ss!1+κHDμHDψHDsexp−1+κHDμHDψHDφ4−mBH+κBHμBHmBH−mBH1ΓμBH∑i=1∞mBHiμBHii!μBH+i−1!mBHκBHμBHmBH+κBHμBHmBHi×1−∑s=0μBH+i−1φ3ss!1+κBHμBHψBHsexp−1+κBHμBHψBHφ3L×mHD+κHDμHDmHD−mHD1ΓμHD∑i=1∞mHDiμHDii!μHD+i−1!mHDκHDμHDmHD+κHDμHDmHDi×1−∑s=0μHD+i−1φ4ss!1+κHDμHDψHDsexp−1+κHDμHDψHDφ4,where φ3=Θ/ρBH and φ4=Θ/ρBH−Θξ1PJ/σHD2.Proof.
See Appendix A.


## 3.3. IP Analysis

IP means the probability that the capacity of the eavesdropping channel is greater thanRS, which is written as
(27)Pint=PrCHE>Rs.For the precise expressions of IP under theκ-μ shadowed fading, the two scenes will be given below.


## 3.4. Nonideal Condition

Substituting (20) into (27), we obtain
(28)Pint.i.i.dni=PrQρHEqh∧HEq2ρHEqσeHEq2+ξ2PJ/σHEq2+ρHEqfHEq2h∧HEq2+σeHEq2+1>Θ=1−Fh∧HEq2φ5.where φ5=ΞσeHEq2ρHEq+ξ2PJ/σHEq2+ρHEqfHEq2+1/ρHEq−Ξξ2PJ/σHEq2+ρHEqfHEq2, Θ=22RS−1, and Ξ=Θ/Q. Substituting (14) into (28), we can get the exact expression of OP as
(29)Pint.i.i.dni=1−mHEq+κHEqμHEqmHEq−mHEq1ΓμHEq∑i=1∞mHEqiμHEqii!mHEqκHEqμHEqmHEq+κHEqμHEqmHEqi×μHEq+i−1!1−∑s=0μHEq+i−1φ5ss!1+κHEqμHEqψHEqsexp−1+κHEqμHEqψHEqφ5.where Ξ<ρHEqσHEq2/ξ2PJ.


## 3.5. Ideal Condition

Similar to the derivation of (29), we obtain
(30)Pintid=1−mHEq+κHEqμHEqmHEq−mHEq1ΓμHEq∑i=1∞mHEqiμHEqii!mHEqκHEqμHEqmHEq+κHEqμHEqmHEqi×μHEq+i−1!1−∑s=0μHEq+i−1φ6ss!1+κHEqμHEqψHEqsexp−1+κHEqμHEqψHEqφ6,where φ6=Ξ/ρHEq−Ξξ2PJ/σHEq2, Θ=22RS−1, and Ξ=Θ/Q and we set Ξ<ρHEqσHEq2/ξ2PJ.


## 4. Asymptotic Analysis

To further verify and investigate the theoretical derivation, we give the asymptotic analysis of the high SNR approximation of OP. It can be divided into the following two conditions: nonideal case and ideal case.Theorem 2.
The precise expressions of the asymptotic analysis of the high SNR approximation of OP are as follows:(1)
Nonideal case
WhenσeXY2=Ω/1+δρΩ, equation (42) can be obtained from Appendix B. When σeXY2=a,
(31)Pout∞,ni2=I5+I6−I5I6,where a is a constant, 1−ΘfBH2>0, and 1−Θξ1/2−ΘfHD2>0.
(2)
Ideal case(32)Pout∞,id=I7+I8−I7I8.Proof.
See Appendix B.


## 5. Numerical Analysis

In this section, we exhibit the comparison charts between theoretical derivations and Monte Carlo simulations in Figures2–7. As can be seen from the graphs, the numerical results coincide well with the simulation, which confirms the validity of the deduced results. In the simulation, our parameters are set as follows: κXY=κ, mXY=m, μXY=μ, Rs=0.5, PB=PH=Pg, PJ=PB/2, σeXY2=σe2, σXY2=1, and ψXY=1. It should be noted that although the theoretical formulas of IP and op contain infinite series, it will not affect the simulation results, because the theoretical formulas will converge when the number of cycles is greater than 45.Figure 2
OP and IP versusτ for different f (ξ1,ξ2=0.05,0.95, P=5dB, σe2=0.1, κ=1, m=2, μ=2, L=2, and Q=2).Figure 3
OP and IP versusP for different ξ1,ξ2 (L=2, κ=1, m=2, μ=2, τ=0.8, and Q=2).Figure 4
OP and IP versusL for different σe2 and P (ξ1,ξ2=0.05,0.95, f=0.1, κ=1, m=2, μ=2, and τ=0.8).Figure 5
OP and IP versusP for different f (ξ1,ξ2=0.05,0.95, τ=0.8, σe2=0.1, f=0,0.1,0.2, κ=1, m=2, μ=2, L=2, and Q=2).Figure 6
OP and IP versusP for different σe2 (ξ1,ξ2=0.05,0.95, τ=0.8, L=2, κ=1, m=2, μ=2, Q=2, and f=0.1).Figure 7
OP versusP for different L and μ (ξ1,ξ2=0.05,0.95, τ=0.8, κ=1, m=2, L=1,2, and μ=2,3).Figure2 describes the variation of OP and IP with the power distribution factor τ and distortion noise f. It can be concluded that as τ increases, OP gradually becomes smaller, while IP continues to rise. And the increase of the distortion noise parameter will reduce the reliability and improve the antieavesdropping ability.Figure3 illustrates the changes of OP and IP with transmit power under different artificial quantization coefficients. Both ideal situations (σe2=0, f=0) and nonideal situations (σe2=0.1, f=0.1) are considered. It can be seen that as ξ1 augments, OP gradually aggrandizes, and the reliability of the system decreases. And due to distortion, the value of OP under ideal conditions is always lower than that with RHIs. In addition, increasing the value of ξ2 will raise the IP, which means that the security of the channel will be reduced.Figure4 plots the impact of different antenna numbers L and CEEs on OP. We clearly find that the increase of transmission power can enhance the transmission performance, while the augment of CEEs will impair the reliability. When the number of antennas is less than 4 dB, the OP decreases as the number of antennas adds. When L is greater than 4 dB, the OP changes less, which shows that the application of OTAS to improve the reliability of the system is more suitable when the number of antennas is small.Figure5 presents the variation of OP and IP with transmission power when RHIs and CEEs exist at the same time. The figure shows that with the increases of f, OP increases while IP decreases. The results illustrate that RHIs are detrimental to the reliability of the system. And OP has an error floor under nonideal conditions. In addition, there is an optimal transmission power, which can balance the security and reliability in the communication process.Figure6 illustrates the change of OP and IP with the transmit power under different CEE conditions. The approximate curves in Figure 6 are drawn based on (31) and (32). We can see that in nonideal situations, with the increase of σe2, the reliability of the channel decreases and there exists an error floor. And the value of OP under the nonideal case is always greater than under the ideal case. In addition, compared with OP, CEEs have less impact on IP. This manifests that CEEs play an important role in the reliability but has little impact on the security.Figure7 plots the variation of OP under the conditions of different antenna numbers L and parameters μ. In this simulation, we set an ideal case (σe2=0, f=0) and a nonideal case (σe2=0.1, f=0.1). It can be concluded that due to the existence of fixed CEEs, there exists an error floor independent of transmission power under nonideal condition. And because of the existence of CEEs and RHIs, the nonideal situation of OP has been worse than the ideal situation. At the same time, we can see that the augment of L and μ can make the system more reliable.


## 6. Conclusion

This paper mainly studies the influence over theκ-μ shadowed fading when RHIs and CEEs exist simultaneously. Meanwhile, we used the multiantenna selection scheme on the transmitting terminal and discussed the multiantenna receiving technology on the eavesdropping transmitting terminal. In addition, we employed the DF method to transmit the destination signal and then verified the correctness of the derivation through simulation We discover that large f, large ξ, large σe2, and small L can reduce the reliability. In addition, small ξ, large f, and large σe2 can improve the antieavesdropping ability on the κ-μ shadowed network. Finally, it is concluded that RHIs and CEEs can reduce the reliability and enhance its security, because RHIs and CEEs often appear in the practical communication network. It is very meaningful to do performance analysis on this premise. As an extension of this work, we will study the scenario of multiple base stations and multiple receivers with different relay protocols.



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*Source: 8593474-2022-04-13.xml*