# Security-Reliability Tradeoff Analysis of Untrusted Full-Duplex Relay Networks

Authors: Xingang Zhang; Dechuan Chen; Jin Li; Zhipeng Wang; Xiaotan Li
Journal: Wireless Communications and Mobile Computing
 (2022)

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## Abstract

In this work, we investigate the physical layer security problem of wireless cooperative network, where the communication from a source to a destination is assisted by an untrusted full-duplex amplify-and-forward (AF) relay. In order to realize a positive secrecy rate, cooperative jamming is exploited at the destination. The secrecy outage probability (SOP), connection outage probability (COP), and effective secrecy throughput (EST) are, respectively, derived in closed-form expressions for the fixed gain relaying (FGR) scheme and the variable gain relaying (VGR) scheme. Subsequently, we conduct an asymptotic analysis for both the relaying schemes in the high signal-to-noise ratio (SNR) region to offer valuable insights into practical design. Theory and simulation results demonstrate that the self-interference caused by full-duplex transmission is harmful for the reliability performance, while it is beneficial to the security performance. Moreover, the EST of the considered system increases as the self-interference level decreases, which implies that the self-interference deteriorates an overall performance in terms of the security-reliability tradeoff.


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## Body

## 1. Introduction

Due to the broadcast nature of wireless transmission, the communications between legitimate users are particularly vulnerable to be overheard by malicious users [1, 2]. Traditionally, cryptographic mechanism in the upper layers of the network based on computational complexity has been employed to guarantee secure communication in wireless networks [3]. However, as the computing power of malicious users is continuously enhancing (e.g., by adopting the quantum computing technology), cryptographic mechanism is being increasingly challenged. Unlike the cryptographic mechanism, physical layer security, which explores the time-varying properties of physical channels to achieve perfect security, has attracted considerable attention from academia and industry [4, 5].In recent years, full-duplex relay techniques, which can transmit and receive simultaneously in the same frequency band, have been exploited to improve physical layer security performance of wireless networks. The authors in [6] proposed a newly full-duplex jamming relay scheme, which has better secrecy performance than the half duplex scheme. For full-duplex two-way relay networks, [7] exploited artificial noise to strengthen the sum secrecy rate. In [8], a power allocation strategy between data and jamming was developed to effectively improve the tradeoff between security and reliability, where both the relay and the destination operated in full-duplex mode. Subsequently, full-duplex techniques were extended to secure multirelay networks [9] and nonorthogonal multiple access (NOMA) networks [10].The works in [6–10] were focused on the security of full-duplex relay networks, where the eavesdroppers are external nodes in addition to legitimate nodes. However, even in the absence of external eavesdroppers, the security caused by legitimate nodes in the network may still be a concern [11–13]. For example, the relay may be untrusted, which means that they can possibly try to eavesdrop the source’s confidential information. The untrusted relay is also a potential eavesdropper, even though it complies with the communication protocols to assist the source forwarding information to the destination [14–16]. In practice, untrusted relay scenario may occur in Internet of Things (IoT), government intelligence networks, and device-to-device (D2D) communications, where not all users do have the same security clearance.Recently, a few works about untrusted full-duplex relay networks have been considered in the context of physical layer security. Based on the source jamming scheme, [17] derived the secrecy outage probability for untrusted full-duplex relay networks. The impact of outdated channel state information (CSI) on the ergodic secrecy rate of untrusted full-duplex relay networks was investigated in [18]. The authors of [19] proposed an artificial noise-aided secure transmission scheme to confuse the untrusted full-duplex relay and external eavesdroppers. In [20], an optimal beamforming scheme was design to maximize the secrecy sum rate for full-duplex multiple-input multiple-output two-way untrusted relay networks. The aforementioned contributions are mainly concerned on enhancing the secrecy rate of untrusted full-duplex relay networks without paying much attention to the communication reliability. In particular, the impact of self-interference caused by full-duplex transmission on the security-reliability tradeoff is indeed worthy of our attention.Motivated by the above considerations, we explore the physical layer security of untrusted full-duplex relay networks, where a source operates in half duplex mode and an untrusted relay and a destination operate in a full-duplex mode. In addition, the source equipped with multiple antennas uses a maximal ratio transmission (MRT) scheme to increase reliability. Depending on the availability of the CSI, the relay amplifies and forwards the source’s signal with a fixed-gain or variable-gain factor. Different from the source-based jamming scheme in [17], we employ destination-assisted jamming for secrecy improvement. Compared with [18–20], we consider the outage performance of the untrusted full-duplex relay network, where the secrecy outage probability (SOP) is used to evaluate the security performance and the connection outage probability (COP) is used to characterize the reliability performance. The main contributions of this paper are summarized as follows:
(i)
We first present exact expressions for the SOP and COP of the fixed gain relaying (FGR) scheme and the variable gain relaying (VGR) scheme and then derive closed-form expressions for the EST of both the relaying schemes, which provide an efficient means to evaluate the reliability and security performance comprehensively(ii)
To gain further insights, we conduct an asymptotic analysis in the high signal-to-noise ratio (SNR) region for the SOP, COP, and EST. Based on the asymptotic analysis, we find that there exists a EST floor for both the relaying schemes in the high SNR region and the floor of the VGR scheme is higher than that of the FGR scheme(iii)
The results demonstrate the intuition that the self-interference caused by full-duplex transmission is harmful for the reliability performance, while it is beneficial to the security performance. In addition, the EST of the considered system increases as the self-interference level decreases, which implies that the self-interference deteriorates an overall performance in terms of the security-reliability tradeoffThe remainder of this paper is organized as follows. Section2 gives the system model and describes the key metric in evaluating the performance of physical layer security, including the SOP, COP, and EST. In Section 3, we present the closed form expressions of the SOP, COP, and EST for the VGR scheme and the FGR scheme and analyze the corresponding asymptotic behavior in the high SNR region. Numerical results and conclusions are, respectively, provided in Sections 4 and 5.


## 2. System Model

We consider an untrusted full-duplex relay network shown in Figure1, where a source S communicates with a destination D via an untrusted relay R. The source has Ns antennas and operates in the half-duplex mode, while the destination and the relay are equipped with dual antennas and operate in the full-duplex mode. It means that the destination and the relay can receive and send data concurrently. The system considered in this work is applicable to numerous practical full-duplex relay scenarios, for instance, the wireless sensor networks and IoT. Moreover, the direct link between the source and the destination is unavailable, due to severe shadow fading or significant obstacle [21]. Thus, source-destination communication can be performed only through the relay. In addition, all channels are modeled as quasistatic block fading channels, following the Rayleigh distribution. As in [13, 15], we assume that the channel between the relay and the destination is reciprocal.Figure 1
System model, whereS, R, and D denote the source, untrusted relay, and destination, respectively.Because the relay is untrusted, the source wishes to keep the information secret from the relay while simultaneously making use of it to forward the information. To maintain the confidentiality of the source information, the destination sends a jamming signal to the relay when the source transmits information to the relay. Thus, the received signal at the relay can be expressed as(1)yri=hsrwsrxsi+hdrxdi+hrrxri+nri,where hsr is the 1×Ns channel vector between the source and the relay and its entries follow independent and identically distributed (i.i.d.) complex Gaussian distribution with zero mean and variance γ¯SR. wsr=hsr†/hsr is the transmit precoding vector at the source. hdr is the channel coefficient between the relay and the destination with parameter γ¯RD. hrr is the loopback interference channel coefficient between transmitting and receiving antennas at the relay. xsi is the confidential signal transferred from the source at time slot i and satisfies Exsi2=βP, where E· represents the expectation operation. xdi is the jamming signal transferred from the destination at time slot i and satisfies Exdi2=1−βP. Here, β∈0,1 is the power allocation factor between the source and the destination. xri is the loopback interference signal transferred from the relay at time slot i and satisfies Exri2=P. nri is the additive white Gaussian noises (AWGN) at the relay with zero mean and variance N0. Since xri is known to the relay, it can apply multiple stage interference cancellation methods to alleviate the loopback interference. Thus, the received signal at the relay after compensation can be expressed as
(2)y^ri=hsrwsrxsi+hdrxdi+Iri+nri,where Iri can be modeled as a Gaussian random variable with mean zero and variance lr2P [18, 22]. lr corresponds to the self-interference cancellation capability at the relay.Then, the relay amplifies the signaly^ri with factor G and forwards it to the destination. The two most common factors employed at the relay are so-called fixed gain factor and variable gain factor. In the FGR scheme, the factor is constant based on the statistical CSI, resulting in an output signal with variable power. In the VGR scheme, the factor is variable depended on the instantaneous CSI, resulting in an output signal with fixed power. The factors of both FGR and VGR schemes can be given by
(3)G=EPβPhsr2+1−βPhdr2+lr2P+N0,withFGR,PβPhsr2+1−βPhdr2+lr2P+N0,withVGR.Hence, the received signal at the destination is given by(4)ydi=hrdxri+hddxdi+ndi=hrdGy^ri+hddxdi+ndi,where hdd is the loopback interference channel coefficient between transmitting and receiving antennas at the destination and ndi is the AWGN at the destination with zero mean and variance N0. Analogously, the received signal at the destination after self-interference cancellation can be written as
(5)y^di=hrdGhsrwsrxsi+hrdGIri+hrdGnri+Idi+ndi,where Idi can be modeled as a Gaussian random variable with mean zero and variance ld21−βP. ld corresponds to the self-interference cancellation capability at the destination.According to (2), the achievable eavesdropping rate with the FGR scheme and the VGR scheme can be obtained as
(6)CRFGR=CRVGR=log21+βλhsr21−βλhrd2+lr2λ+1,where λ=P/N0 denotes the transmit SNR. According to (3) and (5), the achievable data rate with the FGR scheme and the VGR scheme can be, respectively, obtained as
(7)CDFGR=log21+βλ2hsr2hrd2λ+λ2lr2hrd2+C,(8)CDVGR=log21+βλ2hsr2hrd2C1hsr2+C2hrd2+C3,where C=1+lr2λ+βλNsγ¯SR+1−βλγ¯RD1+ld2λ1−β, C1=βλ1+ld2λ1−β, C2=λ+λ2lr2+λ1−β1+ld2λ1−β, and C3=1+lr2λ1+ld2λ1−β.As discussed in [23], when the achievable eavesdropping rate is larger than the positive rate difference between the codeword transmission rate R0 and the confidential information rate Rs, there will be a secrecy outage event. When the achievable data rate is less than R0, there will be a connection outage event. Thus, we can formulate the SOP and COP of our system as
(9)Pso=PrCR>R0−Rs,(10)Pco=PrCD<R0.The SOP and COP give practical insights into security and reliability performance of the transmission system, respectively. To establish the direct relationship between security and reliability, we adopt effective secrecy throughput (EST) to measure the overall efficiency of our system, which is given by(11)ς=Rs·PrCR<R0−Rs,CD>R0.


## 3. Secrecy Performance Analysis

In this section, we endeavor to analyze both the reliability and security performance comprehensively for the untrusted full-duplex relay network. Specifically, we first derive closed form expressions of the SOP, COP, and EST for the FGR scheme and the VGR scheme. Then, the asymptotic analysis of the SOP, COP, and EST is provided to reveal additional insights on the secrecy performance.


### 3.1. Fixed Gain Relaying

#### 3.1.1. Secrecy Outage Probability

Combining (6) and (9), the SOP of the untrusted full-duplex relay network with the FGR scheme can be given by
(12)PsoFGR=Prβλhsr21−βλhrd2+lr2λ+1>t1=Prhsr2>1−βt1hrd2β+lr2t1λ+t1βλ,where t1=2R0−Rs−1. Noting that X=hsr2 is a central chi-square distribution random variable with 2Ns degrees of freedom, its cumulative distribution function (CDF) is given by
(13)FXx=1−e−x/γ¯SR∑m=0Ns−1xmm!γ¯SRm.On the other hand,Y=hrd2 is an exponential variable with probability density function (PDF) given by
(14)fYy=1γ¯RDe−y/γ¯RD.Based on (13) and (14), we can rewrite (12) as
(15)PsoFGR=∫0∞1−FX1−βt1yβ+lr2t1λ+t1βλfYydy=∑m=0Ns−1e−t1lr2λ+t1/βλγ¯SRm!γ¯RDγ¯SRmβmλm∫0∞e−1−βt1/βγ¯SR+1/γ¯RDy×t1+t1lr2λ+1−βt1λymdy.Then, usingt1+t1lr2λ+1−βt1λym=∑n=0mmnt1+t1lr2λm−n1−βnt1nλnyn, we can calculate PsoFGR as
(16)PsoFGR=∑m=0Ns−1∑n=0mt1lr2λ+t1m−n1−βnt1nλnm−n!γ¯RDγ¯SRmβmλm×1−βt1βγ¯SR+1γ¯RD−n−1e−t1lr2λ+t1/βλγ¯SR.From (16), we find that the SOP increases as the power allocation factor β increases from 0 to 1. This is because that increasing β results in decreasing transmission power of the jamming signal. Moreover, the SOP decreases as lr increases, which means that the self-interference caused by full-duplex transmission is beneficial to the security performance.


#### 3.1.2. Connection Outage Probability

Combining (7) and (10), the COP of the untrusted full-duplex relay network with the FGR scheme can be given by
(17)PcoFGR=Prβλ2hsr2hrd2λ+λ2lr2hrd2+C<t2=Prhsr2<t21+λlr2βλ+t2Cβλ2hrd2,where t2=2R0−1. Based on (13) and (14), we can rewrite (17) as
(18)PcoFGR=∫0∞FXt21+λlr2βλ+t2Cβλ2yfYydy=1−∑m=0Ns−1e−t2λlr2+t2/βλγ¯SRm!γ¯RDγ¯SRm∫0∞t2+t2λlr2βλ+t2Cβλ2ym×e−t2C/βλ2γ¯SRy−y/γ¯RDdy.Then, with the help of the binomial theorem and ([24], 3.471.9), the exact COP of the untrusted full-duplex relay network with the FGR scheme can be derived as
(19)PcoFGR=1−∑m=0Ns−1∑n=0mmn2t2mCn1+λlr2m−nm!βmλm+nγ¯RDγ¯SRme−t2λlr2+t2/βλγ¯SR×t2Cγ¯RDβλ2γ¯SR1−n/2K1−n4t2Cβλ2γ¯SRγ¯RD,where K1−n· is the Bessel functions of imaginary argument. From (19), we find that the COP decreases as the power allocation factor β increases from 0 to 1. The main reason is that when β increases, the achievable data rate is larger which makes reliability better. In addition, the COP increases as lr or ld increases, which means that the self-interference caused by full-duplex transmission is harmful for the reliability performance.


#### 3.1.3. Effective Secrecy Throughput

Combining (6), (7), and (11), the EST of the untrusted full-duplex relay network with the FGR scheme can be given by
(20)ςFGR=RsPrhsr2<1−βt1hrd2β+lr2t1λ+t1βλ,hsr2>t21+λlr2βλ+t2Cβλ2hrd2.Since1−βt1hrd2/β+lr2t1λ+t1/βλ and t21+λlr2/βλ+t2C/βλ2hrd2 have only one point of intersection when hrd2>0, and the intersection is u1. Thus, expression (20) can be expressed as
(21)ςFGR=Rs∫u1∞FX1−βt1yβ+lr2t1λ+t1βλfYydy−Rs∫u1∞FXt21+λlr2βλ+t2Cβλ2yfYydy=∑m=0Ns−1∑n=0mmnRst2mCn1+λlr2m−ne−t2+t2λlr2/βλγ¯SRm!γ¯SRmβmγ¯RDλm+n×∫u1∞1yne−t2C/βλ2γ¯SRy+y/γ¯RDdy−∑m=0Ns−1∑n=0mmn×Rst1m1−βne−t11+λlr2/βλγ¯SRm!γ¯SRmβmγ¯RD1+λlr2λm−n×∫u1∞yne−1−βt1/βγ¯SR+1/γ¯RDydy,where u1=−b1+b12−4a1c1/2a1, a1=t1λ21−β, c1=−t2C, and b1=λt1−t2lr2λ+1. Then, using [24, 1.211.1], [24, 3.351.2], and [24, 3.381.3], the exact EST of the untrusted full-duplex relay network with the FGR scheme can be derived as
(22)ςFGR=∑m=0Ns−1∑n=0m∑k=0∞mn−1kRsCn+k1+λlr2m−nm!k!γ¯RDn+kλm+n+2k×t2βγ¯SRm+ke−t2+t2λlr2/βλγ¯SRΓ1−n−k,u1γ¯RD−∑m=0Ns−1∑n=0mmnRst1m1−βne−t11+λlr2/βλγ¯SRm!γ¯SRmβmγ¯RD×1+λlr2λm−n1−βt1βγ¯SR+1γ¯RD−n−1×Γn+1,1−βt1u1βγ¯SR+u1γ¯RD.From (22), we find that the EST of the considered system is small, when the confidential information rate Rs is small. At the other extreme, when the confidential information rate Rs is too large, the secrecy performance of the considered system is poor which also results in small EST. Hence, there exists an optimal Rs to maximize the EST of the considered system. Furthermore, the EST increases as the self-interference level decreases, which means that the self-interference deteriorates an overall performance in terms of the security-reliability tradeoff.


#### 3.1.4. Asymptotic Behavior

To extract additional insights on the FGR scheme, we now investigate the high SNR asymptotic behavior of the SOP, COP, and EST of the considered system. According (16), we have
(23)limλ⟶∞PsoFGR=∑m=0Ns−1∑n=0mt1lr2m−n1−βnt1ne−t1lr2/βγ¯SRm−n!γ¯RDγ¯SRmβm×t1−t1ββγ¯SR+1γ¯RD−n−1.From (19), we have
(24)limλ⟶∞PcoFGR=1−∑m=0Ns−1∑n=0mmn2t2mC¯nlr2m−2ne−t2lr2/βγ¯SRm!γ¯RDγ¯SRmβm×t2C¯γ¯RDβγ¯SR1−n/2K1−n4t2C¯βγ¯SRγ¯RD,where C¯=ld21−βlr2+βNsγ¯SR+γ¯RD1−β. According to (22), we have
(25)limλ⟶∞ςFGR=∑m=0Ns−1∑n=0m∑k=0∞mn−1kRst2m+klr2m−2nC¯n+km!k!γ¯SRm+kγ¯RDn+kβm+k×e−t2lr2/βγ¯SRΓ1−n−k,u2γ¯RD−∑m=0Ns−1∑n=0mmn×Rst1m1−βnlr2m−2nm!γ¯SRmβmγ¯RDe−t1lr2/βγ¯SR1−βt1βγ¯SR+1γ¯RD−n−1Γn+1,1−βt1u2βγ¯SR+u2γ¯RD,where u2=−lr2t1−t2+lr4t1−t22+4t1t2C¯1−β/2t11−β. It is interesting to note that when transmit power is sufficiently large, the security and reliability of the untrusted full-duplex relay network are still depend on the power allocation factor. Thus, the power allocation factor enables a tradeoff between reliability and security for the untrusted full-duplex relay network. From the numerical results, we find that there exists an optimal power allocation factor maximizing the EST.


### 3.2. Variable Gain Relaying

#### 3.2.1. Secrecy Outage Probability

The gain factor employed at the relay determines the quality of signal reception at the destination and has no influence on the eavesdropping ability at the relay. Therefore, the SOP of the VGR scheme is the same as the FGR scheme and it is omitted here.


#### 3.2.2. Connection Outage Probability

Combining (8) and (10), the COP of the untrusted full-duplex relay network with the VGR scheme can be given by
(26)PcoVGR=Prβλ2hsr2hrd2C1hsr2+C2hrd2+C3<t2=Prβλ2hrd2−t2C1hsr2<t2C2hrd2+t2C3.Based on (13) and (14), we can rewrite (26) as
(27)PcoVGR=∫0C5fYydy+∫C5∞FXt2C2y+t2C3βλ2y−t2C1fYydy=1−∑m=0Ns−1e−t2C2/βλ2γ¯SRm!γ¯RDγ¯SRmt2C2βλ2m∫C5∞1+C4y−C5m×e−t2C2C4/βλ2γ¯SRy−C5+y/γ¯RDdy,where C4=C3/C2+t2C1/βλ2, and C5=t2C1/βλ2. Then, let y~=y−C5, and use [24, 1.211.1] and [24, 3.471.9], the exact COP of the untrusted full-duplex relay network with the VGR scheme can be derived as
(28)PcoVGR=1−∑m=0Ns−1∑n=0mmn2C4ne−t2C2/βλ2γ¯SR−C5/γ¯RDm!γ¯RDγ¯SRmt2C2βλ2m×t2C2C4γ¯RDβλ2γ¯SR1−n/2K1−n4t2C2C4βλ2γ¯SRγ¯RD.WhenNs⟶∞, we have hsr2≈aNsγ¯SR, where a follows the law of large numbers. Thus, from (26), we clarify that increasing the number of antennas at the source unboundedly is not helpful because the channel between the relay and the destination will finally become the bottleneck and dominate the achievable data rate.


#### 3.2.3. Effective Secrecy Throughput

Combining (6), (8), and (11), the EST of the untrusted full-duplex relay network with the VGR scheme can be given by
(29)ςVGR=RsPrhsr2<1−βt1hrd2β+lr2t1λ+t1βλ,hsr2>t2C2hrd2+t2C3βλ2hrd2−t2C1,hrd2>t2C1βλ2.Since1−βt1hrd2/β+lr2t1λ+t1/βλ and t2C2hrd2+t2C3/βλ2hrd2−t2C1 have only one point of intersection when hrd2>0, and the intersection is u3. Thus, the expression (29) can be expressed as
(30)ςVGR=Rs∫u4∞FX1−βt1yβ+lr2t1λ+t1βλfYydy−Rs∫u4∞FXt2C2y+t2C3βλ2y−t2C1fYydy=∑m=0Ns−1∑n=0mmnRsC4ne−t2C2/βλ2γ¯SR−C5/γ¯RDm!γ¯SRmγ¯RDt2C2βλ2m×∫u4−C5∞1yne−t2C2C4/βλ2γ¯SRy−y/γ¯RDdy−∑m=0Ns−1∑n=0mmn×Rst1m1−βne−t11+λlr2/βλγ¯SRm!γ¯SRmβmγ¯RD1+λlr2λm−n×∫u4∞yne−1−βt1/βγ¯SR+1/γ¯RDydy,where u4=maxu3,t2C1/βλ2, u3=−b2+b22−4a2c2/2a2, a2=t1βλ31−β, b2=t1βλ21+lr2λ−t1t2λC11−β−βλt2C2, and c2=−βλt2C3−t1t2C11+lr2λ. Then, using [24, 1.211.1], [24, 3.351.2] and [24, 3.381.3], the exact EST of the untrusted full-duplex relay network with the VGR scheme can be derived as
(31)ςVGR=∑m=0Ns−1∑n=0m∑k=0∞mn−1kRsC4n+ke−t2C2/βλ2γ¯SR−C5/γ¯RDm!k!γ¯SRm+kγ¯RDn+k×t2C2βλ2m+kΓ1−n−k,u4−C5γ¯RD−∑m=0Ns−1∑n=0mmnRst1m1−βne−t11+λlr2/βλγ¯SRm!γ¯SRmβmγ¯RD×1+λlr2λm−n1−βt1βγ¯SR+1γ¯RD−n−1×Γn+1,1−βt1u4βγ¯SR+u4γ¯RD.The derived exact results in (31) provide an efficient way to holistically examine the security and reliability performance with arbitrary system parameters, i.e., the number of antennas at the source and the self-interference cancellation capability at the relay and destination. Additionally, it is worth noting that the EST of the considered system will tend toward zero when Ns⟶∞. An intuitive explanation to this phenomenon is that, more antennas at the source would bring us better reliability performance, but at the same time, the grade of the security is severely degraded. Thus, the designers have to carefully take into account the number of antennas at the source.


#### 3.2.4. Asymptotic Behavior

To get more insight, we further study the asymptotic behavior of the COP and EST for the VGR scheme under high SNR case. According to (28), we have
(32)limλ⟶∞PcoVGR=1−∑m=0Ns−1∑n=0mmn2t2mC7mC9ne−t2C7/βγ¯SR−C10/γ¯RDm!βmγ¯RDγ¯SRm×t2C7C9γ¯RDβγ¯SR1−n/2K1−n4t2C7C9βγ¯SRγ¯RD,where C6=βld21−β, C7=lr2+ld21−β2, C8=lr2ld21−β, C9=C8/C7+t2C6/β, and C10=t2C6/β. From (31), we have
(33)limλ⟶∞ςVGR=∑m=0Ns−1∑n=0m∑k=0∞mn−1kRsC9n+ke−t2C7/βγ¯SR−C10/γ¯RDm!k!γ¯SRm+kγ¯RDn+k×t2C7βm+kΓ1−n−k,u6−C10γ¯RD−∑m=0Ns−1∑n=0mmnRst1m1−βnlr2m−2nm!γ¯SRmβmγ¯RD×e−t1lr2/βγ¯SR1−βt1βγ¯SR+1γ¯RD−n−1×Γn+1,1−βt1u6βγ¯SR+u6γ¯RD,where u6=maxu5,t2C6/β, u5=−b3+b32−4a3c3/2a3, a3=t1β1−β, b3=t1βlr2−βt2C7−t1t2C61−β, and c3=−βt2C8−t2t1C6lr2.From the asymptotic behaviour of the SOP, COP, and EST, we find that the continued boosting of the transmit power will not always improve the security and reliability performance of the considered system. This is due to the fact that the self-interference caused by full-duplex transmission becomes a bottleneck of improving the security and reliability performance at the high SNR region.


## 3.1. Fixed Gain Relaying

### 3.1.1. Secrecy Outage Probability

Combining (6) and (9), the SOP of the untrusted full-duplex relay network with the FGR scheme can be given by
(12)PsoFGR=Prβλhsr21−βλhrd2+lr2λ+1>t1=Prhsr2>1−βt1hrd2β+lr2t1λ+t1βλ,where t1=2R0−Rs−1. Noting that X=hsr2 is a central chi-square distribution random variable with 2Ns degrees of freedom, its cumulative distribution function (CDF) is given by
(13)FXx=1−e−x/γ¯SR∑m=0Ns−1xmm!γ¯SRm.On the other hand,Y=hrd2 is an exponential variable with probability density function (PDF) given by
(14)fYy=1γ¯RDe−y/γ¯RD.Based on (13) and (14), we can rewrite (12) as
(15)PsoFGR=∫0∞1−FX1−βt1yβ+lr2t1λ+t1βλfYydy=∑m=0Ns−1e−t1lr2λ+t1/βλγ¯SRm!γ¯RDγ¯SRmβmλm∫0∞e−1−βt1/βγ¯SR+1/γ¯RDy×t1+t1lr2λ+1−βt1λymdy.Then, usingt1+t1lr2λ+1−βt1λym=∑n=0mmnt1+t1lr2λm−n1−βnt1nλnyn, we can calculate PsoFGR as
(16)PsoFGR=∑m=0Ns−1∑n=0mt1lr2λ+t1m−n1−βnt1nλnm−n!γ¯RDγ¯SRmβmλm×1−βt1βγ¯SR+1γ¯RD−n−1e−t1lr2λ+t1/βλγ¯SR.From (16), we find that the SOP increases as the power allocation factor β increases from 0 to 1. This is because that increasing β results in decreasing transmission power of the jamming signal. Moreover, the SOP decreases as lr increases, which means that the self-interference caused by full-duplex transmission is beneficial to the security performance.


### 3.1.2. Connection Outage Probability

Combining (7) and (10), the COP of the untrusted full-duplex relay network with the FGR scheme can be given by
(17)PcoFGR=Prβλ2hsr2hrd2λ+λ2lr2hrd2+C<t2=Prhsr2<t21+λlr2βλ+t2Cβλ2hrd2,where t2=2R0−1. Based on (13) and (14), we can rewrite (17) as
(18)PcoFGR=∫0∞FXt21+λlr2βλ+t2Cβλ2yfYydy=1−∑m=0Ns−1e−t2λlr2+t2/βλγ¯SRm!γ¯RDγ¯SRm∫0∞t2+t2λlr2βλ+t2Cβλ2ym×e−t2C/βλ2γ¯SRy−y/γ¯RDdy.Then, with the help of the binomial theorem and ([24], 3.471.9), the exact COP of the untrusted full-duplex relay network with the FGR scheme can be derived as
(19)PcoFGR=1−∑m=0Ns−1∑n=0mmn2t2mCn1+λlr2m−nm!βmλm+nγ¯RDγ¯SRme−t2λlr2+t2/βλγ¯SR×t2Cγ¯RDβλ2γ¯SR1−n/2K1−n4t2Cβλ2γ¯SRγ¯RD,where K1−n· is the Bessel functions of imaginary argument. From (19), we find that the COP decreases as the power allocation factor β increases from 0 to 1. The main reason is that when β increases, the achievable data rate is larger which makes reliability better. In addition, the COP increases as lr or ld increases, which means that the self-interference caused by full-duplex transmission is harmful for the reliability performance.


### 3.1.3. Effective Secrecy Throughput

Combining (6), (7), and (11), the EST of the untrusted full-duplex relay network with the FGR scheme can be given by
(20)ςFGR=RsPrhsr2<1−βt1hrd2β+lr2t1λ+t1βλ,hsr2>t21+λlr2βλ+t2Cβλ2hrd2.Since1−βt1hrd2/β+lr2t1λ+t1/βλ and t21+λlr2/βλ+t2C/βλ2hrd2 have only one point of intersection when hrd2>0, and the intersection is u1. Thus, expression (20) can be expressed as
(21)ςFGR=Rs∫u1∞FX1−βt1yβ+lr2t1λ+t1βλfYydy−Rs∫u1∞FXt21+λlr2βλ+t2Cβλ2yfYydy=∑m=0Ns−1∑n=0mmnRst2mCn1+λlr2m−ne−t2+t2λlr2/βλγ¯SRm!γ¯SRmβmγ¯RDλm+n×∫u1∞1yne−t2C/βλ2γ¯SRy+y/γ¯RDdy−∑m=0Ns−1∑n=0mmn×Rst1m1−βne−t11+λlr2/βλγ¯SRm!γ¯SRmβmγ¯RD1+λlr2λm−n×∫u1∞yne−1−βt1/βγ¯SR+1/γ¯RDydy,where u1=−b1+b12−4a1c1/2a1, a1=t1λ21−β, c1=−t2C, and b1=λt1−t2lr2λ+1. Then, using [24, 1.211.1], [24, 3.351.2], and [24, 3.381.3], the exact EST of the untrusted full-duplex relay network with the FGR scheme can be derived as
(22)ςFGR=∑m=0Ns−1∑n=0m∑k=0∞mn−1kRsCn+k1+λlr2m−nm!k!γ¯RDn+kλm+n+2k×t2βγ¯SRm+ke−t2+t2λlr2/βλγ¯SRΓ1−n−k,u1γ¯RD−∑m=0Ns−1∑n=0mmnRst1m1−βne−t11+λlr2/βλγ¯SRm!γ¯SRmβmγ¯RD×1+λlr2λm−n1−βt1βγ¯SR+1γ¯RD−n−1×Γn+1,1−βt1u1βγ¯SR+u1γ¯RD.From (22), we find that the EST of the considered system is small, when the confidential information rate Rs is small. At the other extreme, when the confidential information rate Rs is too large, the secrecy performance of the considered system is poor which also results in small EST. Hence, there exists an optimal Rs to maximize the EST of the considered system. Furthermore, the EST increases as the self-interference level decreases, which means that the self-interference deteriorates an overall performance in terms of the security-reliability tradeoff.


### 3.1.4. Asymptotic Behavior

To extract additional insights on the FGR scheme, we now investigate the high SNR asymptotic behavior of the SOP, COP, and EST of the considered system. According (16), we have
(23)limλ⟶∞PsoFGR=∑m=0Ns−1∑n=0mt1lr2m−n1−βnt1ne−t1lr2/βγ¯SRm−n!γ¯RDγ¯SRmβm×t1−t1ββγ¯SR+1γ¯RD−n−1.From (19), we have
(24)limλ⟶∞PcoFGR=1−∑m=0Ns−1∑n=0mmn2t2mC¯nlr2m−2ne−t2lr2/βγ¯SRm!γ¯RDγ¯SRmβm×t2C¯γ¯RDβγ¯SR1−n/2K1−n4t2C¯βγ¯SRγ¯RD,where C¯=ld21−βlr2+βNsγ¯SR+γ¯RD1−β. According to (22), we have
(25)limλ⟶∞ςFGR=∑m=0Ns−1∑n=0m∑k=0∞mn−1kRst2m+klr2m−2nC¯n+km!k!γ¯SRm+kγ¯RDn+kβm+k×e−t2lr2/βγ¯SRΓ1−n−k,u2γ¯RD−∑m=0Ns−1∑n=0mmn×Rst1m1−βnlr2m−2nm!γ¯SRmβmγ¯RDe−t1lr2/βγ¯SR1−βt1βγ¯SR+1γ¯RD−n−1Γn+1,1−βt1u2βγ¯SR+u2γ¯RD,where u2=−lr2t1−t2+lr4t1−t22+4t1t2C¯1−β/2t11−β. It is interesting to note that when transmit power is sufficiently large, the security and reliability of the untrusted full-duplex relay network are still depend on the power allocation factor. Thus, the power allocation factor enables a tradeoff between reliability and security for the untrusted full-duplex relay network. From the numerical results, we find that there exists an optimal power allocation factor maximizing the EST.


## 3.1.1. Secrecy Outage Probability

Combining (6) and (9), the SOP of the untrusted full-duplex relay network with the FGR scheme can be given by
(12)PsoFGR=Prβλhsr21−βλhrd2+lr2λ+1>t1=Prhsr2>1−βt1hrd2β+lr2t1λ+t1βλ,where t1=2R0−Rs−1. Noting that X=hsr2 is a central chi-square distribution random variable with 2Ns degrees of freedom, its cumulative distribution function (CDF) is given by
(13)FXx=1−e−x/γ¯SR∑m=0Ns−1xmm!γ¯SRm.On the other hand,Y=hrd2 is an exponential variable with probability density function (PDF) given by
(14)fYy=1γ¯RDe−y/γ¯RD.Based on (13) and (14), we can rewrite (12) as
(15)PsoFGR=∫0∞1−FX1−βt1yβ+lr2t1λ+t1βλfYydy=∑m=0Ns−1e−t1lr2λ+t1/βλγ¯SRm!γ¯RDγ¯SRmβmλm∫0∞e−1−βt1/βγ¯SR+1/γ¯RDy×t1+t1lr2λ+1−βt1λymdy.Then, usingt1+t1lr2λ+1−βt1λym=∑n=0mmnt1+t1lr2λm−n1−βnt1nλnyn, we can calculate PsoFGR as
(16)PsoFGR=∑m=0Ns−1∑n=0mt1lr2λ+t1m−n1−βnt1nλnm−n!γ¯RDγ¯SRmβmλm×1−βt1βγ¯SR+1γ¯RD−n−1e−t1lr2λ+t1/βλγ¯SR.From (16), we find that the SOP increases as the power allocation factor β increases from 0 to 1. This is because that increasing β results in decreasing transmission power of the jamming signal. Moreover, the SOP decreases as lr increases, which means that the self-interference caused by full-duplex transmission is beneficial to the security performance.


## 3.1.2. Connection Outage Probability

Combining (7) and (10), the COP of the untrusted full-duplex relay network with the FGR scheme can be given by
(17)PcoFGR=Prβλ2hsr2hrd2λ+λ2lr2hrd2+C<t2=Prhsr2<t21+λlr2βλ+t2Cβλ2hrd2,where t2=2R0−1. Based on (13) and (14), we can rewrite (17) as
(18)PcoFGR=∫0∞FXt21+λlr2βλ+t2Cβλ2yfYydy=1−∑m=0Ns−1e−t2λlr2+t2/βλγ¯SRm!γ¯RDγ¯SRm∫0∞t2+t2λlr2βλ+t2Cβλ2ym×e−t2C/βλ2γ¯SRy−y/γ¯RDdy.Then, with the help of the binomial theorem and ([24], 3.471.9), the exact COP of the untrusted full-duplex relay network with the FGR scheme can be derived as
(19)PcoFGR=1−∑m=0Ns−1∑n=0mmn2t2mCn1+λlr2m−nm!βmλm+nγ¯RDγ¯SRme−t2λlr2+t2/βλγ¯SR×t2Cγ¯RDβλ2γ¯SR1−n/2K1−n4t2Cβλ2γ¯SRγ¯RD,where K1−n· is the Bessel functions of imaginary argument. From (19), we find that the COP decreases as the power allocation factor β increases from 0 to 1. The main reason is that when β increases, the achievable data rate is larger which makes reliability better. In addition, the COP increases as lr or ld increases, which means that the self-interference caused by full-duplex transmission is harmful for the reliability performance.


## 3.1.3. Effective Secrecy Throughput

Combining (6), (7), and (11), the EST of the untrusted full-duplex relay network with the FGR scheme can be given by
(20)ςFGR=RsPrhsr2<1−βt1hrd2β+lr2t1λ+t1βλ,hsr2>t21+λlr2βλ+t2Cβλ2hrd2.Since1−βt1hrd2/β+lr2t1λ+t1/βλ and t21+λlr2/βλ+t2C/βλ2hrd2 have only one point of intersection when hrd2>0, and the intersection is u1. Thus, expression (20) can be expressed as
(21)ςFGR=Rs∫u1∞FX1−βt1yβ+lr2t1λ+t1βλfYydy−Rs∫u1∞FXt21+λlr2βλ+t2Cβλ2yfYydy=∑m=0Ns−1∑n=0mmnRst2mCn1+λlr2m−ne−t2+t2λlr2/βλγ¯SRm!γ¯SRmβmγ¯RDλm+n×∫u1∞1yne−t2C/βλ2γ¯SRy+y/γ¯RDdy−∑m=0Ns−1∑n=0mmn×Rst1m1−βne−t11+λlr2/βλγ¯SRm!γ¯SRmβmγ¯RD1+λlr2λm−n×∫u1∞yne−1−βt1/βγ¯SR+1/γ¯RDydy,where u1=−b1+b12−4a1c1/2a1, a1=t1λ21−β, c1=−t2C, and b1=λt1−t2lr2λ+1. Then, using [24, 1.211.1], [24, 3.351.2], and [24, 3.381.3], the exact EST of the untrusted full-duplex relay network with the FGR scheme can be derived as
(22)ςFGR=∑m=0Ns−1∑n=0m∑k=0∞mn−1kRsCn+k1+λlr2m−nm!k!γ¯RDn+kλm+n+2k×t2βγ¯SRm+ke−t2+t2λlr2/βλγ¯SRΓ1−n−k,u1γ¯RD−∑m=0Ns−1∑n=0mmnRst1m1−βne−t11+λlr2/βλγ¯SRm!γ¯SRmβmγ¯RD×1+λlr2λm−n1−βt1βγ¯SR+1γ¯RD−n−1×Γn+1,1−βt1u1βγ¯SR+u1γ¯RD.From (22), we find that the EST of the considered system is small, when the confidential information rate Rs is small. At the other extreme, when the confidential information rate Rs is too large, the secrecy performance of the considered system is poor which also results in small EST. Hence, there exists an optimal Rs to maximize the EST of the considered system. Furthermore, the EST increases as the self-interference level decreases, which means that the self-interference deteriorates an overall performance in terms of the security-reliability tradeoff.


## 3.1.4. Asymptotic Behavior

To extract additional insights on the FGR scheme, we now investigate the high SNR asymptotic behavior of the SOP, COP, and EST of the considered system. According (16), we have
(23)limλ⟶∞PsoFGR=∑m=0Ns−1∑n=0mt1lr2m−n1−βnt1ne−t1lr2/βγ¯SRm−n!γ¯RDγ¯SRmβm×t1−t1ββγ¯SR+1γ¯RD−n−1.From (19), we have
(24)limλ⟶∞PcoFGR=1−∑m=0Ns−1∑n=0mmn2t2mC¯nlr2m−2ne−t2lr2/βγ¯SRm!γ¯RDγ¯SRmβm×t2C¯γ¯RDβγ¯SR1−n/2K1−n4t2C¯βγ¯SRγ¯RD,where C¯=ld21−βlr2+βNsγ¯SR+γ¯RD1−β. According to (22), we have
(25)limλ⟶∞ςFGR=∑m=0Ns−1∑n=0m∑k=0∞mn−1kRst2m+klr2m−2nC¯n+km!k!γ¯SRm+kγ¯RDn+kβm+k×e−t2lr2/βγ¯SRΓ1−n−k,u2γ¯RD−∑m=0Ns−1∑n=0mmn×Rst1m1−βnlr2m−2nm!γ¯SRmβmγ¯RDe−t1lr2/βγ¯SR1−βt1βγ¯SR+1γ¯RD−n−1Γn+1,1−βt1u2βγ¯SR+u2γ¯RD,where u2=−lr2t1−t2+lr4t1−t22+4t1t2C¯1−β/2t11−β. It is interesting to note that when transmit power is sufficiently large, the security and reliability of the untrusted full-duplex relay network are still depend on the power allocation factor. Thus, the power allocation factor enables a tradeoff between reliability and security for the untrusted full-duplex relay network. From the numerical results, we find that there exists an optimal power allocation factor maximizing the EST.


## 3.2. Variable Gain Relaying

### 3.2.1. Secrecy Outage Probability

The gain factor employed at the relay determines the quality of signal reception at the destination and has no influence on the eavesdropping ability at the relay. Therefore, the SOP of the VGR scheme is the same as the FGR scheme and it is omitted here.


### 3.2.2. Connection Outage Probability

Combining (8) and (10), the COP of the untrusted full-duplex relay network with the VGR scheme can be given by
(26)PcoVGR=Prβλ2hsr2hrd2C1hsr2+C2hrd2+C3<t2=Prβλ2hrd2−t2C1hsr2<t2C2hrd2+t2C3.Based on (13) and (14), we can rewrite (26) as
(27)PcoVGR=∫0C5fYydy+∫C5∞FXt2C2y+t2C3βλ2y−t2C1fYydy=1−∑m=0Ns−1e−t2C2/βλ2γ¯SRm!γ¯RDγ¯SRmt2C2βλ2m∫C5∞1+C4y−C5m×e−t2C2C4/βλ2γ¯SRy−C5+y/γ¯RDdy,where C4=C3/C2+t2C1/βλ2, and C5=t2C1/βλ2. Then, let y~=y−C5, and use [24, 1.211.1] and [24, 3.471.9], the exact COP of the untrusted full-duplex relay network with the VGR scheme can be derived as
(28)PcoVGR=1−∑m=0Ns−1∑n=0mmn2C4ne−t2C2/βλ2γ¯SR−C5/γ¯RDm!γ¯RDγ¯SRmt2C2βλ2m×t2C2C4γ¯RDβλ2γ¯SR1−n/2K1−n4t2C2C4βλ2γ¯SRγ¯RD.WhenNs⟶∞, we have hsr2≈aNsγ¯SR, where a follows the law of large numbers. Thus, from (26), we clarify that increasing the number of antennas at the source unboundedly is not helpful because the channel between the relay and the destination will finally become the bottleneck and dominate the achievable data rate.


### 3.2.3. Effective Secrecy Throughput

Combining (6), (8), and (11), the EST of the untrusted full-duplex relay network with the VGR scheme can be given by
(29)ςVGR=RsPrhsr2<1−βt1hrd2β+lr2t1λ+t1βλ,hsr2>t2C2hrd2+t2C3βλ2hrd2−t2C1,hrd2>t2C1βλ2.Since1−βt1hrd2/β+lr2t1λ+t1/βλ and t2C2hrd2+t2C3/βλ2hrd2−t2C1 have only one point of intersection when hrd2>0, and the intersection is u3. Thus, the expression (29) can be expressed as
(30)ςVGR=Rs∫u4∞FX1−βt1yβ+lr2t1λ+t1βλfYydy−Rs∫u4∞FXt2C2y+t2C3βλ2y−t2C1fYydy=∑m=0Ns−1∑n=0mmnRsC4ne−t2C2/βλ2γ¯SR−C5/γ¯RDm!γ¯SRmγ¯RDt2C2βλ2m×∫u4−C5∞1yne−t2C2C4/βλ2γ¯SRy−y/γ¯RDdy−∑m=0Ns−1∑n=0mmn×Rst1m1−βne−t11+λlr2/βλγ¯SRm!γ¯SRmβmγ¯RD1+λlr2λm−n×∫u4∞yne−1−βt1/βγ¯SR+1/γ¯RDydy,where u4=maxu3,t2C1/βλ2, u3=−b2+b22−4a2c2/2a2, a2=t1βλ31−β, b2=t1βλ21+lr2λ−t1t2λC11−β−βλt2C2, and c2=−βλt2C3−t1t2C11+lr2λ. Then, using [24, 1.211.1], [24, 3.351.2] and [24, 3.381.3], the exact EST of the untrusted full-duplex relay network with the VGR scheme can be derived as
(31)ςVGR=∑m=0Ns−1∑n=0m∑k=0∞mn−1kRsC4n+ke−t2C2/βλ2γ¯SR−C5/γ¯RDm!k!γ¯SRm+kγ¯RDn+k×t2C2βλ2m+kΓ1−n−k,u4−C5γ¯RD−∑m=0Ns−1∑n=0mmnRst1m1−βne−t11+λlr2/βλγ¯SRm!γ¯SRmβmγ¯RD×1+λlr2λm−n1−βt1βγ¯SR+1γ¯RD−n−1×Γn+1,1−βt1u4βγ¯SR+u4γ¯RD.The derived exact results in (31) provide an efficient way to holistically examine the security and reliability performance with arbitrary system parameters, i.e., the number of antennas at the source and the self-interference cancellation capability at the relay and destination. Additionally, it is worth noting that the EST of the considered system will tend toward zero when Ns⟶∞. An intuitive explanation to this phenomenon is that, more antennas at the source would bring us better reliability performance, but at the same time, the grade of the security is severely degraded. Thus, the designers have to carefully take into account the number of antennas at the source.


### 3.2.4. Asymptotic Behavior

To get more insight, we further study the asymptotic behavior of the COP and EST for the VGR scheme under high SNR case. According to (28), we have
(32)limλ⟶∞PcoVGR=1−∑m=0Ns−1∑n=0mmn2t2mC7mC9ne−t2C7/βγ¯SR−C10/γ¯RDm!βmγ¯RDγ¯SRm×t2C7C9γ¯RDβγ¯SR1−n/2K1−n4t2C7C9βγ¯SRγ¯RD,where C6=βld21−β, C7=lr2+ld21−β2, C8=lr2ld21−β, C9=C8/C7+t2C6/β, and C10=t2C6/β. From (31), we have
(33)limλ⟶∞ςVGR=∑m=0Ns−1∑n=0m∑k=0∞mn−1kRsC9n+ke−t2C7/βγ¯SR−C10/γ¯RDm!k!γ¯SRm+kγ¯RDn+k×t2C7βm+kΓ1−n−k,u6−C10γ¯RD−∑m=0Ns−1∑n=0mmnRst1m1−βnlr2m−2nm!γ¯SRmβmγ¯RD×e−t1lr2/βγ¯SR1−βt1βγ¯SR+1γ¯RD−n−1×Γn+1,1−βt1u6βγ¯SR+u6γ¯RD,where u6=maxu5,t2C6/β, u5=−b3+b32−4a3c3/2a3, a3=t1β1−β, b3=t1βlr2−βt2C7−t1t2C61−β, and c3=−βt2C8−t2t1C6lr2.From the asymptotic behaviour of the SOP, COP, and EST, we find that the continued boosting of the transmit power will not always improve the security and reliability performance of the considered system. This is due to the fact that the self-interference caused by full-duplex transmission becomes a bottleneck of improving the security and reliability performance at the high SNR region.


## 3.2.1. Secrecy Outage Probability

The gain factor employed at the relay determines the quality of signal reception at the destination and has no influence on the eavesdropping ability at the relay. Therefore, the SOP of the VGR scheme is the same as the FGR scheme and it is omitted here.


## 3.2.2. Connection Outage Probability

Combining (8) and (10), the COP of the untrusted full-duplex relay network with the VGR scheme can be given by
(26)PcoVGR=Prβλ2hsr2hrd2C1hsr2+C2hrd2+C3<t2=Prβλ2hrd2−t2C1hsr2<t2C2hrd2+t2C3.Based on (13) and (14), we can rewrite (26) as
(27)PcoVGR=∫0C5fYydy+∫C5∞FXt2C2y+t2C3βλ2y−t2C1fYydy=1−∑m=0Ns−1e−t2C2/βλ2γ¯SRm!γ¯RDγ¯SRmt2C2βλ2m∫C5∞1+C4y−C5m×e−t2C2C4/βλ2γ¯SRy−C5+y/γ¯RDdy,where C4=C3/C2+t2C1/βλ2, and C5=t2C1/βλ2. Then, let y~=y−C5, and use [24, 1.211.1] and [24, 3.471.9], the exact COP of the untrusted full-duplex relay network with the VGR scheme can be derived as
(28)PcoVGR=1−∑m=0Ns−1∑n=0mmn2C4ne−t2C2/βλ2γ¯SR−C5/γ¯RDm!γ¯RDγ¯SRmt2C2βλ2m×t2C2C4γ¯RDβλ2γ¯SR1−n/2K1−n4t2C2C4βλ2γ¯SRγ¯RD.WhenNs⟶∞, we have hsr2≈aNsγ¯SR, where a follows the law of large numbers. Thus, from (26), we clarify that increasing the number of antennas at the source unboundedly is not helpful because the channel between the relay and the destination will finally become the bottleneck and dominate the achievable data rate.


## 3.2.3. Effective Secrecy Throughput

Combining (6), (8), and (11), the EST of the untrusted full-duplex relay network with the VGR scheme can be given by
(29)ςVGR=RsPrhsr2<1−βt1hrd2β+lr2t1λ+t1βλ,hsr2>t2C2hrd2+t2C3βλ2hrd2−t2C1,hrd2>t2C1βλ2.Since1−βt1hrd2/β+lr2t1λ+t1/βλ and t2C2hrd2+t2C3/βλ2hrd2−t2C1 have only one point of intersection when hrd2>0, and the intersection is u3. Thus, the expression (29) can be expressed as
(30)ςVGR=Rs∫u4∞FX1−βt1yβ+lr2t1λ+t1βλfYydy−Rs∫u4∞FXt2C2y+t2C3βλ2y−t2C1fYydy=∑m=0Ns−1∑n=0mmnRsC4ne−t2C2/βλ2γ¯SR−C5/γ¯RDm!γ¯SRmγ¯RDt2C2βλ2m×∫u4−C5∞1yne−t2C2C4/βλ2γ¯SRy−y/γ¯RDdy−∑m=0Ns−1∑n=0mmn×Rst1m1−βne−t11+λlr2/βλγ¯SRm!γ¯SRmβmγ¯RD1+λlr2λm−n×∫u4∞yne−1−βt1/βγ¯SR+1/γ¯RDydy,where u4=maxu3,t2C1/βλ2, u3=−b2+b22−4a2c2/2a2, a2=t1βλ31−β, b2=t1βλ21+lr2λ−t1t2λC11−β−βλt2C2, and c2=−βλt2C3−t1t2C11+lr2λ. Then, using [24, 1.211.1], [24, 3.351.2] and [24, 3.381.3], the exact EST of the untrusted full-duplex relay network with the VGR scheme can be derived as
(31)ςVGR=∑m=0Ns−1∑n=0m∑k=0∞mn−1kRsC4n+ke−t2C2/βλ2γ¯SR−C5/γ¯RDm!k!γ¯SRm+kγ¯RDn+k×t2C2βλ2m+kΓ1−n−k,u4−C5γ¯RD−∑m=0Ns−1∑n=0mmnRst1m1−βne−t11+λlr2/βλγ¯SRm!γ¯SRmβmγ¯RD×1+λlr2λm−n1−βt1βγ¯SR+1γ¯RD−n−1×Γn+1,1−βt1u4βγ¯SR+u4γ¯RD.The derived exact results in (31) provide an efficient way to holistically examine the security and reliability performance with arbitrary system parameters, i.e., the number of antennas at the source and the self-interference cancellation capability at the relay and destination. Additionally, it is worth noting that the EST of the considered system will tend toward zero when Ns⟶∞. An intuitive explanation to this phenomenon is that, more antennas at the source would bring us better reliability performance, but at the same time, the grade of the security is severely degraded. Thus, the designers have to carefully take into account the number of antennas at the source.


## 3.2.4. Asymptotic Behavior

To get more insight, we further study the asymptotic behavior of the COP and EST for the VGR scheme under high SNR case. According to (28), we have
(32)limλ⟶∞PcoVGR=1−∑m=0Ns−1∑n=0mmn2t2mC7mC9ne−t2C7/βγ¯SR−C10/γ¯RDm!βmγ¯RDγ¯SRm×t2C7C9γ¯RDβγ¯SR1−n/2K1−n4t2C7C9βγ¯SRγ¯RD,where C6=βld21−β, C7=lr2+ld21−β2, C8=lr2ld21−β, C9=C8/C7+t2C6/β, and C10=t2C6/β. From (31), we have
(33)limλ⟶∞ςVGR=∑m=0Ns−1∑n=0m∑k=0∞mn−1kRsC9n+ke−t2C7/βγ¯SR−C10/γ¯RDm!k!γ¯SRm+kγ¯RDn+k×t2C7βm+kΓ1−n−k,u6−C10γ¯RD−∑m=0Ns−1∑n=0mmnRst1m1−βnlr2m−2nm!γ¯SRmβmγ¯RD×e−t1lr2/βγ¯SR1−βt1βγ¯SR+1γ¯RD−n−1×Γn+1,1−βt1u6βγ¯SR+u6γ¯RD,where u6=maxu5,t2C6/β, u5=−b3+b32−4a3c3/2a3, a3=t1β1−β, b3=t1βlr2−βt2C7−t1t2C61−β, and c3=−βt2C8−t2t1C6lr2.From the asymptotic behaviour of the SOP, COP, and EST, we find that the continued boosting of the transmit power will not always improve the security and reliability performance of the considered system. This is due to the fact that the self-interference caused by full-duplex transmission becomes a bottleneck of improving the security and reliability performance at the high SNR region.


## 4. Numerical Results

In this section, we present simulation and numerical results to illustrate tendencies of the performance of the untrusted full-duplex relay network. Unless otherwise stated, we set the codeword transmission rate,R0=2 bits/sec/Hz, and the confidential information rate Rs=1 bits/sec/Hz. Also, we set lr=ld=lave. It is evident from Figures 2–7 that the derived closed form expressions of SOP, COP, and EST agree well with their Monte Carlo simulation points, which indicates the accuracy of our theoretical analysis stated above.Figure 2
The outage probability of the FGR scheme and the VGR scheme versus the transmit SNR.Figure 3
The outage probability of the FGR scheme and the VGR scheme versus the self-interference cancellation capability.Figure 4
The EST of the FGR scheme and the VGR scheme versus the transmit SNR.Figure 5
The EST of the FGR scheme and the VGR scheme versus the power allocation factor.Figure 6
The EST of the FGR scheme and the VGR scheme versus the number of antennas at the source.Figure 7
The EST of the FGR scheme and the VGR scheme versus the self-interference cancellation capability.Figure2 shows the SOP and COP versus the transmit SNR of the FGR scheme as well as the VGR scheme, where Ns=2, β=0.5, lave=−15dB, γ¯SR=0 dB, and γ¯RD=5 dB. We first observe that the SOP of the FGR and VGR schemes increases as the transmit SNR increases, whereas the COP of the FGR and VGR schemes decrease as the transmit SNR increases. This is due to the fact that an increasing transmit SNR will lead to a higher achievable eavesdropping rate and achievable data rate. Second, we observe that the SOP and COP of the FGR and VGR schemes converge to a constant when the transmit SNR is sufficiently large. Thus, we can conclude that the transmit SNR is not the larger the better. Moreover, we observe that the COP of the VGR scheme outperforms the FGR scheme at the high SNR region.Figure3 shows the SOP and COP versus the self-interference cancellation capability of the FGR scheme as well as the VGR scheme, where Ns=2, β=0.5, λ=30, γ¯SR=0 dB, and γ¯RD=0 dB. We observe that the SOP of the FGR and VGR schemes decreases as the self-interference cancellation capability decreases, whereas the COP of the FGR and VGR schemes increase as the self-interference cancellation capability decreases. The reason is that as the self-interference cancellation capability decreases, the loopback interference at the destination and the relay will be stronger. Thus, the self-interference cancellation capability enables a tradeoff between security and reliability of the untrusted full-duplex relay network.Figure4 depicts the EST as a function of the transmit SNR, where Ns=2, β=0.5, lave=−10 dB, and γ¯SR=0 dB. We first observe that the EST of the FGR scheme and the VGR scheme converge to a constant when the transmit SNR tends to infinity. This phenomenon is due to the fact that the EST is constrained by the self-interference caused by full-duplex transmission in the high SNR regime. Second, we observe that the EST of the VGR scheme is better than the FGR scheme in the high SNR region. Moreover, we observe that for a given transmit SNR, the EST of both the FGR scheme and the VGR scheme with γ¯RD=10 dB outperform the EST with γ¯RD=0 dB. This is because that the improved quality of the second hop channel is beneficial for secure transmission.Figure5 examines the impact of power allocation factor on the EST performance, where Ns=2, λ=20 dB, łave=−15 dB, and γ¯SR=0 dB. We observe that when the power allocation factor is either extremely small or large, the EST of the FGR scheme and the VGR scheme approach zero. This is because when the power allocation factor is small, it is hard to establish a reliable link from the source to the destination. On the other hand, when the power allocation factor is large, the power of the jamming signal at the destination is poor and secure transmission is impossible. Thus, there exists an optimal power allocation factor maximizing the EST. Moreover, the optimal power allocation factor gradually increases as γ¯RD increases. It indicates that the destination does not need more power to transmit jamming when the channel between the relay and the destination is better.Figure6 illustrates the EST of the FGR scheme and the VGR scheme as a function of the number of antennas at the source, with λ=20 dB, β=0.5, lave=−15 dB, and γ¯SR=0 dB. We observe that the EST of the FGR scheme and the VGR scheme first increase to a peak value and then decrease as a function of the number of antennas at the source. This phenomenon is due to the fact that increasing the number of antennas at the source leads to the enhancement of the reliability performance, but the degradation of the security performance. Second, we observe that the performance gain of the VGR scheme is much more pronounced for small Ns, and gradually diminishes when Ns becomes large.Figure7 plots the EST of the FGR scheme and the VGR scheme as a function of the self-interference cancellation capability, with Ns=2, β=0.5, λ=20, and γ¯SR=0 dB. It can be seen from the figure that the EST of the FGR and VGR schemes decrease as the self-interference cancellation capability decreases. From Figure 3, we know that the self-interference cancellation capability has a positive effect on the secure transmission but incurs a negative effect on the reliable transmission. Thus, we can conclude that the self-interference caused by full-duplex transmission does more harm than good for the untrusted full-duplex relay network. When the the level of self-interference is strong, the EST is close to zero, which suggests that both reliable and secure transmission can not be achieved.


## 5. Conclusion

In this paper, we investigated the security-reliability tradeoff of an untrusted full-duplex relay network over the Rayleigh fading channels. Based on the availability of the CSI, we considered two types of relaying scheme, namely, the FGR scheme and the VGR scheme. The closed form expressions of the SOP, COP, and EST were, respectively, derived, which quantitatively reveal the relationship between the system performances and the number of antennas at the source, power allocation factor, and self-interference cancellation capability, as well as other various parameters. In addition, simple asymptotic results for the high SNR region, i.e.,λ⟶∞, were also provided to offer valuable insights into practical design. Analytical and numerical results demonstrated that the self-interference caused by full-duplex transmission is harmful for the reliability performance, while it is beneficial to the security performance. Moreover, the EST of the considered system increases as the self-interference level decreases, which implies that the self-interference deteriorates an overall performance in terms of the security-reliability tradeoff.



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*Source: 2419430-2022-07-11.xml*