# On the Secrecy Outage Performance of Cooperative NOMA-Assisted Hybrid Satellite-Terrestrial Networks

Authors: Bingjing Li; Ding Xu; Bailing Chen; Ishtiaq Ahmad
Journal: Wireless Communications and Mobile Computing
 (2022)

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## Abstract

This paper investigates the secrecy outage performance of a downlink cooperative nonorthogonal multiple access- (NOMA-) assisted hybrid satellite-terrestrial network, where a satellite source communicates with two terrestrial users based on NOMA, in the presence of a terrestrial eavesdropper. We assume that there is no direct link between the satellite source and the far terrestrial user, and the near terrestrial user acts as a relay to forward the information to the far terrestrial user. The near user is assumed to support both half-duplex mode and the full-duplex mode for information relaying. We derive the closed-form exact expressions for the secrecy outage probability (SOP) of the near user, as well as the closed-form approximate expressions for the SOP of the far user based on the Gaussian-Chebyshev quadrature. Theoretical results are validated by conducting simulations. The impacts of various system parameters on the performance comparison between the half-duplex mode and full-duplex mode are investigated. Particularly, the full-duplex mode is shown to outperform the half-duplex mode when the received signal-to-noise ratio (SNR) at the near user is low or the residual self-interference is low.


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## Body

## 1. Introduction

Integrated/hybrid satellite-terrestrial networks have been proposed to tackle the limitations of standalone satellite networks and terrestrial networks [1]. The main difference between integrated and hybrid satellite-terrestrial networks is that the terrestrial component of an integrated network is controlled by the satellite resource and network management system and uses the same frequency bands as the satellite component, while the terrestrial component of a hybrid network is independent from the satellite component and does not necessarily use the same satellite frequency [2]. In this paper, we do not put restrictions on the relationship between the terrestrial and satellite components in the satellite-terrestrial networks and thus focus on the hybrid satellite-terrestrial networks.Hybrid satellite-terrestrial networks may face some major challenges, such as the lack of availability of line-of-sight (LoS) link due to obstacles; shadowing effects or weather factors will degrade the communication quality between the terrestrial users and the satellites [3]. In this regard, cooperative technology is an effective method to overcome these challenges by the cooperation between network nodes [4]. Specifically, cooperative technology can be divided into two types: one employs dedicated relays to forward the information to the users with inferior channel condition, and the other one relies on the users with better channel condition to forward the information to the users with inferior channel condition [4]. Most of the existing studies on hybrid satellite-terrestrial networks applied cooperative technology based on dedicated relays to forward the messages from the satellites to the terrestrial users, and such networks are called hybrid satellite-terrestrial relay networks [5, 6].Meanwhile, nonorthogonal multiple access (NOMA) is a promising technique for improving the spectral efficiency of satellite-terrestrial networks [7, 8]. NOMA uses nonorthogonal transmission technology to superpose information of multiple users in the same time and frequency domain, while the receivers use successive interference cancellation (SIC) technology to decode their information [7, 9]. A key feature of NOMA technology is that users with better channel conditions can have prior information of other users. By using this prior knowledge, cooperative NOMA (CNOMA) has been utilized to further improve the performance of far users with inferior channel conditions [10–13]. Recently, a large number of studies have applied NOMA on terrestrial users in hybrid satellite-terrestrial relay networks [14, 15]. For example, [14] investigated the outage probability of a NOMA-assisted hybrid satellite-terrestrial relay network and derived the closed-form expression for the outage probability of each NOMA user.Due to the wide coverage of satellite communications, there are serious security problems for hybrid satellite-terrestrial networks [16–18]. In this respect, physical layer security (PLS) has become a promising technology that complements and significantly improves the security of wireless networks [18] and has been applied in hybrid satellite-terrestrial networks to enhance the security of terrestrial users. It is noted that although applying CNOMA to hybrid satellite-terrestrial networks can improve the performance of far users, almost all existing studies have considered CNOMA with dedicated relays. However, when dedicated relays are not available, cooperation among users needs to be considered. To the best of the authors’ knowledge, there is no work on the secrecy outage performance analysis of CNOMA-assisted hybrid satellite-terrestrial networks with user cooperation. Although the secrecy performance of CNOMA-assisted terrestrial networks has been investigated in existing literature such as [11], the current results cannot be directly applied to the hybrid satellite-terrestrial networks, since the channel environments of the hybrid satellite-terrestrial networks are much more complicated than those of the terrestrial networks [19]. This research gap motivates the work in this paper.The main contributions of this paper are summarized as follows:(i)
We consider a CNOMA-assisted hybrid satellite-terrestrial network with a satellite source, one far terrestrial user, and one near terrestrial user, in the presence of a terrestrial eavesdropper. It is assumed that there is no direct link between the satellite source and the far terrestrial user, and thus, the satellite source sends the information of both users to the near terrestrial user, and then, the near terrestrial user acts as a relay to forward the information to the far terrestrial user under the half-duplex relaying mode or the full-duplex relaying mode. It is also assumed that the eavesdropper wiretaps not only the information broadcast by the satellite source but also the information forwarded by the near terrestrial user(ii)
Under the half-duplex relaying mode and the full-duplex relaying mode of the near user, the closed-form exact expressions for the secrecy outage probability (SOP) of the near terrestrial user are derived, and the closed-form approximate expressions for the SOP of the far terrestrial user are derived based on the Gaussian-Chebyshev quadrature(iii)
Theoretical results are validated by Monte Carlo simulation results. It is shown that the full-duplex relaying mode outperforms the half-duplex relaying mode when the received signal-to-noise ratio (SNR) at the near terrestrial user is low or the residual full-duplex self-interference is lowThe rest of this paper is organized as follows. Section2 introduces the related work. In Section 3, we present the system model. In Sections 4 and 5, the SOPs under the half-duplex mode and the full-duplex mode are investigated, respectively. In Section 6, Monte Carlo simulations are provided to verify our theoretical results, followed by conclusions in Section 7.


## 2. Related Work

Hybrid satellite-terrestrial relay networks have attracted a lot of research attention. In [6], the outage performance of a hybrid satellite-terrestrial relay network was investigated, and the outage probabilities under two cooperative relaying strategies were theoretically derived. In [20], a multiuser hybrid satellite-terrestrial relay network with opportunistic scheduling was considered, and the approximate as well as the asymptotic expressions for the outage probability were derived. In [21], an uplink satellite-terrestrial relay network with multiple decode-and-forward terrestrial relays was considered, and the closed-form expressions for the outage probability and the throughput were derived under the partial relay selection scheme. In [22], the performance of cognitive-uplink fixed satellite service and terrestrial fixed service was investigated, and an analytical expression for the network capacity was derived. In [23], a satellite and aerial-integrated network with rate-splitting multiple access (RSMA) was considered, and an iterative penalty function-based algorithm was proposed to solve the problem of maximizing the system sum rate. In [24], a secure beamforming scheme for a RSMA-based cognitive satellite-terrestrial network in the presence of multiple eavesdroppers was proposed to maximize the secrecy-energy efficiency.NOMA or CNOMA has been applied to hybrid satellite-terrestrial networks as well. Specifically, in [15], a NOMA-assisted hybrid satellite-terrestrial relay network with small-cell users and macrocell users was considered, and the closed-form expressions for the outage probabilities were derived. In [25], a satellite multicast network was considered to share the millimeter wave spectrum with a terrestrial network employing NOMA technology, and the system sum rate was maximized by optimizing the beamforming vectors and the power coefficients. In [16], the effect of hardware impairments on the secrecy performance of a NOMA-assisted hybrid satellite-terrestrial relay network was investigated with colluding or noncolluding eavesdroppers, and the closed-form expressions for the SOP under the partial relay selection scheme were derived. In [26], the performance of a NOMA-assisted hybrid satellite-terrestrial relay network was investigated under the partial relay selection scheme and imperfect SIC, and the closed-form expressions for the outage probability and the ergodic capacity were derived. Besides NOMA and CNOMA, there are other emerging technologies such as reconfigurable intelligent surface (RIS) [27, 28], which have been considered to improve the performance of hybrid satellite-terrestrial networks. For example, in [27], the beamforming was designed to minimize the total transmit power under the user rate constraints for a RIS-aided hybrid satellite-terrestrial relay network.Physical layer security has been studied in CNOMA networks. In [29], the physical layer security of a two-user CNOMA network, where the strong user acts as a relay for the weak user, in the presence of an eavesdropper, was studied, and the closed-form expressions for SOPs of the two users were derived. The work in [30] extended the work in [29] to a multiantenna scenario. In [31], the secrecy performance of a CNOMA network based on half-duplex and full-duplex amplify-and-forward and decode-and-forward relaying protocols in general κ-μ fading channels was investigated. Physical layer security is also a hot research topic in hybrid satellite-terrestrial networks. In [17], the impacts of the relay selection and user scheduling scheme on the secrecy performance for a hybrid satellite-terrestrial relay network with multiple terrestrial relays were investigated. In [18], a downlink hybrid satellite-terrestrial relay network with multiple terrestrial users with the help from multiple cooperative relays, in the presence of multiple eavesdroppers, was considered; the secrecy performances under amplify-and-forward and decode-and-forward protocols were investigated. In [32], the secrecy performance for a NOMA-assisted hybrid satellite-terrestrial network consisting of a satellite source and multiple terrestrial users with the help from a terrestrial base station was investigated.To our best knowledge, current works on the secrecy performance of CNOMA-assisted hybrid satellite-terrestrial network networks such as [17, 18, 32] are all based on dedicated cooperative relays. However, when dedicated relays are unavailable, the cooperation among users is desirable. Such consideration motivates the work in this paper to investigate the secrecy performance of CNOMA-assisted hybrid satellite-terrestrial networks with user cooperation.


## 3. System Model

We consider a CNOMA-based hybrid satellite-terrestrial network consisting of a geosynchronous earth orbit (GEO) satellite source (S) and two terrestrial users (near user D1 and far user D2), in the presence of an eavesdropper (E), as shown in Figure 1. The direct transmission link between S and D2 is assumed to be absent due to shadow fading caused by raining, fog, or other obstacles [18, 20]. We assume that the near user D1 serves as a decoded and forwarding relay to help in transmitting information to the far user D2, while E wiretaps the information broadcast by the satellite source S and the information forwarded by the near user D1 (it is assumed that the eavesdropper equips two receivers: one for decoding the satellite signals and the other one for decoding the terrestrial signals, to eavesdrop the information of both the satellite source and the near user [33]). Let hS1, hSE, h12, and h1E denote the channel coefficients from S to D1, from S to E, from D1 to D2, and from D1 to E, respectively (the channel state information (CSI) is often assumed to be available for resource scheduling. This can be realized by channel training and estimation [34]. In particular, in the training phase, D1 broadcasts pilot signal; S and D2 receive the pilot signal for channel estimation and then feed the CSI back to D1). Additionally, all the links in the network are considered to be independent and flat fading. Similar to the existing works [16, 35], perfect successive interference cancellation (SIC) is assumed to be available. It is assumed that delay-sensitive services are carried by Di,i∈1,2, and there is a minimum rate denoted as Ri,i∈1,2 required to guarantee the quality of service (QoS) for Di,i∈1,2.Figure 1
System model.The satellite links fromS to D1 and from S to E are assumed to follow the shadowed Rician fading model [20, 21, 36], which is the most commonly used channel model in land mobile satellite communication systems [37]. Under the shadowed Rician fading model, the channel coefficient hi,i∈S1,SE can be written as
(1)hi=Cifi,i∈S1,SE,where Ci is the scaling parameter including various practical effects such as the free space loss (FSL) and the antenna gain and fi is the small-scale fading. The Ci can be expressed as [37]
(2)Ci=λ4πGS,iGidi2+d02,i∈S1,SE,where λ is the wavelength, di is the distance between the terrestrial user and the center of the satellite beam, d0≈35786 km is the height of the GEO satellite, and Gi and GS,i denote the receive antenna gain and the satellite beam gain of the terrestrial user, respectively.According to [38], Gi,i∈S1,SE can be expressed as
(3)Gi=G¯max,0∘<β<1∘,32−25logβ,1∘<β<48∘,−10,48∘<β<180∘,where G¯max is the maximum antenna gain at the boresight and β is the off-boresight angle. Denote by θ the angle between the terrestrial receiver and the satellite beam and by θ¯ the 3 dB angle. The expression for GS,i,i∈S1,SE is given by [37]
(4)GS,i=GmaxJ1u2u+36J3uu32,i∈S1,SE,where Gmax denotes the maximal beam gain, u=2.07123sinθ/sinθ¯, and J1. and J3. represent the first-kind bessel functions of orders 1 and 3, respectively. To attain the best beam gain, we have θ⟶0 [39], and thus, GS,i≈Gmax and Ci=Cimax,i∈S1,SE, where Cimax=λ/4πGmaxGi/d2+d02.Under the shadowed Rician fading model, the probability distribution function (PDF) offi2,i∈S1,SE can be described as [16, 21]
(5)ffi2x=αie−βix1F1mi;1;δix,i∈S1,SE,where αi=1/2bi2bimi/2bimi+Ωi, βi=1/2bi, δi=Ωi/2bi2bimi+Ωi, Ωi is the average power of the LOS component, 2bi is the average power of the multipath component, mi is the Nakagami parameter, and 1F1a;b;x denotes the confluent hypergeometric function [40].The terrestrial links fromD1 to D2 and from D1 to E are assumed to follow the Rayleigh fading model. Thus, the PDF and the cumulative distribution function (CDF) of hi2,i∈12,1E are given by
(6)fhi2x=1λie−x/λi,Fhi2x=1−e−x/λi,respectively, where λi is the average channel gain.It is assumed that all the nodes exceptD1 are equipped with a single antenna [16, 17, 41]. It is also assumed that D1 is equipped with one or two antennas. Specifically, D1 works under the half-duplex mode when only one antenna is available and works under the full-duplex mode when two antennas are available, where one antenna is for transmitting and the other one for receiving. Next, we discuss these two modes.


### 3.1. Half-Duplex Mode

WhenD1 works under the half-duplex mode, the entire transmission time is divided into two phases. In the first time phase, S transmits the superimposed signal (considering hardware impairments is not the focus of this paper. Interested readers may refer to [39, 42]) Psα1x1+Psα2x2 to D1 with Exi2=1,i=1,2, where Ps is the transmit power of S and αi is the power allocation coefficient for xi. The received signal at D1 is given by
(7)y1HD=hS1Psα1x1+Psα2x2+n1,where n1~CN0,σ12 denotes the additive white Gaussian noise (AWGN) at D1. It is assumed that D1 adopts SIC to decode the received message [16, 26]. Specifically, x2 is decoded first, and then, x1 is decoded by cancelling the x2 from the received message (in order to successfully perform SIC, a minimum received power difference is required to distinguish between the desired signal and the interference signal [43]. This means that the difference between the two power allocation coefficients α1 and α2 shall be large enough such that SIC can be successfully performed at D1. It is noted that the performance analysis in this paper does not put restriction on the values of α1 and α2. Thus, our analysis is still valid when the values of α1 and α2 are set to satisfy the SIC requirement). The signal-to-interference-plus-noise ratio (SINR) at D1 for decoding x2 is
(8)γ1x2,HD=hS12Psα2hS12Psα1+σ12=γS1α2γS1α1+1,where γS1=γ¯S1fS12 and γ¯S1=CS1max2Ps/σ12. Then, after applying SIC, the SNR at D1 to decode its own signal x1 is given by
(9)γx1,HD=hS12Psα1σ12=γS1α1.Meanwhile, since there is a direct link betweenE and S, the E can also receive the information broadcast by S. Thus, the message received by E is expressed as
(10)yE,1HD=hSEPsα1x1+Psα2x2+nE,where nE~CN0,σE2 denotes the AWGN at E. Here, we consider a worst-case scenario, where E has multiuser detection capability and adopts parallel interference cancellation (PIC) to decode the messages of D1 and D2 [16, 44, 45]. Thus, the received SNRs at E to decode the messages x1 and x2 in the second time phase are
(11)γEx1,HD=hSE2Psα1σE2=γSEα1,γE,1x2,HD=hSE2Psα2σE2=γSEα2,respectively, where γSE=γ¯SEfSE2 and γ¯SE=CSEmax2Ps/σE2.In the second time phase,D1 forwards the information to D2, where E can also intercept. The received message at D2 and E are
(12)y2HD=h12PRx2+n2,yE,2HD=h1EPRx2+nE,respectively, where n2~CN0,σ22 denotes the AWGN at D2 and PR is the transmit power of D1. Thus, the SNRs at D2 and E for decoding x2 in the second time phase can be written as
(13)γ2x2,HD=h122PRσ22=γ12,γE,2x2,HD=h1E2PRσE2=γ1E,respectively, where γ12=PRh122/σ22 and γ1E=PRh1E2/σE2. Note that γ12 and γ1E follow exponential distributions with mean values γ¯12 and γ¯1E, respectively, where γ¯12=PR/σ22 and γ¯1E=PR/σE2.The end-to-end SNR atD2 for decoding x2 is the minimum SNR in the two time phases, which can be written as [26]
(14)γx2,HD=minγ1x2,HD,γ2x2,HD.As forE to decode x2, a worst-case scenario is considered, where E adopts the maximal ratio combining (MRC) to combine the x2 received in the first and the second time phases. Thus, the end-to-end SNR at E for decoding x2 is
(15)γEx2,HD=γE,1x2,HD+γE,2x2,HD.


### 3.2. Full-Duplex Mode

WhenD1 works under the full-duplex mode, during the whole transmission time, S can transmit with the power Ps and the power allocation coefficients α1 and α2, while D1 can forward the information to D2 with transmit power PR. The message received by D1 can be expressed as
(16)y1FD=hS1Psα1x1+Psα2x2+xLI+n1,where xLI is the residual self-interference signal. We assume that multiple analog and digital self-interference cancellation techniques are adopted to suppress the self-interference. Accordingly, xLI can be modelled as a Gaussian distributed signal which is independent from the transmitted signal due to various affecting factors in the self-interference cancellation procedures [46]. Thus, for the convenience of later analysis, we denote the power of xLI as ζPR with 0≤ζ≤1 to identify the relationship between the self-interference signal power and the transmit signal power [47–49].It is assumed thatD1 decodes x2 first, and the SINR at D1 for decoding x2 is
(17)γ1x2,FD=hS12Psα2hS12Psα1+ζPR+σ12=γS1α2γS1α1+ϖ,where ϖ=ζPR+σ12/σ12. After applying SIC to cancel the x2 from the received message, the D1 can proceed to decode x1, where the SNR for decoding x1 is
(18)γx1,FD=hS12Psα1ζPR+σ12=γS1α1ϖ.SinceD1 relays x2 to D2 with transmit power PR, the message received by D2 is
(19)y2FD=h12PRx2+n2,and the SNR for decoding x2 at D2 is given by
(20)γ2x2,FD=h122PRσ22=γ12.Thus, the end-to-end SNR atD2 for decoding x2 is the minimum SNR that can be achieved at D1 and D2, which is written as
(21)γx2,FD=minγ1x2,FD,γ2x2,FD.The message received atE can be written as
(22)yE,1FD=hSEPsα1x1+Psα2x2+h1EPRx2+nE.It is assumed thatE adopts PIC to decode x1 and x2, and thus, the SINRs for decoding x1 and x2 are given by
(23)γEx1,FD=hSE2Psα1σE2=γSEα1,(24)γEx2,FD=γE,1x2,FD+γE,2x2,FD,respectively, where
(25)γE,1x2,FD=hSE2Psα2σE2=γSEα2,γE,2x2,FD=h1E2PRσE2=γ1E.


## 3.1. Half-Duplex Mode

WhenD1 works under the half-duplex mode, the entire transmission time is divided into two phases. In the first time phase, S transmits the superimposed signal (considering hardware impairments is not the focus of this paper. Interested readers may refer to [39, 42]) Psα1x1+Psα2x2 to D1 with Exi2=1,i=1,2, where Ps is the transmit power of S and αi is the power allocation coefficient for xi. The received signal at D1 is given by
(7)y1HD=hS1Psα1x1+Psα2x2+n1,where n1~CN0,σ12 denotes the additive white Gaussian noise (AWGN) at D1. It is assumed that D1 adopts SIC to decode the received message [16, 26]. Specifically, x2 is decoded first, and then, x1 is decoded by cancelling the x2 from the received message (in order to successfully perform SIC, a minimum received power difference is required to distinguish between the desired signal and the interference signal [43]. This means that the difference between the two power allocation coefficients α1 and α2 shall be large enough such that SIC can be successfully performed at D1. It is noted that the performance analysis in this paper does not put restriction on the values of α1 and α2. Thus, our analysis is still valid when the values of α1 and α2 are set to satisfy the SIC requirement). The signal-to-interference-plus-noise ratio (SINR) at D1 for decoding x2 is
(8)γ1x2,HD=hS12Psα2hS12Psα1+σ12=γS1α2γS1α1+1,where γS1=γ¯S1fS12 and γ¯S1=CS1max2Ps/σ12. Then, after applying SIC, the SNR at D1 to decode its own signal x1 is given by
(9)γx1,HD=hS12Psα1σ12=γS1α1.Meanwhile, since there is a direct link betweenE and S, the E can also receive the information broadcast by S. Thus, the message received by E is expressed as
(10)yE,1HD=hSEPsα1x1+Psα2x2+nE,where nE~CN0,σE2 denotes the AWGN at E. Here, we consider a worst-case scenario, where E has multiuser detection capability and adopts parallel interference cancellation (PIC) to decode the messages of D1 and D2 [16, 44, 45]. Thus, the received SNRs at E to decode the messages x1 and x2 in the second time phase are
(11)γEx1,HD=hSE2Psα1σE2=γSEα1,γE,1x2,HD=hSE2Psα2σE2=γSEα2,respectively, where γSE=γ¯SEfSE2 and γ¯SE=CSEmax2Ps/σE2.In the second time phase,D1 forwards the information to D2, where E can also intercept. The received message at D2 and E are
(12)y2HD=h12PRx2+n2,yE,2HD=h1EPRx2+nE,respectively, where n2~CN0,σ22 denotes the AWGN at D2 and PR is the transmit power of D1. Thus, the SNRs at D2 and E for decoding x2 in the second time phase can be written as
(13)γ2x2,HD=h122PRσ22=γ12,γE,2x2,HD=h1E2PRσE2=γ1E,respectively, where γ12=PRh122/σ22 and γ1E=PRh1E2/σE2. Note that γ12 and γ1E follow exponential distributions with mean values γ¯12 and γ¯1E, respectively, where γ¯12=PR/σ22 and γ¯1E=PR/σE2.The end-to-end SNR atD2 for decoding x2 is the minimum SNR in the two time phases, which can be written as [26]
(14)γx2,HD=minγ1x2,HD,γ2x2,HD.As forE to decode x2, a worst-case scenario is considered, where E adopts the maximal ratio combining (MRC) to combine the x2 received in the first and the second time phases. Thus, the end-to-end SNR at E for decoding x2 is
(15)γEx2,HD=γE,1x2,HD+γE,2x2,HD.


## 3.2. Full-Duplex Mode

WhenD1 works under the full-duplex mode, during the whole transmission time, S can transmit with the power Ps and the power allocation coefficients α1 and α2, while D1 can forward the information to D2 with transmit power PR. The message received by D1 can be expressed as
(16)y1FD=hS1Psα1x1+Psα2x2+xLI+n1,where xLI is the residual self-interference signal. We assume that multiple analog and digital self-interference cancellation techniques are adopted to suppress the self-interference. Accordingly, xLI can be modelled as a Gaussian distributed signal which is independent from the transmitted signal due to various affecting factors in the self-interference cancellation procedures [46]. Thus, for the convenience of later analysis, we denote the power of xLI as ζPR with 0≤ζ≤1 to identify the relationship between the self-interference signal power and the transmit signal power [47–49].It is assumed thatD1 decodes x2 first, and the SINR at D1 for decoding x2 is
(17)γ1x2,FD=hS12Psα2hS12Psα1+ζPR+σ12=γS1α2γS1α1+ϖ,where ϖ=ζPR+σ12/σ12. After applying SIC to cancel the x2 from the received message, the D1 can proceed to decode x1, where the SNR for decoding x1 is
(18)γx1,FD=hS12Psα1ζPR+σ12=γS1α1ϖ.SinceD1 relays x2 to D2 with transmit power PR, the message received by D2 is
(19)y2FD=h12PRx2+n2,and the SNR for decoding x2 at D2 is given by
(20)γ2x2,FD=h122PRσ22=γ12.Thus, the end-to-end SNR atD2 for decoding x2 is the minimum SNR that can be achieved at D1 and D2, which is written as
(21)γx2,FD=minγ1x2,FD,γ2x2,FD.The message received atE can be written as
(22)yE,1FD=hSEPsα1x1+Psα2x2+h1EPRx2+nE.It is assumed thatE adopts PIC to decode x1 and x2, and thus, the SINRs for decoding x1 and x2 are given by
(23)γEx1,FD=hSE2Psα1σE2=γSEα1,(24)γEx2,FD=γE,1x2,FD+γE,2x2,FD,respectively, where
(25)γE,1x2,FD=hSE2Psα2σE2=γSEα2,γE,2x2,FD=h1E2PRσE2=γ1E.


## 4. Secrecy Outage Probability under Half-Duplex Mode

The SOP, defined as the probability that the secrecy rate is below a given threshold [41, 50, 51], is investigated. Under the half-duplex mode, the SOP of Di,i∈1,2 is expressed as
(26)Piout,HD=Pr12log1+γxi,HD−12log1+γExi,HD+<Ri,where .+=max0,..


### 4.1. SOP of the Near User

First, the SOP ofD1 is studied. According to the definition in (26), the SOP of D1 can be expressed as
(27)P1out,HD=Pr12log1+γx1,HD−log1+γEx1,HD<12log1+γ¯1HD=Pr1+γx1,HD1+γEx1,HD<1+γ¯1HD=Prγx1,HD<γ¯1HD+γEx1,HD1+γ¯1HD=∫0∞‍fγEx1,HDy×Fγx1,HDγ¯1HD+y1+γ¯1HDdy.where γ¯iHD=22Ri−1 denotes the SINR for the target secrecy rate denoted as Ri for Di under the half-duplex mode.In order to obtain a closed-form expression forP1out,HD, we first deduce the CDF of γx1,HD and the PDF of γEx1,HD. Specifically, the CDF of γx1,HD can be obtained as
(28)Fγx1,HDx=PrγS1α1<x=FγS1xα1.From (5), by supposing γi=γ¯ifi2,i∈S1,SE, the PDF of γi can be expressed as
(29)fγix=αiγ¯ie−βi/γ¯ix1F1mi;1;δiγ¯ix.Under integermi, the PDF of γi can be rewritten as [16, 21, 32]
(30)fγix=αi∑k=0mi−11−mik−δikk!2γ¯ik+1xkexp−Δix,where Δi=βi−δi/γ¯i and ·k is the Pochhammer symbol [40]. Then, the CDF of γi,i∈S1,SE can be calculated as
(31)Fγix=1−αi∑k=0mi−1∑t=0k1−mik−δikk!γ¯ik+1t!Δik−t+1xtexp−Δix.By using (31), the CDF of γx1,HD in (28) can be rewritten as
(32)Fγx1,HDx=1−αS1∑k=0mS1−1∑t=0k1−mS1k−δS1kk!γ¯S1k+1t!ΔS1k−t+1xα1texp−ΔS1xα1.By following similar procedures, the PDF ofγEx1,HD can be obtained as
(33)fγEx1,HDx=1α1fγSExα1=αSEα1∑k=0mSE−11−mSEk−δSEkk!2γ¯SEk+1xα1kexp−ΔSExα1.By substituting (32) and (33) into (27), the P1out,HD can be obtained as
(34)P1out,HD=∫0∞αSEα1∑k1=0mSE−11−mSEk1−δSEk1k1!2γ¯SEk1+1yα1k1exp−ΔSEyα1×1−αS1∑k=0mS1−1∑t=0k1−mS1k−δS1kk!γ¯S1k+1t!ΔS1k−t+1γ¯1HD+y1+γ¯1HDα1t×exp−ΔS1γ¯1HD+y1+γ¯1HDα1dy.By definingAk,t=αS1∑k=0mS1−1∑t=0k1−mS1k−δS1k/k!γ¯S1k+1t!ΔS1k−t+1 and Bk1=αSE∑k1=0mSE−11−mSEk1−δSEk1/k1!2γ¯SEk1+1 and with the help of [40], we can simplify (34) as
(35)P1out,HD=1−∫0∞Ak,tBk1α1yα1k1exp−ΔSEyα11α1t×∑l=0ttlγ¯1HDt−l1+γ¯1HDlyl×exp−ΔS1γ¯1HD+y1+γ¯1HDα1dy=1−∫0∞Ak,tBk1α1k1+t+1∑l=0ttlγ¯1HDt−l1+γ¯1HDl×exp−ΔS1γ¯1HDα1yk1+lexp−ΔSE+ΔS11+γ¯1HDα1ydy.Through some simple calculations of (35), the closed-form expression for the SOP of D1 can be obtained as
(36)P1out,HD=1−Ak,tBk1α1k1+t+1∑l=0ttlγ¯1HDt−l1+γ¯1HDlexp−ΔS1γ¯1HDα1×k1+l!ΔSE+ΔS11+γ¯1HDα1−k1+l−1.Remark 1.
Whenγ¯S1 approaches to infinite, the CDF of γS1 can be approximated as [16, 26]
(37)FγS1x≈αS1γ¯S1x+ox,where ox means the infinitesimal of higher order for x. Thus, the P1out,HD in the high SNR region is approximated as
(38)P1out,HD,∞=∫0∞Bk1α1yα1k1exp−ΔSEyα1αS1γ¯1HD+y1+γ¯1HDγ¯S1α1dy=Bk1αS1γ¯1HDα1γ¯S1ΔSEk1+1k1!+Bk1αS11+γ¯1HDγ¯S1ΔSEk1+2k1+1!.
It is seen from (38) that as γ¯S1⟶∞, the P1out,HD,∞ approaches to zero. Therefore, the SOP of D1 does not saturate when the SNR increases.


### 4.2. SOP of the Far User

Then, we derive the SOP ofD2. The SOP of D2 can be expressed as
(39)P2out,HD=Pr12log1+γx2,HD−log1+γEx2,HD<12log1+γ¯2HD=∫0∞fγEx2,HDy×Fγx2,HDγ¯2HD+y1+γ¯2HDdy.To simplify (39), the CDF of γx2,HD and the PDF of γEx2,HD are first derived. The CDF of γx2,HD is expressed as
(40)Fγx2,HDx=Prminγ1x2,HD,γ2x2,HD<x=1−Prminγ1x2,HD,γ2x2,HD≥x=1−Prγ1x2,HD≥x⏟J1Prγ2x2,HD≥x⏟J2,where J1 can be written as
(41)J1=PrγS1α2γS1α1+1≥x=PrγS1α2−xα1≥x.Note that whenα2−xα1≤0, the inequality γS1α2−xα1≥x cannot be satisfied. Thus, we have J1=0 and Fγx2,HD=1. Otherwise, when α2−xα1>0, the J1 and J2 can be evaluated as
(42)J1=PrγS1≥xα2−xα1=1−FγS1xα2−xα1=Ak,txα2−xα1texp−ΔS1xα2−xα1,J2=exp−xγ¯12,respectively. Based on the above analysis, the CDF of γx2,HD is given by
(43)Fγx2,HDx=1−Ak,txα2−xα1texp−ΔS1xα2−xα1−xγ¯12,x<α2α1,1,x≥α2α1.As for the PDF ofγEx2,HD, it can be obtained as [40]
(44)fγEx2,HDz=∫0zfγE,1x2,HDxfγE,2x2,HDz−xdx=∫0z1α2fγSExα2fγ1Ez−xdx=Ck1exp−zγ¯1Ek1!μk1+1−exp−μz∑t1=0k1k1!t1!zt1μk1−t1+1,where Ck1=αSE/γ¯1E∑k1=0mSE−11−mSEk1−δSEk1/k1!2γ¯SEk1+1α2k1+1 and μ=ΔSEγ¯1E−α2/α2γ¯1E.By substituting (43) and (44) into (39), the P2out,HD is obtained as given by (45), where QHD=α2−α1γ¯2HD/α11+γ¯2HD.(45)P2out,HD=∫0QHDfγEx2,HDy1−Ak,tγ¯2HD+y1+γ¯2HDα2−γ¯2HD+y1+γ¯2HDα1texp−ΔS1γ¯2HD+y1+γ¯2HDα2−γ¯2HD+y1+γ¯2HDα1exp−γ¯2HD+y1+γ¯2HDγ¯12dy+∫QHD∞fγEx2,HDydy=1−∫0QHDfγEx2,HDyAk,tγ¯2HD+y1+γ¯2HD/α2−γ¯2HD+y1+γ¯2HDα1texp−ΔS1γ¯2HD+y1+γ¯2HD/α2−γ¯2HD+y1+γ¯2HDα1exp−γ¯2HD+y1+γ¯2HD/γ¯12dy⏟H1.To the best of authors’ knowledge, it is hard to derive the exact expression forH1 in (45). Thus, we adopt the Gaussian-Chebyshev quadrature to approximate H1 as [52]
(46)P2out,HD≈1−QHD2∑i=1KωiH1yi,where H1yi is given by (47), K is the number of terms, ωi is the Gaussian weight, which is given in Table (25.4) of [52], yi=QHDxi/2+QHD/2, and xi is the ith zero of Legendre polynomials.
(47)H1y=Ak,tCk1exp−yγ¯1Ek1!μk1+1−exp−μy∑t1=0k1k1!t1!yt1μk1−t1+1×γ¯2HD+y1+γ¯2HDα2−γ¯2HD+y1+γ¯2HDα1texp−ΔS1γ¯2HD+y1+γ¯2HDα2−γ¯2HD+y1+γ¯2HDα1exp−γ¯2HD+y1+γ¯2HDγ¯12.Remark 2.
In the high SNR region, i.e.,γ¯S1⟶∞ and γ¯12⟶∞, we have γ1x2,HD≈α2/α1 and γx2,HD=α2/α1. Thus, P2out,HD can be approximated as
(48)P2out,HD,∞=Prα2α1<γ¯2HD+γEx2,HD1+γ¯2HD=Prα2−α1γ¯2HDα11+γ¯2HD<γEx2,HD=∫QHD∞Ck1exp−y/γ¯1Ek1!/μk1+1dy⏟J3−∫QHD∞Ck1exp−y/γ¯1Eexp−μy∑t1=0k1k1!/t1!yt1/μk1−t1+1dy⏟J4,where J3 and J4 can be obtained as
(49)J3=Ck1γ¯1Ek1!μk1+1exp−QHDγ¯1E,J4=Ck1∑t1=0k11μk1−t1+1∑l=0t1k1!l!QHDl1+μγ¯1E/γ¯1Et1−l+1,respectively. It can be shown that, unlike D1, the SOP of D2 saturates in the high SNR region.Remark 3.
Note that whenα1≥1/1+γ¯2HD, we have QHD≤0, and thus, the SOP of user D2 is 1. This means that the power allocation coefficient for D1 shall be smaller than 1/1+γ¯2HD; otherwise, D2 will be always in secrecy outage.


## 4.1. SOP of the Near User

First, the SOP ofD1 is studied. According to the definition in (26), the SOP of D1 can be expressed as
(27)P1out,HD=Pr12log1+γx1,HD−log1+γEx1,HD<12log1+γ¯1HD=Pr1+γx1,HD1+γEx1,HD<1+γ¯1HD=Prγx1,HD<γ¯1HD+γEx1,HD1+γ¯1HD=∫0∞‍fγEx1,HDy×Fγx1,HDγ¯1HD+y1+γ¯1HDdy.where γ¯iHD=22Ri−1 denotes the SINR for the target secrecy rate denoted as Ri for Di under the half-duplex mode.In order to obtain a closed-form expression forP1out,HD, we first deduce the CDF of γx1,HD and the PDF of γEx1,HD. Specifically, the CDF of γx1,HD can be obtained as
(28)Fγx1,HDx=PrγS1α1<x=FγS1xα1.From (5), by supposing γi=γ¯ifi2,i∈S1,SE, the PDF of γi can be expressed as
(29)fγix=αiγ¯ie−βi/γ¯ix1F1mi;1;δiγ¯ix.Under integermi, the PDF of γi can be rewritten as [16, 21, 32]
(30)fγix=αi∑k=0mi−11−mik−δikk!2γ¯ik+1xkexp−Δix,where Δi=βi−δi/γ¯i and ·k is the Pochhammer symbol [40]. Then, the CDF of γi,i∈S1,SE can be calculated as
(31)Fγix=1−αi∑k=0mi−1∑t=0k1−mik−δikk!γ¯ik+1t!Δik−t+1xtexp−Δix.By using (31), the CDF of γx1,HD in (28) can be rewritten as
(32)Fγx1,HDx=1−αS1∑k=0mS1−1∑t=0k1−mS1k−δS1kk!γ¯S1k+1t!ΔS1k−t+1xα1texp−ΔS1xα1.By following similar procedures, the PDF ofγEx1,HD can be obtained as
(33)fγEx1,HDx=1α1fγSExα1=αSEα1∑k=0mSE−11−mSEk−δSEkk!2γ¯SEk+1xα1kexp−ΔSExα1.By substituting (32) and (33) into (27), the P1out,HD can be obtained as
(34)P1out,HD=∫0∞αSEα1∑k1=0mSE−11−mSEk1−δSEk1k1!2γ¯SEk1+1yα1k1exp−ΔSEyα1×1−αS1∑k=0mS1−1∑t=0k1−mS1k−δS1kk!γ¯S1k+1t!ΔS1k−t+1γ¯1HD+y1+γ¯1HDα1t×exp−ΔS1γ¯1HD+y1+γ¯1HDα1dy.By definingAk,t=αS1∑k=0mS1−1∑t=0k1−mS1k−δS1k/k!γ¯S1k+1t!ΔS1k−t+1 and Bk1=αSE∑k1=0mSE−11−mSEk1−δSEk1/k1!2γ¯SEk1+1 and with the help of [40], we can simplify (34) as
(35)P1out,HD=1−∫0∞Ak,tBk1α1yα1k1exp−ΔSEyα11α1t×∑l=0ttlγ¯1HDt−l1+γ¯1HDlyl×exp−ΔS1γ¯1HD+y1+γ¯1HDα1dy=1−∫0∞Ak,tBk1α1k1+t+1∑l=0ttlγ¯1HDt−l1+γ¯1HDl×exp−ΔS1γ¯1HDα1yk1+lexp−ΔSE+ΔS11+γ¯1HDα1ydy.Through some simple calculations of (35), the closed-form expression for the SOP of D1 can be obtained as
(36)P1out,HD=1−Ak,tBk1α1k1+t+1∑l=0ttlγ¯1HDt−l1+γ¯1HDlexp−ΔS1γ¯1HDα1×k1+l!ΔSE+ΔS11+γ¯1HDα1−k1+l−1.Remark 1.
Whenγ¯S1 approaches to infinite, the CDF of γS1 can be approximated as [16, 26]
(37)FγS1x≈αS1γ¯S1x+ox,where ox means the infinitesimal of higher order for x. Thus, the P1out,HD in the high SNR region is approximated as
(38)P1out,HD,∞=∫0∞Bk1α1yα1k1exp−ΔSEyα1αS1γ¯1HD+y1+γ¯1HDγ¯S1α1dy=Bk1αS1γ¯1HDα1γ¯S1ΔSEk1+1k1!+Bk1αS11+γ¯1HDγ¯S1ΔSEk1+2k1+1!.
It is seen from (38) that as γ¯S1⟶∞, the P1out,HD,∞ approaches to zero. Therefore, the SOP of D1 does not saturate when the SNR increases.


## 4.2. SOP of the Far User

Then, we derive the SOP ofD2. The SOP of D2 can be expressed as
(39)P2out,HD=Pr12log1+γx2,HD−log1+γEx2,HD<12log1+γ¯2HD=∫0∞fγEx2,HDy×Fγx2,HDγ¯2HD+y1+γ¯2HDdy.To simplify (39), the CDF of γx2,HD and the PDF of γEx2,HD are first derived. The CDF of γx2,HD is expressed as
(40)Fγx2,HDx=Prminγ1x2,HD,γ2x2,HD<x=1−Prminγ1x2,HD,γ2x2,HD≥x=1−Prγ1x2,HD≥x⏟J1Prγ2x2,HD≥x⏟J2,where J1 can be written as
(41)J1=PrγS1α2γS1α1+1≥x=PrγS1α2−xα1≥x.Note that whenα2−xα1≤0, the inequality γS1α2−xα1≥x cannot be satisfied. Thus, we have J1=0 and Fγx2,HD=1. Otherwise, when α2−xα1>0, the J1 and J2 can be evaluated as
(42)J1=PrγS1≥xα2−xα1=1−FγS1xα2−xα1=Ak,txα2−xα1texp−ΔS1xα2−xα1,J2=exp−xγ¯12,respectively. Based on the above analysis, the CDF of γx2,HD is given by
(43)Fγx2,HDx=1−Ak,txα2−xα1texp−ΔS1xα2−xα1−xγ¯12,x<α2α1,1,x≥α2α1.As for the PDF ofγEx2,HD, it can be obtained as [40]
(44)fγEx2,HDz=∫0zfγE,1x2,HDxfγE,2x2,HDz−xdx=∫0z1α2fγSExα2fγ1Ez−xdx=Ck1exp−zγ¯1Ek1!μk1+1−exp−μz∑t1=0k1k1!t1!zt1μk1−t1+1,where Ck1=αSE/γ¯1E∑k1=0mSE−11−mSEk1−δSEk1/k1!2γ¯SEk1+1α2k1+1 and μ=ΔSEγ¯1E−α2/α2γ¯1E.By substituting (43) and (44) into (39), the P2out,HD is obtained as given by (45), where QHD=α2−α1γ¯2HD/α11+γ¯2HD.(45)P2out,HD=∫0QHDfγEx2,HDy1−Ak,tγ¯2HD+y1+γ¯2HDα2−γ¯2HD+y1+γ¯2HDα1texp−ΔS1γ¯2HD+y1+γ¯2HDα2−γ¯2HD+y1+γ¯2HDα1exp−γ¯2HD+y1+γ¯2HDγ¯12dy+∫QHD∞fγEx2,HDydy=1−∫0QHDfγEx2,HDyAk,tγ¯2HD+y1+γ¯2HD/α2−γ¯2HD+y1+γ¯2HDα1texp−ΔS1γ¯2HD+y1+γ¯2HD/α2−γ¯2HD+y1+γ¯2HDα1exp−γ¯2HD+y1+γ¯2HD/γ¯12dy⏟H1.To the best of authors’ knowledge, it is hard to derive the exact expression forH1 in (45). Thus, we adopt the Gaussian-Chebyshev quadrature to approximate H1 as [52]
(46)P2out,HD≈1−QHD2∑i=1KωiH1yi,where H1yi is given by (47), K is the number of terms, ωi is the Gaussian weight, which is given in Table (25.4) of [52], yi=QHDxi/2+QHD/2, and xi is the ith zero of Legendre polynomials.
(47)H1y=Ak,tCk1exp−yγ¯1Ek1!μk1+1−exp−μy∑t1=0k1k1!t1!yt1μk1−t1+1×γ¯2HD+y1+γ¯2HDα2−γ¯2HD+y1+γ¯2HDα1texp−ΔS1γ¯2HD+y1+γ¯2HDα2−γ¯2HD+y1+γ¯2HDα1exp−γ¯2HD+y1+γ¯2HDγ¯12.Remark 2.
In the high SNR region, i.e.,γ¯S1⟶∞ and γ¯12⟶∞, we have γ1x2,HD≈α2/α1 and γx2,HD=α2/α1. Thus, P2out,HD can be approximated as
(48)P2out,HD,∞=Prα2α1<γ¯2HD+γEx2,HD1+γ¯2HD=Prα2−α1γ¯2HDα11+γ¯2HD<γEx2,HD=∫QHD∞Ck1exp−y/γ¯1Ek1!/μk1+1dy⏟J3−∫QHD∞Ck1exp−y/γ¯1Eexp−μy∑t1=0k1k1!/t1!yt1/μk1−t1+1dy⏟J4,where J3 and J4 can be obtained as
(49)J3=Ck1γ¯1Ek1!μk1+1exp−QHDγ¯1E,J4=Ck1∑t1=0k11μk1−t1+1∑l=0t1k1!l!QHDl1+μγ¯1E/γ¯1Et1−l+1,respectively. It can be shown that, unlike D1, the SOP of D2 saturates in the high SNR region.Remark 3.
Note that whenα1≥1/1+γ¯2HD, we have QHD≤0, and thus, the SOP of user D2 is 1. This means that the power allocation coefficient for D1 shall be smaller than 1/1+γ¯2HD; otherwise, D2 will be always in secrecy outage.


## 5. Secrecy Outage Probability under Full-Duplex Mode

This section derives the SOPs of all the users under the full-duplex mode. The SOP forDi,i∈1,2 can be expressed as
(50)Piout,FD=Prlog1+γxi,FD−log1+γExi,FD+<Ri.


### 5.1. SOP of the Near User

First, the SOP of the near userD1 under the full-duplex mode is studied. The SOP of D1 can be rewritten as
(51)P1out,FD=∫0∞fγEx1,FDy×Fγx1,FDγ¯1FD+y1+γ¯1FDdy.where γ¯1FD=2Ri−1 is the SNR for the target secrecy rate Ri for Di under the full-duplex mode.From (18), (23), (30), and (31), the CDF of γx1,FD and the PDF of γEx1,FD can be derived as
(52)Fγx1,FDx=1−Ak,tϖxα1texp−ΔS1ϖxα1,(53)fγEx1,FDx=Bk1α1xα1k1exp−ΔSExα1,respectively. By inserting (52) and (53) into (51), we get
(54)P1out,FD=∫0∞Bk1α1yα1k1exp−ΔSEyα1×1−Ak,tϖγ¯1FD+y1+γ¯1FDα1t×exp−ΔS1ϖγ¯1FD+y1+γ¯1FDα1=1−Ak,tBk1ϖtα1k1+t+1∑l=0ttlγ¯1FDt−l1+γ¯1FDlk1+l!×exp−ΔS1ϖγ¯1FDα1ΔSE+ΔS1ϖ1+γ¯1FDα1−k1+l−1.Remark 4.
In the high SNR region, i.e.,γ¯S1⟶∞ and γ¯12⟶∞, the SOP of D1 can be approximated as
(55)P1out,FD,∞=Bk1ζαS1γ¯1FDα1ΔSEk1+1k1!+Bk1ζαS11+γ¯1FDΔSEk1+2k1+1!.It can be shown that the SOP of userD1 saturates in the high SNR region. This is because the residual self-interference restricts the achievable secrecy rate of D1 under the full-duplex mode.


### 5.2. SOP of the Far User

Then, we investigate the SOP ofD2, which can be expressed as
(56)P2out,FD=∫0∞fγEx2,FDy×Fγx2,FDγ¯2FD+y1+γ¯2FDdy.Similar to (43), the CDF of γx2,FD can be derived as
(57)Fγx2,FDx=1−Ak,tϖxα2−xα1texp−ΔS1ϖxα2−xα1−xγ¯12,x<α2α1,1,x≥α2α1.The PDF ofγEx2,FD is derived as
(58)fγEx2,FDz=Ck1exp−zγ¯1Ek1!μk1+1−exp−μz∑t1=0k1k1!t1!zt1μk1−t1+1.By substituting (57) and (58) into (56), the P2out,FD can be derived as given by (59), where QFD=α2−α1γ¯2FD/α11+γ¯2FD.
(59)P2out,FD=1−∫0QFDfγEx2,FDyAk,tϖγ¯2FD+y1+γ¯2FD/α2−γ¯2FD+y1+γ¯2FDα1texp−ΔS1ϖγ¯2FD+y1+γ¯2FD/α2−γ¯2FD+y1+γ¯2FDα1exp−γ¯2FD+y1+γ¯2FD/γ¯12dy⏟H2.Since it is hard to derive an exact closed-form expression forP2out,FD, similar to the half-duplex mode, with the help of Gaussian-Chebyshev quadrature, P2out,FD in (59) can be approximated as [52]
(60)P2out,FD≈1−QFD2∑i=1KωiH2yi,where H2yi is given by (61) and yi=QFDxi/2+QFD/2.
(61)H2y=Ak,tCk1exp−yγ¯1Ek1!μk1+1−exp−μy∑t1=0k1k1!t1!yt1μk1−t1+1×ϖγ¯2FD+y1+γ¯2FDα2−γ¯2FD+y1+γ¯2FDα1texp−ΔS1ϖγ¯2FD+y1+γ¯2FDα2−γ¯2FD+y1+γ¯2FDα1exp−γ¯2FD+y1+γ¯2FDγ¯12.Remark 5.
Whenγ¯S1⟶∞ and γ¯12⟶∞, the P2out,FD can be approximated as
(62)P2out,FD,∞=1−QFD2∑i=1KωiH3yi.where
(63)H3y=A′k,tCk1exp−yγ¯1Ek1!μk1+1−exp−μy∑t1=0k1k1!t1!yt1μk1−t1+1×ζγ¯2FD+y1+γ¯2FDα2−γ¯2FD+y1+γ¯2FDα1t×exp−Δ′S1ζγ¯2FD+y1+γ¯2FDα2−γ¯2FD+y1+γ¯2FDα1,A′k,t=αS1∑k=0mS1−1∑t=0k1−mS1k−δS1kk!t!Δ′S1k−t+1,and Δ′S1=βS1−δS1. It can be shown that the SOP of D2 saturates in the high SNR region, which is caused by the residual self-interference.Remark 6.
Note that whenα1≥1/1+γ¯2FD, we can have P2out,FD=1. This means that D2 will be always in secrecy outage when the power allocation coefficient for D1 is not smaller than 1/1+γ¯2FD. Since γ¯2HD>γ¯2FD, we have 1/1+γ¯2FD>1/1+γ¯2HD. Thus, based on the analysis in Remark 3, the threshold for the power allocation coefficient for D1 under the half-duplex mode is larger than that under full-duplex mode. This indicates that more power can be allocated to D1 under the full-duplex mode than that under the half-duplex mode, provided that the SOP of D2 is not 1.


## 5.1. SOP of the Near User

First, the SOP of the near userD1 under the full-duplex mode is studied. The SOP of D1 can be rewritten as
(51)P1out,FD=∫0∞fγEx1,FDy×Fγx1,FDγ¯1FD+y1+γ¯1FDdy.where γ¯1FD=2Ri−1 is the SNR for the target secrecy rate Ri for Di under the full-duplex mode.From (18), (23), (30), and (31), the CDF of γx1,FD and the PDF of γEx1,FD can be derived as
(52)Fγx1,FDx=1−Ak,tϖxα1texp−ΔS1ϖxα1,(53)fγEx1,FDx=Bk1α1xα1k1exp−ΔSExα1,respectively. By inserting (52) and (53) into (51), we get
(54)P1out,FD=∫0∞Bk1α1yα1k1exp−ΔSEyα1×1−Ak,tϖγ¯1FD+y1+γ¯1FDα1t×exp−ΔS1ϖγ¯1FD+y1+γ¯1FDα1=1−Ak,tBk1ϖtα1k1+t+1∑l=0ttlγ¯1FDt−l1+γ¯1FDlk1+l!×exp−ΔS1ϖγ¯1FDα1ΔSE+ΔS1ϖ1+γ¯1FDα1−k1+l−1.Remark 4.
In the high SNR region, i.e.,γ¯S1⟶∞ and γ¯12⟶∞, the SOP of D1 can be approximated as
(55)P1out,FD,∞=Bk1ζαS1γ¯1FDα1ΔSEk1+1k1!+Bk1ζαS11+γ¯1FDΔSEk1+2k1+1!.It can be shown that the SOP of userD1 saturates in the high SNR region. This is because the residual self-interference restricts the achievable secrecy rate of D1 under the full-duplex mode.


## 5.2. SOP of the Far User

Then, we investigate the SOP ofD2, which can be expressed as
(56)P2out,FD=∫0∞fγEx2,FDy×Fγx2,FDγ¯2FD+y1+γ¯2FDdy.Similar to (43), the CDF of γx2,FD can be derived as
(57)Fγx2,FDx=1−Ak,tϖxα2−xα1texp−ΔS1ϖxα2−xα1−xγ¯12,x<α2α1,1,x≥α2α1.The PDF ofγEx2,FD is derived as
(58)fγEx2,FDz=Ck1exp−zγ¯1Ek1!μk1+1−exp−μz∑t1=0k1k1!t1!zt1μk1−t1+1.By substituting (57) and (58) into (56), the P2out,FD can be derived as given by (59), where QFD=α2−α1γ¯2FD/α11+γ¯2FD.
(59)P2out,FD=1−∫0QFDfγEx2,FDyAk,tϖγ¯2FD+y1+γ¯2FD/α2−γ¯2FD+y1+γ¯2FDα1texp−ΔS1ϖγ¯2FD+y1+γ¯2FD/α2−γ¯2FD+y1+γ¯2FDα1exp−γ¯2FD+y1+γ¯2FD/γ¯12dy⏟H2.Since it is hard to derive an exact closed-form expression forP2out,FD, similar to the half-duplex mode, with the help of Gaussian-Chebyshev quadrature, P2out,FD in (59) can be approximated as [52]
(60)P2out,FD≈1−QFD2∑i=1KωiH2yi,where H2yi is given by (61) and yi=QFDxi/2+QFD/2.
(61)H2y=Ak,tCk1exp−yγ¯1Ek1!μk1+1−exp−μy∑t1=0k1k1!t1!yt1μk1−t1+1×ϖγ¯2FD+y1+γ¯2FDα2−γ¯2FD+y1+γ¯2FDα1texp−ΔS1ϖγ¯2FD+y1+γ¯2FDα2−γ¯2FD+y1+γ¯2FDα1exp−γ¯2FD+y1+γ¯2FDγ¯12.Remark 5.
Whenγ¯S1⟶∞ and γ¯12⟶∞, the P2out,FD can be approximated as
(62)P2out,FD,∞=1−QFD2∑i=1KωiH3yi.where
(63)H3y=A′k,tCk1exp−yγ¯1Ek1!μk1+1−exp−μy∑t1=0k1k1!t1!yt1μk1−t1+1×ζγ¯2FD+y1+γ¯2FDα2−γ¯2FD+y1+γ¯2FDα1t×exp−Δ′S1ζγ¯2FD+y1+γ¯2FDα2−γ¯2FD+y1+γ¯2FDα1,A′k,t=αS1∑k=0mS1−1∑t=0k1−mS1k−δS1kk!t!Δ′S1k−t+1,and Δ′S1=βS1−δS1. It can be shown that the SOP of D2 saturates in the high SNR region, which is caused by the residual self-interference.Remark 6.
Note that whenα1≥1/1+γ¯2FD, we can have P2out,FD=1. This means that D2 will be always in secrecy outage when the power allocation coefficient for D1 is not smaller than 1/1+γ¯2FD. Since γ¯2HD>γ¯2FD, we have 1/1+γ¯2FD>1/1+γ¯2HD. Thus, based on the analysis in Remark 3, the threshold for the power allocation coefficient for D1 under the half-duplex mode is larger than that under full-duplex mode. This indicates that more power can be allocated to D1 under the full-duplex mode than that under the half-duplex mode, provided that the SOP of D2 is not 1.


## 6. Numerical Results

In this section, we verify our theoretical results via conducting simulations. The system parameters and channel fading parameters are presented in Tables1 and 2, respectively, where FHS denotes heavy shadowing and AS denotes average shadowing [16, 18, 37].Table 1
System parameters.
ParametersValueSatellite orbitGEOFrequency bandf=2 GHz3 dB angleθ¯=0.8°Maximal beam gainGmax=48 dBThe antenna gainGk,Sk=4 dBTable 2
SR fading channel parameters.
ShadowingmkbkΩkFrequent heavy shadowing (FHS)10.0630.0007Average shadowing (AS)50.2510.279Besides, we assumeγ¯S1=γ¯12=γ¯=30 dB, γ¯SE=γ¯1E=γ¯E=−5 dB, R1=R2=0.5 bits/s/Hz, ζ=−30 dB, and K=20 [18, 32, 49].Figure2 compares the SOPs with FHS and AS for the near user D1 under different values of γ¯. The theoretical results are shown to match well with the simulation results, which verifies the correctness of the theoretical results. As expected, the secrecy outage performance with AS is better than that with FHS. It is seen that compared to the half-duplex mode, the secrecy outage performance under the full-duplex mode is better when γ¯ is small. When γ¯ is large, it is seen that the secrecy outage performance under the full-duplex mode saturates and underperforms that under the half-duplex mode. This is consistent with our analysis in Remarks 1 and 4 that the SOP of D1 under the half-duplex mode does not saturates in the high SNR region, while it saturates under the full-duplex mode. This indicates that the near user prefers the full-duplex mode when the SNR is low and prefers the half-duplex mode when the SNR is high.Figure 2
SOP of the near userD1 versus γ¯.Figure3 compares the SOPs with FHS and AS for the far user D2 under different values of γ¯. It is seen that the secrecy outage performance with AS is better than that with FHS, which is expected. When γ¯ is small, it is seen that the SOP under the full-duplex mode is lower than that under the half-duplex mode. It is also seen that when γ¯ is large, the SOPs under the full-duplex mode and the half-duplex mode both saturate, which is consistent with the analysis in Remarks 2 and 5. Besides, it is seen that compared to the half-duplex mode, the SOP under the full-duplex mode is higher with FHS and is lower with AS. This indicates that for the far user, the half-duplex mode is preferred only when the SNR is high and the channel fading is severe.Figure 3
SOP of the far userD2 versus γ¯.Figures4 and 5 plot the SOPs of D1 and D2 against ζ, respectively. It is seen that the SOP under the full-duplex mode increases as ζ increases, which is expected since higher interference under the full-duplex mode exists with a larger ζ. It is also seen that compared to the half-duplex mode, the SOP under the full-duplex mode is lower when ζ is small and is higher when ζ is large. This indicates that the full-duplex mode is preferred by both the near user and the far user only when ζ is small enough.Figure 4
SOP of the near userD1 versus ζ.Figure 5
SOP of the far userD2 versus ζ.Figure6 plots the SOP versus different values of γ¯E. It is seen that as γ¯E increases, the SOP increases. This is because the achievable rate at E is higher with a larger value of γ¯E. It is also seen that the impact of γ¯E on the SOP of D2 is severer than that on the SOP of D1. This is because E can overhear the message of D2 broadcast not only by the satellite source but also by the near user D1. Besides, it is seen that for D1, the SOP under the full-duplex mode is lower than that under the half-duplex mode when γ¯E is small and is higher than that under the half-duplex mode when γ¯E is large. It is also seen that for D2, the SOP under the full-duplex mode is always lower than that under the half-duplex mode. This indicates that γ¯E has no impact on the secrecy outage performance comparison between the half-duplex mode and the full-duplex mode for the far user.Figure 6
SOP versusγ¯E.Figure7 plots the SOP versus different values of α1. It is seen that as α1 increases, the SOP of D1 decreases, while the SOP of D2 increases. This indicates that allocating more power to the users can lead to lower SOP. When α1 is larger than a certain threshold, it is seen that the SOP of D2 is constant at 1, which is consistent with the analysis in Remarks 3 and 6. It is also seen that such threshold under the full-duplex mode is larger than that under the half-duplex mode, which is also consistent with the analysis in Remark 6. And it is seen that under the given settings, the full-duplex mode always outperforms the half-duplex mode whatever the value of α1 is. This indicates that α1 has no impact on the secrecy outage performance comparison between the half-duplex mode and the full-duplex mode for both the near user and the far user. Besides, it is shown that the secrecy outage performance of D2 with FHS in the high α1 region is better than that with AS. This is explained as follows. When α1 is large enough, the SNR for decoding x2 at D2 is constrained by the SNR for decoding x2 at D1. When the channel condition becomes worse, although the desired signal for decoding x2 at D1 is weakened, the interference signal is also weakened, which slows down the decrease in the SNR, whereas due to the use of PIC, the interference signal for decoding x2 at E is assumed to be perfectly cancelled. Therefore, the channel condition has severer impact on the SNR at E than that at D2 when α1 is large enough, which results in better secrecy outage performance of D2 with FHS compared to that with AS in the high α1 region.Figure 7
SOP versusα1.


## 7. Conclusion

This paper investigates the SOP of a downlink CNOMA-assisted hybrid satellite-terrestrial network, where there is no direct link between the satellite source and the far terrestrial user, and the near terrestrial user acts as a relay to forward the information to the far terrestrial user under the half-duplex mode or the full-duplex mode. The closed-form exact expressions for the SOP of the near user and the closed-form approximate expressions for the SOP of the far user are derived. It is shown that the theoretical results match well with the simulation results. It is also shown that compared to the half-duplex mode, the full-duplex mode is superior when the received SNR at the near user is low or the residual self-interference is low.



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