# Secrecy Analysis of NOMA-Based Integrated Satellite-Multiple UAV Networks under Nonideal Hardware and Colluding Scheme

Authors: Yuanyuan Wang; Feng Zhou; Rugang Wang; Xueling Wang; Kefeng Guo
Journal: Wireless Communications and Mobile Computing
 (2022)

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## Abstract

This paper studies the secrecy outage probability (SOP) of nonorthogonal multiple access (NOMA)-based integrated satellite-multiple unmanned aerial vehicle networks (ISMUAVNs) under nonideal hardware and colluding scheme. Colluding scheme is defined as that the eavesdroppers cooperate with each other to overhear the legitimate users’ signals. On this foundation, we obtain the closed-form expression for the SOP for the considered NOMA-based ISMUAVNs along with the partial unmanned aerial vehicle (UAV) selection scheme. To gain more things in the high signal-to-noise ratio (SNR) regime, we derive the asymptotic expression for the SOP, which gives fast ways to evaluate the benefit of the NOMA scheme and the impact of nonideal hardware impairments on the SOP. Finally, Monte Carlo simulation results are provided to show the correctness of the analytical results. In addition, with the help of representative results, some important and interesting findings are obtained.


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## Body

## 1. Introduction

The integration of satellite communication (SatCom) networks with the terrestrial mobile networks is one of the key elements for the fifth-generation (5G) and beyond 5G networks due to enhanced capability and extended coverage [1]. However, the coverage region may be restricted by the obstacles and shadowing between the terrestrial users and satellite, which leads to the unavailability of the line-of-sight (LOS) links [2]. Under this situation, the terrestrial relay is utilized to overcome this challenge such as fading, shadowing, and path loss [3], which leads to the formation of the integrated satellite-unmanned aerial vehicle networks (ISUAVNs). Particularly, the main idea of ISUAVNs is to utilize terrestrial relays/unmanned aerial vehicles (UAVs) to forward the satellite signal, which has been a good framework for the SatCom, they have great effects on the fixed satellite service and high-speed broadband mobile satellite service, and SiriusXM is the real system which relies on the foundation of this architecture [4].As the wide coverage of the satellite beam, there are often many wireless communication nodes in one satellite beam, which result in the suitable selection scheme for the wireless communication nodes [1]. In [5], the outage probability (OP) was analyzed with the opportunistic wireless node scheduling scheme for the integrated satellite-terrestrial relay networks (ISTRNs). In [6], the OP was investigated for the ISTRNs under the consideration of cognitive technology and multiple terrestrial transmission nodes.With the breakthrough of UAV technology, UAV communication has been widely utilized in many temporary large-scale activities and natural disasters [7]. Total power constraint-beamforming and per-antenna power constraint-beamforming schemes were proposed to optimize the energy efficiency of integrated satellite-UAV terrestrial networks in [7]. The authors of [8] investigated the UAV system with mobile edge computing, where the energy expenditure of users was minimized under assignment delay demands and resource limitations. In [9], the game theory was applied to optimize the resources’ allocation of the UAV communication network for more rewards, and a multiagent reinforcement learning framework was proposed to find the optimal policy. The potential game, mean-field game, Stackelberg game, graphical game, and coalition game were utilized to solve the resource management problem of large-scale UAVs in [10]. The authors of [11] designed a decaying deep Q-network-based algorithm to minimize the energy consumption, in which the trajectory of the UAV, phase shifts of the reconfigurable intelligent surfaces, power allocation strategy, and dynamic decoding sequence were considered.To alleviate the spectrum shortage and meet the requirements of improved spectral efficiency of ISTRNs, nonorthogonal multiple access (NOMA) has been viewed as the new paradigm of multiple access techniques [12–14], which transmits signals simultaneously in the same time/frequency resource block. On this foundation, successive interference cancellation (SIC) is used at the receivers to distinguish the users’ message by different power levels [15]. In NOMA systems, the message for the weaker was first decoded by the stronger users in which case, the stronger can be regarded as the relay to help the user with poor channel environment to enhance the system performance [16, 17]. In [18], the energy efficiency, ergodic capacity, and outage probability (OP) of NOMA-based downlink SatCom were investigated. In [19], the cooperative transmission method was analyzed for the NOMA-based SatCom. In [15], the authors derived the closed-form expression of OP in the presence of secondary networks.As presented before, the satellite beam has a wide coverage, which results in the security problem in reality for the ISTRNs [20]; physical layer security (PLS) is considered as a new method to present the security problem of the wireless transmission networks. The major idea of PLS is to keep the secrecy transmission rate in the case of no interception under malicious nodes. In [21], the authors listed enough problems in the SatCom and investigated the secrecy performance. In [22], the authors investigated the SOP and ASC with the maximum user scheduling scheme for the SatCom. In [23], the authors analyzed the impact of imperfect channel stat information (CSI) for SatCom with multiple legitimate users and multiple eavesdroppers. In [24, 25], the ASC and SOP for the ISTRNs with the opportunistic relaying scheme were analyzed, respectively.It is mentioned that, in real wireless communication systems, the transmission nodes are often not ideal owing to several reasons [26, 27]. In [28], the authors summarized all the reasons and proposed a general nonideal hardware model to research the impact of nonideal hardware on the wireless communication networks. In [29], the authors investigated the performance for the amplify-and-forward multiple relay networks in the presence of nonideal hardware and available direct transmission link. In [30], the OP was analyzed with the nonideal hardware and cognitive technology existing in the ISTRNs.Until now, the impact of nonideal hardware on the NOMA-based SatCom was only investigated in several former papers, such as [31–34]. In [31], the authors analyzed the SOP for the considered SatCom with nonideal hardware and channel estimation errors (CEEs). In [32], the impact of nonideal hardware was investigated for the ISTRNs. In [33], Guo analyzed the effect of nonideal hardware on the ISTRNs with multiple terrestrial relays under the partial wireless node selection scheme.In this paper, we investigate the secrecy NOMA-based integrated satellite-multiple unmanned aerial vehicle networks (ISMUAVNs) in the presence of multiple eavesdroppers and HIs by utilizing the partial UAV selection scheme. In detail, the main contributions of this paper are shown in the following:(i)
Firstly, we establish a NOMA-based secrecy ISMUAVN with nonideal hardware. The partial UAV selection scheme with low implementation complexity is applied in the considered system.(ii)
Secondly, the closed-form expression for SOP of the considered network in the presence of nonideal hardware and the proposed UAV selection scheme is derived.(iii)
Finally, to get insight of the SOP at high signal-to-noise ratios (SNRs), the asymptotic expression for SOP of the considered secrecy ISMUAVNs is also derived.The rest of this paper is given in the following. Section2 introduces the system model and formulates the secrecy problem of the considered secrecy ISMUAVNs. Section 3 provides the detailed analysis for the SOP. In Section 3, the secrecy diversity order and secrecy coding gain are analyzed along with the asymptotic analysis. In Section 4, numerical Monte Carlo (MC) results are obtained. In Section 5, the summarization is presented.


### 1.1. Notations

⋅ denotes the absolute value of a complex scalar, exp⋅ is the exponential function, E⋅ represents the expectation operator, and CNa,b is the complex Gaussian distribution of a random vector a and covariance matrix b.


## 1.1. Notations

⋅ denotes the absolute value of a complex scalar, exp⋅ is the exponential function, E⋅ represents the expectation operator, and CNa,b is the complex Gaussian distribution of a random vector a and covariance matrix b.


## 2. System Model

As plotted in Figure1, in this paper, we investigate NOMA-based secrecy ISMUAVNs, which consist of a satellite S, namely, Alice, multiple UAVs Rp,p∈1,…,M, two destinations (namely, D1 and D2), and N eavesdroppers. Decode-and-forward (DF) protocol is utilized at R. Two time slots are used for the total transmission. The direct transmission link is not considered between S and Dii∈1,229, for some reasons (Owing to some practical reasons, such as heavy fading, shadowing, or obstacle blockage, no direct transmission link is assumed in this paper.). In general, all transmission nodes in the considered NOMA-based ISMUAVNs are equipped with one antenna, respectively (In this paper, only two users are considered; however, the derived results can also be suitable for multiple users, which will be investigated in the near future. In this paper, the single antenna is assumed; however, the derived results will also be suitable for the case with multiple antennas.).Figure 1
Illustration of the system model.For the first time slot, the signalst is forwarded by S with Est2=1 to pth R. The received signal at pth R is derived as(1)yRpt=fSRp,kPSst+ηSt+nRpt,where fSRp,k represents the channel coefficient modeled as shadowed-Rician (SR) fading (In this paper, the perfect CSI is assumed, and our results will be as the branch for the case with imperfect CSI.) and PS denotes the transmit power from S. When applying the NOMA scheme, the signal is shown as st=a1px1t+a2px2t, where x1t and x2t present the target transmitted signal to D1 and D2, respectively. Besides, it is assumed that a1p>a2p for the reason that the channel D1 suffers heavy fading than that of D2. On this foundation, much power should be allocated to D1. a1p and a2p represent the transmission power allocation (PA) factors for D1 and D2, respectively. In addition, a1p2+a2p2=1. ηSt is the distortion noise due to nonideal hardware with ηSt∼CN0,a1p2k1p2+a2p2k2p2. k1p and k2p denote the nonideal levels for nonideal hardware which are defined in [3]. nRpt represents the additive white Gaussian noise (AWGN) at pth R distributed as nRpt∼CN0,δRp2.When the signal is received byR, SIC works as follows: x1t is firstly decoded and then deleted from the obtained signal [12]. Thereafter, x2t is decoded from the left signal (In this paper, perfect SIC is assumed. Besides, different users are distinguished by the channel quality. We assume that the nearby user has a better channel quality than that of the distant users which has been assumed in the literature studies.).Hence, the signal-to-interference-plus-noise-and-distortion ratio (SINDR) of detectingx1t at pth R is given by(2)γRp,1=γ1pa1p2γ1pa1p2k1p2+a2p2+a2p2k2p2+1,where γ1p=fSRp,k2PS/δRp2.Moreover, we can obtain the signal-to-noise-and-distortion ratio (SNDR) of detectingx2t at pth R as(3)γRp,2=γ1pa2p2γ1pa1p2k1p2+a2p2k2p2+1.As announced before, the partial UAV selection scheme [35] is adopted for the considered system model, and the SINDR of x1t and x2t during the first transmission link is, respectively, shown as(4)γR,1=maxp∈1,…,MγRp,1,(5)γR,2=maxp∈1,…,MγRp,2.For the second time slot, the selectedpth R forwards received signal st to D1 and D2, respectively; thus, the signal received at Di can be represented as(6)yDpit=hRpDiPRpξ1px1t+ξ2px2t+ηpt+nDit,where hRpDi denotes the channel coefficient between pth R and Di which undergoes Rayleigh fading, PRp represents the transmit power for pth R, xit denotes the expected signal, ξjpj∈1,2 which suits ξ1p>ξ2p, and ξ1p2+ξ2p2=1 represents the PA factors at pth R, respectively (We apply the same assumption which means the channel quality of user D1 is worse than that of user D2.). nDit denotes the AWGN at Di distributed as nDit∼CN0,δDi2. ηpt with ηpt∼CN0,ξ1p2kp12+ξ2p2kp22 represents the distortion noise which is caused by nonideal hardware. ξ1p along with ξ2p denotes the nonideal levels of nonideal hardware [35]. Therefore, D2 performs SIC to decode x1t (while x2t is treated as the cochannel noise).From (6), the SINDR of x1t sent by pth R at D2 is represented as(7)γDp1,D2=γ2pξ1p2γ2pξ1p2k1p2+ξ2p2+ξ2p2k2p2+1,where γ2p=hRpD22PRp/δD22.γDp1,D2 must be satisfied as γDp1,D2≥γth due to the same condition like (2). D2 deleted the decoded information from the received signal by observing its own signal; hence, the final SNDR for signal x2t forwarded by pth R is given by(8)γDp2,D2=γ2pξ2p2γ2pξ1p2k1p2+ξ2p2k2p2+1.Then,x1t is decoded by D1, while x2t is regarded as the serial interference; the SINDR received at D1 from pth R can be obtained as(9)γDp1,D1=γ3pξ1p2γ3pξ1p2k1p2+ξ2p2+ξ2p2k2p2+1,where γ3p=hRpD12PRp/δD12.At the same time, due to the fact that the signal sent to relays can be received by all eavesdroppers, the heard signal of thejth eavesdropper can be represented as(10)yEpjt=hRpEjPRpξ1px1t+ξ2px2t+ηpt+nEjt,where hRpEj represents the channel coefficient between pth R and the jth Eve which is modeled as Rayleigh fading. nEjt with nEjt∼CN0,δEj2 denotes the AWGN at the jth Eve.Through the whole paper, we assume that parallel interference cancellation (PIC) is available at each eavesdropper [36, 37]. Hence, the received SNR of ith D’s expected messages at the jth Eve can be shown as(11)λDpiEj=ξip2γDpiEjγDpiEjξ1p2k1p2+ξ2p2k2p2+1,where γDpiEj=hRpEj2PRp/δEj2.Due to the cooperation of all eavesdroppers, we can get the SNR of all Eves’ link as(12)γDpiE=∑j=1NλDpiEj.The authors in [1] introduced the definition of the secrecy capacity; with the help of (7), (8), (9), and (12), we can obtain the secrecy capacity for the ith signal as(13)CSpiρ=CBpiρ−CDpiE+,ρ∈D1,D2,where x+=Δmaxx,0, CBpiρ=log21+γDpi,ρ, and CDpiE=log21+γDpiE (CBpiρ means the secrecy capacity at the ρ user to detect the ith signal from pth R.).


## 3. Performance Analysis

### 3.1. The Channel Model

#### 3.1.1. Satellite Channel Model

As mentioned before, the satellite-UAV links are considered to be modeled as SR fading, so the probability density function (PDF) ofγ1p can be shown as(14)fγ1px=α1pγ¯1pe−β1/γ¯11F1m1p;1;δ1pγ¯1px,x>0,where 1F1a;b;x denotes the confluent hypergeometric function [38]. γ¯1p represents the average SNR between Alice and pth R, α1p=2b1pm1p/2b1pm1p+Ω1pm1p/2b1p, β1p=1/2b1p, and δ1p=Ω1p/2b1p2b1pm1p+Ω1p with m1p≥0, 2b1p, and Ω1p being the fading severity parameter ranging from 0 to ∞, the average power of the multipath component, and the average power of the LOS component, respectively. In this paper, we assume that m1p is an integer, and the PDF and cumulative distribution function (CDF) of γ1p are, respectively, written as(15)fγ1px=α1p∑k1=0m1p−11−m1pk1−δ1pk1k1!2γ¯1pk1+1xk1exp−Δ1px,Fγ1px=1−α1p∑k1=0m1p−1∑t=0k11−m1pk1−δ1pk1k1!γ¯1pk1+1t!Δ1pk1−t+1xte−Δ1px,where Δ1p=β1p−δ1p/γ¯1p and ⋅k denotes the Pochhammer symbol [38].


#### 3.1.2. Terrestrial Channel Model

In the former section, we assume the channel model betweenR and intended users or eavesdroppers which undergoes independent and identically distributed (i.i.d) Rayleigh fading. Thus, the CDF and PDF of γUU∈2p,3p,DiEj are, respectively, shown as(16)FγU=1−e−x/γ¯Uand(17)fγU=1γ¯Ue−x/γ¯U,where γ¯U is the average channel gain.By utilizing (11) and (12) along with [5], the CDF and PDF of γDpiE are, respectively, derived as(18)FγDpiEx=1−∑υ=0N−11υxγ¯DpiEξip2−xγ¯DpiEξ1p2k1p2+ξ2p2k2p2υe−x/γ¯DpiEξip2−xξ1p2k1p2+ξ2p2k2p2,fγDpiEx=ξ1p2γ¯DpiE−Nξip2−xξ1p2k1p2+ξ2p2k2p22N−1!xγ¯DpiEξip2−xγ¯DpiEξ1p2k1p2+ξ2p2k2p2N−1×exp−xγ¯DpiEξip2−xξ1p2k1p2+ξ2p2k2p2.


### 3.2. Secrecy Outage Probability

SOP refers to the probability of reaching the nonnegative target confidentiality rate. In this section, the SOP of the NOMA-based ISMUAVN describes the confidentiality behavior, and it can be expressed as(19)Poutγ0=P1γ0+P2D2γ0−P1γ0P2D2γ0,where(20)P1γ0=P1D1γ0+P1D2γ0−P1D1γ0P1D2γ0and(21)P2D2γ0=P12D2γ0+P22D2γ0−P12D2γ0P22D2γ0,where(22)P1ργ0=P11ργ0+P21ργ0−P11ργ0P21ργ0.By using (4) and (5), P1iρ and P2iρ are, respectively, presented as(23)P1iργ0=PrγRp,i≤γ0M,i∈1,2,P2iργ0=PrCSpiρ≤C0=∫0∞∫0γ0+y1+γ0fγDpi,ρxfγDpiEydxdy.Also,C0=log21+γ0.In the following, to make the analysis easier,γ¯Dp1E=γ¯Dp2E=γ¯E is assumed, and the detailed analysis of SOP for D1 and D2 is, respectively, given in the following theorems.Theorem 1.
AtD1, the closed-form expression for P1D1 by detecting x1t is given by(24)P1D1γ0=P11D1γ0+P21D1γ0−P11D1γ0P21D1γ0,where(25)P11D1γ0=1−∑k1=0m1p−1∑t=0k1α1p1−m1pk1−δ1pk1k1!γ¯1pk1+1t!Δ1pk1−t+1×e−Δ1pγ0/a1p2−γ0Bγ0a1p2−γ0BtMand(26)P21D1γ0=1−H12γ¯E2N−1N−1!∑i=1K1wiH1zi.In (25), B=a1p2k1p2+a2p2+a2p2k2p2. In (26),(27)H1z=ξ1p2zN−1ξ1p2−zCN+1exp−zξ1p2γ¯Eξ1p2−zCexp−γ0+z1+γ0γ¯3pξ1p2−γ0+z1+γ0γ¯3pA,where K1 denotes the number of terms, H1=minξ1p2/C,ξ1p2−Aγ0/A+Aγ0, A=ξ1p2k1p2+ξ2p2+ξ2p2k2p2, C=ξ1p2k1p2+ξ2p2k2p2, zi=ti+1 presents the ith zero of Legendre polynomials, and wi denotes the Gaussian weight, which is given in Table (25.4) of [39].With the similar method, atD2, the exact expression of P1D2 to decode x1t can be expressed as(28)P1D2γ0=P11D2γ0+P21D2γ0−P11D2γ0P21D2γ0,where P11D2γ0=P11D1γ0,(29)P21D2γ0=1−H22γ¯E2N−1N−1!∑i=1K2wiH2zi,and(30)H2z=ξ1p2zN−1ξ1p2−zCN+1exp−zξ1p2γ¯Eξ1p2−zCexp−γ0+z1+γ0γ¯2pξ1p2−γ0+z1+γ0γ¯2pA,where H2=minξ1p2/C,ξ1p2−Aγ0/A+γ0A.Proof.
See Appendix A.Theorem 2.
x2t is only detected by D2; therefore, the exact expression of P2D2 to decode x1t can be represented as(31)P2D2γ0=P12D2γ0+P22D2γ0−P12D2γ0P22D2γ0,where(32)P12D2γ0=1−∑k1=0m1p−1∑t=0k1α1p1−m1pk1−δ1pk1k1!γ¯1pk1+1t!Δ1pk1−t+1×e−Δ1pγ0/a2p2−γ0Cγ0a2p2−γ0CtMand(33)P22D2γ0=1−H32γ¯E2N−1N−1!∑i=1K3wiH3zi,where(34)H3z=ξ2p2zN−1ξ2p2−zCN+1exp−zξ2p2γ¯Eξ2p2−zC−γ0+z1+γ0γ¯2pξ2p2−γ0+z1+γ0γ¯2pC.Also,H3=minξ2p2/C,ξ2p2−Cγ0/C+γ0C.Proof.
See Appendix B.


### 3.3. Asymptotic SOP

In the behind description, to derive deeper insights of the impact of the significant system parameters of the considered NOMA-based ISMUAVN in the high SNR regime, we show the following analysis. According to (16) and (15), when γ¯1p⟶∞, we can derive(35)Fγ1px≈α1pγ¯1px+ox,where ox denotes the infinitesimal of x in the higher order. Similarly, when γ¯U⟶∞, (17) can be rewritten as(36)FγUx≈xγ¯U+ox.Proposition 1.
AtD1, the asymptotic SOP for decoding x1t of our proposed system is represented as(37)P1D1∞γ0=1+α1pγ¯1pγ0a1p2−γ0BM−∑i=1K1ziN−1H1wiξ1p22γ¯E2N−1N−1!×e−zi/ξ1p2γ¯Eξ1p2−ziCξ1p2−ziCN+11−γ0+zi1+γ0γ¯3pξ1p2−γ0+zi1+γ0γ¯3pA.AtD2, the asymptotic SOP for decoding x1t of our proposed system is obtained as(38)P1D2∞γ0=1+α1pγ¯1pγ0a1p2−γ0BM−∑i=1K2ziN−1H2wiξ1p22γ¯E2N−1N−1!×e−zi/ξ1p2γ¯Eξ1p2−ziCξ1p2−ziCN+11−γ0+zi1+γ0γ¯2pξ1p2−γ0+zi1+γ0γ¯2pA.x2t is only detected by D2; therefore, the asymptotic expression of SOP for decoding x2t for our proposed system is shown as(39)P2D2∞γ0=1+α1pγ¯1pγ0a2p2−γ0CM−∑i=1K3ziN−1H3wiξ2p22γ¯E2N−1N−1!×e−zi/ξ2p2γ¯Eξ2p2−ziCξ2p2−ziCN+11−γ0+zi1+γ0γ¯2pξ2p2−γ0+zi1+γ0γ¯2pC.Proof.
At first, by replacing (15) and (16) with (35) and (36) and then with the same method of Appendixes A and B, the exact equations for the asymptotic analysis (namely, (37), (38), and (40)) are derived.
In this sequel, we obtain the diversity order along with the coding gain. By lettingγ¯1p=γ¯2p=γ¯3p=γ¯, (22) can be expressed as(40)Pout∞γ0≈Ga1γ¯Gd,where Gd=minM,1 denotes the secrecy diversity order, and the secrecy coding gain of the considered NOMA-based ISMUAVN includes two cases, i.e., when M≤1,(41)Ga=2α1pγ0a1p2−γ0B+α1pγ0a2p2−γ0C+2∑i=1K1ziN−1H1wiξ1p2e−zi/ξ1p2γ¯Eξ1p2−ziCγ0+zi1+γ02γ¯E2N−1N−1!ξ1p2−ziCN+1ξ1p2−γ0+zi1+γ0A+∑i=1K3ziN−1H3wiξ2p2e−zi/ξ2p2γ¯Eξ2p2−ziCγ0+zi1+γ02γ¯E2N−1N−1!ξ2p2−ziCN−1ξ2p2−γ0+zi1+γ0C.
WhenM>1,(42)Ga=2∑i=1K1ziN−1H1wiξ1p2e−zi/ξ1p2γ¯Eξ1p2−ziCγ0+zi1+γ02γ¯E2N−1N−1!ξ1p2−ziCN+1ξ1p2−γ0+zi1+γ0A+∑i=1K3ziN−1H3wiξ2p2e−zi/ξ2p2γ¯Eξ2p2−ziCγ0+zi1+γ02γ¯E2N−1N−1!ξ2p2−ziCN−1ξ2p2−γ0+zi1+γ0C.


## 3.1. The Channel Model

### 3.1.1. Satellite Channel Model

As mentioned before, the satellite-UAV links are considered to be modeled as SR fading, so the probability density function (PDF) ofγ1p can be shown as(14)fγ1px=α1pγ¯1pe−β1/γ¯11F1m1p;1;δ1pγ¯1px,x>0,where 1F1a;b;x denotes the confluent hypergeometric function [38]. γ¯1p represents the average SNR between Alice and pth R, α1p=2b1pm1p/2b1pm1p+Ω1pm1p/2b1p, β1p=1/2b1p, and δ1p=Ω1p/2b1p2b1pm1p+Ω1p with m1p≥0, 2b1p, and Ω1p being the fading severity parameter ranging from 0 to ∞, the average power of the multipath component, and the average power of the LOS component, respectively. In this paper, we assume that m1p is an integer, and the PDF and cumulative distribution function (CDF) of γ1p are, respectively, written as(15)fγ1px=α1p∑k1=0m1p−11−m1pk1−δ1pk1k1!2γ¯1pk1+1xk1exp−Δ1px,Fγ1px=1−α1p∑k1=0m1p−1∑t=0k11−m1pk1−δ1pk1k1!γ¯1pk1+1t!Δ1pk1−t+1xte−Δ1px,where Δ1p=β1p−δ1p/γ¯1p and ⋅k denotes the Pochhammer symbol [38].


### 3.1.2. Terrestrial Channel Model

In the former section, we assume the channel model betweenR and intended users or eavesdroppers which undergoes independent and identically distributed (i.i.d) Rayleigh fading. Thus, the CDF and PDF of γUU∈2p,3p,DiEj are, respectively, shown as(16)FγU=1−e−x/γ¯Uand(17)fγU=1γ¯Ue−x/γ¯U,where γ¯U is the average channel gain.By utilizing (11) and (12) along with [5], the CDF and PDF of γDpiE are, respectively, derived as(18)FγDpiEx=1−∑υ=0N−11υxγ¯DpiEξip2−xγ¯DpiEξ1p2k1p2+ξ2p2k2p2υe−x/γ¯DpiEξip2−xξ1p2k1p2+ξ2p2k2p2,fγDpiEx=ξ1p2γ¯DpiE−Nξip2−xξ1p2k1p2+ξ2p2k2p22N−1!xγ¯DpiEξip2−xγ¯DpiEξ1p2k1p2+ξ2p2k2p2N−1×exp−xγ¯DpiEξip2−xξ1p2k1p2+ξ2p2k2p2.


## 3.1.1. Satellite Channel Model

As mentioned before, the satellite-UAV links are considered to be modeled as SR fading, so the probability density function (PDF) ofγ1p can be shown as(14)fγ1px=α1pγ¯1pe−β1/γ¯11F1m1p;1;δ1pγ¯1px,x>0,where 1F1a;b;x denotes the confluent hypergeometric function [38]. γ¯1p represents the average SNR between Alice and pth R, α1p=2b1pm1p/2b1pm1p+Ω1pm1p/2b1p, β1p=1/2b1p, and δ1p=Ω1p/2b1p2b1pm1p+Ω1p with m1p≥0, 2b1p, and Ω1p being the fading severity parameter ranging from 0 to ∞, the average power of the multipath component, and the average power of the LOS component, respectively. In this paper, we assume that m1p is an integer, and the PDF and cumulative distribution function (CDF) of γ1p are, respectively, written as(15)fγ1px=α1p∑k1=0m1p−11−m1pk1−δ1pk1k1!2γ¯1pk1+1xk1exp−Δ1px,Fγ1px=1−α1p∑k1=0m1p−1∑t=0k11−m1pk1−δ1pk1k1!γ¯1pk1+1t!Δ1pk1−t+1xte−Δ1px,where Δ1p=β1p−δ1p/γ¯1p and ⋅k denotes the Pochhammer symbol [38].


## 3.1.2. Terrestrial Channel Model

In the former section, we assume the channel model betweenR and intended users or eavesdroppers which undergoes independent and identically distributed (i.i.d) Rayleigh fading. Thus, the CDF and PDF of γUU∈2p,3p,DiEj are, respectively, shown as(16)FγU=1−e−x/γ¯Uand(17)fγU=1γ¯Ue−x/γ¯U,where γ¯U is the average channel gain.By utilizing (11) and (12) along with [5], the CDF and PDF of γDpiE are, respectively, derived as(18)FγDpiEx=1−∑υ=0N−11υxγ¯DpiEξip2−xγ¯DpiEξ1p2k1p2+ξ2p2k2p2υe−x/γ¯DpiEξip2−xξ1p2k1p2+ξ2p2k2p2,fγDpiEx=ξ1p2γ¯DpiE−Nξip2−xξ1p2k1p2+ξ2p2k2p22N−1!xγ¯DpiEξip2−xγ¯DpiEξ1p2k1p2+ξ2p2k2p2N−1×exp−xγ¯DpiEξip2−xξ1p2k1p2+ξ2p2k2p2.


## 3.2. Secrecy Outage Probability

SOP refers to the probability of reaching the nonnegative target confidentiality rate. In this section, the SOP of the NOMA-based ISMUAVN describes the confidentiality behavior, and it can be expressed as(19)Poutγ0=P1γ0+P2D2γ0−P1γ0P2D2γ0,where(20)P1γ0=P1D1γ0+P1D2γ0−P1D1γ0P1D2γ0and(21)P2D2γ0=P12D2γ0+P22D2γ0−P12D2γ0P22D2γ0,where(22)P1ργ0=P11ργ0+P21ργ0−P11ργ0P21ργ0.By using (4) and (5), P1iρ and P2iρ are, respectively, presented as(23)P1iργ0=PrγRp,i≤γ0M,i∈1,2,P2iργ0=PrCSpiρ≤C0=∫0∞∫0γ0+y1+γ0fγDpi,ρxfγDpiEydxdy.Also,C0=log21+γ0.In the following, to make the analysis easier,γ¯Dp1E=γ¯Dp2E=γ¯E is assumed, and the detailed analysis of SOP for D1 and D2 is, respectively, given in the following theorems.Theorem 1.
AtD1, the closed-form expression for P1D1 by detecting x1t is given by(24)P1D1γ0=P11D1γ0+P21D1γ0−P11D1γ0P21D1γ0,where(25)P11D1γ0=1−∑k1=0m1p−1∑t=0k1α1p1−m1pk1−δ1pk1k1!γ¯1pk1+1t!Δ1pk1−t+1×e−Δ1pγ0/a1p2−γ0Bγ0a1p2−γ0BtMand(26)P21D1γ0=1−H12γ¯E2N−1N−1!∑i=1K1wiH1zi.In (25), B=a1p2k1p2+a2p2+a2p2k2p2. In (26),(27)H1z=ξ1p2zN−1ξ1p2−zCN+1exp−zξ1p2γ¯Eξ1p2−zCexp−γ0+z1+γ0γ¯3pξ1p2−γ0+z1+γ0γ¯3pA,where K1 denotes the number of terms, H1=minξ1p2/C,ξ1p2−Aγ0/A+Aγ0, A=ξ1p2k1p2+ξ2p2+ξ2p2k2p2, C=ξ1p2k1p2+ξ2p2k2p2, zi=ti+1 presents the ith zero of Legendre polynomials, and wi denotes the Gaussian weight, which is given in Table (25.4) of [39].With the similar method, atD2, the exact expression of P1D2 to decode x1t can be expressed as(28)P1D2γ0=P11D2γ0+P21D2γ0−P11D2γ0P21D2γ0,where P11D2γ0=P11D1γ0,(29)P21D2γ0=1−H22γ¯E2N−1N−1!∑i=1K2wiH2zi,and(30)H2z=ξ1p2zN−1ξ1p2−zCN+1exp−zξ1p2γ¯Eξ1p2−zCexp−γ0+z1+γ0γ¯2pξ1p2−γ0+z1+γ0γ¯2pA,where H2=minξ1p2/C,ξ1p2−Aγ0/A+γ0A.Proof.
See Appendix A.Theorem 2.
x2t is only detected by D2; therefore, the exact expression of P2D2 to decode x1t can be represented as(31)P2D2γ0=P12D2γ0+P22D2γ0−P12D2γ0P22D2γ0,where(32)P12D2γ0=1−∑k1=0m1p−1∑t=0k1α1p1−m1pk1−δ1pk1k1!γ¯1pk1+1t!Δ1pk1−t+1×e−Δ1pγ0/a2p2−γ0Cγ0a2p2−γ0CtMand(33)P22D2γ0=1−H32γ¯E2N−1N−1!∑i=1K3wiH3zi,where(34)H3z=ξ2p2zN−1ξ2p2−zCN+1exp−zξ2p2γ¯Eξ2p2−zC−γ0+z1+γ0γ¯2pξ2p2−γ0+z1+γ0γ¯2pC.Also,H3=minξ2p2/C,ξ2p2−Cγ0/C+γ0C.Proof.
See Appendix B.


## 3.3. Asymptotic SOP

In the behind description, to derive deeper insights of the impact of the significant system parameters of the considered NOMA-based ISMUAVN in the high SNR regime, we show the following analysis. According to (16) and (15), when γ¯1p⟶∞, we can derive(35)Fγ1px≈α1pγ¯1px+ox,where ox denotes the infinitesimal of x in the higher order. Similarly, when γ¯U⟶∞, (17) can be rewritten as(36)FγUx≈xγ¯U+ox.Proposition 1.
AtD1, the asymptotic SOP for decoding x1t of our proposed system is represented as(37)P1D1∞γ0=1+α1pγ¯1pγ0a1p2−γ0BM−∑i=1K1ziN−1H1wiξ1p22γ¯E2N−1N−1!×e−zi/ξ1p2γ¯Eξ1p2−ziCξ1p2−ziCN+11−γ0+zi1+γ0γ¯3pξ1p2−γ0+zi1+γ0γ¯3pA.AtD2, the asymptotic SOP for decoding x1t of our proposed system is obtained as(38)P1D2∞γ0=1+α1pγ¯1pγ0a1p2−γ0BM−∑i=1K2ziN−1H2wiξ1p22γ¯E2N−1N−1!×e−zi/ξ1p2γ¯Eξ1p2−ziCξ1p2−ziCN+11−γ0+zi1+γ0γ¯2pξ1p2−γ0+zi1+γ0γ¯2pA.x2t is only detected by D2; therefore, the asymptotic expression of SOP for decoding x2t for our proposed system is shown as(39)P2D2∞γ0=1+α1pγ¯1pγ0a2p2−γ0CM−∑i=1K3ziN−1H3wiξ2p22γ¯E2N−1N−1!×e−zi/ξ2p2γ¯Eξ2p2−ziCξ2p2−ziCN+11−γ0+zi1+γ0γ¯2pξ2p2−γ0+zi1+γ0γ¯2pC.Proof.
At first, by replacing (15) and (16) with (35) and (36) and then with the same method of Appendixes A and B, the exact equations for the asymptotic analysis (namely, (37), (38), and (40)) are derived.
In this sequel, we obtain the diversity order along with the coding gain. By lettingγ¯1p=γ¯2p=γ¯3p=γ¯, (22) can be expressed as(40)Pout∞γ0≈Ga1γ¯Gd,where Gd=minM,1 denotes the secrecy diversity order, and the secrecy coding gain of the considered NOMA-based ISMUAVN includes two cases, i.e., when M≤1,(41)Ga=2α1pγ0a1p2−γ0B+α1pγ0a2p2−γ0C+2∑i=1K1ziN−1H1wiξ1p2e−zi/ξ1p2γ¯Eξ1p2−ziCγ0+zi1+γ02γ¯E2N−1N−1!ξ1p2−ziCN+1ξ1p2−γ0+zi1+γ0A+∑i=1K3ziN−1H3wiξ2p2e−zi/ξ2p2γ¯Eξ2p2−ziCγ0+zi1+γ02γ¯E2N−1N−1!ξ2p2−ziCN−1ξ2p2−γ0+zi1+γ0C.
WhenM>1,(42)Ga=2∑i=1K1ziN−1H1wiξ1p2e−zi/ξ1p2γ¯Eξ1p2−ziCγ0+zi1+γ02γ¯E2N−1N−1!ξ1p2−ziCN+1ξ1p2−γ0+zi1+γ0A+∑i=1K3ziN−1H3wiξ2p2e−zi/ξ2p2γ¯Eξ2p2−ziCγ0+zi1+γ02γ¯E2N−1N−1!ξ2p2−ziCN−1ξ2p2−γ0+zi1+γ0C.


## 4. Numerical Results

The correctness of the performance analysis of our considered system is verified by Monte Carlo simulations. In general, we assumeδR2=δD12=δD22=δEj2=1, K =  M = 8, and γ¯1p=γ¯2p=γ¯3p=γ¯. The channel parameters are given in Table 1 [5]. Different power coefficient combinations are assumed as follows: scenario 1: a12=ξ12 = 0.9 and a22=ξ22 = 0.1; scenario 2: a12=ξ12 = 0.8 and a22=ξ22 = 0.2; scenario 3: a12=ξ12 = 0.7 and a22=ξ22 = 0.3 (In this paper, the fixed power allocation factors are considered; however, the results derived can also be utilized for our future work when power allocation is studied.).Table 1
Channel parameters.
Shadowingm1pb1pΩ1pFrequent heavy shadowing (FHS)10.0630.0007Average shadowing (AS)50.2510.279Infrequent light shadowing (ILS)100.1581.29As proved in [40], a satellite channel model is expected to be general and applicable for a wide range of elevation angles, under which the satellite can be observed. In this regard, the most common approach to evaluate the impact of elevation angle on channel statistical parameters is based on the transformation from empirical expression. According to Abdi et al. [40], the maximum elevation angle is around 80°, while the minimum elevation angle is considered usually around 20° in order to tackle the geographical terrain affects. Hence, this is particularly useful when we apply a set of data parameters with moderate variation to a model with specific shadowing and infrequent light shadowing conditions (i.e., frequent heavy shadowing, average shadowing, and infrequent light shadowing). Please note that, for specific shadowing conditions, different parameters have been employed to cover a range of elevation angles in many existing works [5].Figure2 illustrates the SOP versus γ¯ and k by setting γ¯E = 1 dB, γ0 = 0 dB, N = 2, scenario 1, and M = 3 under different shadowing fading. We can find that when γ¯ grows to be a fixed value, the SOP tends to be constant. Besides, it can be found that the SOP descends with the increase of the nonideal hardware level. In addition, it should be noted that the SOP under different channel conditions is consistent for high SNR regimes, which means that channel conditions will seriously affect the SOP.Figure 2
The SOP versusγ¯ and k by setting γ¯E = 1 dB, γ0 = 0 dB, N = 2, scenario 1, and M = 3 under different shadowing fading.Figure3 plots the SOP versus γ¯ for different N and k with γ0 = 0 dB, γ¯E = 1 dB, M = 3, and scenario 1 under the AS scenario. It can be found that, with N being larger, the SOP will be larger because more eavesdroppers are utilized to steal the legitimate signals. Similar to Figure 2, the SOP will be a fixed value for big enough γ¯. This is due to the fact that an upper bound of the SINR is existed to the legitimate user, which is related to (2), (3), (7), (8), and (9).Figure 3
The SOP versusγ¯ for different N and k with γ0 = 0 dB, γ¯E = 1 dB, M = 3, and scenario 1 under the AS scenario.Figure4 examines the SOP versus γ¯ for different M and k with γ¯E = 1 dB, γ0 = 0 dB, N = 2, and scenario 1 under the AS scenario. We may obtain that when M grows larger, the SOP will be lower. Figure 5 illustrates the SOP versus γ¯ for different scenarios and k with γ¯E = 1 dB, γ0 = 0 dB, N = 2, and M = 3 under the AS scenario. We can obtain that, with the gap of the power allocation factors becoming larger, the SOP will be lower; this is because the communication quality of the worse user is ensured. At last, we find that the secrecy performance will deteriorate with the nonideal hardware level.Figure 4
The SOP versusγ¯ for different M and k with γ¯E = 1 dB, γ0 = 0 dB, N = 2, and scenario 1 under the AS scenario.Figure 5
The SOP versusγ¯ for different scenarios and k with γ¯E = 1 dB, γ0 = 0 dB, N = 2, and M = 3 under the AS scenario.


## 5. Conclusions

In this paper, we studied the secrecy outage probability of NOMA-based ISMUAVNs in which nonideal hardware is considered and the partial UAV selection scheme is adopted. The exact closed-form expression of SOP for our proposed system was derived. Moreover, in the high SNR regime, we derived the asymptotic SOP. To get more insights, the secrecy coding gain and secrecy diversity order were obtained. The impacts of NOMA technology, nonideal hardware, and the partial UAV selection scheme on the performance of the considered system were revealed by theoretical analysis.



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