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

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.


Introduction
e 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]. communication network for more rewards, and a multiagent reinforcement learning framework was proposed to find the optimal policy. e potential game, mean-field game, Stackelberg game, graphical game, and coalition game were utilized to solve the resource management problem of largescale UAVs in [10]. e 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][13][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. e 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][32][33][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. e 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-tonoise ratios (SNRs), the asymptotic expression for SOP of the considered secrecy ISMUAVNs is also derived.
e rest of this paper is given in the following. Section 2 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 CN(a, b) is the complex Gaussian distribution of a random vector a and covariance matrix b.

System Model
As plotted in Figure 1, in this paper, we investigate NOMAbased secrecy ISMUAVNs, which consist of a satellite (S), namely, Alice, multiple UAVs (R p , p ∈ 1, . . . , M { }), two destinations (namely, D 1 and D 2 ), and N eavesdroppers. Decode-and-forward (DF) protocol is utilized at R. Two time slots are used for the total transmission. e direct transmission link is not considered between S and D i i ∈ 1, 2 { } 29, 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.).

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For the first time slot, the signal s(t) is forwarded by S with E[|s(t)| 2 ] � 1 to pth R. e received signal at pth R is derived as where f SR p ,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 P S denotes the transmit power from S. When applying the NOMA scheme, the signal is shown as s(t) � a 1p x 1 (t) + a 2p x 2 (t), where x 1 (t) and x 2 (t) present the target transmitted signal to D 1 and D 2 , respectively. Besides, it is assumed that a 1p > a 2p for the reason that the channel D 1 suffers heavy fading than that of D 2 . On this foundation, much power should be allocated to D 1 . a 1p and a 2p represent the transmission power allocation (PA) factors for D 1 and D 2 , respectively. In addition, . k 1p and k 2p denote the nonideal levels for nonideal hardware which are defined in [3]. n R p (t) represents the additive white Gaussian noise (AWGN) at pth R distributed as n R p (t) ∼ CN(0, δ 2 R p ). When the signal is received by R, SIC works as follows: x 1 (t) is firstly decoded and then deleted from the obtained signal [12]. ereafter, x 2 (t) 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-anddistortion ratio (SINDR) of detecting x 1 (t) at pth R is given by where c 1p � |f SR p ,k | 2 P S /δ 2 R p . Moreover, we can obtain the signal-to-noise-and-distortion ratio (SNDR) of detecting x 2 (t) at pth R as As announced before, the partial UAV selection scheme [35] is adopted for the considered system model, and the SINDR of x 1 (t) and x 2 (t) during the first transmission link is, respectively, shown as For the second time slot, the selected pth R forwards received signal s(t) to D 1 and D 2 , respectively; thus, the signal received at D i can be represented as where h R p D i denotes the channel coefficient between pth R and D i which undergoes Rayleigh fading, P R p represents the transmit power for pth R, x i (t) denotes the expected signal, ξ jp j ∈ 1, 2 { } which suits ξ 1p > ξ 2p , and ξ 2 1p + ξ 2 2p � 1 represents the PA factors at pth R, respectively (We apply the same assumption which means the channel quality of user D 1 is worse than that of user D 2 .).
) represents the distortion noise which is caused by nonideal hardware. ξ 1p along with ξ 2p denotes the nonideal levels of nonideal hardware [35]. erefore, D 2 performs SIC to decode x 1 (t) (while x 2 (t) is treated as the cochannel noise).
From (6), the SINDR of x 1 (t) sent by pth R at D 2 is represented as where c 2p � |h R p D 2 | 2 P R p /δ 2 D 2 . c D p1 ,D 2 must be satisfied as c D p1 ,D 2 ≥ c th due to the same condition like (2). D 2 deleted the decoded information from the received signal by observing its own signal; hence, the final SNDR for signal x 2 (t) forwarded by pth R is given by Figure 1: Illustration of the system model.

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is regarded as the serial interference; the SINDR received at D 1 from pth R can be obtained as where c 3p � |h R p D 1 | 2 P R p /δ 2 D 1 . At the same time, due to the fact that the signal sent to relays can be received by all eavesdroppers, the heard signal of the jth eavesdropper can be represented as where h R p E j represents the channel coefficient between pth R and the jth Eve which is modeled as Rayleigh fading. n E j (t) with n E j (t) ∼ CN(0, δ 2 E j ) denotes the AWGN at the jth Eve. rough 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 where c D pi E j � |h R p E j | 2 P R p /δ 2 E j . Due to the cooperation of all eavesdroppers, we can get the SNR of all Eves' link as e 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 where [x] + � Δ max[x, 0], C B pi ρ � log 2 (1 + c D pi ,ρ ), and C D pi E � log 2 (1 + c D pi E ) (CBpiρ means the secrecy capacity at the ρ user to detect the ith signal from pth R.).

e 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) of c 1p can be shown as where 1 F 1 (a; b; x) denotes the confluent hypergeometric function [38]. c 1p represents the average SNR between Alice and pth R, and Ω 1p 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 m 1p is an integer, and the PDF and cumulative distribution function (CDF) of c 1p are, respectively, written as where Δ 1p � β 1p − δ 1p /c 1p and (·) k denotes the Pochhammer symbol [38].

Terrestrial Channel Model.
In the former section, we assume the channel model between R and intended users or eavesdroppers which undergoes independent and identically distributed (i.i.d) Rayleigh fading. us, the CDF and PDF of c U (U ∈ 2p, 3p, D i E j ) are, respectively, shown as and where c U is the average channel gain. By utilizing (11) and (12) along with [5], the CDF and PDF of c D pi E are, respectively, derived as 4 Wireless Communications and Mobile Computing 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 where and where P 1ρ c 0 � P 11ρ c 0 + P 21ρ c 0 − P 11ρ c 0 P 21ρ c 0 .
By using (4) and (5), P 1iρ and P 2iρ are, respectively, presented as Also, C 0 � log 2 (1 + c 0 ). In the following, to make the analysis easier, c D p1 E � c D p2 E � c E is assumed, and the detailed analysis of SOP for D 1 and D 2 is, respectively, given in the following theorems. Theorem 1. At D 1 , the closed-form expression for P 1D 1 by detecting x 1 (t) is given by where and In (25), B � a 2 1p k 2 1p + a 2 2p + a 2 2p k 2 2p . In (26),

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where K 1 denotes the number of terms, H 1 � min(ξ 2 1p /C, (ξ 2 1p − Ac 0 )/(A + Ac 0 )), , z i � t i + 1 presents the ith zero of Legendre polynomials, and w i denotes the Gaussian weight, which is given in Table (25.4) of [39].
Proof. See Appendix B.

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 c 1p ⟶ ∞, we can derive where o(x) denotes the infinitesimal of x in the higher order. Similarly, when c U ⟶ ∞, (17) can be rewritten as Proposition 1. At D 1 , the asymptotic SOP for decoding x 1 (t) of our proposed system is represented as At D 2 , the asymptotic SOP for decoding x 1 (t) of our proposed system is obtained as 6 Wireless Communications and Mobile Computing x 2 (t) is only detected by D 2 ; therefore, the asymptotic expression of SOP for decoding x 2 (t) for our proposed system is shown as 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 letting c 1p � c 2p � c 3p � c, (22) can be expressed as where G d � min(M, 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, When M > 1,
As 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°i n 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]. Figure 2 illustrates the SOP versus c and k by setting c E � 1 dB, c 0 � 0 dB, N � 2, scenario 1, and M � 3 under different shadowing fading. We can find that when c 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 3 plots the SOP versus c for different N and k with c 0 � 0 dB, c 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 Wireless Communications and Mobile Computing steal the legitimate signals. Similar to Figure 2, the SOP will be a fixed value for big enough c. is 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 4 examines the SOP versus c for different M and k with c E � 1 dB, c 0 � 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 c for different scenarios and k with c E � 1 dB, c 0 � 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

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. e 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. e impacts of NOMA technology, nonideal hardware, and the partial UAV selection scheme on the performance of the considered system were revealed by theoretical analysis.

AS:
Average shadowing ASC: Average secrecy capacity AWGN: Additive white Gaussian noise CDF: Cumulative distribution function CSI: Channel state information DF: Decode and forward 5G: Fifth generation FHS: Frequency heavy shadowing ILS: Infrequent light shadowing ISMUAVNs: Integrated satellite-multiple unmanned aerial vehicle networks ISTRNs: Integrated satellite-terrestrial relay networks LOS: Line of sight MC: Monte Carlo NOMA: Nonorthogonal multiple access OP: Outage probability PDF: Probability density function PIC: Parallel interference cancellation PLS: Physical layer security SatCom: Satellite communication SIC: Successive interference cancellation SINDR: Signal-to-interference-plus-noise-anddistribution ratio SNDR: Signal-to-noise-and-distortion ratio SNR: Signal-to-noise ratio SNRs: Signal-to-noise ratios SOP: Secrecy outage probability SR: Shadowed-Rician UAV: Unmanned aerial vehicle.