# On the Performance of the Unified NOMA Framework for Satellite-Terrestrial Networks with Hardware Impairments

Authors: Jinlong Zhao; Xinwei Yue; Shaoli Kang; Wanwei Tang
Journal: Wireless Communications and Mobile Computing
 (2022)

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## Abstract

A unified nonorthogonal multiple access (NOMA) framework for satellite-terrestrial network is investigated over Shadowed-Rician fading channels, which can be applied to both power domain NOMA (PD-NOMA) and code domain NOMA (CD-NOMA). Under the realistic assumption of hardware impairments (HIs) and imperfect successive interference cancellation (SIC), the achievable outage probability of the paired users is analyzed first, and the closed-form expressions of exact and asymptotic outage probability are derived. Next, the ergodic sum rate is investigated in different levels of HIs and residual interference. In addition, the energy efficiency in the delay-tolerant transmission mode is studied to characterize the performance of the satellite-terrestrial network. Finally, the Monte Carlo simulation results demonstrate the accuracy of theoretical derivation and showcase the effects of a wide range of satellite channel parameters and the realistic assumption on outage probability, ergodic rate, and energy efficiency.


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## Body

## 1. Introduction

The satellite-terrestrial network has been envisioned as one of the research focuses to provide global service to users in future wireless communication networks [1]. A number of recently published studies have demonstrated its superiority over traditional satellite communications. Nonorthogonal multiple access (NOMA) is an effective approach to provide massive connectivity and is considered a new paradigm for the multiple access technology [2–4]. Therefore, introducing NOMA technology into the satellite-terrestrial network can improve spectrum efficiency and resource utilization [5, 6].The application of NOMA technology to satellite communications is a promising way to further expand the integration of space and earth. The NOMA-based hybrid satellite-terrestrial relay system was investigated, in which the outage probability, asymptotic outage probability, and achievable rate were derived in [7]. In order to analyze the multilayer heterogeneous satellite network, the authors in [8] proposed a NOMA-enabled two-layer geostationary earth orbit/low earth orbit (LEO) satellite network, where the ergodic capacity of the proposed system was investigated in the case of coexistence of frequencies. In [9], a dedicated half-duplex (HD) terrestrial decode-and-forward (DF) relay was used to assist a user with weak channel condition to complete communications with the satellite. Moreover, in [10], the authors extended the work of [9] to investigate an amplify-and-forward multiantenna relay scenario and performed the performance analyses for fixed gain and variable gain relaying protocols, in which the exact and asymptotic outage probability expressions were obtained. In order to guarantee the transmission delay and transmission rate for satellite industrial Internet of things, the authors in [11] proposed the satellite-terrestrial integrated industrial Internet of things and optimized the power distribution ratio of terrestrial users, thus achieving the purpose of reducing transmission cost. Considering the various delay quality of service (QoS) requirements for satellite Internet of things (S-IoT) applications, the authors in [12] investigated the delay-limited performance of NOMA-based downlink S-IoT networks by analyzing the effective capacity and proved the effects of different power distribution schemes on the proposed networks. In [13], the authors investigated the NOMA-assisted integrated satellite terrestrial networks with the partial relay selection scheme under imperfect successive interference cancellation (SIC), which can improve the spectral efficiency and expand the transmission coverage area. In [14], uplink and downlink NOMA were considered in intersatellite communication networks, where the optimum control methods were employed to delay suppression, reduce bit error rate, and increase throughput. The authors in [15] adopted an unmanned aerial vehicle using the DF protocol to assist satellite communicated with the terrestrial NOMA users; specifically, the outage probability and system throughput in delay-limited transmission mode were deduced to validate the superiority of the proposed system. In order to maximize the system capacity and energy efficiency, the NOMA architecture-enabled space-terrestrial satellite network was investigated, in which the satellite and terrestrial base stations provided services in a collaborative manner for users, and the NOMA scheme was only implemented for terrestrial networks [16]. More recently, a great deal of works have focused on the performance of hybrid satellite-terrestrial networks in cognitive radio (CR) scenarios [17–20]. In [17], the NOMA-based cognitive hybrid satellite-terrestrial overlay networks were proposed, in which the outage probability performance was investigated. One of the assumptions was that the power distribution scheme of NOMA users was determined by instantaneous channel conditions. The partial secondary network selection schemes were analyzed for NOMA-based overlay cognitive integrated satellite-terrestrial relay networks in [18], which proved the superiority of the proposed scheme. In [19], the authors derived the closed-form expression of outage probability for the primary satellite network, and the secondary transmitter was regarded as a relay to assist the primary transmission. Furthermore, the authors of [20] investigated the performance of the secondary network in terms of outage probability, sum throughput, and ergodic rate in which the nearby NOMA user operated in full-duplex mode and adopted DF relay protocol to improve the performance of the remote NOMA user.Unfortunately, for some practical reasons such as quantization error, phase noise, in-phase/quadrature imbalance, and limitations on transceiver performance, the transmission nodes are vulnerable to hardware impairments (HIs) [21, 22]. However, the HIs from different sources have a certain randomness, resulting in these impairments which cannot be eliminated satisfactorily [23]. In [24–26], the authors proposed a realistic scenario in which the effects of HIs and imperfect channel state information (CSI) on the performance of simultaneous wireless information and power transfer NOMA relaying network were considered. The authors in [27] studied the outage performance of the NOMA-based integrated satellite-terrestrial relay network in the presence of HIs. In [28], the authors derived a closed-from expression for the outage probability in the NOMA-based satellite communication network considering HIs and channel estimation errors. The authors of [29] investigated the impacts of HIs on the secrecy performance of the NOMA-based integrated satellite multiple-terrestrial relay network.However, existing NOMA schemes can be classified into two categories: power domain NOMA (PD-NOMA) and code domain power NOMA (CD-NOMA) according to the characteristics of resource mapping and spread spectrum mode. This division method is not conducive to the unification of the NOMA standard. Therefore, in order to facilitate the unification of NOMA technical solution, NOMA is divided into single-carrier NOMA and multicarrier NOMA according to the number of resource element (RE) or subcarrier occupied by users. CD-NOMA is a typical multicarrier NOMA, which uses a sparse matrix to map the data flow of multiple users to multiple REs or subcarriers. PD-NOMA can be regarded as a single carrier NOMA to realize user reuse by power adjustment. Therefore, from this perspective, PD-NOMA can be considered a special case of CD-NOMA, and CD-NOMA can be regarded as a special extension of PD-NOMA. While the aforementioned works have provided us with a better understanding of the application of PD-NOMA technology in satellite-terrestrial networks, the analysis for the unified NOMA framework-based satellite-terrestrial networks is far from well understood, which inspires the construction of this paper. In addition, from the perspective of practical application, it is of great significance to investigate the effects of HIs and imperfect SIC on satellite-terrestrial networks. In this paper, we propose a unified NOMA framework for satellite-terrestrial networks and analyze the performance of the proposed system with HIs and imperfect SIC under Shadowed-Rician fading channels. The essential contributions of this paper are summarized as follows:(1)
We propose a unified NOMA technology framework for satellite-terrestrial networks, which integrates two design patterns, PD-NOMA and CD-NOMA. A sparse matrix is applied to map the data stream ofK users to N REs, and the unified NOMA framework can be transformed into PD-NOMA scheme (N=1) and CD-NOMA scheme (N>1) according to the different numbers of RE occupied by users(2)
Considering two imperfections, HIs and imperfect SIC, we investigate the outage performance of the proposed system and derive the closed-form expressions of exact outage probability for the paired NOMA users. More specifically, the asymptotic outage probability is provided to get more insights into the high SNR region. Additionally, the impacts of a wide range of satellite parameters on network performance are studied(3)
We study the ergodic rate of the unified NOMA framework for the satellite-terrestrial network, and the effects of HIs and residual interference on ergodic rate are discussed. In addition, the energy efficiency in the delay-tolerant transmission mode is investigated to characterize the performance of the proposed system(4)
Numerical results validate the theoretical analysis and the advantages of the satellite-terrestrial network adopting the NOMA scheme. The HIs and residual interference have significant negative effects on the outage performance, ergodic sum rate, and energy efficiency. Moreover, a pair of terrestrial users is able to achieve better system performance as the number of REs increasesThe remainder of this article is organized as follows. In Section2, the unified NOMA framework for satellite-terrestrial network is presented and the instantaneous SINR is derived. Section 3 provides the exact and asymptotic outage performance for a pair of terrestrial NOMA users. Then, in Section 4, the ergodic sum rate of satellite-terrestrial NOMA networks is analyzed. Section 5 obtains the numerical Monte Carlo results to verify the theoretical analysis. This paper is summarized in Section 6, and the mathematical proof is shown in Appendix.The main mathematical symbols used in this paper are shown as follows:FX· and fX· denote the cumulative distribution function (CDF) and the probability density function (PDF) of a random variable X, respectively; E· is the expectation operator. ·22 denotes the Euclidean two norms of a vector; ⊙ is the dot product of two matrix elements.


## 2. System Model

Consider a unified downlink NOMA scenario for the satellite-terrestrial network, where the satellite (S) transmits the superimposed signals to the K terrestrial users directly or with the aid of the DF relay (R), as depicted in Figure 1. In particular, S maps the data flows of K users into N time/frequence REs by using a sparse spread spectrum matrix GN×K, where gk=g1kg2k⋯gNkT denotes the RE indication assigned to Dk and gk is the k-th column of the matrix GN×K. It is noteworthy that, depending on the number of RE occupied by users, the unified NOMA scheme can be converted into the PD-NOMA scheme (N=1) and CD-NOMA scheme (N>1). The channel vector pertaining to the satellite link (S⟶Dk) is denoted by hks=h1kh2k⋯hNkT, where hnk represents the channel response between the n-th RE of Dk and S, and each channel response is independent and follows the Shadowed-Rician fading distribution, whereas the channel vector for the terrestrial link (R⟶Dk) is represented by hkr=h1krh2kr⋯hNkrT, where hnkr represents the channel response between the n-th RE of Dk and R with hnkr~CN0,1. Meanwhile, all the receiving nodes are disturbed by additive white Gaussian noise (AWGN) with zero mean and average power of N0. We assume that S and terrestrial nodes are equipped with a single antenna and K terrestrial users are divided into K/2 orthogonal paired user groups, where two prepaired users are ordered according to their respective channel gain [19, 30]. To facilitate analysis, it is assumed that Dp and Dq are randomly selected (user selection can further improve system performance, which inspires us to investigate more effective user selection strategies in future work) to execute the NOMA scheme [31], and the satellite link condition of Dq is worse than that of Dp due to the influence of shadow fading. In addition, due to the propagation characteristics and large transmission delay of the satellite channel, it is difficult to obtain perfect CSI for the satellite channel, which has been studied in [6, 32]. However, the main aim of this paper is to investigate the performance of the unified NOMA framework for satellite-terrestrial networks with HIs and imperfect SIC. Therefore, perfect CSI is assumed in this paper, and the results of this paper will be the basis for future research on imperfect CSI.Figure 1
System model.Two phases are used throughout the transmission process. In the first phase,S sends the superposed information X=gpρξapPSxp+gqρξaqPSxq to R and terrestrial users, where the destination user’s data flow is spread over a column of GN×K. In addition, PS is the transmit power at S, ξ=r,p,q, xu denotes the message transmitted to Du with unit power, and u=p,q. ap and aq are the power allocation coefficients for Dp and Dq that satisfy ap+aq=1 and aq>ap to guarantee the users’ fairness. ρξ=GSGξφξχξ, GS is the gain of the satellite antenna, Gξφξ=GξJ1uξ/2uξ+36J3uξ/uξ3 denotes the beam gain of node ξ, Gξ is the antenna gain of node ξ, and φξ denotes the included angle between ξ and beam center in relation to the satellite [32]. uξ=2.07123sinφξ/sinφξ3dB, φξ3dB denotes the 3 dB angle, and Jι· is the first class-modified Bessel function with order ι. Moreover, χξ=c/4πfcdξ2 is the free space loss between the satellite and node ξ, where c, fc, and dξ denote the speed of light, operating frequency, and the distance between the satellite and node ξ, respectively. Accordingly, the received signals at R and terrestrial users can be given by
(1)yξs=wξhξs⊙X+gηηξs+wξnξs,where nξs~CN0,N0IN denotes the AWGN vector for the link between the satellite and node ξ and IN is an identity matrix of size N. ηξs~CN0,ksξ2ρξPS is the HIs between S and node ξ, where ksξ2 is the level of HIs [29, 32], gη=g1ηg2η⋯gNηT, and gnη denotes the HI indication. Since maximal ratio combiner (MRC) is adopted at Dk over N REs, we have that wξ=hξsH/hξs, where wξ represents the receive weight vector. Assuming ρphps22>ρqhqs22 and applying the SIC principle, the instantaneous received signal to interference and noise ratio (SINR) of R decoding xq can be given by
(2)ΓR,xq1=hrs⊙gq22aqρrγhrs⊙gp22apρrγ+hrs⊙gη22ksr2ρrγ+1,where γ=PS/N0. By applying the SIC scheme, R begins decoding xp after decoding xq. However, taking into consideration the influence of imperfect SIC, the instantaneous received SINR of R decoding xp can be represented as
(3)ΓR,xp1=hrs⊙gp22apρrγhrs⊙gq22κraqρrγ+hrs⊙gη22ksr2ρrγ+1,where κr represents the level of residual interference from the previous SIC process and 0≤κr≤1 (it is worth noting that we use the imperfect SIC model in [33, 34], assuming that κr is a constant. However, κr obeys the Gaussian distribution in some complicated cases [35, 36], which is beyond the scope of our research, but we will consider it in future work).Similar to (2) and (3), we can express the instantaneous received SINRs via the S-Dp link and S-Dq link at respective nodes as
(4)ΓDp,xq1=hps⊙gq22aqρpγhps⊙gp22apρpγ+hps⊙gη22ksp2ρpγ+1,(5)ΓDp,xp1=hps⊙gp22apρpγhps⊙gq22κ1aqρpγ+hps⊙gη22ksp2ρpγ+1,(6)ΓDq,xq1=hps⊙gq22aqρqγhps⊙gp22apρqγ+hps⊙gη22ksq2ρqγ+1,where κ1 represents the level of residual interference from xq and 0≤κ1≤1.In the second phase,R transmits the superimposed signal Y=gpΠubpPRxp+gqΠubqPRxq to Dp and Dq with power PR and power allocation coefficient bu, where bp+bq=1, Πu=1/duΔ, du denotes the distances from R to Du, and Δ is the pathloss exponent. By spreading the terrestrial user’s data stream over a column of the matrix GN×K, the received signal at Du is represented as
(7)yur=wuhur⊙Y+gηηur+wunur,where wu=hurH/hur, the distortion noise from R to Du is denoted as ηur~CN0,kru2ΠuPR, and nur~CN0,N0IN is the AWGN at Du.Assume thatdq>dp and γ=PR/N0; the instantaneous SINR of Dq decoding its own message can be given by
(8)ΓDq,xq2=hqr⊙gq22bqΠqγhqr⊙gp22bpΠqγ+hqr⊙gη22krq2Πqγ+1.Following the SIC scheme,Dp decodes xq firstly with the received SINR as
(9)ΓDp,xq2=hpr⊙gq22bqΠpγhpr⊙gp22bpΠpγ+hpr⊙gη22krp2Πpγ+1.Subsequently, by subtracting the interfering term associated withxq from ypr, the received SINR of Dp decoding its own message can be given by
(10)ΓDp,xp2=hpr⊙gp22bpΠpγhpr⊙gq22κ2bqΠpγ+hpr⊙gη22krp2Πpγ+1,where κ2 represents the level of residual interference from the preceding SIC cycle and 0≤κ2≤1.


## 3. Channel Statistical Properties

Assuming that the channel link from satellite to then-th RE of terrestrial node ξ is modeled as Shadowed-Rician fading distributions, the PDF and CDF of hnξ2 are given by [32]
(11)fhnξ2x=αξ∑i=0mξ−1ζixie−βξ−δξx,(12)Fhnξ2x=1−αξ∑i=0mξ−1i!ζi∑j=0iβξ−δξ−i+1−jj!×xje−βξ−δξx,withαξ=2bξmξmξ/2bξ2bξmξ+Ωξmξ, βξ=1/2bξ, and δξ=Ωξ/2bξ2bξmξ+Ωξ, where 2bξ, mξ, and Ωξ denote the average power of the multipath component, the shape parameter of Nakagami-m, and the average power of the line-of-sight component, respectively. ζi=−1i1−mξiδξi/i!2 and ·i denotes the Pochhammer symbol ([37], p. xliii). Additionally, the terrestrial link from relay to the n-th RE of node u undergoes identically distributed (i.i.d.) Rayleigh fading distribution.Lemma 1.
LetYξ=∑j=1Nh~jξ2 be sum of N squared identically distributed Shadowed-Rician random variables; the PDF of Yξ is given by
(13)fYξNy=αξN∑i1=0mξ−1⋯∑iN=0mξ−1∏j=1Nζij∏k=1N−1Bik+1+1,∑s=1kis+kyΞe−βξ−δξy,where Ξ=∑j=1Nij+N−1 and B·,· denotes the beta function ([37], Eq. (8.384.1)).Proof.
See AppendixA.
Assume that all of the terrestrial links follow the Rayleigh fading distribution andgp, gq, and gη have the same column weights; we derive Zu=hur⊙gp22=∑j=1Nhjurgju2. With the aid of Eq. (7) of [38], the CDF and PDF of Zu are given by
(14)FZuz=1−e−z∑k=0N−1zkk!,(15)fZuz=1N−1!zN−1e−z,respectively.


## 4. Outage Probability Analysis

### 4.1. Exact Outage Probability

In this section, we derive the closed-form expressions for the exact outage probabilities. The outage event is defined as the instantaneous SINR is inferior to the predefined thresholdϕu, where ϕu=22Ru−1 and Ru is the predefined rate for Du. Assume that the terrestrial users process the data received from S and R nodes by selection combining. Thus, the outage probability of Dp can be formulated as
(16)Pout,p=1−PrΓR,xq1>ϕq,ΓR,xp1>ϕp,ΓDp,xq2>ϕq,ΓDp,xp2>ϕp1−PrΓDp,xq1>ϕq,ΓDp,xp1>ϕp.Theorem 2.
The expression for the outage probability ofDp for N>1 is given by
(17)Pout,pCD=1−αrN∑i1=0mr−1⋯∑iN=0mr−1∏j=1Nζij∏k=1N−1Bik+1,Θk∑t=0ΞΞ!t!×ε1te−βr−δrε1βr−δrΞ−t+1e−ε3∑k=0N−1ε3kk!1−αpN∑i1=0mp−1⋯∑iN=0mp−1∏j′=1Nζij′×∏k′=1N−1Bik′+1,Θk′∑v=0ΞΞ!v!ε2ve−βp−δpε2βp−δpΞ−v+1,
whereε1=maxψ1,ψ2, ψ1=ϕq/ρrγaq−ϕqap+ksr2, ψ2=ϕp/ρrγap−ϕpκraq+ksr2, ε2=maxψ3,ψ4, ψ3=ϕq/ρpγaq−ϕqap+ksp2, ψ4=ϕp/ρpγap−ϕpκ1aq+ksp2, ε3=maxψ5,ψ6, ψ5=ϕq/Πpγbq−ϕqbp+krp2, ψ6=ϕp/Πpγbp−ϕpκ2bq+krp2, and Θk=∑s=1kis+k. (17) is obtained in the conditions of ap>maxκraq+ksr2ϕp,κ1aq+ksp2ϕp, aq>maxap+ksr2ϕq,ap+ksp2ϕq, bp>κ2bq+krp2ϕp, and bq>bp+krp2ϕq.Proof.
See AppendixB.Corollary 3.
The outage probability expression forDp with N=1 is given by
(18)Pout,pPD=1−αr∑i=0mr−1i!ζi∑j=0iβr−δr−i+1−jj!ε1je−βr−δrε1−ε3×1−αp∑i′=0mp−1i′!ζi′∑k=0i′βp−δp−i′+1−kε2kk!e−βp−δpε2.Proof.
By substituting (2)–(5), (9), and (10) into (16) and then by some algebraic application, we can obtain the outage probability expression for Dp by means of (12). The proof is completed.Next, we focus attention on calculating the outage probability ofDq. Dq falls into outage if it cannot decode its own signal at both phases or R cannot decode Dq’s signal in the first phase. Hence, the outage probability of Dq can be formulated as
(19)Pout,q=PrΓDq,xq1<ϕq1−PrΓR,xq1>ϕq,ΓDq,xq2>ϕq.Theorem 4.
The expression for the outage probability ofDq for N>1 is given by
(20)Pout,qCD=1−αrN∑i1=0mr−1⋯∑iN=0mr−1∏j=1Nζij∏k=1N−1Bik+1,Θk∑θ=0ΞΞ!θ!×λ1θe−βr−δrλ1βr−δrΞ−θ+1e−λ3∑l=0N−1λ3ll!1−αqN∑i1=0mq−1⋯∑iN=0mq−1∏j′=1Nζij′×∏k′=1N−1Bik′+1,Θk′∑μ=0ΞΞ!μ!λ2μe−βq−δqλ2βq−δqΞ−μ+1,
whereλ1=ϕq/ρrγaq−ap+ksr2ϕq, λ2=ϕq/ρqγaq−ap+ksq2ϕq, and λ3=ϕq/Πqγbq−bp+krq2ϕq. The above variables need to satisfy the conditions that bq>bp+krq2ϕq and aq>maxap+ksr2ϕq,ap+ksq2ϕq.Proof.
See AppendixC.Corollary 5.
The outage probability expression forDq with N=1 is given by
(21)Pout,qPD=1−αr∑i=0mr−1i!ζi∑j=0iβr−δr−i+1−jj!λ1je−βr−δrλ1−λ3×1−αq∑i′=0mq−1i′!ζi′∑s=0i′βq−δq−i′+1−sλ2ss!e−βq−δqλ2.


### 4.2. Asymptotic Outage Probability

In order to reduce computational complexity and better understand the effects of key system indicators and channel parameters on the performance of the proposed system at a high SNR regime, the closed-form expressions of asymptotic outage probabilities forDp and Dq are derived. As y approaches infinity, the first summation term in equation (13) has a greater effect on the system performance than the other terms, which are approximately 0. Thus, we only retain the first summation term in equation (13) and employ the Maclaurin series expansion. The approximate expression of PDF for Yξ can be expressed as
(22)fYξy=αξNN−1!yN−1+oyN−1,N>1,where oyN−1 is a higher-order infinitesimal of y. Based on (22), the approximate expression CDF of Yξ for N>1 at the high SNR regime can be expressed as
(23)FYξy=αξNN!yN+oyN,N>1.In addition, we can easily acquire the approximate expression CDF ofYξ for N=1 at the high SNR regime which can be expressed as [32]
(24)FYξy=αξy,N=1.In the following, the approximate expression CDF ofZu will be derived. By applying the Taylor series expansion, ez can be expanded as ez=∑k=0∞zk/k!. Thereby, ∑k=0N−1zk/k! in (25) can be replaced by Y=ez−∑k=N∞zk/k!. By substituting Y into (25), when z⟶0, the approximate expression CDF of Zu for N>1 can be expressed as
(25)FZuz=1N!zN,N>1.For the special case ofN=1, when z⟶0, FZuz with the approximation of e−z≈1−z can be expressed as
(26)FZuz=zΩ¯0,N=1,where Ω¯0 denotes the channel mean power.By substituting (23)–(26) into (B.1) and (C.1), respectively, we can get the asymptotic outage probability of Dp and Dq at the high SNR regime as follows:
(27)Pout,p∞=αpNε2Nϖ1−1−αrNε1Nϖ1−ε3NN!,N>1,αpε21−1−αrε11−ε3Ω¯0,N=1,(28)Pout,q∞=αqNλ2Nϖ1−1−αrNλ1Nϖ1−λ3NN!,N>1,αqλ21−1−αrλ11−λ3Ω¯0,N=1,respectively, where ϖ=N−1!.According to the conclusions of (27) and (28), the diversity orders of Dp and Dq in CD-NOMA (N>1) and PD-NOMA (N=1) are both 3N/three.


## 4.1. Exact Outage Probability

In this section, we derive the closed-form expressions for the exact outage probabilities. The outage event is defined as the instantaneous SINR is inferior to the predefined thresholdϕu, where ϕu=22Ru−1 and Ru is the predefined rate for Du. Assume that the terrestrial users process the data received from S and R nodes by selection combining. Thus, the outage probability of Dp can be formulated as
(16)Pout,p=1−PrΓR,xq1>ϕq,ΓR,xp1>ϕp,ΓDp,xq2>ϕq,ΓDp,xp2>ϕp1−PrΓDp,xq1>ϕq,ΓDp,xp1>ϕp.Theorem 2.
The expression for the outage probability ofDp for N>1 is given by
(17)Pout,pCD=1−αrN∑i1=0mr−1⋯∑iN=0mr−1∏j=1Nζij∏k=1N−1Bik+1,Θk∑t=0ΞΞ!t!×ε1te−βr−δrε1βr−δrΞ−t+1e−ε3∑k=0N−1ε3kk!1−αpN∑i1=0mp−1⋯∑iN=0mp−1∏j′=1Nζij′×∏k′=1N−1Bik′+1,Θk′∑v=0ΞΞ!v!ε2ve−βp−δpε2βp−δpΞ−v+1,
whereε1=maxψ1,ψ2, ψ1=ϕq/ρrγaq−ϕqap+ksr2, ψ2=ϕp/ρrγap−ϕpκraq+ksr2, ε2=maxψ3,ψ4, ψ3=ϕq/ρpγaq−ϕqap+ksp2, ψ4=ϕp/ρpγap−ϕpκ1aq+ksp2, ε3=maxψ5,ψ6, ψ5=ϕq/Πpγbq−ϕqbp+krp2, ψ6=ϕp/Πpγbp−ϕpκ2bq+krp2, and Θk=∑s=1kis+k. (17) is obtained in the conditions of ap>maxκraq+ksr2ϕp,κ1aq+ksp2ϕp, aq>maxap+ksr2ϕq,ap+ksp2ϕq, bp>κ2bq+krp2ϕp, and bq>bp+krp2ϕq.Proof.
See AppendixB.Corollary 3.
The outage probability expression forDp with N=1 is given by
(18)Pout,pPD=1−αr∑i=0mr−1i!ζi∑j=0iβr−δr−i+1−jj!ε1je−βr−δrε1−ε3×1−αp∑i′=0mp−1i′!ζi′∑k=0i′βp−δp−i′+1−kε2kk!e−βp−δpε2.Proof.
By substituting (2)–(5), (9), and (10) into (16) and then by some algebraic application, we can obtain the outage probability expression for Dp by means of (12). The proof is completed.Next, we focus attention on calculating the outage probability ofDq. Dq falls into outage if it cannot decode its own signal at both phases or R cannot decode Dq’s signal in the first phase. Hence, the outage probability of Dq can be formulated as
(19)Pout,q=PrΓDq,xq1<ϕq1−PrΓR,xq1>ϕq,ΓDq,xq2>ϕq.Theorem 4.
The expression for the outage probability ofDq for N>1 is given by
(20)Pout,qCD=1−αrN∑i1=0mr−1⋯∑iN=0mr−1∏j=1Nζij∏k=1N−1Bik+1,Θk∑θ=0ΞΞ!θ!×λ1θe−βr−δrλ1βr−δrΞ−θ+1e−λ3∑l=0N−1λ3ll!1−αqN∑i1=0mq−1⋯∑iN=0mq−1∏j′=1Nζij′×∏k′=1N−1Bik′+1,Θk′∑μ=0ΞΞ!μ!λ2μe−βq−δqλ2βq−δqΞ−μ+1,
whereλ1=ϕq/ρrγaq−ap+ksr2ϕq, λ2=ϕq/ρqγaq−ap+ksq2ϕq, and λ3=ϕq/Πqγbq−bp+krq2ϕq. The above variables need to satisfy the conditions that bq>bp+krq2ϕq and aq>maxap+ksr2ϕq,ap+ksq2ϕq.Proof.
See AppendixC.Corollary 5.
The outage probability expression forDq with N=1 is given by
(21)Pout,qPD=1−αr∑i=0mr−1i!ζi∑j=0iβr−δr−i+1−jj!λ1je−βr−δrλ1−λ3×1−αq∑i′=0mq−1i′!ζi′∑s=0i′βq−δq−i′+1−sλ2ss!e−βq−δqλ2.


## 4.2. Asymptotic Outage Probability

In order to reduce computational complexity and better understand the effects of key system indicators and channel parameters on the performance of the proposed system at a high SNR regime, the closed-form expressions of asymptotic outage probabilities forDp and Dq are derived. As y approaches infinity, the first summation term in equation (13) has a greater effect on the system performance than the other terms, which are approximately 0. Thus, we only retain the first summation term in equation (13) and employ the Maclaurin series expansion. The approximate expression of PDF for Yξ can be expressed as
(22)fYξy=αξNN−1!yN−1+oyN−1,N>1,where oyN−1 is a higher-order infinitesimal of y. Based on (22), the approximate expression CDF of Yξ for N>1 at the high SNR regime can be expressed as
(23)FYξy=αξNN!yN+oyN,N>1.In addition, we can easily acquire the approximate expression CDF ofYξ for N=1 at the high SNR regime which can be expressed as [32]
(24)FYξy=αξy,N=1.In the following, the approximate expression CDF ofZu will be derived. By applying the Taylor series expansion, ez can be expanded as ez=∑k=0∞zk/k!. Thereby, ∑k=0N−1zk/k! in (25) can be replaced by Y=ez−∑k=N∞zk/k!. By substituting Y into (25), when z⟶0, the approximate expression CDF of Zu for N>1 can be expressed as
(25)FZuz=1N!zN,N>1.For the special case ofN=1, when z⟶0, FZuz with the approximation of e−z≈1−z can be expressed as
(26)FZuz=zΩ¯0,N=1,where Ω¯0 denotes the channel mean power.By substituting (23)–(26) into (B.1) and (C.1), respectively, we can get the asymptotic outage probability of Dp and Dq at the high SNR regime as follows:
(27)Pout,p∞=αpNε2Nϖ1−1−αrNε1Nϖ1−ε3NN!,N>1,αpε21−1−αrε11−ε3Ω¯0,N=1,(28)Pout,q∞=αqNλ2Nϖ1−1−αrNλ1Nϖ1−λ3NN!,N>1,αqλ21−1−αrλ11−λ3Ω¯0,N=1,respectively, where ϖ=N−1!.According to the conclusions of (27) and (28), the diversity orders of Dp and Dq in CD-NOMA (N>1) and PD-NOMA (N=1) are both 3N/three.


## 5. Ergodic Sum Rate and Energy Efficiency

### 5.1. Ergodic Rate

The ergodic sum rate is an important indicator of system performance, which depends on the conditions of the channel links and the decoding process at all the terrestrial nodes. Hence, in this section, the ergodic sum rate of the proposed system is investigated.The achievable rate ofDp and Dq can be given by
(29)R¯p=12log21+minΓR,xp1,ΓDp,xp1,ΓDp,xp2,(30)R¯q=12log21+minΓR,xq1,ΓDp,xq1,ΓDq,xq1,ΓDp,xq2,ΓDq,xq2,respectively.Thus, the ergodic rate ofDp can be written as
(31)Rperg=E12log21+minΓR,xp1,ΓDp,xp1,ΓDp,xp2.Since it is difficult to obtain the exact expression of the ergodic sum rate, we derive the approximate ergodic sum rate.Theorem 6.
The approximate expression for the ergodic rate ofDp for N>1 is given by
(32)RpCD≈12log21+minX1apρrγX1ρrγκraq+ksr2+1,X2apρpγX2ρpγκ1aq+ksp2+1,bpΠpγNΠpγκ2bq+krp2N+1,
whereX1=αrN∑i1=0mr−1⋯∑iN=0mr−1∏j=1Nζij∏k=1N−1Bik+1,ΘkΞ+1!βr−δr−Ξ−2 and X2=αpN∑i1=0mp−1⋯∑iN=0mp−1∏j′=1Nζij′∏k′=1N−1Bik′+1,Θk′Ξ+1!βp−δp−Ξ−2.Proof.
See AppendixD.Corollary 7.
The approximate expression for the ergodic rate ofDp for N=1 is given by
(33)RpPD≈12log21+minX¯1apρrγX¯1ρrγκraq+ksr2+1,X¯2apρpγX¯2ρpγκ1aq+ksp2+1,bpΠpγΠpγκ2bq+krp2+1,
whereX¯1=αr∑i=0mr−1ζii+1!βr−δr−i−2 and X¯2=αp∑i=0mp−1ζii+1!βp−δp−i−2.In what follows, the ergodic rate ofDq can be written as
(34)Rperg=E12log21+minΓR,xq1,ΓDp,xq1,ΓDq,xq1,ΓDp,xq2,ΓDq,xq2.By substituting (2), (4), (6), (8), and (9) into (34), after some mathematical operations, the approximate expression for the ergodic rate of Dq for N>1 can be formulated as
(35)RqCD≈12log21+minΛ1,Λ2.In (35), Λ1 and Λ2 can be expressed as
(36)Λ1=minX1aqρrγX1ρrγap+ksr2+1,X2aqρpγX2ρpγap+ksp2+1,X3aqρqγX3ρqγap+ksq2+1,(37)Λ2=minbqΠpγNΠpγbp+krp2N+1,bqΠqγNΠqγbp+krp2N+1,where X3=αqN∑i1=0mq−1⋯∑iN=0mq−1∏j′′=1Nζij′′∏k′′=1N−1Bik′′+1,Θk′′Ξ+1!βq−δq−Ξ−2.Due to the proof process being similar to (32), we will not elaborate on it.For the special case withN=1, the approximate expression for the ergodic rate of Dq is calculated as
(38)RqPD≈12log21+minΛ¯1,Λ¯2.Λ¯1 and Λ¯2 in (38) can be expressed as
(39)Λ¯1=minX¯1aqρrγX¯1ρrγap+ksr2+1,X¯2aqρpγX¯2ρpγap+ksp2+1,X¯3aqρqγX¯3ρqγap+ksq2+1,(40)Λ¯2=minbqΠpγΠpγbp+krp2+1,bqΠqγΠqγbp+krp2+1,respectively, where X¯3=αq∑i=0mq−1ζii+1!βq−δq−i−2.


### 5.2. Energy Efficiency

Energy efficiency is another important index to evaluate the system performance. According to the definition in [39], energy efficiency can be expressed as
(41)ηEE=RsumνPs+Pc,where Pc is the fixed energy consumed by the transceiver and ν denotes the efficiency of the power amplifier. Rsum is the ergodic sum rate, which can be obtained from (32), (33), (35), and (38).


## 5.1. Ergodic Rate

The ergodic sum rate is an important indicator of system performance, which depends on the conditions of the channel links and the decoding process at all the terrestrial nodes. Hence, in this section, the ergodic sum rate of the proposed system is investigated.The achievable rate ofDp and Dq can be given by
(29)R¯p=12log21+minΓR,xp1,ΓDp,xp1,ΓDp,xp2,(30)R¯q=12log21+minΓR,xq1,ΓDp,xq1,ΓDq,xq1,ΓDp,xq2,ΓDq,xq2,respectively.Thus, the ergodic rate ofDp can be written as
(31)Rperg=E12log21+minΓR,xp1,ΓDp,xp1,ΓDp,xp2.Since it is difficult to obtain the exact expression of the ergodic sum rate, we derive the approximate ergodic sum rate.Theorem 6.
The approximate expression for the ergodic rate ofDp for N>1 is given by
(32)RpCD≈12log21+minX1apρrγX1ρrγκraq+ksr2+1,X2apρpγX2ρpγκ1aq+ksp2+1,bpΠpγNΠpγκ2bq+krp2N+1,
whereX1=αrN∑i1=0mr−1⋯∑iN=0mr−1∏j=1Nζij∏k=1N−1Bik+1,ΘkΞ+1!βr−δr−Ξ−2 and X2=αpN∑i1=0mp−1⋯∑iN=0mp−1∏j′=1Nζij′∏k′=1N−1Bik′+1,Θk′Ξ+1!βp−δp−Ξ−2.Proof.
See AppendixD.Corollary 7.
The approximate expression for the ergodic rate ofDp for N=1 is given by
(33)RpPD≈12log21+minX¯1apρrγX¯1ρrγκraq+ksr2+1,X¯2apρpγX¯2ρpγκ1aq+ksp2+1,bpΠpγΠpγκ2bq+krp2+1,
whereX¯1=αr∑i=0mr−1ζii+1!βr−δr−i−2 and X¯2=αp∑i=0mp−1ζii+1!βp−δp−i−2.In what follows, the ergodic rate ofDq can be written as
(34)Rperg=E12log21+minΓR,xq1,ΓDp,xq1,ΓDq,xq1,ΓDp,xq2,ΓDq,xq2.By substituting (2), (4), (6), (8), and (9) into (34), after some mathematical operations, the approximate expression for the ergodic rate of Dq for N>1 can be formulated as
(35)RqCD≈12log21+minΛ1,Λ2.In (35), Λ1 and Λ2 can be expressed as
(36)Λ1=minX1aqρrγX1ρrγap+ksr2+1,X2aqρpγX2ρpγap+ksp2+1,X3aqρqγX3ρqγap+ksq2+1,(37)Λ2=minbqΠpγNΠpγbp+krp2N+1,bqΠqγNΠqγbp+krp2N+1,where X3=αqN∑i1=0mq−1⋯∑iN=0mq−1∏j′′=1Nζij′′∏k′′=1N−1Bik′′+1,Θk′′Ξ+1!βq−δq−Ξ−2.Due to the proof process being similar to (32), we will not elaborate on it.For the special case withN=1, the approximate expression for the ergodic rate of Dq is calculated as
(38)RqPD≈12log21+minΛ¯1,Λ¯2.Λ¯1 and Λ¯2 in (38) can be expressed as
(39)Λ¯1=minX¯1aqρrγX¯1ρrγap+ksr2+1,X¯2aqρpγX¯2ρpγap+ksp2+1,X¯3aqρqγX¯3ρqγap+ksq2+1,(40)Λ¯2=minbqΠpγΠpγbp+krp2+1,bqΠqγΠqγbp+krp2+1,respectively, where X¯3=αq∑i=0mq−1ζii+1!βq−δq−i−2.


## 5.2. Energy Efficiency

Energy efficiency is another important index to evaluate the system performance. According to the definition in [39], energy efficiency can be expressed as
(41)ηEE=RsumνPs+Pc,where Pc is the fixed energy consumed by the transceiver and ν denotes the efficiency of the power amplifier. Rsum is the ergodic sum rate, which can be obtained from (32), (33), (35), and (38).


## 6. Numerical Results

In this section, we provide numerical results to verify the analysis result and further evaluate the performance of the satellite-terrestrial networks. Assume that the type of satellite is low-orbit earth satellite, satellite links obey the Shadowed-Rician fading distribution, and channel parameters are shown in Table1 [29, 40]. In this simulations, we set that the power allocation coefficients for Dp and Dq are ap=bp=0.25 and aq=bq=0.75, the distances from R to terrestrial users are dp=0.5 and dq=1, and the pathloss exponent of the terrestrial link is Δ=2. In addition, let the terrestrial channel mean power Ω¯0=1. The target rates are Rp=1 bit/s/Hz and Rq=0.5 bit/s/Hz [32], respectively. The Monte Carlo simulation parameters are depicted in Table 2 [5].Table 1
The table of Shadowed-Rician fading channel parameters.
Shadowing conditionbkmkΩkFrequent heavy shadowing (FHS)0.06320.0005Average shadowing (AS)0.25150.279Infrequent light shadowing (ILS)0.158101.29Table 2
The table of simulation parameters.
Carrier frequency1 GHzSatellite antenna gain24.3 dBiUser antenna gain3.5 dBi3 dB angle0.4°The initial angle between the beam center and user0.1°The distance from the satellite to the user1000 kmFigure2 illustrates the effects of the transmit SNR on the outage probability with different numbers of REs, where the satellite links undergo FHS. In this example, we set κ=κr=κ1=κ2 and k2=ksξ2=kru2. For comparison purposes, we choose the PD-NOMA case (N=1) as the benchmark. It can be seen that the calculated result curves perfectly match well with the Monte Carlo simulation curves, and the asymptotic outage probabilities are approximated with the calculated results in the medium-high SNR region. The outage probabilities of Dp and Dq decrease as the value of transmit SNR increases. Additionally, the outage performance improves significantly when the number of RE increases from N=1 to N=2 and N=3. The reason for this phenomenon is that CD-NOMA (N>1) has a higher diversity order than PD-NOMA, which is consistent with the analysis of (27) and (28). Thus, this important observation also indicates that the diversity gains of the paired NOMA users are closely related to the number of REs.Figure 2
Outage probability versus the transmit SNR.Figure3 shows the impacts of the transmit SNR and HIs on the outage probabilities of paired users. In this simulation, we set k2=0,0.01,0.02,0.03, κ=0, and N=2. We can observe that the exact analytical and asymptotic results are well aligned to the Monte Carlo results. From the outage probability versus SNR curves, it can also be seen that the outage performances for Dp and Dq increase as the transmit SNR increases. In particular, different levels of HIs are set to be from 0.01 to 0.03. As can be observed from the figure, the outage performances of paired NOMA users are becoming worse with the value of k2 increasing from 0 to 0.03. Another important observation is that the outage probability of Dp is more sensitive to HIs than that of Dq.Figure 3
Outage probability versus the transmit SNR.Figure4 depicts the outage probabilities of paired NOMA users versus the imperfect SIC and transmit SNR in three different satellite shadowing conditions. For comparison, the outage probabilities of paired users experiencing FHS fading are selected as the benchmark. It is observed that the outage performance for both NOMA users improves as the degree of shadow fading experienced by satellite links decreases. The reason for this phenomenon is that the paired users can achieve a better outage performance when they undergo a better shadowing scenario. Additionally, the effect of residual interference from the SIC process on the outage probability is focused, where satellite links are subject to FHS. In this simulation, we set κ=0,0.01,0.03 and k2=0. Apparently, the outage performance of the Dp will be significantly degraded if the level of residual interference κ increases by 0.03 from zero. Therefore, it is necessary to develop a reasonable SIC scheme in a practical scenario.Figure 4
Outage probability versus the transmit SNR.Figure5 reveals the outage probability of the considered system against the transmit SNR and power allocation factor ap under the FHS scenario. We assume that the system performance is analyzed under ideal conditions (k2=κ=0) and set the transmit SNR at S and R to be 30 dB. As can be seen from Figure 5, the outage probability decreases for Dp and increases for Dq as the value of ap increases. However, the outage performance of Dp starts to decline once ap goes beyond 0.4. This phenomenon indicates that the increase in ap, the corresponding power allocation factor aq=1−ap, will decrease and will affect the Dp decoding xq in the SIC process when ap>0.4. Further, there exists a value ap=0.25 that enables NOMA systems to yield better outage performance. Consequently, to improve the outage performance and fairness of paired users, ap can be set between 0.17 and 0.33 when γ=30 dB.Figure 5
Outage probability versus power allocation coefficientap.Figure6 illustrates the ergodic sum rate versus the transmit SNR with different numbers of REs (i.e., N=1, N=2, and N=3). In this simulation, we set κ=0 and k2=0, and all satellite links undergo FHS fading. A special case, N=1 (PD-NOMA), is selected as a benchmark to evaluate the ergodic rate of the proposed system. The curves of ergodic rate for Dp and Dq can be plotted according to (32), (33), (35), and (38). As can be observed from the figure, the approximate analysis results are close to the Monte Carlo simulation results. Apparently, as the number of REs N increases, the ergodic rates of the paired NOMA users gradually increase, and the slope of the curve in the case of N>1 is steeper than that in the case of N=1. This observation means that CD-NOMA (N>1) is capable of outperforming PD-NOMA on the ergodic sum rate. From different perspectives, the spectral efficiency of the proposed system is negligible in the low SNR regime, regardless of the value of N.Figure 6
Ergodic sum rate versus the transmit SNR.In Figure7, the ergodic sum rates of paired NOMA users versus the transmit SNR are presented in three different levels of HIs. In this example, we set N=2, κ=0, and k2=0,0.01,0.03. As illustrated, the approximate analysis results are basically consistent with the numerical simulation. In order to facilitate comparison, the curves of ergodic sum rate under ideal conditions (k2=0) is taken as a benchmark. As the level of HIs k2 increases from 0 to 0.03, the ergodic rates of Dp and Dq are getting worse and exacerbating. In addition, it can be also observed that the ergodic rate of Dp is more sensitive to HIs than that of Dq, which is corresponding to the analysis results in Figure 3. Therefore, the design of the effective compensation algorithm and calibration method is of great significance to alleviate the influences of HIs on the performance of NOMA-based satellite-terrestrial networks in practical application.Figure 7
Ergodic sum rate versus the transmit SNR.Figure8 illustrates the ergodic rate versus the transmit SNR with different levels of residual interference from the SIC cycle. In this simulation, we set the number of REs N=2 and the level of HIs k2=0. Then, the cases of residual interference level κ=0.01, κ=0.02, and κ=0.03 are considered for the proposed system. As can be observed, the ergodic sum rate increases as the transmit SNR increases. Moreover, the ergodic rate of Dp decreases significantly when the level of residual interference increases from 0 to 0.01, 0.02, and 0.03. Hence, it is necessary to design a reasonable multiuser reception algorithm to improve the performance of NOMA-based satellite-terrestrial networks in practical scenarios.Figure 8
Ergodic sum rate versus the transmit SNR.To illustrate the effects of different levels of HIs and different levels of residual interference and power distribution factors on the energy efficiency, Figure9 depicts the energy efficiency versus the transmit SNR in delay-tolerant transmission mode, where k2=0,0.01,0.02,0.03 and κ=0,0.01,0.02,0.03. The power amplifier and fixed energy loss are set as ν=2 and Pc=50 W, respectively [6, 39]. As can be seen from Figure 9(a), when k2 increases from 0 to 0.03, the energy efficiency of the proposed system decreases significantly, and the transmit SNR that enables the considered system to obtain the maximum energy efficiency moves to the right. This happens because the ergodic rate achieved by the terrestrial users decreases under the influence of HIs, which is consistent with the analysis results in Figure 7. In Figure 9(b), the energy efficiency under perfect SIC condition is obviously better than that under imperfect SIC condition, which also corresponds to the analysis in Figure 8. In addition, it is observed that the energy efficiency of the considered system is improved with the increase in the power distribution factor for Dp. Although the energy efficiency can be improved to a certain extent by increasing the power allocation factor of Dp, it will greatly sacrifice the performance of Dq, thus losing the fairness of users in the weak channel. This phenomenon indicates that optimizing the power distribution factor can further enhance the performance of the paired NOMA users. Moreover, as illustrated in Figures 9(a) and 9(b), energy efficiency increases with the number of REs.Figure 9
Energy efficiency versus the transmit SNR.
(a)(b)


## 7. Conclusion

This paper proposed a unified NOMA framework for the satellite-terrestrial network under Shadowed-Rician fading channels. Considering the realistic scenario with HIs and imperfect SIC, the performance of the considered system has been characterized in terms of outage probability, ergodic sum rate, and energy efficiency. More specifically, the closed-form expressions of exact outage probability for the paired users were derived. In order to gain more insights, the asymptotic outage probability was provided in the high SNR region. The analytical results showed that a pair of NOMA users was able to achieve better outage performance as the number of REs increased, and the CD-NOMA was capable of outperforming the PD-NOMA on the outage performance. Furthermore, the approximate expression of ergodic sum rate was obtained due to the integral being difficult to solve. We can observe that the HIs and residual interference had significant negative effects on ergodic sum rate, and the ergodic rate ofDp is more sensitive to HIs than that of Dq. The energy efficiency of the considered system with imperfect SIC and HIs has been discussed in delay-tolerant transmission mode. Finally, the Monte Carlo simulation was presented to verify the theoretical derivation. The unified NOMA framework applied to the satellite-terrestrial network provides a novel design idea, which is a relatively promising research topic.



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