# Ergodic Capacity for NOMA-Enabled ISATNs with Transmit Antenna Selection and Imperfect CSI and SIC

Authors: Haifeng Shuai; Kefeng Guo; Shibing Zhu
Journal: Wireless Communications and Mobile Computing
 (2023)

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## Abstract

Integrated satellite-aerial-terrestrial networks (ISATNs) expand the breadth and depth of seamless connections, which has been recognized as one of the key facilities of next-generation networks. This paper investigates the impact of transmit antenna selection, imperfect channel state information (CSI), and successive interference cancellation (SIC) on ergodic capacity (EC) of nonorthogonal multiple access- (NOMA-) enabled ISATNs, where a high altitude platform (HAP) assists the satellite transmission through decode-and-forward relaying protocol. Besides, two selection schemes are proposed to balance system complexity and transmission efficiency. Imperfect CSI and SIC are considered to impose detrimental effects on system performance. In addition, analytical expressions for EC of NOMA-enabled ISATNs in the presence of the above imperfections are derived. Finally, the Monte Carlo simulations corroborate analytical results and reveal the impacts of imperfections on integrated satellite-aerial-terrestrial networks.


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## Body

## 1. Introduction

Integrated satellite-aerial-terrestrial networks (ISATNs) have been considered as an indispensable component in next generation wireless communication systems to service remote areas and improve users service quality [1, 2]. However, ISATNs require a vast amount of spectrum resources to be in service of a large amount of data and massive users. Under this condition, nonorthogonal multiple access (NOMA) scheme is introduced into ISATNs [3, 4]. NOMA scheme adopts power domain access to serve multiple users with the same resource block, thus it can enhance the spectral efficiency of the wireless systems compared to orthogonal multiple access (OMA) scheme [5–7]. Under the urgent demands for high spectrum efficiency and ubiquitous connectivity, the NOMA-enabled ISATN has bright foreground.


### 1.1. Related Works

The explosive growth of wireless access requirements for large-scale access users in a wide coverage range put forward high requirements for the current terrestrial network [8]. On this basis, the aerial networks, satellite networks, and current terrestrial network are combined, that is, integrated satellite-aerial-terrestrial networks [9]. The research in ISATNs has attracted global attention from academia and industry. In [10], two unmanned aerial vehicles (UAVs) were deployed as decode-and-forward (DF) relays to assist the communications of two terrestrial users, where outage probability (OP) was derived to validate the performance of system. The authors of [11] analyzed the outage performance of a cooperative satellite-aerial-terrestrial network, and the exact and asymptotic expressions for OP were obtained. In [12], two classical cooperative protocols, i.e., amplify-and-forward (AF) and DF were compared in a downlink hybrid satellite-terrestrial relay network by deriving the accurate expression of secrecy outage probability. Ergodic capacity (EC) is another important performance metric for the wireless network. In [13], the authors obtained the closed form expressions for OP and EC of an integrated satellite-terrestrial relay network to reveal the performance of the considered system.Due to the scarcity of spectrum resources, the application of NOMA scheme, which can serve multiusers simultaneously, is necessary to improve the resource utilization of wireless networks [14]. In a two-way relay NOMA system, the authors of [15] analyzed the outage behavior and ergodic rates of the system. In [16], the reliability and security of an ambient backscatter NOMA system were analyzed by deriving the expression of OP and intercept probability. The authors of [17] introduced NOMA scheme in hybrid satellite-terrestrial relay networks, so that the performance of OP was enhanced. The authors of [18] compared NOMA with OMA in an underlay cognitive hybrid satellite-terrestrial and verified the superiority of NOMA.Multiple-antenna technique is widely deployed in wireless communication system to improve the transmission diversity [19]. On the other hand, multiple-antenna scenario brings greater overhead and system complexity, and even waste resources [20]. Therefore, in order to balance system performance and complexity, transmit antenna selection (TAS) was proposed [21]. In this regard, it is necessary to adopt TAS scheme in the ISATN with multiple-antenna. In [22], the authors have studied the outage behavior of the integrated satellite-HAP-terrestrial network. The authors of [23] analyzed the antenna selection scheme in a multiantenna two-way relay system, and OP was derived to evaluate the performance of the system.Owing to the severe channel fading, it is hard to acquire perfect channel state information (CSI), and channel estimation errors (CEEs) arise during channel estimation process [24]. In [25], the impact of CEEs on the performance for spectrum sharing DF multiple-relay network was analyzed. The authors of [26] investigated the impact of imperfect CSI on cooperative NOMA system, and novel closed-form and asymptotic expressions for the OP, EC, and energy efficiency (EE) were derived. In addition, successive interference cancellation (SIC) in NOMA scheme may not decode the superimposed signal perfectly as a result of errors due to poor synchronization and propagation receiver [27]. In [28], imperfect SIC was considered to obtain the adaptive power allocation coefficient in a cooperative full duplex NOMA system. In [29], the authors obtained the secrecy outage probability for a unified NOMA framework under perfect SIC and imperfect SIC to analyze the physical layer security of NOMA system.


### 1.2. Research Gap and Contributions

The previous paper of us [30] is the first in open technical papers that investigated the joint effect of imperfect CSI and SIC on the performance of land mobile satellite system. Different from [30], in this paper, a HAP which works as an aerial half-duplex decode-and-forward (DF) relay is employed in the system and the TAS scheme is adopted at the transmitter of HAP. Try the authors’ best knowledge, there is no prior work that analyzed the ergodic capacity (EC) for NOMA-enabled ISATNs with TAS scheme and imperfections.Inspired by the aforementioned observations, in this paper, EC of NOMA-enabled ISATN with TAS schemes and imperfect CSI and SIC is investigated. Particularly, our main contributions are summarized as follows:(1)
Firstly, considering the characteristics of HAP, a novel architecture of NOMA-enabled integrated-satellite-aerial-terrestrial network is established, where the HAP is employed as a half-duplex DF relay to assist the transmission between satellite and user(2)
Secondly, on the foundation of imperfect CSI and SIC, the transmit antenna selection schemes based on the quality of service of different users are proposed. Moreover, the channel responses of Nakagami-m fading and shadowed-Rician (SR) fading under multiantenna and TAS schemes are derived(3)
Finally, the expressions of EC for NOMA-enabled ISATN under TAS schemes and imperfections are evaluated by utilization of Meijer-G functions. Numerical simulations are demonstrated to validate the preponderance of the considered system


### 1.3. Organization

The rest of this paper is organized as follows. In Section II, the system model and problem formulation are discussed. Then, the TAS schemes and EC are obtained in Section III. In Section IV, Monte Carlo simulations are given to reveal the performance of the considered system. Finally, the conclusions are included in Section V.Notations: E· represents the expectation. · denotes the absolute value. pk represents the Pochhammer symbol ([31], p.xliii). Γ· is gamma function ([31], 8.310.1). Gp,qm,n·· represents Meijer-G function ([31], 9.301).


## 1.1. Related Works

The explosive growth of wireless access requirements for large-scale access users in a wide coverage range put forward high requirements for the current terrestrial network [8]. On this basis, the aerial networks, satellite networks, and current terrestrial network are combined, that is, integrated satellite-aerial-terrestrial networks [9]. The research in ISATNs has attracted global attention from academia and industry. In [10], two unmanned aerial vehicles (UAVs) were deployed as decode-and-forward (DF) relays to assist the communications of two terrestrial users, where outage probability (OP) was derived to validate the performance of system. The authors of [11] analyzed the outage performance of a cooperative satellite-aerial-terrestrial network, and the exact and asymptotic expressions for OP were obtained. In [12], two classical cooperative protocols, i.e., amplify-and-forward (AF) and DF were compared in a downlink hybrid satellite-terrestrial relay network by deriving the accurate expression of secrecy outage probability. Ergodic capacity (EC) is another important performance metric for the wireless network. In [13], the authors obtained the closed form expressions for OP and EC of an integrated satellite-terrestrial relay network to reveal the performance of the considered system.Due to the scarcity of spectrum resources, the application of NOMA scheme, which can serve multiusers simultaneously, is necessary to improve the resource utilization of wireless networks [14]. In a two-way relay NOMA system, the authors of [15] analyzed the outage behavior and ergodic rates of the system. In [16], the reliability and security of an ambient backscatter NOMA system were analyzed by deriving the expression of OP and intercept probability. The authors of [17] introduced NOMA scheme in hybrid satellite-terrestrial relay networks, so that the performance of OP was enhanced. The authors of [18] compared NOMA with OMA in an underlay cognitive hybrid satellite-terrestrial and verified the superiority of NOMA.Multiple-antenna technique is widely deployed in wireless communication system to improve the transmission diversity [19]. On the other hand, multiple-antenna scenario brings greater overhead and system complexity, and even waste resources [20]. Therefore, in order to balance system performance and complexity, transmit antenna selection (TAS) was proposed [21]. In this regard, it is necessary to adopt TAS scheme in the ISATN with multiple-antenna. In [22], the authors have studied the outage behavior of the integrated satellite-HAP-terrestrial network. The authors of [23] analyzed the antenna selection scheme in a multiantenna two-way relay system, and OP was derived to evaluate the performance of the system.Owing to the severe channel fading, it is hard to acquire perfect channel state information (CSI), and channel estimation errors (CEEs) arise during channel estimation process [24]. In [25], the impact of CEEs on the performance for spectrum sharing DF multiple-relay network was analyzed. The authors of [26] investigated the impact of imperfect CSI on cooperative NOMA system, and novel closed-form and asymptotic expressions for the OP, EC, and energy efficiency (EE) were derived. In addition, successive interference cancellation (SIC) in NOMA scheme may not decode the superimposed signal perfectly as a result of errors due to poor synchronization and propagation receiver [27]. In [28], imperfect SIC was considered to obtain the adaptive power allocation coefficient in a cooperative full duplex NOMA system. In [29], the authors obtained the secrecy outage probability for a unified NOMA framework under perfect SIC and imperfect SIC to analyze the physical layer security of NOMA system.


## 1.2. Research Gap and Contributions

The previous paper of us [30] is the first in open technical papers that investigated the joint effect of imperfect CSI and SIC on the performance of land mobile satellite system. Different from [30], in this paper, a HAP which works as an aerial half-duplex decode-and-forward (DF) relay is employed in the system and the TAS scheme is adopted at the transmitter of HAP. Try the authors’ best knowledge, there is no prior work that analyzed the ergodic capacity (EC) for NOMA-enabled ISATNs with TAS scheme and imperfections.Inspired by the aforementioned observations, in this paper, EC of NOMA-enabled ISATN with TAS schemes and imperfect CSI and SIC is investigated. Particularly, our main contributions are summarized as follows:(1)
Firstly, considering the characteristics of HAP, a novel architecture of NOMA-enabled integrated-satellite-aerial-terrestrial network is established, where the HAP is employed as a half-duplex DF relay to assist the transmission between satellite and user(2)
Secondly, on the foundation of imperfect CSI and SIC, the transmit antenna selection schemes based on the quality of service of different users are proposed. Moreover, the channel responses of Nakagami-m fading and shadowed-Rician (SR) fading under multiantenna and TAS schemes are derived(3)
Finally, the expressions of EC for NOMA-enabled ISATN under TAS schemes and imperfections are evaluated by utilization of Meijer-G functions. Numerical simulations are demonstrated to validate the preponderance of the considered system


## 1.3. Organization

The rest of this paper is organized as follows. In Section II, the system model and problem formulation are discussed. Then, the TAS schemes and EC are obtained in Section III. In Section IV, Monte Carlo simulations are given to reveal the performance of the considered system. Finally, the conclusions are included in Section V.Notations: E· represents the expectation. · denotes the absolute value. pk represents the Pochhammer symbol ([31], p.xliii). Γ· is gamma function ([31], 8.310.1). Gp,qm,n·· represents Meijer-G function ([31], 9.301).


## 2. System Model and Problem Formulation

As shown in Figure1, we consider a power domain NOMA-enabled ISATN, which comprises a geosynchronous earth orbit (GEO) satellite (S), a HAP which works as an aerial DF relay (HAP is quasi-stationary suspended at an altitude of 20 km, hence, the impacts of the scattering and multipath are low.), and two terrestrial users (Ui), i∈1,2 (Two-user group NOMA situation has the authentication of the Third Generation Partnership Project (3GPP), which can improve the spectrum efficiency of the system. Meanwhile, the derivations in this paper can be extended to any NOMA pair with more users [7]). Due to the high buildings, heavy fading, and other obstacles, thus terrestrial users cannot directly receive the satellite signal. Hence, HAP works as a relay which receives and forwards the satellite signal to destinations. It is assumed that the aerial relay and terrestrial destinations are located in the same identical GEO satellite beam. Moreover, HAP uses Nr receive antennas and Nt transmit antennas to form a uniform planar array (UPA), which can improve array gain. In addition, the terrestrial destinations are assumed to be small devices, such as IoT sensors, whose structures can only deploy single antenna [30, 32].Figure 1
System model of NOMA-enabled ISATN.


### 2.1. Signal Model

The whole transmission comprises two time slots. In the first time slot, the GEO satellite broadcasts a superimposed signal to the HAP which is given by(1)s=a1x1+a2x2,where xi denotes the target signal of two terrestrial users Ui, obeying Exi2=1. In NOMA scheme, ai represents the power allocation coefficient of different signal satisfying a1+a2=1. It is assumed that requirements for the quality of service (QoS) of U1 is higher than U2, (U1 requires fast and low-rate connections, such as IoT sensors. U2 is a high-resolution online game or video user which requires a continuous high-speed connection [20].) hence, U1 is allocated with more power, i.e., a1>a2. At HAP, the received signal can be denoted as
(2)yA=PShSAwHaax1+a2x2+wHnA,where hSA is the channel response of satellite to aerial link, which follows SR fading. To obtain the best system performance, maximum ratio combining (MRC) scheme is utilized in the considered system model which obeys w=hSA/hSA. w is the MRC weight vector at HAP due to Nr receive antennas. PS denotes the transmit power of satellite. nA is the addictive white Gaussian noise (AWGN) vector at aerial relay modeled as nH~CN0,σSA2INr×1.In the next time slot, HAP relay utilizes DF and SIC protocol to decode and forward the received signal to the two terrestrial destinations [20, 32]. Notably, TAS scheme is adopted in HAP to balance the system performance and complexity. The received signal at Ui can be denoted as
(3)yUi=PAhAj,Uia1x1+a2x2+nUi,where hAj,Ui is the channel response of HAP-to-Ui links modeled as Nakagami-m fading. PA represents the transmit power at aerial relay. nUi expresses the AWGN at Ui distributed as nUi~CN0,σAj,Ui2.In the channel estimation process, imperfect CSI and the limited estimation technology will result in CEEs inevitably. Utilizing the similar method in [7], CEE is often calculated by linear minimum mean-square error (MMSE). On this foundation, the estimated channel can be expressed as
(4)h~T=hT−eT,where T∈SA,Aj,Ui. eT~CN0,v¯e denotes the CEEs. With the help of [7], the variance can be given by
(5)v¯e=EhT2−Eh~T2=1LTV¯T+1,where V¯T=EVT=PTEhT2/σT2, PT is the power of training symbols satisfying PT=1−ϕPtotal, Ptotal is the total transmission power and ϕ∈0,1 is the power coefficient. LT represents the pilot symbols length for channel estimation.In the signal decoding under NOMA scheme, imperfect SIC is considered for weak performance of receiver [33]. Therefore, when the receiver decodes the signal x2 with less power, the residual signal x1 which does not completely eliminate is viewed as interference [30]. At the same time, considering CEEs, the signal-to-interference-plus-noise ratio (SINR) of different signals at aerial relay is denoted, correspondingly, as
(6)γA−1=ϕa1γSAϕa2γSA+ϕv¯e+1,(7)γA−2=ϕa2γSAϕξa1γSA+ϕv¯e+1,where γSA=Ptotalh~SA2/σSA2=γ¯SAh~SA2, ξ∈0,1 represents imperfect SIC coefficient.Similarly, the SINR of two signals atUi can be given by
(8)γ1=ϕa1γAj,U1ϕa2γAj,U1+ϕv¯e+1,γ2−1=ϕa1γAj,U2ϕa2γAj,U2+ϕv¯e+1,γ2=ϕa2γAj,U2ϕξa1γAj,U2+ϕv¯e+1,where γAj,Ui=Ptotalh~Aj,Ui2/σAj,Ui2=γ¯Aj,Uih~Aj,Ui2.


### 2.2. Channel Models

#### 2.2.1. Satellite to Aerial Link

The fading amplitude of SR channel responsehl,1≤l≤Nr can be expressed as [34].
(9)hl=CSAgl,where gl represents the channel coefficient of SR fading. CSA denotes the path-loss factor in the satellite to aerial link that can be defined as
(10)CSA=λGS,SAGA,SA4πdSAKBτB,where λ represents carrier wavelength. dSA is the distance between satellite and aerial. KB=1.380649×10−23J/K is the Boltzmann constant. τ denotes the transmission noise temperature. B is signal bandwidth. GS,SA is the satellite antenna gain which can be denoted as
(11)GS,SA=GmaxSJ1k2k+36J3kk32,where GmaxS is the maximum satellite antenna gain. k=2.07123sinη/sinη3dB, η denotes the angle of the satellite center beam and the UPA of HAP. Ji denotes the first-kind Bessel function of order i. GA,SA is the HAP receive antenna gain which is defined as
(12)GA,SA≃GmaxA,0∘<φ<1∘,32−25logφ,1∘<φ<48∘,−10,48∘<φ<180∘,where GmaxA is the maximum gain of aerial relay receive antenna. φ is the off-boresight angle.Without loss of generality, we assume that satellite to aerial link follows independent and identically distributed (i.i.d.) SR fading distribution [7]. According to γSA=γ¯SACSA2gSA2 and gSA=g1,g2,⋯,gNrT, the probability density function (PDF) of γSA can be shown by [34].
(13)fγSAx=∑k1=0m−1⋯∑kNr=0m−1ΞNrxΛNr−1e−ΔSAxγ¯SAΛNr,where
(14)ΞNrΔ=∏j=1Nrαjζkj∏j=1Nr−1B∑l=1jj+kl,1+kj+1,and B.,. represents beta function [[31], Eq. 8.384.1]. ζkj=−1kj1−mkjδkj/kj!2, ·kj denotes the Pochhammer symbol. ΛNrΔ=∑j=1Nrkj+Nr, ΔSA=β−δ/γ¯SA. αΔ=2bm/2bm+Ωm/2b,β=Δ1/2b,δ=ΔΩ/2b2bm+Ω, m∈0,∞, Ω, and 2b are the fading severity parameter, average power of the line-of-sight (LOS) component, and the multipath component, respectively.By utilizing the similar method of [[31], Eq. 3.351.2], the cumulative distribution function (CDF) of γSA can be obtained as
(15)FγSAx=1−∑k1=0m−1⋯∑kNr=0m−1∑t=0ΛNr−1ΞNrΛNr−1!t!ΔSAΛNr−tγ¯SAΛNrxte−ΔSAx.


#### 2.2.2. Aerial toUi Links

The channel of aerial toUi links undergoes Nakagami-m fading, which can be given by [32].
(16)hAj,Ui=CAj,UigAj,Ui∂φi,θi,where gAj,Ui represents the channel coefficient of Nakagami-m fading. CAj,Ui is the path-loss factor in the aerial to Ui links that can be expressed as
(17)CAj,Ui=GAj+GUi+1220lgλ/4π−10ςlgdAUi,where GAj and GUi are the antenna gain of HAP relay and users, correspondingly. ς represents the path loss value and dAUi is the distance between aerial and users.Moreover, we set a spatial coordinate system with aerial relay as the origin in Figure1, θi∈0,2π is the azimuth between aerial and users, and φi∈0,π/2 is the elevation from aerial to users. ∂φi,θi shows the array steering matrix [32, 35], which is defined as
(18)∂φi,θi=∂xφi,θi⊗∂yφi,θi,where ∂xφi,θi and ∂yφi,θi are the horizontal and vertical antennas steering vectors, which can be given by
(19)∂xφi,θi=e−j2πlv/λ1−Nh+1/2cosφicosθi,⋯,e−j2πlv/λNh−Nh+1/2cosφicosθiT,∂yφi,θi=e−j2πlh/λ1−Nv+1/2cosφicosθi,⋯,e−j2πlh/λNv−Nv+1/2cosφicosθiT,where Nh×Nv=Nt, Nh, and lh are the number and interelement distance of horizontal antennas. Nv and lv are the number and interelement distance of vertical directions.Finally, on the basis of three-dimensional coordinates,φi and θi are, respectively, expressed as
(20)φi=arctan,θi=arccosxixi2+yi2,yi≥0,2π−arccoszixi2+yi2,yi<0,According toγAj,Ui=γ¯Aj,UiCAj,Ui2∂φi,θi2gAj,Ui2, the PDF and CDF of γAj,Ui are given by
(21)fγAj,Uix=ΘimiΓmixmi−1e−Θix,FγAj,Uix=1−e−Θix∑k=0mi−1Θikxkk!,whereΘi=mi/Ωiγ¯Aj,Ui. mi is the fading severity factor and Ωi denotes the average power.


## 2.1. Signal Model

The whole transmission comprises two time slots. In the first time slot, the GEO satellite broadcasts a superimposed signal to the HAP which is given by(1)s=a1x1+a2x2,where xi denotes the target signal of two terrestrial users Ui, obeying Exi2=1. In NOMA scheme, ai represents the power allocation coefficient of different signal satisfying a1+a2=1. It is assumed that requirements for the quality of service (QoS) of U1 is higher than U2, (U1 requires fast and low-rate connections, such as IoT sensors. U2 is a high-resolution online game or video user which requires a continuous high-speed connection [20].) hence, U1 is allocated with more power, i.e., a1>a2. At HAP, the received signal can be denoted as
(2)yA=PShSAwHaax1+a2x2+wHnA,where hSA is the channel response of satellite to aerial link, which follows SR fading. To obtain the best system performance, maximum ratio combining (MRC) scheme is utilized in the considered system model which obeys w=hSA/hSA. w is the MRC weight vector at HAP due to Nr receive antennas. PS denotes the transmit power of satellite. nA is the addictive white Gaussian noise (AWGN) vector at aerial relay modeled as nH~CN0,σSA2INr×1.In the next time slot, HAP relay utilizes DF and SIC protocol to decode and forward the received signal to the two terrestrial destinations [20, 32]. Notably, TAS scheme is adopted in HAP to balance the system performance and complexity. The received signal at Ui can be denoted as
(3)yUi=PAhAj,Uia1x1+a2x2+nUi,where hAj,Ui is the channel response of HAP-to-Ui links modeled as Nakagami-m fading. PA represents the transmit power at aerial relay. nUi expresses the AWGN at Ui distributed as nUi~CN0,σAj,Ui2.In the channel estimation process, imperfect CSI and the limited estimation technology will result in CEEs inevitably. Utilizing the similar method in [7], CEE is often calculated by linear minimum mean-square error (MMSE). On this foundation, the estimated channel can be expressed as
(4)h~T=hT−eT,where T∈SA,Aj,Ui. eT~CN0,v¯e denotes the CEEs. With the help of [7], the variance can be given by
(5)v¯e=EhT2−Eh~T2=1LTV¯T+1,where V¯T=EVT=PTEhT2/σT2, PT is the power of training symbols satisfying PT=1−ϕPtotal, Ptotal is the total transmission power and ϕ∈0,1 is the power coefficient. LT represents the pilot symbols length for channel estimation.In the signal decoding under NOMA scheme, imperfect SIC is considered for weak performance of receiver [33]. Therefore, when the receiver decodes the signal x2 with less power, the residual signal x1 which does not completely eliminate is viewed as interference [30]. At the same time, considering CEEs, the signal-to-interference-plus-noise ratio (SINR) of different signals at aerial relay is denoted, correspondingly, as
(6)γA−1=ϕa1γSAϕa2γSA+ϕv¯e+1,(7)γA−2=ϕa2γSAϕξa1γSA+ϕv¯e+1,where γSA=Ptotalh~SA2/σSA2=γ¯SAh~SA2, ξ∈0,1 represents imperfect SIC coefficient.Similarly, the SINR of two signals atUi can be given by
(8)γ1=ϕa1γAj,U1ϕa2γAj,U1+ϕv¯e+1,γ2−1=ϕa1γAj,U2ϕa2γAj,U2+ϕv¯e+1,γ2=ϕa2γAj,U2ϕξa1γAj,U2+ϕv¯e+1,where γAj,Ui=Ptotalh~Aj,Ui2/σAj,Ui2=γ¯Aj,Uih~Aj,Ui2.


## 2.2. Channel Models

### 2.2.1. Satellite to Aerial Link

The fading amplitude of SR channel responsehl,1≤l≤Nr can be expressed as [34].
(9)hl=CSAgl,where gl represents the channel coefficient of SR fading. CSA denotes the path-loss factor in the satellite to aerial link that can be defined as
(10)CSA=λGS,SAGA,SA4πdSAKBτB,where λ represents carrier wavelength. dSA is the distance between satellite and aerial. KB=1.380649×10−23J/K is the Boltzmann constant. τ denotes the transmission noise temperature. B is signal bandwidth. GS,SA is the satellite antenna gain which can be denoted as
(11)GS,SA=GmaxSJ1k2k+36J3kk32,where GmaxS is the maximum satellite antenna gain. k=2.07123sinη/sinη3dB, η denotes the angle of the satellite center beam and the UPA of HAP. Ji denotes the first-kind Bessel function of order i. GA,SA is the HAP receive antenna gain which is defined as
(12)GA,SA≃GmaxA,0∘<φ<1∘,32−25logφ,1∘<φ<48∘,−10,48∘<φ<180∘,where GmaxA is the maximum gain of aerial relay receive antenna. φ is the off-boresight angle.Without loss of generality, we assume that satellite to aerial link follows independent and identically distributed (i.i.d.) SR fading distribution [7]. According to γSA=γ¯SACSA2gSA2 and gSA=g1,g2,⋯,gNrT, the probability density function (PDF) of γSA can be shown by [34].
(13)fγSAx=∑k1=0m−1⋯∑kNr=0m−1ΞNrxΛNr−1e−ΔSAxγ¯SAΛNr,where
(14)ΞNrΔ=∏j=1Nrαjζkj∏j=1Nr−1B∑l=1jj+kl,1+kj+1,and B.,. represents beta function [[31], Eq. 8.384.1]. ζkj=−1kj1−mkjδkj/kj!2, ·kj denotes the Pochhammer symbol. ΛNrΔ=∑j=1Nrkj+Nr, ΔSA=β−δ/γ¯SA. αΔ=2bm/2bm+Ωm/2b,β=Δ1/2b,δ=ΔΩ/2b2bm+Ω, m∈0,∞, Ω, and 2b are the fading severity parameter, average power of the line-of-sight (LOS) component, and the multipath component, respectively.By utilizing the similar method of [[31], Eq. 3.351.2], the cumulative distribution function (CDF) of γSA can be obtained as
(15)FγSAx=1−∑k1=0m−1⋯∑kNr=0m−1∑t=0ΛNr−1ΞNrΛNr−1!t!ΔSAΛNr−tγ¯SAΛNrxte−ΔSAx.


### 2.2.2. Aerial toUi Links

The channel of aerial toUi links undergoes Nakagami-m fading, which can be given by [32].
(16)hAj,Ui=CAj,UigAj,Ui∂φi,θi,where gAj,Ui represents the channel coefficient of Nakagami-m fading. CAj,Ui is the path-loss factor in the aerial to Ui links that can be expressed as
(17)CAj,Ui=GAj+GUi+1220lgλ/4π−10ςlgdAUi,where GAj and GUi are the antenna gain of HAP relay and users, correspondingly. ς represents the path loss value and dAUi is the distance between aerial and users.Moreover, we set a spatial coordinate system with aerial relay as the origin in Figure1, θi∈0,2π is the azimuth between aerial and users, and φi∈0,π/2 is the elevation from aerial to users. ∂φi,θi shows the array steering matrix [32, 35], which is defined as
(18)∂φi,θi=∂xφi,θi⊗∂yφi,θi,where ∂xφi,θi and ∂yφi,θi are the horizontal and vertical antennas steering vectors, which can be given by
(19)∂xφi,θi=e−j2πlv/λ1−Nh+1/2cosφicosθi,⋯,e−j2πlv/λNh−Nh+1/2cosφicosθiT,∂yφi,θi=e−j2πlh/λ1−Nv+1/2cosφicosθi,⋯,e−j2πlh/λNv−Nv+1/2cosφicosθiT,where Nh×Nv=Nt, Nh, and lh are the number and interelement distance of horizontal antennas. Nv and lv are the number and interelement distance of vertical directions.Finally, on the basis of three-dimensional coordinates,φi and θi are, respectively, expressed as
(20)φi=arctan,θi=arccosxixi2+yi2,yi≥0,2π−arccoszixi2+yi2,yi<0,According toγAj,Ui=γ¯Aj,UiCAj,Ui2∂φi,θi2gAj,Ui2, the PDF and CDF of γAj,Ui are given by
(21)fγAj,Uix=ΘimiΓmixmi−1e−Θix,FγAj,Uix=1−e−Θix∑k=0mi−1Θikxkk!,whereΘi=mi/Ωiγ¯Aj,Ui. mi is the fading severity factor and Ωi denotes the average power.


## 2.2.1. Satellite to Aerial Link

The fading amplitude of SR channel responsehl,1≤l≤Nr can be expressed as [34].
(9)hl=CSAgl,where gl represents the channel coefficient of SR fading. CSA denotes the path-loss factor in the satellite to aerial link that can be defined as
(10)CSA=λGS,SAGA,SA4πdSAKBτB,where λ represents carrier wavelength. dSA is the distance between satellite and aerial. KB=1.380649×10−23J/K is the Boltzmann constant. τ denotes the transmission noise temperature. B is signal bandwidth. GS,SA is the satellite antenna gain which can be denoted as
(11)GS,SA=GmaxSJ1k2k+36J3kk32,where GmaxS is the maximum satellite antenna gain. k=2.07123sinη/sinη3dB, η denotes the angle of the satellite center beam and the UPA of HAP. Ji denotes the first-kind Bessel function of order i. GA,SA is the HAP receive antenna gain which is defined as
(12)GA,SA≃GmaxA,0∘<φ<1∘,32−25logφ,1∘<φ<48∘,−10,48∘<φ<180∘,where GmaxA is the maximum gain of aerial relay receive antenna. φ is the off-boresight angle.Without loss of generality, we assume that satellite to aerial link follows independent and identically distributed (i.i.d.) SR fading distribution [7]. According to γSA=γ¯SACSA2gSA2 and gSA=g1,g2,⋯,gNrT, the probability density function (PDF) of γSA can be shown by [34].
(13)fγSAx=∑k1=0m−1⋯∑kNr=0m−1ΞNrxΛNr−1e−ΔSAxγ¯SAΛNr,where
(14)ΞNrΔ=∏j=1Nrαjζkj∏j=1Nr−1B∑l=1jj+kl,1+kj+1,and B.,. represents beta function [[31], Eq. 8.384.1]. ζkj=−1kj1−mkjδkj/kj!2, ·kj denotes the Pochhammer symbol. ΛNrΔ=∑j=1Nrkj+Nr, ΔSA=β−δ/γ¯SA. αΔ=2bm/2bm+Ωm/2b,β=Δ1/2b,δ=ΔΩ/2b2bm+Ω, m∈0,∞, Ω, and 2b are the fading severity parameter, average power of the line-of-sight (LOS) component, and the multipath component, respectively.By utilizing the similar method of [[31], Eq. 3.351.2], the cumulative distribution function (CDF) of γSA can be obtained as
(15)FγSAx=1−∑k1=0m−1⋯∑kNr=0m−1∑t=0ΛNr−1ΞNrΛNr−1!t!ΔSAΛNr−tγ¯SAΛNrxte−ΔSAx.


## 2.2.2. Aerial toUi Links

The channel of aerial toUi links undergoes Nakagami-m fading, which can be given by [32].
(16)hAj,Ui=CAj,UigAj,Ui∂φi,θi,where gAj,Ui represents the channel coefficient of Nakagami-m fading. CAj,Ui is the path-loss factor in the aerial to Ui links that can be expressed as
(17)CAj,Ui=GAj+GUi+1220lgλ/4π−10ςlgdAUi,where GAj and GUi are the antenna gain of HAP relay and users, correspondingly. ς represents the path loss value and dAUi is the distance between aerial and users.Moreover, we set a spatial coordinate system with aerial relay as the origin in Figure1, θi∈0,2π is the azimuth between aerial and users, and φi∈0,π/2 is the elevation from aerial to users. ∂φi,θi shows the array steering matrix [32, 35], which is defined as
(18)∂φi,θi=∂xφi,θi⊗∂yφi,θi,where ∂xφi,θi and ∂yφi,θi are the horizontal and vertical antennas steering vectors, which can be given by
(19)∂xφi,θi=e−j2πlv/λ1−Nh+1/2cosφicosθi,⋯,e−j2πlv/λNh−Nh+1/2cosφicosθiT,∂yφi,θi=e−j2πlh/λ1−Nv+1/2cosφicosθi,⋯,e−j2πlh/λNv−Nv+1/2cosφicosθiT,where Nh×Nv=Nt, Nh, and lh are the number and interelement distance of horizontal antennas. Nv and lv are the number and interelement distance of vertical directions.Finally, on the basis of three-dimensional coordinates,φi and θi are, respectively, expressed as
(20)φi=arctan,θi=arccosxixi2+yi2,yi≥0,2π−arccoszixi2+yi2,yi<0,According toγAj,Ui=γ¯Aj,UiCAj,Ui2∂φi,θi2gAj,Ui2, the PDF and CDF of γAj,Ui are given by
(21)fγAj,Uix=ΘimiΓmixmi−1e−Θix,FγAj,Uix=1−e−Θix∑k=0mi−1Θikxkk!,whereΘi=mi/Ωiγ¯Aj,Ui. mi is the fading severity factor and Ωi denotes the average power.


## 3. Performance Analysis

Based on the requirements of different users, we propose two TAS schemes to enhance the performance of our proposed NOMA-enabled ISATN and reduce system complexity. Then, the expressions for EC are derived to manifest the impacts of different schemes under imperfect CSI and SIC.


### 3.1. Transmit Antenna Selection (TAS) Schemes

Two TAS schemes, i.e., TAS1 and TAS2 schemes, are presented to improve the performance of differentUi and reduce the complexity of the whole system. For TAS1 scheme, U1 needs better service quality, thus, the optimal antenna j∗ is selected to transmit the superimposed signal to U1, which can be expressed as
(22)j∗=argmax⏟j∈1,2,⋯,NtγAj,U1,meanwhile, a random antenna is chosen to transmit the signal toU2.For TAS2 scheme,U2 needs better service quality, hence, we choose the best antenna j∗ to transmit the superimposed signal to U2, which can be denoted as
(23)j∗=argmax⏟j∈1,2,⋯,NtγAj,U2,meanwhile, a random antenna is selected to transmit the signal toU1.From the above two schemes and with the help of multinomial theorem [21], the CDF of Nakagami-m fading is derived as
(24)FγAj∗,Uix=FγAj,UixNt=∑t=0NtNtt−e−ΘixtΦx,whereΦx=∑ti≥0,t0+t1+⋯+tmi−1=tt!∏0≤l≤miΘilxl/l!tl/t0!t1!⋯tmi−1!.Through derivations and order statistics, the PDF of Nakagami-m fading under TAS scheme can be obtained as
(25)fγAj∗,Uix=NtfγAj,UixFγAj,UixNt−1.


### 3.2. Ergodic Capacity

EC is a key merit to analyze the performance of the considered ISATN. The EC of the ISATN defines the time average of the capacity on the satellite to aerial link and aerial to users' links [30]. Due to the ISATN consists two phases, the overall EC can be denoted as
(26)EC=minECSA,ECAU,whereECSA and ECAU are the EC of the two links, respectively.In the first time slot,ECSA is defined as the sum of average instantaneous mutual information of the end-to-end SINR at HAP, according to (6) and (7), the expression can be written as.
(27)ECSA=12Elog21+γA−1+Elog21+γA−2,where1/2 is generated because of half duplex DF relaying protocol.After necessary manipulation transformation,ECSA can be obtained as
(28)ECSA=12ln2ElnγSAϕϕv¯e+1+1−ElnγSAϕa2ϕv¯e+1+1+ElnγSAϕa2+ξa1ϕv¯e+1+1−ElnγSAϕξa1ϕv¯e+1+1.LetγSA=x, γSAϕ/ϕv¯e+1=z1, γSAϕa2/ϕv¯e+1=z2, γSAϕa2+ξa1/ϕv¯e+1=z3, and γSAϕξa1/ϕv¯e+1=z4, with the help of probability transformation formula, the PDF of z1 can be calculated as
(29)fz1z=ϕv¯e+1ϕfxϕv¯e+1ϕz.On the basis of (13), (29) can be reexpressed as
(30)fz1z=∑k1=0m−1⋯∑kNt=0m−1ΞNtϕv¯e+1ϕΛNt−1zΛNt−1e−ΔSAϕv¯e+1/ϕzγ¯SAΛNt.Furthermore, the EC of the ISATN can be rewritten as(31)ECSA=12ln2×∫0∞lnz+1fz1zdz⏟S1−∫0∞lnz+1fz2zdz⏟S2+∫0∞lnz+1fz3zdz⏟S3−∫0∞lnz+1fz4zdz⏟S4.With the help of [[36], Eq.8.4.6.5], then we utilize Meijer-G function [[31], Eq. 9.301] to further deduce EC, namely,
(32)ln1+z=G2212z1,110.By substituting (30) and (32) into S1, with the aid of [[36], Eq.2.24.3.1] and [[36], Eq.8.2.2.14], S1 can be expressed as
(33)S1=∑k1=0m−1⋯∑kNt=0m−1ΞNtγ¯SAΛNtΔSAΛNtG2331ϕv¯e+1ΔSAϕ0,1ΛNt,1,0.With the similar way, we can also get the expressions ofS2, S3, and S4 as follows:
(34)S2=∑k1=0m−1⋯∑kNt=0m−1ΞNtγ¯SAΛNtΔSAΛNtG2331ϕv¯e+1ΔSAϕa20,1ΛNt,1,0,S3=∑k1=0m−1⋯∑kNt=0m−1ΞNtγ¯SAΛNtΔSAΛNtG2331ϕv¯e+1ΔSAϕa2+ξa10,1ΛNt,1,0,S4=∑k1=0m−1⋯∑kNt=0m−1ΞNtγ¯SAΛNtΔSAΛNtG2331ϕv¯e+1ΔSAϕξa10,1ΛNt,1,0.By using them, the final expression ofECSA can be expressed as
(35)ECSA=∑k1=0m−1⋯∑kNt=0m−1ΞNt1γ¯SAΛNtΔSAΛNtG2331ϕv¯e+1ϕΔSA0,1ΛNt,1,0−G2331ϕv¯e+1ϕa2ΔSA0,1ΛNt,1,0+G2331ϕv¯e+1ϕa2+ξa1ΔSA0,1ΛNt,1,0−G2331ϕv¯e+1ϕξa1ΔSA0,1ΛNt,1,0.Applying the same method, the expression ofECAU under TAS1 scheme can be derived as
(36)ECAUTAS1=12ln21Γm2G2331ϕv¯e+1ϕΘ20,1m2,0,0−G2331ϕv¯e+1ϕa2Θ20,1m2,0,0+G2331ϕv¯e+1ϕa2+ξa1Θ20,1m2,0,0−G2331ϕv¯e+1ϕξa1Θ20,1m2,0,0+NtΘ1m1Γm1∑t=0Nt−1Nt−1t−1t∑ti≥0,t0+t1+⋯+tm1−1=tt!t0!t1!⋯tm1−1!∏0≤l≤m1Θ1ll!tl×Θ1t+1−l×tl+m1G2331Θ1t+1ϕv¯e+1ϕ0,1l×tl+m1,0,0−G2331Θ1t+1ϕv¯e+1ϕa20,1l×tl+m1,0,0.Finally, by substituting (35) and (36) into (26), the overall EC of the NOMA-enabled ISATNs under TAS1 is obtained.Applying the similar derivations, the expression of overall EC under TAS2 can be obtained. Owing to the limitations of page length, this step is omitted.


## 3.1. Transmit Antenna Selection (TAS) Schemes

Two TAS schemes, i.e., TAS1 and TAS2 schemes, are presented to improve the performance of differentUi and reduce the complexity of the whole system. For TAS1 scheme, U1 needs better service quality, thus, the optimal antenna j∗ is selected to transmit the superimposed signal to U1, which can be expressed as
(22)j∗=argmax⏟j∈1,2,⋯,NtγAj,U1,meanwhile, a random antenna is chosen to transmit the signal toU2.For TAS2 scheme,U2 needs better service quality, hence, we choose the best antenna j∗ to transmit the superimposed signal to U2, which can be denoted as
(23)j∗=argmax⏟j∈1,2,⋯,NtγAj,U2,meanwhile, a random antenna is selected to transmit the signal toU1.From the above two schemes and with the help of multinomial theorem [21], the CDF of Nakagami-m fading is derived as
(24)FγAj∗,Uix=FγAj,UixNt=∑t=0NtNtt−e−ΘixtΦx,whereΦx=∑ti≥0,t0+t1+⋯+tmi−1=tt!∏0≤l≤miΘilxl/l!tl/t0!t1!⋯tmi−1!.Through derivations and order statistics, the PDF of Nakagami-m fading under TAS scheme can be obtained as
(25)fγAj∗,Uix=NtfγAj,UixFγAj,UixNt−1.


## 3.2. Ergodic Capacity

EC is a key merit to analyze the performance of the considered ISATN. The EC of the ISATN defines the time average of the capacity on the satellite to aerial link and aerial to users' links [30]. Due to the ISATN consists two phases, the overall EC can be denoted as
(26)EC=minECSA,ECAU,whereECSA and ECAU are the EC of the two links, respectively.In the first time slot,ECSA is defined as the sum of average instantaneous mutual information of the end-to-end SINR at HAP, according to (6) and (7), the expression can be written as.
(27)ECSA=12Elog21+γA−1+Elog21+γA−2,where1/2 is generated because of half duplex DF relaying protocol.After necessary manipulation transformation,ECSA can be obtained as
(28)ECSA=12ln2ElnγSAϕϕv¯e+1+1−ElnγSAϕa2ϕv¯e+1+1+ElnγSAϕa2+ξa1ϕv¯e+1+1−ElnγSAϕξa1ϕv¯e+1+1.LetγSA=x, γSAϕ/ϕv¯e+1=z1, γSAϕa2/ϕv¯e+1=z2, γSAϕa2+ξa1/ϕv¯e+1=z3, and γSAϕξa1/ϕv¯e+1=z4, with the help of probability transformation formula, the PDF of z1 can be calculated as
(29)fz1z=ϕv¯e+1ϕfxϕv¯e+1ϕz.On the basis of (13), (29) can be reexpressed as
(30)fz1z=∑k1=0m−1⋯∑kNt=0m−1ΞNtϕv¯e+1ϕΛNt−1zΛNt−1e−ΔSAϕv¯e+1/ϕzγ¯SAΛNt.Furthermore, the EC of the ISATN can be rewritten as(31)ECSA=12ln2×∫0∞lnz+1fz1zdz⏟S1−∫0∞lnz+1fz2zdz⏟S2+∫0∞lnz+1fz3zdz⏟S3−∫0∞lnz+1fz4zdz⏟S4.With the help of [[36], Eq.8.4.6.5], then we utilize Meijer-G function [[31], Eq. 9.301] to further deduce EC, namely,
(32)ln1+z=G2212z1,110.By substituting (30) and (32) into S1, with the aid of [[36], Eq.2.24.3.1] and [[36], Eq.8.2.2.14], S1 can be expressed as
(33)S1=∑k1=0m−1⋯∑kNt=0m−1ΞNtγ¯SAΛNtΔSAΛNtG2331ϕv¯e+1ΔSAϕ0,1ΛNt,1,0.With the similar way, we can also get the expressions ofS2, S3, and S4 as follows:
(34)S2=∑k1=0m−1⋯∑kNt=0m−1ΞNtγ¯SAΛNtΔSAΛNtG2331ϕv¯e+1ΔSAϕa20,1ΛNt,1,0,S3=∑k1=0m−1⋯∑kNt=0m−1ΞNtγ¯SAΛNtΔSAΛNtG2331ϕv¯e+1ΔSAϕa2+ξa10,1ΛNt,1,0,S4=∑k1=0m−1⋯∑kNt=0m−1ΞNtγ¯SAΛNtΔSAΛNtG2331ϕv¯e+1ΔSAϕξa10,1ΛNt,1,0.By using them, the final expression ofECSA can be expressed as
(35)ECSA=∑k1=0m−1⋯∑kNt=0m−1ΞNt1γ¯SAΛNtΔSAΛNtG2331ϕv¯e+1ϕΔSA0,1ΛNt,1,0−G2331ϕv¯e+1ϕa2ΔSA0,1ΛNt,1,0+G2331ϕv¯e+1ϕa2+ξa1ΔSA0,1ΛNt,1,0−G2331ϕv¯e+1ϕξa1ΔSA0,1ΛNt,1,0.Applying the same method, the expression ofECAU under TAS1 scheme can be derived as
(36)ECAUTAS1=12ln21Γm2G2331ϕv¯e+1ϕΘ20,1m2,0,0−G2331ϕv¯e+1ϕa2Θ20,1m2,0,0+G2331ϕv¯e+1ϕa2+ξa1Θ20,1m2,0,0−G2331ϕv¯e+1ϕξa1Θ20,1m2,0,0+NtΘ1m1Γm1∑t=0Nt−1Nt−1t−1t∑ti≥0,t0+t1+⋯+tm1−1=tt!t0!t1!⋯tm1−1!∏0≤l≤m1Θ1ll!tl×Θ1t+1−l×tl+m1G2331Θ1t+1ϕv¯e+1ϕ0,1l×tl+m1,0,0−G2331Θ1t+1ϕv¯e+1ϕa20,1l×tl+m1,0,0.Finally, by substituting (35) and (36) into (26), the overall EC of the NOMA-enabled ISATNs under TAS1 is obtained.Applying the similar derivations, the expression of overall EC under TAS2 can be obtained. Owing to the limitations of page length, this step is omitted.


## 4. Numerical Results

In this section, Monte Carlo (MC) simulations are performed to verify our theoretical derivations. In general, we setγ¯SA=γ¯Aj,Ui=γ¯, m1=m2=mU, and Ω1=Ω2=1. Frequent heavy shadowing (FHS), average shadowing (AS), and infrequent light shadowing (ILS) are given as m,b,Ω=1,0.063,0.0007, 5,0.251,0.279, and 10,0.158,1.29, respectively. The rest system values are shown in Table 1 [32].Table 1
System parameters.
Parameter nameParameter valueHAP height20 kmFrequency2 GHzBandwidth15 MHzNoise temperature300 KdB angle0.5°Beam gain48 dBAntenna gain4 dBFigure2 depicts the EC versus γ¯ with different TAS designs under ξ=0.01,L=5,σ=0.75,mU=1 and ILS. Firstly, we can obviously see that the analytical results are in line with simulation results, which verifies the effectiveness of the theoretical derivations. Apparently, Figure 2(a) shows that the EC under TAS2 scheme is superior to that under TAS1 scheme, which can be explained by the fact that the selected best antenna under TAS2 scheme is employed to transmit superimposed signal to the user with less power in NOMA scheme. Thus, the overall achievable system performance is enhanced. In addition, Figure 2(b) plots the overall EC of the system improves with transmit antenna number. The more antennas the system has, the more suitable antenna can be selected to transmit signals. Besides, EC tends to be steady with the increase of γ¯.Figure 2
EC versusγ¯ with different TAS designs. (a) EC with different TAS schemes. (b) EC with different transmit antenna numbers.
(a)(b)Figure3 compares the EC versus γ¯ with different SR fading and Nakagami-m fading parameters under ξ=0.01,L=5,σ=0.75,Nt=6 and TAS1 scheme. It can be clearly seen in Figure 3 that the overall EC enhances gradually along with the improvement of SR fading channel conditions. However, the performance of overall EC is limited due to the second time slot. When mU=1, the upper limit of EC is 4.544. When mU=2, the upper limit of EC is 6.382. Likewise, the overall EC improves in high SNR regime with the improvement of Nakagami-m fading channel conditions.Figure 3
EC versus different fading parameters. (a) Nakagami-m: mU=1. (b) Nakagami-m: mU=2.
(a)(b)Figure4 shows the EC versus γ¯ with different imperfections under Nt=6,mU=1, FHS, and TAS1 scheme. From Figure 4(a), it can be seen that the EC is sensitive to the different imperfect SIC conditions. The upper limit of EC is 5.612 under perfect SIC. When ξ=0.1, the upper limit of EC is only 2.057. The EC of the considered ISATN becomes worse with the increase of ξ, which can be explained the NOMA users employ SIC technique to decode the superimposed signal at the receiver. When the SIC does not work perfectly, the reception performance of the system directly decline, leading to the reduction of the overall EC for the system. Besides, from Figure 4(b), as the growth of L and ϕ, the CEEs of the NOMA-enabled ISATN get smaller, which results in improvement of EC. However, when the average SNR γ¯ increases and exceeds 40 dB, the impact of imperfect CSI is reduced, and several curves gradually overlap.Figure 4
EC versus different imperfections. (a) EC with different imperfect SIC. (b) EC with different CEEs conditions.
(a)(b)


## 5. Conclusions

In this paper, the EC performance of a NOMA-enabled ISATN was investigated with two TAS schemes under imperfect CSI and SIC. Firstly, we deployed a NOMA-enabled ISATN with the help of HAP. Next, two TAS schemes were proposed to balance system performance and complexity in detail. By assuming that imperfect CSI and SIC occurred in the considered system, exact closed-form expressions for overall EC of the ISATN were obtained. Finally, through the MC simulations, we validated the superiority of the performance for our proposed TAS2 schemes. Besides, imperfect CSI and SIC seriously degraded the performance of the system, which is instructive for engineering practice.



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*Source: 1613190-2023-02-06.xml*