# Joint Decoding Order and Power Allocation Design for a NOMA-Based Overlay Cognitive Integrated Satellite-Terrestrial Relay Network

Authors: Rui Liu; Kefeng Guo; Kang An; Shibing Zhu; Haifeng Shuai
Journal: Wireless Communications and Mobile Computing
 (2022)

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## Abstract

Due to the shortage of spectrum resources, a nonorthogonal multiple access- (NOMA-) based overlay cognitive integrated satellite-terrestrial relay network is established in this paper, where cognitive radio (CR) and NOMA scheme are utilized to enhance spectral efficiency. In this system, a secondary terrestrial transmitter assists the transmission of primary signal and transmits its own signal simultaneously. To maximize the achievable rate of a secondary network on the premise of ensuring the quality of service of primary network, the decoding order and power allocation are jointly optimized. Particularly, we first derive the optimal power allocation factor for a given decoding order. After searching all different decoding orders, the globally optimal joint design of the decoding order and power allocation is obtained. To gain deep insights, the exact and asymptotic expressions for outage probability of both primary and secondary networks are derived. Numerical results demonstrate the accuracy of our derivation and indicate the impacts of key parameters on the proposed system.


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## Body

## 1. Introduction

Although the standard of the next-generation wireless network has not been unified, the introduction of satellite communication (SatCom) into the existing wireless communication network has become the consensus of experts and engineers [1, 2], which has the capabilities of all-weather operation, long transmission distance, and seamless coverage. In addition, the integrated satellite-terrestrial relay networks (ISTRNs) are becoming the promising architecture of the next-generation communication network and the Internet of things (IoT), which can offer comprehensive and convenient services [3, 4]. With the rapid development of ISTRN, the establishment of large-scale networks and the exponential growth of the number for terminals will cause serious shortage of spectrum resources. To solve the above problem, the application of cognitive radio (CR) and nonorthogonal multiple access (NOMA) into ISTRN have been regarded as effective approaches to improve spectrum utilization and deal with large-scale users [5, 6]. In general, CR enables unauthorized users to reasonably utilize the authorized spectrum [7, 8]. On the other hand, the NOMA technique allows different users to multiplex the same time/code/frequency [9, 10].


### 1.1. Related Works

Over recent years, the exploration of ISTRN has been investigated in various research areas. Based on fractional programming, a power allocation optimization scheme for downlink land mobile satellite (LMS) was proposed in [11], and the optimal average energy efficiency (EE) was derived. The authors of [12] investigated the performance of the LMS system, which considered the inherent hardware impairments (HIs) and jamming signal. In [13], the authors discussed the physical layer security of ISTRN, in which the secrecy outage probability (SOP) and average security capacity were derived under multiple illegal eavesdroppers and a threshold-based user scheduling scheme. A variety of methods to enhance the physical layer security and reliability of ISTRN were discussed in [14], and a kind of effective achievable rate and a new beamforming (BF) scheme were developed. The authors of [15] carried out joint BF and optimization for reconfigurable intelligent surface- (RIS-) aided ISTRN.To deal with the problem of spectrum shortage caused by increasingly scarce spectrum resources, many existing works have considered the combination of CR and ISTRN. The authors of [16] investigated the cognitive ISTRN with multiple secondary networks, and the outage probability (OP) of the primary network was minimized by the proposed secondary network selection scheme. In [17], the authors proposed a secure BF scheme to deal with the situation of multiple eavesdroppers in rate-splitting multiple access-based cognitive ISTRN. In the case of full channel state information (CSI) and statistical CSI, the authors proposed two BF schemes to maximize the instantaneous rate of terrestrial users in cognitive ISTRN by using hybrid zero forcing and partial zero forcing methods in [18]. The authors of [19] utilized terrestrial base stations and cooperative terminals to enhance the secrecy performance of the cognitive ISTRN system. On this basis, a joint artificial noise produce and cooperative interference BF scheme was proposed.For the sake of solving the service request of ever-increasing large-scale users, NOMA is considered a significant scheme in ISTRN [20]. Joint influence of channel estimation error and HIs on the security performance of cognitive integrated satellite-terrestrial relay networks (CISTRN) was investigated in [21]. The authors of [22] investigated the security performance of ISTRN with the NOMA scheme, where the HIs and multiple relays were considered and the SOP of the system was derived. In [23], the performance of ISTRN which considered cache and NOMA was studied, the OP and hit probability of the considered system were derived based on stochastic geometry. The performance of NOMA-based ISTRN with content delivery was discussed in [24], where the transmission delay can be reduced.


### 1.2. Motivations

The system performance of the NOMA-based system can be enhanced by designing the decoding order or power allocation scheme [25–27]. The authors of [25] minimized the overall OP under the cases of different decoding orders. In [26], to maximize the achievable rate of the secondary network with energy harvesting, the power allocation and time portion were optimized. In [27], the fair throughput for the NOMA-based system and orthogonal multiple access- (OMA-) based system was maximized. However, the above works cannot be simply utilized to optimize the performance of a priority-based NOMA system. In the traditional NOMA system, power is always allocated based on channel conditions. However, the priority-based NOMA system should guarantee the quality of service (QoS) of users with high priority. In fact, the QoS of high-priority users can be guaranteed and the system performance can be improved by optimizing the power allocation. On this basis, the optimization of the decoding order can make the system achieve optimal performance. A few existing works have investigated the joint optimization of the decoding order and power allocation [28–30], but the QoS of high-priority users was not strictly considered. Moreover, the authors of [31] considered the QoS of high-priority users, but the method was carried out in terrestrial networks, and there is no similar work in ISTRN.Motivated by the above discussion, a NOMA-based overlay CISTRN is considered. As far as we know, no prior work has discussed the NOMA-based overlay CISTRN with joint optimization of the decoding order and power allocation.


### 1.3. Our Contributions

The main contributions of this paper can be summarized as follows:(1)
Firstly, an overlay CISTRN with the NOMA scheme is established, in which the secondary network assists the transmission of the primary network in exchange for the opportunity to utilize the authorized spectrum of the primary network to transmit its own signals simultaneously. Besides, the satellite signal cannot reach the primary user and secondary user due to some severe shadowing(2)
Secondly, to improve the achievable performance of the secondary network, the decoding order and power allocation are optimized with the aim of maximizing the achievable rate of the secondary network under the predefined constraints of the primary satellite network. Specifically, the closed-form expressions for the optimal decoding order and power allocation factors are derived, which is of great significance for the study of NOMA-based CISTRN(3)
Finally, basing on the joint optimal design of the decoding order and power allocation, the exact OPs of both the primary network and secondary network are obtained. To get further insights, the asymptotic expressions and diversity order (DO) of the two networks are also derived. In addition, the numerical results validate our theoretical analysis. On this foundation, the impacts of many key parameters on the system performance are revealed


### 1.4. Paper Organization

The structure of the rest of this paper is as follows. Section2 provides the system model and problem formulation. The joint decoding order and power allocation design are given in Section 3. The performance of the proposed system is discussed in Section 5. Section 4 gives the numerical results and reveals the impacts of key parameters on system performance. Finally, the full text summary is given in Section 6.Notations.
CNa,b denotes the Gaussian distribution with mean a and variance b. E· is the expectation operator, and Γ· is the gamma function (Eq. 8.339.1 in [32]). The probability density function (PDF) and the cumulative distribution function (CDF) of variable z are, respectively, represented by fz. and Fz.. .n is the Pochhammer symbol (p. xliii in [32]). Kv· stands for the second kind modified Bessel function (Eq. 8.432.6 in [32]).


## 1.1. Related Works

Over recent years, the exploration of ISTRN has been investigated in various research areas. Based on fractional programming, a power allocation optimization scheme for downlink land mobile satellite (LMS) was proposed in [11], and the optimal average energy efficiency (EE) was derived. The authors of [12] investigated the performance of the LMS system, which considered the inherent hardware impairments (HIs) and jamming signal. In [13], the authors discussed the physical layer security of ISTRN, in which the secrecy outage probability (SOP) and average security capacity were derived under multiple illegal eavesdroppers and a threshold-based user scheduling scheme. A variety of methods to enhance the physical layer security and reliability of ISTRN were discussed in [14], and a kind of effective achievable rate and a new beamforming (BF) scheme were developed. The authors of [15] carried out joint BF and optimization for reconfigurable intelligent surface- (RIS-) aided ISTRN.To deal with the problem of spectrum shortage caused by increasingly scarce spectrum resources, many existing works have considered the combination of CR and ISTRN. The authors of [16] investigated the cognitive ISTRN with multiple secondary networks, and the outage probability (OP) of the primary network was minimized by the proposed secondary network selection scheme. In [17], the authors proposed a secure BF scheme to deal with the situation of multiple eavesdroppers in rate-splitting multiple access-based cognitive ISTRN. In the case of full channel state information (CSI) and statistical CSI, the authors proposed two BF schemes to maximize the instantaneous rate of terrestrial users in cognitive ISTRN by using hybrid zero forcing and partial zero forcing methods in [18]. The authors of [19] utilized terrestrial base stations and cooperative terminals to enhance the secrecy performance of the cognitive ISTRN system. On this basis, a joint artificial noise produce and cooperative interference BF scheme was proposed.For the sake of solving the service request of ever-increasing large-scale users, NOMA is considered a significant scheme in ISTRN [20]. Joint influence of channel estimation error and HIs on the security performance of cognitive integrated satellite-terrestrial relay networks (CISTRN) was investigated in [21]. The authors of [22] investigated the security performance of ISTRN with the NOMA scheme, where the HIs and multiple relays were considered and the SOP of the system was derived. In [23], the performance of ISTRN which considered cache and NOMA was studied, the OP and hit probability of the considered system were derived based on stochastic geometry. The performance of NOMA-based ISTRN with content delivery was discussed in [24], where the transmission delay can be reduced.


## 1.2. Motivations

The system performance of the NOMA-based system can be enhanced by designing the decoding order or power allocation scheme [25–27]. The authors of [25] minimized the overall OP under the cases of different decoding orders. In [26], to maximize the achievable rate of the secondary network with energy harvesting, the power allocation and time portion were optimized. In [27], the fair throughput for the NOMA-based system and orthogonal multiple access- (OMA-) based system was maximized. However, the above works cannot be simply utilized to optimize the performance of a priority-based NOMA system. In the traditional NOMA system, power is always allocated based on channel conditions. However, the priority-based NOMA system should guarantee the quality of service (QoS) of users with high priority. In fact, the QoS of high-priority users can be guaranteed and the system performance can be improved by optimizing the power allocation. On this basis, the optimization of the decoding order can make the system achieve optimal performance. A few existing works have investigated the joint optimization of the decoding order and power allocation [28–30], but the QoS of high-priority users was not strictly considered. Moreover, the authors of [31] considered the QoS of high-priority users, but the method was carried out in terrestrial networks, and there is no similar work in ISTRN.Motivated by the above discussion, a NOMA-based overlay CISTRN is considered. As far as we know, no prior work has discussed the NOMA-based overlay CISTRN with joint optimization of the decoding order and power allocation.


## 1.3. Our Contributions

The main contributions of this paper can be summarized as follows:(1)
Firstly, an overlay CISTRN with the NOMA scheme is established, in which the secondary network assists the transmission of the primary network in exchange for the opportunity to utilize the authorized spectrum of the primary network to transmit its own signals simultaneously. Besides, the satellite signal cannot reach the primary user and secondary user due to some severe shadowing(2)
Secondly, to improve the achievable performance of the secondary network, the decoding order and power allocation are optimized with the aim of maximizing the achievable rate of the secondary network under the predefined constraints of the primary satellite network. Specifically, the closed-form expressions for the optimal decoding order and power allocation factors are derived, which is of great significance for the study of NOMA-based CISTRN(3)
Finally, basing on the joint optimal design of the decoding order and power allocation, the exact OPs of both the primary network and secondary network are obtained. To get further insights, the asymptotic expressions and diversity order (DO) of the two networks are also derived. In addition, the numerical results validate our theoretical analysis. On this foundation, the impacts of many key parameters on the system performance are revealed


## 1.4. Paper Organization

The structure of the rest of this paper is as follows. Section2 provides the system model and problem formulation. The joint decoding order and power allocation design are given in Section 3. The performance of the proposed system is discussed in Section 5. Section 4 gives the numerical results and reveals the impacts of key parameters on system performance. Finally, the full text summary is given in Section 6.Notations.
CNa,b denotes the Gaussian distribution with mean a and variance b. E· is the expectation operator, and Γ· is the gamma function (Eq. 8.339.1 in [32]). The probability density function (PDF) and the cumulative distribution function (CDF) of variable z are, respectively, represented by fz. and Fz.. .n is the Pochhammer symbol (p. xliii in [32]). Kv· stands for the second kind modified Bessel function (Eq. 8.432.6 in [32]).


## 2. System Model and Problem Formulation

As shown in Figure1, a NOMA-based overlay CISTRN is considered in our paper, which consists of a primary satellite source (S) and a secondary terrestrial transmitter (R) with their corresponding receivers (U and D). It is assumed that there is no direct satellite (DS) link between S and U due to severe shadowing effects, such as obstacles. Therefore, the secondary network is asked for assisting the transmission from S to U as a relay [6]. As a reward, R utilizes the authorized spectrum of S to communicate with D. The NOMA scheme is adopted in R to transmit the expected signals of U and D simultaneously, and the two-user pair is considered in our paper to balance the complexity and effectiveness. Each node works in half-duplex mode and applies an omnidirectional antenna to reduce the complexity of the system [16], and the decode-and-forward (DF) protocol is employed by R. Specifically, the channel coefficients of S-R link, R-U link, and R-D link are denoted by hsr, hru, and hrd, respectively. Moreover, the satellite-terrestrial channel experiences Shadowed-Rician (SR) fading, and the terrestrial channels undergo Nakagami-m fading. Without loss of generality, it is assumed that the channel coefficients are variable but remain constant in a time slot (furthermore, the perfect hardware impairments and decoding errors are assumed in our proposed system to simplify the analysis. Besides, the perfect CSI can be obtained by feedback and training from users). In addition, all receivers sustain additive white Gaussian noise (AWGN) with CN0,σ2.Figure 1
System model.


### 2.1. Problem Formulation

There are two phases in the entire transmission process, which is shown in Figure2.Figure 2
The slotted structure of the system.During the first phase,S transmits its signal to R, and the signal received by R is represented by
(1)ysr=Pshsrxs+nr,where Ps is the transmit power of S, xs denotes the satellite message with Exs2=1, and nr is AWGN. From (1), the achievable rate of xs at R can be expressed as
(2)Rr=12log21+γsr,where γsr=γ¯shsr2 and γ¯s=Ps/σ2 is the signal-to-noise ratio (SNR) of the satellite. The target rate of the primary network is denoted as Rthu. To decode the primary signal successfully, the following constraint must be satisfied, namely, Rr≥Rthu.In the second phase,R integrated primary and its own signal with the superposition coding technique (SCT), and the signal received by U and D can be expressed by
(3)yra=hraPrβxs+Pr1−βxr+na,where a=r,d, Pr is the transmit power of R, xr denotes the expected signal of D with Exr2=1, β is the power allocation factor with 0<β≤1, and na represents AWGN.At the receiving ends, there are two decoding orders. The first is thatU decodes the desired signal directly, and D performs successive interference cancellation (SIC). The second is that D detects its expected signal, while U adopts SIC. Besides, the achievable rates of xs at U and xr at D under the conditions of two decoding orders can be given as follows, respectively.


#### 2.1.1. The First Decoding Order

In this order,U decodes xs while xr is regarded as the serial interference. Then, the achievable rate of xs at U can be expressed as
(4)Ru1=12log21+βγru1−βγru+1,where γru=γ¯rhru2, γ¯r=Pr/σ2 is the SNR of the secondary transmitter, and the tag “1” indicates the first decoding order.Recalling to the properties of the NOMA scheme, SIC is adopted atD; thus, xs is decoded by D with xr as the cochannel interference (CCI); then, the decoded message is removed from yrd. Finally, D decodes xr from the remaining signal. Hence, the achievable rates of xs and xr at D can be, respectively, represented as
(5)Ru⟶d1=12log21+βγrd1−βγrd+1,Rd1=12log21+1−βγrd,where γrd=γ¯rhrd2.


#### 2.1.2. The Second Decoding Order

In the second decoding order, SIC is applied atU, and xs is treated as the serial interference at both U and D firstly. Therefore, the achievable rates of xr at U and D are expressed as, respectively,
(6)Rd⟶u2=12log21+1−βγruβγru+1,(7)Rd2=12log21+1−βγrdβγrd+1,where the tag “2” indicates the second decoding order.Ifxr can be decoded by U successfully, the achievable rate of xs at U can be represented as
(8)Ru2=12log21+βγru.


### 2.2. Channel Model

In this section, the statistical characteristics of SR fading and Nakagami-m fading are provided. Firstly, the PDF of hsr is given by [33]
(9)fhsr2x=αsre−βsrx1F1msr;1;δsrx,where αsrΔ=1/2bsr2bsrmsr/2bsrmsr+Ωsrmsr, βsrΔ=1/2bsr, and δsrΔ=Ωsr/2bsr2bsrmsr+Ωsr. msr>0 denotes the Nakagami-m parameter, and 2bsr and Ωsr are the average power of the multipath component and that of the line-of-sight (LOS) component, respectively.Whenmsr is an integer, with the help of Eq. 07.20.03.009.01 and Eq. 07.02.03.0014.01 in [34], we can get
(10)1F1msr;1;δsrx=e−δsrx∑n=0msr−1−δsrn1−msrn!2xn.By taking (10) into (9), it can be rewritten as
(11)fhsr2x=αsre−βsr−δsrx∑n=0msr−1‍−δsrn1−msrnn!2xn.Owing toγsr=γ¯shsr2, the PDF of γsr can expressed as
(12)fγsrx=αsr∑n=0msr−1‍ζnxne−Δsrx,where ζn=−δsrn1−msrn/n!2γ¯sn+1 and Δsr=βsr−δsr/γ¯s.With the utilization of Eq. 3.351.2 in [32], we can get CDF of γsr as
(13)Fγsrx=1−αsr∑n=0msr−1‍∑t=0n‍n!ζnt!Δsrn−t+1xte−Δsrx,In addition, the PDF and CDF ofγra are expressed as, respectively [35],
(14)fγrax=1ΓmraΛramraxmra−1e−Ξrax,(15)Fγrax=1−e−Λrax∑p=0mra−1‍1p!Λrapxp,where Λra=mra/Ωraγ¯r, mra is the fading severity parameter, and Ωra=dra−ω denotes the average fading power, in which dra is the distance from R to U or D and ω denotes the path loss exponent.


## 2.1. Problem Formulation

There are two phases in the entire transmission process, which is shown in Figure2.Figure 2
The slotted structure of the system.During the first phase,S transmits its signal to R, and the signal received by R is represented by
(1)ysr=Pshsrxs+nr,where Ps is the transmit power of S, xs denotes the satellite message with Exs2=1, and nr is AWGN. From (1), the achievable rate of xs at R can be expressed as
(2)Rr=12log21+γsr,where γsr=γ¯shsr2 and γ¯s=Ps/σ2 is the signal-to-noise ratio (SNR) of the satellite. The target rate of the primary network is denoted as Rthu. To decode the primary signal successfully, the following constraint must be satisfied, namely, Rr≥Rthu.In the second phase,R integrated primary and its own signal with the superposition coding technique (SCT), and the signal received by U and D can be expressed by
(3)yra=hraPrβxs+Pr1−βxr+na,where a=r,d, Pr is the transmit power of R, xr denotes the expected signal of D with Exr2=1, β is the power allocation factor with 0<β≤1, and na represents AWGN.At the receiving ends, there are two decoding orders. The first is thatU decodes the desired signal directly, and D performs successive interference cancellation (SIC). The second is that D detects its expected signal, while U adopts SIC. Besides, the achievable rates of xs at U and xr at D under the conditions of two decoding orders can be given as follows, respectively.


### 2.1.1. The First Decoding Order

In this order,U decodes xs while xr is regarded as the serial interference. Then, the achievable rate of xs at U can be expressed as
(4)Ru1=12log21+βγru1−βγru+1,where γru=γ¯rhru2, γ¯r=Pr/σ2 is the SNR of the secondary transmitter, and the tag “1” indicates the first decoding order.Recalling to the properties of the NOMA scheme, SIC is adopted atD; thus, xs is decoded by D with xr as the cochannel interference (CCI); then, the decoded message is removed from yrd. Finally, D decodes xr from the remaining signal. Hence, the achievable rates of xs and xr at D can be, respectively, represented as
(5)Ru⟶d1=12log21+βγrd1−βγrd+1,Rd1=12log21+1−βγrd,where γrd=γ¯rhrd2.


### 2.1.2. The Second Decoding Order

In the second decoding order, SIC is applied atU, and xs is treated as the serial interference at both U and D firstly. Therefore, the achievable rates of xr at U and D are expressed as, respectively,
(6)Rd⟶u2=12log21+1−βγruβγru+1,(7)Rd2=12log21+1−βγrdβγrd+1,where the tag “2” indicates the second decoding order.Ifxr can be decoded by U successfully, the achievable rate of xs at U can be represented as
(8)Ru2=12log21+βγru.


## 2.1.1. The First Decoding Order

In this order,U decodes xs while xr is regarded as the serial interference. Then, the achievable rate of xs at U can be expressed as
(4)Ru1=12log21+βγru1−βγru+1,where γru=γ¯rhru2, γ¯r=Pr/σ2 is the SNR of the secondary transmitter, and the tag “1” indicates the first decoding order.Recalling to the properties of the NOMA scheme, SIC is adopted atD; thus, xs is decoded by D with xr as the cochannel interference (CCI); then, the decoded message is removed from yrd. Finally, D decodes xr from the remaining signal. Hence, the achievable rates of xs and xr at D can be, respectively, represented as
(5)Ru⟶d1=12log21+βγrd1−βγrd+1,Rd1=12log21+1−βγrd,where γrd=γ¯rhrd2.


## 2.1.2. The Second Decoding Order

In the second decoding order, SIC is applied atU, and xs is treated as the serial interference at both U and D firstly. Therefore, the achievable rates of xr at U and D are expressed as, respectively,
(6)Rd⟶u2=12log21+1−βγruβγru+1,(7)Rd2=12log21+1−βγrdβγrd+1,where the tag “2” indicates the second decoding order.Ifxr can be decoded by U successfully, the achievable rate of xs at U can be represented as
(8)Ru2=12log21+βγru.


## 2.2. Channel Model

In this section, the statistical characteristics of SR fading and Nakagami-m fading are provided. Firstly, the PDF of hsr is given by [33]
(9)fhsr2x=αsre−βsrx1F1msr;1;δsrx,where αsrΔ=1/2bsr2bsrmsr/2bsrmsr+Ωsrmsr, βsrΔ=1/2bsr, and δsrΔ=Ωsr/2bsr2bsrmsr+Ωsr. msr>0 denotes the Nakagami-m parameter, and 2bsr and Ωsr are the average power of the multipath component and that of the line-of-sight (LOS) component, respectively.Whenmsr is an integer, with the help of Eq. 07.20.03.009.01 and Eq. 07.02.03.0014.01 in [34], we can get
(10)1F1msr;1;δsrx=e−δsrx∑n=0msr−1−δsrn1−msrn!2xn.By taking (10) into (9), it can be rewritten as
(11)fhsr2x=αsre−βsr−δsrx∑n=0msr−1‍−δsrn1−msrnn!2xn.Owing toγsr=γ¯shsr2, the PDF of γsr can expressed as
(12)fγsrx=αsr∑n=0msr−1‍ζnxne−Δsrx,where ζn=−δsrn1−msrn/n!2γ¯sn+1 and Δsr=βsr−δsr/γ¯s.With the utilization of Eq. 3.351.2 in [32], we can get CDF of γsr as
(13)Fγsrx=1−αsr∑n=0msr−1‍∑t=0n‍n!ζnt!Δsrn−t+1xte−Δsrx,In addition, the PDF and CDF ofγra are expressed as, respectively [35],
(14)fγrax=1ΓmraΛramraxmra−1e−Ξrax,(15)Fγrax=1−e−Λrax∑p=0mra−1‍1p!Λrapxp,where Λra=mra/Ωraγ¯r, mra is the fading severity parameter, and Ωra=dra−ω denotes the average fading power, in which dra is the distance from R to U or D and ω denotes the path loss exponent.


## 3. Joint Decoding Order and Power Allocation Design

In this section, by jointly designing the decoding order and power allocation factor, we maximize the achievable rate of the secondary network under the predefined constraints of the primary network. Firstly, we find the optimal power allocation factor under the two decoding orders. Then, the optimal allocation factors are utilized to obtain the corresponding achievable rates. By comparing the calculated achievable rates under different decoding orders, the desired decoding order corresponding to the optimal achievable rate can be obtained.


### 3.1. The First Decoding Order

Under the first decoding order, the optimization problem can be formulated as follows.Problem 1.
(16)maxβRd1=12log21+1−βγrd,(17)s.t.minRu1,Ru⟶d1≥Rthu,(18)0<β≤1,where (17) guarantees that the primary signal can be decoded correctly and SIC can be performed successfully. Besides, (17) can be reexpressed as
(19)β≥ku1+γruγru1+ku,β≥ku1+γrdγrd1+ku,where ku=22Rthu−1.
From (18), we can obtain γ~≥ku with γ~=minγru,γrd. It is worth noting that if the constraints of (17) and (18) cannot be satisfied, all power is allocated to transmit the message of the primary network.
It can be observed thatRd1 is a decreasing function of β; thus, the optimal power allocation factor can be expressed as
(20)β∗=ku1+γ~γ~1+ku,γ~≥ku,1,else.


### 3.2. The Second Decoding Order

Under the second decoding order, the optimization problem can be formulated as follows.Problem 2.
(21)maxβRd2=12log21+1−βγrdβγrd+1,(22)s.t.Rd⟶u2≥Rthd,(23)Ru2≥Rthu,(24)0<β≤1,where Rthd is the target rate of the secondary network, (22) and (23) ensure that the primary network can decode the expected signal successfully, and we can get
(25)kuγru≤β≤γru−kdγru1+kd,where kd=22Rthd−1. Moreover, to ensure the feasible region of this problem, not an empty set, γru−kd/γru1+kd≥ku/γru should be guaranteed. Hence, we can get γru≥k=ku+kd+kukd.
From (21), we can obtain that Rd2 is a strict decreasing function of β. Thus, the optimal power allocation factor is expressed as
(26)β†=kuγru,γru≥k.Only whenγru≥k can the system work under the second order. Thus, we just need to compare the optimal achievable rates under the first and second order with γru≥k. After comparison, we find that when γru>γrd, Rd2β†>Rd1β∗. Therefore, the joint optimal design of the decoding order and power allocation is given by
(27)λ,β=2,β†,γru>maxγrd,k,1,β∗,else,where λ represents the decoding order.According to the above joint optimization, we can get the achievable rates ofxs and xr at U and D as, respectively,
(28)Ru=Rthu,γru>ku,12log21+γru,else,(29)Rd=12log21+Ξ1,γru>maxγrd,k,12log21+Ξ2,else,where Ξ1=γru−kuγrd/kuγrd+γru and Ξ2=γ~−kuγrd/γ~1+ku.Remark.
The above derivations show that the user with low priority should be decoded firstly when the channel gain of the high-priority user is larger than that of the low-priority user, as well as the specific value. In great majority of existing works on the NOMA system, the high-priority users should be decoded firstly, which is diverse with our observation.


## 3.1. The First Decoding Order

Under the first decoding order, the optimization problem can be formulated as follows.Problem 1.
(16)maxβRd1=12log21+1−βγrd,(17)s.t.minRu1,Ru⟶d1≥Rthu,(18)0<β≤1,where (17) guarantees that the primary signal can be decoded correctly and SIC can be performed successfully. Besides, (17) can be reexpressed as
(19)β≥ku1+γruγru1+ku,β≥ku1+γrdγrd1+ku,where ku=22Rthu−1.
From (18), we can obtain γ~≥ku with γ~=minγru,γrd. It is worth noting that if the constraints of (17) and (18) cannot be satisfied, all power is allocated to transmit the message of the primary network.
It can be observed thatRd1 is a decreasing function of β; thus, the optimal power allocation factor can be expressed as
(20)β∗=ku1+γ~γ~1+ku,γ~≥ku,1,else.


## 3.2. The Second Decoding Order

Under the second decoding order, the optimization problem can be formulated as follows.Problem 2.
(21)maxβRd2=12log21+1−βγrdβγrd+1,(22)s.t.Rd⟶u2≥Rthd,(23)Ru2≥Rthu,(24)0<β≤1,where Rthd is the target rate of the secondary network, (22) and (23) ensure that the primary network can decode the expected signal successfully, and we can get
(25)kuγru≤β≤γru−kdγru1+kd,where kd=22Rthd−1. Moreover, to ensure the feasible region of this problem, not an empty set, γru−kd/γru1+kd≥ku/γru should be guaranteed. Hence, we can get γru≥k=ku+kd+kukd.
From (21), we can obtain that Rd2 is a strict decreasing function of β. Thus, the optimal power allocation factor is expressed as
(26)β†=kuγru,γru≥k.Only whenγru≥k can the system work under the second order. Thus, we just need to compare the optimal achievable rates under the first and second order with γru≥k. After comparison, we find that when γru>γrd, Rd2β†>Rd1β∗. Therefore, the joint optimal design of the decoding order and power allocation is given by
(27)λ,β=2,β†,γru>maxγrd,k,1,β∗,else,where λ represents the decoding order.According to the above joint optimization, we can get the achievable rates ofxs and xr at U and D as, respectively,
(28)Ru=Rthu,γru>ku,12log21+γru,else,(29)Rd=12log21+Ξ1,γru>maxγrd,k,12log21+Ξ2,else,where Ξ1=γru−kuγrd/kuγrd+γru and Ξ2=γ~−kuγrd/γ~1+ku.Remark.
The above derivations show that the user with low priority should be decoded firstly when the channel gain of the high-priority user is larger than that of the low-priority user, as well as the specific value. In great majority of existing works on the NOMA system, the high-priority users should be decoded firstly, which is diverse with our observation.


## 4. Performance Analysis

In this section, the exact and asymptotic expressions of OP for both the primary network and secondary network are obtained based on the derivations of achievable rates, where OP is defined as the probability of the achievable rate lower than the target rate. Besides, we derive the DOs to provide more insights.


### 4.1. Primary Network

The primary network will be interrupted if the achievable rate ofxs at S or U is lower than Rthu; thus, the OP can be expressed as
(30)Poutp=1−PrRr≥RthuPrRu≥Rthu=1−1−Fγsrku1−Fγruku.Then, by taking (13) and (15) into (30), we can get
(31)Poutp=1−αsr∑n=0msr−1‍∑t=0n‍n!ζnt!Δsrn−t+1∑p=0mru−1‍1p!Λrupkup+te−Δsr+Λruku.To derive the asymptotic OP, we first give the asymptotic CDF ofγsr and γra in the high SNR regime as [35]
(32)Fγsrx=αsrγ¯sx,(33)Fγrax=1Γmra+1Λramraxmra.Finally, we substitute (32) and (33) into (30); the asymptotic expression for OP of the primary system can be expressed as
(34)Poutp,∞≃αsrγ¯sku,mru>1,αsrγ¯sku+ΛrumrukumruΓmru+1,mru=1.According to (34), the DO of the primary network can be expressed as DOp=min1,mru.


### 4.2. Secondary Network

In the secondary network, OP means the probability of achievable rates ofxs at R lower than Rthu or that of xr at D lower than Rthd, which can be given by
(35)Pouts=1−PrRr≥Rthu⏟T1PrRd≥Rthu⏟T2.We can easily get(36)T1=αsr∑n=0msr−1‍∑t=0n‍n!ζnt!Δsrn−t+1kute−Δsrku.From (29), T2 can be reexpressed as
(37)T2=Prγru≥maxγrd,k,Ξ1≥kd+Prγru≤maxγrd,k,Ξ2≥kd=Prγru≥k,γru≥γrd≥γrukdγru−k+kd+Prγrd≥k,γrd≥γru≥γrdkdγrd−k+ku.To simplify the calculation, some mathematical processing is conducted for (37) as follows [31].
(38a)T2=Prγru≥k,γru≥γrd≥γrukdγru−k+kd(38b)+Prγru≥k,γrd≥γru≥γrukdγru−k+kd(38c)+Prγrd≥k,γrd≥γru≥γrdkdγrd−k+ku(38d)+Prγrd≥k,γru≥γrd≥γrdkdγrd−k+ku(38e)−Prγru≥k,γrd≥γru≥γrukdγru−k+kd(38f)−Prγrd≥k,γru≥γrd≥γrdkdγrd−k+ku.By merging (38a) and (38b), (38c) and (38d), and (38e) and (38f), we can get
(39)T2=Prγru≥k,γrd≥γrukdγru−k+kd⏟Q1+Prγrd≥k,γru≥γrdkdγrd−k+ku⏟−Prγru≥k,γrd≥k⏟Q3Q2.Similar to the derivation ofPoutp, with the utilization of (15), we can obtain
(40)Q3=1−Fγrdk1−Fγruk=e−Λrd+Λruk∑p1=0mrd−1‍∑p2=0mru−1‍1p1!1p2!Λrdp1Λrup2kp1+p2.Then, the derivation ofQ1 is carried on, which is given by
(41)Q1=∫k+∞‍1−Fγrdxkdx−k+kdfγruxdx.Letx−k+kd=t; (46) can be rewritten as
(42)Q1=∫0+∞‍1−Fγrdt+k−kdkdtfγrut+k−kddt⏟Q11−∫0kd‍1−Fγrdt+k−kdkdtfγrut+k−kddt⏟Q12.By substituting (14) and (15) into Q11, with the help of Eq. 1.111 and Eq. 3.471.9 in [32], the expression of Q11 can be derived as
(43)Q11=2ΛrumruΓmrue−k−kdΛru+Λrdkd∑p1=0mrd−1‍1p1!Λrdkdp1∑p2=0mru−1+p1‍mru−1+p1p2×k−kdmru−1+p1−p2Λrdkdk−kdΛrup2−p1+1/2Kp2−p1+12ΛruΛrdkdk−kd.Lett=u+1/2kd; Q12 can be rewritten as
(44)Q12=kd2∫−11‍1−Fγrdu+1kd+2k−2kdu+1fγruu+1kd+2k−2kd2du.By utilizing the Gaussian-Chebyshev quadrature [33], we can get
(45)Q12≈πkdΛrumruIΓmrue−k−kdΛru+Λrdkd∑p1=0mrd−1‍1p1!Λrdkdp1∑p2=0mru−1+p1‍mru−1+p1p2×k−kdmru−1+p1−p2∑i1=1I‍1−ui12e−2k−2kdΛrd/ui1+1+Λrukdui1+1/2ui1+12p2−p1kdp2,where ui=cos2i−1π/2I and I is the Gaussian-Chebyshev approximation.By taking similar derivations ofQ11 and Q12, Q21 and Q22 are given by, respectively,
(46)Q21=2ΛrdmrdΓmrde−k−kuΛrd+Λruku∑p1=0mru−1‍1p1!Λrukup1∑p2=0mrd−1+p1mrd−1+p1p2‍×k−kumrd−1+p1−p2Λrukuk−kuΛrdp2−p1+1/2Kp2−p1+12ΛruΛrdkuk−ku.(47)Q22≈πkuΛrdmrdIΓmrde−k−kuΛrd+Λruku∑p1=0mru−1‍1p1!Λrukup1∑p2=0mrd−1+p1‍mrd−1+p1p2×k−kumrd−1+p1−p2∑i2=1I‍1−ui22e−2k−2kuΛru/ui2+1+Λrdkuui2+1/2ui2+12p2−p1kup2.Therefore, the expression of OP for the secondary network is given by(48)Pouts=1−αsr∑n=0msr−1‍∑t=0n‍n!ζnt!Δsrn−t+1kute−Δsrkue−k−kdΛru+Λrdkd∑p1=0mrd−1‍1p1!Λrdkdp1×∑p2=0mru−1+p1‍mru−1+p1p2k−kdmru−1+p1−p22ΛrumruΓmruΛrdkdk−kdΛrup2−p1+1/2×Kp2−p1+12ΛruΛrdkdk−kd−πkdΛrumruIΓmru∑i1=1I‍1−ui12e−2k−2kdΛrd/ui1+1+Λrukdui1+1/2×ui1+12p2−p1kdp2+e−k−kuΛrd+Λruku∑p1=0mru−1‍1p1!Λrukup1∑p2=0mrd−1+p1mrd−1+p1p2×k−kumrd−1+p1−p22ΛrdmrdΓmrdΛrukuk−kuΛrdp2−p1+1/2Kp2−p1+12ΛruΛrdkuk−ku−πkuΛrdmrdIΓmrd∑i2=1I‍1−ui22e−2k−2kuΛru/ui2+1+Λrdkuui2+1/2×ui2+12p2−p1kup2‍−e−Λrd+Λruk∑p1=0mrd−1‍∑p2=0mru−1‍1p1!1p2!Λrdp1Λrup2kp1+p2.Similar to the asymptotic OP of the primary network, we can easily obtain the asymptotic OP of the secondary network as(49)Pouts,∞=αsrγ¯sku,mru>1,mrd>1,αsrγ¯sku+ΛrumrukmruΓmru+1,mru=1,mrd>1,αsrγ¯sku+ΛrdmrdkmrdΓmrd+1,αsrγ¯sku+ΛrumrukmruΓmru+1+ΛrdmrdkmrdΓmrd+1,mrd=1,mru>1,mru=mrd=1.In addition, the DO of the secondary network isDOs=min1,mru,mrd.


## 4.1. Primary Network

The primary network will be interrupted if the achievable rate ofxs at S or U is lower than Rthu; thus, the OP can be expressed as
(30)Poutp=1−PrRr≥RthuPrRu≥Rthu=1−1−Fγsrku1−Fγruku.Then, by taking (13) and (15) into (30), we can get
(31)Poutp=1−αsr∑n=0msr−1‍∑t=0n‍n!ζnt!Δsrn−t+1∑p=0mru−1‍1p!Λrupkup+te−Δsr+Λruku.To derive the asymptotic OP, we first give the asymptotic CDF ofγsr and γra in the high SNR regime as [35]
(32)Fγsrx=αsrγ¯sx,(33)Fγrax=1Γmra+1Λramraxmra.Finally, we substitute (32) and (33) into (30); the asymptotic expression for OP of the primary system can be expressed as
(34)Poutp,∞≃αsrγ¯sku,mru>1,αsrγ¯sku+ΛrumrukumruΓmru+1,mru=1.According to (34), the DO of the primary network can be expressed as DOp=min1,mru.


## 4.2. Secondary Network

In the secondary network, OP means the probability of achievable rates ofxs at R lower than Rthu or that of xr at D lower than Rthd, which can be given by
(35)Pouts=1−PrRr≥Rthu⏟T1PrRd≥Rthu⏟T2.We can easily get(36)T1=αsr∑n=0msr−1‍∑t=0n‍n!ζnt!Δsrn−t+1kute−Δsrku.From (29), T2 can be reexpressed as
(37)T2=Prγru≥maxγrd,k,Ξ1≥kd+Prγru≤maxγrd,k,Ξ2≥kd=Prγru≥k,γru≥γrd≥γrukdγru−k+kd+Prγrd≥k,γrd≥γru≥γrdkdγrd−k+ku.To simplify the calculation, some mathematical processing is conducted for (37) as follows [31].
(38a)T2=Prγru≥k,γru≥γrd≥γrukdγru−k+kd(38b)+Prγru≥k,γrd≥γru≥γrukdγru−k+kd(38c)+Prγrd≥k,γrd≥γru≥γrdkdγrd−k+ku(38d)+Prγrd≥k,γru≥γrd≥γrdkdγrd−k+ku(38e)−Prγru≥k,γrd≥γru≥γrukdγru−k+kd(38f)−Prγrd≥k,γru≥γrd≥γrdkdγrd−k+ku.By merging (38a) and (38b), (38c) and (38d), and (38e) and (38f), we can get
(39)T2=Prγru≥k,γrd≥γrukdγru−k+kd⏟Q1+Prγrd≥k,γru≥γrdkdγrd−k+ku⏟−Prγru≥k,γrd≥k⏟Q3Q2.Similar to the derivation ofPoutp, with the utilization of (15), we can obtain
(40)Q3=1−Fγrdk1−Fγruk=e−Λrd+Λruk∑p1=0mrd−1‍∑p2=0mru−1‍1p1!1p2!Λrdp1Λrup2kp1+p2.Then, the derivation ofQ1 is carried on, which is given by
(41)Q1=∫k+∞‍1−Fγrdxkdx−k+kdfγruxdx.Letx−k+kd=t; (46) can be rewritten as
(42)Q1=∫0+∞‍1−Fγrdt+k−kdkdtfγrut+k−kddt⏟Q11−∫0kd‍1−Fγrdt+k−kdkdtfγrut+k−kddt⏟Q12.By substituting (14) and (15) into Q11, with the help of Eq. 1.111 and Eq. 3.471.9 in [32], the expression of Q11 can be derived as
(43)Q11=2ΛrumruΓmrue−k−kdΛru+Λrdkd∑p1=0mrd−1‍1p1!Λrdkdp1∑p2=0mru−1+p1‍mru−1+p1p2×k−kdmru−1+p1−p2Λrdkdk−kdΛrup2−p1+1/2Kp2−p1+12ΛruΛrdkdk−kd.Lett=u+1/2kd; Q12 can be rewritten as
(44)Q12=kd2∫−11‍1−Fγrdu+1kd+2k−2kdu+1fγruu+1kd+2k−2kd2du.By utilizing the Gaussian-Chebyshev quadrature [33], we can get
(45)Q12≈πkdΛrumruIΓmrue−k−kdΛru+Λrdkd∑p1=0mrd−1‍1p1!Λrdkdp1∑p2=0mru−1+p1‍mru−1+p1p2×k−kdmru−1+p1−p2∑i1=1I‍1−ui12e−2k−2kdΛrd/ui1+1+Λrukdui1+1/2ui1+12p2−p1kdp2,where ui=cos2i−1π/2I and I is the Gaussian-Chebyshev approximation.By taking similar derivations ofQ11 and Q12, Q21 and Q22 are given by, respectively,
(46)Q21=2ΛrdmrdΓmrde−k−kuΛrd+Λruku∑p1=0mru−1‍1p1!Λrukup1∑p2=0mrd−1+p1mrd−1+p1p2‍×k−kumrd−1+p1−p2Λrukuk−kuΛrdp2−p1+1/2Kp2−p1+12ΛruΛrdkuk−ku.(47)Q22≈πkuΛrdmrdIΓmrde−k−kuΛrd+Λruku∑p1=0mru−1‍1p1!Λrukup1∑p2=0mrd−1+p1‍mrd−1+p1p2×k−kumrd−1+p1−p2∑i2=1I‍1−ui22e−2k−2kuΛru/ui2+1+Λrdkuui2+1/2ui2+12p2−p1kup2.Therefore, the expression of OP for the secondary network is given by(48)Pouts=1−αsr∑n=0msr−1‍∑t=0n‍n!ζnt!Δsrn−t+1kute−Δsrkue−k−kdΛru+Λrdkd∑p1=0mrd−1‍1p1!Λrdkdp1×∑p2=0mru−1+p1‍mru−1+p1p2k−kdmru−1+p1−p22ΛrumruΓmruΛrdkdk−kdΛrup2−p1+1/2×Kp2−p1+12ΛruΛrdkdk−kd−πkdΛrumruIΓmru∑i1=1I‍1−ui12e−2k−2kdΛrd/ui1+1+Λrukdui1+1/2×ui1+12p2−p1kdp2+e−k−kuΛrd+Λruku∑p1=0mru−1‍1p1!Λrukup1∑p2=0mrd−1+p1mrd−1+p1p2×k−kumrd−1+p1−p22ΛrdmrdΓmrdΛrukuk−kuΛrdp2−p1+1/2Kp2−p1+12ΛruΛrdkuk−ku−πkuΛrdmrdIΓmrd∑i2=1I‍1−ui22e−2k−2kuΛru/ui2+1+Λrdkuui2+1/2×ui2+12p2−p1kup2‍−e−Λrd+Λruk∑p1=0mrd−1‍∑p2=0mru−1‍1p1!1p2!Λrdp1Λrup2kp1+p2.Similar to the asymptotic OP of the primary network, we can easily obtain the asymptotic OP of the secondary network as(49)Pouts,∞=αsrγ¯sku,mru>1,mrd>1,αsrγ¯sku+ΛrumrukmruΓmru+1,mru=1,mrd>1,αsrγ¯sku+ΛrdmrdkmrdΓmrd+1,αsrγ¯sku+ΛrumrukmruΓmru+1+ΛrdmrdkmrdΓmrd+1,mrd=1,mru>1,mru=mrd=1.In addition, the DO of the secondary network isDOs=min1,mru,mrd.


## 5. Numerical Results

In this section, numerical results are provided to prove the correctness of our theoretical derivation. The simulation tool is MATLAB 2019a. Besides, the impacts of key parameters are revealed. Particularly, we setγ¯sr=γ¯ru=γ¯rd=γ¯, ω=3, and I=20 [3]. Moreover, the system and channel parameters of SR fading are provided in Table 1 [12, 36].Table 1
Simulation parameters.
Parameter nameParameter valueSatelliteGEOCarrier frequency2 GHz3 dB angle0.4°Maximal satellite gain48 dBMaximal relay gain4 dBCarrier bandwidth15 MHzNoise temperature300°Variance of AWGN1Frequent heavy shadowing (FHS)1,0.063,0.0007Average shadowing (AS)5,0251,0.279Infrequent light shadowing (ILS)10,0.158,1.29Figure3 illustrates that the achievable rates versus different target rates of the primary system with setting mru=mrd=2, dru=drd=2m, and Rthd=0.5bps/Hz. It can be observed that our theoretical analysis is consistent with the Monte Carlo (MC) simulations, which demonstrates the accuracy of the analytical results. We can find that the achievable rate of the primary network increases with γru and reaches a fixed value, namely, Rthu. Noting that when Ru achieves Rthu, Rd begins to increase. This is due to the fact that if the primary network cannot correctly decode its expected signal, a secondary transmitter allocates all power to the primary network. Besides, Rd decreases with stricter target rates. In a high SNR regime, it can be clearly found that the achievable rate of the secondary network with our method is higher than that with the method in [6], which proves that our method is superior to the fixed power allocation method. Moreover, in a low SNR regime, the achievable rate of the secondary network is lower than that in [6]; it is because our method should ensure the QoS of the primary network.Figure 3
The achievable rates versus different target rates of the primary system.Figure4 depicts that OP versus different SR fading parameters with mru=mrd=2, dru=drd=2m, and Rthu=Rthd=0.5bps/Hz. It can be observed that the primary network outperforms the secondary network. This is because we give higher priority to the primary network in the optimal design for ensuring its QoS. Moreover, the curves in Figure 3 display that lighter SR channel fading can lead to better system performance.Figure 4
OP versus different SR fading parameters.Figure5 shows that OP versus different Nakagami-m fading parameters with setting dru=drd=2m and Rthu=Rthd=0.5bps/Hz. We can also observe that the performance of the primary network is superior to that of the secondary network. Besides, the larger Nakagami-m fading parameters mean lighter channel fading, which can improve the system performance.Figure 5
OP versus different Nakagami-m fading parameters.Figure6 plots the OP versus different distances from R to U with mru=mrd=2 and Rthu=Rthd=0.5bps/Hz,AS. It can be clearly seen that whether it is the primary network or the secondary network, the increase in the distance between the secondary transmitter R and the primary user U improves the OP of the system. It is not difficult to understand that longer distance degrades the communication quality.Figure 6
OP versus different distances fromR to U.However, in Figure7, which illustrates the OP versus different distances from R to D with mru=mrd=2 and Rthu=Rthd=0.5bps/Hz,AS, the distance between the secondary transmitter and the secondary user only affects the OP of the secondary network. It indicates that the OP of the primary network is not related to drd, while that of the secondary network will rise with the increase in both dru and drd.Figure 7
OP versus different distances fromR to D.Figure8 depicts the OP versus different target rates with setting mru=mrd=2 and dru=drd=2m,AS. We can find that the larger target rate of the primary network will increase the OP of both the primary network and the secondary network. It can be explained by the fact that a larger target rate means higher communication requirements, which is more difficult to achieve. Besides, the stricter target rate of the secondary network only reduces the system performance of the secondary network. Like drd, Rthd has no effect on the performance of the primary network.Figure 8
OP versus different target rates.


## 6. Conclusions

In this paper, we established an overlay CISTRN with the NOMA scheme, where the secondary network multiplexed spectrum resources by accessing the spectrum of the primary network. In exchange, the secondary network helped the information transmission of the primary network as a relay. To improve the system performance of the secondary network, we jointly optimized the decoding order and power allocation, where the QoS of the primary network has been ensured. In addition, we derived the exact OP of both the primary network and secondary network. To get further insights, the asymptotic OPs and DOs were provided. From the theoretical results, it can be observed that when the channel coefficient of the high-priority user is higher than that of the low-priority user as well as a fixed value, the system will decode the message of the low-priority user first, which will make the system have better performance. This conclusion is diverse from great majority of existing works on a NOMA-based system. Finally, our theoretical results were validated by simulation results, and the impacts of key system parameters on optimal design as well as system performance were revealed.



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