# Energy Efficiency Optimization Algorithm of CR-NOMA System Based on SWIPT

Authors: Zhen Wang; Qi Zhu
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

This paper proposes a system EE (energy efficiency) optimization algorithm based on OPS (On-off Power Splitting) strategy for SWIPT (Simultaneous Wireless Information and Power Transfer) two-way relay assisted CR-NOMA (Cognitive Radio Non-Orthogonal Multiple Access) network. The system capacity expression of the secondary users with the OPS strategy is derived. Under the constraints of harvested energy and quality of service (QoS) of the users, the optimization problem with the goal of maximizing energy efficiency is constructed. In this paper, the NP-Hard problem is transformed into three subproblems about relay power, NOMA coefficient, and segmentation coefficient, which are solved by golden section algorithm, monotonicity decision function, and genetic algorithm. The simulation results show that compared with the PS (Power Splitting) and TS (Time Switching) strategies, the OPS strategy can significantly improve the transmission energy efficiency of the system.


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## Body

## 1. Introduction

Devices in energy-constrained wireless communication systems, such as wireless sensor networks, wireless positioning networks, and the Internet of Things are mostly battery-powered, and the limitations of battery-powered capacity and inconvenient replacement greatly limit the performance of the system [1]. Energy harvesting is an advanced technology that has emerged in recent years. By collecting renewable resources in the surrounding environment to provide nodes with the necessary working energy and extending the survival time of energy-constrained wireless networks, the goal of achieving green communications is achieved [2]. In addition to harvesting energy from renewable energy sources such as solar energy, wind energy, and geothermal energy, there are near-field power transmission using inductive, capacitive, or resonant coupling, and far-field wireless power transmission via RF (radio frequency), wireless powering of devices using near-field Inductive Power Transfer, has become a reality with several commercially available products and standards. However, its range is severely limited (less than one meter). On the other hand, far-field Wireless Power Transfer (WPT) via RF (as in wireless communication) could be used over longer ranges, which is the main focus of this paper [3]. Due to the advantages of low cost and stable power, WPT will be widely used in the Internet of Things and various types of mobile terminals and has attracted widespread attention in the academic and industrial circles.SWIPT is a product of the combination of wireless energy transmission and wireless information transmission (WIT), which can realize energy harvesting and information transmission at the same time. Nasir et al. [4] proposed two energy harvesting strategies for time switching and power splitting for energy harvesting relay networks and derived the throughput expression of the system. Nasir et al. [5] proposed continuous time acquisition strategy and discrete time acquisition strategy for TS strategy and compared the system outage probability of the two strategies by analyzing the probability of system interruption. Zhou et al. [6] designed a universal receiver that separates the received signals with adjustable power ratios for energy harvesting and information decoding and derives the rate-energy ratio of time switching, static power splitting, and on-off power splitting. Huang and Larsson [7] proposed a power control algorithm for SWIPT-based Orthogonal Frequency Division Multiplex (OFDM) networks to optimize system throughput in this scenario. Zhou et al. [8] studied the design of the SWIPT acquisition strategy in the downlink multiuser OFDM system; TS was used at the receiver for time-division multiplexed information transmission; PS was used at the receiver for OFDM-based information transmission, and the segmentation coefficients of the two strategies are optimized.NOMA allows multiple users to share the same channel resources in the same cell at the same time, which makes it possible to obtain higher spectral efficiency than traditional orthogonal multiple access (OMA) technology. Zaidi et al. [9] deduced the expressions of user’s outage probability and system throughput for the NOMA system based on SWIPT to evaluate the performance of the system. Bariah et al. [10] deduced the bit error rate expression of all users under PS strategy for the cooperative relay NOMA network based on SWIPT. Tang et al. [11] aimed at the joint power allocation problem in the NOMA system based on PS strategy. In order to simplify the problem, the energy harvested by the user was converted into the throughput, and the sum of the converted throughput and the user’s transmission rate was defined as the new objective function, so [11] can give the optimization algorithm for user power and segmentation coefficient. Hedayati and Kim [12] aimed at the joint power allocation problem in the SWIPT-NOMA system based on the TS strategy and proposed an energy efficiency optimization algorithm that satisfies the requirements of transmit power limit, harvested energy, and quality of service. Tang [13] aimed at the joint power allocation problem in the SWIPT-NOMA system based on the TS strategy and proposed an energy efficiency optimization algorithm that satisfies the requirements of transmit power limit, harvested energy, and quality of service. Chang [14] investigated resource allocation algorithm design for an OFDM-based NOMA system empowered by wireless power transfer. Jameel [15] employed deep learning-based optimization to solve the problem of power distribution in SWIPT-NOMA network.Cognitive radio technology can improve the spectrum utilization of the system by sharing the spectrum between different systems. In [16], a closed-form expression of the outage probability was derived for CR-NOMA networks based on SWIPT. Yang et al. [17] aimed at the SWIPT-based NOMA network, respectively, under the two strategies of fixed allocation and optimized allocation of the NOMA coefficient, and the user’s outage probability was derived. Wang [18] derived the approximate expression of energy efficiency based on SWIPT-based CR-NOMA system and gave the relay power and time slot allocation scheme with the energy efficiency maximization as the optimization goal. Zhao et al. [19] proposed a two-way relay CR-NOMA system based on PS strategy and gave an optimization algorithm with energy efficiency as the optimization goal under the constraints of harvested energy and quality of service, but it did not consider the two typical strategies of TS and OPS.This paper proposes a system energy efficiency optimization algorithm using the OPS strategy for a two-way relay CR-NOMA system based on SWIPT. The main innovations of this paper are as follows:(i)
The system capacity expressions with the TS strategy and the OPS strategy are derived, respectively. Under the constraints of harvested energy and quality of service, an optimization problem with the goal of maximizing energy efficiency is constructed.(ii)
The objective function is a multiobjective optimized NP-Hard problem. In this paper, the original problem is transformed into three subproblems about relay power, NOMA coefficient, and segmentation coefficient, which are solved by the golden section algorithm, the monotonicity determination method of the function, and the genetic algorithm; then, the solutions of three subproblems are jointly optimized by alternating iterative algorithm.(iii)
The simulation results show that compared with the PS and TS strategies, the OPS strategy can significantly improve the transmission energy efficiency of the system. Since the OPS strategy can control time division and power division at the same time, thereby ensuring that the system completes the communication process with the lowest energy requirements, OPS strategy is better than PS and TS strategy when the power limit of different users and the distance between the near user and the relay are different.The following content of this article is arranged as follows. The system model of the SWIPT-based two-way relay assisted CR-NOMA network is presented in Section2. We describe the optimization problem with the goal of maximizing energy efficiency and give the optimization algorithm in Section 3. The simulation results and analyses are given in Section 4. Finally, we draw our conclusions in Section 5.


## 2. System Model

The system model of a two-way relay assisted CR-NOMA system in this paper is shown in Figure1 [19], which includes a primary transmitter (PT), a primary receiver (PR), a two-way relay (TWR), and four secondary users of RF energy harvesting ability, among them, SU1 and SU3 and SU2 and SU4 are each a pair of sending and receiving users, and SU1 and SU3 are closer to TWR and SU3 and SU4 are farther from TWR.Figure 1
System model of two-way relay CR-NOMA.The communication process of the system is divided into two phases. In the first phase, four secondary users send signals to TWR at the same time. TWR receives and decodes the mixed signal in turn, which is subject to interference from the primary transmitter. In the second phase, TWR forwards the superimposed signalsb1Prx1+b2Prx2 and b3Prx3+b4Prx4 to the two sets of NOMA secondary users SU1 and SU2 and SU3 and SU4 with a transmit power of Pr and causes interference to the primary receiver, where xi is the signal that TWR forwards to the secondary user SUi and bi is the power allocation coefficient in NOMA. Because SU2 and SU4 are remote users, in order to ensure the fairness of NOMA, b1<b2 and b3<b4, which means x1 and x3 are weak signals and x2 and x4 are strong signals. In the second phase, the secondary user receives and decodes the superimposed signal forwarded by the TWR, which completes the collection of the RF signal energy sent by the TWR at the same time. Channel state information (CSI) is assumed to be perfectly known at all terminals. Note that the CSI at the receivers can be obtained from the channel estimation of the downlink pilots, while CSI at the transmitter can be acquired via uplink channel estimation in the TDD mode [13].


### 2.1. Time Switching

The system model of TS is shown in Figure2, where the front 1−λT/2 is the first phase, and the rest is the second phase, and T is the length of a communication process of the system. TS realizes two processes of energy collection and information transmission by dividing time, namely, energy harvesting is completed during the front λT and information transmission is completed during the rear 1−λT/2.Figure 2
System model of time switching.For ease of calculation, it is assumed that the channel gains of the two groups of users are completely symmetrical. We take SU1 and SU2 as examples, the received signals and harvested energy are given by(1)yiTS=hRib1Prx1+b2Prx2+ni,EiTS=ηλPrhRi2T,where hRi is the link channel gain from TWR, η is the efficiency of energy collection, ni is Gaussian white noise (AWGN) at SUi, and the power is σ2. In practice, EH circuits usually result in a nonlinear end-to-end wireless power transfer [20, 21], and for the convenience of calculation, according to [13, 19], this paper assumes power conversion efficiency is independent of the input power level of the EH circuit and employs a linear energy harvesting model.SU1 is a near user and needs to decode the strong signal first and then the weak signal required. The information rates of the two signals are given by(2)R1⟶2TS=1−λ2log21+b2PrhR12σ2+b1PrhR12,R1TS=1−λ2log21+b1Prσ2hRi2.SU2 can directly decode the required strong signal, and its rate is expressed as follows:(3)R2TS=1−λ2log21+b2PrhR22σ2+b1PrhR22.All the energy harvested by the secondary users is used to complete the signal transmission in the first phase. The maximum average transmission power is obtained by(4)PiTS=EiTS1−λ/2T=2λ1−ληPrhRi2.The signal rate received by the primary receiver in the second phase is given by(5)RpTS=1−λ2log21+PthPP2PrhRP2+σP2,where hRP and hPP are the channel gains between TWR and PR and the primary transmitter and primary receiver in the second phase, respectively, and np is Gaussian white noise at the primary receiver in the second phase, its power is σP2.


### 2.2. On-Off Power Splitting

The system model of OPS is shown in Figure3. The division of time slots is the same as that of TS, the first 1−λT/2 is the first phase, and the rest is the second phase; the difference is that OPS combines the characteristics of PS. During the rear 1−λT/2 in the second phase, OPS divides Pr into two parts ρPr and 1−ρPr; ρPr is used for energy harvesting and 1−ρPr is used to complete the information transmission between TWR and SU.Figure 3
System model of on-off power splitting.The signals received by the secondary users and the energy harvested are given by(6)yiOPS=1−ρhRib1Prx1+b2Prx2+ni,EiOPS=λ+ρ1−λ2ηPrhRi2T.The information rates of the strong and weak signals at SU1 are obtained by(7)R1⟶2OPS=1−λ2log21+1−ρb2PrhR12σ2+1−ρb1PrhR12,R1OPS=1−λ2log21+1−ρb1PrhR12σ2.The strong signal information rate at SU2 is expressed as follows:(8)R2OPS=1−λ2log21+1−ρb2PrhR22σ2+1−ρb1PrhR22.The maximum transmit power of the secondary user is given by(9)PiOPS=EiOPS1−λ/2T=2λ1−λ+ρηPrhRi2.The information rate of the primary receiver is the same as the rate under the TS strategy, which is given by(10)RpOPS=RpTS=1−λ2log21+PthPP2PrhRP2+σP2.


## 2.1. Time Switching

The system model of TS is shown in Figure2, where the front 1−λT/2 is the first phase, and the rest is the second phase, and T is the length of a communication process of the system. TS realizes two processes of energy collection and information transmission by dividing time, namely, energy harvesting is completed during the front λT and information transmission is completed during the rear 1−λT/2.Figure 2
System model of time switching.For ease of calculation, it is assumed that the channel gains of the two groups of users are completely symmetrical. We take SU1 and SU2 as examples, the received signals and harvested energy are given by(1)yiTS=hRib1Prx1+b2Prx2+ni,EiTS=ηλPrhRi2T,where hRi is the link channel gain from TWR, η is the efficiency of energy collection, ni is Gaussian white noise (AWGN) at SUi, and the power is σ2. In practice, EH circuits usually result in a nonlinear end-to-end wireless power transfer [20, 21], and for the convenience of calculation, according to [13, 19], this paper assumes power conversion efficiency is independent of the input power level of the EH circuit and employs a linear energy harvesting model.SU1 is a near user and needs to decode the strong signal first and then the weak signal required. The information rates of the two signals are given by(2)R1⟶2TS=1−λ2log21+b2PrhR12σ2+b1PrhR12,R1TS=1−λ2log21+b1Prσ2hRi2.SU2 can directly decode the required strong signal, and its rate is expressed as follows:(3)R2TS=1−λ2log21+b2PrhR22σ2+b1PrhR22.All the energy harvested by the secondary users is used to complete the signal transmission in the first phase. The maximum average transmission power is obtained by(4)PiTS=EiTS1−λ/2T=2λ1−ληPrhRi2.The signal rate received by the primary receiver in the second phase is given by(5)RpTS=1−λ2log21+PthPP2PrhRP2+σP2,where hRP and hPP are the channel gains between TWR and PR and the primary transmitter and primary receiver in the second phase, respectively, and np is Gaussian white noise at the primary receiver in the second phase, its power is σP2.


## 2.2. On-Off Power Splitting

The system model of OPS is shown in Figure3. The division of time slots is the same as that of TS, the first 1−λT/2 is the first phase, and the rest is the second phase; the difference is that OPS combines the characteristics of PS. During the rear 1−λT/2 in the second phase, OPS divides Pr into two parts ρPr and 1−ρPr; ρPr is used for energy harvesting and 1−ρPr is used to complete the information transmission between TWR and SU.Figure 3
System model of on-off power splitting.The signals received by the secondary users and the energy harvested are given by(6)yiOPS=1−ρhRib1Prx1+b2Prx2+ni,EiOPS=λ+ρ1−λ2ηPrhRi2T.The information rates of the strong and weak signals at SU1 are obtained by(7)R1⟶2OPS=1−λ2log21+1−ρb2PrhR12σ2+1−ρb1PrhR12,R1OPS=1−λ2log21+1−ρb1PrhR12σ2.The strong signal information rate at SU2 is expressed as follows:(8)R2OPS=1−λ2log21+1−ρb2PrhR22σ2+1−ρb1PrhR22.The maximum transmit power of the secondary user is given by(9)PiOPS=EiOPS1−λ/2T=2λ1−λ+ρηPrhRi2.The information rate of the primary receiver is the same as the rate under the TS strategy, which is given by(10)RpOPS=RpTS=1−λ2log21+PthPP2PrhRP2+σP2.


## 3. Problem Formulation

Assume that the channel gains of the two groups of users are completely symmetrical, that is,hR1=hR3 and hR2=hR4, and the system capacity is the sum of the information rates of the four secondary users, which is given by(11)Rtot=2R1+R2.Define the energy consumption of the system as the difference between the energy consumed by the system and the energy collected by the secondary user, which is expressed as follows:(12)Etot=PlT+PrT2−2E1+E2,where T2 is the length of the second phase and Pl is the circuit power loss.This paper needs to maximize the energy efficiency of the system by optimizing the relay power, the NOMA coefficient, and the division coefficient. The optimization problem can be expressed as follows:(13)maxPr,b3,ρ,ληEE=RtotEtots.t.RP≥ithR1⟶2≥rthPtot≥PsPr≤Pmax0≤ρ≤1,0≤λ≤10≤b1≤0.5,b2=1−b1.Among them,ith is the information rate threshold of the primary receiver, rth is the minimum information rate that satisfies the signal-to-interference-and-noise ratio of the strong signals decoded by SU1 and SU3, Pmax is the maximum threshold of the relay transmission power, and Ptot is the sum of the transmission power of the secondary users, its threshold is Ps.This problem is a multivariable optimized NP-Hard problem, which has high complexity and cannot be solved directly. As stated in [22], for any optimization problems with multiple variables, we can analyze and solve the problem over some variables, regarding the rest as constants; then, solve the problem over the remaining variables. Therefore, we will separate Pr, b3, and ρ or λ when developing the optimization algorithm so as to overcome the difficulty. Furthermore, this paper decomposes it into three optimization subproblems about relay power, NOMA coefficient, and partition coefficient, regarding the remaining variables as constants when solving the suboptimization problem of each variable, which are obtained by(14)SP1:maxPrηEE=RtotEtots.t.RP≥ithR1⟶2≥rthPtot≥PsPr≤Pmax,(15)SP2:maxb1ηEE=RtotEtots.t.R1⟶2≥rth0≤b1≤0.5,(16)SP3:maxρ,ληEE=RtotEtots.t.RP≥ithR1⟶2≥rthPtot≥Ps0≤ρ,λ≤1.


### 3.1. Optimization of TWR Transmission Power

Because the expressions ofηEE, RP, R1⟶2, and Ptot of the TS and OPS strategies are inconsistent, the constraints and objective functions of Pr for both strategies will be given separately:(17)maxPrηEETS=1−λlog21+b1PrhR12/σ2+1−λlog21+b2PrhR22/σ2+b1PrhR22PlT+Pr1+λ/2T−2ληPrhR12+hR22T,1−λ2log21+PthPP2PrhRP2+σP2≥ith,1−λ2log21+b2PrhR12σ2+b1PrhR12≥rth,4λ1−ληPrhR12+hR22≥Ps,Pr≤Pmax,maxPrηEEOPS=1−λlog21+1−ρb1PrhR12/σ2+1−λlog21+1−ρb2PrhR22/σ2+1−ρb1PrhR22PlT+Pr1+λ/2T−2ληPrhR12+hR22T−1−λρηPrhR12+hR22T,1−λ2log21+PthPP2PrhRP2+σP2≥ith,1−λ2log21+1−ρb2PrhR12σ2+1−ρb1PrhR12≥rth,4λ1−λ+2ρηPrhR12+hR22≥Ps,Pr≤Pmax.The fractional objective function makes the problem neither linear nor convex and thus difficult to solve straightforwardly. By analyzing the objective function, this paper gets the following proposition.Proposition 1.
The objective function in equation (14) is strictly quasi-concave function in the TWR transmission power Pr.Proof.
Take the objective function of TS strategy as an example, and its objective function can be written as follows:(18)ηEEPr=1−λlog21+b1PrhR12/σ2+log21+b2PrhR22/σ2+b1PrhR221+λ/2T−2ληhR12+hR22TPr+PlT=FUPrFLPr.
Since the remaining variables are all considered constant at this time,FLPr is a linear function with respect to Pr. FUPr is the concave function of Pr, to prove that, the second-order partial derivative of FUPr with respect to Pr is required. The first-order partial derivative of FUPr can be denoted as follows:(19)TS:∂FUPr∂Pr=1−λln2b1hR12σ2+b1PrhR12+hR22σ2+PrhR22−b1hR22σ2+b1PrhR22.
Furthermore, the second-order partial derivative ofFUPr can be obtained as follows:(20)TS:∂2FUPr∂Pr2=1−λln2−b1hR122σ2+b1PrhR122−hR24σ2+PrhR222+b1hR222σ2+b1PrhR222=1−λln21σ2/b1hR22+12−1σ2/b1hR12+12−hR24σ2+PrhR222.
SincehR12>hR22, the right side of the equation is all negative, ∂2FUPr/∂Pr2<0. Thus, FUPr is the concave function with respect to Pr. Therefore, the objective function is strictly quasi-concave function for Pr [22]. Furthermore, this paper can prove that the objective function is first monotonically nonincrease and then monotonically nondecrease. The proof can be easily obtained by ∂ηEEPr/∂Pr|Pr⟶0≤0 and ∂ηEEPr/∂Pr|Pr⟶∞>0 [23]. The process of proving the OPS strategy is similar to the above process and is omitted here.
According to Proposition1, an unique global optimal solution exists and the optimal point can be obtained by using the bisection method [22]. Since the Dinkelbach method [24] has been widely applied to solve nonlinear fractional optimization problem, we apply it to tackle our EE maximization problem. Particularly, we convert the fractional objective function into a subtractive form of numerator and denominator on the basis of the following proposition.Proposition 2.
ForFUPr≥0 and FLPr≥0, the maximum achievable EE q∗=ηEE∗ satisfies the following equation:(21)maxPr≥0FUPr−q∗FLPr=FUPr∗−q∗FLPr∗=0,where FUPr is the numerator of the objective function, FLPr is the denominator of the objective function, and q∗=FUPr∗/FLPr∗.Proof.
Please refer to [24] for the proof of Proposition 1.Proposition2 shows that the original complex fractional objective function can be transformed into an equivalent optimization problem with subtraction. After such a transformation, this paper presents an algorithm to solve the equivalent optimization problem that satisfies the conditions in Proposition 1, which can be concluded in Table 1.Table 1
Golden section algorithm for optimal relay power.
Algorithm: golden section algorithm of optimal relay power1. Initialize the remaining variables and constant termsη=80%, b1=0.5, λTS=0.3, λOPS=0.2, ρPS=0.5, ρOPS=0.5, T=1.2. Determine toleranceδ, β=0.618, α=0.382, the initial interval is a,b, where a=maxχ1,χ2, b=minχ3,Pmax.3. Whileb−a>δ:4.t1=a+αb−a, t2=a+βb−a, ηEE1=ηEEt1, ηEE2=ηEEt2.5. IfηEEt2<ηEEt1, then b=t2, t2=t1, ηEEt2=ηEEt1, t1=α+αb−a, ηEE1=ηEEt1,6. Elsea=t1, t1=t2, ηEEt1=ηEEt2, t2=α+βb−a, ηEE2=ηEEt2,7. End,t∗=t1+t2/2.8. The optimal valuet∗ is the optimal relay power Pr∗ that meets the maximum energy efficiency.In order to ensure the feasibility of the algorithm in Table2, it is necessary to prove its convergence. First, this paper proves that qn will increase in each iteration. Note that q∗ satisfies the optimal condition in Proposition 2, i.e., FUPr∗−q∗FLPr∗=φq∗=0. Suppose that Prn is the optimal power allocation, n is the number of iterations, and qn≠q∗ and qn+1≠q∗ represent the corresponding optimal energy efficiency. It has been proved in [23] that φqn>0 and φqn+1>0 hold. In addition, in the algorithm of Table 2, qn+1=FUPrn/FLPrn. So, this article can obtain the following expression:(22)φqn=FUPrn−qnFLPrn=qn+1FLPrn−qnFLPrn=qn+1−qnFLPrn.Table 2
Iterative algorithm based on the Dinkelbach method.
Algorithm: iterative algorithm based on the Dinkelbach method1. Initializen=0, Prn=1 and qn=0, set the stopping criterion ε;2. WhileFUPrn−qnFLPrn>ε3. For givenqn, solve (20) to obtain the transmission power Prn;4. Letn=n+1 and qn=FUPrn−1/FLPrn−1;5. The optimal energy efficiencyηEE∗=q∗ is qn.SinceFLPrn represents the energy consumed, there is always FLPrn>0, so qn+1>qn. Then, we prove that qn converges to the optimal value q∗ if the number of iterations is large enough. Since there is a limit value q∗ and qn will increase every iteration, when the number of iterations n is large enough, qn+1⟶qn and φqn+1⟶0 hold, and the optimality condition in Proposition 1 can be met.After the above proof, the suboptimization problem SP1 can be converted into the following equation:(23)maxPrFUPr−qFLPr,maxχ1,χ2≤Pr≤minχ3,Pmax,χ1TS=σ2RthTSb2hR12−b1RthTShR12,χ2TS=1−λPs4ληhR12+hR22,χ3TS=PthPP2−σp2IthTShRP2IthTS,χ1OPS=σ2RthOPS1−ρb2hR12−b1RthOPShR12,χ2OPS=1−λPs4λ+2−2λρηhR12+hR22,χ3OPS=PthPP2−σp2IthOPShRP2IthOPS,where RthOPS=RthTS=22rth/1−λ−1 and IthOPS=IthTS=22ith/1−λ−1.Considering the remaining two variables as constants, the original problem becomes a problem of finding the maximum value in the specified interval. This paper uses the golden section algorithm to find the optimal relay power. However, the golden section method can only be used to solve the extreme points in the single-peak interval. Therefore, it is necessary to prove that the objective function is a convex function.According to Proposition1, FUPr is the concave function with respect to Pr. Since FLPr is a linear function of Pr, its second-order partial derivative is 0. Therefore, the positive and negative properties of the second derivative of FUPr−qFLPr are consistent with FUPr. We can conclude that, for a fixed parameter q, the transformation form of objective function FUPr−qFLPr is strictly concave in Pr.After proving that the objective function is a concave function, this paper can use the golden section algorithm to satisfy the value ofPr with the greatest energy efficiency. The specific algorithm steps are shown in Table 1.


### 3.2. Optimization of NOMA Coefficient

In SP2,b1 is included only in the Rtot term, so the original question SP2 can be simplified as follows:(24)maxb1Rtots.t.R1⟶2≥rth0≤b1≤0.5.We simplifyRtot according to b1+b2=1 and take RtotOPS as an example, which is given by(25)RtotOPS=1−λlog21+1−ρb1PrhR12σ2+1−λlog21+1−ρb2PrhR22σ2+1−ρb1PrhR22,=1−λlog2σ2+1−ρb1PrhR12−1−λlog2σ2+1−ρb1PrhR22−1−λlog2σ2+1−λlog2σ2+1−ρPrhR22.Since the last term in equation (25) is a constant term independent of b1, the partial derivative of RtotOPS for b1 is expressed as follows:(26)∂RtotOPS∂b1=1−λPrhR12ln2σ2+1−ρb1PrhR12−1−λPrhR22ln2σ2+1−ρb1PrhR22=1−λln21/σ2/PrhR12+1−ρb1−1σ2/PrhR22+1−ρb1.SincehR12>hR22, σ2/hR12<σ2/hR22, and ∂RtotOPS/∂b1>0, so RtotOPS is a monotonically increasing function of b1, and the objective function takes the maximum value when b1 is maximum. Similarly, the monotonicity of RtotTS in TS strategy also exists.According toR1⟶2≥rth in SP2, the constraint on b1 can be obtained as follows:(27)b1OPS≤PrhR12−σ2RthOPS1−ρ1+RthOPSPrhR12.Therefore, the optimal values ofb1 in the TS and OPS strategies are given by(28)b1TS∗=minPrhR12−σ12RthTS1+RthTSPrhR12,0.5,b1OPS∗=minPrhR12−σ12RthOPS1−ρ1+RthOPSPrhR12,0.5.


### 3.3. Optimization of Partition Coefficient

For the TS strategy, SP3 is a univariate optimization problem aboutλ, and for the OPS strategy, SP3 is a two-variable joint optimization problem about ρ and λ. Referring to the process of solving the optimal transmission power in this paper, this paper also applies the Dinkelbach algorithm to the suboptimization problem SP3, and equation (16) can be converted into the following expression:(29)maxρ,λFUρ,λ−uFLρ,λTS:ps4ηPrhR12+hR22+Ps≤λ≤min1−2rthlog21+b2PrhR12/σ2+b1PrhR12,1−2ithlog21+PthPP2/PrhRP2+σP2,OPS:1−λ2log21+1−ρb2PrhR12σ2+1−ρb1PrhR12≥rth4λ1−λ+2ρηPrhR12+hR22≥Ps0≤ρ≤1,0≤λ≤1−2ithlog21+PthPP2/PrhRP2+σP2,where FUρ,λ=Rtot, FLρ,λ=Etot, and u=FUρ,λ/FLρ,λ.The specific steps for applying Dinkelbach algorithm in this section are similar to Table2, un=FUρn−1,λn−1/FLρn−1,λn−1, and are not given here. The proof process of convergence is also given above and the proof process here is similar to it, so it will not be repeated.Genetic algorithm can solve multivariate optimization problems such as equation (29). This paper uses genetic algorithms to solve these two types of problems. The steps of using genetic algorithm to solve the optimization problem of partition coefficients are as follows. First, M random numbers are generated within the constraint interval satisfied by the partition coefficients of the TS and OPS strategies given in equation (29), and they are converted into a series of “chromosomes” by binary coding. These randomly generated chromosomes are composed of individuals’ initial population. Second, let the objective function in SP3 be the fitness function for evaluating the chromosomes, select the highest fitness individual as the best individual, and select M/2 pairs of maternal chromosomes according to the ratio of the fitness of each chromosome to the total fitness of all chromosomes in the population. Third, we randomly select a codeword on the M/2 pair of maternal chromosomes, perform crossover and mutation operations based on the crossover probability pc and mutation probability pm to generate M new individuals to form a new population, and update the optimal individual according to the individual fitness in the new population. Crossover is the exchange of codewords corresponding to two chromosomes, and mutation is negation of the codeword. Repeat the above process until the number of iterations reaches the preset algebra G, and the optimal individual at this time is the optimal solution of SP3. The specific implementation process is shown in Table 3.Table 3
Genetic algorithm of optimal partition coefficient.
Algorithm: genetic algorithm with optimal partition coefficient1. Initialize the remaining variables and constant termsη=80%, b1=0.5, Pr=1, T=1.2. Set the number of individuals in the genetic algorithm population M, maximum evolution algebraG, cross probability Pc, mutation probability Pm, initial algebra g = 1.3. Randomly generate M individuals who meet the constraints, calculate the fitness of the initial population according to the objective function and select the individual with the highest fitness as the best individual.4. Whileg < G+1:5. Select M/2 to cross the mother chromosomes according to the probability of crossing to generate M new individuals,6. Make these M new individuals mutate according to the set mutation probability,7. Let the M chromosomes be a new population and update the most adaptive individual in the population,8. Letg = g + 1.9. The best individual found is the segmentation coefficientsρ∗ and λ∗ that satisfy the maximum energy efficiency


### 3.4. Alternate Iteration Algorithm for Joint Optimization

After solving the three suboptimization problems, this paper uses an alternating iterative algorithm to jointly optimize the optimal values of the three suboptimization problems. The specific steps are shown in Table4.Table 4
Joint optimal energy efficiency optimization algorithm.
Algorithm: Joint optimal energy efficiency optimization algorithm1. Initialize the remaining variables and constant termsη=80%, b11=0.5, λTS1=0.3, λOPS1=0.2, ρOPS1=0.5, T=1 and calculate ηEE12. Leti=1, ηEE0=ηEE1−κ.3. WhileηEEi−ηEEi−1≥κ:4. Leti=i+1,5. Letq0=ηEEi−1, substitute the remaining variables b1i−1, λi−1 and ρi−1 to solve the optimization problem (25) to obtain Pri and qi∗ according to Tables 2 and 1.6. Letu0=qi∗, substitute the remaining variables b1i−1 and Pri to solve the optimization problem (28) to obtain λi, ρi and ui∗ according to Tables 3 and 2 with the iteration conditions replaced with un=FUρn−1,λn−1/FLρn−1,λn−1.7. Substituteλi, ρi and Pri into equation (28) to obtain b1i;8. CalculateηEEi according b1i, λi, ρi and Pri;9. End and get the bestPr, b1, λ, ρ and ηEE.It is also provable that the energy efficiencyηEE increases in each iteration shown in Table 4 and the convergence of the complete algorithm exists. In each iteration, since the Dinkelbach algorithm is used to solve the transmission power problem and the partition coefficient problem, its convergence has been proved above, and the optimal energy efficiency obtained after solving the optimal transmission power is taken as the initial energy efficiency for solving the optimal partition coefficient, and the objective function is a monotonically increasing function of b1, which ensures that the value of ηEE will increase after each iteration.In addition, Proposition2 has proved that the objective function in the conversion form equation (21) of SP1 is a concave function about Pr. The local optimum of the unimodal function is the global optimal value; the objective function of SP2 is the monotone increasing function of b1; the global optimal solution is at the endpoint; the solution method of SP3 is genetic algorithm, it is a global optimization algorithm [25], so the joint optimal energy efficiency optimization algorithm in this paper can guarantee the global optimality of the obtained solution.This paper also provides the complexity analysis of the joint energy efficiency optimization algorithm. The solution process of SP1 and SP3 both use the Dinkelbach method to convert the objective function into a subtraction form; the computational complexity for the Dinkelbach method-based algorithm with stopping criteriaε is Ologε−1 [24]. Moreover, we have proved that the objective function of the subtraction form in SP1 is a concave function; the computational complexity of solving the concave function is Ologε−1 [26], where ε is the error tolerance for algorithm termination. The optimization process of the NOMA coefficient in SP2 only requires the partial derivative of b1 for the objective function, whose computational complexity is O1. The solution process of SP3 employs genetic algorithm; the computational complexity of the genetic algorithm is OG×M×N, which is related to its maximum evolutionary algebra G, the number of individuals in the population M, and the number of variables N [25]. The joint energy efficiency optimization algorithm in this paper nests the Dinkelbach algorithm twice in each iteration, including the golden section method and the genetic algorithm, so the overall computational complexity is Ologκ−1logε−1logδ−1+G×M×N, κ, ε, and δ are the error tolerance of the joint energy efficiency optimization algorithm, Dinkelbach algorithm, and golden section method, respectively.


## 3.1. Optimization of TWR Transmission Power

Because the expressions ofηEE, RP, R1⟶2, and Ptot of the TS and OPS strategies are inconsistent, the constraints and objective functions of Pr for both strategies will be given separately:(17)maxPrηEETS=1−λlog21+b1PrhR12/σ2+1−λlog21+b2PrhR22/σ2+b1PrhR22PlT+Pr1+λ/2T−2ληPrhR12+hR22T,1−λ2log21+PthPP2PrhRP2+σP2≥ith,1−λ2log21+b2PrhR12σ2+b1PrhR12≥rth,4λ1−ληPrhR12+hR22≥Ps,Pr≤Pmax,maxPrηEEOPS=1−λlog21+1−ρb1PrhR12/σ2+1−λlog21+1−ρb2PrhR22/σ2+1−ρb1PrhR22PlT+Pr1+λ/2T−2ληPrhR12+hR22T−1−λρηPrhR12+hR22T,1−λ2log21+PthPP2PrhRP2+σP2≥ith,1−λ2log21+1−ρb2PrhR12σ2+1−ρb1PrhR12≥rth,4λ1−λ+2ρηPrhR12+hR22≥Ps,Pr≤Pmax.The fractional objective function makes the problem neither linear nor convex and thus difficult to solve straightforwardly. By analyzing the objective function, this paper gets the following proposition.Proposition 1.
The objective function in equation (14) is strictly quasi-concave function in the TWR transmission power Pr.Proof.
Take the objective function of TS strategy as an example, and its objective function can be written as follows:(18)ηEEPr=1−λlog21+b1PrhR12/σ2+log21+b2PrhR22/σ2+b1PrhR221+λ/2T−2ληhR12+hR22TPr+PlT=FUPrFLPr.
Since the remaining variables are all considered constant at this time,FLPr is a linear function with respect to Pr. FUPr is the concave function of Pr, to prove that, the second-order partial derivative of FUPr with respect to Pr is required. The first-order partial derivative of FUPr can be denoted as follows:(19)TS:∂FUPr∂Pr=1−λln2b1hR12σ2+b1PrhR12+hR22σ2+PrhR22−b1hR22σ2+b1PrhR22.
Furthermore, the second-order partial derivative ofFUPr can be obtained as follows:(20)TS:∂2FUPr∂Pr2=1−λln2−b1hR122σ2+b1PrhR122−hR24σ2+PrhR222+b1hR222σ2+b1PrhR222=1−λln21σ2/b1hR22+12−1σ2/b1hR12+12−hR24σ2+PrhR222.
SincehR12>hR22, the right side of the equation is all negative, ∂2FUPr/∂Pr2<0. Thus, FUPr is the concave function with respect to Pr. Therefore, the objective function is strictly quasi-concave function for Pr [22]. Furthermore, this paper can prove that the objective function is first monotonically nonincrease and then monotonically nondecrease. The proof can be easily obtained by ∂ηEEPr/∂Pr|Pr⟶0≤0 and ∂ηEEPr/∂Pr|Pr⟶∞>0 [23]. The process of proving the OPS strategy is similar to the above process and is omitted here.
According to Proposition1, an unique global optimal solution exists and the optimal point can be obtained by using the bisection method [22]. Since the Dinkelbach method [24] has been widely applied to solve nonlinear fractional optimization problem, we apply it to tackle our EE maximization problem. Particularly, we convert the fractional objective function into a subtractive form of numerator and denominator on the basis of the following proposition.Proposition 2.
ForFUPr≥0 and FLPr≥0, the maximum achievable EE q∗=ηEE∗ satisfies the following equation:(21)maxPr≥0FUPr−q∗FLPr=FUPr∗−q∗FLPr∗=0,where FUPr is the numerator of the objective function, FLPr is the denominator of the objective function, and q∗=FUPr∗/FLPr∗.Proof.
Please refer to [24] for the proof of Proposition 1.Proposition2 shows that the original complex fractional objective function can be transformed into an equivalent optimization problem with subtraction. After such a transformation, this paper presents an algorithm to solve the equivalent optimization problem that satisfies the conditions in Proposition 1, which can be concluded in Table 1.Table 1
Golden section algorithm for optimal relay power.
Algorithm: golden section algorithm of optimal relay power1. Initialize the remaining variables and constant termsη=80%, b1=0.5, λTS=0.3, λOPS=0.2, ρPS=0.5, ρOPS=0.5, T=1.2. Determine toleranceδ, β=0.618, α=0.382, the initial interval is a,b, where a=maxχ1,χ2, b=minχ3,Pmax.3. Whileb−a>δ:4.t1=a+αb−a, t2=a+βb−a, ηEE1=ηEEt1, ηEE2=ηEEt2.5. IfηEEt2<ηEEt1, then b=t2, t2=t1, ηEEt2=ηEEt1, t1=α+αb−a, ηEE1=ηEEt1,6. Elsea=t1, t1=t2, ηEEt1=ηEEt2, t2=α+βb−a, ηEE2=ηEEt2,7. End,t∗=t1+t2/2.8. The optimal valuet∗ is the optimal relay power Pr∗ that meets the maximum energy efficiency.In order to ensure the feasibility of the algorithm in Table2, it is necessary to prove its convergence. First, this paper proves that qn will increase in each iteration. Note that q∗ satisfies the optimal condition in Proposition 2, i.e., FUPr∗−q∗FLPr∗=φq∗=0. Suppose that Prn is the optimal power allocation, n is the number of iterations, and qn≠q∗ and qn+1≠q∗ represent the corresponding optimal energy efficiency. It has been proved in [23] that φqn>0 and φqn+1>0 hold. In addition, in the algorithm of Table 2, qn+1=FUPrn/FLPrn. So, this article can obtain the following expression:(22)φqn=FUPrn−qnFLPrn=qn+1FLPrn−qnFLPrn=qn+1−qnFLPrn.Table 2
Iterative algorithm based on the Dinkelbach method.
Algorithm: iterative algorithm based on the Dinkelbach method1. Initializen=0, Prn=1 and qn=0, set the stopping criterion ε;2. WhileFUPrn−qnFLPrn>ε3. For givenqn, solve (20) to obtain the transmission power Prn;4. Letn=n+1 and qn=FUPrn−1/FLPrn−1;5. The optimal energy efficiencyηEE∗=q∗ is qn.SinceFLPrn represents the energy consumed, there is always FLPrn>0, so qn+1>qn. Then, we prove that qn converges to the optimal value q∗ if the number of iterations is large enough. Since there is a limit value q∗ and qn will increase every iteration, when the number of iterations n is large enough, qn+1⟶qn and φqn+1⟶0 hold, and the optimality condition in Proposition 1 can be met.After the above proof, the suboptimization problem SP1 can be converted into the following equation:(23)maxPrFUPr−qFLPr,maxχ1,χ2≤Pr≤minχ3,Pmax,χ1TS=σ2RthTSb2hR12−b1RthTShR12,χ2TS=1−λPs4ληhR12+hR22,χ3TS=PthPP2−σp2IthTShRP2IthTS,χ1OPS=σ2RthOPS1−ρb2hR12−b1RthOPShR12,χ2OPS=1−λPs4λ+2−2λρηhR12+hR22,χ3OPS=PthPP2−σp2IthOPShRP2IthOPS,where RthOPS=RthTS=22rth/1−λ−1 and IthOPS=IthTS=22ith/1−λ−1.Considering the remaining two variables as constants, the original problem becomes a problem of finding the maximum value in the specified interval. This paper uses the golden section algorithm to find the optimal relay power. However, the golden section method can only be used to solve the extreme points in the single-peak interval. Therefore, it is necessary to prove that the objective function is a convex function.According to Proposition1, FUPr is the concave function with respect to Pr. Since FLPr is a linear function of Pr, its second-order partial derivative is 0. Therefore, the positive and negative properties of the second derivative of FUPr−qFLPr are consistent with FUPr. We can conclude that, for a fixed parameter q, the transformation form of objective function FUPr−qFLPr is strictly concave in Pr.After proving that the objective function is a concave function, this paper can use the golden section algorithm to satisfy the value ofPr with the greatest energy efficiency. The specific algorithm steps are shown in Table 1.


## 3.2. Optimization of NOMA Coefficient

In SP2,b1 is included only in the Rtot term, so the original question SP2 can be simplified as follows:(24)maxb1Rtots.t.R1⟶2≥rth0≤b1≤0.5.We simplifyRtot according to b1+b2=1 and take RtotOPS as an example, which is given by(25)RtotOPS=1−λlog21+1−ρb1PrhR12σ2+1−λlog21+1−ρb2PrhR22σ2+1−ρb1PrhR22,=1−λlog2σ2+1−ρb1PrhR12−1−λlog2σ2+1−ρb1PrhR22−1−λlog2σ2+1−λlog2σ2+1−ρPrhR22.Since the last term in equation (25) is a constant term independent of b1, the partial derivative of RtotOPS for b1 is expressed as follows:(26)∂RtotOPS∂b1=1−λPrhR12ln2σ2+1−ρb1PrhR12−1−λPrhR22ln2σ2+1−ρb1PrhR22=1−λln21/σ2/PrhR12+1−ρb1−1σ2/PrhR22+1−ρb1.SincehR12>hR22, σ2/hR12<σ2/hR22, and ∂RtotOPS/∂b1>0, so RtotOPS is a monotonically increasing function of b1, and the objective function takes the maximum value when b1 is maximum. Similarly, the monotonicity of RtotTS in TS strategy also exists.According toR1⟶2≥rth in SP2, the constraint on b1 can be obtained as follows:(27)b1OPS≤PrhR12−σ2RthOPS1−ρ1+RthOPSPrhR12.Therefore, the optimal values ofb1 in the TS and OPS strategies are given by(28)b1TS∗=minPrhR12−σ12RthTS1+RthTSPrhR12,0.5,b1OPS∗=minPrhR12−σ12RthOPS1−ρ1+RthOPSPrhR12,0.5.


## 3.3. Optimization of Partition Coefficient

For the TS strategy, SP3 is a univariate optimization problem aboutλ, and for the OPS strategy, SP3 is a two-variable joint optimization problem about ρ and λ. Referring to the process of solving the optimal transmission power in this paper, this paper also applies the Dinkelbach algorithm to the suboptimization problem SP3, and equation (16) can be converted into the following expression:(29)maxρ,λFUρ,λ−uFLρ,λTS:ps4ηPrhR12+hR22+Ps≤λ≤min1−2rthlog21+b2PrhR12/σ2+b1PrhR12,1−2ithlog21+PthPP2/PrhRP2+σP2,OPS:1−λ2log21+1−ρb2PrhR12σ2+1−ρb1PrhR12≥rth4λ1−λ+2ρηPrhR12+hR22≥Ps0≤ρ≤1,0≤λ≤1−2ithlog21+PthPP2/PrhRP2+σP2,where FUρ,λ=Rtot, FLρ,λ=Etot, and u=FUρ,λ/FLρ,λ.The specific steps for applying Dinkelbach algorithm in this section are similar to Table2, un=FUρn−1,λn−1/FLρn−1,λn−1, and are not given here. The proof process of convergence is also given above and the proof process here is similar to it, so it will not be repeated.Genetic algorithm can solve multivariate optimization problems such as equation (29). This paper uses genetic algorithms to solve these two types of problems. The steps of using genetic algorithm to solve the optimization problem of partition coefficients are as follows. First, M random numbers are generated within the constraint interval satisfied by the partition coefficients of the TS and OPS strategies given in equation (29), and they are converted into a series of “chromosomes” by binary coding. These randomly generated chromosomes are composed of individuals’ initial population. Second, let the objective function in SP3 be the fitness function for evaluating the chromosomes, select the highest fitness individual as the best individual, and select M/2 pairs of maternal chromosomes according to the ratio of the fitness of each chromosome to the total fitness of all chromosomes in the population. Third, we randomly select a codeword on the M/2 pair of maternal chromosomes, perform crossover and mutation operations based on the crossover probability pc and mutation probability pm to generate M new individuals to form a new population, and update the optimal individual according to the individual fitness in the new population. Crossover is the exchange of codewords corresponding to two chromosomes, and mutation is negation of the codeword. Repeat the above process until the number of iterations reaches the preset algebra G, and the optimal individual at this time is the optimal solution of SP3. The specific implementation process is shown in Table 3.Table 3
Genetic algorithm of optimal partition coefficient.
Algorithm: genetic algorithm with optimal partition coefficient1. Initialize the remaining variables and constant termsη=80%, b1=0.5, Pr=1, T=1.2. Set the number of individuals in the genetic algorithm population M, maximum evolution algebraG, cross probability Pc, mutation probability Pm, initial algebra g = 1.3. Randomly generate M individuals who meet the constraints, calculate the fitness of the initial population according to the objective function and select the individual with the highest fitness as the best individual.4. Whileg < G+1:5. Select M/2 to cross the mother chromosomes according to the probability of crossing to generate M new individuals,6. Make these M new individuals mutate according to the set mutation probability,7. Let the M chromosomes be a new population and update the most adaptive individual in the population,8. Letg = g + 1.9. The best individual found is the segmentation coefficientsρ∗ and λ∗ that satisfy the maximum energy efficiency


## 3.4. Alternate Iteration Algorithm for Joint Optimization

After solving the three suboptimization problems, this paper uses an alternating iterative algorithm to jointly optimize the optimal values of the three suboptimization problems. The specific steps are shown in Table4.Table 4
Joint optimal energy efficiency optimization algorithm.
Algorithm: Joint optimal energy efficiency optimization algorithm1. Initialize the remaining variables and constant termsη=80%, b11=0.5, λTS1=0.3, λOPS1=0.2, ρOPS1=0.5, T=1 and calculate ηEE12. Leti=1, ηEE0=ηEE1−κ.3. WhileηEEi−ηEEi−1≥κ:4. Leti=i+1,5. Letq0=ηEEi−1, substitute the remaining variables b1i−1, λi−1 and ρi−1 to solve the optimization problem (25) to obtain Pri and qi∗ according to Tables 2 and 1.6. Letu0=qi∗, substitute the remaining variables b1i−1 and Pri to solve the optimization problem (28) to obtain λi, ρi and ui∗ according to Tables 3 and 2 with the iteration conditions replaced with un=FUρn−1,λn−1/FLρn−1,λn−1.7. Substituteλi, ρi and Pri into equation (28) to obtain b1i;8. CalculateηEEi according b1i, λi, ρi and Pri;9. End and get the bestPr, b1, λ, ρ and ηEE.It is also provable that the energy efficiencyηEE increases in each iteration shown in Table 4 and the convergence of the complete algorithm exists. In each iteration, since the Dinkelbach algorithm is used to solve the transmission power problem and the partition coefficient problem, its convergence has been proved above, and the optimal energy efficiency obtained after solving the optimal transmission power is taken as the initial energy efficiency for solving the optimal partition coefficient, and the objective function is a monotonically increasing function of b1, which ensures that the value of ηEE will increase after each iteration.In addition, Proposition2 has proved that the objective function in the conversion form equation (21) of SP1 is a concave function about Pr. The local optimum of the unimodal function is the global optimal value; the objective function of SP2 is the monotone increasing function of b1; the global optimal solution is at the endpoint; the solution method of SP3 is genetic algorithm, it is a global optimization algorithm [25], so the joint optimal energy efficiency optimization algorithm in this paper can guarantee the global optimality of the obtained solution.This paper also provides the complexity analysis of the joint energy efficiency optimization algorithm. The solution process of SP1 and SP3 both use the Dinkelbach method to convert the objective function into a subtraction form; the computational complexity for the Dinkelbach method-based algorithm with stopping criteriaε is Ologε−1 [24]. Moreover, we have proved that the objective function of the subtraction form in SP1 is a concave function; the computational complexity of solving the concave function is Ologε−1 [26], where ε is the error tolerance for algorithm termination. The optimization process of the NOMA coefficient in SP2 only requires the partial derivative of b1 for the objective function, whose computational complexity is O1. The solution process of SP3 employs genetic algorithm; the computational complexity of the genetic algorithm is OG×M×N, which is related to its maximum evolutionary algebra G, the number of individuals in the population M, and the number of variables N [25]. The joint energy efficiency optimization algorithm in this paper nests the Dinkelbach algorithm twice in each iteration, including the golden section method and the genetic algorithm, so the overall computational complexity is Ologκ−1logε−1logδ−1+G×M×N, κ, ε, and δ are the error tolerance of the joint energy efficiency optimization algorithm, Dinkelbach algorithm, and golden section method, respectively.


## 4. Numerical Results

This paper simulates the proposed energy efficiency optimization algorithm with the OPS strategy so that to analyze and compare the system performance of the energy efficiency optimization algorithm of the PS and TS strategy; the calculation of the optimal energy efficiency of the PS strategy comes from the algorithm in [19]. The simulated system model is shown in Figure 1. The parameters are set as follows: η=80%, Pl=20dBm, Pt=30dBm, ith=0.1bit/s/Hz, rth=0.2bit/s/Hz, and T=1 [19]. The channel gain between the two-way relay and the secondary user is a random variable obeying the complex Gaussian distribution, namely, hRi∼CN0,1/diα with the average path loss α=2 [19, 27]. di is the distance from the secondary user to the two-way relay TWR, and d1=d3=2m and d2=d4=10m [19].Figure4 shows the energy efficiency of the three energy harvesting strategies at different user power thresholds Ps. It can be seen from the figure that the energy efficiency of the three harvesting strategies decreases as Ps increases. This is because the increase of Ps indicates that the system needs to allocate more resources to complete energy harvesting, so resources allocated for information transmission are correspondingly reduced, resulting in a reduction in energy efficiency. The energy efficiency of the OPS strategy is better than the other two strategies. According to the system model analysis of the OPS strategy, OPS can simultaneously control the two partition coefficients of time and power and regulate the ratio of collected energy and information transmission, which is more flexible than the other two strategies so that it can get the highest energy efficiency.Figure 4
The energy efficiency vs. secondary user power threshold.Figure5 shows the energy efficiency of the three energy harvesting strategies when the distance between the near user and the relay is different. The distance between the remote user and the relay is the same, that is, d2=d4=10m. It can be seen from the figure that the energy efficiency of the three collection strategies decreases with the increase of d1 and d3. This is because as the distance between the near user and the relay increases, the information rate of the secondary user decreases and the demand for energy collection increases, resulting in energy efficiency decreasing. The energy efficiency of the OPS strategy is better than the other two strategies which is because, with the increase of the distance between the near user and the relay, the OPS strategy can simultaneously control the time division and power partition to regulate the ratio of collected energy and information transmission. With the increase of d1, the energy efficiency difference of the three strategies is gradually narrowing, which indicates that the energy efficiency impact of the three strategies is gradually decreasing.Figure 5
The energy efficiency vs. the distance between the near user and TWR.Figure6 shows the energy consumption of the three strategies at different user power thresholds Ps. It can be seen from the figure that the energy consumption Etot of the three strategies rises with the increase of the user power limit Ps, which is because as the power limit of secondary users increases, the resource requirements for harvesting energy increase, and the relay power and the partition coefficient increase correspondingly, leading to an increase in energy consumption. The OPS strategy consumes less energy than the PS and TS strategies because the OPS strategy can control both time division and power division to ensure that the system completes the communication process with the lowest energy requirements.Figure 6
The energy consumption vs. secondary user power threshold.Figure7 shows the system capacity of the OPS strategy with different Ps at η=0.8 and η=0.6. It can be seen from the figure that the system capacity of the OPS strategy decreases with the increase of Ps. As Ps increases, the system needs to allocate more resources to harvest energy, and the resources allocated for information transmission are correspondingly reduced, at the same time, the decrease in the information rate of the secondary users results in a decrease in system capacity. The system capacity at η=0.8 is higher than the system capacity at η=0.6 because higher energy collection efficiency means lower energy requirements, OPS strategy can regulate more resources for information transmission, and the corresponding system capacity would be higher.Figure 7
The sum rate vs. secondary user power threshold whenη=0.8 and η=0.6.Figure8 shows the system capacity of the OPS strategy when relay power Pr=0.2W and Pr=0.1W at different b1. It can be seen from the figure that the system capacity of the OPS strategy increases with the increase of b1, which is because the near user has a higher channel gain than the far user, increasing the NOMA power allocation coefficient that TWR forwards to the near user will significantly increase the information rate of the near user, resulting in a corresponding increase in system capacity. The system capacity at Pr=0.2W is higher than the system capacity at Pr=0.1W because increasing the relay power increases the signal-to-interference-and-noise ratio of the secondary user’s received signal, and the increase in the information rate causes the system capacity to increase.Figure 8
The sum rate vs. NOMA coefficient whenPr=0.2W and Pr=0.1W.


## 5. Conclusion

This paper proposes a system energy efficiency optimization algorithm using the OPS strategy for a two-way relay CR-NOMA system based on SWIPT. The system capacity expressions with the TS strategy and the OPS strategy are derived, respectively. Under the constraints of harvested energy and quality of service, an optimization problem with the goal of maximizing energy efficiency is constructed. The objective function is a multiobjective optimized NP-Hard problem. In this paper, the original problem is transformed into three subproblems about relay power, NOMA coefficient, and segmentation coefficient, which are solved by the golden section algorithm, the monotonicity determination method of the function, and the genetic algorithm; then, the solutions of three subproblems are jointly optimized by alternating iterative algorithm. The simulation results show that compared with the PS and TS strategies, the OPS strategy can significantly improve the transmission energy efficiency of the system. Since the OPS strategy can control time division and power division at the same time, thereby ensuring that the system completes the communication process with the lowest energy requirements, OPS strategy is better than PS and TS strategy when the power limit of different users and the distance between the near user and the relay are different.



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*Source: 6207989-2020-06-29.xml*