Joint Time and Power Allocation Algorithm in NOMA Relaying Network

Nonorthogonal multiple access (NOMA) is one of the promising access techniques in 5G network. )e application of relay in NOMA system is a hotspot in recent research. NOMA-based cooperative relay network can achieve a higher spectral efficiency and a lower outage probability. In this paper, we analyse the performance of the two-hop DF relay NOMA network scenario, where the number of cell edge users is more than the cell center user, and obtained the closed-form expression of the user’s ergodic rates and outage probabilities under the high signal-to-noise (SNR) ratio. )en, we establish an optimization model to maximize the system rates, and a joint optimal time and power allocation algorithm based on the exhaustive search and the binary algorithm is proposed. Simulation results show that the proposed scheme can outperform exiting scheme in terms of achieving a higher ergodic sum rate, a lower outage probability under the premise of fairness.


Introduction
As the demand for smart terminals and new mobile services continues to grow, wireless transmission rates will increase exponentially and the 4G system will be difficult to meet the communications requirements of high-speed, low-latency.Besides, the unexpected excessive energy consumption of the fourth-generation (4G) and pre-4G wireless networks causes serious carbon dioxide emissions.To achieve green wireless networks, the fifth-generation (5G) wireless networks are expected to significantly increase the network energy efficiency while guaranteeing the quality of service (QoS) for time-sensitive multimedia wireless traffics [1].NOMA has been recognized as one of the key technologies of the fifthgeneration mobile communication (5G) for higher spectral efficiency [2][3][4].e key idea of NOMA is accommodating multiple users in the same frequency band but each user has a different power.Compared with the traditional OMA system, NOMA can achieve a higher spectral efficiency [5,6].Since cooperative relay can improve system capacity and expand network coverage [7], NOMA-based cooperative relay network has become a hotspot in wireless field research.
A preliminary study has been conducted on the resource allocation of the NOMA relay network.In [8], the strong users work as the relay, and the proposed cooperative scheme is strong user decode and transmits the weak user's signals.e ergodic sum rate and outage probability of this cooperative NOMA scheme are analysed.In [9], a dedicated relay node is used to provide services for users equipped with multiple antennas, and the literature obtains the lower bound of the outage probability.In [10], the author studied the outage performance of the NOMA system that relay operates in amplify-and-forward (AF) strategy and derived the exact approximation of the outage probability.In [11], the author has derived the system outage probability and the ergodic sum rate of the scenario where users are directly communication with the base station (BS) and the relay.e relay operates in decode-and-forward (DF) strategy.In [12], the time and power allocation of two-hop relay using DF protocol is studied.e closed-form expression of the outage probability is derived, and the optimal time allocation for minimizing the outage probability is obtained but without considering the fairness of the system and the user communicating directly with the base station.In [13], the author analyses the scenario that the user communicates either directly with the base station or through relay with the base station and derivate the system outage probability and the expression of user ergodic rate.e theoretical and simulation results show that the ergodic sum rate of cooperative NOMA system can be significantly improved compared with noncooperation.However, the paper only analyses the case where the number of users in the cell center is equal to the number of users at the cell edge, and the time shared by the two hops is not optimized.In [14], the author analyses the performance of the scenario, where the number of cell edge users are more than the cell center users by introducing time sharing technology.However, the user is considered to communicate with the BS through the relay, and the strategy of equal time slot division and fixed power allocation factor is adopted, and the fairness of the system is not considered.
In this paper, for the NOMA relay scenario where the cell edge users are more than the cell center users and the channel of each user is significantly different, we use the method of dividing the time slot to transmit the information of the user and derive the expressions of the system's ergodic sum rate and the outage probability.Under the condition of system fairness index factor, the optimization problem of maximizing system rates is constructed.In order to get the solution to this problem, we proposed an optimal time and power allocation algorithm based on exhaustive search and binary algorithm.e algorithm can obtain the optimal time and power factor allocation strategy under different fairness index factors.In this allocation policy, we obtained the maximum rates of the system.e main contributions of this paper are as follows: (1) Modelled the scenario that the number of cell center users is lower than users at the edge (2) Derived the ergodic rate and outage probability of the system under the new scenario (3) Proposed the joint and power allocation algorithm to maximize the system sum rate e rest of the paper is organized as follows: section 2 gives the system model of this paper, section 3 deduces the system performance, including the derivation of the system's ergodic rates and the outage probability, section 4 proposes an optimal allocation algorithm for time and power factor, section 5 performs simulation analysis to analyse the effect of fairness factors on the overall system rates, and section 6 summarizes the full text.

System Model
Figure 1 is the proposed system model for this paper, which contains one base station (BS), one relay (R), and four users (UE 1 , UE 2 , UE 3 , and UE 4 ).We clarify that UE 1 is the cell center user directly connected to the base station and has better channel conditions.UE 2 , UE 3 , and UE 4 are cell edge users that need to forward information through the relay which operates in the half-duplex mode using DF strategy.
e h b 1 and h b r indicate the channel coefficients from the BS to the UE 1 , from the BS to the relay.h r 2 , h r 3 , and h r 4 denote the channel coefficients from the relay to each cell edge user.Channels are independent of each other and are subject to Rayleigh fading; we model these channels as , and h r 4 ∼ CN(0, λ r 4 ).We assume that UE 2 and UE 4 have similar channel conditions and the same variance, then λ r 2 � λ r 4 .e channel condition of UE 3 is significantly worse than other users.In the transmission process, relay forwards signals to UE 2 , UE 3 , and UE 4 in NOMA strategy.We divide a transmission time slot into four subslots, and cell center users are paired with different cell edge users in different subslots for information transmission.
It is assumed that one time slot is divided into four subslots which are denoted as t 1 , t 2 , t 3 , and t 4 , respectively, and the channel state in one subslot does not change.
(1) During the t 1 subslot, the BS transmits x N (t where n (•) is the additive white Gaussian noise (AWGN) at each node.
When receiving the superimposed signal, UE 1 needs to apply the SIC to obtain its own signal after decoding the signals of UE 2 and UE 3 .It can be seen from [13] that the optimal decode order is the user with the worst channel condition decode first.In this method, UE 1 first decodes the signal of UE 3 , and the decoded signal to interference and noise ratio (SINR) is given by where ρ s is the transmit signal-to-noise ratio (SNR) of the base station, ρ s � P s /σ 2 , P s is the transmission power of the BS, and σ 2 is the variance of Gaussian additive white noise.
After decoding the signal of the UE 3 , the SIC is applied to remove the signal of UE 3 and then the UE 2 is decoded, and the decoded SINR is Assuming that UE 2 signal can successfully decode, after applying SIC, the SNR of UE 1 is given by 2 Mobile Information Systems After receiving the superimposed signal, the UE 2 first decodes the UE 3 signal and the decoded SINR is given by When SIC is applied at UE 2 , the signal of UE 3 has been removed and the received SNR of the UE 2 is where ρ r is the transmit SNR of the relay, ρ r � P r /σ 2 , and P r is the transmission power of the relay.UE 3 has the worst channel condition and directly decodes its own signal, and the decoded SINR is given by After receiving the superimposed signal, UE 1 first decodes the signal of UE 3 and then decodes the signal of UE 4 .SIC is applied to remove the two signals, and finally UE 1 decodes its own signal.e relay decodes the signals of UE 3 and UE 4 in the same manner, and similar to the t 1 subslot analysis, the SINR of the UE 1 decodes signals of UE 3 and UE 4 can be obtained as follows: Assuming that the relay successfully decodes the two-user signal, the SINR of UE 1 after applying the SIC is given by e SINR of decoding UE 3 and UE 4 at the relay is given by (4) In the t 4 subslot, the situation is similar to the t 2 subslot, we can obtain the results as follows: Relay  Since the user channel conditions are unchanged in one time slot and the channel conditions of user 2 and user 4 are similar, then After applying SIC technique, the achievable data rates of UE 1 in the t 1 subslot is given by Because UE 2 must be decoded at UE 1 for SIC and the capacity of DF relaying is dominated by the weakest link, the achievable data rates of UE 2 is given by [15] e achievable data rates of UE 3 is given by In the t 3 ∼t 4 subslot, the system status is similar to t 1 ∼t 2 , then

Performance Analysis
In this section, we analyse the system's ergodic rates and the outage probabilities.e closed-form expression of user's ergodic rates in high SNR is derived, and the outage probability of each user and system is obtained.
3.1.Ergodic Rates.Ergodic rates refer to the time average of the maximum information rates of a random channel in a fast fading state.
e system ergodic sum rate is given by where C 1 , C 2 , C 3 , and C 4 indicate the ergodic rates of users 1, 2, 3, and 4, respectively.e ergodic rates of the UE i in the subslot t is given by where ω is the SINR of UE i and F W (ω) and f W (ω) are the distribution function (CDF) and the probability density function (PDF) of ω. 1 is given by where Using the definition of the distribution function, there is Since channel h b 1 ∼ CN(0, λ b 1 ) obeys the complex Gaussian distribution, according to the literature [16], Taking (25) into equation ( 23) and the ergodic rates of UE 1 in t 1 subslot is given by where Ei(x) � x −∞ e t /tdt.From equation (26), we know that the ergodic rates of UE 1 is decided by t 1 , λ b 1 , ρ s , and a 1 , so the ergodic rates of UE 1 is given by Assuming that the SINR of UE 2 is c R 2 , UE 2 transmits its signal via the relay in the subslot t 1 and subslot t 2 , because the SINR is decided by the weakest link of the relay, we can obtain that c R 2 � min(c ). e ergodic rates of UE 2 is given by 4

Mobile Information Systems
To get the C 2 , we need to obtain the CDF of F c R 2 (ω).Since the channels h b r , h b 1 , and h r 2 are independent of each other, F c R 2 can be derived as follows: where e expression of F c R 2 and C 2 are given by In order to obtain the closed-form expression of equation (32), a high SNR approximation is used.When ρ s ⟶ ∞ and ρ r ⟶ ∞, equation (29) can be rewrite as follows: e closed-form expression of C 2 is given by (34) 3.1.3.Ergodic Rates of UE 3 .Assuming that the SINR of UE 3 is c R 3 , it decided by the weakest link of the relay, that is c R 3 � min(c R 3 ).We can obtain the ergodic rates of UE 3 in the subslots t 1 and t 2 as follows: Since the channels h b r , h b 1 , and h r 2 are independent of each other, the F c R 3 is given by where (ω), and 3⟶1 , and c

respectively, and we can obtain
Mobile Information Systems Taking equations (37)-( 39) into (36), the are given by When ρ s ⟶ ∞ and ρ r ⟶ ∞, equation (36) can be rewritten as follows:  . ( 3 , we can obtain Combine a and b, we can get Since t 2 � t 4 , we can obtain that 3 combined with (16).erefore, the ergodic rates of UE 3 is given by Taking C 1 , C 2 , C 3 , and C 4 into (21), the system ergodic sum rate is given by

Outage Probability.
e outage probability is the probability of the outage event happened in the communication.When the user's reachable rates are lower than target rates, the outage event happened.In this paper, the outage probability of the cell center user is determined by its channel state.e outage probability of the cell edge users are determined by the two-hop system, and the state of the previous hop will affect the state of the next hop [17].
(1) Outage probability of UE 1 Assumed that E O 1 t 1 and E O 1 t 3 , respectively, represent the outage event happened of UE 1 in t 1 and t 3 subslots, the outage probability in t 1 time slot is P 1,(t 1 ) For UE 1 , if it cannot detect the signal of UE 2 and UE 3 or the throughput does not reach the target rates R Q in the t 1 time slot, the communication will interrupted.Denote these three cases as , and , respectively, we can obtain the outage probabil- Mobile Information Systems where R where DFO 1 is given by Since the channel state does not change in the time slots t 1 and t 3 , the relationship erefore, the outage probability of UE 1 is given by 8 Mobile Information Systems (2) Outage probability of UE 2 e UE 2 communicates with the BS through the relay in the t 1 and t 2 sub-time slots, and its outage probability is related to each hop of the two-hop system.If UE 2 cannot be successfully decoded in the relay or cannot decode the signal of UE 3 , communication will be interrupted.If UE 2 can be decoded in the relay, but in the second hop transmission rate does not reach the target rate, the outage event will also occur.Denote the events that UE 2 cannot successfully decode at relay, UE 2 cannot decode the signal of the UE 3 , and in the second hop, the communication rates of UE 2 does not reach the target rate R Q , respectively, as and E O U 2 .e outage probability P 2 DFO of UE 2 can be obtained as follows: where β 2 � 2 R Q /t 2 − 1, and P 2 DFO is given by where R

Mobile Information Systems
From the previous analysis, the outage probability of UE 3 is given by (4) Outage probability of UE 4 Since the power allocation factor unchanged in one time slot, obviously the power allocation factor of the UE 4 in the t 4 subslot is the same as the UE 2 in the t 2 subslot.erefore, the outage probability is the same for both, we can obtain e overall system outage event is defined the any user in the system cannot achieve reliable detection, which means the overall outage probability is defined as follows [16]: Substituting equations ( 53), (56), (58), and (59) into (60), we can obtain the system outage probability.

Time and Power Optimization Allocation Algorithm
In this paper, we consider maximizing the system sum rate under the premise of the system fairness.Since the system sum rate in NOMA system is mainly determined by the cell center user [18,19], it is necessary to allocate time and power resources to the cell center user as much as possible.However, due to the system fairness constrain, sufficient resources should allocate to the cell edge users to ensure users communication does not interrupt and meets the corresponding communication service quality.erefore, the system sum rate is affected by the fairness factor F ′ : When the transmitting power of the base station is fixed, the more time and power resources the edge users are allocated, the higher the fairness index of the system will be, but the overall rate of the system will decrease.erefore, the sum rates maximum is contradictory to the system fairness, and we need to compromise on the fairness and rates of the system.e system sum rate is maximized under the condition of the cell edge user's outage probability and system fairness limit.e optimization problem can be expressed as follows: T slot , k, and F ′ , respectively, represent the time-slot length, the user outage probability target, and the given system target fairness factor.
e system fairness index is measured by introducing the fairness index factor.In literature [20], the fairness index factor expression is given by From the above analysis, the system sum rate maximization is negatively correlated with the system fairness.erefore, it is necessary to consider how to schedule time and power to maximize the system sum rate while ensuring the given fairness factor.Since the system sum rate is a piecewise function and the objective function contains multiple parameters such as time and power allocation factor, the KKT method [21] cannot be used to find the optimal value in our proposed scenario.
In this paper, we propose the joint optimal time power allocation algorithm based on exhaustive search and binary algorithm to obtain the optimal solution: firstly, the user's power allocation factor is obtained by the following method, UE 1 is the cell center user, and the allocated power is small.Since the user power allocation scheme of the t 1 ∼t 2 time slot is the same as the t 3 ∼t 4 time slot, the time slot t 1 ∼t 2 is taken as Figure 3 shows the ergodic sum rate under different fairness index factors.We have compared the system throughput optimized by Algorithm 1 with the equal time slot allocation scheme.It can be seen that the smaller the fairness index factor F ′ is, the larger the system ergodic sum rate is.e larger the F ′ is, the smaller the system ergodic sum rate is.When F ′ � 0.4, the ergodic sum rate optimized by the proposed algorithm is greater than the equal-time transmission [11] at any SNR.When F ′ � 0.5 and ρ s < 45 dB, the ergodic sum rate is greater than equal-time transmission, and when ρ s > 45 dB, the fairness factor of equal-time transmission is low, and the ergodic sum rate is higher than that in this paper.When F � 0.6, the value of the intersection of the ergodic sum rate by the proposed algorithm and the equal-time transmission becomes smaller.
Figure 4 shows the relationship between user outage probability and ρ s when F ′ � 0.5.It can be seen that the user outage probability decreases as ρ s increases.Since different power allocation methods are adopted for different ρ s , the logarithm of the outage probability and ρ s are nonlinear.
Figure 5 shows a comparison of the system outage probability of this paper and the equal-time transmission strategy.e algorithm proposed in this paper has a lower system outage probability when ρ s is low.When ρ s is high, the probability of two systems is close.
Figure 6 shows the relationship between the fairness index factor and ρ s .It shows that in the case of equal-time    Mobile Information Systems transmission, the fairness index factor will gradually decrease as the ρ s increases.However, the proposed algorithm enables the system fairness index factor maintained near the given fairness factor F ′ .Figure 7 shows the relationship between user rates and F ′ when ρ s � 50 dB.It can be seen that the UE 1 rates and the system sum rate are negatively correlated with F ′ , and users 2, 3, and 4 are positively correlated with F ′ .e rates of UE 1 is always the largest, as F increases, and these rates of increase in users 2, 3, and 4 slow down.Because when F ′ increases, the power and time resources allocated to the cell edge users rise, and the cell center user decreases accordingly.Besides, the throughput difference between users decreases.Since the cell center user has the greatest impact on system sum rate, although the cell edge users rates increase, the system sum rate still drops.

Conclusion
In this paper, we study the time and power optimization allocation algorithm in the NOMA relay network.Firstly, the scenario of two-hop DF relay is established.Based on this model, the expressions of user outage probability and ergodic rates are derived.Secondly, an optimization model for maximizing the system rates is constructed.irdly, the joint time and power factor allocation algorithm is proposed to obtain the solution of the model.e simulation result shows, under the premise of considering the fairness of the system, the system rates optimized by the proposed algorithm is significantly improved compared with the equaltime allocation.

Figure 2 :Figure 3 :
Figure 2: Relationship between user rate and SNR (ρ s ).e users' rates are positively correlated with ρ s .

Figure 4 :
Figure 4: Relationship between user outage probability and ρ s .
x 3 (t 1 ) to the UE 1 and the relay, where x 1 (t 1 ), x 2 (t 1 ), and x 3 (t 1 ) are data symbols for UE 1 , UE 2 , and UE 3 with 1. P s is the transmission power of BS, and a � h r 2 x N t 2  + n 2 , y 3 t 2  � h r 3 x N t 2  + n 3 .( are the power allocation factors of UE 3 and UE 4 in the t 4 sub-time slot, respectively. 4 3.1.1.Ergodic Rates of UE 1 .In the t 1 subslot, assuming that UE 1 successfully decodes the UE 2 and UE 3 signals, using (22), the C t 1 and F c t 2 46)3.1.4.Ergodic Rates of UE 4 .It can be seen from (34) that C 2 is related to t 2 , λ r 2 , ρ r , a

Table 1 :
System performance parameters.