Abstract
Nonorthogonal multiple access (NOMA) is one of the promising access techniques in 5G network. The 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. Then, 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.
1. 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 fifth-generation mobile communication (5G) for higher spectral efficiency [2–4]. The 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. The 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. The relay operates in decode-and-forward (DF) strategy. In [12], the time and power allocation of two-hop relay using DF protocol is studied. The 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. The 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. The 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. The 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
The 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.
2. System Model
Figure 1 is the proposed system model for this paper, which contains one base station (BS), one relay (R), and four users (, , , and ). We clarify that is the cell center user directly connected to the base station and has better channel conditions. , , and are cell edge users that need to forward information through the relay which operates in the half-duplex mode using DF strategy. The and indicate the channel coefficients from the BS to the , from the BS to the relay. , , and 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 . We assume that and have similar channel conditions and the same variance, then . The channel condition of is significantly worse than other users. In the transmission process, relay forwards signals to , , and 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 t1, t2, t3, and t4, respectively, and the channel state in one subslot does not change.(1)During the t1 subslot, the BS transmits to the and the relay, where , , and are data symbols for , , and with . is the transmission power of BS, and , , and are the power allocation coefficients, where and . The received signals at and relay are given bywhere is the additive white Gaussian noise (AWGN) at each node.
When receiving the superimposed signal, needs to apply the SIC to obtain its own signal after decoding the signals of and . It can be seen from [13] that the optimal decode order is the user with the worst channel condition decode first. In this method, first decodes the signal of , and the decoded signal to interference and noise ratio (SINR) is given bywhere is the transmit signal-to-noise ratio (SNR) of the base station, , is the transmission power of the BS, and is the variance of Gaussian additive white noise.
After decoding the signal of the , the SIC is applied to remove the signal of and then the is decoded, and the decoded SINR is
Assuming that signal can successfully decode, after applying SIC, the SNR of is given by
At relay R, is decoded first and then is decoded. The SINR of decoding and are, respectively, given by(2)In the t2 subslot, the relay first regenerates the superimposed signal of and . Then, the relay transmits it to users. is the transmission power of relay and and are the power allocation coefficients where and . The received signals at and are given by
After receiving the superimposed signal, the first decodes the signal and the decoded SINR is given by
When SIC is applied at , the signal of has been removed and the received SNR of the iswhere is the transmit SNR of the relay, , and is the transmission power of the relay.
has the worst channel condition and directly decodes its own signal, and the decoded SINR is given by(3)In the t3 subslot, the BS transmits the superimposed signals of the users 1, 3, and 4. The signal is , where , , and are the power allocation coefficients for users 1, 3, and 4, where and . The received signals at and relay are given by
After receiving the superimposed signal, first decodes the signal of and then decodes the signal of . SIC is applied to remove the two signals, and finally decodes its own signal. The relay decodes the signals of and in the same manner, and similar to the t1 subslot analysis, the SINR of the decodes signals of and can be obtained as follows:
Assuming that the relay successfully decodes the two-user signal, the SINR of after applying the SIC is given by
The SINR of decoding and at the relay is given by(4)In the t4 subslot, the situation is similar to the t2 subslot, we can obtain the results as follows:where and are the power allocation factors of and in the t4 sub-time slot, respectively.
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 in the t1 subslot is given by
Because must be decoded at for SIC and the capacity of DF relaying is dominated by the weakest link, the achievable data rates of is given by [15]
The achievable data rates of is given by
In the t3∼t4 subslot, the system status is similar to t1∼t2, then
3. Performance Analysis
In this section, we analyse the system’s ergodic rates and the outage probabilities. The 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. The system ergodic sum rate is given bywhere , , , and indicate the ergodic rates of users 1, 2, 3, and 4, respectively.
The ergodic rates of the in the subslot t is given bywhere is the SINR of and and are the distribution function (CDF) and the probability density function (PDF) of .
3.1.1. Ergodic Rates of
In the t1 subslot, assuming that successfully decodes the and signals, using (22), the is given bywhere is the CDF of the . Using the definition of the distribution function, there is
Since channel obeys the complex Gaussian distribution, according to the literature [16], obeys the exponential distribution of . The distribution function of is given by
Taking (25) into equation (23) and the ergodic rates of in t1 subslot is given bywhere .
From equation (26), we know that the ergodic rates of is decided by , , , and . Since t1 = t3, we can obtain that , so the ergodic rates of is given by
3.1.2. Ergodic Rates of
Assuming that the SINR of is , transmits its signal via the relay in the subslot t1 and subslot t2, because the SINR is decided by the weakest link of the relay, we can obtain that . The ergodic rates of is given by
To get the , we need to obtain the CDF of . Since the channels , , and are independent of each other, can be derived as follows:where , , and are the CDF of the , , and . The CDFs are derived separately as follows:
The expression of and are given by
In order to obtain the closed-form expression of equation (32), a high SNR approximation is used. When and , equation (29) can be rewrite as follows:
The closed-form expression of is given by
3.1.3. Ergodic Rates of
Assuming that the SINR of is , it decided by the weakest link of the relay, that is . We can obtain the ergodic rates of in the subslots t1 and t2 as follows:
Since the channels , , and are independent of each other, the is given bywhere , , and represent the CDF of , , and , respectively, and we can obtain
Taking equations (37)–(39) into (36), the and are given by
When and , equation (36) can be rewritten as follows:
The closed-form expression of is given bywhere is determined by and , since is an monotonically increasing function about x, we only need to discuss and .(a)If , we can obtain(b)If , we can obtain
Combine a and b, we can get
Since t2 = t4, we can obtain that combined with (16). Therefore, the ergodic rates of is given by
3.1.4. Ergodic Rates of
It can be seen from (34) that is related to , , , , , and . Combined with (16), the is given by
Taking into (21), the system ergodic sum rate is given by
3.2. Outage Probability
The 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. The 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
Assumed that and , respectively, represent the outage event happened of in t1 and t3 subslots, the outage probability in t1 time slot is and in t3 time slot is . For , if it cannot detect the signal of and or the throughput does not reach the target rates in the t1 time slot, the communication will interrupted. Denote these three cases as , , and respectively, we can obtain the outage probability as follows:where , , and .
The expressions , , and are given bywhere . The is given by
Similarly, is given by
Since the channel state does not change in the time slots t1 and t3, the relationship is established. Therefore, the outage probability of is given by(2)Outage probability of
The communicates with the BS through the relay in the t1 and t2 sub-time slots, and its outage probability is related to each hop of the two-hop system. If cannot be successfully decoded in the relay or cannot decode the signal of , communication will be interrupted. If 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 cannot successfully decode at relay, cannot decode the signal of the , and in the second hop, the communication rates of does not reach the target rate , respectively, as , , and . The outage probability of can be obtained as follows:where , , and .
The expressions , , and are given bywhere , and is given by(3)Outage probability of
Assumed that and , respectively, indicate the event that interrupts in t1∼t2 and t3∼t4 subtime slots. The corresponding outage probability is and ; obviously, the equation is established. Taking for example, here we denote the event that cannot be decoded at the relay as , and the rates at the second hop does not reach the target rate as . The is given bywhere and .
From the previous analysis, the outage probability of is given by(4)Outage probability of
Since the power allocation factor unchanged in one time slot, obviously the power allocation factor of the in the t4 subslot is the same as the in the t2 subslot. Therefore, the outage probability is the same for both, we can obtain
The overall system outage event is defined as the event that 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.
4. 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. Therefore, the system sum rate is affected by the fairness factor : 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. Therefore, the sum rates maximum is contradictory to the system fairness, and we need to compromise on the fairness and rates of the system.
The system sum rate is maximized under the condition of the cell edge user’s outage probability and system fairness limit. The optimization problem can be expressed as follows:where , , , . , , and , respectively, represent the time-slot length, the user outage probability target, and the given system target fairness factor.
The 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. Therefore, 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, is the cell center user, and the allocated power is small. Since the user power allocation scheme of the t1∼t2 time slot is the same as the t3∼t4 time slot, the time slot t1∼t2 is taken as an example. According to equation (16), if , then , , , and . Let take values in a certain step in the interval . When , searches in . and are also obtained in this way, so that all the power allocation factor list satisfying the conditions can be obtained; secondly, according to each value set of power allocation factors in , the corresponding user outage probability can be obtained. If the user outage probability satisfies the given threshold value, the corresponding time set of each user when the outage probability is satisfied can be further calculated. The feasible time allocation list L can be obtained by taking the intersection of L1∼L4. Binary algorithm is then employed to find the optimal transmission time at a certain fairness factor index premise; finally, the sorting algorithm is used to find the power allocation factor corresponding to the maximum system rates. This process shown in Algorithm 1.
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When acquiring , the complexity of this algorithm to calculate , , and is , and the complexity to calculate and is . Therefore, the algorithm complexity of obtaining the power allocation factor is . The calculation of time list L and system capacity can be completed in constant time, and the complexity is . The complexity of the time allocation list under the feasible power set is .
After the feasible power and time solution set is obtained, the sorting algorithm is used to obtain the optimal time and power allocation scheme. The complexity of this step is . The system complexity is . When the exhaustive search method is used, the corresponding system complexity is , and it can be seen that , so the proposed algorithm can effectively reduce the computational time complexity.
5. Numerical and Simulation Results
The simulation scenario is shown in Figure 1. It includes a base station (BS), a relay (R), and four users (). The relay operates in the half-duplex mode using DF strategy. is the cell center user, and denotes the channel coefficients from the BS to . , , and are cell edge users, and , , and denote the channel coefficients from the relay to each cell edge user. We model these channels as , , , , and . The cell center user is more concerned with the ergodic rates, and the cell edge users are more concerned with the outage probability; we set different target outage probability for different users, and the simulation parameters are shown in Table 1.
Figure 2 shows the theoretical and simulated rates of the user when . It can be seen from the figure that the simulation value agrees with the theoretical value. When increases, the rates of each user gradually increase. rates increase rapidly at low SNR, and the growth become slower due to the constrain of system fairness index factor at high SNR. Users 2, 3, and 4 are the cell edge users, and the growth rate is slow.
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 is, the larger the system ergodic sum rate is. The larger the is, the smaller the system ergodic sum rate is. When , the ergodic sum rate optimized by the proposed algorithm is greater than the equal-time transmission [11] at any SNR. When and , the ergodic sum rate is greater than equal-time transmission, and when , 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 when . It can be seen that the user outage probability decreases as increases. Since different power allocation methods are adopted for different , the logarithm of the outage probability and are nonlinear.
Figure 5 shows a comparison of the system outage probability of this paper and the equal-time transmission strategy. The algorithm proposed in this paper has a lower system outage probability when is low. When is high, the probability of two systems is close.
Figure 6 shows the relationship between the fairness index factor and . It shows that in the case of equal-time transmission, the fairness index factor will gradually decrease as the increases. However, the proposed algorithm enables the system fairness index factor maintained near the given fairness factor .
Figure 7 shows the relationship between user rates and when . It can be seen that the rates and the system sum rate are negatively correlated with , and users 2, 3, and 4 are positively correlated with . The rates of is always the largest, as F increases, and these rates of increase in users 2, 3, and 4 slow down. Because when 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.
6. 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. Thirdly, the joint time and power factor allocation algorithm is proposed to obtain the solution of the model. The 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 equal-time allocation.
Data Availability
All data included in this study are available upon request by contact with the corresponding author.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
This work was supported by the National Natural Science Foundation of China (61571234 and 61631020).