Research Article | Open Access

# Joint Time and Power Allocation Algorithm in NOMA Relaying Network

**Academic Editor:**Wenchi Cheng

#### 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 [2â€“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 userâ€™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 systemâ€™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 *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 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 *t*_{2} 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 *t*_{3} 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 *t*_{1} 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 *t*_{4} subslot, the situation is similar to the *t*_{2} subslot, we can obtain the results as follows:where and are the power allocation factors of and in the *t*_{4} 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 *t*_{1} 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 *t*_{3}âˆ¼*t*_{4} subslot, the system status is similar to *t*_{1}âˆ¼*t*_{2}, then

#### 3. Performance Analysis

In this section, we analyse the systemâ€™s ergodic rates and the outage probabilities. The closed-form expression of userâ€™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 *t*_{1} 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 *t*_{1} subslot is given bywhere .

From equation (26), we know that the ergodic rates of is decided by , , , and . Since *t*_{1}â€‰=â€‰*t*_{3}, 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 *t*_{1} and subslot *t*_{2}, 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 *t*_{1} and *t*_{2} 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)â€“(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 *t*_{2}â€‰=â€‰*t*_{4}, 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 userâ€™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 *t*_{1} and *t*_{3} subslots, the outage probability in *t*_{1} time slot is and in *t*_{3} time slot is . For , if it cannot detect the signal of and or the throughput does not reach the target rates in the *t*_{1} 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 *t*_{1} and *t*_{3}, 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 *t*_{1} and *t*_{2} 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 by