Power Allocation with Proportional Fairness in Downlink NOMA System

Hanyu Zheng, Hai Li and Shujuan Hou

(School of Information and Electronics, Beijing Institute of Technology, Beijing 100081, China)

Abstract: The non-orthogonal multiple access (NOMA) has been regarded as a candidate radio access technology in the 5th generation (5G) mobile networks. In this paper, the problem of power allocation is considered with proportional fairness for downlink NOMA. Our objective is to maximize the logarithmic throughput of the system, which achieves a good tradeoff between the system throughput and user fairness. The approximate optimal solution of this proposed method can be found by the means of quasi-Newton method. For any number of users K, the numerical results verify the accuracy and convergence of our proposed method. Compared to the exhaustive search method, the computational complexity is significantly reduced when the two methods achieve the same accuracy. Furthermore, we show that as the number of users increases, the number of iterations of our method increases with an approximately linear trend.

Key words: non-orthogonal multiple access (NOMA); power allocation; proportional fairness; logarithmic throughput

The trend towards development of 5th generation (5G) mobile networks has attracted a huge amount of attention from the world. In order to meet the demand of growing data services and accommodate the tremendous amount of mobile devices, non-orthogonal multiple access (NOMA) has been proposed as a promising technology which has the potential to tackle this problem[1]. Typically, the existing NOMA solutions can be roughly divided into two categories[2]; code-domain NOMA and power-domain NOMA. Some new schemes based on code-domain NOMA have been proposed, such as MC-LDSMA and SCMA[3-4]. Meanwhile, a great deal of literature has analyzed the capability and performance of power-domain NOMA with successive interference cancellation (SIC) receivers[5].This is the main concern of this paper.

It has been proved that NOMA can achieve higher capacity region compared to orthogonal multiple access (OMA)[6]. For the problem of power allocation, it is worth mentioning that the authors in Ref.[7] use game theory to model the interaction between the base station and multiple users as a Stackelberg game which outperforms the uniform power allocation scheme. In Ref. [8], the authors propose an efficient “relax-then-adjust” algorithm and provideresults of performance evaluation for power minimization in NOMA. The authors of Ref.[9] investigate an optimization problem of minimizing the total transmittal power under quality of service requirements in a downlink OFDM-based NOMA system.

When considering the scheme of power allocation in NOMA system, we need to ensure that each user is connected and obtains a certain data rate no matter how the channel condition is. Hence, the fairness issue has to be taken into consideration during scheduling. In Ref.[10], the authors classify and analyze the fairness issues in wireless networking research. In Ref. [11], the authors introduce maximizing geometric mean user throughput as the policy for power allocation. Their method exploits tree-based search which can reduce the computational complexity by discarding the redundant power allocation ratio combinations. In Ref. [12], the author analyzes the power allocation schemes for both max-sum rate and max-min rate based on proportional fairness. In Ref.[13], the authors generalize the previous work in Ref.[14] into the multi-user NOMA system and propose a user set tree structure to enhance the efficiency of user selection.

Primarily, in resource constrained wireless systems, throughput and fairness are conflicting objectives. It is known that proportional fairness strikes a good balance between system throughput and user fairness[15]. However, to the best of our knowledge, there are limited works about power allocation with proportional fairness in downlink NOMA system, most of which care about the case of 2 users sharing the same subchannel. For the situation of multi-users (K>2), the methods in existing literature always have a large computational load. Hence, in this paper, we investigate the proportional fairness power allocation problem in downlink NOMA systems to achieve tradeoff between throughput and fairness. The primary contributions of this paper can be summarized as follows.

① Unlike previous literature such as Refs. [14,16], which only tackle the case of 2 users sharing the same subchannel, we propose a proportional fairness power allocation scheme which is aimed at the situation ofKusers multiplexed on the same subchannel.

② Due to the existence of inter-user interference, the power allocation problem is non-convex. Quasi-Newton method[17]is applied to solve the non-convex optimization problem. We prove that the formulated optimization problem has only one local maximum point, which guarantees the convergence of quasi-Newton method.

1 System Model and Problem Formulation

1.1 System model

We consider a downlink NOMA based cellular network servingKusers in a single cell. These users are multiplexed in the power domain and share the same carrier. The channel coefficient from the BS to the userkis denoted byhk. Without loss of generality, we assume that the channel gains are sorted in the descending order, i.e., |h1|2>|h2|2>…>|hK|2>0. Furthermore, the receiver Gaussian noise powerσ2of each user is assumed to be equal.

The BS broadcasts the superposition ofKsignals to itsKusers. Based on the principle of NOMA, the user who has a higher channel coefficient can decode the signals of other users with lower channel coefficients for interference cancellation. Consequently, the noise of the userkcontains both the additive Gaussian noise and the inter-user interferences from the users whose number is less thank. We set the total transmit power of the BS asP. For simplicity, we first make the following definition as

(1)

It is obvious thatφ1>φ2>…>φK>0. Hence, the data rate of userkcan be expressed as

(2)

whereαiis the power assignment ratio of useri. This should satisfy the following conditions

(3)

Sinceαi>0, it means that each user needs to be allocated some power even if it has a poor channel condition.

1.2 Problem formulation

In NOMA systems, if we simply attempt to improve the system throughput without taking into concern the fairness amongKusers, all of the power will be allocated to the single user with the best channel condition[18]. Hence, it is important to develop appropriate policies to maintain fairness and specify the optimization objective[19]. In the following, we will introduce the proportional fairness policy for problem formulation.

Proportional fairness makes a good compromise between system throughput and user fairness. In this paper, we aim to maximize the logarithmic sum of user data rates under the total transmit power constraint in order to achieve proportional fairness power allocation. Therefore, this optimization problem can be formulated as

(4)

(5)

whereαis converted from aK×1 column vector to a (K-1)×1 one.

2 Optimality Conditions

Since the power allocation problem is non-convex, Quasi-Newton method may not converge to the global maximum point. However, if Eq.(5) has one and only one local maximum point in the definition domain, the convergence to the global maximum point can be ensured. Hence, in the following, we first prove that the derivative of Eq.(5) has only one stationary point in the definition domain. Then we demonstrate that this stationary point is a maximum point.

(6)

For the sake of simplicity, we defineri,jas

(7)

andr0,1=rK,K=0. Then, we can get the partial derivative of the objective functionf(α) as

(8)

(9)

(10)

(11)

for all 1≤k≤K-1. Eq.(11) can also be converted as

Rk+1/Rk=rk,k+1/rk,k

(12)

In order to demonstrate that Eq.(12) has a unique solution, we define two new functions as

(13)

When we regardλk+1andλk-1as constants, it is obvious thatga(λk) is a strictly monotone increasing function aboutλk, andgb(λk) is a strictly monotone decreasing function ofλk. Let us concern the value ofga(λ1) andgb(λ1) as

(14)

We notice that for a certainλ2

λ1→0,ga(λ1)

λ1→λ2,ga(λ1)>gb(λ1)

(15)

(16)

For the stationary pointα*, since we knowrk,k>rk,k+1and Eq.(11), it is obvious that

(17)

(18)

Therefore, the Hessian matrix off(α) can be expressed as

(19)

To prove H(f) is negative definite at the stationary point, we need to ensure that for ?k, 1≤k≤K-1, we have

(20)

On account of

(21)

(22)

where z=α*+tp for somet∈(0,1). Since z∈D, we have pTH(z)p<0, and thereforef(α*+p)0, the definition domain is an open-interval. That is to say,α*is the optimum solution for the problem Eq.(5).

As a result of the complicated Hessian matrix off(α), it is hard to figure out theα*directly. Quasi-Newton method has a good performance on unconstrained nonlinear optimization problems. The most popular quasi-Newton algorithm is the BFGS method, named for its discoverers Broyden, Fletcher, Goldfarb, and Shanno. It has been proved in Ref. [17] that the BFGS method can exhibit a superior convergence rate in solving non-linear problems.Thus, the problem Eq. (5) can be solved by BFGS method. Those details are not discussed here.

3 Numerical Results

In this section, we provide some numerical results to validate the performance of our proposed power allocation method. The simulation parameters are listed in Tab.1. In the first simulation scenario, we consider 3 users sharing the same downlink NOMA channel. We compare the performance of the proposed approach to exhaustive search method in order to check the convergence and accuracy of our method.

In Fig.1, we use exhaustive search method to find a power allocation scheme which can get the maximum logarithmic throughput based on this scenario. Thex-axis andy-axis display the power allocation portion for user 1 and user 2. We can obtain the allocation ratio of User 3 fromα3=1-(α1+α2). Thez-axis exhibits the logarithmic throughput of each allocation scheme. Due to the large computing load, we first search a rough approximation in Fig.1a. Then we find the higher region in Fig.1a for further searching. The point with the highest logarithmic throughput has been marked in Fig.1b.

Tab.1 Simulation parameters

Fig.1 Power allocation scheme with exhaustire search method

Fig.2 Comparison of the logarithmic throughput performance of the proposed method with exhaustive search

To illustrate the accuracy and convergence of the proposed method, some simulation results are presented in Fig.2. Fig.2 displays the comparison of the logarithmic throughput performance of the proposed method with exhaustive search. The simulation results reveal that the proposed method can converge to the solution of the exhaustive search within several iterations. For the situation ofK=3, the allocation scheme calculated by the proposed method is the same as the result in Fig.1. These two methods both obtain the logarithmic throughput at 0.523 3. When these two methods achieve the same precision (10-6), the complexity of the proposed method is much lower. Since the total power is fixed, the data rate of each user will be reduced when the number of users increases. Therefore, the logarithmic throughput will be negative if the data rate of the new user is less than 1.

Fig.3 Number of iterations for multi-users case

The fairness plays an important role in the NOMA system as each user needs to be allocated some data rate. Hence, in the second scenario, we mainly focus on the performance of fairness among users in our proposed scheme. The fairness index (FI) which is mentioned in Ref. [15] is adopted for comparison. This is defined as

(23)

We compare our scheme (PF-in-NOMA) with the following schemes: ① max-sum-rate scheme in NOMA system and ② proportional fairness scheme in OMA system (PF-in-OMA). For the max-sum-rate scheme, we only optimize the throughput of the system. Thus, the user with the best channel condition will occupy the total power. And the purpose of the latter scheme is to allocate time slots to the users with proportional fairness[15]. We use Monte Carlo simulations to analyze the performance of these three schemes.

In Fig.4, we compare the fairness of the three schemes with respect to the number of users. The two schemes with proportional fairness both reach higher FIs. Since the max-sum-rate scheme allocates the total power to only one user, it has the lowest FI which is inversely proportional to the number of users. On account of the importance of fairness in NOMA system, the proposed scheme with the highest FI has a more superior performance compared to the other two schemes. In Fig.5, the throughput of the three schemes is displayed. The throughput of our proposed scheme is always larger than PF-in-OMA even though these two schemes have similar FIs. This implies that NOMA outperforms OMA. Although the max-sum-rate scheme achieves the highest throughput of all, it sacrifices the data rate of the other users, leading to a lower FI. In summary, our proposed scheme achieves a good tradeoff between the throughput and fairness.

Fig.4 User fairness comparison between the proposed scheme and other two schemes

Fig.5 System throughput comparison between the proposed scheme and other two schemes

4 Conclusion

This paper focused on the problem of power allocation for downlink NOMA. To find a compromise between the throughput and fairness, we maximized the logarithmic throughput of the system based on quasi-Newton method, which acted as proportional fairness. The simulation results demonstrate that the optimum solution solved by quasi-Newton method can achieve high accuracy with several iterations. When compared to exhaustive search method, our scheme significantly reduces the computational complexity. The numerical results also show that as the number of users grows, the number of iterations of our proposed method increases with an approximately linear trend. In our future work, the issues of user pairing and subcarrier allocation in multi-carrier circumstance will be taken into consideration.