Power Allocation for Sum Rate Maximization in Non ... - IEEE Xplore

2015 IEEE 26th International Symposium on Personal, Indoor and Mobile Radio Communications - (PIMRC): Mobile and Wireless Networks

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Power Allocation for Sum Rate Maximization in Non-Orthogonal Multiple Access System Ziad Qais Al-Abbasi and Daniel K. C. So, School of Electrical and Electronic Engineering The University of Manchester, United Kingdom Email: [email protected]; [email protected]

Abstract—Non-orthogonal multiple access (NOMA) can increase the spectral efficiency and could play an important role in improving the capacity of 5G networks. In this paper, an optimization problem is formulated to maximize the overall sum rate in a sub-carrier based NOMA system. The optimal transmission power of each user is obtained based on the users’ instantaneous channel state information under the total power and the minimum rate constraints. Moreover, two closed-form suboptimal solutions are also proposed for a two-user scenario to reduce the complexity of the optimal solution. The suboptimal approaches are also extended to multiuser scenario by pairing users for subbands transmission. Simulation results show that the derived sub-optimal solutions provide better performance than the orthogonal frequency division multiple access (OFDMA) in terms of coverage probability and sum rate. Moreover, the results show that the proposed sub-optimal solutions achieve a comparable results to the optimal one with lower complexity. Index Terms—Orthogonal Frequency Division Multiple Access (OFDMA), Non-Orthogonal Multiple Access (NOMA), power allocation, sum rate maximization, coverage probability

I. I NTRODUCTION

R

ECENTLY, many research works focused on tackling the potential challenges for future mobile networks in 2020 and beyond. Due to the anticipated higher quality of service (QoS) and better multimedia services, it has been predicted that the traffic demand in the near future will be around 15 to 30 times of the current one [1], [2]. There are a number of solutions proposed to cope with this evolution; one of these is the non-orthogonal multiple access (NOMA) proposed to enhance the efficiency of the air interface as well as making efficient use of available resources [1]–[3]. In general, multiple access techniques can be categorized into orthogonal and non-orthogonal techniques. The signals in the first category are made to be orthogonal to their counterparts that leads to no cross correlation between these signals, such as orthogonal frequency multiple access (OFDMA) which is widely used in 4G networks. On the other hand, nonorthogonal signals will have a cross correlation among them, and NOMA is a promising candidate for 5G wireless networks [2], [4]–[8]. NOMA offers good fairness as it allows multiple users to be served simultaneously using the whole spectrum and provides better spectral efficiency, throughput, and fairness than OFDMA [9]. So far, resource allocation for orthogonal multiple access techniques have been studied intensively in literature. For instance, in [10], the dynamic resource allocation scheme based

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on maximizing the sum capacity with proportional fairness of OFDMA system is proposed. The authors in [11] investigated the energy efficiency maximization of OFDMA system taking into account the data rates and bit error rate (BER) constraints to ensure proportional fairness among the users. Both articles depicted the importance of fairness constraints to help users in achieving their target rates. The subcarrier allocation in both [10] and [11] was based on the access probability between the users and their serving units, which is calculated using each user’s channel to noise ratio. On the other hand, although NOMA has recently drawn more attention, studies on the resource allocation aspects is till in its early stage. The authors in [5] analyzed NOMA with user’s index based fixed power allocation (UFPA) in a scenario with user target rate and another with opportunistic rate allocation. It is proved analytically that NOMA is better than OFDMA in terms of the achievable sum rate and the coverage probability. The authors in [6] and [7] presented how NOMA could improve the capacity and the throughput of the network over orthogonal access approaches. In particular, [7] presented several power allocation schemes: the full search power allocation (FSPA), the fractional transmit power allocation (FTPA), and the fixed power allocation (FPA) which was also presented in [12]. The FSPA achieves the best performance but suffers from high computational complexity. The FPA has the lowest complexity but have poor performance. The FTPA is a balance between the two, but required a prior computer simulation to determine a specific parameter to obtain the best performance. These techniques will be compared to the proposed schemes in this paper. Other works on NOMA such as [8] and [9] do not consider power allocation at all. In this paper, we investigate the sum rate maximization problem for a subcarrier based NOMA system. By dividing the spectrum into subcarriers power allocation can be performed for each subcarrier to maximize the rate. We mainly focus on a two-user scenario because although NOMA allows multiple users, the sum rate gain diminishes when the number of users is more than two [6]. The optimal solution is presented but with excessive complexity. Two low complexity power allocation schemes with closed form solutions are presented, and shown to achieve performance that is close to the optimal one. The scenario with more than two users are also considered by applying the suboptimal solutions to a subband based transmissions. The rest of this paper is organized as follows. Section II

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discusses the system model of the subcarrier based NOMA system. Section III presents the sum rate maximization problem and the optimal and the two sub-optimal solutions. The simulation scenarios and results are presented in Section V.

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II. SYSTEM MODEL

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We consider the downlink of a small cell serving U cellular users with a total bandwidth of WT and Nsc subcarriers. For OFDMA, the set of subcarriers allocated to one user will not be used by another as depicted in Fig. 1 - a. The advantage of such procedure is the robustness against multiple access interference. However, the drawbacks here are the inefficient use of the radio spectrum, sensitivity to frequency offset that could affect the orthogonality, and susceptibility to co-channel interference [13]. NOMA differs from OFDMA in that all users share the whole bandwidth as in Fig. 1 - b. Thus there is no need for subcarrier allocation with NOMA as all the users will be eligible to use all the subcarriers. In addition, the users are multiplexed in the power domain at the transmitter side and are being separated by successive interference removal at the receiver side as illustrated in Fig. 2.

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ZĞĐĞŝǀĞƌŽĨƚǁŽƵƐĞƌƐEKD;,Ϯ͕Ŷ х,ϭ͕ŶͿ

Fig. 2. Successive interference cancellation in a receiver of NOMA with two users (H2,n > H1,n ) [6], [7]

The successive interference cancellation (SIC) at the receiver side is a superposition scheme where the signal is the linear combination of the users signals [5], [6], [9]. Using the two-user scenario as an example illustrated in Fig. 2, the user with the worst channel conditions (denoted as user 1) treats the signal of the user with the better channel as noise and decodes its data from the received signal. On the other hand, the user with the better channel (denoted as user 2) performs SIC as it first decodes the data of the other user and then cancel it from the received signal. The interference-cancelled received signal is then used to decode the data for this user [5], [9]. For the case of subcarrier based NOMA with two users (U = 2), the ergodic rate for user 1 and user 2 are respectively R1 = B sc

Nsc

log2 (1 + γ1,n )

(2)

log2 (1 + γ2,n )

(3)

n=1

R2 = B sc

Nsc n=1

P

Fig. 1. Power spectrum of OFDMA and NOMA

Suppose that the BS is transmitting the signal Xu,n to the u-th user (u = {1, 2, ...U }) and at the n-th subcarrier (n = {1, 2, ...Nsc }) with transmission power Pu,n . The received signal by user u at the n-th subcarrier is given by [14]: Yu =

U

Pi,n Hi,n Xi,n + Ni

|H

|2 g

P

|H

|2

1,n u 2,n where γ1,n = P2,n |H1,n and γ2,n = 2,n are 2 B sc N0 1,n | g u +B sc N0 the received SINR at the n-th subcarrier of user 1 and user 2 WT is the sub-carrier bandwidth, and N0 respectively, B sc = N sc represents the noise power spectral density. H u = ξ|hu,n |2 d−υ represents the channel gain that includes the Rayleigh fading as the |hu,n |2 , the Log-normal shadowing factor ξ, and d−υ u distance between the user and the serving base station with υ being the path loss exponent. From equations (2) and (3) it could be noted that the user power has a direct effect not only on its achieved data rate but also on the other users SINR, which reflects the importance of power allocation in NOMA.

(1)

III. PROBLEM FORMULATION AND PROPOSED POWER ALLOCATION

i=1

where Hi,n represents the channel gain between the BS and the user at the n-th subcarrier and Ni represents the additive white Gaussian noise. From equation (1), it is clear that each user will receive a signal with its data and the data intended for the other users. Therefore, successive interference cancellation (SIC) is applied at NOMA receiver so each user could extract its respective data as shown in Fig. 2.

The aim of this work is to devise power allocation schemes to maximize the sum rate of NOMA system. However, it is important to guarantee a minimum rate for each user to provide some degree of fairness. Therefore, the optimization problem is to provide a minimum rate to each user whilst maximizing the overall sum rate within the limits of Pt as the total transmission power. Taking into account the two-user

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scenario and each one uses Nsc subcarriers, the sum rate for NOMA could be expressed as Nsc

Rsum = Bsc

log2 (1 + γ1,n ) + Bsc

n=1

Nsc

Setting each of these equations to zero and solving (9) for the Lagrange variable ψ we obtain − (γ2,n )2 γ1,n + − ψ = Bsc P2,n (1 + γ2,n ) P1,n (1 + γ1,n ) Bsc (γ2,n )2 |H1,n |2 τ + . N0 Φ (1 + γ1,n ) ΦP2,n (1 + γ2,n )

log2 (1 + γ2,n ) . (4)

n=1

Mathematically, the problem is formulated as maximize

Rsum Nsc U

Subject to

(5) Pu,n ≤ PT

Next, by substituting (13) in (10) and solving for P1,n we obtain

(6)

u=1 n=1

Pu,n ≥ 0, ∀u, ∀n 0 ≥ Ru ≥ rmin

(7) (8)

where (6) and (7) represent the total power constraints, and (8) represents the minimum rate constraints to maintain fairness among users with rmin as the minimum target rate.

P1,n = −

Bsc N0

|H1,n |2 + |H2,n |2 τ + |H2,n |2 − |H1,n |2 Φ . 2|H2,n |2 |H1,n |2 Nsc τ (14)

At this point, the solution is optimal; however, solving for P2,n by using (11) would be Nsc

P2,n = Pt +

(15)

n=1

A. Optimal Solution Taking into account the objective function in (4) and using one of the optimization methods in [15], the Lagrangian function of the optimization problem in (5) could be expressed as F = Bsc

Nsc

log2 (1 + γ1,n ) + Bsc

n=1

−ψ

N sc

P1,n +

n=1

Nsc

Nsc

log2 (1 + γ2,n ) −

P2,n − Pt

−τ

n=1

Nsc

Bsc

Bsc

Nsc

n=1

n=1

−

Φ

P2,n + P1,n =

log2 (1 + γ2,n )

Nsc − (γ2,n )2 dF γ1,n + − ψ− = Bsc dP1,n P2,n (1 + γ2,n ) P1,n (1 + γ1,n ) n=1

n=1

|h1,n |2 Bsc (γ2,n )2 + No Φ (1 + γ1,n ) ΦP2,n (1 + γ2,n )

Nsc

Pt + (17) Nsc 2 2 2 2 Bsc N0 |H1,n | + |H2,n | τ + |H2,n | − |H1,n | Φ . 2|H2,n |2 |H1,n |2 Nsc τ

(9)

This means that the total power for all subcarriers will be the same; however, the subcarrier received by each user will be allocated a share of this power which is not necessarily the same. The power share of each subcarrier at each user will be decided by the close form solution derived below. Using (17) to solve (12) for the second Lagrange variable τ we obtain √

τ =

Nsc

γ2,n γ2,n τ dF + Bsc −ψ = Bsc dP2,n P ΦP 2,n (1 + γ2,n ) 2,n (1 + γ2,n ) n=1 n=1 (10) Nsc Nsc dF P2,n − P1,n = Pt − dψ n=1 n=1

dF dτ

=

Bsc Bsc

Nsc

log2 [1 + γ2,n ] − Φ n=1 Nsc log2 [1 + γ1,n ] n=1

Φ

(16)

P2,n =

where ψ, τ and Φ represent the Lagrange multipliers and the target data rate respectively. Differentiating against P1,n , P2,n , ψ, and τ , respectively, we obtain

τ

Pt Pt −→ P2,n = − P1,n . Nsc Nsc

Using (16) to solve (15) for P2,n we obtain

Φ

|H1,n |2 + |H2,n |2 τ + |H2,n |2 − |H1,n |2 Φ . 2|H2,n |2 |H1,n |2 Nsc τ

Solving this requires the use of computationally intensive numerical solutions. In the following subsections, we present two low complexity sub-optimal closed form solutions.

n=1

Nsc

The first suboptimal approach is that for each subcarrier, the total transmitted power is set to the same, i.e.,

log2 (1 + γ1,n )

n=1

Nsc Bsc N0

B. Equal Subcarrier Power Allocation (ESPA)

n=1 Nsc

(13)

.

(11)

Nsc Bsc N0 Φ |H1,n |2 − |H2,n |2

. 4Pt (|H2,n |2 )2 |H1,n |2 + Nsc Bsc N0 (|H1,n |2 + |H2,n |2 )2 (18)

Finally, substituting (18) into (14), the sub-optimal power per subcarrier for user 1 and user 2 is found to be, P1,n = −

(12) −

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Bsc N0 |H1,n |2 + |H2,n |2 2|H1,n |2 |H2,n |2

−

Nsc Bsc N0 4Pt (|H2,n |2 )2 |H1,n |2 + Nsc Bsc N0 (|H1,n |2 + |H2,n |2 )2 2|H1,n |2 |H2,n |2 Nsc

(19)


4

Table I S IMULATION PARAMETERS .

2Pt |H2,n |2 |H1,n |2 + Nsc Bsc N0 |H1,n |2 + |H2,n |2

−

P2,n = − 2|H1,n |2 |H2,n |2 Nsc Nsc Bsc N0 4Pt (|H2,n |2 )2 |H1,n |2 + Nsc Bsc N0 (|H1,n |2 + |H2,n |2 )2 2|H1,n |2 |H2,n |2 Nsc

Parameter Name No. of Subcarriers (N sc ) Total Transmission Power (Pt ) Total Bandwidth (WT ) Shadowing standard deviation Noise Power Density (No ) Cell Radius Path Loss Exponent (υ)

(20)

respectively.

Value 128 10 dBm 5 MHz 8 dB -174 dBm / Hz 50 m 3

C. Average Channel Based Power Allocation (ACPA) The proposed ESPA is a closed form solution, which has significantly lower complexity than the optimal solution. However, it requires one power allocation calculation for each subcarrier. To further reduce the complexity, we propose a second approach that use the average channel gain for power allocation. Inother words, for each user u the average channel Nsc |H |2 gain Gu = n=1Nsc u,n will be used to determine the power to be allocated. Applying this approach to (19) and (20), the sub-optimal power per subcarrier for user 1 is

−

P1,n = −

Bsc N0 (G1 + G2 ) − 2G1 G2

Nsc Bsc N0 4Pt (G2 )2 G1 + Nsc Bsc N0 (G1 + G2 )2 2G1 G2 Nsc

compared to the proposed low complexity ESPA and ACPA schemes, and other existing schemes such as FPA, FSPA and FTPA in [7], and UFPA in [5]. Also the OFDMA system with optimal power allocation will be compared to show the advantage of NOMA. In Fig. 3, the performance curves are illustrated in terms of the sum rate for total power constraints that vary from 10 dBm to 20 dBm. From this figure, it is clear that NOMA with optimal power allocation performs significantly better than OFDMA as it achieves higher sum rate. In addition, the performance of the proposed sub-optimal methods only have small degradation from the optimal solution but with much lower complexity.

(21)

and those of user 2 is

4.5

2Pt G2 G1 + Nsc Bsc N0 (G1 + G2 ) − 2G1 G2 Nsc

Nsc Bsc N0 4Pt (G2 )2 G1 + Nsc Bsc N0 (G1 + G2 )2

4

P2,n =

2G1 G2 Nsc

. (22)

Sum Rate (Mbps)

−

3.5

Equations (21) and (22) represent the power allocated based on the average channel gain (ACPA). This method offers simplicity over the ESPA method, and will also be compared to the optimal solution in the next section.

3 2.5

Optimal NOMA ACPA ESPA FSPA UFPA FPA FTPA Optimal OFDMA

2 1.5 1

D. Extension to Multiuser Case To investigate the scenario of U users, we divide the total bandwidth into a number of subbands having the same number of subcarriers. A maximum of two users will be multiplexed per each subband. The power across the subbands will be equally allocated while within each subband the power will be allocated using the proposed schemes. The total transmission power across all subbands must equals the total power Pt so that the total power constraint in (6) is maintained. IV. SIMULATION AND RESULTS We consider a downlink scenario consisting of a small cell at the centre and two cellular users uniformly distributed within the circular coverage area of radius R = 50m. The wireless channel is modeled as a six-path frequency selective fading channel using the ITU pedestrian - B model where the average power of the multi-path are [0 dB, -0.9 dB, -4.9 dB, -8 dB, -7.8 dB, -23.9 dB]. Table I depicts the simulation parameters that are used in the simulation. The optimal power allocation for NOMA are numerically solved and the performance is

0.5 0 10

12

14 16 Total Power (dBm)

18

20

Fig. 3. Sum rate against the maximum transmission power

Next, the coverage probability is simulated which is defined as the probability that a user achieves a rate that exceeds the target rate. The simulation parameters and set up are all the same as the previous one with a total transmission power of 20 dBm. Fig. 4 shows a comparison in terms of the coverage probability for different target rate. Again the optimal NOMA shows a superior performance against the optimal OFDMA. This means that NOMA can achieve a higher sum rate as well as a higher coverage probability than OFDMA. It also depicts how close the performance of the sub-optimal methods are to the optimal one. From these two figures, it is clear that the proposed methods are better than the other existing methods. In particular, the ESPA method provides better and closer

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performance to the optimal one than the ACPA method. This is because ESPA allocates power on a per-subcarrier basis, while ACPA does so based on the average gain of the whole subband. These figures are also clear evidence that NOMA is better than OFDMA in utilizing both the power and the bandwidth. Optimal NOMA ACPA ESPA FSPA UFPA FTPA FPA Optimal OFDMA

1 0.9

Coverage Probability

0.8 0.7 0.6

R EFERENCES

0.5 0.4 0.3 0.2 0.1 0

0.6

0.8

1 1.2 1.4 Target Data Rate (Mbps)

1.6

1.8

Fig. 4. Coverage Probability against various rate constraints for a total power of 20 dBm

Finally, in Fig. 5 we investigate the scenario of U users as described in Section III-D, with our proposed methods the simulation is carried out for 16 subbands. From this figure, it is clear that the proposed methods are better than the other existing methods and is close to the optimal scheme. 5 Optimal NOMA ACPA ESPA FSPA UFPA FTPA FPA

4.5 4

Sum Rate (Mbps)

subcarrier for each user. It also presents the comparisons between the optimal and the two proposed methods with optimal OFDMA in terms of the coverage probability and the sum rate. Simulation results show that NOMA provides better performance than OFDMA. Moreover, the proposed sub-optimal methods achieve comparable performance to the optimal one with the advantage of lower complexity and also better than the other NOMA existing schemes. Amongst the two proposed low complexity methods with closed form solution, the one that allocate power based on equal subcarrier power performs slightly better than the channel average based power allocation method at the expense of slightly higher complexity.

3.5 3 2.5 2 1.5 1 0.5

5

10

15 20 Number of Users

25

30

Fig. 5. Sum rate against the number of users for total power of 20 dBm

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V. CONCLUSION This paper investigates the power allocation problem for subcarrier based NOMA system. Two sub-optimal power allocation methods are proposed to allocate the power per

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