Fairness Comparison of Uplink NOMA and OMA

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1 Fairness Comparison of Uplin N and Zhiqiang Wei, Jiajia Guo, Derric Wing Kwan Ng, and Jinhong Yuan arxiv:7.4959v [cs.it] 5 Mar 7 Abstract In this paper, we compare the resource allocation fairness of uplin communications between non-orthogonal multiple access (N) schemes and orthogonal multiple access () schemes. Through characterizing the contribution of the individual user data rate to the system sum rate, we analyze the fundamental reasons that N offers a more fair resource allocation than that of in asymmetric channels. Furthermore, a fairness indicator metric based on Jain s index is proposed to measure the asymmetry of multiuser channels. More importantly, the proposed metric provides a selection criterion for choosing between N and for fair resource allocation. Based on this discussion, we propose a hybrid N- scheme to further enhance the users fairness. Simulation results confirm the accuracy of the proposed metric and demonstrate the fairness enhancement of the proposed hybrid N- scheme compared to the conventional and N schemes. I. INTRODUCTION In the upcoming 5th generation (5G) wireless networs, non-orthogonal multiple access (N) has been recognized as a promising consideration of multiple access scheme to accommodate more users and to improve the spectral efficiency[], [], [], [4], [5]. A preliminary version of N, multiuser superposition transmission (MUST) scheme, has been proposed in the rd generation partnership project long-term evolution advanced (GPP-LTE-A) networs[6]. The principal idea of N is to exploit the power domain for multiuser multiplexing and to utilize successive interference cancellation (SIC) to harness interuser interference (IUI). In contrast to conventional orthogonal multiple access () schemes [7], [8], N enables simultaneous transmission of multiple users on the same degrees of freedom (DOF) via superposition coding with different power levels. Meantime, by exploiting the received power disparity, advanced signal processing techniques, e.g., SIC, can be adopted to retrieve the desired signals at the receiver. It has been proved that N can increase the system spectral efficiency substantially compared to the conventional schemes[9], [], []. As a result, N is able to support massive connections, to reduce communication latency, and to increase system spectral efficiency. Most of existing wors focused on downlin N systems [9], [], [], []. However, N inherently exists Zhiqiang Wei, Jiajia Guo, Derric Wing Kwan Ng, and Jinhong Yuan are with the School of Electrical Engineering and Telecommunications, the University of New South Wales, Australia ( zhiqiang.wei@student.unsw.edu.au; jiajia.guo@student.unsw.edu.au; w..ng@unsw.edu.au; j.yuan@unsw.edu.au). Derric Wing Kwan Ng is supported under Australian Research Council Discovery Early Career Researcher Award funding scheme (project number DE77). This wor is partially supported by Australia Research Council (ARC) Discovery Project DP in uplin communications, where electromagnetic waves are naturally superimposed with different received power at a receiving base station (BS). Besides, SIC decoding is generally more affordable for BSs than mobile users. The authors in [] compared N and in the uplin from the perspective of spectral-power efficiency. Most recently, the authors in [4], [5] designed a resource allocation algorithm based on the maximum lielihood (ML) receiver at the BS. On the other hand, another ey feature of N is to offer fairness provisioning in resource allocation. In contrast to systems where users with poor channel conditions may temporarily suspended from service, N allows users with disparate channel conditions being served simultaneously. In [6], a power allocation scheme was proposed to provide the max-min fairness to users in an uplin N system. In [7], the authors studied a proportional fair based scheduling scheme for non-orthogonal multiplexed users. In [8], [9], power allocation with fairness consideration was investigated for single antenna and multiple antennas N downlin systems, respectively. Despite some preliminary wors [6], [7], [8], [9], [], [] have already considered fairness in resource allocation, it is still unclear why and when N offers a more fair resource allocation than that of. In this paper, we aim to compare the fairness in resource allocation of uplin between N and. To this end, a selection criterion is proposed for determining whether N or should be used given current channel state information. Through characterizing the contribution of achievable data rate of individual users to the system sum rate, we explain the underlying reasons that N is more fair in resource allocation than that of in asymmetric channels. Furthermore, for two-user N systems, we propose a closed-form fairness indicator metric to determine when N is more fair than. In addition, a simple hybrid N- scheme which adaptively chooses N and according to the proposed metric is proposed to further enhance the users fairness. Numerical results are shown to verify the accuracy of our proposed metric and to demonstrate the fairness enhancement of the proposed hybrid N- scheme. The rest of the paper is organized as follows. In Section II, we present the uplin N system model and discuss the capacity regions of N and. In Section III, the reason of N being more fair than is analyzed. Besides, a closed-form fairness indicator metric and a hybrid N- scheme are proposed. Simulation results are In a two-user N system, there are at most two users multiplexing on the same DOF to reduce the computational complexity and delay incurred by SIC at receivers.

2 Base station UserK User User Fig.. The system model for uplin N with one BS and K users. presented and analyzed in Section IV. Finally, Section V concludes this paper. Notations used in this paper are as follows. The circularly symmetric complex Gaussian distribution with mean µ and varianceσ is denoted bycn(µ,σ ); stands for distributed as ; C denotes the set of all complex numbers; denotes the absolute value of a complex scalar; Pr{ } denotes the probability of a random event. II. SYSTEM MODEL In this section, we present an uplin N system model and introduce the capacity regions of N and. A. System Model We consider an uplin N system with one singleantenna BS and K single-antenna users, as shown in Figure. All the K users are transmitting within a single subcarrier with the same maximum transmit power P. For the N scheme, K users are multiplexed on the same subcarrier with different received power levels, while for the scheme,k users are utilizing the subcarrier via the time-sharing strategy []. For the N scheme, the received signal at the BS is given by y = K p h s +v. () = where h C denotes the channel coefficient between the BS and user {,...,K}, s denotes the modulated symbol for user, p denotes the transmit power of user, and v CN(,σ ) denotes the additive white Gaussian noise (AWGN) at the BS andσ is the noise power. Without loss of generality, we assume that h h h K. B. Capacity Region It is well nown that the scheme with the optimal DOF allocation and the N scheme with the optimal power allocation can achieve the same system sum rate in uplin transmission[], [], as shown in Figure. Here, the optimal resource allocation for both N and schemes is in the sense of maximizing the system sum rate. To simplify the notations, we focus on the system with K users multiplexing on a single subcarrier. This case will generalized to the case of multi-carrier systems in Section III-D and Section IV. R (bit/s/hz) A C Optimal point of h h = Capacity region of Capacity region of N Optimal point for Optimal point for N h h = R (bit/s/hz) D B Optimal point of N Fig.. The capacity region of N and with one BS and two users for a single channel realization. The transmit power of both users is P = dbm. For the curve of h h =, we have h σ = h σ = db. For the curve of h h =, we have h σ = 8 db and h σ = 8 db. To facilitate the following presentation, we define α as a time-sharing factor for user, where K α =. Particularly, = the optimal DOF allocation of the scheme, i.e., point C and point F in Figure, can be achieved by []: α = h,. () K h i Note that α can also be interpreted as the normalized channel gain of user. In other words, the optimal DOF allocation for the scheme is to share the subcarrier with the time duration proportional to their normalized channel gains, whereas it relies on adaptive time allocation according to the instantaneous channel realizations. We note that the optimal DOF allocation is obtained with all the users transmitting with their maximum transmit power P since there is no IUI in the scheme. On the other hand, power allocation of N that achieves the corner points, i.e., point A, point B, point D, and point E in Figure, can be obtained by simply setting p = P,, and performing SIC at the BS[], [4]. Any rate pairs on the line segments between the corner points can be achieved via a timesharing strategy. It can be observed from Figure that N with a time-sharing strategy always outperforms, both in the sense of spectral efficiency and user fairness, since the capacity region of is a subset of that of N. We note that N without the time-sharing strategy can only achieve the corner points in the capacity region, which might be less fair than in some cases. In this paper, we study the users fairness of the N scheme without time-sharing and the scheme with an adaptive DOF allocation. Both schemes achieve the same system sum rate but results in different users fairness. Intuitively, F E

3 in Figure, for symmetric channel with h =, h at point C is more fair than N since both users have the same individual data rate. However, for an asymmetric channel with h =, it can be observed that N at h the optimal point D is more fair than at the optimal point F. Therefore, it is interesting to unveil the reasons for fairness enhancement of N in asymmetric channels and to derive a quantitative fairness indicator metric for determining when N is more fair than. III. FAIRNESS COMPARISON OF N AND In this section, we first present the adopted Jain s fairness index[5] for quantifying the notion of resource allocation fairness. Then, we characterize the contribution of individual user data rate to the system sum rate and investigate the underlying reasons of N being more fair than. Subsequently, for a two-user N system, a closed-form fairness indicator metric is derived from Jain s index [5] to determine whether using N or for any pair of users on a single subcarrier. Furthermore, a hybrid N- scheme is proposed which employs N or adaptively based on the proposed metric. A. Jain s Fairness Index In this paper, we adopt the Jain s index[5] as the fairness measurement in the following ( K ) R = J =, () K K (R ) = where R denotes the individual rate of user. Note that K J. A scheme with a higher Jain s index is more fair and it achieves the maximum when all the users obtain the same individual data rate. B. Fairness Analysis For the optimal resource allocation of both N and schemes discussed in Section II-B, it is easily to obtain the sum rate and individual data rates for both schemes as follows: K K sum = R sum = = log ( + P = σ R ) K h i, (4) = log + P h, and (5) P h i +σ R = α R sum, (6) where Rsum N and Rsum denote the system sum rate for N and schemes with the optimal resource allocation, respectively, and and R denote the R 5 R 4 R R R g(x) f(x) 4 Rsum N = Rsum α α α 4 α α Accumulative normalized channel gain φ Fig.. An illustration of system sum rate versus the accumulative normalized channel gains for the N and with K = 5 uplin users. The sum rates of the N scheme and the scheme are denoted by the green double-side arrow. The individual rates of the N scheme and the scheme are denoted by the red line segments and the blac line segments, respectively. individual data rate for user in N and schemes, respectively. For the N scheme, we first define the accumulative normalized channel gain as φ = α i, = {,,K}, φ =, and then rewrite the achievable rate of user as ( ) R N =log + P φ K σ h i ( ) log + P φ K σ h i. (7) The first term in (7) denotes the sum rate of a system with users and the second term denotes the counterpart of a system with users. In other words, the contribution of user to the system sum rate depends on the difference of a logarithm function with respect to (w.r.t.) φ and φ. For notational simplicity and without loss of generality, we define the logarithm function as with = P σ K g(x) = log (+x), x, (8) h i and = g(φ ) g(φ ). (9) On the other hand, for the scheme, it can be observed from (6) that R has a linear relationship with Rsum and the slope w.r.t. the system sum rate is determined by the normalized channel gain α = φ φ. Similarly, the contribution of user to the system sum rate depends on the Rate (bit/s/hz)

4 difference of a linear function of φ and φ, where the linear function is given by f (x) = log (+)x, x, and R = f (φ ) f (φ ). () Figure illustrates the linear and logarithmic increments of the system data rate w.r.t. the accumulative channel gain for and N, respectively, with K = 5 uplin users. It can be observed that the N and schemes have the same system sum rate but contributed by different date rates of individual users. In particular, the N scheme achieves a more fair resource allocation than that of the scheme since all the users are allocated with similar individual rates. In fact, the fairness of resource allocation in N inherits from the logarithmic mapping of g(φ ) w.r.t. the accumulative channel gain φ. The first and second derivatives of g(φ ) are increasing and decreasing w.r.t. φ, respectively. The larger normalized channel gain α, the slower g(φ ) increasing with φ, which results in a smaller individual rate compared to that of the scheme. On the other hand, a smaller normalized channel gain α would result in a higher increasing rate of g(φ ) with φ, when a higher individual rate is obtained compared to that of the scheme. For instance, considering the weaest user and the strongest user with their normalized channel gain α and α K, respectively, R N is raised up by the logarithm function g(x) compared to R, while RK N is reduced compared to RK. Remar : Note that for symmetric channels, linear mapping of the scheme is more fair than the N scheme. However, the probability that all the users have the same channel gains is quite small, especially for a system with a large number of users. C. Fairness Indicator Metric In practice, most of N schemes assume that there are at most two users multiplexing via the same DOF[6], [], [], which can reduce both the computational complexity and decoding delay at the receiver. Therefore, we focus on the fairness comparison of N and with K = in this section. We aim to find a simple metric to determine when N is more fair than for any pair of users, which is fundamentally important for user scheduling design in the system with multiple DOF and multiple users. The fairness indicator metric is proposed in the following theorem. Theorem : Given a pair of users with their channel realizations h h, the N scheme is more fair in the sense of Jain s fairness index if and only if W ( h h β β, () (+) + log(+) where β = log(+) and W(x) is the Lambert W function. In the high SNR regime, i.e.,, we have the high SNR approximation of β as ) β W (log(+)). () log(+) Proof : Since both the N and schemes have the same sum rate, we need to compare the sum of square of individual rates (SSR), i.e., SSR = (R ), in the = denominator of (). The scheme with a smaller SSR would be more fair in terms of Jain s index. For the scheme, we have SSR = (log (+)) ( α +α ) = (log (+)) ( +α α ), () where α since we assume h h. For the N scheme, the SS can be given by SS =(log (+α )) +(log (+) log (+α )) =(log (+)) +(log (+α )) log (+)log (+α ). (4) Note that a trivial solution for SSR = SS is given with α =, which corresponds to a single user scenario. In addition, at α =, i.e., h = h, we have SSR < SS as observed from the capacity region in Figure. Further, SSR is a monotonic decreasing function of α within α, while SS is a monotonic decreasing function of α within α + and it is increasing with α within + α. Also, from Figure, we can observe that SSR > SS for an arbitrary small positive α. Therefore, there is a unique intersection of SSR and SS atα = β in the range of < α <. Before the intersection, i.e., < α < β, N is more fair, while after the intersection, i.e.,β < α <, is more fair. Solving the equation of SSR = SS within < α <, we obtain ( W β = ) (+) + log(+) log(+). (5) Furthermore, with α β, we have h β, which completes the proof for the sufficiency of the proposed fairness indicator metric. For the necessity, since the intersection of SSR and SS within < α < is unique, the only region within < α < where SSR > SS is < α < β. In other words, N is more fair only if < α < β, which completes the proof for the necessity of the proposed metric. Remar : Note that the proposed fairness indicator metric only depends on the parameter defined in (9). As a result, the metric depends on the instantaneous channel gains. Compared to the Jain s index, our proposed metric is more insightful which connects and N. Particularly, for the high SNR approximation (), we can observe that β decreases with the increasing maximum transmit power since the Lambert W function in the numerator increases slower than that of the denominator. Therefore, the probability of N being more fair will decrease when increasing the maximum transmit power, which will be verified in the simulations. h β

5 Pr{J N J } Pr{ h β h β } Approx., Pr{ h h β β } N Hybrid N- Probability PDF Maximum transmit power P o (dbm) Fig. 4. The probability of N being more fair than versus the maximum transmit power, P. 5 5 User rate (bit/s/hz) Fig. 5. The PDF of user rate for the N scheme, the scheme, and the hybrid N- scheme. D. A Hybrid N- Scheme The proposed fairness indicator metric in Theorem provides a simple way to determine if N is more fair than, and would serve as a criterion for user scheduling design for systems with multi-carrier serving multiple users. In particular, for an arbitrarily user scheduling strategy, we propose an adaptive hybrid scheme which decides each pair of users on each subcarrier in choosing either the scheme or the N scheme to enhance users fairness. Instead of using the N scheme or the scheme across all the subcarriers, this hybrid N- scheme can enhance the user fairness substantially. Note that the fairness performance can be further improved if it is jointly designed with the user scheduling. It will be considered in the future wor. IV. SIMULATION RESULTS In this section, we adopt simulations to verify the effectiveness of the proposed metric and to evaluate the proposed hybrid N- scheme. A single cell with a BS located at the center with a cell radius of 4 m is considered. There are N F = 8 subcarriers in the system and N F numbers of users are randomly paired on all the subcarriers. All the N F users are randomly and uniformly distributed in the cell. We set the noise power in each subcarrier at the BS as σ = 9 dbm. The GPP path loss model in urban macro cell scenario is adopted in our simulations[7]. Figure 4 depicts the probability of N being more fair than versus the maximum transmit } power, P. It can be observed that Pr { h h β β matches well with Pr{J N J }. In other words, our proposed fairness indicator metric can accurately predict if N is more fair than. Also, } for the high SNR approximation β in (), { h β h β Pr closely matches with the simulation results. In addition, we can observe that the N scheme has a high probability (.75.8) of being more fair than that of the scheme in terms of Jain s index. This is due CDF N Hybrid N- 5 5 User rate (bit/s/hz) Fig. 6. The CDF of the N scheme, the scheme, and the hybrid N- scheme. to the fact that the probability of asymmetric channels is much larger than that of symmetric channels. On the other hand, the probability of N being more fair is decreasing with the maximum transmit power as discussed in Remar. This is because the N scheme is interference-limited in the high transmit power regime. Specifically, the strong user (with higher received power) will face a large amount of interference, while the wea user (with lower received power) is interference-free owing to the SIC decoding. As a result, in the high transmit power regime, the wea user can achieve a much higher data rate than that of the strong user, which may result in a less fair resource allocation than that of. Even though, N is still more fair than with a probability of about.75 in the high transmit power regime. Figure 5 shows the probability density function (PDF) of user rate for a multi-carrier system with a random pairing strategy. Three multiple access schemes are compared, includ-

6 ing the N scheme, the scheme, and the proposed hybrid N- scheme. It can be observed that the individual data rate distribution of the N scheme is more concentrated than that of the scheme, which means that the N scheme offers a more fair resource allocation than the scheme. Further, the individual rate distribution of the hybrid N- scheme is more concentrated than that of the N scheme. In fact, our proposed hybrid N- scheme can better exploit the channel gains relationship via the adaptive selection between N and according to the fairness indicator metric. Actually, for the three multiple access schemes, we have J N =.76 J =.6, and J Hybrid =.9, where J Hybrid denotes the Jain s index for the hybrid N- scheme. In addition, the cumulative distribution function (CDF) of user rate is more of interest in practice, which is illustrated in Figure 6. We can observe that the th-percentile the user rate, which is closely related to fairness and user experience, increased about bit/s/hz compared to that of the N scheme. This shows that our proposed hybrid N- scheme can significantly improve the performance of low-rate users and therefore elevate the quality of user experience. V. CONCLUSION In this paper, we investigated the resource allocation fairness of the N and schemes in uplin. The fundamental reason of N being more fair than in asymmetric multiuser channels was analyzed through characterizing the contribution of data rate of each user to the system sum rate. It is the logarithmic mapping between the normalized channel gains and the individual data rates that exploits the channel gains asymmetry to enhance the users fairness in the N scheme. Based on this observation, we proposed a quantitative fairness indicator metric for two-user N systems which determines if N offers a more fair resource allocation than. In addition, we proposed a hybrid N- scheme that adaptively choosing between N and based on the proposed metric to further improve the users fairness. Numerical results demonstrated that our proposed metric can accurately predict when N is more fair than. Besides, compared to the conventional N and schemes, the proposed hybrid N- scheme can substantially enhance the users fairness. REFERENCES [] L. Dai, B. Wang, Y. Yuan, S. Han, I. Chih-Lin, and Z. Wang, Nonorthogonal multiple access for 5G: solutions, challenges, opportunities, and future research trends, IEEE Commun. Mag., vol. 5, no. 9, pp. 74 8, Sep. 5. [] Z. Ding, Y. Liu, J. Choi, Q. Sun, M. Elashlan, C. L. I, and H. V. Poor, Application of non-orthogonal multiple access in LTE and 5G networs, IEEE Commun. Mag., vol. 55, no., pp. 85 9, Feb. 7. [] Z. Wei, J. Yuan, D. W. K. Ng, M. Elashlan, and Z. 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