I Introduction
Recently, nonorthogonal multiple access (NOMA) has drawn significant attention in both industry and academia as a promising multiple access technique for the fifthgeneration (5G) wireless newtworks[1, 2, 3, 4]. The principal of NOMA is to exploit the power domain for multiuser multiplexing together with superposition coding and to alleviate interuser interference (IUI) via successive interference cancellation (SIC) techniques[3]. Towards the evolution of 5G, the industrial community has proposed various forms of NOMA as potential multiple access technologies[5].
In contrast to conventional orthogonal multiple access (OMA) schemes [6], NOMA can serve multiple users via the same degree of freedom (DOF) and achieve a higher spectral efficiency[7, 8, 9]. In particular, in singleantenna systems, it has been proved that singleinput singleoutput NOMA (SISONOMA) can provide a higher ergodic sumrate than that of SISOOMA with a fixed resource allocation scheme[10]. Furthermore, the superior performance of SISONOMA over SISOOMA with an optimal resource allocation design in singleantenna systems was shown in [11]. In [12, 13], the authors firstly revealed that the sources of performance gain of NOMA over OMA are twofold: nearfar diversity gain and angle diversity gain when there are multiple antennas. However, analytical results for quantifying the ergodic sumrate gain (ESG) of NOMA over OMA has not been reported yet. More importantly, the potential gain of NOMA over OMA has not been well understood to provide some system design insights.
To achieve a higher spectral efficiency, the concept of NOMA has been extended to multiantenna systems, namely multipleinput multipleoutput NOMA (MIMONOMA), based on the signal alignment technique [14] and the concept of quasidegradation [15]. In particular, it has been shown that there is a performance gain of MIMONOMA over multipleinput multipleoutput OMA (MIMOOMA)[14, 15]. However, when we extend NOMA from singleantenna systems to multiantenna systems, it is still unclear, if the same ESG can be achieved. Moreover, how the ESG of NOMA over OMA changes from singleantenna to multiantenna systems is still an open and interesting problem and deserves our efforts to explore. The answers to these questions above are the key to unlock the potential in applying NOMA to the future 5G communications systems.
This paper aims to deepen the understanding on the ESG of NOMA over OMA in uplink singlecell systems. In particular, we quantify the ESG of NOMA over OMA in both singleantenna and multiantenna systems. Furthermore, performance analyses are provided to study the improvement of ESG when extending NOMA from singleantenna systems to multiantenna systems. Simulation results confirm the accuracy of our derived performance analyses and demonstrate some interesting insights which are summarized in the following:

In both singleantenna and multiantenna scenarios, the ESGs of NOMA over OMA saturate for large numbers of users and sufficiently high signaltonoise ratio (SNR).

In singleantenna scenario, we identify two types of nearfar gains[13] brought by NOMA and reveal their different behaviors. In particular, the largescale nearfar gain via exploiting the largescale fading increases with the cell size, while the smallscale nearfar gain arising from the smallscale fading is a constant of nat/s/Hz in Rayleigh fading channels.

As applying NOMA in multiantenna systems, the ESG of SISONOMA over SISOOMA can be amplified fold when the base station is equipped with antennas owing to the spatial degrees of freedom[13].
Notations used in this paper are as follows. Boldface capital and lower case letters are reserved for matrices and vectors, respectively.
denotes the transpose of a vector or matrix and denotes the Hermitian transpose of a vector or matrix. denotes the set of all matrices with complex entries; denotes the absolute value of a complex scalar or the determinant of a matrix, anddenotes Euclidean norm of a complex vector. The circularly symmetric complex Gaussian distribution with mean
and variance
is denoted by .Ii System Model
Iia System Model
We consider an uplink^{1}^{1}1We restrict ourselves to the uplink NOMA communications, as the reception of NOMA is more affordable for the base station. singlecell NOMA system with a base station (BS) and users, as shown in Fig. 1. The cell is modeled as two concentric ringshaped discs. The BS is located at the center of the ringshaped discs with the inner radius of and outer radius of , wherein all the users are scattered uniformly. For the NOMA scheme, all the users are multiplexed on the same frequency band and time slot, while for the OMA scheme, users utilize the frequency or time resources orthogonally. Without loss of generality, we consider a frequency division multiple access (FDMA) as a typical OMA scheme.
As shown in Fig. 1, we consider two kinds of typical communication systems:

SISONOMA and SISOOMA: the BS is equipped with a singleantenna () and all the users are singleantenna devices.

MIMONOMA and MIMOOMA: the BS is equipped with a multiantenna array () and all the users are singleantenna devices with .
Note that, in the second case, we still denote them as MIMONOMA and MIMOOMA even each user is equipped with a singleantenna, because there are multiple users uploading their independent data streams to the BS.
The received signal at the BS is given by
(1) 
where , denotes the power allocation for user , denotes the normalized modulated symbol for user with , and denotes the additive white Gaussian noise (AWGN) at the BS with zero mean and covariance matrix of . For the system power budget, we consider a sumpower constraint for all the users, i.e.,
(2) 
where is the maximum transmit power for all the users. Note that, the sumpower constraint is a commonly adopted assumption in the literature to simplify the performance analysis for uplink communications, e.g. [16, 17, 13]. In fact, the sumpower constraint is a reasonable assumption for practical cellular communication systems, where a total transmit power limitation is usually imposed to prevent a large intercell interference.
The channel vector between user and the BS is modeled as
(3) 
where denotes the Rayleigh fading coefficients, i.e., , denotes distance between user and the BS with the unit of meter, and denotes the path loss exponent^{2}^{2}2In this paper, we ignore the lognormal shadowing to simplify our performance analysis. Note that, the lognormal shadowing only introduces an additional scalar multiplication to the channel model in (3). Although the lognormal shadowing may change the resulting channel distribution of , the distancebased channel model is sufficient to characterize the nearfar gain [13] discussed in this paper.. We note as the channel matrix between all the users and the BS. When , denotes the corresponding channel coefficient of user in singleantenna systems. We assume that the channel coefficients are independent and identically distributed (i.i.d.) over all the users and antennas. Since this paper aims to provide some theoretical insights about the performance gain of NOMA over OMA, we assume that perfect channel state information (CSI) is known at the BS. Without loss of generality, we assume , i.e., users are indexed based on their channel gains.
IiB Signal Detection and Resource Allocation Strategy
To facilitate the performance analysis, we focus on the following wellknown and effective signal detection and resource allocation strategies.
IiB1 Signal detection
For SISONOMA, we adopt the commonly used SIC [18] decoding at the BS, since it is capacity achieving for singleantenna systems[19]. On the other hand, for SISOOMA, as all the users are separated by orthogonal frequency subbands and thus the simple singleuser detection (SUD) can be used to achieve the optimal performance. For MIMONOMA, the minimum mean square error and successive interference cancellation (MMSESIC) decoding is an appealing reception technique since it is capacity achieving [19] with an acceptable computational complexity for a finite number of antennas at the BS. On the other hand, FDMA zero forcing (FDMAZF) is adopted for MIMOOMA, which is known as capacity achieving in the high SNR regime[19]. In particular, owing to the extra spatial DOF induced by multiple antennas at the BS, all the users can be divided into groups^{3}^{3}3Without loss of generality, we consider the case with as an integer in this paper. with each group containing users. Then, ZF is utilized to handle the interuser interference within each group and FDMA is utilized to separate all the groups on orthogonal frequency subbands.
IiB2 Resource allocation strategy
We consider an equal resource allocation (ERA) strategy for both NOMA and OMA scheme. In particular, equal power allocation is adopted for NOMA schemes, i.e., . On the other hand, equal power and frequency allocation is adopted for OMA schemes, i.e., and or , where and denotes the normalized frequency allocation for user in SISOOMA and for group in MIMOOMA, respectively. Note that the ERA is a typical but effective strategy utilized in some application scenarios requiring a limited system overhead, e.g. machinetype communications (MTC).
We note that the user grouping design is involved in the considered MIMOOMA system with FDMAZF. Fortunately, for any user grouping strategy, the ergodic sumrate of MIMOOMA remains the same, since all the users have i.i.d. channel distributions and can access the identical resource. Therefore, a random user grouping strategy is adopted for the MIMOOMA system with FDMAZF. In particular, we randomly select users for each group on each frequency subband and denote the corresponding composite channel matrix of th group as . On the other hand, we consider a fixed SIC and MMSESIC decoding order at the BS as for SISONOMA and MIMONOMA, respectively^{4}^{4}4Note that the SIC and MMSESIC decoding order at the BS do not affect the system sumrate in uplink SISONOMA systems[20] and uplink MIMONOMA systems[19], respectively.. In addition, to provide some theoretical insights about the performance gain of NOMA over OMA, we assume that there is no error propagation during SIC and MMSESIC decoding at the BS.
Iii ESG of SISONOMA over SISOOMA
In this section, we first derive the ergodic sumrate of SISONOMA and SISOOMA. Then, the ESG of SISONOMA over SISOOMA is discussed asymptotically.
Iiia Ergodic Sumrate
The instantaneous achievable data rate of user of the considered SISONOMA system is given by:
(4) 
while the instantaneous achievable data rate of user in the considered SISOOMA system is given by:
(5) 
Under the ERA strategy defined above, we have the instantaneous sumrate of SISONOMA and SISOOMA given by
(6)  
(7) 
respectively. It can be observed that in the low SNR regime with , the performance gain of SISONOMA over SISOOMA vanishes.
Since all the users are scattered uniformly in two concentric rings with the inner radius of and outer radius of
, the cumulative distribution function (CDF) of the channel gain
is given by^{5}^{5}5Since all the users have i.i.d. channel distributions, the subscript might be dropped without causing notational confusion in this paper.(8) 
where ,
, denotes the probability density function (PDF) for the random distance
. With the GaussianChebyshev quadrature approximation[21], the CDF and PDF of are given by(9)  
(10) 
respectively, where
(11) 
are approximation parameters and denotes the number of integral approximation terms.
Based on (6), we have the asymptotic ergodic sumrate of the considered SISONOMA system with as follows:
(12) 
where denotes the average channel power gain and it is given by
(13) 
Note that the equality in (IIIA) only holds asymptotically for due to . Otherwise, for a finite number of users , the asymptotic ergodic sumrate in (IIIA) serves as an upper bound for the actual ergodic sumrate, i.e.,
, owing to the concavity of the logarithm function and the Jensen’s inequality. In the simulations, we show that the asymptotic analysis in (
IIIA) is accurate even for a finite value of and becomes tight with increasing .IiiB ESG in Singleantenna Systems
Based on (IIIA) and (IIIA), the asymptotic ESG of SISONOMA over SISOOMA with can be given as follows:
(15) 
To obtain more insights, in the high SNR regime with , we approximate the asymptotic ESG in (IIIB) as
(16) 
where denotes the largescale nearfar gain and it is given by
(17) 
The EulerMascheroni constant, [21], denotes the smallscale nearfar gain in Rayleigh fading channels. The approximation in (16) is obtained by applying [21].
There are three important observations from (16). Firstly, it can be observed that the asymptotic ESG in (16) is not a function of but a function of and . It implies that the asymptotic ESG saturates for a sufficiently large . In fact, the sumrates of both SISONOMA systems in (6) and SISOOMA systems in (7) increase very slowly with , which results in a saturated ESG in the high SNR regime.
Secondly, based on the weighted arithmetic and geometric means (AMGM) inequality
[22], we can show that . In particular, for the extreme case that all the users are randomly deployed on a circle, i.e., , we have according to (IIIA) and and thus the performance gain achieved by NOMA is bounded by below with nat/s/Hz. Otherwise, the asymptotic ESG is larger than for a general cell deployment . As a result, in the asymptotic case with and , SISONOMA can provide at least nat/s/Hz spectral efficiency gain over SISOOMA for an arbitrary user deployment. In fact, the minimum asymptotic ESG gain arises from the smallscale Rayleigh fading since we force all the users have the same largescale fading when .Thirdly, the dominating component of the asymptotic ESG (16) originates from function given in (17), which actually characterizes the nearfar gain due to largescale fading. To visualize the largescale nearfar gain, we illustrate the asymptotic ESG in (16) versus and in Fig. 2. We can observe that, when , the minimum asymptotic ESG nat/s/Hz due to the smallscale nearfar gain can be obtained. Besides, when , the asymptotic ESG is much larger than due to the largescale nearfar gain . It can be observed that in (17) is actually a function of . It implies that, the larger , the larger and the larger ESG. In fact, for a larger , the heterogeneity in the largescale fading among users becomes higher and the SISONOMA can exploit the nearfar diversity more efficiently to improve the sumrate performance.
Remark 1
Note that, in the literature [23, 8], it has been shown that two users with a large distance difference or channel gain difference are preferred to be paired. It is consistent with our conclusion in this paper, where a larger implies a higher ESG of NOMA over OMA. However, it is worth to note that this is the first work which quantifies the ESG of NOMA over OMA and identifies two kinds of nearfar gains in ESG.
Iv ESG of MIMONOMA over MIMOOMA
In this section, we first derive the ergodic sumrate of MIMONOMA and MIMOOMA. Then, the ESG of MIMONOMA over MIMOOMA is discussed asymptotically.
Iva Ergodic Sumrate
The instantaneous achievable data rate of user of the considered MIMONOMA system with the MMSESIC detection is given by[19]:
(18) 
For the considered MIMOOMA system, the instantaneous achievable data rate of user is given by [19]
(19) 
where vector denotes the normalized ZF detection vector for user with , which is obtained based on the pseudoinverse of the channel matrix in the th user group[19].
With the ERA strategy, the instantaneous sumrate of MIMONOMA and MIMOOMA are obtained by
(20)  
(21) 
respectively.
In fact, MMSESIC is capacity achieving [19] and (20) is the channel capacity with the deterministic channel matrix [24]. It is difficult to obtain a closedform expression for the channel capacity above due to the determinant of summation of matrices in (20). To provide more insights, in the following theorem, we consider an asymptotically tight upper bound for the achievable sumrate in (20) with .
Theorem 1
proof 1
The main idea of proof is based on the bounding techniques in [17]. Due to the page limitation, we omit the detailed proof here and leave it for the journal version.
Given the distance from a user to the BS as , its channel gain
follows the Gamma distribution
[25], whose PDF and CDF are given by(24) 
respectively, where
denotes the PDF of a random variable with a Gamma distribution,
denotes the Gamma function, and denotes the lower incomplete Gamma function.Then, the CDF of the channel gain is given by
(25) 
Again, with the GaussianChebyshev quadrature approximation[21], the CDF and PDF of are given by
(26) 
respectively, where and are given by (IIIA).
According to (23), the asymptotic ergodic sumrate of MIMONOMA with is given by
(27)  
where denotes the average channel gain and it is given by
(28) 
Remark 2
Comparing (IIIA) and (27), we can observe that the performance of a MIMONOMA system is asymptotically equivalent to that of a SISONOMA system with fold orthogonal frequency bands and an equivalent channel gain of , when there is a sufficiently large number of users. On the other hand, for but a finite , the number of antennas at the BS is much smaller compared to the number of users. In such a case, the performance of this system approaches the one with a singleantenna BS. In addition, with a sufficiently large number of users, the received signals at the BS fully span the dimensional signal space[17]. Therefore, MIMONOMA with MMSESIC can fully utilize the system spatial DOF, , and its performance can be approximated by that of a SISONOMA system with fold frequency bands.
IvB ESG in Multiantenna Systems
Based on (27) and (29), the asymptotic ESG of MIMONOMA over MIMOOMA with can be obtained by:
(30) 
To reveal more insights, similar to equation (16), we consider the asymptotic ESG of MIMONOMA over MIMOOMA in the high SNR regime as follows
(31) 
where is given in (17).
Comparing (16) and (31), we have
(32) 
which implies that the asymptotic ESG of SISONOMA over SISOOMA is amplified by times when antennas are deployed at the BS. In fact, for , the received signals fully span in the dimensional signal space[17], which enables MIMONOMA and MIMOOMA to fully exploit the system spatial DOF, . In addition, to suppress the interuser interference in MIMOOMA, there is a factor of power loss on average within each group due to the ZF projection[19]. Therefore, we have an additional power gain of MIMONOMA over MIMOOMA in the third term in (32).
V Simulations
We use simulations to verify our derived analytical results. The inner cell radius is m and the outer cell radius is given by m. The number of users ranges from to and the number of antennas equipped at the BS is ^{6}^{6}6In this work, we focus on the multiantenna system with a finite number of antennas at the BS. The performance gain of NOMA over OMA with a massiveantenna array at the BS will be considered in our future work.. The path loss exponent is according to the 3GPP path loss model[26]. To characterize the system SNR in simulations, we define the received sum SNR at the BS as follows:
(33) 
where and are given by (13) and (28), respectively. The total transmit power is adjusted adaptively for different cell sizes to satisfy in (33) ranging from dB to dB. All the simulation results in this paper are averaged over both smallscale fading and largescale fading.
Fig. 3 illustrates the ESG of NOMA and OMA versus the number of users. We consider for SISONOMA and SISOOMA in Fig. 3(a) and for MIMONOMA and MIMOOMA in Fig. 3(b). In both singleantenna and multiantenna scenarios, we observe a higher ESG of NOMA over OMA in the high SNR case, e.g. dB. In addition, for both cases with and , the ESGs increase with the number of users monotonically and approach the derived asymptotic ESG expressions in (IIIB) and (IVB). It implies that the ESG saturates with increasing and thus the nearfar gain can be captured by a finite number of users. Comparing Fig. 3(a) and Fig. 3(b), we can observe that a substantial improvement in ESG when applying NOMA in multiantenna systems due to the extra spatial DOF as predicted in (32).
Fig. 4 depicts the ESG of NOMA over OMA versus , in both singleantenna and multiantenna scenarios. We can observe that the asymptotic analyses of ESG in (IIIB) and (IVB) matches simulation results in all the considered cases, particularly for the case with a low . With increasing the SNR, the ESGs monotonically approach the asymptotic analyses in (16) and (31) for the cases of and , respectively. In particular, in singleantenna scenario, when all the users are randomly distributed on a circle with m, we can observe an ESG about nat/s/Hz at dB. This verifies the accuracy of the derived smallscale nearfar gain in (16). Besides, in both singleantenna and multiantenna systems, we can observe that a larger results in a larger performance gain owing to the increased largescale nearfar gain. In addition, it can be observed that the ESG increases faster for the case with a larger . In other words, a higher largescale nearfar gain enables NOMA to utilize the power more efficiently.
Fig. 5 illustrates the ESG of MIMONOMA over MIMOOMA versus the number of antennas . It can be observed that the simulation results follow our asymptotic analyses derived in (IVB) and (31) closely, especially for the case with a small . More importantly, as predicted in (31), a larger enables a larger increasing rate in the ESG with respective to the number of antennas , due to the increased largescale nearfar gain .
Vi Conclusion and Discussion for Multicell Systems
In this paper, we investigated the ESG brought by NOMA over OMA in both singleantenna and multiantenna systems via asymptotic performance analyses for a sufficiently large number of users in the high SNR regime. For singleantenna systems, the ESG of NOMA over OMA was quantified and two types of nearfar gains were identified in the derived ESG, i.e., the largescale nearfar gain and the smallscale nearfar gain. The largescale nearfar gain increases with the cell size, while the smallscale nearfar gain is a constant of nat/s/Hz in Rayleigh fading channels. Furthermore, we unveiled that the ESG of SISONOMA over SISOOMA can be amplified by times when equipping antennas at the BS, owing to the extra spatial DOF offered by additional antennas.
In a multicell system, the system SNR in (33) should be redefined as follows:
(34) 
where characterizes the intercell interference[13]. Due to this interference, NOMA might work at a low SNR regime and the performance gain of NOMA over OMA in multicell systems will be considered in our future work.
Vii Acknowledgement
The authors would like to appreciate Prof. Ping Li from City University of Hong Kong for valuable discussion during this work.
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