I Introduction
The networked world we live in has revolutionized our daily life. Wireless communications has become one of the disruptive technologies and it is one of the best business opportunities of the future[2, 3]. In particular, the development of wireless communications worldwide fuels the massive growth in the number of wireless communication devices and sensors for emerging applications such as smart logistics & transportation, environmental monitoring, energy management, safety management, and industry automation, just to name a few. It is expected that in the InternetofThings (IoT) era [4], there will be billion wireless communication devices connected worldwide with a connection density up to a million devices per [5, 6]. The massive number of devices and explosive data traffic pose challenging requirements, such as massive connectivity [7] and ultrahigh spectral efficiency for future wireless networks[2, 3]. As a result, compelling new technologies, such as massive multipleinput multipleoutput (MIMO)[8, 9], nonorthogonal multiple access (NOMA)[10, 11, 12, 13, 14], and millimeter wave (mmWave) communications[15, 16, 17, 18, 19] etc. have been proposed to address the aforementioned issues. Among them, NOMA has drawn significant attention both in industry and in academia as a promising multiple access technique. The principle of powerdomain NOMA is to exploit the users’ power difference for multiuser multiplexing together with superposition coding at the transmitter, while applying successive interference cancelation (SIC) at the receivers for alleviating the interuser interference (IUI)[12]. In fact, the industrial community has proposed up to 16 various forms of NOMA as the potential multiple access schemes for the forthcoming fifthgeneration (5G) networks[20].
Compared to the conventional orthogonal multiple access (OMA) schemes, NOMA allows users to simultaneously share the same resource blocks and hence it is beneficial for supporting a large number of connections in spectrally efficient communications. The concept of nonorthogonal transmissions dates back to the 1990s, e.g. [21, 22], which serves as a foundation for the development of the powerdomain NOMA. Indeed, NOMA schemes relying on nonorthogonal spreading sequences have led to popular code division multiple access (CDMA) arrangements, even though eventually the socalled orthogonal variable spreading factor (OVSF) code was selected for the global thirdgeneration (3G) wireless systems[23, 24, 25, 26]. To elaborate a little further, the spectral efficiency of CDMA was analyzed in [23]. In [24], the authors compared the benefits and deficiencies of three typical CDMA schemes: singlecarrier directsequence CDMA (SC DSCDMA), multicarrier CDMA (MCCDMA), and multicarrier DSCDMA (MC DSCDMA). Furthermore, a comparative study of OMA and NOMA was carried out in [26]. It has been shown that NOMA possesses a spectralpower efficiency advantage over OMA[26] and this theoretical gain can be realized with the aid of the interleave division multiple access (IDMA) technique proposed in [27]. Despite the initial efforts on the study of NOMA, the employment of NOMA in practical systems has been developing relatively slowly due to the requirement of sophisticated hardware for its implementation. Recently, NOMA has rekindled the interests of researchers as a benefit of the recent advances in signal processing and silicon technologies[28, 29]. However, existing contributions, e.g. [21, 22, 26], have mainly focused their attention on the NOMA performance from the information theoretical point of view, such as its capacity region[21, 22] and power region[26]. The recent work in [30] intuitively explained the source of performance gain attained by NOMA over OMA via simulations. The authors of [31, 32] surveyed the stateoftheart research on NOMA and offered a highlevel discussion of the challenges and research opportunities for NOMA systems. However, to the best of our knowledge, there is a paucity of literature on the comprehensive analysis of the achievable ergodic sumrate gain (ESG) of NOMA over OMA relying on practical signal detection techniques. Furthermore, the ESG of NOMA over OMA in different practical scenarios, such as singleantenna, multiantenna, and massive antenna array aided systems relying on singlecell or multicell deployments has not been compared in the open literature.
As for singleantenna systems, several authors have analyzed the performance of NOMA from different perspectives, e.g.[33, 34, 35, 36]. More specifically, based on the achievable rate region, Xu et al. proved in [33]
that NOMA outperforms time division multiple access (TDMA) with a high probability in terms of both its overall sumrate and the individual userrate. Furthermore, the ergodic sumrate of singleinput singleoutput NOMA (SISONOMA) was derived and the performance gain of SISONOMA over SISOOMA was demonstrated via simulations by Ding
et al.[34]. Upon relying on their new dynamic resource allocation design, Chen et al. [35] proved that SISONOMA always outperforms SISOOMA using a rigorous optimization technique. In [36], Yang et al. analyzed the outage probability degradation and the ergodic sumrate of SISONOMA systems by taking into account the impact of partial channel state information (CSI). As a further development, efficient resource allocation was designed for NOMA systems by Sun et al. [37] as well as by Wei et al. [38] under the assumptions of perfect CSI and imperfect CSI, respectively. The simulation results in [37] and [38] quantified the performance gain of NOMA over OMA in terms of its spectral efficiency and power efficiency, respectively. The aforementioned contributions studied the performance of NOMA systems or discussed the superiority of NOMA over OMA in different contexts. However, the analytical results quantifying the ESG of SISONOMA over SISOOMA has not been reported at the time of writing. More importantly, the source of the performance gain of NOMA over OMA has not been well understood and the impact of specific system parameters on the ESG, such as the number of NOMA users, the signaltonoise ratio (SNR), and the cell size, have not been revealed in the open literature.To achieve a higher spectral efficiency, the concept of NOMA has also been amalgamate with multiantenna systems, resulting in the notion of multipleinput multipleoutput NOMA (MIMONOMA), for example, by invoking the signal alignment technique of Ding et al. [39] and the quasidegradationbased precoding design of Chen et al. [40]. Although the performance gain of MIMONOMA over MIMOOMA has indeed been shown in [39, 40] with the aid of simulations, the performance gain due to additional antennas has not been quantified analytically. Moreover, how the ESG of NOMA over OMA increases upon upgrading the system from having a single antenna to multiple antennas is still an open problem at the time of writing, which deserves our efforts to explore. The answers to these questions can shed light on the practical implementation of NOMA in future wireless networks. On the other hand, there are only some preliminary results on applying the NOMA principle to massiveMIMO systems. For instance, Zhang et al. [41] investigated the outage probability of massiveMIMONOMA (mMIMONOMA). Furthermore, Ding and Poor [42] analyzed the outage performance of mMIMONOMA relying on realistic limited feedback and demonstrated a substantial performance improvement for mMIMONOMA over mMIMOOMA. Upon extending NOMA to a mmWave massiveMIMO system, the capacity attained in the highSNR regime and lowSNR regime were analyzed by Zhang et al. [43]. Yet, the ESG of mMIMONOMA over mMIMOOMA remains unknown and the investigation of mMIMONOMA has the promise attaining NOMA gains in largescale systems in the networks of the near future.
On the other hand, although singlecell NOMA has received significant research attention [33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43], the performance of NOMA in multicell scenarios remains unexplored but critically important for practical deployment, where the intercell interference becomes a major obstacle[44]. Centralized resource optimization of multicell NOMA was proposed by You in [45], while a distributed power control scheme was studied in [46]. The transmit precoder design of MIMONOMA aided multicell networks designed for maximizing the overall sum throughput was proposed by Nguyen et al. [47] and a computationally efficient algorithm was proposed for achieving a locally optimal solution. Despite the fact that the simulation results provided by [44, 45, 46, 47] have demonstrated a performance gain for applying NOMA in multicell cellular networks, the analytical results quantifying the ESG of NOMA over OMA for multicell systems relying on singleantenna, multiantenna, and massiveMIMO arrays at the BSs have not been reported in the open literature. Furthermore, the performance gains disseminated in the literature have been achieved for systems having a high transmit power or operating in the highSNR regime. However, a high transmit power inflicts a strong intercell interference, which imposes a challenge for the design of intercell interference management. Therefore, there are many practical considerations related to the NOMA principle in multicell systems, while have to be investigated.
Considered system setup  Main results  [30]  [34]  [35]  [39, 40]  [42]  [45, 46, 47]  This work  


Outage probability  
Proof of superiority  
Ergodic sumrate  
Ergodic sumrate gain  
Numerical results  

Outage probability  
Proof of superiority  
Ergodic sumrate  
Ergodic sumrate gain  
Numerical results  

Outage probability  
Proof of superiority  
Ergodic sumrate  
Ergodic sumrate gain  
Numerical results  
Multicell systems  Outage probability  
Proof of superiority  
Ergodic sumrate  
Ergodic sumrate gain  
Numerical results 
In summary, the comparison of our work with the most pertinent existing contributions in the literature is shown in Table I. Although the existing treatises have investigated the system performance of NOMA from different perspectives, such as the outage probability [34, 39, 40, 42] and the ergodic sumrate [34], in various specifically considered system setups, no unified analysis has been published to discuss the performance gain of NOMA over OMA. To fill this gap, our work offers a unified analysis on the ergodic sumrate gain of NOMA over OMA in singleantenna, multiantenna, and massiveMIMO systems with both singlecell and multicell deployments.
This paper aims for providing answers to the above open problems and for furthering the understanding of the ESG of NOMA over OMA in the uplink of communication systems. To this end, we carry out the unified analysis of ESG in singleantenna, multiantenna and massiveMIMO systems. We first focus our attention on the ESG analysis in singlecell systems and then extend our analytical results to multicell systems by taking into account the intercell interference (ICI). We quantify the ESG of NOMA over OMA relying on practical signal reception schemes at the base station for both NOMA as well as OMA and unveil its behaviour under different scenarios. Our simulation results confirm the accuracy of our performance analyses and provide some interesting insights, which are summarized in the following:

In all the cases considered, a high ESG can be achieved by NOMA over OMA in the highSNR regime, but the ESG vanishes in the lowSNR regime.

In the singleantenna scenario, we identify two types of gains attained by NOMA and characterize their different behaviours. In particular, we show that the largescale nearfar gain achieved by exploiting the distancediscrepancy between the base station and users increases with the cell size, while the smallscale fading gain is given by an EulerMascheroni constant[48] of nat/s/Hz in Rayleigh fading channels.

When applying NOMA in multiantenna systems, compared to the MIMOOMA utilizing zeroforcing detection, we analytically show that the ESG of SISONOMA over SISOOMA can be increased by fold, when the base station is equipped with antennas and serves a sufficiently large number of users .

Compared to MIMOOMA utilizing a maximum ratio combining (MRC) detector, an
fold degrees of freedom (DoF) gain can be achieved by MIMONOMA. In particular, the ESG in this case increases linearly with the system’s SNR quantified in dB with a slope of
in the highSNR regime. 
For massiveMIMO systems with a fixed ratio between the number of antennas, , and the number of users, , i.e., , the ESG of mMIMONOMA over mMIMOOMA increases linearly with both and using MRC detection.

In practical multicell systems operating without joint cell signal processing, the ESG of NOMA over OMA is degraded due to the existence of ICI, especially for a small cell size with a dense cell deployment. Furthermore, no DoF gain can be achieved by NOMA in multicell systems due to the lack of joint multicell signal processing to handle the ICI. In other words, all the ESGs of NOMA over OMA in singleantenna, multiantenna, and massiveMIMO multicell systems saturate in the highSNR regime.

For both singlecell and multicell systems, a large cell size is always beneficial to the performance of NOMA. In particular, in singlecell systems, the ESG of NOMA over OMA is increased for a larger cell size due to the enhanced largescale nearfar gain. For multicell systems, a larger cell size reduces the ICI level, which prevents a severe ESG degradation.
The 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. represents the set of all matrices with complex entries. denotes the absolute value of a complex scalar or the determinant of a matrix, denotes Euclidean norm of a complex vector, is the ceiling function which returns the smallest integer greater than the input value, anddenotes the expectation over the random variable
. The circularly symmetric complex Gaussian distribution with mean
and variance
is denoted by .Ii System Model
Iia System Model
We first consider the uplink^{1}^{1}1We restrict ourselves to the uplink NOMA communications[49], as advanced signal detection/decoding algorithms of NOMA are more affordable at the base station. of a singlecell^{2}^{2}2We first focus on the ESG analysis for singlecell systems, which serves as a building block for the analyses for multicell systems presented in Section VI. NOMA system with a single base station (BS) supporting users, as shown in Fig. 1. The cell is modeled by a pair of two concentric ringshaped discs. The BS is located at the center of the ringshaped discs with the inner radius of and outer radius of , where all the users are scattered uniformly within the two concentric ringshaped discs. 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 frequency division multiple access (FDMA) as a typical OMA scheme.
In this paper, we consider three typical types of communication systems:

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

MIMONOMA and MIMOOMA: the BS is equipped with a multiantenna array () and all the users have a singleantenna associated with .

Massive MIMONOMA (mMIMONOMA) and massiveMIMOOMA (mMIMOOMA): the BS is equipped with a largescale antenna array (), while all the users are equipped with a single antenna, associated with , i.e., the number of antennas at the BS is lower than the number of users , but with a fixed ratio of .
IiB Signal and Channel Model
The signal received at the BS is given by
(1) 
where , denotes the power transmitted by user , is the normalized modulated symbol of user with , and represents the additive white Gaussian noise (AWGN) at the BS with zero mean and covariance matrix of . To emphasize the impact of the number of users on the performance gain of NOMA over OMA, we fix the total power consumption of all the uplink users and thus we have
(2) 
where is the maximum total transmit power for all the users. Note that the sumpower constraint is a commonly adopted assumption in the literature[50, 51, 30] for simplifying the performance analysis of uplink communications. In fact, the sumpower constraint is a reasonable assumption for practical cellular communication systems, where a total transmit power limitation is intentionally imposed to limit the ICI.
The uplink (UL) channel vector between user and the BS is modeled as
(3) 
where denotes the Rayleigh fading coefficients, i.e., , is the distance between user and the BS in the unit of meter, and represents the path loss exponent^{3}^{3}3In this paper, we ignore the impact of shadowing to simplify our performance analysis. Note that, shadowing only introduces an additional power factor to in the channel model in (3). Although the introduction of shadowing may change the resulting channel distribution of , the distancebased channel model is sufficient to characterize the largescale nearfar gain exploited by NOMA, as will be discussed in this paper.. We denote the UL channel matrix between all the users and the BS by . Note that the system model in (1) and the channel model in (3) include the cases of singleantenna and massiveMIMO aided BS associated with and , respectively. For instance, 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 for providing some insights concerning the performance gain of NOMA over OMA, we assume that perfect UL CSI knowledge is available at the BS for coherent detection.
IiC Signal Detection and Resource Allocation Strategy
To facilitate our performance analyses, we focus our attention on the following efficient signal detection and practical resource allocation strategies.
IiC1 Signal detection
NOMA system  Reception technique  OMA system  Reception technique 

SISONOMA  SIC  SISOOMA  FDMASUD 
MIMONOMA  MMSESIC  MIMOOMA  FDMAZF, FDMAMRC 
mMIMONOMA  MRCSIC  mMIMOOMA  FDMAMRC 
The signal detection techniques adopted in this paper for NOMA and OMA systems are shown in Table II, which are detailed in the following.
For SISONOMA, we adopt the commonly used successive interference cancelation (SIC) receiver [52] at the BS, since its performance approaches the capacity of singleantenna systems[22]. On the other hand, given that all the users are separated orthogonally by different frequency subbands for SISOOMA, the simple singleuser detection (SUD) technique can be used to achieve the optimal performance.
For MIMONOMA, the minimum mean square error criterion based successive interference cancelation (MMSESIC) constitutes an appealing receiver algorithm, since its performance approaches the capacity [22] at an acceptable computational complexity for a finite number of antennas at the BS. On the other hand, two types of signal detection schemes are considered for MIMOOMA, namely FDMA zero forcing (FDMAZF) and FDMA maximum ratio combining (FDMAMRC). Exploiting the extra spatial degrees of freedom (DoF) attained by multiple antennas at the BS, ZF can be used for multiuser detection (MUD), as its achievable rate approaches the capacity in the highSNR regime[22]. In particular, all the users are categorized into groups^{4}^{4}4Without loss of generality, we consider the case with as an integer in this paper. with each group containing users. Then, ZF is utilized for handling the interuser interference (IUI) within each group and FDMA is employed to separate all the groups on orthogonal frequency subbands. In the lowSNR regime, the performance of ZF fails to approach the capacity [22], thus a simple lowcomplexity MRC scheme is adopted for single user detection on each frequency subband. We note that there is only a single user in each frequency subband of our considered FDMAMRC aided MIMOOMA systems, i.e., no user grouping.
With a massive number of UL receiving antennas employed at the BS, we circumvent the excessive complexity of matrix inversion involved in ZF and MMSE detection by adopting the lowcomplexity MRCSIC detection [53] for mMIMONOMA systems and the FDMAMRC scheme for mMIMOOMA systems. Given the favorable propagation property of massiveMIMO systems[54], the orthogonality among the channel vectors of multiple users holds fairly well, provided that the number of users is sufficiently lower than the number of antennas. Therefore, we can assign users to every frequency subband and perform the simple MRC detection while enjoying negligible IUIs in each subband. In this paper, we consider a fixed ratio between the group size and the number of antennas, namely, , and assume that the abovementioned favorable propagation property holds under the fixed ratio considered.
IiC2 Resource allocation strategy
To facilitate our analytical study in this paper, we consider an equal resource allocation strategy for both NOMA and OMA schemes. In particular, equal power allocation is adopted for NOMA schemes^{5}^{5}5As shown in [55], allocating a higher power to the user with the worse channel is not necessarily required in NOMA[55].. On the other hand, equal power and frequency allocation is adopted for OMA schemes. Note that the equal resource allocation is a typical selected strategy for applications bearing only a limited system overhead, e.g. machinetype communications (MTC).
We note that beneficial user grouping design is important for the MIMOOMA system relying on FDMAZF and for the mMIMOOMA system using FDMAMRC. In general, finding the optimal user grouping strategy is an NPhard problem and the performance analysis based on the optimal user grouping strategy is generally intractable. Furthermore, the optimal SIC decoding order of NOMA in multiantenna and massiveMIMO systems is still an open problem in the literature, since the channel gains on different antennas are usually diverse. To avoid tedious comparison and to facilitate our performance analysis, we adopt a random user grouping strategy for the OMA systems considered and a fixed SIC decoding order for the NOMA systems investigated. In particular, we randomly select and users for each group on each frequency subband for the MIMOOMA and mMIMOOMA systems, respectively. For NOMA systems, without loss of generality, we assume , that the users are indexed based on their channel gains, and the SIC/MMSESIC/MRCSIC decoding order^{6}^{6}6Note that, in general, the adopted decoding order is not the optimal SIC decoding order for maximizing the achievable sumrate of the considered MIMONOMA and mMIMONOMA systems. at the BS is . Additionally, to unveil insights about the performance gain of NOMA over OMA, we assume that there is no error propagation during SIC/MMSESIC/MRCSIC 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 asymptotic ESG of SISONOMA over SISOOMA is discussed under different scenarios.
Iiia Ergodic Sumrate of SISONOMA and SISOOMA
When decoding the messages of user , the interferences imposed by users have been canceled in the SISONOMA system by SIC reception. Therefore, the instantaneous achievable data rate of user in the SISONOMA system considered is given by:
(4) 
On the other hand, in the SISOOMA system considered, user is allocated to a subband exclusively, thus there is no interuser interference (IUI). As a result, the instantaneous achievable data rate of user in the SISOOMA system considered is given by:
(5) 
with and denoting the power allocation and frequency allocation of user . Note that we consider a normalized frequency bandwidth for both the NOMA and OMA schemes in this paper, i.e., . Under the identical resource allocation strategy, i.e., for and , we have the instantaneous sumrate of SISONOMA and SISOOMA given by
(6)  
(7) 
respectively.
Given the instantaneous sumrates in (6) and (7), firstly we have to investigate the channel gain distribution before embarking on the derivation of the corresponding ergodic sumrates. Since all the users are scattered uniformly across the pair of concentric rings between the inner radius of and the outer radius of in Fig. 1
, the cumulative distribution function (CDF) of the channel gain
^{7}^{7}7As mentioned before, we assumed that the channel gains of all the users are ordered as in Section IIC2. However, as shown in (6), the system sumrate for the considered SISONOMA system is actually independent of the SIC decoding order. Therefore, we can safely assume that all the users have i.i.d. channel distribution, which does not affect the performance analysis results. In the sequel of this paper, the subscript is dropped without loss of generality. is given by(8) 
where ,
, denotes the probability density function (PDF) for the random distance
. With the GaussianChebyshev quadrature approximation[48], the CDF and PDF of can be approximated by(9)  
(10) 
respectively, where the parameters in (9) and (10) are:
(11) 
while denotes the number of terms for integral approximation. The larger , the higher the approximation accuracy becomes.
Based on (6), the ergodic sumrate of the SISONOMA system considered is defined as:
(12) 
where the expectation is averaged over both the largescale fading and smallscale fading in the overall channel matrix . For a large number of users, i.e., , the sum of channel gains of all the users within the in (12
) becomes a deterministic value due to the strong law of large number, i.e.,
, where denotes the average channel power gain and it is given by(13) 
Therefore, the asymptotic ergodic sumrate of the SISONOMA system considered is given by
(14) 
where the equality is due to the bounded convergence theorem[56] and owing to the finite channel capacity. Note that for a finite number of users , the asymptotic ergodic sumrate in (IIIA) serves as an upper bound for the actual ergodic sumrate in (12), i.e., we have , owing to the concavity of the logarithmic function and the Jensen’s inequality. In the Section VII
, we will show that the asymptotic analysis in (
IIIA) is also accurate for a finite value of and becomes tighter upon increasing .Similarly, based on (7), we can obtain the ergodic sumrate of the SISOOMA system as follows:
(15) 
where denotes the order exponential integral[48]. The equality in (IIIA) is obtained since all the users have i.i.d. channel distributions. Note that in contrast to SISONOMA, in (IIIA) is applicable to SISOOMA supporting an arbitrary number of users.
IiiB ESG in Singleantenna Systems
Comparing (IIIA) and (IIIA), the asymptotic ESG of SISONOMA over SISOOMA with can be expressed as follows:
(16) 
Then, in the highSNR regime, we can approximate the asymptotic ESG^{8}^{8}8Under the sumpower constraint, the system SNR directly depends on the total system power budget , and thus the system SNR and are interchangeably in this paper. in (IIIB) by applying [48] as
(17) 
where is given by
(18) 
and is the EulerMascheroni constant[48]
. Based on the weighted arithmetic and geometric means (AMGM) inequality
[57], we can observe that . This implies that and SISONOMA provides a higher asymptotic ergodic sumrate than SISOOMA in the system considered.To further simplify the expression of ESG, we consider path loss exponents in the range of in (IIIA), which usually holds in urban environments[58]. As a result, . Hence, in (18) can be further simplified as follows:
(19) 
where . The normalized cell size of is the ratio between the outer radius and the inner radius , which also serves as a metric of the path loss discrepancy.
We can see that the asymptotic ESG of SISONOMA over SISOOMA in (17) is composed of two components, i.e., and . As observed in (19), the former component of only depends on the normalized cell size of instead of the absolute values of and . In fact, it can characterize the largescale nearfar gain attained by NOMA via exploiting the discrepancy in distances among NOMA users. Interestingly, for the extreme case that all the users are randomly deployed on a circle, i.e., , we have , , and . In other words, the largescale nearfar gain disappears, when all the users are of identical distance away from the BS. With the aid of , we can observe in (17) that the performance gain achieved by NOMA is a constant value of nat/s/Hz. Since all the users are set to have the same distance when , the minimum asymptotic ESG arising from the smallscale Rayleigh fading is named as the smallscale fading gain in this paper. In fact, in the asymptotic case associated with and , SISONOMA provides at least nat/s/Hz spectral efficiency gain over SISOOMA for an arbitrary cell size in Rayleigh fading channels. Additionally, we can see that the ESG of SISONOMA over SISOOMA is saturated in the highSNR regime. This is because the instantaneous sumrates of both the SISONOMA system in (6) and the SISOOMA system in (7) increase logarithmically with .
To visualize the largescale nearfar gain, we illustrate the asymptotic ESG in (17) versus and in Fig. 2. We can observe that when , the largescale nearfar gain disappears and the asymptotic ESG is bounded from below by its minimum value of nat/s/Hz due to the smallscale fading gain. Additionally, for different values of and but with a fixed , SISONOMA offers the same ESG compared to SISOOMA. This is because as predicted in (19), the largescale nearfar gain only depends on the normalized cell size . More importantly, we can observe that the largescale nearfar gain increases with the normalized cell size . In fact, for a larger normalized cell size , the heterogeneity in the largescale fading among users becomes higher and SISONOMA attains a higher nearfar gain, hence improving the sumrate performance.
Remark 1
Note that it has been analytically shown in [59] that two users with a large distance difference (or equivalently channel gain difference) are preferred to be paired. This is consistent with our conclusion in this paper, where a larger normalized cell size enables a higher ESG of NOMA over OMA. However, [59] only considered a pair of two NOMA users. In this paper, we analytically obtain the ESG of NOMA over OMA for a more general NOMA system supporting a large number of UL users. More importantly, we identify two kinds of gains in the ESG derived and reveal their different behaviours.
Iv ESG of MIMONOMA over MIMOOMA
In this section, the ergodic sumrates of MIMONOMA and MIMOOMA associated with FDMAZF as well as FDMAMRC are firstly analyzed. Then, the asymptotic ESGs of MIMONOMA over MIMOOMA with FDMAZF and FDMAMRC detection are investigated.
Iva Ergodic Sumrate of MIMONOMA with MMSESIC
Let us consider that an antenna BS serves singleantenna nonorthogonal users relying on MIMONOMA. The BS employs MMSESIC detection for retrieving the messages of all the users. The instantaneous achievable data rate of user in the MIMONOMA system relying on MMSESIC detection^{9}^{9}9The derivation of individual rates in (20) for MMSESIC detection of MIMONOMA is based on the matrix inversion lemma:
(20) 
As a result, the instantaneous sumrate of MIMONOMA is obtained as
(21) 
In fact, MMSESIC is capacityachieving [22] and (21) is the channel capacity for a given instantaneous channel matrix [60]. In general, it is a challenge to obtain a closedform expression for the instantaneous channel capacity above due to the determinant of the summation of matrices in (21). To provide some insights, in the following theorem, we consider an asymptotically tight upper bound for the achievable sumrate in (21) associated with .
Theorem 1
proof 1
Please refer to Appendix A for the proof of Theorem 1.
Now, given the instantaneous achievable sumrate obtained in (23), we proceed to calculate the ergodic sumrate. Given the distance from a user to the BS as , the channel gain
follows the Gamma distribution
[61], whose conditional PDF and CDF are given by^{10}^{10}10Similar to (9) and (10), we can safely assume that all the users have i.i.d. channel distribution within the cell and drop the subscript in (24), since the system sumrate in (21) is independent of the MMSESIC decoding order[22].(24) 
respectively, where denotes the PDF of a random variable obeying a Gamma distribution, denotes the Gamma function, and denotes the lower incomplete Gamma function. Then, the CDF of the channel gain can be obtained by
(25) 
By applying the GaussianChebyshev quadrature approximation[48], the CDF and PDF of can be written as
(26) 
respectively, where and are given in (IIIA).
According to (23), given the equal resource allocation strategy, i.e., , the asymptotic ergodic sumrate of MIMONOMA associated with can be obtained as follows:
(27)  
where denotes the average channel gain, which is given by
(28) 
Remark 2
Comparing (IIIA) and (27), we can observe that for a sufficiently large number of users, the considered MIMONOMA system is asymptotically equivalent to a SISONOMA system with fold increases in DoF and an equivalent average channel gain of in each DoF. Intuitively, when the number of UL receiver antennas at the BS, , is much smaller than the number of users, , which corresponds to the extreme asymmetric case of MIMONOMA, the multiantenna BS behaves asymptotically in the same way as a singleantenna BS. Additionally, when , due to the diverse channel directions of all the users, the received signals fully span the dimensional signal space[51]. Therefore, MIMONOMA using MMSESIC reception can fully exploit the system’s spatial DoF, , and its performance can be approximated by that of a SISONOMA system with fold DoF.
IvB Ergodic Sumrate of MIMOOMA with FDMAZF
Upon installing more UL receiver antennas at the BS, ZF can be employed for MUD and the MIMOOMA system using FDMAZF can accommodate users on each frequency subband. As mentioned before, we adopt a random user grouping strategy for the MIMOOMA system using FDMAZF detection, where we randomly select users as a group and denote the composite channel matrix of the th group by . Then, the instantaneous achievable data rate of user in the MIMOOMA system is given by
(29) 
where denotes the normalized frequency allocation for the th group. The vector denotes the normalized ZF detection vector for user with , which is obtained based on the pseudoinverse of the composite channel matrix in the th user group[22].
Given the equal resource allocation strategy, i.e., and , the instantaneous sumrate of MIMOOMA using FDMAZF can be formulated as:
(30) 
Since and , we have [22]. As a result, in (30) has an identical distribution with , i.e., its CDF and PDF are given by (9) and (10), respectively. Therefore, the ergodic sumrate of the MIMOOMA system considered can be expressed as:
(31)  
IvC Ergodic Sumrate of MIMOOMA with FDMAMRC
The instantaneous achievable data rate of user in the MIMOOMA system using the FDMAMRC receiver is given by
(32) 
Upon adopting the equal resource allocation strategy, i.e., and , the instantaneous sumrate of MIMOOMA relying on FDMAMRC is obtained by
(33) 
Averaging over the channel fading, we arrive at the ergodic sumrate of MIMOOMA using FDMAMRC as
(34) 
with given by
(35) 
where denotes the Meijer Gfunction. The equality in (IVC) is obtained due to and the equality in (IVC) is based on Equation (3) in [62]. Now, the ergodic sumrate of MIMOOMA using FDMAMRC can be written as
(36) 
Note that, the ergodic sumrate in (36) is applicable to an arbitrary number of users and an arbitrary SNR, but it is too complicated to offer insights concerning the ESG of MIMONOMA over MIMOOMA. Hence, based on (IVC), we derive the asymptotic ergodic sumrate of MIMOOMA with FDMAMRC in the lowSNR regime with as follows:
(37) 
On the other hand, in the highSNR regime, based on (IVC), the asymptotic ergodic sumrate of MIMOOMA using FDMAMRC is given by
(38) 
IvD ESG in Multiantenna Systems
By comparing (27) and (31), we have the asymptotic ESG of MIMONOMA over MIMOOMA relying on FDMAZF as follows:
(39) 
To unveil some insights, we consider the asymptotic ESG in the highSNR regime as follows
(40) 
where denotes the largescale nearfar gain given in (18).
Remark 3
The identified two kinds of gains in ESG of the singleantenna system in (17) are also observed in the ESG of MIMONOMA over MIMOOMA using FDMAZF in (40). Moreover, it can be observed that both the largescale nearfar gain and the smallscale fading gain are increased by times as indicated in (40). In fact, upon comparing (17) and (40), we have
(41) 
which implies that the asymptotic ESG of MIMONOMA over MIMOOMA is times of that in singleantenna systems, when there are UL receiver antennas at the BS. In fact, for , the heterogeneity in channel directions of all the users allows the received signals to fully span across the dimensional signal space. Hence, MIMONOMA and MIMOOMA using FDMAZF can fully exploit the system’s maximal spatial DoF . Furthermore, we have an additional power gain of in the second term in (41). This is due to a factor of average power loss within each group for ZF projection to suppress the IUI in the MIMOOMA system considered[22].
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