On the Performance Gain of NOMA over OMA in Uplink Communication Systems

03/05/2019 ∙ by Zhiqiang Wei, et al. ∙ University of Southampton UNSW 0

In this paper, we investigate and reveal the ergodic sum-rate gain (ESG) of non-orthogonal multiple access (NOMA) over orthogonal multiple access (OMA) in uplink cellular communication systems. A base station equipped with a single-antenna, with multiple antennas, and with massive antenna arrays is considered both in single-cell and multi-cell deployments. In particular, in single-antenna systems, we identify two types of gains brought about by NOMA: 1) a large-scale near-far gain arising from the distance discrepancy between the base station and users; 2) a small-scale fading gain originating from the multipath channel fading. Furthermore, we reveal that the large-scale near-far gain increases with the normalized cell size, while the small-scale fading gain is a constant, given by γ = 0.57721 nat/s/Hz, in Rayleigh fading channels. When extending single-antenna NOMA to M-antenna NOMA, we prove that both the large-scale near-far gain and small-scale fading gain achieved by single-antenna NOMA can be increased by a factor of M for a large number of users. Moreover, given a massive antenna array at the base station and considering a fixed ratio between the number of antennas, M, and the number of users, K, the ESG of NOMA over OMA increases linearly with both M and K. We then further extend the analysis to a multi-cell scenario. Compared to the single-cell case, the ESG in multi-cell systems degrades as NOMA faces more severe inter-cell interference due to the non-orthogonal transmissions. Besides, we unveil that a large cell size is always beneficial to the ergodic sum-rate performance of NOMA in both single-cell and multi-cell systems. Numerical results verify the accuracy of the analytical results derived and confirm the insights revealed about the ESG of NOMA over OMA in different scenarios.

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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 Internet-of-Things (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 ultra-high spectral efficiency for future wireless networks[2, 3]. As a result, compelling new technologies, such as massive multiple-input multiple-output (MIMO)[8, 9], non-orthogonal 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 power-domain 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 inter-user 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 fifth-generation (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 non-orthogonal transmissions dates back to the 1990s, e.g. [21, 22], which serves as a foundation for the development of the power-domain NOMA. Indeed, NOMA schemes relying on non-orthogonal spreading sequences have led to popular code division multiple access (CDMA) arrangements, even though eventually the so-called orthogonal variable spreading factor (OVSF) code was selected for the global third-generation (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: single-carrier direct-sequence CDMA (SC DS-CDMA), multicarrier CDMA (MC-CDMA), and multicarrier DS-CDMA (MC DS-CDMA). Furthermore, a comparative study of OMA and NOMA was carried out in [26]. It has been shown that NOMA possesses a spectral-power 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 state-of-the-art research on NOMA and offered a high-level 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 sum-rate 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 single-antenna, multi-antenna, and massive antenna array aided systems relying on single-cell or multi-cell deployments has not been compared in the open literature.

As for single-antenna 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 sum-rate and the individual user-rate. Furthermore, the ergodic sum-rate of single-input single-output NOMA (SISO-NOMA) was derived and the performance gain of SISO-NOMA over SISO-OMA was demonstrated via simulations by Ding

et al.[34]. Upon relying on their new dynamic resource allocation design, Chen et al. [35] proved that SISO-NOMA always outperforms SISO-OMA using a rigorous optimization technique. In [36], Yang et al. analyzed the outage probability degradation and the ergodic sum-rate of SISO-NOMA 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 SISO-NOMA over SISO-OMA 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 signal-to-noise 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 multi-antenna systems, resulting in the notion of multiple-input multiple-output NOMA (MIMO-NOMA), for example, by invoking the signal alignment technique of Ding et al. [39] and the quasi-degradation-based precoding design of Chen et al. [40]. Although the performance gain of MIMO-NOMA over MIMO-OMA 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 massive-MIMO systems. For instance, Zhang et al. [41] investigated the outage probability of massive-MIMO-NOMA (mMIMO-NOMA). Furthermore, Ding and Poor [42] analyzed the outage performance of mMIMO-NOMA relying on realistic limited feedback and demonstrated a substantial performance improvement for mMIMO-NOMA over mMIMO-OMA. Upon extending NOMA to a mmWave massive-MIMO system, the capacity attained in the high-SNR regime and low-SNR regime were analyzed by Zhang et al. [43]. Yet, the ESG of mMIMO-NOMA over mMIMO-OMA remains unknown and the investigation of mMIMO-NOMA has the promise attaining NOMA gains in large-scale systems in the networks of the near future.

On the other hand, although single-cell NOMA has received significant research attention [33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43], the performance of NOMA in multi-cell scenarios remains unexplored but critically important for practical deployment, where the inter-cell interference becomes a major obstacle[44]. Centralized resource optimization of multi-cell NOMA was proposed by You in [45], while a distributed power control scheme was studied in [46]. The transmit precoder design of MIMO-NOMA aided multi-cell 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 multi-cell cellular networks, the analytical results quantifying the ESG of NOMA over OMA for multi-cell systems relying on single-antenna, multi-antenna, and massive-MIMO 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 high-SNR regime. However, a high transmit power inflicts a strong inter-cell interference, which imposes a challenge for the design of inter-cell interference management. Therefore, there are many practical considerations related to the NOMA principle in multi-cell systems, while have to be investigated.

Considered system setup Main results [30] [34] [35] [39, 40] [42] [45, 46, 47] This work
Single-antenna
single-cell systems
Outage probability
Proof of superiority
Ergodic sum-rate
Ergodic sum-rate gain
Numerical results
Multi-antenna
single-cell systems
Outage probability
Proof of superiority
Ergodic sum-rate
Ergodic sum-rate gain
Numerical results
Massive-MIMO
single-cell systems
Outage probability
Proof of superiority
Ergodic sum-rate
Ergodic sum-rate gain
Numerical results
Multi-cell systems Outage probability
Proof of superiority
Ergodic sum-rate
Ergodic sum-rate gain
Numerical results
TABLE I: Comparison of This Work with Literature for the Results of Performance Gain of NOMA over OMA

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 sum-rate [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 sum-rate gain of NOMA over OMA in single-antenna, multi-antenna, and massive-MIMO systems with both single-cell and multi-cell 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 single-antenna, multi-antenna and massive-MIMO systems. We first focus our attention on the ESG analysis in single-cell systems and then extend our analytical results to multi-cell systems by taking into account the inter-cell 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 high-SNR regime, but the ESG vanishes in the low-SNR regime.

  • In the single-antenna scenario, we identify two types of gains attained by NOMA and characterize their different behaviours. In particular, we show that the large-scale near-far gain achieved by exploiting the distance-discrepancy between the base station and users increases with the cell size, while the small-scale fading gain is given by an Euler-Mascheroni constant[48] of nat/s/Hz in Rayleigh fading channels.

  • When applying NOMA in multi-antenna systems, compared to the MIMO-OMA utilizing zero-forcing detection, we analytically show that the ESG of SISO-NOMA over SISO-OMA can be increased by -fold, when the base station is equipped with antennas and serves a sufficiently large number of users .

  • Compared to MIMO-OMA utilizing a maximum ratio combining (MRC) detector, an

    -fold degrees of freedom (DoF) gain can be achieved by MIMO-NOMA. In particular, the ESG in this case increases linearly with the system’s SNR quantified in dB with a slope of

    in the high-SNR regime.

  • For massive-MIMO systems with a fixed ratio between the number of antennas, , and the number of users, , i.e., , the ESG of mMIMO-NOMA over mMIMO-OMA increases linearly with both and using MRC detection.

  • In practical multi-cell 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 multi-cell systems due to the lack of joint multi-cell signal processing to handle the ICI. In other words, all the ESGs of NOMA over OMA in single-antenna, multi-antenna, and massive-MIMO multi-cell systems saturate in the high-SNR regime.

  • For both single-cell and multi-cell systems, a large cell size is always beneficial to the performance of NOMA. In particular, in single-cell systems, the ESG of NOMA over OMA is increased for a larger cell size due to the enhanced large-scale near-far gain. For multi-cell 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, and

denotes the expectation over the random variable

. The circularly symmetric complex Gaussian distribution with mean

and variance

is denoted by .

Ii System Model

Fig. 1: The system model of the single-cell uplink communication with one base station and users.

Ii-a System Model

We first consider the uplink111We 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 single-cell222We first focus on the ESG analysis for single-cell systems, which serves as a building block for the analyses for multi-cell 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 ring-shaped discs. The BS is located at the center of the ring-shaped discs with the inner radius of and outer radius of , where all the users are scattered uniformly within the two concentric ring-shaped 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:

  • SISO-NOMA and SISO-OMA: the BS is equipped with a single-antenna () and all the users also have a single-antenna.

  • MIMO-NOMA and MIMO-OMA: the BS is equipped with a multi-antenna array () and all the users have a single-antenna associated with .

  • Massive MIMO-NOMA (mMIMO-NOMA) and massive-MIMO-OMA (mMIMO-OMA): the BS is equipped with a large-scale 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 .

Ii-B 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 sum-power constraint is a commonly adopted assumption in the literature[50, 51, 30] for simplifying the performance analysis of uplink communications. In fact, the sum-power 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 exponent333In 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 distance-based channel model is sufficient to characterize the large-scale near-far 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 single-antenna and massive-MIMO aided BS associated with and , respectively. For instance, when , denotes the corresponding channel coefficient of user in single-antenna 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.

Ii-C 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.

Ii-C1 Signal detection

NOMA system Reception technique OMA system Reception technique
SISO-NOMA SIC SISO-OMA FDMA-SUD
MIMO-NOMA MMSE-SIC MIMO-OMA FDMA-ZF, FDMA-MRC
mMIMO-NOMA MRC-SIC mMIMO-OMA FDMA-MRC
TABLE II: Signal Detection Techniques for NOMA and OMA Systems

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 SISO-NOMA, we adopt the commonly used successive interference cancelation (SIC) receiver [52] at the BS, since its performance approaches the capacity of single-antenna systems[22]. On the other hand, given that all the users are separated orthogonally by different frequency subbands for SISO-OMA, the simple single-user detection (SUD) technique can be used to achieve the optimal performance.

For MIMO-NOMA, the minimum mean square error criterion based successive interference cancelation (MMSE-SIC) 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 MIMO-OMA, namely FDMA zero forcing (FDMA-ZF) and FDMA maximum ratio combining (FDMA-MRC). Exploiting the extra spatial degrees of freedom (DoF) attained by multiple antennas at the BS, ZF can be used for multi-user detection (MUD), as its achievable rate approaches the capacity in the high-SNR regime[22]. In particular, all the users are categorized into groups444Without 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 inter-user interference (IUI) within each group and FDMA is employed to separate all the groups on orthogonal frequency subbands. In the low-SNR regime, the performance of ZF fails to approach the capacity [22], thus a simple low-complexity 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 FDMA-MRC aided MIMO-OMA 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 low-complexity MRC-SIC detection [53] for mMIMO-NOMA systems and the FDMA-MRC scheme for mMIMO-OMA systems. Given the favorable propagation property of massive-MIMO 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 above-mentioned favorable propagation property holds under the fixed ratio considered.

Ii-C2 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 schemes555As 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. machine-type communications (MTC).

We note that beneficial user grouping design is important for the MIMO-OMA system relying on FDMA-ZF and for the mMIMO-OMA system using FDMA-MRC. In general, finding the optimal user grouping strategy is an NP-hard problem and the performance analysis based on the optimal user grouping strategy is generally intractable. Furthermore, the optimal SIC decoding order of NOMA in multi-antenna and massive-MIMO 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 MIMO-OMA and mMIMO-OMA systems, respectively. For NOMA systems, without loss of generality, we assume , that the users are indexed based on their channel gains, and the SIC/MMSE-SIC/MRC-SIC decoding order666Note that, in general, the adopted decoding order is not the optimal SIC decoding order for maximizing the achievable sum-rate of the considered MIMO-NOMA and mMIMO-NOMA 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/MMSE-SIC/MRC-SIC decoding at the BS.

Iii ESG of SISO-NOMA over SISO-OMA

In this section, we first derive the ergodic sum-rate of SISO-NOMA and SISO-OMA. Then, the asymptotic ESG of SISO-NOMA over SISO-OMA is discussed under different scenarios.

Iii-a Ergodic Sum-rate of SISO-NOMA and SISO-OMA

When decoding the messages of user , the interferences imposed by users have been canceled in the SISO-NOMA system by SIC reception. Therefore, the instantaneous achievable data rate of user in the SISO-NOMA system considered is given by:

(4)

On the other hand, in the SISO-OMA system considered, user is allocated to a subband exclusively, thus there is no inter-user interference (IUI). As a result, the instantaneous achievable data rate of user in the SISO-OMA 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 sum-rate of SISO-NOMA and SISO-OMA given by

(6)
(7)

respectively.

Given the instantaneous sum-rates in (6) and (7), firstly we have to investigate the channel gain distribution before embarking on the derivation of the corresponding ergodic sum-rates. 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

777As mentioned before, we assumed that the channel gains of all the users are ordered as in Section II-C2. However, as shown in (6), the system sum-rate for the considered SISO-NOMA 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 Gaussian-Chebyshev 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 sum-rate of the SISO-NOMA system considered is defined as:

(12)

where the expectation is averaged over both the large-scale fading and small-scale 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 sum-rate of the SISO-NOMA 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 sum-rate in (III-A) serves as an upper bound for the actual ergodic sum-rate 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 (

III-A) is also accurate for a finite value of and becomes tighter upon increasing .

Similarly, based on (7), we can obtain the ergodic sum-rate of the SISO-OMA system as follows:

(15)

where denotes the -order exponential integral[48]. The equality in (III-A) is obtained since all the users have i.i.d. channel distributions. Note that in contrast to SISO-NOMA, in (III-A) is applicable to SISO-OMA supporting an arbitrary number of users.

Iii-B ESG in Single-antenna Systems

Comparing (III-A) and (III-A), the asymptotic ESG of SISO-NOMA over SISO-OMA with can be expressed as follows:

(16)

Then, in the high-SNR regime, we can approximate the asymptotic ESG888Under the sum-power constraint, the system SNR directly depends on the total system power budget , and thus the system SNR and are interchangeably in this paper. in (III-B) by applying [48] as

(17)

where is given by

(18)

and is the Euler-Mascheroni constant[48]

. Based on the weighted arithmetic and geometric means (AM-GM) inequality

[57], we can observe that . This implies that and SISO-NOMA provides a higher asymptotic ergodic sum-rate than SISO-OMA in the system considered.

To further simplify the expression of ESG, we consider path loss exponents in the range of in (III-A), 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 SISO-NOMA over SISO-OMA 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 large-scale near-far 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 large-scale near-far 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 small-scale Rayleigh fading is named as the small-scale fading gain in this paper. In fact, in the asymptotic case associated with and , SISO-NOMA provides at least nat/s/Hz spectral efficiency gain over SISO-OMA for an arbitrary cell size in Rayleigh fading channels. Additionally, we can see that the ESG of SISO-NOMA over SISO-OMA is saturated in the high-SNR regime. This is because the instantaneous sum-rates of both the SISO-NOMA system in (6) and the SISO-OMA system in (7) increase logarithmically with .

To visualize the large-scale near-far gain, we illustrate the asymptotic ESG in (17) versus and in Fig. 2. We can observe that when , the large-scale near-far gain disappears and the asymptotic ESG is bounded from below by its minimum value of nat/s/Hz due to the small-scale fading gain. Additionally, for different values of and but with a fixed , SISO-NOMA offers the same ESG compared to SISO-OMA. This is because as predicted in (19), the large-scale near-far gain only depends on the normalized cell size . More importantly, we can observe that the large-scale near-far gain increases with the normalized cell size . In fact, for a larger normalized cell size , the heterogeneity in the large-scale fading among users becomes higher and SISO-NOMA attains a higher near-far gain, hence improving the sum-rate performance.

Fig. 2: The asymptotic ESG in (17) under equal resource allocation versus and with and .
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 MIMO-NOMA over MIMO-OMA

In this section, the ergodic sum-rates of MIMO-NOMA and MIMO-OMA associated with FDMA-ZF as well as FDMA-MRC are firstly analyzed. Then, the asymptotic ESGs of MIMO-NOMA over MIMO-OMA with FDMA-ZF and FDMA-MRC detection are investigated.

Iv-a Ergodic Sum-rate of MIMO-NOMA with MMSE-SIC

Let us consider that an -antenna BS serves single-antenna non-orthogonal users relying on MIMO-NOMA. The BS employs MMSE-SIC detection for retrieving the messages of all the users. The instantaneous achievable data rate of user in the MIMO-NOMA system relying on MMSE-SIC detection999The derivation of individual rates in (20) for MMSE-SIC detection of MIMO-NOMA is based on the matrix inversion lemma:

Interested readers are referred to [22] for a detailed derivation. is given by[22]:

(20)

As a result, the instantaneous sum-rate of MIMO-NOMA is obtained as

(21)

In fact, MMSE-SIC is capacity-achieving [22] and (21) is the channel capacity for a given instantaneous channel matrix [60]. In general, it is a challenge to obtain a closed-form 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 sum-rate in (21) associated with .

Theorem 1

For the MIMO-NOMA system considered in (1) relying on MMSE-SIC detection, given any power allocation strategy , the achievable sum-rate in (21) is upper bounded by

(22)

This upper bound is asymptotically tight, when , i.e.,

(23)
proof 1

Please refer to Appendix A for the proof of Theorem 1.

Now, given the instantaneous achievable sum-rate obtained in (23), we proceed to calculate the ergodic sum-rate. 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 by101010Similar 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 sum-rate in (21) is independent of the MMSE-SIC 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 Gaussian-Chebyshev quadrature approximation[48], the CDF and PDF of can be written as

(26)

respectively, where and are given in (III-A).

According to (23), given the equal resource allocation strategy, i.e., , the asymptotic ergodic sum-rate of MIMO-NOMA associated with can be obtained as follows:

(27)

where denotes the average channel gain, which is given by

(28)
Remark 2

Comparing (III-A) and (27), we can observe that for a sufficiently large number of users, the considered MIMO-NOMA system is asymptotically equivalent to a SISO-NOMA 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 MIMO-NOMA, the multi-antenna BS behaves asymptotically in the same way as a single-antenna BS. Additionally, when , due to the diverse channel directions of all the users, the received signals fully span the -dimensional signal space[51]. Therefore, MIMO-NOMA using MMSE-SIC reception can fully exploit the system’s spatial DoF, , and its performance can be approximated by that of a SISO-NOMA system with -fold DoF.

Iv-B Ergodic Sum-rate of MIMO-OMA with FDMA-ZF

Upon installing more UL receiver antennas at the BS, ZF can be employed for MUD and the MIMO-OMA system using FDMA-ZF can accommodate users on each frequency subband. As mentioned before, we adopt a random user grouping strategy for the MIMO-OMA system using FDMA-ZF 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 MIMO-OMA 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 sum-rate of MIMO-OMA using FDMA-ZF 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 sum-rate of the MIMO-OMA system considered can be expressed as:

(31)

Iv-C Ergodic Sum-rate of MIMO-OMA with FDMA-MRC

The instantaneous achievable data rate of user in the MIMO-OMA system using the FDMA-MRC receiver is given by

(32)

Upon adopting the equal resource allocation strategy, i.e., and , the instantaneous sum-rate of MIMO-OMA relying on FDMA-MRC is obtained by

(33)

Averaging over the channel fading, we arrive at the ergodic sum-rate of MIMO-OMA using FDMA-MRC as

(34)

with given by

(35)

where denotes the Meijer G-function. The equality in (IV-C) is obtained due to and the equality in (IV-C) is based on Equation (3) in [62]. Now, the ergodic sum-rate of MIMO-OMA using FDMA-MRC can be written as

(36)

Note that, the ergodic sum-rate 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 MIMO-NOMA over MIMO-OMA. Hence, based on (IV-C), we derive the asymptotic ergodic sum-rate of MIMO-OMA with FDMA-MRC in the low-SNR regime with as follows:

(37)

On the other hand, in the high-SNR regime, based on (IV-C), the asymptotic ergodic sum-rate of MIMO-OMA using FDMA-MRC is given by

(38)

Iv-D ESG in Multi-antenna Systems

By comparing (27) and (31), we have the asymptotic ESG of MIMO-NOMA over MIMO-OMA relying on FDMA-ZF as follows:

(39)

To unveil some insights, we consider the asymptotic ESG in the high-SNR regime as follows

(40)

where denotes the large-scale near-far gain given in (18).

Remark 3

The identified two kinds of gains in ESG of the single-antenna system in (17) are also observed in the ESG of MIMO-NOMA over MIMO-OMA using FDMA-ZF in (40). Moreover, it can be observed that both the large-scale near-far gain and the small-scale 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 MIMO-NOMA over MIMO-OMA is -times of that in single-antenna 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, MIMO-NOMA and MIMO-OMA using FDMA-ZF 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 MIMO-OMA system considered[22].

Comparing (27) and (36), the asymptotic ESG of MIMO-NOMA over MIMO-OMA with FDMA-MRC is obtained by:

(42)

Then, based on (27) and (37), the asymptotic ESG of MIMO-NOMA over MIMO-OMA with FDMA-MRC in the low-SNR regime is given by