Downlink and Uplink Non-Orthogonal Multiple Access in ... - IEEE Xplore

This article has been accepted for publication This is theinauthor's a futureversion issue ofofthis an article journal,that buthas hasbeen not been published fully edited. in this journal. ContentChanges may change wereprior madetotofinal thispublication. version by the Citation publisher information: prior to publication. DOI 10.1109/JSAC.2017.2724646, IEEE Journal The final version of on record Selected is available Areas in at Communications http://dx.doi.org/10.1109/JSAC.2017.2724646 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS

1

Downlink and Uplink Non-Orthogonal Multiple Access in a Dense Wireless Network Zekun Zhang, Student Member, IEEE, Hanjian Sun, Student Member, IEEE, and Rose Qingyang Hu, Senior Member, IEEE

Abstract—To address the ever increasing high data rate and connectivity requirements in the next generation 5G wireless network, novel radio access technologies (RATs) are actively explored to enhance the system spectral efficiency and connectivity. As a promising RAT for 5G cellular networks, nonorthogonal multiple access (NOMA) has attracted extensive research attentions. Compared with orthogonal multiple access (OMA) that has been widely applied in existing wireless communication systems, NOMA possesses the potential to further improve system spectral efficiency and connectivity capability. This paper develops analytical frameworks for NOMA downlink and uplink multi-cell wireless systems to evaluate the system outage probability and average achievable rate. In the downlink NOMA system, two different NOMA group pairing schemes are considered, based on which theoretical results on outage and achievable data rates are derived. In the uplink NOMA, revised back-off power control scheme is applied and outage probability and per UE average achievable rate are derived. As wireless networks turn into more and more densely deployed, inter-cell interference has become a dominant capacity limiting factor but has not been addressed in most of the existing NOMA studies. In this paper a stochastic geometry approach is used to model a dense wireless system that supports NOMA on both uplink and downlink, based on which analytical results are derived either in pseudo-closed forms or succinct closed forms and are further validated by simulations. Numerical results demonstrate that NOMA can bring considerable system-wide performance gain compared to OMA on both uplink and downlink when properly designed. Index Terms—5G, average achievable rate, dense wireless system, downlink, inter-cell interference, NOMA, outage probability, Poisson Point Process, stochastic geometry, uplink

I. I NTRODUCTION

I

N the existing wireless communications systems like 4G LTE, orthogonal multiple access (OMA) schemes such as orthogonal frequency division multiple access (OFDMA) and single carrier frequency division multiple access (SCFDMA) have been widely used [1] [2]. While OMA can effectively minimize inter-user interference with a relatively low implementation complexity, its spectral efficiency and connectivity capability need to be further improved [3]. In order to meet demands such as very high data rates and tremendous connectivities required by the next generation (5G) cellular network [4], new radio access technologies (RATs) are actively pursued and explored. Zekun Zhang, Haijian Sun and Rose Qingyang Hu are with the Department of Electrical and Computer Engineering, Utah State University, Logan, UT 84322 USA (e-mail: [email protected], [email protected], [email protected]).

Non-orthogonal multiple access (NOMA), recognized as a promising candidate RAT in 5G wireless system, has received tremendous attentions lately. In contrast to OMA, NOMA allows multiple users to use one frequency/time resource at the same time and offers many advantages such as improving spectral efficiency, enhancing connectivity, providing higher cell-edge throughput, and reducing transmission latency [5]– [9]. NOMA can be classified into two categories, namely code domain NOMA (CD-NOMA) and power domain NOMA (PDNOMA). CD-NOMA utilizes different codes on the same resource to achieve multiplexing gain while PD-NOMA assigns users with distinct power levels to maximize the performance. CD-NOMA can obtain a spreading gain at the cost of more consumed bandwidth [8]. In this paper, we focus on PDNOMA, which multiplexes users on the power domain. At the transmitter side, messages for multiple users are superposed by Superposition Coding (SC) [10]. At receiver side, Successive Interference Cancellation (SIC) [11] [12] is utilized to extract the intended message. Extensive research has been done on the power domain NOMA. In [13], authors investigated power allocation (PA) techniques in the downlink NOMA that ensure fairness for users. In [6], the analytical results of outage probability and achievable sum-rate are derived for downlink NOMA in a single cell scenario. In [14] authors investigated the impact of user pairing on the performance of NOMA systems and demonstrated that the performance gain of NOMA over OMA can be further enlarged by selecting users whose channel conditions are more distinctive. Authors in [15] analyzed uplink NOMA performance in a single cell scenario and proposed a power back-off scheme to obtain diverse arrived powers. In [16], authors proposed a novel dynamic power allocation scheme for 2 users, which can meet various quality of service requirements in both downlink and uplink. In most of the existing work on NOMA [6] [14]–[17], intercell interference is either not considered or simply treated as a constant value. Further, the typical cell under analysis is normally assumed to have a regular shape such as a circle. With these assumptions, the interference value must be empirically estimated offline and the accuracy is compromised. Furthermore, the lack of a realistic interference model makes it difficult to analyze the impact of some key system parameters/features, such as transmit power, base station (BS)/user equipment (UE) densities, and power control scheme, on the overall NOMA system performance [18]. As a dense wireless network tends to be interference limited, a novel analytical framework explicitly considering the impact of inter-cell in-

0733-8716 (c) 2017 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information. Copyright (c) 2017 IEEE. Personal use is permitted. For any other purposes, permission must be obtained from the IEEE by emailing [email protected].

This article has been accepted for publication This is theinauthor's a futureversion issue ofofthis an article journal,that buthas hasbeen not been published fully edited. in this journal. ContentChanges may change wereprior madetotofinal thispublication. version by the Citation publisher information: prior to publication. DOI 10.1109/JSAC.2017.2724646, IEEE Journal The final version of on record Selected is available Areas in at Communications http://dx.doi.org/10.1109/JSAC.2017.2724646 2

IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS

terference is desired to analyze the performance of NOMA in a dense network. In this paper, we develop an analytical framework to evaluate the performance of a wireless system with NOMA on both downlink and uplink. The system model consists of densely deployed cells and inter-cell interference is explicitly modeled. Stochastic geometry approach is taken to model the deployment of BSs and UEs on a 2-D plane and to analyze the overall system performance. More specifically, Poisson Point Process (PPP) is used to model the locations of BSs and UEs due to its tractable yet rather accurate results [18], [19]. The main contributions of this work are summarized in the following. 1) This paper provides a complete multi-cell NOMA system model for analysis and performance evaluation on both uplink and downlink. 2) Extended from previous work [20] on downlink, significant improvements are made in the following. a) SIC error propagation during the decoding process is considered in this paper. Moreover, outage probability is added in the analysis. b) In [20], the members within a NOMA pair are randomly selected from all UEs. In this work, a selective pairing scheme is proposed and analyzed. The comparison between random pairing and selective pairing is also presented. 3) By using the same system model in downlink, the uplink NOMA performance is also analyzed in terms of outage probability and average achievable rate. To the best of our knowledge, this work is the first to provide analytical results for uplink NOMA system in a dense multi-cell scenario. Owning to the tractability of the PPP model, all results are derived in closed form or a pseudo-closed form. 4) All the analytical results are validated by system level simulations. The rest of the paper is organized as follows. In Section II, both uplink and downlink NOMA system models are developed. In Section III, downlink NOMA system performance on outage probability and per UE average achievable rate are analyzed. Two NOMA pairing schemes, namely random pairing and selective pairing, are used and compared. Section IV presents the analysis on uplink NOMA system performance on outage probability and per UE average achievable rate based on revised back-off power control scheme. Section V gives the numerical system performance results from both analysis and simulation. The paper concludes with Section VI. II. S YSTEM M ODEL In this paper we consider a dense multi-cell wireless system that supports NOMA on both downlink and uplink. Both BS and UE are equipped with one antenna. The set of BSs, denoted as Φb , are deployed in the Euclidean plane according to a PPP model with a density λb [19]. The system assumes a frequency reuse factor 1, hence the same frequency resources are used in all the cells. The radio resources are partitioned into a number of subbands and resources are allocated in

the unit of subband. For notation simplicity, the bandwidth of each subband is normalized to 1 and the analysis of user performance focuses on a typical subband by assuming flat fading channels across subbands. The NOMA study in this paper assumes a group size of two. Existing results show that NOMA with more than two UEs may provide a better performance gain [20]. However, considering processing complexity for SIC receivers, especially when SIC error propagation is considered, 2-UE NOMA is actually more practical in reality [21]. This assumption is also consistent with specifications in 3GPP LTE Advanced [22]. UE locations are assumed to follow another PPP Φu with a density λu . Φu is independent on Φb . We assume λu >> λb so that a sufficient number of UEs can always be found to form a NOMA group in each cell. A UE is associated with the nearest BS and is located in the Voronoi cell of its associated BS. NOMA system performance is analyzed on both downlink and uplink. A. Downlink NOMA System Model The downlink NOMA system is shown in Fig. 1. Without loss of generality, the analysis is performed in a typical cell denoted as BS0 . Based on Slivnyak’s Theorem [23], due to the stationarity of Φb , the typical cell can reflect the spatially averaged performance of the entire system. Two different pairing schemes are investigated in the downlink NOMA system. The first scheme is based on random pairing, in which 2 UEs are randomly selected to form a NOMA group. The second scheme is based on selective pairing. The first UE has an signal-to-interference-plus-noise ratio (SINR) above threshold T1 and the second UE has an SINR below threshold T2 , T2 ≤ T1 . In both pairing schemes, UE with a better normalized channel gain is denoted as U E1 and UE with a worse normalized channel gain is denoted as U E2 . The detailed definition of normalized channel gain is provided later. The power allocation strategy for downlink NOMA can be classified into two categories, namely fixed power allocation and dynamic power allocation [21]. In a fixed power allocation scheme, the downlink power allocated to a UE is predefined and remains unchanged [6] [14]. In contrast, a dynamic power allocation adapts power allocation based on instantaneous channel information [21]. In this paper, we adopt the fixed power allocation strategy due to the fact that it can achieve a suboptimal performance without excessive signaling overhead required by dynamic power allocation strategy [24] [25]. Pb denotes the total transmit power on a subband. The powers allocated to U E1 and U E2 can be expressed as P1d = ǫPb and P2d = (1 − ǫ)Pb respectively, where ǫ ∈ (0, 0.5) is a NOMA power control parameter. The transmitted signals to U E1 and U E2 are expressed as x1 and x2 respectively, with E[|xi |2 ] = 1. Since U E1 and U E2 form a NOMA group, x1 and x2 are encoded as the composite signal at the BS [6], q q (1) x = P1d x1 + P2d x2 . Thus the received signal at U Ei , i ∈ {1, 2}, can be represented as q yi = hi ri−α x + ni , (2)


This article has been accepted for publication This is theinauthor's a futureversion issue ofofthis an article journal,that buthas hasbeen not been published fully edited. in this journal. ContentChanges may change wereprior madetotofinal thispublication. version by the Citation publisher information: prior to publication. DOI 10.1109/JSAC.2017.2724646, IEEE Journal The final version of on record Selected is available Areas in at Communications http://dx.doi.org/10.1109/JSAC.2017.2724646 ZHANG et al.: DOWNLINK AND UPLINK NON-ORTHOGONAL MULTIPLE ACCESS IN A DENSE WIRELESS NETWORK

3

B. Uplink NOMA System Model

BS

UE1 Desired Signal

UE2 Interference

Figure 1. System model for downlink NOMA system. All the other cells generate inter-cell interference to UEs under analysis though only interference from one BS is noted on the graph to make graph succinct.

where ni denotes the additive noise plus inter-cell interference. As U E1 has a better channel condition, U E1 first decodes x2 and removes it from the received composite signal y1 , based on which U E1 can further decode x1 . U E2 directly decodes x2 by treating x1 as interference. In reality, SIC decoding in the first step may not be successful and thus error is carried over to the next level decoding, which is called SIC error propagation. The achievable rates of U E1 and U E2 on each subband in a downlink NOMA by considering SIC error are given as τ1d = log2 1 +

h1 r1−α P1d c1 P1d , −α d = log2 1 + βc1 P2d + 1 I1 + βh1 r1 P2 (3)

and τ2d = log2 1 +

h2 r2−α P2d c2 P2d . −α d = log2 1 + c2 P1d + 1 I2 + h2 r2 P1 (4)

hi is the Rayleigh fading gain between BS0 and U Ei and it follows an exponential distribution with mean 1, i ∈ {1, 2}. It is assumed that all hi , ∀i are i.i.d. and are reciprocal on uplink and downlink. ri is the distance between U Ei to BS0 . P −α α is the path-loss exponent. Ii = j∈Φb \BS0 gi,j Ri,j Pb is the cumulative downlink inter-cell interference from all other BSs to U Ei , where gi,j is the Rayleigh fading gain from BSj , j ∈ Φb \ BS0 , to U Ei and it also follows an exponential distribution with mean 1. Φb \ BS0 represents the set of all BSs excluding BS0 . It is assumed that all {gi,j } are i.i.d. and independent on Φb \ BS0 . Ri,j is the distance from interfering h r −α BSj to U Ei . The normalized channel gain ci = i Iii is defined as the complete channel gain including path-loss and fast fading normalized by inter-cell interference. β ∈ [0, 1] denotes the fraction of NOMA interference due to SIC error propagation [26]. Noise can be safely neglected in a dense interference-limited wireless system.

In the uplink, inter-cell interference comes from all the UEs in other cells sharing the same subband, as shown in Fig. 2. When modeling inter-cell interference, we assume the system is fully loaded and all the cells perform a 2-UE uplink NOMA with the same power control scheme. The locations of two UEs that form a NOMA group in each cell are randomly selected among UEs associated to that cell. For instance, there are N UEs in a single cell (N >> 2 as we assume λu >> λb ) and we can randomly select 2 out of N UEs in this cell to form a NOMA group on the subband under consideration. One of these two UEs is treated as U E1 and the other one is treated as U E2 . The locations of U E1 s in each cell form a distribution Φ1 and the locations of U E2 s in each cell form a distribution Φ2 . Both Φ1 and Φ2 depend on Φb . However, as validated in many existing work [18] [27], Φ1 and Φ2 can be approximated as PPP with respective densities λ1 = λb and λ2 = λb . The accuracy of this approximation can be validated by simulation results later. In the uplink, a receiver BS normally has much more capable hardware and advanced algorithms than a UE, so a perfect SIC is assumed at BSs. Distance based proportional power control has been widely applied, in which the transmit power of U Ei is inversely proportional to distance, i.e., P0 ∗ riα . P0 is the target received power and is set the same for all UEs. As the channel model consists of path-loss and Rayleigh fading, the actual received power at BS0 is P0 ∗ hi . Within a 2-UE NOMA group, UE with the higher Rayleigh fading gain is denoted as U E1 and UE with the smaller Rayleigh fading gain is denoted as U E2 , h1 > h2 . In this power control scheme, the diversity of the received power only depends on Rayleigh fading and all the UEs have the same averaged received power. Nevertheless, this level of difference on the received powers by using distance based proportional power control may not be sufficient enough to distinguish UEs within the same NOMA group [15]. Therefore in this paper, a revised power back-off scheme is used for uplink power control. The original power back-off scheme was proposed in [15]. In the revised version, we define a back-off step ρ, ρ ∈ (0, 1], and the transmit power of U E2 is set as P2u = ρ ∗ P0 ∗ r2α . No back-off is applied to U E1 . Thus its power is set as P1u = P0 ∗ r1α . The received powers of U E1 and U E2 are P0 ∗h1 and ρ∗P0 ∗h2 respectively. Since h1 > h2 , the received powers are more distinctive with ρ < 1. Notice that the original scheme in [15] applies back-off to the UE that has a longer transmit distance to save power. Due to the independency between path-loss and Rayleigh fading, in [15] the decoding order can be decided only when the UE that has a longer transmit distance also has a smaller Rayleigh fading gain, which may not be true in reality. In the revised scheme, U E1 is always decoded first because we apply back-off based on the fading channel gain. Also since we are considering a dense network, the extreme case that transmit power may exceed the UE’s hardware capacity is not taken into account in this paper. Uplink NOMA allows 2 UEs to transmit on the same subband. The received signal at BS0 using revised power




Table I L IST OF K EY N OTATIONS

BS

UE1

UE2

Desired Signal

Interference

Figure 2. System model for uplink NOMA system. All the UEs in other cells using the same subband generate inter-cell interference to BS0 . Only interference from one cell is noted to make graph succinct.

(5)

where w denotes the additive noise plus inter-cell interference. With SIC at BS, x1 is decoded first by treating x2 as interference. Afterwards, x1 is removed from y0 and then x2 can be decoded. It is assumed that x1 is always decoded first in this paper. Otherwise the rate of U E2 is extremely low. As a result, the achievable per subband rate of each UE in uplink NOMA are given as h1 ρh2 + I0,1 + I0,2 + ρh2 τ2u = log2 1 + . 2 I0,1 + I0,2 + Pσ 0

σ2 P0

,

(6) (7)

P −α u In the above equations, I0,1 = j∈Φ1 /UE1 g1,j R1,j P1,j and P −α u g R P are accumulated inter-cell I0,2 = j∈Φ2 /UE2 2,j 2,j 2,j interference from Φ1 /U E1 and Φ2 /U E2 respectively. g1,j , R1,j , g2,j and R2,j are the corresponding Rayleigh fading u α u α gain and interfering distance. P1,j = r1,j and P2,j = ρ ∗ r2,j represent the respective UE transmit powers in Φ1 /U E1 and Φ2 /U E2 divided by P0 . r1,j and r2,j are the corresponding u transmit distances. Notice that P2,j has a back-off step ρ. σ 2 2 denotes noise power which is constant and additive. Pσ 0 is the inverse of uplink arrived signal-to-noise ratio (SNR). The key notations used in this paper are listed in Table. I

III. D OWNLINK NOMA

PPP constituted by BSs (UEs) Density of BSs (UEs) PPP constituted by U Ei s in other cells Density of U Ei s in each cell Selecting threshold for U Ei Transmit power of BS on a subband Downlink NOMA power control parameter Downlink NOMA transmit power allocated to U Ei Rayleigh fading gain between BS0 and U Ei Distance between BS0 and U Ei Path-loss exponent Normalized channel gain Cumulative inter-cell interference received at U Ei Fraction of remaining NOMA interference Transmit power of U E1 Target received power at BS Back-off step for P2u Cumulative inter-cell interference from Φi Noise power

A. Downlink NOMA with random pairing

back-off scheme is expressed as p p y0 = P0 h1 x1 + ρP0 h2 x2 + w,

τ1u = log2 1 +

Φb (Φu ) λb (λu ) Φi /U Ei λi Ti Pb ǫ Pid hi ri α ci Ii β Piu P0 ρ I0,i σ2

SYSTEM ANALYSIS

The performance of downlink NOMA system is analyzed in the metrics of outage probability and average achievable rate. We first present the study on NOMA with the random pairing scheme, then followed by the study on NOMA with the selective paring scheme.

The outage probability and average achievable rate for a downlink NOMA system with random pairing are analyzed in this subsection. To facilitate the derivation, the cumulative distribution function (CDF) of the normalized channel gain for a randomly selected UE is first presented. 1) CDF of normalized channel gain for a randomly selected UE: The normalized channel gain of a randomly selected UE −α is c = hrI . The CDF of c is given in [20] as Fc (C) = 1 −

1 + (CPb )

2 α

1 R∞

1 2 α (CPb )− α 1+u 2

du

.

(8)

For a special case with α = 4, (8) can be further simplified to α=4

Fc (C) = 1 −

1+

√

CPb ( π2

1 . 1 − arctan( √CP ))

(9)

b

2) Outage performance of NOMA with random pairing: In this subsection the outage probabilities of U E1 and U E2 are derived as well as the overall outage probability in a 2UE downlink NOMA case. The analysis extends our previous work in [20] with following major improvements. First we consider imperfect SIC in the analytical while in [20] a perfect SIC is assumed. Second, in this work, we evaluate the performance of each UE in term of outage probability while in [20] the distribution of post process signal-to-interference ratio (SIR) is studied. We also provide the overall NOMA system performance, which is not considered in [20]. Two UEs are randomly selected among all UEs associated with the tagged BS and are marked as U Ea and U Eb . Denote the normalized channel gains of U Ea and U Eb as ca and cb . Rank the channel gains and let U E1 = {U Ei |U Ei ∈ {U Ea , U Eb }, ci = max(ca , cb )} and U E2 = {U Ei |U Ei ∈ {U Ea , U Eb }, ci = min(ca , cb )}. Let z = max(x, y) and w = min(x, y). The CDF of z and w can be expressed as



Fz (z) = Fxy (z, z) and Fw (w) = Fx (w)+Fy (w)−Fxy (w, w) [28]. Thus CDFs of c1 and c2 can be derived in the following. Fc1 (C) = Fca cb (C, C) = Fc (C)2 , Fc2 (C) = Fca (C) + Fcb (C) − Fca cb (C, C) = 2Fc (C) − Fc (C)2 ,

(10) (11)

where we use the facts that {ca , cb } are i.i.d. and Fc (C) is given in (8). The outage probability is defined as the probability that τik , k ∈ {d, u}, fails to meet the defined quality of service (QoS) requirement, which is defined as the target data rate τ¯i in this paper. We first derive the respective outage probability for U E1 and U E2 , and then we provide the outage probability for the overall system. d For U E1 , the rate for U E1 to decode U E2 ’s message τ1→2 must be greater than the QoS requirement of U E2 τ¯2 so that it is able to remove U E2 ’s signal from interference. Thus the outage probability of U E1 can be evaluated as follows: d pd1 (τ¯1 , τ¯2 ) = 1 − P[τ1d > τ¯1 , τ1→2 > τ¯2 ]

c1 P1d c1 P2d = 1 − P[ γ , > > γ2 ] 1 βc1 P2d + 1 c1 P1d + 1 ( Pd Pd 1, if γ1 ≥ βP1d or γ2 ≥ P2d ; = (12) 2 1 Fc1 max(θ1 , θ2 ) , otherwise.

γ1 = 2τ¯1 − 1, γ2 = 2τ¯2 − 1, θ1 = P d −γγ1 βP d , and θ2 = 1 1 2 γ2 are used to simplify the expression. Fc1 (C) is given P2d −γ2 P1d in (10). The outage probability of U E2 is given as

i h c Pd 2 2 γ > pd2 (τ¯2 ) = 1 − P[τ2d > τ¯2 ] = 1 − P 2 c2 P1d + 1 ( d P 1, if γ2 ≥ P2d ; = (13) 1 Fc2 (θ2 ), otherwise. In addition to the individual UE outage probability, we also derive the overall outage probability of NOMA UEs as pdtotal (τ¯1 , τ¯2 )   1, if γ1 ≥  =

P1d βP2d

or γ2 ≥ Pd

P2d ; P1d

5

3) Average Achievable Rate of NOMA with Random Pairing: It is assumed that each UE can reach Shannon bound for d their instantaneous SIR. Notice that τ1→2 > τ2d is always true as c1 > c2 . Therefore, the average achievable rate of U E1 can be given as h1 r1−α P1d )] I1 + βh1 r1−α P2d = E[log2 (1 + (βP2d + P1d )c1 )] − E[log2 (1 + βP2d c1 )]. (15)

d τ1,avg = E[log2 (1 +

The first expectation in (15) can be computed as E[log2 (1 + (βP2d + P1d )c1 )] Z (a) P[log2 (1 + (βP2d + P1d )c1 ) > t]dt = Z t>0 2t − 1 (1 − Fc1 ( = ))dt, βP2d + P1d t>0

(16)

R where (a) is met as E[X] = t>0 P(X > t)dt for a positive random variable X. By setting P1d = 0 in (16), the second expectation in (15) can be acquired. Then the complete result d is of τ1,avg d τ1,avg =

Z

Fc1 (

t>0

2t − 1 2t − 1 )dt. ) − F ( c 1 βP2d βP2d + P1d

(17)

By following the same way given above, the average achievable rate of U E2 is h2 r2−α P2d )] I2 + h2 r2−α P1d = E[log2 (1 + (P1d + P2d )c2 )] − E[log2 (1 + P1d c2 )] Z 2t − 1 2t − 1 = Fc2 ( )dt. (18) ) − F ( c 2 P1d P2d + P1d t>0

d τ2,avg = E[log2 (1 +

The framework developed here can also be applied to more general order based pairing schemes [14]. For instance, the performance of pairing U Ev and U Ew that are selected from M UEs, 1 ≤ v ≤ w ≤ M , can be derived by following the same approach.

Pd

Fc2 (θ2 ), if γ1 < βP1d , γ2 < P2d , θ2 > θ1 ;  2 1   Fc (θ1 )2 + 2Fc (θ2 ) − 2Fc (θ1 )Fc (θ2 ), otherwise. (14)

The detailed derivation of (14) is provided in Appendix A. By comparing (12) and (13) one can observe that pd2 (τ¯2 ) is only affected by τ¯2 . However, pd1 (τ¯1 , τ¯2 ) is affected by both τ¯1 and τ¯2 . This is due to the fact that U E1 needs to decode x2 at first in order to decode its own message x1 . Therefore, failing to decode x2 also causes outage at U E1 . Also one can see that the outage probability always equals 1 if τ¯1 or τ¯2 is not set appropriately. After selecting reasonable values for τ¯1 and τ¯2 , pd1 (τ¯1 , τ¯2 ) is only determined by either τ¯1 or τ¯2 , whereas pdtotal (τ¯1 , τ¯2 ) is affected by both τ¯1 and τ¯2 simultaneously as shown in (14).

B. Downlink NOMA with Selective Pairing The previous analysis is based on random pairing for the two UEs in a NOMA group. The results in [14] show that the performance gain of NOMA can be further enlarged by selecting NOMA UEs deliberately. In this subsection, we provide the analytical results when selective pairing scheme is used. Instead of selecting UEs based on the order of their channel gain like what is applied in [14], NOMA selects UEs based on the actual channel gain values. More specifically, UE whose normalized channel gain is above a pre-determined threshold T1 can be selected as U E1,s while UE whose normalized channel gain is below another threshold T2 , T2 ≤ T1 , is selected as U E2,s .




1) Outage Probability with Selective Pairing: Denoted by c1,s the normalized channel gain of U E1,s , the CDF of c1,s is calculated as Fc1,s (C) = P[c1,s < C] = P[c < C|c > T1 ] ( 0, if C < T1 ; (a) = P[cT1 ] otherwise. P[c>T1 ] , ( 0, if C < T1 ; = Fc (C)−Fc (T1 ) otherwise. 1−Fc (T1 ) ,

(19)

(20)

By substituting Fc1 (C) in (12) with Fc1,s (C) and substituting Fc2 (C) in (13) by Fc2,s (C), the outage probabilities of U E1,s and U E2,s respectively become pd1,s (τ¯1 , τ¯2 ) = 1 − P[τ1,s > τ¯1 , τ1,s→2 > τ¯2 ]

c1,s P1d c1,s P2d = 1 − P[ > γ1 , > γ2 ] d βc1,s P2 + 1 c1,s P1d + 1  d d 1, if γ1 ≥ P1d or γ2 ≥ P2d ; βP2 P1 = Fc max(θ1 , θ2 ) , otherwise. 1,s  Pd Pd  1, if γ1 ≥ βP1d or γ2 ≥ P2d ;   2 1 d d = 0, if γ1 < P1d and γ2 < P2d and max(θ1 , θ2 ) < T1 ; βP2 P1    Fc (max(θ1 ,θ2 ))−Fc (T1 ) , otherwise. 1−Fc (T1 ) (21) pd2,s (τ¯2 ) = 1 − P[τ2,s > τ¯2 ] ( Pd 1, if γ2 ≥ P2d ; = 1 Fc2,s (θ2 ), otherwise.  1, if γ ≥ P2d or θ > T ; 2 2 2 P1d = F (θ )  c 2 , otherwise.

(22)

Fc (T2 )

The definitions of τ1,s , τ1,s→2 , and τ2,s are the same as defined earlier on. As pd1,s (τ¯1 , τ¯2 ) and pd2,s (τ¯2 ) are independent, the total system outage probability with selective pairing is pdtotal,s (τ¯1 , τ¯2 ) = 1 − P[τ1,s > τ¯1 , τ2,s > τ¯2 ]

= 1 − P[τ1,s > τ¯1 ]P[τ2,s > τ¯2 ]

= 1 − (1 − pd2,s (τ¯2 ))P[τ1,s > τ¯1 ],

h c Pd i 1,s 1 P[τ1,s > τ¯1 ] = P > γ 1 βc P d + 1 ( 1,s 2 Pd 0, if γ1 ≥ βP1d ; = 2 1 − Fc1,s (θ1 ), otherwise.

(24)

By summarizing the equations above, we get the complete result of pdtotal,s (τ¯1 , τ¯2 ) as

In (a) Bayes’ rule is applied. Similarly the CDF of U E2,s ’s normalized channel gain is calculated as Fc2,s (C) = P[c2,s < C] = P[c < C|c < T2 ] ( 1, if C > T2 ; = P[c0 Z log2 ((βP2d +P1d )T1 +1) Fc ( 2t −1 ) − Fc (T1 ) βP2d dt = 1 − Fc (T1 ) log2 (βP2d T1 +1) t 2t −1 Z ∞ Fc ( 2βP−1 d ) − Fc ( βP d +P d ) 2 2 1 + dt. (26) d d 1 − F (T ) c 1 log2 ((βP2 +P1 )T1 +1) Similarly, substituting Fc1 (C) in (18) with Fc1,s (C), the average achievable rate of U E2,s with selective pairing is Z 2t − 1 2t − 1 d τ2,s,avg = Fc2,s ( ) − Fc2,s ( d )dt d P1 P2 + P1d t>0 Z log2 (P1d T2 +1) F ( 2t −1 ) − F ( 2t −1 ) c Pd c P d +P d 1 2 1 dt = F (T ) c 2 0 t −1 Z log2 ((P1d +P2d )T2 +1) Fc ( P2d +P d) 2 1 + 1− dt. (27) Fc (T2 ) log2 (P1d T2 +1) C. NOMA Power Control with Selective Pairing In [16] authors point out that a fixed power allocation based NOMA can not strictly meet the predefined QoS. For example, in a fixed power allocation based NOMA, the rate of a poor channel UE can be lower than that in OMA. However, selective pairing NOMA has the freedom to set the values of P1d and P2d so that the performance of all UEs can be better than that in OMA. Assuming a perfect SIC, the rate of U E1,s is τ1,s = o log2 (1+c1,s P1d ) when using NOMA. τ1,s = 12 log2 (1+c1,s Pb ) is the rate when using OMA assuming the transmit power and resource are equally allocated to two UEs. In order to o guarantee τ1,s ≥ τp,1 , the following constraint needs to be met. 1 log2 (1 + c1,s P1d ) ≥ log2 (1 + c1,s Pb ), 2



When γ1 ρ ≥ 1, p¯u1 (τ¯1 ) becomes

which is equivalent to p 1 + c1,s Pb − 1 d . P1 ≥ c1,s

(28)

Since c1,s > T1 , (28) is always true when P1d > c2,s P2d c2,s P1d +1

√

which is equivalent to p 1 + c2,s Pb − 1 ≤ . c2,s

(29)

(29) is always true when P1d < T1 ≥ T2 , both (28) and (29) can be √ satisfied simultaneously when setting 1+PTb1T1 −1 < P1d < √ 1+Pb T2 −1 , i.e., T2 √ √ 1 + Pb T1 − 1 1 + Pb T2 − 1 h2 , the complementary of the outage probability of U E1 is given by i h h1 > τ¯1 p¯u1 (γ1 ) = P log2 1 + σ2 ρh2 + I0,1 + I0,2 + P0 2

= P[h1 > max(h2 , γ1 (ρh2 + I0,1 + I0,2 +

σ ))]. P0 (35)

σ2 )] P[h1 > γ1 (ρh2 + I0,1 + I0,2 + P0 Z ∞ Z ∞ = E[ fh1 h2 (h1 , h2 )dh1 dh2 ] γ1 ρ>1

=

h2 =0

h1 =γ1 (ρh2 +I0,1 +I0,2 )

γ1 σ 2 2 e− P0 LI0,1 (γ1 )LI0,2 (γ1 ). = 1 + γ1 ρ

(36)

In last step we use fh1 h2 (h1 , h2 ) = 2e−h1 −h2 , based on order statistics [29]. LI0,1 (s) and LI0,2 (s) are given in (31) and (32) respectively. When γ1 ρ < 1, 2 h γ1 (I0,1 + I0,2 + σP0 ) i P h1 > h2 , h2 > 1 − γ1 ρ h σ2 + P h1 > γ1 (ρh2 + I0,1 + I0,2 + ), P0 2 γ1 (I0,1 + I0,2 + Pσ 0 ) i h2 < 1 − γ1 ρ Z Z ∞ 2 fh1 h2 (h1 , h2 )dh1 dh2 + =E γ1 (I0,1 +I0,2 + σ )

γ1 ρh2

σ2 h1 >γ1 (ρh2 +I0,1 +I0,2 + P 0

fh1 h2 (h1 , h2 )dh1 dh2 )

2γ1 σ2 2γ1 2γ1 γ1 ρ − 1 − LI0,1 ( )LI0,2 ( )e (1−γ1 ρ)P0 = 1 + γ1 ρ 1 − γ1 ρ 1 − γ1 ρ γ σ2 2 − 1 LI0,1 (γ1 )LI0,2 (γ1 )e P0 . + 1 + γ1 ρ

(37)

By summarizing (36) (37) and using pu1 (γ1 ) = 1 − p¯u1 (γ1 ), one can get the outage probability of U E1 as pu1 (γ1 ) =  γ σ2 − 1P  2  0 L if γ1 ρ ≥ 1; e 1 −  I0,1 (γ1 )LI0,2 (γ1 ), 1+γ1 ρ   2γ σ2 − (1−γ1 ρ)P γ1 ρ−1 2γ1 2γ1 1 0 1 − L ( )L ( )e I0,2 1−γ1 ρ 1+γ1 ρ I0,1 1−γ1 ρ    γ1 σ 2   − 2 L (γ )L (γ )e− P0 , if γ ρ < 1. I0,2 1 1 1+γ1 ρ I0,1 1

(38)

To decode U E2 ’s message, U E1 ’s message must be decoded successfully first. After U E1 ’s message is removed from composed signal, U E2 ’s message can be decoded. Therefore, the complementary of the outage probability of U E2 can be expressed as h1 1, 2 ) > τ¯ ρh2 + I0,1 + I0,2 + σP0 i ρh2 ) > τ ¯ log2 (1 + 2 2 I0,1 + I0,2 + Pσ 0

h p¯u2 (γ1 , γ2 ) = P log2 (1 +

= P[h1 > max(h2 , γ1 (ρh2 + I0,1 + I0,2 + h2 >

γ2 σ2 )]. (I0,1 + I0,2 + ρ P0

σ2 )), P0 (39)




When γ1 ρ ≥ 1, p¯u2 (γ1 , γ2 ) is

B. Average Achievable Rate for Uplink NOMA u The average achievable rate τ1,avg for uplink NOMA U E1 can be expressed in form of outage probability as given below.

p¯u2 (γ1 , γ2 ) = P[h1 > γ1 (ρh2 + I0,1 + I0,2 +

σ2 ), P0

i h1 2 h2 ρ + I0,1 + I0,2 + Pσ 0 Z i h h1 (a) t > 2 − 1 dt P = 2 h2 ρ + I0,1 + I0,2 + Pσ 0 t>0 Z = p¯u1 (2t − 1)dt, (44)

h u τ1,avg = E log2 1 +

2

σ γ2 )] h2 > (I0,1 + I0,2 + ρ P 0 Z Z h =E 2 h2 =

γ2 ρ

σ (I0,1 +I0,2 + P ) 0

fh1 h2 (h1 , h2 )dh1 dh2 =

2 LI (A)LI0,2 (A)e 1 + γ1 ρ 0,1

A = γ1 +

γ2 ρ

2

h1 =γ1 (ρh2 +I0,1 +I0,2 + σ P )

i

0

t>0

2 − Aσ P0

.

+ γ1 γ2 . When γ1 ρ < 1 and

(40) γ1 1−γ1 ρ

>

γ2 ρ ,

p¯u2 (γ1 , γ2 )

γ1 , h1 > h2 ]+ 1 − γ1 ρ γ2 γ1 σ2 σ2 P[ (I0,1 + I0,2 + ) < h2 < ), (I0,1 + I0,2 + ρ P0 1 − γ1 ρ P0 σ2 h1 > γ1 (ρh2 + I0,1 + I0,2 + )] P0 2γ1 σ2 2γ1 2γ1 γ1 ρ − 1 − LI0,1 ( )LI0,2 ( )e (1−γ1 ρ)P0 = γ1 ρ + 1 1 − γ1 ρ 1 − γ1 ρ Aσ2 2 (41) + LI0,1 (A)LI0,2 (A)e− P0 1 + γ1 ρ = P[h2 >

Finally, when γ1 ρ < 1 and

γ1 1−γ1 ρ

h2 ] (I0,1 + I0,2 + ρ P0 2γ2 2γ2 − 2γρP2 σ2 0 . = LI0,1 ( (42) )LI0,2 ( )e ρ ρ

p¯u2 (γ1 , γ2 ) = P[h2 >

Summarizing (40) (41) (42) and using pu2 (γ1 , γ2 ) = 1 − p¯u2 (γ1 , γ2 ), one can obtain the outage probability of U E2 as pu2 (γ1 , γ2 ) =  2 − Aσ 2  P0  1 − 1+γ (A)e L (A)L , if γ1 ρ ≥ 1; I I  0,2 0,1 1ρ    2γ σ2  1 − γ1 ρ−1 LI ( 2γ1 )LI ( 2γ1 )e− (1−γ11 ρ)P0 0,1 1−γ1 ρ 0,2 1−γ1 ρ γ1 ρ+1 2  − 2 L (A)L (A)e− Aσ γ1 P0  , if γ1 ρ < 1, 1−γ ≥ γρ2 ;  I0,2 1+γ1 ρ I0,1  1ρ  2   1 − L ( 2γ2 )L ( 2γ2 )e− 2γρP2 σ0 , if γ ρ < 1, γ1 < γ2 . I0,1 ρ I0,2 ρ 1 1−γ1 ρ ρ (43) When α = 4, both pu1 (γ1 ) and pu2 (γ1 , γ2 ) can be expressed in closed forms. Unlike downlink NOMA system, in uplink NOMA system the outage probability of U E1 (good channel UE) is only affected by its own target rate τ¯1 . Whereas the outage probability of U E2 (poor channel UE) is affected by both τ¯1 and τ¯2 due to the fact that a BS needs to decode the message from U E1 at first. More details about the relationship between outage probability and target rate are discussed in section V.

R where in (a) again we use E[X] = t>0 P(X > t)dt. Similarly u the average achievable rate τ2,avg for uplink NOMA U E2 is derived as h i h2 ρ u τ2,avg = E log2 1 + σ2 I0,1 + I0,2 + P0 Z h i h2 ρ t P = > 2 − 1 dt 2 I0,1 + I0,2 + Pσ 0 t>0 Z p¯u2 (0, 2t − 1)dt. (45) = t>0

V. N UMERICAL P ERFORMANCE R ESULTS In this section, system performance is numerically evaluated based on both analytical models and simulations. For all the results, the density of BS is set as λb = 10−3 /m2 (corresponding to a hexagon grid with a radius 6.2 m) and path-loss exponent α = 4. The bandwidth of one subband is normalized to 1. As comparisons to NOMA, OMA results from simulations are also presented. For a fair comparison to NOMA, OMA gives each of the two UEs in the NOMA group half unit of the resource. For the downlink OMA, BS transmits with half of its full power to each UE on its dedicated resource. For the uplink OMA, each UE transmits on its dedicated resource subject to the uplink proportional power control. A. Downlink NOMA Performance Results BS power Pb is also normalized to 1 and the power control parameter ǫ for downlink NOMA is set to 0.2 unless otherwise mentioned. Fig. 3 shows U E1 outage probability pd1 (τ¯1 , τ¯2 ), U E2 outage probability pd2 (τ¯2 ), and overall system outage probability pdtotal (τ¯1 , τ¯2 ) vs. different target rates τ¯1 and τ¯2 . Fig. 3(a) presents the impact of τ¯1 on the outage probability when τ¯2 is fixed at 0.2 bits/s/subband. Fig. 3(b) fixes τ¯1 at 0.1 bits/s/subband to demonstrate the impact of τ¯2 . β is set to 0, i.e., perfect SIC, in both cases. From the figure one can see the analytical results match the simulation results very well, which validates the accuracy of the analysis. In Fig. 3(a), pd1 (τ¯1 , τ¯2 ) remains constant when τ¯1 is below 0.06, due to the fact that U E1 needs to decode the signal intended to U E2 first before it can decode the signal for itself. When τ¯1 is below 0.06, the outage is always caused by failing to decode x2 . This outcome can also be explained by looking into the definition of pd1 (τ¯1 , τ¯2 ). Recall that in (12) d pd1 (τ¯1 , τ¯2 ) = 1 − P[τ1d > τ¯1 , τ1→2 > τ¯2 ], which is a function



τ¯2 is fixed to 0.2 bits/s/subband NOMA NOMA NOMA NOMA NOMA NOMA

Outage Probability

0.5

τ¯2 is fixed to 0.2 bits/s/subband

0.8

OMA UE1

U E1 Analytical Result U E1 Simulation U E2 Analytical Result U E2 Simulation Overall Analytical Result Overall Simulation

NOMA UE1 ǫ=0.1

0.7

NOMA UE1 ǫ=0.3 NOMA UE1 ǫ=0.5

0.6

Outage Probability

0.6

9

0.4

0.3

0.2

0.5 0.4 0.3 0.2

0.1 0.1 0

0

0.1

0.2

0.3

0.4

0

0.5

0

0.1

τ¯1 (bits/s/subband)

(a) Outage probability for random pairing NOMA when τ¯2 is fixed to 0.2 bits/s/subband

0.5

0.8

0.4


NOMA U E2 ǫ=0.1

0.7

NOMA U E2 ǫ=0.3 NOMA U E2 ǫ=0.5

0.6

Outage Probability

Outage Probability

0.4

OMA U E2 NOMA NOMA NOMA NOMA NOMA NOMA

0.5

0.3

0.2

0.5 0.4 0.3 0.2 0.1

0.1

0 0

0.3

(a) Outage probability of U E1 . τ¯2 is fixed to 0.2 bits/s/subband


0.6

0.2


0

0.1

0.2

0.3

0.4

0

0.1

0.2

0.3

0.4

0.5


0.5


(b) Outage probability of U E2

(b) Outage probability for random pairing NOMA when τ¯1 is fixed to 0.1 bits/s/subband

Figure 4. Downlink outage probability comparison between random pairing NOMA and OMA

Figure 3. downlink outage probability for random pairing NOMA

d of both τ¯1 and τ¯2 . Given a fixed τ¯2 , τ1→2 > τ¯2 can guarantee d d τ1 > τ¯1 when τ¯1 is small and p1 (τ¯1 , τ¯2 ) can be rewritten d to pd1 (τ¯1 , τ¯2 ) = 1 − P[τ1→2 > τ¯2 ], which is not a function

of τ¯1 anymore and thus keeps a constant when τ¯1 is small. The curve of pdtotal (τ¯1 , τ¯2 ) overlaps with pd2 (τ¯2 ) when τ¯1 is small due to a similar reason declared above. Recall that in (47) pdtotal (τ¯1 , τ¯2 ) = 1 − P[τ1d > τ¯1 , τ2d > τ¯2 ], which can be rewritten to pdtotal (τ¯1 , τ¯2 ) = 1 − P[τ2d > τ¯2 ] when τ¯1 is small. Therefore, we have pdtotal (τ¯1 , τ¯2 ) = pd2 (τ¯2 ) and thus their curves completely overlap with each other when τ¯1 is small. pd2 (τ¯2 ) is a flat line in Fig. 3(a) as τ¯2 is fixed and it is not influenced by τ¯1 .

Fig. 3(b) in turn fixes τ¯1 at 0.1 bits/s/subband to investigate the impact of τ¯2 . One can observe that pd1 (τ¯1 , τ¯2 ) remains constant at first and then increases in the same way in Fig. 3(a), due to a similar reason that makes pd1 (τ¯1 , τ¯2 ) constant in Fig. 3(a). When τ¯2 is small, pd1 (τ¯1 , τ¯2 ) can be rewritten as 1 − P[τ1d > τ¯1 ], which is not affected by τ¯2 . As τ¯2 goes up, both τ¯1 and τ¯2 will impact pd1 (τ¯1 , τ¯2 ) and pd1 (τ¯1 , τ¯2 ) starts to increase along with τ¯2 . pd2 (τ¯2 ) completely overlaps with pdtotal (τ¯1 , τ¯2 ) after τ¯2 exceeds a certain value, which is Pd consistent with the result in (14). When τ¯2 satisfies γ2 < P2d 1

and θ2 > θ1 (both γ2 and θ2 are functions of τ¯2 ), pdtotal (τ¯1 , τ¯2 ) becomes a function of τ¯2 . By comparing Fig. 3(a) and Fig. 3(b), one can discover that for a given τ¯2 , τ¯1 does not affect pd1 (τ¯1 , τ¯2 ) or pdtotal (τ¯1 , τ¯2 ) within a certain range of τ¯1 . However, τ¯2 always affects the outage probability regardless of the value of τ¯1 . Fig. 4 compares the outage performance between random pairing NOMA and OMA. Only analytical numerical results are presented as simulation results always match the analytical results very well on the downlink. In Fig. 4(a) one can observe that with a smaller value of ǫ, which means a smaller transmit power is allocated to U E1 , pd1 (τ¯1 , τ¯2 ) is always higher than the OMA outage probability. Even with ǫ = 0.5, pd1 (τ¯1 , τ¯2 ) is still higher than the OMA outage probability when τ¯1 is relatively small. This is due to the fact that U E1 needs to decode x2 first and outage occurs regardless of the τ¯1 value if x2 fails to be decoded. Also one can observe that the lower bound of pd1 (τ¯1 , τ¯2 ), which is determined by successfully decoding x2 , goes up along with ǫ. This is because more transmit power allocated to U E1 also means less power allocated to U E2 and thus it is more difficult for U E1 to decode x2 . Therefore, increasing ǫ does not necessarily help improve pd1 (τ¯1 , τ¯2 ). In Fig. 4(b), with a




OMA NOMA NOMA NOMA NOMA NOMA NOMA

2.8 2.6 2.4


1 ǫ=0.1 ǫ=0.1 ǫ=0.3 ǫ=0.3 ǫ=0.5 ǫ=0.5

Analytical Result Simulation Analytical Result Simulation Analytical Result Simulation

2.2 2 1.8

NOMA NOMA NOMA NOMA NOMA NOMA

0.9 0.8

Outage Probability

Average Achievable Rate (bits/s/subband)

3

0.7


0.6 0.5 0.4 0.3

1.6

0.2 1.4

0.1 1.2

0

0.02

0.04

0.06

0.08

0

0.1

β d Figure 5. Average achievable rate of downlink NOMA UEs (sum of τ1,avg d and τ2,avg ) with imperfect SIC

0

0.5

1

1.5


(a) Outage probability of U E1 . τ¯2 is fixed to 0.2 bits/s/subband τ¯1 is fixed to 0.1 bits/s/subband

1 0.9 0.8

Outage Probability

smaller ǫ, i.e., more transmit power is allocated to U E2 , pd2 (τ¯2 ) decreases. From Fig. 4, one can conclude that pd2 (τ¯2 ) always improves with more allocated power to U E2 . However, for U E1 , pd1 (τ¯1 , τ¯2 ) is affected by both τ¯1 and τ¯2 . More allocated power to U E1 can result in an even worse outage probability for U E1 . In Fig. 5 the impact of imperfect SIC on NOMA is investigated. The impact of imperfect SIC is crucial on the average achievable rate of NOMA. When β = 0.06, i.e., 6% inter-user NOMA interference fails to be eliminated, the gain of NOMA completely vanishes in the presented cases. As we know, a greater ǫ value means more power is allocated to U E1 . One can observe that with a greater ǫ, the gain of NOMA over OMA is higher. So a greater ǫ makes it more resistant to the impact of imperfect SIC. But when ǫ < 0.1, even with a perfect SIC, NOMA does not show any gain over OMA. Fig. 6 and Fig. 7 present the performance of downlink NOMA with selective pairing. The thresholds are set as T1 = 3 dB and T2 = 0 dB respectively. Fig. 6 shows how τ¯1 and τ¯2 impact the outage probability. In Fig. 6(a), the outage probability of U E1 can be as low as 0 and the overall outage probability is the same as pd2,s (τ¯2 ) when pd1,s (τ¯1 , τ¯2 ) is 0, since the channel gain of U E1 in selective pairing has a lower bound c1,s > T1 . By looking into (21), one can see that given a fixed τ¯2 selective pairing can have zero outage for τ¯1 that satisfies max(θ1 , θ2 ) < T1 . Similarly, in Fig. 6(b), pd1,s (τ¯1 , τ¯2 ) also has zero outage when τ¯2 < 1.1 bits/s/subband for a fixed τ¯1 . pd2,s (τ¯2 ) has an upper bound, which is due to the channel threshold c2,s < T2 . By summarizing Fig. 6(a) and Fig. 6(b), one can conclude that zero outage is possible for selective pairing by choosing τ¯1 and τ¯1 deliberately while outage always has a non-zero probability for the random pairing case. Fig. 7 compares the outage performance between selective pairing NOMA and OMA. In Fig. 7(a), one can find that by setting ǫ properly, the outage probability of U E1 in selective pairing NOMA can always be lower than that in OMA. However, such a value of ǫ does not exist for random pairing NOMA, as shown in Fig. 4(a). By observing Fig. 7(a) and Fig. 7(b) together, one can find that when setting ǫ = 0.4, which satisfies (30), both U E1 and U E2 can have better outage

0.7 NOMA NOMA NOMA NOMA NOMA NOMA

0.6 0.5 0.4


0.3 0.2 0.1 0

0

0.5

1

1.5


(b) Outage probability of U E2 . τ¯1 is fixed to 0.1 bits/s/subband Figure 6. Downlink outage probability for selective pairing NOMA

performance than using OMA. Therefore, for selective pairing NOMA, performance gain over OMA can be guaranteed with a simple fixed power allocation scheme, whereas dynamic power allocation is required to achieve such a performance gain in random pairing NOMA [16]. Fig. 8 compares the performance of selective pairing NOMA and random pairing NOMA. It is not fair to compare them by absolute achievable rate as they each select different UEs for NOMA, i.e., UEs in random pairing NOMA are selected randomly whereas UEs in selective pairing NOMA are selected based on thresholds. Therefore, the comparison between two NOMA pairing schemes is made on the gain of average achievable rate over OMA, i.e., the difference of achievable rate obtained by NOMA and OMA for the same group of UEs. T2 is fixed at 0 dB and we focus on the affect of T1 . By increasing T1 , the performance of selective pairing can be further improved. This outcome is consistent with the conclusion drawn in [14], which states that the performance gain of NOMA over OMA can be further enlarged by selecting UEs whose channel conditions are more distinctive. Although it improves the performance of selective pairing, increasing T1 also makes it harder to find a qualified U E1 . In practice, there is a trade off between the performance of a selected pair and the number of pairs that can be selected. Another observation one can obtain from Fig.




1

NOMA U E1 ǫ=0.4

0.8

NOMA U E1 ǫ=0.5

Outage Probability

0.7 0.6 0.5 0.4 0.3

0.7

0.4 0.3 0.2 0.1

0.5

1

0

1.5

0

0.2


1 0.9

0.8

0.8

0.7

0.7

Outage Probability

Outage Probability

1

0.6 0.5 0.4 0.3

0

OMA U E2 NOMA U E2 ǫ=0.1 NOMA U E2 ǫ=0.4 NOMA U E2 ǫ=0.5

0

0.5

0.6

0.8

1

(a) Outage probability of U E1 .

0.9

0.1

0.4


(a) Outage probability of U E1 . τ¯2 is fixed to 0.2 bits/s/subband

0.2


0.5

0.1 0

ρ=0.2 ρ=0.2 ρ=0.5 ρ=0.5 ρ=0.8 ρ=0.8

0.6

0.2

0

OMA U E1 NOMA U E1 NOMA U E1 NOMA U E1 NOMA U E1 NOMA U E1 NOMA U E1

0.9

NOMA U E1 ǫ=0.1

0.8

Outage Probability

1

OMA U E1

0.9

11

1

0.6 0.5 OMA U E2 NOMA U E2 NOMA U E2 NOMA U E2 NOMA U E2 NOMA U E2 NOMA U E2 NOMA U E2 NOMA U E2

0.4 0.3 0.2 0.1

1.5


0

0

0.2

0.4

ρ=0.2 ρ=0.2 ρ=0.5 ρ=0.5 ρ=0.5 ρ=0.5 ρ=0.8 ρ=0.8

τ¯1 τ¯1 τ¯1 τ¯1 τ¯1 τ¯1 τ¯1 τ¯1

=0.2 =0.2 =0.2 =0.2 =0.5 =0.5 =0.2 =0.2

Analytical Result Simulation Analytical Result Simulation Analytical Result Simulation Analytical Result Simulation

0.6

0.8

1




Figure 7. Downlink outage probability comparison between selective pairing NOMA and OMA

Figure 9. Uplink outage probability comparison between NOMA and OMA

T2 is fixed to 0 dB


1.4

Random Pairing ǫ=0.2 Selective Pairing ǫ=0.2 Selective Pairing ǫ=0.2 Random Pairing ǫ=0.4 Selective Pairing ǫ=0.4 Selective Pairing ǫ=0.4

1.2

Analytical Result Simulation Analytical Result Simulation

1

0.8

0.6

0.4

0.2

0

1

2

3

4

5

T1 (dB)

Figure 8. Comparison of downlink average achievable rate gain over OMA between random pairing NOMA and selective paring NOMA.

8 is that allocating more transmit power to U E1 can boost the total rate and selective pairing NOMA can benefit more from that than random pairing NOMA. B. Uplink NOMA Performance Results Fig. 9 compares the uplink outage probability between NOMA with different back-off steps ρ and OMA. The arrived

SNR Pσ20 is set to 30 dB. We assume OMA UE under comparison uses the same power as its NOMA counterpart. Each OMA UE is allocated half of the subband resource and its interference plus noise becomes 12 (P0 I0,1 + P0 I0,2 + σ 2 ). The small gap between the simulation and the analytical results comes from the assumption that Φ1 and Φ2 are approximated to PPPs, which in reality are not. Fig. 9(a) shows that for U E1 , the outage probability of NOMA outperforms OMA when ρ is small, as a smaller ρ results in a less inter-user interference from U E2 in the same cell and also less inter-cell interference from U E2 s in other cells (I0,2 ). The outage probability of U E2 is given in Fig. 9(b). U E2 ’s outage performance degrades as ρ gets small. Although a smaller ρ decreases the inter-cell interference to U E2 , the loss due to a lower transmit power is more significant. Fig. 9(b) also shows that increasing τ¯1 can increase the outage probability of U E2 . In uplink NOMA, BS needs to decode the message intended to U E1 first. Increasing τ¯1 results in a higher outage for U E1 , which consequently impacts U E2 signal decoding. Fig. 10 compares the average achievable rates of U E1 and U E2 with different ρ values. One can observe that the average achievable rate goes up with the arrived SNR level. However, after the arrived SNR level reaches a certain value, this gain disappears due to the fact that the uplink system turns into





1.4

ρ=0.2 ρ=0.2 ρ=0.5 ρ=0.5 ρ=0.8 ρ=0.8

1.2


1

0.8

0.6 U E1

0.4

0.2

0

U E2

0

5

10

15

20

Arrived SNR (dB)

The study shows that selective pairing can offer a better performance gain over OMA than the random pairing scheme. Moreover, selective pairing NOMA is able to use a fixed power allocation scheme to achieve the prominent performance gain, which can only be realized by a dynamic power allocation in order based NOMA scheme. Analytical results of uplink NOMA are derived with the same system model for the downlink scenario and a revised back-off uplink power control scheme is used. The study shows that increasing arrived SNR does not bring significant performance gain once the arrived SNR reaches a certain level. This paper systematically investigates outage probability and achievable data rates for both uplink and downlink NOMA in a dense wireless network with various system settings.

Figure 10. Uplink average achievable rate of NOMA vs. different arrived SNR.

A PPENDIX A interference limited. Again the slight gap between simulation and analytical results comes from the approximation of the UE distribution as PPP, with which the analytical results become tight lower bounds. Fig. 11 presents the comparison between

d pdtotal (τ¯1 , τ¯2 ) = 1 − P[τ1d > τ¯1 , τ1→2 > τ¯2 , τ2d > τ¯2 ].


1.4

(46)

d Notice that τ1→2 > τ¯2 is always true when τ2d > τ¯2 due to c1 > c2 . Therefor (46) can be simplified to pdtotal (τ¯1 , τ¯2 ) = 1 − P[τ1d > τ¯1 , τ2d > τ¯2 ]. The derivation proceeds as

1.3 1.2 1.1 1

pdtotal (τ¯1 , τ¯2 )

0.9 NOMA sum ρ=0.2 OMA sum ρ=0.2 NOMA sum ρ=0.5 OMA sum ρ=0.5 NOMA sum ρ=0.8 OMA sum ρ=0.8

0.8 0.7 0.6

The overall outage probability of NOMA UEs is defined as

0

5

10

15

20

Arrived SNR (dB)

Figure 11. Uplink average achievable rate comparison between NOMA and OMA.

uplink NOMA and OMA on the average achievable rate. Clearly the sum rate of NOMA outperforms OMA for all SNR scenarios. And the gain of NOMA over OMA is more significant when the value of ρ is small, which means the gain of U E1 is much more significant than the loss of U E2 . However, in Fig. 10, it shows that the gap between the rates of U E1 and U E2 also goes up as the value of ρ gets small, leading to a poorer fairness. Thus when selecting the value of ρ, both sum rate and fairness need to be evaluated.

= 1 − P[τ1d > τ¯1 , τ2d > τ¯2 ] i h c Pd c2 P2d 1 1 γ , γ > > =1−P 1 2 βc1 P2d + 1 c2 P1d + 1 ( d P Pd 1, if γ1 ≥ βP1d or γ2 ≥ P2d ; = (47) 2 1 1 − P[c1 > θ1 , c2 > θ2 ], otherwise. As c1 > c2 , P[c1 > θ1 , c2 > θ2 ] can be simplified to P[c2 > θ2 ] = 1 − Fc2 (θ2 ) when θ1 < θ2 . When θ1 ≥ θ2 , P[c1 > θ1 , c2 > θ2 ] can be derived as P[c1 > θ1 , c2 > θ2 ] Z ∞ Z c1 = fc1 c2 (c1 , c2 )dc1 dc2 2 Zc1∞=θ1 Zc2c=θ 1 = 2fc (c1 )fc (c2 )dc1 dc2 c1 =θ1

c2 =θ2

VI. C ONCLUSIONS

= 1 − Fc (θ1 )2 − 2Fc (θ2 ) + 2Fc (θ1 )Fc (θ2 ),

In this paper, we developed a theoretical framework to analyze NOMA downlink and uplink system performance. Analytical results are derived by using stochastic geometry approach in a dense wireless network environment. For the downlink NOMA system, two different pairing schemes, namely random pairing and selective pairing, are investigated.

where fc1 c2 (c1 , c2 ) is the joint probability density function (PDF) of c1 and c2 , and it can be acquired by using the knowledge of order statistics as fc1 c2 (c1 , c2 ) = 2fc (c1 )fc (c2 ) [29]. By summarizing the results above, the complete expression of pdtotal (τ¯1 , τ¯2 ) is acquired.

(48)



A PPENDIX B The Laplace transform of I0,1 , LI0,1 = EI0,1 [e given as LI0,1 (s) h −α u = E exp(−1(P1,j R1,j < 1) h =E

Y

j∈Φ1 /UE1

exp −

X

j∈Φ1 /UE1

−α u 1(P1,j R1,j

−sI0,1

], is

i −α u sP1,j g1,j R1,j )

−α u < 1)sP1,j g1,j R1,j

Z ∞ −sP1u gx−α = exp − 2πλ1 E (1 − e )xdx 1 (P u ) α Z ∞1 1 (b) xdx = exp − 2πλ1 E xα 1 (P1u ) α 1 + sP1u Z ∞ 1 2 (c) u α = exp − πλ1 E α (sP1 ) dy 2 s− α 1 + y 2 Z ∞ 2 2 1 = exp − πλ1 s α E[(P1u ) α ] α dy . 2 s− α 1 + y 2 (a)

i

(49)

1(·) is an indicator function that equals 1 when the condition in the parentheses is satisfied and equals 0 otherwise. The −α u condition P1,j R1,j < 1 defined here reflects the fact that u α P1,j = r1,j and r1,j < R1,j since all UEs are associated to the nearest BS. (a) follows from the probability generating Q functional (PGFL)R of PPP [30], which states that E[ x∈Φ f (x)] = exp(−λ R2 (1 − f (x))dx). (b) follows from 2 g ∼ exp(1) and (c) is acquired by using y = xu 2 . By using (sP1 ) α

the PDF of the distance between a UE and its associated BS 2 given ealier, E[(P1u ) α ] can be computed as Z ∞ 2 2 E[(P1u ) α ] = (rα ) α fr (r)dr Z0 ∞ 2 1 . (50) = r2 · 2πλb re−πλb r dr = πλ b 0 2

Notice that the value of E[(P1u ) α ] is not affected by α. LI0,2 (s) can be derived in a similar way. LI0,2 (s) h −α u = E exp(−1(P2,j R2,j < ρ)

2

2

= exp − πλ2 s α E[(P2u ) α ] 2

where E[(P2u ) α ] =

X

j∈Φ2 /UE2 ∞

Z

2 (ρs)− α

i −α u sP2,j g2,j R2,j )

1 α dy , 1+y2

(51)

2 1 α πλb ρ .

R EFERENCES [1] 3GPP TS36.300, “Evolved Universal Terrestrial Radio Access (E-UTRA) and Evolved Universal Terrestrial Radio Access Network (E- UTRAN),” Overall description. [2] 3GPP TR36.913 (V8.0.0), “Requirements for further advancements for E-UTRA (LTE-Advanced),” Jun. 2008. [3] R. Q. Hu and Y. Qian, “An Energy Efficient and Spectrum Efficient Wireless Heterogeneous Network Framework for 5G Systems,” IEEE Communications Magazine, vol.52, no.5, pp.94-101, May. 2014. [4] J. G. Andrews, S. Buzzi, W. Choi, S. V. Hanly, A. Lozano, A. C. K. Soong, and J. C. Zhang, “What will 5G be?” IEEE Journal on Selected Areas in Communications, vol. 32, pp. 1065-1082, Jun. 2014.

13

[5] P. Xu, Y. Yuan, Z. Ding, X. Dai, and R. Schober, “On the Outage Performance of Non-Orthogonal Multiple Access With 1-bit Feedback,” IEEE Transactions on Wireless Communications, vol. 15, no. 10, pp. 6716-6730, Oct 2016. [6] Z. Ding, Z. Yang, P. Fan, and H. V. Poor, “On the performance of nonorthogonal multiple access in 5G systems with randomly deployed users,” IEEE Signal Process. Letters, vol. 21, no. 12, pp. 1501-1505, Dec. 2014. [7] Y. Liu, Z. Ding, M. Elkashlan, and H. V. Poor, “Cooperative nonorthogonal multiple access with simultaneous wireless information and power transfer,” IEEE Journal on Selected Areas in Communications, vol. 34, no. 4, pp. 938-953, Apr. 2016. [8] L. Dai, B. Wang, Y. Yuan, S. Han, C. L. I, and Z. Wang, “Nonorthogonal multiple access for 5G: Solutions, challenges, opportunities, and future research trends,” IEEE Communications Magazine, vol. 53, no. 9, pp. 74-81, Sep. 2015. [9] S. M. Riazul Islam, Nurilla Avazov, Octavia A. Dobre, and Kyung-Sup Kwak,“Power-Domain Non-Orthogonal Multiple Access (NOMA) in 5G Systems: Potentials and Challenges,” IEEE Communications Surveys & Tutorials, vol. PP, Issue. 99, 2016. [10] T. Cover, “Broadcast channels,” IEEE Transactions on Information Theory, vol. 18, no. 1, pp. 2-14, Jan 1972. [11] S. Sen, N. Santhapuri, R. Choudhury, and S. Nelakuditi, “Successive Interference Cancellation: A Back-of-the-Envelope Perspective,” Proc. Ninth ACM Special Interest Group Data Comm. (SIGCOMM) Workshop Hot Topics in Networks (HOTNETS), 2010. [12] Mazen O. Hasna et al., “Performance analysis of mobile cellular systems with successive co-channel interference cancellation,” IEEE Transactions on Wireless Communications, vol. 2, issue 1, pp. 29-40, February 2003. [13] S. Timotheou and I. Krikidis, “Fairness for non-orthogonal multiple access in 5G systems,” IEEE Signal Process. Letters, vol. 22, no. 10, pp. 1647-1651, Oct. 2015. [14] Z. Ding, P. Fan, and H. V. Poor, “Impact of User Pairing on 5G NonOrthogonal Multiple Access Downlink Transmissions,” IEEE Transactions on Vehicular Technology, vol. 65, no. 8, Aug. 2016. [15] N. Zhang, J. Wang, G. Kang, and Y. Liu, “Uplink Nonorthogonal Multiple Access in 5G Systems,” IEEE Communications Letters, vol. 20, no. 3, Mar. 2016. [16] Z. Yang, Z. Ding, P. Fan and Naofal Al-Dhahir, “A General Power Allocation Scheme to Guarantee Quality of Service in Downlink and Uplink NOMA Systems,” accepted to IEEE Transactions on Wireless Communications, Citation information: DOI 10.1109/TWC.2016.2599521 [17] Z. Yang, Z. Ding, P. Fan, and G. K. Karagiannidis, “On the performance of 5G non-orthogonal multiple access systems with partial channel information,” IEEE Transactions on Communications, vol. 64, no. 2, pp. 654-667, Feb. 2016. [18] T. D. Novlan, H. S. Dillon, and J. G. Andrews, “Analytical modeling of uplink cellular networks,” IEEE Transactions on Wireless Communications, vol. 12, no. 6, pp. 2669-2679, June 2013. [19] J. G. Andrews, F. Baccelli, and R. Ganti, “A tractable approach to coverage and rate in cellular networks,” IEEE Transactions on Communications, vol. 59, no. 11, pp. 3122-3134, Nov. 2011. [20] Z. Zhang, H. Sun, R. Q. Hu, and Y. Qian, “Stochastic Geometry Based Performance Study on 5G Non-Orthogonal Multiple Access Scheme,” IEEE Global Communications Conference (GLOBECOM) 2016. Dec. 2016 [21] F. Liu, Petri Ma´yh´ynen and M. Petrova, “Proportional fairness-based power allocation and user set selection for downlink NOMA systems,” IEEE International Conference on Communications (ICC) 2016, May 2016. [22] 3rd Generation Partnership Project (3GPP), “Study on downlink multiuser superposition transmission for LTE,” Shanghai, China, Mar. 2015. [23] D. Stoyan, W. Kendall and J. Mecke, Stochastic Geometry and Its Applications. 1995, Wiley. [24] N. Otao, Y. Kishiyama, and K. Higuchi, “Performance of Nonorthogonal Access with SIC in Cellular Downlink Using Proportional Fair-Based Resource Allocation,” International Symposium on Wireless Communication Systems (ISWCS) 2012, pp. 476-480, Aug. 2012. [25] A. Benjebbour, A. Li, Y. Saito, Y. Kishiyama, A. Harada, and T. Nakamura, “System-Level Performance of Downlink NOMA for Future LTE Enhancements,” IEEE Global Communications Conference (GLOBECOM) 2013, Dec. 2013. [26] H. Sun, B. Xie, R. Q. Hu and G. Wu, “Non-orthogonal Multiple Access with SIC Error Propagation in Downlink Wireless MIMO Networks,” in Proc. IEEE Vehicular Technology Conference (VTC) Fall 2016, invited paper. Sep. 2016. [27] H. ElSawy, E. Hossain, and M. S. Alouini, “Analytical Modeling of Mode Selection and Power Control for Underlay D2D Communication




in Cellular Networks,” IEEE Transactions on Wireless Communications, vol. 62, no. 11, pp.4147-4161, Nov. 2014. [28] A. Papoulis, Probability, Random Variables, and Stochastic Processes. Fourth Edition. [29] H. A. David, Order statistics. John Wiley & Sons Inc, 1970. [30] M. Haenggi, Stochastic geometry for wireless networks. Cambridge University Press, 2012.

Zekun Zhang received the B.S. degree in electronics and information engineering from Beihang University, Beijing, China, in 2012. He is currently pursuing the Ph.D. degree with Department of Electrical and Computer Engineering, Utah State University. His current research interests include Deviceto-Device communication, Wireless Heterogeneous Networks and Non-Orthogonal Multiple Access Systems.

Haijian Sun [S’ 14] received the M. S. and B. S. degrees from Xidian University, Xi’an, China, in 2014 and 2011, respectively. He is currently pursuing the Ph. D. degree in the department of electrical and computer engineering at Utah State University, Logan, UT, USA. His research interests include MIMO, nonorthogonal multiple access, SWIPT, wearable communications, and 5G PHY.

Rose Qingyang Hu [S’95, M’98, SM’06] ([email protected]) is a Professor of Electrical and Computer Engineering Department at Utah State University. She received her B.S. degree from University of Science and Technology of China, her M.S. degree from New York University, and her Ph.D. degree from the University of Kansas. She has more than 10 years of R&D experience with Nortel, Blackberry and Intel as a technical manager, a senior wireless system architect, and a senior research scientist, actively participating in industrial 3G/4G technology development, standardization, system level simulation and performance evaluation. Her current research interests include next-generation wireless communications, wireless system design and optimization, green radios, Internet of Things, Cloud computing/fog computing, multimedia QoS/QoE, wireless system modeling and performance analysis. She has published over 170 papers in top IEEE journals and conferences and holds numerous patents in her research areas. Prof. Hu is an IEEE Communications Society Distinguished Lecturer Class 2015-2018 and the recipient of Best Paper Awards from IEEE Globecom 2012, IEEE ICC 2015, IEEE VTC Spring 2016, and IEEE ICC 2016. She is currently serving on the editorial boards for IEEE Transactions on Wireless Communications, IEEE Transactions on Vehicular Technology, IEEE Communications Magazine, IEEE Wireless Communications Magazine, IEEE Internet of Things journal. She has also been 9 times guest editors for IEEE Communications Magazine, IEEE Wireless Communications Magazine, and IEEE Network Magazine. Prof. Hu is a senior member of IEEE.
