Outage Performance of NOMA in Downlink SDMA ... - IEEE Xplore

IEEE ICC 2017 Wireless Communications Symposium

Outage Performance of NOMA in Downlink SDMA Systems with Limited Feedback Qian Yang∗ , Tong-Xing Zheng∗ , Hui-Ming Wang∗ , Hao Deng† , Yi Zhang∗ , Xiayi Qiu∗ , and Pengcheng Mu∗ ∗ School

of Electronic and Information Engineering, Xi’an Jiaotong University Ministry of Education Key Lab for Intelligent Networks and Network Security, Xi’an Jiaotong University Xi’an 710049, Shaanxi, P. R. China † School of Physics and Electronics, Henan University, Kaifeng 475004, Henan, P. R. China Email: qian-yang@outlook.com

Abstract—In this paper, the outage performance of nonorthogonal multiple access (NOMA) is investigated in a downlink space division multiple access (SDMA) network with a multiantenna base station and randomly deployed users. The NOMA technology is concurrently exploited with SDMA to further improve spectral efficiency. With limited channel state information (CSI) feedback taken into account, an analytical framework is proposed to evaluate outage performance for a given user. An expression for the outage probability is derived in closed form. Moreover, the diversity order and the effect of the number of feedback bits on the outage performance of NOMA are analyzed. Numerical results are demonstrated to verify our analytical findings and show that different from the perfect CSI case there always exists performance floor in the considered network due to limited feedback.

I. I NTRODUCTION Non-orthogonal multiple access (NOMA), as a promising and enabling technology in the fifth generation (5G) networks, has received increasing research interests nowadays. One of the most prominent features brought by NOMA is the superior spectral efficiency, which plays an indispensable role in the evolution of 5G wireless communications [1]. The NOMA technology has been widely investigated under single-antenna systems for spectral efficiency promotion [2]– [4]. In [2], the performance of NOMA is studied under a cellular downlink scenario with randomly deployed users. In [3], the power allocation problem in NOMA downlink systems is studied from a fairness standpoint under both instantaneous and average channel state information (CSI). The optimal power allocation in terms of energy efficiency maximization is derived in [4]. In addition, little literature is available concerning performance analysis with imperfect CSI. The implementation of NOMA in multi-antenna broadcast systems, also known as space division multiple access (SDMA) systems, is of great importance, since spectral efficiency and performance can be further improved with the introduced additional degree of freedom (DoF) [1], [5], [6]. The weight vector is designed to maximize the sum rate in a multi-user MIMO (MU-MIMO) NOMA system in [7]. In [8], a MU-MIMO NOMA system is considered, where the minimum-power multicast beamforming for multi-resolution broadcast is derived. In [9], the users in the MIMO system

978-1-4673-8999-0/17/$31.00 ©2017 IEEE

are randomly grouped into multiple clusters, and NOMA principles are exploited by intra-cluster users to suppress intracluster interference once the MIMO system is decomposed into separate single-antenna NOMA systems. To relax the assumption of the demand on the antenna number at receivers in [9], the authors in [10] propose a novel MIMO-NOMA framework for downlink and uplink transmission using the concept of signal alignment. The application of NOMA in Massive-MIMO systems is investigated in [11]. It should be pointed out that the required CSI overhead at the transmitter in [9]–[11] can be small, since the receivers in the considered MU-MIMO systems equip with multiple antennas and thus the inter-group interference can be well suppressed at the receiver. However, when it comes to the MU-MIMO system with single-antenna receivers, in most of current works such as [7], [8] perfect CSI is assumed to be available at the transmitter for beamforming and power allocation, which is impractical especially when the antenna number at the transmitter is large. In practice, collecting CSI at a base station usually relies on the limited feedback from its users especially in frequencydivision duplexed (FDD) systems where the channel reciprocity is not guaranteed [12], [13]. However, few work has been done on the study of NOMA performance in a SDMA system with limited feedback. In [11], only one-bit feedback for user ordering is investigated under a Massive-MIMONOMA system. In [14], user selection and power allocation schemes are proposed to improve the sum rate of a MUMIMO system with limited feedback. However, the work in [14] confines the number of users in each beam to two. Most importantly, a comprehensive analysis concerning the outage performance of NOMA in multi-antenna SDMA systems with limited feedback is still missing, which motivates our work. In this paper, we consider a downlink single-cell cellular network with a multi-antenna base station and randomly deployed users. The NOMA technology is integrated with SDMA to simultaneously serve multiple users under a general limited feedback framework. We give a comprehensive study on the outage performance of NOMA and the effect of the number of feedback bits under the considered system. Notations: AT and AH represent the transpose and Hermitian transpose of a matrix A, respectively. The factorial of a


B. Feedback and Schedule Model 6'0$

6DPH FRGHERRN

' 'LIIHUHQW W FRGHERRN

6DPH FRGHERRN

%DVHVWDWLRQ 6FKHGXOHG8VHUV ,GOH8VHUV

Fig. 1. The considered downlink cellular network with a multi-antenna base station and multiple single-antenna users. The NOMA technology is exploited to simultaneously serve multiple users under each spatial beam of SDMA.

n! non-negative integer n is denoted by n!, and nk = (n−k)!k! . X ∼ Exp(λ) denotes the exponential distributed random variable with rate λ, X ∼ Gamma(M, α) denotes the Gammadistributed random variable with shape M and scale α, and X ∼ Beta(a, b) denotes the Beta-distributed random variable with parameters a and b. Γ(x) is the Gamma function [15, 8.310], and γ(α, x) is the lower incomplete gamma function d [15, 8.350.1]. X = Y means that the two random variables X and Y are equal in distribution. II. S YSTEM M ODELS In this section, the basic elements consisting of our system model will be described. The main idea of our scheme is to exploit NOMA to simultaneously serve more users in each spatial beam of SDMA under a cellular network. The details will be presented in the following subsections. A. Network Model Consider a downlink single-cell cellular network with M transmit antennas equipped at the base station and W ≥ M single-antenna users. The base station aims to simultaneously serve multiple scheduled users through broadcast. We assume that the users are uniformly distributed within the round cell with radius D, and the base station locates at its center as shown in Fig. 1. Therefore, the distances between the base station and the users follow the independent and identically distributed (i.i.d.) distribution with the cumulative distribution function (CDF) given by x2 , 0 ≤ x ≤ D. (1) D2 In the considered cellular network, the cell is assumed to be divided into G disjoint sectors where the scheduled users in each sector as a group are served by a common beam of SDMA as shown in Fig. 1. The detailed schedule strategy will be introduced in Section II-B. Over the conventional SDMA framework, the NOMA technology is concurrently exploited to simultaneously serve the users within each group for the further promotion of spectral efficiency. Fd (x) =

In this work, before data transmission each user is assumed to have perfect CSI through downlink channel estimation, and then each user feeds the CSI back to the base station through an error-free but limited-rate feedback channel, which is rather common in FDD systems [12]. Denote the channel from the base station to the k-th user −α/2 in the n-th group as hn,k = gn,k dn,k , where gn,k is the corresponding Rayleigh fading gain, α is the path-loss factor, and dn,k denotes the distance from the base station to the user. To reduce the overhead of channel quantization and feedback, we only consider the quantization of channel ñ,k = gn,k /gn,k , which direction information (CDI), i.e., g is critical in the beamforming design of SDMA. Furthermore, we assume that the same codebook for CDI quantization is shared by the users within the same group, while the users in different groups use different and independent codebooks as shown in Fig. 1. Note that this assumption can be easily satisfied with low overhead in practice. Once a user in a group receives its codebook from the base station, the codebook can be shared within the group via decentralized ways such as device-to-device (D2D) transmission and thus the overhead of the base station is reduced. Assuming that the number of feedback bits is B, then the size of the codebook consisting of M -dimensional unit-norm vectors is N = 2B . Denoting the codebook in the n-th group as Cn = {cn1 , cn2 , . . . , cnN },

1 ≤ n ≤ G,

(2)

the channel quantization for the k-th user in the n-th group is to find the nearest vector from codebook Cn satisfying H j = arg max |˜ gn,k cnj | 1≤j≤N

(3)

in terms of maximum inner product [12], [13]. After channel quantization, each user feeds the selected index j back to the base station. To simultaneously serve multiple users in different groups using SDMA, the base station first schedules a part of the users in each group which have the similar channel direction, and then the scheduled users in each group are served by a common beam during data transmission.1 For analytical simplicity, we assume that K users feeding back the same quantized index are randomly scheduled in each group and thus the distances from the base station to the scheduled users still follow the i.i.d. distribution with the CDF given in (1). Since the base station has the full knowledge of the codebooks in all the groups, the common beamformer shared by the K scheduled users in the n-th group can be designed by ˆn = cnj viewing the users’ channel directions as the same g in each group for 1 ≤ n ≤ G. In the subsequent performance analysis, we will use the well-known quantization cell approximation (QCA) and the codebook design based on random vector quantization (RVQ) 1 Based on the assumed group division among users, the users in the same group actually have the similar channel direction in a high probability since they are in the same sector and have close directions.


for analytical tractability as in [12], [13]. The main idea of the QCA is that each quantization cell can be approximated by a Voronoi region of a spherical cap with the area 2−B in a unit sphere [13]. If we decompose the actual channel direction of the k-th user in the n-th group as ñ,k = cos θn,k · g ˆn + sin θn,k · en,k , g

(4)

where θn,k denotes the angle between the actual CDI gn,k ˆn , and en,k denotes the error vector and the quantized CDI g ˆn , the CDF of isotropically distributed in the nullspace of g sin2 θn,k resulted from QCA is given by [12], [13] 2B xM −1 , 0 ≤ x ≤ δ, Fsin2 θn,k (x) = (5) 1, x ≥ δ, B

with δ = 2− M −1 . The QCA and RVQ are shown in [13] to yield very close performance, which means that the results derived in this work will reflect the actual performance for any well-designed codebook. C. Data Transmission Model In the conventional SDMA scheme, only one user in each spatial beam is served and the multiple access is implemented over spatial domain while power domain is generally neglected. In this paper, power domain is further exploited and K scheduled users in each beam (group) are simultaneously served by NOMA. In the NOMA scheme, successive interference cancellation (SIC) is exploited to suppress the intra-group interference. The decoding order of SIC in single-antenna systems is easy to figure out, but the optimal one in multi-antenna systems is difficult to obtain [7]. In this work, in terms of the channel quality information (CQI) we assume that the base station has the perfect knowledge about the distances of the users dn,k due to their rather steady locations. Therefore, here we sort the users within a group for performing SIC according to dn,k as in [16, Section II-B]. Without loss of generality, the distances between the users in the n-th group and the base station are sorted as dn,1 ≥ dn,2 ≥ . . . ≥ dn,K for 1 ≤ n ≤ G. Based on the NOMA and SDMA protocol, the base station simultaneously serves all the scheduled users in G groups and its transmit signal is characterized by x=

√

=

G √ wn P n=1

n=1 K

β n,k sn,k

yn,k =

H P βn,k hH n,k wn sn,k + hn,k wn

K

P βn,j sn,j

j=1,j=k

intra-group interference

+

√

G

P

¯m +nn,k , hH n,k wm s

m=1,m=n

(7)

inter-group interference

where nn,k is zero-mean additive white Gaussian noise (AWGN) with variance σ 2 . To suppress the intra-group interference in (7), SIC is employed at each user. To be specific, in the n-th group the k-th user first detects the i-th user’s message (i < k) and then removes the decoded message from its observation in a successive manner for i = 1, 2, . . . , k − 1. Then the k-th user detects its own message by regarding the i-th user’s message (i > k) as noise. Therefore, the signalto-interference-plus-noise ratio (SINR) for detecting the i-th user’s message (1 ≤ i ≤ k) by the k-th user is denoted by SINRin,k = |hH n,k wn |

2 |hH n,k wn | βn,i (,8) G H 2 j=i+1 βn,j +Λ m=1,m=n |hn,k wm | +1/ρ

K 2

where Λ = 1/G and ρ = P/σ 2 denotes the transmit signalto-noise ratio (SNR). For the specific design of precoding matrix W in SDMA, we assume that zero-forcing (ZF) beamforming is adopted to suppress the inter-group interference in (7) for analytical tractability. To fully exploit spatial DoFs provided by multiple antennas equipped at the base station, we only consider the case where G = M groups are simultaneously served by SDMA and leave the more general case G < M for future work. The base station uses the quantized CSI {ˆ g m }G m=1 to design the beamformer for each group, according to ZF the beamformer for group n is designed to satisfy H ˆm wn = 0, g

G √ P W¯s = P wn s¯n

ensure fairness among users within a group. Moreover, we assume that the transmit K power is equally allocated among different groups, i.e., k=1 βn,k = 1/G for 1 ≤ n ≤ G. The received signal at the k-th user in the n-th group is given by

∀m = n, 1 ≤ m ≤ G.

(9)

III. O UTAGE A NALYSIS

,

(6)

k=1

where P is the total transmit power; W = [w1 , w2 , . . . , wG ] ∈ CM ×G and ¯s = [¯ s1 , s¯2 , . . . , s¯G ]T ∈ CG×1 are the collections of the unit-norm beamforming vectors and the information bearing signals for all the G groups, respectively; βn,k and sn,k are the power allocation factor and unit-power information bearing signal for the k-th user in the n-th group, respectively. According to the principle of NOMA, we usually have βn,1 ≥ βn,2 ≥ . . . ≥ βn,K to

In this section, the outage performance for the considered SDMA-NOMA system is analyzed. The outage probability serves as an important metric in delay-sensitive communications where the transmitter transmits the message at a fixed data rate. In the sequel, we assume that the power allocation and transmit rates in NOMA are the same among all the G groups for the ease of analysis. Therefore, we only focus on the performance analysis of the users in one group and omit the group subscript n for notational simplicity. According to (8), the corresponding rate for the k-th user to decode the i-th


user’s message is denoted as Cki = log(1 + SINRin,k ). Note that the rates for the users to decode their own messages are given by {Ckk }K k=1 . By denoting the target rate (quality of ˜ k }K , an service, QoS) for the K users in a group as {R k=1 outage event will occur at the k-th user (1 ≤ k ≤ K) when either one of the following two constraints are not satisfied: ˜ i for i < k; 2) the user 1) the SIC is successful, i.e., Cki ≥ R can successfully decode the message targeted for itself, i.e., ˜k . Ckk ≥ R Since the obtained CSI is not perfect, under ZF beamforming the inter-group interference is still present. By leveraging (4) and (9), (8) can be recast as βX i SINRik = , (10) K j=i+1 βj X + ΛY + Z/ρ G 2 where Y = gn,k 2 sin2 θn,k m=1,m=n |eH n,k wm | , X = H 2 α |gn,k wn | , and Z = dn,k . The outage probability of the kth user in a group is then calculated as k out i ˜ Pk = 1 − Pr C k ≥ Ri = 1 − Pr (a)

= Pr

⎧i=1 k ⎨ ⎩

i=1

X ΛY + Z/ρ

⎫ ⎬

βX i ≥ φi ⎭ βj X + ΛY + Z/ρ (11) < ηk ,

K j=i+1

˜

where φi = 2Ri − 1, ci = β −φ φiK , and ηk = i i j=i+1 βj max1≤i≤k ci . Note that the step (a) in (11) is obtained by assuming βi > φi

K

√ ∞ M −1−1 √ where τ1 = a1 n=0 (1+ηk Λabn2 )G+n−1 , a1 = G−2+ , M −1 n 1 1 i a2 = δ 1 − √M −1 , bn = n i=1 (1 − a1 ) bn−i for n = 1, 2, . . . with b0 = 1, and G = M from the full-loaded ZF-SDMA assumption. ñ,k and wn are inProof: According to [13], since g dependent and isotropically distributed in CM ×1 , we have H wn |2 ∼ Beta(1, M − 1) and thereby X ∼ Exp(1). Z |˜ gn,k is a function of the k-th user’s distance dn,k . The probability distribution function (PDF) of the ordered distance from the base station to its k-th farthest user is obtained by order statistics [19] as K k−1 fdn,k (x) = k Fd (x)K−k (1 − Fd (x)) fd (x) k k−1 2(K−k)+1 K x x2 = 2k 1 − k D2(K−k+1) D2 k−1 x2(K+i−k)+1 K k−1 (−1)i 2(K+i−k+1) (14) = 2k i k i=0 D

for 0 ≤ x ≤ D, where fd (x) is the corresponding PDF of the unordered CDF in (1). Since we have assumed that X is independent of Y , once we know PDF fY (y) the outage probability in (11) reduces to Pout k = Pr {X − ηk ΛY < (ηk /ρ)Z} D ∞ α = 1 − e−ηk Λy−(ηk /ρ)z fY (y)dy fdn,k (z)dz 0 0 D ηk α = 1 − τ1 fdn,k (z)e− ρ z dz 0

βj ,

∀1 ≤ i ≤ k,

(12)

j=i+1

otherwise the k-th user’s outage probability is always one [2]. The computation of the probability in (11) is challenging due to the following two reasons. On one hand, the norm of the channel gn,k 2 appears both in the signal term X and the inter-group interference term Y , which makes the numerator and denominator in (11) deal with. On the Gcoupled and hard to 2 other hand, the term m=1,m=n |eH w | in Y involves the n,k m sum of G − 1 random variables, of which the CDF is difficult to figure out. Here we adopt the similar method in [17], [18] by imposing the independence on the signal and inter-group interference terms, i.e., X and Y , to obtain an approximate result, which is shown to be reasonable in [17]. The following theorem gives an approximate closed-form expression of the outage probability in (11). Theorem 1: The outage probability of the k-th user in a group in (11) can be approximated as k−1 (−1)i K k−1 out Pk = 1 − 2kτ1 k i=0 i αD2(K+i−k+1) α2 (K+i−k+1) ρ 2 ηk α , (13) × γ (K + i − k + 1), D ηk α ρ

k−1 (−1)i K k−1 = 1 − 2kτ1 k i=0 i αD2(K+i−k+1) α2 (K+i−k+1) ρ 2 ηk × γ (K + i − k + 1), Dα , (15) ηk α ρ where

τ1 =

∞ 0

fY (y)e−ηk Λy dy

(16)

and the last equality in (15) follows from [15, 3.381.8]. To fulfill the proof, now our aim reduces to find the PDF of random variable Y . From [13], we know that en,k and wm are independent and isotropically distributed in ˆn . ThereC(M −1)×1 whose hyperplane is orthogonal to g 2 fore, we have |eH w | ∼ Beta(1, M − 2). Additionally, m n,k gn,k 2 sin2 θn,k ∼ Gamma(M − 1, δ) is satisfied from [13, Lemma 1]. Thus, the random variable Y can be represented G−1 d as Y = Gamma(M − 1, δ) i=1 Beta(1, M − 2). Since we d know that Gamma(M −1, δ)Beta(1, M −2) = Gamma(1, δ) [17], Y is actually the sum of G − 1 Gamma(1, δ) random variables correlated by a common Gamma(M − 1, δ) factor. It is not hard to see that the correlation coefficient between any two addends in Y is the same and given by M1−1 . By


0

10

applying the method proposed in [20], we find the exact PDF of Y given by ∞

˜1 = R ˜2 = 1 bits/s/Hz R

Outage Probability

bn y G+n−2 e−y/a2 , y ≥ 0, (17) G+n−1 a Γ(G + n − 1) n=0 2 √ M −1−1 √ 1 √ , and bn is , a = δ 1 − where a1 = G−2+ 2 M −1 M −1 recursively obtained by 1, n = 0, bn = 1 n (18) i (1 − a ) b , n = 1, 2, . . . . 1 n−i i=1 n fY (y) = a1

˜1 = R ˜2 = 0.5 bits/s/Hz R −1

10

Substituting (17) into (16) yields τ1 = a 1

∞

bn , (1 + ηk Λa2 )G+n−1 n=0

(19)

where the equality follows from [15, 3.326.2]. By substituting (19) into (15), the proof is completed. By rewriting the lower incomplete gamma function in the series form [15, 8.354.1], (13) is changed to ∞ k−1 k − 1 K out Pk = 1 − kτ1 k n=0 i=0 i n (−1)i+n Dnα ηρk × . (20) n! (K + i − k + 1 + nα/2) O( ρ1 )

terms in (20), we obtain the approxiBy neglecting the mation of the outage probability for the k-th user in the high SNR regime as 1 α Pρ→∞ (21) k,out = 1 − τ1 + τ1 τ2 D ηk , ρ where we have used the fact that the CDF Fdn,k (D) = k−1 k−1 (−1)i k K K+i−k+1 = 1 from (14), and τ2 = k i K i=0 k−1 k−1 (−1)i k k i=0 K+i−k+1+α/2 . i From (21), we know that under this situation there exists outage probability floor P∞ k,c = 1 − τ1 , which means that the outage probability remains the constant P∞ k,c even if the transmit power approaches infinity. In this case, there is no diversity order. From Theorem 1, we see that τ1 increases with an increase in B, which means that when we increase the number of feedback bits the outage floor is reduced while the outage probability becomes larger at low SNR. Additionally, ˜ i }k ηk is an increasing function of the QoS constraints {R i=1 and thus can be viewed as an indicator for the QoS. It can be easily seen from (21) that the outage floor becomes higher when the QoS constraint becomes more stringent. As the number of feedback bits goes to infinity, we find that τ1 → 1 since when B → ∞ we successively have δ → ∞ 0, a2 → 0, and τ1 → a1 n=0 bn = 1, where the last equality follows from the fact that the integral of PDF (17) is equal to one. When we investigate the diversity order and further let the SNR approach infinity now, the outage probability in (21) becomes 1 = τ2 D α η k . (22) Pρ,B→∞ k,out ρ

0

Conventional OMA, User 1 NOMA, User 1 − Simulation NOMA, User 1 − Analytical NOMA, User 1 − High SNR Approx. Conventional OMA, User 2 NOMA, User 2 − Simulation NOMA, User 2 − Analytical NOMA, User 2 − High SNR Approx. 10

20

30

40

50

SNR (dB)

Fig. 2. The outage probability versus transmit SNR ρ under different multiple access technologies. The number of feedback bits is B = 10 bits, and the ˜ 2 = 0.5 bits/s/Hz or 1 bits/s/Hz. ˜1 = R target rates are either R

The outage probability floor vanishes now and all the users have the same diversity order equal to one. It is straightforward to see from (22) that the outage probability increases if either the QoS constraint at the user becomes more stringent or the cellular size becomes larger. IV. S IMULATION R ESULTS Numerical results are presented in this section to verify the derived analytical expressions in the considered system. In addition, a comparison between the proposed NOMA scheme and the conventional orthogonal multiple access (OMA) scheme is given. To be specific, we consider a time division multiple access (TDMA) counterpart implemented under each spatial beam, where the base station serves one user in each group per time slot. The outage probability of the k-th user in each group under the OMA scheme is then ˜ k } for 1 ≤ k ≤ K, where given by P◦k = Pr{Ck◦ < R 2 Λ|hH 1 n,k wn | ◦ Ck = . (23) log 1 + G 2 K Λ m=1,m=n |hH n,k wm | + 1/ρ The simulation settings are as follows, unless otherwise specified: The cell radius is D = 10 m with the pass-loss exponent α = 2. The sector (group) number in the cell and the antenna number at the base station are G = M = 3. The number of users in each group is K = 2, and the power allocation factors in each group are βi = μ1 (2K−i+1 − 1) for K 1 ≤ i ≤ K where μ is set to ensure i=1 βi = Λ. All the simulation results to be shown are averaged over 105 trials. In Fig. 2, the outage performance of NOMA and OMA is compared. One can see that for both the two schemes there always exists performance floor due to the limited feedback. It is shown that under the given power allocation, whether NOMA is superior to conventional OMA for a specific user in a group depends on the distance from the base station to this user. For the user with farther distance (User 1), the outage probability achieved by NOMA is lower than that of conventional OMA, whereas the situation is opposite for the


0

10

61640005, the Foundation for the Author of National Excellent Doctoral Dissertation of China under Grant 201340, and the Young Talent Support Fund of Science and Technology of Shaanxi Province under Grant 2015KJXX-01.

Outage Probability

B = 8 bits

R EFERENCES

−1

10

B = 16 bits

−2

10

0

NOMA, User 1 − Simulation NOMA, User 1 − Analytical NOMA, User 1 − High SNR Approx. NOMA, User 2 − Simulation NOMA, User 2 − Analytical NOMA, User 2 − High SNR Approx. 10

20

30

40

50

SNR (dB)

Fig. 3. The outage probability versus transmit SNR ρ under different numbers ˜ 2 = 0.5 bits/s/Hz. ˜1 = R of feedback bits. The target rates are R

user with nearer distance (User 2). The reason behind this is two-fold. On one hand, User 1 benefits from the more fair power allocation policy in NOMA which allows the user with weak channel condition to be allocated with more power. On the other hand, User 2 with less allocated power actually has more stringent constraints compared with User 1 since the additional SIC constraint in NOMA has to be satisfied, which makes NOMA inferior to OMA under this scenario. Fig. 2 also shows the fairness provided by NOMA. Specifically, under ˜1 = R ˜ 2 = 0.5 bits/s/Hz the two users achieve target rates R the same outage performance with transmit SNR around 20 dB even though the two users have different large-scale path loss, and this fairness can be further adjusted by power allocation. Fig. 3 shows the effect of the number of feedback bits on the outage performance of NOMA. We can see that the outage probability largely depends on the number of feedback bits. With larger B the outage probability floor approaches vanished as predicted in Section III. Under large number of feedback bits (B = 16 bits), it can be observed from Fig. 3 that the two users approximately have the same diversity order equal to one as expected. Moreover, according to Fig. 3 the derived analytical expression is shown to coincide well with the simulation results especially in the low-to-moderate SNR range (SNR from 0 dB to 35 dB). V. C ONCLUSION In this paper, we have investigated the outage performance in a downlink SDMA-NOMA cellular network with limited feedback for the first time. A closed-form expression of the outage probability in the considered network has been derived. It has been shown that there always exists outage probability floor due to limited feedback, and the outage floor vanishes with all users’ diversity orders equal to one when the number of feedback bits goes to infinity. ACKNOWLEDGMENT This work was partially supported by the National Natural Science Foundation of China under Grant 61671364 and Grant

[1] 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 Commun. Mag., vol. 53, no. 9, pp. 74–81, Sep. 2015. [2] Z. Ding, Z. Yang, P. Fan, and H. V. Poor, “On the performance of non-orthogonal multiple access in 5G systems with randomly deployed users,” IEEE Signal Process. Lett., vol. 21, no. 12, pp. 1501–1505, Dec. 2014. [3] S. Timotheou and I. Krikidis, “Fairness for non-orthogonal multiple access in 5G systems,” IEEE Signal Process. Lett., vol. 22, no. 10, pp. 1647–1651, Oct. 2015. [4] Y. Zhang, H.-M. Wang, T.-X. Zheng, and Q. Yang, “Energy-efficient transmission design in non-orthogonal multiple access,” IEEE Trans. Veh. Technol., 2016, to appear. [5] F. Gao, T. Cui, and A. Nallanathan, “On channel estimation and optimal training design for amplify and forward relay network,” IEEE Trans. Wireless Commun., vol. 7, no. 5, pp. 1907–1916, May 2008. [6] T. Cui, F. Gao, T. Ho, and A. Nallanathan, “Distributed space-time coding for two-way wireless relay networks,” IEEE Trans. Signal Processing, vol. 57, no. 2, pp. 658–671, Feb. 2009. [7] M. F. Hanif, Z. Ding, T. Ratnarajah, and G. K. Karagiannidis, “A minorization-maximization method for optimizing sum rate in the downlink of non-orthogonal multiple access systems,” IEEE Trans. Signal Process., vol. 64, no. 1, pp. 76–88, Jan. 2016. [8] J. Choi, “Minimum power multicast beamforming with superposition coding for multiresolution broadcast and application to NOMA systems,” IEEE Trans. Commun., vol. 63, no. 3, pp. 791–800, Mar. 2015. [9] Z. Ding, F. Adachi, and H. V. Poor, “The application of MIMO to nonorthogonal multiple access,” IEEE Trans. Wireless Commun., vol. 15, no. 1, pp. 537–552, Jan. 2016. [10] Z. Ding, R. Schober, and H. V. Poor, “A general MIMO framework for NOMA downlink and uplink transmission based on signal alignment,” IEEE Trans. Wireless Commun., vol. 15, no. 6, pp. 4438–4454, Jun. 2016. [11] Z. Ding and H. V. Poor, “Design of massive-MIMO-NOMA with limited feedback,” IEEE Signal Process. Lett., vol. 23, no. 5, pp. 629–633, May 2016. [12] N. Jindal, “MIMO broadcast channels with finite-rate feedback,” IEEE Trans. Inf. Theory, vol. 52, no. 11, pp. 5045–5060, Nov. 2006. [13] T. Yoo, N. Jindal, and A. Goldsmith, “Multi-antenna downlink channels with limited feedback and user selection,” IEEE J. Sel. Areas Commun., vol. 25, no. 7, pp. 1478–1491, Sep. 2007. [14] S. Liu and C. Zhang, “Downlink non-orthogonal multiple access system with limited feedback channel,” in Proc. Int. Wireless Commun. Signal Process. (WCSP) Conf., Nanjing, China, Oct. 2015. [15] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 7th ed. New York, NY: Academic, 2007. [16] Z. Yang, Z. Ding, P. Fan, and G. K. Karagiannidis, “On the performance of non-orthogonal multiple access systems with partial channel information,” IEEE Trans. Commun., vol. 64, no. 2, pp. 654–667, Feb. 2016. [17] M. Kountouris and J. G. Andrews, “Downlink SDMA with limited feedback in interference-limited wireless networks,” IEEE Trans. Wireless Commun., vol. 11, no. 8, pp. 2730–2741, Aug. 2012. [18] J. Park, N. Lee, J. G. Andrews, and R. W. Heath, “On the optimal feedback rate in interference-limited multi-antenna cellular systems,” IEEE Trans. Wireless Commun., vol. 15, no. 8, pp. 5748–5762, Aug. 2016. [19] H. A. David and H. N. Nagaraja, Order Statistics, 3rd ed. Hoboken, NJ: Wiley, 2003. [20] M. S. Alouini, A. Abdi, and M. Kaveh, “Sum of gamma variates and performance of wireless communication systems over Nakagami-fading channels,” IEEE Trans. Veh. Technol., vol. 50, no. 6, pp. 1471–1480, Nov. 2001.