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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 non-
orthogonal multiple access (NOMA) is investigated in a downlink
space division multiple access (SDMA) network with a multi-
antenna base station and randomly deployed users. The NOMA
technology is concurrently exploited with SDMA to further im-
prove 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. INTRODUCTION
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 (S-
DMA) systems, is of great importance, since spectral effi-
ciency 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
are randomly grouped into multiple clusters, and NOMA
principles are exploited by intra-cluster users to suppress intra-
cluster 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 frequency-
division duplexed (FDD) systems where the channel reci-
procity 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-MIMO-
NOMA system. In [14], user selection and power allocation
schemes are proposed to improve the sum rate of a MU-
MIMO 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: ATand AHrepresent the transpose and Hermi-
tian transpose of a matrix A, respectively. The factorial of a
IEEE ICC 2017 Wireless Communications Symposium
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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.
non-negative integer nis denoted by n!, and n
k=n!
(n−k)!k!.
X∼Exp(λ)denotes the exponential distributed random vari-
able with rate λ,X∼Gamma(M,α)denotes the Gamma-
distributed random variable with shape Mand scale α, and
X∼Beta(a, b)denotes the Beta-distributed random variable
with parameters aand b.Γ(x)is the Gamma function [15,
8.310], and γ(α, x)is the lower incomplete gamma function
[15, 8.350.1]. Xd
=Ymeans that the two random variables X
and Yare equal in distribution.
II. SYSTEM MODELS
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
Fd(x)= x2
D2,0≤x≤D. (1)
In the considered cellular network, the cell is assumed to be
divided into Gdisjoint 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.
B. Feedback and Schedule Model
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
in the n-th group as hn,k =gn,kd−α/2
n,k , where gn,k is
the corresponding Rayleigh fading gain, αis 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
direction information (CDI), i.e., ˜
gn,k =gn,k/gn,k , which
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=2
B. 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 Cnsatisfying
j=arg max
1≤j≤N|˜
gH
n,kcnj |(3)
in terms of maximum inner product [12], [13]. After channel
quantization, each user feeds the selected index jback 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.1For
analytical simplicity, we assume that Kusers 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 Kscheduled users in the n-th group can be designed by
viewing the users’ channel directions as the same ˆ
gn=cnj
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)
1Based 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.
IEEE ICC 2017 Wireless Communications Symposium
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−Bin a unit
sphere [13]. If we decompose the actual channel direction of
the k-th user in the n-th group as
˜
gn,k =cosθn,k ·ˆ
gn+sinθn,k ·en,k,(4)
where θn,k denotes the angle between the actual CDI gn,k
and the quantized CDI ˆ
gn, and en,k denotes the error vector
isotropically distributed in the nullspace of ˆ
gn, the CDF of
sin2θn,k resulted from QCA is given by [12], [13]
Fsin2θn,k (x)=2BxM−1,0≤x≤δ,
1,x≥δ, (5)
with δ=2
−B
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 ne-
glected. In this paper, power domain is further exploited and
Kscheduled 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 Ggroups and
its transmit signal is characterized by
x=√PW¯
s=√P
G
n=1
wn¯sn
=√P
G
n=1 wnK
k=1 βn,ksn,k ,(6)
where Pis the total transmit power; W=
[w1,w2,...,wG]∈CM×Gand ¯
s=[¯s1,¯s2,...,¯sG]T∈
CG×1are 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
ensure fairness among users within a group. Moreover, we
assume that the transmit power is equally allocated among
different groups, i.e., K
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
yn,k =Pβ
n,khH
n,kwnsn,k +hH
n,kwn
K
j=1,j=kPβ
n,j sn,j
intra-group interference
+√P
G
m=1,m=nhH
n,kwm¯sm
inter-group interference
+nn,k,(7)
where nn,k is zero-mean additive white Gaussian noise
(AWGN) with variance σ2. To suppress the intra-group inter-
ference 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 signal-
to-interference-plus-noise ratio (SINR) for detecting the i-th
user’s message (1≤i≤k)bythek-th user is denoted by
SINRi
n,k =
|hH
n,kwn|2βn,i
|hH
n,kwn|2K
j=i+1 βn,j +ΛG
m=1,m=n|hH
n,kwm|2+1/ρ ,(8)
where Λ=1/G and ρ=P/σ2denotes the transmit signal-
to-noise ratio (SNR).
For the specific design of precoding matrix Win 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=Mgroups are simultaneously served by
SDMA and leave the more general case G<Mfor future
work. The base station uses the quantized CSI {ˆ
gm}G
m=1 to
design the beamformer for each group, according to ZF the
beamformer for group nis designed to satisfy
ˆ
gH
mwn=0,∀m=n, 1≤m≤G. (9)
III. OUTAGE ANALYSIS
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 communica-
tions 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 Ggroups
for the ease of analysis. Therefore, we only focus on the
performance analysis of the users in one group and omit the
group subscript nfor notational simplicity. According to (8),
the corresponding rate for the k-th user to decode the i-th
IEEE ICC 2017 Wireless Communications Symposium
user’s message is denoted as Ci
k= log(1 + SINRi
n,k). Note
that the rates for the users to decode their own messages are
given by {Ck
k}K
k=1. By denoting the target rate (quality of
service, QoS) for the Kusers in a group as {˜
Rk}K
k=1,an
outage event will occur at the k-th user (1≤k≤K) when
either one of the following two constraints are not satisfied:
1) the SIC is successful, i.e., Ci
k≥˜
Rifor i<k; 2) the user
can successfully decode the message targeted for itself, i.e.,
Ck
k≥˜
Rk.
Since the obtained CSI is not perfect, under ZF beamform-
ing the inter-group interference is still present. By leveraging
(4) and (9), (8) can be recast as
SINRi
k=βiX
K
j=i+1 βjX+ΛY +Z/ρ
,(10)
where Y=gn,k2sin2θn,k G
m=1,m=n|eH
n,kwm|2,X=
|gH
n,kwn|2, and Z=dα
n,k. The outage probability of the k-
th user in a group is then calculated as
Pout
k=1−Pr k
i=1
Ci
k≥˜
Ri
=1−Pr ⎧
⎨
⎩
k
i=1
βiX
K
j=i+1 βjX+ΛY +Z/ρ ≥φi⎫
⎬
⎭
(a)
=Pr
X
ΛY +Z/ρ <η
k,(11)
where φi=2
˜
Ri−1,ci=φi
βi−φiK
j=i+1 βj, and ηk=
max1≤i≤kci. Note that the step (a) in (11) is obtained by
assuming
βi>φ
i
K
j=i+1
βj,∀1≤i≤k, (12)
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,k2appears both in the signal term Xand the
inter-group interference term Y, which makes the numerator
and denominator in (11) coupled and hard to deal with. On the
other hand, the term G
m=1,m=n|eH
n,kwm|2in Yinvolves the
sum of G−1random 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., Xand 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
Pout
k=1−2kτ1K
kk−1
i=0 k−1
i(−1)i
αD2(K+i−k+1)
×ρ
ηk2
α(K+i−k+1)
γ2
α(K+i−k+1),ηk
ρDα,(13)
where τ1=a1∞
n=0
bn
(1+ηkΛa2)G+n−1,a1=√M−1−1
G−2+√M−1,
a2=δ1−1
√M−1,bn=1
nn
i=1(1 −a1)ibn−ifor
n=1,2,... with b0=1, and G=Mfrom the full-loaded
ZF-SDMA assumption.
Proof: According to [13], since ˜
gn,k and wnare in-
dependent and isotropically distributed in CM×1,wehave
|˜
gH
n,kwn|2∼Beta(1,M −1) and thereby X∼Exp(1).Z
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
fdn,k (x)=kK
kFd(x)K−k(1 −Fd(x))k−1fd(x)
=2kK
kx2(K−k)+1
D2(K−k+1) 1−x2
D2k−1
=2kK
kk−1
i=0 k−1
i(−1)ix2(K+i−k)+1
D2(K+i−k+1) (14)
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−ηkΛY < (ηk/ρ)Z}
=D
0∞
01−e−ηkΛy−(ηk/ρ)zαfY(y)dyfdn,k (z)dz
=1−τ1D
0
fdn,k (z)e−ηk
ρzαdz
=1−2kτ1K
kk−1
i=0 k−1
i(−1)i
αD2(K+i−k+1)
×ρ
ηk2
α(K+i−k+1)
γ2
α(K+i−k+1),ηk
ρDα,(15)
where
τ1=∞
0
fY(y)e−ηkΛydy (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 wmare independent and isotropically distributed in
C(M−1)×1whose hyperplane is orthogonal to ˆ
gn. There-
fore, we have |eH
n,kwm|2∼Beta(1,M −2). Additionally,
gn,k2sin2θn,k ∼Gamma(M−1,δ)is satisfied from [13,
Lemma 1]. Thus, the random variable Ycan be represented
as Yd
= Gamma(M−1,δ)G−1
i=1 Beta(1,M −2). Since we
know that Gamma(M−1,δ)Beta(1,M−2) d
= Gamma(1,δ)
[17], Yis 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 Yis the same and given by 1
M−1.By
IEEE ICC 2017 Wireless Communications Symposium
applying the method proposed in [20], we find the exact PDF
of Ygiven by
fY(y)=a1
∞
n=0
bnyG+n−2e−y/a2
aG+n−1
2Γ(G+n−1),y≥0,(17)
where a1=√M−1−1
G−2+√M−1,a2=δ1−1
√M−1, and bnis
recursively obtained by
bn=1,n=0,
1
nn
i=1(1 −a1)ibn−i,n=1,2,.... (18)
Substituting (17) into (16) yields
τ1=a1
∞
n=0
bn
(1 + ηkΛa2)G+n−1,(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
Pout
k=1−kτ1K
k∞
n=0
k−1
i=0 k−1
i
×
(−1)i+nDnα ηk
ρn
n!(K+i−k+1+nα/2).(20)
By neglecting the O(1
ρ)terms in (20), we obtain the approxi-
mation of the outage probability for the k-th user in the high
SNR regime as
Pρ→∞
k,out =1−τ1+τ1τ2Dαηk
1
ρ,(21)
where we have used the fact that the CDF Fdn,k(D)=
kK
kk−1
i=0 k−1
i(−1)i
K+i−k+1 =1from (14), and τ2=
kK
kk−1
i=0 k−1
i(−1)i
K+i−k+1+α/2.
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,ceven if the
transmit power approaches infinity. In this case, there is no
diversity order. From Theorem 1, we see that τ1increases
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,
ηkis an increasing function of the QoS constraints {˜
Ri}k
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→1since when B→∞we successively have δ→
0,a
2→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
Pρ,B→∞
k,out =τ2Dαηk
1
ρ.(22)
0 10 20 30 40 50
10−1
100
SNR (dB)
Outage Probability
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.
˜
R1=˜
R2= 1 bits/s/Hz
˜
R1=˜
R2=0.5 bits/s/Hz
Fig. 2. The outage probability versus transmit SNR ρunder different multiple
access technologies. The number of feedback bits is B=10bits, and the
target rates are either ˜
R1=˜
R2=0.5bits/s/Hz or 1 bits/s/Hz.
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. SIMULATION RESULTS
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 (O-
MA) 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
given by P◦
k=Pr{C◦
k<˜
Rk}for 1≤k≤K, where
C◦
k=1
Klog 1+ Λ|hH
n,kwn|2
ΛG
m=1,m=n|hH
n,kwm|2+1/ρ.(23)
The simulation settings are as follows, unless otherwise
specified: The cell radius is D=10m 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=1
μ(2K−i+1 −1) for
1≤i≤Kwhere μis set to ensure K
i=1 βi=Λ. All the
simulation results to be shown are averaged over 105trials.
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
IEEE ICC 2017 Wireless Communications Symposium
0 10 20 30 40 50
10−2
10−1
100
SNR (dB)
Outage Probability
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.
B=16bits
B=8bits
Fig. 3. The outage probability versus transmit SNR ρunder different numbers
of feedback bits. The target rates are ˜
R1=˜
R2=0.5bits/s/Hz.
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
target rates ˜
R1=˜
R2=0.5bits/s/Hz the two users achieve
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 Bthe outage probability floor approaches
vanished as predicted in Section III. Under large number of
feedback bits (B=16bits), 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
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.
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