1

Antenna Selection in MIMO Non-Orthogonal
Multiple Access Systems

arXiv:1612.04943v2 [] 22 Nov 2017

Yuehua Yu, He Chen, Yonghui Li, Zhiguo Ding, Lingyang Song, and Branka Vucetic

Abstract—This paper considers the antenna selection (AS)
problem for a MIMO non-orthogonal multiple access (NOMA)
system. In particular, we develop new computationally-efficient
AS algorithms for two commonly-used scenarios: NOMA with
fixed power allocation (F-NOMA) and NOMA with cognitive
radio-inspired power allocation (CR-NOMA), respectively. For
the F-NOMA scenario, a new max-max-max antenna selection
(A3 -AS) scheme is firstly proposed to maximize the system sumrate. This is achieved by selecting one antenna at the base
station (BS) and corresponding best receive antenna at each
user that maximizes the channel gain of the resulting strong
user. To improve the user fairness, a new max-min-max antenna
selection (AIA-AS) scheme is subsequently developed, in which
we jointly select one transmit antenna at BS and corresponding
best receive antennas at users to maximize the channel gain of
the resulting weak user. For the CR-NOMA scenario, we propose
another new antenna selection algorithm, termed maximumchannel-gain-based antenna selection (MCG-AS), to maximize
the achievable rate of the secondary user, under the condition
that the primary user’s quality-of-service requirement is satisfied.
The asymptotic closed-form expressions of the average sumrate for A3 -AS and AIA-AS and that of the average rate
of the secondary user for MCG-AS are derived, respectively.
Numerical results demonstrate that the AIA-AS provides better
user-fairness, while the A3 -AS achieves a near-optimal sum-rate
in F-NOMA systems. For the CR-NOMA scenario, MCG-AS
achieves a near-optimal performance in a wide signal-to-noiseratio regime. Furthermore, all the proposed AS algorithms yield
a significant computational complexity reduction, compared to
exhaustive search-based counterparts.
Index Terms—Multiple-input multiple-output (MIMO), nonorthogonal multiple access (NOMA), antenna selection (AS).

I. I NTRODUCTION
The non-orthogonal multiple access (NOMA) technique has
emerged as a promising solution to significantly improve the
spectral efficiency of the next-generation wireless networks
[2]–[4]. By superimposing the information of multiple users
in the power domain, multiple users can be served within
the same time/frequency/code domain. Different from the conventional water-filling power allocation strategy, the NOMA
technique generally allocates more transmit power to the users
with the poor channel conditions (weak users). In this case,
these users can decode their higher-power-level signals directly
by treating others’ signals as noise. In contrast, those users
Y. Yu, H. Chen, Y. Li, and B. Vucetic are with the
School of Electrical and Information Engineering, The University
of
Sydney,
Sydney,
NSW
2006,
Australia
(email:
{yuehua.yu, he.chen, yonghui.li, branka.vucetic}@sydney.edu.au).
Z. Ding is with the School of Computing and Communications, Lancaster
University, Lancaster LA1 4YW, U.K. (email: z.ding@lancaster.ac.uk).
L. Song is with the School of Electrical Engineering and
Computer Science, Peking University, Beijing 100871, China (e-mail:
lingyang.song@pku.edu.cn).

with the better channel conditions (strong users) adopt the
successive interference cancellation (SIC) technique for signal
detection. It has been demonstrated that both the throughput
performance and user fairness can be significantly improved in
NOMA systems compared to conventional orthogonal multiple
access (OMA) systems [2], [5].
Initial efforts on the design and analysis of the NOMA
technique focused on single-antenna systems, see e.g., [6]–
[12]. Recently, multiple antennas have been used in NOMA
systems (MIMO-NOMA) to exploit the spatial degrees of
freedom [13]–[17]. Although the system capacity can be potentially scaled up with the number of antennas, this superior
performance comes at the price of expensive RF chains, high
power consumption and computational complexity required
for signal processing at both the transmitter and receivers. To
avoid the high hardware costs, demanding power consumption
and heavy computational burden while preserving the diversity
and throughput benefits from MIMO, the antenna selection
(AS) technique has been recognized as an effective solution
[18]–[20]. There are only a few papers that considered the AS
problem for MIMO-NOMA systems in the open literature.
Specifically, a transmit AS (TAS) algorithm was proposed
in [21], and a joint TAS and user scheduling algorithm was
considered in [22]. However, both [21] and [22] only focused
on the TAS design at the base station (BS) side and there were
no analytical characterization of the system performance for
the proposed algorithms.
To our best knowledge, joint AS at both the BS and users
for MIMO-NOMA systems is still an open problem. Though
there have been some published results on joint AS schemes in
conventional MIMO-OMA systems, they cannot be extended
to MIMO-NOMA systems directly. This is mainly because
there is severe inter-user interference in MIMO-NOMA systems while the signals are transmitted in an interference-free
manner in MIMO-OMA systems. A global optimal solution
of this problem requires an exhaustive search (ES) over
all possible antenna combinations, and its complexity would
become unacceptable when the numbers of antennas at all
terminals become large. Motivated by this, in this paper,
we focus on the design and analysis of the low-complexity
joint AS algorithms for a typical MIMO-NOMA downlink
scenario, where a small number of users (e.g., two users) are
grouped together to perform NOMA and different groups are
admitted into the network in a OMA manner [2], [14], [23].
In particular, two commonly used power allocation policies:
NOMA with fixed power allocation (F-NOMA) and NOMA
with cognitive radio-inspired power allocation (CR-NOMA)
[23]–[25], are considered respectively. In F-NOMA, the set

2

of power allocation coefficients is predefined and dynamically
allocated to users in each resource block. In contrast, in CRNOMA, the quality-of-service (QoS) requirements of users
are satisfied with different priorities. Generally, the user with
lower priority (i.e., secondary user, SU) is opportunistically
served only when the QoS of user with higher priority (i.e.,
primary user, PU) is guaranteed. In this case, the set of power
allocation coefficients in CR-NOMA is not predetermined
but depending on the QoS requirement of the PU and the
instantaneous channel order of users. Note that the design of
low-complexity AS algorithm for MIMO-NOMA systems is
fundamentally different from that for conventional MIMOOMA systems. This is because multiple channel matrices
with different distributions need to be considered concurrently
to perform the antenna selection in NOMA, while only the
channel status of one user is required for OMA. Furthermore,
selecting different antenna not only affects the instantaneous
channel gains of individual users, but also easily leads to
different order of instantaneous channel gains of all users.
This means that the information decoding order at the user
side and the resulting expressions of the signal-to-interferenceplus-noise (SINR) of different users are not predetermined
but largely affected by the antenna selection process, which
makes the design and analysis of antenna selection for MIMONOMA system non-trivial. The main contributions of this
paper are summarized as follows:
•

•

•

•

For the F-NOMA scenario, a max-max-max AS (A3 -AS)
algorithm is firstly designed to maximize the system sumrate. This is achieved by selecting one antenna at the BS
and corresponding best receive antenna at each user that
maximize the channel gain of the resulting strong user. To
improve the user fairness, a new max-min-max antenna
selection (AIA-AS) scheme is subsequently developed, in
which one transmit antenna at BS and corresponding best
receive antennas at users are jointly selected to maximize
the channel gain of the resulting weak user.
For the CR-NOMA scenario, a joint antenna selection
scheme called maximum-channel-gain-based AS (MCGAS) algorithm is first proposed to maximize the achievable rate of the SU subject to the QoS requirements of
the PU, and two simplified versions of MCG-AS, i.e.,
primary-user-based AS (PU-AS) and second-user-base
AS (SU-AS), are subsequently presented to further reduce
the computational complexity of MCG-AS.
Asymptotic closed-form expressions for the average sumrates (average rate of the SU) of the proposed algorithms
in F-NOMA (CR-NOMA) systems are derived for high
SNR scenarios, respectively.
The computational complexities for all the proposed joint
AS algorithms are considered, and are shown to be
significantly lower than that of the ES-based schemes.
Extensive simulation results illustrate that both A3 -AS
and AIA-AS in F-NOMA systems yield significant performance gains over the OMA ES-based scheme and
the random selection scheme. In particular, A3 -AS can
achieve the near-optimal system sum-rate while AIAAS can provide better user fairness. In CR-NOMA,

the MCG-AS scheme can achieve the near-optimal rate
performance in a wide SNR regime. In contrast, the rate
performance of both PU-AS and SU-AS is significantly
affected by the distances between the BS and users.
Specifically, the PU-AS (SU-AS) scheme can approach
the optimal performance when the primary (secondary)
user is closer to the BS.
The rest of the paper is organized as follows. In Section II,
the considered MIMO-NOMA downlink model is introduced,
and the joint AS optimization problems for both F-NOMA
and CR-NOMA scenarios are formulated, respectively. In
Section III, two new joint AS schemes for F-NOMA systems
are proposed and their achievable sum-rate are analysed. In
Section IV, a computationally-efficient joint AS algorithm and
its two simplified versions are developed and analysed for CRNOMA systems. Finally, simulation results and comparisons
between the proposed algorithms and benchmark schemes are
given in Section V, and conclusions are drawn in Section VI.
Notations: C m×n represents the set of all m×n matrices. All
bold uppercase letters represent matrices, and all Calligraphic
letters represent sets. Pr(·) denotes the probability of an event,
log (·) denotes the logarithm function based on 2, and |·| and
E[·] denote the absolute value and the expectation operation,
respectively. FX (x) and fX (x) represent the cumulative density function (CDF) and the probability density function (PDF)
of a random variable X, respectively. O is usually used in the
efficiency analysis of algorithms and q(x) = O(p(x)) when
q(x)
| = c, 0 < c < ∞.
lim | p(x)

x→∞

II. S YSTEM

MODEL

Consider a downlink communication scenario with one BS
and multiple users. As discussed in [23], it is impractical to
ask all the users in a system to perform NOMA due to the
strong co-channel interference and heavy system overhead for
user coordination. One promising alternative is to construct a
hybrid multiple access system, wherein the users are divided
into multiple small groups, and NOMA is implemented within
each group while OMA is performed between different groups.
Without loss of generality, in this paper, we follow [2], [14],
[23] and assume two users1 are paired in each group randomly
to perform NOMA and different groups are admitted into the
network in a time-division multiple access manner. There is no
doubt that some other delicately designed user pairing schemes
may further improve the rate performance of the NOMA
implementation, however, it is out of the scope of this paper.
We consider that the BS is equipped with N antennas,
while the two selected users (termed as UE1 and UE2 ) are
equipped with M and K antennas, respectively. The channel
matrix from the BS to UE1 (UE2 ) is denoted by H ∈ C N ×M
(G ∈ C N ×K ). We assume that the channels between the BS
and users undergo spatially independent flat Rayleigh fading.
The entries of H (G), e.g., h̃nm (g̃nk ), can be modelled
as independent and identically distributed (i.i.d.) complex
Gaussian random variables, where h̃nm (g̃nk ) represents the
channel coefficient between the nth antenna of the BS and the
1 Note that the two-user setting is how NOMA is implemented in LTEAdvanced.

3

mth (kth) antenna of UE1 (UE2 ). For notation simplicity, we
define hnm = |h̃nm |2 and gnk = |g̃nk |2 .
As in [26], [27], we assume that the BS selects one (e.g.,
nth) out of N available antennas to transmit information, while
the users select one (e.g., mth and kth) out of M and K
available antennas to receive messages, respectively. In this
case, only one RF chain is needed at each node such that the
hardware cost and complexity can be reduced. Moreover, only
the partial channel state information (CSI), i.e., the channel
amplitudes, is needed instead of the full CSI at the BS, which
costs much less feedback bandwidth2.
Let δn,m,k denote the channel order indicator, defined as

1, if hnm ≥ gnk ,
δn,m,k =
(1)
0, if hnm < gnk .

user’s rates. Note that the Jains fairness index is a performance
metric rather than an approach to guarantee the fairness of the
considered MIMO-NOMA systems. To improve user fairness,
the interested readers are referred to various power allocation
schemes for NOMA, e.g., ordered power allocation [29], maxmin power allocation [30], and α-fairness [31], etc.
In the following subsections we formulate the joint AS
optimization problems for two MIMO-NOMA scenarios, i.e.,
F-NOMA and CR-NOMA, respectively.
A. Joint AS Optimization Problem for F-NOMA Systems
In F-NOMA, fixed power allocation coefficients are predefined. In this case, we can formulate the following optimization
problem to maximize the achievable system sum-rate

P1 : {n∗ , m∗ , k ∗ } =
arg max Rsum (hnm , gnk ) , (7)
According to the NOMA principle, the BS broadcasts the
n∈N ,m∈M,k∈K
signals superimposed in the power domain as
where N = {1, 2, · · · , N }, M = {1, 2, · · · , M }, K =
√
√
√
√
x = δn,m,k ( bs1 + as2 ) + (1 − δn,m,k )( as1 + bs2 ), (2) {1, 2, · · · , K} and the achievable sum-rate is given by
2

where si denotes the signal to UEi with E[|si | ] = 1, and a
and b are the power allocation coefficients with a + b = 1.
Without loss of generality, we assume that a > b to guarantee
that more power is allocated to the instantaneous weak user.
The received signals at users are given by
p
p
(3)
y1 = Ps h̃nm x + n1 , y2 = Ps g̃nk x + n2 ,
where Ps is the transmit power at the BS, and ni is the
complex additive white Gaussian noise with variance σi2 . For
notational simplicity, we hereafter assume σ12 = σ22 = σ 2 .
When δn,m,k = 1, UE2 is the weak user and UE1 is the
strong user, hence the power level of s2 is larger than that
of s1 . In this case, UE2 decodes s2 directly by treating s1 as
noise. In contrast, UE1 first decodes s2 and subtracts it via
SIC, and then decodes its own s1 without interference. For
the case δn,m,k = 0, the decoding order is reversed. By using
the fact that the channels are ordered, it can be easily verified
that SIC can be implemented successfully, and the following
two rates can be achievable to UE1 and UE2 , respectively:

Rsum (hnm , gnk ) = R1 + R2
= log (1 +

ρbγns )


+ log 1 +

aγnw
bγnw + 1/ρ



,(8)

where γns = max (hnm , gnk ) denotes the channel gain of the
strong user and γnw = min (hnm , gnk ) denotes the channel
gain of the weak user. It is worth pointing out that for a
given time slot, which user is the strong (weak) user is not
predefined, but depending on the channel fluctuations and
antenna selection result. This uncertainty needs to be carefully
considered when designing the antenna selection algorithm.
B. Joint AS Optimization for CR-NOMA Systems

In the considered CR-NOMA downlink scenario, without
loss of generality, we treat UE2 as the primary user and UE1
as the secondary user. Specifically, UE1 is opportunistically
served when the QoS of UE2 is achieved. Hereafter we use
R1c and R2c to respectively denote the achievable rate of UE1
!
and UE2 , where the superscript (·)c is used to distinguish the
ahnm
,(4) parameters in CR-NOMA systems from those in F-NOMA
R1 = δn,m,k log (1 + ρbhnm )+(1−δn,m,k ) log 1+
bhnm + ρ1
systems. Mathematically, the achievable rate of the secondary
!
user, R2c , should satisfy R2c ≥ Rth , where Rth is the QoS
agnk
+(1−δ
)
log
(1+ρbg
)
,
(5)
R2 = δn,m,k log 1 +
n,m,k
nk
threshold of UE2 . By noting that the power allocation coefbgnk + ρ1
ficient
0 ≤ bcn,m,k ≤ 1, we can express the valid range of
 2
c
where ρ = Ps σ is the transmit SNR.
bn,m,k as below:

To evaluate the user fairness, the Jain’s fairness index [28]


ε

,
1
,
if δn,m,k = 0,
min
is adopted in this paper, which can be expressed as
 ρgnk

bcn,m,k =
(9)
ρg
−ε
2
2
2
nk
 max
η = (R1 + R2 ) /2(R1 + R2 ).
(6)
ρgnk (ε+1) , 0 , if δn,m,k = 1,

We can observe that the Jain’s fairness index is bounded
between 0 and 1 with the maximum achieved by equalling
2 It is worth noting that for the case of slow fading (i.e., the channel coherent
time is long such that the time used for channel estimation can be neglected
compared to the duration of each transmission block), the obtained results in
this paper can serve as a tight performance bound for the case of considering
channel estimation overhead. When the channels change relatively quickly,
the impact of channel estimation overhead can be significant. However,
incorporating the channel estimation cost into the considered MIMO-NOMA
system will require the definition of new system performance metric, the joint
optimization of channel estimation overhead, antenna selection, and power
allocation, which could constitute a new full paper and thus has been left as
a future work.

where ε = 2Rth − 1, and we then have acn,m,k = 1 − bcn,m,k .
By substituting (9) into (4), we can observe that the
achievable rate of the secondary user, R1c , is zero when
δn,m,k = 0, bcn,m,k = 1 or δn,m,k = 1, bcn,m,k = 0. In
other words, when the QoS of UE2 cannot be satisfied, no
power would be allocated to UE1 and hence R1c = 0. For the
considered high SNR scenarios, i.e., ρ → ∞, the expression
of bcn,m,k given in (9) can be simplified as
(
ε
if δn,m,k = 0,
ρgnk ,
c
(10)
bn,m,k =
ρgnk −ε
ρgnk (ε+1) , if δn,m,k = 1.

4

By substituting (10) into (4), the achievable rate of the
secondary user (i.e., R1c ) can be represented as

ρhnm
εhnm
+
(ε + 1)gnk
ε+1


ρhnm gnk
gnk
+
+ (1 − δn,m,k ) log
εhnm + gnk
εhnm + gnk


ρ→∞
ρhnm
≈ δn,m,k log
ε+1


ρhnm gnk
.
(11)
+ (1 − δn,m,k ) log
εhnm + gnk


R1c (hnm , gnk ) = δn,m,k log 1 −

Now we can formulate the following joint AS problem for
CR-NOMA systems:
∗

∗

∗

P2 : {n , m , k }

=

arg max

F-NOMA

SYSTEMS

In this section, two low-complexity joint AS algorithms
are developed to maximize the system sum-rate for F-NOMA
systems without and with the consideration of user fairness.
The closed-form expressions for the average sum-rate of these
two proposed algorithms are derived in high SNR regime.
A. Proposed AS Algorithms for F-NOMA Systems
We can readily verify that the sum-rate Rsum defined in
(8) is a monotonically increasing function of the channel
gains of both the strong and weak user (i.e., γns and γnw ).
Unfortunately, γns and γnw cannot be maximized at the same
time in most cases. This is because both the two users need
to share the same transmit antenna of the BS and it is very
less likely that selecting one of the transmit antennas can
lead to the maximum channel gains of both users. Based on
this observation, we propose two new joint AS schemes for
the considered F-NOMA systems in this subsection, termed
max-max-max AS (A3 -AS) and max-min-max AS (AIA-AS),
respectively. The key idea of A3 -AS (AIA-AS) is to make γnw
(γns ) as large as possible under the condition that γns (γnw ) is
maximized. More details are elaborated as below.
1) A3 -AS: A3 -AS mainly consists of three stages.
• Stage 1. For a given transmit antenna, we firstly find
the best receive antennas for UE1 and UE2 , respectively.
Mathematically, we need to find out the largest element
hmax
(gnmax ) in each row of H (G) as follows:
n
hmax
n

=

gnmax

=

•

≥ Rth .

It is straightforward to see that both P1 and P2 are NP-hard
problems, which mean finding the optimal combinations of
antennas at both the BS and users may require an exhaustive
search with the complexity of O (N M K). This becomes
unaffordable when N , M and K become large. Motivated
by this, in the subsequent two sections we will develop joint
AS algorithms for F-NOMA and CR-NOMA systems with
significantly reduced computational complexity, respectively.
FOR

max
γns = max(hmax
), n ∈ N .
n , gn

R1c (hnm , gnk )(12)

n∈N ,m∈M,k∈K
s.t. R2c (hnm , gnk )

III. A NTENNA S ELECTION

•

max (hn1 , · · · , hnM ) ,

max (gn1 , · · · , gnK ) ,

(13)
(14)

max
for n ∈ N . Each pair (hmax
) is then treated as one
n , gn
AS candidate. The set of all the N pairs can be written
(1)
max
as SA3 = {(hmax
, g1max ), · · · , (hmax
1
N , gN )}.
(1)
Stage 2. For a given pair from SA3 , the one with larger
channel gain is set as the strong user while the other one
is set as the weak user. In order to maximize the channel
gain of the strong user, we need to find out all its potential
channel gains. Mathematically, we need to find out the
max
larger element of (hmax
) as follows:
n , gn

(15)
(2)
SA3

The set of the N larger elements are denoted by
=
s
{γ1s , · · · , γN
}.
Stage 3. After obtaining the set of the potential channel
gains of the strong user, we thirdly find out the largest
(2)
one from SA3 to maximize γns . Mathematically, we have
s
s
n ∈ N.
γA
3 = max(γn ),

(16)

s
γA
3

Recall that
coming from UE1 or UE2 is not predetermined, we then use (n∗A3 , m∗A3 ) to denote the original row
s
and column indexes of γA
3 when it lies in H. In this case,
∗
the nA3 th antenna at the BS and the m∗A3 th antenna at UE1
∗
are selected. We use kA
3 to denote the original column index
w
max
∗
of γA3 = gn∗ 3 . Therefore the kA
3 th antenna at UE2 can be
A
s
selected concurrently. For the case that γA
3 lies in G, the
selected antenna indexes can be obtained in a similar way.
In A3 -AS, the channel gain of the instantaneous strong user
s
γA3 is maximized. However, the achievable rate of the instantaneous weak user cannot be guaranteed and may be worse
for some cases. Recall that the Jain’s fairness index decreases
when the gap between the achievable rates of users enlarges. In
order to improve the user fairness, we subsequently develop a
AIA-AS scheme, which aims to improve the rate performance
of the instantaneous weak user and to narrow the gap of the
rates
users. Similar to the A3 -AS algorithm, the AIA-AS
2) of
AIA-AS:
scheme also has three stages. The main difference between
A3 -AS and AIA-AS lies in the second and the third stages.
In particular, A3 -AS constructs the set of all potential channel
gains for the strong user and then maximizes γns . In contrast,
AIA-AS tries to obtain the set of all potential channel gains
for the weak user and then to maximize γnw . The three stages
of AIA-AS are elaborated as follow.
(1)
=
• Stage
1.
Construct
the
set
SAIA
max
{(hmax
, g1max ), · · · , (hmax
1
N , gN )} as (13) and (14).
• Stage 2. In order to maximize the channel gain of the
weak user, we need to find out all the potential channel
(1)
gains of the weak user from SAIA . Mathematically, we
need to find out the smaller element in each pair of
max
(hmax
), that is,
n , gn
max
γnw = min(hmax
), n ∈ N .
n , gn

•

(17)

The set containing these N smaller elements is denoted
(2)
w
}.
by SAIA = {γ1w , · · · , γN
Stage 3. After obtaining the set of the potential channel
gains of the weak user, we thirdly find out the largest one
(2)
from SAIA to maximize γnw . Mathematically, we have
w
γAIA
= max(γnw ), n ∈ N .

(18)

5

w
Similarly, we can see that γAIA
coming from UE1 or UE2
∗
∗
is not predefined. We use (nAIA , kAIA
) to denote the original
w
row and column indexes of γAIA
when it lies in G. In this
∗
case, the n∗AIA th antenna at the BS and the kAIA
th antenna at
UE2 are selected, respectively. Meanwhile, we use m∗AIA to
s
denote the original column index of γAIA
= hmax
. Therefore
n∗
AIA
∗
the mAIA th antenna at UE1 can be selected simultaneously.
w
For the case that γAIA
lies in H, the selected antenna indexes
can be obtained in a similar way.
3) Computational Complexities Comparisons: As mentioned before, the complexity of the optimal selection algorithm achieved by the ES-based scheme in F-NOMA systems
is as high as O (N M K). When the number of antennas at
each node becomes large, e.g., up to hundreds of antennas
[32], the computational burden would increase significantly.
The two new AS algorithms for F-NOMA systems dramat
ically reduce the selection complexity to O N (M + K + 3) ,
where the main computation lies in sorting the channel gains.
For the case N = M = K, we can find that the complexities
of A3 -AS and AIA-AS are approximately O(N 2 ), which is
an order of magnitude lower compared to that of the optimal
ES scheme, which is O(N 3 ).

B. Performance Analysis of the New Algorithms for F-NOMA
In this subsection, we analyse the average sum-rate of the
proposed joint AS algorithms, i.e., A3 -AS and AIA-AS in FNOMA systems, respectively.
Assuming flat Rayleigh fading, hnm is then an exponentially
distributed random variable. When x ≥ 0, the CDF and PDF
of hnm are respectively given by
Fh (x) = 1 − e−Ωh x , fh (x) = Ωh e−Ωh x ,

Rsum (γns )

ρ→∞

≈

log(1 + bργns ) + log (1/b) .

(25)

Subsequently, we perform the asymptotic analysis for the average sum-rates of A3 -AS and AIA-AS algorithms, respectively.
1) Average Sum-Rate Analysis for A3 -AS: Let hmax =
max (hnk ) and g max = max (gnk ) for n ∈ N , m ∈ M and
s
k ∈ K. Recall that in A3 -AS, γA
3 is actually the relatively
max
max
larger element of h
and g
. In order to obtain the
s
distribution of γA
3 , at first we derive the CDF and PDF of
hmax and g max respectively as follows:
XN M
µi,N M e−iΩh x ,
(26)
Fhmax (x) =
i=0
XN M
iΩh µi,N M e−iΩh x ,
(27)
fhmax (x) = −
i=1
XN K
µj,N K e−jΩg x ,
(28)
Fgmax (x) =
j=0
XN K
jΩg µi,N K e−jΩg x .
(29)
fgmax (x) = −
j=1

s
max max
Then the CDF and PDF of γA
,g
) in A3 -AS
3 = max(h
can be calculated as follows

Fγ s 3 (x) = Pr (max(hmax , g max ) < x)
A

= Pr (hmax ≤ g max < x) + Pr(g max < hmax < x) ,(30)
Z x
Z x
fγ s 3 (x) = fgmax (x)
fhmax (y)dy + fhmax (x)
fgmax (y)dy
A

0

=

(19)

NM
NK
XX

0

µi,NM µj,NK iΩh e−iΩh x

i=1 j=1

where Ωh = 1/E[hnm ]. Similarly, when x ≥ 0 we have the
CDF and PDF of gnk as follows,
Fg (x) = 1 − e−Ωg x , fg (x) = Ωg e−Ωg x ,

allocation coefficient b. In this case, we mainly focus on the
contribution of the instantaneous strong user to the sum-rate.
Mathematically, we have the approximation of the system
instantaneous sum-rate given in (8) as follows:

(20)

where Ωh = 1/E[gnk ].
Recall that both the two proposed AS schemes for F-NOMA
need to find the maximum element in each row of H and G
(i.e., hmax
and gnmax ). We thus first obtain the distribution
n
functions of hmax
for x ≥ 0 as below,
n
M (c1 ) XM
µi,M e−iΩh x , (21)
=
(x) = 1 − e−Ωh x
Fhmax
n
i=0
XM
iΩh µi,M e−iΩh x ,
(22)
fhmax
(x) = −
n
i=1

where µi,M = (−1)i Mi and the expansion step (c1 ) is
conducted based on the Binomial theorem.
Similarly, the CDF and PDF of gnmax for x ≥ 0 are given by
K XK
µj,K e−jΩg x , (23)
Fgnmax (x) = 1 − e−Ωg x
=
j=0
XK
jΩg µj,K e−jΩg x .
(24)
fgnmax (x) = −
j=1

From the sum-rate expression of the considered F-NOMA
system in (8), we can observe that the achievable rate of the
instantaneous
weak UE approaches a constant when ρ → ∞,


w
ρ→∞
aγn
≈ log 1 + ab = log (1/b) with
i.e., log 1 + bγ w +1/ρ
n
a + b = 1, which is only related to the predefined power


+ jΩg e−jΩg x − (iΩh + jΩg )e−(iΩh +jΩg )x .

(31)

We now can analyse the average sum-rate of A3 -AS for high
SNR scenarios and obtain Proposition 1 as below.
s
3
Proposition 1. With the PDF of γA
3 in A -AS, i.e., fγ s (x),
A3
when ρ → ∞, the asymptotic closed-form expression for the
average sum-rate of the A3 -AS algorithm is given by
3

A
R̄sum
≈

+

1
(ln ρ − C)
ln 2
NK
NM
XX
(−1)i+j
i=1 j=1

NM
i

ln 2



NK
j



ln



iΩh + jΩg
ijΩh Ωg



(32)

Proof: See Appendix A.
A3
As can be observed from (32), the system sum-rate R̄sum
is
an increasing function of the transmit SNR ρ as expected. Interestingly, it is not affected by the power allocation coefficient
b. The reason behind this observation is that when ρ → ∞,
Rsum ≈ log(γ s ρ), in which the power allocation factor b has
been removed. It will be verified via the computer simulations
in section V.
2) Average Sum-rate Analysis for AIA-AS: In this subsection, we derive the asymptotic average sum-rate of the proAIA
posed AIA-AS algorithm, i.e., R̄sum
, for high SNR scenarios.
According to (25), we need to know the PDF of the channel
gain of the instantaneous strong user who shares the transmit
s
antenna with the weak user, i.e., fγAIA
(x). With the help of

6

hence R1c increases. In contrast, the contribution of hnm to R1c
does not lie in the power domain, but can boost R1c directly
Lemma 1. In the AIA-AS algorithm, the PDF of the channel via providing a better channel gain. However, it is not easy to
s
tell which one, gnk or hnm , has greater contribution to R1c .
gain of the selected strong user, i.e., γAIA
, is given by
In this case, we propose to maximize the contribution of the
M X
K X
X

larger one from gnk and hnm . Accordingly, a low-complexity
s
Cℓ tℓ ζij ψ(iΩh , jΩg ) + ψ(jΩg , iΩh ) , (33)
fγAIA
(x) =
AS algorithm, referred to as MCG-AS, is developed for the
i=1 j=1 ℓ
considered CR-NOMA system. More specifically, the MCGwhere
AS algorithm consists of four stages as below.
(N − 1)!
• Stage 1. Find out the best transmit and receive candifor ℓ0 + · · · + ℓM K = N − 1,
Cℓ =
ℓ0 ! · · · ℓM K !
dates for UE1 (UE2 ) which can maximize hnm (gnk ).
Y
tℓ =
(−µi,M µj,K )ℓij ,
Mathematically, we have
the law of full probability and the multinomial theorem, the
s
PDF of fγAIA
(x) is provided in Lemma 1 as below.

1≤i≤M
1≤j≤K

ζij

=

ψ(θ1 , θ2 )

=

ξℓ

=

N ijΩh Ωg µi,M µj,K ,
 −θ2 x

e
−1
e−(θ2 +ξℓ )x − 1
e−θ1 x
−
,
θ2
θ2 + ξ ℓ
K
M X
X
(iΩh + jΩj )ℓij .
i=1 j=1

Proof: See Appendix B.
s
Provided the PDF of γAIA
in AIA-AS algorithm, we can
approximate the average sum-rate achieved by AIA-AS in
Proposition 2 as below.
s
Proposition 2. With the PDF of γAIA
, when ρ → ∞,
AIA
the asymptotic average sum-rate of AIA-AS, i.e., R̄sum
, is
given by
M

AIA
R̄sum
≈ log

K

1 X X X Cℓ tℓ
+
(T1 + T2 + T3 + T4 ) , (34)
b
ln 2
i=1 j=1 ℓ

where Tn , n = {1, 2, 3, 4} are given by
T1

=

T3

=

ξℓ ζ̃ij
ξℓ ζ̃ij
χ(jΩg ), T2 =
χ(iΩh ),
φi
φj
ζij φij,2 χ(φij,1 )
, T4 = −ζ̃ij χ(iΩh + jΩg ),
φi φj φij,1

in which, ζ̃ij = N µi,M µj,K , φi = iΩh + ξℓ , φj = jΩg + ξℓ ,
φij,1 = iΩh + jΩg + ξℓ , φij,2 = iΩh + jΩg + 2ξℓ , χ(x) =
x
C + ln bρ
, and the constant C is the Euler’s constant.
Proof: See Appendix C.
IV. A NTENNA S ELECTION FOR CR-NOMA

SYSTEMS

In this section, CR-NOMA is considered, and a computationally efficient joint AS algorithm, i.e., MCG-AS, is
proposed to maximize the achievable rate of secondary user
UE1 . To further reduce the computational complexity, two
simplified versions of MCG-AS, i.e., PU-AS and SU-AS, are
proposed. In the high SNR regime, the asymptotic expressions
of average achievable rates of UE1 using MCG-AS, PU-AS
and SU-AS are then derived, respectively.
A. Proposed AS Algorithms for CR-NOMA Systems
By observing (9)-(11), we can see that the channel gain of
the primary user UE2 (i.e., gnk ) and that of the secondary user
UE1 (i.e., hnm ) both make contributions to R1c but in different
ways in CR-NOMA systems. In particular, if gnk increases,
less power would be needed to satisfy the QoS requirement of
UE2 . In other words, more power can be allocated to UE1 and

hmax
g max

•

=
=

max (hn1 , · · · , hN M ) ,
max (gn1 , · · · , gN K ) .

(35)
(36)

Stage 2. Compare and find out the larger one from hmax
and g max and set it as the channel gain of the strong user.
That is,
s
γMCG
= max(hmax , g max ).
(37)

s
Denote the row index of γMCG
as n∗MCG . Then, the
∗
nMCG th antenna at the BS can be selected.
• Stage 3. Since the weak user share the same transmit
antenna with the strong user, we should find out the
w
largest element γMCG
for the weak user from the same
∗
nMCG th row. Mathematically, we have

s
max(gn∗MCG 1 , · · · , gn∗MCG K ), if γMCG
= hmax ,
w
γMCG=
s
max(hn∗MCG 1 , · · · , hn∗MCG M ), if γMCG = g max .
(38)
s
For the case that γMCG
= hmax , we denote the original
s
w
∗
column indexes of γMCG
and γMCG
as m∗MCG and kMCG
,
∗
respectively. Then, the mMCG th antenna at UE1 and the
∗
kMCG
th antenna at UE2 can be selected concurrently. For
s
the case that γMCG
= g max , the corresponding antenna
indexes at users can be obtained similarly.
c
c
• Stage 4. Compute bn∗ ,m∗ ,k∗ and an∗ ,m∗ ,k∗ by substituts
w
ing γMCG and γMCG to (10).
We also realize that, when the primary user UE2 is much
closer to the BS than the secondary user UE1 , it is likely
that g max is larger than hmax . In this case, MCG-AS can be
simplified into the PU-AS scheme, in which the AS can be
performed by maximizing the channel gain of the primary
user UE2 . In this sense, more power can be allocated to the
secondary user UE1 . In contrast, when UE2 is much farther
from the BS than UE1 , it is likely that g max is smaller than
hmax . In this case, MCG-AS can be reduced to the SUAS scheme, which aims to maximize the channel gain of
the secondary user UE1 directly. We elaborate the principles
and key stages of the simplified PU-AS and SU-AS in the
subsequent two subsections.
1) PU-AS: The key idea of PU-AS is to select the antenna
pair from the BS and the primary user UE2 concurrently,
which provides the maximum channel gain for UE2 . In this
case, less power is needed for UE2 to satisfy its QoS requirement and more power can thus be allocated to the secondary
user UE1 to increase the corresponding R1c . The main three
stages of PU-AS scheme are described as below.

7

TABLE I: Complexity comparison in CR-NOMA

CR-NOMA-ES
MCG-AS
PU-AS
SU-AS

•

Stage 1. Find out the best transmit and receive antennas
for UE2 to maximize its channel gain. Mathematically,
that is to find out the largest element g max from G,
g

•

Computational Complexity

O NMK

O N (M + K)
 +2
O NK + M
O NM + K

max

= max (gi1 , · · · , gN K ) .

R̄1MCG = Pr (hmax < g max ) R̄1P +Pr (hmax ≥ g max ) R̄1S , (43)

(39)

Denote the row and column indexes of g max as n∗P and
∗
kP
, respectively. Then, the n∗P th antenna at the BS and
∗
the kP
th antenna at UE2 are selected simultaneously.
Stage 2. For the n∗P th selected transmit antenna, find out
the best receive antenna for UE1 who shares the same
transmit antenna with UE2 . Mathematically, find out the
∗
largest element hmax
n∗ in the nP th row of H. That is,
P

hmax
= max(hn∗P 1 , · · · , hn∗P M ).
n∗
P

•

(40)

Denote the column index of hmax
as m∗P . Then, the m∗P th
n∗
P
antenna at UE1 is selected.
Stage 3. Calculate bcn∗ ,m∗ ,k∗ and acn∗ ,m∗ ,k∗ by substituting the value of g max and hmax
n∗ to (10).
P

2) SU-AS: The key idea of SU-AS is to first select the antenna pair from the BS and secondary user UE1 concurrently,
which provides the maximum channel gain for UE1 . Similarly,
SU-AS consists of three stages.
•

Stage 1. Find out the largest element hmax from H.
hmax = max (hi1 , · · · , hN M ) .

•

•

(41)

Denote the row and column indexes of hmax as n∗S and
m∗S , respectively. Then, the n∗S th antenna at the BS and
the m∗S th antenna at UE1 are selected simultaneously.
Stage 2. Find out the largest element gnmax
in the n∗S th
∗
S
row of G. Mathematically,
gnmax
= max(gn∗S 1 , · · · , gn∗S K ).
∗
S

B. Average Rate Analysis of the SU in CR-NOMA
As discussed above, since the QoS of the primary user UE2
is guaranteed in CR-NOMA systems, we thus only focus on
the analysis of the average rate of the secondary user UE1 ,
which is served opportunistically in CR-NOMA networks.
For the proposed MCG-AS scheme, by carefully analysing
its antenna selection procedure, we can find that the MCG-AS
scheme would become SU-AS when hmax ≥ g max and would
simplify to PU-AS otherwise. In this case, the average rate of
UE1 in MCG-AS (i.e., R̄1MCG ) can be calculated according to
the law of total probability as below:

(42)

Denote the column index of gnmax
as kS∗ . Then, the kS∗ th
∗
S
antenna at UE2 is selected.
Stage 3. Obtain bcn∗ ,m∗ ,k∗ and acn∗ ,m∗ ,k∗ by substituting
the value of hmax and gnmax
to (10).
∗
S

3) Computational Complexities Comparisons: Similar to
F-NOMA systems, the complexity of the optimal selection
algorithm achieved by the ES-based scheme in CR-NOMA
systems is also as high as O (N M K). In contrast, all the
three proposed AS algorithms, i.e., MCG-AS, PU-AS and
SU-AS, can dramatically reduce the selection complexities as
illustrated in Table I. Furthermore, the complexities of PUAS and SU-AS are lower than that of MCG-AS. For the case
N = M = K, the computational burdens of PU-AS and SU-AS
are nearly half of that of MCG-AS.

where R̄1P and R̄1S denote the average rate of UE1 in PU-AS
and SU-AS, respectively, and the probability of the event that
hmax ≥ g max is given by
Pr (hmax ≥ g max ) =
=

Z

∞

fhmax (x)

0
NK
NM
XX
i=1 j=1

Z

x

fgmax (y)dydx
0
 NK 
(−1)i+j NM
jΩg
i
j
iΩh + jΩg

.

(44)

Accordingly, Pr (hmax < g max ) = 1 − Pr (hmax ≥ g max ).
Therefore, the analytical expression of R̄1MCG can be obtained
if we can respectively derive the expressions of R̄1P and R̄1S .
We first calculate R̄1P . Recall that in the PU-AS scheme,
the maximum element of G, i.e., g max , is first selected, and
then the maximum element in the corresponding row of H is
subsequently selected. By using the i.i.d. property of the all
the elements in H, the PDF of hmax
can be derived by using
n∗
P
fhmax
(x).
In
this
case,
by
given
the
PDF
of g max in (28) and
n
max
that of hn∗ in (22), an approximated expression of R̄1P can
P
be can be given in Proposition 3 as below.
Proposition 3. With the QoS requirement of the primary user
Rth and the distributions of g max and hmax
, the average
n∗
P
rate of the secondary user UE1 in PU-AS, i.e., R̄1P , can be
approximated in the high SNR regime as below:
R̄1P

≈

M X
NK
X
(−1)i+j
i=1 j=1



M
i

ln 2



NK
j



(ε + 1)jΩg
iΩh
εjΩg
ln
− ln
−C
iΩh − εjΩg
iΩh + jΩg
ρ



. (45)

Proof: See Appendix D.
After obtaining R̄1P , we thus turn to calculate R̄1S for SU-AS.
Recall that in SU-AS, the maximum element of H (i.e., hmax )
is selected first, followed by the corresponding maximum
element in the n∗S row of G (i.e., gnmax
∗ ). Similarly, considering
S
the i.i.d. property of the all the elements in G, the PDF of gnmax
∗
S
max
max
can be characterized by fgn (x). Given the PDF of h
in
(26) and that of gnmax
in (24), the average rate of UE1 in
∗
S
SU-AS, i.e., R̄1S , is given by Proposition 4 as below.
Proposition 4. With the QoS requirement of the primary
UE and the distributions of hmax and gnmax
∗ , respectively, the
S
S
average rate of UE1 in SU-AS, i.e., R̄1 , can be approximated
in the high SNR regime as below:
R̄1S

≈

K
NM
XX
i=1 j=1



(−1)i+j

NM
i

ln 2



K
j




εjΩg
(ε + 1)jΩg
iΩh
ln
− ln
− C . (46)
iΩh − εjΩg
iΩh + jΩg
ρ

8

25

22.5

23
22
21
20
F-NOMA-ES (optimal)
F-NOMA-RA
OMA-ES
A 3 -AS (simulation)
AIA-AS (simulation)
A 3 -AS (analytical)
AIA-AS (analytical)

19
18
17
16
0

4

8

12

16

Average sum-rate (bps/Hz)

Average sum-rate (bps/Hz)

24

21.5

21

20.5

20

19.5

20

2

Transmit power at the base station (dBm)

Proof: Similar to Proposition 3, R̄1S can be obtained by
resolving the integral (11) over hmax and gnmax
∗ . Comparing
S
the distributions of hmax and gnmax
in Proposition 4 to those
∗
S
of hmax
and g max in Proposition 3, we can observe that the
n∗
P
only difference between them lies in the corresponding upper
limits of the summations. In this case, we can calculate R̄1S
by replacing the upper limits of i and j in (45) by N M and
K, respectively.
Finally, by substituting (44)-(46) into (43), we can obtain
an approximated closed-from expression for R̄1MCG .
Remark 1. When the primary user UE2 is much closer to the
BS than the secondary user UE1 , the event that g max > hmax
occurs with a large probability. In this case, the MCG-AS
scheme will reduce to the PU-AS scheme and thus R̄1MCG ≈
R̄1P . On the other hand, when UE2 is much further from the
BS than UE1 , it is more likely that g max ≤ hmax . In this
case, the MCG-AS scheme degrades to SU-AS, and we have
the approximation R̄1MCG ≈ R̄1S .
V. N UMERICAL P ERFORMANCE E VALUATIONS
In this section, the performance of the proposed joint AS
algorithms for both F-NOMA and CR-NOMA systems is
evaluated by using computer simulations and comparing to
the benchmarks. In all simulations, we set M = K = 2,
α
Ωh = dα
1 , Ωg = d2 , d1 (d2 ) is the distance between the BS
and UE1 (UE2 ), the path-loss exponent α = 3 and the variance
of the noise is set as σ 2 = −110dBm.
A. Numerical Results for F-NOMA Systems
Fig. 1 illustrates how the transmit power Ps at the BS affects
the system average sum-rate R̄sum . As can be observed from
Fig. 1, when Ps increases, R̄sum increases for all the schemes.
Moreover, the performance of the proposed A3 -AS and AIAAS schemes is much better than that of the random AS scheme
in F-NOMA scenarios (F-NOMA-RA). Furthermore, the A3 AS scheme can achieve the same performance as that of the
optimal ES scheme in F-NOMA scenarios (F-NOMA-ES) but
with a much lower computational complexity. We should note
that the analytical results match the simulation results for both
A3 -AS and AIA-AS, which validate our theoretical analysis
in Sec. III. It is also worth pointing out that all the NOMA
schemes outperform the ES scheme in OMA systems (OMAES) over the entire SNR region. For simplicity, we only show
the analytical results in the following discussions.

3

4

5

6

7

8

Number of antennas at the base station, N

Fig. 2: Average sum-rate vs. N in F-NOMA, d1 = 80m, d2 =
200m, a = 0.6, b = 0.4, Ps = 10dBm.
22.5

Average sum-rate (bps/Hz)

Fig. 1: Average sum-rate vs. transmit power in F-NOMA, N =
2, d1 = 80m, d2 = 200m, a = 0.6, b = 0.4.

F-NOMA-ES (optimal)
F-NOMA-RA
OMA-ES
A 3 -AS (analytical)
AIA-AS (analytical)

22

22

21.5

21

20.5

20

19.5
80

F-NOMA-ES (optimal)
F-NOMA-RA
OMA-ES
A 3 -AS (analytical)
AIA-AS (analytical)
100

120

140

160

180

200

Distance of UE from the base station, d (m)
2

2

Fig. 3: Average sum-rate vs. d2 in F-NOMA, N = 2, d1 =
80m, a = 0.6, b = 0.4, Ps = 10dBm.
Fig. 2 illustrates how the number of antennas N at the BS
influences the average sum-rate R̄sum . We can see from this
figure that the sum-rates of F-NOMA-RA and AIA-AS keep
constant when N increases. For the F-NOMA-RA scheme, this
is because it does not properly utilize the multiple-antenna
setting but selects one antenna at each node randomly. The
reason why AIA-AS keeps constant is because it guarantees
the performance of the weak user with the poor channel gain
γ w , but not the strong user with the better channel condition
γ s , which contributes the most to R̄sum . In contrast, the
average sum-rate of A3 -AS increases along with N , and A3 AS achieves the same performance as that of the optimal FNOMA-ES scheme. Again, all the proposed NOMA schemes
outperform the OMA-ES scheme in the entire region.
Fig. 3 depicts how the distances between the BS and users
influence R̄sum for the schemes in F-NOMA systems. Take a
constant d1 and a variable d2 for example. We can observe
that when d2 increases, R̄sum decreases for all the schemes in
F-NOMA systems. We also note that both A3 -AS and AIAAS outperform the F-NOMA-RA and the OMA-ES schemes,
and again A3 -AS achieves almost the same performance as
F-NOMA-ES. There is a crossing between the curves for FNOMA-RA and OMA-ES. The reason for this is that in OMAES, when d2 is much larger than d1 , the energy and frequency
resources allocated to UE2 are wasted since they contribute
very little to R̄sum .
Fig. 4 demonstrates how the power allocation coefficient b
affects the R̄sum for various AS schemes in F-NOMA systems.
Interestingly we can see that for all the F-NOMA schemes the
average sum-rate keeps almost constant when b increases. The
main reason is that Rsum ≈ log(γ s ρ) when ρ → ∞; that is,
the sum-rate is not affected by the value of b.

21.8

19

21.6

18.5

21.4

21
F-NOMA-ES (optimal)
F-NOMA-RA
A 3 -AS (analytical)
AIA-AS (analytical)

20.8
20.6
20.4
20.2
20
0.5

CR-NOMA-ES
CR-RA
PU-AS (simulation)
PU-AS (analysis)
SU-AS (simulation)
SU-AS (analysis)
MCG-AS (simulation)
MCG-AS (analysis)

18
17.5

1

21.2

Average rate of UE (bps/Hz)

Average sum-rate (bps/Hz)

9

17
16.5
16
15.5
15

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

Power allocation coefficient, a

14.5
120

Fig. 4: Average sum-rate vs. a in F-NOMA, N = 2, d1 =
80m, d2 = 200m, b = 1 − a, Ps = 10dBm.

140

160

180

200

220

240

260

280

The distance between the BS and UE , d (m)
1

1

Fig. 6: Average rate of UE1 vs. d1 in CR-NOMA, N = 4, d2 =
200m, Rth = 5bps/Hz, Ps = 20dBm.

1
20

0.95

16

0.85

0.8

0.75
F-NOMA-ES (optimal)
F-NOMA-RA
A 3 -AS (simulation)
AIA-AS (simulation)

0.7

0.65
0.5

0.6

0.7

0.8

Average rate of UE1 (bps/Hz)

Jain's fairness index

18

0.9

CR-NOMA-ES
CR-NOMA-RA
PU-AS
SU-AS
MAG-AS

14
12
10
0

4

8

12

16

20

Transmit power at BS (dBm), d =100m, d =200m
1

20

2

CR-NOMA-ES
CR-NOMA-RA
PU-AS
SU-AS
MAG-AS

15

0.9

Power allocation coefficient, a

Fig. 5: Jain’s fairness index vs. b in F-NOMA, N = 4, d1 =
80m, d2 = 200m, b = 1 − a, Ps = 20dBm.
Although the system sum-rate performance of A3 -AS is
shown to be slightly better than that of AIA-AS, regarding
the fairness between UE1 and UE2 , we can observe in Fig. 5
that AIA-AS can provide better user fairness than A3 -AS. In
other words, in practice AIA-AS would be a better trade-off
between the system throughput and user fairness.
B. Numerical Results for CR-NOMA Systems
In the considered two-user CR-NOMA down-link scenario,
without loss of generality, we treat UE2 as the PU and UE1
as the SU. Fig. 6 illustrates how the locations of users affect
the average rate of UE1 . We now fix the location of the
primary user UE2 (d2 = 200m) and place the secondary user
UE1 at various locations. As can be observed from Fig. 6,
when d1 increases, the average rate R̄1c of UE1 decreases
for all the proposed AS schemes in CR-NOMA systems.
Interestingly, there is a crossing for the curves of PU-AS
and SU-AS around d1 = d2 = 200m. R̄1c achieved by SUAS is larger than that of PU-AS when d1 ≤ 200m and the
situation is reverse when d1 > 200m. The reason is that
when UE1 is closer to the BS, the channel gain of UE1
contributes more to R̄1c , but when UE2 is closer to the BS,
the channel gain of UE2 is more dominant to R̄1c as more
power is allocated to the secondary UE. Furthermore, MCGAS, which takes advantage of PU-AS and SU-AS, can achieve
almost the same performance as the optimal ES scheme in CRNOMA scenarios (CR-NOMA-ES) for all the location settings
but with much lower computational complexity, and all the
proposed schemes outperform the random selection scheme in
CR-NOMA scenarios (CR-NOMA-RA). It is worth pointing
out that the analytical results match the simulation results for
all the proposed CR-NOMA AS schemes, which validates our

10

5
0

4

8

12

16

20

Transmit power at BS (dBm), d1 =300m, d2 =200m

Fig. 7: Average rate of UE1 vs. Ps in CR-NOMA, N =
4, Rth = 5bps/Hz.
theoretical analysis in Sec. IV. For brevity, we only show the
analytical results in the following discussions.
Fig. 7 illustrates how the transmit power at the BS influences
the achievable R̄1c . We can see that when Ps increases, R̄1c increases for all the schemes in CR-NOMA systems. Moreover,
all the proposed AS schemes outperform CR-NOMA-RA. In
particular, when UE1 is closer to the BS, the performance
of SU-AS is better than that of PU-AS. In contrast, when
UE2 is closer to the BS, the performances of SU-AS and PUAS are inverted. Again MCG-AS achieves the near-optimal
performance as CR-NOMA-ES in the entire region.
Fig. 8 illustrates how the number of antennas at the BS
affects the achievable R̄1c . When UE1 is closer to the BS, the
curve for SU-AS increases along with N . This is because in
this case, the channel gain of UE1 contributes more to R̄1c
and hnk is maximized in SU-AS. In contrast, the curve for
PU-AS keeps constant even when N increases. The reason
for this is that PU-AS aims at maximizing the channel gain
of UE2 instead of UE1 . That is, the channel gain of UE1 is
selected in a similar way to find out the maximum element
from a random row of H and hnk is affected by M but not
N . Interestingly, when UE1 is farther from the BS than UE2 ,
the aforementioned situation is inverted. Furthermore, all the
proposed schemes outperform CR-NOMA-RA, and MCG-AS
can achieve a near-optimal performance in the entire region.
Fig. 9 demonstrates how the QoS requirement Rth affects the achievable R̄1c . We can see when Rth increases,
the performances decrease for all the AS schemes in CRNOMA. This is because when Rth increases, more power is
allocated to UE2 to meet the higher QoS requirement and

10

20
CR-NOMA-ES
CR-NOMA-RA
PU-AS
SU-AS
MCG-AS

19.5

Average rate of UE1 (bps/Hz)

19
18.5
18

17.5
1

2

3

4

5

6

7

8

The number of antennas at BS (N), d1=100m, d 2=200m
17

16
CR-NOMA-ES
CR-NOMA-RA
PU-AS
SU-AS
MCG-AS

15

14
1

2

3

4

5

6

7

8

The number of antennas at BS (N), d1=300m, d 2=200m

Fig. 8: Average rate of UE1 vs. N in CR-NOMA, Ps =
20dBm, Rth = 5bps/Hz.
22

CR-NOMA-ES
CR-NOMA-RA
PU-AS
SU-AS
MCG-AS

20

Average rate of UE1 (bps/Hz)

asymptotic closed-form expressions for the average sum-rates
in F-NOMA systems and average rates of the secondary user
in CR-NOMA systems were provided. Numerical simulations
demonstrated that in F-NOMA systems, both A3 -AS and AIAAS yield significant performance gains over the random AS
scheme and the OMA scheme with exhaustive search-based
AS. Furthermore, A3 -AS achieves the near-optimal sum-rate
performance while AIA-AS provides better user fairness. In
CR-NOMA scenarios, MCG-AS can achieve a near-optimal
performance with much reduced complexity in the entire
region. The complexities of the PU-AS and SU-AS is further
lower than that of the MCG-AS scheme, while they only
achieve a near-optimal performance when one user is much
closer to the base station that the other.

18
16
14
3

3.5

4

4.5

5

5.5

6

6.5

7

7.5

8

The QoS requirement of UE 2 (bps/Hz), d1=100m, d 2=200m.

P ROOF

A PPENDIX A
OF THE P ROPOSITION

1

When ρ → ∞, we can attain the asymptotic closed-form
expression for the average sum-rate of the A3 -AS algorithm
as follows:

18

3

Z ∞
(c2 )
1
1
+
log(1 + bρs x)fγ s 3 (x)dx = log
A
b
b
0



NK
NM
X X µi,NM µj,NK
iΩh +jΩg
iΩh + jΩg
e bρ Ei −
ln 2
bρ
i=1 j=1




jΩg
iΩh
iΩh
jΩg
−e bρ Ei −
− e bρ Ei −
bρ
bρ



NM
NK
(c3 )
iΩh + jΩg
1 X X µi,NM µj,NK
ln
≈ log +
b
ln 2
bρ
i=1 j=1





iΩh
jΩg
1
− ln
− ln
− C = log
bρ
bρ
b


 
NK
NM
XX
µi,NM µj,NK
iΩh + jΩg
+
+ ln (bρ) − C
ln
ln 2
ijΩh Ωg
i=1 j=1
NM
NK
X X µi,NM µj,NK  iΩh + jΩg 
(c4 ) (ln ρ − C)
=
.(47)
+
ln
ln 2
ln 2
ijΩh Ωg
i=1 j=1

A
R̄sum
≈ log

16
14

CR-NOMA-ES
CR-NOMA-RA
PU-AS
SU-AS
MCG-AS

12
10
3

3.5

4

4.5

+
5

5.5

6

6.5

7

7.5

8

The QoS requirement of UE 2 (bps/Hz), d1=300m, d 2=200m.

Fig. 9: Average rate of UE1 vs. the QoS requirement of UE2
in CR-NOMA, N = 4, Ps = 20dBm.
less power can be allocated to UE1 . In this case, R̄1c would
decrease when Rth increases. Again, all the proposed AS
schemes outperform CR-NOMA-RA, and MCG-AS achieves
a near-optimal performance in the entire region. Moreover, the
performance of PU-AS is better than that of SU-AS when UE2
is closer to the BS, and it is opposite when UE1 is closer to
the BS.
As discussed above, MCG-AS can provide a near-optimal
average rate performance for the SU in CR-NOMA networks
by taking advantage of the instantaneous channel conditions
of all the participants in systems. However, once the networkaccessing priority and the near-far relationship of the users in
CR-networks are determined, the PU-AS or SU-AS algorithms
would be better choices. This is because the computational
complexities of PU-AS and SU-AS are further lower than
that of MCG-AS, and a near-optimal average rate performance
for the secondary UE can be achieved by PU-AU when the
primary user is much closer to the BS, and that is achieved
by SU-AS when the secondary user is much closer to the BS.
VI. C ONCLUSIONS
This paper studied the joint antenna selection (AS) problem for a classical MIMO-NOMA downlink communication
scenario. Two computationally efficient AS algorithms, named
A3 -AS and AIA-AS, were proposed to maximize the system
sum-rate of F-NOMA systems without and with the consideration of user fairness, respectively. Meanwhile, a lowcomplexity AS algorithm (i.e., MCG-AS) and its two simplified versions (i.e., PU-AS and SU-AS) were developed to
maximize the rate of the SU in CR-NOMA systems. The

Specifically, the step (c2 ) is obtained according to [33, Eq.
(4.337.2)], the step (c3 ) is obtained by applying the approxx→0
imation Ei(x) ≈ C + ln |x| [34] and P
the step (c4 ) follows

n
the property of the binomial coefficient i=0 (−1)i ni = 0.
A PPENDIX B
P ROOF OF THE L EMMA 1

In the first stage of AIA-AS, the maximum elements of
each row from H and G, i.e., hmax
and gnmax , and their
n
distributions are obtained according to (21)-(24). In the second stage of AIA-AS, the relatively smaller element γnw =
max
min(hmax
) in each row for n ∈ N is found out. Thus,
n , gn
the CDF of γnw for x ≥ 0 can be calculated as follows:
Fγnw (x)

=
=
=
+

Pr {min(hmax
, gnmax ) < x}
n
max
max
Pr (h
< gn < x) + Pr (gnmax < hmax
< x)
n
Z x n
Z y
fgnmax (y)
fhmax
(z)dzdy
n
Z0 x
Z0 y
fhmax
(y)
fgnmax (z)dzdy
n
0

=

1−

0

M X
K
X
i=1 j=1

µi,M µj,K e−(iΩh +jΩg )x .

(48)

11

w
In the third stage of AIA-AS, γAIA
= max(γnw ) for n ∈ N
w
is selected. In the case that γAIA lies in the n∗AIA row, we
first define γ̂ w = max
(γnw ) and obtain the CDF of γ̂ w
∗

in which, φj = jΩg + ξℓ , φij,1 = iΩh + jΩg + ξℓ , and φij,2 =
iΩh + jΩg + 2ξℓ .
A PPENDIX D

as follows:

Recall that the achievable rate of the secondary user in (11)
consists of two terms. Here we can calculate the asymptotic
P
average
 P  rate of UE1 in PU-AS, which is denoted by R̄1 =
E R1 , in two parts.
P
The first part of (11), i.e., R̄1,1
, can be obtained by calculating the following integral for δn∗ ,m∗ ,k∗ = 1:

(c6 )

Fγ̂ w (x) = [Fγnw (x)]N −1 =

X

Cℓ tℓ e−ξℓ x ,

(49)

ℓ

where the step (c6 ) is expanded according to the Multinomial theorem. Specifically, ℓ0 + · · · + ℓMK
N − 1,
 = (N
N −1
= ℓ0 !···ℓ−1)!
the multinomial coefficient Cℓ = ℓ0 ,···
,
,ℓM K
MK !
PM PK
Q
tℓ = 1≤i≤M (−µi,M µj,K )ℓij and ξℓ = i=1 j=1 (iΩh +
1≤j≤K

jΩj )ℓij .
s
Next we need to obtain the CDF and PDF of γAIA
=
max
max
∗
w
max(hn∗ , gn∗ ) which lies in the same nAIA row as γAIA
.
AIA
AIA
By applying some algebraic manipulations, we have
s
FγAIA
(x)

=
=
+

and
s
fγAIA
(x)

P ROOF

n6=nAIA

=
+

n
o
max
w
w
∗
∗
Pr max(hmax
,
g
)
<
x,
γ
≥
γ̂
n∗
n
n
AIA
AIA
AIA
n
w
max
max
N Pr(γ̂ < gn∗AIA < hn∗AIA < x)
o
∗
Pr(γ̂ w < hmax
< gnmax
< x) ,
(50)
n∗
AIA
AIA

Z x
Z y
N fhn max (x)
fgn max (y)
fγ̂ w (z)dzdy

Z x 0
Z y 0
fgn max (x)
fhn max (y)
fγ̂ w (z)dzdy ,
0

P
R̄1,1

=

ℓ

where T1 is given by
Z ∞
ξℓ ζij ln(1 + bρx)e−jΩg x
T1 = −
dx
iΩh φi
0


g
ξℓ ζ̃ij jΩ
jΩg (c8 ) ξℓ ζ̃ij
(c7 )
≈
e bρ Ei −
χ (jΩg ) , (52)
=
φi
bρ
φi

∞

Z

x

log

0
0
M X
NK
X



P ROPOSITION 3

ρx
ε+1



fgmax (y)fhn max (x)dydx

Z ∞ 

iΩh µi,M µj,NK
ρx
e−iΩh x dx
ln
ln
2
ε
+
1
0
i=1 j=1


Z ∞ 
ρx
−(iΩh +jΩg )x
−
ln
e
dx
ε+1
0
Z ∞
M NK
(ε+1)iΩh y
y=ρx/(ε+1) X X i(ε + 1)Ωh µi,M µj,NK
ρ
ln ye−
dy
=
ρ ln 2
0
i=1 j=1

Z ∞
(ε+1)(iΩh +jΩg )y
ρ
dy
−
ln ye−
=

0




NK
M X
X
µi,M µj,NK
i(ε + 1)Ωh
−C − ln
ln 2
ρ
i=1 j=1


(ε+1)(iΩh +jΩg )

iΩh ln
ρ
iΩh C
+

iΩh + jΩg
iΩh + jΩg

(c9 )

=

0

s
and by applying some algebraic operations, finally fγAIA
(x)
A PPENDIX C
can be obtained as in (33).
P ROOF OF THE P ROPOSITION 2

s
s
Given the PDF of γAIA
i.e., fγAIA
(x) , we
 can approximate
AIA
according to (25)
the average sum-rate of AIA-AS i.e., R̄sum
as below:
Z ∞
1
AIA
s
log(1 + bρx)fγAIA
(x)dx
(51)
R̄sum
≈ log +
b
0
M K
1 X X X Cℓ tℓ
(T1 + T2 + T3 + T4 ) ,
= log +
b i=1 j=1
ln 2

Z

OF THE

+

NK
M X
X
µi,M µj,NK
{− ln (iΩh )
ln 2
i=1 j=1


 

ρ
− C + iΩh ln (iΩh + jΩg ) 
jΩg ln ε+1

=

+

iΩh + jΩg

.



Specifically, the step (c9 ) is obtained with the help of [33, Eq.
(4.331.1)].
P
The second part of (11), i.e., R̄1,2
, can be obtained by
calculating the following integral for δn∗ ,m∗ ,k∗ = 0:
P
R̄1,2
=

=

+

Z

∞

Z

∞

log

Z0 ∞ Zx∞

|0
Z

x

∞

Z



ρxy
εx + y



(x)dx
fgmax (y)dyfhmax
n

(x)dx
log (ρx) fgmax (y)dyfhmax
n
{z
}
T5

∞

log (y) fgmax (y)dyfhmax (x)dx

n
x
|,
in which ζ̃ij = N µi,M µj,K , φi = iΩh +ξℓ , χ(x) = C+ln | bρ
x
{z
}
|0
Ei(x) is the Exponential integral function and C ≈ 0.577 is
T6
Z ∞Z ∞
the Euler’s constant. Specifically, the step (c7 ) is obtained with
−
(x) dx, (54)
log (εx + y) fgmax (y)dyfhmax
n
the help of [33, Eq. (4.337.2)], and the step (c8 ) is obtained
0
x
{z
}
|
by using the approximation of ex ≈ 1 and Ei(x) ≈ C + ln x
T7
when x → 0 [34]. Similarly,
in which,
Z ∞
ξℓ ζij ln(1 + bρx)e−iΩh x
ξℓ ζ̃ij
R ∞R ∞
NK
M X
X
T2 = −
dx =
χ(iΩh ),
ijΩh Ωg µi,M µj,NK 0 x ln (ρx) e−jΩg y dye−iΩh x dx
jΩ
φ
φ
T
=
g j
j
5
0
ln 2
Z ∞
i=1 j=1
ζij φij,2 χ(φij,1 )
ζij φij,2 ln(1 + bρx)e−φij,1 x
Z
dx =
, X
T3 = −
NK
M X
iΩh µi,M µj,NK ∞
φi φj
φi φj φij,1
0
ln (ρx) e−(iΩh +jΩg )x dx
=
Z ∞
ln
2
−(iΩh +jΩg )x
0
ζij (iΩh + jΩg ) ln(1 + bρx)e
i=1 j=1
T4 =
dx
M X
NK
X
ijΩ
Ω
h g
0
(c10 )
µi,M µj,NK (−iΩh (C + ln(iΩh + jΩg ) − ln ρ))

= −ζ̃ij χ(iΩh + jΩg ),

=

i=1 j=1

ln 2 (iΩh + jΩg )

, (55)

(53)

12

where the step (c10 ) is obtained with the help of the substitution t = ρx and that of [33, Eq. (4.331.1)]. Similarly,

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