Antenna Selection in Full-Duplex Cooperative NOMA Systems

Abstract

We investigate the problem of antenna selection (AS) in full-duplex (FD) cooperative non-orthogonal multiple access (NOMA) systems, where a multi-antenna FD relay assists transmission from a multi-antenna base station (BS) to a far user, while at the same, the BS transmits to a near user. Specifically, based on the end-to-end signal-to-interference-plus-noise ratio at the near and far users, two AS schemes to select a single transmit antenna at both the BS and the relay, respectively, as well as a single receive antenna at relay are proposed. In order to study the ergodic sum rate and outage probability of these AS schemes, we have derived closed-form expressions assuming Rayleigh fading channels. The sum rate and outage probability of the AS schemes are also compared with the optimum selection scheme that maximizes the performance as well as with a random AS scheme. Our results show that the proposed AS schemes can deliver a near-optimal performance for near and far users, respectively.

Full text

arXiv:1802.04218v1 [cs.IT] 12 Feb 2018
Antenna Selection in Full-Duplex Cooperative
NOMA Systems
Mohammadali Mohammadi†, Zahra Mobini†, Himal A. Suraweera‡, and Zhiguo Ding§
†Faculty of Engineering, Shahrekord University, Iran
‡Department of Electrical and Electronic Engineering, University of Peradeniya, Sri Lanka
§School of Computing and Communications, Lancaster University, United Kingdom
Email: {m.a.mohammadi, z.mobini}@eng.sku.ac.ir, himal@ee.pdn.ac.lk, z.ding@lancaster.ac.uk
Abstract—We investigate the problem of antenna selection
(AS) in full-duplex (FD) cooperative non-orthogonal multiple
access (NOMA) systems, where a multi-antenna FD relay
assists transmission from a multi-antenna base station (BS) to
a far user, while at the same, the BS transmits to a near user.
Specifically, based on the end-to-end signal-to-interference-
plus-noise ratio at the near and far users, two AS schemes
to select a single transmit antenna at both the BS and the
relay, respectively, as well as a single receive antenna at relay
are proposed. In order to study the ergodic sum rate and
outage probability of these AS schemes, we have derived
closed-form expressions assuming Rayleigh fading channels.
The sum rate and outage probability of the AS schemes
are also compared with the optimum selection scheme that
maximizes the performance as well as with a random AS
scheme. Our results show that the proposed AS schemes can
deliver a near-optimal performance for near and far users,
respectively.
I. INTRO DUC TIO N
Each new generation of wireless communication sys-
tems has been designed to support the demands of in-
creased traffic and data throughput with spectral efficiency
as a key factor. To this end, non-orthogonal multiple
access (NOMA) principle has been recognized as a key
technology that can improve the spectral efficiency of
5G wireless systems. In contrast to traditional orthogonal
multiple access (OMA) techniques, NOMA multiplexes
signals between users with strong/weak channel conditions
(a.k.a. near/far users) in the power domain and apply
successive interference cancellation (SIC) at the receivers
to remove the inter-user interference [1]–[3].
On the other hand, multiple-input multiple-output
(MIMO) technology that offers substantial performance
gains has become an integral part of modern communi-
cation systems. However, the benefits of MIMO come at
the price of increased computational complexity and cost
of hardware radio frequency chains that scales with the
number of antennas [4]. In this context, antenna selection
(AS) schemes with low implementation complexity and
perform close to traditional MIMO systems have been
touted as a practical solution in the literature. There is
a sizable matured body of work on AS for different
MIMO systems. To this end, AS in combination with
NOMA has received interest in the recent literature [5]–
[7], however the topic is still in infancy. Especially, the
complexity of deciding on AS solutions in NOMA systems
are exacerbated due to the complicated nature of near/far
user performance criterion [5].
In this paper, we analyze the AS performance of a full-
duplex (FD) cooperative NOMA system. FD is another
promising technology considered for 5G implementation.
Furthermore, the marriage between FD and NOMA can
boost the performance as confirmed in [8]–[11] so far. The
main bottleneck for FD operation is the self-interference
(SI) [12]–[14]. Thus, AS should consider the effect of SI
during the selection of strong channels toward the near/far
users. This makes the AS problem in FD NOMA systems
a much more complicated affair than in half-duplex (HD)
NOMA systems [5]. Specifically, for the considered FD
NOMA relay system, we propose AS schemes to achieve
near/far user end-to-end (e2e) signal-to-interference noise
ratio (SINR) maximization and study the sum rate and
outage probability, respectively. Our contributions can be
summarized as follows:
•Two low complexity AS schemes, i.e., max-U1 AS
scheme and max-U2 AS scheme are proposed to
maximize the e2e SINR at the near and far user,
respectively. The performance of the FD cooperative
NOMA system with the two AS schemes is analyzed
by deriving exact ergodic sum rate and outage prob-
ability expressions.
•Our findings reveal that max-U1 AS scheme can sig-
nificantly improve the system sum-rate, while max-
U2 AS scheme can provide better user fairness.
In particular, max-U1 AS scheme can achieve near
optimum rate performance in the entire SNR range.
Notation: We use E{X}to denote the expected value
of the random variable (RV) X; its probability density
function (pdf) and cumulative distribution function (cdf)
are fX(·)and FX(·)respectively; CN (µ, σ2)denotes
a circularly symmetric complex Gaussian RV Xwith
mean µand variance σ2and Ei(x) = Rx
−∞
et
tdt is the
exponential integral function [15, Eq. (8.211.1)].
II. SYS TE M MO DE L
Consider a two user NOMA downlink system where
U1 (near user) communicates directly with the base station
(BS), while U2 (far user) requires the assistance of a multi-
antenna FD relay, Ras shown in Fig. 1. Both U1 and
U2 are equipped with a single antenna each, the BS is
equipped with NTantennas, while Ris equipped with two

group of antennas: MRreceive antennas and MTtransmit
antennas.
In order to keep the implementation complexity low, we
assume that the BS and Rperform single AS [4]. To be
precise, the BS selects one (e.g. i-th) out of NTavailable
transmit antennas, while Rselects one (e.g. j-th) out of
MRavailable antennas to receive signals. Moreover, one
antenna (e.g. k-th) out of MTtransmit antennas is selected
at Rto forward the BS signal to U2.
We assume that all channels experience independent
Rayleigh fading and that they remain constant over one
transmission slot. The channel between the j-th receive
and the i-th transmit antenna from terminal Xto terminal
Y, is denoted by hi,j
XY ∼ C N (0, σ2
XY)where X∈ {S,R}
and Y∈ {R,U1,U2}.
A. Transmission Protocol
According to the NOMA concept [2], [3], the BS
transmits a combination of intended messages to both
users as
s[n] = pPSa1x1[n] + pPSa2x2[n],(1)
where xi,i∈ {1,2}denotes the information symbol
intended for Ui, and aidenotes the power allocation
coefficient, such that a1+a2= 1 and a1< a2.
Since Ris FD capable, it transmits the decoded symbol
x2[n−τ], where τ≥1accounts for the time delay caused
by FD processing at R[13]. Therefore, the receive signal
at Rcan be written as [8]
yR[n] = hi,j
SRs[n] + pPRhk,j
RR x2[n−τ] + nR[n],(2)
where PRis the relay transmit power and nR[n]∼
CN (0, σ 2
n)is the additive white Gaussian noise (AWGN)
at R. We assume imperfect SI cancellation due to FD
operation at Rand model the elements of the MR×MT
residual SI channel HRR = [hk,j
RR ]as independent identi-
cally distributed (i.i.d) C N (0, σ2
SI)RVs [13].
The information intended for U2 is decoded at Rwith
SIC treating the symbol of U1 as interference [8]. Hence,
the SINR at Rcan be written as
γR=a2γi,j
SR
a1γi,j
SR +γk,j
SI + 1,(3)
where γi,j
SR =ρS|hi,j
SR|2and γk,j
SI =ρR|hk,j
RR |2with ρS=
PS
σ2
nand ρR=PR
σ2
n.
At the same time, U1 receives the following signal
y1[n]= hi
SU1s[n]+pPRhk
RU1x2[n−τ]+n1[n],(4)
where n1[n]∼ CN (0, σ 2
n)is the AWGN at U1.
Based on (4), the SINR of U2 observed at U1 can be
written as
γ12 =a2γi
SU1
a1γi
SU1 +γk
RU1 + 1,(5)
where γi
SU1 =ρS|hi
SU1|2and γk
RU1 =ρR|hk
RU1|2.
It is assumed that the symbol, x2[n−τ], is priory
known to U1 and thus U1 can be removed [8]. However,
by considering realistic imperfect interference cancellation
wherein U1 cannot perfectly remove x2[n−τ], we model
hk
RU1 ∼ CN (0, k1σ2
RU1)as the inter-user interference
channel where the parameter k1presents the strength of
,
RR
k j
h
R
T
M
R
M
hϭ
hϮ
1
SU
i
h
,
i j
SR
h
1
RU
k
h
2
RU
k
h
^
Fig. 1. FD Cooperative NOMA system with antenna selection.
inter-user interference [8]. If U1 perfectly cancels the U2’s
signal, the SINR at U1 is given by
γ1=a1γi
SU1
γk
RU1 + 1 .(6)
Moreover, the received signal at U2, from Rcan be
written as
y2[n] = pPRhk
RU2x2[n−τ] + n2[n],(7)
where n2[n]∼ C N (0, σ2
n2)denotes the AWGN at U2.
Hence, the signal-to-noise ratio (SNR) at U2 is given by
γk
RU2 =PR
σ2
n2
|hk
RU2|2.(8)
The e2e SINR at U2 can be expressed as
γ2=min a2γi
SU1
a1γi
SU1 +γk
RU1 +1 ,a2γi,j
SR
a1γi,j
SR +γk,j
SI +1 , γk
RU2
!.
(9)
B. Antenna Selection Schemes
We now propose two AS schemes for the described FD
cooperative NOMA system where joint selection of single
transmit and receive antennas at the BS and Rbased on e2e
SINRs at the near user, U1 and far user, U2 is performed.
1) max-U1 AS Scheme: This scheme selects antennas
first to maximizes the e2e SINR at U1, (6) and next
with remaining AS choices tries to maximize also e2e
SINR at U2. Therefore, the particular AS scheme can be
mathematically expressed as
{i∗, k∗}= arg max
1≤i≤NT,1≤k≤MT
a1γi
SU1
γk
RU1 + 1
j∗= arg max
1≤j≤MR
a2γi∗,j
SR
a1γi∗,j
SR +γk∗,j
SI + 1 .(10)
2) max-U2 AS Scheme: The max-U2 AS scheme
achieves e2e SINR maximization at U2 according to
{i∗, j∗, k∗}= arg max
1≤i≤NT,1≤j≤MR,
1≤k≤MT
min a2γi
SU1
a1γi
SU1 +γk
RU1 + 1,
a2γi,j
SR
a1γi,j
SR +γk,j
SI + 1 , γ k
RU2!.(11)
According to (11) there is no degree-of-freedom (in
terms of AS) available for maximizing the e2e SINR at

U1. Therefore, in terms of the near user SINR, max-U2
AS scheme is inferior to that of the max-U1 AS scheme.
It worth pointing out that other AS schemes, for exam-
ple with criterion such as to maximize the near/far user
performance subject to a pre-defined far/near performance
are also possible to design. Analyzing such AS schemes
whose performance remain in between max-U1 and max-
U2 AS schemes is relegated to the journal version of this
work.
In order to implement max-U1 and max-U2 AS
schemes, the BS can transmit a pilot signal and Rcan
decide on the best antenna indexes to be used at the BS and
its receive side. Rcan also transmit a pilot signal to U1 and
U2 from each of the relay antenna. Upon reception of the
pilot, U1 (in max-U1 AS scheme) and U2 (in max-U2 AS
scheme) can next feedback the antenna index that Rmust
used in subsequent information transmission. Moreover,
in max-U1 AS scheme, upon reception of the BS pilot
signal, U1 can decide the best antenna and feedback the
corresponding index to the BS to commence information
transmission.
III. PER FO RMA NCE ANA LYS IS
In this section, we investigate the performance of pro-
posed AS schemes in terms of the achievable ergodic sum
rate and outage probability. The derived results will enable
us to examine the benefits of the proposed AS schemes.
A. Ergodic Sum Rate
The ergodic achievable sum rate of the FD cooperative
NOMA system is given by
Rsum =RAS
U1 +RAS
U2 ,(12)
with
RAS
U1 =E{log2(1 + γ1,AS)},(13a)
RAS
U2 =E{log2(1 + γ2,AS)},(13b)
where γ1,AS and γ2,AS denote the e2e SINR at the U1
and U2, corresponding to the specific AS scheme with
AS ∈ {S1,S2}. We use S1 to refer to the max-U1 AS
scheme and S2 to refer to the max-U2 AS scheme.
Before we proceed, it is useful, to note that for a
nonnegative RV X, since E{X}=R∞
t=0 Pr(X > t)dt,
the ergodic achievable rate for near user (u= 1) or far
user (u= 2) can be written as
RAS
Uu=1
ln2 Z∞
0
1−Fγu,AS (x)
1 + xdx, (14)
where Fγu,AS (z) = Pr(γu,AS ≤z)is the cdf of the RV
γu,AS.
In the sequel, we present key results for the near and
far user ergodic rates due to the proposed AS schemes.
Proposition 1. The ergodic achievable rates of U1 and
U2 of max-U1 AS scheme, are respectively given by
RS1
U1 =NT
ln2
NT−1
X
p=0
(−1)pNT−1
p
(p+ 1) (p+1) ¯γRU1
MTa1¯γSU1 −1(15)
×e1
¯γRU1 Ei−1
¯γRU1 −e(p+1)
a1¯γSU1 Ei−(p+ 1)
a1¯γSU1 
and
RS1
U2 =MRNT
ln2 Z∞
0
e−x
¯γRU2
1 + x
×
NT−1
X
p=0
(−1)pNT−1
pe−(p+1)x
¯γSU1 (a2−a1x)
(p+ 1) 1 + ¯γRU1
MT¯γSU1
(p+1)x
(a2−a1x)
×
MR−1
X
q=0
(−1)qMR−1
qe−(q+1)x
¯γSR (a2−a1x)
(q+ 1) 1 + ¯γSI
¯γSR
(q+1)x
(a2−a1x)dx, (16)
where ¯γSR =ρSσ2
SR,¯γSU1 =ρSσ2
SU1,¯γRU1 =ρRk1σ2
RU1,
¯γRU2 =ρRσ2
RU2 and ¯γSI =ρRσ2
SI.
Proof: According to (14), the ergodic achievable rate
can be calculated from the cdf of (6) and (9). Let us start
with the ergodic achievable rate of U1. By invoking (10),
the ratio a1γi
SU1
γk
RU1+1 is maximized when the strongest BS-
U1 channel and the weakest R-U1 channel are selected.
Therefore, the cdf of γ1,S1 can be evaluated as
Fγ1,S1 (x) = Z∞
0
FA((y+ 1)x/a1)fB(y)dy,
where Ais a RV defined as the maximum out of NT
exponentially distributed independent RVs, while Bis the
minimum out of MTexponentially distributed indepen-
dent RVs. Substituting the required cdf and the pdf and
simplifying yields
Fγ1,S1 (x)= 1 −NT
NT−1
X
p=0
(−1)pNT−1
pe−(p+1)x
a1¯γSU1
(p+ 1) 1 + (p+1)¯γRU1x
MTa1¯γSU1 .(17)
Next, substituting (17) into (14), then with the help of the
integration identity in [15, Eq. (3.352.4)], and after some
algebraic manipulations, we arrive at (15).
We now turn our attention to evaluate RS1
U2. To this end,
the cdf of γ2,S1 can be expressed as
Fγ2,S1 (x) = Pr (min (γ12,S1, γR,S1 , γRU2,S1)< x)
= 1 −Pr (γ12,S1 > x)Pr (γR,S1 > x)
×Pr (γRU2,S1 > x).(18)
According to the criterion in (10), for x < a2
a1, we can
write
Pr (γ12,S1 >x) = 1 −Z∞
0
Fγi
SU1 (y+ 1)x
a2−a1xfγk
RU1 (y)dy
=NTMT
¯γRU1
NT−1
X
p=0
(−1)pNT−1
p
p+ 1 e−(p+1)x
¯γSU1 (a2−a1x)
×Z∞
0
e−(p+1)yx
¯γSU1 (a2−a1x)e−MTy
¯γRU1 dy
=NT
NT−1
X
p=0
(−1)pNT−1
pe−(p+1)x
¯γSU1 (a2−a1x)
(p+ 1) 1 + ¯γRU1
¯γSU1
(p+1)x
MT(a2−a1x),(19)
where the second equality follows since fγk
RU1 (y) =
MT
¯γRU1 e−MTy
¯γRU1 and Fγi
SU1 (x) = (1 −e−x
¯γSU1 )MTcan be written
as
Fγi
SU1 (x) = 1 −NT
NT−1
X
p=0
(−1)pNT−1
p
p+ 1 e−(p+1)x
¯γSU1 .(20)

Moreover, based on (10), for the selected transmit
antennas at the BS and R, the ratio a2γi∗,j
SR
a1γi∗,j
SR +γk∗,j
SI +1 can be
maximized when the strongest BS-Rchannel and weakest
SI channel are selected. However, theses two channels are
coupled with each other through the selected antenna at
the Rinput. Therefore, it is difficult, if not impossible, to
find the cdf of a2γi∗,j
SR
a1γi∗,j
SR +γk∗,j
SI +1 . Alternatively, we propose
to select the receive antenna at Rsuch that γi∗,j
SR is maxi-
mized, which will be shown to be a good approximation
across the entire SNR range in Section IV (cf. Optimum
AS (U2)). Therefore, we have
Pr (γR,S1 > x) = 1 −Z∞
0
FA(y+ 1)x
a2−a1xfB(y)dy,
for x < a2
a1, where Ais a RV defined as the largest out of
MRexponentially distributed independent RVs, and since
SI link is ignored, Bis an exponentially distributed RV
with parameter ¯γSI . Substituting the required cdf and the
pdf and simplifying yields
Pr (γR,S1 >x) = MR
MR−1
X
q=0
(−1)qMR−1
qe−(q+1)x
¯γSR (a2−a1x)
(q+1) 1+ ¯γSI
¯γSR
(q+1)x
(a2−a1x).
(21)
Finally, since the R-U2 link is ignored, we have
Pr(γRU2,S1 > x) = e−x
¯γRU2 . To this end, pulling everything
together, we obtain
Fγ2,S1 (x) = 1 −MRNTe−x
¯γRU2
×
NT−1
X
p=0
(−1)pNT−1
pe
−(p+1)
¯γSU1 (a2
x−a1)
(p+ 1) 1 + ¯γRU1
¯γSU1
(p+1)
MT(a2
x−a1)
×
MR−1
X
q=0
(−1)qMR−1
qe−(q+1)x
¯γSR (a2−a1x)
(q+ 1) 1 + ¯γSI
¯γSR
(q+1)x
(a2−a1x).(22)
Having obtained the cdf of γ2,S1 , the ergodic rate of U2
can be obtained by employing (14). For x > a2
a1it can be
readily checked that Fγ2,S1 (x) = 1 and hence RS1
U2 = 0.
Proposition 2. The ergodic achievable rates of U1 and
U2 of max-U2 AS scheme, are respectively given by
RS2
U1 =1
ln2
a1¯γSU1
(¯γRU1 −a1¯γSU1)(23)
×e1
¯γRU1 Ei−1
¯γRU1 −e1
a1¯γSU1 Ei−1
a1¯γSU1 ,
and
RS2
U2 =MTNT
ln2 Z∞
0
e−x
¯γSU1 (a2−a1x)
1 + ¯γRU1
¯γSU1
x
(a2−a1x)(1 + x)
×
NT−1
X
p=0
(−1)pNT−1
pe−(p+1)x
¯γSR (a2−a1x)
(p+ 1) 1 + ¯γSI
MR¯γSR
(p+1)x
(a2−a1x)
×
MT−1
X
q=0
(−1)qMT−1
qe−(q+1)x
¯γRU2
(q+ 1) dx. (24)
Proof: The proof follows similar steps to the proof
of Proposition 1 and thus only an outline is presented.
According to (11) the e2e SINR at U2 is maximized when
each term inside in the minimum function is maximized.
Therefore, a transmit antenna at Ris selected such that
γk
RU2 is maximized, i.e., γk
RU2 is the largest of MTexpo-
nential RVs with parameter ¯γRU2 . Moreover, since SI is
the main source of performance degradation in FD mode,
for a particular transmit antenna at R, the best receive
antenna at the Ris selected such that the SI strength
is minimized [16]. Hence, γk∗,j
SI is the minimum of MR
exponential RVs with parameter ¯γSI . Finally, for given k∗
and j∗, a best transmit antenna at the BS is selected such
that γRis maximized. Hence, a single transmit antenna
at the BS is selected such that the SINR at BS-Ris
maximized for the i∗-th receive antenna at R.
Collecting the required cdfs and pdfs and simplifying
yields
Fγ1,S2 (x) = 1 −e−x
a1¯γSU1
1 + ¯γRU1
a1¯γSU1 x,(25)
and
Fγ2,S2 (x) = 1 −MTNT
e−x
¯γSU1 (a2−a1x)
1 + ¯γRU1
¯γSU1
x
(a2−a1x)
×
NT−1
X
p=0
(−1)pNT−1
pe−(p+1)c
¯γSR (a2−a1x)
(p+ 1) 1 + ¯γSI
¯γSR
(p+1)x
MR(a2−a1x)
×
MT−1
X
q=0
(−1)qMT−1
qe−(q+1)x
¯γRU2
(q+ 1) ,(26)
respectively. To this end, by invoking (14), (25), and (26)
after some algebraic manipulations, we arrive at the de-
sired result. 
B. Outage Probability
Outage probability is a key metric used to measure the
event that the data rate supported by instantaneous channel
realizations is less than a targeted user rate. Therefore, the
outage probability is an important performance metric to
characterizes the performance of NOMA systems [3].
The following Propositions present exact closed-form
expressions for the outage probability of max-U1 and max-
U2 AS schemes
Proposition 3. The outage probability of U1 with max-U1
and max-U2 AS scheme, are respectively given by
PS1
out,1=1−NT
NT−1
X
p=0
(−1)pNT−1
pe−(p+1)ζ
¯γSU1
(p+ 1) 1 + ¯γRU1
¯γSU1
(p+1)ζ
MT,(27)
and
PS2
out,1= 1 −e−ζ
¯γSU1
1 + ¯γRU1
¯γSU1 ζ,(28)
where ζ= max θ2
a2−a1θ2,θ1
a1,θ1= 2R1−1and θ2=
2R2−1with R1and R2being the transmission rates at
U1 and U2.
Proof: An outage event at U1 occurs when it cannot
decode the intended signal for U2, or when it can decode

0 5 10 15 20 25
0
1
2
3
4
5
6
7
ρS=ρR (dB)
Near User Acheivable Rate (bit/sec/Hz)
NOMA: max−U1 scheme
NOMA: Optimum AS
NOMA: Optimum AS (U2)
NOMA: max−U2 scheme
NOMA: Random AS
0 5 10 15 20 25
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
ρS=ρR (dB)
Far User Acheivable Rate (bit/sec/Hz)
NOMA: Optimum AS (U2)
NOMA: Optimum AS
NOMA: max−U2 scheme
NOMA: max−U1 scheme
NOMA: Random AS
Fig. 2. Ergodic rate of near and far users with different AS schemes
(NT=MR=MT= 4,σ2
SI = 0.3).
it but fails to decode its own signal. Therefore, the outage
probability of U1 can be derived as
PAS
out,1=1−Pr (γ12,AS > θ2, γ1,AS > θ1
),
=Pr γi
SU1
γk
RU1 + 1 < ζ
=Fγ1,AS (a1ζ).(29)
To this end, the desired result is obtained by evaluat-
ing (17) and (25) at a1ζ.
Proposition 4. The outage probability of U2 with max-U1
and max-U2 AS scheme, are respectively given by
PS1
out,2=1−MRe−θ2
¯γRU2
MR−1
X
q=0
(−1)qMR−1
qe−(q+1)θ2
¯γSR (a2−a1θ2)
(q+1) 1 + ¯γSI
¯γSR
(q+1)θ2
a2−a1θ2,
(30)
and
PS2
out,2= 1 −MTNT
MT−1
X
q=0
(−1)qMT−1
qe−(q+1)θ2
¯γRU2
(q+ 1) ,
×
NT−1
X
p=0
(−1)pNT−1
pe−(p+1)θ2
¯γSR (a2−a1θ2)
(p+1) 1+ ¯γSI
MR¯γSR
(p+1)θ2
(a2−a1θ2).(31)
Proof: An outage event at U2 occurs if Rfails to decode
the intended message to U2, or Rcan decode U2 signal
but U2 fails to decode its message. Therefore, the outage
probability of U2 can be expressed as
PAS
out,2= 1 −Pr (γR,AS > θ2, γRU2,AS > θ2)
= 1 −Pr (γR,AS > θ2)Pr (γRU2,AS > θ2).(32)
To this end by substituting the corresponding probabilities
presented in subsection III-A into (32), we arrive at (30)
and (31). 
0 5 10 15 20 25
0
1
2
3
4
5
6
7
8
9
ρS=ρR (dB)
Average Sum Rate (bit/sec/Hz)
NOMA: Optimum AS
NOMA: max−U1 scheme
NOMA: Optimum AS (U2)
NOMA: max−U2 Scheme
NOMA: Random AS
OMA: Random AS
Simulation
Fig. 3. Ergodic sum rate with different AS schemes (NT=MR=
MT= 4,σ2
SI = 0.3).
IV. NUM E RI CAL RE SULTS AN D DISC USS IO N
In this section, we present numerical results to quantify
the performance gains when max-U1 and max-U2 AS
schemes are adopted in the considered FD cooperative
NOMA system. We set a1= 0.25,a2= 0.75 and
k1= 0.01.
Fig. 2 shows the ergodic rates of U1 and U2 with the
proposed AS schemes. We have also plotted the curves for
i) Optimum AS scheme: that performs an exhaustive search
of all possible combinations to determine the antenna
subset in order to maximize the ergodic sum rate, ii)
Optimum AS (U2) scheme : This scheme aims to maximize
the e2e SINR at U2 in an optimal sense and performs an
exhaustive search of all possible combinations to deter-
mine the optimum antenna subset in order to maximize
the e2e SINR at U2, and iii) Random AS scheme: that
performs random AS at the BS and relay input/output. It
can be observed that the max-U1 AS scheme is able to
improve the ergodic rate of both U1 and U2, while max-
U2 AS scheme, only increases the ergodic rate of U2. The
max-U1 AS scheme provides the best performance for U1,
while the performances of the max-U2 AS scheme and
random AS scheme for U1 are almost identical. However,
both max-U1 AS scheme and max-U2 AS scheme are able
to increase the ergodic rate of U2. Moreover, we see that
with increasing transmit power at BS and R, max-U2 AS
scheme achieves the same performance as Optimum AS
scheme.
In Fig. 3 the ergodic sum rate with different AS schemes
is illustrated. It is evident that the ergodic achievable sum
rate with the proposed AS schemes is higher than that
of OMA FD relay system. In all transmit power regimes
the performance gap between the max-U1 AS scheme
and Optimum AS scheme is negligible. Moreover, the
difference between the max-U1 and Optimum AS (U2)
enlarges when transmit power at the BS and Ris increased
and remains constant for medium-to-high transmit power
values.
Fig. 4 shows the Jain’s fairness index versus transmit
power and for different AS schemes. This index is a

0 5 10 15 20 25
0.65
0.7
0.75
0.8
0.85
0.9
0.95
1
ρS=ρR (dB)
Jains Fairness Index
NOMA: max−U2 Scheme
NOMA: Optimum AS (U2)
NOMA: Optimum AS
NOMA: Random AS
NOMA: max−U1Scheme
Fig. 4. Jain’s fairness index versus transmit power (NT=MR=
MT= 4,σ2
SI = 0.3).
0 5 10 15
10−4
10−3
10−2
10−1
100
PS=PR (dBW)
Outage Probability
Pout,1
S1
Pout,2
S2 , σ2
SI=0.1
Pout,2
S2 , σ2
SI=0.3
Pout,2
S1 , σ2
SI=0.1
Pout,1
S2
Pout,1
Rd
Pout,2
S1 , σ2
SI=0.3
Pout,2
Rd , σ2
SI=0.1
Pout,2
Rd , σ2
SI=0.3
Simulation
Fig. 5. Outage probability versus transmit power (NT=MR=MT=
4,R1=R2= 0.5bps/Hz).
bounded continuous function and is the most used quan-
titative measure to study the fairness in wireless systems.
The Jain’s fairness index for the considered dual-users
scenario can be expressed as [17]
J=(RAS
U1 +RAS
U2 )2
2((RAS
U1 )2+ (RAS
U2 )2),
which its range is the interval [1
2,1]. In this interval, J=
1
2corresponds to the least fair allocation in which only
one user receives a non-zero rate, and J= 1 corresponds
to the fairest allocation in which both near and far user
receive the same rate. From Fig. 4 We see that in the
low SNR region, Optimum AS scheme achieves the best
user fairness out of all AS schemes. However, max-U2
AS scheme can provide a better user fairness in medium-
to-high SNR regimes and hence can balance the tradeoff
between the ergodic rate and user fairness.
Fig. 5 shows the outage probability of the AS schemes
with different residual SI strengths. The max-U1 AS
scheme provides the best outage performance for U1 and
improves the outage performance of U2. Moreover, the
max-U2 AS scheme can significantly improve the outage
performance of U2, while it exhibits the same performance
as Random AS scheme for U1. Finally, due to inter-user
interference at U1 and SI at Rour results show that all
AS schemes suffer from zero-order diversity. However, the
proposed AS schemes can reduce the error floor.
V. CO NC LUS ION
In this paper, we have studied the AS problem for a
FD cooperative NOMA system. Two low complexity AS
schemes, namely max-U1 AS scheme and max-U2 AS
scheme were proposed to maximize the e2e SINR at the
near and far user, respectively. The performance of both
AS schemes have been characterized in terms of the er-
godic sum rate and outage probability. Our results revealed
that the max-U1 AS scheme achieves near optimum sum-
rate performance while the max-U2 AS scheme exhibits a
better user fairness. Moreover, the outage performance of
the near user can be significantly improved by using the
max-U1 AS scheme, while the outage probability of the
far user can be effectively improved via the max-U2 AS
scheme.
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