Outage Performance of Underlay CR-NOMA Networks

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Outage Performance of Underlay CR-NOMA
Networks
Sultangali Arzykulov∗†, Galymzhan Nauryzbayev◦, Theodoros A. Tsiftsis†and Mohamed Abdallah◦
∗School of Engineering, Nazarbayev University, Astana, Kazakhstan
†School of Electrical & Information Engineering, Jinan University, 519070 Zhuhai, China
◦Division of Information and Computing Technology, College of Science and Engineering,
Hamad Bin Khalifa University, Qatar Foundation, Doha, Qatar
Email: sultangali.arzykulov@nu.edu.kz; nauryzbayevg@gmail.com; theo tsiftsis@jnu.edu.cn; moabdallah@hbku.edu.qa
Abstract—In this paper, we study the outage probability (OP) of
the cooperative underlay cognitive radio (CR) networks consisted
of a secondary relay node and Knon-orthogonal multiple access
(NOMA) secondary destination users (SDUs) under imperfect
channel state information (CSI) conditions. We first derive gen-
eralized closed-form expressions for the outage probability (OP)
of NOMA-enabled SDUs. The numerical results show that the
considered system model outperforms cooperative orthogonal
multiple access in terms of the OP. Finally, the obtained analytical
expressions are corroborated by Monte Carlo simulations.
Index Terms—Cognitive radio (CR), decode-and-forward (DF),
non-orthogonal multiple access (NOMA), outage probability (OP).
I. INTRODUCTION
COGNITIVE radio (CR) is a promising technique that
can improve the spectrum efficiency (SE) of wireless
communication networks (WCNs) [1]. The concept of the CR
is to allow secondary networks (SNs) to share the spectrum
of primary networks (PN). Particularly, underlay CR enables
the SN access the frequency band of the PN if secondary
transmitters do not cause tolerable interference to the PN [2].
Another promising technique that can not only improve
the SE, but also can increase user connectivity is the non-
orthogonal multiple access (NOMA). In contrast to the conven-
tional orthogonal multiple access (OMA), NOMA users can be
served simultaneously through non-orthogonal resources, i.e.,
frequency, time or power [3]. The key concept of NOMA is that
a base station broadcasts a superimposed signal with various
power allocation (PA) factors to all NOMA receivers nodes.
At the receiver side, the user with better channel condition
performs successive interference cancellation (SIC) to remove
interference caused by users with poor channel conditions [4].
Recently, 3GPP Long-Term Evolution has adopted downlink
NOMA which is called multiple user superposition transmis-
sion (MUST) [5]. The authors in [6] showed that the NOMA
technique provides higher SE compared with the OMA. In [7],
the performance of NOMA in a cellular downlink scenario with
randomly deployed users was investigated. It was shown that
NOMA achieves better ergodic sum rates compared to OMA.
However, it was also deduced that the outage probability (OP)
of NOMA crucially depends on the appropriate choice of the
PA factors. Moreover, a cooperative NOMA was studied in [8],
where users with good channel gains operated as relays for
users with poor channel conditions to strengthen their signals.
The authors concluded that a cooperative NOMA outperforms
both the conventional cooperative OMA and non-cooperative
NOMA. In additional, cooperative NOMA with a dedicate relay
node was studied in [9] and [10]. The approximated OP derived
in [9], where it revealed that cooperative NOMA performs
better than cooperative OMA regards coding gain. The work
in [10] investigated NOMA with a two-way relay which can
support both uplink and downlink efficient data exchange. From
the obtained results, the authors concluded that the proposed
scheme achieves better sum rate performance compared to that
of OMA scheme. Furthermore, in [11], it was shown that the
application of NOMA into CR networks can further improve
SE.
In this paper, we study a dowlink cooperative underlay CR-
NOMA model consisting of KNOMA secondary destination
users (SDUs) over Rayleigh fading channels imperfect channel
state information (CSI) conditions. Moreover, we consider
an interference temperature constraint (ITC) at the primary
receiver (P R) for the secondary source. Exact generalized
closed-form expressions for the OP of the SDUs are derived
and verified by Monte-Carlo simulations. Moreover, we find
optimal PA factors for NOMA users with different distances.
Additionally, the cooperative CR-NOMA is compared with the
conventional cooperative CR-OMA in terms of the OP to show
the supremacy of the former.
The remainder of this paper is organized as follows. Section
II describes the system model with evaluated achievable rates.
Section III presents the OP analysis of KNOMA SDUs while,
in Section IV, numerical results are presented and discussed.
Finally, Section V concludes the paper.
II. SY ST EM MO DE L
Consider a system model with a downlink dual-hop detect-
and-forward (DF) underlay CR-NOMA network consisting of
aP R and a SN with a source (S), a relay (R) operating in
half-duplex mode and KNOMA SDUs (D1, ..., DK−1, DK) as
shown in Fig. 1. Channels, gχ, with ∀χ∈ {SP, S R, 1, ..., K −
1, K}, between nodes follow Rayleigh distribution and by

assuming imperfect CSI and minimum mean square estimation
error model, can be written as [12]
gχ= ˆgχ+ ˜gχ,(1)
where ˆgχis the estimated channel coefficient with CN(0, σ2
ˆgχ)
and ˜gχdenotes the error term, which can be modeled as a com-
plex Gaussian distributed random variable with CN(0, σ2
˜gχ),
where σ2
˜gχis the variance. The corresponding distances be-
tween nodes are denoted by dSP , dSR and d1, ..., dK−1, dK.
Additionally, the interference from the PN to the SN’s users is
denoted by PI1.
Scauses interference to P R, while Rdoes not interfere
with P R due its remoteness. Thus, the SN communication is
available only if P R does not receive harmful interference from
S. Hence, the following transmit power constraint is imposed
at S[2]
PS≤min Idτ
SP
|gSP |2,´
PS,(2)
where ´
PSstands for the maximum average transmit power
level at Swhile Idenotes the ITC at P R and τis the path-
loss exponent. Considering the above power restriction, the
superimposed signal PK
p=1 pαpPSxpis conveyed from Sto
KSDUs via the assistance of Rwithin two time periods (TPs),
where xp, with E(|xp|2)=1, is the intended message for Dp
and αpis the PA factor, with PK
p=1 αp= 1. Without loss of
generality, it is assumed that the channel gains of destination
NOMA users are ordered as g1< ... < gK−1< gKand, thus,
α1> ... > αK−1> αK, which means that the channel of DK
is stronger than that of DK−1and a lower power portion is
allocated to DK.
Then, in TP 1, Rreceives the following signal
yR= (ˆgSR + ˜gS R) ΨS
K
X
p=1
√αpxp+PI+nR
=ˆgSR ΨS
K
X
p=1
√αpxp+ ˜gSR ΨS
K
X
p=1
√αpxp+PI+nR
| {z }
effective noise
,(3)
where ΨS=qPS
dτ
SR
while n(·)∼ CN(0, σ2
(·))stands for the
additive white Gaussian noise (AWGN) at a certain receive
node. For the sake of brevity, we assume that all secondary
receive nodes obtain the same PIfrom the PN. Further, R
implements SIC according to the principle of NOMA [8]. The
decoding order of the users’ messages can be described as
follows: Rdecodes the first user’s message (j= 1) while
treating the other messages (j= 2, ..., K) as a background
noise. Then, the decoded message is removed from the received
signal. In the next stage, a message of the second user is
decoded by treating the rest messages (j= 3, .., K) as a
1CSI of primary transmitters is not available for the SN. Thus, with respect to
the central limit theorem [13], all interference signals from primary transmitters
can be treated as AWGN noise with CN (0, νσ2), where νis the scaling
coefficient of PI.
PR S R
D1
DK
gSP
gSR
g1
gK
.
.
.
Interference from toS PR
Secondary link
Interference from PN to S N
D2
g2
PIPI
PI
PI
PI
Fig. 1. A downlink cooperative underlay CR-NOMA network.
noise, and so on. Finally, Rdecodes the message of the K-
th user without any inter-user interference. Thus, the signal-to-
interference-plus-noise ratio (SINR) and signal-to-noise ratio
(SNR) of decoding xjand xKat Rcan be respectively
expressed as
γR,j =|ˆgSR |2αjρS
|ˆgSR |2ρSΘ + ζRdτ
SR
and (4)
γR,K =|ˆgSR |2αKρS
ζRdτ
SR
,(5)
where ρS=PS
σ2is the source transmit SNR, σ2is the noise
power at each receive node2,ζR=ρSσ2
˜gS R
dτ
SR +ν+ 1 and Θ =
PK
p=j+1 αp. Note that γR,K can be achieved if RR,j ≥ Rthj,
i.e., the SIC is successfully implemented at Rto remove the
message xj, where RR,j and Rthjdenote the received and
targeted data rates at Dj, respectively.
During TP 2, Rforwards the detected superimposed signal
PK
p=1 pPRβp˜xpto all NOMA SDUs, where PRdenotes
the relay transmit power, ˜xpis the detected message of cor-
responding SDU at R;βp, with PK
p=1 βp= 1, satisfying
β1> ... > βK−1> βK, indicates the PA factor at R. Thus,
Djobtains the following superimposed signal
yj= ˆgjsPR
dτ
j
K
X
p=1 pβp˜xp+ ˜gjsPR
dτ
j
K
X
p=1 pβp˜xp+PI+nj
| {z }
effective noise
,
(6)
where ∀j∈ {1, ..., K −1, K }. Furthermore, Dkapplies the
SIC to decode the undesired message of Dj, j < k < K, by
following the same manner as in R, with the SINR given by
γk,j =|ˆgk|2βjρR
|ˆgk|2ρRΦ + ζkdτ
k
,∀k∈ {2, ..., K},(7)
where ρR=PR
σ2denotes the relay transmit SNR, Φ =
PK
p=j+1 βpand ζ(·)=ρRσ2
˜g(·)
dτ
(·)+ν+ 1. Then, when j=kand
j6=K, the user of interest decodes its own signal by treating
2For mathematical tractability and without loss of generality, we assume
σ2
P=σ2
R=σ2
1=σ2
2=... =σ2
K=σ2throughout the paper.

the other messages as a noise, which SINR can be written as
γj,j =|ˆgj|2βjρR
|ˆgj|2ρRΦ + ζjdτ
j
.(8)
Moreover, when j=K, the K-th user detects its own signal
without inter-user interference with the SNR defined as
γK,K =|ˆgK|2βKρR
ζKdτ
K
.(9)
Finally, the achievable rate for the message dedicated to
Dj(j < ∀k≤K)and DKcan be respectively derived as
[14]
Rj=1
2log2(1 + min (γR,j, γk,j , γj,j )) ,(10)
RK=1
2log2(1 + min (γR,K, γK,K )) ,(11)
where all SNR values involved in each min function are
independent RVs.
III. OUTAG E ANALYSIS
This section investigates the OP for the proposed system
model. The message xjis considered to be in an outage if
the achievable rate of xjis below a predefined target rate Rthj
(which corresponding receive SNR threshold is ξj= 22Rthj−1)
[15], [16], i.e., Pout =Pr [Rj<Rthj]. Thus, the OP of xjcan
be expressed as in (12), at the top of the next page, where
t={m(0), m(1) , ..., m(n)},n=K−j,m(0) =j, j + 1, ..., K ,
m(1) =j+ 1, ..., K,m(n)=j+n, ..., K and m(n)6=m(n+1).
For example, in the case when K= 3, the OP of x2(j= 2)
can be obtained from (12) as the next: n= 1,m(0) = 2,3and
m(1) = 3 (m(0) 6=m(1)). Thus, we first sum the cumulative
distribution functions (CDFs) of Fγ2,2and Fγ3,2, then subtract
the product of the both CDFs (since m(0) 6=m(1) →m(0) = 2
and m(1) = 3), which can be written as
P{2}
out (ξ2) =1 −Pr [min (γR,2, γ3,2, γ2,2)> ξ2]
=1 −Pr [γR,2> ξ2]Pr [γ3,2> ξ2]Pr [γ2,2> ξ2]
=Fγ2,2(ξ2) + Fγ3,2(ξ2)−Fγ2,2(ξ2)Fγ3,2(ξ2)
×1−FγR,2(ξ2)+FγR,2(ξ2).(13)
Lemma 1: The CDF of the RV γR,j in (4) can be derived in
its closed form as
FγR,j (ξj)=1−



e
−
ξjζRdτ
SR
2µρ ´
S−ξjζRdτ
SR e
−
µρIdτ
SP +ξjζRdτ
SR
2µρ ´
S
µρIdτ
SP +ξjζRdτ
SR




| {z }
ΛR,j
,
(14)
where uj<αj
Θ, otherwise, FγR,j (ξj)=1.
Proof: See Appendix A. 
Unordered RVs |¯gk|2and |¯gj|2in (7) and (8) follow expo-
nential distribution with unit mean and variance [17], the CDFs
of which can be respectively derived as
F|¯gk|2(ξj) = Pr |¯gk|2<ξjζkdτ
k
ρR=(1−e−ξjζkdτ
k
2ρR, uj<βj
Φ
1,otherwise,
(15)
F|¯gj|2(ξj) = Pr |¯gj|2<ξjζjdτ
j
ρR=


1−e−ξjζjdτ
j
2ρR, uj<βj
Φ
1,otherwise,
(16)
where =βj−ξjΦ. Then, by applying order statistics [18,
Eq. (19)], the CDF of ordered RVs |ˆgk|2and |ˆgj|2can be
accordingly written as
F|ˆgk|2(ξj) = K!
(K−k)!(k−1)!
K−k
X
i=0
(−1)i
k+iM−k
i
| {z }
Lk(i)
×1−e−ξjζkdτ
k
2ρRk+i
,Fγk,j (ξj)and (17)
F|ˆgj|2(ξj) = K!
(K−j)!(j−1)!
K−j
X
i=0
(−1)i
j+iM−j
i
| {z }
Lj(i)
×1−e−ξjζjdτ
j
2ρRj+i
,Fγj,j (ξj).(18)
Furthermore, using (14), (17) and (18), the OP in the example
shown in (13) can be further rewritten in a closed-form as in
(19), at the top of the next page.
Similar to xj, the OP of xKcan be expressed as
P{K}
out =1 −Pr [min (γR,K , γK,K )> ξK]
=FγK,K (ξK) + FγR,K (ξK)−FγK,K (ξK)FγR,K (ξK),
(20)
where ξKis the receive SNR threshold at DK. Now, similar
to Appendix A, the CDF of γR,K can be derived as
FγR,K (ξK)=1−

e
−
ξKζRdτ
SR
2αKρ´
S−ξKζRe
−
αKρIdτ
SP +ξKζRdτ
SR
2αKρ´
S
αKρIdτ
SP /dτ
SR +ξKζR


.
| {z }
ΛR,K
(21)
Moreover, using order statistics for the strongest channel [18,
Eq. (19)], the CDF of the ordered RV |ˆgK|2is derived as
F|ˆgK|2(ξK),FγK,K (ξK) = 1−e−ξKζKdτ
K
2βKρRK
.(22)
Finally, substituting (21) and (22) into (20), the closed-form
OP of xKcan be expressed as
P{K}
out = 1 −ΛR,K 1−1−e−ξKζKdτ
K
2βKρRK!.(23)

P{j}
out =1 −Pr [min (γR,j , γk,j , γj,j )> ξj] = FγR,j (ξj) + 1−FγR,j (ξj)
(−1)nX
∀jY
∀t
Fγt,j (ξj)
(12)
P{2}
out (ξ2) = 




1−ΛR,2+ ΛR,2 P3
p=2 Lp(i)1−e−ξ2ζpdτ
p
2ρRp+i
−Q3
p=2 Lp(i)1−e−ξ2ζpdτ
p
2ρRp+i!, ξ2<α2
α3, ξ2<β2
β3,
1,otherwise.
(19)
Fig. 2. OP vs. transmit SNR for x1and x2.
IV. NUMERICAL RES ULT S
This section presents numerical results on the OP over
Rayleigh fading. As a special case, we consider K= 2
SDUs, i.e., D1and D2. Thus, to focus on the OP, without
loss of generality, we adopt the following system parameters:
dSP =dRP =dSR =d2=d, when dis assumed to be unity;
α1=β1and α2=β2;I= 20 dB; ν= 0.5,σ2
˜gt= 0.001, with
∀t∈ {SP, SR, 1,2}and ξ=ξ1=ξ2= 3 dB.
Fig. 2 illustrates the OP results of x1and x2for perfect and
imperfect CSI cases. It is considered that α1=β1= 0.8,
α2=β2= 0.2,d1= 3d. Additionally, the asymptotic
case without ITC (IdB =∞) is also plotted to analyze the
OP when SUs can transmit with a maximum transmit power.
Moreover, the OP results of the conventional cooperative CR-
OMA are plotted to compare with those of the proposed
cooperative NOMA model. In order to serve both D1and D2,
the CR-NOMA model requires two TPs, whilst cooperative
CR-OMA needs four TPs for the same purpose. Hence, the
data requirement for cooperative CR-OMA is set as twice
higher as for cooperative CR-NOMA for a fair comparison of
the both techniques. The power allocated for each OMA data
Fig. 3. OP vs. PA factors for x1and x2at 20 dB transmit SNR
when d1={1.5d, 3d, 4d}.
transmission at Sis equal to 1
2PS. From the plotted results,
it is clearly seen that x2outperforms x1in terms of the OP
results. It can be explained by the fact that D2implements the
SIC to remove the interference from D1, while D1detects its
data without canceling the interference which results in worse
OP results. Moreover, when IdB =∞, both messages of the
cooperative CR-NOMA obtain better OP compared to that of
cooperative CR-OMA for all SNR values. It is noticed that a
saturation of OP curves for NOMA users begin at lower SNR
levels compared with those of the OMA ones. It is due to the
fact that the transmit power of 1
2PSat Sresults in an increase of
the ITC value at PDregarding (2). In addition, when imperfect
CSI and PN interference are applied, it can be seen that the
OP of both signals show degradation. Moreover, for d1= 3d,
it is noticed that the effect of the imperfect channel on the OP
is not considerable at lower SNRs (below 25 dB). However,
at higher SNRs, the impact of the imperfect CSI on the OP
becomes more significant.
Fig. 3 shows simulated results for the optimal values of PA
factors for x1and x2at 20 dB transmit SNR for different values
of d1. It is noted that x1is in outage for all values of d1when

Fig. 4. OP vs. transmit SNR for x1and x2with optimal PA
factors when d1={1.5d, 3d, 4d}.
Table I. Optimal PA factors for different d1.
d11.5d3d4d
α1/β10.855 0.956 0.976
α2/β20.145 0.044 0.024
u > α1
α2. However, when u < α1
α2, an increase of α1improves
the OP of x1and decrease that of x2. Moreover, it is noticed
that when d1is increased, the OP results become worse. Finally,
the observed optimal PA factors for various values of d1are
illustrated in Table I.
Fig. 4 plots the results on the OP with optimal PA factors
when d1={1.5d, 3d, 4d}. Noticeably, in all d1cases, the OP
of x2performs better than that of x1at SNR levels below
20 dB. However, the OP of x1obtains better results than that
of x2at higher SNRs. Moreover, the OP curve of x1starts
to saturate at higher SNRs when d1is increased. It is due to
the fact that larger α1is needed when D1is located far away
from R. Finally, it can be deduced that the optimal PA factors
improved the OP performance of x1in all d1cases providing
fairness for both users.
In Fig. 5, we demonstrate the OP performance versus receive
SNR threshold for K= 5 users at 20 dB transmit SNR. The
distance between Rand Dk, with ∀k∈ {1, ..., K}, is given by
dk= 1.5K−kand the fixed PA factors are derived by setting
αk=βk=2K−kP
2K−1. From the plot, it can be noticed that the
OP of all messages degrades when the receive SNR threshold
raises. All messages, except x5, reach outage before 3dB, while
the OP of x5achieves the best results at all the SNR values by
achieving outage at 15 dB.
V. CONCLUSION
This paper derived the closed-form expressions for the end-
to-end OP of the DF cooperative underlay CR-NOMA network
Fig. 5. OP vs. receive SNR threshold for K= 5 users.
over Rayleigh fading with KSDUs. Simulation results val-
idated the accuracy of our performance analysis. Moreover,
we compared the OP results of messages for K={2; 5}
users. Furthermore, the considered cooperative NOMA was
superior compared with the cooperative OMA in terms of the
OP. Finally, from the numerical results, it is concluded that
the proper evaluation of PA factors for different distances can
guarantee fairness of the performance of NOMA users.
APPENDIX A
PROO F OF LE MM A 1
The CDF of the RV γR,i can be derived as
FγR,j (uj) =Pr "|ˆgSR |2αjρ´
S
|ˆgSR |2ρ´
SΘ + ζdξ
SR
< uj, ρ ´
S<∆#
+Pr "|ˆgSR |2α1∆
|ˆgSR |2∆Θ + ζdξ
SR
< uj, ρ ´
S>∆#
=Pr "|ˆgSR |2<ujζdξ
SR
µρ ´
S
,|gSP |2<ρIdξ
SP
ρ´
S#
| {z }
A1
+Pr "|ˆgSR |2
|gSP |2<ujζdξ
SR
µρIdξ
SP
,|gSP |2>ρIdξ
SP
ρ´
S#
| {z }
A2
,(24)
where ∆ = ρIdξ
SP
|gSP |2,µ=αj−ξjΘ,ρ´
S=´
PS
σ2denotes
the maximum allowed transmit SNR at Sand ρI=I
σ2
represents the temperature-constraint-to-noise ratio at P R. Note
that |gSP |2and |ˆgSR|2also follow exponential distribution with
unit mean and unit variance. In the term A1of (24), RVs |gSP |2
and |ˆgSR |2are independent from each other. Therefore, A1can

be further rewritten as
A1= 1−e−ujζdξ
SR
2µρ ´
S! 1−e−ρIdξ
SP
2ρ´
S!.(25)
Then, A2in (24) can be further formulated as
A2=Z∞
ρIdξ
SP
ρ´
S
f|gSP |2(y)Zyujζdξ
SR
µρIdξ
SP
0
f|ˆgSR |2(z)dzdy
=e−ρIdξ
SP
2ρ´
S−µρIdξ
SP e−µρIdξ
SP +ujζdξ
SR
2µρ ´
S
µρIdξ
SP +ujζdξ
SR
.(26)
Finally, by substituting (25) and (26) into (24), and after some
algebraic manipulations, the closed-form for the CDF of the
RV γR,j can be derived as in (14). 
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