1

Improved User Fairness in Decode-Forward

Relaying Non-orthogonal Multiple Access Schemes
with Imperfect SIC
Ferdi Kara, Member, IEEE, Hakan Kaya

Abstract—Non-orthogonal multiple access (NOMA) is one of coefficients [4]. The interference mitigation in PD-NOMA
the key technologies to serve in ultra-dense networks with is held by successive interference canceler (SIC) [5]. Due
arXiv:2004.12353v1 [cs.IT] 26 Apr 2020

massive connections which is crucial for Internet of Things (IoT) to its potential for 5G and beyond, NOMA1 has attracted
applications. Besides, NOMA provides better spectral efficiency
compared to orthogonal multiple access (OMA) schemes. How- tremendous attention from researchers where NOMA is widely
ever, in NOMA, successive interference canceler (SIC) should investigated mostly in terms of achievable rate/ergodic ca-
be implemented for interference mitigation and mostly in the pacity (EC) and outage probability (OP) [6]. Besides, since
literature, perfect SIC is assumed for NOMA involved systems. only superposition coding at the transmitter and SICs at the
Unfortunately, this is not the case for practical scenarios and this receivers are required, the integration of NOMA with other
imperfect SIC effect limits the performance of NOMA involved
systems. In addition, it causes unfairness between users. In this physical layer techniques such as cooperative communication,
paper, we introduce reversed decode-forward relaying NOMA (R- mm-wave communication, multi-input-multi-output (MIMO)
DFNOMA) to improve user fairness compared to conventional systems, visible light communication, etc. has also taken a
DFNOMA (C-DFNOMA) which is widely analyzed in literature. remarkable attention [7].
In the analysis, we define imperfect SIC effect dependant to
channel fading and with this imperfect SIC, we derive exact
expressions for ergodic capacity (EC) and outage probability A. Related Works and Motivation
(OP). Moreover, we evaluate bit error performance of proposed One of the most attracted topics is the interplay between
R-DFNOMA and derive bit error probability (BEP) in closed-
NOMA and cooperative communication which is held in
form which has not been also studied well in literature. Then,
we define user fairness index in terms of all key performance three major concepts: 1) cooperative-NOMA [8], [9] where
indicators (KPIs) (i.e., EC, OP and BEP). Based on extensive near users also act as relays for far user, 2) NOMA-based
simulations, all derived expressions are validated, and it is cooperative systems [10], [11] where NOMA is implemented
proved that proposed R-DFNOMA provides better user fairness to increase spectral efficiency of device-to-device commu-
than C-DFNOMA in terms of all KPIs. Finally, we discuss the
nication and 3) relay-assisted/aided-NOMA where relays in
effect of power allocations at the both source and relay on the
performance metrics and user fairness. the network help NOMA users to enhance coverage. This
paper focuses on relay-assisted-NOMA networks. The relay-
Index Terms—NOMA, DF relaying, user fairness, power allo-
assisted-NOMA networks have also been analyzed widely. In
cation, imperfect SIC
these works, an amplify-forward (AF) or a decode-forward
(DF) relay helps source/BS to transmit symbols to the NOMA
I. I NTRODUCTION users [12]. Hybrid DF/AF relaying strategies have been in-
vestigated to improve outage performance of relay-assisted-
I N recent years, exponential increase in connected devices
(i.e., smart-phones, tablets, watches etc.) [1] to the internet
and with the introduce of the Internet of Things (IoT), future
NOMA [13]. Outage and sum-rate performance of relay-
assisted-NOMA networks have also been analyzed whether a
radio networks (FRN) are keen to serve massive users in dense direct link between the source and the users exists [14] or not
networks which is called Massive Machine Type Commu- [15]–[17]. Then, relay-assisted-NOMA networks have been
nication (mMTC) -one of the three major concepts of 5G analyzed in terms of achievable rate and outage performance
and beyond- [2]. Non-orthogonal Multiple Access (NOMA) under different conditions such as buffer aided-relaying [18],
is seen as a strong candidate for mMTC in FRN due to [19], partial channel state information (CSI) at transmitter
its high spectral efficiency and ability to support massive [20] and imperfect CSI at receiver [21] when a single relay
connections [3]. In NOMA, users are assigned into same is located between the source and the users. In addition,
resource block to increase spectral efficiency and the most relay selection schemes have been investigated when multiple
attracted scheme is power domain (PD)-NOMA where users relays are available [22]–[25]. Relay selection schemes are
share the same resource block with different power allocation based on guaranteeing QoS of users and maximizing outage
performance of users. Moreover, two-way relaying strategies
This work is supported by the Scientific and Technological Research where relay operates as a coordinated multi-point (CoMP),
Council of Turkey (TUBITAK) under the 2211-E program
The authors are with Wireless Communication Technologies Labora-
have been investigated in terms of achievable rate and outage
tory (WCTLab), Department of Electrical and Electronics Engineering, performance [26]–[29].
Zonguldak Blent Ecevit University, Zonguldak, 67100, Turkey (e-mail:
{f.kara,hakan.kaya}@beun.edu.tr) 1 NOMA is used for PD-NOMA after this point.
 2

However, in aforementioned either conventional or relay-

assisted NOMA networks, mostly perfect SIC is assumed.
This is not a reasonable assumption when considered fading
channels. To the best of the authors’ knowledge, very limited
studies investigate NOMA involved system with imperfect
SIC. However, in those works, the imperfect SIC effect is
assumed to be independent from the channel fading [27],
[28], [30]. Thus, this strict assumption should be relaxed.
Besides, all the studies with imperfect SIC [27], [28], [30] Fig. 1: The illustration of R-DFNOMA
have been devoted to two-way relaying NOMA systems. To
the best of the authors’ knowledge, relay assisted NOMA
networks have not been analyzed with imperfect SIC effects. the benchmark. In this content, to the best of the authors’
Moreover, once the imperfect SIC is taken into consideration, knowledge, this is also the first study which provides an
it is shown that in downlink NOMA schemes, users encounter overall performance evaluation for any NOMA involved
a performance degradation in bit/symbol error rate (BER/SER) systems. All literature researches have biased on inves-
compared to orthogonal multiple access (OMA) though its tigations for only one or two performance metrics (e.g.,
performance gains in terms of EC and OP [31], [32]. Indeed, EC and/or OP).
this performance degradation may be more severe for one of • We define users fairness in terms of all KPIs (i.e., EC, OP
the users. Hence, the user fairness should be also considered and BEP). Based on extensive simulations, it is proved
in system design. Although this is raised in conventional that proposed R-DFNOMA provides better user fairness
downlink NOMA networks [33] and some studies are devoted compared to C-DFNOMA. Finally, we reveal the effect
to improve user fairness in conventional NOMA networks in of power allocation on user fairness and discuss optimum
terms of EC and OP [34]–[36], to the best of the authors’ power allocation
knowledge, user fairness in terms of BER/SER in conventional
NOMA has been taken into consideration. Moreover, this user C. Organization
unfairness becomes worse in relay-assisted-NOMA systems
due to the effects of two phases (e.g., from source-to-relay The remainder of this paper is as follows. In Section II,
and from relay-to-users). Besides all this, BER performance the proposed R-DFNOMA and the benchmark C-DFNOMA
of relay-assisted-NOMA networks has been only analyzed in schemes are introduced. The detection algorithms at the users
[37] though they have been widely-analyzed in terms of EC and the signal-to-interference plus noise ratio (SINR) defini-
and OP. User fairness also has not been considered for relay- tions are also provided in this section. Then, the performance
assisted-NOMA networks in terms of any key performance analysis for three KPIs (i.e., EC, OP, BER) are derived in Sec-
indicators (KPIs) (e.g., EC, OP, BER). To this end, we analyze tion III and the user fairness indexes for all KPIs are provided.
relay-assisted-NOMA network with imperfect SIC for all In Section IV, all derived expressions are validated via Monte
performance metrics. The user fairness has been also raised Carlo simulations. In addition, performance comparisons are
for relay-assisted-NOMA networks. also revealed in this section. Finally, results are discussed and
the paper is concluded in Section V.

B. Contributions
D. Notation
The main contributions of this paper are as follow:
The list of symbols, notations and abbreviations through this
• We introduce reversed DF relaying NOMA (R- paper is given in Table 1.
DFNOMA) to improve user fairness in conventional
DFNOMA (C-DFNOMA).
II. S YSTEM AND C HANNEL M ODEL
• For a more realistic/practical scenario, we re-define im-
perfect SIC effect as dependant to channel fading coeffi- A. Proposed: Reversed DF Relaying in NOMA
cient. The capacity and outage performances of proposed As shown in Fig. 1, a source (S) communicates with two
R-DFNOMA are investigated with this imperfect SIC destinations (i.e., D1 and D2 ) with the help of a relay (R). The
effect. The exact EC expressions are derived and closed- relay applies decode-forward (DF) strategy in a half-duplex
form upper bounds are provided for EC. Besides, exact mode, thus the total communication occupies two time slots.
OP expressions are derived in closed-forms. All derived We assume that direct links from source to destinations are
expressions match perfectly with simulations. not available due to the high path-loss effects and/or obstacles.
• Contrary to the most of the literature, we also analyze According to their average channel qualities between relay and
the error performance of R-DFNOMA rather than only destinations (i.e., R − D1 and R − D2 , users are defined as near
EC and/or OP performances. Exact bit error probability and far users. We assume that D1 has a better channel than
(BEP) expressions are provided in closed-forms and val- D2 to the relay node (R). In this case, D1 and D2 are denoted
idated via computer simulations. as near and far user, respectively and the system design is
• We evaluate the performances of proposed model in terms handled. In the first phase of communication, source (S)
of all KPIs (i.e., EC, OP and BEP) and compared with implements superposition coding for the base-band modulated
 3

TABLE I: List of Symbols, Notations and Abbreviations µ and τ are the propagation constant and path-loss exponent,
Pi Transmit power at node i = s, r respectively. ds,r is the Euclidean distance between the nodes.
αj Power allocation at the source for user j In (1), α1 and α2 are the power allocation coefficient for the
xj Modulated base-band (IQ) symbol of user j symbol of D1 and D2 , respectively. In order to improve user
Flat fading channel coefficient fairness, in R-DFNOMA, we propose to allocate α1 > α2 in
hi,k
between nodes i and k the first phase where α1 + α2 = 1. Unlike previous works,
ni Additive white Gaussian noise (AWGN) we propose to reverse power allocation coefficient in the first
µ Propagation constant phase (e.g., α1 > α2 ) and conventional power allocation
is proposed in the second phase (e.g., β2 > β1 -will be
τ Path loss exponent
defined below-) whereas in conventional DFNOMA schemes,
di,k Euclidean distance between i and k
they have performed same way in both phases -as defined in
∼ Follows/distributed
benchmark in the next subsection-. Thus, the proposed system
Complex Normal distribution with m mean
CN(m, σ 2 ) model is called as reversed-DFNOMA (R-DFNOMA). This
and σ2 variance in each component
2
reversed power allocation brings also reversed detecting order
|.| Absolute value in the first phase. Since more power is allocated to D1 symbols,
ρi Transmit signal-to-noise ratio (SNR) relay node (R) firstly detects x1 symbols by pretending x2
Signal-to-interference plus noise ratio (SINR) symbols as noise based on the received signal in the first phase.
SI N Ri,k
j for user j between nodes i and k The maximum-likelihood (ML) detection of x1 symbols at the
Absolute square for channel fading between relay is given
γi,k 2
nodes i and k ( hi,k ) p √ 2
Detected/estimated base-band (IQ) symbol x̂1 = argmin ys,r − Ps hs,r α1 x1,k (2)
x̂ j k
of user j at relay
Ξi Imperfect SIC effect coefficient at the node i where x1,k denotes the k th point in the M1 -ary constellation.
βj Power allocation at the relay for user j The received signal-to-interference plus noise ratio (SINR) for
Detected/estimated base-band (IQ) symbol of the x1 symbols at the relay is given by
x̃ j ρs α1 γs,r
user j at destination SI N R1(s,r) = (3)
Rj Achievable (Shannon) rate of user j ρs α2 γs,r + 1
Cj Ergodic capacity (EC) of user j 2
where ρs = Ps/N0 and γs,r = hs,r are defined. On the
Probability density function (PDF)
fZ (z) other hand, a successive interference canceler (SIC) should be
of random variable z
implemented at the relay to detect less-powered x2 symbols.
Cumulative distribution function (CDF)
FZ (z) The ML detection of x2 symbols at the relay is given as
of random variable z
R̂ j Target rate of user j (QoS requirement) 0 p √ 2
x̂2 = argmin ys,r − Ps hs,r α2 x2,k (4)
P j (out) Outage probability (OP) of user j k
e2e End-to-end where
0 p √
P(i,k) (e)
Bit error probability (BEP) of user j ys,r = ys,r − Ps hs,r α1 x̂1 (5)
j between nodes i and k
P j (e|γi,k ) Conditional BEP on γi,k and x2,k denotes the k th point in the M2 -ary constellation.
Marcum-Q function One can easily see that, the remaining signal after SIC highly
Q(.) ∫∞ √ depends on the detection of x1 symbols and unlike previous
Q(z) = 1/ 2π exp(−z 2/2)dz works, it is not reasonable to assume perfect SIC (e.g., no
z interference from x1 symbols). In addition, the interference
Coefficient A of user j between nodes i and k
A(i,k)
j (e) after SIC is a function of γs,r , Ps and α1 , thus the interference
in BEP analysis
cannot be assumed an independent random variable unlike
Proportional fairness index given in [27], [28], [30]. To this end, the SINR for x2 symbols
PFl (out)
l = c, o, e c →EC, o →OP and e →BEP at the relay is given as
ρs α2 γs,r
SI N R2(s,r) = (6)
Ξr ρs α1 γs,r + 1
symbols of destinations (i.e., x1 and x2 ) and transmits it to the
where Ξr defines the imperfect SIC effect coefficient (e.g.,
relay. The received signal by the relay is given as
Ξr = 0 for perfect SIC and Ξr = 1 for no SIC at all).
p √ √
yr = α1 x1 + α2 x2 + nr


Ps hs,r (1)
In the second phase of communication, relay node (R)
where Ps is the transmit power of source. hs,r and nr denote again implements superposition coding for detected x̂1 and x̂2
the complex flat fading channel coefficient between S − R and symbols and broadcasts this total symbol to the destinations.
the additive white
Gaussian noise (AWGN) at the relay. They The received signal by both destinations is given as
follow CN 0, σs,r2 and CN (0, N0 ), respectively. σs,r
2 includes
p p 
yi = Pr hr,i β1 x̂1 + β2 x̂2 + ni i = 1, 2
p
the large-scale fading effects and σs,r = µds,r is defined where
2 −τ (7)
 4

where Pr is the transmit power of relay 2 . hr,i and ni denote The SINRs in the first phase of communication are given as
the complex flat fading channel coefficient between R − Di ρs α2∗ γs,r
and the additive
 white Gaussian noise (AWGN) at the Di . hr,i SI N R2(s,r) = (16)
ρs α1∗ γs,r + 1
∼ CN 0, σr,i 2 and ni ∼ CN (0, N0 ), respectively. We assume
dr,2 ≥ dr,1 , hence D1 has better channel condition and more and
ρs α1∗ γs,r
power allocated to D2 -user with weaker channel condition-, SI N R1(s,r) = . (17)
Ξr ρs α2∗ γs,r + 1
(e.g., β2 > β1 ). Based on received signals, users implement
whether ML or SIC plus ML in order to detect their own The signal detections and the SINRs in the second phase of
symbols. Since more power is allocated for the symbols of C-DFNOMA are the same in R-DFNOMA.
D2 , D2 implements only ML by pretending D1 ’s symbols as
noise and it is given, III. P ERFORMANCE A NALYSIS
2
In this section, we analyze the proposed R-DFNOMA in
p
x̃2 = argmin yr,2 − P2 hr,2 β2 x2,k .
p
(8)
k terms of three KPIs (i.e., EC, OP and BEP) in order to evaluate
The received SINR at the D2 is given as its performance. Then, we define user fairness index for all
three KPIs.
ρr β2 γr,2
SI N R2(r,2) = . (9)
ρr β1 γr,2 + 1
A. Ergodic Capacity (EC)
where ρr = Pr /N0 is defined.
On the other hand, D1 implements SIC in order to detect its Since the proposed model includes a relaying strategy,
own symbols. Thus, it firstly detects x2 symbols and subtract its achievable rate is limited by the weakest link. Hence,
regenerated forms from received signal. The detection process considering both S − R and R − D1 links, the achievable
at the D1 is given as (Shannon) rate of D1 is given as
1 n    o
0 2
R1 = min log2 1 + SI N R1(s,r) , log2 1 + SI N R1(r,1) .
p
x̃1 = argmin yr,1 − Pr hr,1 β1 x1,k
p
(10) 2
k (18)
1
where where 2 exists since the total communication covers two
0 p
yr,1 = yr,1 − Pr hr,1 β2 x̃2 .
p
(11) time slots. The ergodic capacity (EC) of D1 is obtained by
averaging R1 over instantaneous SINRs in (3) and (12). It is
The received SINR after SIC at the D1 is given as given as
ρr β1 γr,1 1
SI N R1(r,1) = (12) C1 =
Ξ1 ρr β2 γr,1 + 1 2
where Ξ1 defines the imperfect SIC effect coefficient at the ∬∞  n o
D1 likewise in relay. log2 1 + min SI N R1(s,r), SI N R1(r,1) fγs, r fγr,1 dγs,r dγr,1
0
(19)
B. Benchmark: Conventional DF Relaying in NOMA
where fγs, r and fγr,1 are probability density functions (PDFs)
In conventional DF relay-aided NOMA (C-DFNOMA) of γs,r and γr,1 , respectively. Let define Z = min {X, Y }, the
schemes, detecting order at both relay and user are the same. cumulative density function (CDF) of Z is given by FZ (z) =
The power allocation in the first phase is arranged as α2∗ > α1∗ . 1 − (1 − FX (z)) (1 − FY (z)) where FX (.) and FY (.) are CDFs of
Hence, the relay node (R) firstly detects x2 symbols and ∫∞
X and Y , respectively [38]. Recalling, log2 (1 + x) fX (x)dx =
implements SIC to detect x1 symbols. To this end, given 0
detection algorithms and SINR definitions eq. (2)-(6) should ∫∞ 1−F
1 X (x)
be re-defined. The detection of x2 symbols at the relay is given ln2 1+x dx [39], with some algebraic manipulations, we
0
p q 2 derive EC of D1 as
x̂2 = argmin ys,r − Ps hs,r α2∗ x2,k (13)  
k z z
∫∞ exp − (α −α z)ρ −
s σs, r (β1 −β2 Ξ1 z)ρr σr,1
2 2
and of x1 symbols 1 1 2
C1 = dz
q 2 2ln2 1+z
+
p
x̂1 = argmin ys,r − Ps hs,r α1∗ x1,k (14) 0
k (20)
where q To the best of the authors’ knowledge, (20) cannot be solved
+
p
ys,r = ys,r − Ps hs,r α2∗ x̂2 . (15) in closed-form analytically. Nevertheless, it can be easily
computed by numerical tools. In addition, we can obtain it in
2 The relay can harvest its energy from received RF signal to transmit the closed-form for high SNR regime. To n this end, we assumeo
signals. However, in this paper this constraint has not been regarded and that ρs, ρr → ∞. In this case, Z = min SI N R1(s,r), SI N R1(r,1)
energy harvesting (EH) models such as linear and non-linear seen as future
in (19) turns out to be limρs ,ρr →∞ Z = min αα21 , Ξβ1 β1 2 . With
n o
researches.
 5

some algebraic simplifications, the upper bound for EC of D1 and

is given by P2 (out) =
1
C1 ≈ log2 η1 (21)
!
2 φ2 φ2
1 − exp − − .
(α2 − α1 Ξr φ2 ) ρs σs,r
2 (β2 − β1 φ2 ) ρr σr,2
2
where η1 = min αα12 , Ξβ1 β1 2 .
n o
(29)
Likewise in capacity analysis of D1 , the achievable rate of
D2 is given by
1 n    o
R2 = min log2 1 + SI N R2(s,r) , log2 1 + SI N R2(r,2) . C. Bit Error Probability (BEP)
2
(22)
Since a cooperative communication is included in R-
and taking the similar steps between (19)-(20), EC of D2 is
DFNOMA, the number of total erroneous bits from source
derived as
  to destination (i.e., end-to-end (e2e)) of users are given as
z z
∫ exp − (α −α Ξ z)ρ σ 2 − (β −β z)ρ σ 2
∞
Ni =
1 2 1 r s s, r 2 1 r r,2
C2 = dz. Ni (xi → x̂i ) + Ni ( x̂i → x̃i ) − Ni (xi → x̂i ) ∩ Ni ( x̂i → x̃i ) ,
2ln2 1+z
0 i = 1, 2
(23)
(30)
Likewise (20), (21) can be easily computed by numerical tools.
where Ni (xi → x̂i ) and Ni ( x̂i → x̃i ) denote the number of
Again in order to obtain upper bound for EC of D2 , if we
erroneous detected bits of Di in the first and second phases,
assume ρs, ρr → ∞, the EC is obtained as
respectively. If erroneous detections have been performed
1 in both phase, this means that correct detection has been
C2 ≈ log2 η2 (24)
2 achieved from source to destinations (e2e). Thus, the set of
β2 intersection of erroneous detections (3rd term) is subtracted
n o
α2
where η2 = min Ξr α1 , β1 .
in (30). Considering all combinations, the BEPs of Di are
given as in (31) (see top of the next page).
Recalling that Ni (xi → x̂i ) and Ni ( x̂i → x̃i ) events are
statistically independent, thus with the law of total probability,
BEPs of users are given as
B. Outage Probability (OP) Pi(e2e) (e) =
   
Pi(s,r) (e) 1 − Pi(r,i) (e) + 1 − Pi(s,r) (e) Pi(r,i) (e) i = 1, 2
The outage event for any user is defined as (32)
 
Pi (out) = P Ri < Ŕi i = 1, 2 (25) (s,r)
where Pi (e) = Mi log1 Mi k=1
Í Mi Í

2
∀l,k Ni xi,k → x̂i,l and
(r,i) M
Pi (e) = Mi log Mi l=1 ∀p,l Ni x̂i,l → x̃i, p denote the
1 Í i
Í

where Ŕi is the target rate of Di . By substituting (18) and (22) 2
into (25), OPs of users are derived as BEPs in the first and second phases, respectively. Thus, the
BEPs in each phases should be firstly derived. Each phase
Pi (out) = of communication can be considered separately. In the first
 
1 n 
(s,r)
 
(r,i)
o
phase of communication, it turns out to be a conventional
P min log2 1 + SI N Ri , log2 1 + SI N Ri < Ŕi
2 downlink NOMA system and the BEPs of x1 symbols will
i = 1, 2. be the same with BEP of far user in downlink NOMA. Since
(26) the superposition is applied, the BEP of far user in NOMA is
highly depended on the chosen constellation pairs (i.e., M1 and
With some algebraic manipulations, OPs of users are derived M2 ) [31], [32]. Nevertheless, the conditional BEP on channel
as conditions is given in the form,
Pi (out) = FZi (φi ) i = 1, 2 (27)
(s, r )
L1 q 
n φi = 2 =
where 2 Ŕi − 1 and F (.) CDF of Z Õ
(s,r) (r,i)
o Zi i P1(s,r) (e|γs, r ) = (s,r)
ς1,q Q (s,r)
2ν1,q ρs γs,r , (33)
min SI N Ri , SI N Ri i = 1, 2 are defined. Recalling q=1
CDF for minimum of two exponential random variables in
(20) and (23), OP of users are derived in the closed-forms as where L1(s,r) , ς1,q
(s,r) (s,r)
and ν1,q coefficients change according to
chosen modulation constellation pairs for x1 and x2 symbols
P1 (out) = [40, Table 1]. For instance, in case M1 = M2 = 4 is used
for both symbols (i.e., x1 and x2 ), L1(s,r) = 2, ς1,q
(s,r)
= 0.5 ∀q
!
φ1 φ1
1 − exp − − (s,r) √ √ 2
(α1 − α2 φ1 ) ρs σs,r (β1 − β2 Ξ1 φ1 ) ρr σr,1 and ν1,q = 2 α1 ∓ α2 (for proof see [41, Appendix A]).
1


2 2

(28) Then, recalling γs,r is exponentially distributed, with the aid

 6

 Mi Õ Õ
Õ 
1
Pi (e) = +





 Ni x i,k → x̂i,l Ni x̂i,l → x̃i, p − Ni x i,k → x̂i,l ∩ Ni x̂i,l → x̃i, p 

Mi log2 Mi 
 k=1 ∀l,k ∀p,l  (31)
 
i = 1, 2

of [42] the average BEP (ABEP) of x1 symbols in the first and

phase is obtained as, (r,1)
(r,1)
v
u
t ν (r,1) ρ σ 2
Õ ς1,q
L1 u
(r,1) 1,q r r,1 ª
P1 (e) = ®.
v ©
(s, r )
­1 − (40)
u
(s,r) t ν (s,r) ρ σ 2
Õ ς1,q
L1 u ­
2 (r,1)
(s,r)
P1 (e) =
©
­1 − 1,q s s,r ª
®. (34) q=1 1 + ν 1,q ρ σ 2 ®
r r,1
2 (s,r)
1 + ν1,q ρs σs,r
2 ®
­ « ¬
q=1
« ¬ where L1(r,1) , L2(s,r) , ς1,q
(r,1) (s,r)
, ς2,q (r,1)
and ν1,q (s,r)
, ν2,q .
On the other hand, x2 symbols in the first phase can be Lastly, substituting (34), (36), (38) and (40) into (32), the
considered as near user symbols in conventional downlink ABEPs of users are derived as in (41) and (42) (see top of the
NOMA, Thus, the conditional BEP should be derived consid- next page).
ering correct and erroneous SIC cases. After summing these
BEPs of two cases, the conditional BEP of x2 symbols in the
first phase is given in the form just as (33) D. User Fairness
(s, r )
L2
Õ q  In this subsection, we define fairness between users’ perfor-
P2(s,r) (e|γs, r ) = (s,r)
ς2,q Q (s,r)
2ν2,q ρs γs,r , (35) mances. In NOMA schemes, since the total power is allocated
q=1 between users, the users have different performances. Due
where L2(s,r) = 5, ς2,q (s,r) (s,r)
= 21 [2, 1, −1, −1, 1] and ν2,q = to the inter-user-interference and the SIC operation, one of
√ √ 2 √ √ 2 √ √ 2 √ √ 2 the users may have better performance than the other. This
α1 ( α2 − α1 ) ( α2 + α1 ) (2 α2 − α1 ) (2 α2 + α1 )
[2, 2 , 2 , 2 , 2 ] are given performance gap can be higher in some performance metrics
for M1 = M2 = 4 [41, Appendix A and B]. By averaging over (e.g. EC and BER).
instantaneous γs,r , the ABEP of x2 symbols in the first phase The performance gap between users should not be increased.
is derived as We use proportional fairness (PF) index to compare users’
performances for all KPIs. For instance, let we firstly consider
(s, r )
v
u
(s,r) u (s,r)
Õ ς2,q
L2 t ν2,q ρs σs,r 2
(s,r)
P2 (e) =
©
­ 1 − ®.
ª
(36) EC. In this case, if the fairness has not been considered,
2 (s,r)
1 + ν ρ σ 2 one of the users may achieve much more EC than the other.
­ ®
q=1 2,q s s,r
« ¬ To alleviate this unfair situation, PF index for EC should be
defined and it is given as
C1
PFc = (43)
C2
In the second phase of communication, more power is which can be easily obtained by substituting (20) and (23) into
allocated to x̂2 symbols. Thus, D2 implements a ML detection (43). One can easily see that optimum value for PFc can be
without SIC so the BEP of x2 symbols in the second phase considered as 1 which means that both user have exactly the
can be easily derived by using (33) as same EC. Nevertheless, this may not be achieved when the
(r,2)
users have different QoS requirements. Thus, fairness index
L2
should be obtained for other KPIs and all three should be
Õ q 
P2(r,2) (e|γr,2 ) = (r,2)
ς2,q Q (r,2)
2ν2,q ρr γr,2 , (37) evaluated together. To this end, fairness indexes for outage
t=1
and error performances are given as
where L2(r,2) , L1(s,r) , ς2,q
(r,2) (s,r)
, ς1,q and ν2,q(r,2) (s,r)
, ν1,q . By P1 (out)
using (34), (36), the ABEP is given as PFo = (44)
P2 (out)
(r,2)
v
u
Õ ς2,q
L2 (r,2) t ν (r,2) ρ σ 2
u and
(r,2) 2,q r r,2 ª P1 (e)
P2 (e) = ®.
©
­1 −
(r,2)
(38) PFe = (45)
q=1
2 ­
1 + ν2,q ρr σr,2 2 ® P2 (e)
which can be computed by substituting (28), (29) into (44)
« ¬
Likewise, the BEP of x1 symbols in the second phase can be and (41), (42) into (45), respectively. It is again clear that the
easily obtained by repeating steps (35), (35). The conditional optimal values for PFo and PFe are also 1. However likewise
BEP and the ABEP are given as in PFc , it may not be always achieved due to the priority in
(r,1)
L1 q  QoS requirements of users. It is noteworthy that in the PF
index for all KPIs, κ and 1/κ have the same meaning. For
Õ
P1(r,1) (e|γr,1 ) = (r,1)
ς1,q Q (r,1)
2ν1,q ρr γr,1 , (39)
q=1 instance, if the PF index for any performance metric has 2
 7

(s, r )
v
u (r,1)
v
u
(s,r) t ν (s,r) ρ σ 2 (r,1) t ν (r,1) ρ σ 2
Õ ς1,q
L1 L1
ς
u  u 
(e2e) 1,q s s,r
 Õ 1,q ­ 1,q r r,1 ª
P1 (e) =
© ª ©
1− 1 − 1−
­ ®  ®
(s,r) (r,1)

q=1
2 ­ 1 + ν1,q ρs σs,r
2 ®
q=1
2 ­ 1 + ν1,q ρr σr,1
2 ®
 
«
(s, r )
v
u
¬ «
 L (r,1) (r,1) v
u
¬ (41)
(s,r) t ν (s,r) ρ σ 2 t ν (r,1) ρ σ 2
Õ ς1,q
L1
ς1,q ©
 u u
1

1,q s s,r ª Õ 1,q r r,1 ª
+ 1 −
©
 ­1 − ® ­1 − ®
2 (s,r) 2 (r,1)
1 + ν1,q ρs σs,r2 1 + ν1,q ρr σr,1 2 ®
 ­ ®  ­
 q=1  q=1
 « ¬ « ¬

(s, r )
v
u (r,2)
v
u
(s,r) t ν (s,r) ρ σ 2 (r,2) t ν (r,2) ρ σ 2
Õ ς2,q
L2 L2
ς
u  u 
(e2e) 2,q s s,r  Õ 2,q 2,q r r,2 ª
P2 (e) =
© ª ©
­1 − ® 1 −
 ­1 − ®
2 (s,r) 2 (r,2)
1 + ν ρ σ 2 ® 1 + ν ρ σ 2 ®
­ ­
q=1 2,q s s,r  q=1 2,q r r,2 
«
(s, r )
v
u
¬
(r,2)
«
v
u
¬ (42)
(s,r) (s,r) (r,2) (r,2)
L
ς2,q © ν2,q ρs σs,r 2 L
ς2,q © ν2,q ρr σr,2 2
 u  u
2 2
 t t
Õ ª Õ
+ 1 −
ª
 ­1 − ® ­1 − ®
2 (s,r) 2 (r,2)
1 + ν ρ σ 2 1 + ν ρ σ 2
 ­ ®  ­ ®
 q=1 2,q s s,r  q=1 2,q r r,2
 « ¬ « ¬

and/or 0.5, this means that one of the users has two times Theo. exact
3
better performance than the other. Theo. upper bound
sim C1 r=-10dB
sim C2 =-10dB
2.5 r
IV. P ERFORMANCE E VALUATION Ergodic capacity (bps/Hz)
sim Csum r
=-10dB
sim C1 r
=-15dB
In this section, we provide validation of the provided 2 sim C2 r
=-15dB
analysis in the previous sections. In addition, we present user sim Csum r
=-15dB

fairness comparisons between proposed R-DFNOMA and C- 1.5

DFNOMA3 . In all simulations, we assume that µ = 10 and

τ = 2. The transmit power of source and relay are assumed 1

to be equal (i.e., Ps = Pr ). In validations of R-DFNOMA,

unless otherwise stated, curves denote theoretical analysis4 0.5

and simulations are demonstrated by markers. Moreover, in

0
all simulations, the imperfect SIC effect coefficients at the 0 5 10 15 20 25 30 35 40
both nodes are assumed to be equal (i.e., Ξr = Ξ1 ). Transmit SNR (dB)

Fig. 2: EC of R-DFNOMA vs ρs when α1 = 0.9, β1 = 0.2,

A. The Effect of Imperfect SIC ds,r = 5, dr,1 = 1 and dr,2 = 3
In this subsection, the distances between the nodes are
assumed to be ds,r = 5, dr,1 = 1 and dr,2 = 3. It can be
100
seen from following figures that all derived expressions match
perfectly with simulations.
In Fig. 2, EC of users and the ergodic sum-rate of the R-
10-1
DFNOMA (Csum = C1 + C2 ) are given for various imperfect
SIC effects. Power allocations are assumed to be α1 = 0.9,
β1 = 0.2. As it is expected, imperfect SIC limits the per-
Outage

10-2
formance of the systems and when it gets higher (i.e., Ξr =
Ξ1 = −10dB), EC of R-DFNOMA becomes worse. The power
allocation at the source and relay are chosen as different values
10-3
for better illustration, otherwise both users’ upper bound would
be the same. In Fig. 3, outage performances of the users are
presented for the same power allocation coefficients. Target 10-4
rates of the users are chosen as Ŕ1 = 0.2, Ŕ2 = 0.1 and 0 5 10 15 20 25
Transmit SNR (dB)
30 35 40

Ŕ1 = 0.75, Ŕ2 = 0.5 as two different QoS requirements. In all

NOMA involved systems, the outage performances of the users Fig. 3: OP of R-DFNOMA vs ρs when α1 = 0.9, β1 = 0.2,
ds,r = 5, dr,1 = 1 and dr,2 = 3
3 In C-DFNOMA, power allocation in the first phase is complement of the
power allocation of R-DFNOMA (i.e., α1∗ = 1 − α1 )
4 In numerical integration for exact EC, the infinity in the upper bounds of
the integrals is changed with 103 not to cause numerical calculation errors. get better with low QoS requirements (e.g., lower target rates).
 8

Likewise, in EC performance, imperfect SIC has a dominant

100
effect on outage performances of users and it may cause users
to be in outage with higher QoS.
10-1

Bit Error Rate (BER)

2.5 C1 , 1
=0.8, 1
=0.2
C2 , 1
=0.8, 1
=0.2
Theo.
Csum, =0.8, =0.2
1 1 D1, 1=0.9, 1=0.2
C1 , =0.9, =0.1 10-2
2 1 1 D2, 1=0.9, 1=0.2
C2 , =0.9, =0.1
Ergodic Capacity (bps/Hz)

1 1
D 1, 1
=0.8, 1=0.1
Csum, 1
=0.9, 1
=0.1
D 2, 1
=0.8, 1=0.1
1.5
10-3

1
10-4
0 5 10 15 20 25 30 35 40
Transmit SNR (dB)
0.5

Fig. 6: BER of R-DFNOMA vs ρs when ds,r = 5, dr,1 = 1

0
and dr,2 = 3
-30 -25 -20 -15 -10 -5 0
r
(dB)

Fig. 4: The effect of imperfect SIC on EC of R-DFNOMA tions, since an actual modulation/demodulation (i.e., QPSK for
when ds,r = 5, dr,1 = 1, dr,2 = 3 and ρs = 20dB Mi = 4) is implemented contrary to EC and OP simulations,
imperfect SIC effect coefficient has not been defined. The
erroneous detected symbols during SIC process will show this
100 effect. Thus, we present simulations for two different power
Dashed Lines: 1=0.8, 1=0.2
Solid Lines: =0.9, 1=0.1
allocation pairs (i.e., α1 = 0.8, β1 = 0.2 and α1 = 0.9,
β1 = 0.1). Just as in previous validations, it is clearly seen
1

in Fig. 6 that derived expressions are perfectly-matched with

simulations.
Outage

10-1
3.5 C-DFNOMA, =-10dB, =0.9, =0.1
r 1 1
R-DFNOMA, r
=-10dB, 1
=0.9, 1
=0.1

3 C-DFNOMA, r
=-10dB, 1
=0.8, 1
=0.2
R-DFNOMA, =-10dB, =0.8, =0.2
Capacity Fairness Index (PFc)

r 1 1
C-DFNOMA, r
=-15dB, 1
=0.9, 1
=0.1
2.5
R-DFNOMA, r
=-15dB, 1
=0.9, 1
=0.1
C-DFNOMA, r
=-15dB, 1
=0.8, 1
=0.2
-2
10 2
-30 -25 -20 -15 -10 -5 0 R-DFNOMA, r
=-15dB, 1
=0.8, 1
=0.2

r
(dB)
1.5
Fig. 5: The effect of imperfect SIC on outage of R-DFNOMA
when ds,r = 5, dr,1 = 1, dr,2 = 3 and ρs = 20dB 1

In order to further investigate the effect of imperfect SIC, 0.5

we present capacity and outage performance of users with the

0
change of imperfect SIC effect coefficient in Fig. 4 and Fig. 0 5 10 15 20 25 30 35 40
Transmit SNR (dB)
5, respectively. The power allocation coefficients at the source
and the relay are assumed to be α1 = 0.8, β1 = 0.2 and Fig. 7: Capacity Fairness Index (PFc ) vs ρs when ds,r = 5
α1 = 0.9, β1 = 0.1. The result are presented for ρs = 20dB. and dr,1 = dr,2 = 2
In bots figures, as expected, with the increase of imperfect
SIC effect, users have worse performance in both EC and
OP. In Fig. 5, the system will have almost 0.5 capacity if
the imperfect SIC effect is higher than −5dB that is too poor B. User Fairness
performance for 20dB SNR. In Fig. 6, for the same imperfect Contrary to commonly belief, NOMA users do not need
channel conditions, users are always in outage for all target to have different channel conditions (stronger and weaker).
rates. If more strict QoS requirements are needed (higher target NOMA users can be chosen among users with similar channel
rates), users will be in outage even for lower imperfect SIC conditions. In this case, user fairness turns out to be more
effects (e.g., between −10dB and −5dB). Nevertheless, all important, since none of them should be served with lower KPI
these are expected results for imperfect SIC case and represent [43], [44]. Thus, user fairness comparisons are more mean-
more practical/reasonable scenarios than perfect SIC. ingful when the users experience similar channel conditions.
In Fig. 6, we present error performances of users in R- To this end, we provide PF index comparisons between R-
DFNOMA for M1 = M2 = 4. In error performance simula- DFNOMA and C-DFNOMA in Fig. 7- Fig. 9 when ds,r = 5
 9

than D1 .5 In addition, it is obvious that R-DFNOMA provides

5
optimum user fairness in high SNR regimes (PFc → 1)
4.5
whereas it cannot be achieved in C-DFNOMA. Once user
4 fairness is considered in terms of outage performance, the
Outage Fairness Index (PFo)

3.5 improvement by proposed R-DFNOMA becomes outstanding

3 C-DFNOMA, r
=-10dB, 1
=0.9, 1
=0.1 in Fig. 8. In all considered scenarios, R-DFNOMA achieves
2.5
R-DFNOMA, r
=-10dB, 1
=0.9, 1
=0.1 about PFo = 0.5 for all SNR region which means D1 has 2
C-DFNOMA, =-10dB, =0.8, =0.2
R-DFNOMA,
r
=-10dB,
1
=0.8,
1
=0.2
times better OP than D2 . However, in C-DFNOMA, with the
2 r 1 1
C-DFNOMA, r
=-15dB, 1
=0.9, 1
=0.1 increase of transmit SNR, user unfairness gets worse. Indeed,
1.5 R-DFNOMA, =-15dB, =0.9, =0.1
C-DFNOMA,
r
=-15dB,
1
=0.8,
1
=0.2
in some scenarios user fairness becomes atrocious with the
increase of SNR. For instance, when α1 = 0.9, β1 = 0.1 and
r 1 1
1
R-DFNOMA, r
=-15dB, 1
=0.8, 1
=0.2

0.5 Ξr = −10dB, PFo → 80 at SN R → 40dB which means

0 D2 has 80 times better OP than D1 although they have same
0 5 10 15 20 25 30 35 40
Transmit SNR (dB)
channel conditions. Likewise user fairness comparisons for EC
and OP, proposed R-DFNOMA is superior to C-DFNOMA
Fig. 8: Outage Fairness Index (PFo ) vs ρs when ds,r = 5, also in terms of error performance in Fig. 9. Furthermore, this
dr,1 = dr,2 = 2 and Ŕ1 = Ŕ2 = 0.5 improvement is significant in some scenarios. For instance,
when α1 = 0.9, β1 = 0.1, with the increase of SNR
(ρs → 40dB), PFe → 2 in R-DFNOMA whereas PFe → 7
in C-DFNOMA which is a superb gain for data reliability of
5 D2 .
4.5 If the comparisons between C-DFNOMA and proposed
C-DFNOMA, =0.9, =0.1
1 1
R-DFNOMA are studied only in terms of fairness indexes
Error Fairness Index (PFe)

4
R-DFNOMA, 1
=0.9, 1
=0.1

3.5 C-DFNOMA, 1
=0.8, 1
=0.2 (e.g., Fig. 7- Fig. 9), the questions may be raised that: Are
R-DFNOMA, =0.8, =0.2
3
1 1 the user fairness indexes be improved by degrading both of
users’ performances? or does the R-DFNOMA has worse
2.5
performance? In order to resolve this concern and to prove
2
that R-DFNOMA achieves better user fairness indexes without
1.5 degrading users’ individual and overall performances, we
1 present performance comparison between C-DFNOMA and R-
0.5 DFNOMA in Fig. 10. The channel conditions are assumed to
0
be same with above comparisons and the power allocations
0 5 10 15 20 25
Transmit SNR (dB)
30 35 40
are fixed as α1 = 0.8, β1 = 0.2. In capacity and outage
comparisons in Fig. 10.a and Fig. 10.b, The imperfect SIC
Fig. 9: Error Fairness Index (PFe ) vs ρs when ds,r = 5 and effect coefficient is Ξr = −15dB and the target rates of users
dr,1 = dr,2 = 2 are Ŕ1 = Ŕ2 = 0.5. As can be easily from Fig. 10 that in R-
DFNOMA, users performances for all KPIs (i.e., EC, OP and
BEP) get closer so that user fairness indexes are improved.
Besides, when this improvement is achieved, proposed R-
DFNOMA does not cause performance decay in either users’
and dr,1 = dr,2 = 2. In comparisons, two different power
or in overall system performances. When the users have similar
allocations scenarios are set α1 = 0.9, β1 = 0.1 and α1 = 0.8,
channel conditions, the overall system performance is limited
β1 = 0.2, respectively. In Fig. 7 and Fig. 8, imperfect SIC
by the minimum performance achieved within users. To this
effect coefficients are set Ξr = −10dB and Ξr = −15dB.
end,in order evaluate both system (e.g., C-DFNOMA and R-
Since the users have similar channel conditions, their target
DFNOMA), we can define performance metrics as
rates are chosen as Ŕ1 = Ŕ2 = 0.5 for PFo . In Fig. 9,
modulation orders are set M1 = M2 = 4. One can easily see C (m) = min {C1, C2 } , m : C − DF NOM A or R − DF NOM A
that proposed R-DFNOMA provides better user fairness than P(m) (out) = max {P1 (out), P2 (out)} ,
C-DFNOMA in terms of all KPIs. In proposed R-DFNOMA,
users’ performance orders are also reversed. In terms of EC P(m) (e) = max {P1 (e), P2 (e)} .
in Fig. 7, D1 has higher EC than D2 in R-DFNOMA, hence (46)
the fairness index is higher than 1 whereas it is vice-versa and the performance comparisons between R-DFNOMA and
in C-DFNOMA. Nevertheless, R-DFNOMA outperforms C-
DFNOMA considering user fairness. For instance, when α1 =
0.9, β1 = 0.1 and Ξr = −10dB, PFc = 1.545 at 15dB SNR in
R-DFNOMA which means D1 has 1.545 times better EC than 5 As discussed in the previous sections, we hereby again note that PF index
D2 at 15dB. However, for the same conditions, PFc = 0.2433 of any performance metrics have the same meaning for κ and 1/κ . It only
in C-DFNOMA, which means D2 has 4.11 times better EC defines which user has better performance.
 10

C1, C-DFNOMA
3 C2, C-DFNOMA 100 100
Csum, C-DFNOMA
C1, R-DFNOMA
2.5
C2, R-DFNOMA

Ergodic Capacity (bps/Hz) Csum, R-DFNOMA 10-1 10-1

Bit Error Rate (BER)

2

Outage
1.5 10-2 10-2

1
D1, C-DFNOMA D1, C-DFNOMA
10-3 10-3
D2, C-DFNOMA D2, C-DFNOMA
0.5 D1, R-DFNOMA
D1, R-DFNOMA
D2, R-DFNOMA D2, R-DFNOMA

0 10-4 10-4
0 10 20 30 40 0 10 20 30 40 0 10 20 30 40
Transmit SNR (dB) Transmit SNR (dB) Transmit SNR (dB)

Fig. 10: Performance Comparisons between C-DFNOMA and R-DFNOMA vs ρs when ds,r = 5, dr,1 = dr,2 = 2, α1 = 0.8 and
β1 = 0.2 a) Capacity when Ξr = −15dB b) Outage when Ξr = −15dB and Ŕ1 = Ŕ2 = 0.5 c) Bit Error Rate

C-DFNOMA are given by

n o
max C (C−DF N OM A), C (R−DF N OM A) , 2 2
n o
Ergodic Capacity (bps/Hz)

min P(C−DF N OM A) (out), P(R−DF N OM A) (out) , (47)

Ergodic Capacity (bps/Hz)

1.5
n o 1.5
min P(C−DF N OM A) (e), P(R−DF N OM A) (e) .
1
Considering performance comparisons in (46)-(47), we can 1

see from Fig. 10 that, C-DFNOMA and R-DFNOMA has 0.5

the same performance in terms of theoretical Shannon rate 0.5

(e.g., Fig. 10.a) where minimum of users’ rates are the 0
1
same in C-DFNOMA and R-DFNOMA. Although it seems
that D1 in C-DFNOMA may have higher achievable rate 0 0
0.8 0.4
in high SNR region, this should be jointly evaluated with 1
0.2 1
0.6 0.2
1
0.6 0.8
performance of SIC receivers. It is proved in [45] that when 0 1 1
0.4

the higher modulation orders (mean higher achievable rate) are

implemented, none of the users’ symbols cannot be detected Fig. 11: EC performance in R-DFNOMA vs α1 and β1 when
and all users have 0.5 BER performance. On the other hand, ds,r = 5, dr,1 = dr,2 = 2, ρs = 30dB and Ξr = −10dB a) D1
R-DFNOMA outperforms C-DFNOMA in terms of outage and b) D2
BER performances. In Fig. 10.b, D2 has the maximum outage
probability in R-DFNOMA (performance limits as given in
(46)) whereas D1 has the maximum in C-DFNOMA and D2 is Ξr = −10dB assumed in EC and OP comparisons and
in R-DFNOMA has lower OP (better performance as given transmit SNR ρs = 30dB is assumed in all figures. Target
in (47)) than D1 in C-DFNOMA. Once the same evaluations rates of users are assumed to be equal to Ŕ1 = Ŕ2 = 0.2.
have been discussed for BER performances, we can easily M1 = M2 = 4 is set in BER simulations. As expected, both
see that D2 in R-DFNOMA (maximum in R-DFNOMA) has power allocation coefficients (source and relay) have reverse
lower BER (better performance as given in (47)) than D1 in effects on the performances of users. Increasing/decreasing α1
C-DFNOMA (maximum in C-DFNOMA). and/or β1 provides performance gain for one of the user and
causes a decay for the other vice-versa. In terms of EC in
C. The Effect of Power Allocations Fig. 11, increase in α1 and/or β1 provide better EC for D1
From above simulations and discussions, it can bee seen and lower EC for D2 . The same discussions are also valid
that power allocations at the both nodes (source and relay) for outage performances of users in Fig. 11. Nevertheless, it
have a remarkable effect on the performance of R-DFNOMA is noteworthy that increasing/decreasing any power allocation
so that on the user fairness. Thus, in order to reveal this coefficient too much causes one of the users to be in always
effect, we present EC, OP and BER performances of users outage. The same performance trends can be seen for also BER
with the respect to power allocations (α1 and β1 ) in Fig. in Fig. 13. However, since an actual modulation/demodulation
11 - Fig. 13. The channel conditions are assumed to be is implemented, the effect of power allocation on the SIC
ds,r = 5 and dr,1 = dr,2 = 2. The imperfect SIC effect performance is more clear in Fig. 13. For instance, increasing
 11

DFNOMA, R-DFNOMA provides better user fairness (close

to 1), whereas user fairness in C-DFNOMA is close to 1.5,
for most of power the allocation pairs. In Fig. 15, user fairness
100 is presented in terms of OP when the users have same QoS
100
requirement ( Ŕ1 = Ŕ2 = 0.2). It can bee seen that users in
10-1 R-DFNOMA have similar outage performance even if power
Outage

10-1 allocation coefficients change. In R-DFNOMA, one of the

Outage
10-2 users may have maximum 6 times better outage performance
10-2 than the other. However, this unfairness may raise to 50 times
10-3
better performance in C-DFNOMA. Lastly, user fairness in
0
terms of error performance is presented in Fig. 16. Similar
0 10-3
0.6 discussions can be seen for also user fairness in terms of BER
0.2
0.6
1 0.8
0.2 1 from Fig. 16. The fairness index may have the maximum 3.5
1
0.4 1 0.8 0.4 in R-DFNOMA whereas the value of 80 in C-DFNOMA. This
1 1 shows that users in R-DFNOMA have similar data reliability,
Fig. 12: Outage performance in R-DFNOMA vs α1 and β1 however in C-DFNOMA, one of the users may outperform
when ds,r = 5, dr,1 = dr,2 = 2, ρs = 30dB, Ξr = −10dB and 80 times the other. Based on the results between Fig. 14 and
Ŕ1 = Ŕ2 = 0.2 a) D1 b) D2 Fig. 16, R-DFNOMA is much more robust to change in power
allocation coefficients in terms of user fairness. For the given
channel conditions, by considering the user fairness in terms of
all KPIs, the optimum power allocation pairs in R-DFNOMA
could be defined α1 ≈ 0.85 and β1 ≈ 0.15 where users have
100 same performance so that user fairness becomes very close to
1.
10-1
10-1 1 0.5
BER
BER

0.9 0.4
10-2 10-2

0.8 0.3
1

1
*
10-3
10-3 1
0.7 0.2
1 0.4
0.8 0.4
0.8 0.2 0.6 0.1
0.2
1 0.6 1 1 0.6 1
0 0
0.5 0
Fig. 13: BER performance in R-DFNOMA vs α1 and β1 when 0 0.1 0.2 0.3 0.4 0 0.1 0.2 0.3 0.4

ds,r = 5, dr,1 = dr,2 = 2, ρs = 30dB a) D1 b) D2 1 1

1 2 3 4 5 0.5 1 1.5 2 2.5

β1 means that higher power is allocated to D1 symbols in the Fig. 14: Capacity Fairness Index (PFc ) vs α1 and β1 when
second phase thereby better error performance is expected. ds,r = 5, dr,1 = dr,2 = 2, ρs = 30dB and Ξr = −10dB a)
Nevertheless, one can easily see that increasing β1 too much R-DFNOMA b) C-DFNOMA
causes a decay in error performance of D1 along with error
performance of D2 . This is explained as follows. Since the
D1 has to implement SIC in the second phase, increasing β1 V. C ONCLUSION
causes erroneous SIC more likely and this pulls down the error In this paper, we introduce reversed decode-forward relay-
performance of D1 . Based on above discussions, it is clear ing NOMA (R-DFNOMA) to improve user fairness in conven-
that power allocation affects users’ performances reversely tional DFNOMA (C-DFNOMA). We consider imperfect SIC
(gain for one and decay for the other), thus it is not possible at detections and according to imperfect SIC effect and we re-
to define a optimum power allocation which offers the best define SINRs at the nodes to provide a more practical scenario.
performances for both users. Nevertheless, by considering user With the imperfect SIC, we investigate the performance of
fairness, a sub-optimum power allocation can be obtained. proposed R-DFNOMA and derive closed-form expressions
To this end, user fairness comparisons between R- for ergodic capacity (EC), outage probability (OP) and bit
DFNOMA and C-DFNOMA with respect to power allocations error probability (BEP). Then, in order to emphasize user
(α1 and β1 ) in Fig. 14 - Fig. 16 for the same conditions fairness, we define fairness indexes for all KPIs (i.e., EC, OP
above comparisons. Based on user fairness for EC in Fig. and BEP). With the extensive simulations, derived expressions
14, although PFc changes within larger range (max. 5) in R- are validated and it is proved that R-DFNOMA outperforms
 12

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