# Improved User Fairness in Decode-Forward Relaying Non-orthogonal Multiple
Access Schemes with Imperfect SIC

Ferdi Kara, Hakan Kaya  This work is supported by the Scientific and
Technological Research Council of Turkey (TUBITAK) under the 2211-E programThe
authors are with Wireless Communication Technologies Laboratory (WCTLab),
Department of Electrical and Electronics Engineering, Zonguldak Bülent Ecevit
University, Zonguldak, 67100, Turkey (e-mail<EMAIL_ADDRESS>
###### Abstract

Non-orthogonal multiple access (NOMA) is one of the key technologies to serve
in ultra-dense networks with massive connections which is crucial for Internet
of Things (IoT) applications. Besides, NOMA provides better spectral
efficiency compared to orthogonal multiple access (OMA) schemes. However, in
NOMA, successive interference canceler (SIC) should be implemented for
interference mitigation and mostly in the literature, perfect SIC is assumed
for NOMA involved systems. Unfortunately, this is not the case for practical
scenarios and this imperfect SIC effect limits the performance of NOMA
involved systems. In addition, it causes unfairness between users. In this
paper, we introduce reversed decode-forward relaying NOMA (R-DFNOMA) to
improve user fairness compared to conventional DFNOMA (C-DFNOMA) which is
widely analyzed in literature. 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 (OP). Moreover,
we evaluate bit error performance of proposed R-DFNOMA and derive bit error
probability (BEP) in closed-form which has not been also studied well in
literature. Then, we define user fairness index in terms of all key
performance indicators (KPIs) (i.e., EC, OP and BEP). Based on extensive
simulations, all derived expressions are validated, and it is proved that
proposed R-DFNOMA provides better user fairness than C-DFNOMA in terms of all
KPIs. Finally, we discuss the effect of power allocations at the both source
and relay on the performance metrics and user fairness.

###### Index Terms:

NOMA, DF relaying, user fairness, power allocation, imperfect SIC

##  I Introduction

In 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 radio networks (FRN) are keen to serve
massive users in dense networks which is called Massive Machine Type
Communication (mMTC) -one of the three major concepts of 5G and beyond- [2].
Non-orthogonal Multiple Access (NOMA) is seen as a strong candidate for mMTC
in FRN due to its high spectral efficiency and ability to support massive
connections [3]. In NOMA, users are assigned into same resource block to
increase spectral efficiency and the most attracted scheme is power domain
(PD)-NOMA where users share the same resource block with different power
allocation coefficients [4]. The interference mitigation in PD-NOMA is held by
successive interference canceler (SIC) [5]. Due to its potential for 5G and
beyond, NOMA111NOMA is used for PD-NOMA after this point. has attracted
tremendous attention from researchers where NOMA is widely investigated mostly
in terms of achievable rate/ergodic capacity (EC) and outage probability (OP)
[6]. Besides, since only superposition coding at the transmitter and SICs at
the receivers are required, the integration of NOMA with other physical layer
techniques such as cooperative communication, mm-wave communication, multi-
input-multi-output (MIMO) systems, visible light communication, etc. has also
taken a remarkable attention [7].

###  I-A Related Works and Motivation

One of the most attracted topics is the interplay between NOMA and cooperative
communication which is held in three major concepts: 1) cooperative-NOMA [8,
9] where near users also act as relays for far user, 2) NOMA-based cooperative
systems [10, 11] where NOMA is implemented to increase spectral efficiency of
device-to-device communication and 3) relay-assisted/aided-NOMA where relays
in the network help NOMA users to enhance coverage. This paper focuses on
relay-assisted-NOMA networks. The relay-assisted-NOMA networks have also been
analyzed widely. In these works, an amplify-forward (AF) or a decode-forward
(DF) relay helps source/BS to transmit symbols to the NOMA users [12]. Hybrid
DF/AF relaying strategies have been investigated to improve outage performance
of relay-assisted-NOMA [13]. Outage and sum-rate performance of relay-
assisted-NOMA networks have also been analyzed whether a direct link between
the source and the users exists [14] or not [15, 16, 17]. Then, relay-
assisted-NOMA networks have been analyzed in terms of achievable rate and
outage performance under different conditions such as buffer aided-relaying
[18, 19], partial channel state information (CSI) at transmitter [20] and
imperfect CSI at receiver [21] when a single relay is located between the
source and the users. In addition, relay selection schemes have been
investigated when multiple relays are available [22, 23, 24, 25]. Relay
selection schemes are based on guaranteeing QoS of users and maximizing outage
performance of users. Moreover, two-way relaying strategies where relay
operates as a coordinated multi-point (CoMP), have been investigated in terms
of achievable rate and outage performance [26, 27, 28, 29].

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] 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.
Moreover, once the imperfect SIC is taken into consideration, it is shown that
in downlink NOMA schemes, users encounter a performance degradation in
bit/symbol error rate (BER/SER) compared to orthogonal multiple access (OMA)
though its performance gains in terms of EC and OP [31, 32]. Indeed, this
performance degradation may be more severe for one of the users. Hence, the
user fairness should be also considered in system design. Although this is
raised in conventional downlink NOMA networks [33] and some studies are
devoted to improve user fairness in conventional NOMA networks in terms of EC
and OP [34, 35, 36], to the best of the authors’ knowledge, user fairness in
terms of BER/SER in conventional NOMA has been taken into consideration.
Moreover, this user unfairness becomes worse in relay-assisted-NOMA systems
due to the effects of two phases (e.g., from source-to-relay and from relay-
to-users). Besides all this, BER performance of relay-assisted-NOMA networks
has been only analyzed in [37] though they have been widely-analyzed in terms
of EC and OP. User fairness also has not been considered for relay-assisted-
NOMA networks in terms of any key performance indicators (KPIs) (e.g., EC, OP,
BER). To this end, we analyze relay-assisted-NOMA network with imperfect SIC
for all performance metrics. The user fairness has been also raised for relay-
assisted-NOMA networks.

###  I-B Contributions

The main contributions of this paper are as follow:

  * •

We introduce reversed DF relaying NOMA (R-DFNOMA) to improve user fairness in
conventional DFNOMA (C-DFNOMA).

  * •

For a more realistic/practical scenario, we re-define imperfect SIC effect as
dependant to channel fading coefficient. The capacity and outage performances
of proposed R-DFNOMA are investigated with this imperfect SIC effect. The
exact EC expressions are derived and closed-form upper bounds are provided for
EC. Besides, exact OP expressions are derived in closed-forms. All derived
expressions match perfectly with simulations.

  * •

Contrary to the most of the literature, we also analyze the error performance
of R-DFNOMA rather than only EC and/or OP performances. Exact bit error
probability (BEP) expressions are provided in closed-forms and validated via
computer simulations.

  * •

We evaluate the performances of proposed model in terms of all KPIs (i.e., EC,
OP and BEP) and compared with the benchmark. In this content, to the best of
the authors’ knowledge, this is also the first study which provides an overall
performance evaluation for any NOMA involved systems. All literature
researches have biased on investigations for only one or two performance
metrics (e.g., EC and/or OP).

  * •

We define users fairness in terms of all KPIs (i.e., EC, OP and BEP). Based on
extensive simulations, it is proved that proposed R-DFNOMA provides better
user fairness compared to C-DFNOMA. Finally, we reveal the effect of power
allocation on user fairness and discuss optimum power allocation

###  I-C Organization

The remainder of this paper is as follows. In Section II, the proposed
R-DFNOMA and the benchmark C-DFNOMA schemes are introduced. The detection
algorithms at the users and the signal-to-interference plus noise ratio (SINR)
definitions are also provided in this section. Then, the performance analysis
for three KPIs (i.e., EC, OP, BER) are derived in Section III and the user
fairness indexes for all KPIs are provided. In Section IV, all derived
expressions are validated via Monte Carlo simulations. In addition,
performance comparisons are also revealed in this section. Finally, results
are discussed and the paper is concluded in Section V.

###  I-D Notation

The list of symbols, notations and abbreviations through this paper is given
in Table 1.

##  II System and Channel Model

###  II-A Proposed: Reversed DF Relaying in NOMA

Figure 1: The illustration of R-DFNOMA

As shown in Fig. 1, a source ($S$) communicates with two destinations (i.e.,
$D_{1}$ and $D_{2}$) with the help of a relay ($R$). The relay applies decode-
forward (DF) strategy in a half-duplex mode, thus the total communication
occupies two time slots. We assume that direct links from source to
destinations are not available due to the high path-loss effects and/or
obstacles. According to their average channel qualities between relay and
destinations (i.e., $R-D_{1}$ and $R-D_{2}$, users are defined as near and far
users. We assume that $D_{1}$ has a better channel than $D_{2}$ to the relay
node ($R$). In this case, $D_{1}$ and $D_{2}$ are denoted as near and far
user, respectively and the system design is handled. In the first phase of
communication, source ($S$) implements superposition coding for the base-band
modulated symbols of destinations (i.e., $x_{1}$ and $x_{2}$) and transmits it
to the relay. The received signal by the relay is given as

TABLE I: List of Symbols, Notations and Abbreviations $P_{i}$ | Transmit power at node $i=s,r$
---|---
$\alpha_{j}$ | Power allocation at the source for user $j$
$x_{j}$ | Modulated base-band (IQ) symbol of user $j$
$h_{i,k}$ | Flat fading channel coefficient
between nodes $i$ and $k$
$n_{i}$ | Additive white Gaussian noise (AWGN)
$\mu$ | Propagation constant
$\tau$ | Path loss exponent
$d_{i,k}$ | Euclidean distance between $i$ and $k$
$\sim$ | Follows/distributed
$CN(m,\sigma^{2})$ | Complex Normal distribution with $m$ mean
and $\frac{\sigma^{2}}{2}$ variance in each component
$\left|.\right|$ | Absolute value
$\rho_{i}$ | Transmit signal-to-noise ratio (SNR)
$SINR_{j}^{i,k}$ | Signal-to-interference plus noise ratio (SINR)
for user $j$ between nodes $i$ and $k$
$\gamma_{i,k}$ | Absolute square for channel fading between
nodes $i$ and $k$ ($\left|h_{i,k}\right|^{2}$)
$\hat{x}_{j}$ | Detected/estimated base-band (IQ) symbol
of user $j$ at relay
$\Xi_{i}$ | Imperfect SIC effect coefficient at the node $i$
$\beta_{j}$ | Power allocation at the relay for user $j$
$\tilde{x}_{j}$ | Detected/estimated base-band (IQ) symbol of
user $j$ at destination
$R_{j}$ | Achievable (Shannon) rate of user $j$
$C_{j}$ | Ergodic capacity (EC) of user $j$
$f_{Z}(z)$ | Probability density function (PDF)
of random variable $z$
$F_{Z}(z)$ | Cumulative distribution function (CDF)
of random variable $z$
$\hat{R}_{j}$ | Target rate of user $j$ (QoS requirement)
$P_{j}(out)$ | Outage probability (OP) of user $j$
$e2e$ | End-to-end
$P_{j}^{(i,k)}(e)$ | Bit error probability (BEP) of user $j$
between nodes $i$ and $k$
$P_{j}(e|\gamma_{i,k})$ | Conditional BEP on $\gamma_{i,k}$
$Q(.)$ | Marcum-Q function
$Q(z)=\int\limits_{z}^{\infty}\nicefrac{{1}}{{\sqrt{2\pi}}}exp(-\nicefrac{{z^{2}}}{{2}})dz$
$A_{j}^{(i,k)}(e)$ | Coefficient $A$ of user $j$ between nodes $i$ and $k$
in BEP analysis
$PF_{l}(out)$ | Proportional fairness index
$l=c,o,e$ $c\to$EC, $o\to$OP and $e\to$BEP

$y_{r}=\sqrt{P_{s}}h_{s,r}\left(\sqrt{\alpha_{1}}x_{1}+\sqrt{\alpha_{2}}x_{2}\right)+n_{r}$
(1)

where $P_{s}$ is the transmit power of source. $h_{s,r}$ and $n_{r}$ denote
the complex flat fading channel coefficient between $S-R$ and the additive
white Gaussian noise (AWGN) at the relay. They follow
$CN\left(0,\sigma_{s,r}^{2}\right)$ and $CN\left(0,N_{0}\right)$,
respectively. $\sigma_{s,r}^{2}$ includes the large-scale fading effects and
$\sigma_{s,r}^{2}=\mu d_{s,r}^{-\tau}$ is defined where $\mu$ and $\tau$ are
the propagation constant and path-loss exponent, respectively. $d_{s,r}$ is
the Euclidean distance between the nodes. In (1), $\alpha_{1}$ and
$\alpha_{2}$ are the power allocation coefficient for the symbol of $D_{1}$
and $D_{2}$, respectively. In order to improve user fairness, in R-DFNOMA, we
propose to allocate $\alpha_{1}>\alpha_{2}$ in the first phase where
$\alpha_{1}+\alpha_{2}=1$. Unlike previous works, we propose to reverse power
allocation coefficient in the first phase (e.g., $\alpha_{1}>\alpha_{2}$ ) and
conventional power allocation is proposed in the second phase (e.g.,
$\beta_{2}>\beta_{1}$ -will be defined below-) whereas in conventional DFNOMA
schemes, they have performed same way in both phases -as defined in benchmark
in the next subsection-. Thus, the proposed system model is called as
reversed-DFNOMA (R-DFNOMA). This reversed power allocation brings also
reversed detecting order in the first phase. Since more power is allocated to
$D_{1}$ symbols, relay node ($R$) firstly detects $x_{1}$ symbols by
pretending $x_{2}$ symbols as noise based on the received signal in the first
phase. The maximum-likelihood (ML) detection of $x_{1}$ symbols at the relay
is given

$\hat{x}_{1}=\operatorname*{argmin}_{k}{\left|y_{s,r}-\sqrt{P_{s}}h_{s,r}\sqrt{\alpha_{1}}x_{1,k}\right|^{2}}$
(2)

where $x_{1,k}$ denotes the $k$ th point in the $M_{1}$-ary constellation. The
received signal-to-interference plus noise ratio (SINR) for the $x_{1}$
symbols at the relay is given by

$SINR_{1}^{(s,r)}=\frac{\rho_{s}\alpha_{1}\gamma_{s,r}}{\rho_{s}\alpha_{2}\gamma_{s,r}+1}$
(3)

where $\rho_{s}=\nicefrac{{P_{s}}}{{N_{0}}}$ and
$\gamma_{s,r}=\left|h_{s,r}\right|^{2}$ are defined. On the other hand, a
successive interference canceler (SIC) should be implemented at the relay to
detect less-powered $x_{2}$ symbols. The ML detection of $x_{2}$ symbols at
the relay is given as

$\hat{x}_{2}=\operatorname*{argmin}_{k}{\left|y_{s,r}^{{}^{\prime}}-\sqrt{P_{s}}h_{s,r}\sqrt{\alpha_{2}}x_{2,k}\right|^{2}}$
(4)

where

$y_{s,r}^{{}^{\prime}}=y_{s,r}-\sqrt{P_{s}}h_{s,r}\sqrt{\alpha_{1}}\hat{x}_{1}$
(5)

and $x_{2,k}$ denotes the $k$ th point in the $M_{2}$-ary constellation. One
can easily see that, the remaining signal after SIC highly depends on the
detection of $x_{1}$ symbols and unlike previous works, it is not reasonable
to assume perfect SIC (e.g., no interference from $x_{1}$ symbols). In
addition, the interference after SIC is a function of $\gamma_{s,r}$, $P_{s}$
and $\alpha_{1}$, thus the interference cannot be assumed an independent
random variable unlike given in [27, 28, 30]. To this end, the SINR for
$x_{2}$ symbols at the relay is given as

$SINR_{2}^{(s,r)}=\frac{\rho_{s}\alpha_{2}\gamma_{s,r}}{\Xi_{r}\rho_{s}\alpha_{1}\gamma_{s,r}+1}$
(6)

where $\Xi_{r}$ defines the imperfect SIC effect coefficient (e.g.,
$\Xi_{r}=0$ for perfect SIC and $\Xi_{r}=1$ for no SIC at all).

In the second phase of communication, relay node (R) again implements
superposition coding for detected $\hat{x}_{1}$ and $\hat{x}_{2}$ symbols and
broadcasts this total symbol to the destinations. The received signal by both
destinations is given as

$y_{i}=\sqrt{P_{r}}h_{r,i}\left(\sqrt{\beta_{1}}\hat{x}_{1}+\sqrt{\beta_{2}}\hat{x}_{2}\right)+n_{i}\quad
i=1,2$ (7)

where $P_{r}$ is the transmit power of relay 222The relay can harvest its
energy from received RF signal to transmit signals. However, in this paper
this constraint has not been regarded and energy harvesting (EH) models such
as linear and non-linear seen as future researches.. $h_{r,i}$ and $n_{i}$
denote the complex flat fading channel coefficient between $R-D_{i}$ and the
additive white Gaussian noise (AWGN) at the $D_{i}$. $h_{r,i}$ $\sim$
$CN\left(0,\sigma_{r,i}^{2}\right)$ and $n_{i}$ $\sim$
$CN\left(0,N_{0}\right)$, respectively. We assume $d_{r,2}\geq d_{r,1}$, hence
$D_{1}$ has better channel condition and more power allocated to $D_{2}$ -user
with weaker channel condition-, (e.g., $\beta_{2}>\beta_{1}$). Based on
received signals, users implement whether ML or SIC plus ML in order to detect
their own symbols. Since more power is allocated for the symbols of $D_{2}$,
$D_{2}$ implements only ML by pretending $D_{1}$’s symbols as noise and it is
given,

$\tilde{x}_{2}=\operatorname*{argmin}_{k}{\left|y_{r,2}-\sqrt{P_{2}}h_{r,2}\sqrt{\beta_{2}}x_{2,k}\right|^{2}}.$
(8)

The received SINR at the $D_{2}$ is given as

$SINR_{2}^{(r,2)}=\frac{\rho_{r}\beta_{2}\gamma_{r,2}}{\rho_{r}\beta_{1}\gamma_{r,2}+1}.$
(9)

where $\rho_{r}=\nicefrac{{P_{r}}}{{N_{0}}}$ is defined.

On the other hand, $D_{1}$ implements SIC in order to detect its own symbols.
Thus, it firstly detects $x_{2}$ symbols and subtract regenerated forms from
received signal. The detection process at the $D_{1}$ is given as

$\tilde{x}_{1}=\operatorname*{argmin}_{k}{\left|y_{r,1}^{{}^{\prime}}-\sqrt{P_{r}}h_{r,1}\sqrt{\beta_{1}}x_{1,k}\right|^{2}}$
(10)

where

$y_{r,1}^{{}^{\prime}}=y_{r,1}-\sqrt{P_{r}}h_{r,1}\sqrt{\beta_{2}}\tilde{x}_{2}.$
(11)

The received SINR after SIC at the $D_{1}$ is given as

$SINR_{1}^{(r,1)}=\frac{\rho_{r}\beta_{1}\gamma_{r,1}}{\Xi_{1}\rho_{r}\beta_{2}\gamma_{r,1}+1}$
(12)

where $\Xi_{1}$ defines the imperfect SIC effect coefficient at the $D_{1}$
likewise in relay.

###  II-B Benchmark: Conventional DF Relaying in NOMA

In conventional DF relay-aided NOMA (C-DFNOMA) schemes, detecting order at
both relay and user are the same. The power allocation in the first phase is
arranged as $\alpha_{2}^{*}>\alpha_{1}^{*}$. Hence, the relay node (R) firstly
detects $x_{2}$ symbols and implements SIC to detect $x_{1}$ symbols. To this
end, given detection algorithms and SINR definitions eq. (2)-(6) should be re-
defined. The detection of $x_{2}$ symbols at the relay is given

$\hat{x}_{2}=\operatorname*{argmin}_{k}{\left|y_{s,r}-\sqrt{P_{s}}h_{s,r}\sqrt{\alpha_{2}^{*}}x_{2,k}\right|^{2}}$
(13)

and of $x_{1}$ symbols

$\hat{x}_{1}=\operatorname*{argmin}_{k}{\left|y_{s,r}^{+}-\sqrt{P_{s}}h_{s,r}\sqrt{\alpha_{1}^{*}}x_{1,k}\right|^{2}}$
(14)

where

$y_{s,r}^{+}=y_{s,r}-\sqrt{P_{s}}h_{s,r}\sqrt{\alpha_{2}^{*}}\hat{x}_{2}.$
(15)

The SINRs in the first phase of communication are given as

$SINR_{2}^{(s,r)}=\frac{\rho_{s}\alpha_{2}^{*}\gamma_{s,r}}{\rho_{s}\alpha_{1}^{*}\gamma_{s,r}+1}$
(16)

and

$SINR_{1}^{(s,r)}=\frac{\rho_{s}\alpha_{1}^{*}\gamma_{s,r}}{\Xi_{r}\rho_{s}\alpha_{2}^{*}\gamma_{s,r}+1}.$
(17)

The signal detections and the SINRs in the second phase of C-DFNOMA are the
same in R-DFNOMA.

##  III Performance Analysis

In this section, we analyze the proposed R-DFNOMA in terms of three KPIs
(i.e., EC, OP and BEP) in order to evaluate its performance. Then, we define
user fairness index for all three KPIs.

###  III-A Ergodic Capacity (EC)

Since the proposed model includes a relaying strategy, its achievable rate is
limited by the weakest link. Hence, considering both $S-R$ and $R-D_{1}$
links, the achievable (Shannon) rate of $D_{1}$ is given as

$R_{1}=\frac{1}{2}\min{\left\\{\log_{2}{\left(1+SINR_{1}^{(s,r)}\right)},\log_{2}{\left(1+SINR_{1}^{(r,1)}\right)}\right\\}}.$
(18)

where $\frac{1}{2}$ exists since the total communication covers two time
slots. The ergodic capacity (EC) of $D_{1}$ is obtained by averaging $R_{1}$
over instantaneous SINRs in (3) and (12). It is given as

$\begin{split}&C_{1}=\frac{1}{2}\\\
&\iint\limits_{0}^{\infty}\log_{2}{\left(1+\min{\left\\{SINR_{1}^{(s,r)},SINR_{1}^{(r,1)}\right\\}}\right)}f_{\gamma_{s,r}}f_{\gamma_{r,1}}d\gamma_{s,r}d\gamma_{r,1}\end{split}$
(19)

where $f_{\gamma_{s,r}}$ and $f_{\gamma_{r,1}}$ are probability density
functions (PDFs) of $\gamma_{s,r}$ and $\gamma_{r,1}$, respectively. Let
define $Z=\min{\left\\{X,Y\right\\}}$, the cumulative density function (CDF)
of $Z$ is given by $F_{Z}(z)=1-\left(1-F_{X}(z)\right)\left(1-F_{Y}(z)\right)$
where $F_{X}(.)$ and $F_{Y}(.)$ are CDFs of $X$ and $Y$, respectively [38].
Recalling,
$\int\limits_{0}^{\infty}\log_{2}{(1+x)}f_{X}(x)dx=\frac{1}{ln2}\int\limits_{0}^{\infty}\frac{1-F_{X}(x)}{1+x}dx$
[39], with some algebraic manipulations, we derive EC of $D_{1}$ as

$\begin{split}C_{1}=\frac{1}{2ln2}\int\limits_{0}^{\infty}\frac{exp\left(-\frac{z}{\left(\alpha_{1}-\alpha_{2}z\right)\rho_{s}\sigma_{s,r}^{2}}-\frac{z}{\left(\beta_{1}-\beta_{2}\Xi_{1}z\right)\rho_{r}\sigma_{r,1}^{2}}\right)}{1+z}dz\end{split}$
(20)

To the best of the authors’ knowledge, (20) cannot be solved in closed-form
analytically. Nevertheless, it can be easily computed by numerical tools. In
addition, we can obtain it in the closed-form for high SNR regime. To this
end, we assume that $\rho_{s},\rho_{r}\to\infty$. In this case,
$Z=\min{\left\\{SINR_{1}^{(s,r)},SINR_{1}^{(r,1)}\right\\}}$ in (19) turns out
to be
$\lim_{\rho_{s},\rho_{r}\to\infty}{Z}=min{\left\\{\frac{\alpha_{1}}{\alpha_{2}},\frac{\beta_{1}}{\Xi_{1}\beta_{2}}\right\\}}$.
With some algebraic simplifications, the upper bound for EC of $D_{1}$ is
given by

$C_{1}\approx\frac{1}{2}\log_{2}{\eta_{1}}$ (21)

where
$\eta_{1}=min{\left\\{\frac{\alpha_{1}}{\alpha_{2}},\frac{\beta_{1}}{\Xi_{1}\beta_{2}}\right\\}}$.

Likewise in capacity analysis of $D_{1}$, the achievable rate of $D_{2}$ is
given by

$R_{2}=\frac{1}{2}\min{\left\\{\log_{2}{\left(1+SINR_{2}^{(s,r)}\right)},\log_{2}{\left(1+SINR_{2}^{(r,2)}\right)}\right\\}}.$
(22)

and taking the similar steps between (19)-(20), EC of $D_{2}$ is derived as

$\begin{split}C_{2}=\frac{1}{2ln2}\int\limits_{0}^{\infty}\frac{exp\left(-\frac{z}{\left(\alpha_{2}-\alpha_{1}\Xi_{r}z\right)\rho_{s}\sigma_{s,r}^{2}}-\frac{z}{\left(\beta_{2}-\beta_{1}z\right)\rho_{r}\sigma_{r,2}^{2}}\right)}{1+z}dz.\end{split}$
(23)

Likewise (20), (21) can be easily computed by numerical tools. Again in order
to obtain upper bound for EC of $D_{2}$, if we assume
$\rho_{s},\rho_{r}\to\infty$, the EC is obtained as

$C_{2}\approx\frac{1}{2}\log_{2}{\eta_{2}}$ (24)

where
$\eta_{2}=min{\left\\{\frac{\alpha_{2}}{\Xi_{r}\alpha_{1}},\frac{\beta_{2}}{\beta_{1}}\right\\}}$.

###  III-B Outage Probability (OP)

The outage event for any user is defined as

$P_{i}(out)=P\left(R_{i}<\acute{R}_{i}\right)\quad i=1,2$ (25)

where $\acute{R}_{i}$ is the target rate of $D_{i}$. By substituting (18) and
(22) into (25), OPs of users are derived as

$\begin{split}&P_{i}(out)=\\\
&P\left(\frac{1}{2}\min{\left\\{\log_{2}{\left(1+SINR_{i}^{(s,r)}\right)},\log_{2}{\left(1+SINR_{i}^{(r,i)}\right)}\right\\}}<\acute{R}_{i}\right)\\\
&\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad
i=1,2.\end{split}$ (26)

With some algebraic manipulations, OPs of users are derived as

$P_{i}(out)=F_{Z_{i}}(\phi_{i})\quad i=1,2$ (27)

where $\phi_{i}=2^{2\acute{R}_{i}}-1$ and $F_{Z_{i}}(.)$ CDF of
$Z_{i}=\min{\left\\{SINR_{i}^{(s,r)},SINR_{i}^{(r,i)}\right\\}}\quad i=1,2$
are defined. Recalling CDF for minimum of two exponential random variables in
(20) and (23), OP of users are derived in the closed-forms as

$\begin{split}&P_{1}(out)=\\\
&1-exp\left(-\frac{\phi_{1}}{\left(\alpha_{1}-\alpha_{2}\phi_{1}\right)\rho_{s}\sigma_{s,r}^{2}}-\frac{\phi_{1}}{\left(\beta_{1}-\beta_{2}\Xi_{1}\phi_{1}\right)\rho_{r}\sigma_{r,1}^{2}}\right)\end{split}$
(28)

and

$\begin{split}&P_{2}(out)=\\\
&1-exp\left(-\frac{\phi_{2}}{\left(\alpha_{2}-\alpha_{1}\Xi_{r}\phi_{2}\right)\rho_{s}\sigma_{s,r}^{2}}-\frac{\phi_{2}}{\left(\beta_{2}-\beta_{1}\phi_{2}\right)\rho_{r}\sigma_{r,2}^{2}}\right).\end{split}$
(29)

###  III-C Bit Error Probability (BEP)

Since a cooperative communication is included in R-DFNOMA, the number of total
erroneous bits from source to destination (i.e., end-to-end (e2e)) of users
are given as

$\begin{split}&N_{i}=\\\
&N_{i}\left(x_{i}\rightarrow\hat{x}_{i}\right)+N_{i}\left(\hat{x}_{i}\rightarrow\tilde{x}_{i}\right)-N_{i}\left(x_{i}\rightarrow\hat{x}_{i}\right)\cap
N_{i}\left(\hat{x}_{i}\rightarrow\tilde{x}_{i}\right),\\\
&\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad
i=1,2\end{split}$ (30)

where $N_{i}\left(x_{i}\rightarrow\hat{x}_{i}\right)$ and
$N_{i}\left(\hat{x}_{i}\rightarrow\tilde{x}_{i}\right)$ denote the number of
erroneous detected bits of $D_{i}$ in the first and second phases,
respectively. If erroneous detections have been performed in both phase, this
means that correct detection has been achieved from source to destinations
(e2e). Thus, the set of intersection of erroneous detections (3rd term) is
subtracted in (30). Considering all combinations, the BEPs of $D_{i}$ are
given as in (31) (see top of the next page).

$\begin{split}P_{i}(e)=\frac{1}{M_{i}\log_{2}{M_{i}}}\left[\sum_{k=1}^{M_{i}}\sum_{\forall
l\neq k}\sum_{\forall p\neq
l}N_{i}\left(x_{i,k}\rightarrow\hat{x}_{i,l}\right)+N_{i}\left(\hat{x}_{i,l}\rightarrow\tilde{x}_{i,p}\right)-N_{i}\left(x_{i,k}\rightarrow\hat{x}_{i,l}\right)\cap
N_{i}\left(\hat{x}_{i,l}\rightarrow\tilde{x}_{i,p}\right)\right]&\\\ \quad
i=1,2&\end{split}$ (31)

Recalling that $N_{i}\left(x_{i}\rightarrow\hat{x}_{i}\right)$ and
$N_{i}\left(\hat{x}_{i}\rightarrow\tilde{x}_{i}\right)$ events are
statistically independent, thus with the law of total probability, BEPs of
users are given as

$\begin{split}&P_{i}^{(e2e)}(e)=\\\
&P_{i}^{(s,r)}(e)\left(1-P_{i}^{(r,i)}(e)\right)+\left(1-P_{i}^{(s,r)}(e)\right)P_{i}^{(r,i)}(e)\quad
i=1,2\end{split}$ (32)

where
$P_{i}^{(s,r)}(e)=\frac{1}{M_{i}\log_{2}{M_{i}}}\sum_{k=1}^{M_{i}}\sum_{\forall
l\neq k}N_{i}\left(x_{i,k}\rightarrow\hat{x}_{i,l}\right)$ and
$P_{i}^{(r,i)}(e)=\frac{1}{M_{i}\log_{2}{M_{i}}}\sum_{l=1}^{M_{i}}\sum_{\forall
p\neq l}N_{i}\left(\hat{x}_{i,l}\rightarrow\tilde{x}_{i,p}\right)$ denote the
BEPs in the first and second phases, respectively. Thus, the BEPs in each
phases should be firstly derived. Each phase of communication can be
considered separately. In the first phase of communication, it turns out to be
a conventional downlink NOMA system and the BEPs of $x_{1}$ symbols will be
the same with BEP of far user in downlink NOMA. Since the superposition is
applied, the BEP of far user in NOMA is highly depended on the chosen
constellation pairs (i.e., $M_{1}$ and $M_{2}$)[31, 32]. Nevertheless, the
conditional BEP on channel conditions is given in the form,

$P_{1}^{(s,r)}(e|_{\gamma_{s,r}})=\sum_{q=1}^{L_{1}^{(s,r)}}\varsigma_{1,q}^{(s,r)}Q\left(\sqrt{2\nu_{1,q}^{(s,r)}\rho_{s}\gamma_{s,r}}\right),$
(33)

where $L_{1}^{(s,r)}$, $\varsigma_{1,q}^{(s,r)}$ and $\nu_{1,q}^{(s,r)}$
coefficients change according to chosen modulation constellation pairs for
$x_{1}$ and $x_{2}$ symbols [40, Table 1]. For instance, in case
$M_{1}=M_{2}=4$ is used for both symbols (i.e., $x_{1}$ and $x_{2}$),
$L_{1}^{(s,r)}=2$, $\varsigma_{1,q}^{(s,r)}=0.5\ \forall q$ and
$\nu_{1,q}^{(s,r)}=\frac{1}{2}\left(\sqrt{\alpha_{1}}\mp\sqrt{\alpha_{2}}\right)^{2}$
(for proof see [41, Appendix A]). Then, recalling $\gamma_{s,r}$ is
exponentially distributed, with the aid of [42] the average BEP (ABEP) of
$x_{1}$ symbols in the first phase is obtained as,

$P_{1}^{(s,r)}(e)=\sum_{q=1}^{L_{1}^{(s,r)}}\frac{\varsigma_{1,q}^{(s,r)}}{2}\left(1-\sqrt{\frac{\nu_{1,q}^{(s,r)}\rho_{s}\sigma_{s,r}^{2}}{1+\nu_{1,q}^{(s,r)}\rho_{s}\sigma_{s,r}^{2}}}\right).$
(34)

On the other hand, $x_{2}$ symbols in the first phase can be considered as
near user symbols in conventional downlink NOMA, Thus, the conditional BEP
should be derived considering correct and erroneous SIC cases. After summing
these BEPs of two cases, the conditional BEP of $x_{2}$ symbols in the first
phase is given in the form just as (33)

$P_{2}^{(s,r)}(e|_{\gamma_{s,r}})=\sum_{q=1}^{L_{2}^{(s,r)}}\varsigma_{2,q}^{(s,r)}Q\left(\sqrt{2\nu_{2,q}^{(s,r)}\rho_{s}\gamma_{s,r}}\right),$
(35)

where $L_{2}^{(s,r)}=5$, $\varsigma_{2,q}^{(s,r)}=\frac{1}{2}[2,1,-1,-1,1]$
and
$\nu_{2,q}^{(s,r)}=[\frac{\alpha_{1}}{2},\frac{\left(\sqrt{\alpha_{2}}-\sqrt{\alpha_{1}}\right)^{2}}{2},\frac{\left(\sqrt{\alpha_{2}}+\sqrt{\alpha_{1}}\right)^{2}}{2},\frac{\left(2\sqrt{\alpha_{2}}-\sqrt{\alpha_{1}}\right)^{2}}{2},\frac{\left(2\sqrt{\alpha_{2}}+\sqrt{\alpha_{1}}\right)^{2}}{2}]$
are given for $M_{1}=M_{2}=4$ [41, Appendix A and B]. By averaging over
instantaneous $\gamma_{s,r}$, the ABEP of $x_{2}$ symbols in the first phase
is derived as

$P_{2}^{(s,r)}(e)=\sum_{q=1}^{L_{2}^{(s,r)}}\frac{\varsigma_{2,q}^{(s,r)}}{2}\left(1-\sqrt{\frac{\nu_{2,q}^{(s,r)}\rho_{s}\sigma_{s,r}^{2}}{1+\nu_{2,q}^{(s,r)}\rho_{s}\sigma_{s,r}^{2}}}\right).$
(36)

In the second phase of communication, more power is allocated to $\hat{x}_{2}$
symbols. Thus, $D_{2}$ implements a ML detection without SIC so the BEP of
$x_{2}$ symbols in the second phase can be easily derived by using (33) as

$P_{2}^{(r,2)}(e|_{\gamma_{r,2}})=\sum_{t=1}^{L_{2}^{(r,2)}}\varsigma_{2,q}^{(r,2)}Q\left(\sqrt{2\nu_{2,q}^{(r,2)}\rho_{r}\gamma_{r,2}}\right),$
(37)

where $L_{2}^{(r,2)}\triangleq L_{1}^{(s,r)}$,
$\varsigma_{2,q}^{(r,2)}\triangleq\varsigma_{1,q}^{(s,r)}$ and
$\nu_{2,q}^{(r,2)}\triangleq\nu_{1,q}^{(s,r)}$. By using (34), (36), the ABEP
is given as

$P_{2}^{(r,2)}(e)=\sum_{q=1}^{L_{2}^{(r,2)}}\frac{\varsigma_{2,q}^{(r,2)}}{2}\left(1-\sqrt{\frac{\nu_{2,q}^{(r,2)}\rho_{r}\sigma_{r,2}^{2}}{1+\nu_{2,q}^{(r,2)}\rho_{r}\sigma_{r,2}^{2}}}\right).$
(38)

Likewise, the BEP of $x_{1}$ symbols in the second phase can be easily
obtained by repeating steps (35), (35). The conditional BEP and the ABEP are
given as

$P_{1}^{(r,1)}(e|_{\gamma_{r,1}})=\sum_{q=1}^{L_{1}^{(r,1)}}\varsigma_{1,q}^{(r,1)}Q\left(\sqrt{2\nu_{1,q}^{(r,1)}\rho_{r}\gamma_{r,1}}\right),$
(39)

and

$P_{1}^{(r,1)}(e)=\sum_{q=1}^{L_{1}^{(r,1)}}\frac{\varsigma_{1,q}^{(r,1)}}{2}\left(1-\sqrt{\frac{\nu_{1,q}^{(r,1)}\rho_{r}\sigma_{r,1}^{2}}{1+\nu_{1,q}^{(r,1)}\rho_{r}\sigma_{r,1}^{2}}}\right).$
(40)

where $L_{1}^{(r,1)}\triangleq L_{2}^{(s,r)}$,
$\varsigma_{1,q}^{(r,1)}\triangleq\varsigma_{2,q}^{(s,r)}$ and
$\nu_{1,q}^{(r,1)}\triangleq\nu_{2,q}^{(s,r)}$.

Lastly, substituting (34), (36), (38) and (40) into (32), the ABEPs of users
are derived as in (41) and (42) (see top of the next page).

$\begin{split}P_{1}^{(e2e)}(e)=&\sum_{q=1}^{L_{1}^{(s,r)}}\frac{\varsigma_{1,q}^{(s,r)}}{2}\left(1-\sqrt{\frac{\nu_{1,q}^{(s,r)}\rho_{s}\sigma_{s,r}^{2}}{1+\nu_{1,q}^{(s,r)}\rho_{s}\sigma_{s,r}^{2}}}\right)\left[1-\sum_{q=1}^{L_{1}^{(r,1)}}\frac{\varsigma_{1,q}^{(r,1)}}{2}\left(1-\sqrt{\frac{\nu_{1,q}^{(r,1)}\rho_{r}\sigma_{r,1}^{2}}{1+\nu_{1,q}^{(r,1)}\rho_{r}\sigma_{r,1}^{2}}}\right)\right]\\\
&+\left[1-\sum_{q=1}^{L_{1}^{(s,r)}}\frac{\varsigma_{1,q}^{(s,r)}}{2}\left(1-\sqrt{\frac{\nu_{1,q}^{(s,r)}\rho_{s}\sigma_{s,r}^{2}}{1+\nu_{1,q}^{(s,r)}\rho_{s}\sigma_{s,r}^{2}}}\right)\right]\sum_{q=1}^{L_{1}^{(r,1)}}\frac{\varsigma_{1,q}^{(r,1)}}{2}\left(1-\sqrt{\frac{\nu_{1,q}^{(r,1)}\rho_{r}\sigma_{r,1}^{2}}{1+\nu_{1,q}^{(r,1)}\rho_{r}\sigma_{r,1}^{2}}}\right)\end{split}$
(41)
$\begin{split}P_{2}^{(e2e)}(e)=&\sum_{q=1}^{L_{2}^{(s,r)}}\frac{\varsigma_{2,q}^{(s,r)}}{2}\left(1-\sqrt{\frac{\nu_{2,q}^{(s,r)}\rho_{s}\sigma_{s,r}^{2}}{1+\nu_{2,q}^{(s,r)}\rho_{s}\sigma_{s,r}^{2}}}\right)\left[1-\sum_{q=1}^{L_{2}^{(r,2)}}\frac{\varsigma_{2,q}^{(r,2)}}{2}\left(1-\sqrt{\frac{\nu_{2,q}^{(r,2)}\rho_{r}\sigma_{r,2}^{2}}{1+\nu_{2,q}^{(r,2)}\rho_{r}\sigma_{r,2}^{2}}}\right)\right]\\\
&+\left[1-\sum_{q=1}^{L_{2}^{(s,r)}}\frac{\varsigma_{2,q}^{(s,r)}}{2}\left(1-\sqrt{\frac{\nu_{2,q}^{(s,r)}\rho_{s}\sigma_{s,r}^{2}}{1+\nu_{2,q}^{(s,r)}\rho_{s}\sigma_{s,r}^{2}}}\right)\right]\sum_{q=1}^{L_{2}^{(r,2)}}\frac{\varsigma_{2,q}^{(r,2)}}{2}\left(1-\sqrt{\frac{\nu_{2,q}^{(r,2)}\rho_{r}\sigma_{r,2}^{2}}{1+\nu_{2,q}^{(r,2)}\rho_{r}\sigma_{r,2}^{2}}}\right)\end{split}$
(42)

###  III-D User Fairness

In this subsection, we define fairness between users’ performances. In NOMA
schemes, since the total power is allocated between users, the users have
different performances. Due to the inter-user-interference and the SIC
operation, one of the users may have better performance than the other. This
performance gap can be higher in some performance metrics (e.g. EC and BER).

The performance gap between users should not be increased. We use proportional
fairness (PF) index to compare users’ performances for all KPIs. For instance,
let we firstly consider EC. In this case, if the fairness has not been
considered, one of the users may achieve much more EC than the other. To
alleviate this unfair situation, PF index for EC should be defined and it is
given as

$PF_{c}=\frac{C_{1}}{C_{2}}$ (43)

which can be easily obtained by substituting (20) and (23) into (43). One can
easily see that optimum value for $PF_{c}$ can be considered as 1 which means
that both user have exactly the same EC. Nevertheless, this may not be
achieved when the users have different QoS requirements. Thus, fairness index
should be obtained for other KPIs and all three should be evaluated together.
To this end, fairness indexes for outage and error performances are given as

$PF_{o}=\frac{P_{1}(out)}{P_{2}(out)}$ (44)

and

$PF_{e}=\frac{P_{1}(e)}{P_{2}(e)}$ (45)

which can be computed by substituting (28), (29) into (44) and (41), (42) into
(45), respectively. It is again clear that the optimal values for $PF_{o}$ and
$PF_{e}$ are also 1. However likewise in $PF_{c}$, it may not be always
achieved due to the priority in QoS requirements of users. It is noteworthy
that in the PF index for all KPIs, $\kappa$ and $\nicefrac{{1}}{{\kappa}}$
have the same meaning. For instance, if the PF index for any performance
metric has $2$ and/or $0.5$, this means that one of the users has two times
better performance than the other.

##  IV Performance Evaluation

In this section, we provide validation of the provided analysis in the
previous sections. In addition, we present user fairness comparisons between
proposed R-DFNOMA and C-DFNOMA333In C-DFNOMA, power allocation in the first
phase is complement of the power allocation of R-DFNOMA (i.e.,
$\alpha_{1}^{*}=1-\alpha_{1})$. In all simulations, we assume that $\mu=10$
and $\tau=2$. The transmit power of source and relay are assumed to be equal
(i.e., $P_{s}=P_{r}$). In validations of R-DFNOMA, unless otherwise stated,
curves denote theoretical analysis444In numerical integration for exact EC,
the infinity in the upper bounds of the integrals is changed with $10^{3}$ not
to cause numerical calculation errors. and simulations are demonstrated by
markers. Moreover, in all simulations, the imperfect SIC effect coefficients
at the both nodes are assumed to be equal (i.e., $\Xi_{r}=\Xi_{1}$).

###  IV-A The Effect of Imperfect SIC

In this subsection, the distances between the nodes are assumed to be
$d_{s,r}=5$, $d_{r,1}=1$ and $d_{r,2}=3$. It can be seen from following
figures that all derived expressions match perfectly with simulations.

Figure 2: EC of R-DFNOMA vs $\rho_{s}$ when $\alpha_{1}=0.9$, $\beta_{1}=0.2$,
$d_{s,r}=5$, $d_{r,1}=1$ and $d_{r,2}=3$ Figure 3: OP of R-DFNOMA vs
$\rho_{s}$ when $\alpha_{1}=0.9$, $\beta_{1}=0.2$, $d_{s,r}=5$, $d_{r,1}=1$
and $d_{r,2}=3$

In Fig. 2, EC of users and the ergodic sum-rate of the R-DFNOMA
($C_{sum}=C_{1}+C_{2}$) are given for various imperfect SIC effects. Power
allocations are assumed to be $\alpha_{1}=0.9$, $\beta_{1}=0.2$. As it is
expected, imperfect SIC limits the performance of the systems and when it gets
higher (i.e., $\Xi_{r}=\Xi_{1}=-10dB$), EC of R-DFNOMA becomes worse. The
power allocation at the source and relay are chosen as different values 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 rates of the users are chosen as
$\acute{R}_{1}=0.2$, $\acute{R}_{2}=0.1$ and $\acute{R}_{1}=0.75$,
$\acute{R}_{2}=0.5$ as two different QoS requirements. In all NOMA involved
systems, the outage performances of the users get better with low QoS
requirements (e.g., lower target rates). Likewise, in EC performance,
imperfect SIC has a dominant effect on outage performances of users and it may
cause users to be in outage with higher QoS.

Figure 4: The effect of imperfect SIC on EC of R-DFNOMA when $d_{s,r}=5$,
$d_{r,1}=1$, $d_{r,2}=3$ and $\rho_{s}=20dB$ Figure 5: The effect of imperfect
SIC on outage of R-DFNOMA when $d_{s,r}=5$, $d_{r,1}=1$, $d_{r,2}=3$ and
$\rho_{s}=20dB$

In order to further investigate the effect of imperfect SIC, we present
capacity and outage performance of users with the change of imperfect SIC
effect coefficient in Fig. 4 and Fig. 5, respectively. The power allocation
coefficients at the source and the relay are assumed to be $\alpha_{1}=0.8$,
$\beta_{1}=0.2$ and $\alpha_{1}=0.9$, $\beta_{1}=0.1$. The result are
presented for $\rho_{s}=20dB$. 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 performance for $20dB$ SNR. In Fig. 6,
for the same imperfect channel conditions, users are always in outage for all
target rates. If more strict QoS requirements are needed (higher target
rates), users will be in outage even for lower imperfect SIC effects (e.g.,
between $-10dB$ and $-5dB$). Nevertheless, all these are expected results for
imperfect SIC case and represent more practical/reasonable scenarios than
perfect SIC.

Figure 6: BER of R-DFNOMA vs $\rho_{s}$ when $d_{s,r}=5$, $d_{r,1}=1$ and
$d_{r,2}=3$

In Fig. 6, we present error performances of users in R-DFNOMA for
$M_{1}=M_{2}=4$. In error performance simulations, since an actual
modulation/demodulation (i.e., QPSK for $M_{i}=4$) 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 effect. Thus,
we present simulations for two different power allocation pairs (i.e.,
$\alpha_{1}=0.8$, $\beta_{1}=0.2$ and $\alpha_{1}=0.9$, $\beta_{1}=0.1$). Just
as in previous validations, it is clearly seen in Fig. 6 that derived
expressions are perfectly-matched with simulations.

Figure 7: Capacity Fairness Index ($PF_{c}$) vs $\rho_{s}$ when $d_{s,r}=5$
and $d_{r,1}=d_{r,2}=2$ Figure 8: Outage Fairness Index ($PF_{o}$) vs
$\rho_{s}$ when $d_{s,r}=5$, $d_{r,1}=d_{r,2}=2$ and
$\acute{R_{1}}=\acute{R}_{2}=0.5$ Figure 9: Error Fairness Index ($PF_{e}$) vs
$\rho_{s}$ when $d_{s,r}=5$ and $d_{r,1}=d_{r,2}=2$ Figure 10: Performance
Comparisons between C-DFNOMA and R-DFNOMA vs $\rho_{s}$ when $d_{s,r}=5$,
$d_{r,1}=d_{r,2}=2$, $\alpha_{1}=0.8$ and $\beta_{1}=0.2$ a) Capacity when
$\Xi_{r}=-15dB$ b) Outage when $\Xi_{r}=-15dB$ and
$\acute{R}_{1}=\acute{R}_{2}=0.5$ c) Bit Error Rate

###  IV-B User Fairness

Contrary to commonly belief, NOMA users do not need to have different channel
conditions (stronger and weaker). NOMA users can be chosen among users with
similar channel conditions. In this case, user fairness turns out to be more
important, since none of them should be served with lower KPI [43, 44]. Thus,
user fairness comparisons are more meaningful when the users experience
similar channel conditions. To this end, we provide PF index comparisons
between R-DFNOMA and C-DFNOMA in Fig. 7- Fig. 9 when $d_{s,r}=5$ and
$d_{r,1}=d_{r,2}=2$. In comparisons, two different power allocations scenarios
are set $\alpha_{1}=0.9$, $\beta_{1}=0.1$ and $\alpha_{1}=0.8$,
$\beta_{1}=0.2$, respectively. In Fig. 7 and Fig. 8, imperfect SIC effect
coefficients are set $\Xi_{r}=-10dB$ and $\Xi_{r}=-15dB$. Since the users have
similar channel conditions, their target rates are chosen as
$\acute{R_{1}}=\acute{R}_{2}=0.5$ for $PF_{o}$. In Fig. 9, modulation orders
are set $M_{1}=M_{2}=4$. One can easily see that proposed R-DFNOMA provides
better user fairness than C-DFNOMA in terms of all KPIs. In proposed R-DFNOMA,
users’ performance orders are also reversed. In terms of EC in Fig. 7, $D_{1}$
has higher EC than $D_{2}$ in R-DFNOMA, hence the fairness index is higher
than $1$ whereas it is vice-versa in C-DFNOMA. Nevertheless, R-DFNOMA
outperforms C-DFNOMA considering user fairness. For instance, when
$\alpha_{1}=0.9$, $\beta_{1}=0.1$ and $\Xi_{r}=-10dB$, $PF_{c}=1.545$ at
$15dB$ SNR in R-DFNOMA which means $D_{1}$ has $1.545$ times better EC than
$D_{2}$ at $15dB$. However, for the same conditions, $PF_{c}=0.2433$ in
C-DFNOMA, which means $D_{2}$ has $4.11$ times better EC than $D_{1}$.555As
discussed in the previous sections, we hereby again note that PF index of any
performance metrics have the same meaning for $\kappa$ and
$\nicefrac{{1}}{{\kappa}}$. It only defines which user has better performance.
In addition, it is obvious that R-DFNOMA provides optimum user fairness in
high SNR regimes ($PF_{c}\to 1$) whereas it cannot be achieved in C-DFNOMA.
Once user fairness is considered in terms of outage performance, the
improvement by proposed R-DFNOMA becomes outstanding in Fig. 8. In all
considered scenarios, R-DFNOMA achieves about $PF_{o}=0.5$ for all SNR region
which means $D_{1}$ has $2$ times better OP than $D_{2}$. However, in
C-DFNOMA, with the increase of transmit SNR, user unfairness gets worse.
Indeed, in some scenarios user fairness becomes atrocious with the increase of
SNR. For instance, when $\alpha_{1}=0.9$, $\beta_{1}=0.1$ and $\Xi_{r}=-10dB$,
$PF_{o}\to 80$ at $SNR\to 40dB$ which means $D_{2}$ has $80$ times better OP
than $D_{1}$ although they have same channel conditions. Likewise user
fairness comparisons for EC and OP, proposed R-DFNOMA is superior to C-DFNOMA
also in terms of error performance in Fig. 9. Furthermore, this improvement is
significant in some scenarios. For instance, when $\alpha_{1}=0.9$,
$\beta_{1}=0.1$, with the increase of SNR ($\rho_{s}\to 40dB$), $PF_{e}\to 2$
in R-DFNOMA whereas $PF_{e}\to 7$ in C-DFNOMA which is a superb gain for data
reliability of $D_{2}$.

If the comparisons between C-DFNOMA and proposed R-DFNOMA are studied only in
terms of fairness indexes (e.g., Fig. 7- Fig. 9), the questions may be raised
that: Are the user fairness indexes be improved by degrading both of users’
performances? or does the R-DFNOMA has worse performance? In order to resolve
this concern and to prove that R-DFNOMA achieves better user fairness indexes
without degrading users’ individual and overall performances, we present
performance comparison between C-DFNOMA and R-DFNOMA in Fig. 10. The channel
conditions are assumed to be same with above comparisons and the power
allocations are fixed as $\alpha_{1}=0.8$, $\beta_{1}=0.2$. In capacity and
outage comparisons in Fig. 10.a and Fig. 10.b, The imperfect SIC effect
coefficient is $\Xi_{r}=-15dB$ and the target rates of users are
$\acute{R}_{1}=\acute{R}_{2}=0.5$. 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’
or in overall system performances. When the users have similar channel
conditions, the overall system performance is limited by the minimum
performance achieved within users. To this end,in order evaluate both system
(e.g., C-DFNOMA and R-DFNOMA), we can define performance metrics as

$\begin{split}&C^{(m)}=\min\left\\{C_{1},C_{2}\right\\},\ m:C-DFNOMA\ or\
R-DFNOMA\\\ &P^{(m)}(out)=\max\left\\{P_{1}(out),P_{2}(out)\right\\},\\\
&P^{(m)}(e)=\max\left\\{P_{1}(e),P_{2}(e)\right\\}.\\\ \end{split}$ (46)

and the performance comparisons between R-DFNOMA and C-DFNOMA are given by

$\begin{split}&\max\left\\{C^{(C-DFNOMA)},C^{(R-DFNOMA)}\right\\},\\\
&\min\left\\{P^{(C-DFNOMA)}(out),P^{(R-DFNOMA)}(out)\right\\},\\\
&\min\left\\{P^{(C-DFNOMA)}(e),P^{(R-DFNOMA)}(e)\right\\}.\\\ \end{split}$
(47)

Considering performance comparisons in (46)-(47), we can see from Fig. 10
that, C-DFNOMA and R-DFNOMA has the same performance in terms of theoretical
Shannon rate (e.g., Fig. 10.a) where minimum of users’ rates are the same in
C-DFNOMA and R-DFNOMA. Although it seems that $D_{1}$ in C-DFNOMA may have
higher achievable rate in high SNR region, this should be jointly evaluated
with performance of SIC receivers. It is proved in [45] that when the higher
modulation orders (mean higher achievable rate) are implemented, none of the
users’ symbols cannot be detected and all users have $0.5$ BER performance. On
the other hand, R-DFNOMA outperforms C-DFNOMA in terms of outage and BER
performances. In Fig. 10.b, $D_{2}$ has the maximum outage probability in
R-DFNOMA (performance limits as given in (46)) whereas $D_{1}$ has the maximum
in C-DFNOMA and $D_{2}$ in R-DFNOMA has lower OP (better performance as given
in (47)) than $D_{1}$ in C-DFNOMA. Once the same evaluations have been
discussed for BER performances, we can easily see that $D_{2}$ in R-DFNOMA
(maximum in R-DFNOMA) has lower BER (better performance as given in (47)) than
$D_{1}$ in C-DFNOMA (maximum in C-DFNOMA).

Figure 11: EC performance in R-DFNOMA vs $\alpha_{1}$ and $\beta_{1}$ when
$d_{s,r}=5$, $d_{r,1}=d_{r,2}=2$, $\rho_{s}=30dB$ and $\Xi_{r}=-10dB$ a)
$D_{1}$ b) $D_{2}$ Figure 12: Outage performance in R-DFNOMA vs $\alpha_{1}$
and $\beta_{1}$ when $d_{s,r}=5$, $d_{r,1}=d_{r,2}=2$, $\rho_{s}=30dB$,
$\Xi_{r}=-10dB$ and $\acute{R}_{1}=\acute{R}_{2}=0.2$ a) $D_{1}$ b) $D_{2}$
Figure 13: BER performance in R-DFNOMA vs $\alpha_{1}$ and $\beta_{1}$ when
$d_{s,r}=5$, $d_{r,1}=d_{r,2}=2$, $\rho_{s}=30dB$ a) $D_{1}$ b) $D_{2}$

###  IV-C The Effect of Power Allocations

From above simulations and discussions, it can bee seen that power allocations
at the both nodes (source and relay) have a remarkable effect on the
performance of R-DFNOMA so that on the user fairness. Thus, in order to reveal
this effect, we present EC, OP and BER performances of users with the respect
to power allocations ($\alpha_{1}$ and $\beta_{1}$) in Fig. 11 - Fig. 13. The
channel conditions are assumed to be $d_{s,r}=5$ and $d_{r,1}=d_{r,2}=2$. The
imperfect SIC effect is $\Xi_{r}=-10dB$ assumed in EC and OP comparisons and
transmit SNR $\rho_{s}=30dB$ is assumed in all figures. Target rates of users
are assumed to be equal to $\acute{R}_{1}=\acute{R}_{2}=0.2$. $M_{1}=M_{2}=4$
is set in BER simulations. As expected, both power allocation coefficients
(source and relay) have reverse effects on the performances of users.
Increasing/decreasing $\alpha_{1}$ and/or $\beta_{1}$ provides performance
gain for one of the user and causes a decay for the other vice-versa. In terms
of EC in Fig. 11, increase in $\alpha_{1}$ and/or $\beta_{1}$ provide better
EC for $D_{1}$ and lower EC for $D_{2}$. The same discussions are also valid
for outage performances of users in Fig. 11. Nevertheless, it is noteworthy
that increasing/decreasing any power allocation coefficient too much causes
one of the users to be in always outage. The same performance trends can be
seen for also BER in Fig. 13. However, since an actual modulation/demodulation
is implemented, the effect of power allocation on the SIC performance is more
clear in Fig. 13. For instance, increasing $\beta_{1}$ means that higher power
is allocated to $D_{1}$ symbols in the second phase thereby better error
performance is expected. Nevertheless, one can easily see that increasing
$\beta_{1}$ too much causes a decay in error performance of $D_{1}$ along with
error performance of $D_{2}$. This is explained as follows. Since the $D_{1}$
has to implement SIC in the second phase, increasing $\beta_{1}$ causes
erroneous SIC more likely and this pulls down the error performance of
$D_{1}$. Based on above discussions, it is clear that power allocation affects
users’ performances reversely (gain for one and decay for the other), thus it
is not possible to define a optimum power allocation which offers the best
performances for both users. Nevertheless, by considering user fairness, a
sub-optimum power allocation can be obtained.

To this end, user fairness comparisons between R-DFNOMA and C-DFNOMA with
respect to power allocations ($\alpha_{1}$ and $\beta_{1}$) in Fig. 14 - Fig.
16 for the same conditions above comparisons. Based on user fairness for EC in
Fig. 14, although $PF_{c}$ changes within larger range (max. $5$) in R-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 is presented in terms of OP when the users have same QoS
requirement ($\acute{R}_{1}=\acute{R}_{2}=0.2$). It can bee seen that users in
R-DFNOMA have similar outage performance even if power allocation coefficients
change. In R-DFNOMA, one of the users may have maximum $6$ times better outage
performance than the other. However, this unfairness may raise to $50$ times
better performance in C-DFNOMA. Lastly, user fairness in terms of error
performance is presented in Fig. 16. Similar discussions can be seen for also
user fairness in terms of BER from Fig. 16. The fairness index may have the
maximum $3.5$ in R-DFNOMA whereas the value of $80$ in C-DFNOMA. This shows
that users in R-DFNOMA have similar data reliability, however in C-DFNOMA, one
of the users may outperform $80$ times the other. Based on the results between
Fig. 14 and 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 $\alpha_{1}\approx 0.85$
and $\beta_{1}\approx 0.15$ where users have same performance so that user
fairness becomes very close to $1$.

Figure 14: Capacity Fairness Index ($PF_{c}$) vs $\alpha_{1}$ and $\beta_{1}$
when $d_{s,r}=5$, $d_{r,1}=d_{r,2}=2$, $\rho_{s}=30dB$ and $\Xi_{r}=-10dB$ a)
R-DFNOMA b) C-DFNOMA Figure 15: Outage Fairness Index ($PF_{o}$) vs
$\alpha_{1}$ and $\beta_{1}$ when $d_{s,r}=5$, $d_{r,1}=d_{r,2}=2$,
$\rho_{s}=30dB$, $\Xi_{r}=-10dB$ and $\acute{R}_{1}=\acute{R}_{2}=0.2$ a)
R-DFNOMA b) C-DFNOMA Figure 16: Error Fairness Index ($PF_{e}$) vs
$\alpha_{1}$ and $\beta_{1}$ when $d_{s,r}=5$, $d_{r,1}=d_{r,2}=2$ and
$\rho_{s}=30dB$ a) R-DFNOMA b) C-DFNOMA

##  V Conclusion

In this paper, we introduce reversed decode-forward relaying NOMA (R-DFNOMA)
to improve user fairness in conventional DFNOMA (C-DFNOMA). We consider
imperfect SIC at detections and according to imperfect SIC effect and we re-
define SINRs at the nodes to provide a more practical scenario. With the
imperfect SIC, we investigate the performance of proposed R-DFNOMA and derive
closed-form expressions for ergodic capacity (EC), outage probability (OP) and
bit error probability (BEP). Then, in order to emphasize user fairness, we
define fairness indexes for all KPIs (i.e., EC, OP and BEP). With the
extensive simulations, derived expressions are validated and it is proved that
R-DFNOMA outperforms significantly C-DFNOMA in terms of user fairness when the
users have similar channel conditions. Moreover, user fairness is investigated
with the change of power allocation coefficients and optimum power allocation
is discussed under user fairness constraint. Based on the results and
discussions, it is revealed that reversed power allocation at the source and
changing SIC order at the relay can provide better user fairness and the
system will be more robust to power allocation. This promising result shows
that reversed networks can be implemented in other NOMA and cooperative
involved systems. To this end, analysis and discussions for DF relaying in
this paper can be extended for other strategies. Hence, reversed power
allocation in DFNOMA with full-duplex, DFNOMA with energy harvesting and
DFNOMA with direct links are seen as future research topics.

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|  Ferdi Kara received B.Sc. (with Hons.) in electronics and communication
engineering from Suleyman Demirel University, Turkey in 2011. He received
M.Sc. and PhD degrees in electrical and electronics engineering from Zonguldak
Bulent Ecevit University, Turkey, in 2015 and 2019, respectively. Since 2011,
he works with Wireless Communication Technologies Research Laboratory
(WCTLab), where he now holds Senior Researcher position. He serves as regular
reviewer for reputed IEEE journals and conferences. He was awarded with
Exemplary Reviewer Certificate by IEEE Communications Letters in 2019. His
research area spans within wireless communications specified with NOMA, MIMO
systems, cooperative communication, index modulations, energy harvesting, and
machine learning in physical communication.
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|  Hakan Kaya received B.Sc., M.Sc., and Ph.D degrees in all electrical and
electronics engineering from Zonguldak Karaelmas University, Turkey, in 2007,
2010, and 2015, respectively. Since 2015, he has been working at Zonguldak
Bulent Ecevit University as Assistant Professor and the head of the Wireless
Communication Technologies Research Laboratory (WCTLab). His research
interests are cooperative communication, NOMA, turbo coding and machine
learning.
---|---