I Introduction
In recent years, exponential increase in connected devices (i.e., smartphones, tablets, watches etc.) [VNI] 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 [Andrews2014]. Nonorthogonal 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 [Shirvanimoghaddam2017]. 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 [Dai2015]. The interference mitigation in PDNOMA is held by successive interference canceler (SIC) [Saito2013]. Due to its potential for 5G and beyond, NOMA^{1}^{1}1NOMA is used for PDNOMA 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) [6868214]. 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, mmwave communication, multiinputmultioutput (MIMO) systems, visible light communication, etc. has also taken a remarkable attention [Ding2017].
Ia 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) cooperativeNOMA [7117391, Kara2019] where near users also act as relays for far user, 2) NOMAbased cooperative systems [Kim2015a, Kara2020a] where NOMA is implemented to increase spectral efficiency of devicetodevice communication and 3) relayassisted/aidedNOMA where relays in the network help NOMA users to enhance coverage. This paper focuses on relayassistedNOMA networks. The relayassistedNOMA networks have also been analyzed widely. In these works, an amplifyforward (AF) or a decodeforward (DF) relay helps source/BS to transmit symbols to the NOMA users [8287018]. Hybrid DF/AF relaying strategies have been investigated to improve outage performance of relayassistedNOMA [7556297]. Outage and sumrate performance of relayassistedNOMA networks have also been analyzed whether a direct link between the source and the users exists [Liu2018] or not [8466778, 7870605, 7876764]. Then, relayassistedNOMA networks have been analyzed in terms of achievable rate and outage performance under different conditions such as buffer aidedrelaying [7747506, 7803598], partial channel state information (CSI) at transmitter [8103803] and imperfect CSI at receiver [7752764] 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 [7482785, 8038031, 8329423, 8031883]. Relay selection schemes are based on guaranteeing QoS of users and maximizing outage performance of users. Moreover, twoway relaying strategies where relay operates as a coordinated multipoint (CoMP), have been investigated in terms of achievable rate and outage performance [7417453, 7819537, 8275033, 8316931].
However, in aforementioned either conventional or relayassisted 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 [7819537, 8275033, Im2019]. Thus, this strict assumption should be relaxed. Besides, all the studies with imperfect SIC [7819537, 8275033, Im2019] have been devoted to twoway 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 [Kara2018, Assaf2019]. 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 [Timotheou2015] and some studies are devoted to improve user fairness in conventional NOMA networks in terms of EC and OP [Liu2015a, Liu2016, Liu2016e], 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 relayassistedNOMA systems due to the effects of two phases (e.g., from sourcetorelay and from relaytousers). Besides all this, BER performance of relayassistedNOMA networks has been only analyzed in [Kara2020b] though they have been widelyanalyzed in terms of EC and OP. User fairness also has not been considered for relayassistedNOMA networks in terms of any key performance indicators (KPIs) (e.g., EC, OP, BER). To this end, we analyze relayassistedNOMA network with imperfect SIC for all performance metrics. The user fairness has been also raised for relayassistedNOMA networks.
IB Contributions
The main contributions of this paper are as follow:

We introduce reversed DF relaying NOMA (RDFNOMA) to improve user fairness in conventional DFNOMA (CDFNOMA).

For a more realistic/practical scenario, we redefine imperfect SIC effect as dependant to channel fading coefficient. The capacity and outage performances of proposed RDFNOMA are investigated with this imperfect SIC effect. The exact EC expressions are derived and closedform upper bounds are provided for EC. Besides, exact OP expressions are derived in closedforms. All derived expressions match perfectly with simulations.

Contrary to the most of the literature, we also analyze the error performance of RDFNOMA rather than only EC and/or OP performances. Exact bit error probability (BEP) expressions are provided in closedforms 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 RDFNOMA provides better user fairness compared to CDFNOMA. Finally, we reveal the effect of power allocation on user fairness and discuss optimum power allocation
IC Organization
The remainder of this paper is as follows. In Section II, the proposed RDFNOMA and the benchmark CDFNOMA schemes are introduced. The detection algorithms at the users and the signaltointerference 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.
ID Notation
The list of symbols, notations and abbreviations through this paper is given in Table 1.
Ii System and Channel Model
Iia Proposed: Reversed DF Relaying in NOMA
As shown in Fig. 1, a source () communicates with two destinations (i.e., and ) with the help of a relay (). The relay applies decodeforward (DF) strategy in a halfduplex 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 pathloss effects and/or obstacles. According to their average channel qualities between relay and destinations (i.e., and , users are defined as near and far users. We assume that has a better channel than to the relay node (). In this case, and are denoted as near and far user, respectively and the system design is handled. In the first phase of communication, source () implements superposition coding for the baseband modulated symbols of destinations (i.e., and ) and transmits it to the relay. The received signal by the relay is given as
Transmit power at node  
Power allocation at the source for user  
Modulated baseband (IQ) symbol of user  
Flat fading channel coefficient  
between nodes and  
Additive white Gaussian noise (AWGN)  
Propagation constant  
Path loss exponent  
Euclidean distance between and  
Follows/distributed  
Complex Normal distribution with mean 

and variance in each component  
Absolute value  
Transmit signaltonoise ratio (SNR) 

Signaltointerference plus noise ratio (SINR)  
for user between nodes and  
Absolute square for channel fading between  
nodes and ()  
Detected/estimated baseband (IQ) symbol 

of user at relay  
Imperfect SIC effect coefficient at the node  
Power allocation at the relay for user  
Detected/estimated baseband (IQ) symbol of  
user at destination  
Achievable (Shannon) rate of user  
Ergodic capacity (EC) of user  
Probability density function (PDF)  
Cumulative distribution function (CDF)  
of random variable  
Target rate of user (QoS requirement)  
Outage probability (OP) of user  
Endtoend  
Bit error probability (BEP) of user  
between nodes and  
Conditional BEP on  
MarcumQ function  
Coefficient of user between nodes and  
in BEP analysis  
Proportional fairness index  
EC, OP and BEP 
(1) 
where is the transmit power of source. and denote the complex flat fading channel coefficient between and the additive white Gaussian noise (AWGN) at the relay. They follow and , respectively. includes the largescale fading effects and is defined where and are the propagation constant and pathloss exponent, respectively. is the Euclidean distance between the nodes. In (1), and are the power allocation coefficient for the symbol of and , respectively. In order to improve user fairness, in RDFNOMA, we propose to allocate in the first phase where . Unlike previous works, we propose to reverse power allocation coefficient in the first phase (e.g., ) and conventional power allocation is proposed in the second phase (e.g., 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 reversedDFNOMA (RDFNOMA). This reversed power allocation brings also reversed detecting order in the first phase. Since more power is allocated to symbols, relay node () firstly detects symbols by pretending symbols as noise based on the received signal in the first phase. The maximumlikelihood (ML) detection of symbols at the relay is given
(2) 
where denotes the th point in the ary constellation. The received signaltointerference plus noise ratio (SINR) for the symbols at the relay is given by
(3) 
where and are defined. On the other hand, a successive interference canceler (SIC) should be implemented at the relay to detect lesspowered symbols. The ML detection of symbols at the relay is given as
(4) 
where
(5) 
and denotes the th point in the ary constellation. One can easily see that, the remaining signal after SIC highly depends on the detection of symbols and unlike previous works, it is not reasonable to assume perfect SIC (e.g., no interference from symbols). In addition, the interference after SIC is a function of , and , thus the interference cannot be assumed an independent random variable unlike given in [7819537, 8275033, Im2019]. To this end, the SINR for symbols at the relay is given as
(6) 
where defines the imperfect SIC effect coefficient (e.g., for perfect SIC and for no SIC at all).
In the second phase of communication, relay node (R) again implements superposition coding for detected and symbols and broadcasts this total symbol to the destinations. The received signal by both destinations is given as
(7) 
where is the transmit power of relay ^{2}^{2}2The 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 nonlinear seen as future researches.. and denote the complex flat fading channel coefficient between and the additive white Gaussian noise (AWGN) at the . and , respectively. We assume , hence has better channel condition and more power allocated to user with weaker channel condition, (e.g., ). 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 , implements only ML by pretending ’s symbols as noise and it is given,
(8) 
The received SINR at the is given as
(9) 
where is defined.
On the other hand, implements SIC in order to detect its own symbols. Thus, it firstly detects symbols and subtract regenerated forms from received signal. The detection process at the is given as
(10) 
where
(11) 
The received SINR after SIC at the is given as
(12) 
where defines the imperfect SIC effect coefficient at the likewise in relay.
IiB Benchmark: Conventional DF Relaying in NOMA
In conventional DF relayaided NOMA (CDFNOMA) schemes, detecting order at both relay and user are the same. The power allocation in the first phase is arranged as . Hence, the relay node (R) firstly detects symbols and implements SIC to detect symbols. To this end, given detection algorithms and SINR definitions eq. (2)(6) should be redefined. The detection of symbols at the relay is given
(13) 
and of symbols
(14) 
where
(15) 
The SINRs in the first phase of communication are given as
(16) 
and
(17) 
The signal detections and the SINRs in the second phase of CDFNOMA are the same in RDFNOMA.
Iii Performance Analysis
In this section, we analyze the proposed RDFNOMA 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.
Iiia Ergodic Capacity (EC)
Since the proposed model includes a relaying strategy, its achievable rate is limited by the weakest link. Hence, considering both and links, the achievable (Shannon) rate of is given as
(18) 
where exists since the total communication covers two time slots. The ergodic capacity (EC) of is obtained by averaging over instantaneous SINRs in (3) and (12). It is given as
(19) 
where and are probability density functions (PDFs) of and , respectively. Let define , the cumulative density function (CDF) of is given by where and are CDFs of and , respectively [Devore2002]. Recalling, [Gradshteyn1994], with some algebraic manipulations, we derive EC of as
(20) 
To the best of the authors’ knowledge, (20) cannot be solved in closedform analytically. Nevertheless, it can be easily computed by numerical tools. In addition, we can obtain it in the closedform for high SNR regime. To this end, we assume that . In this case, in (19) turns out to be . With some algebraic simplifications, the upper bound for EC of is given by
(21) 
where .
Likewise in capacity analysis of , the achievable rate of is given by
(22) 
and taking the similar steps between (19)(20), EC of is derived as
(23) 
Likewise (20), (21) can be easily computed by numerical tools. Again in order to obtain upper bound for EC of , if we assume , the EC is obtained as
(24) 
where .
IiiB Outage Probability (OP)
The outage event for any user is defined as
(25) 
where is the target rate of . By substituting (18) and (22) into (25), OPs of users are derived as
(26) 
With some algebraic manipulations, OPs of users are derived as
(27) 
where and CDF of are defined. Recalling CDF for minimum of two exponential random variables in (20) and (23), OP of users are derived in the closedforms as
(28) 
and
(29) 
IiiC Bit Error Probability (BEP)
Since a cooperative communication is included in RDFNOMA, the number of total erroneous bits from source to destination (i.e., endtoend (e2e)) of users are given as
(30) 
where and denote the number of erroneous detected bits of 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 are given as in (31) (see top of the next page).
(31) 
Recalling that and
events are statistically independent, thus with the law of total probability, BEPs of users are given as
(32) 
where and 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 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., and )[Kara2018, Assaf2019]. Nevertheless, the conditional BEP on channel conditions is given in the form,
(33) 
where , and coefficients change according to chosen modulation constellation pairs for and symbols [Kara2019a, Table 1]. For instance, in case is used for both symbols (i.e., and ), , and (for proof see [Kara2019c, Appendix A]). Then, recalling
is exponentially distributed, with the aid of
[Alouini1999] the average BEP (ABEP) of symbols in the first phase is obtained as,(34) 
On the other hand, 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 symbols in the first phase is given in the form just as (33)
(35) 
where , and are given for [Kara2019c, Appendix A and B]. By averaging over instantaneous , the ABEP of symbols in the first phase is derived as
(36) 
In the second phase of communication, more power is allocated to symbols. Thus, implements a ML detection without SIC so the BEP of symbols in the second phase can be easily derived by using (33) as
(37) 
where , and . By using (34), (36), the ABEP is given as
(38) 
Likewise, the BEP of symbols in the second phase can be easily obtained by repeating steps (35), (35). The conditional BEP and the ABEP are given as
(39) 
and
(40) 
where , and .
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).
(41) 
(42) 
IiiD 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 interuserinterference 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
(43) 
which can be easily obtained by substituting (20) and (23) into (43). One can easily see that optimum value for 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
(44) 
and
(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 and are also 1. However likewise in , 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, and have the same meaning. For instance, if the PF index for any performance metric has and/or , 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 RDFNOMA and CDFNOMA^{3}^{3}3In CDFNOMA, power allocation in the first phase is complement of the power allocation of RDFNOMA (i.e., . In all simulations, we assume that and . The transmit power of source and relay are assumed to be equal (i.e., ). In validations of RDFNOMA, unless otherwise stated, curves denote theoretical analysis^{4}^{4}4In numerical integration for exact EC, the infinity in the upper bounds of the integrals is changed with 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., ).
Iva The Effect of Imperfect SIC
In this subsection, the distances between the nodes are assumed to be , and . It can be seen from following figures that all derived expressions match perfectly with simulations.
In Fig. 2, EC of users and the ergodic sumrate of the RDFNOMA () are given for various imperfect SIC effects. Power allocations are assumed to be , . As it is expected, imperfect SIC limits the performance of the systems and when it gets higher (i.e., ), EC of RDFNOMA 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
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