In recent years, exponential increase in connected devices (i.e., smart-phones, 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]. 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 [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 PD-NOMA is held by successive interference canceler (SIC) [Saito2013]. 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) . 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 [Ding2017].
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 [7117391, Kara2019] where near users also act as relays for far user, 2) NOMA-based cooperative systems [Kim2015a, Kara2020a] 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 . Hybrid DF/AF relaying strategies have been investigated to improve outage performance of relay-assisted-NOMA . 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 [Liu2018] or not [8466778, 7870605, 7876764]. Then, relay-assisted-NOMA networks have been analyzed in terms of achievable rate and outage performance under different conditions such as buffer aided-relaying [7747506, 7803598], partial channel state information (CSI) at transmitter  and imperfect CSI at receiver  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, two-way relaying strategies where relay operates as a coordinated multi-point (CoMP), have been investigated in terms of achievable rate and outage performance [7417453, 7819537, 8275033, 8316931].
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 [7819537, 8275033, Im2019]. Thus, this strict assumption should be relaxed. Besides, all the studies with imperfect SIC [7819537, 8275033, Im2019] 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 [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 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 [Kara2020b] 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.
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
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.
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
As shown in Fig. 1, a source () communicates with two destinations (i.e., and ) with the help of a relay (). 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., 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 base-band 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 base-band (IQ) symbol of user|
|Flat fading channel coefficient|
|between nodes and|
|Additive white Gaussian noise (AWGN)|
|Path loss exponent|
|Euclidean distance between and|
Complex Normal distribution withmean
|and variance in each component|
Transmit signal-to-noise ratio (SNR)
|Signal-to-interference plus noise ratio (SINR)|
|for user between nodes and|
|Absolute square for channel fading between|
|nodes and ()|
Detected/estimated base-band (IQ) symbol
|of user at relay|
|Imperfect SIC effect coefficient at the node|
|Power allocation at the relay for user|
|Detected/estimated base-band (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|
|Bit error probability (BEP) of user|
|between nodes and|
|Conditional BEP on|
|Coefficient of user between nodes and|
|in BEP analysis|
|Proportional fairness index|
|EC, OP and BEP|
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 large-scale fading effects and is defined where and are the propagation constant and path-loss 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 R-DFNOMA, 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 reversed-DFNOMA (R-DFNOMA). 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 maximum-likelihood (ML) detection of symbols at the relay is given
where denotes the th point in the -ary constellation. The received signal-to-interference plus noise ratio (SINR) for the symbols at the relay is given by
where and are defined. On the other hand, a successive interference canceler (SIC) should be implemented at the relay to detect less-powered symbols. The ML detection of symbols at the relay is given as
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
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
where 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.. 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,
The received SINR at the is given as
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
The received SINR after SIC at the is given as
where defines the imperfect SIC effect coefficient at the 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 . 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 re-defined. The detection of symbols at the relay is given
and of symbols
The SINRs in the first phase of communication are given as
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 and links, the achievable (Shannon) rate of is given as
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
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
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 . In this case, in (19) turns out to be . With some algebraic simplifications, the upper bound for EC of is given by
Likewise in capacity analysis of , the achievable rate of is given by
and taking the similar steps between (19)-(20), EC of is derived as
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
Iii-B Outage Probability (OP)
The outage event for any user is defined as
where is the target rate of . By substituting (18) and (22) into (25), OPs of users are derived as
With some algebraic manipulations, OPs of users are derived as
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 closed-forms as
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
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).
Recalling that and
events are statistically independent, thus with the law of total probability, BEPs of users are given as
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,
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,
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)
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
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
where , and . By using (34), (36), the ABEP is given as
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
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).
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
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
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 R-DFNOMA and C-DFNOMA333In C-DFNOMA, power allocation in the first phase is complement of the power allocation of R-DFNOMA (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 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 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., ).
Iv-a 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 sum-rate of the R-DFNOMA () 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 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