Multi-points cooperative relay in NOMA system with N-1 DF relaying nodes in HD/FD mode for N user equipments with energy harvesting

electronics Article Multi-Points Cooperative Relay in NOMA System with N-1 DF Relaying Nodes in HD/FD Mode for NUser Equipments with Energy Harvesting Thanh-Nam Tran 1,2,* and Miroslav Voznak 1 1Faculty of Electrical Engineering and Computer Science, Technical University of Ostrava, 17. listopadu 2172/15, 708 33 Ostrava-Poruba, Czech Republic; miroslav[email protected] 2Faculty of Electronics and Telecommunications, Sai Gon University, 220 Tran Binh Trong st., Dict. 5, Ho Chi Minh City, Vietnam *Correspondence: [email protected] Received: 14 December 2018; Accepted: 29 January 2019; Published: 1 February 2019   Abstract: Non-Orthogonal Multiple Access (NOMA) is the key technology promised to be applied in next-generation networks in the near future. In this study, we propose a multi-points cooperative relay (MPCR) NOMA model instead of just using a relay as in previous studies. Based on the channel state information (CSI), the base station (BS) selects a closest user equipment (UE) and sends a superposed signal to this UE as a first relay node. We have assumed that there are N UEs in the network and the N -th UE, which is farthest from BS, has the poorest quality signal transmitted from the BS compared the other UEs. The N -th UE received a forwarded signal from N− 1 relaying nodes that are the UEs with better signal quality. At the i -th relaying node, it detects its own symbol by using successive interference cancellation (SIC) and will forward the superimposed signal to the next closest user, namely the (i+ 1 ) -th UE, and include an excess power which will use for energy harvesting (EH) intention at the next UE. By these, the farthest UE in network can be significantly improved. In addition, closed-form expressions of outage probability for users over both the Rayleigh and Nakagami-mfading channels are also presented. Analysis and simulation results performed by Matlab software, which are presented accurately and clearly, show that the effectiveness of our proposed model and this model will be consistent with the multi-access wireless network in the future. Keywords: cooperative NOMA; multi-points DF relaying nodes; half-duplex; full-duplex; Rayleigh fading channels; Nakagami-mfading channels; energy harvesting 1. Introduction The next-generation network (5G) technology has the advantage of increasing system capacity by superior sharing-spectrum efficiency [ 1 ]. Therefore, multiple users in the network can be served in the same frequency band/time slot and various allocation power coefficients by the key technology is called Non-Orthogonal Multiple Access (NOMA). The is fundamentally different from previous orthogonal access methods, e.g., Orthogonal Multiple Access (OMA) [ 2 ]. In NOMA system, the users with better channel conditions are allocated less transmitting power coefficients. On the other hand, the users with worse channel conditions are allocated more transmitting power coefficients to guarantee the quality of service (QoS) for all users in the system. After receiving a superposed signal, successive interference cancellation (SIC) is done at the end users [ 3 ]. In [ 4 ], the authors investigated the impact of imperfect SIC on the analysis performance of NONA system. Their analysis results showed that even though SIC is not perfect, the performance of the NOMA system is still better than the orthogonal system. A down-link NOMA wireless network was studied in [ 5 ] by considering using a relay for forwarding Electronics 2019,8, 167; doi:10.3390/electronics8020167 www.mdpi.com/journal/electronics Electronics 2019,8, 167 2 of 21 signals to combat the fading effect of the transmission channel. Authors applied to dual-hop relaying systems with decode-and-forward (DF) or amplify-and-forward (AF) protocols [6] . Relay full-duplex (FD) model over the Rayleigh fading channels using the DF protocol investigated the performance by optimizing the transmit power factor [ 7 ]. The study impacts relay selection of cooperative NOMA on the performance system [ 8 ]. The authors in [ 9 ] proposed a novel best cooperative mechanism (BCM) for wireless EH and spectrum sharing in the 5G network. The [10–12] include AF and DF relaying. In [ 12 ], it showed that a dual-hop power line communication (PLC) system can improve the system capacity compared to direct-link (DL) transmission. And M. Rabie et al. [13] proposed using Multi-hop relay instead of using one hop relay or dual-hop relays. The authors investigated the energy efficiency over PLC channels with assuming log-normal fading. The studies [14,15] analyzed the system performance of multi-hop AF/DF relaying over PLC channels in terms of average bit error and ergodic capacity. These studies showed that the system performance can be improved by increasing the number of relaying. In addition, The authors in [ 8 ] studied the impact of relay selection (RS) on system performance. The compared results on two-stage versus max-min RS showed that cooperative NOMA system over Rayleigh fading channels with two-stage RS is better than the max-min one. We hypothesized that there are N users with the N -th user at the far end from BS with the worst channel condition. The QoS of the N -th user can be improved with the cooperation of N− 1 users instead of just receiving only a relay cooperation. At each node, one must perform the best neighbor selection to forward the signal next neighbor. The best selection of neighbors is repeated until the signal reaches the destination. In addition, we also consider EH at UEs. The explosion of the number of wireless devices, radio frequency (RF) EH becomes a potential technology to convert the energy of receiving wireless signal into electricity. Therefore, the MPCR is not only transmitting information but also delivering energy to the users. In Ref. [ 16 – 18 ], only users located close to BS can collect energy. This is because signal reception and energy collection cannot be done simultaneously. Thus, the users need to divide the received signal for EH and information decoding (ID) by using power splitting (PS) or time switching (TS) which was called “received TS” [ 19 , 20 ]. Though the PS approach has been shown to mostly outperform the receive-TS approach, however, the PS is complicated and inefficient for practical implementation. The research results have shown that PS is better than TS, however, PS is more complex and difficult to practical application than TS. In our study, we consider compressing both information and energy in one transmission phase instead of splitting it into two transmission phases as in previous studies. Furthermore, a user faraway from BS can still receive information and collect energy from the nearest relay node. Researchers have made important contributions to the 5G wireless multi-access network. Specially, L. Dai et al. [ 21 ] presented the introduction, development process, and recent research trends on NOMA, comprehensively. Because of the potential application of NOMA in the future, there have been many important research contributions [ 22 – 29 ]. These positive research results are motivations for other researchers to continue to study NOMA improvement. In this study, we focus on MPCR in NOMA network to improve the QoS for the user faraway form BS with poor channel. In terms of contributions in this research, the main contributions include: •The first, this article proposes a down-link side NOMA network with random NUEs. • The next, the MPCR model is proposed to improve QoS for the N th UE with farthest distance from BS among the others users by using N− 1 UEs as DF relaying nodes in HD/FD mode. Each ·UEi relaying node receives and forwards a superposed signal to next hop, namely UEi+1 , which is nearest from UEi . This work will loop until the superposed signal is sent to last UE, namely UEN. •A algorithm for selecting relay nodes in MPCR is also presented clearly in next section. • At UEi with ∀i> 1, the received signal has an excess power that is used for EH to charge the battery with assuming unlimited capacity of the battery. • In additional, this study investigates and finds an outage probability and system throughput for each UE, which are written in closed-form expressions. Electronics 2019,8, 167 3 of 21 • Further, The analysis and simulation results are presented in a clear way by the Monte Carlo simulation (106samples of channels) from the Matlab software to prove our propositions. This article is presented as following. In the next section, namely Experimental Models, we propose system models and analyse two transmission scenarios which are called N− 1 relaying nodes in HD or FD mode. In the third section, we have analyzed the system performance on outage probability and system throughput. In Section 4, we use Matlab software to simulate and results will also be presented in this section. A summary of the results of this study will be presented in Section 5. Notice: In this study, we use a few notations included as •ha,bis a channel from source ato destination b. •αiis an allocation power coefficient for the i-th UE. •yΩ iis the received signal at the i-th UE with Ωprotocol where Ω={HD,FD}. •γΩ i→xj is a signal-to-interference-plus-noise-ratios (SINRs) at i-th UE while the i-th UE decodes xjsymbol. •Pr {.}is a probability. •<ΘΩ i or ℵΘΩ i is an outage probability of the i-th UE with Ω protocol over Rayleigh or Nakagami-m fading channels, respectively. •R∗ iis a bit rate threshold of the i-th UE. 2. Experimental Models In previous studies about NOMA, a direct down-link scenario is considered to serve a number of users in the same time slot. However, in such studies, there are usually a fixed number of users. Therefore, they have not shown the generality of the model. In order to ensure the generality, we have upgraded the model to a random and unpredictable number of users. 2.1. Direct Link Scenario The authors analyzed different NOMA techniques including power domain and code domain [ 22 ]. The role of the power domain is proven to be important in determining the performance of the system through the availability of CSI [ 23 ]. The BS send a superposed signal S to all UEs in the same power domain and same time slot as following S=pP0 N ∑ j=1pαjxj. (1) Thus, the received signal at the i-th UE, ∀i∈{1, . . . , N}, would be expressed as following yDir i=h0,ipP0 N ∑ j=1pαjxj+ni, (2) where h0,i is denoted as the channels from BS to each the i -th UE over Rayleigh or Nakagami-mfading channel. Furthermore, N is a random number of UEs joined to network, αj in rule with N ∑ j=1 αj= 1 is an allocation power coefficient for each UE and P0 is the transmission power of BS. ni is denoted as the additive white Gaussian noise (AWGN) of the i -th UE, where ni∼CN (0, N0) with zero mean, variance N0and i∈{1, . . . , N}. It is important to notice that the channel coefficient from BS to each UE, in paired, is expressed as h0,iin our expressions. Electronics 2019,8, 167 4 of 21 In direct link scenario, the first user in the nearest distance from the BS with the strongest channel conditions was ordered first in the channel gain list. Furthermore, the list is in decreasing order as following h0,1 >h0,2 >. . . >h0,i>. . . >h0,N−1>h0,N. (3) According to the NOMA theory, users with the worst signal quality should be given priority to allocate the highest transmitting power factor. Another assumption in terms of the NOMA characteristics, we have assumed that the BS already owns the CSI of all UEs fully. In a previous study [ 30 ], the authors considered that CSI is available to the system and used to determine the decoding order of user’s data. The authors in [ 31 ] studied how NOMA performance depends on power allocation techniques to ensure fairness for users under instantaneous CSI and average CSI. The superimposed signals are sent to the UEs in the same power domain with different power coefficients, in the hope of ensuring system performance and ensuring service quality fairness for all users. Therefore, the list of allocation power factors is arranged in descending order for each UE in the network as α1<α2<. . . <αi<. . . <αN−1<αN. (4) In Figure 1, the UEN is farthest from the BS. Thus, the xN symbol is allocated the strongest power factor. Therefore, xN symbol will be first decoded at all UEs in the network by applying SIC [ 3 ]. Furthermore, the order of decoding is done sequentially according to the reversed list of power factor allocations presented in (4) expression. The Signal-to-interference-plus-noise ratios (SINRs) of all UEs have been expressed as γDir i→xj=|h0,i|2ρ0αj |h0,i|2ρ0 j−1 ∑ k=1 αk+1, (5) where i∈{1, . . . , N}and j∈{N, . . . , i}. In a special case at the UE1 , after it decoded xj symbols with j∈ {N , . . . , 2 } by using (5), UE1decodes its own symbol x1with only AWGN n1as γDir 1→x1=|h0,1|2ρ0α1. (6) Furthermore, ρ0in (5) or (6) is signal-to-noise ratio (SNR) which can be calculated by ρi=Pi N0 , (7) where i∈{0, . . . , N−1}, e.g., ρ0=P0/N0with P0is the transmitting power of the BS. The achievable instantaneous bit rate of the i -th UE when it decodes xj symbol with xj∈{xN, . . . , xi} is shown by RDir i→xj=1 2log21+γDir i→xj, (8) where i∈{1, . . . , N} and j∈{N, . . . , i} . If i6=j6= 1, and γDir i→xj is given by (5) then. Else if i=j= 1, and γDir i→xjis given by (6) then. 2.2. N −1DF Relaying Nodes Scenario On the other hand, the system model in [ 13 ] has only one relaying to improve the QoS of UEs which are faraway from the BS. We propose a improved model with using a MPCR model instead of using only one user as a relay device. See in Figure 1, there are N users in the network with descending order channel conditions with the N -th UE has the poorest signal compared to the other UEs. The Figure 1a,b are N− 1 HD relaying nodes model and N− 1 FD relaying nodes models, Electronics 2019,8, 167 5 of 21 respectively. In FD mode, the relays are impacted by the loop interference channels, which themselves affected the system’s performance. This study investigates the system performance on MPCR in HD or FD mode for N users over Rayleigh or Nakagamim fading channels. Previous studies on the NOMA system used a cooperative relay to improve system performance compared to a direct transmission system. The contributions of previous studies [ 30 – 32 ] are the motivation for this research to continue to improve system performance. SUEi h0,1 h1,2 UE2 hi-1,i hN-1,N UE1 UEN (a)DF relaying nodes in HD mode. SUEi h0,1 h1,2 UE2 hi-1,i hN-1,N UE1 UEN (b)DF relaying nodes in FD mode. Figure 1. The NOMA system with N−1 relaying nodes in HD/FD mode. Z. Ding et al. [ 8 ] proposed the relay selection method to choose the best relay with the best channel condition by using two-stage relay selection protocol which outperforms versus max-min relay selection protocol. There is a difference compared model in [ 8 ] versus our model. The authors consider selection a best relay in N relays to serve for two other users [ 8 ]. In our proposed model, Figure 1, all of the N− 1 UEs can be selected for relaying node. A selected relay node set is initialized empty v=∅, and a first relaying node can be selected by v1=max nRΩ i→x1>R∗ 1o, (9) where RΩ i→x1is given by (22), and v1has been added into v=v∪v1then. BS sends a superposed signal to the closest distance user with strongest channel condition, namely UE1 in the Figure 1a,b, after BS selected UE1 as a relay successfully. It is important to point out the difference. In this study, each relay node has a single or a twin antenna and works in HD or FD mode. The received signals at the UE1in HD or FD mode are respectively the same like (2) or (10) as yFD 1=h0,1pP0 N ∑ j=1pαjxj+hLI,1pP0˜ x1+n1, (10) where hLI,1 is the loop interference channel generated by the itself transmitter antenna, and n1 is the AWGN noise of the device UE1. In case the UE1 is working in HD relaying mode, UE1 decodes its own symbol by applying (5) and (6), respectively. On the other hand, the UE1 is working in FD relaying mode, UE1 decodes xj symbol with j∈{N, . . . , 2}or j=1 by applying SINRs in (11a) or (11b), respectively, γFD 1→xj ∆ =|h0,1|2ρ0αj |h0,1|2ρ0 j−1 ∑ k=1 αk+|hLI,1|2ρ1+1 (11a) ∧ =|h0,1|2ρ0α1 |hLI,1|2ρ1+1. (11b) Then, the UE1 sends a mixed signal, namely S1 in (13), to the next UE which is next nearest relay node, namely UE2. The second relay node can be selected by applying (9) as Electronics 2019,8, 167 6 of 21 v2=max nRΩ i→x2>R∗ 2,i={1, . . . , N},i/∈vo, (12) where RΩ i is also given by (22) and not being contained in v set which is a selected relay nodes set. We removed UEi with i∈v from the relays selection because the signal could be sent back to the previous relay node and the superposed signal is unable to send to the UEN . Furthermore, the v2 is also added into v then. Note that the nearest neighbor represented in [ 33 , 34 ] are neighbors closest to the BS. However, the authors in [ 35 ] have extended the definition of nearest neighbor as the device can set up the transmission channel in the best condition compared to the other devices. A mixed signal is sent to the next relay node as expressed S1=pP1 √α1x∅+ N ∑ j=2pαjxj!, (13) where x∅is an empty information symbol which was also namely x1decoded at the UE1. The received signals at the UE2 in both HD and FD relaying modes are expressed as, respectively, yHD 2=h1,2pP1 √α1x∅+ N ∑ j=2pαjxj!+n2, (14) and yFD 2=h1,2pP1 √α1x∅+ N ∑ j=2pαjxj! +hLI,2pP2˜ x2+n2, (15) where h1,2 is the channel from UE1 to UE2 , P1 is denoted as transmitting power at UE1 , and hLI,2 is loop interference channel from transmitting antenna to receiving one at UE2 . Specially, the x1 symbol existed in (2) and (10) but it was replaced by x∅ symbol in (14) and (15). Because x1 was previously decoded and removed from the mixed signal by UE1 . Therefore, the x∅ symbol does not contain information and becomes a redundancy in the mixed signal. This paper will use excess power of x∅ symbol for EH purposes as is described in the next section. The SINRs for decoding xj symbol and its own x2 symbol at UE2 in both HD and FD relaying modes can be expressed, respectively, as following γHD 2→xj ∆ =|h1,2|2ρ1αj |h1,2|2ρ1 j−1 ∑ k=2 αk+1 (16a) ∧ =|h1,2|2ρ1α2, (16b) and γFD 2→xj ∆ =|h1,2|2ρ1αj |h1,2|2ρ1 j−1 ∑ k=2 αk+|hLI,2|2ρ2+1 (17a) ∧ =|h1,2|2ρ1α2 |hLI,2|2ρ2+1, (17b) where (16a) and (17a) with j∈{N, . . . , 3}, or (16b) and (17b) with j=2. Electronics 2019,8, 167 7 of 21 After UE2 decoded its own symbol, it selects a next relay node and sends a new superposed signal to next nearest UE, namely UE3 . This work will loop until a superposed signal is sent to the farthest UE, namely UENin Figure 1. Proposition 1. In this study, we propose a EH model to use excess power in the mixed signals for purposing EH as Figure 2. As expressing in (18) and (19), the received signals at the i -th UE, where i∈{2, . . . , N} , have an empty x∅ symbol with no information. Thus, the transmit power coefficients of each empty symbol can be harvested. In previous studies, the power for EH was transmitted to users on different time slots or on different antennas on the receivers. However, in this study, we use only one antenna for receiving both signals and energy from the transmitter. Nth x symbol (i+1)th x symbol ithx symbol Decode Forward UEi hLI,i hi-1,i hi,i+1 Mix ∑ α via k={1,i-1} Energy Harvesting k Figure 2. DF protocol and EH protocol at the i-th UE node. In general, the received signals at the UEi in both HD and FD relaying nodes can be rewritten by, respectively yHD i=hi−1,ipPi−1 i−1 ∑ l=1 √αlx∅+ N ∑ k=i √αkxk!+ni, (18) and yFD i=hi−1,ipPi−1 i−1 ∑ l=1 √αlx∅+ N ∑ k=i √αkxk! +hLI,ipPi˜ xi+ni, (19) where yHD i and yFD i are denoted as receiving signals at the UEi node, hi−1,i is the channel from previous node to current node, Pi−1 and Pi are transmitting power of previous UE and current UE, respectively. It is important to notice that i−1 ∑ l=1 αl+N ∑ k=i αk=1. The SINRs of each the i -th UE relaying node for detecting xj symbol in HD and FD modes are expressed as, respectively γHD i→xj ∆ =|hi−1,i|2ρi−1αj |hi−1,i|2ρi−1 j−1 ∑ k=i αk+1 , (20a) ∧ =|hi−1,i|2ρi−1αi, (20b) Electronics 2019,8, 167 8 of 21 and γFD i→xj ∆ =|hi−1,i|2ρi−1αj |hi−1,i|2ρi−1 j−1 ∑ k=i αk+|hLI,i|2ρi+1 , (21a) ∧ =|hi−1,i|2ρi−1αi |hLI,i|2ρi+1, (21b) where both (20a) and (21a) are with i∈{1, . . . , N} and j∈{N, . . . , i+1} . Furthermore, both (20b) and (21b) are with i=j. In NOMA theory, reachable instantaneous bit rate can be calculated by RΩ i→xj=1 2log21+γΩ i→xj, (22) where Ω={HD,FD} , i∈{1, . . . , N} , and j∈{N, . . . , i} . If i6=j , and γΩ i→xj is given by (20a) or (21a) then. Else if i=j, and γΩ i→xjis given by (20b) or (21b) then. A selected relay node can be performed by vi=max nRΩ i→xj>R∗ j,i∈{1, . . . , N},i/∈vo. (23) Furthermore, a selected relay nodes set vafter the signal has been sent to the UENincluded v=v1∪v2∪. . . ∪vN−1. (24) 3. The System Performance Analysis In this section, we evaluate the performance of the system that we have proposed based on outage probability and system throughput, in order. 3.1. Outage Probability In terms of investigating outage probability, the outage probability is defined as the occurrence of the stop transmitting event if any instantaneous bit rate in (8) or (22) cannot reach minimum bit rate thresholds. The probability density function (PDF) and cumulative distribution function (CDF) of Rayleigh distribution are shown by, respectively, f|ha,b|2(x)=1 σ2 a,b e−x σ2 a,bdx, (25) and F|ha,b|2(x)=1−e−x σ2 a,b, (26) where ha,b2 are random independent variables namely x in PDF and CDF, respectively, with a and b are source and destination of channels, and σ2 a,bis mean of channel with σ2 a,b=Ehha,b2i. In general, the PDF and CDF over nakagami-mfading channels can be expressed, respectively, f|ha,b|2(x)= m σ2 a,b!mxm−1 Γ(m)e−mx σ2 a,b, (27) Electronics 2019,8, 167 9 of 21 and F|ha,b|2(x) = γm,mx σ2 a,b Γ(m) =1−e−mx σ2 a,b m−1 ∑ j=0 mx σ2 a,b!j1 j!. (28) In direct link scenario, outage event occurs if UEi , where i∈{1, . . . , N} , cannot decode xj symbol, where j∈{N, . . . , i} . The outage probability for each of the joining UE in NOMA system is expressed as ΘDir i=1− i ∏ j=N Pr RDir i→xj>R∗ j. (29) where RDir i→xjis given by (8) and R∗ jis bit rate threshold of UEj. By applying the CDF in (25) and (27), the (29) is solved and it can be rewritten in closed-form as <ΘDir i=1− i ∏ j=N e−R∗∗ j χjρ0σ2 0,i, (30) and ℵΘDir i=1−i ∏ j=N      m σ2 0,i!m  m σ2 0,i!−m Γ(m)+ R∗∗ j χjρ0!m mR∗∗ j χjρ0σ2 0,i!−m Γ m,mR∗∗ j χjρ0σ2 0,i!−Γ(m)!  Γ(m)     ,(31) where Γ(.) and Γ(., .) are gamma function and gamma incomplete function, respectively. Furthermore, R∗∗ j=22R∗ j− 1. It is important to notice that (30) and (31) are with the users over Rayleigh and Nakagami-mfading channels, respectively. In addition, χjin both (30) and (31) is given by χj ∆ =αj−R∗∗ j j−1 ∑ k=1 αk(32a) χj∧ =α1, (32b) where (32a) is with ∀i, and j∈{N, . . . , 2}then. Furthermore, Equation (32b) is with i=j=1 then. Remark 1. Base on the proposed model with N− 1relaying nodes as in Figure 1, this study investigates the outage probabilities of N UE nodes in both HD and FD modes as ΘΩ i=      1− i−1 ∏ l=1 Pr RΩ l→xi>R∗ i | {z } η       and        1− i ∏ j=N Pr RΩ i→xj>R∗ j | {z } µ        , (33) where η is the successful probability to detect xi symbol at previous UEs and µ is the successful probability to detect xj symbol at the i -th UE. In a special case of the i -th UE with i= 1, It is important to notice that η in (33) is equal with zero and the (33) becomes the same with (29). In (33), η and µ are also solved by applying the CDF and gotten closed-form outage probability of each UE node over Rayleigh fading channel on both HD and FD modes as, respectively, Electronics 2019,8, 167 16 of 21 -10 -5 0 5 10 15 20 25 30 0 0.1 0.2 0.3 0.4 0.5 0.6 Throughput of 3 UEs (a)N=3 UEs in network. -10 -5 0 5 10 15 20 25 30 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 Throughput of 4 UEs (b)N=4 UEs in network. Figure 6. The system throughput results of the users over Rayleigh fading channels. On the other hand, this study analyzes the impact of the allocation power factor α4 on the fourth user’s throughput with variable α4∈{0.1, ..., 0.9} values instead fixing α4= 0.48. As shown in Figure 7, higher grid lines are better results than the other ones. In this case, the instantaneous bit rate threshold of UE4 is R∗ 4= 0.12 bps/Hz. In low SNRs, e.g., SNR = 0 db, the system throughput results in all scenarios being approximately zero. On the other hand, although the SNRs have been increased, e.g., SNR = 10 dB, the system throughput results are still approximately zero if the power factor, namely α4 , is still in low, e.g., α4= 0.1. However, with α4= 0.4 and SNR is still held in 10 dB, the system throughput results of UE4 in both three HD relaying nodes and three FD relaying nodes in MPCR scenarios are improved and reach their bit rate threshold. The Figure 6b showed that at SNR in 10 dB and α4= 0.48, the UE4 reach its bit rate threshold, approximately. Another e.g., in paired α4= 0.5 and SNR = 0 dB, UE4 also reach its bit rate threshold in Figure 7. By this analysis, we can find pairs of values α4and SNR where UE4can reach the threshold R∗ 4=0.12 bps/Hz. -10 -5 0 0 0.9 0.8 5 0.02 0.7 10 0.6 0.04 15 0.5 0.4 20 0.3 0.06 System Throughput of 4th UE 25 0.2 30 0.1 0.08 0.1 0.12 0 0.02 0.04 0.06 0.08 0.1 via Direct link (a) via N-1 HD relaying nodes (b) via N-1 FD relaying nodes (c) Simulation (a) Simulation (b) Simulation (c) Throughput of 4th UE Figure 7. The throughput of the 4th UE over Rayleigh fading channels with α4={0.1, . . . , 0.9} and SNRs ={−10, . . . , 30}dB. The system throughput of the users in N− 1 HD relaying nodes over both Rayleigh and Nakagamim scenarios were analyzed, compared and presented in Figure 8a. In Figure 8a, there are N= 3 UEs over Rayleigh fading channels and Nakagamim fading channels with solid lines and dashed ones, respectively. This is because of the results of ΘHD 1>ΘHD 2>ΘHD 3 as shown in Figure 5a . By applying (40) , we get PHD 1<PHD 2<PHD 3 with low SNRs. With increasing SNRs, the system throughput of each UE changes, e.g., SNR = 30 dB, PHD 1>PHD 2>PHD 3 and reach their bit rate thresholds R∗ i. Electronics 2019,8, 167 17 of 21 The similarly results also happen in N− 1 FD relaying nodes scheme as shown in Figure 8b. Specifically, because the users over Nagamim fading channels have better outage probability results than the ones over the Rayleigh fading channels as shown in Figure 5b, in some SNRs, e.g., SNR =10 dB then ℵΘFD i<<ΘFD i . Therefore, ℵPFD i><PFD i where ℵ and < were denoted as Nakagamim and Rayleigh fading channels, respectively, after applying (40). These results proved that the Nakagamim channel is better than the Rayleigh channel. However, when SNRs are increasing, the users have the throughput results approximately and close to the thresholds ℵPHD i≈ <PHD i≈R∗ i. -10 -5 0 5 10 15 20 25 30 0 0.1 0.2 0.3 0.4 0.5 0.6 System Throughput of 3 UEs (a)N−1 HD relaying nodes. -10 -5 0 5 10 15 20 25 30 0 0.1 0.2 0.3 0.4 0.5 0.6 System Throughput of 3 UEs (b)N−1 FD relaying nodes. Figure 8. Comparison of the system throughput results of Rayleigh versus Nakagami-mvia m=2. 4.3. N UEs with N −1HD/FD Relaying Nodes As modeling Figure 1a,b, the proposed Proposition 3 can investigate the system performance with N UEs where N is a random and big number. Because of the limited power of our personal computers, this study only investigates and presents cases where there are only three or four users, N={3, 4} , in the system. However, the results presented do not show all the advantages of proposing algorithm. Thus, we are increasing the limit the number user with bigger number N . As shown in Figure 9a,b, there are 9 UEs in the network. By applying Proposition 3, we investigated the outage probability of the UEs in the network over both Rayleigh and Nakagamim fading channels. For e.g., in N− 1 HD relaying nodes scenario, the outage probability of the first UE, namely UE1 , can be calculated by (34) or (28) over Rayleigh or Nakagamim fading channels with m= 2, respectively, where η= 0. Another e.g., in FD scenario, the outage probability of last UEs, namely UE9 , over Rayleigh or Nakamagmim fading channels can be computed by (35) or (39), respectively. With the number of users is greater than nine UEs, N> 9, the results of the analysis are difficult to observe in the figure and it needs more time for the simulation. Therefore, we end the investigation with up to nine users in network. -10 -5 0 5 10 15 20 25 30 10-14 10-12 10-10 10-8 10-6 10-4 10-2 100 Outage Probability of 9 UEs (a)9 UEs in 8 HD relaying nodes model. -10 -5 0 5 10 15 20 25 30 10-14 10-12 10-10 10-8 10-6 10-4 10-2 100 Outage Probability of 9 UEs (b)9 UEs in 8 FD relaying nodes model. Figure 9. Comparison of the outage probability results of Rayleigh versus Nakagami-mfading channels. Electronics 2019,8, 167 18 of 21 5. Conclusions In this study, we proposed a novel NOMA network model with N− 1 relaying nodes instead of using only one relay as in previous studies. A superposed signal would be sent through N− 1 relaying nodes before it reaches the farthest UE which is denoted by UEN . The closed-form expressions of N− 1 HD/FD relaying nodes scenarios over Rayleigh/Nakagamim fading channels are also presented along with an explanation for the corresponding processing. By presenting results in the figures, our proposed models with N− 1 HD/FD relaying nodes are effective for applying to the cooperator NOMA network in the next generation of wireless telecommunications. Author Contributions: T.-N.T. is the first author who proposed the main idea, analyzed and simulated the system, and presented the writing—original draft preparation, writing—review and editing, visualization. M.V. is the second who has experience in wireless communication research. He has made a supervision, review, and given the first author some useful comments and funding acquisition for this research. All authors read and approved the final manuscript. Funding: This research received no external funding. Acknowledgments: We would like to extend special thanks to the Reviewers for their comments and suggestions to improve this article. Conflicts of Interest: We declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results. Abbreviations The following abbreviations are used in this manuscript: No. Abbreviations Full description 1 AWGNs Additive white Gaussian noises 2 BS Base station 3 CDF Cummuative distribution function 4 CSI Channel state information 5 FD Full-duplex 6 Fig. Figure 7 HD Half-duplex 8 MPCR Multi-Point Cooperative Relay 9 NOMA non-orthogonal multiple access 10 PDF Probability density function 11 QoS Quality of service 12 S Source 13 SIC Successive interference cancellation 14 SINR Signal-to-interference-plus-noise ratio 15 SNR Signal-to-noise ratio 16 UEs User Equipments Appendix A Proof of N−1HD relaying nodes scenario: The condition for occurrence of the outage events has been presented in (33). By submitting (22), where Ω=HD , into (33), we can get a expression for computing the outage probability of each UE in N−1 HD relaying nodes scenario as following ΘHD i= 1− i−1 ∏ l=1 Pr hl−1,l2>R∗∗ i χiρl−1!and 1− i ∏ j=N Pr |hi−1,i|2> R∗∗ j χjρi−1!!. (A1) Electronics 2019,8, 167 19 of 21 The (A1) can be rewritten in experimental integral by applying the PDF (25) of Rayleigh distributions as <ΘHD i=   1−i−1 ∏ l=1 ∞ R R∗∗ i χiρl−1 1 σ2 l−1,l e−x σ2 l−1,ldx   and      1−i ∏ j=N ∞ R R∗∗ j χjρi−1 1 σ2 i−1,i e−x σ2 i−1,idx     .(A2) The (A2) can be solved and expressed as (34). On the other hand, the (A2) can be written with the PDF (27) of Nakagami-mfading channels as following ℵΘHD i=   1−i−1 ∏ l=1 ∞ R R∗∗ i χiρl−1 m σ2 l−1,lmxm−1 Γ(m)e−mx σ2 l−1,ldx   and      1−i ∏ j=N ∞ R R∗∗ j χjρi−1 1 σ2 i−1,imxm−1 Γ(m)e−mx σ2 i−1,idx     .(A3) and after the (A3) was solved, it can be expressed as (38). Proof of N−1FD relaying nodes scenario: Similarly, by submitting (22) with Ω=FD into (33), we can get an expression for computing the outage probability of each UE in N− 1 FD relaying nodes scenario ΘFD i= 1−i−1 ∏ l=1 Pr hl−1,l2> R∗∗ i|hLi,l|2ρl+1 χiρl−1,hLi,l2>0!! 1−i ∏ j=N Pr |hi−1,i|2> R∗∗ j|hLi,i|2ρi+1 χjρi−1,|hLi,i|2>0!!. (A4) The (A4) is also rewritten in experimental integral by applying the PDF of Rayleigh or Nakagami-m fading which are respectively (25) or (27), respectively, as <ΘHD i=   1−i−1 ∏ l=1 ∞ R0 ∞ R R∗∗ i(yρl+1) χiρl−1 1 σ2 l−1,lσ2 LI,l e− x σ2 l−1,l +y σ2 LI,l!dxdy         1−i ∏ j=N ∞ R0 ∞ R R∗∗ j(yρi+1) χjρi−1 1 σ2 i−1,iσ2 LI,i e− x σ2 i−1,i +y σ2 LI,i!dxdy     , (A5) and ℵΘFD i=   1−i−1 ∏ l=1 ∞ R0 ∞ R R∗∗ i(yρl+1) χiρl−1 m2 σ2 l−1,lσ2 LI,lm(xy)m−1 (Γ(m))2e−m x σ2 l−1,l +y σ2 LI,l!dxdy         1−i ∏ j=N ∞ R0 ∞ R R∗∗ j(yρi+1) χjρi−1 m2 σ2 i−1,iσ2 LI,im(xy)m−1 (Γ(m))2e−m x σ2 i−1,i +y σ2 LI,i!dxdy     . 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