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Performance analysis of powersplitting relaying protocol in SWIPT based cooperative NOMA systems
EURASIP Journal on Wireless Communications and Networking volume 2021, Article number: 110 (2021)
Abstract
This paper investigates a relay assisted simultaneous wireless information and power transfer (SWIPT) for downlink in cellular systems. Cooperative nonorthogonal multiple access (CNOMA) is employed along with power splitting protocol to enable both energy harvesting (EH) and information processing (IP). A downlink model consists of a base station (BS) and two users is considered, in which the near user (NU) is selected as a relay to forward the received signal from the BS to the far user (FU). Maximum ratio combining is then employed at the FU to combine both the signals received from the BS and NU. Closed form expressions of outage propability, throughput, ergodic rate and energy efficiency (EE) are firstly derived for the SWIPT based CNOMA considering both scenarios of with and without direct link between the BS and FU. The impacts of EH time, EH efficiency, powersplitting ratio, source data rate and distance between different nodes on the performance are then investigated. The simulation results show that the CNOMA with direct link achieves an outperformed performance over CNOMA without direct link. Moreover, the performance of CNOMA with direct link is also higher than that for OMA. Specifically, (1) the outage probability for CNOMA in both direct and relaying link cases is always lower than that for OMA. (2) the outage probability, throughput and ergodic rate vary according to \(\beta\), (3) the EE of both users can obtain in SNR range of from \(10\) to 5 dB and it decreases linearly as SNR increases. Numerical results are provided to verify the findings.
Introduction
Nonorthogonal multiple access (NOMA) has recently been shown as one of the potential candidates for 5G and beyond based wireless networks to overcome the limitations of the current technologies such as energy efficiency, latency and user fairness [1,2,3]. One of the critical features of NOMA techniques is that multiple users are permitted to use the same resources in time, frequency and/or code domain [4]. It means that a strong user, i.e. a NU, is given a lower power allocation factor than a weak user, i.e. a FU, to ensure user fairness [1, 5,6,7]. Two key techniques applied in NOMA consist of superposition coding (SC) [2] and successive interference cancellation (SIC) [1, 2]. As an extended version of NOMA, cooperative NOMA (CNOMA) [8, 9] exploits a user with better channel conditions, namely a relaying user, to assist to forward the information to another user with poor channel conditions. Therefore, it can increase the coverage region of BS and improve the performance of NOMA systems.
Radio frequency (RF) based energy harvesting (EH) can help solve energy constraint issues in mobile devices, wireless sensors as well as the relayingacted nodes of wireless communication networks [10, 11]. At relay nodes, the energy harvesting is normally performed in the first phase of signal transmitting time block. This harvested energy is dedicated for: i) consuming at the relay and ii) forwarding the decoded information to the destination.
The combination of simultaneous wireless information and power transfer (SWIPT) and CNOMA in 5G systems has demonstrated an outperformed energy efficiency and coverage area over OMA [7, 12]. More, by forwarding the information to far users, the relay based SWIPT CNOMA can improve the integrity and reliability of the transmitted data for weak users [13]. Powersplitting protocol (PSR) and timeswitching protocol (TSR) are exploited at SWIPT based relaying nodes to harvest energy and process information [5, 6, 14, 15]. In [16], the sum throughput of users in SWIPT based CNOMA system was studied. Closedform and closedform approximate expressions of outage probability were achieved. In [17], two protocols based on SWIPT, namely CNOMASWIPTPS and CNOMASWIPTTS, were proposed. The effectiveness of the proposed schemes was demonstrated over OMA and the work in [18]. In [19], a SWIPT based CNOMA system was investigated. A joint design for the power allocation coefficients and the PS factor was proposed to improve the system performance. The derivation of analytical expressions for the outage probabilities of near and far users was also provided. In [20], a PSR based SWIPT for CNOMA was studied. Compared to the protocol in [21], this protocol can considerably reduce the outage probability of the strong users and increase the system throughput. In [22], the outage probability and throughput of the proposed TSR protocol was superior to the normal TSR protocol.
There are two main data forwarding schemes in relayassisted CNOMA, including decodeandforward (DF) and amplifyandforward (AF) [1]. Furthermore, in relay based CNOMA, far users normally receive the transmitted signal which is forwarded from relay nodes [23,24,25,26,27]. This is because there are some obstacles on the propagation [5, 6, 28]. However, in system models without obstacle, these far users can receive signals from both relay and BS, namely therefore relay based CNOMA with direct links [25, 29,30,31]. In [29], a dynamic DF based CNOMA scheme for downlink transmission was proposed. The outage probability of the proposed scheme was derived by applying point process theory. In [32], three cooperative relaying schemes were proposed in a DF based CNOMA system. The system performance for the proposed schemes was superior to the cooperative DF relaying without direct links and multiple user superposition transmission without relaying. In [33], a DF relay aimed CNOMA system with direct link between BS and weak user was studied. In [34], a system cooperative devicetodevice systems with NOMA in which the BS can communicate simultaneously with all users was considered. Two decoding strategies, namely single signal decoding scheme and maximum ratio combining (MRC) decoding scheme, were proposed. The numerical results showed that the ergodic sum rate as well as outage probability achieve better than the conventional NOMA schemes. The authors in [35] proposed a protocol to permit the BS to adaptively switch between direct and indirect modes in CNOMA system with two users. The analytical results demonstrated that the proposed protocol overwhelmed the conventional CNOMA protocol. In [36], the outage performance of dual DF based SWIPT NOMA system with direct link was presented.
The use of relays for forwarding information from sources to destinations and harvesting RF energy has been investigated in the current technologies such as OFDMA, SWIPT/WPT [37,38,39]. In [37], a relaying selection scheme, namely OFDMA relaying selection, was proposed for OFDM multihop cooperative networks with L relays and M hops (\(M,\,L\ge 2\)). The endtoend outage performance of the proposed approach was evaluated and compared to that of the OFDM relaying selection approach. In [38], a relaying selection scheme was investigated in a twohop relayassisted multiuser OFDMA network with K fixed relays and L users (\(2\le L\le K\)), where the endnodes exploited the SWIPT mechanism based on the power splitting (PS) technique. This relaying selection is to optimize the PS ratio of the end nodes as well as the relay, carrier, and power assignment so that the sumrate of the system was maximized under the harvested energy and transmitted power constraints. In [39], a survey of the SWIPT and WPT assisted energy harvesting techniques was presented. The survey provided a detailed description of various potential emerging technologies for the fifth generation (5G) communications with SWIPT/WPT.
In this paper, we investigate a wireless communication system model which can ensure the user fairness by allocating power and harvest the RF energy from the source, namely CNOMA based system model. We combine SWIPT and CNOMA in our model system to study its performance metric in terms of the outage probability, throughput and energy efficiency. The investigated model consists one base station and two users where one user acts as a relaying user, another user is a FU. The BS simultaneously broadcasts the superposed coding signals to both users and thus the FU also receives the signal from BS. Based on NOMA mechanism, the FU with poor channel conditions is allocated more power than the NU with strong channel conditions. Moreover, the SIC process is performed at the NU which acts as the relaying user. After receiving the transmitted signal, the relaying user decodes the FU\(^{'}\)s signal and its own signal utilizing SIC. The decoded signal of the FU at the relaying user is then forwarded to the FU using DF protocol. The relaying user employs PSR protocol in its communication process. Involving in signal processing at relay node, delay limited transmission (DLT) and delay tolerant transmission (DTT) modes can be exploited at this node [14]. The DLT mode refers to the block wise received signal decoding mechanism at the destination node while the DTT mode refers to the storage of the received data block in the buffer of the destination node prior to data decoding. The key contributions of our work in this paper are sumarized as follows:

Closedform expressions of the performance, i.e., outage probability, throughput, ergodic rate and EE, are derived for the PSR protocol with DLT and DTT modes and direct link. This performance of the system model with direct link is compared to that for CNOMA without direct link as well as OMA. The simulation results show that the CNOMA with direct link achieves a better performance than that for the CNOMA without direct link and OMA.

The impacts of above mentioned parameters on the direct link are evaluated via the numerical simulation results to realize the changes of the performance. These impacts are as a background for choosing the suitable values of the parameters for system model to achieve the tradeoff among terms of the performance as well as users.
The rest of paper is organized as follows. Section 2 presents the detail of the proposed system model and assumptions. Section 3 analyzes the performance parameters including outage probability, throughput, ergodic rate and EE. Section 4 discusses the simulation results. Finally, Sect. 5 gives the main conclusions.
Methods
In this section, we investigate the combination of the CNOMA based system model and PSR based SWIPT technique. From the system model, we analyze and derive the performance metric in terms of outage probability, throughput, ergodic rate, and energy efficiency under constraints of direct and relay links among source, destination, and relay. We then utilize the Monte Carlo numerical approach to simulate and verify the analytical results. Furthermore, the performance metric is compared between CNOMA and OMA schemes, a direct link and relay link to clarify which scheme is superior.
System model
Figure 1 describes the model system with one source S and two users \(D_{1}\) and \(D_{2}\). These two users receive the transmitted signal from the source S. Because \(D_{2}\) is far from S, so \(D_{1}\) also helps S forward the information to \(D_{2}\). According to [40], the BS first broadcasts a threshold signal to all users in its coverage. One bit, i.e., 0 or 1, is used by the users to give the feedbacks of their received signal strength to the BS by comparing its own signal to the threshold signal. The threshold value therefore needs to be chosen carefully. In [41], optional thresholds for systems with different power constraints were also developed. After receiving the feedback bits from the users, the BS decides which user is a strong or weak user and sends this information to the users. In this model, the energy is harvested from the RF signal received at the relay. The harvested energy is stored in the battery which is a finite energy source. A part of this energy is used for the operation of the relay, while the remaining part is stored in the battery. It is assumed that the capacity of the battery is finite. Specifically, to maintain the operation, \(D_1\) harvests the energy from S by employing the PSR protocol, while the DF scheme is employed to decode and forward the information from S to \(D_2\). This system model can also be applied in wireless sensor and mobile networks where the nodes and/or users experience urban with high buildings e.g., some system models in [42,43,44,45,46,47].
Energy harvesting and data transmission protocol
Figure 2 shows the power splitting (PS) protocol for EH and IP. Specifically, in the first T/2, S sends data to both \(D_{1}\) and \(D_{2}\) and \(D_{1}\) harvests the energy from S with a part of the received signal power of \(\beta {P_S}\). Because the data is already sent to \(D_{2}\) in the first T/2, it is not necessary to resend the same information to \(D_{2}\) in the second time slot, unless S wants to send new information but it is beyond the scope of this work. In other words, \(D_{2}\) only decodes the information from \(D_{1}\) with a part of the remaining signal power of \(\left( {1\beta }\right) {P_S}\) in the second time slot and employs MRC with the one received from S in the first time slot.
Table 1 lists the definition of the parameters used in the model and through the paper.
Energy harvesting at \(D_{1}\)
We know that the amount of harvested energy from RF signal is small. However, the main power of the relay is the battery and the harvested energy is stored in the battery. Following the same approach as in [48], in this paper, we assumed that \(D_{1}\) uses the total harvested energy to relay its detected message of \(D_{2}\) and this harvested energy is sufficient for data transmission and processing.
The observation at \(D_{1}\) is determined based on SC as follows
The assumptions are given that \(E[{x_1^2}] = E[{x_2^2}] = 1\), and, without loss of generality, \({\alpha _2}>{\alpha _1}>0\) satisfying \({\alpha _1}+{\alpha _2}=\)1.
As shown in Fig 2, \(D_1\) only harvests the energy from S during the first T/2. The harvested energy at \(D_1\) is thus computed by
From (2), we understand that the operation of the energy harvesters can occur in the nonlinear region. Several works have also investigated the nonlinearity of practical energy harvesting circuit [49, 50]. However, to overcome this challenge in our work for practical problems as well as to scope with our study, several EH circuits can be placed in parallel to yield a sufficiently large linear conversion region [51, 52].
The harvested energy is dissipated at \(D_1\) and used to forward the decoded data to \(D_2\). The transmitted power at \(D_1\) obtaining from the harvest energy EH is determined by
We can see that Eqs. (2) and (3) represent for linear energy harvesting circuits. However, as mentioned above, the nonlinear region can appear in the operation of the energy harvester. Various studies in the recent trends have also emphasized on the linear architecture of the energy harvester [43, 44]. Therefore, to solve the nonlinear issue of the EH circuits based on linear region, we can yield a sufficiently large linear conversion region via parallel EH circuits.
Information processing at \(D_1\) and \(D_2\)
In fact, the NOMA scheme is only enabled when the strong/weak users can be identified. Hence, channel state information (CSI) of the links from S to \(D_{1}\) and from \(D_{1}\) to \(D_{2}\) is critical. In order to share these CSI, there are different approaches [53, 54], among which a typical one is to use pilot sequences.
Based on NOMA principle, \(D_{2}\) is allocated more power than \(D_{1}\). By applying SIC [14], \(D_{1}\) decodes both signal \(x_{2}\) and its own signal \(x_{1}\). It is assumed that the SIC is perfect. From (1), the received signal to interference plus noise ratio (SINR) at \(D_{1}\) to detect \(x_{2}\) of \(D_{2}\) is determined by
The interference is not in the received signal at \(D_1\) after SIC process. Therefore, the received SNR at \(D_{1}\) to detect its own message \(x_{1}\) is given by
Over direct link, the signal at \(D_{2}\) is given by
Therefore, the received SINR at \(D_{2}\) to detect \(x_2\) for the direct link is given by
Over the relaying link, the decoded signal \(x_2\) at \(D_1\) is forwarded to \(D_{2}\). Thus, the received signal at \(D_{2}\) can be expressed as
Substituting Eq. (3) into Eq. (8), we obtain
The received SNR at \(D_2\) over the relaying link is thus expressed by
At \(D_{2}\), the signals from both relaying and direct links are combined by employing the MRC mechanism. The combined SINR can be obtained by
Performance Analysis
This section presents the analysis of the performance of the system model in which closedform expressions of the outage probability, throughput, ergodic rate and EE are determined in DTT and DLT modes.
Outage performance
Outage probability at \(D_{1}\)
User \(D_1\) is not in outage when it can decode both signals \(x_1\) and \(x_2\) received from the BS. The outage probability at \(D_1\) is thus obtained by
where, \(\gamma _{t{h_1}} = {2^{2{R_1}}}  1\) and \(\gamma _{t{h_2}} = {2^{2{R_2}}}  1\) represent the threshold SNRs at \(D_{1}\) for detecting signals \(x_1\) and \(x_2\), respectively.
Theorem 1
The outage probability at \(D_{1}\) is given by
where, \({\theta _1} = \max ({\tau _1},{\nu _1}),{\tau _1} = \frac{{\gamma _{th2}}}{{\rho {\psi _I}({\alpha _2}  {\alpha _1}\gamma _{th2})}}\) and \({\nu _1} = \frac{{\gamma _{th1}}}{{{a_1}{\psi _I}\rho }}\) with \({\alpha _2} > {\alpha _1}\gamma _{th2}.\)
Proof
From (12), the outage probability at \(D_{1}\) can be determined by
Applying the following equation
Eq. (14) can be obtained as follows
The proof is completed. \(\square\)
Corollary 1
From (15), the outage probability at \(D_{1}\) for high SNR \(\rho \rightarrow \infty\) is expressed by
Proof
From (12), when \(\rho \rightarrow \infty\), the outage probability at \(D_{1}\) with \(1  {e^{x}} \approx x\) is given by
The proof is completed. \(\square\)
Based on (15) and \({\alpha _2} > {\alpha _1}\gamma _{th2}\), \(P_{D_1}\) depends on \(\tau _{1}\) and the random variable \(\Omega _{1}\) (\(h_1^2\)). The closer the d, the lower the \(P_{D_1}\). This means that a better transmission quality can be achieved, and vice versa.
Outage probability at \(D_{2}\) for no direct link
Since \(D_1\) can not detect \(x_2\) as well as \(D_2\) can not recover the forwarded information from \(D_1\), the \(D_2\) is in outage. Hence, the outage probability at \(D_2\) is derived as (see (18)). By calculating \(J_2\) and \(J_3\), the outage probability for no direct link is determined by
Theorem 2
The outage probability at \(D_{2}\) can be obtained by
Proof
Considering the Rayleigh fading channel, \(J_2\) can be given by
and \(J_3\) can be expressed as (see(21)).
The outage probability at \(D_2\) is given by
\(\square\)
Corollary 2
The outage probability at \(D_{2}\) for high SNR can be determined as (see(23)), where \({K_1}(.)\) is the first order modified Bessel function of the second kind [55, Eq.(3.324.1)].
Outage probability at \(D_{2}\) for User Relaying with Direct Link
When \(x_{2}\) can be detected at \(D_{1}\) but the SINR is smaller than the target SNR after MRC or both \(D_{1}\) and \(D_{2}\) can not detect \(x_{2}\), the outage probability will occur at \(D_{2}\) and is given by (see(24))
Theorem 3
The outage probability at \(D_{2}\) can be given by (see(25))
Proof
From (24), the outage probability at \(D_{2}\) is determined by (see(26)), (see(27)), (see(28))
\(\square\)
Throughput for DLT mode
User relaying without direct link
With a given constant R, the transmitted information of the source node depends on the outage probability performance due to wireless fading channels. Therefore, the throughput of the system is determined by
where \(P_{D_1}\) and \(P_{{D_2},nodir}\) can be achieved from (15) and (19), respectively.
User relaying with direct link
The throughput of system is given by
where \(P_{{D_1}}\) and \(P_{{D_2},dir}\) can be achieved from (15) and (25), respectively.
Ergodic rate for DTT mode
Ergodic rate at \(D_{1}\)
The achievable rate at \(D_{1}\) where \(D_{1}\) can detect \(x_{2}\) is given by
Theorem 4
The ergodic rate at \(D_{1}\) is determined by
where Ei(.) indicates the exponential integral function [55, Eq.(3.354.4)].
Proof
See Appendix 1. \(\square\)
Ergodic rate at \(D_{2}\) for User Relaying Without Direct Link
Since \(x_2\) needs to be detected at both \(D_1\) and \(D_2\), the achievable rate at \(D_{2}\) is given by
Theorem 5
The ergodic rate at \(D_{2}\) is given by
Proof
See Appendix 2 . \(\square\)
Remark 1
The ergodic rate in the asymptotic expression at \(D_{2}\) for high SNR region \(\rho \rightarrow \infty\) is obtained by
From the analytical result in (35), this expression can be deployed by
Proof
See Appendix 3. \(\square\)
Ergodic rate at \(D_{2}\) for user relaying with direct link
The ergodic rate at \(D_{2}\) is given by
Theorem 6
From (37), the ergodic rate at \(D_{2}\) can be computed by (see(38))
Proof
See Appendix 4. \(\square\)
Remark 2
The ergodic rate in the asymptotic expression at \(D_{2}\) for high SNR region \(\rho \rightarrow \infty\) is given by
From (39), this expression can be deployed by
Proof
See Appendix 5. \(\square\)
Ergodic rate of the system for user relaying without direct link
The ergodic rate of system is determined by
where \(R_{{D_1}}\) and \(R_{{D_2},nodir}\) can be obtained from (32) and (34), respectively.
Ergodic rate of the system for user relaying with direct link
The ergodic rate of system is thus expressed by
where \(R_{{D_1}}\) and \(R_{{D_2},dir}\) can be obtained from (32) and (38), respectively.
Energy efficiency
The EE can be determined as the ratio of the total data rate over the total consumed power in entire network, which is given by \(\mathrm{{EE}} \buildrel \Delta \over = \frac{R}{{{P_S} + {P_r}}}\). The energy efficiency of user relaying systems can be given as
where \(\phi \in \left( {t,r} \right)\), denotes the system energy efficiency in DLT mode and DTT mode, respectively.
Results and discussion
Simulation parameters and scenarios
In this section, Monte Carlo based simulation scenarios are performed in Matlab to verify the derived analytical results. The simulation parameters of the system model are listed in Table 2. In addition, we use the conventional OMA as a counterpart for comparison. The operation of this scheme is described as follows. During the first phase of time block, the information \(x_{1}\) is transmitted to \(D_{1}\) by S. During the second phase of time block, the information \(x_{2}\) is sent to \(D_{1}\) by S. Lastly, during the third phase of time block, \(D_{1}\) decodes and then forwards \(x_{2}\) to \(D_{2}\).
Outage probability versus SNR and \(\beta\)
Figure 3 describes the relation between the outage probability and SNR of from −10 to 40 dB in case of \(\beta =0.7\). As shown in the figure, User 1 obtains a higher outage probability than User 2. In particular, with increasing SNR, the outage probability of both users decreases approximately linear. As a result, the gap width of these curves is bigger more and more. It means that the User 2 achieves a higher outage probability than User 1 in high SNR regime. Compared with OMA in high SNR regime, it is seen that the outage probability of User 1 for CNOMA without direct link is lower. Besides, the outage probability of User 1 for OMA is also higher than that for CNOMA in high SNR. Moreover, the User 2 for CNOMA achieves a lower outage probability than that for OMA in both no direct and direct link cases. However, compared between direct link and no direct link, the CNOMA scheme with direct link achieves a lower outage probability than CNOMA scheme without direct link. The indicated curves for these cases correspond with the makerred line and blue line in the figure. Obviously, the outage probability for CNOMA with direct link is lowest as compared to both CNOMA without direct link and OMA. It can be explained that the received information at User 2 in case of direct link existence between the BS and User 2 includes the one sent from BS and other one sent from the relay user 1. Thus, dropped package percentage at User 2 is lower as compared to the case of no direct link existence. In addition, we can base on the Eqs. (13), (14), (16), (18), (19), (23), (24), (25) to explain for these features. Therefore, it is shown that the CNOMA scheme with direct link has a low outage probability over the CNOMA without direct link as well as OMA.
Similarly, Fig. 4 shows the outage probability versus \(\beta\), where beta is from 0 to 1. It is observed that the outage probability of the User 2 for CNOMA scheme with direct link is the lowest among the curves for CNOMA without direct link and OMA for both users. For comparison of the outage probability of User 1 for both CNOMA and OMA, we can see that the curve for CNOMA is always below that for OMA and thus the probability of User 1 for the CNOMA achieves better than that for OMA. Moreover, the outage probabilty between direct and no direct links is different in each user. This is shown in the maker red curve and the maker green curve in the figure corresponding with direct OMA and without direct OMA and that the outage probability for direct OMA is much lower than that for without direct OMA. Besides, the probability of User 2 for OMA with direct link is lower than that for CNOMA without direct link. This means that the OMA scheme is better than the CNOMA without direct link. However, the outage probability of User 2 for CNOMA with direct link is lower than that for OMA with direct link. Therefore, we can conclude that the CNOMA with direct link outperforms the CNOMA without direct link as well as OMA. These can be explained similarly to Fig. 3 as well as based on Eqs. (12), (13), (16), (18), (19), (23), (24), (25). In addition, it can be seen that when the \(\beta\) is in range of from 0.1 to 0.65, the outage probability of User 2 for CNOMA with direct link is considerably lower as compared to others. In range of from 0.65 to approximately 0.9 and from 0 to 0.1, this probability decreases and increases gradually, respectively. As a result, the gap width between the curve of the outage probability of User 2 for CNOMA with direct link and the others achieves the biggest in range of 0.1 to 0.65. It means that we can choose \(\beta\) in this range to obtain a better outage probability for the system model.
Throughput and ergodic versus SNR and \(\beta\)
Figure 5 decribes the throughput versus \(\beta\) in from 0 to 1. From the figure, it is shown that the throughput of User 1 for CNOMA is considerably higher than that for OMA. However, this throughput trends to decrease quickly when the \(\beta\) towards to 1. This is because that as the power splitting ratio \(\beta\) increases, the power allocated to User 1 is lower, thus resulting in a decreased throughput of User 1. Furthermore, the throughput of User 2 for CNOMA with direct link is the highest among the graphs illustrating for User 2 in cases of CNOMA and OMA. We can see that the maker blue line of User 2 for CNOMA with direct link is higher and approximately constant as compared that for CNOMA without direct link as well as OMA, i.e. the maker green line and the maker magenta line, respectively. Besides, the throughput of User 2 for these three cases is almost constant in range of \(\beta\) from about 0.1 to 0.8. This means that we can choose a suitable value of \(\beta\) to satify the tradeoff between the throughput of User 1 and the throughput of User 2 for the system model. Additionally, the figure also shows that the throughput of User 2 for CNOMA is lower than that for CNOMA without direct link. Thus, the CNOMA scheme with direct link is superior to the CNOMA without direct link and CNOMA.
Figure 6 plots the ergodic rate as a function of \(\beta\) in from 0 to 1. The figure shows that the ergodic rate of User 1 for CNOMA is considerably higher than that for OMA. However, these curves degrade quickly when the \(\beta\) towards to 1. The reason is similar to Fig. 4. It can be also explained according to Eqs. (32), (34), (36), (38). Besides, the ergodic rate of User 2 for CNOMA with direct link is the highest as compared to that of User 2 in cases of CNOMA and OMA. Specifically, the ergodic rate of User 2 for CNOMA with direct link is about four times higher than that for OMA. Futhermore, the ergodic rate of User 2 is almost constant in range of \(\beta\) from about 0.1 to 0.9. Therefore, one can choose a suitable value of \(\beta\) to satify the tradeoff between the ergodic rate of User 1 and the ergodic rate of User 2 for the system model. From Eqs. (32), (34), (36), (38), the ergodic rate of User 2 depends on the \(\beta\) less than that of User 1. Thus, the CNOMA scheme with direct link outperforms the CNOMA without direct link and OMA.
Energy efficiency
Figure 7 illustrates the energy efficiency according to SNR from 10 to 40 dB. It is shown that the EE for CNOMA with direct link achieves much higher than that for CNOMA without direct link and OMA. In particular, with the SNR range of from 10 to 5 dB, this energy efficiency is large but linearly decreases as the SNR value increases. When SNR range is from 5 to 40 dB, the energy efficiency for both CNOMA with and without direct link and OMA asymptotically decreases to 0. Therefore, it can be concluded that the CNOMA scheme with direct link provides a better EE as compared among the schemes in low SNR regime.
Therefore, the simulation results are in accordant with theoretical calculations.
Conclusion
In this paper, a proposed EH scheme for SWIPT CNOMA has been presented. The closedform expressions of the performance were derived. The analytical results shown that the CNOMA with direct link obtained a better outage probability over the OMA as well as CNOMA without direct link. Numerical results provided that CNOMA with direct link outperformed throughput and ergodic rate than OMA. Besides, the CNOMA with direct link obtained a higher EE performance than the CNOMA without direct link and OMA in low SNR region. The proposed system model can be applied for CNOMA based wireless sensor networks where relaying sensor nodes can harvest RF energy from BSs to maintain their operation and assist to forward information to other sensor nodes. Futhermore, we can develope the system using multiple antennas or combine relay selection with multiple antennas to enhance the performance of the system.
Availability of data and materials
Data sharing not applicable to this article as no datasets were generated or analysed during the current study.
Abbreviations
 Acronym:

Definition
 SWIPT:

Simultaneous wireless information and power transfer
 CNOMA:

Cooperative nonorthogonal multiple access
 EH:

Energy harvesting
 IP:

Information processing
 BS:

Base station
 NU:

Near user
 FU:

Far user
 OP:

Outage propability
 EE:

Energy efficiency
 RF:

Radiofrequency
 PSR:

Powersplitting relaying
 DF:

Decodeandforward
 SC:

Superposition coding
 SIC:

Successive interference cancellation
 OMA:

Orthogonal multiple access
 TSR:

Timeswitching relaying
 DLT:

Delaylimited transmission
 DTT:

Delaytolerant transmission
 MRC:

Maximum ratio combining
 SNR:

Signaltonoise ratio
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HQT: Conceptualization, methodology, software, formal analysis, investigation. HQT: Data curation, WritingOriginal draft preparation. HQT, CVP and QTV validation, resources. HQT, CVP and QTV: WritingReviewing and Editing. CVP and QTV supervision. All authors read and approved the final manuscript.
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Appendices
Appendices
Appendix 1: Proof of Theorem 1
In this appendix, the proof of (32) is presented. To achieve this closedform expression, the ergodic rate of \(D_{1}\) for HD NOMA can be expressed as
The cumulative distribution function (CDF) of X is computed as follows
Substituting (42) into (43), the ergodic rate of \({D_1}\) can be derived as
We can derive (32). The proof is completed.
Appendix 2: Proof of Theorem 2
In this appendix, the proof starts by giving the ergodic rate of \(D_{2}\) as follows
The CDF of X is calculated as follows (see(44))
\(I_{3}\) and \(I_{4}\) are given by (see(45)) and (see(46))
And \({I_{41}}\) and \({I_{42}}\) are computed by (see(47)) and (see(48))
And
where U(x) is unit step function as
From (47) and (48), we have (46). Substituting (45) and (46) into (44), the CDF of X is given by (see(47)).
By replacing (49) into (33), we can obtain (34).
The proof is completed.
Appendix 3: Proof of Remark 1
The proof starts by giving the ergodic rate of \(D_{2}\) for the high SNR regime as follows
The CDF of X is computed as follows
where U(x) is unit step function as
Substituting (51) into (50), we can obtain \(R_{{D_2},PSR}^{\infty }\).
The proof is completed.
Appendix 4: Proof of Theorem 6
In this appendix, the proof starts by giving the ergodic rate at \({D_2}\) for direck link as follows
The CDF of X is calculated as follows (see (53)).
\({I_{61}}\) is computed by (see(54)).
where U(x) is unit step function as
Substituting (54) into (53), we can achieve (see(55)).
Substituting (55) into (52), we can achieve \(R_{D_2}\).
The proof is completed.
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Tran, H.Q., Phan, C.V. & Vien, QT. Performance analysis of powersplitting relaying protocol in SWIPT based cooperative NOMA systems. J Wireless Com Network 2021, 110 (2021). https://doi.org/10.1186/s13638021019819
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DOI: https://doi.org/10.1186/s13638021019819
Keywords
 Nonorthogonal multiple access (NOMA)
 Energy harvesting (EH)
 Information processing (IP)
 Radiofrequency (RF)
 Powersplitting relaying (PSR)
 Decodeandforward (DF)