PSR versus TSR Relaying Protocols: Leveraging Full-DuplexDF and Energy Harvesting for SWIPT in NOMA Systems

In this study, we introduce a cooperative non-orthogonal multiple access (NOMA) system model for simultaneous wireless information and power transfer (SWIPT) with the assistance of a relay node, all within the context of full-duplex (FD) communication mode. Within this framework, we consider the utilization of two distinct protocols: the power splitting protocol (PSR) and the time switching protocol (T-SR). These protocols are designed to accommodate both delay-limited-transmission (DLT) and delay-tolerant-transmission (DTT) modes, enabling efﬁcient energy harvesting and information processing at the relay user. The paper also presents comprehensive closed-form expressions for various performance metrics, including outage probability, throughput, ergodic rate, and energy efﬁciency (EE). Furthermore, we explore the impact of key parameters such as energy harvesting time, power splitting ratio, energy harvesting efﬁciency, source data rate, and inter-node distance across normal and high signal-to-noise ratio (SNR) regimes. Our ﬁndings demonstrate that enhanced energy harvesting efﬁciency leads to improved overall system performance. Notably, our results underscore the superiority of NOMA over conventional orthogonal multiple access schemes, particularly in terms of energy efﬁciency.


Introduction
The rapid expansion of wireless communication has prompted the exploration of innovative solutions to address the limitations encountered in evolving generations of mobile networks (nG, where n = 1, 2, 3, 4).These limitations encompass factors like spectral efficiency, energy efficiency, and latency, among others [1]- [3].Non-Orthogonal Multiple Access (NOMA) has emerged as a promising contender to mitigate these constraints.
NOMA introduces a mechanism that enables multiple users to share the same time and frequency resources effectively.Within NOMA, three distinct categories exist: power domain NOMA, code domain NOMA, and hybrid domain NOMA.In the context of power domain NOMA, the allocation of power factors to individual users is contingent upon their respective channel conditions [2].In simpler terms, users in proximity to the base station (BS), often referred to as "near users," are assigned lower power allocations compared to users located farther away from the BS, known as "far users."This approach is adopted to ensure fairness among users [1], [4].Power domain NOMA incorporates two fundamental techniques: Superposition Coding (SC) [2] and Successive Interference Cancellation (SIC) [1], [2].These techniques play pivotal roles in enhancing the efficiency and performance of power domain NO-MA systems.
Full-duplex (FD) communications represent a significant advancement in wireless technology, with the primary objective of enhancing spectral efficiency and thereby improving throughput within the same bandwidth, as opposed to the traditional Half-duplex (HD) mechanism.In half-duplex communication, transmission and reception occur in different time slots or frequency subbands [5].In contrast, FD NOMA enables the simultaneous transmission and reception of information over the same frequency resources and within a single time frame.This concurrent operation results in a potential doubling of the achievable spectral efficiency [5].Consequently, FD NOMA has garnered considerable attention as one of the promising techniques for the development of 5G and future wireless communication systems [6].However, a key challenge associated with FD wireless communication is the substantial power difference between the Self-Interference (SI) generated by a device's own transmissions and the desired signal received from a source [5], [7].This SI can significantly degrade the performance of the FD mode.Nonetheless, in certain scenarios, the SI may be tolerable [8].Finding ways to mitigate and manage this self-interference is a crucial focus in the advancement of FD communication technology.Cooperative communication employing high-performance, HD relaying mode has garnered significant attention within the wireless communication community.This approach is valued for its ability to efficiently harness power and bandwidth resources while simultaneously maximizing the capacity of wireless links, all achieved through hardware simplicity [9], [10].Furthermore, the integration of wireless power transfer and energy harvesting has become a focal point of research within the realm of cooperative relaying systems [11].In the context of energy-constrained wireless cooperative communication systems, a concept known as simultaneous wireless information and power transfer (SWIPT) comes into play.SWIPT is strategically employed at relay nodes to accomplish a dual purpose: firstly, to harvest energy for the relay's consump-tion, and secondly, to facilitate Message Processing (MP) for the ultimate destination node.This synergy between power transfer and information exchange represents a promising avenue for enhancing the efficiency and sustainability of cooperative wireless communication systems.In our work, we have leveraged two essential protocols for SWIPT, namely the Power Splitting Protocol (PSR) and the Time Switching Protocol (TSR) [12]- [14].Additionally, in a related study [15], a SWIPT-enhanced cooperative NOMA system was introduced, demonstrating the versatility of SWIPT in various communication scenarios.Furthermore, our investigation in [16] delved into a FD communication system featuring a single FD multi-antenna BS and multiple HD single-antenna users.Simulation results indicated the superiority of our proposed scheme compared to our previous work, particularly in terms of achieving higher sum rates.Moreover, in [17], researchers proposed the use of Amplify-and-Forward (AF) FD relays within millimeter-wave-based wireless backhaul links connecting cellular base stations.This approach presents another application of FD technology in improving wireless communication infrastructure.In the context of [7], a novel adaptation scheme was introduced that combined beamforming, energy harvesting (EH), and cooperative NOMA techniques.This scheme was designed to efficiently accommodate a larger number of users within each beamforming vector.To provide a comprehensive analysis, closed-form expressions were derived for the outage probability (OP) for both weak and strong users.This research aimed to optimize the utilization of resources to enhance the performance of wireless communication systems.In a separate study discussed in [18], the focus was on investigating optimal resource allocation strategies within a wireless-powered communication network.The research aimed to identify the most efficient methods for allocating resources, such as spectrum, power, or bandwidth, to maximize the overall system performance in a network where devices are powered wirelessly.This research is crucial for designing and optimizing wireless-powered communication systems for various applications.In the exploration of SIC at the High Altitude Platform (HAP), both scenarios with perfect and imperfect SIC were thoroughly investigated.These investigations aimed to derive optimal solutions for time and power allocation in order to optimize the performance of the communication system, considering the presence of self-interference.These findings contribute to enhancing the efficiency of communication systems by addressing the challenges posed by self-interference, whether it can be fully canceled or not.Furthermore, in another research endeavor discussed in [19], a FD aided cooperative NOMA scheme was introduced.The primary objective of this scheme was to enhance the outage performance of the weak user in device-to-device (D2D) communication scenarios.By leveraging FD technology and NOMA principles, this scheme aimed to improve the reliability and efficiency of communication for weaker users in D2D setups, which is a critical aspect of modern wireless communication systems.The results obtained from the proposed scheme showcased its superior performance in terms of outage compared to both Orthogonal Multiple Access (O-MA) and conventional NOMA methods.This indicates that the proposed approach is capable of achieving more reliable and efficient communication compared to traditional techniques.In another study discussed in [20], a dual-hop FD relaying network was examined, where two relaying strategies, Decode-and-Forward (DF) and AF, were considered.This research aimed to provide a comprehensive understand-ing of the system's performance by deriving exact closed-form expressions for the outage probability.Such analytical insights are crucial for evaluating and optimizing the reliability of dual-hop FD relay networks, especially when considering different relay strategies such as DF and AF [21]- [23].The FD scenario has been demonstrated to outperform HD relaying systems, highlighting its advantages in terms of improved communication efficiency and capacity.Additionally, in studies conducted by the authors of [4], [24]- [26], the concept of FD Cooperative NOMA with SWIP-T was explored.This research likely delved into the integration of FD technology, NOMA, and SWIPT to investigate how these techniques can synergize and enhance the performance of cooperative communication systems.Furthermore, the PSR and TSR protocols employ two distinct data transmission modes, namely Delay-Limited-Transmission (DLT) and Delay-Tolerant-Transmission (DTT) modes [27].It's important to note that the DLT mode does not allow for the buffering of received data during the transmission process, whereas the DTT mode permits data buffering at the receiver.Consequently, in the DLT mode, data is decoded incrementally, while in the DTT mode, data is decoded using a block-wise signal decoding mechanism.These different modes cater to various communication scenarios and requirements within the context of the PSR and TSR protocols.In this research paper, we delve into the analysis of a FD relaying system that integrates SWIPT within a cooperative NOMA framework.The system configuration comprises a single BS and two users, namely the "close user" and the "far user," all of which operate in FD communication mode.Our investigation centers around a scenario where there is no direct communication link between the BS and the far user, primarily because of the presence of an obstructing obstacle or physical barrier.This challenging scenario adds complexity to the system design and necessitates specialized strategies for information and power transfer in a cooperative NOMA setup.In our research, we undertake a comprehensive analysis by conducting simulations and comparisons between two prominent protocols, namely the PSR and the TSR, as well as between NOMA and OMA schemes.We base our evaluations on a range of crucial performance metrics, including outage probability, throughput, ergodic rate, and energy efficiency.Additionally, we consider the influence of several critical parameters, such as the power splitting ratio, EH duration, EH efficiency, inter-node distance, and source data rate.The primary contributions of our work in this paper can be summarized as follows: -In our study, we introduce a novel approach that bridges PSR and TSR protocols, incorporating DTT and DLT modes, and employs FD-based SWIPT for cooperative NOMA, all without the need for a direct link.-We have formulated closed-form expressions for the OP, throughput, ergodic rate, and energy efficiency (EE) in the context of FD cooperative NOMA utilizing the PSR and TSR protocols within both DLT and DTT modes.-Our study encompasses a comprehensive comparative analysis that spans multiple performance metrics, including OP, throughput (TP), ergodic rate (ER), and EE, across different operational modes.Specifically, we investigate and contrast these metrics within the context of FD mode, NOMA, and OMA.-In our research, we assess the influence of various parameters, including EH time and power allocation coefficients, on the FD cooperative NOMA system.The paper's structure is outlined as follows: In Section II, we introduce the complete system model and make explicit our assumptions.Section III is dedicated to the analysis of performance parameters, encompassing outage probability, ergodic rate, throughput, and energy efficiency.Section IV presents and discusses the simulation results.Lastly, in Section V, we present the primary conclusions drawn from our study.Figure 1 illustrates the system model under examination.This model comprises a source node labeled as B and two users, namely R 1 and R 2 .In this setup, R 1 , the closer user, serves as a relaying user, while R 2 , the more distant user, receives assistance from R 1 for communication with S. R 1 employs the DF protocol to decode the information and forward it to R 2 .Additionally, R 1 utilizes the PSR and TSR protocols to harvest radio-frequency (RF) energy and process information transmitted from the source node B. The objective is for B to establish communication with both R 1 and R 2 to facilitate the transfer of their respective information.Our assumption considers the presence of an obstacle obstructing direct communication between B and R 2 .Consequently, B is unable to transmit data directly to R 2 .Instead, R 1 takes on the role of a user relaying, aiding B in transmitting data to R 2 .In the context of FD communication, both B and R 2 are equipped with a single antenna each, whereas R 1 possesses one receive antenna and one transmit antenna.We denote c 1 and c 2 as the complex channel coefficients representing the channels from B → R 1 and from R 1 → R 2 , respectively.The power gains, denoted as |c 1 | 2 and |c 2 | 2 , are assumed to follow exponential probability distributions.Specifically, the expected values of these power gains are defined as

SYSTEM MODEL
, respectively.In this context, the expectation operation, denoted as E[.], signifies the average or mean value of the specified quantity.The LI is represented as a Rayleigh fading channel with a coefficient c LI , and Ω LI denotes the average power associated with this channel.

EH at R 1
At R 1 , we sequentially evaluate two EH protocols, which include the PSR-based R 1 and the TSR-based R 1 .

EH at PSR
Fig. 2: The PSR Protocol for energy harvesting and message transmission paradigm.
Figure 2 illustrates the communication block diagram employing the PSR protocol for EH and MP at R 1 within the total time block T .The power of the received signal at R 1 is represented as P. We assume that during the first half of T , B transmits information to R 1 , leaving the remaining time, T /2, for transmitting information from R 1 to R 2 .In the PSR protocol, we establish two layers: the EH layer and the MP layer.The signals utilized for energy harvesting are as follows: By employing the superposition of transmitted signals at B, similar to the NOMA scheme [22], the observation at R 1 can be expressed as follows: Here, P S represents the transmission power at B, b 1 and b 2 are power allocation coefficients associated with the data symbols x 1 and x 2 that are intended to be sent from B to R 1 and R 2 , respectively, n R 1 signifies the additive white Gaussian noise (AWGN) at R 1 , which has zero mean and a variance of σ 2 , Ψ I , and Ψ I represents the MP coefficient in both the PSR and TSR protocols, and In accordance with the assumptions provided: Furthermore, without loss of generality, we assume: b 2 > b 1 > 0 satisfying the condition b 1 +b 2 =1.Under the PSR protocol, R 1 divides the received power into two parts: i) harvested energy and ii) MP energy.Let β , where 0 < β < 1, represent the power splitting ratio.The energy harvested at R 1 can be calculated as follows: The equation provided describes the energy harvested at D 1 during the EH phase.In this equation: η, where 0 < η < 1, represents the EH efficiency at the energy receiver.This efficiency factor accounts for the losses in the rectifier and the EH circuitry, indicating how effectively the harvested energy is converted and utilized.It's worth noting that all the energy harvested during the EH phase is entirely consumed at R 1 while forwarding the decoded signal to R 2 , ensuring efficient utilization of the harvested energy.Figure 3 illustrates the key parameters involved in the TSR protocol for EH and MP at the relay.The energy harvested at R 1 can be determined as follows: It is important to note that all the energy harvested during the EH phase is fully utilized by R 1 while forwarding the signal from B to R 2 .This efficient utilization ensures that the harvested energy contributes directly to the relay operation and the communication process without waste.
In the TSR protocol: T represents the total duration of the time block during which information is transmitted from B to R 2 .The parameter α, where 0 < α < 1, signifies the fraction of the time block during which R 1 harvests energy from the signal transmitted by B. Specifically, the first sub-block of time, αT , is dedicated to the EH process and the exchange of information between B and R 1 .The remaining portion of the time block, (1 − α) T , is allocated for transmitting information.Within this duration, (1 − α) T /2 is utilized for the transmission of information from B to R 1 , and the remaining half, (1 − α) T /2, is used for receiving information at R 2 .This partitioning of time ensures efficient EH and communication processes within the TSR protocol.

MP at R 1 and R 2
Following the NOMA principle, R 2 is allocated more power compared to R 1 .After receiving the signal from B, R 1 performs decoding of the signal x 2 and subsequently decodes its own signal x 1 using SIC [23].From equation ( 1), the signal-tointerference-plus-noise ratio (SINR) at R 1 , which is used to detect x 2 transmitted by R 2 , can be expressed as: Indeed, ρ, where ρ ∆ = P S σ 2 , represents the transmit signal-to-noise ratio (SNR).After the SIC process, where all interference is effectively eliminated from the received signal at R 1 , the received SNR at R 1 for detecting its own message x 1 can be expressed as: Meanwhile, the decoded signal x 2 at R 1 is relayed to R 2 .The received signal at R 2 can be expressed as follows: In the provided equation, Ψ E represents the EH coefficient in both the PSR and TSR protocols, and The received SNR at R 2 can be expressed as: 3 PERFORMANCE ANALYSIS OF THE PSR PROTOCOL

OP at D 1
In the NOMA protocol, R 1 is not in outage when it can successfully decode both x 1 and x 2 received from B. Therefore, the OP at R 1 can be expressed as follows: Here, In this context, R 1 and R 2 represent the target rates for the detection of x 1 and x 2 at R 1 .It is worth noting that we have made an important discovery concerning the OP at R 1 .
Theorem 1 The OP at D 1 is given by where Proof See Appendix A.
Corollary 1 As the SNR approaches infinity, in the high SNR regime (ρ → ∞), the OP at R 1 in the context of FD NOMA can be derived as follows: (12) Proof See Appendix B.

OP at R 2
Much like the R 1 node, the OP at R 2 is influenced by the following factors: R 1 's inability to receive x 2 or R 1 's ability to detect x 2 when R 2 cannot detect it.

Theorem 2
The OP at R 2 can be expressed as: Based on equation (13), we can implement the following approach, as outlined in equation (14).
Proof See Appendix C.
Corollary 2 In the high SNR regime, as ρ tends towards infinity, the OP for R 2 in the context of FD NOMA can be characterized as follows (refer to equation 15):

TP for Delay-limited Transmission Mode
In this operational mode, the S node transmits information at a consistent rate denoted as R, contingent upon the performance of the OP arising from wireless fading channels.The total system TP of the FD transmission mode within the NOMA system is determined by: where P FD R 1 ,PSR and P FD R 2 ,PSR can be obtained from (10) and (13), respectively.

ER at R 1
In the event that R 1 possesses the capability to detect x 2 , one can express the attainable rate of R 1 as follows: By employing the integration-by-parts technique, we can rephrase this expression as follows: The ER for R 1 in the NOMA system under FD transmission mode can be derived using the following theorem: Corollary 3 As the SNR approaches infinity (ρ → ∞) in the case of FD NOMA, the ER at R 1 is determined as follows:

ER at R 2
Given that the signal x 2 must be detected at both R 1 and R 2 , the achievable rate for R 2 in the FD transmission mode of the NOMA system can be expressed as: The ER for R 2 in the FD transmission mode of the NOMA system can be derived from equation (20) using the following theorem.
Theorem 4 The ER at R 2 is represented as: Proof See Appendix E.

Corollary 4
As the SNR approaches infinity (ρ → ∞) in the context of FD NOMA, the ergodic rate at R 2 is determined as follows:

ER of the system
The system's ER in the FD transmission mode of the NOMA system is defined as: The values of R FD R 1 ,PSR and R FD R 2 ,PSR can be derived from equations (19) and (23), respectively.

OP at R 1
In the NOMA protocol, R 1 is typically capable of detecting both the x 1 and x 2 signals under normal conditions.However, there may be scenarios where R 1 is unable to receive either the x 1 or x 2 signals.Consequently, the OP at R 1 can be formulated as: Theorem 5 The OP at R 1 for FD NOMA is represented as: where Proof See Appendix A.

Corollary 5
In the limit of a high SNR, as ρ approaches infinity, the OP at R 1 for FD NOMA is determined as follows:

OP at R 2
Similar to the R 1 node, the OP at R 2 is influenced by the following reasons: either R 1 cannot receive x 2 , or R 1 can detect x 2 while R 2 cannot detect x 2 .

Theorem 6
The OP at R 2 can be expressed as: We can apply the following deployment based on equation (28), as shown in equation (29): Proof See Appendix C.

Corollary 6
As the SNR tends to infinity (ρ → ∞) in the context of FD NOMA, the OP at R 2 can be determined as indicated in equation (30):

TP for Delay-limited Transmission Mode
In this mode, the B node transmits information at a constant rate of R, which depends on the system's performance in terms of the OP resulting from wireless fading channels.The total system TP for the FD transmission mode in the NOMA system is expressed as: The values of P FD R 1 ,T SR and P FD R 2 ,T SR can be derived from equations ( 26) and (29), respectively.

ER at R 1
If R 1 has the capability to detect x 2 , we can express the achievable rate at R 1 as follows: The ER at R 1 for the FD transmission mode in the NOMA system can be determined using the theorem presented below: Theorem 7 The ER at R 1 for FD NOMA is expressed as follows: Corollary 7 In the limit of a high SNR, as ρ approaches infinity, the OP at R 1 for FD NOMA is determined as shown in the following manner:

ER at R 2
As x 2 must be detected by both R 1 and R 2 , the achievable rate at R 2 in the FD transmission mode of the NOMA system can be expressed as: Theorem 8 The ER at R 2 is defined as: Proof See Appendix E.

Corollary 8
In the limit of a high SNR, as ρ approaches infinity, the OP at R 1 for FD NOMA is determined as follows:

ER of the system
The system's ER in the FD transmission mode of the NOMA system can be determined through:

Energy efficiency
In our analysis, we intend to evaluate EE while incorporating user relaying for FD NOMA systems.EE is quantified as the ratio of the total achievable data rate to the total power consumption across the entire network, and it is given by the formula: EE ∆ = R P S +P r , where: R represents the total data rate, which is the cumulative TP from B to R 1 and from R 1 to R 2 , P S denotes the transmitted power at the source node B, P r signifies the power consumption at node R 1 .This EE metric allows us to assess the system's efficiency in terms of data transmission relative to the power expended in the network.The EE of user relaying in FD NOMA systems can be expressed as: where ϕ ∈ (l,t) , X ∈ (T SR, PSR).

SIMULATION RESULTS
Simulation results have been conducted using the Matlab tool to validate and demonstrate the analytical expressions presented in the previous sections.This empirical verification helps ensure the accuracy and reliability of the theoretical findings by comparing them with real-world performance data obtained through simulation experiments.By doing so, you can assess how well the analytical models align with the actual behavior of the FD NOMA system under various scenarios and conditions.Without sacrificing the generality of our analysis, we make the following assumptions regarding distance normalization in our setup: Specifically, the distance between BS and R 2 is set as Here, 'd' represents the normalized distance between B and R 1 , which is set to d = 0.3, and 'm' denotes the pathloss exponent, set at m = 2. Our NOMA scheme employs power allocation coefficients b 1 = 0.2 and b 2 = 0.8 for R 1 and R 2 , correspondingly.Moreover, we define our target rates as R 1 = 2 bps for D 1 and R 2 = 1 bps for R 2 .For comparison purposes in the simulation, we employ the performance of the conventional OMA scheme as a benchmark.In the OMA scheme, the following sequence of events takes place: Initially, B transmits information x 1 to user relay R 1 during the first time slot, and subsequently, in the second time slot, B transmits x 2 to R 1 .Finally, R 1 decodes and forwards the information x 2 to R 2 during the third time slot.This benchmark scenario allows us to assess and contrast the performance of the FD NOMA system against the more traditional OMA approach.We begin our analysis by examining the OP of the system, which is illustrated in Figures 4 and 5.These figures depict the OP for two users in the context of the PSR protocol, with a focus on its relationship with SNR and parameters β and α, respectively.Figure 4 illustrates that as SNR increases, the OP for both users generally decreases.However, this relationship is not linear, resulting in curves with inflection points on the graph.Specifically, the OP experiences a significant reduction in the SNR range of -10 to 30 dB.This trend suggests that as SNR grows, noise diminishes, leading to a higher likelihood of successful data transmission and, consequently, lower outage probabilities.Furthermore, in the SNR range of 30 to 40 dB, the OP for both users, under different protocols, remains relatively constant.This stability is due to the signal strength being substantial enough to overcome any noise interference.Comparing the PSR and TSR protocols, it becomes evident that the PSR protocol achieves a lower OP for both User 1 and User 2.Moreover, for both FD NOMA and FD OMA schemes, the OP for User 2 is lower than that for User 1.Additionally, FD NOMA exhibits a superior OP performance compared to FD OMA.These observations highlight the effectiveness of the PSR protocol in minimizing outage probabilities, particularly in high-SNR scenarios, and the advantages of FD NOMA over FD OMA in achieving lower outage probabilities.In a similar fashion, Figure 5 illustrates the OP with respect to β = α for both FD NOMA and OMA for two users, considering both the PSR and TSR protocols.The OP for both users exhibits gradual variations, resulting in inflection points on the graph.These changes are indicative of how different values of β and α affect the OP.Observations from Figure 5 reveal that the OP for User 1 is highest when using FD OMA with TSR, whereas the OP for User 2 is lowest when employing FD NOMA with PSR.Across these curves, it's evident that FD NOMA consistently outperforms FD OMA in terms of OP.Furthermore, the PSR protocol generally yields a lower OP compared to the TSR protocol.It's worth noting that the OP of User 1 remains relatively constant in the high SNR regime, hovering around 10 − 1.This behavior is attributed to the fact that β and α have minimal impact on the OP of User 1 in the high SNR regime, as indicated by equation (12).These outcomes can also be comprehensively explained by considering the mathematical expressions provided in equations ( 10)-( 15) and ( 25)-(30).In summary, the figures demonstrate how different relay protocols (PSR and TSR), user parameters (β and α), and SNR levels influence the OP for FD NOMA and FD OMA, with FD NOMA consistently delivering superior performance in minimizing outage probabilities.Figure 6 illustrates the relationship between TP and the parameter β , which varies in the range of 0 to 1, where β represents α.The graph demonstrates consistently high TP for User 1 in the context of FD NOMA, regardless of whether it is in PSR or TSR mode, with PSR yielding the highest TP.In contrast, User 2 in FD OMA experiences the lowest TP when operating in TSR mode.For User 1 in both FD NOMA and O-MA, as β increases, there is a noticeable decline in TP, particularly in the β range of 0.7 to 1, where this decline is more pronounced.Conversely, User 2's TP tends to remain relatively constant as β increases, gradually decreasing as β approaches 1.This behavior can be attributed to the fact that when β is small, User 1 receives a larger share of allocated power from the source, leading to higher TP.User 2, on the other hand, relies on User 1 to assist in receiving information from the source.In a comparative analysis between PSR and TSR, as well as between FD OMA and NO-MA, it becomes evident that PSR consistently outperforms TSR, while FD NOMA exhibits higher TP compared to FD OMA. Figure 7 portrays the ER's dependence on β , where β corresponds to α and ranges from 0 to 1.Much like Figure 6, this graph reaffirms that the ER for PSR surpasses that for TSR.Additionally, it's evident that FD NOMA consistently exhibits a superior ER compared to FD MOMA.These distinctions can be comprehended by referencing equations ( 17)-( 24) and ( 32)-(38).
In Figure 8 align perfectly with the results obtained from Monte Carlo simulations.

CONCLUSION
In this research paper, our focus was on investigating the performance of FD users under two distinct protocols: PSR and TSR for applications in wireless EH and MP.We employed a DF relaying network with two cooperative relaying protocols, namely PSR and TSR.We successfully derived closed-form expressions for the OP and ER pertaining to two users in our system.Our analytical findings reveal that, within the typical SNR range, the PSR protocol consistently exhibits superior performance compared to the TSR protocol.Additionally, we observed that NOMA offers better EE performance in comparison to conventional OMA at R 2 .Furthermore, our research delved into determining expressions for the achievable TP, ergodic sum rate, and EE for FD users operating under the PSR and TSR protocols in the context of wireless EH and MP.These findings contribute to a comprehensive understanding of the system's performance and offer valuable insights for the design and optimization of wireless communication systems in EH and MP scenarios.We have As per its definition, I 1 represents the inverse or opposing event to R 1 and can be calculated through the following means: By employing the probability density function (PDF) denoted as f W (.) for a random variable W , we can substitute equation (41) into equation (40).This substitution allows us to derive equations ( 11) and (26), thus concluding our proof.

Appendix B -Proofs of Corollary 1 and Corollary 5
In the scenario where the SNR reaches exceedingly high values, as ρ approaches infinity, we will observe:
(44) As ρ approaches infinity, the calculation process is detailed in equation (46).When dealing with high SNR, the OP for D 2 is determined as follows: I 2 is computed as shown in equation (47), and I 3 is calculated according to equation (48).By substi- In this appendix, we will establish the proof for equations (22) and (36).To derive these closed-form expressions, we can express the ER at R 2 for FD NOMA as follows: We set G 1 = min The cumulative distribution function (CDF) of W is computed using the method outlined in equation (50).G 11 is determined through the calculation procedure provided in equation (51).
Where U(w) is unit step function as By substituting the expression from equation (51) into equation (50), we can derive the result as presented in equation (52).By inserting equation (52) into equation (49), we are able to derive equations ( 22) and (36).This concludes the proof.

Fig. 7 :
Fig. 7: The ER of two users for the PSR and the TSR protocols versus β = α.

Fig. 8 :
Fig. 8: EE of two users for the PSR and the TSR protocols.