# Cancel-Decode-Encode Processing on Two-Way Cooperative NOMA Schemes in Realistic Conditions

Authors: Thu-Thuy Thi Dao; Pham Ngoc Son
Journal: Wireless Communications and Mobile Computing
 (2021)

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## Abstract

This paper considers the effects of perfect/imperfect successive interference cancellation (SIC) and perfect/imperfect ` information (CSI) in a multiple-relay two-way cooperative network using nonorthogonal multiple access (NOMA) and digital network coding (DNC). In this model, a relay is selected by maximizing estimated channel gains to enhance the decoding capacity of the nearer source and minimize the collection time of imperfect CSI. Spectrum utilization efficiency is enhanced two times by a mixture of the SIC and DNC techniques at the selected relay (called as the SIC-2TS protocol). The system performance is considered through analysis of the exact and asymptotic expressions of the system outage probabilities and throughput. The major thing is exposed as the proposed SIC-2TS protocol can reach the best performance at optimal positions of the selected relay. Besides, the system throughput of the proposed protocol outperforms a SIC-utilized two-way relaying scheme without the DNC (called as the SIC-3TS protocol) and a conventional two-way scheme (called as the CONV-4TS protocol) for all signal-to-noise ratio regions. Lastly, the validity of the analytical expressions is verified by the Monte Carlo simulation results.


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## Body

## 1. Introduction

Recently, wireless networks have rising challenges in enhancing system throughput and spectrum efficiency owing to the increasing user devices and increasing various Internet of Things applications. A key technology for the fifth-generation wireless network to solve these challenges is NOMA technology because of its attainments to help grow spectral efficiency, enlarge connections, decrease access latency, and increase the users’ fairness [1–3]. Power domain NOMA uses the superposition coding to allocate different power levels for transmitted signals to the multiusers at the same time, frequency, and code domains. At receivers, the successive interference cancellation method is applied to decode the received signals [2, 3]. However, unexpected errors in decoding when using SIC still occur due to the complexity scale and error propagation, leading to the near user enduring a residual interference signal and the NOMA system performance impacted by this imperfect SIC (ipSIC) [3–6]. In [7], the authors investigated the reliability and security of the ambient backscatter NOMA systems, where the source was aimed at communicating with two NOMA users in the presence of an eavesdropper. The authors in [7] considered a more practical case that nodes and backscatter devices suffer from in-phase and quadrature-phase imbalance.Besides, cooperative communication has also been widely studied because its spatial diversity advantage helps to reduce fading, widen coverage, and increase communication preciseness [6, 8–10]. In conventional cooperative communications, relaying nodes apply the decode-and-forward (DF) method or the amplify-and-forward (AF) method to process their received and transmitted signals [6, 11]. The DF method is better because it decodes received signals at the relay, then reencodes them for forwarding to the destination so it does not amplify noises in received signals like the AF.Cooperative models show that the selection of the best-relaying devices, including partial relay selection and opportunistic relay selection, is necessary to improve system performance [5, 11–19]. These methods are based on the collection of channel state information to select the optimal relay to support communication. The partial relay selection does not offer the full spatial diversity, but it is not as complicated as the full relay selection, and it is useful for applications in industrial IoT and wireless sensor networks. In most practical applications, CSIs cannot be perfectly measured and there are some mismatch, known as imperfect CSIs (ipCSIs) [12, 14, 16, 19–22]. The mismatch can happen due to the feedback delays of the CSIs [12, 14, 16, 19, 22] or the faults in the CSI estimating process [20, 21]. One-way NOMA and cooperative NOMA (CNOMA) networks with the SIC have been widely discussed to increase spectral efficiency gain, improve secure performance, enlarge system energy efficiency, and enhance significantly sum throughput in several studies [1, 4, 23–25]. Besides, two-way cooperative networks have advantages in using frequency spectral efficiency over one-way networks because two sources are able to both transmit and receive signals. Moreover, network coding has the advantages of compressing data and high spectral efficiency, and it plays a crucial role in two-way relay networks [5].In this paper, we propose a two-way cooperative network with a cluster of DF relays and two sources in which the relay will be chosen in the setup phase. The relay selection method in our work helps to enhance the decoding capacity of the nearer source and decrease the collection time of the CSIs than the opportunistic relay selection in [5]. To achieve higher spectral efficiency, we use the NOMA protocol for uplink and the DNC technique for downlink. At the selected relay, the SIC is used to decode sequentially the received signals; next, the DNC is applied to create a new encoded signal and then, this signal is transmitted back to sources, called as the SIC-2TS protocol. Moreover, this paper also investigates the effect of realistic conditions as the ipCSIs and the ipSIC on the system performance. Afterward, the system performance of the SIC-2TS protocol is evaluated based on the analysis expressions of the outage probabilities and the system throughput. Lastly, we compare the proposed protocol with the conventional two-way DF protocol, denoted as CONV-4TS protocol, and the SIC-utilized two-way relaying without the DNC, denoted as the SIC-3TS protocol.The contributions in this paper are summarized as follows. Firstly, the exact and asymptotic expressions of the outage probabilities and throughput of two sources in the proposed scheme are analyzed and verified under the overall effects of pSIC/ipSIC and pCSIs/ipCSI conditions. Secondly, the exact and asymptotic closed-form expressions are proved valid by the simulation results. Next, the simulation results show that the ipSIC and the ipCSI conditions significantly affect the system performance. Fourthly, the performance of the SIC-2TS protocol is improved by the increased number of relays as well as the perfect operations of the SIC process and the CSI estimations. Moreover, the performance of SIC-2TS can attain the best level at optimal locations of the selected relay and the power suitable coefficients of two sources. Last but not the least, the system throughput of the proposed method outperforms the conventional CONV-4TS and SIC-3TS protocols in both cases of pCSIs and ipCSIs for all SNR regions.The rest of our paper is organized as follows. Section Section2 shows some related works. Section 3 describes the system model. Section 4 analyzes the system performance of the SIC-2TS, SIC-3TS, and CONV-4TS protocols. Section 5 shows the results and respective discussions. Finally, section 6 summarizes contributions in this paper.Notations used in this paper are listed in Table1.Table 1
Notation table.
NotationMeaningNNumber of relaysfX.Probability density function (PDF) ofXFX.Cumulative distribution function (CDF) ofX⊕XOR operationmax.Find the maximum valueargmax.Find an element to achieve the maximum valueαkPower allocation coefficient of the sourceSkPr.Probability operationγtTarget signal-to-interference-plus-noise ratio (SINR)OPSkOutage probability at the sourceSkTPThroughput (bits/s/Hz)


## 2. Related Works

In recent researches, specific two-way CNOMA (TWR CNOMA) networks have been investigated to take benefit on system performance. The performance of the NOMA-based two-way relaying network for uplink and downlink of two users or two groups in the perfect SIC (pSIC) and ipSIC conditions and Rayleigh fading was analyzed with a half-duplex DF relay in [26, 27] and a full-duplex DF relay in [28]. The works in [26–28] showed that two-way NOMA is superior to two-way orthogonal multiple access (OMA) in terms of outage probability in low signal-to-noise ratio (SNR) regimes. In [29], the joint effects of in-phase and quadrature-phase imbalance and ipSIC on the performance of TWR CNOMA networks over the Rician fading channels were studied. Besides, the realistic assumptions of the residual hardware impairments or ipCSIs of two-way or multiway CNOMA networks have also been considered in the articles [21, 22, 30].Moreover, DNC and NOMA techniques can be combined to decreasing transmission time between devices and improve the performance system [5, 31, 32]. In [31, 32], the authors combined NOMA and DNC techniques in a two-way DF relay cooperative scheme confirming that performance in this proposed asymmetric scheme had better spectrum utilization efficiency than the traditional two-way DF OMA scheme, the two-way DF with only using the CNOMA, and the two-way relaying system with OMA in the uplink and DNC in the downlink. The authors in [31, 32] only used a DF relay to support communication. The system performance in [31] was considered in the case of the ideal conditions. In [5], the system performance was investigated in perfect CSI (pCSI) conditions with the opportunistic relay selection.


## 3. System Model

A cooperative two-way network has two sourcesS1 and S2 and a closed group of N half-duplex DF relays Ri with i∈1,2,⋯N, as depicted in Figure 1. This system model can be applied for data transmission in heterogeneous cellular networks. The sources and the relays in the SIC-2TS are single antenna and HD devices. We assume that the direct link between the two sources does not exist due to severe fading and path loss, and the information exchange can be performed only via relays [32–34]; the relays are close together as a cluster [17] so the link distances between each source and the relays are identical; hence, we denote d1 and d2 as the normalized distances between S1 and Ri and between S2 and Ri, respectively. We also assume that the flat and block Rayleigh fading channels with the fading coefficients for links Sk⟶Ri and Ri⟶Sk are denoted as hSkRi and hRiSk, respectively, where k∈1,2. In addition, perfect knowledge of all links is assumed at the receivers by channel estimators that are error-free [35], and the ipCSIs are only caused by the feedback delay with a time-variant channel representation and is described by [12, 14, 16, 20, 22, 32, 35]
(1)h∧ν=ρνhν+1−ρν2εν,where ν∈SkRi,RiSk. Here, εν stand for the estimation errors and h∧ν describes the estimated CSIs. εν and h∧ν are independent complex Gaussian random variables (RVs) with zero means and variances λν,εν,h^ν~CN0,λν [14, 16]. And the correlation coefficients ρν0≤ρν≤1 are constants (where ρν=1 denotes no delay effect), characterizing the average quality of channel estimations [14, 16, 32]. For uncomplicated, we assume that ρν=ρ for all the similar devices, ρ=1 for perfect CSIs, and ρ<1 for ipCSIs [12, 14, 16, 20, 32]. The estimated channel gains gSkRi=h∧SkRi2 and gRiSk=h∧RiSk2 are exponentially distributed RVs with PDFs fgSkRix=fgRiSkx=1/λke−x/λk, and CDFs FgSkRix=FgRiSkx=1−e−x/λk, where λk=λSkRi=λRiSk=dk−β and β is a path-loss exponent [15].Figure 1
Two-way cooperative model of the SIC-2TS protocol.Prior to transmitting data, the setup phase is performed firstly by request and feedback messages through the cooperative medium access control (MAC) protocol [8]. The nearer source, denoted as Sn, n∈1,2, receives all channel coefficients from it to the relays under effect of feedback delay and performs the cooperative relay selection. The SIC-2TS protocol uses two time slots for signal communication. In the first slot, two sources S1 and S2 send concurrently their signals x1 and x2, respectively, to all relays, and at the selected relay, the SIC technique is applied to decode the received signals. In the second slot, the DNC technique is used to create a new signal x=x1⊕x2 [5, 15, 32] and the selected relay sends it back to two sources S1 and S2 with transmit power PR.The signal which the relays receive in the first time slot is a weighted sum ofx1 and x2 as follows:
(2)yRi=α1PShS1Rix1+α2PShS2Rix2+nRi,where α1PS and α2PS are transmit powers to carry x1 and x2, respectively; α1 and α2 are power allocation coefficients to fairness between two sources (a lower power should be given to the source which is nearer relays) [28], 0<α1,α2<1; and nRi is the AWGNs with the variance N0 at the nodes Ri.Substituting (1) into (2), the received signal is given by
(3)yRi=α1PSh∧S1Ri−1−ρ2εS1Riρx1+α2PSh∧S2Ri−1−ρ2εS2Riρx2+nRi=α1PSh∧S1Rix1+α2PSh∧S2Rix2−α1PS1−ρ2ρεS1Rix1−α2PS1−ρ2ρεS2Rix2+nRi.Due to the symmetry of the proposed system model in Figure1, without loss of generality, assume that Sn and Sf are near and far sources from the relays, respectively, where n,f∈1,2 and n≠f. Applying the SIC technique [5, 6, 24, 27, 32], firstly, the relays decode the signal xn of the nearby source which has better average channel quality while the signal xf is considered as interference. The received signal-to-interference-plus-noise ratio (SINR) for detecting xn is given by
(4)γSnRi⟶xn∣dn≤df=αnPSh∧SnRi2/ρ2αfPSh∧SfRi2/ρ2+αnPSλSnRi+αfPSλSfRi1−ρ2/ρ2+N0=αnγh∧SnRi2αfγh∧SfRi2+γαnλn+αfλf1−ρ2+ρ2=αnh∧SnRi2αfh∧SfRi2+αnλn+αfλf1−ρ2+ρ2/γ⏟ϕ1=αngSnRiαfgSfRi+ϕ1,where γ is the transmit SNR, γ=PS/N0.In this paper, the relay selection method is used by maximizing estimated channel gains to enhance the decoding capacity of the nearer source. This method has an outstanding advantage in minimizing the collection time of ipCSIs. The relay selection criterion based on the estimated channel gains has been used in [36–38] to achieve the better performance. From (4), the selected relay Rb is expressed as follows:
(5)Rb=argmaxi=1⋯NgSnRi.After decodingxn successfully, the relay Rb deletes the component containing the xn signal in (3); then, it decodes the xf signal and the received SINR for detecting xf is given by
(6)γSfRb⟶xf∣dn≤df=αfPSh∧SfRb2/ρ2εPShRb2+αnPSλn+αfPSλf1−ρ2/ρ2+N0=αfγh∧SfRb2εγhRb2ρ2+γαnλn+αfλf1−ρ2+ρ2=αfgSfRbερ2gRb+ϕ1,where hRb is a remaining interference signal with zero mean and variance Ω at the relay Rb [27], gRb=hRb2 is exponentially distributed RVs with the PDF as fgRbx=1/Ωe−x/Ω, and CDF FgRbx=1−e−x/Ω [27, 28]. ε=0 and ε=1 correspond to pSIC and ipSIC at the relay Rb, respectively [5, 28].In the second time slot, at the relayRb, the signal x=x1⊕x2 is synthesized and transmitted to two sources. And the received signal at the source Sk, k∈1,2, is described as follows:
(7)yRbSk=PRhRbSkx+nSk=PRh∧RbSkx−PR1−ρ2ρεRbSkx+nSk,where nSk is the AWGNs at the sources Sk with the variance N0.Next, thex signal is detected at the two sources with the SINR as follows:
(8)γRbSk⟶x=PRh∧RbSk2/ρ2PRλk1−ρ2/ρ2+N0=ηγgRbSkηγλk1−ρ2+ρ2,where η=PR/PS and η>0.Remark 1.
If the proposed SIC-2TS protocol operates without the DNC at the selected relay, the signalsx1 and x2 are sent sequences by the Rb to the sources S2 and S1 in two different time slots (the second and third time slots). The received signals and the corresponding SINRs are expressed identically as formulas (7) and (8) in which the symbol x is changed to x1 (to send S2) and x2 (to send S1). We denoted the SIC-2TS protocol in this case (without the DNC) as the SIC-3TS protocol to distinguish in the rest of this paper.We also investigate a conventional two-way CONV-4TS protocol using four time slots with relay selection. This protocol’s transmission process is as follows:S1→1stx1Rb1→2ndx1S2→3rdx2Rb2→4thx2S1.Firstly, the data signalx1 is transmitted from the source S1 to all the relays; the received signal and the SINR at the relay Ri are described by
(9)yCS1Ri=α1PShS1Rix1+nRi=α1PSh∧S1Ri−1−ρ2εS1Riρx1+nRi=α1PSh∧S1Rix1−α1PS1−ρ2ρεS1Rix1+nRi,(10)γS1Ri=α1PSh∧S1Ri2/ρ2α1PSλ11−ρ2/ρ2+N0=α1γgS1Riα1λ1γ1−ρ2+ρ2.Secondly, the relayRb1 is selected as by formula Rb1=argmaxi=1⋯NgS1Ri and then Rb1 decodes and transmits the signal x1 to the source S2. The received signal and the SINR at the source S2 are described by
(11)yCRb1S2=PRhRb1S2x1+nS2=PRh∧Rb1S2x1−PR1−ρ2ρεRb1S2x1+nS2,(12)γRb1S2=PRh∧Rb1S22/ρ2PRλ21−ρ2/ρ2+N0=ηγgRb1S2ηγλ21−ρ2+ρ2.In the same way, in the third and fourth time slots, the sourceS2 transmits the signal x2 to the source S1 via the best relay Rb2. We have the SINRs to decode the signal x2 at the relay Ri and the source S1 as follows:
(13)γS2Ri=αfPSh∧S2Ri2/ρ2α2PSλ21−ρ2/ρ2+N0=α2γgS2Riα2λ2γ1−ρ2+ρ2,(14)γRb2S1=PRh∧Rb2S12/ρ2PRλ11−ρ2/ρ2+N0=ηγgRb2S1ηγλ11−ρ2+ρ2.


## 4. Performance Analysis

Section IV presents expressions of the outage probability and throughput for the protocols. We assume that the outage occurs at the nodesRi and Sk if their SINRs are less than a predefined target γt. Conversely, these nodes decode signals successfully.


### 4.1. Outage Probability Analysis

#### 4.1.1. The Proposed SIC-2TS Protocol

(1) The Outage Probability at the SourceSffor theSn→xnRb→xSfLink. The outage of the system occurs in this link when the relay Rb fails to decode the signal xn or it decodes successfully the signal xn but the source Sf fails to decode the signal x. Besides, the outage probability can also be calculated by the complementary event of the success transmission probability. The successful transmission is the signal xn, and the signal x is received and decoded successfully at the Rb and the source Sf, respectively [32]. At the source Sf, the outage probability of the signal xn can be described as
(15)OPSfdn≤df=PrγSnRb⟶xn∣dn≤df<γt+PrγSnRb⟶xn∣dn≤df≥γt,γRbSf⟶x<γt=1−PrγSnRb⟶xn∣dn≤df≥γt,γRbSf⟶x≥γt.A point to remark is thatγSnRb⟶xn∣dn≤df≥γt and γRbSf⟶x≥γt are separate events. Thus, the OPSfdn≤df can be given by
(16)OPSfdn≤df=1−PrγSnRb⟶xn∣dn≤df≥γt×PrγRbSf⟶x≥γt.Lemma 2.
The probabilityPrγSnRb⟶xn∣dn≤df≥γt is calculated by
(17)PrγSnRb⟶xn∣dn≤df≥γt=1−λn∑p=0NCNp−1pe−pϕ3/λnλn+pϕ2λf,where ϕ2=γtαf/αn, ϕ3=γtϕ1/αn, and CNp=N!/p!N−p!.Proof.
See the proof in “AppendixA.”
The probabilityPrγRbSf⟶x≥γt is solved as
(18)PrγRbSf⟶x≥γt=1−PrγRbSf⟶x<γt=1−PrgRbSf<γtλf1−ρ2+γtρ2ηγ=1−FgRbSfγtλf1−ρ2+γtρ2ηγ=e−γt1−ρ2+ρ2/λfηγ.
Substituting (17) and (18) into (16), the outage probability at the source Sf is obtained as
(19)OPSfdn≤df=1−1−λn∑p=0NCNp−1pe−pϕ3/λnλn+pϕ2λfe−γt1−ρ2+ρ2/λfηγ.(2) The Outage Probability at the SourceSnfor theSf→xfRb→xSnLink. The outage of the system occurs in this link when the signal xn is not decoded successfully at relay Rb; or it is decoded successfully but the signal xf is not decoded successfully at relay Rb; or both the signals xn and xf are decoded successfully at the relay Rb but the source Sn decodes unsuccessfully the signal x. Conversely, the success transmission of the signal xf occurs when the relay Rb and the source Sn decode successfully the signals xn,xf and the signal x, respectively. At the source Sn, the outage probability of the signal xf can be described as
(20)OPSndn≤df=PrγSnRb⟶xn∣dn≤df<γt+PrγSnRb⟶xn∣dn≤df≥γt,γSfRb⟶xf∣dn≤df<γt+PrγSnRb⟶xn∣dn≤df≥γt,γSfRb⟶xn∣dn≤df≥γt,γRbSn⟶x<γt=1−PrγSnRb⟶xn∣dn≤df>γt,γSfRb⟶xf∣dn≤df>γt,γRbSn⟶x>γt=1−PrγSnRb⟶xn∣dn≤df≥γt,γSfRb⟶xf∣dn≤df≥γt×PrγRbSn⟶x≥γt.Lemma 3.
The probabilityPrγSnRb⟶xn∣dn≤df≥γt,γSfRb⟶xf∣dn≤df≥γt is calculated by
(21)PrγSnRb⟶xn∣dn≤df≥γt,γSfRb⟶xf∣dn≤df≥γt=λfe−ϕ5/λfλf+ϕ4Ω−∑p=0Nλn2λfCNp−1pe−pϕ3+ϕ5ϕ2/λn+ϕ5/λfλn+pϕ2λfλnλf+ϕ4Ωλn+pϕ5ϕ2λf,where ϕ4=γtερ2/αf and ϕ5=γtϕ1/αf.Proof.
See the proof in “AppendixB.”
The final probabilityPrγRbSn⟶x≥γt in (20) is answered as
(22)PrγRbSn⟶x≥γt=1−PrγRbSn⟶x<γt=1−PrgRbSn<γtλn1−ρ2+γtρ2ηγ=e−γt1−ρ2+ρ2/λnηγ.
By substituting (21) and (22) into (20), the outage probability at the source Sn is solved as
(23)OPSndn≤df=1−λfe−ϕ5/λfλf+ϕ4Ω−∑p=0Nλn2λfCNp−1pe−pϕ3+ϕ5ϕ2/λn+ϕ5/λfλn+pϕ2λfλnλf+ϕ4Ωλn+pϕ5ϕ2λf×e−γt1−ρ2+ρ2/λnγ.Corollary 4.
Whenγ⟶+∞, we obtain asymptotic expression as
(24)OPSfdn≤dfγ⟶+∞=1−1−λn∑p=0NCNp−1pe−pϕ6/λnλn+pϕ2λfe−γt1−ρ2,(25)OPSndn≤dfγ⟶+∞=1−λfe−ϕ7/λfλf+ϕ4Ω−∑p=0Nλn2λfCNp−1pe−pϕ6+ϕ7ϕ2/λn+ϕ7/λfλn+pϕ2λfλnλf+ϕ4Ωλn+pϕ7ϕ2λfe−γt1−ρ2,where ϕ6=γtαnλn+αfλf1−ρ2/αn and ϕ7=γtαnλn+αfλf1−ρ2/αf.Remark 5.
In the SIC-3TS protocol, the outage probabilities and the asymptotic expressions of the sourcesSf and Sn are also identified as closed-form formulas (19), (23), (24), and (25), respectively.


#### 4.1.2. The CONV-4TS Protocol

The outage probability at the sourceS2 for the S1→1stx1Rb1→2ndx1S2 link in the CONV-4TS protocol is described as follows:
(26)OPS2C=1−Prγs1Rb1≥γt,γRb1S2≥γt=1−Prγs1Rb1≥γt×PrγRb1S2≥γt=1−1−Prγs1Rb1<γt×1−PrγRb1S2<γt.SubstitutingγS1Rb1 and γRb1S2 in (10) and (12) into ((26), we obtain
(27)OPS2C=1−1−Prα1γgS1Rb1α1λ1γ1−ρ2+ρ2<γt1−PrγgRb1S2γλ21−ρ2+ρ2<γt=1−1−PrgS1Rb1<γtλ11−ρ2+γtρ2/α1γ⏟ϕ8×1−PrgRb1S2<γtλ21−ρ2+γtρ2/γ⏟ϕ9=1−1−FgS1Rb1ϕ81−FgRb1S2ϕ9=1−1−1−e−ϕ8/λ1Ne−ϕ9/λ2=1−1−∑p=0NCNp−1pe−pϕ8/λ1e−ϕ9/λ2.Similarly, the outage probability at the sourceS1 for the S2→3rdx2Rb2→4thx2S1 link is expressed as
(28)OPS1C=1−PrγS2Rb2≥γt,γRb2S1≥γt=1−PrγS2Rb2≥γt×PrγRb2S1≥γt.By substitutingγS2Rb2 and γRb2S1 in (13) and (14), respectively, into (28) and after some manipulations as finding the outage probability OPS2C, we get a final result as
(29)OPS1C=1−1−∑p=0NCNp−1pe−pϕ10/λ2e−ϕ11/λ1, where ϕ10=γtλ21−ρ2+γtρ2/α2γ and ϕ11=γtλ11−ρ2+γtρ2/γ.Corollary 6.
Whenγ⟶∞, asymptotic expressions of the outage probabilities at the sources S2 and S1 are obtained as
(30)OPS2Cγ⟶+∞=1−1−∑p=0NCNp−1pe−pγt1−ρ2e−γt1−ρ2,OPS1Cγ⟶+∞=1−1−∑p=0NCNp−1pe−pγt1−ρ2e−γt1−ρ2.


### 4.2. Throughput Analysis

The system throughput of the SIC-2TS, SIC-3TS, and the CONV-4TS protocols is obtained in the following, respectively [39]:
(31)TPSIC−2TSdn≤df=121−OPSndn≤dfRt+121−OPSfdn≤dfRt,(32)TPSIC−3TSdn≤df=131−OPSndn≤dfRt+131−OPSfdn≤dfRt,(33)TPCONV−4TS=141−OPS1CRt+141−OPS2CRt,where 1/2, 1/3, and 1/4 denote that the protocols SIC-2TS, SIC-3TS, and CONV-4TS work in two, three, and four time slots, respectively; Rt=log21+γt (bits/s/Hz) [40].Remark 7.
The SIC-3TS protocol without the DNC operates in three time slots to send two data; thus, the throughput is obtained by the formula (32) where the outage probabilities OPSndn≤df and OPSfdn≤df are taken from the proposed SIC-2TS protocol.


## 4.1. Outage Probability Analysis

### 4.1.1. The Proposed SIC-2TS Protocol

(1) The Outage Probability at the SourceSffor theSn→xnRb→xSfLink. The outage of the system occurs in this link when the relay Rb fails to decode the signal xn or it decodes successfully the signal xn but the source Sf fails to decode the signal x. Besides, the outage probability can also be calculated by the complementary event of the success transmission probability. The successful transmission is the signal xn, and the signal x is received and decoded successfully at the Rb and the source Sf, respectively [32]. At the source Sf, the outage probability of the signal xn can be described as
(15)OPSfdn≤df=PrγSnRb⟶xn∣dn≤df<γt+PrγSnRb⟶xn∣dn≤df≥γt,γRbSf⟶x<γt=1−PrγSnRb⟶xn∣dn≤df≥γt,γRbSf⟶x≥γt.A point to remark is thatγSnRb⟶xn∣dn≤df≥γt and γRbSf⟶x≥γt are separate events. Thus, the OPSfdn≤df can be given by
(16)OPSfdn≤df=1−PrγSnRb⟶xn∣dn≤df≥γt×PrγRbSf⟶x≥γt.Lemma 2.
The probabilityPrγSnRb⟶xn∣dn≤df≥γt is calculated by
(17)PrγSnRb⟶xn∣dn≤df≥γt=1−λn∑p=0NCNp−1pe−pϕ3/λnλn+pϕ2λf,where ϕ2=γtαf/αn, ϕ3=γtϕ1/αn, and CNp=N!/p!N−p!.Proof.
See the proof in “AppendixA.”
The probabilityPrγRbSf⟶x≥γt is solved as
(18)PrγRbSf⟶x≥γt=1−PrγRbSf⟶x<γt=1−PrgRbSf<γtλf1−ρ2+γtρ2ηγ=1−FgRbSfγtλf1−ρ2+γtρ2ηγ=e−γt1−ρ2+ρ2/λfηγ.
Substituting (17) and (18) into (16), the outage probability at the source Sf is obtained as
(19)OPSfdn≤df=1−1−λn∑p=0NCNp−1pe−pϕ3/λnλn+pϕ2λfe−γt1−ρ2+ρ2/λfηγ.(2) The Outage Probability at the SourceSnfor theSf→xfRb→xSnLink. The outage of the system occurs in this link when the signal xn is not decoded successfully at relay Rb; or it is decoded successfully but the signal xf is not decoded successfully at relay Rb; or both the signals xn and xf are decoded successfully at the relay Rb but the source Sn decodes unsuccessfully the signal x. Conversely, the success transmission of the signal xf occurs when the relay Rb and the source Sn decode successfully the signals xn,xf and the signal x, respectively. At the source Sn, the outage probability of the signal xf can be described as
(20)OPSndn≤df=PrγSnRb⟶xn∣dn≤df<γt+PrγSnRb⟶xn∣dn≤df≥γt,γSfRb⟶xf∣dn≤df<γt+PrγSnRb⟶xn∣dn≤df≥γt,γSfRb⟶xn∣dn≤df≥γt,γRbSn⟶x<γt=1−PrγSnRb⟶xn∣dn≤df>γt,γSfRb⟶xf∣dn≤df>γt,γRbSn⟶x>γt=1−PrγSnRb⟶xn∣dn≤df≥γt,γSfRb⟶xf∣dn≤df≥γt×PrγRbSn⟶x≥γt.Lemma 3.
The probabilityPrγSnRb⟶xn∣dn≤df≥γt,γSfRb⟶xf∣dn≤df≥γt is calculated by
(21)PrγSnRb⟶xn∣dn≤df≥γt,γSfRb⟶xf∣dn≤df≥γt=λfe−ϕ5/λfλf+ϕ4Ω−∑p=0Nλn2λfCNp−1pe−pϕ3+ϕ5ϕ2/λn+ϕ5/λfλn+pϕ2λfλnλf+ϕ4Ωλn+pϕ5ϕ2λf,where ϕ4=γtερ2/αf and ϕ5=γtϕ1/αf.Proof.
See the proof in “AppendixB.”
The final probabilityPrγRbSn⟶x≥γt in (20) is answered as
(22)PrγRbSn⟶x≥γt=1−PrγRbSn⟶x<γt=1−PrgRbSn<γtλn1−ρ2+γtρ2ηγ=e−γt1−ρ2+ρ2/λnηγ.
By substituting (21) and (22) into (20), the outage probability at the source Sn is solved as
(23)OPSndn≤df=1−λfe−ϕ5/λfλf+ϕ4Ω−∑p=0Nλn2λfCNp−1pe−pϕ3+ϕ5ϕ2/λn+ϕ5/λfλn+pϕ2λfλnλf+ϕ4Ωλn+pϕ5ϕ2λf×e−γt1−ρ2+ρ2/λnγ.Corollary 4.
Whenγ⟶+∞, we obtain asymptotic expression as
(24)OPSfdn≤dfγ⟶+∞=1−1−λn∑p=0NCNp−1pe−pϕ6/λnλn+pϕ2λfe−γt1−ρ2,(25)OPSndn≤dfγ⟶+∞=1−λfe−ϕ7/λfλf+ϕ4Ω−∑p=0Nλn2λfCNp−1pe−pϕ6+ϕ7ϕ2/λn+ϕ7/λfλn+pϕ2λfλnλf+ϕ4Ωλn+pϕ7ϕ2λfe−γt1−ρ2,where ϕ6=γtαnλn+αfλf1−ρ2/αn and ϕ7=γtαnλn+αfλf1−ρ2/αf.Remark 5.
In the SIC-3TS protocol, the outage probabilities and the asymptotic expressions of the sourcesSf and Sn are also identified as closed-form formulas (19), (23), (24), and (25), respectively.


### 4.1.2. The CONV-4TS Protocol

The outage probability at the sourceS2 for the S1→1stx1Rb1→2ndx1S2 link in the CONV-4TS protocol is described as follows:
(26)OPS2C=1−Prγs1Rb1≥γt,γRb1S2≥γt=1−Prγs1Rb1≥γt×PrγRb1S2≥γt=1−1−Prγs1Rb1<γt×1−PrγRb1S2<γt.SubstitutingγS1Rb1 and γRb1S2 in (10) and (12) into ((26), we obtain
(27)OPS2C=1−1−Prα1γgS1Rb1α1λ1γ1−ρ2+ρ2<γt1−PrγgRb1S2γλ21−ρ2+ρ2<γt=1−1−PrgS1Rb1<γtλ11−ρ2+γtρ2/α1γ⏟ϕ8×1−PrgRb1S2<γtλ21−ρ2+γtρ2/γ⏟ϕ9=1−1−FgS1Rb1ϕ81−FgRb1S2ϕ9=1−1−1−e−ϕ8/λ1Ne−ϕ9/λ2=1−1−∑p=0NCNp−1pe−pϕ8/λ1e−ϕ9/λ2.Similarly, the outage probability at the sourceS1 for the S2→3rdx2Rb2→4thx2S1 link is expressed as
(28)OPS1C=1−PrγS2Rb2≥γt,γRb2S1≥γt=1−PrγS2Rb2≥γt×PrγRb2S1≥γt.By substitutingγS2Rb2 and γRb2S1 in (13) and (14), respectively, into (28) and after some manipulations as finding the outage probability OPS2C, we get a final result as
(29)OPS1C=1−1−∑p=0NCNp−1pe−pϕ10/λ2e−ϕ11/λ1, where ϕ10=γtλ21−ρ2+γtρ2/α2γ and ϕ11=γtλ11−ρ2+γtρ2/γ.Corollary 6.
Whenγ⟶∞, asymptotic expressions of the outage probabilities at the sources S2 and S1 are obtained as
(30)OPS2Cγ⟶+∞=1−1−∑p=0NCNp−1pe−pγt1−ρ2e−γt1−ρ2,OPS1Cγ⟶+∞=1−1−∑p=0NCNp−1pe−pγt1−ρ2e−γt1−ρ2.


## 4.1.1. The Proposed SIC-2TS Protocol

(1) The Outage Probability at the SourceSffor theSn→xnRb→xSfLink. The outage of the system occurs in this link when the relay Rb fails to decode the signal xn or it decodes successfully the signal xn but the source Sf fails to decode the signal x. Besides, the outage probability can also be calculated by the complementary event of the success transmission probability. The successful transmission is the signal xn, and the signal x is received and decoded successfully at the Rb and the source Sf, respectively [32]. At the source Sf, the outage probability of the signal xn can be described as
(15)OPSfdn≤df=PrγSnRb⟶xn∣dn≤df<γt+PrγSnRb⟶xn∣dn≤df≥γt,γRbSf⟶x<γt=1−PrγSnRb⟶xn∣dn≤df≥γt,γRbSf⟶x≥γt.A point to remark is thatγSnRb⟶xn∣dn≤df≥γt and γRbSf⟶x≥γt are separate events. Thus, the OPSfdn≤df can be given by
(16)OPSfdn≤df=1−PrγSnRb⟶xn∣dn≤df≥γt×PrγRbSf⟶x≥γt.Lemma 2.
The probabilityPrγSnRb⟶xn∣dn≤df≥γt is calculated by
(17)PrγSnRb⟶xn∣dn≤df≥γt=1−λn∑p=0NCNp−1pe−pϕ3/λnλn+pϕ2λf,where ϕ2=γtαf/αn, ϕ3=γtϕ1/αn, and CNp=N!/p!N−p!.Proof.
See the proof in “AppendixA.”
The probabilityPrγRbSf⟶x≥γt is solved as
(18)PrγRbSf⟶x≥γt=1−PrγRbSf⟶x<γt=1−PrgRbSf<γtλf1−ρ2+γtρ2ηγ=1−FgRbSfγtλf1−ρ2+γtρ2ηγ=e−γt1−ρ2+ρ2/λfηγ.
Substituting (17) and (18) into (16), the outage probability at the source Sf is obtained as
(19)OPSfdn≤df=1−1−λn∑p=0NCNp−1pe−pϕ3/λnλn+pϕ2λfe−γt1−ρ2+ρ2/λfηγ.(2) The Outage Probability at the SourceSnfor theSf→xfRb→xSnLink. The outage of the system occurs in this link when the signal xn is not decoded successfully at relay Rb; or it is decoded successfully but the signal xf is not decoded successfully at relay Rb; or both the signals xn and xf are decoded successfully at the relay Rb but the source Sn decodes unsuccessfully the signal x. Conversely, the success transmission of the signal xf occurs when the relay Rb and the source Sn decode successfully the signals xn,xf and the signal x, respectively. At the source Sn, the outage probability of the signal xf can be described as
(20)OPSndn≤df=PrγSnRb⟶xn∣dn≤df<γt+PrγSnRb⟶xn∣dn≤df≥γt,γSfRb⟶xf∣dn≤df<γt+PrγSnRb⟶xn∣dn≤df≥γt,γSfRb⟶xn∣dn≤df≥γt,γRbSn⟶x<γt=1−PrγSnRb⟶xn∣dn≤df>γt,γSfRb⟶xf∣dn≤df>γt,γRbSn⟶x>γt=1−PrγSnRb⟶xn∣dn≤df≥γt,γSfRb⟶xf∣dn≤df≥γt×PrγRbSn⟶x≥γt.Lemma 3.
The probabilityPrγSnRb⟶xn∣dn≤df≥γt,γSfRb⟶xf∣dn≤df≥γt is calculated by
(21)PrγSnRb⟶xn∣dn≤df≥γt,γSfRb⟶xf∣dn≤df≥γt=λfe−ϕ5/λfλf+ϕ4Ω−∑p=0Nλn2λfCNp−1pe−pϕ3+ϕ5ϕ2/λn+ϕ5/λfλn+pϕ2λfλnλf+ϕ4Ωλn+pϕ5ϕ2λf,where ϕ4=γtερ2/αf and ϕ5=γtϕ1/αf.Proof.
See the proof in “AppendixB.”
The final probabilityPrγRbSn⟶x≥γt in (20) is answered as
(22)PrγRbSn⟶x≥γt=1−PrγRbSn⟶x<γt=1−PrgRbSn<γtλn1−ρ2+γtρ2ηγ=e−γt1−ρ2+ρ2/λnηγ.
By substituting (21) and (22) into (20), the outage probability at the source Sn is solved as
(23)OPSndn≤df=1−λfe−ϕ5/λfλf+ϕ4Ω−∑p=0Nλn2λfCNp−1pe−pϕ3+ϕ5ϕ2/λn+ϕ5/λfλn+pϕ2λfλnλf+ϕ4Ωλn+pϕ5ϕ2λf×e−γt1−ρ2+ρ2/λnγ.Corollary 4.
Whenγ⟶+∞, we obtain asymptotic expression as
(24)OPSfdn≤dfγ⟶+∞=1−1−λn∑p=0NCNp−1pe−pϕ6/λnλn+pϕ2λfe−γt1−ρ2,(25)OPSndn≤dfγ⟶+∞=1−λfe−ϕ7/λfλf+ϕ4Ω−∑p=0Nλn2λfCNp−1pe−pϕ6+ϕ7ϕ2/λn+ϕ7/λfλn+pϕ2λfλnλf+ϕ4Ωλn+pϕ7ϕ2λfe−γt1−ρ2,where ϕ6=γtαnλn+αfλf1−ρ2/αn and ϕ7=γtαnλn+αfλf1−ρ2/αf.Remark 5.
In the SIC-3TS protocol, the outage probabilities and the asymptotic expressions of the sourcesSf and Sn are also identified as closed-form formulas (19), (23), (24), and (25), respectively.


## 4.1.2. The CONV-4TS Protocol

The outage probability at the sourceS2 for the S1→1stx1Rb1→2ndx1S2 link in the CONV-4TS protocol is described as follows:
(26)OPS2C=1−Prγs1Rb1≥γt,γRb1S2≥γt=1−Prγs1Rb1≥γt×PrγRb1S2≥γt=1−1−Prγs1Rb1<γt×1−PrγRb1S2<γt.SubstitutingγS1Rb1 and γRb1S2 in (10) and (12) into ((26), we obtain
(27)OPS2C=1−1−Prα1γgS1Rb1α1λ1γ1−ρ2+ρ2<γt1−PrγgRb1S2γλ21−ρ2+ρ2<γt=1−1−PrgS1Rb1<γtλ11−ρ2+γtρ2/α1γ⏟ϕ8×1−PrgRb1S2<γtλ21−ρ2+γtρ2/γ⏟ϕ9=1−1−FgS1Rb1ϕ81−FgRb1S2ϕ9=1−1−1−e−ϕ8/λ1Ne−ϕ9/λ2=1−1−∑p=0NCNp−1pe−pϕ8/λ1e−ϕ9/λ2.Similarly, the outage probability at the sourceS1 for the S2→3rdx2Rb2→4thx2S1 link is expressed as
(28)OPS1C=1−PrγS2Rb2≥γt,γRb2S1≥γt=1−PrγS2Rb2≥γt×PrγRb2S1≥γt.By substitutingγS2Rb2 and γRb2S1 in (13) and (14), respectively, into (28) and after some manipulations as finding the outage probability OPS2C, we get a final result as
(29)OPS1C=1−1−∑p=0NCNp−1pe−pϕ10/λ2e−ϕ11/λ1, where ϕ10=γtλ21−ρ2+γtρ2/α2γ and ϕ11=γtλ11−ρ2+γtρ2/γ.Corollary 6.
Whenγ⟶∞, asymptotic expressions of the outage probabilities at the sources S2 and S1 are obtained as
(30)OPS2Cγ⟶+∞=1−1−∑p=0NCNp−1pe−pγt1−ρ2e−γt1−ρ2,OPS1Cγ⟶+∞=1−1−∑p=0NCNp−1pe−pγt1−ρ2e−γt1−ρ2.


## 4.2. Throughput Analysis

The system throughput of the SIC-2TS, SIC-3TS, and the CONV-4TS protocols is obtained in the following, respectively [39]:
(31)TPSIC−2TSdn≤df=121−OPSndn≤dfRt+121−OPSfdn≤dfRt,(32)TPSIC−3TSdn≤df=131−OPSndn≤dfRt+131−OPSfdn≤dfRt,(33)TPCONV−4TS=141−OPS1CRt+141−OPS2CRt,where 1/2, 1/3, and 1/4 denote that the protocols SIC-2TS, SIC-3TS, and CONV-4TS work in two, three, and four time slots, respectively; Rt=log21+γt (bits/s/Hz) [40].Remark 7.
The SIC-3TS protocol without the DNC operates in three time slots to send two data; thus, the throughput is obtained by the formula (32) where the outage probabilities OPSndn≤df and OPSfdn≤df are taken from the proposed SIC-2TS protocol.


## 5. Numerical Results and Discussion

In this section, the outage probabilities and system throughput of three protocols SIC-2TS, SIC-3TS, and CONV-4TS are analyzed and evaluated. The exactness of the asymptotic and exact theory extractions is validated by Monte Carlo simulations (simulated results are shown by the marker point in all figures). We default the threshold SINR asγt=1 and the path-loss exponent as β=3 in all the analyses and evaluations. From Figures 2–5, the distance d1 has smaller value and d2=1−d1.Figure 2
Outage probabilities of the source nodesS1 and S2 in the proposed SIC-2TS protocol versus PS/N0 (dB) when Ω=−10 (dB), d1=0.4,d2=1−d1, N∈2,6, ρ∈0.92,1, ε=0, and the power allocation coefficients α1=α2=η=1.Figure 3
Outage probabilities of the source nodesS1 and S2 in the proposed SIC-2TS protocol versus PS/N0 (dB) when Ω=−10 (dB), d1=0.4,d2=1−d1, N∈2,6, ρ=1, ε∈0,1, and the power allocation coefficients α1=α2=η=1.Figure 4
Outage probabilities of the source nodesS1 and S2 in the proposed SIC-2TS and CONV-4TS protocols versus PS/N0 (dB) when Ω=−10 (dB), d1=0.4,d2=1−d1, N=6, pSIC-pCSIs ε=0,ρ=1, ipSIC-ipCSIs ε=1,ρ=0.92, and the power allocation coefficients α1=α2=η=1.Figure 5
Throughput of the SIC-2TS, SIC-3TS, and CONV-4TS protocols versusPS/N0 (dB) when Ω=−10 (dB), d1=0.4, d2=1−d1, N=6, ε=0, ρ∈0.92,1, and the power allocation coefficients α1=α2=η=1.In Figure2, we examine the outage probabilities of the two sources S1 and S2 in the proposed SIC-2TS protocol as a function of the PS/N0 (dB) with assuming perfect CSIs ρ=1 when Ω=−10 (dB), d1=0.4, d2=1−d1, N∈2,6, and α1=α2=η=1 in both pSIC case (ε=0) and ipSIC case (ε=1). Figure 2 shows that the outage probabilities of the source S2 are equal in the pSIC case and the ipSIC case as formula (19). The outage probabilities of the source S1 in the pSIC case are higher than those of the source S2 at the low PS/N0 (dB) regions, and they move to the same saturation values at the high PS/N0 (dB) regions. The outage probabilities of the source S1 in the ipSIC case are higher than those in the pSIC case with every PS/N0 (dB) due to adding the residual interference signals to the SINR of the signal x2 at the relay as in formulas (6). Furthermore, the system diversity capacity increases because of using the relay selection methods as in (5) and (17) so the system performance of the proposed SIC-2TS protocol is better when the number of relays increases. Finally, the asymptotic and exact theory analysis lines of the outage probabilities also coincide well with their Monte Carlo simulation lines.Figure3 illustrates the outage probabilities of the sources S1 and S2 in the proposed SIC-2TS protocol as a function of PS/N0 (dB) in both ideal (pSIC-pCSIs) and practical (ipSIC- ipCSIs) conditions when Ω=−10 (dB), d1=0.4,d2=1−d1, N=6, and the power allocation coefficients α1=α2=η=1. Figure 3 shows that the outage probabilities of the two sources with the pSIC- pCSI condition are better than with the ipSIC-ipCSI condition. In the pSIC-pCSI condition, the outage probabilities of the two source nodes have a small difference. But in the ipSIC-ipCSI case, the system outage probability for the source nodes S1 is a lot higher. These results happen because the SIC technique is used to decode the signal at the relay to make the signal of the farther source more influenced in imperfect cases. In order to have fairness, meaning the two sources can have the nearly same system outage probability in the ipSIC-ipCSI condition, we can provide the higher transmit power for the farther source by changing the transmit power coefficients (α1,α2) in formula (2). Finally, the asymptotic and exact theory analysis lines of the outage probabilities also coincide well with their Monte Carlo simulation lines.In Figure4, we consider the outage probabilities of the two sources S1 and S2 in the proposed SIC-2TS and CONV-4TS protocols as a function of PS/N0 (dB) with assuming perfect SIC ε=0 when Ω=−10 (dB), d1=0.4,d2=1−d1, N=6, and α1=α2=η=1 [5, 13] in both pCSI case (ρ=1) and ipCSI case (ρ=0.92). Considering the SIC-2TS protocol in Figure 4, firstly, the outage probabilities of two sources in the pCSI case are smaller than those in the ipCSI case and all of them have the floor values when PS/N0 (dB) is large. Secondarily, the outage probabilities of the source S1 has higher than the source S2. Thirdly, if the PS/N0 (dB) has enough large value the outage probability of the two sources will be equal in the pCSI condition, but the source S1 outage probabilities are always bigger than the outage probabilities of the source S2 at all PS/N0 (dB) values in the ipCSI condition. Those SIC-2TS protocol results occur because the negative effects of imperfect CSIs lead to channel gain coefficients decrease as formula (1); and in case of d1≤d2, decoding the signal x2 is decided by the SIC technique as (4) and (6) so the SINR of the x2 is affected by the ipCSIs of both links S1⟶Ri and S2⟶Ri. Moreover, we also see that the CONV-4TS protocol has a smaller outage probabilities than the SIC-2TS protocol in both the pCSI and the ipCSI conditions, but this conventional protocol will take a lot of time and energy to transmit the signals. Lastly, the asymptotic and exact theory analysis lines of the outage probabilities coincide well with their Monte Carlo simulation lines.Figure5 plots the system throughput for the SIC-2TS, SIC-3TS, and CONV-4TS protocols as a function of PS/N0 (dB) with pCSIs/ipCSIs ρ∈0.92,1 when Ω=−10 (dB), d1=0.4,d2=1−d1, N=6, and the power allocation coefficients α1=α2=η=1 for case d1≤d2 in formulas (31), (32), and (33), respectively. We can see that the proposed SIC-2TS protocol has the ability to achieve higher throughput than the CONV-4TS and SIC-3TS protocols in all PS/N0(dB) for both pCSI and ipCSI cases because it combines the NOMA, SIC, and DNC techniques to help degrade the number of the time slot of the transmission between two sources. In addition, the interference parts on the received SINRs are skipped in the case of pCSIs so the throughput of protocols in this condition is always better than that in the ipCSI condition. Furthermore, the SIC-2TS protocol throughput converges at the same value in the high PS/N0 (dB) regions PS/N0>15dB. Finally, the exact theory values of the system throughput of three protocols fix well the Monte Carlo simulations.Figure6 demonstrates the system throughput of the SIC-2TS and CONV-4TS protocols versus d1 in cases of pCSIs/ipCSIs ρ∈0.92,1 when PS/N0=5dB, Ω=−10 (dB), N=6, and the power allocation coefficients α1=α2=η=1. Figure 6 shows that the SIC-2TS protocol has the system throughput higher than the CONV-4TS protocol in the pCSI case. But in the ipCSI case, its throughput is only better when the distances d1 are about from 0.3 to 0.7. Moreover, the throughput of the SIC-2TS protocol reaches the highest values at optimal locations of the selected relay as d1=0.4 (in the pCSIs) and d1=0.45 (in the ipCSIs). Besides, the CONV-4TS protocol has the highest system throughput when the relay is at an equidistant point of the two sources (d1=0.5). Lastly, in the perfect CSI ρ=1, the throughput of the two protocols is always better than in the imperfect CSI ρ=0.92 case.Figure 6
Throughput of the SIC-2TS and CONV-4TS protocols versusd1 when PS/N0=5 (dB), Ω=−10dB, d2=1−d1, N=6, ε=0, ρ∈0.92,1, and the power allocation coefficients α1=α2=η=1.Figure7 observes the throughput of the proposed SIC-2TS protocol versus α1 and d1. The scopes of α1 and d1 are from 0.05 to 0.95. The throughput of the proposed SIC-2TS protocol is the highest at about 0.8853 when joint pairs α1,d1=0.4,0.5 and α1,d1=0.6,0.5. Table 2 shows the detail of the maximum throughput value corresponding to the distance d1 and the power coefficient α1 of the S1. The coefficients α1 and α2 help to adjust the transmit powers of the source nodes; a smaller power is set to the source node nearer to the relay cluster and higher power for the farther source node. The transmit power allocation can achieve the best throughput performance for the proposed SIC-2TS protocol.Figure 7
Throughput of the SIC-2TS protocol as a function ofα1 and d1 when PS/N0=5 (dB), Ω=−10 (dB), d2=1−d1, N=6, ε=0, ρ=1, and the power allocation coefficients α2=1−α1 and η=1.Table 2
The maximum throughput values corresponding to the distancesd1 and the power allocation coefficient α1.
d10.10.20.30.40.50.60.70.80.9α10.10.10.30.50.30.50.70.90.9Maximum throughput0.78390.84130.87150.88530.87540.88530.87150.84130.7839Figure8 presents the throughput of the proposed SIC-2TS protocol as functions of ρ and d1. The range of ρ is from 0.8 to 1, and the range of d1 is set from 0.05 to 0.95. The power coefficients α1 and α2 also vary according to distance d1 and d2, respectively, to achieve the best throughput performance as mentioned in Figure 7. It is seen that a small range decrease in ρ will result in a large range reduction in throughput so obviously, it is necessary to consider the effect of feedback delay when examining a real system. In other words, channel error estimation in cooperation networks becomes principally important and any ineffective estimation can have detrimental consequences for system performance and it should be not omitted when surveying a cooperation network model. Furthermore, the relative distance between the two sources and the relay cluster also affects different throughput decreases as ρ decreases. When the distance d1 is in the range 0.3:0.4 and 0.6:0.7, the throughput performance of the system is affected by reduction less than the rest. At last, the larger the number of the relay is, the larger the throughput; therefore, this shows the advantage of using multiple relays.Figure 8
Throughput of the SIC-2TS protocol as a function ofd1 and ρ when PS/N0=5dB, Ω=−10dB, d2=1−d1, N=1,6, ε=0, ρ=0, α1=d1, α2=1−α1, and η=1.


## 6. Conclusion

In this article, a two-way cooperative NOMA model with two sources and multiple relaying nodes under the reality conditions as the ip/pCSIs and the ip/pSIC is studied. In the proposed protocol, a relay was selected in the setup phase by the MAC layer protocol to enhance the decoding capacity of the nearer source and minimize the collection time of imperfect CSIs. Spectrum utilization efficiency was improved by using the SIC and DNC techniques at the selected relay. In order to analyze and evaluate the system performance, exact and asymptotic closed-form outage probabilities and throughput expressions were considered and demonstrated by the Monte Carlo simulations. Our results showed that the performance of the proposed SIC-2TS protocol is significantly improved by the increased number of relays as well as the perfect operations of the SIC process and the CSI estimations. Besides, the system performance is decreased in the ipSIC and the ipCSI conditions. The noteworthy thing is found as the proposed SIC-2TS protocol can reach the best performance at optimal locations of the relay cluster and suitable values of power coefficients. In the pCSI condition, the proposed SIC-2TS protocol always has the system performance much better than the CONV-4TS and SIC-3TS protocols. However, in the ipCSI condition, the SIC-2TS protocol only performs better if the distances from two sources to the relay cluster are not very different. Finally, the analysis expressions of the outage probabilities and system throughput are validated by the Monte Carlo simulations.



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*Source: 8828443-2021-04-28.xml*