# Exploiting Impacts of Intercell Interference on SWIPT-Assisted Non-Orthogonal Multiple Access

Authors: Thanh-Luan Nguyen; Dinh-Thuan Do
Journal: Wireless Communications and Mobile Computing
 (2018)

---

## Abstract

In this paper, we examine the influence of intercell interference (ICI) on the system outage behavior with important derived results in the proposed model of simultaneous wireless information and power transfer (SWIPT) together with the nonorthogonal multiple access (NOMA) using the amplify-and-forward protocol. We derive the closed-form expression of coverage probability for two NOMA users as a function of the signal-to-interference-plus-noise ratio (SINR). To fully take into account the effect of ICI, we adopt more practical parameters to evaluate the optimal power splitting coefficient regarding energy harvesting system performance analysis. Furthermore, to consider a more practical scenario, based on the fact that the number of ICI sources can affect wireless powered relays, we investigate the average outage probability by considering impacts of the reasonable number of participating ICI.


---

## Body

## 1. Introduction

In recent years, as a significant technique in forthcoming 5G transmission, nonorthogonal multiple access (NOMA) has more attraction due to its opportunity to improve substantial spectrum efficiency (SE) [1]. Considering advantages of power domain allocation to gain simultaneous access for diversified data streams to network, NOMA is evaluated as having better performance than the conventional orthogonal multiple access (OMA) [2]. Concentrating on the SE improvement, many earlier works regarding NOMA scheme are primarily focused. For example, single-input, single-output (SISO) schemes are studied for deployment in cooperative NOMA [3, 4]. In principle, by distributing multiple users into the different power domains, signal intending for the NOMA scheme at the source superimposes the symbol data to serve multiple users at destination. At the receiver, in order to detect multiplexed users’ information, successive interference cancellation (SIC) is acquired to eliminate the interference term. In addition to spectrum employment, other metrics need be considered, i.e., user fairness evaluating to multiple users in NOMA. Unlike traditional water-filling power allocation, NOMA users with better channel qualities are allocated less power while more power is allocated to users with poor channel qualities to highlight an enhanced trade-off between model throughput and user fairness. As a result, the same frequency and spreading codes are deployed in a lot of user equipment simultaneously; however they are distinguished by different power levels. Such principle results in improved spectral efficiency and ensured user fairness. The multiple-input-multiple-output (MIMO) NOMA designs are investigated in terms of the ergodic capacity maximization and an optimal power allocation strategy was then proposed as in [5]. The outage performance and the ergodic achievable rate are metrics to consider system performance of cellular downlink NOMA communications. Capacity is another metric to consider performance of NOMA as in [6]. Recently, stochastic geometry networks have been employed in NOMA to exhibit the physical layer security as in [7]. As an extension of [7], single-antenna and multiple-antenna stochastic geometry networks were investigated in two proposed schemes to increase the secrecy performance [8]. In [9], the popular metrics including optimal designs of decoding order, transmission rates, and power allocated to each user are studied in a new design of NOMA under secrecy considerations.Regarding the idea of combining NOMA with wireless powered networks, NOMA can be deployed with the wireless power transfer; i.e., the simultaneous wireless information and power transfer (SWIPT) technique was presented together with relaying network and its outage performance is considered in [10]. As a promising technology, to prolong the lifetime of energy-constrained users, wireless power transfer is facilitated in wireless relaying networks to forward a signal to far users [11–13]. To permit energy-limited users scavenge energy and information from the transmitted radio frequency (RF) signals, two famous policies, namely, time switching (TS) and power splitting (PS) receiver architecture, are introduced as in [14]. As a scenario of green NOMA, a SWIPT-assisted cooperative NOMA (SWIPT-CNOMA) network is studied as a combination between the NOMA and SWIPT scheme [15], and such novel scheme can be employed to wireless sensor or cellular networks. In such topology, it is possible to avoid lifetime limitation of the energy-constrained NOMA user that acts as a relay which can be able to harvest energy from the received signals. The main benefit of the SWIPT NOMA can be appreciated in the scenario where the relay employs that harvested energy rather than itself to forward the signal to the far NOMA users. In addition, to serve the near NOMA users with strong channel conditions, a PS protocol was employed in the SWIPT-CNOMA system [15]. By revealing the received information in the PS protocol, energy is first harvested and then used to help the signal forwarded to serve the far NOMA users. However, a shortcoming of the PS protocol [15] can be raised in which the relay always retains a quiet slot for some target data rate requirements. Considering the locations of NOMA users, three NOMA user selection policies are performed, including random near NOMA user and random far NOMA user (RNRF) selection, nearest near NOMA user and nearest far NOMA user (NNNF) selection, and nearest near NOMA user and farthest far NOMA user (NNFF) selection. In order to obtain the better outage performance, the authors in [16] considered a best-near best-far NOMA user collection scheme.However, there are few works that focus on interference effects on the NOMA system. Regarding the influences of cochannel interference (CCI) on the system performance, many outcomes in the literature showed that the aggressive frequency reuse is considered as main reason for CCI in the conventional cellular relaying systems [17–19]. However, FD schemes are more vulnerable to CCI due to the higher frequency reuse, and such characterization exhibits comparison between full-duplex (FD) transmission mode and traditional half-duplex (HD) transmission mode. In addition, in a multicell FD relay base stations meet influences with a much higher CCI due to adjacent cells compared with its HD counterparts [20–22]. As an alternative method, MIMO FD relaying transmission was introduced to examine the harmful effect of CCI on the system performance and such system can be applied in theoretical and practical applications. In [20], to evaluate the average spectral efficiency, a stochastic geometry is conveyed to consider the situation in which the base stations and user equipment operate in small cell network FD mode with FD mode (i.e., the dedicated antennas for transmission and reception are equipped in such FD nodes). In [21, 22], the authors studied the outage probability of a decode-and-forward FD relay with single-antenna nodes in FD relaying subject to CCI. It is worth noting that all these studies are restricted by employing a single antenna in each node of relaying networks. Therefore, deploying multiple receive and transmit antennas at the FD relay is a powerful scheme to eliminate both the CCI and loop interference channels at the FD relay and hence reliability and capacity are achieved.However, to the best of our knowledge, the impact of ICI on the performance of CNOMA with capability of energy harvesting has yet to be fully pressed. Such analysis motivates us to find impacts of ICI on the SWIPT-assisted NOMA.The primary contributions of our paper are summarized as follows:(i)
Different from the system model presented in [10, 14] where the relay only harvests energy from one source (i.e., base station), by deploying ICI as an extra resource to feed energy for relay, we propose a new SWIPT NOMA protocol under impacts of ICI to enhance the level of harvested energy to prolong the lifetime of NOMA systems(ii)
We derive closed-form expressions for the outage probability at NOMA users, when considering the interference channel schemes(iii)
The optimal power splitting factor is derived to obtain the maximum coverage probability at each user in SWIPT NOMAThe rest of the paper is structured as follows. In Section2, the system model for studying cooperative SWIPT NOMA is introduced. In Section 3, new analytical expressions are derived for the outage probability when the proposed scheme is used. The optimal power splitting factor for SWIPT can be shown in Section 4. Numerical results are offered in Section 5, which is followed by the conclusion in Section 6.


## 2. System Model

Consider a SWIPT NOMA network with the help of an energy-constrained relay for transmission to two representative NOMA users considering impacts of external interferers to the relay. In particular, Figure1 illustrates the proposed system model, where one source node (s), i.e., the main base station, one EH-assisted relay (r), and two users are considered. The relay utilizes the amplify-and-forward (AF) relaying protocol to transmit the superimposed signals from the source to each user and the power splitting (PS) protocol is adopted for energy harvesting. The channel between node x and node y is assumed to experience Rayleigh fading where Ωxy=E[|hxy|2] is the average channel power gain. Each node is equipped with a single half-duplex antenna.Figure 1
System model.Assume that the relay is affected byN external interferers. The channel coefficient between the n-th interferer and the relay is modeled as Nakagami-m fading with shape factor mn and variance of Ωn=Egn2, i.e., gn~Γ(mn,Ωn/mn). All given channels are assumed to be independent and not necessarily identically distributed. The transmission between the source and two users is divided into two phases. In the first phase, the source broadcasts a superimposed mixture of signals, i.e., xs=α1x1+α2x2, where xi, i∈1,2, is the intended signal for user ui, α1+α2=1.Assuming that the relay utilizes the power splitting (PS) protocol for energy harvesting [11], thus the received signal at the relay node is given by (1)yr=1-βPsxshsr+∑n=1N1-βPngnxn+nr,where β∈0,1 is the power splitting ratio, Ps is the transmit power of the source, xn and Pn are signal and the transmit power of the nth interferer, respectively, and nr is the zero mean additive white Gaussian noise (AWGN) at the relay with variance of σ2. Specifically, the harvested energy at the relay node is obtained as(2)Eh=ηβPshsr2+∑n=1NPngn2T2,where η∈0,1 is a coefficient representing the efficiency of the harvesting circuitry and T is the block time for the transmission from the source node to both users. Therefore, the transmit power at the relay is obtained as (3)Pr=EhT/2=ηβPshsr2+∑n=1NPngn2.In the second phase, the relay multiplies the received signal with an amplifying gain G and then forwards the amplified signal to both users. Hence, the received signal at useri is given by(4)yi=PryrhruiG+ni=Prhrui1-βPsxshsrG+Prhrui∑n=1N1-βPngnxnG+PrhruinrG+ni,where ni is the zero mean additive white Gaussian noise (AWGN) at user ui with variance of σ2. The amplifying factor due to the AF protocol at the relay is given by(5)G=11-βPshsr2+1-β∑n=1NPngn2+σ2.Remark 1.
Without loss of generality, we assume that useru1 has better channel quality than user u2; i.e., hru22≤hru12. Hence, due to NOMA, user 1 is encouraged to priorly decode x2 before detecting its own signal by applying successive interference cancellation (SIC), while user 2 can directly detect x2.Due to Remark1, the instantaneous signal-to-interference-plus-noise ratio (SINR) at user 1 to detect user 2’s signal is given by(6)γ1→2=α2Pshsr2α1Pshsr2+∑n=1NPngn2+σ2/PrG2/1-βhru12+σ2/1-β.Lemma 2.
The termσ2/PrG2 can be approximated in the high signal-to-noise ratio (SNR) region, in which the source transmits with a relatively large power, as(7)σ2PrG2≃1-βσ2ηβ.Proof.
By substituting (3) and (5) into σ2/PrG2 we have(8)σ2PrG2=1-βPshsr2+1-β∑n=1NPngn2σ2ηβPshsr2+∑n=1NPngn2+σ4ηβPshsr2+∑n=1NPngn2.
Note that, in the high SNR region, the second term, i.e.,σ4/ηβPshsr2+∑n=1NPngn2, can be neglected. Hence, after ignoring this part, we can simply achieve the result of Lemma 2.
Therefore, the approximated instantaneousγ1→2 is obtained by(9)γ1→2≈α2ψsrα1ψsr+ψI+1/ψru1+1/1-β,where ψsr≜Pshsr2/σ2, ψI≜∑n=1NPngn2/σ2, and ψru1≜ηβhru12. If γ1→2>22R2-1, where R2 (bits/s/Hz) is the target data rate for u2, user 1 can successfully carry out the successive interference cancellation (SIC) technique to detect user 2’s signal and remove this signal from y1. Assuming perfect SIC, the approximated instantaneous SINR for user 1 to detect its own signal is given by(10)γ1≈α1ψsrψI+1/ψru1+1/1-β.
Note that ifγ1>22R1-1, where R1 (bits/s/Hz) is the target data rate for u1, user 1 can successfully decode its own message. Similarly, the SINR at user 2 to detect its signal is obtained as(11)γ2=α2Pshsr2α1Pshsr2+∑n=1NPngn2+σ2/PrG2/1-βhru22+σ2/1-β.
Defineψru2≜ηβhru22; the approximated γ2 achieved by adopting Lemma 2 is given by(12)γ2≈α2ψsrα1ψsr+ψI+1/ψru2+1/1-β.


## 3. Performance Analysis

### 3.1. Preliminaries

In this section, some preliminary results are presented in terms of the solutions of some PDF and/or CDF calculation, which will be frequently invoked in the analysis. LetY=∑n=1NXn be the sum of N independent and not necessarily identically distributed gamma random variables (RVs), i.e., Xn~Γmn,Ωn/mn, where Ωn=EXn.Theorem 3.
The approximated probability density function (PDF) ofY follows the gamma distribution, where mI is the shape factor and ΩI is the variance, which is given by [23–25](13)fY1γ≈1ΓmIμImIγmI-1exp⁡-γμI,γ≥0,where μI=ΩI/mI, ΩI=∑n=1NΩn. The shape parameter, mI, is computed by using moment-base estimators, which is given by(14)mI=ΩI2EΦ2-ΩI2,where the first term in the denominator, E[Φ2], can be obtained with the help of (15)EΦk0=∑knn=1N-1k0EXN2kN-1∏t=1Nkt-1ktEXN-12kt-1-kt,in which(16)EXnk=Γmn+k/2ΓmnΩnmnk/2,where ∑knn=1N-1k0‍ is a short-hand representation of ∑k1=0k0∑k2=0k1⋯∑kN-1=0kN-1⋯ and nk denotes the binomial coefficient. In addition, when mn is a nonnegative integer, the exact PDF and CDF of Y are obtained through Theorem 4 as below.Theorem 4.
Without loss of generality, we assume thatμ1≥μ2≥…≥μN, where μn=Ωn/mnn=1…N. Therefore, the exact probability density function (PDF) of Y, where mn∈Z+, is given by (17)fY2γ=∑p=1P∑q=1mpRp,qSq-1!μpqγq-1exp-γμp,γ≥0,in which S=μnn=1N is a vector containing all μn’s. Without loss of generality, assume that vector S contains P distinct elements; each element is denoted by μp, where p=1,2,…,P. Intuitively, mp is the sum of all mn’s n≥1, in which the subscript is the index of the n-th element of S, i.e., μn, that is identical to μp. Specifically, the value of mp is determined through Figure 2 and the below description.Figure 2
A realization betweenμp and the elements in vector S.In Figure2, τp is the number of elements identical to μp and sp≜∑k=1pτk and sp-1+1 denote the indices of the last and the first element identical to μp, respectively. Therefore, mp=∑k=sp-1+1spmk. Subsequently, Rp,qS is obtained by(18)Rp,qS=1mp-q!μpmp-q×dmp-qdυmp-q∏k=1,k≠pP11+μkυmkυ=-1/μp.Proof.
See AppendixA.The closed-form expression of (18) is given in Appendix B in order to find the values of Rp,qS without requiring any differentiation manipulation. In addition, Remark 5 is given in order to provide some insights of Theorem 4.Remark 5.
Theorem4 is priorly defined for mn∈Z+; however in case of mn∈R+ one can still utilize Theorem 4 to find the sum of N i.n.i.d. gamma RVs as long as it satisfies mp∈Z+. This statement can be understood by observing the agreement between the simulation curves and analytical curves in Section 5. Further, it is noticed from Appendix A that (A.3) can also extend to the case of mn∈R+ without any modification. However, in order to adopt partial fraction decomposition (PFD) to achieve the closed-form expression for fY2γ in the third equality of (A.4), mp∈Z+ must be a positive integer, which can be easily achieved by assuming mn∈Z+ is also a positive integer.
Furthermore, in order to support the analysis in Section3.2 we introduce Theorem 6 as below.Theorem 6.
LetΙYκ≜EYe-κY; its closed-form expression is obtained as(19)ΙψIκ=∫0∞e-κγfYtγdγ(20)=1+μIκ-mIt=1∑p=1P∑q=1mpRp,qS1+μpκqt=2.Proof.
Using the given PDF ofψI in (13) for t=1 and (17) for t=2, with the help of [26, Eq. 3.351.3], we can easily achieve the above results.
In addition, when the interferences are statistically independent and identically distributed (i.i.d.), i.e.,μ1=μ2=…=μN=μ, then P=1 and mp=∑n=1Nmn≜m; ΙYκ becomes(21)ΙψIκ=1+μκ-m.
WhenN=0, i.e., Y=0, it immediately follows thatΙψIκ=1.


### 3.2. Performance Analysis on Outage Probability at Each User

In this section, the analytical result for the coverage probability of userui, denoted by O-i, is given in closed form. Hence, the outage probability at user ui is Oi=1-O-i. It is worth noticing that the results in this section are obtained when α2>τ2α1 and if α2<τ2α1 the coverage probability of each user becomes zero. Subsequently, the coverage probability of user 1 is defined as the probability that this user successfully decodes both x2 and x1 which is mathematically given by(22)O-1=Prγ1→2>τ2,γ1>τ1,where τ2≜22R2-1 and τ1≜22R1-1. However, deriving the exact O-1 in closed form is not tractable due to the complexity of (6); thus we then derive the approximated coverage probability in the high SNR region by substituting (9) and (10) into (22). Note that this assumption is widely used in literature. Subsequently, this probability is approximately obtained in closed form as(23)O-1≃ΙψIΤ1μsrexp⁡-Τ1μsr11-β×∑k=13-1k-12Τ1μsrμkK12Τ1μsrμk,where μsr=PsΩsr/σ2, μru1=ηβΩru1, μru2=ηβΩru2, Τ1=max⁡τ2/α2-τ2α1,τ1/α1, μ1=μru1, μ2=μru1μru2/μru2+μru1, and μ3=μru2, ΙψIκ is obtained in Theorem 6 and Theorem 4 by setting ΩI=∑n=1NPnΩn/σ2 for t=1 and mI=∑n=1NPnΩn/σ2, Kvz is the vth order modified Bessel function of the second kind [26], and α2>τ2α1.Proof.
See AppendixC.Hence, the coverage probability of user 2 is defined as the probability that this user successfully decodes its own message,x2, and can be represented mathematically by(24)O-2=Prγ2>τ2.Subsequently, the closed-form expression for the approximatedO-2 is given by(25)O-2≃ΙψIΤ2μsrexp-Τ2μsr11-β2Τ2μsrμ2K12Τ2μsrμ2,where Τ2=τ2/α2-τ2α1 and α2>τ2α1.Proof.
Substituting (12) into (22), we then obtain(26)O-2≃Pr⁡ψsr>Τ2ψI+1ψru2+11-β≃exp⁡-Τ2μsr11-β∫0∞exp⁡-Τ2μsrxfψItxdx×∫0∞exp⁡-Τ2μsr1yfψru2ydy.
In addition, the first integral is obtained by adopting (20) when κ=Τ2/μsr. Recall that hru22≤hru12 due to Remark 1; thus ψru2≤ψru1. Using order statistic in [27], the PDF of the ordered ψru2 is given by(27)fψru2y=1μ2exp⁡-yμ2,y≥0.
Therefore, the closed-form expression for the second integral is obtained by using (27) and [26, Eq. 3.471.9]. Hence, the proof is done.


## 3.1. Preliminaries

In this section, some preliminary results are presented in terms of the solutions of some PDF and/or CDF calculation, which will be frequently invoked in the analysis. LetY=∑n=1NXn be the sum of N independent and not necessarily identically distributed gamma random variables (RVs), i.e., Xn~Γmn,Ωn/mn, where Ωn=EXn.Theorem 3.
The approximated probability density function (PDF) ofY follows the gamma distribution, where mI is the shape factor and ΩI is the variance, which is given by [23–25](13)fY1γ≈1ΓmIμImIγmI-1exp⁡-γμI,γ≥0,where μI=ΩI/mI, ΩI=∑n=1NΩn. The shape parameter, mI, is computed by using moment-base estimators, which is given by(14)mI=ΩI2EΦ2-ΩI2,where the first term in the denominator, E[Φ2], can be obtained with the help of (15)EΦk0=∑knn=1N-1k0EXN2kN-1∏t=1Nkt-1ktEXN-12kt-1-kt,in which(16)EXnk=Γmn+k/2ΓmnΩnmnk/2,where ∑knn=1N-1k0‍ is a short-hand representation of ∑k1=0k0∑k2=0k1⋯∑kN-1=0kN-1⋯ and nk denotes the binomial coefficient. In addition, when mn is a nonnegative integer, the exact PDF and CDF of Y are obtained through Theorem 4 as below.Theorem 4.
Without loss of generality, we assume thatμ1≥μ2≥…≥μN, where μn=Ωn/mnn=1…N. Therefore, the exact probability density function (PDF) of Y, where mn∈Z+, is given by (17)fY2γ=∑p=1P∑q=1mpRp,qSq-1!μpqγq-1exp-γμp,γ≥0,in which S=μnn=1N is a vector containing all μn’s. Without loss of generality, assume that vector S contains P distinct elements; each element is denoted by μp, where p=1,2,…,P. Intuitively, mp is the sum of all mn’s n≥1, in which the subscript is the index of the n-th element of S, i.e., μn, that is identical to μp. Specifically, the value of mp is determined through Figure 2 and the below description.Figure 2
A realization betweenμp and the elements in vector S.In Figure2, τp is the number of elements identical to μp and sp≜∑k=1pτk and sp-1+1 denote the indices of the last and the first element identical to μp, respectively. Therefore, mp=∑k=sp-1+1spmk. Subsequently, Rp,qS is obtained by(18)Rp,qS=1mp-q!μpmp-q×dmp-qdυmp-q∏k=1,k≠pP11+μkυmkυ=-1/μp.Proof.
See AppendixA.The closed-form expression of (18) is given in Appendix B in order to find the values of Rp,qS without requiring any differentiation manipulation. In addition, Remark 5 is given in order to provide some insights of Theorem 4.Remark 5.
Theorem4 is priorly defined for mn∈Z+; however in case of mn∈R+ one can still utilize Theorem 4 to find the sum of N i.n.i.d. gamma RVs as long as it satisfies mp∈Z+. This statement can be understood by observing the agreement between the simulation curves and analytical curves in Section 5. Further, it is noticed from Appendix A that (A.3) can also extend to the case of mn∈R+ without any modification. However, in order to adopt partial fraction decomposition (PFD) to achieve the closed-form expression for fY2γ in the third equality of (A.4), mp∈Z+ must be a positive integer, which can be easily achieved by assuming mn∈Z+ is also a positive integer.
Furthermore, in order to support the analysis in Section3.2 we introduce Theorem 6 as below.Theorem 6.
LetΙYκ≜EYe-κY; its closed-form expression is obtained as(19)ΙψIκ=∫0∞e-κγfYtγdγ(20)=1+μIκ-mIt=1∑p=1P∑q=1mpRp,qS1+μpκqt=2.Proof.
Using the given PDF ofψI in (13) for t=1 and (17) for t=2, with the help of [26, Eq. 3.351.3], we can easily achieve the above results.
In addition, when the interferences are statistically independent and identically distributed (i.i.d.), i.e.,μ1=μ2=…=μN=μ, then P=1 and mp=∑n=1Nmn≜m; ΙYκ becomes(21)ΙψIκ=1+μκ-m.
WhenN=0, i.e., Y=0, it immediately follows thatΙψIκ=1.


## 3.2. Performance Analysis on Outage Probability at Each User

In this section, the analytical result for the coverage probability of userui, denoted by O-i, is given in closed form. Hence, the outage probability at user ui is Oi=1-O-i. It is worth noticing that the results in this section are obtained when α2>τ2α1 and if α2<τ2α1 the coverage probability of each user becomes zero. Subsequently, the coverage probability of user 1 is defined as the probability that this user successfully decodes both x2 and x1 which is mathematically given by(22)O-1=Prγ1→2>τ2,γ1>τ1,where τ2≜22R2-1 and τ1≜22R1-1. However, deriving the exact O-1 in closed form is not tractable due to the complexity of (6); thus we then derive the approximated coverage probability in the high SNR region by substituting (9) and (10) into (22). Note that this assumption is widely used in literature. Subsequently, this probability is approximately obtained in closed form as(23)O-1≃ΙψIΤ1μsrexp⁡-Τ1μsr11-β×∑k=13-1k-12Τ1μsrμkK12Τ1μsrμk,where μsr=PsΩsr/σ2, μru1=ηβΩru1, μru2=ηβΩru2, Τ1=max⁡τ2/α2-τ2α1,τ1/α1, μ1=μru1, μ2=μru1μru2/μru2+μru1, and μ3=μru2, ΙψIκ is obtained in Theorem 6 and Theorem 4 by setting ΩI=∑n=1NPnΩn/σ2 for t=1 and mI=∑n=1NPnΩn/σ2, Kvz is the vth order modified Bessel function of the second kind [26], and α2>τ2α1.Proof.
See AppendixC.Hence, the coverage probability of user 2 is defined as the probability that this user successfully decodes its own message,x2, and can be represented mathematically by(24)O-2=Prγ2>τ2.Subsequently, the closed-form expression for the approximatedO-2 is given by(25)O-2≃ΙψIΤ2μsrexp-Τ2μsr11-β2Τ2μsrμ2K12Τ2μsrμ2,where Τ2=τ2/α2-τ2α1 and α2>τ2α1.Proof.
Substituting (12) into (22), we then obtain(26)O-2≃Pr⁡ψsr>Τ2ψI+1ψru2+11-β≃exp⁡-Τ2μsr11-β∫0∞exp⁡-Τ2μsrxfψItxdx×∫0∞exp⁡-Τ2μsr1yfψru2ydy.
In addition, the first integral is obtained by adopting (20) when κ=Τ2/μsr. Recall that hru22≤hru12 due to Remark 1; thus ψru2≤ψru1. Using order statistic in [27], the PDF of the ordered ψru2 is given by(27)fψru2y=1μ2exp⁡-yμ2,y≥0.
Therefore, the closed-form expression for the second integral is obtained by using (27) and [26, Eq. 3.471.9]. Hence, the proof is done.


## 4. Optimal Power Splitting Factor Problem

In this section, the maximum coverage probability at a single user is obtained by adjusting the power splitting factor to maximize the SNRs/SINRs given in Section3. Specifically, the optimal power splitting factor to achieve the maximum SNRs/SINRs at user ui is given by(28)βiop=argmax0<β<1⁡φi,where φ1≜min⁡γ1→2,γ1 and φ2≜γ2. Since β only appears in the denominator of φi we can rewrite (28) as(29)βiop=argmin0<β<1⁡ψiβ≜1ηβhrui2+11-β.Lemma 7.
The functionψiβ is convex.Proof.
Since1/ηβhrui2 and 1/1-β are convex functions of β∈0,1 and the sum of two convex functions is also convex, thus ψiβ is convex.
From Lemma7, it is clear that (30) has a unique solution for βiop which is obtained by solving (∂/∂β)ψiβ=0. Subsequently, after some simple algebraic manipulations, the optimal energy harvesting factor is given as below: (30)βiop=11+ηhrui2.
Substituting (30) into (9), (10), and (12), the optimal coverage probability of user ui is given approximately as(31)O-iop≃ΙψIΤiμsrexp⁡-Τiμsr∑k=i4-i-1k+i--1×ωi,kK1ωi,k-πΤiμsrηωi,kexp-ωi,k,where ωi,k=2Τi/μsrηΩk, Ω1=Ωru1, Ω2=Ωru1Ωru2/Ωru1+Ωru2, Ω3=Ωru2, and i-=2/i.Proof.
See AppendixD.Note that the relay can only choose to tuneβ to achieve the best performance for a single user in a specific block time, T; thus the other user may not reach maximum coverage probability during that block time. In this case, the coverage probability of the other user is given approximately by(32)O-i-≃ΙψIΤi-μsrexp⁡-Τi-μsr∑ki-=i-4-i-∑ki=i4-i-1ki-+i-1-1ki+i--1×1-Τi-μsrπηΩkiωi-,ki-K1ωi-,ki--Τi-μsrΩki-πΩkiηK0ωi-,ki-,where ωi-,ki-=2Τi-/μsrηΩki-.Proof.
See AppendixE.


## 5. Numerical Results

In this section, in terms of the outage probability of both users in the cooperative SWIPT NOMA network, we present representative numerical results to demonstrate the performance assessments. In the considered SWIPT NOMA network under impacts of ICI, we set the energy conversion efficiency of SWIPT asη=1. We define SNR≜PsΩsr/σ2 as the average SNR at the transmitter (source); assume that Pn=PI and define INR≜∑n=1NPIΩn/σ2 as the average interference-to-noise ratio. Ωsr=1, Ωru1=1.5, and Ωru2=0.5. From Figures 3–8, the number of external interferers is set to 2 (N=2) with m1=1.2, m2=1.8, Ω1=1.2, and Ω2=1.8; the power allocation coefficients are α1=0.1 and α2=0.9. The below results are obtained by averaging 500,000 separated simulations under a MATLAB environment.Figure 3
Outage probability at user 1 vs. the energy harvesting ratioβ, where SNR=30dB and INR=-10dB.Figure 4
Outage probability at user 2 vs. the energy harvesting ratioβ, where SNR=30dB and INR=-10dB.Figure 5
Optimal outage probability at user 1 vs. the average SNR.Figure 6
Optimal outage probability at user 2 vs. the average SNR.Figure 7
Optimal outage probability vs. user 1’s target data rate.Figure 8
Optimal outage probability vs. user 2’s target data rate.Firstly, we consider the results in Figures3 and 4. The curves denoted by (O) can be obtained from (23) with ΙψIκ in (20) achieved at t=1 whereas the curves denoted by (X) can be obtained with ΙψIκ obtained at t=2(20). As a clear observation, the simulation lines obtained via (22) and (24), respectively, strictly match with the analytical lines for both t=1 and t=2, which confirms the accuracy of our derivation. Furthermore, Figure 3 shows the outage probability for user 1, as increasing R1 to serve this user with higher data rate or higher quality of service and outage performance will be the worse case. In such result, the black curve is R1=1.0 bits/s/Hz while the red curve is R1=1.5 bits/s/Hz. One can observe that a lower outage probability is achieved by changing approximate β=0.6→0.8. The figures also demonstrate that as β is very high or very low, a higher outage event can occur due to the low harvested energy at the relay where it makes impacts on the end-to-end SNR of the system. Similarly, outage probability (not optimal) is shown for user 2 in Figure 4. It can be observed that as R2 is increased then outage performance will be worse; in this case, we set R2=0.5 bits/s/Hz as the black curves and the red ones corresponding with R2=1.0 bits/s/Hz.In Figure5, the outage probabilities for user 1 are achieved by varying INR levels and are shown as functions of the average SNR. The simulation curves are obtained by adopting (22) with β defined in (28) while the analytical curves are obtained from (31) with i=1. As can be seen from the figure, NOMA with an INR level equal to −10 dB outperforms the other remaining scenarios, since it can ensure that the outage is achievable by all the users as controlling impacts of ICI.Figure6 demonstrates the outage performance for user 2 versus SNR with different INR levels. Similarly, the simulation curves can be obtained via (24) with the optimal β in (28) while the analytical curves are obtained from (31) with i=2. One can observe that the proposed method achieves the lowest outage since it has the lowest impacts on the system among three scenarios. The figure also demonstrates the existence of the outage ceilings in the low SNR region (i.e., SNR less than 10 dB). This is due to the fact that the system probability is approaching an outage event and such outage is determined only by the SNR while other parameters do not affect outage. It is worth noting that increasing SNR can improve the outage; however, for the case INR=-10dB, the throughput is the lowest among three scenarios. This is because, in the latter case, the lower impact for information processing at user 2, and hence an outage, will be improved. Therefore, we see that it is important to select appropriate ICI when designing practical NOMA downlink transmission systems.Figures7 and 8 illustrate the optimal outage probability of both users versus the target rates, i.e., R1 and R2, in two different time blocks. The red and blue solid lines are both obtained from (31) when i=1 (ensuring QoS at user 1) and i=2 (ensuring QoS at user 2), respectively, while the simulation results for user 1 and user 2 are obtained by substituting β=1+ηhru12-1 into (22) and β=1+ηhru22-1 into (24), respectively. It is worth noting that, in the same signal block, the relay only satisfies QoS criteria for one user only; however we intend to compare the optimal outage performance each user can achieve. Firstly, one can observe that the outage probabilities of user 2 and user 1 can increase as R2 and R1 increase, respectively. The reason is that increasing R2 and R1 can lead to the higher threshold of decoding and therefore can result in more outage. More specifically, the outage probability at user 2 increases as R2 increases but it remains constant varying R1 as depicted in Figure 8, where the solid line overlaps the dashed line. The reason is that the performance of this user only depends on the data rate of x2 but not that of x1. In addition, the outage probability at user 1 depends highly on Τ1; thus if Τ1=τ1/α1 increasing R2 will not affect the outage probability at this user resulting in a segment of straight blue lines in Figure 8 and a portion of the blue curves in Figure 7, where the solid curve overlaps the dashed curve. However, the system performance can be decreased in case of Τ1=τ2/α2-τ2α1.Figure9 presents the optimal outage probability at both users as considering on increasing number of interferers. The solid curves are obtained through (31) representing the optimal outage probability of user i(i=1,2) whereas the dashed curves are achieved via (32) denoting the outage probability of user i when the relay focuses on optimizing the QoS of the other user, i.e., user i-. In such experiment, ΙψIκ can be extracted from (21) as m1=…=mN=1 and Ω1=…=ΩN=1 (i.i.d. scenario). It is confirmed that if we try to optimize the outage probability for a user then another user cannot achieve maximum performance. This situation is equivalent to the case of requirement for maximum QoS for the selected user (i.e., user 1) resulting in system performance of the remaining user (i.e., user 2) being dropped.Figure 9
Optimal outage probability vs. number of external interferers.


## 6. Conclusions

In this paper, the application of simultaneously wireless information and power transfer (SWIPT) in nonorthogonal multiple access (NOMA) under the influence of external interferers has been investigated. In particular, we derived the closed-form expression of outage probability in the SWIPT NOMA protocol. In order to provide a complete framework, we illustrate system performance through simulation results used to address the impacts of the target rates of the near and far users and the number of interference sources and turn in evaluating the outage performance of the proposed protocol. In terms of outage probability, new analytical results have been derived to conclude the system efficiency. Such numerical results have been demonstrated to corroborate our analysis. We conclude that, by carefully choosing the parameters of the network regarding external interferers, acceptable system performance can be ensured by applying the SWIPT NOMA protocol in practical networks.



---

*Source: 2525492-2018-11-25.xml*