# CR-NOMA Networks over Nakagami-m Fading: CSI Imperfection Perspective

Authors: Anh-Tu Le; Dinh-Thuan Do
Journal: Wireless Communications and Mobile Computing
 (2021)

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## Abstract

There are high demands on both massive connections and high spectrum efficiency, and the cognitive radio-based nonorthogonal multiple access (CR-NOMA) system is developed to satisfy such demand. The unreal situation of CR-NOMA is considering the perfect channel state information (CSI) in receivers. This paper indicates impacts of imperfect CSI on outage and throughput performance. In particular, we focus on performance of the secondary network related to the imperfect CSI, and we derive closed-form expressions of outage probability and throughput for the downlink in such a CR-NOMA system. Particularly, a general form of Nakagami-m fading channel is adopted to examine the impact of fading on the performance of the CR-NOMA system. As the main achievement, we conduct extensive simulations and provide analyses to demonstrate the outage performance of the CR-NOMA system with CSI imperfections.


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## Body

## 1. Introduction

Considering as a promising multiple access technique, nonorthogonal multiple access (NOMA) has been proposed due to the higher traffic adaptation and optimized spectral efficiency (SE) [1]. By sharing frequency/code/time resources, the NOMA-based networks serve multiple users allocated various power allocation (PA) coefficients. In particular, the users with weak channel qualities are required more power compared with the users possessing stronger channel conditions. The strong users benefit from successive interference cancellation (SIC) to detect their own messages [1]. By introducing a cooperative diversity gain, cooperative NOMA (co-NOMA) has been studied in various scenarios to highlight the performance of NOMA [2–5]. To enhance the reliability performance, a co-NOMA scheme was investigated in [5], in which relay nodes are assigned as the users with better channel conditions (namely, NOMA-strong users).To further improve the SE, an underlay cognitive radio (CR) network is considered as another technique that allows secondary users (SUs) to transmit only if they do not result in reducing performance of primary users (PUs) related to harmful interference [6]. To address some of the 5G challenges including spectrum efficiency and massive connectivity, it is necessary to study the combination of NOMA with CR networks while maintaining reliable transmission of primary and secondary users. In various circumstances of the underlay CR-NOMA developed in [7–10], a superimposed signal is sent by the secondary transmitter to the secondary NOMA destination users. The authors concluded that if the PA factors and target rates are accurately chosen, the performance of NOMA users can be enhanced.The outage performance of CR NOMA systems has been investigated extensively in the literature [11–15]. Particularly, Nakagami-m fading channels are adopted in suitable systems such as [11, 15]. The authors in [16] studied overlay CR-inspired NOMA by introducing expressions of the outage probability. A similar scenario was performed in [17]; however, different from [16], only the best secondary user is chosen to forward the signal to the primary user. To assist in transmission between secondary NOMA users and a secondary base station (SBS), many relays are deployed [18]. In such a system, it is beneficial as the selected relay with the strongest link with the SBS is required to forward signals to the secondary receivers [18]. However, these papers assumed that channel state information (CSI) is available and exact estimation at the receivers. In practice, imperfect CSI occurs as the main reason of degraded performance. It motivates us to study the new system performance analysis for CR-NOMA over Nakagami-m fading. In particular, we evaluate numerous new challenges when compared to previous works. These main challenges are summarized as follows.
(i)
A cooperative scheme needs to be deployed in the context of NOMA transmission. Such a NOMA scheme has been recognized as a crucial approach to further improve spectral efficiency in CR-NOMA, especially for users with poor channel condition(ii)
Spectrum sharing is required at the secondary network in CR-NOMA; then, transmit power constraint is necessary since the SUs are permitted to reuse the licensed spectrum bands belonging to the PU. Spectrum sharing performs well under the conditions that the interference to the PU is guaranteed to be less than an assigned threshold(iii)
Outage performance of CR-NOMA network depends on the estimated channels at receiver. However, imperfect CSI make crucial impacts on SNR and then outage probability becomes worseTo address these challenges, we propose the CR-NOMA under a scenario of imperfect CSI by considering the Nakagami-m fading channel model. The proposed system in this research is challenging and more realistic compared with [19]. On the comprehensiveness, we implement our proposed method through simulations. We summarize the main contributions of this research as follows.
(i)
It need be improved performance of far NOMA users in CR-NOMA. Due to poor channel condition, the far user requires to assign a higher power allocation factor. Particularly, the secondary network is affected by interference from the primary nodes; then, relaying scheme should be implemented to achieve expected system improvement(ii)
Impacts of interference among two networks in CR-NOMA are considered. Degraded performance can be accepted under the limitation of interference satisfied. In particular, outage probability and throughput are the two most important metrics and we derive the concrete expressions of these metrics under the constraints of transmit power and separated target rates of NOMA users(iii)
Simulation results show the performance of the proposed CR-NOMA model using Nakagami-m fading channel models with respect to different variable conditions. We mainly focus on the trade-off between the system outage performance and other characteristic parameters such as level of CSI imperfections, total transmit power, and target rates. The results illustrate that the system outage behavior has a satisfied value within a certain range


## 2. System Model

We examine a general NOMA model as a system model shown in Figure1 wherein a downlink of CR-NOMA scenario. The relay is necessary to serve two NOMA users U1 and U2 which can operate in the secondary network (SN) containing secondary source (BS) intends and relay R. Moreover, the relay operates in a decode-and-forward (DF) mode. It is noted that such SN deals with interference to the primary transmitter (PT) and primary destinations (PD) are operated in the primary network (PN). The transmit source BS in the SN is limited by transmit power of the PN. It is assumed that all nodes are equipped single antenna and operated in half-duplex mode. The Nakagami-m fading channel coefficients are adopted in the considered system with parameter m. In such a CR-NOMA system, we denote wireless channels from node a to node b as gab. In the second hop of the SN, we denote g1 and g2 denote the links from relay to user U1 and U2, respectively. To guarantee NOMA fairness characteristic, let a1 and a2 be the power allocation factors. It is noted that we can assume that a1>a2 with a1+a2=1 [20, 21]. nR, n1, and n2 are denoted as Additive White Gaussian Noise (AWGN) with variance of N0.Figure 1
System model of CR-NOMA.In a real scenario, the imperfect CSI-related channels are known at the receivers; it is given by [22]
(1)gj=g^j+ej,where j∈SR,1,2, g^j represents the estimated channel factor, and ej stands for the channel estimation error. Moreover, IP~CN0,N0μ is denoted as the interference from the PN to SN, and μ is the scaling coefficient of IP. The CSI of primary transmitters is not available at the secondary receivers as reported in [19].To enable functions of CR in the context of the CR-NOMA system, the transmit power of secondary nodeQ is restricted as [19] PQ≤minIth/gQP2,P̄Q, Q∈S,R. We denote P̄Q as the maximum average allowed transmit power, with Ith denoted as the interference temperature constraint (ITC) at node PD.There are two phases in transmission at the SN. The received signal at the relayR in the first phase is computed by
(2)ySR=g^SR+eSRPSa1x1+a2x2+IP+nR=g^SRPSa1x1+a2x2+eSRPSa1x1+a2x2+IP+nR⏟effectivenoise.To further evaluate performance, the signal-to-interference-plus-noise ratio (SINR) and signal-to-noise ratio (SNR) before and after performing SIC can be obtained to detect signalx1 and x2, respectively, at R and they are given by
(3)γR,1=PSg∧SR2a1PSg∧SR2a2+PSeSR2a1+a2+N0μ+N0=PS/N0g∧SR2a1PS/N0g∧SR2a2+PS/N0eSR2+μ+1=g∧SR2a1ρSg∧SR2a2ρS+eSR2ρS+μ+1,(4)γR,2=g∧SR2a2ρSeSR2ρS+μ+1,where ρS=PS/N0 is defined as the transmit SNR at the BS.In the second phase, the received signals inUi, i∈1,2, can be expressed by
(5)yi=g^i+eiPRa1x1+a2x2+IP+ni=g^iPRa1x1+a2x2+eiPRa1x1+a2x2+IP+ni.Considering signal atU1, the SINR to detect its signal x1 is given by
(6)γU1,1=g∧12a1ρRg∧12a2ρR+ρRe12+μ+1.Similarly, to detect signalx1, x2, we compute the corresponding SINR at user U2 as
(7)γU2,1=g∧22a1ρRg∧22a2ρR+ρRe22+μ+1,(8)γU2,2=g∧22a2ρRρRe22+μ+1.


## 3. Performance Analysis

To compute the outage probability (OP) for usersUi, these functions are necessary, and they include the probability density function (PDF) and the cumulative distribution function (CDF) of dedicated Nakagami-m channel, respectively, as [15]
(9)fgj2x=xmj−1e−x/ΩjΓmjΩjmj,(10)Fgj2x=1−Γmj,x/ΩjΓmj=γmj,x/ΩjΓmj=1−e−x/Ωj∑i=0mj−11i!xΩji,where Γα=∫0∞xα−1e−xdx, γα,x=∫0xxα−1e−xdx, and Γα,x=∫x∞xα−1e−xdx represent for the Gamma function, the lower and upper incomplete Gamma function, respectively ([23], Equations 8.310.1, 8.350.1, and 8.350.2).It is noted that the PDF ofej2 is given as fej2x=xmej−1e−x/Ωej/ΓmejΩejmej, in which Ω=λ/m, m, and λ stand for the fading severity factor and mean, respectively. Regarding Nakagami fading, we also assume m is an integer number.


### 3.1. Outage Probability of UserU1

The OP is determined by SNR, i.e.,γthi=22Rthi−1 with Rthi being the target rate at user Ui,i=1,2, and its unit is BPCU (BPCU is short for bit per channel use). Then, it can be computed the OP of U1 as [19]
(11)Pout1=1−PrminγR,1,γU2,1,γU1,1>γth1=1−PrγR,1>γth1⏟Ψ1PrγU2,1>γth1⏟Ψ2PrγU1,1>γth1⏟Ψ3.Lemma.
The first term of (11) Ψ1 is formulated by
(12)Ψ1=∑iSR=0mSR−1∑kSR=0iSRmeSR+iSR−kSR−1meSR−1ΘkSRΩ̄SRmeSRβ1iSR1−ΓmSP,ρI/ρ̄SΩSP/ΓmSPe−β1Θ/ΩSRkSR!ΩSRkSRβ1+Ω̄SRmeSR+iSR−kSR+∑iSR=0mSR−1∑kSR=0iSRmeSR+iSR−kSR−1meSR−1β1iSRΩ̄SRmeSRkSR!ΓmSPβ1+Ω̄SRmeSR+iSR−kSR×ΓmSP+kSR,β1Θ̄ΩSP+ΩSRρIΩSRΩSPρ̄SΩSRΘ̄ΩSPmSPβ1+ΩSRΘ̄ΩSP−mSP−kSR.Proof.
See the appendix.
Base on (7), we can write Ψ2 as
(13)Ψ2=Prg∧22a1ρRg∧22a2ρR+ρRe22+μ+1>γth1=Prg∧22a1ρ̄Rg∧22a2ρ̄R+e22ρ̄R+μ+1>γth1,ρ̄R<ρIgRP2+Prg∧22a1ρIg∧22a2ρI+e22ρI+gRP2μ+1>γth1,ρ̄R>ρIgRP2.
With the help of (9), (10), and some transform, it can be expressed as
(14)Ψ2=∑i1=0m1−1∑k1=0i1me1+i1−k1−1me1−1Θk1Ω̄1me1β1i11−ΓmRP,ρI/ρ̄RΩRP/ΓmRPe−β1Θ/Ω1k1!Ω1k1β1+Ω̄1me1+i1−k1+∑i1=0m1−1∑k1=0i1me1+i1−k1−1me1−1β1i1Ω̄1me1k1!ΓmRPβ1+Ω̄1me1+i1−k1×ΓmRP+k1,β1Θ̄ΩRP+Ω1ρIΩ1ΩRPρ̄RΩ1Θ̄ΩRPmRPβ1+Ω1Θ̄ΩRP−mRP−k1,where Ω̄i=Ωi/Ωei. Similarly, putting (6) into (11) and the close form of Ψ3 can be obtained as
(15)Ψ3=∑i2=0m2−1∑k2=0i2me2+i2−k2−1me2−1Θk2Ω̄2me2β1i21−ΓmRP,ρI/ρ̄RΩRP/ΓmRPe−β1Θ/Ω2k2!Ω2k2β1+Ω̄2me2+i2−k2+∑i2=0m2−1∑k2=0i2me2+i2−k2−1me2−1β1i2Ω̄2m2k2!ΓmRPβ1+Ω̄1me2+i2−k2×ΓmRP+k2,β1Θ̄ΩRP+Ω2ρIΩ2ΩRPρ̄RΩ2Θ̄ΩRPmRPβ1+Ω2Θ̄ΩRP−mRP−k2.
Putting the result of Lemma (12), (14), and (15) into (11), the OP of x1 is written as
(16)Pout1=1−Ψ1×Ψ2×Ψ3,if:γth1<a1a2,1,otherwise.
□


### 3.2. Outage Probability ofx2

Similarly, it can be obtained outage probability forx2 as
(17)Pout2=1−PrminγR,2,γU2,2>γth2=1−PrγR,2>γth2⏟Ψ4PrγU2,2>γth2⏟Ψ5.Then, with the (4), Ψ4 can be expressed as
(18)Pout2=Prg∧SR2a2ρSe22ρS+μ+1>γth1=Prg∧SR2a2ρ̄SeSR2ρ̄S+μ+1>γth1,ρ̄S<ρIgSP2+Prg∧SR2a2ρIeSR2ρI+gSP2μ+1>γth1,ρ̄S>ρIgSP2.Similarly, the close form ofΨ4 is expressed as
(19)Ψ4=∑iSR=0mSR−1∑kSR=0iSRmeSR+iSR−kSR−1meSR−1ΘkSRΩ̄SRmeSRβ2iSR1−ΓmSP,ρI/ρ̄SΩSP/ΓmSPe−β2Θ/ΩSRkSR!ΩSRkSRβ2+Ω̄SRmeSR+iSR−kSR+∑iSR=0mSR−1∑kSR=0iSRmeSR+iSR−kSR−1meSR−1β2iSRΩ̄SRmeSRkSR!ΓmSPβ2+Ω̄SRmeSR+iSR−kSR×ΓmSP+kSR,β2Θ̄ΩSP+ΩSRρIΩSRΩSPρ̄SΩSRΘ̄ΩSPmSPβ2+ΩSRΘ̄ΩSP−mSP−kSR,where β2=γth2/α2. Then, Ψ5 is written as
(20)Ψ5=∑i2=0m2−1∑k2=0i2me2+i2−k2−1me2−1Θk2Ω̄2me2β2i21−ΓmRP,ρI/ρ̄RΩRP/ΓmRPe−β2Θ/Ω2k2!Ω2k2β2+Ω̄2me2+i2−k2+∑i2=0m2−1∑k2=0i2me2+i2−k2−1me2−1β2i2Ω̄2m2k2!ΓmRPβ2+Ω̄1me2+i2−k2×ΓmRP+k2,β2Θ̄ΩRP+Ω2ρIΩ2ΩRPρ̄RΩ2Θ̄ΩRPmRPβ2+Ω2Θ̄ΩRP−mRP−k2.Finally, the closed-form outage probability ofx2 is formulated as
(21)Pout2=1−Ψ4×Ψ5.


### 3.3. Asymptotic Analysis

To obtain the insight about CR-NOMA systems, the asymptotic outage probability is presented in high SNR regimesρ̄=ρ̄S=ρ̄R⟶∞ based on the outage probability. In high SNR, we use e−x≈1−x. Moreover, we can write the incomplete Gamma function as
(22)γa,bx=x⟶0bxaa.Base on analysis, the asymptotic outage probability ofx1 is obtained as
(23)Pout,∞1=1−Ψ1∞×Ψ2∞×Ψ3∞,where Ψ1∞, Ψ2∞, and Ψ3∞ are expressed, respectively, as
(24)Ψ1∞=∑iSR=0mSR−1∑kSR=0iSRmeSR+iSR−kSR−1meSR−1ΘkSRΩ̄SRmeSRβ1iSRβ1+Ω̄SRkSR−meSR−iSRΓmSP+1kSR!ΩSRkSR1−β1ΘΩSRρIρ̄SΩSPmSP+∑iSR=0mSR−1∑kSR=0iSRmeSR+iSR−kSR−1meSR−1ΓmSP+kSRβ1iSRΩ̄SRmeSRkSR!ΓmSPβ1+Ω̄SRmeSR+iSR−kSRΩSRΘ̄ΩSPmSP×1−1ΓmSP+kSR+1β1Θ̄ΩSP+ΩSRρIΩSRΩSPρ̄SmSP+kSRβ1+ΩSRΘ̄ΩSP−mSP−kSR,Ψ2∞=∑i1=0m1−1∑k1=0i1me1+i1−k1−1me1−1Θk1Ω̄1me1β1i1β1+Ω̄1k1−me1−i1ΓmRP+1k1!Ω1k11−β1ΘΩ1ρIρ̄RΩRPmRP+∑i1=0m1−1∑k1=0i1me1+i1−k1−1me1−1ΓmRP+k1β1i1Ω̄1me1k1!ΓmRPβ1+Ω̄1me1+i1−k1Ω1Θ̄ΩRPmRPβ1+Ω1Θ̄ΩRP−mRP−k1×1−1ΓmRP+k1+1β1Θ̄ΩRP+Ω1ρIΩ1ΩRPρ̄RmRP+k1,Ψ3∞=∑i2=0m2−1∑k2=0i2me2+i2−k2−1me2−1Θk2Ω̄2me2β1i2β1+Ω̄2k2−me2−i2k2!Ω2k2ΓmRP+11−β1ΘΩ2ρIρ̄RΩRPmRP+∑i2=0m2−1∑k2=0i2me2+i2−k2−1me2−1ΓmRP+k2β1i2Ω̄2m2k2!ΓmRPβ1+Ω̄1me2+i2−k2Ω2Θ̄ΩRPmRPβ1+Ω2Θ̄ΩRP−mRP−k2×1−1ΓmRP+k2+1β1Θ̄ΩRP+Ω2ρIΩ2ΩRPρ̄RmRP+k2.Similarly, the asymptotic outage probability ofx2 is expressed as
(25)Pout,∞1=1−Ψ4∞×Ψ5∞,where
(26)Ψ4∞=∑iSR=0mSR−1∑kSR=0iSRmeSR+iSR−kSR−1meSR−1ΘkSRβ2+Ω̄SRkSR−meSR−iSRkSR!ΩSRkSRΓmSP+1Ω̄SR−meSRβ2−iSR1−β2ΘΩSRρIρ̄SΩSPmSP+∑iSR=0mSR−1∑kSR=0iSRmeSR+iSR−kSR−1meSR−1ΓmSP+kSRβ2iSRΩ̄SRmeSRkSR!ΓmSPβ2+Ω̄SRmeSR+iSR−kSRβ2+ΩSRΘ̄ΩSP−mSP−kSR×1−1ΓmSP+kSR+1β2Θ̄ΩSP+ΩSRρIΩSRΩSPρ̄SmSP+kSRΩSRΘ̄ΩSPmSP,Ψ5∞=∑i2=0m2−1∑k2=0i2me2+i2−k2−1me2−1Θk2Ω̄2me2β2i2β2+Ω̄2k2−me2−i2k2!Ω2k2ΓmRP+11−β2ΘΩ2ρIρ̄RΩRPmRP+∑i2=0m2−1∑k2=0i2me2+i2−k2−1me2−1β2i2Ω̄2m2ΓmRP+k2k2!ΓmRPβ2+Ω̄1me2+i2−k2Ω2Θ̄ΩRPmRPβ2+Ω2Θ̄ΩRP−mRP−k2×1−1ΓmRP+k2+1β2Θ̄ΩRP+Ω2ρIΩ2ΩRPρ̄RmRP+k2.


### 3.4. Consideration on Throughput

The achieved outage probability leads to computations on throughput to indicate the ability of transmission at fixed target rates. As an extra metric, overall throughput can be given as(27)Ttotal=1−Pout1Rth1+1−Pout2Rth2.


## 3.1. Outage Probability of UserU1

The OP is determined by SNR, i.e.,γthi=22Rthi−1 with Rthi being the target rate at user Ui,i=1,2, and its unit is BPCU (BPCU is short for bit per channel use). Then, it can be computed the OP of U1 as [19]
(11)Pout1=1−PrminγR,1,γU2,1,γU1,1>γth1=1−PrγR,1>γth1⏟Ψ1PrγU2,1>γth1⏟Ψ2PrγU1,1>γth1⏟Ψ3.Lemma.
The first term of (11) Ψ1 is formulated by
(12)Ψ1=∑iSR=0mSR−1∑kSR=0iSRmeSR+iSR−kSR−1meSR−1ΘkSRΩ̄SRmeSRβ1iSR1−ΓmSP,ρI/ρ̄SΩSP/ΓmSPe−β1Θ/ΩSRkSR!ΩSRkSRβ1+Ω̄SRmeSR+iSR−kSR+∑iSR=0mSR−1∑kSR=0iSRmeSR+iSR−kSR−1meSR−1β1iSRΩ̄SRmeSRkSR!ΓmSPβ1+Ω̄SRmeSR+iSR−kSR×ΓmSP+kSR,β1Θ̄ΩSP+ΩSRρIΩSRΩSPρ̄SΩSRΘ̄ΩSPmSPβ1+ΩSRΘ̄ΩSP−mSP−kSR.Proof.
See the appendix.
Base on (7), we can write Ψ2 as
(13)Ψ2=Prg∧22a1ρRg∧22a2ρR+ρRe22+μ+1>γth1=Prg∧22a1ρ̄Rg∧22a2ρ̄R+e22ρ̄R+μ+1>γth1,ρ̄R<ρIgRP2+Prg∧22a1ρIg∧22a2ρI+e22ρI+gRP2μ+1>γth1,ρ̄R>ρIgRP2.
With the help of (9), (10), and some transform, it can be expressed as
(14)Ψ2=∑i1=0m1−1∑k1=0i1me1+i1−k1−1me1−1Θk1Ω̄1me1β1i11−ΓmRP,ρI/ρ̄RΩRP/ΓmRPe−β1Θ/Ω1k1!Ω1k1β1+Ω̄1me1+i1−k1+∑i1=0m1−1∑k1=0i1me1+i1−k1−1me1−1β1i1Ω̄1me1k1!ΓmRPβ1+Ω̄1me1+i1−k1×ΓmRP+k1,β1Θ̄ΩRP+Ω1ρIΩ1ΩRPρ̄RΩ1Θ̄ΩRPmRPβ1+Ω1Θ̄ΩRP−mRP−k1,where Ω̄i=Ωi/Ωei. Similarly, putting (6) into (11) and the close form of Ψ3 can be obtained as
(15)Ψ3=∑i2=0m2−1∑k2=0i2me2+i2−k2−1me2−1Θk2Ω̄2me2β1i21−ΓmRP,ρI/ρ̄RΩRP/ΓmRPe−β1Θ/Ω2k2!Ω2k2β1+Ω̄2me2+i2−k2+∑i2=0m2−1∑k2=0i2me2+i2−k2−1me2−1β1i2Ω̄2m2k2!ΓmRPβ1+Ω̄1me2+i2−k2×ΓmRP+k2,β1Θ̄ΩRP+Ω2ρIΩ2ΩRPρ̄RΩ2Θ̄ΩRPmRPβ1+Ω2Θ̄ΩRP−mRP−k2.
Putting the result of Lemma (12), (14), and (15) into (11), the OP of x1 is written as
(16)Pout1=1−Ψ1×Ψ2×Ψ3,if:γth1<a1a2,1,otherwise.
□


## 3.2. Outage Probability ofx2

Similarly, it can be obtained outage probability forx2 as
(17)Pout2=1−PrminγR,2,γU2,2>γth2=1−PrγR,2>γth2⏟Ψ4PrγU2,2>γth2⏟Ψ5.Then, with the (4), Ψ4 can be expressed as
(18)Pout2=Prg∧SR2a2ρSe22ρS+μ+1>γth1=Prg∧SR2a2ρ̄SeSR2ρ̄S+μ+1>γth1,ρ̄S<ρIgSP2+Prg∧SR2a2ρIeSR2ρI+gSP2μ+1>γth1,ρ̄S>ρIgSP2.Similarly, the close form ofΨ4 is expressed as
(19)Ψ4=∑iSR=0mSR−1∑kSR=0iSRmeSR+iSR−kSR−1meSR−1ΘkSRΩ̄SRmeSRβ2iSR1−ΓmSP,ρI/ρ̄SΩSP/ΓmSPe−β2Θ/ΩSRkSR!ΩSRkSRβ2+Ω̄SRmeSR+iSR−kSR+∑iSR=0mSR−1∑kSR=0iSRmeSR+iSR−kSR−1meSR−1β2iSRΩ̄SRmeSRkSR!ΓmSPβ2+Ω̄SRmeSR+iSR−kSR×ΓmSP+kSR,β2Θ̄ΩSP+ΩSRρIΩSRΩSPρ̄SΩSRΘ̄ΩSPmSPβ2+ΩSRΘ̄ΩSP−mSP−kSR,where β2=γth2/α2. Then, Ψ5 is written as
(20)Ψ5=∑i2=0m2−1∑k2=0i2me2+i2−k2−1me2−1Θk2Ω̄2me2β2i21−ΓmRP,ρI/ρ̄RΩRP/ΓmRPe−β2Θ/Ω2k2!Ω2k2β2+Ω̄2me2+i2−k2+∑i2=0m2−1∑k2=0i2me2+i2−k2−1me2−1β2i2Ω̄2m2k2!ΓmRPβ2+Ω̄1me2+i2−k2×ΓmRP+k2,β2Θ̄ΩRP+Ω2ρIΩ2ΩRPρ̄RΩ2Θ̄ΩRPmRPβ2+Ω2Θ̄ΩRP−mRP−k2.Finally, the closed-form outage probability ofx2 is formulated as
(21)Pout2=1−Ψ4×Ψ5.


## 3.3. Asymptotic Analysis

To obtain the insight about CR-NOMA systems, the asymptotic outage probability is presented in high SNR regimesρ̄=ρ̄S=ρ̄R⟶∞ based on the outage probability. In high SNR, we use e−x≈1−x. Moreover, we can write the incomplete Gamma function as
(22)γa,bx=x⟶0bxaa.Base on analysis, the asymptotic outage probability ofx1 is obtained as
(23)Pout,∞1=1−Ψ1∞×Ψ2∞×Ψ3∞,where Ψ1∞, Ψ2∞, and Ψ3∞ are expressed, respectively, as
(24)Ψ1∞=∑iSR=0mSR−1∑kSR=0iSRmeSR+iSR−kSR−1meSR−1ΘkSRΩ̄SRmeSRβ1iSRβ1+Ω̄SRkSR−meSR−iSRΓmSP+1kSR!ΩSRkSR1−β1ΘΩSRρIρ̄SΩSPmSP+∑iSR=0mSR−1∑kSR=0iSRmeSR+iSR−kSR−1meSR−1ΓmSP+kSRβ1iSRΩ̄SRmeSRkSR!ΓmSPβ1+Ω̄SRmeSR+iSR−kSRΩSRΘ̄ΩSPmSP×1−1ΓmSP+kSR+1β1Θ̄ΩSP+ΩSRρIΩSRΩSPρ̄SmSP+kSRβ1+ΩSRΘ̄ΩSP−mSP−kSR,Ψ2∞=∑i1=0m1−1∑k1=0i1me1+i1−k1−1me1−1Θk1Ω̄1me1β1i1β1+Ω̄1k1−me1−i1ΓmRP+1k1!Ω1k11−β1ΘΩ1ρIρ̄RΩRPmRP+∑i1=0m1−1∑k1=0i1me1+i1−k1−1me1−1ΓmRP+k1β1i1Ω̄1me1k1!ΓmRPβ1+Ω̄1me1+i1−k1Ω1Θ̄ΩRPmRPβ1+Ω1Θ̄ΩRP−mRP−k1×1−1ΓmRP+k1+1β1Θ̄ΩRP+Ω1ρIΩ1ΩRPρ̄RmRP+k1,Ψ3∞=∑i2=0m2−1∑k2=0i2me2+i2−k2−1me2−1Θk2Ω̄2me2β1i2β1+Ω̄2k2−me2−i2k2!Ω2k2ΓmRP+11−β1ΘΩ2ρIρ̄RΩRPmRP+∑i2=0m2−1∑k2=0i2me2+i2−k2−1me2−1ΓmRP+k2β1i2Ω̄2m2k2!ΓmRPβ1+Ω̄1me2+i2−k2Ω2Θ̄ΩRPmRPβ1+Ω2Θ̄ΩRP−mRP−k2×1−1ΓmRP+k2+1β1Θ̄ΩRP+Ω2ρIΩ2ΩRPρ̄RmRP+k2.Similarly, the asymptotic outage probability ofx2 is expressed as
(25)Pout,∞1=1−Ψ4∞×Ψ5∞,where
(26)Ψ4∞=∑iSR=0mSR−1∑kSR=0iSRmeSR+iSR−kSR−1meSR−1ΘkSRβ2+Ω̄SRkSR−meSR−iSRkSR!ΩSRkSRΓmSP+1Ω̄SR−meSRβ2−iSR1−β2ΘΩSRρIρ̄SΩSPmSP+∑iSR=0mSR−1∑kSR=0iSRmeSR+iSR−kSR−1meSR−1ΓmSP+kSRβ2iSRΩ̄SRmeSRkSR!ΓmSPβ2+Ω̄SRmeSR+iSR−kSRβ2+ΩSRΘ̄ΩSP−mSP−kSR×1−1ΓmSP+kSR+1β2Θ̄ΩSP+ΩSRρIΩSRΩSPρ̄SmSP+kSRΩSRΘ̄ΩSPmSP,Ψ5∞=∑i2=0m2−1∑k2=0i2me2+i2−k2−1me2−1Θk2Ω̄2me2β2i2β2+Ω̄2k2−me2−i2k2!Ω2k2ΓmRP+11−β2ΘΩ2ρIρ̄RΩRPmRP+∑i2=0m2−1∑k2=0i2me2+i2−k2−1me2−1β2i2Ω̄2m2ΓmRP+k2k2!ΓmRPβ2+Ω̄1me2+i2−k2Ω2Θ̄ΩRPmRPβ2+Ω2Θ̄ΩRP−mRP−k2×1−1ΓmRP+k2+1β2Θ̄ΩRP+Ω2ρIΩ2ΩRPρ̄RmRP+k2.


## 3.4. Consideration on Throughput

The achieved outage probability leads to computations on throughput to indicate the ability of transmission at fixed target rates. As an extra metric, overall throughput can be given as(27)Ttotal=1−Pout1Rth1+1−Pout2Rth2.


## 4. Simulation Result

The main parameters in simulation are provided in detail. First, the fixed power allocation factors are assigned for two NOMA users,a1=0.8, a2=0.2. In addition, we set channel gains as λSR=1, λSP=λRP=0.1, λ1=1, λ2=2, Rth=Rth1=Rth2=1 (BPCU), λe=λeSR=λe1=λe2, m=mSR=m1=m2=mSP=mRP, μ=0.1, and ρI=10 dB except for specific cases.In Figure2, we consider the OP versus transmit SNR at the SN. Different OP values are seen for two users at the SN due to different power allocation factors assigned to Ui. As the main achievement, the exact matching can be seen between analytical results and simulation results. Secondly, the OP of two users are better than the OMA case in the whole range of SNR ρ̄, which confirms the benefit of the CR-NOMA model. In addition, the OP can be improved as increasing of transmit SNR at the BS, which confirms the necessity of improving the quality of transmit signal.Figure 2
The OP of two users versus transmit SNR withm=2 and λe=0.01.In Figure3, two cases of ρ̄ which makes impacts on the OP of two users versus the transmit SNR at secondary source are studied. It can be seen clearly that ρ̄=30 dB is a better case. The main reason is that ρ̄=30 dB leads to higher SINR and then OP will be improved.Figure 3
Outage performance versus transmitρI varying ρ̄ with m=2 and λe=0.01.Figure4 plots OP of two users in different fading parameters; m=1, m=3, and m=3 are the better outage behavior. It can be seen clearly that m=3 leads to a better channel which plays an important role to improve the performance of destinations. The main reason is that main expressions of SINR depend on channel gains. Therefore, higher channel gains result in higher SINR and the corresponding outage performance can be enhanced. When m=3, such outage performance meets saturation lines sooner for the case of m=1, which confirms that the outage performance can not be adjusted by increasing SNR at the base station at high SNR region.Figure 4
Outage performance versus SNR varyingm with λe=0.01.Moreover, other trend of OP is similar as previous cases.In Figure5, the OP decreases significantly for the case of Rth=0.5 and λe=0.01. Two factors make influence on OP performance which is confirmed in this figure. It is important to examine OP under the impact of imperfect CSI. Therefore, λe related to the level of imperfect CSI is controlling parameter to target best outage performance for two users. The lower target rate Rth is required at lower CSI imperfection CSI that provides better outage performance. It can be observed the lowest case is the OP of user U1 that corresponds to λe=0.01 and Rth=0.5.Figure 5
The OP versus transmit SNR as varyingλe and Rth with m=2.Regarding impacts of power allocation factors on OP performance, Figure6 requires we perform simulation for case of a1>0.5. Figure 5 shows the different trends of OPs for two users Ui when we change power factor as 0.5<a1<1. Looking at case a1<0.7, the OP of the user U2 is better than that of U1. However, opposite trend happens as a1>0.7. The main reason is that power level allocated to each user is strictly related to the OP.Figure 6
The OP versus power allocation factora1 with m=2 and λe=0.01.To look on throughput performance,m fading parameter makes influence on the performance of two users of such a CR-NOMA, all observations as Figure 7. It can be observed that the highest throughput occurs at λe=0.01 and m=3. The ceiling throughput is also seen at a high region on transmit SNR as SNR is greater than 25 dB. Interestingly, the existence of Rth corresponds to maximal throughput as simulation reported in Figure 8.Figure 7
Throughput of system versus transmit SNR at BS with varyingλe and m.Figure 8
Throughput of system versusRth with varying ρ̄=ρI.


## 5. Conclusions

In this paper, we have studied the impact of imperfect CSI on the CR-NOMA as a possible realistic scenario. Two users are evaluated in terms of outage performance to show advantages of a developed CR-NOMA system wherein NOMA users cooperate with the relay nodes. In particular, we derived the closed-form expressions of outage probabilities and throughput to show performance gaps among two NOMA users. Our simulation analysis confirms that the target rates and power allocation factors are main impacts on these metrics. The further simulation results show that CR-NOMA yields better outage performance once we limit imperfect CSI at low level and proper power allocation factor achieved.



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*Source: 9915974-2021-06-23.xml*