# Exploiting Opportunistic Scheduling for Physical-Layer Security in Multitwo User NOMA Networks

Authors: Kyusung Shim; Beongku An
Journal: Wireless Communications and Mobile Computing
 (2018)

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## Abstract

In this paper, we address the opportunistic scheduling in multitwo user NOMA system consisting of one base station, multinear user, multifar user, and one eavesdropper. To improve the secrecy performance, we propose the users selection scheme, called best-secure-near-user best-secure-far-user (BSNBSF) scheme. The BSNBSF scheme aims to select the best near-far user pair, whose data transmission is the most robust against the overhearing of an eavesdropper. In order to facilitate the performance analysis of the BSNBSF scheme in terms of secrecy outage performance, we derive the exact closed-form expression for secrecy outage probability (SOP) of the selected near user and the tight approximated closed-form expression for SOP of the selected far user, respectively. Additionally, we propose the descent-based search method to find the optimal values of the power allocation coefficients that can minimize the total secrecy outage probability (TSOP). The developed analyses are corroborated through Monte Carlo simulation. Comparisons with the random-near-user random-far-user (RNRF) scheme are performed and show that the proposed scheme significantly improves the secrecy performance.


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## Body

## 1. Introduction

To support the 5G-and-beyond-5G communication networks, called future communication networks, nonorthogonal multiple access (NOMA) has been considered as a solution to increase the spectral efficiency by superposing multiple users in resource domain (e.g., time and frequency) [1–3]. More specifically, the downlink scenario of the multitwo user NOMA networks consisting of one base station, a near user (that has strong channel state information (CSI)), and a far user (that has low CSI) is actively considered. The base station simultaneously transmits message to both users using the superposition coding; then the transmit power of the far user message is allocated more than that of the near user message. Under the NOMA scenario, the near user first subtracts the far user message using the successive interference cancellation (SIC) technique [4]; after that the near user decodes its own message. Different from the near user, the far user directly decodes its own message from the received signal without the interference elimination process since the far user message has allocated higher transmit power than that of the near user message.The NOMA system can allocate the multiple information on the transmit signal using the superposition coding. Thus, the NOMA system has an advantage in terms of spectrum efficiency compared to that of the OMA system [5–7]. However, this point is a disadvantage in terms of the security [8, 9]. For example, if the eavesdropper succeeds in the interception of one NOMA signal, then the eavesdropper can obtain multiple user information [10]. Thus, the security issue is more important in the NOMA system. Physical layer security (PLS) is one of the possible solutions against the attack of the malicious user. More specifically, from the information-theoretical aspect, the message can be confidentially transmitted if the main channel (between legitimate users) and the eavesdropper channel (between a legitimate user and an eavesdropper) can be controlled  so that the legitimate user can decode the received message successfully while the eavesdropper is not able to decode their intercepted message [11]. It is shown that PLS is on the cutting edge of the technique of security in wireless communication to prevent eavesdropping attacks [12–14].In PLS, the secrecy outage probability is defined as the probability that the achievable system secrecy capacity of the near/far user drops below a predefined target secrecy rate as in [15]. From the secrecy outage probability, we can measure the degree of security on the system. For example, the low secrecy outage probability system means that the system is more robust compared to that of the high secrecy outage probability system in the viewpoints of security. To improve the PLS performance, the CSI-based opportunistic scheduling [16], that is, the source (destination), communicates with one scheduled destination (source) in a certain resource block (e.g., time and frequency). It has been recognized as an attractive scheduling scheme [17–19] since the time-varying nature of wireless channels is exploited. Thus, opportunistic scheduling is expected as one of techniques to enhance the secrecy performance on NOMA system.Indeed, the authors of [20] studied the security performance of single-input-single-output (SISO) NOMA system. The authors confirmed that the secrecy sum rate of SISO NOMA system has better performance than that of the OMA system. Liu etal. [21] compared the multiple-input-multiple-output (MIMO) NOMA system and MIMO OMA system. Additionally, the authors proved that MIMO NOMA system is better than MIMO OMA system in terms of sum channel capacity. In [22], the authors proposed the transmit antenna selection (TAS) scheme to improve the secrecy performance for downlink multiple-input-single-output (MISO) NOMA system. The first proposed antenna selection scheme is the optimal antenna selection scheme for near user and far user, respectively, while the second proposed antenna selection scheme is the suboptimal antenna selection scheme for the main channel performance of the near user and far user, respectively. The authors of [23] addressed the PLS of large-scale NOMA system. This network consisted of the multiple-antenna base station; the legitimate users were uniformly distributed within a disc, and the eavesdroppers were distributed in an infinite two-dimensional plane via a homogeneous Poisson point process (PPP). The authors derived new exact and asymptotic analyses for the secrecy outage probability (SOP). Li etal. [24] optimized the secure beamforming and power allocation design on the MISO NOMA system. From the numerical results, the authors demonstrated that their proposed scheme outperformed that of the conventional OMA system. The authors of [25] designed a secure NOMA system under secrecy constraints where a transmitter sends a signal to multiusers in the presence of an eavesdropper. The authors minimized the total transmit power subject to the QoS and secrecy constraints. The authors of [26] exploited the artificial-noise-aided secure transmission for NOMA full-duplex relay network. The authors proposed the novel joint NOMA and artificial-noise-aided full-duplex relay (NOMA-ANFDR) scheme to enhance the physical security. Specifically, this paper derived the SOP and secrecy throughput. The NOMA-ANFDR outperformed the joint NOMA and artificial-noise half-duplex relay (NOMA-ANHDR) scheme.Very recently, in [8], the authors proposed the cooperative NOMA system to enhance PLS performance. The authors considered the NOMA system including a base station, two users (near user and far user), a relay, and an eavesdropper. The authors derived the closed-form and asymptotic analysis for security outage probability and strictly positive secrecy capacity, respectively. The authors of [27] designed a secure massive NOMA system with multiclusters. The authors studied the interuser interference in the designed NOMA system and proposed the power control and power allocation algorithm for maximizing the secrecy capacity. In [28], the authors studied the security impact in the multiuser visible lighting communication (VLC) [29] with NOMA system. The authors considered both single eavesdropper and multiple eavesdroppers scenarios when the one transmitter communicates with multiple users. The authors derived the exact secrecy outage probability in this network. The author in [30] proposed the relay-supported cooperative NOMA system. The proposed system helped to transmit data using the multiple antenna equipped with relay. In a certain time block, this system selected the transmitted antenna in the relay based on the legitimate channel state information. The authors derived the exact closed-form expression in the proposed system. Lei etal. [31] proposed the NOMA system consisting of multiple-antenna base station, multidestinations equipped with multiple antennas, and multieavesdroppers equipped with multiple antennas. Then, the authors proposed and analyzed the max-min based transmit antenna selection scheme when the eavesdroppers work independently or collude.In the related works, the authors proposed the several schemes to improve the PLS performance. However, the opportunistic scheduling in the multitwo user NOMA network was not considered in the literature. Thus, in this paper, we propose a new users selection scheme to improve the PLS performance in downlink multitwo user NOMA system using the opportunistic scheduling in the presence of one eavesdropper. Compared with the aforementioned works, the main contribution and feature of this paper can be summarized as follows.(i)
In this paper, we propose a new near/far user selection scheme on the multiple-two user NOMA system. The proposed scheme, denoted by best-secure-near-user best-secure-far-user (BSNBSF) scheme, aims to select the best near-far user pairs, whose data transmission is the most robust combat against the eavesdropper’s attack. More specifically, in a certain resource block, we take into account the main/eavesdropper channel state information to select the received user pairs.(ii)
We drive novel expressions for the selected user pairs, respectively, to estimate the secrecy performance achieved by the proposed scheme. The analyses are not reported in the literature. More specifically, for the selected near user case, we derive an exact closed-form expression on the SOP, whereas, for the selected far user case, we obtain a tight approximated closed-form expression on the SOP, respectively.(iii)
We present other results, called the asymptotic SOP, for the proposed scheme. These expressions provide the insight into the behavior of the average far user channel power gain (λSF).(iv)
We conduct the optimal analysis of the power allocation coefficient (θN). More specifically, the proposed optimal analysis finds the optimal power allocation coefficient value to make the total SOP minimum on the proposed system based on the descent method.(v)
Through the numerical results, we demonstrate that the BSNBSF scheme significantly enhances the secrecy performance on the multitwo user NOMA system compared to that of the benchmarked scheme. More specifically, in the case of the SOP for the selected near user, the floor occurs and the selected far user shows the convex pattern as a function of the transmit SNR. Additionally, the secrecy sum throughput of the BSNBSF scheme shows better performance than that of the RNRF scheme.The rest of this paper is organized as follows. In Section2, the proposed system model and user pair selection scheme are described. Section 3 investigates the closed-form and asymptotic analyses of the multitwo user NOMA systems. Section 4 presents the optimal analysis of the power allocation coefficients. Section 5 presents some illustrative numerical results based on the insightful discussion. Monte Carlo simulations are shown to corroborate the proposed analyses. Finally, Section 6 concludes the paper.Notations:  X~CN(0,σ2) denotes a circularly symmetric complex Gaussian random variable X with zero mean and variance σ2; Pr(·) is the probability; fX(·) and FX(·) represent the probability density function (PDF) and cumulative distribution function (CDF) of the random variable X, respectively.


## 2. System Model

Let us consider a downlink multitwo user NOMA system including a base station (S), a set of K near users, N={Ni∣i=1,2,…,K}, and a set of M far users, F={Fj∣j=1,2,…,M}, and an eavesdropper (E), as shown in Figure 1. More specifically, the near users can perfectly use the SIC technique to subtract the far xF [22]. Both the legitimate and the illegitimate receivers are equipped with a single antenna and operate in half-duplex mode. In this paper, hXY and hXY2, where X∈{S}, Y∈N∪F∪{E}, present the channel coefficient and the corresponding channel gain of X→Y, respectively. Assuming that all wireless channels exhibit Rayleigh block fading channel, hXY can be modeled as independent and identically distributed (i.i.d.) complex Gaussian random variable with zero mean and variance λXY. Thus, the corresponding channel gain, hXY2, is an exponential random variable with probability density function (PDF), fhXY2(z) = 1/λXYexp(-z/λXY), if z≥0; otherwise, fhXY2(z) = 0 as in [32–34]. More specifically, the average channel gain can be written as λXY=(dXY/d0)-ϵL, where L is the reference signal power attenuation, dXY denotes the distance between X and Y, d0 presents the reference distance, and ϵ is the path-loss exponent [33]. And the channel noise is followed by ωY~CN(0,σY2). We also assume that the source perfectly knows the channel state information (CSI) of all legitimate users and eavesdropper, as in [11, 35].Figure 1
Schematic illustration of the multitwo user NOMA system with the presence of an eavesdropper.


### 2.1. Communication Process

In this subsection, we present in detail the communication process in two-user NOMA system. Assuming thatNi and Fj are selected to receive its data from the S in a certain time slot. From the principle of NOMA, the messages xN and xF that will be allocated to θN and θF, respectively, are superposed as θNxN+θFxF and then broadcasted by S, where θN and θF denote the power allocation coefficients. We suppose that hSNi2>hSFj2, and set 0<θN<θF and θN+θF=1 as in [21].At the near user N i, the received signal is given by(1)ySNi=PSθNhSNixN+PSθFhSNixF+ωNi,where PS denotes the transmit power of the S. Because of the power allocation coefficient condition, the near user needs to subtract the component, xF, from the ySNi using the SIC process [36]. The signal-to-interference-plus-noise-ratio (SINR) of the eliminated component, xF, is expressed as(2)γSNixF=PSθFhSNi2PSθNhSNi2+σSNi2.After the SIC process, the Ni archives its own message, xN, from the received signal in which signal-to-noise-ratio (SNR) is obtained as(3)γSNixN=PSθNhSNi2σSNi2.At the far user F j, the received signal is given by(4)ySFj=PSθFhSFjxF+PSθNhSFjxN+ωFj.Different from the near user, the Fj directly decodes the SINR from the received signal because of the power allocation coefficient condition. The received SINR at Fj to decode xF is given by(5)γSFjxF=PSθFhSFj2PSθNhSFj2+σSFj2.Meanwhile, the eavesdropper can intercept the signal due to the broadcast nature of wireless medium. Thus, the received signal atE can be written as(6)ySE=PSθNhSExN+PSθFhSExF+ωE.Different from the legitimate user, we assume that the E has enough ability to distinguish each message from the received signal. Thus, the SINRs of the received signal are given by(7)γSExF=PSθFhSE2PSθNhSE2+σSE2,and(8)γSExN=PSθNhSE2σSE2,respectively.In physical-layer security, the secrecy capacity means the difference between main channel capacity and eavesdropper channel capacity. Thus, in two-user NOMA system, the secrecy capacities ofxN and xF are given by [22, 37](9)Cs,Ni=log21+γSNixN-log21+γSExN+,(10)Cs,Fj=log21+γSFjxF-log21+γSExF+,respectively, where [x]+=max⁡{x,0}.


### 2.2. The Proposed Best-Secure-Near-User Best-Secure-Far-User (BSNBSF) Scheme

The proposed user selection process is conducted through the channel state information (CSI) estimation/calculation system. Thus, this process is carried out before the data communication process as in [11, 35]. In this paper, we propose the BSNBSF user selection scheme to maximize the secrecy capacity at Ni and Fj, respectively. The proposed scheme can be mathematically expressed as(11)Nb=argmaxi∈Nlog21+γSNixN1+γSExN,and(12)Fb=argmaxj∈Flog21+γSFjxF1+γSExF.(11) and (12) mean that the selected users are the best secrecy capacity of a certain time slot.


## 2.1. Communication Process

In this subsection, we present in detail the communication process in two-user NOMA system. Assuming thatNi and Fj are selected to receive its data from the S in a certain time slot. From the principle of NOMA, the messages xN and xF that will be allocated to θN and θF, respectively, are superposed as θNxN+θFxF and then broadcasted by S, where θN and θF denote the power allocation coefficients. We suppose that hSNi2>hSFj2, and set 0<θN<θF and θN+θF=1 as in [21].At the near user N i, the received signal is given by(1)ySNi=PSθNhSNixN+PSθFhSNixF+ωNi,where PS denotes the transmit power of the S. Because of the power allocation coefficient condition, the near user needs to subtract the component, xF, from the ySNi using the SIC process [36]. The signal-to-interference-plus-noise-ratio (SINR) of the eliminated component, xF, is expressed as(2)γSNixF=PSθFhSNi2PSθNhSNi2+σSNi2.After the SIC process, the Ni archives its own message, xN, from the received signal in which signal-to-noise-ratio (SNR) is obtained as(3)γSNixN=PSθNhSNi2σSNi2.At the far user F j, the received signal is given by(4)ySFj=PSθFhSFjxF+PSθNhSFjxN+ωFj.Different from the near user, the Fj directly decodes the SINR from the received signal because of the power allocation coefficient condition. The received SINR at Fj to decode xF is given by(5)γSFjxF=PSθFhSFj2PSθNhSFj2+σSFj2.Meanwhile, the eavesdropper can intercept the signal due to the broadcast nature of wireless medium. Thus, the received signal atE can be written as(6)ySE=PSθNhSExN+PSθFhSExF+ωE.Different from the legitimate user, we assume that the E has enough ability to distinguish each message from the received signal. Thus, the SINRs of the received signal are given by(7)γSExF=PSθFhSE2PSθNhSE2+σSE2,and(8)γSExN=PSθNhSE2σSE2,respectively.In physical-layer security, the secrecy capacity means the difference between main channel capacity and eavesdropper channel capacity. Thus, in two-user NOMA system, the secrecy capacities ofxN and xF are given by [22, 37](9)Cs,Ni=log21+γSNixN-log21+γSExN+,(10)Cs,Fj=log21+γSFjxF-log21+γSExF+,respectively, where [x]+=max⁡{x,0}.


## 2.2. The Proposed Best-Secure-Near-User Best-Secure-Far-User (BSNBSF) Scheme

The proposed user selection process is conducted through the channel state information (CSI) estimation/calculation system. Thus, this process is carried out before the data communication process as in [11, 35]. In this paper, we propose the BSNBSF user selection scheme to maximize the secrecy capacity at Ni and Fj, respectively. The proposed scheme can be mathematically expressed as(11)Nb=argmaxi∈Nlog21+γSNixN1+γSExN,and(12)Fb=argmaxj∈Flog21+γSFjxF1+γSExF.(11) and (12) mean that the selected users are the best secrecy capacity of a certain time slot.


## 3. Secrecy Outage Performance Analysis

In this section, the performance investigation of the proposed user selection scheme in terms of secrecy outage probability (SOP) is presented. Because the wireless channels undergo i.i.d. Rayleigh fading, for the sake of notational convenience, we assume thatλSN1=λSN2=…=λSNi=λSN, λSF1=λSF2=…=λSFj=λSF.We also assume the all nodes have the same noise variance. Letγ=PS/σ2 present the transmit SNR as in [11, 22]. The SOP of a user can be defined as the probability that the instantaneous secrecy capacity of the user falls below a predefined target data rate [23]. Thus, the SOPs at Nb and Fb are obtained as(13)PSOP,Nb=PrCs,Nb<Rth,Nb,and(14)PSOP,Fb=PrCs,Fb<Rth,Fb,where Rth,Nb and Rth,Fb denote the target data rate at Nb and Fb, respectively.


### 3.1. The Exact Secrecy Outage Probability

At the selected near user, the SOP of Nb can be expressed as(15)PSOP,Nb=Prlog21+γθNhSNb21+γθNhSE2<Rth,Nb.Because all wireless channels are assumed to be independent, (15) can be rewritten as(16)PSOP,Nb=Prmaxi∈N1+γθNhSNi21+γθNhSE2<γth,Nb,where γth,Nb≜2Rth,Nb. As we can observe that the events of the probability in (16) are not mutually exclusive because they include the same components hSE2, therefore conditioning on hSE2 = z, the PSOP,Nb can be further expressed as(17)PSOP,Nb=∫0∞Pr⋂i=1K1+γθNhSNi21+γθNz<γth,NbfZzdz.Since the assumption that all wireless channels are identical, (17) can be further expressed as(18)PSOP,Nb=∫0∞∏i=1KPrhSNi2<γth,Nb-1γθN+γth,Nbz︸ΨfZzdz.For the sake of notational convenience, let Xi≜hSNi2; Ψ in (18) can be rewritten as(19)Ψ=∫0γth,Nb-1/γθN+γth,NbzfXixdx.After some algebra manipulations, Ψ can be obtained as(20)Ψ=1-exp⁡-1λSNγth,Nb-1γθN+γth,Nbz.Plugging (20) into (18) and making use of the binomial theorem [38, Eq. (1.111)](21)a+bN=∑k=0NNkaN-kbk,PSOP,Nb in (18) can be further expressed as(22)PSOP,Nb=∑n=0KKn-1n1λSEexp⁡-nγth,Nb-1λSNγθN∫0∞exp⁡-nγth,NbλSN+1λSEzdz.After some algebra manipulations and making use of the fact that ∫0∞exp(-1/ax)dx=a [38, Eq. (3.310)], consequently, the exact closed-form expression for SOP of the selected near user can be obtained as(23)PSOP,Nb=∑n=0KKn-1nλSNnγth,NbλSE+λSNexp⁡-nγth,Nb-1λSNγθN.At the selected far user, the SOP of Fb can be expressed as(24)PSOP,Fb=Prlog21+γθFhSFb2/γθNhSFb2+11+γθFhSE2/γθNhSE2+1<Rth,Fb.Similar to the case of the selected near user, all wireless channels are independent and identical. And the events of the probability in (24) are not mutually exclusive because they include the same components γθFhSE2/γθNhSE2+1. Therefore, conditioning on γθFhSE2/γθNhSE2+1 = t, (24) can be further expressed as(25)PSOP,Fb=∫0∞∏j=1MPrγθFhSFj2γθNhSFj2+1<γth,Fb-1+γth,Fbt︸ΦfTtdt.In order to further simplify the integral (25), the following lemma enables us to characterize the SINR at far user and eavesdropper to decode xF.Lemma 1.
Suppose thatU≜γθFρ/γθNρ+1ρ∈hSFj2,hSE2; the cumulative distribution function (CDF) and probability density function (PDF) can be expressed as(26)FUu=1-φλU,u,if0≤u<θFθN,1,ifθFθN≤u, and(27)fUu=θFλUγθF-θNu2φλU,u,if0≤u<θFθN,0,ifθFθN≤u,respectively, where φ(α,t)=exp(-t/αγ(θF-θNt)); λU represents the average channel power gain.Proof.
The CDF ofU can be written as(28)FUu=PrγθFρ<γθNρ+1u. After some basic manipulations, (28) can be rewritten as(29)FUu=Prρ<uγθF-θNu,if ρ<θF/θN; otherwise, FU(u)=1. After some calculation steps, the PDF of U can be obtained as presented in equation (27). This completes the proof of Lemma 1.For the sake of notational convenience, letYj≜hSFj2. After some algebra manipulations, since Yj is a nonnegative random variable and constants are strictly positive constants, Φ in (25) can be further obtained as(30)Φ=PrYj<γth,Fb-1+γth,FbtγθF-θNγth,Fb-1+γth,Fbt=∫0γth,Fb-1+γth,Fbt/γθF-θNγth,Fb-1+γth,FbtfYjydx=1-φλSF,γth,Fb-1+γth,Fbt,if1θNγth,Fb-1>t,1,if1θNγth,Fb-1≤t.By plugging (27) and (30) into (25), since θF/θN≥1/γth,FbθN-1, the PSOP,Fb can be further expressed as(31)PSOP,Fb=∫01/γth,FbθN-1∏j=1M1-φλSF,γth,Fb-1+γth,FbtθFλSEγθF-θNt2φλSE,tdt+∫1/γth,FbθN-1θF/θNθFλSEγθF-θNt2φλSE,tdt.Similar to (20), we rely on the binomial theorem [38, eq. (1.111)]. Consequently, (31) can be further written as(32)PSOP,Fb=∫01/γth,FbθN-1∑k=0MMk-1kexp⁡-kγth,Fb-1+γth,FbtλSFγθF-θNγth,Fb-1+γth,Fbt·θFλSEγθF-θNt2exp⁡-tλSEγθF-θNtdt+∫1/γth,FbθN-1θF/θNθFλSEγθF-θNt2exp⁡-tλSEγθF-θNtdt.To the best of the authors’ knowledge, it is very difficult to obtain the exact closed-form expression of (32). Thus, in this paper, we approximate (32) using Gaussian-Chebyshev quadrature [39, eq. (25.4.38)]. First, to utilize the Gaussian-Chebyshev quadrature, the range of (32) can be coordinated as(33)PSOP,Fb=β1∫-11∑k=0MMk-1kexp⁡-kγth,Fb-1+γth,Fbβ1x+β1λSFγθF-θNγth,Fb-1+γth,Fbβ1x+β1·θFλSEγθF-θNβ1x+β12exp⁡-β1x+β1λSEγθF-θNβ1x+β1dx+β2∫-11θFλSEγθF-θNβ2x+β32exp⁡-β2x+β3λSEγθF-θNβ2x+β3dx,where β1=(1/γth,FbθN)-1/2, β2=(θF/θN)-(1/γth,FbθN)+1/2, β3=(θF/θN)+(1/γth,FbθN)-1/2. Next, to approximate PSOP,Fb in (33), (33) can be transformed using the fact that(34)∫-11fx1-x21-x2dx=∑i=1Nwi1-τi2fxi,where wi=π/N, τi=cos⁡(2i-1/2Nπ), and N is the number of terms, respectively. Plugging (33) into (34), and after some manipulations, eventually, the tight approximated closed-form expression for SOP of the selected far user can be obtained as(35)
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### 3.2. The Asymptotic Analysis

In this subsection, we derive the asymptotic SOP in the high SINR regime, which is mathematically described asλSF→∞ and λSN=βλSF(β>1) [21, 40]. This expression gives insight into the behavior of the secrecy outage for high SINR. Using the Taylor’s series approximation of the term, that is, exp⁡(-a/x)≈1-a/x, and after some algebraic manipulations, the asymptotic SOP of the selected near user can be obtained as(36)PSOP,Nbasym=∑n=0KKn-1nβλSFnγth,Nb+βλSF1-nγth,Nb-1γθNβλSF.Similar to the asymptotic SOP for the selected near user, the asymptotic SOP for selected far user can be expressed as(37)PSOP,Fbasym=β1∑i=1NπN1-τi2∑k=0MMk-1kθFλSEγθF-θNβ1τi+β12exp⁡-β1τi+β1λSEγθF-θNβ1τi+β1·1-kγth,Fb-1+γth,Fbβ1τi+β1γθF-θNγth,Fb-1+γth,Fbβ1τi+β1λSF+β2∑i=1NπN1-τi2θFλSEγθF-θNβ2τi+β32exp⁡-β2τi+β3λSEγθF-θNβ2τi+β3.


## 3.1. The Exact Secrecy Outage Probability

At the selected near user, the SOP of Nb can be expressed as(15)PSOP,Nb=Prlog21+γθNhSNb21+γθNhSE2<Rth,Nb.Because all wireless channels are assumed to be independent, (15) can be rewritten as(16)PSOP,Nb=Prmaxi∈N1+γθNhSNi21+γθNhSE2<γth,Nb,where γth,Nb≜2Rth,Nb. As we can observe that the events of the probability in (16) are not mutually exclusive because they include the same components hSE2, therefore conditioning on hSE2 = z, the PSOP,Nb can be further expressed as(17)PSOP,Nb=∫0∞Pr⋂i=1K1+γθNhSNi21+γθNz<γth,NbfZzdz.Since the assumption that all wireless channels are identical, (17) can be further expressed as(18)PSOP,Nb=∫0∞∏i=1KPrhSNi2<γth,Nb-1γθN+γth,Nbz︸ΨfZzdz.For the sake of notational convenience, let Xi≜hSNi2; Ψ in (18) can be rewritten as(19)Ψ=∫0γth,Nb-1/γθN+γth,NbzfXixdx.After some algebra manipulations, Ψ can be obtained as(20)Ψ=1-exp⁡-1λSNγth,Nb-1γθN+γth,Nbz.Plugging (20) into (18) and making use of the binomial theorem [38, Eq. (1.111)](21)a+bN=∑k=0NNkaN-kbk,PSOP,Nb in (18) can be further expressed as(22)PSOP,Nb=∑n=0KKn-1n1λSEexp⁡-nγth,Nb-1λSNγθN∫0∞exp⁡-nγth,NbλSN+1λSEzdz.After some algebra manipulations and making use of the fact that ∫0∞exp(-1/ax)dx=a [38, Eq. (3.310)], consequently, the exact closed-form expression for SOP of the selected near user can be obtained as(23)PSOP,Nb=∑n=0KKn-1nλSNnγth,NbλSE+λSNexp⁡-nγth,Nb-1λSNγθN.At the selected far user, the SOP of Fb can be expressed as(24)PSOP,Fb=Prlog21+γθFhSFb2/γθNhSFb2+11+γθFhSE2/γθNhSE2+1<Rth,Fb.Similar to the case of the selected near user, all wireless channels are independent and identical. And the events of the probability in (24) are not mutually exclusive because they include the same components γθFhSE2/γθNhSE2+1. Therefore, conditioning on γθFhSE2/γθNhSE2+1 = t, (24) can be further expressed as(25)PSOP,Fb=∫0∞∏j=1MPrγθFhSFj2γθNhSFj2+1<γth,Fb-1+γth,Fbt︸ΦfTtdt.In order to further simplify the integral (25), the following lemma enables us to characterize the SINR at far user and eavesdropper to decode xF.Lemma 1.
Suppose thatU≜γθFρ/γθNρ+1ρ∈hSFj2,hSE2; the cumulative distribution function (CDF) and probability density function (PDF) can be expressed as(26)FUu=1-φλU,u,if0≤u<θFθN,1,ifθFθN≤u, and(27)fUu=θFλUγθF-θNu2φλU,u,if0≤u<θFθN,0,ifθFθN≤u,respectively, where φ(α,t)=exp(-t/αγ(θF-θNt)); λU represents the average channel power gain.Proof.
The CDF ofU can be written as(28)FUu=PrγθFρ<γθNρ+1u. After some basic manipulations, (28) can be rewritten as(29)FUu=Prρ<uγθF-θNu,if ρ<θF/θN; otherwise, FU(u)=1. After some calculation steps, the PDF of U can be obtained as presented in equation (27). This completes the proof of Lemma 1.For the sake of notational convenience, letYj≜hSFj2. After some algebra manipulations, since Yj is a nonnegative random variable and constants are strictly positive constants, Φ in (25) can be further obtained as(30)Φ=PrYj<γth,Fb-1+γth,FbtγθF-θNγth,Fb-1+γth,Fbt=∫0γth,Fb-1+γth,Fbt/γθF-θNγth,Fb-1+γth,FbtfYjydx=1-φλSF,γth,Fb-1+γth,Fbt,if1θNγth,Fb-1>t,1,if1θNγth,Fb-1≤t.By plugging (27) and (30) into (25), since θF/θN≥1/γth,FbθN-1, the PSOP,Fb can be further expressed as(31)PSOP,Fb=∫01/γth,FbθN-1∏j=1M1-φλSF,γth,Fb-1+γth,FbtθFλSEγθF-θNt2φλSE,tdt+∫1/γth,FbθN-1θF/θNθFλSEγθF-θNt2φλSE,tdt.Similar to (20), we rely on the binomial theorem [38, eq. (1.111)]. Consequently, (31) can be further written as(32)PSOP,Fb=∫01/γth,FbθN-1∑k=0MMk-1kexp⁡-kγth,Fb-1+γth,FbtλSFγθF-θNγth,Fb-1+γth,Fbt·θFλSEγθF-θNt2exp⁡-tλSEγθF-θNtdt+∫1/γth,FbθN-1θF/θNθFλSEγθF-θNt2exp⁡-tλSEγθF-θNtdt.To the best of the authors’ knowledge, it is very difficult to obtain the exact closed-form expression of (32). Thus, in this paper, we approximate (32) using Gaussian-Chebyshev quadrature [39, eq. (25.4.38)]. First, to utilize the Gaussian-Chebyshev quadrature, the range of (32) can be coordinated as(33)PSOP,Fb=β1∫-11∑k=0MMk-1kexp⁡-kγth,Fb-1+γth,Fbβ1x+β1λSFγθF-θNγth,Fb-1+γth,Fbβ1x+β1·θFλSEγθF-θNβ1x+β12exp⁡-β1x+β1λSEγθF-θNβ1x+β1dx+β2∫-11θFλSEγθF-θNβ2x+β32exp⁡-β2x+β3λSEγθF-θNβ2x+β3dx,where β1=(1/γth,FbθN)-1/2, β2=(θF/θN)-(1/γth,FbθN)+1/2, β3=(θF/θN)+(1/γth,FbθN)-1/2. Next, to approximate PSOP,Fb in (33), (33) can be transformed using the fact that(34)∫-11fx1-x21-x2dx=∑i=1Nwi1-τi2fxi,where wi=π/N, τi=cos⁡(2i-1/2Nπ), and N is the number of terms, respectively. Plugging (33) into (34), and after some manipulations, eventually, the tight approximated closed-form expression for SOP of the selected far user can be obtained as(35)
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## 3.2. The Asymptotic Analysis

In this subsection, we derive the asymptotic SOP in the high SINR regime, which is mathematically described asλSF→∞ and λSN=βλSF(β>1) [21, 40]. This expression gives insight into the behavior of the secrecy outage for high SINR. Using the Taylor’s series approximation of the term, that is, exp⁡(-a/x)≈1-a/x, and after some algebraic manipulations, the asymptotic SOP of the selected near user can be obtained as(36)PSOP,Nbasym=∑n=0KKn-1nβλSFnγth,Nb+βλSF1-nγth,Nb-1γθNβλSF.Similar to the asymptotic SOP for the selected near user, the asymptotic SOP for selected far user can be expressed as(37)PSOP,Fbasym=β1∑i=1NπN1-τi2∑k=0MMk-1kθFλSEγθF-θNβ1τi+β12exp⁡-β1τi+β1λSEγθF-θNβ1τi+β1·1-kγth,Fb-1+γth,Fbβ1τi+β1γθF-θNγth,Fb-1+γth,Fbβ1τi+β1λSF+β2∑i=1NπN1-τi2θFλSEγθF-θNβ2τi+β32exp⁡-β2τi+β3λSEγθF-θNβ2τi+β3.


## 4. Optimal Analysis of the Power Allocation CoefficientsθN

In this section, we carry out the optimal analysis of the power allocation coefficientθN. Specifically, using the descent algorithm [41], we propose a descent-based algorithm (Algorithm 1) for finding the optimal point of θN, denoted as θN∗, which results from the minimum value of the total secrecy outage probability (TSOP) as follows.Algorithm 1:The Descent-based Algorithm for Finding θN∗.
Input: system parameters such as γth,Nb, γth,Fb, λSN, λSF, λSE, θN, θF, γ; and the given optimization parameters such as step size,st = 10-2, search length, Δk = 1 Output: θ N ∗ (1) Assign iteration index to the initial: k←1 (2) Choose a starting point of the searching optimal value: θNk←10-9 (3) whilePTSOP(θNk)>PTSOP(θN+stΔk)do (4) Update the searching optimal value: θNk+1←θNk+stΔk, k←k+1 (5) end while (6) returnθNkIn our paper, the optimization algorithm is applied to obtain the optimal power allocation coefficient ofθN∈(0,0.5) that minimizes the TSOP. The process of the proposed optimization algorithm can be explained as follows.(i)
At the starting pointθNk, we determine a descent direction satisfying PTSOP(θNk)>PTSOP(θNk+1), where k presents the index of the iteration.(ii)
We update the searching optimal value, that is,θNk+1=θNk+stΔk, where st means the step size. If θNk<θNk+1, we can obtain PTSOP(θNk)>PTSOP(θNk+1); otherwise θNk is optimal.(iii)
As we can see, whenθNk is its optimal value, PTSOP(θNk) becomes the smallest value. Therefore, the iteration process of updating θNk will be stopped when PTSOP(θNk)<PTSOP(θNk+1).Please note that the proposed algorithm is operated on the system parameters collected through CSI estimation process before data transmission process.


## 5. Numerical Results

In this section, we present the representative numerical results to illustrate the achievable performance of the proposed scheme. Monte Carlo simulation results are generated to validate the developed analysis. In simulation setting, we assume that positions of the sourceS, the cluster of near users, the cluster of far users, and the eavesdropper E are randomly deployed satisfying some given distance constraints. Specifically, we set that the distance between S and the cluster near users is dSN=10m, the distance between S and the cluster of far users is dSF=20m, and the distance between S and eavesdropper is dSE=30m, respectively. It is noted that although multiple near or far users are located at the same location, their channel characteristics are different from one to another. The power allocation coefficients of the near user (θN) and far user (θF) are 0.2 and 0.8, respectively. Additionally, the reference distance d0=1m, and power degradation at d0 is L=30 (dB), the path-loss exponent ϵ=2.7.Figure2 presents the performance comparison of the BSNBSF scheme and random near user and random far user (RNRF) scheme, in which the number of the near users is 3, and the number of the far users is 3, respectively. As can be seen in Figure 2, the SOP of the BSNBSF scheme is lower than that of random selection scheme. The reason is that the proposed scheme considers the secrecy channel capacity to select the near user and far user, respectively. When the transmit power increases, the SOP of the selected near user is increased until it reaches a performance floor. Different from the case of the selected near user, the SOP of the selected far user is plotted as in convex pattern with respect to the transmit SNR.Figure 2
Performance comparison between the proposed BSNBSF scheme and the RNRF scheme with secrecy outage probability as a function of the transmit SNR, whereθN=0.2, Rth,Nb=Rth,Fb=0.1 bps/Hz.Next, we investigate the TSOP of the proposed scheme and RNRF scheme, which can be mathematically defined as [23](38)PTSOP=1-1-PSOP,Nb1-PSOP,Fb.Figure 3 illustrates the impact of the transmit SNR and the number of near and far users on the performance of the proposed scheduling scheme. As can be seen in Figure 3, increasing the number of near and far users does not improve the performance of the RNRF while the performance of the proposed scheme is improved when the number of the near and far users is higher. The reason is that the BSNBSF scheme exploits the difference in channel conditions between users to select the best near and far users pair.Figure 3
Illustration of the impact of the number of near users and far users as a function of the transmit SNR, whereRth,Fb=Rth,Nb=0.1.In Figure4, we plot the SOP as a function of the transmit SNR with different numbers of the near user and the far user, respectively. As can be observed in Figure 4, the SOP decreases when the number of the near user increases. The reason is that the security performance increases when more nodes take into account the data transmission. In contrast to the case of near user, when the number of the far user increases, the secrecy performance is impacted  a little bit. One of possible reasons is that the channel condition of the far user is not so much stronger than that of near user. Thus, security performance does not have much effect on the number of far participants. Additionally, we can observe that the results of the asymptotic analysis tightly approximate the closed-form analysis as a function of the increase of average far user channel power gain.Figure 4
Illustration of the impact of the number of near users and far users on the secrecy outage probability as a function of the average far user channel power gains, where transmitSNR=30 dBm, Rth,Fb=Rth,Nb=0.1 bps/Hz.Figure5 illustrates the TSOP as a function of the distance between the base station and eavesdropper with different numbers of the near users and the far users. As can be seen in Figure 5, when the distance between the base station and eavesdropper increases, the TSOP decreases. The reason is that, when the distance between the base station and eavesdropper increases, it is difficult for the eavesdropper to intercept the information between the legitimate users. Additionally, we can observe that the proposed scheme has better secrecy performance than the benchmarked scheme with the same transmit power.Figure 5
Illustration of the total secrecy outage probability as a function of the distance between base station and eavesdropper with different numbers of the near users and the far users, where transmitSNR=30 dBm, Rth,Fb=Rth,Nb=0.1 bps/Hz.Figure6 presents the TSOP as a function of the secrecy target data rate, Rth (bps/Hz), with different numbers of the near users and the far users. As can be seen in Figure 6, the total secrecy outage probability increases when the secrecy target data rate increases. This means that if the base station is allowed to transmit with a higher secrecy data rate (in order to obtain higher secrecy throughput), the data transmission will be vulnerable to the eavesdropper.Figure 6
Illustration of the total secrecy outage probability as a function of the secrecy target data rate,Rth (bps/Hz), with different numbers of the near users and far users, where transmit SNR=30 dBm.We investigate the effect on power allocation mechanism on the secrecy performance of the proposed scheduling scheme as shown in Figure7, where the total secrecy outage probability of the selected user pair is plotted as a function of the transmit power and power allocation coefficient of the near user, θN. It is noted that θF=1-θN. As can be observed, the TSOP poses a complicated convex characteristic with respect to θN. More specifically, the total secrecy outage probability is a convex function with respect to θN when the transmit SNR is less than 30 dBm or greater than 45 dBm under our setting, while it is not a convex function when the transmit power is in the range from 30 dBm to 45 dBm. Hence, we apply the optimization algorithm for finding the minimum power allocation coefficient of the near user, θN, when the transmit SNR is less than 30 dBm.Figure 7
The total secrecy outage probability of the selected users pair as a function of the transmit SNR and the power allocation coefficients of the near user, withK=3, M=3, Rth,Nb=Rth,Fb=0.1 bps/Hz.Figure8 illustrates the TSOP with different transmit SNR as a function of the power allocation coefficient of the near user, denoted as θN. When the transmit SNR increases, the TSOP decreases. Additionally, there is the optimal value of θN where the TSOP is minimum as presented in Figure 8. It is noted that the minimum TSOPs are obtained using the proposed descent-based search method.Figure 8
Illustration of the total secrecy outage probability with different transmit power as a function of the power allocation for the near user (θN), with K=5, M=5, Rth,Nb=Rth,Nb=0.1 bps/Hz.Finally, we investigate the secrecy performance improvements of the proposed scheme in terms of the secrecy sum throughput as a function of the transmit SNR. The secrecy sum throughput can be defined as [42](39)Tsum=1-PSOP,NbRth,Nb+1-PSOP,FbRth,Fb.As can be observed in Figure 9, comparing the proposed scheme with the RNRF scheme, the proposed scheme significantly improves the secrecy sum throughput. More specifically, when the number of the near and far users increases, the secrecy sum throughput increases. Different from the proposed scheme, the RNRF scheme does not affect the number of the near and far users. The reason is that, when the near and far users are selected in the proposed system, we consider the channel condition of the main channel and eavesdropper channel. However, the benchmarked scheme does not consider that condition to select the near and far users, respectively.Figure 9
Illustration of the secrecy sum throughput with the different numbers of the near users and far users as a function of the transmit power, whereRth,Fb=Rth,Nb=0.1 bps/Hz.


## 6. Conclusion

In this paper, we propose the opportunistic scheduling scheme to enhance the secrecy performance in multitwo user NOMA system. The proposed BSNBSF scheme aims to improve the physical layer security of the considered NOMA system. Specifically, the BSNBSF scheme selects the user pair by estimating both main channel and eavesdropper channel characteristics to select the most robust near and far users, respectively. The exact closed-form expression for the SOP of the selected near user and the tight approximated closed-form expression for the SOP of the selected far user are presented. In order to get more insightful results, the asymptotic results for the SOP of the selected near user and the selected far user are obtained. Moreover, we propose the optimization algorithm, called the descent-based search algorithm, to find the optimal value ofθN that minimizes the TSOP when the transmit SNR is less than 35 dBm. The obtained results are verified by the computer simulation. From the numerical results, the proposed scheme provides better secrecy performance compared to that of the RNRF scheme. Additionally, the increasing number of participant near and/or far users improves the robustness of the BSNBSF scheme. The descent-based search algorithm has been proposed to find out the optimal value of θN∗ that minimizes the secrecy performance. The BSNBSF scheme can also support better performance than the RNRF scheme in terms of the secrecy sum throughput.



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*Source: 2797824-2018-10-23.xml*