arXiv:1811.03220v2 [cs.IT] 1 Sep 2019 NM)ntokoe Nakagami- over network multipl non-orthogonal (NOMA) assisted multiple-relay a of mance omncto ytm,rlyslcinshm,sceyou secrecy scheme, selection probability relay systems, communication oe loainotistesm D sTMRC. as dynamic SDO with same scheme power the OSRS fixed obtains with the allocation schemes and power RS SDO the zero all obtain scheme compared that allocation and performance proposed scheme secrecy TMRC our the the i that enhance the systems significantly demonstrate Furthermore, schemes NOMA can results cooperative the allocations. The of all power (SDO) obtained. for order dynamic diversity SOP clo and secrecy the asymptotic derive fixed the also ODRS with we for insights, and expressions validated more and TSRS, form get derived To are OSRS, simulations. scheme TMRC proposed via the with the along ou of schemes security for the (SOP) requirement for secre expressions probability secrecy closed-form the different the and users, between NOMA users correlation wit two users of the decod NOMA capacity the Considering successfully to that power. signals relay forward equal source the the all from combin which signals relays in multiple scheme traditional with (TMRC) performan systems outage analyze NOMA secrecy the the are examine of performance also we outage benchmark, secrecy selec a As the sche relay and (ODRS) selection single proposed relay are dual two-step optimal and scheme, scheme, (TSRS) (OSRS) NOMA by transmit power-domain selection opti relays i.e., via relay schemes, relay (RS) users decode-and-forward selection selected two relay all Three the the technology. to to Then signal users strategy. superposition two mapping des super the message the to broadcasts station of station base base the signal the slot, first from the signals At tion. transmit to utilized are rba(-al [email protected] slim.alouini 23955 [email protected]; Thuwal (KAUST), (e-mail: Technology Arabia and Science of University emi:[email protected]). Universit(e-mail: Southwest Processing, Information Intelligent hn emi:lijcuteuc;[email protected]) [email protected]; Cho Te Telecommunications, (e-mail: Communications and China Posts Mobile of of University Lab Chongqing Key Chongqing & Engineering eso h dcto iityo hn,CogigUniversi Chongqing China, of (e-mail:[email protected]). Ministry China. Education 400044, the of tems imran.ansari@gla (e-mail: Kingdom United 8QQ, G12 Glasgow th with deal to technologies promising most the of one as ered Works Related and Background A. .H akadM-.Aoiiaewt ES iiin igAb King Division, CEMSE with are Alouini M.-S. and Park K.-H. .Pni ihCogigKyLbrtr fNnierCircuit Nonlinear of Laboratory Key Chongqing with is Pan G. .Goi ihteKyLbrtr fOteetoi Technolo of Optoelectronic University of Laboratory Engineering, Key the of with School is Guo the Y. with is Ansari S. I. Inform and Communication of School with are Yang Z. and Lei H. received. Manuscript Abstract ne Terms Index o-rhgnlmlil ces(OA a enconsid- been has (NOMA) access multiple Non-orthogonal erc uaeAayi o oprtv NOMA Cooperative for Analysis Outage Secrecy Ti ae osdr h erc uaeperfor- outage secrecy the considers paper —This Nnotooa utpeacs,cooperative access, multiple —Non-orthogonal ytm ihRlySlcinScheme Selection Relay with Systems .I I. NTRODUCTION ogin e,Zxa ag iHn ak ma hfiu Ans Shafique Imran Park, Ki-Hong Yang, Zixuan Lei, Hongjiang oga u,GoegPn n oae-lmAlouini Mohamed-Slim and Pan, Gaofeng Guo, Yongcai m aigcanl.Toslots Two channels. fading ,Cogig400715, Chongqing y, @kaust.edu.sa). . gig400065, ngqing y Chongqing ty, sgow.ac.uk). 60,Saudi -6900, yadSys- and gy a single mal chnology, position Glasgow, access e dullah and s tina- ation tage tage sed- tion two me, ing cy ce to d. h e e s s s eevdsgas n hndcdstesgasfritself. for th signals from the it decodes cancels then user, and weaker signals, the firstly to systems received NOMA sent in signal user the stronger decodes the node, source tran the signal and by superposition bandwidth ultra-connectivity, the high receiving fairness, After as flexibility. user NOMA such merits, better schemes, superior efficiency, access obtain Compared multiple can [1]-[9]. systems orthogonal network mobile traditional i (5G) to fairness generation user enhance fifth and the resources bandwidth of shortage srly oipoetepromneo hs ekrusers Ding utilized weaker [11]. those be [10], of conditions can performance channel the users poor in the improve with stronger users to for the relays messages stronger as therefore the the decode users, Since firstly weaker destinations. always the systems also NOMA technology at but diversity utilized coverage when network’s performance is the system extends the only enhances not that nique ln ihisegdcsmrt n Pwr netgtdin investigated were OP and sum-rate [15]. ergodic systems its power NOMA novel with relaying a along dual-hop Moreover, for [14]. scheme of [13], (PA) bound [12], allocation lower in and derived exact and were the systems OP multiple for NOMA expressions cooperative with closed-form the example, the relay in For utilized (AF) was nodes. antennas amplify-and-forward relay dedicated dedicated cooperat a with the derived on systems were focused throughput literatures NOMA many system Recently, closed-form and [11]. the OP in and the proposed for was expressions protocol a pow user Furthermore, NOMA and [10]. transfer on information in wireless based simultaneous complexity cooperative approach cooperativ system new an reduce of to proposed order pairing and diversity systems, (OP), NOMA probability outage the bandadterslssoe htF easoutperform relays FD that Liu weaker showed was relays. OP (HD) the results for half-duplex to expression the closed-form signals and The forward [18]. obtained to and [17] utilized in in working was user user mode stronger FD the the better example, and For efficiency [16]. spectral higher performance obtain to systems NOMA ekruesi 1] 2] h efrac fF cooperativ FD of the performance of The performance [20]. dedi- [19], the the The in improve users become to scenarios. weaker utilized will many was relay in users FD systems weaker cated NOMA the interfer of as of bottleneck user b performance stronger signals the its the decodes then for user signals allocatio weaker treating the power directly Since optimal [18]. the in for policies obtaining expression thereby systems closed-form NOMA the cooperative FD and HD for oprtv omncto sapriual trciete attractive particularly a is communication Cooperative uldpe F)rlyas a enuiie ncooperativ in utilized been has also relay (FD) Full-duplex tal. et ari nlzdteP problems PA the analyzed tal. et analyzed smitted ence, high ch- ive er n y n e e e e 1 2
NOMA systems over Nakagami-m channels was investigated The secrecy performance of a NOMA system with multiple in [19] and the analytical expressions for OP and ergodic eavesdroppers was investigated while zero-forcing and mini- rate were derived. In [20], a sharing FD relay was utilized mum mean-square error decoding schemes were utilized on in NOMA systems with two source-destination pairs and the the legitimate destinations in [39]. A new joint subcarrier ergodic rate, OP, and outage capacity were investigated while (SC) assignment and PA scheme was proposed to improve both perfect and imperfect self-interference cancellation (SIC) the security of the two-way relay NOMA systems in [40]. schemes were considered, respectively. The secrecy performance of a cooperative NOMA system Relay selection (RS) technique is an effective scheme in with a dedicated AF/DF relay was investigated and the closed- making full use of space diversity with low implementation form expressions for the SOP were obtained in [41]. In some complexity and it can straightforwardly improve the spectral scenarios, the relay in cooperative NOMA systems is untrusted efficiency of cooperative systems [21], [22]. Considering the and curious to decode the users’ messages. Two new HD quality of service (QoS) requirements for the two users are nonorthogonal AF schemes were proposed for cooperative different, a two-stage single-relay-selection strategy and dual- NOMA systems with an untrusted relay and the secrecy relay-selection with fixed power allocation (FPA) was studied rate maximization-based PA scheme was discussed in [42]. in [23] and [24], respectively, in order to obtain the OP. The Arafa et al. proposed two new relaying schemes to resist the results showed that the two-stage strategy achieves a better OP untrusted relay and the secrecy performance of cooperative and diversity gain. Combining DF and AF relaying, Yang et al. NOMA systems with new relaying schemes was analyzed and extended the two-stage RS scheme with dynamic power allo- compared in [43]. Feng et al. proposed an artificial-noise (AN) cation (DPA) and derived the exact and asymptotic analytical aided scheme to build up secure cooperative NOMA systems expressions for OP in [25]. In [26], considering whether the with a FD relay and the closed-form expression for the SOP relay can correctly decode two users’ messages, two optimal was derived in [44]. The secrecy performance of cooperative RS schemes were proposed for cooperative NOMA systems NOMA systems in which the stronger user behaves as a FD with FPA and DPA at the relays, respectively. Yue et al., in relay was investigated in [45] and the SOP was analyzed with [27], studied the performance of cooperative NOMA systems an assumption of the imperfect SIC. The security-reliability with two RS schemes in which the relay operates in either tradeoff for both cooperative and non-cooperative NOMA FD or HD mode. Modeling the spatial topology of relays with schemes was analyzed and analytical expressions for SOP homogeneous poisson point process (PPP), the performance of were derived in [46]. cooperative NOMA systems with two-stage RS scheme was analyzed in [28] and a closed-form approximation for the OP B. Contributions was obtained. To deal with the security issues followed by the explosive In many works, such as [32], [34], [41], the worst-case increase in cellular data, physical layer security (PLS) is scenario was considered, in which it is assumed that the emerging as one of the most promising ways to ensure eavesdroppers have powerful detection capabilities and they secure communication basing on the time varying nature of can extract the message from the signals transmitted from the wireless channels [29], [30]. A new optimal PA strategy the resource. Under this assumption, one can realize that the was proposed to maximize the secrecy sum rate of NOMA secrecy capacity of both/all the users in NOMA systems are systems and the results confirmed that a significant secrecy not independent [34]. The major contributions of this paper performance for NOMA systems was obtained in [31]. In are as follows, [32], the security performance of a NOMA-based large-scale 1) In this work, we analyze a cooperative NOMA system network is considered, where both the NOMA users and with multiple relays and an eavesdropper employing eavesdroppers were modeled by homogeneous PPPs. In [33], RS schemes to enhance the secrecy performance. Three we investigated the secrecy outage performance of a multiple- RS (OSRS, TSRS, and ODRS) schemes are proposed. input single-output (MISO) NOMA systems with transmit For the purpose of comparison, we also investigate antenna selection (TAS) schemes and the closed-form ex- the secrecy performance of the NOMA systems with pressions for secrecy outage probability (SOP) were derived. traditional multiple relays combining (TMRC) scheme The results demonstrated that the proposed DPA scheme can in which all the relays that successfully decode signals achieve non-zero secrecy diversity order (SDO) for all the from the source forward signals to the NOMA users TAS schemes. The secrecy outage performance of multiple- with equal power. Considering the correlation between input multiple-output (MIMO) NOMA systems with multiple the secrecy capacity of two users, the closed-form ex- legitimate and illegitimate receivers was studied in [34] and pressions for the SOP under different RS schemes are the closed-form expressions for SOP were subsequently ob- derived and validated via simulations. tained. Feng et al. proposed in [35] a new PA to maximize 2) Different secrecy QoS is considered in our work, i.e., the secrecy rate of the stronger user while guaranteeing the different secrecy threshold rates are requested for the non-secure transmission rate requirement to the weaker user, two NOMA users. Although the weaker user cannot but the secrecy performance of the weaker user was not perform SIC, either the stronger or the weaker user will considered. Secrecy beamforming schemes were proposed for be the bottleneck of the cooperative NOMA system. MISO-NOMA systems, cognitive MISO-NOMA systems, and Our results demonstrate that this depends on the power MIMO-NOMA systems in [36], [37], and [38], respectively. allocation parameters and secrecy rate thresholds. 3
where ~ ∈ {S, Rk} signifies the transmitter, λ¯ ∈ R1 U1 {Rk,U1,U2, E} denotes the receiver, Γ (·) is Gamma function, Rk ~ ~ GU as defined by (8.310.1) of [47], mλ¯ and Ωλ¯ are the fading S parameters and average channel power gains, respectively. G Rk GU Rk R S k U2 To make analysis simple, it is assumed that the links of the
Rk GE first hop are independent and identically distributed (i.i.d.), which means mS m , S . The same assumption Rk = R ΩRk =ΩR R K E is made to the links between Rk and each receiver in the second hop, which means mRk mRk m , Rk , U1 = U2 = U ΩU1 = Ω1 Rk , mRk m , and Rk . ΩU2 =Ω2 E = E ΩE =ΩE Fig. 1. System model demonstrating a cooperative NOMA system with one Two time slots are assumed to utilize to the communication resource (S), multiple relays (Rk, k = 1, ··· ,K), two legitimate user (U1 and U2), and an illegitimate eavesdropper (E). between S and the NOMA users. In the first time slot, S broadcasts the superposition signal to all relays by message mapping strategy, which is utilized and proved optimal to 3) To obtain more insights, we also derive the closed-form achieve the minimal common OP in [26]. During the second expressions for the asymptotic SOP under different RS time slot, a relay is selected using the RS scheme proposed schemes with FPA and DPA. Furthermore, the SDO below to send superposition signal to the two NOMA users. of cooperative NOMA systems are derived. The results We assume all the relays are close enough such that the relative demonstrate that all the RS schemes with FPA obtain distance between the two users and the relay is determined. zero SDO and the OSRS scheme achieves the same SDO To make the representation clear and simple, we use as TMRC scheme. GRk = Gk. It is also assumed that U has a better channel Ui i 1 4) Differing from [32], [34], [41], wherein it has been k k condition than U2 G1 > G2 , which is adopted in many assumed that the secrecy capacity for the stronger and NOMA studies, e.g., [5], [16], [17], [18], [26], [33]. Then the weaker users is independent, correlation between the signal-to-interference-noise ratio (SINR) of the NOMA users secrecy capacity of the two users and different secrecy can be written as requirement for the two users is considered in this work, γk = α ρGk, (3) which is more practical. 1 1 1 α ρGk C. Organization γk = 2 2 , (4) 2 α ρGk This paper is organized as follows, in Section II, the system 1 2 +1 model and RS schemes are introduced. In Section III, we ana- where αi (i =1, 2) represents the power allocation coeffi- lyze the security outage performance of the proposed system. cients at the kth relay, α1 + α2 =1, α1 > α2, and ρ signifies The asymptotic SOP for cooperative NOMA systems with the transmit signal to noise ratio (SNR). FPA and DPA are derived in Sections IV and V, respectively. Numerical and simulation results are presented in Section VI III. SECRECY OUTAGE PROBABILITY ANALYSIS to demonstrate the security performance of the system. Finally, The set of relays that can correctly decode mixed signals Section VII concludes the paper. from S can be expressed as
II. SYSTEM MODEL ∆ 1 S th th Φ = k :1 ≤ k ≤ K, log 1+ ρSG ≥ R + R , As shown in Fig. 1, we consider a cooperative downlink 2 2 Rk 1 2 NOMA system that consists of a base station (S), K (K ≥ 1) (5) where the factor 1 arises from the fact that two slots are DF HD relays (Rk, k = 1, ··· ,K), and two users (U1 and 2 required to complete the data transmission, ρS denotes the U2). An eavesdropper (E) wants to wiretap the information th through decoding the received signals. It is assumed that all transmit SNR at S, and Ri (i =1, 2) is the data rate nodes are equipped with a single antenna and the direct link threshold for Ui. Thus the SOP of cooperative NOMA systems between S and both users are unavailable due to deep fading. can be written as K The communication links between S and all the receivers (U1, U2, and E) are relayed by Rk. Pout = Pr (|Φ| = n) PΦn , (6) n=0 The probability density function (PDF) and the cumulative X distribution function (CDF) of the channel power gains of link where PΦn denotes the SOP under the condition that there between the source node ~ and the destination node λ¯ can be are n relays that correctly decode the mixed signals. One can expressed by easily have PΦ0 =1.
~ ~ It is assumed that all the links between the source and the m mλ¯ −1 ~ k − λ¯ x k Ω~ mλ¯ x relays are i.i.d., so we have FG~ (x)=1 − e λ¯ ~ , (1) λ¯ Ω k! k=0 λ¯ n S n S K−n X Pr {|Φ| = n} = CK Pr GRk ≥ η Pr GRk < η ~ ~ ~ m m m −1 ~ λ¯ λ¯ n n K−n x λ¯ m − ~ x λ¯ Ω = CK χ (1− χ) , fG~ (x)= ~ ~ e λ¯ , (2) λ¯ Γ m Ω (7) λ¯ λ¯ 4
TMRC s TMRC TMRC TMRC Pr Cs,2 > R1 = Pr ρ1G2 1 − θ2α1 − θ2α1α2ρ1GE >θ2 − 1+ θ2α2ρ1GE ∆ (13) =Λ = Pr Λ > 0, GTMRC| >δ GTMRC{z } 2 2 E TMRC TMRC TMRC = Pr GE
τ 2Rth+2Rth E n m! e 1 2 −1 λE mE where C = , η = , χ = where βE = , λE= , and τE = nmE. m n!(m−n)! ρS Γ(τE ) ΩE mR−1 k k We can express the secrecy connection probability (SCP) of −λRη λRη mR e , and λR = . k! ΩR U1, which is the complementary of SOP, as k=0 P TMRC s TMRC TMRC Pr Cs,1 > R1 = Pr G1 >δ1 GE , (12) A. Traditional Multiple Relays Combining Scheme s θ1−1 2R where δ1 (x)= b1 + θ1x, b1= , and θ 1 = e 1 . As a benchmark, the traditional multiple relays combining α1ρ1 Similarly, we obtain the SCP of U2 as (13), given at the (TMRC) scheme is presented in this subsection, where all the 1−θ2α1 α2 top of this page, where a1 = , δ2 (x)= c1 + , n relays can successfully decode and forward the signal to both ρ1α1α2θ2 d1−e1x c = − 1 , d = α ρ (1 − α θ ), e = ρ2α2α θ , and 1 α1ρ1 1 1 1 1 2 1 1 1 2 2 the users with equal power. Both the legitimate and illegitimate s θ = e2R2 . receivers combine their received signals with maximal ratio 2 Remark 1: One can easily find that secrecy outage would combining (MRC) scheme to maximize their SINR. To make occur at U when 1 − θ α < 0. This means that in order to fair comparison, it is assumed that the total transmit power at 2 2 1 ensure a secure NOMA system (U ), there is a constraint for these relays is given by P . Then the P with this scheme 2 R Φn the power allocation coefficients, which is expressed as can be expressed as 1 − s 1 − s TMRC s TMRC s α ≤ = e 2R2 or α > 1 − =1 − e 2R2 . (14) PΦ = Pr C < R or C < R ||Φ| = n , (8) 1 2 n s,1 1 s,2 2 θ2 θ2 TMRC TMRC 1+γi Based on (12) and (13), P in this case can be rewritten where C means the secrecy capac- Φn s,i = ln 1+γTMRC E,i as (15), given at the top of this page. TMRC TMRC TMRC ity of Ui, i = 1, 2,γ1 = ρ1α1G1 , γ2 = TMRC n To facilitate the following analysis, we define ρ1α2G2 TMRC k TMRC TMRC , Gi = Gi , γE,i signifies the SNR a 1+ρ1α1G2 h k=1 b−1 −fx− 1−qx s g (a,b,c,r,q,f,h,k,j)= x e at E when Ui is wiretapped,PRi is the secrecy rate threshold 0 PR 2 Z for U , ρ = 2 , and σ is the noise power. j i 1 nσ r The CDF of GTMRC can be expressed as [48] × (1 + cx)k 1+ dx. i 1 − qx τD −1 (16) λk −λix i k FGTMRC (x)=1 − e x , (9) To the authors’ best knowledge, it’s very difficult to obtain i k! k=0 the closed-form expression of g (a,b,c,r,q,f,h,k,j). Here X mU by making use of Gaussian-Chebyshev quadrature from eq. where i =1, 2, λi = , and τU = nmU . Ωi As similar to [32], [34], and [41], we assume that E has (25.4.30) of [49], we obtain enough capabilities to detect multiuser data. Then the SNR at N a b afsi 2h b−1 − 2 − 2−aqs E can be expressed as g (a,b,c,r,q,f,h,k,j)= wis e i 2 i i=1 TMRC TMRC X j γE,i = ρ1αiGE , (10) acs k 2r × 1+ i 1+ , n TMRC k 2 2 − aqsi where i =1, 2 and GE = GE. Similarly, the PDF of (17) k=1 TMRC where N is the number of terms, si = ti +1, ti is the ith zero GE can be expressed as [48]P of Legendre polynomials, wi is the Gaussian weight, which is τE −1 −λE x f TMRC (x)= βEx e , (11) given in Table (25.4) of [49]. GE 5
m m m Pout,2 =1 − Pr {GE
OSRS n PΦn = (Pr {Xm < 1}) n = (1 − Pr {Xm > 1}) m m n G1 G2 m = 1 − Pr min m , m ≥ 1, GE m m m m m = 1 − Pr {G1 ≥ δ3 (GE ) , G2 ≥ δ4 (GE ) , GE Substituting (9) and (11) into (15) and with some simple minimum secrecy capacity of two users. Here the selection algebraic manipulations, we have criterion is to minimize the SOP of the cooperative NOMA system. τU −1 τU −1 k j TMRC −λ1b1−λ2c1 λ1 λ2Ξ1 Remark 3 P =1 − βEe , (18) : There is another important difference between Φn k!j! s s k=0 j=0 MMTAS and OSRS, i.e., it is assumed that R1 = R2 in X X s s MMTAS. However, in the OSRS scheme, R1 and R2 can where Ξ = bkcj g a , τ , θ1 , α2 , e1 , λ θ + λ , λ2α2 ,k,j . 1 1 1 1 E b1 d1c1 d1 1 1 E d1 be different. MMTAS is a special case of OSRS when it is s s assumed that R1 = R2 and the secrecy capacities of the two B. Optimal Single Relay Selection Scheme users are independent. The SOP of the cooperative NOMA system conditioned on In this subsection, we propose OSRS scheme to minimize n can be written as (23), given at the top of this page. the overall SOP of the proposed cooperative NOMA system. |Φ| = Based on (12) and (13), we have Based on (12), one can easily obtain the SOP for U1 with mth relay as m m m m m m m ∆1 = Pr {G1 ≥ δ3 (GE ) , G1 ≥ δ4 (GE ) , GE n TSRS j i s k s p s PΦ = Cn Pr min Cs,1 > R1, max Cs,1 < R1, max Cs,2 < R2 n 1≤i≤j 1≤k≤n−j 1≤p≤j j=1 Xn j i s p s k s = Cn Pr min Cs,1 > R1, max Cs,2 < R2 Pr max Cs,1 < R1 1≤i≤j 1≤p≤j 1≤k≤n−j j=1 X j n−j n j i s i s k s (27) = CnPr Cs,1 > R1, Cs,2 < R2 Pr Cs,1 < R1 j=1 X ∆ ∆ =∆2 =∆3 n = (∆2 + ∆3|) {z } | {z } i s i s i s n = Pr Cs,1 > R1, Cs,2 < R2 + Pr Cs,1 < R1 i s i s n = Pr Cs,1 < R1 or Cs,1 < R 1 i s i s n = 1 −Pr Cs,1 > R1, Cs,2 > R 2 J k s k s n PΦn = 1 − Pr Cs,1 > R1 ,Cs,2 > R2 k n α2ρ3G k 2 1+ α ρ G 1+ 1+α ρ Gk = 1 − Pr ln 1 3 1 > Rs, ln 1 3 2 > Rs α ρ Gk 1 α ρ Gk 2 1 3 E 2 3 E 1+ 1+ρ H 1+ 1+ρ H 4 E 4 E (31) n wu 1 = 1 − Pr Gk >ℓ + θ Y, Gk > w + ,Y < 1 1 2 1 − vY v ∆4 | {z } i i 1 1+α1ρ2G1 where Cs,1 = ln i signifying the secrecy capac- power of all the relays is constrained to PR. It is assumed that 2 1+α1ρ2GE ity for U1 from Riand i ∈ Φ. αJ (0 ≤ αJ < 1) portion is utilized to transmit the jamming At the second step, the relay to maximize the secrecy signals, then the SINR at E can be written as capacity of U2 is selected, i.e., k J αiρ3GE ∗ j γE,i = , (29) j = argmax Cs,2 , (26) 1+ ρ4HE j∈Ψ (1−αJ)PR αJPR j where i , , ρ , ρ , and H α2ρ2G = 1 2 3 = σ2 4 = σ2 E = 1+ 2 k j +1 j 1 α1ρ2G2 max G . where C = ln signifies the secrecy ca- ¯ E s,2 2 α ρ Gj k∈Φ 1+ 2 2 E We assume that both users are aware of the jamming signals, pacity for U2 from Rj (j ∈ Ψ). which means jamming signals do not influence the SINR at The SOP with this scheme can be achieved as (27), given both users [32], [36], [37]. It must be noted that the special at the top of the this page. case of n = K or αJ = 0, ODRS scheme reduces to OSRS Remark 4: An interesting result can be observed from (27) scheme. Then the SOP with ODRS scheme is expressed as that the expression for the SOP under TSRS scheme is the K−1 same as that under OSRS. ODRS J OSRS Pout = Pr (|Φ| = n) PΦn + PΦK , (30) n=0 D. Optimal Dual Relay Selection Scheme X where P OSRS is given in (23). ΦK In this subsection, a new RS scheme named ODRS scheme Similar to (27), the SOP of the cooperative NOMA system is proposed, in which one relay is selected to transmit signals on condition that |Φ| = n (n wu 1 ∆ = Pr Gk >ℓ + θ Y, Gk > w + ,Y < 4 1 1 2 1 − vy v 1 v wu = Pr Gk >ℓ + θ Y, Gk > w + f (y) dy 1 1 2 1 − vy Y 0 Z 1 (34) v wu = 1 − FGk (ℓ + θ1y) 1 − FGk w + fY (y) dy 1 2 1 − vy Z0 m −1 m −1 m −1 k U U E δλpλq Ξ = e−λ1ℓ−λ2wϕ 1 2 3 0 p!q! p=0 q=0 k S j=0 X X X=0 XE X a1 TMRC,∞ ∞ ∞ α2 P =1 − 1 − F (b1 + θ1x) 1 − F c1 + fG (x) dx Φn G1 G2 d − e x E Z0 1 1 τU CkθkbτU −kΥ (k + τ , λ a ) τU 1 1 E E 1 τU α2 e1 =1 − FG (a1)+ ϕ1βE + ϕ2βEc g a1, τE , 0, , , λE, 0, 0, τU E k+τE 1 c d d k λE 1 1 1 X=0 τU τU θ1 α2 e1 − ϕ1ϕ2βEb1 c1 g a1, τE, , , , λE, 0, τU , τU b1 c1d1 d1 (37) mE +1 N IV. ASYMPTOTIC SECRECY OUTAGE PROBABILITY where SE = (n1, ··· ,nmE +1) ∈ np = K − n − 1 , ( p=1 ) ANALYSIS P To obtain more insights, in this section we analyze the m +1 −2 n E λp p A = (K−n−1)! − E , B = asymptotic SOP in the higher SINR regime. Similar to [33] mE +1 (p−2)! Q (nq )! p=2 and [34], we assume that Ω2 → ∞, Ω1 = ε1Ω2 (ε1 > 1), and q=1 Q mE +1 mE +1 ΩR = ε2Ω2, where ε1 and ε2 are constant. k np (p − 2), and C =1+ np. The asymptotic CDF of Gv (v ∈{1, 2}) can be expressed p=2 p=2 as [48] PProof : See Appendix A. P ∞ τU τU F k (x)= ϕvx + O (x ) , (35) Lemma 2: The PDF of Y is given by Gv τ τ m −1 k m U m U E −λE y U U where ϕ = τ , ϕ = τU , and O (·) denotes δe 1 (ε1Ω2) U (τU )! 2 τ f (y)= ϕ Ω2 ( U )! Y 0 ς+1 higher order terms. Then the asymptotic SOP of cooperative k S j=0 (ρ4y + C) (33) X=0 XE X NOMA systems is obtained as k+1 k k−1 × ρ4λEy + Dy − Cky , K m (K−n)λ E ∞ n K−n mR(K−n) ∞ E P C ϕ η P , (36) where ϕ = , ς = B + mE +j, δ = out = K R Φn 0 Γ(mE ) − Cj Aρj λk ς (ς−1)! n=0 k 4 E X , and D = CλE − ρ4k + ρ4ς. m k! mR R ∞ where ϕR = m . The P for all the RS schemes Proof : See Appendix B. (ε2Ω2) R (mR)! Φn Similar to (24), based on (12) and (13), we obtain the is given as follows. closed-form expression of ∆4 as (34), given at the top of this q 1 page, where Ξ3 = w h v ,p,q,λ1θ1 + λE, λ2wu,u,v,ℓ and A. Traditional Multiple Relays Combining Scheme a c b u Based on (15), we have the closed-form for SOP with h (a,b,c,f,r,u,v,ℓ)= (ℓ + θ1y) 1+ TMRC scheme as (37), given at the top of this page, where 0 1 − vy Z Υ (·, ·) is the lower incomplete Gamma function, as defined ρ λ yk+1 + Dyk − Ckyk−1 4 E by (8.350.1) of [47]. × ς+1 r dy. fy+ 1−vy ,∞ (ρ4y + C) e Remark 5: From (37), one can easily obtain P TMRC Φn ≈ Similar to (17), utilizing Gaussian-Chebyshev quadrature, we 1 − FGE (a1) when ϕi → 0 (i ∈{1, 2}) with Ω2 → ∞. have N c a b u B. Optimal Single Relay Selection Scheme h (a,b,c,f,r,u,v,ℓ)= wi(ℓ + θ1sj ) 1+ 2 1 − vsj Based on (23), we obtain i=1 X ρ λ sk+1 + Dsk − Cksk−1 P OSRS,∞ = (1 − ∆∞)n, (38) 4 E j j j Φn 1 × r , fsj + − ς+1 1 vsj ∞ (ρ4sj + C) e where ∆1 is given in (39) at the top of the next page, mU mU a mU mU where sj = (ti + 1). where ϕ3 = mU and ϕ4 = mU . Similarly, 2 (ε1Ω2) (mU )! Ω2 (mU )! 8 a2 ∞ ∞ ∞ ∆ = 1 − F m (δ (x)) 1 − F m (δ (x)) f m (x) dx 1 G1 3 G2 4 GE 0 Z mU m −k Ck θkb U k m , λ a mU 1 2 Υ ( + E E 2) mU α2 e2 = FG (a2) − ϕ3βE − ϕ4βEc g a2,mE, 0, , , λE, 0, 0,mU (39) E k+mE 2 c d d k λE 2 2 2 X=0 θ α e + ϕ ϕ β bmU cmU g a ,m , 1 , 2 , 2 , λ , 0,m ,m 3 4 E 2 2 2 E b c d d E U U 2 2 2 2 1 v ∞ ∞ ∞ wu ∆4 = 1 − FGk (ℓ + θ1y) 1 − FGk w + fY (y) dy 1 2 1 − vy 0 Z λ mE −1 k − E mE −1 k δe v 1 =1 − ϕ0 ς − ϕ0ϕ3 δh ,mU , 0, λE, 0,u,v,ℓ ρ4 k v k S j=0 C + v v k S j=0 X=0 XE X X=0 XE X m −1 k m −1 k E 1 E 1 − ϕ ϕ wmU δh , 0,m , λ , 0,u,v,ℓ + ϕ ϕ ϕ wmU δh ,m ,m , λ , 0,u,v,ℓ 0 4 v U E 0 3 4 v U U E k=0 SE j=0 k=0 SE j=0 X X X X X X (42) τ −k τU Ck θk bDPA U k τ , λ aDPA TMRC,∞,DPA DPA τU 1 1 Υ + E E 1 P =1 − FGE a1 + ϕ1βE Φn λk+τE k=0 E X DPA DPA τ α e + ϕ β cDPA U g aDPA, τ , 0, 2 , 1 , λ , 0, 0, τ (44) 2 E 1 1 E cDPAdDPA dDPA E U 1 1 1 DPA DPA τ τ θ α e − ϕ ϕ β bDPA U cDPA U g aDPA, τ , 1 , 2 , 1 , λ , 0, τ , τ 1 2 E 1 1 1 E bDPA cDPAdDPA dDPA E U U 1 1 1 1 OSRS,∞ n we obtain P F a when ϕ V. ASYMPTOTIC SECRECY OUTAGE PROBABILITY Φn ≈ (1 − GE ( 2)) i → 0 (i ∈{1, 2}) with Ω2 → ∞. ANALYSIS WITH DYNAMIC POWER ALLOCATION SCHEME Since there is a ceiling for the SINR of U2, which equals to α2 , there is a floor for the SOP of NOMA systems when the C. Optimal Dual Relay Selection Scheme α1 transmit SNR increases. A new DPA scheme was proposed in Based on (30), we obtain the asymptotic SOP for coopera- [33] and the PA coefficients are given as tive NOMA systems with ODRS scheme as DPA 1 α = − K−1 1 1+µλ ̟ −2 ODRS,∞ K−n − ODRS,∞ OSRS,∞ µλ ̟ , (43) P = Cn ϕ ηmR(K n)P + P . DPA 2 out K R Φn ΦK α2 = −̟ 1+µλ2 n=0 X (40) where 0 <̟< 1and µ > 1. It was testified that non-zero Based on (31), we have DPA α2 −̟ diversity order can be obtained due to αDPA = µλ2 → 0 ODRS,∞ ∞ n 1 P , (41) with Ω2 → ∞ in [33] and [34]. In this section, the asymp- Φn = (1 − ∆4 ) totic SOP is analyzed while this DPA scheme is utilized. −̟ 2 −̟ ∞ 1+µλ −θ2µλ −θ2 where ∆4 was shown in (42), shown at the top of this page. DPA ( 2 ) 2 Based on (43), we have ai = −̟ , θ2µλ2 ρi Similarly, we have − µλ ̟ bDPA = θ1−1 1+ µλ−̟ , cDPA = − 1+ 2 , dDPA = n i ρi 2 i ρi i −̟ −̟ 2 mE −1 k 1+µλ2 −θ2 ρi θ2µλ ρ λ −ς ( ) DPA 2 i ,∞ E ρ4 − 2 , e = − 3 , and i =1, 2. ODRS −k − v ̟ i ̟ P ≈ ϕ0 δv e C + , (1+µλ2 ) (1+µλ2 ) Φn v k S j=0 X=0 XE X when ϕi → 0 (i ∈{1, 2}) with Ω2 → ∞. A. Traditional Multiple Relays Combining Scheme with Dy- Remark 6 namic Power Allocation Scheme ∞: Based on the definition of SDO as Gd = − lim ln(Pout) , we obtain the SDO for the three RS schemes ln(Ω2) Ω2→∞ Substitute (43) into (37), we achieve the closed-form ex- TMRC OSRS ODRS as Gd = Gd = Gd =0 because the first term is pression for asymptotic SOP under TMRC-DPA scheme as independent of Ω2 in (37), (39), and (42), respectively. (44), shown at the top of this page. 9 m −k mU Ck θk bDPA U k m , λ aDPA ∞,DPA DPA mU 1 2 Υ + E E 2 ∆ = FE a2 − ϕ3βE 1 λk+mE k=0 E X DPA DPA m α e − ϕ β cDPA U g aDPA,m , 0, 2 , 2 , λ , 0, 0,m (47) 4 E 2 2 E cDPAdDPA dDPA E U 2 2 2 DPA DPA m m θ α e + ϕ ϕ β bDPA U cDPA U g aDPA,m , 1 , 2 , 2 , λ , 0,m ,m 3 4 E 2 2 2 E bDPA cDPAdDPA dDPA E U U 2 2 2 2 mE −1 k λ −ς ∞,DPA − E ρ4 DPA −k ∆ =1 − ϕ δe vDPA C + v 4 0 vDPA k S j=0 X=0 XE X m −1 k E 1 − ϕ ϕ δh ,m , 0, λ , 0,uDPA, vDPA,ℓDPA 0 3 vDPA U E k=0 S j=0 X XE X (51) mE −1 k mU 1 − ϕ ϕ wDPA δh , 0,m , λ , 0,uDPA, vDPA,ℓDPA 0 4 vDPA U E k=0 SE j=0 X X X mE −1 k mU 1 + ϕ ϕ ϕ wDPA δh ,m ,m , λ , 0,uDPA, vDPA,ℓDPA 0 3 4 vDPA U U E k=0 SE j=0 X X X Then the SDO of the cooperative NOMA systems in this C. Optimal Dual Relay Selection with Dynamic Power Allo- case is obtained as cation Scheme Similar to (40), we write the asymptotic SOP for the TMRC,DPA Gd = min {mR (K − n)+ τU (1 − ̟)} cooperative NOMA system under ODRS-DPA scheme as 0≤n≤K K−1 = min {KmR − n (mR − mU (1 − ̟))} 0≤n≤K P ODRS,∞,DPA Cn ϕK−nηmR(K−n)P ODRS,∞,DPA out = K R Φn KmU (1 − ̟), mR >mU (1 − ̟) n=0 = X ∞ + P OSRS, ,DPA. ( KmR, mR ∞,DPA where ∆4 is shown in (51) at the top of this page, B. Optimal Single Relay Selection Scheme with Dynamic DPA DPA DPA 1 DPA α1 α2 θ2ρ3 DPA where w = − DPA , v = DPA , u = Power Allocation Scheme α1 ρ3 1−α1 θ2 DPA α2 DPA θ1−1 αDPAθ − , and ℓ = αDPAρ . Similar to (38), we have 1 2 1 1 3 Then the SDO of the cooperative NOMA systems in this n case is obtained as (52) at the top of the next page, where OSRS,∞,DPA ∞,DPA PΦ = 1 − ∆1 , (46) H1 = mD (1 − ̟) − ̟ and H2 = mD (1 − ̟)+ ̟ (K − 1). n Remark 8 : Based on (52), one can realize that using a ∞,DPA single relay to send jamming signals may reduce the SDO where ∆1 is given by (47), shown at the top of this page. of cooperative NOMA systems in some conditions. Then the SDO of the cooperative NOMA systems in this case is obtained as VI. NUMERICAL RESULTS AND DISCUSSIONS OSRS,DPA In this section, we present numerical results and Monte- Gd = min {mR (K − n)+ nmD (1 − ̟)} 0≤n≤K (48) Carlo simulations to testify our analysis. The main adopted TMRC,DPA parameters are set to P = P = P , Rth = 0.2 nat per = Gd . S R 1 th 2 channel use, R2 = 0.1 nat per channel use, and σ = 1. In all the figures, ‘Sim’ denotes the simulation results1. One Remark 7: Based on (45) and (48), one can observe the can observe that the analysis results match perfectly with SDO of cooperative NOMA systems under both TMRC-DPA simulation resutls and the secrecy performance with OSRS and OSRS-DPA schemes equals to the minimum SDO of dual hops. Moreover, it is a full diversity order based on the results 1We omit the results for TSRS scheme since it is exactly the same as that in [33] and [34]. for OSRS. 10 ODRS,DPA Gd = min min {mR (K − n)+ nH1} ,K(1 − ̟)mD 0≤n≤K−1 = min min {KmR + n (H1 − mR)} ,K(1 − ̟)mD 0≤n≤K−1 min {KmD (1 − ̟)+ mR − H2,KmD(1 − ̟)} , mR >H1 = (52) ( min {KmR,KmD(1 − ̟)} , mR KmR, mR 100 in the higher-ρ region. It is demonstrated in Fig. 2 that the SOP is enhanced by increasing K due to the improved diversity of the cooperative NOMA systems. One can also find that the -1 10 SOP depends merely on the RS scheme and is independent of K in the high-ρ region. Fig. 3 demonstrates that the SOP 10-2 with larger m outperforms the ones with smaller m as larger m signifies that the quality of wireless channel is better. However, the SOP with larger m underperforms that with the smaller m 10-3 in the high-ρ region. The reason is that the SOP of cooperative NOMA systems will benefit from the stronger fading over eavesdropping channels when the SINR of U2 approaches the 10-4 upper bound. -15 -10 -5 0 5 10 15 20 25 s s Figs. 4 and 5 present the effect on different R1 and R2, respectively. As we know, increasing the secrecy rate threshold Fig. 2. SOP for various K with R1 = 0.1, R2 = 0.2, mU = mE = mR = deteriorates the secrecy outage performance of wireless com- 2 = 0 5 Ω = 12 Ω =Ω = 10 Ω = −5 , αJ . , 1 dB, R 2 dB, and E dB. munications. Now, from Figs. 4 and 5, we can observe that s R1 has significant effect on the SOP of NOMA systems in 100 s low-ρ region. Different R1 has significant effect on the SOP of NOMA systems in low-ρ region but does not influence the SOP in high-ρ region. The opposite effect is observed s for R2. This proves that in the low-ρ region the bottleneck 10-1 in secrecy performance of NOMA systems is due to U1. In high-ρ region, weaker user must be given more attention. One can also observe from Fig. 4 that the gap between two RS schemes in high-ρ region is larger than those in low-ρ region 10-2 s and the gap in low-ρ region is decreased as R1 decreases. The impact of PA parameter on the SOP is investigated in Figs. 6 - 8. We can observe that the SOP decreases as α1 decreases. In other words, the secrecy performance is enhanced -15 -10 -5 0 5 10 15 20 25 by enlarging the difference between allocated power to the two users, which is consistent with the results presented in Fig. 3. SOP for various m with R1 = 0.1, R2 = 0.2, mU = mE = mR = Figs. 4 and 5. Fig. 8 presents the SOP for various αJ. One m, α1 = 0.2, αJ = 0.5, Ω1 = 12 dB, ΩR =Ω2 = 10 dB, and ΩE = −5 dB. can observe that the ODRS scheme enhances the secrecy outage performance of cooperative NOMA systems relative to OSRS scheme because the SNR at eavesdropper is degraded. always outperforms that of TMRC and in some scenarios Furthermore, the SOP is firstly improved with increasing αJ. the ODRS scheme can obtain the optimal secrecy outage However, it gets worse with increasing αJ since the transmit performance. power is reduced. Obviously, there is an optimal αJ, which Figs. 2 - 6 present the SOP for various ρ with FPA. One can be easily obtained via one dimensional search based on can easily observe that the secrecy performance of cooperative the results of our work. NOMA systems is enhanced and then becomes worse by im- The SOP for various Ω2 with FPA is presented in Figs. 9 proving the transmit SNR, which is similar to the conclusions - 11. One can observe that the asymptotic SOP approaches in [33] and [34]. This is because the SINR of U2 has a ceiling the exact ones at high-Ω2 region. The SDO of the cooperative 11 0 100 10 10-1 Rs . 1 = 0 5 10-1 10-2 out P 10-3 10-2 ODRS Sim OSRS Sim 10-4 Rs . 1 = 0 1 TMRC Sim Analysis -15 -10 -5 0 5 10 15 20 25 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 ρ (dB) s s = 0 1 = 0 2 = = Fig. 4. SOP for various R1 with R2 = 0.2, K = 2, mU = mE = mR = 2, Fig. 7. SOP for various α1 and K with R1 . , R2 . , mU mE α1 = 0.2, αJ = 0.5, Ω1 = 12 dB, ΩR =Ω2 = 10 dB, and ΩE = −5 dB. mR = m, αJ = 0.5, Ω1 = 12 dB, ΩR =Ω2 = 10 dB, and ΩE = −5 dB. 100 10-1 10-1 Rs . 2 = 0 5 10-2 out P -3 10-2 10 Rs . 2 = 0 1 ODRS Sim OSRS Sim TMRC Sim 10-4 Analysis 10-3 -15 -10 -5 0 5 10 15 20 25 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 ρ (dB) s s = 0 1 = 0 2 Fig. 5. SOP for various R2 with R1 = 0.2, K = 2, mU = mE = mR = 2, Fig. 8. SOP for various αJ and K and m with R1 . , R2 . , α1 = 0.2, αJ = 0.5, Ω1 = 12 dB, ΩR =Ω2 = 10 dB, and ΩE = −5 dB. mU = mE = mR = m, α1 = 0.2, Ω1 = 12 dB, ΩR =Ω2 = 10 dB, and ΩE = −5 dB. 100 100 10-1 10-2 10-2 10-4 10-3 10-6 10-4 -15 -10 -5 0 5 10 15 20 25 10-8 -10 -5 0 5 10 15 20 25 30 35 Fig. 6. SOP for various α1 with R1 = 0.1, R2 = 0.2, K = 2, mU = = = 2 = 0 5 Ω = 12 Ω = Ω = 10 mE mR , αJ . , 1 dB, R 2 dB, and Fig. 9. SOP for various α1 with R1 = 0.1, R2 = 0.2, K = 2, αJ = 0.5, Ω = −10 E dB. P = 10 dB, mU = mE = mR = 2, ε1 = 1.5, ε2 = 2, and ΩE = −10 dB. NOMA systems is zero since the SOP will be a constant as Ω2 increases. The results in Figs. 9 and 10 prove that increasing Figs. 12 - 14 present the SOP for various Ω2 under DPA K and / or decreasing α1 enhances the SOP of cooperative scheme. One can observe that DPA scheme achieves a non- NOMA systems. The results in Fig. 11 demonstrate that we zero secrecy diversity order, which depends on the relay can obtain better secrecy outage performance under a better number, the fading parameters of S −Rk and Sk −Ui. Fig. 14 channel condition for both TMRC and OSRS schemes. testifies that the SDO of cooperative NOMA systems equal to 12 100 100 10-2 10-2 10-4 10-4 10-6 10-8 10-6 10-10 -10 -5 0 5 10 15 20 25 30 0 5 10 15 20 25 30 35 40 45 50 55 1 = 0 1 2 = 0 2 Fig. 10. SOP for various K with R1 = 0.1, R2 = 0.2, mU = mE = Fig. 13. SOP with DPA scheme for various µ with R . , R . , = 0 5 = 10 = = = 2 = 3 1 = 1 5 2 = 2 mR = 2, P = 15 dB, α1 = 0.2, αJ = 0.5, ε1 = 1.5, ε2 = 2, and αJ . , P dB, mU mE mR , K , ε . , ε , = 0 2 Ω = −5 ΩE = −12 dB. ̟ . , and E dB. 100 VII. CONCLUSION In this work, three RS schemes were proposed to enhance -2 10 the secrecy performance of cooperative NOMA systems. We investigated the secrecy outage performance of these schemes 10-4 under FPA and DPA considering the presence of dependence between the two NOMA users with different secrecy rate thresholds. The closed-form expressions for the exact and 10-6 asymptotic SOP were obtained and verified by Monte Carlo simulation results. Moreover, the SDO for all the RS schemes were derived. The results demonstrate that relative to TMRC 10-8 scheme, OSRS and ODRS schemes enhance the security of -10 -5 0 5 10 15 20 25 30 35 the cooperative NOMA systems. OSRS scheme achieves the same SDO as TMRC scheme. The jamming signal influences Fig. 11. SOP for various m with R1 = 0.1, R2 = 0.2, mU = mE = the SDO of ODRS scheme. There are optimal PA coefficients mR = m, αJ = 0.3, K = 3, P = 10 dB, ε1 = 1.5, ε2 = 2, and between jamming and signal transmissions, which can be Ω = −10 E dB. easily obtained via one dimensional search based on the results of our work. 100 10-2 APPENDIX A k The PDF of HE = max GE is obtained as k∈Φ¯ 10-4 K−n−1 fHE (z) = (K − n) (FGE (z)) fGE (z) , (53) 10-6 where FGE (z) and fGE (z) refer to the CDF and PDF, respectively, of Gk , which is easily obtained by utilizing (11). 10-8 E Based on multi-nomial theorem, we obtain the pdf of HE as mE 10-10 (K − n)λ 0 5 10 15 20 25 30 35 40 45 50 55 E B+mE−1 −CλEz fHE (z)= Az e , (54) Γ (mE) S XE Fig. 12. SOP with DPA scheme for various K with R1 = 0.1, R2 = 0.2, = 0 5 = 10 = = = 2 = 1 5 = 2 mE +1 αJ . , P dB, mU mE mR , ε1 . , ε2 , where S n , ,n N n K n , ̟ = 0.1, µ = 5, and ΩE = −5 dB E = ( 1 ··· mE +1) ∈ p = − − 1 . ( p=1 ) P m +1 −2 n − − E λp p A = (K n 1)! − E , B = the minimum SDO of two hops. One can observe from Fig. mE +1 (p−2)! Q (nq )! p=2 14 (c) that a single relay is utilized to send jamming signals q=1 Q influences the SDO of the cooperative NOMA system in some mE +1 mE +1 np (p − 2), and C =1+ np. scenarios. p=2 p=2 P P 13 0 100 100 10 ̟ = 0.1 ̟ = 0.1 Analysis ̟ = 0.3 ̟ = 0.3 Asymptotic 10-5 m m U = 1 U = 1 mU = 1 mU = 3 mR = 3 -5 m -5 10 R = 3 10 mR = 3 mR = 2 10-10 mU = 3 mU = 3 out out out m mR = 1 m U = 3 P P R = 1 P mR = 3 10-15 10-10 10-10 10-20 10-25 25 30 35 40 45 50 25 30 35 40 45 50 25 30 35 40 45 50 Ω2 (dB) Ω2 (dB) Ω2 (dB) (a) SOP with TMRC scheme (b) SOP with OSRS scheme (c) SOP with ODRS scheme with ̟ = 0.15 Fig. 14. SOP for DPA scheme with R1 = 0.1, R2 = 0.2, αJ = 0.5, K = 3, mE = 2, ε1 = 1.5, ε2 = 2, µ = 8, and ΩE = −5 dB. APPENDIX B [4] S. M. R. Islam, N. Avazov, O. A. Dobre, and K. S. Kwak, “Power-domain non-orthogonal multiple access (NOMA) in 5G systems: Potentials and Based on (32) and utilizing (1.111) and (3.351.3) of [47], challenges,” IEEE Commun. Surveys Tuts., vol. 19, no. 2, pp. 721 - 742, the CDF of Y can be obtained as (55) at the bottom of this 2nd Quart. 2017. m (K−n)λ E page, where ϕ = E , ς = B + m + j, and δ = [5] Y. Liu, Z. Qin, M. Elkashlan, Z. Ding, A. Nallanathan, and L. Hanzo, Γ(mE ) E − “Nonorthogonal multiple access for 5G and beyond,” Proc. IEEE, vol. Cj Aρj λk ς (ς−1)! k 4 E . 105, no. 12, pp. 2347-2381, Dec. 2017. k! [6] Z. Ding, X. Lei, G. K. Karagiannidis, R. Schober, J. H. Yuan, and V. K. Using integration by parts, and applying some algebra, we Bhargava, “A survey on non-orthogonal multiple access for 5G networks: derive the PDF of Y as (33). Research challenges and future trends,” IEEE J. Sel. Areas Commun., vol. 35, no. 10, pp. 2181-2195, Oct. 2017. REFERENCES [7] Y. Liu, H. Xing, C. Pan, A. Nallanathan, M. Elkashlan, and L. Hanzo, “Multiple antenna assisted non-orthogonal multiple access,” IEEE Wire- [1] L. Dai, B. Wang, Y. Yuan, S. Han, C.-L. I, and Z. Wang, “Non-orthogonal less Commun., vol. 25, no. 2, pp. 17 - 23, Apr. 2018. multiple access for 5G: solutions, challenges, opportunities, and future [8] D. Wan, M. Wen, F. Ji, H. Yu, and F. Chen, “Non-orthogonal multiple research trends,” IEEE Commun. Mag., vol. 53, no. 9, pp. 74-81, Sept. access for cooperative communications: Challenges, opportunities, and 2015. trends,” IEEE Wireless Commun., vol. 25, no. 2, pp. 109-117, Apr. 2018. [2] Y. Wang, B. Ren, S. Sun, S. Kang and X. Yue, “Analysis of non- [9] F. Zhou, Y. Wu, Y.-C. Liang, Z. Li, Y. Wang, and K.-K. Wong, “State orthogonal multiple access for 5G,” China Communications, vol. 13, no. of the art, taxonomy, and open issues on cognitive radio networks with Supplement2, pp. 52-66, N/A 2016. NOMA,” IEEE Wireless Commun., vol. 25, no. 2, pp. 100-108, Apr. 2018. [3] Z. Ding, Y. Liu, J. Choi, Q. Sun, M. Elkashlan, I. C. L, and H. V. Poor, [10] Z. Ding, M. Peng, and H. V. Poor, “Cooperative non-orthogonal multiple “Application of non-orthogonal multiple access in LTE and 5G networks,” access in 5G systems,” IEEE Commun. Lett., vol. 19, no. 8, pp. 1462- IEEE Commun. Mag., vol. 55, no. 2, pp. 185-191, Feb. 2017. 1465, Aug. 2015. k GE FY (y)=Pr [11] Y. Liu, Z. Ding, M. Elkashlan, and H. V. Poor, “Cooperative non- [35] Y. Feng, S. Yan, and Z. Yang. “Secure transmission to the stronger user orthogonal multiple access with simultaneous wireless information and with optimal power allocation in NOMA,” arXiv:1807.00935, Jul. 2018, power transfer,” IEEE J. Sel. Areas Commun., vol. 34, no. 4, pp. 938- [Online]: https://arxiv.org/abs/1807.00935 953, Apr. 2016. [36] L. Lv, Z. Ding, Q. Ni, and J. Chen, “Secure MISO-NOMA transmission [12] J. Men and J. Ge, “Non-orthogonal multiple access for multiple-antenna with artificial noise,” IEEE Trans. Veh. Technol., vol. 67, no. 7, pp. 6700- relaying networks,” IEEE Commun. Lett., vol. 19, no. 10, pp. 1686-1689, 6705, Jul. 2018. Oct. 2015. [37] F. Zhou, Z. Chu, H. Sun, R. Q. Hu, and L. Hanzo, “Artificial noise [13] J. Men, J. Ge, and C. Zhang, “Performance analysis of non-orthogonal aided secure cognitive beamforming for cooperative MISO-NOMA using multiple access for relaying networks over Nakagami-m fading channels,” SWIPT,” IEEE J. Sel. Areas Commun., vol. 36, no. 4, pp. 918-931, Apr. IEEE Trans. Veh. Technol., vol. 66, no. 2, pp. 1200-1208, Feb. 2017. 2018. [14] Y. Xiao, L. Hao, Z. Ma, Z. Ding, Z. Zhang, and P. Fan, “Forwarding [38] Z. Zhu, Z. Chu, F. Zhou, H. Niu, Z. Wang, and I. Lee, “Secure strategy selection in dual-hop NOMA relaying systems,” IEEE Commun. beamforming designs for secrecy MIMO SWIPT systems,” IEEE Wireless Lett., vol. 22, no. 8, pp. 1644-1647, Aug. 2018. Commun. Lett., vol. 7, no. 3, pp. 424-427, Jun. 2018. [15] W. Duan, M. Wen, Z. Xiong, and M. H. Lee, “Two-stage power [39] K. Jiang, T. Jing, Y. Huo, F. Zhang, and Z. Li, “SIC-based secrecy allocation for dual-hop relaying systems with non-orthogonal multiple performance in uplink NOMA multi-eavesdropper wiretap channels,” access,” IEEE Access, vol. 5, pp. 2254-2261, Mar. 2017. IEEE Access, vol. 6, pp. 19664-19680, Apr. 2018. [16] C. Zhong and Z. Zhang, “Non-Orthogonal multiple access with coop- [40] H. Zhang, N. Yang, K. Long, M. Pan, G. K. Karagiannidis, and V. erative full-duplex relaying,” IEEE Commun. Lett., vol. 20, no. 12, pp. C. M. Leung, “Secure communications in NOMA system: Subcarrier 2478-2481, Dec. 2016. assignment and power allocation,” IEEE J. Sel. Areas Commun., vol. 36, [17] Z. Zhang, Z. Ma, M. Xiao, Z. Ding, and P. Fan, “Full-duplex device-to- no. 7, pp. 1441-1452, Jul. 2018. device aided cooperative non-orthogonal multiple access,” IEEE Trans. [41] J. Chen, L. Yang, and M. Alouini, “Physical layer security for coop- Veh. Technol., vol. 66, no. 5, pp. 4467 - 4471, May 2016. erative NOMA systems,”IEEE Trans. Veh. Technol., vol. 67, no. 5, pp. [18] G. Liu, X. Chen, Z. Ding, Z. Ma, and F. R. Yu, “Hybrid half-duplex/full- 4645-4649, May. 2018. duplex cooperative non-orthogonal multiple access with transmit power [42] W. Jiang, Y. Gong, Q. Xiao, and Y. Liao, “Rate maximization for adaptation,” IEEE Trans. Wireless Commun., vol. 17, no. 1, pp. 506-519, untrusted relay networks with nonorthogonal cooperative transmission Jan. 2018. protocols,” IEEE Trans. Veh. Technol., vol. 67, no. 7, pp. 6325-6339, [19] T. M. C. Chu and H.-J. Zepernick, “Performance of a non-orthogonal Jul. 2018. multiple access system with full-duplex relaying,” IEEE Commun. Lett., [43] A. Arafa, W. Shin, M. Vaezi, and H. V. Poor. “Secure relaying vol. 22, no. 10, pp. 2084-2087, Oct. 2018. in non-orthogonal multiple access: Trusted and untrusted scenarios,” [20] M. F. Kader, S. Y. Shin, and V. C. M. Leung, “Full-duplex non- arXiv:1808.07864, Aug. 2018, [Online]: https://arxiv.org/abs/1808.07864 orthogonal multiple access in cooperative relay sharing for 5G systems,” [44] Y. Feng, Z. Yang, and S. Yan, “Non-orthogonal multiple access and IEEE Trans. Veh. Technol., vol. 67, no. 6, pp. 5831-5840, Jul. 2018. artificial-noise aided secure transmission in FD relay networks,” in Proc. 2017 IEEE Globecom Workshops (GC Wkshps) [21] Y. Zou, X. Wang, and W. Shen, “Optimal relay selection for physical- , Singapore, Dec. 2017, layer security in cooperative wireless networks,” IEEE J. Sel. Areas pp. 1-6. Commun., vol. 31, no. 10, pp. 2099-2111, Oct. 2013. [45] C. Liu, L. Zhang, M. Xiao, Z. Chen, and S. Li, “Secrecy performance analysis in downlink NOMA systems with cooperative full-duplex relay- [22] H. Lei, H. Zhang, I. S. Ansari, Z. Ren, G. Pan, K. A. Qaraqe, and M.- ing,” in Proc. 2018 IEEE International Conference on Communications S. Alouini, “On secrecy outage of relay selection in underlay cognitive Workshops (ICC Workshops), Kansas City, MO, USA, Jul. 2018, pp. 1-6. radio networks over Nakagami-m fading channel,” IEEE Trans. Cogn. [46] B. Li, X. Qi, K. Huang, Z. Fei, F. Zhou, and R. Q. Hu, “Security- Commun. Netw., vol. 3, no. 4, pp. 614 - 627, Dec. 2017. reliability tradeoff analysis for cooperative NOMA in cognitive radio [23] Z. Ding, H. Dai, and H. V. Poor, “Relay selection for cooperative networks,” IEEE Trans. Commun., doi: 10.1109/tcomm.2018.2873690, NOMA,” IEEE Wireless Commun. Lett., vol. 5, no. 4, pp. 416-419, Aug. Oct. 2018. 2016. [47] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and [24] J. Zhao, Z. Ding, P. Fan, Z. Yang, and G. K. Karagiannidis, “Dual relay Products: Academic press, 2014. selection for cooperative NOMA with distributed space time coding,” [48] H. Lei, C. Gao, I. S. Ansari, Y. Guo, Y. Zou, G. Pan, and K. Qaraqe, IEEE Access, vol. 6, pp. 20440-20450, Apr. 2018. “Secrecy outage performance of transmit antenna selection for MIMO [25] Z. Yang, Z. Ding, Y. Wu, and P. Fan, “Novel relay selection strategies underlay cognitive radio systems over Nakagami-m channels,” IEEE for cooperative NOMA,” IEEE Trans. Veh. Technol., vol. 66, no. 11, pp. Trans. Veh. Technol., vol. 66, no. 3, pp. 2237-2250, Mar. 2017. 10114-10123, Nov. 2017. [49] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions [26] P. Xu, Z. Yang, Z. Ding, and Z. Zhang, “Optimal relay selection schemes with Formulas, Graphs, and Mathematical Tables, 9th. New York, NY, for cooperative NOMA,” IEEE Trans. Veh. Technol., vol. 67, no. 8, pp. USA: Dover Press, 1972. 7851-7855, Aug. 2018. [27] X. Yue, Y. Liu, S. Kang, A. Nallanathan, and Z. Ding, “Spatially random relay selection for full/half-duplex cooperative NOMA networks,” IEEE Trans. Commun., vol. 66, no. 8, pp. 3294 - 3308, Aug. 2018. [28] J. Chen, L. Yang, and M.-S. Alouini, “Performance analysis of coop- erative NOMA schemes in spatially random relaying networks,” IEEE Access, vol. 6, pp. 33159-33168, Jul. 2018. [29] Y. Zou, J. Zhu, X. Wang, and L. Hanzo, “A survey on wireless security: Technical challenges, recent advances, and future trends,” Proc. IEEE, vol. 104, no. 9, pp. 1727-1765, May 2016. [30] H.-M. Wang and T.-X. Zheng, Physical Layer Security in Random Cellular Networks. Singapore: Springer, 2016. [31] Y. Zhang, H. Wang, Q. Yang, and Z. Ding, “Secrecy sum rate maxi- mization in non-orthogonal multiple access,” IEEE Commun. Lett., vol. 20, no. 5, pp. 930-933, May. 2016. [32] Y. Liu, Z. Qin, M. Elkashlan, Y. Gao, and L. Hanzo, “Enhancing the physical layer security of non-orthogonal multiple access in large-scale networks,”IEEE Trans. Wireless Commun., vol. 16, no. 3, pp. 1656-1672, Mar. 2017. [33] H. Lei, J. Zhang, K. Park, P. Xu, I. S. Ansari, G. Pan, B. Alomair, and M.-S. Alouini, “On secure NOMA systems with transmit antenna selection schemes,”IEEE Access, vol. 5, pp. 17450-17464, Sep. 2017. [34] H. Lei, J. Zhang, K.-H. Park, P. Xu, Z. Zhang, G. Pan, and M.-S. Alouini, “Secrecy outage of max-min TAS scheme in MIMO-NOMA systems,”IEEE Trans. Veh. Technol., vol. 67, no. 8, pp. 6981-6990, Aug. 2018. arXiv:1811.03220v2 [cs.IT] 1 Sep 2019 nvriy hnqn 004 hn.(e-mail:ycguo@cqu. China. 400044, Chongqing University, [email protected]). Posts [email protected]). of [email protected]; University Chongqing Technology, Communications etme ,2019 4, September kihong.park@kau (e-mail: Arabia Saudi 23955-6900, Thuwal [email protected]). (e-mail: 400715, Chongqing OASseswt ea eeto Scheme Selection Relay with Systems NOMA .Pni ihCogigKyLbrtr fNnierCircuit Nonlinear of Laboratory Key Chongqing with is Pan G. Technolo Optoelectronic of of Laboratory Key University the with Engineering, is of Guo Y. School the with is Ansari S. I. Inform and Communication of School with are Yang Z. and Lei H. received. Manuscript .H akadM-.Aoiiaewt ES iiin igAb King Division, CEMSE with are Alouini M.-S. and Park K.-H. utpeacs NM)ntokoe Nakagami- over network (NOMA) access multiple rpsdOR,TR,adOR cee ln ihteTR sch TMRC the with along secu schemes the ODRS for and expressions TSRS, closed-form OSRS, proposed the users, NOMA two o for capacity secrecy the between si correlation forward the source Considering the from signals combinin decode successfully relays that multiple traditional with exa systems also NOMA we benchmark, the a As analyzed. are performance ( outage selection relay dual po optimal se and relay via scheme, (TSRS) single selection users optimal i.e., two schemes, the (RS) selection to relay signal superposition m by transmits relays decode-and-forward relay all to users slo two first the the of At signal destination. to station base the from signals erc uaeAayi o Cooperative for Analysis Outage Secrecy ogin e,Zxa ag iHn ak ma hfiu Ans Shafique Imran Park, Ki-Hong Yang, Zixuan Lei, Hongjiang hspprcniestesceyotg efrac famul a of performance outage secrecy the considers paper This oga u,GoegPn n oae-lmAlouini Mohamed-Slim and Pan, Gaofeng Guo, Yongcai Abstract edu.cn). teus;[email protected]). st.edu.sa; m yadSseso h dcto iityo hn,Chongqing China, of Ministry Education the of Systems and gy n eeomnctos hnqn 005 hn (e-mail: China 400065, Chongqing Telecommunications, and n nelgn nomto rcsig otws Unive Southwest Processing, Information Intelligent and s aigcanl.Tososaeuiie otransmit to utilized are slots Two channels. fading lso,GagwG28Q ntdKndm(e-mail: Kingdom United 8QQ, G12 Glasgow Glasgow, ,tebs tto racsstesuperposition the broadcasts station base the t, eto OR)shm,tose igerelay single two-step scheme, (OSRS) lection ulhUiest fSineadTcnlg (KAUST), Technology and Science of University dullah w sr n ifrn erc requirement secrecy different and users two f DS cee r rpsdadtesecrecy the and proposed are scheme, ODRS) nl oteNM sr iheulpower. equal with users NOMA the to gnals to niern hnqn e a fMobile of Lab Key Chongqing & Engineering ation saempigsrtg.Te h selected the Then strategy. mapping essage TR)shm nwihalterelay the all which in scheme (TMRC) g e-oanNM ehooy Three technology. NOMA wer-domain ietesceyotg efrac of performance outage secrecy the mine iyotg rbblt SP fthe of (SOP) probability outage rity il-ea sitdnon-orthogonal assisted tiple-relay m r eie n validated and derived are eme ari DRAFT rsity, 1 2 via simulations. To get more insights, we also derive the closed-form expressions for the asymptotic SOP for all the schemes with fixed and dynamic power allocations. Furthermore, the secrecy diversity order (SDO) of cooperative NOMA systems is obtained. The results demonstrate that our proposed schemes can significantly enhance the secrecy performance compared to the TMRC scheme and that all the RS schemes with fixed power allocation obtain zero SDO and the OSRS scheme with dynamic power allocation obtains the same SDO as TMRC. Index Terms Non-orthogonal multiple access, cooperative communication systems, relay selection scheme, se- crecy outage probability I. INTRODUCTION A. Background and Related Works Non-orthogonal multiple access (NOMA) has been considered as one of the most promising technologies to deal with the shortage of bandwidth resources and enhance user fairness in the fifth generation (5G) mobile network [1]-[9]. Compared to traditional orthogonal multiple access schemes, NOMA systems can obtain superior merits, such as high bandwidth efficiency, better user fairness, ultra-connectivity, and high flexibility. After receiving the superposition signal transmitted by the source node, the stronger user in NOMA systems firstly decodes the signal sent to the weaker user, cancels it from the received signals, and then decodes the signals for itself. Cooperative communication is a particularly attractive technique that not only extends the network’s coverage but also enhances the system performance when diversity technology is utilized at the destinations. Since the stronger users in NOMA systems always firstly decode the messages for the weaker users, therefore the stronger users can be utilized as relays to improve the performance of those weaker users with poor channel conditions [10], [11]. Ding et al. analyzed the outage probability (OP), diversity order of cooperative NOMA systems, and proposed an approach based on user pairing to reduce system complexity in [10]. Furthermore, a new cooperative simultaneous wireless information and power transfer NOMA protocol was proposed and the closed-form expressions for the OP and system throughput were derived in [11]. Recently, many literatures focused on the cooperative NOMA systems with dedicated relay nodes. For example, a dedicated amplify-and-forward (AF) relay with multiple antennas was September 4, 2019 DRAFT 3 utilized in the cooperative NOMA systems and the closed-form expressions for the exact and lower bound of OP were derived in [12], [13], [14]. Moreover, a novel power allocation (PA) scheme for dual-hop relaying NOMA systems along with its ergodic sum-rate and OP were investigated in [15]. Full-duplex (FD) relay also has been utilized in cooperative NOMA systems to obtain higher spectral efficiency and better performance [16]. For example, the stronger user working in the FD mode was utilized to forward signals to the weaker user in [17] and [18]. The closed-form expression for OP was obtained and the results showed that FD relays outperform half-duplex (HD) relays. Liu et al. analyzed the PA problems for HD and FD cooperative NOMA systems thereby obtaining the closed-form expression for the optimal power allocation policies in [18]. Since the weaker user decodes its signals by directly treating signals for the stronger user as interference, then the performance of the weaker users will become the bottleneck of NOMA systems in many scenarios. The dedicated FD relay was utilized to improve the performance of the weaker users in [19], [20]. The performance of FD cooperative NOMA systems over Nakagami-m channels was investigated in [19] and the analytical expressions for OP and ergodic rate were derived. In [20], a sharing FD relay was utilized in NOMA systems with two source- destination pairs and the ergodic rate, OP, and outage capacity were investigated while both perfect and imperfect self-interference cancellation (SIC) schemes were considered, respectively. Relay selection (RS) technique is an effective scheme in making full use of space diversity with low implementation complexity and it can straightforwardly improve the spectral efficiency of cooperative systems [21], [22]. Considering the quality of service (QoS) requirements for the two users are different, a two-stage single-relay-selection strategy and dual-relay-selection with fixed power allocation (FPA) was studied in [23] and [24], respectively, in order to obtain the OP. The results showed that the two-stage strategy achieves a better OP and diversity gain. Combining DF and AF relaying, Yang et al. extended the two-stage RS scheme with dynamic power allocation (DPA) and derived the exact and asymptotic analytical expressions for OP in [25]. In [26], considering whether the relay can correctly decode two users’ messages, two optimal RS schemes were proposed for cooperative NOMA systems with FPA and DPA at the relays, respectively. Yue et al., in [27], studied the performance of cooperative NOMA systems with two RS schemes in which the relay operates in either FD or HD mode. Modeling the spatial topology of relays with homogeneous poisson point process (PPP), the performance of cooperative NOMA systems with two-stage RS scheme was analyzed in [28] and a closed-form September 4, 2019 DRAFT 4 approximation for the OP was obtained. To deal with the security issues followed by the explosive increase in cellular data, physical layer security (PLS) is emerging as one of the most promising ways to ensure secure commu- nication basing on the time varying nature of the wireless channels [29], [30]. A new optimal PA strategy was proposed to maximize the secrecy sum rate of NOMA systems and the results confirmed that a significant secrecy performance for NOMA systems was obtained in [31]. In [32], the security performance of a NOMA-based large-scale network is considered, where both the NOMA users and eavesdroppers were modeled by homogeneous PPPs. In [33], we investigated the secrecy outage performance of a multiple-input single-output (MISO) NOMA systems with transmit antenna selection (TAS) schemes and the closed-form expressions for secrecy outage probability (SOP) were derived. The results demonstrated that the proposed DPA scheme can achieve non-zero secrecy diversity order (SDO) for all the TAS schemes. The secrecy outage performance of multiple-input multiple-output (MIMO) NOMA systems with multiple legitimate and illegitimate receivers was studied in [34] and the closed-form expressions for SOP were subsequently obtained. Feng et al. proposed in [35] a new PA to maximize the secrecy rate of the stronger user while guaranteeing the non-secure transmission rate requirement to the weaker user, but the secrecy performance of the weaker user was not considered. Secrecy beamforming schemes were proposed for MISO-NOMA systems, cognitive MISO-NOMA systems, and MIMO-NOMA systems in [36], [37], and [38], respectively. The secrecy performance of a NOMA system with multiple eavesdroppers was investigated while zero-forcing and minimum mean-square error decoding schemes were utilized on the legitimate destinations in [39]. A new joint subcarrier (SC) assignment and PA scheme was proposed to improve the security of the two-way relay NOMA systems in [40]. The secrecy performance of a cooperative NOMA system with a dedicated AF/DF relay was investigated and the closed-form expressions for the SOP were obtained in [41]. In some scenarios, the relay in cooperative NOMA systems is untrusted and curious to decode the users’ messages. Two new HD nonorthogonal AF schemes were proposed for cooperative NOMA sys- tems with an untrusted relay and the secrecy rate maximization-based PA scheme was discussed in [42]. Arafa et al. proposed two new relaying schemes to resist the untrusted relay and the secrecy performance of cooperative NOMA systems with new relaying schemes was analyzed and compared in [43]. Feng et al. proposed an artificial-noise (AN) aided scheme to build up secure cooperative NOMA systems with a FD relay and the closed-form expression for the September 4, 2019 DRAFT 5 SOP was derived in [44]. The secrecy performance of cooperative NOMA systems in which the stronger user behaves as a FD relay was investigated in [45] and the SOP was analyzed with an assumption of the imperfect SIC. The security-reliability tradeoff for both cooperative and non-cooperative NOMA schemes was analyzed and analytical expressions for SOP were derived in [46]. B. Contributions In many works, such as [32], [34], [41], the worst-case scenario was considered, in which it is assumed that the eavesdroppers have powerful detection capabilities and they can extract the message from the signals transmitted from the resource. Under this assumption, one can realize that the secrecy capacity of both/all the users in NOMA systems are not independent [34]. The major contributions of this paper are as follows, 1) In this work, we analyze a cooperative NOMA system with multiple relays and an eaves- dropper employing RS schemes to enhance the secrecy performance. Three RS (OSRS, TSRS, and ODRS) schemes are proposed. For the purpose of comparison, we also in- vestigate the secrecy performance of the NOMA systems with traditional multiple relays combining (TMRC) scheme in which all the relays that successfully decode signals from the source forward signals to the NOMA users with equal power. Considering the correla- tion between the secrecy capacity of two users, the closed-form expressions for the SOP under different RS schemes are derived and validated via simulations. 2) Different secrecy QoS is considered in our work, i.e., different secrecy threshold rates are requested for the two NOMA users. Although the weaker user cannot perform SIC, either the stronger or the weaker user will be the bottleneck of the cooperative NOMA system. Our results demonstrate that this depends on the power allocation parameters and secrecy rate thresholds. 3) To obtain more insights, we also derive the closed-form expressions for the asymptotic SOP under different RS schemes with FPA and DPA. Furthermore, the SDO of cooperative NOMA systems are derived. The results demonstrate that all the RS schemes with FPA obtain zero SDO and the OSRS scheme achieves the same SDO as TMRC scheme. 4) Differing from [32], [34], [41], wherein it has been assumed that the secrecy capacity for the stronger and weaker users is independent, correlation between the secrecy capacity of September 4, 2019 DRAFT 6 R1 U1 GRk U S G Rk GU Rk R S k U2 Rk GE R K E Fig. 1. System model demonstrating a cooperative NOMA system with one resource (S), multiple relays (Rk, k = 1, · · · , K), two legitimate user (U1 and U2), and an illegitimate eavesdropper (E). the two users and different secrecy requirement for the two users is considered in this work, which is more practical. C. Organization This paper is organized as follows, in Section II, the system model and RS schemes are intro- duced. In Section III, we analyze the security outage performance of the proposed system. The asymptotic SOP for cooperative NOMA systems with FPA and DPA are derived in Sections IV and V, respectively. Numerical and simulation results are presented in Section VI to demonstrate the security performance of the system. Finally, Section VII concludes the paper. II. SYSTEM MODEL As shown in Fig. 1, we consider a cooperative downlink NOMA system that consists of a base station (S), K (K ≥ 1) DF HD relays (Rk, k =1, ··· ,K), and two users (U1 and U2). An eavesdropper (E) wants to wiretap the information through decoding the received signals. It is assumed that all nodes are equipped with a single antenna and the direct link between S and both users are unavailable due to deep fading. The communication links between S and all the receivers (U1, U2, and E) are relayed by Rk. The probability density function (PDF) and the cumulative distribution function (CDF) of the channel power gains of link between the source node ~ and the destination node λ¯ can be expressed by ~ ~ m mλ¯ −1 ~ k − λ¯ x k Ω~ mλ¯ x FG~ (x)=1 − e λ¯ ~ , (1) λ¯ Ω k! k λ¯ X=0 September 4, 2019 DRAFT 7 ~ ~ ~ − ~ m¯ m mλ¯ 1 λ − λ¯ x x mλ¯ Ω~ fG~ (x)= ~ ~ e λ¯ , (2) λ¯ Γ(m ) Ω λ¯ λ¯ where ~ ∈ {S, Rk} signifies the transmitter, λ¯ ∈ {Rk, U1, U2, E} denotes the receiver, Γ(·) is ~ ~ Gamma function, as defined by (8.310.1) of [47], mλ¯ and Ωλ¯ are the fading parameters and average channel power gains, respectively. To make analysis simple, it is assumed that the links of the first hop are independent and identically distributed (i.i.d.), which means mS m , S . The same assumption is made Rk = R ΩRk = ΩR Rk Rk to the links between Rk and each receiver in the second hop, which means mU1 = mU2 = mU , Rk Rk Rk Rk ΩU1 = Ω1, ΩU2 = Ω2, mE = mE, and ΩE = ΩE. Two time slots are assumed to utilize to the communication between S and the NOMA users. In the first time slot, S broadcasts the superposition signal to all relays by message mapping strategy, which is utilized and proved optimal to achieve the minimal common OP in [26]. During the second time slot, a relay is selected using the RS scheme proposed below to send superposition signal to the two NOMA users. We assume all the relays are close enough such that the relative distance between the two users and the relay is determined. To make the representation clear and simple, we use GRk Gk. It is also assumed that U Ui = i 1 k k has a better channel condition than U2 G1 >G2 , which is adopted in many NOMA studies, e.g., [5], [16], [17], [18], [26], [33]. Then the signal-to- interference-noise ratio (SINR) of the NOMA users can be written as k k γ1 = α1ρG1, (3) k k α2ρG2 γ2 = k , (4) α1ρG2 +1 where αi (i =1, 2) represents the power allocation coefficients at the kth relay, α1 + α2 = 1, α1 >α2, and ρ signifies the transmit signal to noise ratio (SNR). III. SECRECY OUTAGE PROBABILITY ANALYSIS The set of relays that can correctly decode mixed signals from S can be expressed as ∆ 1 S th th Φ = k :1 ≤ k ≤ K, log 1+ ρS G ≥ R + R , (5) 2 2 Rk 1 2 September 4, 2019 DRAFT 8 1 where the factor 2 arises from the fact that two slots are required to complete the data trans- th mission, ρS denotes the transmit SNR at S, and Ri (i =1, 2) is the data rate threshold for Ui. Thus the SOP of cooperative NOMA systems can be written as K Pout = Pr (|Φ| = n) PΦn , (6) n=0 X where PΦn denotes the SOP under the condition that there are n relays that correctly decode the mixed signals. One can easily have PΦ0 =1. It is assumed that all the links between the source and the relays are i.i.d., so we have n S n S K−n Pr {|Φ| = n} = CK Pr GR ≥ η Pr GR < η k k (7) n n K−n = CKχ (1 − χ) , m − 2Rth+2Rth R 1 k k n m! e 1 2 −1 −λRη λRη mR where C = , η = , χ = e , and λR = . m n!(m−n)! ρS k! ΩR k=0 P A. Traditional Multiple Relays Combining Scheme As a benchmark, the traditional multiple relays combining (TMRC) scheme is presented in this subsection, where all the n relays can successfully decode and forward the signal to both the users with equal power. Both the legitimate and illegitimate receivers combine their received signals with maximal ratio combining (MRC) scheme to maximize their SINR. To make fair comparison, it is assumed that the total transmit power at these relays is given by PR. Then the PΦn with this scheme can be expressed as TMRC s TMRC s PΦn = Pr Cs,1 < R1 or Cs,2 < R2 ||Φ| = n , (8) TMRC TMRC 1+γi TMRC TMRC where Cs,i = ln TMRC means the secrecy capacity of Ui, i =1, 2, γ1 = ρ1α1G1 , 1+γE,i TMRC n TMRC ρ1α2G2 TMRC k TMRC γ = TMRC , G = G , γ signifies the SNR at E when Ui is wiretapped, 2 1+ρ1α1G i i E,i 2 k=1 s PR 2 Ri is the secrecy rate threshold forPUi, ρ1 = nσ2 , and σ is the noise power. TMRC The CDF of Gi can be expressed as [48] − τD 1 λk −λix i k FGTMRC (x)=1 − e x , (9) i k! k X=0 mU where i =1, 2, λi = , and τU = nmU . Ωi As similar to [32], [34], and [41], we assume that E has enough capabilities to detect multiuser data. Then the SNR at E can be expressed as TMRC TMRC γE,i = ρ1αiGE , (10) September 4, 2019 DRAFT 9 n TMRC k TMRC where i =1, 2 and GE = GE. Similarly, the PDF of GE can be expressed as [48] k=1 P τE −1 −λE x f TMRC (x)= βEx e , (11) GE τE λE mE where βE = , λE= , and τE = nmE. Γ(τE ) ΩE We can express the secrecy connection probability (SCP) of U1, which is the complementary of SOP, as TMRC s TMRC TMRC Pr Cs,1 > R1 = Pr G1 > δ1 GE , (12) θ − s where δ (x)= b + θ x,b = 1 1 , and θ = e2R1 . 1 1 1 1 α1ρ1 1 Similarly, we obtain the SCP of U2 as TMRC s TMRC TMRC TMRC Pr Cs,2 > R1 = Pr ρ1G2 1 − θ2α1 − θ2α1α2ρ1GE > θ2 − 1+ θ2α2ρ1GE ∆ =Λ TMRC| TMRC{z } = Pr Λ > 0,G2 > δ2 GE TMRC TMRC TMRC = Pr GE < a1,G2 > δ2 GE , (13) where a = 1−θ2α1 , δ (x) = c + α2 , c = − 1 , d = α ρ (1 − α θ ), e = ρ2α2α θ , 1 ρ1α1α2θ2 2 1 d1−e1x 1 α1ρ1 1 1 1 1 2 1 1 1 2 2 2Rs and θ2 = e 2 . Remark 1: One can easily find that secrecy outage would occur at U2 when 1 − θ2α1 < 0. This means that in order to ensure a secure NOMA system (U2), there is a constraint for the power allocation coefficients, which is expressed as 1 s 1 s −2R2 −2R2 α1 ≤ = e or α2 > 1 − =1 − e . (14) θ2 θ2 Based on (12) and (13), PΦn in this case can be rewritten as TMRC TMRC s TMRC s PΦn = Pr Cs,1 < R1 or Cs,2 < R2 ||Φ| = n TMRC s TMRC s =1 − Pr Cs,1 > R1 ,Cs,2 > R2 ||Φ| = n TMRC TMRC TMRC TMRC TMRC =1 − Pr G1 > δ1 GE ,G1 > δ2 GE ,GE < a1 (15) a1 TMRC TMRC =1 − Pr G > δ1 (x) Pr G > δ2 (x) f TMRC (x) dx 1 2 GE Z0 a1 =1 − 1 − F TMRC (δ1 (x)) 1 − F TMRC (δ2 (x)) f TMRC (x) dx. G1 G2 GE 0 Z To facilitate the following analysis, we define a j b− −fx− h k r g (a,b,c,r,q,f,h,k,j)= x 1e 1−qx (1 + cx) 1+ dx. (16) 1 − qx Z0 September 4, 2019 DRAFT 10 To the authors’ best knowledge, it’s very difficult to obtain the closed-form expression of g (a,b,c,r,q,f,h,k,j). Here by making use of Gaussian-Chebyshev quadrature from eq. (25.4.30) of [49], we obtain N j a b − afsi − 2h acs k 2r b−1 2 2−aqs i g (a,b,c,r,q,f,h,k,j)= wisi e i 1+ 1+ , (17) 2 2 2 − aqsi i=1 X where N is the number of terms, si = ti +1, ti is the ith zero of Legendre polynomials, wi is the Gaussian weight, which is given in Table (25.4) of [49]. Substituting (9) and (11) into (15) and with some simple algebraic manipulations, we have τ −1 τ −1 U U λkλj Ξ P TMRC =1 − β e−λ1b1−λ2c1 1 2 1 , (18) Φn E k!j! k j=0 X=0 X where Ξ = bkcj g a , τ , θ1 , α2 , e1 ,λ θ + λ , λ2α2 ,k,j . 1 1 1 1 E b1 d1c1 d1 1 1 E d1 B. Optimal Single Relay Selection Scheme In this subsection, we propose OSRS scheme to minimize the overall SOP of the proposed cooperative NOMA system. Based on (12), one can easily obtain the SOP for U1 with mth relay as m m Pout,1 =1 − Pr {G1 > δ1 (GE )} m m = Pr {G1 < δ3 (GE )} (19) Gm = Pr 1 < 1 , δ (Gm) 3 E θ1−1 PR where δ (x)= b + θ x, b = , and ρ = 2 . 3 2 1 2 α1ρ2 2 σ Similarly, based on (13), we can express SOP for U2 as m m m Pout,2 =1 − Pr {GE < a2,G2 > δ4 (GE )} m m m m =1 − (Pr {GE < a2} − Pr {GE < a2,G2 < δ4 (GE )}) m m m m = Pr {GE < a2,G2 < δ4 (GE )} +1 − Pr {GE < a2} (20) m m m m m = Pr {G2 < δ4 (GE ) |GE < a2 } Pr {GE < a2} + Pr {GE > a2} Gm = Pr 2 < 1 |Gm < a Pr {Gm < a } + Pr {Gm > a } , δ (Gm) E 2 E 2 E 2 4 E where a = 1−θ2α1 , δ (x)= c + α2 , c = − 1 , d = α ρ − θ α2ρ , and e = θ α2α ρ2. 2 θ2α1α2ρ2 4 2 d2−e2x 2 α1ρ2 2 1 2 2 1 2 2 2 1 2 2 m One can observe that when GE > a2 the secrecy outage would occur at U2, which means the cooperative NOMA system is not secure. September 4, 2019 DRAFT 11 For the mth relay, we define m m G1 G2 m min m , m , GE < a2 δ3(G ) δ4(G ) Xm = E E . (21) m 0, GE ≥ a2 Then to maximize the secrecy performance the relay is selected with the following criterion ∗ m = argmax (Xm) . (22) m∈Φ Remark 2: It should be noted that the selection scheme in (22) is different from the max- min transmit antenna selection scheme (MMTAS) proposed in [34], which is to maximize the minimum secrecy capacity of two users. Here the selection criterion is to minimize the SOP of the cooperative NOMA system. Remark 3: There is another important difference between MMTAS and OSRS, i.e., it is s s s s assumed that R1 = R2 in MMTAS. However, in the OSRS scheme, R1 and R2 can be different. s s MMTAS is a special case of OSRS when it is assumed that R1 = R2 and the secrecy capacities of the two users are independent. The SOP of the cooperative NOMA system conditioned on |Φ| = n can be written as OSRS n PΦn = (Pr {Xm < 1}) n = (1 − Pr {Xm > 1}) m m n G1 G2 m = 1 − Pr min m , m ≥ 1,GE < a2 (23) δ3 (GE ) δ4 (GE ) n m m m m m = 1 − Pr {G1 ≥ δ3 (GE ) ,G2 ≥ δ4 (GE ) ,GE < a2} . ∆ =∆1 Based on (12) and (13), we| have {z } m m m m m ∆1 = Pr {G1 ≥ δ3 (GE ) ,G1 ≥ δ4 (GE ) ,GE < a2} a2 = 1 − FGm (δ3 (x)) 1 − FGm (δ4 (x)) fGm (x) dx 1 2 E (24) Z0 mU −1 mU −1 j λk λ = β e−λ1b2−λ2c2 1 2 Ξ , E k! j! 2 k j=0 X=0 X where Ξ = bkcj g a , m , θ1 , α2 , e2 ,λ θ + λ , λ2α2 ,k,j . 2 2 2 2 E b2 d2c2 d2 1 1 E d2 September 4, 2019 DRAFT 12 C. Two-Step Single Relay Selection Scheme In some scenarios stated in [23], [24], the QoS requirements for the two users are different. We can easily obtain the similar conclusion: the secrecy QoS for one user is higher than that of the other user. In this subsection, a new two-step RS (TSRS) scheme is proposed for such scenarios and the follwing two purposes will be realized simultaneously. One is to ensure there is no secrecy outage for the user that has lower secrecy QoS requirement and the other is to serve the user with higher secrecy QoS requirement with a secrecy rate as large as possible. The TSRS scheme is presented as follows. In the first step, the following subset is built in the relays by focusing on U1’s target secrecy rate i s Ψ= i : i ∈ Φ,Cs,1 > R1 , (25) i i 1 1+α1ρ2G1 where Cs,1 = ln i signifying the secrecy capacity for U1 from Ri and i ∈ Φ. 2 1+α1ρ2GE At the second step, the relay to maximize the secrecy capacity of U2 is selected, i.e., ∗ j j = argmax Cs,2 , (26) j∈Ψ j α ρ G 1+ 2 2 2 j +1 j 1 α1ρ2G2 where C = ln j signifies the secrecy capacity for U from R (j ∈ Ψ). s,2 2 α ρ G 2 j 1+ 2 2 E The SOP with this scheme can be achieved as n TSRS j i s k s p s PΦ = Cn Pr min Cs,1 > R1, max Cs,1 < R1, max Cs,2 < R2 n 1≤i≤j 1≤k≤n−j 1≤p≤j j=1 X n j i s p s k s = Cn Pr min Cs,1 > R1, max Cs,2 < R2 Pr max Cs,1 < R1 1≤i≤j 1≤p≤j 1≤k≤n−j j=1 X j n−j n j i s i s k s = CnPr Cs,1 > R1,Cs,2 < R2 Pr Cs,1 < R1 j=1 ∆ ∆ X =∆2 =∆3 i | s i {z s }i | s {zn } = Pr Cs,1 > R1,Cs,2 < R2 + Pr Cs,1 < R1 i s i s n = Pr Cs,1 < R1 or Cs,1 < R1 i s i s n = 1 − Pr Cs,1 > R1,Cs,2 > R2 . (27) Remark 4: An interesting result can be observed from (27) that the expression for the SOP under TSRS scheme is the same as that under OSRS. September 4, 2019 DRAFT 13 D. Optimal Dual Relay Selection Scheme In this subsection, a new RS scheme named ODRS scheme is proposed, in which one relay is selected to transmit signals to NOMA users among the ones that can decode the mixed signals and another relay is selected to transmit AN among the ones that can not decode the source signals. In order to deteriorate the SINR at E, we select the jamming relay with the following criterion as ∗ k k = argmax GE , (28) k∈Φ¯ where Φ¯ means the complement of Φ. For the sake of a fair comparison with TMRC and OSRS schemes, the total transmit power of all the relays is constrained to PR. It is assumed that αJ (0 ≤ αJ < 1) portion is utilized to transmit the jamming signals, then the SINR at E can be written as k J αiρ3GE γE,i = , (29) 1+ ρ4HE (1−αJ)PR αJPR k where i =1, 2, ρ3 = σ2 , ρ4 = σ2 , and HE = max GE . k∈Φ¯ We assume that both users are aware of the jamming signals, which means jamming signals do not influence the SINR at both users [32], [36], [37]. It must be noted that the special case of n = K or αJ =0, ODRS scheme reduces to OSRS scheme. Then the SOP with ODRS scheme is expressed as K−1 P ODRS n P J P OSRS, (30) out = Pr (|Φ| = ) Φn + ΦK n=0 X where P OSRS is given in (23). ΦK Similar to (27), the SOP of the cooperative NOMA system on condition that |Φ| = n (n J k s k s n PΦn = 1 − Pr Cs,1 > R1 ,Cs,2 > R2 k n α2ρ3G2 k 1+ k 1+ α ρ G 1+α1ρ3G = 1 − Pr ln 1 3 1 > Rs, ln 2 > Rs α ρ Gk 1 α ρ Gk 2 1+ 1 3 E 1+ 2 3 E 1+ρ4HE 1+ρ4HE (31) n wu 1 = 1 − Pr Gk >ℓ + θ Y,Gk >w + ,Y < , 1 1 2 1 − vY v ∆4 Gk where ℓ = θ1−1 , w = −| 1 , u = α2 , v = {zα1α2θ2ρ3 , and Y = E} . α1ρ3 α1ρ3 α1θ2−1 1−a1θ2 1+ρ4HE +1 September 4, 2019 DRAFT 14 To achieve the analytical expression of ∆4, the PDF of HE and Y are given in Lemma 1 and Lemma 2, respectively. Lemma 1: Given as Φ¯ = K − n (n m +1 m +1 −2 n E E λp p N (K−n−1)! E where SE = (n1, ··· , nm +1) ∈ np = K − n − 1 , A = +1 − , E mE (p−2)! p=1 Q (nq )! p=2 P q=1 Q mE +1 m E +1 B = np (p − 2), and C =1+ np. p=2 p=2 ProofP : See Appendix A. P Lemma 2: The PDF of Y is given by m −1 k E −λEy δe k+1 k k−1 fY (y)= ϕ0 ς+1 (ρ4λEy + Dy − Cky ), (33) k S j=0 (ρ4y + C) X=0 XE X mE j j k−ς (K−n)λE CkAρ4λE (ς−1)! where ϕ = , ς = B + mE + j, δ = , and D = CλE − ρ k + ρ ς. 0 Γ(mE ) k! 4 4 Proof : See Appendix B. Similar to (24), based on (12) and (13), we obtain the closed-form expression of ∆4 as wu 1 ∆ = Pr Gk >ℓ + θ Y,Gk >w + ,Y < 4 1 1 2 1 − vy v 1 v wu = Pr Gk >ℓ + θ Y,Gk >w + f (y) dy 1 1 2 1 − vy Y Z0 1 (34) v wu = 1 − F k (ℓ + θ1y) 1 − F k w + fY (y) dy G1 G2 1 − vy Z0 m −1 m −1 m −1 k U U E δλpλqΞ = e−λ1ℓ−λ2wϕ 1 2 3 , 0 p!q! p=0 q=0 k S j=0 X X X=0 XE X q 1 where Ξ3 = w h v ,p,q,λ1θ1 + λE,λ2wu,u,v,ℓ and a c k+1 k k−1 b u ρ4λEy + Dy − Cky h (a,b,c,f,r,u,v,ℓ)= (ℓ + θ1y) 1+ ς+1 fy+ r dy. 1 − vy ρ y C e 1−vy Z0 ( 4 + ) Similar to (17), utilizing Gaussian-Chebyshev quadrature, we have N c k+1 k k−1 a b u ρ4λEsj + Dsj − Cksj h (a,b,c,f,r,u,v,ℓ)= wi(ℓ + θ1sj) 1+ r , ς+1 fsj + − 2 1 − vsj 1 vsj i=1 (ρ4sj + C) e X a where sj = 2 (ti + 1). September 4, 2019 DRAFT 15 IV. ASYMPTOTIC SECRECY OUTAGE PROBABILITY ANALYSIS To obtain more insights, in this section we analyze the asymptotic SOP in the higher SINR regime. Similar to [33] and [34], we assume that Ω2 →∞, Ω1 = ε1Ω2 (ε1 > 1), and ΩR = ε2Ω2, where ε1 and ε2 are constant. k The asymptotic CDF of Gv (v ∈{1, 2}) can be expressed as [48] ∞ τU τU F k (x)= ϕvx + O (x ) , (35) Gv τU τU mU mU where ϕ1 = τU , ϕ2 = τU , and O (·) denotes higher order terms. Then the (ε1Ω2) (τU )! Ω2 (τU )! asymptotic SOP of cooperative NOMA systems is obtained as K ∞ n K−n mR(K−n) ∞ Pout = CKϕR η PΦn , (36) n=0 m X mR R ∞ where ϕR = m . The P for all the RS schemes is given as follows. (ε2Ω2) R (mR)! Φn A. Traditional Multiple Relays Combining Scheme Based on (15), we have the closed-form for SOP with TMRC scheme as a1 α P TMRC,∞ =1 − 1 − F ∞ (b + θ x) 1 − F ∞ c + 2 f (x) dx Φn G1 1 1 G2 1 d − e x GE Z0 1 1 τU k k τU −k CτU θ1 b1 Υ(k + τE,λEa1) =1 − FG (a1)+ ϕ1βE E k+τE k λE X=0 (37) α e + ϕ β cτU g a , τ , 0, 2 , 1 ,λ , 0, 0, τ 2 E 1 1 E c d d E U 1 1 1 θ α e − ϕ ϕ β bτU cτU g a , τ , 1 , 2 , 1 ,λ , 0, τ , τ , 1 2 E 1 1 1 E b c d d E U U 1 1 1 1 where Υ(·, ·) is the lower incomplete Gamma function, as defined by (8.350.1) of [47]. Remark 5 TMRC,∞ : From (37), one can easily obtain PΦn ≈ 1 − FGE (a1) when ϕi → 0 (i ∈{1, 2}) with Ω2 →∞. B. Optimal Single Relay Selection Scheme Based on (23), we obtain OSRS,∞ ∞ n PΦn = (1 − ∆1 ) , (38) September 4, 2019 DRAFT 16 ∞ where ∆1 is given as a2 ∞ ∞ ∞ ∆ = 1 − F m (δ (x)) 1 − F m (δ (x)) f m (x) dx 1 G1 3 G2 4 GE 0 Z mU Ck θkbmU −k k m ,λ a mU 1 2 Υ( + E E 2) = FG (a2) − ϕ3βE E k+mE k λE X=0 (39) α e − ϕ β cmU g a , m , 0, 2 , 2 ,λ , 0, 0, m 4 E 2 2 E c d d E U 2 2 2 θ α e + ϕ ϕ β bmU cmU g a , m , 1 , 2 , 2 ,λ , 0, m , m , 3 4 E 2 2 2 E b c d d E U U 2 2 2 2 mU mU mU mU OSRS,∞ n m m where ϕ3 = U and ϕ4 = U . Similarly, we obtain PΦn ≈ (1 − FGE (a2)) (ε1Ω2) (mU )! Ω2 (mU )! when ϕi → 0 (i ∈{1, 2}) with Ω2 →∞. C. Optimal Dual Relay Selection Scheme Based on (30), we obtain the asymptotic SOP for cooperative NOMA systems with ODRS scheme as K−1 P ODRS,∞ Cn ϕK−nηmR(K−n)P ODRS,∞ P OSRS,∞. (40) out = K R Φn + ΦK n=0 X Based on (31), we have ODRS,∞ ∞ n PΦn = (1 − ∆4 ) , (41) ∞ where ∆4 was shown as 1 v ∞ ∞ ∞ wu ∆4 = 1 − F k (ℓ + θ1y) 1 − F k w + fY (y) dy G1 G2 1 − vy Z0 λ mE −1 k − E mE −1 k δe v 1 =1 − ϕ − ϕ ϕ δh , mU , 0,λE, 0,u,v,ℓ 0 C ρ4 ς vk 0 3 v k S j=0 + v k S j=0 X=0 XE X X=0 XE X m −1 k E 1 − ϕ ϕ wmU δh , 0, m ,λ , 0,u,v,ℓ 0 4 v U E k S j=0 X=0 XE X m −1 k E 1 + ϕ ϕ ϕ wmU δh , m , m ,λ , 0,u,v,ℓ . 0 3 4 v U U E k=0 SE j=0 X X X (42) Similarly, we have n mE −1 k λ −ς ODRS,∞ −k − E ρ4 P ≈ ϕ δv e v C + , Φn 0 v k S j=0 ! X=0 XE X when ϕi → 0 (i ∈{1, 2}) with Ω2 →∞. September 4, 2019 DRAFT 17 ∞ Remark 6: Based on the definition of SDO as G = − lim ln(Pout) , we obtain the SDO for d ln(Ω2) Ω2→∞ TMRC OSRS ODRS the three RS schemes as Gd = Gd = Gd = 0 because the first term is independent of Ω2 in (37), (39), and (42), respectively. V. ASYMPTOTIC SECRECY OUTAGE PROBABILITY ANALYSIS WITH DYNAMIC POWER ALLOCATION SCHEME Since there is a ceiling for the SINR of U , which equals to α2 , there is a floor for the SOP 2 α1 of NOMA systems when the transmit SNR increases. A new DPA scheme was proposed in [33] and the PA coefficients are given as DPA 1 α = −̟ 1 1+µλ 2 , (43) −̟ DPA µλ2 α2 = −̟ 1+µλ2 DPA α2 where 0 <̟< 1 and µ> 1. It was testified that non-zero SDO can be obtained due to DPA = α1 −̟ µλ2 → 0 with Ω2 → ∞ in [33] and [34]. In this section, the asymptotic SOP is analyzed 2 −̟ − −̟− DPA (1+µλ2 ) θ2µλ2 θ2 DPA while this DPA scheme is utilized. Based on (43), we have ai = −̟ , bi = θ2µλ2 ρi − −̟ − ̟ 1+µλ −θ2 ρ ̟ 2 θ1−1 −̟ DPA 1+µλ2 DPA ( 2 ) i DPA θ2µλ2 ρi 1+ µλ , c = − , d = − 2 , e = − 3 , and i =1, 2. ρi 2 i ρi i ̟ i ̟ (1+µλ2 ) (1+µλ2 ) A. Traditional Multiple Relays Combining Scheme with Dynamic Power Allocation Scheme Substitute (43) into (37), we achieve the closed-form expression for asymptotic SOP under TMRC-DPA scheme as − τU Ck θk bDPA τU k k τ ,λ aDPA TMRC,∞,DPA DPA τU 1 1 Υ + E E 1 P =1 − FG a + ϕ1βE Φn E 1 λk+τE k=0 E X DPA DPA τ α e + ϕ β cDPA U g aDPA, τ , 0, 2 , 1 ,λ , 0, 0, τ 2 E 1 1 E cDPAdDPA dDPA E U 1 1 1 DPA DPA τ τ θ α e − ϕ ϕ β bDPA U cDPA U g aDPA, τ , 1 , 2 , 1 ,λ , 0, τ , τ . 1 2 E 1 1 1 E bDPA cDPAdDPA dDPA E U U 1 1 1 1 (44) Then the SDO of the cooperative NOMA systems in this case is obtained as TMRC,DPA Gd = min {mR (K − n)+ τU (1 − ̟)} 0≤n≤K = min {KmR − n (mR − mU (1 − ̟))} 0≤n≤K (45) KmU (1 − ̟), mR > mU (1 − ̟) = KmR, mR < mU (1 − ̟) = K min {mU (1 − ̟) , mR} . September 4, 2019 DRAFT 18 B. Optimal Single Relay Selection Scheme with Dynamic Power Allocation Scheme Similar to (38), we have n OSRS,∞,DPA ∞,DPA PΦn = 1 − ∆1 , (46) ∞,DPA where ∆1 is given by − mU Ck θk bDPA mU k k m ,λ aDPA ∞,DPA DPA mU 1 2 Υ + E E 2 ∆ = FE a − ϕ3βE 1 2 λk+mE k=0 E X DPA DPA m α e − ϕ β cDPA U g aDPA, m , 0, 2 , 2 ,λ , 0, 0, m 4 E 2 2 E cDPAdDPA dDPA E U 2 2 2 DPA DPA m m θ α e + ϕ ϕ β bDPA U cDPA U g aDPA, m , 1 , 2 , 2 ,λ , 0, m , m . 3 4 E 2 2 2 E bDPA cDPAdDPA dDPA E U U 2 2 2 2 (47) Then the SDO of the cooperative NOMA systems in this case is obtained as OSRS,DPA Gd = min {mR (K − n)+ nmD (1 − ̟)} 0≤n≤K (48) TMRC,DPA = Gd . Remark 7: Based on (45) and (48), one can observe the SDO of cooperative NOMA systems under both TMRC-DPA and OSRS-DPA schemes equals to the minimum SDO of dual hops. Moreover, it is a full SDO based on the results in [33] and [34]. C. Optimal Dual Relay Selection with Dynamic Power Allocation Scheme Similar to (40), we write the asymptotic SOP for the cooperative NOMA system under ODRS- DPA scheme as K−1 P ODRS,∞,DPA Cn ϕK−nηmR(K−n)P ODRS,∞,DPA P OSRS,∞,DPA. (49) out = K R Φn + ΦK n=0 X Similar to (41), we have n ODRS,∞,DPA ∞,DPA PΦn = 1 − ∆4 , (50) September 4, 2019 DRAFT 19 ∞,DPA where ∆4 is shown as mE −1 k λ −ς ∞,DPA − E ρ4 DPA −k ∆ =1 − ϕ δe vDPA C + v 4 0 vDPA k=0 SE j=0 X X X m −1 k E 1 − ϕ ϕ δh , m , 0,λ , 0,uDPA, vDPA,ℓDPA 0 3 vDPA U E k S j=0 X=0 XE X mE −1 k m 1 − ϕ ϕ wDPA U δh , 0, m ,λ , 0,uDPA, vDPA,ℓDPA 0 4 vDPA U E k=0 SE j=0 X X X mE −1 k m 1 + ϕ ϕ ϕ wDPA U δh , m , m ,λ , 0,uDPA, vDPA,ℓDPA , 0 3 4 vDPA U U E k=0 SE j=0 X X X (51) DPA DPA DPA DPA 1 DPA α1 α2 θ2ρ3 DPA α2 DPA θ1−1 where w = − DPA , v = DPA , u = DPA , and ℓ = DPA . α1 ρ3 1−α1 θ2 α1 θ2−1 α1 ρ3 Then the SDO of the cooperative NOMA systems in this case is obtained as ODRS,DPA Gd = min min {mR (K − n)+ nH1} ,K(1 − ̟)mD 0≤n≤K−1 = min min {KmR + n (H1 − mR)} ,K(1 − ̟)mD 0≤n≤K−1 min {KmD (1 − ̟)+ mR − H2, KmD(1 − ̟)} , mR >H1 = min {KmR, KmD(1 − ̟)} , mR KmR, mR VI. NUMERICAL RESULTS AND DISCUSSIONS In this section, we present numerical results and Monte-Carlo simulations to testify our th analysis. The main adopted parameters are set to PS = PR = P , R1 = 0.2 nat per channel th 2 use, R2 =0.1 nat per channel use, and σ =1. In all the figures, ‘Sim’ denotes the simulation September 4, 2019 DRAFT 20 100 10-1 10-2 10-3 10-4 -15 -10 -5 0 5 10 15 20 25 Fig. 2. SOP for various K with R1 = 0.1, R2 = 0.2, mU = mE = mR = 2, αJ = 0.5, Ω1 = 12 dB, ΩR = Ω2 = 10 dB, and ΩE = −5 dB. 100 10-1 10-2 -15 -10 -5 0 5 10 15 20 25 Fig. 3. SOP for various m with R1 = 0.1, R2 = 0.2, mU = mE = mR = m, α1 = 0.2, αJ = 0.5, Ω1 = 12 dB, ΩR = Ω2 = 10 dB, and ΩE = −5 dB. results1. One can observe that the analysis results match perfectly with simulation resutls and the secrecy performance with OSRS always outperforms that of TMRC and in some scenarios the ODRS scheme can obtain the optimal secrecy outage performance. Figs. 2 - 6 present the SOP for various ρ with FPA. One can easily observe that the secrecy performance of cooperative NOMA systems is enhanced and then becomes worse by improving the transmit SNR, which is similar to the conclusions in [33] and [34]. This is because the SINR of U2 has a ceiling in the higher-ρ region. It is demonstrated in Fig. 2 that the SOP is enhanced 1We omit the results for TSRS scheme since it is exactly the same as that for OSRS. September 4, 2019 DRAFT 21 100 Rs . 1 = 0 5 10-1 out P 10-2 ODRS Sim OSRS Sim Rs . 1 = 0 1 TMRC Sim Analysis -15 -10 -5 0 5 10 15 20 25 ρ (dB) s s Fig. 4. SOP for various R1 with R2 = 0.2, K = 2, mU = mE = mR = 2, α1 = 0.2, αJ = 0.5, Ω1 = 12 dB, ΩR = Ω2 = 10 dB, and ΩE = −5 dB. by increasing K due to the improved diversity of the cooperative NOMA systems. One can also find that the SOP depends merely on the RS scheme and is independent of K in the high-ρ region. Fig. 3 demonstrates that the SOP with larger m outperforms the ones with smaller m as larger m signifies that the quality of wireless channel is better. However, the SOP with larger m underperforms that with the smaller m in the high-ρ region. The reason is that the SOP of cooperative NOMA systems will benefit from the stronger fading over eavesdropping channels when the SINR of U2 approaches the upper bound. s s Figs. 4 and 5 present the effect on different R1 and R2, respectively. As we know, increasing the secrecy rate threshold deteriorates the secrecy outage performance of wireless communications. s Now, from Figs. 4 and 5, we can observe that R1 has significant effect on the SOP of NOMA s systems in low-ρ region. Different R1 has significant effect on the SOP of NOMA systems in low-ρ region but does not influence the SOP in high-ρ region. The opposite effect is observed s for R2. This proves that in the low-ρ region the bottleneck in secrecy performance of NOMA systems is due to U1. In high-ρ region, weaker user must be given more attention. One can also observe from Fig. 4 that the gap between two RS schemes in high-ρ region is larger than those s in low-ρ region and the gap in low-ρ region is decreased as R1 decreases. The impact of PA parameter on the SOP is investigated in Figs. 6 - 8. We can observe that the SOP decreases as α1 decreases. In other words, the secrecy performance is enhanced by enlarging the difference between allocated power to the two users, which is consistent with the results presented in Figs. 4 and 5. Fig. 8 presents the SOP for various αJ. One can observe September 4, 2019 DRAFT 22 100 10-1 Rs . 2 = 0 5 out P 10-2 Rs . 2 = 0 1 ODRS Sim OSRS Sim TMRC Sim Analysis 10-3 -15 -10 -5 0 5 10 15 20 25 ρ (dB) s s Fig. 5. SOP for various R2 with R1 = 0.2, K = 2, mU = mE = mR = 2, α1 = 0.2, αJ = 0.5, Ω1 = 12 dB, ΩR = Ω2 = 10 dB, and ΩE = −5 dB. 100 10-1 10-2 10-3 10-4 -15 -10 -5 0 5 10 15 20 25 Fig. 6. SOP for various α1 with R1 = 0.1, R2 = 0.2, K = 2, mU = mE = mR = 2, αJ = 0.5, Ω1 = 12 dB, ΩR = Ω2 = 10 dB, and ΩE = −10 dB. that the ODRS scheme enhances the secrecy outage performance of cooperative NOMA systems relative to OSRS scheme because the SNR at eavesdropper is degraded. Furthermore, the SOP is firstly improved with increasing αJ. However, it gets worse with increasing αJ since the transmit power is reduced. Obviously, there is an optimal αJ, which can be easily obtained via one dimensional search based on the results of our work. The SOP for various Ω2 with FPA is presented in Figs. 9 - 11. One can observe that the asymptotic SOP approaches the exact ones at high-Ω2 region. The SDO of the cooperative NOMA systems is zero since the SOP will be a constant as Ω2 increases. The results in Figs. 9 and 10 prove that increasing K and / or decreasing α1 enhances the SOP of cooperative NOMA September 4, 2019 DRAFT 23 100 10-1 10-2 10-3 10-4 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 Fig. 7. SOP for various α1 and K with R1 = 0.1, R2 = 0.2, mU = mE = mR = m, αJ = 0.5, Ω1 = 12 dB, ΩR = Ω2 = 10 dB, and ΩE = −5 dB. 10-1 10-2 10-3 10-4 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Fig. 8. SOP for various αJ and K and m with R1 = 0.1, R2 = 0.2, mU = mE = mR = m, α1 = 0.2, Ω1 = 12 dB, ΩR = Ω2 = 10 dB, and ΩE = −5 dB. systems. The results in Fig. 11 demonstrate that we can obtain better secrecy outage performance under a better channel condition for both TMRC and OSRS schemes. Figs. 12 - 14 present the SOP for various Ω2 under DPA scheme. One can observe that DPA scheme achieves a non-zero secrecy diversity order, which depends on the relay number, the fading parameters of S − Rk and Sk − Ui. Fig. 14 testifies that the SDO of cooperative NOMA systems equal to the minimum SDO of two hops. One can observe from Fig. 14 (c) that a single relay is utilized to send jamming signals influences the SDO of the cooperative NOMA system in some scenarios. September 4, 2019 DRAFT 24 100 10-2 10-4 10-6 10-8 -10 -5 0 5 10 15 20 25 30 35 Fig. 9. SOP for various α1 with R1 = 0.1, R2 = 0.2, K = 2, αJ = 0.5, P = 10 dB, mU = mE = mR = 2, ε1 = 1.5, ε2 = 2, and ΩE = −10 dB. 100 10-2 10-4 10-6 -10 -5 0 5 10 15 20 25 30 Fig. 10. SOP for various K with R1 = 0.1, R2 = 0.2, mU = mE = mR = 2, P = 15 dB, α1 = 0.2, αJ = 0.5, ε1 = 1.5, ε2 = 2, and ΩE = −12 dB. VII. CONCLUSION In this work, three RS schemes were proposed to enhance the secrecy performance of co- operative NOMA systems. We investigated the secrecy outage performance of these schemes under FPA and DPA considering the presence of dependence between the two NOMA users with different secrecy rate thresholds. The closed-form expressions for the exact and asymptotic SOP were obtained and verified by Monte Carlo simulation results. Moreover, the SDO for all the RS schemes were derived. The results demonstrate that relative to TMRC scheme, OSRS and ODRS schemes enhance the security of the cooperative NOMA systems. OSRS scheme achieves the same SDO as TMRC scheme. The jamming signal influences the SDO of ODRS September 4, 2019 DRAFT 25 100 10-2 10-4 10-6 10-8 -10 -5 0 5 10 15 20 25 30 35 Fig. 11. SOP for various m with R1 = 0.1, R2 = 0.2, mU = mE = mR = m, αJ = 0.3, K = 3, P = 10 dB, ε1 = 1.5, ε2 = 2, and ΩE = −10 dB. 100 10-2 10-4 10-6 10-8 10-10 0 5 10 15 20 25 30 35 40 45 50 55 Fig. 12. SOP with DPA scheme for various K with R1 = 0.1, R2 = 0.2, αJ = 0.5, P = 10 dB, mU = mE = mR = 2, = 1 5 = 2 = 0 1 = 5 Ω = −5 ε1 . , ε2 , ̟ . , µ , and E dB . scheme. There are optimal PA coefficients between jamming and signal transmissions, which can be easily obtained via one dimensional search based on the results of our work. APPENDIX A k The PDF of HE = max GE is obtained as k∈Φ¯ K−n−1 fHE (z)=(K − n)(FGE (z)) fGE (z) , (53) September 4, 2019 DRAFT 26 100 10-2 10-4 10-6 10-8 10-10 0 5 10 15 20 25 30 35 40 45 50 55 Fig. 13. SOP with DPA scheme for various µ with R1 = 0.1, R2 = 0.2, αJ = 0.5, P = 10 dB, mU = mE = mR = 2, K = 3, ε1 = 1.5, ε2 = 2, ̟ = 0.2, and ΩE = −5 dB. 0 100 100 10 ̟ = 0.1 ̟ = 0.1 Analysis ̟ = 0.3 ̟ = 0.3 Asymptotic 10-5 m m U = 1 U = 1 mU = 1 mU = 3 mR = 3 -5 m -5 10 R = 3 10 mR = 3 mR = 2 10-10 mU = 3 mU = 3 out out out m mR = 1 m U = 3 P P R = 1 P mR = 3 10-15 10-10 10-10 10-20 10-25 25 30 35 40 45 50 25 30 35 40 45 50 25 30 35 40 45 50 Ω2 (dB) Ω2 (dB) Ω2 (dB) (a) SOP with TMRC scheme (b) SOP with OSRS scheme (c) SOP with ODRS scheme with ̟ = 0.15 Fig. 14. SOP for DPA scheme with R1 = 0.1, R2 = 0.2, αJ = 0.5, K = 3, mE = 2, ε1 = 1.5, ε2 = 2, µ = 8, and ΩE = −5 dB. k where FGE (z) and fGE (z) refer to the CDF and PDF, respectively, of GE, which is easily obtained by utilizing (11). Based on multi-nomial theorem, we obtain the pdf of HE as (K − n)λmE f (z)= E AzB+mE −1e−CλE z, (54) HE Γ(m ) E S XE m +1 m +1 −2 n E E λp p N (K−n−1)! E where SE = (n1, ··· , nm +1) ∈ np = K − n − 1 , A = +1 − , E mE (p−2)! p=1 Q (nq )! p=2 P q=1 Q mE +1 m E +1 B = np (p − 2), and C =1+ np. p=2 p=2 P P September 4, 2019 DRAFT 27 APPENDIX B Based on (32) and utilizing (1.111) and (3.351.3) of [47], the CDF of Y can be obtained as Gk F (y)=Pr E REFERENCES [1] L. Dai, B. Wang, Y. Yuan, S. Han, C.-L. I, and Z. Wang, “Non-orthogonal multiple access for 5G: solutions, challenges, opportunities, and future research trends,” IEEE Commun. Mag., vol. 53, no. 9, pp. 74-81, Sept. 2015. [2] Y. Wang, B. Ren, S. Sun, S. Kang and X. Yue, “Analysis of non-orthogonal multiple access for 5G,” China Communications, vol. 13, no. Supplement2, pp. 52-66, N/A 2016. [3] Z. Ding, Y. Liu, J. Choi, Q. Sun, M. Elkashlan, I. C. L, and H. V. Poor, “Application of non-orthogonal multiple access in LTE and 5G networks,” IEEE Commun. Mag., vol. 55, no. 2, pp. 185-191, Feb. 2017. [4] S. M. R. Islam, N. Avazov, O. A. Dobre, and K. S. Kwak, “Power-domain non-orthogonal multiple access (NOMA) in 5G systems: Potentials and challenges,” IEEE Commun. Surveys Tuts., vol. 19, no. 2, pp. 721 - 742, 2nd Quart. 2017. [5] Y. Liu, Z. Qin, M. Elkashlan, Z. Ding, A. Nallanathan, and L. Hanzo, “Nonorthogonal multiple access for 5G and beyond,” Proc. IEEE, vol. 105, no. 12, pp. 2347-2381, Dec. 2017. [6] Z. Ding, X. Lei, G. K. Karagiannidis, R. Schober, J. H. Yuan, and V. K. Bhargava, “A survey on non-orthogonal multiple access for 5G networks: Research challenges and future trends,” IEEE J. Sel. Areas Commun., vol. 35, no. 10, pp. 2181-2195, Oct. 2017. [7] Y. Liu, H. Xing, C. Pan, A. Nallanathan, M. Elkashlan, and L. Hanzo, “Multiple antenna assisted non-orthogonal multiple access,” IEEE Wireless Commun., vol. 25, no. 2, pp. 17 - 23, Apr. 2018. [8] D. Wan, M. Wen, F. Ji, H. Yu, and F. Chen, “Non-orthogonal multiple access for cooperative communications: Challenges, opportunities, and trends,” IEEE Wireless Commun., vol. 25, no. 2, pp. 109-117, Apr. 2018. [9] F. Zhou, Y. Wu, Y.-C. Liang, Z. Li, Y. Wang, and K.-K. Wong, “State of the art, taxonomy, and open issues on cognitive radio networks with NOMA,” IEEE Wireless Commun., vol. 25, no. 2, pp. 100-108, Apr. 2018. September 4, 2019 DRAFT 28 [10] Z. Ding, M. Peng, and H. V. Poor, “Cooperative non-orthogonal multiple access in 5G systems,” IEEE Commun. Lett., vol. 19, no. 8, pp. 1462-1465, Aug. 2015. [11] Y. Liu, Z. Ding, M. Elkashlan, and H. V. Poor, “Cooperative non-orthogonal multiple access with simultaneous wireless information and power transfer,” IEEE J. Sel. Areas Commun., vol. 34, no. 4, pp. 938-953, Apr. 2016. [12] J. Men and J. Ge, “Non-orthogonal multiple access for multiple-antenna relaying networks,” IEEE Commun. Lett., vol. 19, no. 10, pp. 1686-1689, Oct. 2015. [13] J. Men, J. Ge, and C. Zhang, “Performance analysis of non-orthogonal multiple access for relaying networks over Nakagami- m fading channels,” IEEE Trans. Veh. Technol., vol. 66, no. 2, pp. 1200-1208, Feb. 2017. [14] Y. Xiao, L. Hao, Z. Ma, Z. Ding, Z. Zhang, and P. Fan, “Forwarding strategy selection in dual-hop NOMA relaying systems,” IEEE Commun. Lett., vol. 22, no. 8, pp. 1644-1647, Aug. 2018. [15] W. Duan, M. Wen, Z. Xiong, and M. H. Lee, “Two-stage power allocation for dual-hop relaying systems with non- orthogonal multiple access,” IEEE Access, vol. 5, pp. 2254-2261, Mar. 2017. [16] C. Zhong and Z. Zhang, “Non-Orthogonal multiple access with cooperative full-duplex relaying,” IEEE Commun. Lett., vol. 20, no. 12, pp. 2478-2481, Dec. 2016. [17] Z. Zhang, Z. Ma, M. Xiao, Z. Ding, and P. Fan, “Full-duplex device-to-device aided cooperative non-orthogonal multiple access,” IEEE Trans. Veh. Technol., vol. 66, no. 5, pp. 4467 - 4471, May 2016. [18] G. Liu, X. Chen, Z. Ding, Z. Ma, and F. R. Yu, “Hybrid half-duplex/full-duplex cooperative non-orthogonal multiple access with transmit power adaptation,” IEEE Trans. Wireless Commun., vol. 17, no. 1, pp. 506-519, Jan. 2018. [19] T. M. C. Chu and H.-J. Zepernick, “Performance of a non-orthogonal multiple access system with full-duplex relaying,” IEEE Commun. Lett., vol. 22, no. 10, pp. 2084-2087, Oct. 2018. [20] M. F. Kader, S. Y. Shin, and V. C. M. Leung, “Full-duplex non-orthogonal multiple access in cooperative relay sharing for 5G systems,” IEEE Trans. Veh. Technol., vol. 67, no. 6, pp. 5831-5840, Jul. 2018. [21] Y. Zou, X. Wang, and W. Shen, “Optimal relay selection for physical-layer security in cooperative wireless networks,” IEEE J. Sel. Areas Commun., vol. 31, no. 10, pp. 2099-2111, Oct. 2013. [22] H. Lei, H. Zhang, I. S. Ansari, Z. Ren, G. Pan, K. A. Qaraqe, and M.-S. Alouini, “On secrecy outage of relay selection in underlay cognitive radio networks over Nakagami-m fading channel,” IEEE Trans. Cogn. Commun. Netw., vol. 3, no. 4, pp. 614 - 627, Dec. 2017. [23] Z. Ding, H. Dai, and H. V. Poor, “Relay selection for cooperative NOMA,” IEEE Wireless Commun. Lett., vol. 5, no. 4, pp. 416-419, Aug. 2016. [24] J. Zhao, Z. Ding, P. Fan, Z. Yang, and G. K. Karagiannidis, “Dual relay selection for cooperative NOMA with distributed space time coding,” IEEE Access, vol. 6, pp. 20440-20450, Apr. 2018. [25] Z. Yang, Z. Ding, Y. Wu, and P. Fan, “Novel relay selection strategies for cooperative NOMA,” IEEE Trans. Veh. Technol., vol. 66, no. 11, pp. 10114-10123, Nov. 2017. [26] P. Xu, Z. Yang, Z. Ding, and Z. Zhang, “Optimal relay selection schemes for cooperative NOMA,” IEEE Trans. Veh. Technol., vol. 67, no. 8, pp. 7851-7855, Aug. 2018. [27] X. Yue, Y. Liu, S. Kang, A. Nallanathan, and Z. Ding, “Spatially random relay selection for full/half-duplex cooperative NOMA networks,” IEEE Trans. Commun., vol. 66, no. 8, pp. 3294 - 3308, Aug. 2018. [28] J. Chen, L. Yang, and M.-S. Alouini, “Performance analysis of cooperative NOMA schemes in spatially random relaying networks,” IEEE Access, vol. 6, pp. 33159-33168, Jul. 2018. [29] Y. Zou, J. Zhu, X. Wang, and L. Hanzo, “A survey on wireless security: Technical challenges, recent advances, and future trends,” Proc. IEEE, vol. 104, no. 9, pp. 1727-1765, May 2016. [30] H.-M. Wang and T.-X. Zheng, Physical Layer Security in Random Cellular Networks. Singapore: Springer, 2016. September 4, 2019 DRAFT 29 [31] Y. Zhang, H. Wang, Q. Yang, and Z. Ding, “Secrecy sum rate maximization in non-orthogonal multiple access,” IEEE Commun. Lett., vol. 20, no. 5, pp. 930-933, May. 2016. [32] Y. Liu, Z. Qin, M. Elkashlan, Y. Gao, and L. Hanzo, “Enhancing the physical layer security of non-orthogonal multiple access in large-scale networks,”IEEE Trans. Wireless Commun., vol. 16, no. 3, pp. 1656-1672, Mar. 2017. [33] H. Lei, J. Zhang, K. Park, P. Xu, I. S. Ansari, G. Pan, B. Alomair, and M.-S. Alouini, “On secure NOMA systems with transmit antenna selection schemes,”IEEE Access, vol. 5, pp. 17450-17464, Sep. 2017. [34] H. Lei, J. Zhang, K.-H. Park, P. Xu, Z. Zhang, G. Pan, and M.-S. Alouini, “Secrecy outage of max-min TAS scheme in MIMO-NOMA systems,”IEEE Trans. Veh. Technol., vol. 67, no. 8, pp. 6981-6990, Aug. 2018. [35] Y. Feng, S. Yan, and Z. Yang. “Secure transmission to the stronger user with optimal power allocation in NOMA,” arXiv:1807.00935, Jul. 2018, [Online]: https://arxiv.org/abs/1807.00935 [36] L. Lv, Z. Ding, Q. Ni, and J. Chen, “Secure MISO-NOMA transmission with artificial noise,” IEEE Trans. Veh. Technol., vol. 67, no. 7, pp. 6700-6705, Jul. 2018. [37] F. Zhou, Z. Chu, H. Sun, R. Q. Hu, and L. Hanzo, “Artificial noise aided secure cognitive beamforming for cooperative MISO-NOMA using SWIPT,” IEEE J. Sel. Areas Commun., vol. 36, no. 4, pp. 918-931, Apr. 2018. [38] Z. Zhu, Z. Chu, F. Zhou, H. Niu, Z. Wang, and I. Lee, “Secure beamforming designs for secrecy MIMO SWIPT systems,” IEEE Wireless Commun. Lett., vol. 7, no. 3, pp. 424-427, Jun. 2018. [39] K. Jiang, T. Jing, Y. Huo, F. Zhang, and Z. Li, “SIC-based secrecy performance in uplink NOMA multi-eavesdropper wiretap channels,” IEEE Access, vol. 6, pp. 19664-19680, Apr. 2018. [40] H. Zhang, N. Yang, K. Long, M. Pan, G. K. Karagiannidis, and V. C. M. Leung, “Secure communications in NOMA system: Subcarrier assignment and power allocation,” IEEE J. Sel. Areas Commun., vol. 36, no. 7, pp. 1441-1452, Jul. 2018. [41] J. Chen, L. Yang, and M. Alouini, “Physical layer security for cooperative NOMA systems,”IEEE Trans. Veh. Technol., vol. 67, no. 5, pp. 4645-4649, May. 2018. [42] W. Jiang, Y. Gong, Q. Xiao, and Y. Liao, “Rate maximization for untrusted relay networks with nonorthogonal cooperative transmission protocols,” IEEE Trans. Veh. Technol., vol. 67, no. 7, pp. 6325-6339, Jul. 2018. [43] A. Arafa, W. Shin, M. Vaezi, and H. V. Poor. “Secure relaying in non-orthogonal multiple access: Trusted and untrusted scenarios,” arXiv:1808.07864, Aug. 2018, [Online]: https://arxiv.org/abs/1808.07864 [44] Y. Feng, Z. Yang, and S. Yan, “Non-orthogonal multiple access and artificial-noise aided secure transmission in FD relay networks,” in Proc. 2017 IEEE Globecom Workshops (GC Wkshps), Singapore, Dec. 2017, pp. 1-6. [45] C. Liu, L. Zhang, M. Xiao, Z. Chen, and S. Li, “Secrecy performance analysis in downlink NOMA systems with cooperative full-duplex relaying,” in Proc. 2018 IEEE International Conference on Communications Workshops (ICC Workshops), Kansas City, MO, USA, Jul. 2018, pp. 1-6. [46] B. Li, X. Qi, K. Huang, Z. Fei, F. Zhou, and R. Q. Hu, “Security-reliability tradeoff analysis for cooperative NOMA in cognitive radio networks,” IEEE Trans. Commun., doi: 10.1109/tcomm.2018.2873690, Oct. 2018. [47] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products: Academic press, 2014. [48] H. Lei, C. Gao, I. S. Ansari, Y. Guo, Y. Zou, G. Pan, and K. Qaraqe, “Secrecy outage performance of transmit antenna selection for MIMO underlay cognitive radio systems over Nakagami-m channels,” IEEE Trans. Veh. Technol., vol. 66, no. 3, pp. 2237-2250, Mar. 2017. [49] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th. New York, NY, USA: Dover Press, 1972. September 4, 2019 DRAFTH2 = min {Km , (K − 1) (m (1 − ̟) − ̟)+ m ,Km (1 − ̟)} R D R D
H2 = min {KmR, (K − 1) (mD (1 − ̟) − ̟)+ mR, KmD(1 − ̟)} , where H1 = mD (1 − ̟) − ̟ and H2 = mD (1 − ̟)+ ̟ (K − 1). Remark 8: Based on (52), one can realize that using a single relay to send jamming signals may reduce the SDO of cooperative NOMA systems in some conditions.