Performance Analysis of Cooperative NOMA Schemes in Spatially Random Relaying Networks

Transcription

1 Performance Analysis of Cooperative NOMA Schemes in Spatially Random Relaying Networks Item Type Article Authors Chen, Jianchao; Yang, Liang; Alouini, Mohamed-Slim Citation Chen J, Yang L, Alouini M-S 08 Performance Analysis of Cooperative NOMA Schemes in Spatially Random Relaying Networks. IEEE Access 6: Available: dx.doi.org/0.09/access Eprint version Publisher's Version/PDF DOI 0.09/ACCESS Publisher Institute of Electrical and Electronics Engineers IEEE Journal IEEE Access Rights This is under the open access policy. Download date /0/09 0::34 Link to Item

2 Received May 4, 08, accepted June 8, 08, date of publication June 3, 08, date of current version July 6, 08. Digital Object Identifier 0.09/ACCESS Performance Analysis of Cooperative NOMA Schemes in Spatially Random Relaying Networks JIANCHAO CHEN, LIANG YANG, AND MOHAMED-SLIM ALOUINI, Fellow, IEEE School of Information Engineering, Guangdong University of Technology, Guangzhou 50006, China CEMSE Division, KAUST, Thuwal 386, Saudi Arabia Corresponding author: Liang Yang This work was supported in part by the National Natural Science Foundation of China under Grant 66760, in part by the Department of Education of Guangdong Province under Grant 06KZDXM050, and in part by the Open Research Fund of National Mobile Communications Research Laboratory, Southeast University, under Grant 08D0. ABSTRACT In this paper, we propose a general framework to investigate cooperative non-orthogonal multiple access NOMA using two-stage relay selection TSRS in spatially random relaying networks. More specifically, we consider both amplify-and-forward and decode-and-forward protocols and compare the performance between them. From practical consideration, we adopt a stochastic geometry-based model and assume that the spatial topology of relays is modeled by using homogeneous Poisson point process PPP. Based on such a setting, an effective coverage area of the relays modeled by using homogeneous PPP in cooperative NOMA systems is developed and performance comparison between TSRS and the conventional max min RS scheme is also presented. According to the locations of the NOMA users, we develop the complete strategies for calculating the effective coverage area of the relays. Furthermore, in the high signal-to-noise ratio regime, asymptotic expressions are provided to show that the outage probability tends to a constant which is only related to the density of homogeneous PPP and the effective coverage area of the relays. For a given outage probability, we reveal the relationship between the shortest and longest radii of the effective district of the relays. Finally, Monte Carlo simulations are provided to verify the accuracy of the analytical results. INDEX TERMS Amplify-and-forward AF, decode-and-forward DF, homogeneous Poisson point process PPP, non-orthogonal multiple access NOMA, two-stage relay selection TSRS. I. INTRODUCTION Recently, non-orthogonal multiple access NOMA as one of the promising key techniques of fifth generation 5G wireless networks has attracted significant attention ] 3]. NOMA exploits the power domain to implement multipleaccess schemes, which is different from the conventional orthogonal multiple access OMA schemes, such as frequency division multiple access FDMA. Comparing to the traditional OMA techniques, the advantage of NOMA system is that the multiple receivers sharing a same resource slot e.g., time/frequency can be allocated and served by power resource blocks, which increases the spectrum resources serving the receivers. Due to its superior spectral efficiency, NOMA systems have been proved that it is capable of combining with many wireless communication techniques and enhancing the system performance. For example, in a multi-user environment, the technique of cooperative transmission, an important application adopted by NOMA, can form a virtual multiple-input multiple-output MIMO system to transmit data cooperatively. Therefore, the technique of cooperative transmission for NOMA can enhance the communication reliability for the users who are in poor channels 4]. Generally, cooperative NOMA techniques can be categorized as two types. Due to the application of successive interference cancellation SIC at the near user, the near user knows the prerequisite for decoding the far user. Hence, one category is that the near user can act as an assisted relay forwarding signals to the far user. Do et al. 5] studied the outage performance of relaying cooperative NOMA systems with simultaneous wireless information and power transfer SWIPT technique at the near users to power their relaying operations. Another general application of the cooperative NOMA technique is exploiting dedicated relays to process cooperative transmission 6], 7]. Liang et al. 6] studied a cooperative NOMA scenario with the help of an VOLUME 6, IEEE. Translations and content mining are permitted for academic research only. Personal use is also permitted, but republication/redistribution requires IEEE permission. See for more information. 3359

3 amplify-and-forward AF relay and derived an approximate outage probability expression. Liau et al. 7] combined a virtual full-duplex relaying technique with cooperative NOMA systems, where the operation of full-duplex relay using the successive relaying SR technique was achieved by using two half-duplex AF relays. Since more relays bring higher diversity gains, applying multi-relay network to NOMA techniques has been considered in 8] 0]. Specifically, Ding et al. 8] analyzed the outage performance of a decodeand-forward DF relaying NOMA system and proposed a two-stage relay selection TSRS method. In 9], an optimal joint user and relay selection algorithm in cooperative NOMA networks was proposed to achieve the transmissions from multiple users to two destinations via multiple AF relays. In 0], two kinds of two-stage DF and AF relay selection schemes for cooperative NOMA have been proposed and their results show that the performance of the proposed schemes in 0] outperforms the scheme proposed in 8], but it cannot achieve the optimal outage performance. More recently, some stochastic geometry models have been considered in NOMA systems. For instance, in ], to improve the security of a random network, the physical layer security of NOMA systems in large-scale networks was investigated and a protected zone around the source node was adopted. Applying random beamforming to mmwave- NOMA systems was proposed in ], where a stochastic geometry model was used for characterizing the performance of transmission. Zhou et al. 3] proposed a dynamic DFbased cooperative NOMA strategy for downlink transmission with spatially random users. A pair of RS schemes for either full-duplex or half-duplex NOMA networks was investigated in 4], where the locations of the relays in the network were modeled by a stochastic geometry. As mentioned above, although the cooperative NOMA has been extensively considered in the literature, the cooperative NOMA systems with randomly roaming relays are still lack of sufficient consideration. In particular, the results in 8] 0] are only applicable to the static models where the locations of the relays are assumed to be static. Liu et al. ], Ding et al. ], and Zhou et al. 3], assumed that the users in their considered NOMA systems are spatially random. To the best of the authors knowledge, few works have used the stochastic geometry model to model the locations of the multiple relays in cooperative NOMA networks. We note that Yue et al. 4] have proposed a model that the relays are uniformly distributed in a given area. However, the effective coverage district of the spatially random relays in NOMA networks is still not reported in the literature. In practical scenarios, it is unrealistic to assume that the coverage district of the relays is fixed when the locations of NOMA users are undefined. This is because the relays are generally deployed between the source and the destination. Otherwise this relaying model makes no sense. Hence, it is necessary to consider a dynamic coverage district of spatially random relays related to the locations of the users. Motivated by this, in this work, we investigate the performance of cooperative NOMA systems with a random number of relays randomly deployed in a two dimensional coverage area determined by the locations of the NOMA users, where the relays are modeled by a homogeneous poisson point process PPP. With this system model, our analytical results can be applicable to some more practical cases. The main contributions of this work can be summarized as follows: A closed-form approximation for the outage probability is derived, which is shown to closely match with the simulation results at high signal-to-noise ratio SNR. The performance comparison between the conventional max-min RS scheme and TSRS is also provided. It shows that the outage performance of TSRS outperforms that of the max-min RS scheme. Moreover, an asymptotic analysis is provided and it is shown that an error floor exists in the AF/DF-based outage probability performance and the zero diversity happens. To obtain an effective district of the relays, we propose the constraints related to the radius of the relays in cellular networks, which depends on the locations of the users. Additionally, we present comprehensive analysis for the areas of the effective district of the relays and further plot the effective district of the spatially random relays to imitate the practical scenario. Finally, our results show that increasing the density of PPP and the acreage of the relaying zone results in improving the outage probability significantly due to the enlargement of the average number of the relays. Since noises at the first time slot are amplified by AF-based relays, we show that the DF-based outage performance is superior to the AF mode. In order to obtain more insights, we analyze the performance of cooperative NOMA systems at high SNR regimes. It is observed that the impact of the channels between the relays and the poor user on the system performance can almost be omitted at high SNRs. II. SYSTEM AND CHANNEL MODELS Consider a NOMA system including one base station BS, a random number of relays, and two users UE and UE, which are equipped with single antennas. From practical consideration, we adopt the stochastic geometry-based model and assume that the locations of the relays are modeled by homogeneous PPP, denoted by with density λ. As shown in Fig., our communication networks can be illustrated as a coordinate system, where the BS is the center of the circle with radius R D, the relays are assumed to be distributed in this circle and are located in x, y, and UE and UE are respectively located in x, y and x, y, which are out of the circle. In addition, in order to ensure the channels between the relays and the NOMA users without being impaired by path loss due to obstacles, it is assumed that the distance from the BS to the relay should be larger than a secure radius R d. We further assume that there are no direct paths between BS and the two users due to the significant path loss between them. Also, due to the long distance between the BS and user, 3360 VOLUME 6, 08

4 FIGURE. Illustration of a NOMA-based network including one BS, randomly distributed relays, and two NOMA users. it can be viewed as an outage event if the distance from the relay to the user is larger than that of user s direct link, which can be formulated as the following conditions: x + y = r, x + y = r, x + y = r, R d r R D, SE = x x + y y x + y, x x + y y x + y, where SE denotes the effective area of the relays, r, r and r are the radiuses of the relay, UE and UE, respectively. For multi-relay NOMA systems, obtaining perfect channel state information CSI of the users results in high system overheads. However, in this paper, UE can be served for small packet transmission and UE can be served opportunistically 5] because the user ordering is only affected by the users quality of service QoS requirements instead of the knowledge of CSI, which means that the relays do not need to acquire the perfect CSI of the users. The small-scale fading coefficients of BS-relay n R n, R n -UE and R n -UE channels are denoted by h n, g n,, and g n,, respectively, which are assumed to be subject to independent Rayleigh fading. During the first time slot, the source broadcasts the superimposed mixture, α s + α s, to the relays, where s i i =, is the unit power signal received by user i and α i are the power allocation coefficients. To fulfill UE s QoS requirements, we set that α α and define the power allocation coefficients satisfy the equation that α + α = 5]. Therefore, the received signal at the relay of the set can be expressed as y n,r = Ld n,0 h n α s + α s P s + ω n,r, where n, Ld n,0 = d α n,0 is the path loss, d n,0 = r is the distance from BS to R n, α is the path-loss factor usually satisfying α 4, P s is the transmit power of BS, and ω n,r is the additive white Gaussian noise. In order to investigate the impacts of the relaying protocols on the performance of NOMA systems, the analytical frameworks for both AF and DF are depicted as the following subsections. A. AMPLIFY-AND-FORWARD During the second time slot, the R n amplifies and forwards its received signals to the two users. Therefore, the signal received at user i is y AF n,i = βy n,r g n,i + ω n,i, 3 where β is the amplifying factor given by β = P n P s dn,0 α h 6], ω n +N n,i denotes the additive Gaussian noise 0 with zero mean and variance N 0, P n is the transmit power of R n. For simplifying the calculations, we assume that P n = P s. Then, β = dn,0 α h n + ρ, where ρ = P s /N 0 = P n /N 0 denotes the average SNR. The conditions for UE and UE to decode s and s are given by, respectively, AF log + γn, R AF, log + Zn, R, 4 where γ AF n, = g n, dn,0 α h n α g n, dn,0 α h n α + ρ g n, + ρ d α n,0 h n + ρ the signal-to-interference-plus-noise ratio SINR, Z AF ρd α n,0 h n g n, α g n, +d α n,0 h n + ρ denotes n, = denotes the SNR, and R i is the targeted data rate for user i. UE decodes its signal with γn, AF. On the other hand, UE decodes its own signal with Zn, AF under the condition log + γ AF n, R, where γn, AF = g n, d α n,0 h n α. g n, dn,0 α h n α + ρ g n, + ρ d α n,0 h n + ρ B. DECODE-AND-FORWARD For DF scheme, the R n firstly decodes its received superimposed message from BS and then re-encodes and sends it to the destination. Then, the received signals at UEi can be expressed as y DF n,i = g n,i α s + α s P n + ω n,i, 5 Combining and 5, the condition that UEi decodes s and UE decodes s can be expressed as DF log + γ where γn,i DF = min min n,i R, d α n,0 h n α, dn,0 α h n α + ρ ρd α n,0 h n α, ρ g n, α. DF log + Z n, R, 6, Zn, DF = g n,i α g n,i α + ρ C. RELAY SELECTION STRATEGIES Notice that the relay selection scheme can significantly improve the performance of NOMA systems. In the next, we present the relay selection strategies as follows: TSRS STRATEGY 8] According to the QoS requirements of NOMA users, the optimal RS strategy can be expressed as ϑ = max n, log + γ j n, R or log + γ j n, R or log + Z j n, R. 7 VOLUME 6,

5 where j = AF/DF. The main motivation of this relay selection scheme is to achieve two targets simultaneously. First we should ensure UE s targeted data rate can be achieved, and another one is to serve UE with a rate as large as possible. MAX-MIN RELAY SELECTION STRATEGY 7] This traditional relay selection case can be written as max min Z j n,, Z j n,, n S r. 8 where S r = log + γ j n, R, log + γ j n, R, n, r SE. As shown in the simulation results in Section IV, TSRS has a better system performance than the max-min relay selection scheme, which is in agreement with the result in 8]. III. PERFORMANCE ANALYSIS A. OUTAGE PROBABILITY ANALYSIS In this section, we derive the outage probability expressions for our considered NOMA systems. The overall outage event ϑ can be defined as ϑ = ϑ ϑ, 9 where ϑ denotes the event that s cannot be decoded by either of the two users successfully, i.e., ϑ = γ j n, <f R ] γ j n, <f R ], n, 0 where f R = R. The term ϑ can be described as the event that the UE cannot decode s correctly even though s can be decoded successfully by the two users, i.e., ϑ = Z j n, < f R ] γ j n, f R ] γ j n, f R ], n. Therefore, the overall outage probability can be expressed as P out = Pr ϑ + Pr ϑ. B. AMPLIFY-AND-FORWARD We assume that the locations of the relays in a circle following the PPPs and the channels of the relays are ordered. Then, according to 0, we have Pr ϑ = E Fϑ AF R, 3 n, where Fϑ AF R = Prmin γn, AF, γ n, AF >f R. Remark : We note that F ϑ R is intractable because the considered random events, γ j n, and γ AF n,, are correlated. Therefore, a closed-form lower bound on Fϑ AF R in the high SNR can be calculated as d α Fϑ AF n,0 R Pr max h n, g n, < T, d α n,0 max h n, g n, <, 4 T where T = R ρα α R ]. Assume that h n, g n, and g n, are independent Rayleigh distributions with parameters λ n,0, λ n,, and λ n,, respectively. Note that α > α R is required. Otherwise, P ϑ =. With above expressions and using the method derived in 8], we can obtain a lower bound on Pr ϑ as Pr ϑ E e ωt 5 n, where ω = d α n,0 λ n,0 + λ n, + λ n,. On the other hand, Pr ϑ can be calculated as Pr ϑ = PrE, S r > 0, 6 where E denotes the event that s cannot be decoded by UE, S r is the size of subset S r. Then, 6 can be rewritten as PrE, S r > 0 = Pr log + ρd α n,0 h n g n, α g n, +d α n,0 h n + ρ < R, S r >0. 7 and Assuming S r > 0 and defining x n = log + ρd α n,0 h n g n, α g n, + d α n,0 h n + ρ, 8 we have x n = max x n, n S r, 9 Pr ϑ = Prx n < R, S r > 0 ] = E Prx n < R S r = l Pr S r = l 0 l= Similar to 4, the cumulative distribution function CDF of x n can be derived as Fx = Pr Zn, AF < x n S r, S r =0 d α n,0 > Pr max h n, g n, > max d α n,0 ξ h n, g n, < T = Q + Q, where ξ = x ρα. In, Q can be expressed as d α n,0 Pr h n > g n,, dα n,0 h n > dn,0 α ξ h n < T, g n, < T Pr g n, > max hn dn,0 α, T, T < h n dn,0 α < ξ = Pr h n > dn,0 α T, g n, > T = λ n,0d α n,0 eηt η e ηt e ηξ. 336 VOLUME 6, 08

6 where η = λ n,0 dn,0 α +λ n,. Similar to, Q can be obtained as λ n,e ηt η e ηt e ηξ. Therefore, the CDF expression in is given by Fx = e ηξ T. 3 With 0, Pr ϑ can be evaluated as l Pr ϑ = E FR Pr S r = l, 4 l= where Pr S r = l = l l n=, F ϑ R n=, n= l+, Therefore, 5 can be further approximated as Pr S r = l l l e ωt n=, Fϑ R ]. 5 n= l+, e ωt. 6 With, 5, 4, and 6, the overall outage probability is given in the following lemma. Lemma : Assuming that the relays follow the homogeneous PPP and the effective zone of the relays is a circle with radius R D, the overall outage probability of an AF NOMA system with the TSRS method can be written as P out = Pr ϑ + Pr ϑ l E ] e ωt e ωt FR l l=0 n=, n= l+, = E e ε ], 7 n, where ε = ηt + λ n, T, T ρα. From 7, it is clearly shown that e ε is the outage probability for an arbitrary relay case, which verifies the correctness of the TSRS scheme. By taking similar steps in 9] and applying the generating function 0], 7 can be rewritten as P out = exp λ e ε ] rdr = exp λ S π 0 = R Rmin R d ] re ε dr, 8 where S is the selected area according to and θ = arctany/x denotes the angel of the relay. Lemma : To obtain the area of the effective relays, we need to solve the constraint in. Hence, a condition of r can be obtained as R d r R min, 9 where R min = minr D, x x + y y, x x + y y. Then, we have R min = minr D, r cosθ θ, r cos θ θ by using the polar coordinate transform, where θ i = arctany i /x i denotes the angel of UEi. Proof : The proof is given in APPENDIX A. To obtain the area of the effective relays, we divide this area into three parts. More specifically, we describe these three parts in the following three remarks, respectively. Remark : Firstly, with the assumption R d R min, the valid area of the relays can be ensured according to the interval of θ. Therefore, the available range of the relays can be written as θ 0, θ a π θ b, π, 30 where θ a = minarccosr d /r + θ, arccosr d /r + θ and θ b = minarccosr d /r θ, arccosr d /r θ. Notice that θ a + θ b π. Proof : The proof is given in APPENDIX B. Remark 3: Notice that the value of R min depends on the interval of θ. Hence, we can obtain the value of R min by analyzing the interval of θ. Therefore, in the next we consider two cases: R min = R D and R min = r i cosθ θ i. For R min = R D, it means that R D < minr cosθ θ, r cosθ θ. Similar to 49 in APPENDIX B, the condition of R min = R D is given by θ 0, θ a π θ b, π, 3 where θ a = minarccosr D /r +θ, arccosr D /r +θ and θ b = minarccosr D /r θ, arccosr D /r θ. Obviously, arccosx is a monotonically decreasing function when 0 x. Hence, it can be observed that θ a < θ a and π θ b < π θ b, which means that the interval of θ in 3 is included in 30. Remark 4: Finally, we derive the interval of θ when r cosθ θ > r cosθ θ. After a series of calculations, we have 0, π + θ 3π, 3 θ + θ, π where θ = arctany /x satisfies with θ 0, π, x = x x and y = y y. Accordingly, the result is different when θ π + θ, 3π + θ. Since the locations of x and x are variable, we rotate y = tanθ x to distinguish the areas corresponding to R min. Interestingly, we can observe that y = tan π + θ x and y = tan 3π + θ x can not be covered by the effective distinct of the relays simultaneously due to θ a + θ b π, where x and y denote the vectors. The complete strategies for choosing R min are presented in TABLE I. For example, in Fig. -Fig. 3, we simulate the effective areas of the relays for different cases of TABLE I, where Fig. and Fig. 3 correspond to case 5 and 6, respectively. Proof : The proof is given in APPENDIX C. VOLUME 6,

7 J. Chen et al.: Performance Analysis of Cooperative NOMA Schemes in Spatially Random Relaying Networks TABLE. Different cases for the corresponding areas of the relays. FIGURE. Illustration of the effective area of the relays. RD = 000, = 00, λ = 0.0, UE 900,500, and UE 500,300. FIGURE 3. Illustration of the effective area of the relays. UE 900,500, UE 500, 500 and θ = arctan0.4. By applying, eq ], U can be obtained as Z Z σ Z U = re ε dr re ε dr σ RD! α 0 α, µ 0 α, µrαd, 34 = σ σ ϕ αµ α Furthermore, the integral in 8 can be expressed as Z π Z Rmin Z σ Z RD re ε dr = re ε dr 0 σ R zd U σ4 33 where ϕ = e λn, T +λn, T, µ = λn,0 T, and 0a, b denotes the incomplete gamma function. Similarly, we have Z σ4 U = ϕ σ3 h i 0 α, µrαd 0 α, µr cosθ θ α, 35 αµ α where the values of σ σ6 depend on the locations of the users and are given in TABLE I. Since the different environments result in an unpredictable value of a, we assume an arbitrary value of a in our systems. Z + σ3 Z r cosθ θ Z z U σ6 + σ Z r cosθ θ z U3 re ε dr re ε dr, VOLUME 6, 08

8 U3 ϕ αµ α U ϕ αµ α σ 6 σ 5 Ɣ σ 4 σ 3 Ɣ α, µrα d + σ 6 σ 5 α, µrα d + σ 4 σ 3 n k= n k= W x k Ɣ α, µ σ6 σ 5 r cos x k + σ ] 6 + σ α ] 5 θ 36 W x k Ɣ α, µ σ4 σ 3 r cos x k + σ ] 4 + σ α ] 3 θ 37 However, it is difficult to derive the closed-form expression for 35. By adopting the Gaussian-Chebyshev quadrature ], the approximate expressions for U and U3 are given at the top of this page. With 8, 34, 36, and 37, the over-all system outage probability is given by P out = exp λu + U + U3]. 38 where W x k and x k are the number of terms, the weight factor, and the abscissas for the Gaussian-Chebyshev integration, respectively. C. DECODE-AND-FORWARD Similar to 3, the probability of ϑ when considering DF mode can be written as Pr ϑ = E Fϑ DF R, 39 n, where Fϑ DF R = Prmin γn, DF, n, DF > f R = e ωt. Then, the probability of ϑ is given by Pr ϑ = PrZ DF n, < R, S r > 0, 40 From, we use an approximate method and obtain Zn, AF min ρd α n,0 h n α, ρ g n, α = Zn, DF, which means that the theoretical result of Pr ϑ in the AF scheme is exactly equal to that in the DF mode. Accordingly, the overall outage probability for the DF case is also equal to P out = exp λu + U + U3]. 4 By comparing 38 and 4, we can conclude that the outage performance of AF and DF based NOMA systems at high SNRs are almost the same. Lemma 3: If ρ, T and T reduce to zero. Then, 8 can be asymptotically expressed as P asympt out = exp λ π 0 Rmin R d rdr, 4 where the value of S depends on the coordinates of the users and can be divided into three parts which are provided in TABLE I. For given coordinates of the users, using,. 53. ], we have π 0 Rmin R d rdr = σ σ R D R d + r sinϕ cosϕ + ϕ sinϕ cosϕ ϕ ] + r sinϕ 4 cosϕ 4 + ϕ 4 sinϕ 3 cosϕ 3 ϕ 3 ], 43 where ϕ = σ 3 θ, ϕ = σ 4 θ, ϕ 3 = σ 5 θ, and ϕ 4 = σ 6 θ. From 43, we observe that the asymptotic outage probability tends to a constant only related to the area of the effective district of the relays and the density of the homogeneous PPP when ρ. Furthermore, we can see that the mean number of relays,, is dependent on S and λ. In high SNRs, the outage probability for an arbitrary relay link approximates to zero. Hence, the asymptotic expression of the over-all system outage probability is only affected by S and λ. Lemma 4: From 38, for a given or known outage probability, the relationship between R d and R D can be derived as Ɣ α, µrα D = Ɣ From 44, we have α, αµ α µrα d + σ σ ϕ ] lnpout + U + U3 = ϜR d. 44 λ R D = Ɣ α,ϝr d µ α. 45 Obviously, R D can be expressed as a R d -dependent function when the outage probability is a constant value. Moreover, for fixed outage probability, we note that the area of the effective district of the relays needs to be kept constant when other parameters are unchanging. Accordingly, to keep S unchanging, we should ensure that R D increase as R d increase. IV. NUMERICAL RESULTS In this section, we plot the outage probability comparison between the simulation results and their corresponding analytical results for both AF and DF schemes which have been obtained in Eq. 38 and Eq. 4, respectively. Monte-Carlo simulation is also provided to verify our analytical results. In Fig. 4, we plot the outage probability curves versus ρ for different values of λ. From Fig. 4, we can observe that lower bounds developed in this work are very tight at high SNRs. First, we see that our proposed AF scheme almost can obtain a same system performance with the DF-based scheme at high SNR. It can be observed that the DF-based performance outperforms the AF-based performance due to the noise amplification. However, compared with the DF case, VOLUME 6,

9 of g n,. FIGURE 4. Outage performance comparison between our proposed scheme and other schemes. λ n,0 = λ n, = λ n, =, R D = 0, R d =, UE 9, 5, UE 5, 5, α = 4, R = 0.0 bit per channel use BPCU, R = 0.04 BPCU, and α = the structure of the AF strategy is simple because DF needs the decoding and encoding procedures at the relay node, which results in a higher system complexity. Furthermore, we observe that the TSRS outperforms the conventional maxmin approach proposed in 7], which is in agreement with the result in 8]. As expected, increasing the density of the homogeneous PPP can improve the system performance significantly because it means that the mean number of available relays increases. Furthermore, the outage probability tends to a constant dominated by S and λ when ρ. FIGURE 6. Outage probability for different values of path loss exponent. In Fig. 6, we plot the outage probability versus ρ for different path loss exponents. Increasing the value of path loss exponent affects the system performance significantly. Furthermore, we can see these asymptotic results are not influenced by α when ρ, which verifies the accuracy of Lemma 3. Finally, we find that the path loss decreases significantly with the path loss exponent increasing, which results in the approximated expressions of γn,i AF and Zn, AF equal to γn,i DF and Zn, DF at low SNR regimes. Thus, we can clearly find that the curves for all the schemes become tight. FIGURE 5. Outage performance comparison between Rayleigh fading case and other schemes. In Fig. 5, we plots the outage probability versus ρ for different fading impairing g n,. In particular, we assume that g n, suffers from Rayleigh, exponentiated Weibull EW 3], and generalized-k GK 4] fading. Interestingly, we can observe that the curves whatever adopting DF or AF schemes are very tight when we change the channel fading model of g n,. According to the characteristic of NOMA systems, it can be obtained that γn,i AF γn,i DF α α and Zn, AF Zn, DF min ρd α n,0 h n α, ρ g n, α at the high SNR. Therefore, we can conclude that the throughput of relaying NOMA systems is almost not affected by the characteristics FIGURE 7. Outage probability comparison between our proposed scheme and other schemes versus R d. R D = 0, R = 0.0 BPCU, R = 0.04 BPCU, and ρ = 5 db. In Fig. 7, we plot the outage probability curves versus R d for different values of λ. From Fig. 4, it is clearly shown that increasing λ can result in a higher, i.e., increasing the average of the number of relays. Additionally, we can see that the outage probability of both RS schemes increases as increasing R d. According to the Lemma 3, we note that the value of is influenced by both λ and S. It is known that increasing R d results in decreasing S and leads to a higher path loss, which implies that a larger secure radius causes a lower multi-relay diversity gain and a higher path loss impairing the link between the BS and the relays. Therefore, 3366 VOLUME 6, 08

10 the above reasons significantly deteriorate the throughout of the communication systems. Moreover, the observation obtained in Fig. 7 verifies the correctness of Lemma 3. FIGURE 8. Outage probability comparison between our proposed scheme and other schemes versus R D. R d =, R = 0.0 BPCU, R = 0.04 BPCU, and ρ = 5 db. For comparison, in Fig. 8, we plot the outage probability versus R D for different values of λ. Again, similar observations obtained in Fig. 7 can be found in Fig. 8. For a given R d, increasing R D results in the opposite performance compared to increasing R d. When the density of is small, we can increase R D and improve the performance of the proposed NOMA systems. V. CONCLUSION In this paper, we proposed a general framework to investigate the performance of NOMA systems with the two-stage RS strategy and spatially random relays. Specifically, we proposed a method to calculate the area of the effective district of the relays. Additionally, we show that our results converge to the error floor and obtain the zero diversity. Furthermore, we simplified the analyzing method of the TSRS scheme and compared the performance between AF and DF. Also, we developed the relationship between the area of the effective coverage district of the relays and the locations of the users. Finally, we revealed the impact of the channel gain of the poor user on the outage performance at the high SNR. APPENDIX A PROOF OF LEMMA From, we have r minr D, x x + y y, x x + y y. Utilizing the polar coordinate transform and double angle formula transform, we can obtain xi x + y i y = r cosθr i cosθ i + r sinθr i sinθ i = rr i cosθ θ i. 46 According to, the constraint of x x i +y y i x i +y i can be written as Substituting 47 into, we have r r i cosθ θ i 47 R d r minr D, r cosθ θ, r cosθ θ. 48 APPENDIX B PROOF OF REMARK To ensure the valid area of the relays, we need to derive the interval of θ corresponding to R d R min. With the assumption of R d R D, the event of R d R min can be expressed as R d minr cosθ θ, r cosθ θ. 49 Then, we can further obtain ] ] cosθ θ cosθ θ. 50 r r Since R d /r i > 0, θ θ belongs to the first or fourth quadrants. According to the characteristics of the cosine function, we know that it monotonically increases in the first quadrant and monotonically decreases in the fourth quadrant. Furthermore, the range of arccosx is within π/, π/]. Therefore, the range of θ belongs to 0, arccos r i + θ i ] π arccos r i ] + θ i, π. 5 Thus, the interval of θ for the valid area of the relays is given by 0, arccos R d /r +θ ] π arccos R d /r +θ, π] 0, arccos R d /r + θ ] π arccos R d /r + θ, π] = 0, minarccosr d /r + θ, arccosr d /r + θ ] π minarccosr d /r θ, arccosr d /r θ, π], 5 The proof of 30 is completed. To prove θ a + θ b π, we first have θ sum = arccosr d /r + θ + arccosr d /r + θ + arccosr d /r θ + arccosr d /r θ = arccosr d /r + arccosr d /r ] 53 Then, it can be observed that arccos R d /r i is an acute angle. Therefore, we have arccos R d /r i < π/. Finally, we obtain θ sum >minarccosr d /r + θ, arccosr d /r + θ + minarccosr d /r θ, arccosr d /r θ, 54 From 54, we have θ sum >θ a + θ b. Therefore, with above facts, we obtain θ a + θ b < θ sum < π and complete the proof. APPENDIX C PROOF OF REMARK 4 To obtain the range of θ in 3, we compare r cosθ θ with r cosθ θ. Using the double angle formula, we obtain r cosθ θ r cosθ θ as r cosθ cosθ + r sinθ sinθ r cosθ cosθ r sinθ sinθ VOLUME 6,

11 J. Chen et al.: Performance Analysis of Cooperative NOMA Schemes in Spatially Random Relaying Networks Then, by using the polar coordinate transform, 55 can be rewritten as x x cosθ + y y sinθ q + y cosθ θ 0 56 = x According to the characteristic of the cosine function, we have θ θ, π/ + θ ] 3π/ + θ, π + θ ] 57 In the polar coordinate, 57 can be expressed as θ 0, π/ + θ ] 3π/ + θ, π ] 58 From 57 or 58, we see that the intervals of θ corresponding to r cosθ θ r cosθ θ or r cosθ θ r cosθ θ are π. In APPENDIX B, it has been proved that the interval of θ for the valid area of the district of the relays is less than π. Thus, we can draw some interesting conclusions. For instance, we assume that the boundary of the area obtained in Remark 4 can be viewed as a rotatable linear function rotating around the center and it can be mathematically expressed as y = tan π + θ x or y = tan 3π + θ x. Setting different locations of the NOMA users, we can put this linear function into different areas proposed in Remark and Remark. Hence, the interval of θ corresponding to Rmin can be presented in different strategies developed in TABLE I. REFERENCES ] Y. Saito, Y. Kishiyama, A. Benjebbour, T. Nakamura, A. Li, and K. Higuchi, Non-orthogonal multiple access NOMA for cellular future radio access, in Proc. IEEE Veh. Technol. Conf., Dresden, Germany, Jun. 03, pp. 5. ] Z. Ding, Z. Yang, P. Fan, and H. V. Poor, On the performance of nonorthogonal multiple access in 5G systems with randomly deployed users, IEEE Signal Process. Lett., vol., no., pp , Dec ] S. Timotheou and I. Krikidis, Fairness for non-orthogonal multiple access in 5G systems, IEEE Signal Process. Lett., vol., no. 0, pp , Oct ] Z. Ding, M. Peng, and H. V. Poor, Cooperative non-orthogonal multiple access in 5G systems, IEEE Commun. Lett., vol. 9, no. 8, pp , Aug ] N. T. Do, D. B. da Costa, T. Q. Duong, and B. An, A BNBF user selection scheme for NOMA-based cooperative relaying systems with SWIPT, IEEE Commun. Lett., vol., no. 3, pp , Mar ] X. Liang, Y. Wu, D. W. K. Ng, Y. Zuo, S. Jin, and H. Zhu, Outage performance for cooperative NOMA transmission with an AF relay, IEEE Commun. Lett., vol., no., pp , Nov ] Q. Y. Liau, C. Y. Leow, and Z. Ding, Amplify-and-forward virtual fullduplex relaying based cooperative NOMA, IEEE Wireless Commun. Lett., to be published. 8] Z. Ding, H. Dai, and H. V. Poor, Relay selection for cooperative NOMA, IEEE Wireless Commun. Lett., vol. 5, no. 4, pp , Aug ] D. Deng, L. Fan, X. Lei, W. Tan, and D. Xie, Joint user and relay selection for cooperative NOMA networks, IEEE Access, vol. 5, pp , Aug ] Z. Yang et al., Novel relay selection strategies for cooperative NOMA, IEEE Trans. Veh. Technol., vol. 66, no., pp , Nov. 07. ] 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. 6, no. 3, pp , Mar. 07. ] Z. Ding, P. Fan, and H. V. Poor, Random beamforming in millimeter-wave NOMA networks, IEEE Access, vol. 5, pp , ] Y. Zhou, V. W. S. Wong, and R. Schober, Dynamic decode-and-forward based cooperative NOMA with spatially random users, IEEE Trans. Wireless Commun., vol. 7, no. 5, pp , May 08. 4] X. Yue et al., Spatially random relay selection for full/half-duplex cooperative NOMA networks, IEEE Trans. Commun., to be published. 5] Z. Ding, L. Dai, and H. V. Poor, MIMO-NOMA design for small packet transmission in the Internet of Things, IEEE Access, vol. 4, pp , 06. 6] C. S. Patel, G. L. Stuber, and T. G. Pratt, Statistical properties of amplify and forward relay fading channels, IEEE Trans. Veh. Technol., vol. 55, no., pp. 9, Jan ] Y. Jing and H. Jafarkhani, Single and multiple relay selection schemes and their achievable diversity orders, IEEE Trans. Wireless Commun., vol. 8, no. 3, pp , Mar ] J. N. Laneman, D. N. C. Tse, and G. W. Wornell, Cooperative diversity in wireless networks: Efficient protocols and outage behavior, IEEE Trans. Inf. Theory, vol. 50, no., pp , Dec ] Z. Qin, Y. Liu, Y. Gao, M. Elkashlan, and A. Nallanathan, Wireless powered cognitive radio networks with compressive sensing and matrix completion, IEEE Trans. Commun., vol. 65, no. 4, pp , Apr ] S. N. Chiu, D. Stoyan, W. S. Kendall, and J. Mecke, Stochastic Geometry and Its Applications, nd ed. Hoboken, NJ, USA: Wiley, 996. ] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 6th ed. New York, NY, USA: Academic, 000. ] E. Hildebrand, Introduction to Numerical Analysis. New York, NY, USA: Dover, ] P. Wang et al., Performance analysis for relay-aided multihop BPPM FSO communication system over exponentiated Weibull fading channels with pointing error impairments, IEEE Photon. J., vol. 7, no. 4, Aug. 05, Art. no ] C. Yoon, H. Lee, and J. Kang, Effect of generalized-k fading on the performance of symmetric coordinate interleaved orthogonal designs, IEEE Commun. Lett., vol. 8, no. 4, pp , Apr. 04. JIANCHAO CHEN was born in Zhanjiang, Guangdong, China. He received the B.S. degree from the School of Electronic and Information Engineering, Hengyang Normal University, China, in 06. He is currently pursuing the M.S. degree with the School of Information Engineering, Guangdong University of Technology. His current research interests include nonorthogonal multiple access, cooperative networks, and unmanned aerial vehicle communication. LIANG YANG was born in Shaoyang, Hunan, China. He received the Ph.D. degree in electrical engineering from Sun Yat-sen University, China, in 006. From 006 to 03, he was a Teacher with Jinan University, Guangzhou, China. He joined the Guangdong University of Technology in 03. His current research interests include the performance analysis of wireless communications systems. MOHAMED-SLIM ALOUINI S 94 M 98 SM 03 F 09 was born in Tunis, Tunisia. He received the Ph.D. degree in electrical engineering from the California Institute of Technology, Pasadena, CA, USA, in 998. He was with the Department of Electrical and Computer Engineering, University of Minnesota, Minneapolis, MN, USA, and then with the Electrical and Computer Engineering Program, Texas A&M University at Qatar, Doha, Qatar. Since 009, he has been a Professor of electrical engineering with the Division of Physical Sciences and Engineering, KAUST, Thuwal, Saudi Arabia. His current research interests include the design and performance analysis of wireless communication systems. VOLUME 6, 08