Spatially Random Relay Selection for Full/Half-Duplex Cooperative NOMA Networks

Transcription

1 Spatially Rom Relay Selection for Full/Half-Duplex Cooperative NOMA Networs Xinwei Yue, Student Member, IEEE, Yuanwei Liu, Member, IEEE, Shaoli Kang, Arumugam Nallanathan, Fellow, IEEE, Zhiguo Ding, Senior Member, IEEE Abstract This paper investigates the impact of relay selection RS on the performance of cooperative non-orthogonal multiple access NOMA, where relays are capable of woring in either full-duplex FD or half-duplex HD mode. A number of relays i.e., K relays are uniformly distributed within the disc. A pair of RS schemes are considered insightfully: Single-stage RS SRS scheme; Two-stage RS TRS scheme. In order to characterize the performance of these two RS schemes, new closed-form expressions for both exact asymptotic outage probabilities are derived. Based on analytical results, the diversity orders achieved by the pair of RS schemes for FD/HD cooperative NOMA are obtained. Our analytical results reveal that: i The FD-based RS schemes obtain a zero diversity order, which is due to the influence of loop interference LI at the relay; ii The HD-based RS schemes are capable of achieving a diversity order of K, which is equal to the number of relays. Finally, simulation results demonstrate that: The FD-based RS schemes have better outage performance than HD-based RS schemes in the low signal-to-noise radio SNR region; As the number of relays increases, the pair of RS schemes considered are capable of achieving the lower outage probability; 3 The outage behaviors of FD/HD-based NOMA SRS/TRS schemes are superior to that of rom RS RRS orthogonal multiple access OMA based RS schemes. Index Terms Decode--forward, full/half-duplex, relay selection, non-orthogonal multiple access I. INTRODUCTION With the rapid advancement in the wireless communication technology, the fifth generation 5G mobile communication networs have attracted a great deal of attention 4]. In particular, three major families of new radio NR usage scenarios, i.e., massive machine type communications mmtc, enhanced mobile broadb embb ultra-reliable low-latency communications URLLC are proposed to satisfy the different requirements for 5G networs. To improve system throughput achieve enhanced spectrum efficiency of 5G networs, non-orthogonal multiple access NOMA has been considered to be a promising cidate technique X. Yue is with School of Electronic Information Engineering, Beihang university, Beijing, China xinwei yue@buaa.edu.cn. Y. Liu A. Nallanathan are with School of Electronic Engineering Computer Science, Queen Mary University of London, London, UK {yuanwei.liu, arumugam.nallanathan}@qmul.ac.u. S. Kang is with State Key Laboratory of Wireless Mobile Communications, China Academy of Telecommunication TechnologyCATT, Beijing, China also with School of Electronic Information Engineering, Beihang University, Beijing, China angshaoli@catt.cn. Z. Ding is with the Department of Electrical Engineering, Princeton University, Princeton, USA also with the School of Computing Communications, Lancaster University, U.K. z.ding@lancaster.ac.u. Part of this wor has been submitted to IEEE GLOBECOM 07 ]. identified for 3GPP Long Term Evolution LTE 5]. The core idea of NOMA is able to multiplex additional users in the same physical resource. More specifically, the superposition coding scheme is employed at the transmitting end, where the linear superposition of signals of multiple users is formed to be the transmit signal. The successive interference cancellation SIC procedure is carried out by the receiving end who has the better channel conditions 6]. Furthermore, downlin multiuser superposition transmission scheme MUST 7] which is the special case of NOMA has found application in wireless stard. Hence numerous excellent contributions have surveyed the performance of point-to-point NOMA in wireless networs in 8 ]. To evaluate the performance of downlin NOMA, the closed-form expressions of outage probability ergodic rate for NOMA were derived in 8] by use of the bounded path loss model. Furthermore, the authors of 9] have studied the impact of user pairing on the performance of NOMA, where both the outage performance of fixed power allocation based NOMA F-NOMA cognitive radio based NOMA CR-NOMA schemes were characterized. By considering user grouping decoding order selection, the outage balancing among users was investigated 0], in which the closed-form expressions of optimal decoding order power allocation for downlin NOMA were derived. In ], the authors researched the outage behavior of downlin NOMA for the case where each NOMA user only feed bac one bit of its channel state information CSI to a base station BS. It was shown that NOMA is capable of providing higher fairness for multiple users compared to conventional opportunistic one-bit feedbac. As a further advance, there is a paucity of research treaties on investigating the application of point-to-point NOMA systems. In ], the authors analyzed the outage behavior of largescale underlay CR for NOMA with the aid of stochastic geometry. To emphasize physical layer security PLS, the authors in 3] discussed the PLS issues of NOMA, where the secrecy outage probabilities were derived for both singleantenna multiple-antenna scenarios, respectively. Recently, the NOMA-based wireless cashing strategies were introduced in 4], in which two cashing phases, i.e., content pushing content delivery, are characterized in terms of caching hit probability. Additionally, explicit insights for understing the performance of uplin NOMA have been provided in 5, 6]. In 5], the novel uplin power control protocol was proposed for the single-cell uplin NOMA. In large-scale cellular networs, the performance of multi-cell uplin NOMA was characterized in terms of coverage probability using the

2 theory of Poisson cluster process 6]. Cooperative communication is a promising approach to overcome signal fading arising from multipath propagation as well as obtain the higher diversity 7]. Obviously, combining cooperative communication technique NOMA is the research topic which has spared of wide interest in 8 ]. The concept of cooperative NOMA was initially proposed for downlin transmission in 8], where the nearby user with better channel conditions was viewed as decode--forward DF relay to deliver the information for the distant users. Driven by these, authors in 9] analyzed the achievable data rate of NOMA systems for DF relay over Rayleigh fading channels. On the stpoint of tacling spectrum efficiency energy efficiency, in 0], the application of simultaneous wireless information power transfer SWIPT to NOMA with romly deployed users was investigated using stochastic geometry. In ], NOMA based dual-hop relay systems were addressed, where both statistical CSI instantaneous CSI were considered for the networs. On the other h, the outage performance of NOMA for a variable gain amplify-forward AF relay was characterized over Naagami-m fading channels in ]. With the emphasis on imperfect CSI, authors studied the system outage behavior of AF relay for NOMA networs in 3]. Additionally, the authors of 4] analyzed the outage performance of a fixed gain based AF relay for NOMA systems over Naagami-m fading channels. Above existing contributions on cooperative NOMA are all based on the assumption of half-duplex HD relay, where the communication process was completed in two slots 7]. To further improve the bwidth usage efficiency of system, fullduplex FD relay technology is a promising solution which can simultaneously receive transmit the signal in the same frequency b 5]. Nevertheless, FD operation suffers from residual loop self-interference LI, which is usually modeled as a fading channel 6]. Particularly, FD relay technologies in 7] have been discussed from the view of selfinterference cancellation, protocol design relay selection for 5G networs. To maximize the weighted sum throughput of system, the design of resource allocation algorithm for FD multicarrier NOMA MC-NOMA was investigated in 8], where a FD BS was capable of serving downlin uplin users in the meantime. The recent findings in FD operation considered for cooperative NOMA were surveyed in 9, 30]. The performance of FD device-to-device DD based cooperative NOMA was characterized in terms of outage probability in 9]. Considering the influence of imperfect selfinterference, the authors in 30] investigated the performance of FD-based DF relay for NOMA, where the expressions of outage probability achievable sum rate for two NOMA users were derived. Applying relay selection RS technique to cooperative communication systems is a straightforward effective approach for taing advantages of space diversity improving spectral efficiency. The following research contributions have surveyed the RS schemes for two inds of operation modes: HD FD. For HD mode, the authors of 3] derived the diversity of single RS scheme investigated the complexity of multiple RS scheme by exhaustive search. It was shown that these RS schemes are capable of providing full diversity order. Furthermore, in 3], the ergodic rate was studied with a buffer-aided relay scheme for HD-based single RS networ. Additionally, the application of RS scheme to cognitive DF relay networs was discussed in 33]. For FD mode, assuming the availability of different instantaneous CSI, the authors analyzed the RS problem of AF cooperative system in 34]. It was worth noting that FD-based RS scheme converges to an error floor obtains a zero diversity order. The performance of DF RS scheme was characterized in terms of outage probability for the CR networs in 35]. Very recently, two-stage RS scheme was proposed for HD-based cooperative NOMA in 36], where the RS scheme considered was capable of realizing the maximal diversity order. A. Motivations Contributions While the aforementioned significant contributions have laid a solid foundation for the understing of cooperative NOMA RS techniques, the RS technique for cooperative NOMA networs is far from being well understood. It is worth pointing out that from a practical perspective, the requirements of Internet of Things IoT scenarios, i.e, lin density, coverage enhancement small pacet service are capable of being supported through the RS schemes. One of the best relays is selected from K relays as the BS s helper to forward the information. In 36], the two-stage RS scheme is capable of achieving the minimal outage probability obtaining the maximal diversity order, but only HD-based RS for cooperative NOMA was considered. To the best of our nowledge, there are no existing wors to investigate the RS scheme for FD cooperative NOMA networs. Moreover, the spatial impact of RS on the performance of FD cooperative NOMA was not examined in 36]. Motivated by these, we specifically consider a pair of RS schemes for FD/HD NOMA networs, namely single-stage RS SRS scheme two-stage RS TRS scheme, where the locations of relays are modeled by invoing the uniform distribution. More specifically, In the SRS scheme, the data rate of distant user is ensured to select a relay as its helper to forward the information. In the TRS scheme, on the condition of ensuring the data rate of distant user, we serve the nearby user with data rate as large as possible for selecting a relay. Based on the proposed schemes, the primary contributions can be summarized as follows: We investigate the outage behaviors of two RS schemes i.e., SRS scheme TRS scheme for FD NOMA networs. We derive the closed-form asymptotic expressions of outage probability for FD-based NOMA RS schemes. Due to the influence of residual LI at relays, a pair of FD-based NOMA RS schemes converge to an error floor in the high signal-to-noise radio SNR region provide zero diversity order. We also derive the closed-form expressions of outage probability for two HD-based NOMA RS schemes. To get more insights, the asymptotic outage probabilities of HD-based NOMA RS schemes are derived. We observe that with the number of relays increasing, the lower outage probability can be achieved for HD-based NOMA

3 3 RS schemes. We confirm that the HD-based NOMA RS schemes are capable of providing the diversity order of K, which is equal to the number of relays. 3 We show that the outage behaviors of FD-based NOMA SRS/TRS schemes are superior to that of HD-based NOMA SRS/TRS schemes in the low SNR region rather than in the high SNR region. Furthermore, we confirm that the FD/HD-based NOMA TRS/SRS schemes are capable of providing better outage performance compare to rom RS RRS orthogonal multiple access OMA based RS schemes. Additionally, we analyze the system throughput in delay-limited transmission mode based on the outage probabilities derived. B. Organization Notation The rest of the paper is organized as follows. In Section II, the networ model of the RS schemes for FD/HD NOMA is set up. New analytical approximate expressions of outage probability for the RS schemes are derived in Section III. In Section IV, numerical results are presented for performance evaluation comparison. Section V concludes the paper. The main notations of this paper is shown as follows: E{ } denotes expectation operation; f X F X denote the probability density function PDF the cumulative distribution function CDF of a rom variable X. II. NETWORK MODEL In this section, the networ signal models are presented. Additionally, the criterions of a pair of RS schemes in the networs considered are introduced for FD/HD NOMA. A. Networ Description Consider a downlin cooperative NOMA scenario consisting of one BS, K relays R i with i K a pair of users i.e., the nearby user D distant user D, as shown in Fig.. To reduce the complexity of NOMA system, multiple users can be divided into several groups the NOMA protocol is carried out in each group 9, 37]. The groups between each other are orthogonal. We assume that the BS is located at the origin of a disc, denoted by D the radius of disc is R D. In addition, K relays are uniformly distributed within D 8]. The DF protocol is employed at each relay only one relay is selected to assist BS conveying the information to the NOMA users in each time slot. To enable FD operation, each relay is equipped with one transmit antenna one receive antenna, while the BS users have a single antenna, respectively. All wireless channels in the scenario considered Note that multiple antennas equipped by the BS relays will further suppress the self-interference enhance the performance of the NOMAbased RS schemes. Additionally, more sophisticated assumption of antennas at relay, i.e., omni-directional directional antennas 38, 39] can be developed for further evaluating the performance of the networs considered. However, these are beyond the scope of this treatise. It is assumed that perfect CSI can be obtained, our future wor will relax this idealized assumption. Furthermore, we note that relaxing the setting of Rayleigh fading channels i.e., Naagami-m fading channels considered in 4] will provide a more general system setup, which are set aside for our future wor. R 3 R 4 R K BS R R 5 h Ri h LI R Fig. : An illustration of RS scheme for downlin FD/HD cooperative NOMA networs. are assumed to be independent non-selective bloc Rayleigh fading are disturbed by additive white Gaussian noise with mean power N 0. h SRi CN 0,, h RiD CN 0,, h Ri D CN 0, denote the complex channel coefficient of BS R i, R i D, R i D lins, respectively. d d denote the distance from the BS to D D, respectively. Assuming that an imperfect self-interference cancellation scheme is employed at each relay such as 34] the corresponding LI is modeled as a Rayleigh fading channel with coefficient h LI CN 0, Ω LI. As stated in 36], two NOMA users are classified into the nearby user distant user by their quality of service QoS not sorted by their channel conditions. More particularly, via the assistance of the best relay selected, the QoS requirements of NOMA users can be supported effectively for the IoT scenarios i.e., small pacet business telemedicine service 40]. Hence we assume that D can be served opportunistically D needs to be served quicly for small pacet with a lower target data rate. As a further example, D is to download a movie or carry out some bacground tass so on; D can be a medical health sensor which is to send the pivotal safety information containing in a few bytes, such as blood pressure, pulse heart rates. B. Signal Model During the l-th time slot, l,, 3,..., the BS sends the superposed signal a P s x l] + a P s x l] to the relay on the basis of NOMA principle 8], where x x are the normalized signal for D D, respectively, i.e, E{x } E{x }. a a are the corresponding power allocation coefficients. Practically speaing, to stipulate better fairness QoS requirements between the users 40], we assume that a a with a + a. The LI signal exists at the relay due to it wors in FD mode. Therefore the observation at the ith relay R i is given by y Ri h h h Ri a P s x l] + a P s x l] + h LI ϖpr x LI l l d ] + n Ri, h where h Ri SRi +d α, d SRi is the distance between the SRi BS R i α denotes the path loss exponent. ϖ is the switching operation factor, where ϖ ϖ 0 denote D D

4 4 the relay woring in FD mode HD mode, respectively. According to the practical usage scenarios, we can select the different duplex mode. It is worth noting that in FD mode, it is capable of improving the spectrum efficiency, but will suffer from the LI signals. On the contrary, in HD mode, this situation can be avoided precisely. P s P r denote normalized transmission power i.e., P s P r at the BS R i, respectively. x LI l l d ] denotes the LI signal with E x LI ] an integer l d denotes processing delay at R i with l d. n Ri denotes the Gaussian noise at R i. Based on NOMA protocol, SIC 3 is employed at R i to first decode the signal x of D having a higher power allocation factor, since R i has a less interference-infested signal to decode the signal x of D. Based on this, the received signalto-interference-plus-noise ratio SINR at R i to detect x x are given by γ D R i ρ h Ri a ρ h Ri a + ρϖ h LI +, γ D R i ρ h R i a ρϖ h LI +, 3 respectively, where ρ P s N 0 is the transmit SNR. Assuming that R i is capable of decoding the two NOMA user s information, i.e, satisfying the following conditions, log + γ D R i R D ; log + γ D R i R D, where R D R D are the target rate for D D, respectively. Therefore the observation at D j can be expressed as y Dj h j a P r x l l d ] + a P r x l l d ] + n Dj, 4 h Ri D where h j +d j, d α Ri D j is the distance between Ri Dj R i D j assuming d Ri D j d SRi ; d Ri D j d SR i + d j d SR i d j cos θ i, j,. θ i denotes the angle D j SR i ; n Dj denotes the Gaussian noise at D j. In similar, assuming that SIC can be also invoed successfully by D to detect the signal of D having a higher transmit power, who has less interference. Hence the received SINR at D to detect x can be given by γ D D ρ h a ρ h a +. 5 Then the received SINR at D to detect its own information is given by γ D ρ h a. 6 The received SINR at D to detect x is given by γ D ρ h a ρ h a +. 7 Note that the fixed power allocation coefficients for two NOMA users are considered in the networs. Reasonable 3 In this paper, we assume that perfect SIC is employed, our future wor will relax this assumption. power control optimizing the mode of power allocation can further enhance the performance of the RS schemes, which may be investigated in our future wor. C. Relay Selection Schemes In this subsection, we consider a pair of RS schemes for FD/HD NOMA, which are detailed in the following. Single-stage Relay Selection: Prior to the transmissions, a relay can be romly selected by the BS as its helper to forward the information. The aim of SRS scheme is to maximize the minimum data rate of D for FD/HD NOMA. More specifically, the size of data rate for D depends on three inds of data rates, such as the data rate for the relay R i to detect x ; The data rate for D to detect x ; 3 the data rate for D to detect its own signal x. Among the relays in the networ, based on, 5 7, the SRS scheme activates a relay, i.e., i SRS arg max {min {log + γ D R i, log + γ D D, i log + γ D }, i SR}, 8 where SR denotes the number of relays in the networ. Note that FD/HD-based SRS schemes inherit advantage to ensure the data rate of D, where the application of small pacets can be achieved. Two-stage Relay Selection: The TRS scheme mainly include two stages for FD/HD NOMA: In the first stage, the target data rate of D is to be satisfied; In the second stage, on the condition that the data rate of D is ensured, we serve D with data rate as large as possible. Hence the first stage activates the relays that satisfy the following condition S R {log + γ D R i R D, log + γ D D R D, log + γ D R D, i K}, 9 where the size of SR is defined as SR. Among the relays in SR, the second stage selects a relay to convey the information which can maximize the data rate of D is expressed as i T RS arg max {min {log + γ D R i, i log + γ D }, i SR }. 0 As can be observed from the above explanations, the merit of FD/HD-based TRS schemes is that in addition to guarantee the data rate of D, the BS can support D to carry out some bacground tass, i.e., downloading a movie or multimedia files. III. PERFORMANCE EVALUATION In this section, the performance of this pair of RS schemes are characterized in terms of outage probability as well as the delay-limited throughput for FD/HD NOMA networs.

5 5 A. Single-stage Relay Selection Scheme According to NOMA protocol, the complementary events of outage for SRS scheme can be explained as: The relay i SRS can detect the signal x of D ; while the signal x can be successfully detected at D D, respectively. From the above descriptions, the outage probability of SRS scheme for FD NOMA can be expressed as follows: P F D SRS K i Pr Wi th, where W i min {γ D R i, γ D D, γ D } ϖ. γth F D R D with R D being the target rate of D. The following theorem provides the outage probability of SRS scheme for FD NOMA. Theorem. The closed-form expression of outage probability for FD-based NOMA SRS scheme can be approximated as follows: P F D SRS π N N n ϕ n + e c nτ + ϖρτc n Ω LI γ F D th ρ a a γ F D e +dα τ +dα τ ] K, where ϖ, τ with a > a γth F D. c n th + R D ϕ n + α, ϕn cos n N π N is a parameter to ensure a complexity-accuracy tradeoff. Proof: See Appendix A. Corollary. Upon substituting ϖ 0 into, the approximate expression of outage probability for HD-based NOMA SRS scheme is given by P HD SRS π N N n ϕ n + e τcn e +dα τ +dα τ ] K, 3 γ HD th ρ a a γ HD where τ with a > a γth HD th R D with R D being the target rate of D. B. Two-stage Relay Selection Scheme γ HD th In the case of TRS scheme, the overall outage event can be expressed 36] as follows: φ φ φ, 4 where φ denotes the outage event that relay i T RS cannot detect x, or neither D D can detect the x correctively, φ denotes the outage event that either of i T RS D cannot detect x while three nodes can detect x successfully. As a consequence, the outage probability of TRS scheme for FD NOMA can be expressed as follows: P F D T RS Pr φ + Pr φ. 5 On the basis of analytical results in III-A, the first outage probability in 5 is approximated as Pr φ π N N n ϕ n + e c nτ + ρϖτc n Ω LI e +dα τ +dα τ ] K, 6 where ϖ. In order to calculate the second outage probability, P φ can be further expressed as Pr φ Pr Λ, SR > 0 + Pr Λ, Λ, SR > 0, 7 where Λ denotes the outage event that the relay i T RS cannot detect x Λ denotes the corresponding complementary event of Λ. Λ denotes that D cannot detect x. The first term in the above equation is given by Pr Λ, SR > 0 8 Pr log + γ D Ri T < R D, S RS R > 0. The second term in 7 is given by Pr Λ, Λ, SR > 0 Pr log + γ D < R D, log + γ D Ri T > R D, S RS R > 0. 9 Combining 8 with 9, the second outage probability in 5 can be expressed as Pr φ Pr log + γ D Ri T < R D, S RS R > 0 + Pr log + γ D < R D, log + γ D Ri T > R D, S RS R > 0, 0 where ϖ. To derive the closed-form expression of outage probability for TRS scheme in 0, we define s i min {log + γ D R i, log + γ D }, s i T RS max { s, S R}, respectively. The probability Pr φ can be given by { Pr φ Pr min log + γ D Ri T, RS log + γ D } < R D, SR > 0 Pr s i T RS < R D, SR > 0. 3 The above probability can be further expressed as K Pr φ Pr s i T RS < R D, SR K Pr s i T RS < R D SR Pr SR

6 6 K F R D }{{} Θ Pr SR. 4 }{{} Θ For selecting a relay at rom from SR, denoted by relay i, let us now turn our attention to the derivation of s i s CDF i.e., F R D in the following lemma. Define these two probabilities at the right h side of 4 by Θ Θ, respectively. Lemma. The conditional probability in 4 can be approximated as follows: Θ where π N N M + M + M 3 e +dα τ χe c nτ, 5 n ϕ n +, θ max τ, ξ, ξ γf D th ρa, ζ cn++dα ρϖc n, χ +ρϖτc n Ω LI, ψ α ξ +ρϖξc n Ω LI, α τ T +dα e c n++d ρϖc nω LI, Φ +dα e cn++d ρϖc nω LI, M e +dα θ χe cnτ ψe c nξ, M e +d α τ e ξ +dα e cnτ χ Te ζ Ω LIψ ζ E i Ω LI ψ +Φe ζ Ω LIχ ζ E i Ω LI χ, M 3 e +dα τ e +dα ξ e +dα τ e +dα ξ Te ζ Ω LIψ ζ E i Ω LI ψ + Φe ζ Ω LI χ, γth F D R D with R D being the target rate of D E i is the exponential integral function 4, Eq. 8..]. E i ζ Ω LI χ Proof: See Appendix B. On the other h, there are relays in SR corresponding probability Θ is given by Θ K m K Pr γd Ri th Pr γ D D > γth F D Pr γd K mk + Pr γ D R i > γth F D th the Pr γ D D th Pr γd th. 6 With the aid of Theorem, the above probability can be further approximated as follows: K Θ π N N n ϕ n + e c nτ ] e +dα τ +dα τ K + ρϖτc n Ω LI π N N n ϕ n + e c nτ ] e +dα τ +dα τ. + ρϖτc n Ω LI 7 With the aid of Lemma, combining 6, 4 7 applying some algebraic manipulations, the outage probability of TRS scheme for FD NOMA can be provided in the following theorem. Theorem. The closed-form expression of outage probability for the FD-based NOMA TRS scheme is approximated by 8 at the top of next page. Corollary. For the special case ϖ 0, the approximate expression of outage probability for HD-based NOMA TRS scheme is given by 9 at the top of next page, where ξ γhd th ρa γ HD th R D with R D being the target rate of D. C. Benchmars for SRS TRS schemes In this subsection, we consider the rom relay selection RRS scheme as a benchmar for comparison purposes, where the relay R i is selected romly to help the BS transmitting the information. Note that R i selected maybe not the optimal one for the NOMA RS schemes. In this case, the RRS scheme is capable of being regarded as the special case for SRS/TRS schemes with K, which is independent of the number of relays. As such, for SRS scheme, the outage probability of the RRS scheme for FD/HD NOMA can be easily approximated as P F D,SRS RRS P HD,SRS RRS π N N n ϕ n + e cnτ + ρτc n Ω LI π N ] e +dα τ +dα τ, N n ϕ n + 30 e τ c n ] e +d α τ +d α τ, 3 respectively. Similarly, for TRS scheme, the outage probability of RRS scheme for FD/HD NOMA can be obtained from 8 9 by setting K, respectively. D. Diversity Order Analysis To gain more insights for these two RS schemes, the asymptotic diversity analysis is provided in the high SNR region according to the derived outage probabilities. The diversity order is defined as d lim ρ log P ρ, 3 log ρ where P ρ is the asymptotic outage probability. Single-stage Relay Selection Scheme: Based on the analytical results in, when ρ, we can derive the asymptotic outage probability of SRS scheme for FD NOMA in the following corollary. Corollary 3. The asymptotic outage probability of FD-based NOMA SRS scheme at high SNR is given by P F D, π N ] K ρτcn Ω SRS LI N n ϕ n +. + ρτc n Ω LI 33

7 7 P F D T RS K 0 K e +dα θ χe cnτ ψe c nξ e +dα τ χe c nτ + α + e +d τ e +dα ξ e +dα τ e +dα ξ e +dα τ χe cnτ e +d α τ e +dα ξ e c nτ χ e +dα τ χe c nτ ] χe cnτ ] e +dα τ +dα τ K χe cnτ e +dα τ +dα τ ]. 8 P HD T RS K 0 K c n e +d α +d α +c nτ e +dα +c nξ +c n e +dα τ e τ c n e +dα τ e +dα ξ + e +dα τ e +dα ξ + +dα ] +d e +d α α +cnξ e +dα +cnτ +cn e +dα τ e τ c n e τcn ] K e +dα ++dα ]τ e τ c n ] e +d. α ++dα ]τ 9 Substituting 33 into 3, we can obtain d F D SRS 0. Remar. The diversity order of SRS scheme for FD NOMA is zero, which is the same as the conventional FD RS scheme. Corollary 4. For the special case ϖ 0, the asymptotic outage probability of HD-based NOMA SRS scheme with e x x at high SNR is given by P HD, SRS π N N n ϕ n + τ c n + d α + + d α τ ] K. 34 Substituting 34 into 3, we can obtain d HD SRS K. Remar. The diversity order of SRS scheme for HD NOMA is K, which provides a diversity order equal to the number of the available relays. Two-stage Relay Selection Scheme: As such, we can derive asymptotic outage probability of TRS scheme for FD NOMA in the following corollary. Corollary 5. The asymptotic outage probability of FD-based NOMA TRS scheme at high SNR is given by P F D, T RS K 0 ] K χ ψ χ] K χ χ]. 35 Proof: See Appendix C. Upon substituting 35 into 3, we obtain d F D T RS 0. Remar 3. The zero diversity order of TRS scheme for FD NOMA is obtained, which is the same as the FD-based SRS scheme. Corollary 6. For the special case ϖ 0, the asymptotic outage probability of HD-based NOMA TRS scheme with e x x at high SNR is given by K P HD, T SR K cn ξ τ ] + + d α ξ τ c n 0 + dα τ + + d α ξ + d α τ ] τ c n + ] dα τ ξ τ c n K τ c n. τ c n 36 Upon substituting 36 into 3, we obtain d HD T RS K. Remar 4. The diversity order of TRS scheme for HD cooperative NOMA is K, which is the same as the HD-based SRS scheme. 3 Rom Relay Selection Scheme: For SRS scheme, based on the analytical results in 33 34, the asymptotic outage probability of RRS scheme for FD/HD NOMA with K can be given by N P F D, RRS,SRS π N P HD, RRS,SRS n ϕ n + cn ρτω LI + c n ρτω LI π N, 37 N n ϕ n + τ c n + d α + + d α τ, 38 respectively. For TRS scheme, based on the analytical results in 35 36, the asymptotic outage probability of RRS scheme for FD/HD NOMA with K can be given by P F D, RRS,T RS 0 ] χ ψ χ] χ χ], 39

8 8 Duplex mode RS scheme D Application scenario SRS 0 Small pacet service FD NOMA TRS 0 Bacground tass RRS 0 SRS K Small pacet service HD NOMA TRS K Bacground tass RRS TABLE I: Diversity orders application scenarios for FD/HD-based NOMA RS schemes. P HD, RRS,T SR cn ξ τ ] + + d α ξ τ c n respectively. 0 + dα τ + + d α ξ + d α τ ] τ c n + ] dα τ ξ τ c n τ c n τ c n, 40 Remar 5. Substituting 37, 38 39, 40 into 3, we can observed that the diversity orders of RRS scheme for FD-NOMA HD-NOMA are zero one, respectively. In order to get intuitional insights, as shown in TABLE I, the diversity orders application scenarios of FD/HDbased NOMA RS schemes are summarized to illustrate the comparison between them. For the sae of simplicity, we use D to represent the diversity order. E. Throughput Analysis In this subsection, the delay-limited transmission modes of these RS schemes are investigated for FD/HD NOMA networs. In this mode, the BS sends information at a constant rate the system throughput is subjective to the effect of outage probability. Hence it is significant to discuss the system throughput for delay-limited mode in practical scenarios. Proposition. Based on above explanation, the system throughput of the RS schemes for FD/HD NOMA are given by PΨ F D RD + PΨ F D RD, 4 R F D Ψ R HD Ψ P HD Ψ RD + PΨ HD RD, 4 respectively, where Ψ {SRS, T RS}. R SRS R T RS are system throughputs of single-stage two-stage RS schemes, respectively. IV. NUMERICAL RESULTS In this section, our numerical results are provided for characterizing the outage performance of these two inds of RS schemes. Monte Carlo simulation parameters used in this section are summarized in Table II, 4], in which BPCU is short for bit per channel use. The complexity-vs-accuracy Monte Carlo simulations repeated 0 6 iterations Power allocation coefficients of NOMA a 0., a 0.8 Targeted data rates R D, R D 0. BPCU Pass loss exponent α The radius of a disc region R D m The distance between the BS D 0 m The distance between the BS D m TABLE II: Table of Parameters for numerical results SRS OMA RS scheme FD RRS scheme HD RRS scheme FD Error floor FD SRS scheme HD Asymptotic HD SRS scheme RRS SNR db Fig. : Outage probability versus the transmit SNR for SRS scheme with K ; R D, R D 0. BPCU E{ h LI } 0 db. tradeoff parameter is N 5. Except FD/HD-based NOMA RRS schemes, the performance of OMA-based RS scheme is also shown as a benchmar for comparison, where the total communication process is finished in four slots. In the first slot, the BS sends information x to relay R i send x to R i in the second slot. In the third fourth slot, R i decodes forwards the information x x to D D, respectively. Adding the performance of AF-based RS schemes for comparison will further enrich this paper, but this is beyond the scope of this paper. Note that NOMA users with low target data rate can be applied to the IoT scenarios, i.e., the low energy consumption, small pacet service so on. A. Single-stage Relay Selection Scheme In this subsection, the FD/HD-based NOMA RRS schemes OMA-based RS schemes are regarded as the baselines for comparison purposes. Fig. plots the outage probability of SRS scheme versus S- NR for a simulation setting with K E{ h LI } 0 db. The blac blue solid curves are the SRS scheme for FD/HD NOMA, corresponding to the approximate results derived in 3, respectively. The dash dotted curves represent the approximate outage probabilities of RRS schemes for FD/HD NOMA derived in 30 3, respectively. Obviously, the outage probability curves match precisely with the Monte Carlo simulation results. It is observed that the performance of FD-based NOMA SRS scheme is superior

9 R, R 0. BPCU R R BPCU R,R 0.5 BPCU Error floor Asymptotic FD SRS scheme HD SRS scheme K 0 4 FD Error floor FD SRS scheme HD Asymptotic HD SRS scheme SNR db Fig. 3: Outage probability versus the transmit SNR for SRS scheme with the different target rates; K E{ h LI } 0 db. 0 6 K3 K SNR db Fig. 4: Outage probability versus the transmit SNR for SRS scheme with K, 3, 4; R D, R D 0. BPCU E{ h LI } 0 db. to HD-based NOMA scheme on the condition of low SNR region. The reason is that loop interference is not the dominant impact factor for FD cooperative NOMA in the low SNR region. Moreover, the outage performance of the HD-based NOMA SRS scheme outperforms the HD-based RRS scheme. Another observation is that HD-based NOMA SRS scheme is superior to OMA-based RS scheme. This is due to the fact that HD-based NOMA RS schemes is capable of enhancing the spectral efficiency compared to OMA. The asymptotic outage probability cures of the SRS schemes for FD/HD NOMA are plotted according to the analytical results in 33 34, respectively. One can observe that the asymptotic curves well approximate the analytical performance curves in the high SNR region. It is worth noting that an error floor exists in the FD-based NOMA SRS scheme, which verifies the conclusion in Remar obtain zero diversity order. This is due to the fact that there is the loop interference in FD NOMA. Fig. 3 plots the outage probability of SRS scheme with different target rates. One can observe that adjusting the target rates of NOMA users will affect the outage behaviors of the FD/HD-based SRS schemes. As the value of target rates increases, the superior of FD/HD-based NOMA SRS schemes becomes not obvious. It is worth noting that based on the application requirements of different scenarios, the setting of reasonable target rates for NOMA users is prerequisite. Fig. 4 plots the outage probability of SRS scheme versus SNR for a simulation setting with K, 3, 4 relays E{ h LI } 0 db. As can be seen that the analytical curves perfectly match with the simulation results, while the approximations match the analytical performance curves in the high SNR region. It is shown that the number of relays in the networs considered strongly affect the performance of FD/HD-based NOMA SRS schemes. With the number of relays increasing, the lower outage probability are achieved by this RS scheme. This is because more relays bring higher Error floor Asymptotic FD SRS scheme HD SRS scheme E{ h LI } 0, 5, 0 db SNR db Fig. 5: Outage probability versus the transmit SNR for SRS scheme with K 3; R D, R D 0. BPCU. diversity gains, which improves the reliability of the cooperative networs. Another observation is that the HD-based NOMA SRS scheme provides a diversity order that is equal to the number of the relays K, which verifies the conclusion in Remar. As a further development, Fig. 5 plots the outage probability of SRS scheme versus different values of LI from E{ h LI } 0 db to E{ h LI } 5 db. As observed from the figure, we can see that the value of LI also strongly affect the performance of FD-based SRS scheme for NOMA, while the HD-based SRS scheme is not affected. This is due to the fact that LI is not existent for the HD-based SRS scheme with ϖ 0. As the value of LI becomes larger, the outage performance of the FD-based SRS scheme becomes more worse. In consequence, it is significant to consider the

10 0 System Throughput BPCU K, 3, 4 E{ h LI } 0, 0, 5 db 0. OMA SRS scheme HD SRS scheme FD SRS scheme FD SRS scheme SNR db Fig. 6: System throughput in delay-limited transmission mode versus SNR for the SRS scheme R 0.5,R 0. BPCU FD Error floor FD TRS scheme HD Asymptotic HD TRS scheme R,R 0. BPCU R 0.7,R 0. BPCU SNR db Fig. 8: Outage probability versus the transmit SNR for TRS scheme with different target rates; K 3 E{ h LI } 0 db TRS OMA RS scheme HD RSS scheme FD RSS scheme FD Error floor FD TRS scheme HD Asymptotic HD TRS scheme RRS SNR db Fig. 7: Outage probability versus the transmit SNR for TRS scheme with K 3; R D, R D 0. BPCU E{ h LI } 0 db FD Error floor HD Asymptotic FD TRS scheme HD TRS scheme SNR db Fig. 9: Outage probability versus the transmit SNR for TRS scheme with K, 3, 4; R D, R D 0. BPCU E{ h LI } 0 db. K K4 K3 influence of LI in the practical FD NOMA networs. Fig. 6 plots system throughput versus SNR in delay-limited transmission mode for the different number of relays from K to K 4 with E{ h LI } 0 db. The blue solid red dashed curves represent throughput of SRS scheme for FD/HD NOMA networs which are obtained from 4 4, respectively. One can observe that the FD-based SRS scheme achieves a higher throughput compared to the HDbased SRS scheme for NOMA networs. This is because that the value of LI has a smaller influence for the outage behavior of FD NOMA in the low SNR region. Furthermore, the FD/HD-based NOMA SRS schemes outperform OMA-based RS scheme in terms of system throughput. This is due to the fact that NOMA-based SRS scheme can provide more spec- trum efficiency than OMA-based SRS scheme. As the number of relays becomes larger, the FD/HD-based SRS schemes can improve the system throughput. This phenomenon can be explained as that a lower outage probability can be obtained by the FD/HD-based SRS schemes. In addition, Fig. 6 further give system throughput in delay-limited transmission mode for the different values of LI with K 3. As can be observed that increasing the values of LI from E{ h LI } 0 db to E{ h LI } 5 db reduces the system throughput. This phenomenon indicates that it is of significance to consider the impact of LI for FD-based SRS scheme when designing practical cooperative NOMA systems.

11 E{ h LI } 0, 5, 0 db FD Error floor HD Asymptotic FD TRS scheme HD TRS scheme SNR db Fig. 0: Outage probability versus the transmit SNR for TRS scheme with K 3; R D, R D 0. BPCU. System Throughput BPCU K, 3, 4 E{ h LI } 5, 0, 0 db 0. OMA RS scheme HD TRS scheme FD TRS scheme FD TRS scheme SNR db Fig. : System throughput in delay-limited transmission mode versus SNR for the TRS scheme with R D, R D 0. BPCU. B. Two-stage Relay Selection Scheme In this subsection, except FD/HD-based NOMA RRS scheme, the outage performance of OMA-based RS scheme is also shown as a benchmar for comparison. Fig. 7 plots the outage probability of TRS scheme versus SNR with setting to be K 3 E{ h LI } 0 db. The approximate analytical curves of the TRS schemes for FD/HD NOMA are plotted based on 8 9, respectively. As can be observed from the figure, the analytical curves match perfectly with Monte Carlo simulation results. We confirm that the higher outage performance can be obtained by FD-based NOMA TRS scheme in the low SNR region. This is due to fact that there is a low loop interference for FD-based TRS scheme does not suffer from bwidth-loss influence. One can observe that the outage behaviors of FD/HD-based NOMA TRS schemes outperform OMA-based RS scheme. The asymptotic outage probability curves of FD/HD-based NOMA TRS scheme are plotted according to 35 36, which are practically indistinguishable from the analytical results. It is also observed that the FD-based TRS scheme for NOMA converges to an error floor obtains the zero diversity, which is due to the fact that the loop interference exists at the relay. This phenomenon is confirmed by the insights in Remar 3. However, the HD-based TRS scheme for NOMA overcomes the problem of zero diversity inherent to FD-based scheme. As shown in Fig. 3, Fig. 8 plots the outage probability of TRS scheme with different target rates. It is shown that when the target rates of NOMA users is reduced, the FD/HD-based NOMA TRS schemes is capable of providing better outage performance. We confirm that the IoT scenarios i.e., small pacet service considered can be supported by the NOMAbased RS schemes. Fig. 9 plots the outage probability of TRS scheme versus SNR for a simulation setting to be K, 3, 4 relays E{ h LI } 0 db. We observed that the number of relays affect the performance of TRS scheme. With the number of relays increasing, the superiority of FD/HD-based NOMA TRS schemes is apparent the lower outage probabilities are obtained. We also see that the HD-based RS scheme is capable of achieving a diversity order of K, which confirms the insights in Remar 4. From a practical perspective, it is important to consider multiple relays in the networs when designing the NOMA RS systems. Fig. 0 plots the outage probability of the TRS scheme versus different values of LI from E{ h LI } 0 db to E{ h LI } 0 db. We also can observe that with the value of LI increasing, the superior of outage performance for the FD-based TRS scheme is not existent. Fig. plots system throughput versus SNR in delay-limited transmission mode for the different number of relays from K to K 4 with E{ h LI } 0 db. The solid blac dashed magenta curves represent throughput of TRS for FD/HD NOMA networs which are obtained from 4 4, respectively. We can also observe that FD-based NOMA TRS scheme has a higher throughput than HD-based scheme in the low SNR region. The reason is that the FD-based TRS scheme is capable of achieving a lower outage probability compared to HD-based scheme. Moreover, the throughput of FD/HD-NOMA TRS schemes precedes that of OMA-based RS scheme. Additionally, it is worth pointing out that adjusting the size of target data rate i.e., R D R D will affect the system throughput for delay-limited transmission mode. The main performance of TRS scheme trends follow those in Fig. 6. Additionally, as can be seen from the figure that increasing the values of LI from E{ h LI } 0 db to E{ h LI } 0 db reduces the system throughput the existence of the throughput ceilings in the high SNR region. This is due to the fact that the FD-based TRS scheme converges to the error floor.

12 V. CONCLUSIONS This paper has investigated a pair of RS schemes for FD/HD NOMA networs insightfully. Stochastic geometry has been employed for modeling the locations of relays in the networ. New analytical expressions of outage probability for two RS schemes have been derived. Due to the influence of LI at relay, a zero diversity order has been obtained by these two RS schemes for FD NOMA. Based on the analytical results, it was demonstrated that the diversity orders of HDbased RS schemes were determined by the number of relays in the networs considered. results showed that the FD/HD-based NOMA SRS/TRS schemes are capable of providing better outage behaviors than RRS OMA-based RS schemes. The system throughput of delay-limited transmission mode for FD/HD-based NOMA RS schemes were discussed. The setting of perfect SIC operation my bring about overestimated performance for the RS schemes considered, hence our future treaties will consider the impact of imperfect SIC. Another promising future research direction is to optimize the power allocation between NOMA users, which can further enhance the performance of NOMA-based RS schemes. APPENDIX A: PROOF OF THEOREM Let W i min {γ D R i, γ D D, γ D }, W max {W, W..., W N }, then Pr W < γth F D Pr max {W, W,..., W N } < γ F D K Pr W i < γth F D. A. i th Hence the outage probability of the FD-based SRS scheme only requires Pr W i < γ F D th, which is given by Pr W i < γth F D Pr min {γd R i, γ D D, γ D } < γth F D where ϖ. Define X i h SR i Pr γ D R i > γth F D }{{ } J Pr γ D D th } {{ } J +d α SR i, Y i h R i D +d α R i D, Y i h R i D Pr γd > γth F D }{{ } J 3, A. +d, Z α R i D h LI. As stated in 8, 0] by utilizing the polar coordinate, the CDF F Xi of X i is given by F Xi x R D RD 0 e +rα x rdr. A.3 However, for many communication scenarios α >, A.3 does not have a closed-form solution. In this case, the approximate expression of A.3 can be obtained by using Gaussian- Chebyshev quadrature 43] given by F Xi x π N N n e c n x ϕ n +. A.4 Substituting 7 into A. applying algebraic manipulations, J can be further expressed as follows: J Pr X i < ρϖf Z z + τ, A.5 where f Z z Ω LI. By the virtue of approximate expression of CDF for X i in A.4, J is calculated as J 0 π N Ω LI e z e z Ω LI F Xi ρϖz + τ dz Ω LI N e cnτ n ϕ n +. + ϖρτc n Ω LI A.6 On the condition of d Ri D j d SR i + d j d jd SRi cos θ i d Ri D j d SRi, j,, to further simplify computational complexity, we assume that the distance between R i D j can be approximated as the distance between the BS D j, i.e., d Ri D j d j. It is worth noting through this approximation, the distance d j between the BS D j is a fixed value. Hence we can obtain the corresponding approximate CDF of F Yji i.e., F Yji e +dα j τ. Upon substituting 5 7 into A., J J 3 are approximated by J Pr Y i > τ e +dα τ, J 3 Pr Y i > τ e +dα τ, A.7 A.8 respectively. Combining A.6, A.7, A.8, we can calculate Pr W i < γth F D. Finally, substituting A. into A., we can obtain. The proof is completed. APPENDIX B: PROOF OF LEMMA Based on 4, the conditional probability Θ can be expressed as Θ Pr s i < R D S R Pr min {γ D R i, γ D } < γth F D i SR, SR > 0 Pr γ D R i < γ D, γ D R i < γth F D γ D R i > γth F D } th, γ D D {{ } J + Pr γ D < γ D R i, γ D < γth F D γ D R i > γth F D } th, γ D D {{ } J 3 B. where ϖ γ F D th R D with R D being the target rate of D. According to the definition of conditional probability, J can be expressed as J Pr γ D R i < γ D, γ D R i < γ F D th γ D R i Pr γ D R i th, γ D D > γth F D th, γ D D th. B.

13 3 Define the numerator denominator of J in B. by Ξ Ξ, respectively. Substituting, 3, 5 7 to B. applying some algebraic manipulations, we rewrite Ξ as follows: Ξ Pr X i < Y i ρϖz +, X i < ξ ρϖz +, X i > τ ρϖz +, Y i > τ Pr τ ρϖz + < X i < ξ ρϖz +, Y i > θ + Pr τ ρϖz + < X i < Y i ρϖz +, τ < Y i < ξ 0 f Z z θ f Yi y F Xi ξ ρϖz + F Xi τ ρϖz + ] dydz } {{ } + 0 f Z z ξ τ J f Yi y F Xi y ρϖz + F Xi τ ρϖz + ] dydz, }{{} J 3 B.3 On the basis of Appendix A, for an arbitrary choice of α, we can use Gaussian-Chebyshev quadrature to find the approximation for the CDF of X i in A.4. In addition, d RiD d + d SR i d d SRi cos θ i d Ri D d SRi, we can approximate the distance between R i D as d Ri D d. The approximation for pdf of Y i is given by f Yi y e +dα τ. B.4 Substituting A.4 B.4 into B.3, J J 3 can be calculated as follows: J e +dα θ π N N n ϕ n + e z Ω LI e cnτρϖz+ e c nξρϖz+ dz 0 Ω LI e +dα θ π N N n ϕ n + χe cnτ ψe cnξ], B.5 where χ +ρϖτc n Ω LI ψ +ρϖξc n Ω LI. J 3 π + dα N 0 N n ϕ n + e Ω LI z Ω LI ξ e c nyρϖz+ +d α y dydz τ e cnτρϖz+ +dα y χe c nτ e +dα τ e +dα ξ z z + T 0 z + ζ e ψω LI dz Φ 0 z + ζ e χω LI dz, }{{}}{{} I I B.6 where π N N +d α e cn++dα ξ ρϖc n Ω LI by n ϕ n +, ζ c n++d α ρϖc n T α Φ +dα e cn++d τ ρϖc n Ω LI. By the virtue of 4, Eq ], I I can be given I e I e ζ ψω LI Ei ζ χω LI Ei ζ ψω LI ζ χω LI, B.7, B.8 respectively. Substituting B.7 B.8 into B.6, we can obtain J 3 χe c nτ e +dα τ e ξ +dα Te ζ ψω ζ LI Ei + Φe ζ χω ζ LI Ei ψω LI χω LI. B.9 Applying the results derived in Appendix A, the denominator Ξ for J in B. can be approximated as follows: Ξ e +dα τ χe c nτ. Combining B.5, B.9 B.0, we can obtain J e +d α θ χe cnτ ψe c nξ e +dα τ χe c nτ + χe c nτ e +d α τ e ξ +dα e +dα τ χe cnτ Te ζ ψω LI ζ e +dα τ χe c nτ E i Φe ζ χω LI ζ + e +dα τ χe c nτ E i ψω LI χω LI Similarly as the above derived process, we can obtain e +dα τ e +dα ξ J 3 e +dα τ χe c nτ e +d α τ e ξ +dα e +dα τ χe c nτ Te ζ Ω LIψ ζ E i Ω LIψ + Φe ζ Ω LIχ E i ζ Ω LIχ e +dα τ χe c nτ B.0. B.. B. Combining B. B., we can obtain 5. The proof is completed. APPENDIX C: PROOF OF COROLLARY 5 Based on the derived results in Appendix B, the proof starts by providing the term J with ϖ as follows: J e +dα θ π N N n ϕ n + e cnτ e cnξ. C. + ρτc n Ω LI + ρξc n Ω LI

14 4 To facilitate our asymptotic analysis, when x 0, we use zero order series expansion to approximate the exponential function e x, i.e., e x. Therefore, J can be further approximated as follows: J π N N n ϕ n +. C. + ρτc n Ω LI + ρξc n Ω LI Similar as C., J 3 J 3 can be further approximated by utilizing zero order series expansion as follows: J 3 0, J 3 0, C.3 C.4 respectively. Additionally, the denominator Ξ for J in B., Ξ χ, C.5 Substituting C., C.5, C.3 C.4 into B., the conditional probability Θ can be obtained as follows: Θ χ ψ χ. C.6 Using a similar approximation method as that used to obtain C.6, Θ is given by K Θ χ K χ. C.7 Substituting C.6, C.7 into 4 applying some manipulations, we can obtain 35. The proof is completed. REFERENCES ] X. Yue, Y. Liu, R. Liu, A. Nallanathan, Z. Ding, Full/half-duplex relay selection for cooperative NOMA networs, in Proc. IEEE Global Commun. Conf. GLOBECOM, accepted, Singapore, SG, Dec. 07. ] Q. C. Li, H. Niu, A. T. Papathanassiou, G. 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