A Unified Framework For Non-Orthogonal Multiple Access

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5346IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 66, NO. 11, NOVEMBER 2018A Unified Framework for Non-OrthogonalMultiple AccessXinwei Yue , Zhijin Qin , Member, IEEE, Yuanwei Liu , Member, IEEE,Shaoli Kang, and Yue Chen, Senior Member, IEEEAbstract— This paper proposes a unified framework ofnon-orthogonal multiple access (NOMA) networks. Stochasticgeometry is employed to model the locations of spatially NOMAusers. The proposed unified NOMA framework is capableof being applied to both code-domain NOMA (CD-NOMA)and power-domain NOMA (PD-NOMA). Since the detection ofNOMA users mainly depend on efficient successive interferencecancelation (SIC) schemes, both imperfect SIC (ipSIC) andperfect SIC (pSIC) are taken into account. To characterizethe performance of the proposed unified NOMA framework,the exact and asymptotic expressions of outage probabilitiesas well as delay-limited throughput for CD/PD-NOMA withipSIC/pSIC are derived. In order to obtain more insights,the diversity analysis of a pair of NOMA users (i.e., the nth userand mth user) is provided. Our analytical results reveal that:1) the diversity orders of mth and nth user with pSIC forCD-NOMA are mK and nK, respectively; 2) due to theinfluence of residual interference, the nth user with ipSIC obtainsa zero diversity order; and 3) the diversity order is determinedby the user who has the poorer channel conditions out of thepair. Finally, Monte Carlo simulations are presented to verify theanalytical results: 1) when the number of subcarriers becomeslager, the NOMA users are capable of achieving more steep slopein terms of outage probability and 2) the outage behavior ofCD-NOMA is superior to that of PD-NOMA.Index Terms— A unified framework, imperfect/perfect successive interference cancelation, non-orthogonal multiple access,stochastic geometry.I. I NTRODUCTIONITH the rapid increase of requirement for theInternet-enabled smart devices, applications and services, the fifth generation (5G) mobile communicationWManuscript received October 29, 2017; revised March 12, 2018 andMay 9, 2018; accepted May 13, 2018. Date of publication May 31, 2018;date of current version November 16, 2018. This paper was presented in partat the IEEE ICC 2018 [1]. The associate editor coordinating the review ofthis paper and approving it for publication was M. Di Renzo. (Correspondingauthor: Zhijin Qin.)X. Yue is with the School of Information and Communication Engineering,Beijing Information Science and Technology University, Beijing 100101,China (e-mail: [email protected]).Z. Qin is with the School of Computing and Communications, LancasterUniversity, Lancaster LA1 4YW, U.K. (e-mail: [email protected]).Y. Liu and Y. Chen are with the School of Electronic Engineering andComputer Science, Queen Mary University of London, London E1 4NS, U.K.(e-mail: [email protected]; [email protected]).S. Kang is with the School of Electronic and Information Engineering, Beihang University, Beijing 100191, China, and also with the StateKey Laboratory of Wireless Mobile Communications, China Academy ofTelecommunications Technology (CATT), Beijing 100094, China (e-mail:[email protected]).Color versions of one or more of the figures in this paper are availableonline at http://ieeexplore.ieee.org.Digital Object Identifier 10.1109/TCOMM.2018.2842217networks have sparked a great deal of attention in both academia and industry. The application of new radio scenarios [2],i.e., ultra-reliable and low latency communications, massive machine type communications and enhanced mobilebroadband, aims to satisfy the different requirements for5G networks [3], [4]. In particular, the design of novelmultiple access (MA) is desired to enhance spectrum efficiency and massive connectivity. Non-orthogonal multipleaccess (NOMA) [5] has been identified as one of the keytechnologies in 3GPP Long Term Evolution, which has beenstandard application in downlink multiuser superposition transmission scenarios [6]. The primary feature of NOMA isits capability of achieving the higher spectrum efficiency,in which multiple users’ signals are linearly superposed overdifferent power levels by using the superposition codingscheme [7], and then transmitted in the same time/frequenceresource element (RE). To get the desired signal, multiuser detection algorithm [8], [9] (e.g., successive interferencecancelation (SIC) or message passing algorithm) is carried outat the receiver.Up to now, NOMA techniques have been investigatedextensively. Based on spreading signature of MA, NOMAschemes can be divided into two categories: power-domainNOMA (PD-NOMA) and code-power NOMA (CD-NOMA).1More particularly, the point-to-point PD-NOMA has beensurveyed in detail in [10]–[13]. Two evaluation metrics ofPD-NOMA networks including outage probability and ergodicrate have been proposed in [10], where the outage behaviorsof users and ergodic rate have been discussed by applying stochastic geometry. Furthermore, the impact of userpair with fixed power allocation for PD-NOMA has beencharacterized in terms of outage probability in [11]. It hasbeen shown that when the selected user pairing have moredisparate channel conditions, PD-NOMA is capable of providing more performance gain. From a practical perspective,Yang et al. [12] studied the performance of PD-NOMA forthe two-user case with imperfect channel state information,where the closed-form and approximate expressions of outageprobability and ergodic sum rate were derived, respectively.On the condition that the NOMA users have similar channel1 The superposition of signals for multiple users can be mapped to singlesubcarrier or multiple subcarriers. Driven by this, NOMA can also be classifiedas single carrier NOMA (SC-NOMA) and multi-carrier NOMA (MC-NOMA).More specifically, SC-NOMA and MC-NOMA are equivalent to PD-NOMAand CD-NOMA, respectively.0090-6778 2018 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.See http://www.ieee.org/publications standards/publications/rights/index.html for more information.

YUE et al.: UNIFIED FRAMEWORK FOR NON-ORTHOGONAL MULTIPLE ACCESSconditions, Ding et al. [13] proposed a PD-NOMA basedmulticast-unicast scheme and verified that the spectral efficiency of PD-NOMA based multicast-unicast scheme is higherthan that of orthogonal multiple access (OMA) based one.To evaluate the performance of uplink PD-NOMA, in [14],the coverage probability performance of the NOMA userswas discussed in large scale cellular for uplink PD-NOMAby invoking poisson cluster processes, where both imperfectSIC (ipSIC) and perfect SIC (pSIC) were taken into considered. By applying the concept of NOMA to cooperative communications, cooperative NOMA was first introduced in [15],where the nearby users with better channel conditions wereregarded as decode-and-forward relay to deliver the signalsfor the distant users. To further improve spectrum efficiency,Yue et al. [16] studied the outage behavior and ergodicrate of PD-NOMA, where user relaying can switch betweenfull-duplex mode and half-duplex mode based on applicationrequirements.As adopted by many 5G MA concepts, CD-NOMA mainlyinclude sparse code multiple access (SCMA), pattern divisionmultiple access (PDMA), multi-user sharing access (MUSA),interleave division multiple access (IDMA), etc. Actually,CD-NOMA is viewed as a special extension of PD-NOMA,in which the data streams of multiple users are directlymapped into multiple REs (or K subcarriers) through thesparse matrix/codebook or low density spread sequence. Morespecifically, in [17], the modulation symbols of NOMAusers are directly mapped into sparse codebook by invokingmultidimensional constellation, where a sub-optimal designapproach was proposed to design the sparse codebook ofSCMA. Considering user pair and power sharing, the systemthroughput of heavily loaded networks has been improvedin [18] by adopting SCMA for donwlink transmission scenarios. To perform the green analysis of SCMA scheme,Zhang et al. [19] have analyzed the energy efficiency andoutage behavior by proposing the unified framework in fadingchannels. With the goal of maximizing the ergodic sum rate,an optimal sparse matrix of SCMA system has been designedin [20]. Moreover, the performance of uplink SCMA systemhas been characterized in terms of average symbol error ratewith randomly deployed users in [21].For another special case, the thought of PDMA was first proposed in [22], where the joint design of sparse matrix and SICbased detector has been considered at the transmitting end andreceiving end, respectively. From the perspective of link leveland system level, the evaluated results confirmed that PDMAis capable of achieving the enhanced spectrum efficiency overOMA. In the case of given sparse matrix, a novel link estimation scheme for uplink PDMA systems was proposed in [23]based on interference cancelation receiver. It was shown thatthe proposed estimation scheme can achieve accurate performance compared to conventional method. With the aid ofpSIC, Tang et al. [24] studied the outage behavior of cooperative uplink PDMA systems by employing one fixed dimension of sparse matrix. As the further special cases [25], [26],in [25], the data symbols of each user for MUSA systemsare spread to a set of complex spreading sequences andthen superposed at transmitter. The design of low-correlation5347spreading sequence is to deal with the higher overloading ofusers and to carry out SIC expediently at receiver. Exploitingthe low-rate coded sequence, the bit error rate of IDMAsystems based on semi-analytical technique has been discussedin [26]. Furthermore, the performance of cooperative IDMAsystems is characterized in terms of bit error probabilityin [27]. Very recently, the progresses of CD/PD-NOMA for5G networks have been surveyed in [28]–[30], which havesummarized potentials and challenges from the perspective ofimplementation.A. Motivations and ContributionsWhile the above-mentioned research contributions have laida solid foundation for a good understanding of PD-NOMAand CD-NOMA techniques, a unified framework for NOMAnetworks is far from being well understood. In [10], it isdemonstrated that the diversity order of the sorted NOMAuser, i.e., the m-th user is m, which is directly combined withthe users’ channel conditions. However, only the performanceof PD-NOMA has been discussed. Qin et al. [31] have proposed user association and resource allocation schemes for theunified NOMA enabled heterogeneous ultra-dense networks.Moreover, the above contributions for NOMA networks havecomprehensively concentrated on the assumption of pSIC.In practice, the assumption of pSIC might not be valid atreceiver, since there still exist several potential implementationissues by using SIC, i.e., error propagation and complexityscaling. Hence it is significant to examine the detrimentalimpacts of ipSIC for the unified NOMA framework. To thebest of our knowledge, there is no existing work investigatingthe unified NOMA network performance, which motivates usto develop this treatise. In addition, new connection outageprobability (COP) is defined as an evaluation metric for theunified NOMA framework. The essential contributions of ourpaper are summarized as follows:1) We derive the exact expressions of COP for a pair ofusers, i.e., the n-th user and m-th user in CD-NOMAnetworks. Based on the analytical results, we also derivethe asymptotic COP and obtain the diversity orders.We confirm that the diversity order of the m-th useris equal to mK. Due to the impact of residual interference (RI) from the imperfect cancelation process,the COP of the n-th user with ipSIC for CD-NOMAconverges to an error floor in the high signal-to-noiseratio (SNR) region and obtain a zero diversity order.2) We study the COP of the n-th user with pSIC and derivethe corresponding asymptotic COP for CD-NOMA.On the condition of pSIC, the n-th user is capable ofattaining the diversity order of nK. We confirm thatthe outage performance of the n-th user with pSICis superior to OMA, while the outage performance ofthe m-th user is inferior to OMA. It is shown thatwhen multiple users are served simultaneously, NOMAis capable of providing better fairness.3) We investigate the outage behaviors of the special casePD-NOMA with ipSIC/pSIC for CD-NOMA (K 1).To provide valuable insights, we derive both exact and

5348IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 66, NO. 11, NOVEMBER 2018Fig. 1. An illustration of the unified downlink NOMA transmission networks,where the spatial distributions of NOMA users follow homogeneous BPPs.asymptotic COP of a pair of users for PD-NOMA.We observe that the diversity orders of the n-th userwith ipSIC/pSIC are equal to n and zero, respectively. The m-th user of PD-NOMA obtains the diversityorder of m.4) For the selected user pairing in CD/PD-NOMA networks, we observe that the diversity order is determinedby the user who has the poorer channel conditionsout of the pair. We discuss the system throughput ofCD/PD-NOMA with ipSIC/pSIC in delay-limited transmission mode. When frequency dependent factor η 1,we observe that the outage performance of the n-th userwith ipSIC is superior to OMA in the low SNR region.B. Organization and NotationThe remainder of this paper is organized as follows.In Section II, a unified NOMA framework is presented in thewireless networks, where users are ordered randomly based ontheir channel conditions. In Section III, the exact expressionsof outage probability and delay-limited throughput for a pairof NOMA users are derived. Section IV provides the numerical results to verify the derived analytical results and thenSection V concludes our paper. The proofs of mathematicsare collected in the Appendix.The main notations of this paper are shown as follows:E{·} denotes expectation operation; fX (·) and FX (·) denotethe probability density function (PDF) and cumulative distribution function (CDF) of a random variable X, respecT tively; The superscripts (·) and (·) stand for transpose2and conjugate-transpose operations, respectively; · 2 denotesEuclidean two norm of a vector; diag (·) represents a diagonalmatrix; IK is an K K identity matrix.base station (BS) transmits the information to M randomlyusers. More precisely, the BS directly maps the data streamsof multiple users into K subcarriers or REs by utilizingone sparse spreading matrix GK M (i.e, sparse matrix orcodebook), in which there are a few number of non-zeroentries within it and satisfies the relationship M K.To present straightforward results and analysis, we assume thatthe BS and NOMA users are equipped with a single antenna,3respectively. Furthermore, assuming that the BS is located atthe center of circular cluster denoted as D, with radius RD andthe spatial locations of M users are modeled as homogeneousBinomial point processes (HBPPs) Φl [33], [34]. To facilitateanalysis, we assumed that M users are divided into M/2orthogonal pairs, in which distant user and nearby user canbe distinguished based on their disparate channel conditions.Each pair of users is randomly selected to carry out NOMAprotocol [10]. A bounded path-loss model [33] is employed tomodel the channel coefficients, which is capable of avoiding ofsingularity at small distances from the BS to users. Meanwhile,these wireless links are disturbed by additive white Gaussiannoise (AWGN) with mean power N0 . Without loss of generality, the effective channel gains between the BS and users22over subcarriers are sorted as hM 2 · · · hn 2 · · · 22 hm 2 · · · h1 2 [35], [36] with the assistance of orderstatistics. In this treatise, we focus on the m-th user pairedwith the n-th user for NOMA transmission.B. Signal ModelRegarding the unified NOMA transmission in downlinksingle cell scenario, the BS transmits the superposed signalsto multiple users, where the data stream of each user is spreadover one column of sparse matrix. Hence the observationyϕ [yϕ1 yϕ2 · · · yϕK ]T at the ϕ-th user over K subcarriersis given by yϕ diag (hϕ ) (gn Ps an xn gm Ps am xm ) nϕ , (1)As shown in Fig. 1, we consider a unified downlinkNOMA transmission scenario in a single cell,2 where awhere ϕ (n, m). xn and xm are supposed to be normalizedunity power signals for the n-th and m-th users, respectively,i.e, E{x2n } E{x2m } 1. Assuming the fixed powerallocation coefficients satisfy the condition that am anwith am an 1, which is for fairness considerations.Note that optimal power allocation coefficients [37], [38] arecapable of enhancing the performance of NOMA networks,but it is beyond the scope of this paper. Ps denotes thenormalized transmission power at the BS, i.e., Ps 1.The sparse indicator vector of the ϕ-th user is denoted bygϕ [gϕ1 gϕ2 · · · gϕK ]T , which is one column of GK M .More specifically, gϕk is the subcarrier index, where gϕk 1and gϕk 0 indicate whether the signals are mapped intothe corresponding RE or not. Let hϕ [h̃ϕ1 h̃ϕ2 · · · h̃ϕK ]Tdenotes the channel vector between the BS and ϕ-th userηhoccupying K subcarriers with h̃ϕk 1 dϕkα , where hϕk CN (0, 1) is Rayleigh fading channel gain between the BS andϕ-th user occupying the k-th subcarrier. Additionally, η and α2 It is worth noting that estimating multi-cell scenarios in a unified NOMAframework can further enrich the contents of the paper considered [32], whichis set aside for our future work.3 Note that equipping multiple antennas on the nodes will further enhancethe performance of CD/PD-NOMA networks, but this is beyond the scope ofthis treatise.II. N ETWORK M ODELA. Network Descriptions

YUE et al.: UNIFIED FRAMEWORK FOR NON-ORTHOGONAL MULTIPLE ACCESSare the frequency dependent factor and path loss exponent,respectively. d denotes the distance from BS to ϕ-th user.nϕ CN (0, N0 IK ) denotes AWGN. It is worth noting thatbased on the number of subcarriers, this unified frameworkcan be reduced into CD-NOMA4 (K 1) and PD-NOMA(K 1), respectively. In particular, when K is set to beone, the data streams of multiple users are mapped into onesubcarrier, which can also be selected as a benchmark forCD-NOMA in the following.To maximize the output SNRs and diversity orders,we employ the maximal ratio combiner (MRC) [7] at theϕ-th user over K subcarriers. Note that using MRC is notoptimal but with low computational complexity. Let uϕ (diag(hϕ )gϕ ) diag(hϕ )gϕ , and then the received signal at the ϕ-th usercan be written as ỹϕ uϕ diag (hϕ ) (gn Ps an xn gm Ps am xm ) uϕ nϕ .(2)On the basis of aforementioned assumptions, the signal-plusinterference-to-noise ratio (SINR) at the n-th user to detectthe m-th user’s signal xm is given byγn m ρ diag (hn ) gn 22 an 1γn n 2ρ diag (hn ) gn 2 an2 ρ hI 2 1(4),4 It is worth pointing out that applying multi-dimensional constellations [39],channel coding (i.e., Low-Density Parity-Check (LDPC) codes or Turbocodes) and iterative decoding are capable of providing shaping gain and codinggain, which we may include in our future work.M n M n ( 1)pφn pn p(K 1) !ΩKI p 02ρ diag (hn ) gn 2 an2ρ diag (hn ) gm 2 am 1 y0yK 1 ΩIe(5).The SINR of a typical cell at the m-th NOMA user todecode the information of itself can be expressed asγm 2ρ diag (hm ) gm 2 am2ρ diag (hm ) gn 2 an 1(6).C. Channel Statistical PropertiesIn this subsection, different channel statistical properties arederived under the unified NOMA frameworks [40], which canbe used for deriving the COP in the following sections.Lemma 1: Assuming M users randomly distributed withinthe circular cluster, the CDF Fγm of the m-th user is given byFγm (x) φm U pM m ( 1) bupm p u 1M m p 01 excu ηρ(am xan )K 1 i 0(3),where 0 and 1 denote the pSIC and ipSIC operations,respectively. Note that hI [hI1 hI2 · · · hIK ]T denotes theRI channel vector at K subcarriers with hIk CN (0, ΩI ).On the other hand, the n-th user is not always first detect theinformation of the m-th user and then decode its own signal.At this moment, the n-th user will decode the message ofitself by directly treating the m-th user as interference withoutcarrying out SIC operation. In this case, the correspondingFγipSICnSINR can be expressed as2ρ diag (hn ) gm 2 amPswhere ρ Ndenotes the transmit SNR. For the sake of0brevity, it is assumed that gm and gn have the same columnweights for GK M . The optimization design of sparse matrixand spread sequence is capable of enhancing the performanceof the unified NOMA framework, but this is beyond scope ofthis treatise.By applying SIC technologies [5], the SINR of then-th user, who needs to decode the information of itself isgiven byγn 5349 i 1xcui! ηρ (am xan )m p,(7)M!, where am xan , φm (M m)!(m 1)! (M m)!π2 (θ 1), c,b 1 θ 1 uuuup!(M m p)!2UαRD2u 12 (θu 1) , θu cos2U π and U is a parameterto ensure a complexity-accuracy tradeoff.Proof: See Appendix A.Lemma 2: Assuming M users randomly distributed withinthe circular cluster, the CDF FγipSICof the n-th user withnipSIC is given in (8), as shown at the bottom of this page,where 1.Proof: See Appendix B.Substituting 0 into (8), the CDF FγpSICof the n-th usernwith pSIC is given by UM n M n ( 1)p φnbuFγpSICnpn p u 1p 0 i n pK 1 xcu 1xcu ηρan 1 e. (9)i! ηρani 0M mpIII. P ERFORMANCE E VALUATIONSince the capacity of channel from the BS to thegoal-directed user is less than the target transmission rate,the connection outage will occur [41]. As a consequence,the goal-directed user is incapable of detecting the informationaccurately. In this section, the COP is selected as a metric U u 1 bu 1 e( ρy 1) xcuηρanK 1 i 0 i 1 xcu ( ρy 1)i!ηρann pdy,(8)

5350IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 66, NO. 11, NOVEMBER 2018to evaluate the performance of unified downlink NOMAnetworks. More specially, a pair of NOMA users (i.e., the mth user and n-th user) for CD/PD-NOMA are characterized interms of outage probabilities in the following.A. The COP of the m-th UserThe outage event of the m-th user in the typical cell is thatthe m-th user cannot detect its own information. Hence theCOP of the m-th user for CD-NOMA can be expressed asPm,CD Pr (γm εm ) ,(10)where εm 2Rm 1 and Rm is the target rate of the m-th userin the typical cell.By applying (7), the following theorem provides the COPof the m-th user.Theorem 1: The COP of the m-th user for CD-NOMA isgiven byM m M m ( 1)pPm,CD φmpm pp 0 U i m pK 1 cu 1τ cu τη bu 1 e,i! ηu 1i 0where εn 2Rn 1 with Rn being the target rate at then-th user to detect the m-th user.The following theorem provides the COP of the n-th userwith ipSIC for the downlink CD-NOMA networks.Theorem 2: The COP of the n-th user with ipSIC for EXFin CD-NOMA networks is given by (14), as shown at theεn, ϑ aεnn and 1.bottom of this page, where β ρanProof: See Appendix C.Substituting 0 into (14), the COP of the n-th user withpSIC for EXF in CD-NOMA networks is given by UM n M n ( 1)p pSICbuPn1,CD φnpn p u 1p 0 n pK 1 1 βcu i βcηu 1 e. (15)i! ηi 0Corollary 2: For the special case with K 1, the COP ofthe n-th user with ipSIC for EXF in PD-NOMA networks isgiven by M n pφn M n ( 1) pΩI p 0n p U cu (ϑy β) y e ΩIbu 1 e ηipSICPn1,PD(11)εmwith am εm an .where τ ρ(am εm an )Corollary 1: For the special case with K 1, the COP ofthe m-th user for PD-NOMA is given byPm,P D φm U p τ cuM m ( 1) bu 1 e ηpm p u 1M m p 0m p0n pdy.u 1(16)Substituting 0 into (16), the COP of the n-th user withpSIC for EXF in PD-NOMA networks is given bypSICPn1,PD. φn(12) U n pβcuM n ( 1)p bu 1 e η.pn p u 1M n p 0(17)B. The COP of the n-th User1) Existing Outage Probability Formulation: Considering atwo-user case, the m-th user and n-th user are paired togetherto perform NOMA protocol. The outage for the n-th user canhappen in the following two cases [10], [42]: The n-th user cannot decode the message of them-th user. The n-th user can decode the message of the m-th user,then carries out SIC operations, but cannot decode theinformation of itself.Based the aforementioned descriptions, the COP of then-th user for existing formulation (EXF) can be expressed asPn1,CD Pr {γn m εm } Pr {γn m εm , γn εn } ,(13)ipSICPn1,CDM n M n ( 1)pφn pn p(K 1) ΩKI p 0 y02) Alternative Outage Probability Formulation: However,for the first case, when the decoding process for the m-th userat the n-th user fails, the outage event is not necessarilyhappened. Because the n-th user can still try to decodethe message of itself by treating the m-th user’s signal asinterference without carrying out SIC operations. In otherwords, the previous outage formulation makes the decodingprocedure of the n-th user highly depend the target rateof the m-th user, which ignores one possible case whichcan also support reliable transmission. As such, the alternative outage for the n-th user can happen in the followingtwo cases: The n-th user can not decode the message of them-th user and the message of itself with treating them-th user’s signal as interference.yK 1 ΩIeU u 1 bu1 e cu (ϑy β)ηK 1 i 0 i 1 (ϑy β) cui!ηn pdy.(14)

YUE et al.: UNIFIED FRAMEWORK FOR NON-ORTHOGONAL MULTIPLE ACCESSipSICPn2,CD U i n pK 1 pζcu 1M n ( 1) ζcφnu φnbu 1 e η pn pi!η(K 1) ΩKIp 0u 1i 0 M nUK 1p M n ( 1) 1 (ϑy β) cu wy cu (ϑy β)K 1 ΩIη yebu 1 epn p 0w!ηp 0u 1 U w 0n p K 1M nip M n ( 1) τ cu 1τ cu φnbu 1 e η.pn pi!ηp 0u 1i 0pSIC φnPn2,CDM n M n p 0ipSICPn2,PD φn n p U UK 1p 1 ζcu i M n ( 1) ζcηubu 1 e bupn p u 1i! ηu 1i 0 n p U i K 1K 1 1 βcu i cu 1τ cu βcηu τη 1 e bu 1 ei! ηi! ηu 1i 0i 0 Un pn pU ςcτcM n ( 1)uubu 1 e η bu 1 e η pn p u 1u 1 UM np cu (ϑy β)φn M n ( 1) y e ΩIbu 1 e ηpΩI p 0n p 0u 1M n p 0 n pdy(19)n p . (20) pThe n-th user can decode the message of the m-th user,but cannot detect the information of itself after carryingout SIC operations.By the virtue of previous assumptions, the COP ofthe n-th user for alternative formulation (ALF) can beexpressed asPn2,CD Pr {γn m εm , γn n εn } Pr {γn m εm , γn εn } . (18) The following theorem provides the COP of the n-th userwith ipSIC for the downlink CD-NOMA networks.Theorem 3: The COP of the n-th user with ipSIC for ALFin CD-NOMA networks is given by (19), as shown at the topεnwith an εn am , ζ of this page, where υ ρ(an εn am )min (τ, υ).Proof: See Appendix D.Substituting 0 into (19), the COP of the n-th userwith pSIC for ALF in CD-NOMA networks is given by (20),as shown at the top of this page.Corollary 3: For the special case with K 1, the COP ofthe n-th user with ipSIC for ALF in PD-NOMA networks isgiven by (21).Substituting 0 into (21), as shown at the top ofthis page, the COP of the n-th user with pSIC for ALF inPD-NOMA networks is given bypSICPn2,PD UM n n p M n ( 1)p φn ζcηu bu 1 epn pp 0u 1 U U n p n p βcuτ cu bu 1 e η bu 1 e η. u 15351u 1(22)n pdy(21)Proposition 1: The COP of the selected user pairing withipSIC/pSIC for CD/PD-NOMA are given by ψψ 1 (1 Pm,CD ) 1 Pñ,CD,(23)Pnm,CDand ψψPnm,P 1 (1 P)1 Pm,P DDñ,P D ,(24)respectively, where ψ (ipSIC, pSIC) and ñ (n1, n2).Pm,CD and Pm,P D are given by (11) and (12), respectively.ipSICipSICpSICpSICPn1,CD, Pn1,PD , Pn1,CD and Pn1,P D are given by (14), (15),ipSICipSICpSIC, Pn2,P(16) and (17), respectively. Pn2,CDD , Pn2,CD andpSICPn2,P D are given by (19), (20), (21) and (22), respectively.C. Diversity Order AnalysisTo gain more deep insights, diversity order is usuallyselected to be a matric to evaluate the system performance,which highlights the slope of the curves for outage probabilities varying with the SNRs. Hence the definition of diversityorder is given bylog (P (ρ)),ρ log ρd lim(25)where P (ρ) denotes the asymptotic COP.Corollary 4: Based on analytical result in (11), the asymptotic COP of the m-th user at high SNR for CD-NOMA isgiven by Um bu τ cu KM!1 mK , (26)Pm,CD (M m)!m! u 1 K! ηρwhere represents “be proportional to.”

5352IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 66, NO. 11, NOVEMBER 2018Proof: To facilitate the calculation, define the summa iK 1 τ cu 1τ cution term in (11), i.e., Θ1 1 e η.i! ηi 0 Θ2Applying power series expansion, the summationterm Θ2 i τ cu 1 τ cuηcan be rewritten as Θ2 e . Substitutingi!ηi KΘ2 into Θ1 , when x 0, Θ1 with the approximation of Kτ cu1. As a furthere x 1 x is formulated as Θ1 K!ηdevelopment, substituting Θ1 into (11) and taking the first term (p 0) [43], we obtain (26). Obviously, Pm,CDis a1function of ρ, which is proportional to ρmK . The proof iscompleted.For the special case with K 1, the asymptotic COP ofthe m-th user at high SNR for PD-NOMA is given by Um τ cu M!1 Pm,P D bu m.(27)(M m)!m! u 1ηρRemark 1: Upon substituting (26) and (27) into (25),the diversity orders of the m-th user for CD-NOMA andPD-NOMA are mK and m, respectively.Corollary 5: Based on analytical result in (14), whenρ , the asymptotic COP of the n-th user with ipSIC forEXF in CD-NOMA networks is given byipSIC, Pn1,CD M n M n ( 1)p φny K 1pn p(K 1) ΩK0I p 0 U i K 1 yϑcu 1yϑcu Ωy e Ibu 1 e ηi!ηu 1i 0ipSIC, Pn2,PDipSIC, Pn1,PD M n pφn M n ( 1)pΩI p 0n p U yϑcu y e ΩIbu 1 e η0ndy.u 1(30)Substituting 0 into (30), the asymptotic COP of then-th user at high SNR with pSIC in PD-NOMA networks forEXF is given Un τ cu M!1pSIC, Pn1,P D bu n.(31)(M n)!n! u 0ηρRemark 3: Upon substituting (30) and (31) into (25),the diversity orders of the n-th user with ipSIC/pSIC for EXFin PD-NOMA networks are zero and n, respectively.Corollary 7: The asymptotic COP of the n-th user withipSIC at high SNR for ALF in CD-NOMA networks is givenby (32) as shown at the bottom of this page.Similar to the solvin

5G networks [3], [4]. In particular, the design of novel multiple access (MA) is desired to enhance spectrum effi-ciency and massive connectivity. Non-orthogonal multiple access (NOMA) [5] has been identified as one of the key technologies in 3GPP Long Term Evolution, which has been standard application in downlink multiuser superposition trans-