I Introduction
With the rapid increase of requirement for the Internetenabled smart devices, applications and services, the fifth generation (5G) mobile communication networks have sparked a great deal of attention in both academia and industry. The application of new radio scenarios [2], i.e., ultrareliable and low latency communications, massive machine type communications and enhanced mobile broadband, aims to satisfy the different requirements for 5G networks [3, 4]. In particular, the design of novel multiple access (MA) is desired to enhance spectrum efficiency and massive connectivity. Nonorthogonal 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 transmission scenarios [6]. The primary feature of NOMA is its capability of achieving the higher spectrum efficiency, in which multiple users’ signals are linearly superposed over different power levels by using the superposition coding scheme [7], and then transmitted in the same time/frequence resource element (RE). To get the desired signal, multiuser detection algorithm [8, 9] (e.g., successive interference cancellation (SIC) or message passing algorithm) is carried out at the receiver.
Up to now, NOMA techniques have been investigated extensively. Based on spreading signature of MA, NOMA schemes can be divided into two categories: powerdomain NOMA (PDNOMA) and codepower NOMA (CDNOMA)^{1}^{1}1
The superposition of signals for multiple users can be mapped to single subcarrier or multiple subcarriers. Driven by this, NOMA can also be classified as single carrier NOMA (SCNOMA) and multicarrier NOMA (MCNOMA). More specifically, SCNOMA and MCNOMA are equivalent to PDNOMA and CDNOMA, respectively.
. More particularly, the pointtopoint PDNOMA has been surveyed in detail in [10, 11, 12, 13]. Two evaluation metrics of PDNOMA networks including outage probability and ergodic rate have been proposed in
[10], where the outage behaviors of users and ergodic rate have been discussed by applying stochastic geometry. Furthermore, the impact of user pair with fixed power allocation for PDNOMA has been characterized in terms of outage probability in [11]. It has been shown that when the selected user pairing have more disparate channel conditions, PDNOMA is capable of providing more performance gain. From a practical perspective, the authors in [12] studied the performance of PDNOMA for the twouser case with imperfect channel state information, where the closedform and approximate expressions of outage probability and ergodic sum rate were derived, respectively. On the condition that the NOMA users have similar channel conditions, the authors of [13] proposed a PDNOMA based multicastunicast scheme and verified that the spectral efficiency of PDNOMA based multicastunicast scheme is higher than that of orthogonal multiple access (OMA) based one. To evaluate the performance of uplink PDNOMA, in [14], the coverage probability performance of the NOMA users was discussed in large scale cellular for uplink PDNOMA by invoking poisson cluster processes, where both imperfect SIC (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 were regarded as decodeandforward relay to deliver the signals for the distant users. To further improve spectrum efficiency, the authors of [16] studied the outage behavior and ergodic rate of PDNOMA, where user relaying can switch between fullduplex mode and halfduplex mode based on application requirements.As adopted by many 5G MA concepts, CDNOMA mainly include sparse code multiple access (SCMA), pattern division multiple access (PDMA), multiuser sharing access (MUSA), interleave division multiple access (IDMA), etc. Actually, CDNOMA is viewed as a special extension of PDNOMA, in which the data streams of multiple users are directly mapped into multiple REs (or subcarriers) through the sparse matrix/codebook or low density spread sequence. More specifically, in [17], the modulation symbols of NOMA users are directly mapped into sparse codebook by invoking multidimensional constellation, where a suboptimal design approach was proposed to design the sparse codebook of SCMA. Considering user pair and power sharing, the system throughput of heavily loaded networks has been improved in [18] by adopting SCMA for donwlink transmission scenarios. To perform the green analysis of SCMA scheme, the authors in [19] have analyzed the energy efficiency and outage behavior by proposing the unified framework in fading channels. With the goal of maximizing the ergodic sum rate, an optimal sparse matrix of SCMA system has been designed in [20]. Moreover, the performance of uplink SCMA system has been characterized in terms of average symbol error rate with 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 SIC based detector has been considered at the transmitting end and receiving end, respectively. From the perspective of link level and system level, the evaluated results confirmed that PDMA is capable of achieving the enhanced spectrum efficiency over OMA. In the case of given sparse matrix, a novel link estimation scheme for uplink PDMA systems was proposed in
[23] based on interference cancellation receiver. It was shown that the proposed estimation scheme can achieve accurate performance compared to conventional method. With the aid of pSIC, the authors of [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 systems are spread to a set of complex spreading sequences and then superposed at transmitter. The design of lowcorrelation spreading sequence is to deal with the higher overloading of users and to carry out SIC expediently at receiver. Exploiting the lowrate coded sequence, the bit error rate of IDMA systems based on semianalytical technique has been discussed in [26]. Furthermore, the performance of cooperative IDMA systems is characterized in terms of bit error probability in [27]. Very recently, the progresses of CD/PDNOMA for 5G networks have been surveyed in [28, 29, 30], which have summarized potentials and challenges from the perspective of implementation.Ia Motivations and Contributions
While the abovementioned research contributions have laid a solid foundation for a good understanding of PDNOMA and CDNOMA techniques, a unified framework for NOMA networks is far from being well understood. In [10], it is demonstrated that the diversity order of the sorted NOMA user, i.e., the th user is , which is directly combined with the users’ channel conditions. However, only the performance of PDNOMA has been discussed. In [31], the authors have proposed user association and resource allocation schemes for the unified NOMA enabled heterogeneous ultradense networks. Moreover, the above contributions for NOMA networks have comprehensively concentrated on the assumption of pSIC. In practice, the assumption of pSIC might not be valid at receiver, since there still exist several potential implementation issues by using SIC, i.e., error propagation and complexity scaling. Hence it is significant to examine the detrimental impacts of ipSIC for the unified NOMA framework. To the best of our knowledge, there is no existing work investigating the unified NOMA network performance, which motivates us to develop this treatise. In addition, new connection outage probability (COP) is defined as an evaluation metric for the unified NOMA framework. The essential contributions of our paper are summarized as follows:

We derive the exact expressions of COP for a pair of users, i.e., the th user and th user in CDNOMA networks. Based on the analytical results, we also derive the asymptotic COP and obtain the diversity orders. We confirm that the diversity order of the th user is equal to . Due to the impact of residual interference (RI) from the imperfect cancellation process, the COP of the th user with ipSIC for CDNOMA converges to an error floor in the high signaltonoise ratio (SNR) region and obtain a zero diversity order.

We study the COP of the nth user with pSIC and derive the corresponding asymptotic COP for CDNOMA. On the condition of pSIC, the th user is capable of attaining the diversity order of . We confirm that the outage performance of the th user with pSIC is superior to OMA, while the outage performance of the th user is inferior to OMA. It is shown that when multiple users are served simultaneously, NOMA is capable of providing better fairness.

We investigate the outage behaviors of the special case PDNOMA with ipSIC/pSIC for CDNOMA (). To provide valuable insights, we derive both exact and asymptotic COP of a pair of users for PDNOMA. We observe that the diversity orders of the th user with ipSIC/pSIC are equal to and zero, respectively. The th user of PDNOMA obtains the diversity order of .

For the selected user pairing in CD/PDNOMA networks, we observe that the diversity order is determined by the user who has the poorer channel conditions out of the pair. We discuss the system throughput of CD/PDNOMA with ipSIC/pSIC in delaylimited transmission mode. When frequency dependent factor , we observe that the outage performance of the th user with ipSIC is superior to OMA in the low SNR region.
IB Organization and Notation
The remainder of this paper is organized as follows. In Section II, a unified NOMA framework is presented in the wireless networks, where users are ordered randomly based on their channel conditions. In Section III, the exact expressions of outage probability and delaylimited throughput for a pair of NOMA users are derived. Section IV provides the numerical results to verify the derived analytical results and then Section V concludes our paper. The proofs of mathematics are collected in the Appendix.
The main notations of this paper are shown as follows: denotes expectation operation; and
denote the probability density function (PDF) and cumulative distribution function (CDF) of a random variable
, respectively; The superscripts and stand for transpose and conjugatetranspose operations, respectively;denotes Euclidean two norm of a vector;
represents a diagonal matrix; is an identity matrix.Ii Network Model
Iia Network Descriptions
As shown in Fig. 1, we consider a unified downlink NOMA transmission scenario in a single cell^{2}^{2}2It is worth noting that estimating multicell scenarios in a unified NOMA framework can further enrich the contents of the paper considered [32], which is set aside for our future work., where a base station (BS) transmits the information to randomly users. More precisely, the BS directly maps the data streams of multiple users into subcarriers or REs by utilizing one sparse spreading matrix (i.e, sparse matrix or codebook), in which there are a few number of nonzero entries within it and satisfies the relationship . To present straightforward results and analysis, we assume that the BS and NOMA users are equipped with a single antenna^{3}^{3}3Note that equipping multiple antennas on the nodes will further enhance the performance of CD/PDNOMA networks, but this is beyond the scope of this treatise., respectively. Furthermore, assuming that the BS is located at the center of circular cluster denoted as , with radius and the spatial locations of users are modeled as homogeneous Binomial point processes (HBPPs) [33, 34]. To facilitate analysis, we assumed that users are divided into orthogonal pairs, in which distant user and nearby user can be distinguished based on their disparate channel conditions. Each pair of users is randomly selected to carry out NOMA protocol [10]. A bounded pathloss model [33] is employed to model the channel coefficients, which is capable of avoiding of singularity at small distances from the BS to users. Meanwhile, these wireless links are disturbed by additive white Gaussian noise (AWGN) with mean power . Without loss of generality, the effective channel gains between the BS and users over subcarriers are sorted as [35, 36] with the assistance of order statistics. In this treatise, we focus on the th user paired with the th user for NOMA transmission.
IiB Signal Model
Regarding the unified NOMA transmission in downlink single cell scenario, the BS transmits the superposed signals to multiple users, where the data stream of each user is spread over one column of sparse matrix. Hence the observation at the th user over subcarriers is given by
(1) 
where . and are supposed to be normalized unity power signals for the th and th users, respectively, i.e, . Assuming the fixed power allocation coefficients satisfy the condition that with , which is for fairness considerations. Note that optimal power allocation coefficients [37, 38] are capable of enhancing the performance of NOMA networks, but it is beyond the scope of this paper. denotes the normalized transmission power at the BS, i.e., . The sparse indicator vector of the th user is denoted by , which is one column of . More specifically, is the subcarrier index, where and indicate whether the signals are mapped into the corresponding RE or not. Let denotes the channel vector between the BS and th user occupying subcarriers with , where is Rayleigh fading channel gain between the BS and th user occupying the th subcarrier. Additionally, and are the frequency dependent factor and path loss exponent, respectively. denotes the distance from BS to th user. denotes AWGN. It is worth noting that based on the number of subcarriers, this unified framework can be reduced into CDNOMA^{4}^{4}4It is worth pointing out that applying multidimensional constellations [39], channel coding (i.e., LowDensity ParityCheck (LDPC) codes or Turbo codes) and iterative decoding are capable of providing shaping gain and coding gain, which we may include in our future work. () and PDNOMA (), respectively. In particular, when is set to be one, the data streams of multiple users are mapped into one subcarrier, which can also be selected as a benchmark for CDNOMA in the following.
To maximize the output SNRs and diversity orders, we employ the maximal ratio combiner (MRC) [7] at the th user over subcarriers. Note that using MRC is not optimal but with low computational complexity. Let , and then the received signal at the th user can be written as
(2) 
On the basis of aforementioned assumptions, the signalplusinterferencetonoise ratio (SINR) at the th user to detect the th user’s signal is given by
(3) 
where denotes the transmit SNR. For the sake of brevity, it is assumed that and have the same column weights for . The optimization design of sparse matrix and spread sequence is capable of enhancing the performance of the unified NOMA framework, but this is beyond scope of this treatise.
By applying SIC technologies [5], the SINR of the th user, who needs to decode the information of itself is given by
(4) 
where and denote the pSIC and ipSIC operations, respectively. Note that denotes the RI channel vector at subcarriers with .
On the other hand, the th user is not always first detect the information of the
th user and then decode its own signal. At this moment, the
th user will decode the message of itself by directly treating the th user as interference without carrying out SIC operation. In this case, the corresponding SINR can be expressed as(5) 
The SINR of a typical cell at the th NOMA user to decode the information of itself can be expressed as
(6) 
IiC Channel statistical properties
In this subsection, different channel statistical properties are derived under the unified NOMA frameworks [40], which can be used for deriving the COP in the following sections.
Lemma 1.
Assuming users randomly distributed within the circular cluster, the CDF of the th user is given by
(7) 
where , , , , , and is a parameter to ensure a complexityaccuracy tradeoff.
Proof.
See Appendix A. ∎
Lemma 2.
Assuming users randomly distributed within the circular cluster, the CDF of the th user with ipSIC is given in (8) at the top of next page, where .
Proof.
See Appendix B. ∎
(8) 
Substituting into (8), the CDF of the th user with pSIC is given by
(9) 
Iii Performance evaluation
Since the capacity of channel from the BS to the goaldirected user is less than the target transmission rate, the connection outage will occur [41]. As a consequence, the goaldirected user is incapable of detecting the information accurately. In this section, the COP is selected as a metric to evaluate the performance of unified downlink NOMA networks. More specially, a pair of NOMA users (i.e., the th user and th user) for CD/PDNOMA are characterized in terms of outage probabilities in the following.
Iiia The COP of the th user
The outage event of the th user in the typical cell is that the th user cannot detect its own information. Hence the COP of the th user for CDNOMA can be expressed as
(10) 
where and is the target rate of the th user in the typical cell.
By applying (1), the following theorem provides the COP of the th user.
Theorem 1.
The COP of the th user for CDNOMA is given by
(11) 
where with .
Corollary 1.
For the special case with , the COP of the th user for PDNOMA is given by
(12) 
IiiB The COP of the th user
IiiB1 Existing Outage Probability Formulation
Considering a twouser case, the th user and th user are paired together to perform NOMA protocol. The outage for the th user can happen in the following two cases [10, 42]:

The th user cannot decode the message of the th user.

The th user can decode the message of the th user, then carries out SIC operations, but cannot decode the information of itself.
Based the aforementioned descriptions, the COP of the th user for existing formulation (EXF) can be expressed as
(13) 
where with being the target rate at the th user to detect the th user.
The following theorem provides the COP of the th user with ipSIC for the downlink CDNOMA networks.
Theorem 2.
The COP of the th user with ipSIC for EXF in CDNOMA networks is given by (14), where , and .
(14) 
Proof.
See Appendix C. ∎
Substituting into (14), the COP of the th user with pSIC for EXF in CDNOMA networks is given by
(15) 
Corollary 2.
For the special case with , the COP of the th user with ipSIC for EXF in PDNOMA networks is given by
(16) 
Substituting into (2), the COP of the th user with pSIC for EXF in PDNOMA networks is given by
(17) 
IiiB2 Alternative Outage Probability Formulation
However, for the first case, when the decoding process for the th user at the th user fails, the outage event is not necessarily happened. Because the th user can still try to decode the message of itself by treating the th user’s signal as interference without carrying out SIC operations. In other words, the previous outage formulation makes the decoding procedure of the th user highly depend the target rate of the th user, which ignores one possible case which can also support reliable transmission. As such, the alternative outage for the th user can happen in the following two cases:

The th user can not decode the message of the th user and the message of itself with treating the th user’s signal as interference.

The th user can decode the message of the th user, but cannot detect the information of itself after carrying out SIC operations.
By the virtue of previous assumptions, the COP of the th user for alternative formulation (ALF) can be expressed as
(18) 
The following theorem provides the COP of the th user with ipSIC for the downlink CDNOMA networks.
Theorem 3.
The COP of the th user with ipSIC for ALF in CDNOMA networks is given by (3), where with , .
(19) 
Proof.
See Appendix D. ∎
Substituting into (3), the COP of the th user with pSIC for ALF in CDNOMA networks is given by (IIIB2).
(20) 
Corollary 3.
For the special case with , the COP of the th user with ipSIC for ALF in PDNOMA networks is given by (3).
(21) 
Substituting into (3), the COP of the th user with pSIC for ALF in PDNOMA networks is given by
(22) 
Proposition 1.
The COP of the selected user pairing with ipSIC/pSIC for CD/PDNOMA are given by
(23) 
and
(24) 
respectively, where and . and are given by (1) and (12), respectively. , , and are given by (14), (IIIB1), (2) and (IIIB1), respectively. , , and are given by (3), (IIIB2), (3) and (IIIB2), respectively.
IiiC Diversity Order Analysis
To gain more deep insights, diversity order is usually selected 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 diversity order is given by
(25) 
where denotes the asymptotic COP.
Corollary 4.
Based on analytical result in (1), the asymptotic COP of the th user at high SNR for CDNOMA is given by
(26) 
where represents “be proportional to”.
Proof.
To facilitate the calculation, define the summation term in (1), i.e., . Applying power series expansion, the summation term can be rewritten as . Substituting into , when , with the approximation of is formulated as . As a further development, substituting into (1) and taking the first term [43], we obtain (26). Obviously, is a function of , which is proportional to . The proof is completed. ∎
For the special case with , the asymptotic COP of the th user at high SNR for PDNOMA is given by
(27) 
Remark 1.
Corollary 5.
Based on analytical result in (14), when , the asymptotic COP of the th user with ipSIC for EXF in CDNOMA networks is given by
(28) 
Substituting into (5), the asymptotic COP of the th user with pSIC at high SNR for EXF in CDNOMA networks is given by
(29) 
Remark 2.
Corollary 6.
For the special case with in (5), the asymptotic COP of the th user with ipSIC at high SNR for EXF in PDNOMA networks is given by
(30) 
Substituting into (6), the asymptotic COP of the th user at high SNR with pSIC in PDNOMA networks for EXF is given
(31) 
Remark 3.
Corollary 7.
The asymptotic COP of the th user with ipSIC at high SNR for ALF in CDNOMA networks is given by (7) at the top of next page.
(32) 
Similar to the solving process of (29), substituting into (7), the asymptotic COP of the th user at high SNR with pSIC in CDNOMA networks for ALF is given by
(33) 
Remark 4.
Corollary 8.
For the special case with , the asymptotic COP of the th user at high SNR for ALF in PDNOMA networks is given by (8) at the top of next page.
(34) 
Substituting into (8), the asymptotic COP of the th user at high SNR with pSIC for ALF in PDNOMA networks is given
(35) 
As can be seen from (8), the third term of is a constant and leads directly to the fact that is not proportional to . However, in (IIIC), is proportional to . Hence the observation can be obtained in the following.
Remark 5.
From the above remarks, we can observe that CDNOMA with pSIC is capable of providing a higher diversity order than PDNOMA. Hence we can adjust the size of subcarriers to meet different application requirements. Additionally, we find that due to the impact of RI, CD/PDNOMA with ipSIC obtain zero diversity order. The design of an efficient SIC is important for NOMA networks.
Remark 6.
Under the condition of fixed SNR, the outage probabilities are determined by the SIC used for interference cancellation as well as the number of subcarriers and the size of target rate.
Proposition 2.
The asymptotic COP of the selected user pairing with ipSIC/pSIC for CD/PDNOMA at high SNR are given by
(36) 
and
(37) 
respectively, where and are given by (26) and (27), respectively. , , and are given by (5), (29), (6) and (31), respectively. , , and are given by (7), (IIIC) and (8) and (IIIC), respectively.
Remark 7.
On the basis of conclusions of above corollaries, the diversity orders of the selected user pairing with ipSIC/pSIC for CDNOMA and PDNOMA are zero/ and zero/, respectively. As can be observed that due to the impact of RI for imperfect cancellation process, the selected user pairing with ipSIC for CD/PDNOMA obtain zero diversity order. Additionally, it is shown that the diversity orders of the selected user pairing are determined by the th user.
As shown in TABLE I, the relationship between different factors for CD/PDNOMA, such as, outage probability formulations, SIC schemes and diversity order, are summarized to illustrate the comparison between them. In TABLE I, we use “F” , “S” and “D” to represent outage probability formulation, SIC scheme and diversity order, respectively.
Pairing users  NOMA  F  S  D 
The th user  CDNOMA  ——  ——  
——  ——  
PDNOMA  ——  ——  
——  ——  
The th user  CDNOMA  EXF  ipSIC  
pSIC  
PDNOMA  ALF  ipSIC  
pSIC  
EXF  ipSIC  
pSIC  
ALF  ipSIC  
pSIC 
IiiD Throughput Analysis
In this subsection, the system throughput of the unified NOMA framework is characterized in delaylimited transmission mode. In this mode, the BS transmits information at a constant rate , which is subject to the effect of outage probability due to wireless fading channels.
CDNOMA case
PDNOMA case
Iv Numerical Results
In this section, we focus on investigating a typical pair of users with random pairing. Monte Carlo simulation parameters used in this section are summarized in TABLE II [42, 44], where BPCU is short for bit per channel use and the pass loss exponent aims to simplify simulation analysis. The complexityvsaccuracy tradeoff parameter is set to be and simulation results are denoted by . Additionally, the conventional OMA is selected to be a benchmark for comparison purposes. The target rate for OMA satisfies the relationship with . Note that the setting of smaller target data rate for NOMA users can be applied into the Internet of Things (IoT) scenarios, which require low energy consumption, small packet service and so on.
Monte Carlo simulations repeated  iterations 

Carrier frequency  GHz 
Power allocation coefficients of NOMA  , 
Target data rates  BPCU 
Pass loss exponent  
The radius of the user zone  m 
Fig. 2 plots the COP of a pair of NOMA users (the th and th user) versus the transmit SNR with ipSIC/pSIC, where . In particular, the different values of RI are set to be from dB to dB. The exact analytical curve for the outage probability of the th user is plotted according to (1). Furthermore, the exact analytical curves for the outage probability of the th user with both ipSIC and pSIC for EXF and ALF are plotted based on (14), (3) and (IIIB1), (IIIB2), respectively. Obviously, the exact outage probability curves match perfectly with Monte Carlo simulations results. It is observed that the outage performance of OMA is inferior to the th user with pSIC and superior to the th user. This is due to the fact that NOMA is also capable of providing better fairness since multiple users are served simultaneously, which is the same as the conclusions in [10, 45]. Additionally, as can be observed from figure, the dashed curves represent the asymptotic COP of the th user and th user with pSIC for EXF and ALF, which can be obtained by numerically evaluating (26), (29) and (IIIC). One can observe that the asymptotic outage probabilities are approximated to the analytical results in the high SNR regime. The dotted curves represent the asymptotic outage probabilities of the th user with ipSIC for EXF and ALF, which are calculated from (5) and (7), respectively. It is shown that the outage performance of the th user with ipSIC converges to an error floor and obtain zero diversity order, which verifies the insights in Remark 2 and Remark 3. Due to the influence of RI, the outage behavior of the th user with ipSIC is inferior to OMA. The reason is that the RI signal from imperfect cancellation operation is the dominant impact factor. With the value of RI increasing from dB to dB, the outage behavior of the th user is becoming more worse and deteriorating. More specifically, when
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