With the rapid advancement in the wireless communication technology, the fifth generation (5G) mobile communication networks have attracted a great deal of attention [2, 3, 4]. In particular, three major families of new radio (NR) usage scenarios, i.e., massive machine type communications (mMTC), enhanced mobile broadband (eMBB) and ultra-reliable and low-latency communications (URLLC) are proposed to satisfy the different requirements for 5G networks. To improve system throughput and achieve enhanced spectrum efficiency of 5G networks, non-orthogonal multiple access (NOMA) has been considered to be a promising candidate technique and identified for 3GPP Long Term Evolution (LTE) . 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 . Furthermore, downlink multiuser superposition transmission scheme (MUST)  which is the special case of NOMA has found application in wireless standard.
Hence numerous excellent Contributions have surveyed the performance of point-to-point NOMA in wireless networks in [8, 9, 10, 11]. To evaluate the performance of downlink NOMA, the closed-form expressions of outage probability and ergodic rate for NOMA were derived in  by use of the bounded path loss model. Furthermore, the authors of  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) and cognitive radio based NOMA (CR-NOMA) schemes were characterized. By considering user grouping and decoding order selection, the outage balancing among users was investigated , in which the closed-form expressions of optimal decoding order and power allocation for downlink NOMA were derived. In , the authors researched the outage behavior of downlink NOMA for the case where each NOMA user only feed back 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 feedback. 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 large-scale underlay CR for NOMA with the aid of stochastic geometry. To emphasize physical layer security (PLS), the authors in  discussed the PLS issues of NOMA, where the secrecy outage probabilities were derived for both single-antenna and multiple-antenna scenarios, respectively. Recently, the NOMA-based wireless cashing strategies were introduced in , in which two cashing phases, i.e., content pushing and content delivery, are characterized in terms of caching hit probability. Additionally, explicit insights for understanding the performance of uplink NOMA have been provided in [15, 16]. In , the novel uplink power control protocol was proposed for the single-cell uplink NOMA. In large-scale cellular networks, the performance of multi-cell uplink NOMA was characterized in terms of coverage probability using the theory of Poisson cluster process .
Cooperative communication is a promising approach to overcome signal fading arising from multipath propagation as well as obtain the higher diversity . Obviously, combining cooperative communication technique and NOMA is the research topic which has sparked of wide interest in[18, 19, 20, 21]. The concept of cooperative NOMA was initially proposed for downlink transmission in , where the nearby user with better channel conditions was viewed as decode-and-forward (DF) relay to deliver the information for the distant users. Driven by these, authors in  analyzed the achievable data rate of NOMA systems for DF relay over Rayleigh fading channels. On the standpoint of tackling spectrum efficiency and energy efficiency, in , the application of simultaneous wireless information and power transfer (SWIPT) to NOMA with randomly deployed users was investigated using stochastic geometry. In , NOMA based dual-hop relay systems were addressed, where both statistical CSI and instantaneous CSI were considered for the networks. On the other hand, the outage performance of NOMA for a variable gain amplify-and-forward (AF) relay was characterized over Nakagami- fading channels in . With the emphasis on imperfect CSI, authors studied the system outage behavior of AF relay for NOMA networks in . Additionally, the authors of  analyzed the outage performance of a fixed gain based AF relay for NOMA systems over Nakagami- 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 . To further improve the bandwidth usage efficiency of system, full-duplex (FD) relay technology is a promising solution which can simultaneously receive and transmit the signal in the same frequency band . Nevertheless, FD operation suffers from residual loop self-interference (LI), which is usually modeled as a fading channel . Particularly, FD relay technologies in  have been discussed from the view of self-interference cancellation, protocol design and relay selection for 5G networks. To maximize the weighted sum throughput of system, the design of resource allocation algorithm for FD multicarrier NOMA (MC-NOMA) was investigated in , where a FD BS was capable of serving downlink and uplink users in the meantime. The recent findings in FD operation considered for cooperative NOMA were surveyed in [29, 30]. The performance of FD device-to-device (D2D) based cooperative NOMA was characterized in terms of outage probability in . Considering the influence of imperfect self-interference, the authors in  investigated the performance of FD-based DF relay for NOMA, where the expressions of outage probability and achievable sum rate for two NOMA users were derived.
Applying relay selection (RS) technique to cooperative communication systems is a straightforward and effective approach for taking advantages of space diversity and improving spectral efficiency. The following research contributions have surveyed the RS schemes for two kinds of operation modes: HD and FD. For HD mode, the authors of  derived the diversity of single RS scheme and 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 , the ergodic rate was studied with a buffer-aided relay scheme for HD-based single RS network. Additionally, the application of RS scheme to cognitive DF relay networks was discussed in . For FD mode, assuming the availability of different instantaneous CSI, the authors analyzed the RS problem of AF cooperative system in . It was worth noting that FD-based RS scheme converges to an error floor and obtains a zero diversity order. The performance of DF RS scheme was characterized in terms of outage probability for the CR networks in . Very recently, two-stage RS scheme was proposed for HD-based cooperative NOMA in , where the RS scheme considered was capable of realizing the maximal diversity order.
I-a Motivations and Contributions
While the aforementioned significant contributions have laid a solid foundation for the understanding of cooperative NOMA and RS techniques, the RS technique for cooperative NOMA networks 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, link density, coverage enhancement and small packet service are capable of being supported through the RS schemes. One of the best relays is selected from relays as the BS’s helper to forward the information. In , the two-stage RS scheme is capable of achieving the minimal outage probability and obtaining the maximal diversity order, but only HD-based RS for cooperative NOMA was considered. To the best of our knowledge, there are no existing works to investigate the RS scheme for FD cooperative NOMA networks. Moreover, the spatial impact of RS on the performance of FD cooperative NOMA was not examined in . Motivated by these, we specifically consider a pair of RS schemes for FD/HD NOMA networks, namely single-stage RS (SRS) scheme and two-stage RS (TRS) scheme, where the locations of relays are modeled by invoking 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 and TRS scheme) for FD NOMA networks. We derive the closed-form and 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 and 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 RS schemes. We confirm that the HD-based NOMA RS schemes are capable of providing the diversity order of , which is equal to the number of relays.
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 random RS (RRS) and orthogonal multiple access (OMA) based RS schemes. Additionally, we analyze the system throughput in delay-limited transmission mode based on the outage probabilities derived.
I-B Organization and Notation
The rest of the paper is organized as follows. In Section II, the network model of the RS schemes for FD/HD NOMA is set up. New analytical and approximate expressions of outage probability for the RS schemes are derived in Section III. In Section IV, numerical results are presented for performance evaluation and comparison. Section V concludes the paper.
Ii Network Model
In this section, the network and signal models are presented. Additionally, the criterions of a pair of RS schemes in the networks considered are introduced for FD/HD NOMA.
Ii-a Network Description
Consider a downlink cooperative NOMA scenario consisting of one BS, relays ( with ) and a pair of users (i.e., the nearby user and distant user ), as shown in Fig. 1. To reduce the complexity of NOMA system, multiple users can be divided into several groups and 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 and the radius of disc is . In addition, relays are uniformly distributed within . The DF protocol is employed at each relay and 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 and one receive antenna, while the BS and users have a single antenna111Note that multiple antennas equipped by the BS and relays will further suppress the self-interference and enhance the performance of the NOMA-based RS schemes. Additionally, more sophisticated assumption of antennas at relay, i.e., omni-directional and directional antennas [38, 39] can be developed for further evaluating the performance of the networks considered. However, these are beyond the scope of this treatise., respectively. All wireless channels222It is assumed that perfect CSI can be obtained, our future work will relax this idealized assumption. Furthermore, we note that relaxing the setting of Rayleigh fading channels (i.e., Nakagami- fading channels considered in ) will provide a more general system setup, which are set aside for our future work. in the scenario considered are assumed to be independent non-selective block Rayleigh fading and are disturbed by additive white Gaussian noise with mean power . , , and denote the complex channel coefficient of , , and links, respectively. and denote the distance from the BS to and , respectively. Assuming that an imperfect self-interference cancellation scheme is employed at each relay such as  and the corresponding LI is modeled as a Rayleigh fading channel with coefficient . As stated in 
, two NOMA users are classified into the nearby user and 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 packet business and telemedicine service). Hence we assume that can be served opportunistically and needs to be served quickly for small packet with a lower target data rate. As a further example, is to download a movie or carry out some background tasks and so on; can be a medical health sensor which is to send the pivotal safety information containing in a few bytes, such as blood pressure, pulse and heart rates.
Ii-B Signal Model
During the -th time slot, , the BS sends the superposed signal to the relay on the basis of NOMA principle , where and are the normalized signal for and , respectively, i.e, . and are the corresponding power allocation coefficients. Practically speaking, to stipulate better fairness and QoS requirements between the users , we assume that with . The LI signal exists at the relay due to it works in FD mode. Therefore the observation at the th relay is given by
where , is the distance between the BS and and denotes the path loss exponent. is the switching operation factor, where and denote the relay working in FD mode and 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. and denote normalized transmission power (i.e., ) at the BS and , respectively. denotes the LI signal with and an integer denotes processing delay at with . denotes the Gaussian noise at .
Based on NOMA protocol, SIC333In this paper, we assume that perfect SIC is employed, our future work will relax this assumption. is employed at to first decode the signal of having a higher power allocation factor, since has a less interference-infested signal to decode the signal of . Based on this, the received signal-to-interference-plus-noise ratio (SINR) at to detect and are given by
respectively, where is the transmit SNR.
Assuming that is capable of decoding the two NOMA user’s information, i.e, satisfying the following conditions, 1) ; and 2) , where and are the target rate for and , respectively. Therefore the observation at can be expressed as
where , is the distance between and (assuming ); , . denotes the angle ; denotes the Gaussian noise at .
In similar, assuming that SIC can be also invoked successfully by to detect the signal of having a higher transmit power, who has less interference. Hence the received SINR at to detect can be given by
Then the received SINR at to detect its own information is given by
The received SINR at to detect is given by
Note that the fixed power allocation coefficients for two NOMA users are considered in the networks. Reasonable power control and optimizing the mode of power allocation can further enhance the performance of the RS schemes, which may be investigated in our future work.
Ii-C Relay Selection Schemes
In this subsection, we consider a pair of RS schemes for FD/HD NOMA, which are detailed in the following.
Ii-C1 Single-stage Relay Selection
Prior to the transmissions, a relay can be randomly selected by the BS as its helper to forward the information. The aim of SRS scheme is to maximize the minimum data rate of for FD/HD NOMA. More specifically, the size of data rate for depends on three kinds of data rates, such as 1) the data rate for the relay to detect ; 2) The data rate for to detect ; and 3) the data rate for to detect its own signal . Among the relays in the network, based on (2), (5) and (7), the SRS scheme activates a relay, i.e.,
where denotes the number of relays in the network. Note that FD/HD-based SRS schemes inherit advantage to ensure the data rate of , where the application of small packets can be achieved.
Ii-C2 Two-stage Relay Selection
The TRS scheme mainly include two stages for FD/HD NOMA: 1) In the first stage, the target data rate of is to be satisfied; and 2) In the second stage, on the condition that the data rate of is ensured, we serve with data rate as large as possible. Hence the first stage activates the relays that satisfy the following condition
where the size of is defined as .
Among the relays in , the second stage selects a relay to convey the information which can maximize the data rate of and is expressed as
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 , the BS can support to carry out some background tasks, 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 networks.
Iii-a Single-stage Relay Selection Scheme
According to NOMA protocol, the complementary events of outage for SRS scheme can be explained as: 1) The relay can detect the signal of ; and 2) while the signal can be successfully detected at and , respectively. From the above descriptions, the outage probability of SRS scheme for FD NOMA can be expressed as follows:
where and . with being the target rate of .
The following theorem provides the outage probability of SRS scheme for FD NOMA.
The closed-form expression of outage probability for FD-based NOMA SRS scheme can be approximated as follows:
where , with . , and is a parameter to ensure a complexity-accuracy tradeoff.
See Appendix A. ∎
Upon substituting into (1), the approximate expression of outage probability for HD-based NOMA SRS scheme is given by
where with and with being the target rate of .
Iii-B Two-stage Relay Selection Scheme
In the case of TRS scheme, the overall outage event can be expressed  as follows:
where denotes the outage event that relay cannot detect , or neither and can detect the correctively, and denotes the outage event that either of and cannot detect while three nodes can detect successfully.
As a consequence, the outage probability of TRS scheme for FD NOMA can be expressed as follows:
In order to calculate the second outage probability, can be further expressed as
where denotes the outage event that the relay cannot detect and denotes the corresponding complementary event of . denotes that cannot detect . The first term in the above equation is given by
The second term in (17) is given by
To derive the closed-form expression of outage probability for TRS scheme in (III-B), we define
respectively. The probability can be given by
The above probability can be further expressed as
For selecting a relay at random from , denoted by relay , let us now turn our attention to the derivation of ’s CDF (i.e., ) in the following lemma. Define these two probabilities at the right hand side of (III-B) by and , respectively.
See Appendix B. ∎
On the other hand, there are relays in and the corresponding probability is given by
With the aid of Theorem 1, the above probability can be further approximated as follows:
With the aid of Lemma 1, combining (III-B), (III-B), (25) and (III-B) and applying some algebraic manipulations, the outage probability of TRS scheme for FD NOMA can be provided in the following theorem.
The closed-form expression of outage probability for the FD-based NOMA TRS scheme is approximated by (2) at the top of next page.
For the special case , the approximate expression of outage probability for HD-based NOMA TRS scheme is given by (2) at the top of next page,
where and with being the target rate of .
Iii-C Benchmarks for SRS and TRS schemes
In this subsection, we consider the random relay selection (RRS) scheme as a benchmark for comparison purposes, where the relay is selected randomly to help the BS transmitting the information. Note that 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 , 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
Iii-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
where is the asymptotic outage probability.
Iii-D1 Single-stage Relay Selection Scheme
Based on the analytical results in (1), when , we can derive the asymptotic outage probability of SRS scheme for FD NOMA in the following corollary.
The asymptotic outage probability of FD-based NOMA SRS scheme at high SNR is given by
The diversity order of SRS scheme for FD NOMA is zero, which is the same as the conventional FD RS scheme.
For the special case , the asymptotic outage probability of HD-based NOMA SRS scheme with at high SNR is given by