I Introduction
Each new generation of wireless communication systems has been designed to support the demands of increased traffic and data throughput with spectral efficiency as a key factor. To this end, nonorthogonal multiple access (NOMA) principle has been recognized as a key technology that can improve the spectral efficiency of 5G wireless systems. In contrast to traditional orthogonal multiple access (OMA) techniques, NOMA multiplexes signals between users with strong/weak channel conditions (a.k.a. near/far users) in the power domain and apply successive interference cancellation (SIC) at the receivers to remove the interuser interference [1, 2, 3].
On the other hand, multipleinput multipleoutput (MIMO) technology that offers substantial performance gains has become an integral part of modern communication systems. However, the benefits of MIMO come at the price of increased computational complexity and cost of hardware radio frequency chains that scales with the number of antennas [4]. In this context, antenna selection (AS) schemes with low implementation complexity and perform close to traditional MIMO systems have been touted as a practical solution in the literature. There is a sizable matured body of work on AS for different MIMO systems. To this end, AS in combination with NOMA has received interest in the recent literature [5, 6, 7], however the topic is still in infancy. Especially, the complexity of deciding on AS solutions in NOMA systems are exacerbated due to the complicated nature of near/far user performance criterion [5].
In this paper, we analyze the AS performance of a fullduplex (FD) cooperative NOMA system. FD is another promising technology considered for 5G implementation. Furthermore, the marriage between FD and NOMA can boost the performance as confirmed in [8, 9, 10, 11] so far. The main bottleneck for FD operation is the selfinterference (SI) [12, 13, 14]. Thus, AS should consider the effect of SI during the selection of strong channels toward the near/far users. This makes the AS problem in FD NOMA systems a much more complicated affair than in halfduplex (HD) NOMA systems [5]. Specifically, for the considered FD NOMA relay system, we propose AS schemes to achieve near/far user endtoend (e2e) signaltointerference noise ratio (SINR) maximization and study the sum rate and outage probability, respectively. Our contributions can be summarized as follows:

Two low complexity AS schemes, i.e., maxU1 AS scheme and maxU2 AS scheme are proposed to maximize the e2e SINR at the near and far user, respectively. The performance of the FD cooperative NOMA system with the two AS schemes is analyzed by deriving exact ergodic sum rate and outage probability expressions.

Our findings reveal that maxU1 AS scheme can significantly improve the system sumrate, while maxU2 AS scheme can provide better user fairness. In particular, maxU1 AS scheme can achieve near optimum rate performance in the entire SNR range.
Notation: We use
to denote the expected value of the random variable (RV)
; its probability density function (pdf) and cumulative distribution function (cdf) are
and respectively; denotes a circularly symmetric complex Gaussian RV with meanand variance
and is the exponential integral function [15, Eq. (8.211.1)].Ii System Model
Consider a two user NOMA downlink system where U1 (near user) communicates directly with the base station (BS), while U2 (far user) requires the assistance of a multiantenna FD relay, as shown in Fig. 1. Both U1 and U2 are equipped with a single antenna each, the BS is equipped with antennas, while is equipped with two group of antennas: receive antennas and transmit antennas.
In order to keep the implementation complexity low, we assume that the BS and perform single AS [4]. To be precise, the BS selects one (e.g. th) out of available transmit antennas, while selects one (e.g. th) out of available antennas to receive signals. Moreover, one antenna (e.g. th) out of transmit antennas is selected at to forward the BS signal to U2.
We assume that all channels experience independent Rayleigh fading and that they remain constant over one transmission slot. The channel between the th receive and the th transmit antenna from terminal to terminal , is denoted by where and .
Iia Transmission Protocol
According to the NOMA concept [2, 3], the BS transmits a combination of intended messages to both users as
(1) 
where , denotes the information symbol intended for , and denotes the power allocation coefficient, such that and .
Since is FD capable, it transmits the decoded symbol , where accounts for the time delay caused by FD processing at [13]. Therefore, the receive signal at can be written as [8]
(2) 
where is the relay transmit power and is the additive white Gaussian noise (AWGN) at . We assume imperfect SI cancellation due to FD operation at and model the elements of the residual SI channel as independent identically distributed (i.i.d) RVs [13].
The information intended for U2 is decoded at with SIC treating the symbol of U1 as interference [8]. Hence, the SINR at can be written as
(3) 
where and with and .
At the same time, U1 receives the following signal
(4) 
where is the AWGN at U1.
It is assumed that the symbol, , is priory known to U1 and thus U1 can be removed [8]. However, by considering realistic imperfect interference cancellation wherein U1 cannot perfectly remove , we model as the interuser interference channel where the parameter presents the strength of interuser interference [8]. If U1 perfectly cancels the U2’s signal, the SINR at U1 is given by
(6) 
Moreover, the received signal at U2, from can be written as
(7) 
where denotes the AWGN at U2. Hence, the signaltonoise ratio (SNR) at U2 is given by
(8) 
The e2e SINR at U2 can be expressed as
(9) 
IiB Antenna Selection Schemes
We now propose two AS schemes for the described FD cooperative NOMA system where joint selection of single transmit and receive antennas at the BS and based on e2e SINRs at the near user, U1 and far user, U2 is performed.
IiB1 maxU1 AS Scheme
This scheme selects antennas first to maximizes the e2e SINR at U1, (6) and next with remaining AS choices tries to maximize also e2e SINR at U2. Therefore, the particular AS scheme can be mathematically expressed as
(10) 
IiB2 maxU2 AS Scheme
The maxU2 AS scheme achieves e2e SINR maximization at U2 according to
(11) 
According to (IIB2
) there is no degreeoffreedom (in terms of AS) available for maximizing the
e2e SINR at U1. Therefore, in terms of the near user SINR, maxU2 AS scheme is inferior to that of the maxU1 AS scheme.It worth pointing out that other AS schemes, for example with criterion such as to maximize the near/far user performance subject to a predefined far/near performance are also possible to design. Analyzing such AS schemes whose performance remain in between maxU1 and maxU2 AS schemes is relegated to the journal version of this work.
In order to implement maxU1 and maxU2 AS schemes, the BS can transmit a pilot signal and can decide on the best antenna indexes to be used at the BS and its receive side. can also transmit a pilot signal to U1 and U2 from each of the relay antenna. Upon reception of the pilot, U1 (in maxU1 AS scheme) and U2 (in maxU2 AS scheme) can next feedback the antenna index that must used in subsequent information transmission. Moreover, in maxU1 AS scheme, upon reception of the BS pilot signal, U1 can decide the best antenna and feedback the corresponding index to the BS to commence information transmission.
Iii Performance Analysis
In this section, we investigate the performance of proposed AS schemes in terms of the achievable ergodic sum rate and outage probability. The derived results will enable us to examine the benefits of the proposed AS schemes.
Iiia Ergodic Sum Rate
The ergodic achievable sum rate of the FD cooperative NOMA system is given by
(12) 
with
(13a)  
(13b) 
where and denote the e2e SINR at the U1 and U2, corresponding to the specific AS scheme with . We use to refer to the maxU1 AS scheme and to refer to the maxU2 AS scheme.
Before we proceed, it is useful, to note that for a nonnegative RV , since , the ergodic achievable rate for near user () or far user () can be written as
(14) 
where is the cdf of the RV .
In the sequel, we present key results for the near and far user ergodic rates due to the proposed AS schemes.
Proposition 1.
The ergodic achievable rates of U1 and U2 of maxU1 AS scheme, are respectively given by
(15)  
and
(16) 
where , , , and .
Proof: According to (14), the ergodic achievable rate can be calculated from the cdf of (6) and (9). Let us start with the ergodic achievable rate of U1. By invoking (IIB1), the ratio is maximized when the strongest BSU1 channel and the weakest U1 channel are selected. Therefore, the cdf of can be evaluated as
where is a RV defined as the maximum out of exponentially distributed independent RVs, while is the minimum out of exponentially distributed independent RVs. Substituting the required cdf and the pdf and simplifying yields
(17) 
Next, substituting (17) into (14), then with the help of the integration identity in [15, Eq. (3.352.4)], and after some algebraic manipulations, we arrive at (15).
We now turn our attention to evaluate . To this end, the cdf of can be expressed as
(18) 
According to the criterion in (IIB1), for , we can write
(19) 
where the second equality follows since and can be written as
(20) 
Moreover, based on (IIB1), for the selected transmit antennas at the BS and , the ratio can be maximized when the strongest BS channel and weakest SI channel are selected. However, theses two channels are coupled with each other through the selected antenna at the input. Therefore, it is difficult, if not impossible, to find the cdf of . Alternatively, we propose to select the receive antenna at such that is maximized, which will be shown to be a good approximation across the entire SNR range in Section IV (cf. Optimum AS (U2)). Therefore, we have
for , where is a RV defined as the largest out of exponentially distributed independent RVs, and since SI link is ignored, is an exponentially distributed RV with parameter . Substituting the required cdf and the pdf and simplifying yields
(21) 
Finally, since the U2 link is ignored, we have . To this end, pulling everything together, we obtain
(22) 
Having obtained the cdf of , the ergodic rate of U2 can be obtained by employing (14). For it can be readily checked that and hence .
Proposition 2.
The ergodic achievable rates of U1 and U2 of maxU2 AS scheme, are respectively given by
(23)  
and
(24) 
Proof: The proof follows similar steps to the proof of Proposition 1 and thus only an outline is presented. According to (IIB2) the e2e SINR at U2 is maximized when each term inside in the minimum function is maximized. Therefore, a transmit antenna at is selected such that is maximized, i.e., is the largest of exponential RVs with parameter . Moreover, since SI is the main source of performance degradation in FD mode, for a particular transmit antenna at , the best receive antenna at the is selected such that the SI strength is minimized [16]. Hence, is the minimum of exponential RVs with parameter . Finally, for given and , a best transmit antenna at the BS is selected such that is maximized. Hence, a single transmit antenna at the BS is selected such that the SINR at BS is maximized for the th receive antenna at .
IiiB Outage Probability
Outage probability is a key metric used to measure the event that the data rate supported by instantaneous channel realizations is less than a targeted user rate. Therefore, the outage probability is an important performance metric to characterizes the performance of NOMA systems [3].
The following Propositions present exact closedform expressions for the outage probability of maxU1 and maxU2 AS schemes
Proposition 3.
The outage probability of U1 with maxU1 and maxU2 AS scheme, are respectively given by
(27) 
and
(28) 
where , and with and being the transmission rates at U1 and U2.
Proof: An outage event at U1 occurs when it cannot decode the intended signal for U2, or when it can decode it but fails to decode its own signal. Therefore, the outage probability of U1 can be derived as
(29) 
Proposition 4.
The outage probability of U2 with maxU1 and maxU2 AS scheme, are respectively given by
(30) 
and
(31) 
Proof: An outage event at U2 occurs if fails to decode the intended message to U2, or can decode U2 signal but U2 fails to decode its message. Therefore, the outage probability of U2 can be expressed as
(32) 
To this end by substituting the corresponding probabilities presented in subsection IIIA into (IIIB), we arrive at (30) and (31).
Iv Numerical Results and Discussion
In this section, we present numerical results to quantify the performance gains when maxU1 and maxU2 AS schemes are adopted in the considered FD cooperative NOMA system. We set , and .
Fig. 2 shows the ergodic rates of U1 and U2 with the proposed AS schemes. We have also plotted the curves for i) Optimum AS scheme: that performs an exhaustive search of all possible combinations to determine the antenna subset in order to maximize the ergodic sum rate, ii) Optimum AS (U2) scheme : This scheme aims to maximize the e2e SINR at U2 in an optimal sense and performs an exhaustive search of all possible combinations to determine the optimum antenna subset in order to maximize the e2e SINR at U2, and iii) Random AS scheme: that performs random AS at the BS and relay input/output. It can be observed that the maxU1 AS scheme is able to improve the ergodic rate of both U1 and U2, while maxU2 AS scheme, only increases the ergodic rate of U2. The maxU1 AS scheme provides the best performance for U1, while the performances of the maxU2 AS scheme and random AS scheme for U1 are almost identical. However, both maxU1 AS scheme and maxU2 AS scheme are able to increase the ergodic rate of U2. Moreover, we see that with increasing transmit power at BS and , maxU2 AS scheme achieves the same performance as Optimum AS scheme.
In Fig. 3 the ergodic sum rate with different AS schemes is illustrated. It is evident that the ergodic achievable sum rate with the proposed AS schemes is higher than that of OMA FD relay system. In all transmit power regimes the performance gap between the maxU1 AS scheme and Optimum AS scheme is negligible. Moreover, the difference between the maxU1 and Optimum AS (U2) enlarges when transmit power at the BS and is increased and remains constant for mediumtohigh transmit power values.
Fig. 4 shows the Jain’s fairness index versus transmit power and for different AS schemes. This index is a bounded continuous function and is the most used quantitative measure to study the fairness in wireless systems. The Jain’s fairness index for the considered dualusers scenario can be expressed as [17]
which its range is the interval . In this interval, corresponds to the least fair allocation in which only one user receives a nonzero rate, and corresponds to the fairest allocation in which both near and far user receive the same rate. From Fig. 4 We see that in the low SNR region, Optimum AS scheme achieves the best user fairness out of all AS schemes. However, maxU2 AS scheme can provide a better user fairness in mediumtohigh SNR regimes and hence can balance the tradeoff between the ergodic rate and user fairness.
Fig. 5 shows the outage probability of the AS schemes with different residual SI strengths. The maxU1 AS scheme provides the best outage performance for U1 and improves the outage performance of U2. Moreover, the maxU2 AS scheme can significantly improve the outage performance of U2, while it exhibits the same performance as Random AS scheme for U1. Finally, due to interuser interference at U1 and SI at our results show that all AS schemes suffer from zeroorder diversity. However, the proposed AS schemes can reduce the error floor.
V Conclusion
In this paper, we have studied the AS problem for a FD cooperative NOMA system. Two low complexity AS schemes, namely maxU1 AS scheme and maxU2 AS scheme were proposed to maximize the e2e SINR at the near and far user, respectively. The performance of both AS schemes have been characterized in terms of the ergodic sum rate and outage probability. Our results revealed that the maxU1 AS scheme achieves near optimum sumrate performance while the maxU2 AS scheme exhibits a better user fairness. Moreover, the outage performance of the near user can be significantly improved by using the maxU1 AS scheme, while the outage probability of the far user can be effectively improved via the maxU2 AS scheme.
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