Antenna Selection in MIMO Cognitive RadioInspired NOMA Systems
Abstract
This letter investigates a joint antenna selection (AS) problem for a MIMO cognitive radioinspired nonorthogonal multiple access (CRNOMA) network. In particular, a new computationally efficient joint AS algorithm, namely subsetbased joint AS (SJAS), is proposed to maximize the signaltonoise ratio of the secondary user under the condition that the quality of service (QoS) of the primary user is satisfied. The asymptotic closedform expression of the outage performance for SJAS is derived, and the minimal outage probability achieved by SJAS among all possible joint AS schemes is proved. The provided numerical results demonstrate the superior performance of the proposed scheme.
I Introduction
Nonorthogonal multiple access (NOMA) and cognitive radio (CR) have emerged as efficient techniques to improve the spectral efficiency [1, 2]. By naturally combining the concepts of both NOMA and CR, a cognitive radioinspired NOMA (CRNOMA) scheme was proposed and studied in [3]. In CRNOMA, the unlicensed secondary users (SU) is opportunistically served under the condition that the quality of service (QoS) of the licensed primary users (PU) is satisfied. As a result, the transmit power allocated to the SU is constrained by the instantaneous signaltointerferenceplusnoise ratio (SINR) of the PU. Compared to the conventional CR systems, higher spectral efficiency can be achieved by CRNOMA because both the PU and SU can be served simultaneously using the same spectrum.
Recently, multipleinput multipleoutput (MIMO) techniques have been considered in CRNOMA systems to exploit the spatial degrees of freedom [4]. To avoid high hardware costs and computational burden while preserving the diversity and throughput benefits from MIMO, the antenna selection (AS) problem for MIMO CRNOMA systems has been investigated in [5], wherein the SU was assumed to be rate adaptive and the design criterion was to maximize the SU’s rate subject to the QoS of PU. On the other hand, the outage probability has also been commonly used to quantify the performance of AS for an alternative scenario, wherein users have fixed transmission rates [6]. To the best of the authors’ knowledge, the outageoriented AS schemes for CRNOMA systems have not been studied in open literature.
Motivated by this, the design and analysis of the outageoriented joint AS algorithm for MIMO CRNOMA networks is studied in this letter, which is fundamentally different from that for orthogonal multiple access (OMA) networks. This is because there is severe interuser interference in NOMA scenarios, wherein the signals are transmitted in an interferencefree manner in OMA scenarios. Moreover, the transmit power allocated to the SU in CRNOMA scenarios is constrained by the instantaneous SINR of the PU, which is affected by the antenna selection result. In this case, the joint antenna selection for NOMA networks is coupled with the power allocation design at the BS, which makes the design and analysis of the joint AS problem for CRNOMA networks more challenging. In this letter, we propose a new lowcomplexity joint AS scheme, namely subsetbased joint AS (SJAS), to maximize the signaltonoise ratio (SNR) of the SU under the condition that the QoS of the PU is satisfied. The asymptotic closedform expression of the outage performance for SJAS is derived, and the minimal outage probability achieved by SJAS among all possible joint AS schemes is proved. Numerical results demonstrate the superior performance of the proposed scheme.
Ii System Model and Proposed Joint AS Scheme
Consider a MIMO CRNOMA downlink scenario as [4], wherein two users including one PU and one SU are paired in one group to perform NOMA. We consider that BS, PU and SU are equipped with , and antennas, respectively. We assume that the channels between the BS and users undergo spatially independent flat Rayleigh fading, then the entries of the channel matrix, e.g., (), can be modelled as independent and identically distributed complex Gaussian random variables, where () represents the channel coefficient between the th antenna of the BS and the th (th) antenna of the PU (SU). For notation simplicity, we define and .
As in [7], we consider that the BS selects one (e.g., th) out of its antennas to transmit information, while the users select one (e.g., th and th) out of and available antennas respectively to receive massages. In this sense, only one RF chain is needed at each node to reduce the hardware cost, power consumption and complexity, and only the partial channel state information, i.e., the channel amplitudes, is needed at the BS, which is assumed perfectly known at the BS through the control signalling.
According to the principle of NOMA, the BS broadcasts the superimposed signals , where () denotes the signal to the PU (SU) with , and and are the power allocation coefficients satisfying . Then the received signals at the PU and SU are given by
(1)  
(2) 
where is the transmit power at the BS, and () is the complex additive white Gaussian noise with variance (). For simplicity, we assume .
Following the principle of CRNOMA, is decoded by treating as noise at both users, and may be recovered at the SU when has been successfully subtracted in the SIC procedure. By denoting the transmit SNR as , the received SINR of decoding at the PU is given by
(3) 
Similarly, the received SINR to detect at the SU is given by
(4) 
When is successfully removed, the SNR to detect at the SU is given by
(5) 
Let () denotes the predetermined detecting threshold of (). As the SU is served on the condition that is met, mathematically, and should satisfy the following constraint simultaneously: .
Iia The Formulation of Joint AS Optimization Problem
In order to maximize the received SNR of the SU, we would like to solve the following optimization problem:
(6a)  
(6b)  
(6c) 
where , and is the joint optimization problem of antenna selection and power allocation. Specifically, similar to [8], given the antenna indexes , and , the optimal power allocation strategy can be obtained based on Lemma 1.
Lemma 1.
Given the antenna indexes , and , the optimal power allocation strategy is given by
(7) 
where .
Proof.
Given antenna indexes , and , by substituting (3)(4) into (6b), the power coefficient should satisfy the condition: . Meanwhile, is an increasing function of as shown in (5). In this case, in order to maximize , should take the maximum value in its range. By noting that , we then can express the optimal power allocation coefficient as in (7). The proof is completed. ∎
By substituting (7) into (5) and when , we have
(8) 
otherwise . At this point, the joint optimization problem is simplified into the joint antenna selection problem. It is straightforward to see that finding the optimal antenna indexes may require an exhaustive search (ES) over all possible antenna combinations with the complexity of
IiB Proposed Subsetbased Joint AS (SJAS) Scheme
The aim of SJAS algorithm is to decrease the computational complexity by greatly reducing the searching set, while ensuring the QoS of the PU and maximizing the achievable SNR of the SU. Specifically, SJAS mainly consists of the following three stages.

Stage 1. Build the subset to reduce the search space, where and are the maximumvalue elements in the th row of and , respectively. Mathematically, we have
(9) (10) 
Stage 2. Build the subset by selecting the pairs from , in which each pair ensures the target SINR of the PU can be supported and can be subtracted successfully at the SU. That is,
(11) where and can be obtained by substituting and into (3) and (4), respectively. Specifically, is given in (7) with .

Stage 3. When , select the antenna triple which can maximize the SNR for the SU, i.e.,
(12) Let and denote the original column indexes of and , respectively. That is, the th antenna at the BS, and the th and th antennas at the PU and SU are jointly selected. In contrast, when , the system suffers from an outage.
As mentioned before, the complexity of the ESbased scheme is as high as . In contrast, the complexity of the proposed SJAS scheme is upper bounded by . For the case , we can find that the complexity of SJAS is approximately , which is an order of magnitude lower than of the optimal ESbased scheme.
Iii Performance Evaluation
In this section, we will analyse the system outage performance achieved by SJAS. By using the assumption that channel coefficients are Rayleigh distributed, the cumulative density functions (CDF) and the probability density functions (PDF) of and in can be expressed as in [5],
(13)  
(14)  
(15) 
where , , and and are expanded based on the binomial theorem.
Let denote the event , and denote the event while . As in [9], the overall system outage is defined as the event that any user in the system cannot achieve reliable detection, i.e.,
(16) 
In this case, the asymptotic system outage probability can be obtained according to the following lemma.
Lemma 2.
When the transmit SNR , the system outage probability achieved by SJAS can be approximated as
(17)  
where , , , , , and .
Proof.
we can first calculate the term as
(18)  
where and the CDF of is given by
(19)  
We then turn to the calculation of ,
(20)  
Let for . Since for , the product in (20) can be presented as
(21) 
Then can be further expressed as
(22)  
in which,
(23)  
Where
(24)  
(25) 
Let , , and , then can be obtained as follows,
(26)  
Similarly, when , can be approximated as
(27)  
where .
Remark 1: When approaches infinity, , and approach zero. By using the binomial theorem and the approximation , the system outage probability can be further approximated as follows:
(30)  
where , , and . From (30), we can see that the SJAS scheme can realize a diversity of .
Remark 2: The optimality of the proposed SJAS is illustrated in the following lemma.
Lemma 3.
The proposed SJAS scheme minimizes the system outage probability of the considered MIMO CRNOMA system.
Proof.
This lemma can be proved by contradiction. Suppose there exists another joint AS strategy achieving a lower system outage probability than SJAS. Let denote the antenna triple selected by the new strategy. According to the assumption, it is possible that there is no outage with antennas while an outage occurs with antennas. In this case, the pair must be in to satisfy the target SINR of the PU, i.e., . Recall that the pair is selected according to (12) to maximize . In this case, if the maximized cannot meet , the antennas selected by other scheme which provides smaller cannot meet , either. Therefore, one can conclude that there is NO other joint AS strategies can achieve a lower outage probability than SJAS, which is contradicted to the assumption made earlier. The lemma is proved. ∎
Iv Numerical Studies
In this section, the performance of the proposed SJAS algorithm for MIMO CRNOMA networks is evaluated by Monte Carlo simulations. Let (), where () is the distance between the BS and PU (SU), and the pathloss exponent is set as .
Fig. (a) and (b) compare the received SNR of the SU and the system outage performance between SJAS and other AS strategies. As illustrated in both figures, over the entire SNR region, SJAS outperforms the conventional maxmin scheme, in which the antenna selection is executed under the maxmin criteria, i.e., for all and . Furthermore, the performance of both SJAS and the maxmin scheme are much better than that of random AS, since both SJAS and the maxmin scheme utilize the spatial degrees of freedom brought by the multiple antennas at each node. We also see that the analytical results match the simulation results for SJAS, which validates our theoretical analysis in Sec. III. Moreover, compared to the optimal ES scheme, SJAS can achieve the optimal outage performance as discussed in Remark 2, but with significantly reduced computational complexity. In particular, the corresponding average power allocation coefficient for each scheme is illustrate in Table. I. Again we can find that the SJAS can achieve the same power allocation of the optimal ES scheme.
Transmit power  dBm  dBm  dBm  dBm  dBm 

Random AS  0.0155  0.1491  0.3715  0.5425  0.6359 
Maxmin AS  0.1441  0.5006  0.6417  0.6864  0.7006 
ES AS  0.1418  0.4624  0.5997  0.6641  0.6915 
SJAS  0.1418  0.4624  0.5997  0.6641  0.6915 
V Conclusion
In this letter, we studied the joint AS and power allocation problem for a MIMO CRNOMA system. A computationally efficient SJAS scheme was proposed, and the asymptotic closedform expression for the system outage performance and the diversity order for SJAS were both obtained. Numerical results demonstrated that SJAS can outperform both the conventional maxmin approach and the random selection scheme, and can achieve the optimal performance of the ES algorithm.
Footnotes
 is usually used in the efficiency analysis of algorithms and when .
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