This paper studies a multiple-input single-output non-orthogonal multiple access cognitive radio network relying on simultaneous wireless information and power transfer. A realistic non-linear energy harvesting model is applied and a power splitting architecture is adopted at each secondary user (SU). Since it is difficult to obtain perfect channel state information (CSI) in practice, instead either a bounded or gaussian CSI error model is considered. Our robust beamforming and power splitting ratio are jointly designed for two problems with different objectives, namely that of minimizing the transmission power of the cognitive base station and that of maximizing the total harvested energy of the SUs, respectively. The optimization problems are challenging to solve, mainly because of the non-linear structure of the energy harvesting and CSI errors models. We converted them into convex forms by using semi-definite relaxation. For the minimum transmission power problem, we obtain the rank-2 solution under the bounded CSI error model, while for the maximum energy harvesting problem, a two-loop procedure using a one-dimensional search is proposed. Our simulation results show that the proposed scheme significantly outperforms its traditional orthogonal multiple access counterpart. Furthermore, the performance using the gaussian CSI error model is generally better than that using the bounded CSI error model.
Non-orthogonal multiple access (NOMA) has been recognized as one of the most promising techniques for next-generation wireless communication systems due to its capability of supporting a high spectral efficiency (SE) and massive connectivity . Since its design philosophy may be combined with diverse transceivers, it has drawn tremendous attention in multiple-antenna systems -, in cooperative networks , , in device-to-device (D2D) networks , as well as in downlink and uplink multi-cell networks . In contrast to classic orthogonal multiple access (OMA), NOMA provides simultaneous access to multiple users at the same time and on the same frequency band, for example by using power-domain multiplexing. In order to decrease the mutual interference among different users of power-domain NOMA, successive interference cancellation (SIC) may be applied by the receivers . It has been shown that NOMA is capable of achieving a higher SE and energy efficiency (EE) than OMA -.
As another promising technique of improving the SE, cognitive radio (CR) techniques have also been investigated for decades, where the secondary users (SUs) may access the spectrum bands of the primary users (PUs), as long as the interference caused by SUs is tolerable . According to , in order to implement CR in practice, three operational models have been proposed, namely, opportunistic spectrum access, spectrum sharing, and sensing-based enhanced spectrum sharing. It is envisioned that the combination of NOMA with CR is capable of further improving the SE. As a benefit of its low implementational complexity, spectrum sharing has been widely applied. In -, the authors analyzed the performance of a spectrum sharing CR combined with NOMA. It was shown that the SE can be significantly improved by using NOMA in CR compared to that achieved by using OMA in CR.
On the other hand, the increasing greenhouse gas emissions have become a major concern also in the design of wireless communication networks. According to , cellular networks world-wide consume approximately 60 billion kWh energy per year. Moreover, this energy consumption is explosively increasing due to the unprecedented expansion of wireless networks to support ubiquitous coverage and connectivity. Furthermore, because of the rapid proliferation of Internet of Things (IoT) applications, most battery driven power limited IoT devices become useless if their battery power is depleted. Thus it is critical to use energy in an efficient way or to harness renewable energy sources. As remedy, energy harvesting (EH) exploits the pervasive frequency radio signals for replenishing the batteries . There have been two research thrusts on EH using RF technology. One focuses on wirelessly powered networks, where a so-called harvest-then-transmit protocol is applied . The other one uses simultaneous wireless information and power transfer (SWIPT) -, which is the focus of this paper. The contributions of SWIPT in CR has been extensively studied. Specifically, authors of  considered the optimal beamforming design in a MISO CR downlink network. A similar power splitting structure to that of our work is applied at the user side. Hu et. al , on the other hand, investigated the objective function of EH energy maximization, and a resource allocation problem was formulated to address that goal. Additionally,  considered the underlay scheme in CR network and proposed the optimal beamforming design. To address both the SE and EE, a multiple-input single-output (MISO) NOMA CR using SWIPT is considered based on a practical non-linear EH model. Robust beamforming design problems are studied under a pair of channel state information (CSI) error models. The related contributions and the motivation of our work are summarized as follows.
I-a Related Work and Motivation
The prior contributions related to this paper can be divided into two categories based on the EH model adopted, i.e. the linear - and the non-linear EH model , -. In the linear EH model, the power harvested increases linearly with the input power, while the EH under the non-linear model exhibits more realistic non-linear characteristics especially at the power-tail.
Linear EH model: In , Liu et al. analyzed the performance of a cooperative NOMA system relying on SWIPT, which outperformed OMA. Do et al.  extended  and studied the beneficial effect of the user selection scheme on the performance of a cooperative NOMA system using SWIPT. In , Yang et al. presented a theoretical analysis of two power allocation schemes conceived for a cooperative NOMA system with SWIPT. It was shown that the outage probability achieved under NOMA is lower than that obtained under OMA. Diamantoulakis et al.  studied the optimal resource allocation design of wireless-powered NOMA systems. The optimal power and time allocation were designed for maximizing the max-min fairness among users. In their following work , a joint downlink and uplink scheme was considered in a wireless powered network, followed by comparisons between NOMA and TDMA. The results show that NOMA is more energy efficient in the downlink of SWIPT networks. In order to improve the EE, multiple antennas were applied in a NOMA system associated with SWIPT, and the transmit beamforming and the power splitting factor were jointly optimized for maximizing the transmit rate of users .
The contributions in - investigated conventional wireless NOMA systems, which did not consider the interference between the secondary network and the primary network. Recently, authors of , - studied optimal resource allocation problems in CR associated with SWIPT. In , an optimal transmit beamforming scheme was proposed in a multi-objective optimization framework. It was shown that there are several tradeoffs in CR-aided SWIPT. Based on the work in , the authors proposed a jointly optimal beamforming and power splitting scheme to minimize the transmit power of the base station in multiple-user CR-aided SWIPT . Considering the practical imperfect CSI, Zhou et al.  studied robust beamforming design problems in MISO CR-aided SWIPT, where the bounded and the gaussian CSI error models were applied. It was shown that the performance achieved under the gaussian CSI error model is better than that obtained under the bounded CSI error model. The work in  was then extended to multiple-input multiple-output (MIMO) CR-aided SWIPT in  and , where the bounded CSI error model was applied in  and the gaussian CSI error model was used in  and . In contrast to , -, Zhou et al.  studied robust resource allocation problems in CR-aided SWIPT under opportunistic spectrum access.
Non-linear EH model: In , robust resource allocation schemes were proposed for maximizing the sum transmission rate or the max-min transmission rate of MIMO-assisted wireless powered communication networks, where a practical non-linear EH model is considered. It was shown that a performance gain can be obtained under a practical non-linear EH model over that attained under the linear EH model. In order to maximize the power-efficient and sum-energy harvested by SWIPT systems, Boshkovska et al. designed optimal beamforming schemes in  and . Recently, under the idealized perfect CSI assumption, the rate-energy region was quantified in MIMO systems relying on SWIPT and the practical non-linear EH model in . In order to improve the security of a SWIPT system, a robust beamforming design problem was studied under a bounded CSI error model in . The investigations in , - were performed in the context of conventional SWIPT systems. Recently, Wang et al.  extended a range of classic resource allocation problems into a wireless powered CR counterpart. The optimal channel and power allocation scheme were proposed for maximizing the sum transmission rate.
The resource allocation schemes proposed in - investigated a conventional NOMA system with SWIPT. The mutual interference should be considered and the quality of service (QoS) of the PUs should be protected in NOMA CR. Moreover, the resource allocation schemes proposed in , - are based on the classic OMA scheme. Thus, these schemes are not applicable to NOMA CR with SWIPT due to the difference between OMA and NOMA. Furthermore, an idealized linear EH model was applied in -, which is impractical since the practical power conversion circuit results in a non-linear end-to-end wireless power transfer. Therefore, it is of great importance to design optimal resource allocation schemes for NOMA CR-aided SWIPT based on the practical non-linear EH model.
Although the practical non-linear EH model was applied in , -, the authors of , - considered conventional OMA systems using SWIPT. Moreover, the resource allocation scheme proposed in  is based on OMA and cannot be directly introduced in NOMA CR-aided SWIPT. However, at the time of writing, there is a scarcity of investigations on robust resource allocation design for NOMA CR-aided SWIPT under the practical non-linear EH model. Several challenges have to be addressed to design robust resource allocation schemes for NOMA CR-aided SWIPT. For example, the impact of the CSI error and of the residual interference due to the imperfect SIC should be considered, which makes the robust resource allocation problem quite challenging. Thus, we study robust resource allocation problems in NOMA CR-aided SWIPT.
I-B Contributions of the Paper
Our contribution expands  in three major contexts. Firstly, in this paper, a NOMA MISO CR-aided SWIPT is considered, while a OMA MIMO wireless powered network was used in . Secondly, the work in  relies on the bounded CSI error model, while both the bounded and the gaussian CSI error model are applied in our work. Thirdly, part of our work considers the minimum transmit power as the optimization objective, which is not considered in . Notice that this paper is also an extension from our conference one  which only considered minimizing transmission power under bounded imperfect CSI model. The contributions of our work are hence summarized as follows.
A minimum transmission power problem is formulated under both the bounded and the gaussian CSI error models in a NOMA MISO CR network. The robust beamforming weights and the power splitting ratio are jointly designed. The original problem is hard to solve owing to its non-convex nature arising from the non-linear EH model as well as owing to the imperfect CSI. Hence we transform this problem to a convex one. Finally, we prove that the robust beamforming weights can be found and the rank is lower than two under the bounded CSI error model.
We also consider another optimization problem, where the objective function is based on maximizing the harvested energy. Similarly, this problem is formulated under the above pair of imperfect CSI error models. The non-linear EH model makes the original problem even harder to solve. Nevertheless, we managed to transform it to an equivalent form and applied a two-loop procedure for solving it. The inner loop solves a convex problem, while the outer loop iteratively adjusts the parameters. Furthermore, to decouple the coupled variables, a one-dimensional search algorithm is proposed as well.
Simulation results show the superiority of the proposed scheme over the traditional OMA scheme; the performance gain of NOMA becomes higher when the required data rate at each SU is higher. Moreover, the results also demonstrate that under gaussian CSI error model, the performance is generally better than that under the bounded CSI error model.
The remainder of this paper is organized as follows. The system model is presented in Section II. Section III details our robust beamforming design in the context of our power minimization problems under a pair of imperfect CSI error models. Robust beamforming design in EH maximization problems under both imperfect CSI error models are presented in Section IV. Our simulation results are discussed in Section V. Finally, Section VI concludes the paper.
Notations: Boldface capital letters and boldface lower case letters denote matrices and vectors, respectively. The identity matrix is denoted by ; vec(A) represents the vectorization of matrix A and it is attained by stacking its column vectors. The Hermitian (conjugate) transpose, trace, and rank of a matrix A are represented respectively by , Tr and Rank. denotes the conjugate transpose of a vector . denotes a -by- dimensional complex matrix set. represents that is a Hermitian positive semi-definite (definite) matrix. represents the Euclidean norm of a vector. denotes the absolute value of a complex scalar. represents that is a random vector, which follows a complex Gaussian distribution with mean and covariance matrix . represents the expectation operator. extracts the real part of vector . represents the set of all non-negative real numbers.
Ii System and Energy Harvesting Models
Ii-a System Model
We consider a downlink CR system with one cognitive base station (CBS), one primary base station (PBS), PUs and SUs. The CBS is equipped with antennas, while each user and PBS have a single antenna. It is assumed that the SUs are energy-constrained and energy harvest circuits are used. Specifically, the receiver architecture relies on a power splitting design. Once the signal is detected by the receiver, it will be divided into two parts. One part is used for information detection, while the other part for energy harvesting. Similar structures can be found in , . To better utilize the radio resources, all UEs are allowed to access the same resource simultaneously. To be specific, the PBS sends messages to all PUs, while the CBS communicates with all SUs simultaneously by applying NOMA principles by controlling the interference from the CBS to PUs below a certain level .
Let us denote the set of SUs and PUs as and , respectively. The signal received by the th SU can be expressed as
where is the channel gain between the CBS and the th SU, while is the joint effect of additive white Gaussian noise (AWGN) and interference from the PBS. , where is the power. This interference model represents a worst-case scenario . Furthermore, is the message transmitted to SUs after precoding. According to the NOMA principle, we have:
where is the precoding vector for the -th UE and is the corresponding intended message. Furthermore, is the energy vector allowing us to improve the energy harvesting efficiency at the SUs. We assume that is unitary, i.e. , and obeys the complex Gaussian distribution, i.e. , where is the covariance matrix of .
Likewise, the extra interference arriving from the CBS to the -th PU is
where is the channel gain between the CBS and the -th PU .
Ii-B Non-linear EH Model
Most of the existing literature considered an idealized linear energy harvesting model, where the energy collected by the -th SU is expressed as , is the input power, where is the power splitting factor that controls the amount of received energy allocated to energy harvesting, , while is the energy conversion efficiency factor, . However, measurements relying on real-world testbeds show that a typical energy harvesting model exhibits a non-linear end-to-end characteristic. To be specific, the harvested energy first grows almost linearly with the increase of the input power, and then saturates when the input power reaches a certain level. Several models have been proposed in the literature and one of the most popular ones is , which is formulated as follows:
where is the actual energy harvested from the circuit. Furthermore, represents a function of the input power . Additionally, is the maximum power that a receiver can harvest, while together with characterizes the physical hardware in terms of its circuit sensitivity, limitations, and leakage currents .
On the other hand, the signal received in the -th SU information decoding circuit is
where is the AWGN imposed by the information decoding receiver.
Iii Power Minimization Based Problem Formulation
Since is a composite signal consisting of all SUs’ messages, SIC is applied at the receiver side to detect the received signal. The detection is carried out in the same order of the channel gains, i.e. the SUs with lower channel gain will be decoded first. A pair of imperfect CSI error models are considered, namely a bounded and a gaussian model. We adopt both of these in this paper and assume that all SUs have a perfect knowledge of their own CSI.
Iii-a Bounded CSI Error Model
In this model, we consider a bounded error imposed on the estimated CSI, which can be treated as the worst-case scenario. Specifically, the channels can be modeled as follows.
where and are the estimated channel vectors for and , respectively, while and define the set of channel variations due to estimation errors. The model defines all the uncertainty regions that are confined by power constraints. Furthermore, we use block Rayleigh fading channels, which remain constant within each block, but change from block to block independently.
Iii-A1 NOMA Transmission
Without loss of generality, we sort the estimated channel of SUs in the ascending order, i.e., . According to the SIC principle, SU can detect and remove SU ’s signal, for . Thus, when SU decodes signal , the signals of the previous SUs have already been removed from the composite received signal. Due to channel estimation errors, however, these ) signals may not be completely removed, leaving some residual signals as interference. Therefore, the signal at UE when decoding becomes
Here, the first term is the desired received signal, the second term is the interference due to imperfect channel estimation, and the third term represents the NOMA interference. The corresponding signal-to-interference-plus-noise ratio (SINR) for the -th SU after the SIC applied at the receiver is given by:
Since the signal can be detected at every SU , as long as , there will be a set of SINRs for signal . For CBS, the maximum data rate for SU should be Moreover, the channel estimation error should be considered. The worst-case data rate for SU becomes
Iii-A2 Problem Formulation
In this sub-section, we seek to find the precoding vectors , the energy vector , and the power split ratio , which altogether achieve a satisfactory quality of service (QoS) for all users, and at the same time, they can harvest part of the energy for their future usage. Thus, the problem can be formulated as follows:
Our goal is to minimize the total transmitted power. The constraint ensures that SU does attain the predefined minimum data rate; allows each SU to harvest the amount of energy that at least compensates the static power dissipation ; is the interference limit for the -th PU; represents the maximum transmit power constraint of the BS; in , the power split factor should be in the range of . The optimization problem is hard to solve due to its non-convexity constraints and . Moreover, the realistic imperfect CSI imposes another challenge on the original problem. In the following, we transform the variables.
Let us introduce and . Then in (10b) becomes
For the notational simplicity, we denote the above constraint as . Thus, becomes
Here, comes from the fact that both and are positive semi-definite matrices. The extra constraint that the rank of should be 1 is also non-convex. In what follows, we first reformulate in (12b) according to the -Procedure of .
in (12b) can be reformulated as
where and , and is a slack variable conditioned on .
Given and (11), we have
From the fact that and according to the -Procedure, the lemma is proved. \qed
Similarly, in (12d) can be transformed into
where , and is also a slack variable.
Next, we apply similar manipulations to (12c), which becomes
where is a constant. This condition holds, provided that , which is always true in real systems.
Then, applying the -Procedure to (16), we have the following
Observe that we drop (12h), since it is not a convex term. This relaxation is commonly referred to as the semi-definite relaxation (SDR) technique. For the specific problem in , the following theorem proves that the optimal has a limited rank.
If is feasible, the rank of is always less than or equal to 2.
See Appendix. \qed
The transformed problem is not convex because of the coupling variables in (17) and in the denominator of (13). To be able to take advantage of the CVX software package, we introduce a pair of auxiliary variables. Specifically, let and . In this way, (13), (15), and (17) become convex terms. Then, we have additional constraints for and :
It may be readily verified that this transformation does not change the optimal solution of .
Iii-B Matrix Decomposition
Now we proceed to find the solution of the problem , after which there is one more step to get the original solution for . If yields rank 1, we can simply write . Otherwise, if , we have several optional approaches to extract . To name a few, we list two methodologies here.
Eigen-decomposition. Let us denote two eigenvalues of by and , where . Clearly, , are the corresponding eigenvectors. To get the rank 1 approximation from a rank 2 matrix, we can let the solution of the original problem be , provided it is feasible.
Randomization technique. Similar to eigen-decomposition, we first decompose according to . Then, we let , where the m-th element of is and obeys an independent and uniform distribution within .
The above two methods are essentially the same. If we want to get a more precise result, another scaling factor can be added. Specifically, let us define as the scaling factor yet to be determined. Certainly, the problem can be transformed in terms of and , once we get the optimal value, we can apply either one of the above methods to get a better result. Another point worth noting here is that when the rank of is 2, there only exists the approximation result of , and this approximation always provides an upper bound.
Iii-C Gaussian CSI Error Model
In Section III-A, we introduced a bounded channel model, which defines a confined region for the channel variations, which provides a worst-case estimation. Another commonly used more realistic estimation model assumes that the channel estimation error obeys the Gaussian distribution , which is formulated as follows:
where and are the channel estimation error vectors, while and are the channel vectors estimated at the BS side. Furthermore, and are the covariance matrices of the estimation error vectors.
Even though we apply different channel models, the residual interference due to imperfect CSI estimation affects the message detection similarly to the bounded error model. Thus the achievable data rate expression of SU remains the same except that is in a new set. In contrast to the existing NOMA contributions on imperfect CSI , in this paper we use the above-mentioned gaussian estimation error model to form an optimization problem as follows:
Here, we assume that the probability of having a rate of is higher than , which is a predefined value, and we use the threshold to control the probability. Likewise, and , where and , are used for controlling the outage probability of harvested energy of the th SU and the interference experienced by the -th PU, respectively. is hard to solve owing to its non-convexity, together with constraints , which involve probability and uncertainty. Inspired by , we solve the resulted optimization problem with the aid of approximations by applying Bernstein-type inequalities .
Iii-C1 Bernstein-type Inequality I 
Let , where , , , and . For any , an approximate and convex form of
can be written as
Here, and are slack variables.
In order to use the above Lemma, we have to transform to a standard complex Gaussian vector. Let , where . Substituting it into (11), the convex approximation becomes
where and are slack variables.
Iii-C2 Bernstein-type Inequality II 
Let , where , , , and . For any , an approximate and convex form for
can be written as
where and are slack variables.
We apply Bernstein-type Inequality II to (21d), and let , where is a standard Gaussian vector. We can have the following convex-form approximation.
where and are slack variables.
Lastly, we relax by dropping the constraint that should have rank 1 for now, since it is not a convex one. The relaxed version of the problem is
Likewise, the coupling variables in (24b) and (26b) make a non-convex problem. Thus we can still use the transformation in (19), which converts into an equivalent optimization problem that can be efficiently solved by CVX.
Iv Maximum Harvested Energy Problem Formulation
In contrast to Sections III, where the minimum transmission power problem is considered, in the following we consider the optimization problem of maximizing the total harvested energy. This problem has important real-world applications, since most of the consumer electronics products are battery-driven and thus their energy efficiency is critical. In this section, we first formulate the problem, then we transform it in a convex way so that an existing software package can solve it efficiently. A one-dimensional search algorithm will be used. Furthermore, we also consider our previous pair of channel models.
Iv-a Bounded CSI Error Model
Upon considering the imperfect CSI model used in (6), the maximum total harvested energy of all SUs can be formulated as follows:
The rank operation is not convex, thus we drop the constraint (12h) first, as previously in . Additionally, the objective function relies on a realistic non-linear energy harvesting model, and it is not convex either. Essentially, it is a sum-of-ratio problem, and its global optimization is possible by applying the following transformations:
Furthermore, according to , , if has the optimal solutions and , there exist two sets of vectors and such that the solutions are also optimal for the following equivalent parametric optimization problem:
The optimal solutions and the vectors should satisfy
Now, the objective function has the log-concave form and it can be solved given the sets and . The iterative update of the vector sets can be carried out in the following way. Let us define the function , . The next set of values of and can be updated by solving . Specifically, in the -th iteration, we update them as:
A two-loop algorithm is proposed for solving the problem. The outer loop gives and as the inputs of the inner loop, while the inner loop finds and . Observe that in (33), there is a coupling variable , which is convex with a given . Therefore, in the inner loop, we have to perform a one-dimensional search for as well. The detailed algorithm is formulated in Algorithm 1.