I Introduction
I-a Motivation and Related Work
The application of simultaneous wireless information and power transfer (SWIPT) to CRNs is a promising solution to simultaneously improve the spectrum efficiency (SE) and energy efficiency [1]-[5]. Up to now, the existing SWIPT literature can be categorized into two research directions based on the energy harvesting (EH) model, namely, the linear EH model [3]-[5] and the non-linear EH model [6]-[11]. In [3], we studied robust resource allocation in CRNs with SWIPT and proposed optimal beamforming schemes for minimizing the transmission power of the CBS. In [4], the authors proposed a multi-objective optimization problem for MISO CRNs with SWIPT. It was shown that there are several tradeoffs among the objectives considered in [4], such as the tradeoff between the harvested energy and the secrecy rate. To further improve SE and energy transfer efficiency, the authors studied the precoding design problem in multiple-input multiple-output CRNs with SWIPT [5].
Although the beamforming design problems have been well studied in CRNs with SWIPT under the ideal linear EH model [2]-[5], the proposed beamforming schemes are not suitable or suboptimal in practice due to the non-linear behavior of practical EH circuits [6]-[11]. In order to address this issue, the authors in [6] proposed a practical non-linear EH model and a beamforming scheme for maximizing the total harvested power of all EHRs. It was shown that the beamforming scheme designed under the non-linear EH model and achieves better performance than that achieved under the linear EH model. These investigations were extended to MIMO systems in [7]-[9]. More recently, the authors in [10] and [11] have proposed optimal resource allocation schemes in CRNs with SWIPT under the non-linear EH model.
However, the beamforming design problems in [6]-[11] were formulated in a single objective framework. It over-emphasizes the importance of one metric and cannot achieve a good tradeoff among multiple metrics, especially when there are several conflicting metrics, such as energy efficiency and SE. Thus, in order to achieve a good tradeoff between the transmission power of the CBS and the harvested power of EHRs, unlike the work in [6]-[11], an optimization problem with a weighted sum of multiple objectives is established for MISO CRNs with SWIPT under two non-linear EH models.
I-B Contributions and Organization
The main contributions of our work are: Unlike the existing works in [2]-[5], a robust resource allocation scheme is proposed for CRNs with SWIPT under two practical non-linear EH models and imperfect CSI. The successive convex approximation (SCA) technique and are exploited to solve the formulated challenging non-convex problems. We prove that the solutions obtained using SCA satisfy the Karush-Kuhn-Tucker (KKT) conditions. In addition, our simulation results compare the performances achieved under the two non-linear EH models.
The remain of this letter is organized as follows. Section II presents the system model. The robust resource allocation problem is formulated in Section III. Section IV presents simulation results and Section V concludes this letter.
Ii System Model
In this letter, a downlink MISO CR network with SWIPT is considered, where the primary network coexists with the secondary network under the spectrum sharing paradigm [4]. The primary network consists of a primary base station (PBS) and primary users (PUs). In the secondary network, the CBS transmits information to a secondary user (SU) and simultaneously transfers energy to EHRs. Only one secondary user is assumed in order to reduce the multiple access interference caused to the primary user receivers. This assumption has been widely used [3], [4]. The PBS is equipped with () antennas while the CBS is equipped with () antennas. The SU, PUs and EHRs are each equipped with a single antenna. The signals received at the SU, the th EHR, and the interference signals affecting the th PU from the CBS are denoted by , and , respectively, and given by
(1a) | |||
(1b) | |||
(1c) |
where , and
denote the channel vectors between the CBS and the SU, the
th EHR and the th PU, respectively. Boldface lower case letters represent vectors; represents the conjugate transpose of a vector ; represents a -by- dimensional complex matrix set. In addition, and are the channel vectors from the PBS to the SU and the th EHR, respectively. In (1), and , respectively, denote the corresponding beamforming vectors of the CBS and the PBS while and denote the zero-mean additive white Gaussian noise (AWGN) at the SU and the th EHR, respectively. In (1), and are the data symbols intended for the SU and the PUs, respectively. Without loss of generality, it is assumed that . Similar to [3]-[5], the PBS adopts a constant power transmission policy.A practical spectrum sharing scenario where there is no cooperation among the CBS, PUs and EHRs due to the limited resource is considered. In this case, , , and are imperfect and modeled as , , and , respectively [3]-[5]. , , and
are the estimates of
, , and , and , , , and are the corresponding estimation errors, where , , and denote the uncertainty regions, respectively. , , and are the corresponding radii of the uncertainty regions.The total radio frequency (RF) power received at the th EHR denoted by , can be represented as
(2) |
In this letter, two practical non-linear EH models are considered. According to [6] and [7], the harvested energy in the first model at the th EHR denoted by is given by
(3a) | |||
(3b) |
where represents the maximum harvested power at the th EHR when the EH circuit is satured. Parameters and are related to the specifications of the EH circuit.
Recently, another practical non-linear EH model has been proposed in [8], where the authors considered a sensitivity property in the non-linear EH model. More specifically, the harvested energy is set to zero when the input RF power is smaller than the EHRs’ sensitivity threshold. The non-linear EH model proposed in [8] is given by
(4) |
where is the sensitivity threshold. Based on this sensitivity property, when is smaller than , is 0. The parameters and control the steepness of this function.
Iii Robust Resource Optimization Formulation
To achieve a good tradeoff between the transmission power and the total harvested energy, an optimization problem with a weighted sum of multiple objectives is formulated. Let , and . Since does not affect the beamforming design, can represent the harvested power at the th EHR [6]. The robust resource optimization problem can be formulated as
(5a) | |||
(5b) | |||
(5c) | |||
(5d) | |||
(5e) | |||
(5f) |
where the constant is a weight in the range . In (5a), and indicate the preference of the system designer for the harvested energy maximization and the transmit power minimization, respectively. In the special case, when or , P is equivalent to the harvested energy maximization or the transmit power minimization, respectively, and is the maximum tolerable interference power of the th PU. The constant is the minimum signal-to-interference-noise (SINR) required at the SU. The constraint guarantees that the SINR of the SU is not less than and denotes the maximum transmit power of the CBS.
It is observed that the objective function in P is non-convex. To this end, we introduce auxiliary variables () to transform the non-convex objective function into an equivalent convex form. Concretely, P can be equivalently expressed as
(6a) | |||
(6b) | |||
Although the transformed objective function is linear, the optimization problem P is still non-convex due to the rank-one constraint and the non-convex constraint . To obtain a tractable solution, semi-definite relaxation (SDR) is exploited to tackle the constraint and the SCA technique is applied to address . Thus, P can be solved by iteratively solving P given by
(7a) | ||||
(7b) | ||||
(7c) | ||||
where is a constant at the th iteration of the proposed algorithm, which is shown in Table I and denotes the number of iterations. In order to address the channel uncertainty, the is applied [3]. Let , , and . Thus, the constraints (5b), (5c) and (7c) can be equivalently expressed as
(8) |
(9) |
(10a) | |||
(10b) | |||
where , , , , and are slack variables. It is seen from that . Thus, can be obtained by using one dimensional search method. In this case, P can be solved by solving P given as
(11a) | |||
(11b) |
where is the set of all slack variables. It is seen that P is convex and can be easily solved by using convex optimization tools such as CVX [3].
Theorem 1: For problem P, it is assumed that . The optimal W matrix is unique and its rank is one.
Proof: See Appendix A.
Theorem 2: When the non-linear EH model given by eq. (4) is adopted, the robust resource optimization problem can be also solved by using Algorithm 1 and the optimal W matrix is unique and rank-one.
Algorithm 1: The iterative algorithm for |
1: Setting: |
. |
2: Inputting: |
, , the CSI h, the estimated CSI , , and . |
3: Initialization: |
the iteration index , the maximum accuracy , |
’s initialization , one-dimensional search step . |
4: Optimization: |
for |
repeat |
Solve the problem P by the software CVX, obtain the |
objective value and the intermediate beamformer w, |
where . |
if is satisfied, then |
return the objective value and w. |
else |
Update to according to |
and . |
end if |
until is satisfied. |
end |
return the max and the corresponding w, obtain . |
Iv Simulation Results
The simulation parameters are based on the works in [6] and [8], given as: , , , , dBm, dBm, dBm, , , , , mW, mW, , , , , , , , and . The number of channel realizations is .
Fig. 1(a) depicts the tradeoff between the total harvested power and the transmit power achieved by using the proposed algorithm. The results are obtained with different weights . It is seen that the total harvested power increases with the minimum required SINR for a fixed weight . When SINR is increased above a certain level (such as 35 dB), the harvested power does not change and becomes a constant. The reason is that the harvested power directly depends on the transmit power and the transmit power increases when the SINR is increased. However, the transmit power cannot be higher than the maximum transmit power. On the other hand, for a fixed , the total harvested power increases with the weight . The reason is that the effect of the harvested power increases with the weight .
Fig. 1(b) shows the total harvested power versus the minimum SINR required at the SU under different CSI conditions. It is seen from Fig. 1(b) that the performance achieved under the non-linear EH model given by eq. (3) is better than that obtained by using the model given by eq. (4) since the former model does not consider the sensitivity property of the EH circuit unlike the latter model. It is also seen that the imperfect CSI significantly influences the total harvested power since a higher transmit power is required to overcome the CSI uncertainty. Fig. 1(c) verifies the efficiency of our proposed algorithm 1 since it is seen that it only requires less than 4 iterations to converge.
V Conclusion
A robust resource optimization problem was formulated in MISO CRNs with SWIPT where two practical non-linear EH models and imperfect CSI were investigated. An algorithm based on SCA and was proposed to solve the challenging formulated problem. We proved that a rank-one solution is obtained at each iteration of the proposed algorithm. Our simulation results quantify the effect of the sensitivity threshold on the achievable performance. Comparing the performance under the two practical non-linear EH models provides valuable insights for the design of CRNs with SWIPT.
APPENDIX A
The proof of Theorem 1 is based on the KKT optimal conditions of P. Let , , , , and denote the dual variables with respect to (8), (9), (5d), (10) and (5e), respectively. denotes a N-by-N dimensional Hermitian positive semidefinite matrix set. The partial KKT conditions related to the proof are given as
(12b) | |||
(12c) |
where , is the element of B in the th row of the th column.
Right-multiplying (12a) by W and combining (12b), we get , which implies that
(13) |
Moreover, according to (12a) and (12c), one has
(14) |
Since , one has . Thus,
(15) |
Based on (15), the rank of W is one and the proof is complete.
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