Beamforming Optimization for Information Enhancement Transmission in MISO SWIPT with NOMA System

Miao, Qing; Zhou, Longtao; Wang, Jiongqi

doi:https://doi.org/10.1155/2022/1090582

Journal of Control Science and Engineering

On this page

Abstract Introduction System Model Conclusion Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2022 | Article ID 1090582 | https://doi.org/10.1155/2022/1090582

Beamforming Optimization for Information Enhancement Transmission in MISO SWIPT with NOMA System

Qing Miao,¹Longtao Zhou,²and Jiongqi Wang³

Academic Editor: Giulio Ferro

Received04 Aug 2021

Accepted05 Apr 2022

Published02 Jun 2022

Abstract

This paper mainly considers a communication network with multiple-input single-output (MISO), in which nonorthogonal multiple access (NOMA) decoding technology is adopted, and wireless power-carrying technology is used to enhance the transmission quality of user information in the communication network. Through joint optimization of power splitting ratio and beamforming vector, we aim at minimizing transmission power (base station transmit power and secondary transmit power) while satisfying the quality-of-service requirement of all users and minimum power for secondary transmission. The problem is a nonconvex optimization program, and it is difficult to get the optimal value directly. First, in order to solve the problem for two-user case, the original problem is transformed into a semidefinite programming (SDP) problem, and then, the iterative updating algorithm is used to approximate the optimal value. When the number of users is greater than two, the original problem is transformed into a second-order cone problem, in which we deal with a sequence of second-order cone programming (SOCP). Results verify that the optimal value of the sequence of SOCPs is not increasing and converges to a local optimal value. Detailed simulation experiments indicate that the algorithm improves the performance of NOMA downlink beamforming under the condition of enhancing information and eliminating network interference.

1. Introduction

In order to achieve good system throughput and have features of improvement of spectral efficiency, nonorthogonal multiple access (NOMA) beamforming has received great attention in multiuser systems. NOMA beamforming permits the base station (BS) to apply spatially free overlay coding. Currently, NOMA beamforming has become an important wireless access technology for the 5G wireless networks [1, 2]. In 5G wireless networks, increasing the spectral efficiency of the downlink is an important factor [3–12]. Maximizing energy efficiency is another key objective of 5G networks besides improving the spectrum efficiency. In order to explore more energy networks, simultaneous wireless information and power transfer (SWIPT) technology was first mentioned in [5]. It has caught much attention for years. In [5], it is shown that radio frequency signals could carry information as well as energy at the same time. Motivated by the issue, the authors in [6] studied two kinds of receiving protocols, namely the time switch receiving protocol and the power split receiving protocol. Furthermore, the transmitter design in a multiple-input single-output (MISO) interference channel to increase the energy efficiency of the communication system is considered in [7]. Recent research on SWIPT has concentrated on cooperative transmission protocol [8, 9] in communication systems.

Motivated by the advantages of NOMA and SWIPT as well as the requirements of 5G, researchers propose NOMA cooperation strategies [13–16] that can improve the communication performance of users at the edge of the communication network. Due to the energy storage limitation of relay equipment in the communication network, there is a tradeoff between users decoding their own information directly and forwarding information to other users. SWIPT has been used in the NOMA to reduce the energy constraints, and users who are close to the base station can be used as relays to provide information enhancement for those who have long distance. Furthermore, many scholars have done studies on optimization problems of NOMA with multiple users [17–19]. In [17], considering that the signal-to-interference-plus-noise ratio (SINR) thresholds of the users are equal, a user admission approach is proposed which is optimal with respect to both the sum rate and the maximum number of admitted users. In [18], a joint design of beamforming and power allocation is proposed, which is designed to maximize the sum rate of the users in one group while ensuring the minimum required target rates of the users in the other group. In [19], two beamforming schemes are proposed, and the corresponding optimal power allocation is developed to maximize the sum rate of the downlink system.

In this work, considering cooperative MISO SWIPT with the NOMA system, we do studies on a MISO communication optimization problem with information enhancement (EN) and internetwork interference elimination in the setting of a MISO multiuser downlink transmission system. We mainly consider a total power minimization problem constrained by base station power and user SINR and secondary transmission power constraint under the NOMA scheme. Correspondingly, we propose iterative algorithms based on semidefinite programming (SDP) approximation and second-order cone programming (SOCP) approximation. First, for the case of two users, we use the SDP approximation to deal with. For the multiuser problem with the enhancement of adjacent user information, we use the internetwork interference cancellation technology to eliminate the interference and solve the problem through SOCP approximation.

2. System Model and Problem Formulation

As shown in Figure 1, a downlink MISO transmission system is considered, where the base station of this system has K antennas to transmit information to M users. In other words, a user nearest the base station is a single antenna receiving multiple antenna transmissions, and the other M − 1 users are a single-antenna user. Under general circumstances, we assume that M users in the communication network have different channel conditions, among which the user m with a shorter distance has better channel conditions than the user m − 1 with a longer distance. In this paper, we divide M users into two groups: group 1 is the central user, and group 2 is the edge user. To ensure the quality-of-service requirements of group 2, group 1 is used as an energy harvesting relay to help users in group 2. In this communication system, we use NOMA to improve the efficiency of the spectrum, assuming that the channel is perfect, and the channel state is known.

In this paper, the SWIPT NOMA transmission mainly includes two stages. In the first stage, the base station transmits information to all users. The information received by group 1 is divided into two parts: one part is used to decode information, and the other part is collected by SWIPT. In the second stage, group 1 enhances the information of users in group 2 with the collected energy, and users in group 2 combined the information collected in the two stages to enhance the information.

2.1. Direct Transmission Stage

For the communication network system in this paper, the base station with K antennas transmits signals for M single-antenna users, and the signal transmitted by the base station can be expressed as follows:where _m is the beamforming vector for user m, and s_m is the information symbol. The received signal of user m can be expressed as follows:where h_m is the channel vector, n_m and is the complex Gaussian white noise with mean 0 and variance .

To conduct successive interference cancellation at the users, a decoding sequence should be built. It depends on the users’ power level. Thus, we assume that h_m, m = 1,…,M, follow the model of the Rician channel model [11, 20].where u_m is a Gaussian random vector with mean 0 and covariance and is “the steering vector for a uniform linear array of half-wavelength spacing” [20]. Here, is the angle of departure to user m, and is calculated by [20], where d_m is the distance from the base station to the user m, and is the path loss exponent [20], which is a nonnegative number.

In the paper, we used to denote the user set, which is an ordered set [12, 20]. The shorter the distance between the base station and the user, the stronger the channel condition the user (with a bigger index) has. Thus, under the condition of n ≤ m, u_m is able to decode the information for u_n, under quality-of-service conditions [20].where is expressed asand is given bywhere R_n is the target rate for user u_n.

For user M, it acts as an energy harvesting relay to help users in group 2. As shown in Figure 2, information decoding and energy harvesting are carried out in the architecture. Then, the SINR of user M can be described as

The energy that is harvested by group 1 can be modeled as [21]where Γ denotes the transfer time fraction of the first stage. Suppose that the transfer time of the two stages is the same, that is, . Suppose that all the energy collected by user M is forwarded to user information, and the energy for signal processing, circuit consumption, is ignored. Therefore, the transmitted power of group 1 is shown as

2.2. Cooperative Transmission Stage

In the second stage, the signal transmitted by user M is given bywhere is the beamforming vector of group 1, and s_m is the information transmitted to user m. The received signal by user m of group 2 is expressed asand we getwhere _m is the channel vector between user M and user m(m∈{1…(M − 1)}), and n_m is the complex Gaussian white noise with mean 0 and variance . For user m of group 2, the received SINR can be described as

At the end of the second stage, group 1 decodes the message {s₁, s₂, …, s_M − 1} jointly based on the signals received from the base station and the information received in the first stage. Therefore, the signal-to-noise ratio (SINR) of user m can be written as

In order to enhance the information of user m, we design a special beam vector matrix to ensure that user m is not disturbed by other users. Here, a new K×(M − 2) matrix G^m = [₁,…,_m-1,_m+1,…,_M-1] is defined, where G^m contains the channel vectors from the user M to all the weaker users in the second stage except for those from the user M to user m of group 2. We decompose the beamforming vector _m as _m = U_mq_m, where U_m is normalized and lies in the null space of G^m. Based on the definition of U_m and using the singular value decomposition, we decompose G^m aswhere is the first M − 2 left eigenvectors of , which form an orthogonal basis of , and corresponding the zero eigenvalues represents the last K − M + 2 left eigenvectors of , which form an orthogonal basis of the null space of . Thus, U_m is given by .

Suppose that successive interference cancellation technology is employed for user m to eliminate the interference of other users’ information. Therefore, user m can be decoded, and the information received by user m can be expressed as

For user m of group 2, the SNR can be described as

Therefore, the equivalent SINR at user m can be expressed as

In order to work normally for M users, the received power needs to be greater than or equal to the transmitted power

The system design aims to minimize the total power of base and M user and, meanwhile, guarantee the quality-of-service requirement of all users and decoding requirements of NOMA and constraints on user M ability to work properly in the two stages. Then, problem (P1) can be formulated as

In problem (P1), the objective (20a) is to minimize the total power of the base station and M user in the two stages. Constraints (20b) and (20c) are traditional NOMA constraints. Constraints (20b) and (20c) indicate that the received SINR to decode s_n needs to be no less than the target SINR γ_n. Constraint (20d) is guaranteed to transmission normally for the second stage. Constraint (20e) is the sum of the information received by user m of group 2 in the first stage plus the information received in the second stage, which should be greater than the information decoding threshold γ_m. Constraint (20f) indicates the power splitting ratio.

As for the optimization problem (P1), we can know that it is a nonconvex problem. It is difficult to find the optimal solution directly mainly because the constraint conditions in the optimization problem have quadratic terms. We start with the simple two-user communication network and use SDP and SDR technology to solve the optimization problem and use successive convex approximation iterative approach to gradually approximate the optimal value. Then, we extended to general multiusers, where SOCP techniques are used to solve optimization problems. The iterative algorithm is used to gradually approximate optimal values.

3. An Approximate SDP Algorithm for Two Users

In this part, we consider the two-user case, and the whole problem (P2) can be written as

The above optimization problem (P2) is nonconvex, and we can use SDP and SDR technology to transform the problem (P2) into an equivalent optimization problem [22]. Thus, a successive convex approximation iterative algorithm is used to gradually approach the optimal solution of the optimization problem. We set W₁, W₂, and V as positive semidefinite matrices

Through adopting the SDR technology, problem (P2) can be relaxed as

In problem (P2), tr(·) denotes the trace of the square matrix argument. The curled inequality symbol (and its strict form ) denotes generalized inequality. AB means that A − B is a Hermitian positive semidefinite matrix (AB for positive definiteness).

Proposition 3.1. Assume that is optimal for (23a)–(23g). Then, there always exist , and is the optimal solution of (21a)–(21f).

Proof. See Appendix A.
Note that since the rank-one constraints are removed in problem (P2), the equivalence of optimization problem (P1) and optimization problem (P2) is not able to be guaranteed, so we should prove that the two optimization problems are equivalent. In problem (P2), because the objective function and constraints (23d) and (23e) are nonlinear, it cannot be directly proved the rank-one optimality. By introducing an auxiliary variable y for constraint (23d), we observe thatWe utilize the constrained concave-convex procedure to transform (24a) into a tractable approximation. It is observed that ; we again observed that as the first-order Taylor expansion of the second quadratic term around a given point . Thus, (24a) can be approximated asBy introducing an auxiliary variable into (23e), then constraint (23e) can be equivalently rewritten asFor constraint (26b), the arithmetic-geometric mean inequality can be used to produce an approximately equivalent constraint. For nonnegative variables x, y, z, the arithmetic-geometric mean inequality based approximation of expression can be expressed asso for condition (27), we can see that when , the first equality can be obtained. So constraint (26b) can be approximately given bywhere a⁽ⁿ⁾ is the value of a at the n-th iteration, and it can be determined byTherefore, the problem in the n-th iteration can be given by problem (P3)It is assumed that the problem (P3) is feasible, and accordingly, it is dual feasible. Based on Theorem 3.2 in [23, 24], it can be deduced that problem (P3) always has an optimal solution and satisfiesSince the variables satisfy the condition (31), the equivalence of optimization problem (P1) and optimization problem (P2) can be guaranteed. Therefore, the rank-one constraints of problem (P2) can be dropped, and the two optimization problems are equivalent. Here, we propose an approximate constraint algorithm based on SDP technology. Algorithm 1 is shown as follows.

	Require:
	Ensure: A solution and for problem (21a)–(21f);
(1)	Suppose that is an initial point; set l = 0 and (a large number);
(2)	do;
(3)	solve SDP (30a)–(30c), obtaining optimal solution and optimal value ;
(4)	solve(29), obtaining solution ;
(5)	;
(6)	UNTIL ;
(7)	return.

4. An Approximate SOCP Algorithm for Multiusers

In this part, we establish a second-order cone program-based approximate algorithm for (20a)–(20f). Observe that the optimization problem (20a)–(20f) can be rewritten as a separable quadratic constrained quadratic programming [20]. And the SDP technology for the optimization problem (20a)–(20f) is not tight. Therefore, a new method should be established instead of SDP relaxation. By introducing auxiliary variables {t_mn,t_Mn}, (20a)–(20f) can be rewritten as

It is nonconvex for (32c) and (32d) because of the absolute value, we propose a method of approximation [20] as follows:

In order to design an algorithm for solving (20a)–(20f), we assume that is an iteration starting point. Observe that

It concludes that if

then one has

Therefore, consider the second-order cone programming problem as follows:

For the following analysis, let l: = 1. Solve SOCP problem (37a)–(37c), finding a solution . Observe that , and that

Then, construct another convex restriction of (20a)–(20f) in a similar way based on

Solve problem (39a)–(39c) to get . Set l: = l+1 and again deal with (39a)–(39c), and an iterative process is formulated in the way. Letbe the optimal values. The property about to be demonstrated is that {} is a nonincreasing sequence.

Proposition 4.1. It holds that for .

Proof. The optimal solution for problem (37a)–(37c) with replaced by is feasible for problem (39a)–(39c). In that case, the optimal solution for (39a)–(39c) has the following property:We have an immediate check whether it is feasible for problem (39a)–(39c). Since replacing with is the optimal solution of (37a)–(37c), it is also feasible, so (37c) satisfieswhich imply thatSo we haveand (39a) is also fulfilledfor . It follows from (44), (45) that is feasible for (39a).
For problem (39a)–(39c), the optimal value of each iteration is not increasing. So with the increase in the number of iterations, the optimal value will converge to a local minimum point.
Note that we can equivalently rewrite (39a)–(39c) asFor problem (46a)–(46d), by introducing an auxiliary variable to the constraint (20d), thus, (20d) can be rewritten asSo for formula (47), we deal with it in the same way as (24a)To approximate the quadratic matrix inequality (48), we assume that an initial point is given, and thatfor n = 1,…M, andThis means that is sufficiently close to _n for each n and is sufficiently close to . Then, we haveand by this approximation, we can getFor problem (46a)–(46d), we can introduce an auxiliary variable x_m, so the constraint (20e) can be written asSo for formula (55), we deal with it in the same way as (48)In order to approximate the inequality (56), by constructing a perfect plane, we can getFor formula (56), it can be approximated by the perfect plane method, which can be obtainedTherefore, problem (20a)–(20f) is solved by solving the following optimization problem:Thus, an approximate SOCP algorithm is formulated as in Algorithm 2.

	Require:
	Ensure: A solution for problem (20a)–(20f);
(1)	Suppose that is an initial point; set l = 0 and (a large number);
(2)	repeat;
(3)	solve SOCP (60a)–(60b), obtaining optimal solution ;
(4)	;
(6)	until.

5. Simulations Results

In this part, simulation results are presented to verify the performance of the proposed approximate algorithms for dealing with beamforming optimization problems of the NOMA system with information enhancement and internetwork interference elimination. For the NOMA downlink system, let the number of antennas in the base station be the same as that in user M, and the receiver noise power Watt for all m.

Example 1. Assume that the base station serves M = 2 users (as in problem (21a)–(21f)). There are two channel models, namely channel 1 and channel 2. And for channel 1, suppose that the Rayleigh fading channel h_m follows (i.e., (3) with ; similarly, suppose that _m also follows ; for channel 2, suppose that h_m follows (i.e., (3) with ); similarly, suppose that _m also follows , where d_m is the distance from the base station to user m, is the distance from the user M to user m, is the angle of departure (AoD) between the base station and user m, is the angle of departure between the user M and user m, and denotes the path loss exponent. In the simulation, we set d_M = d₂ = 0.8 m (meter), d₁ = 1.6 m, m, and , and the AoDs in (3) are . The SINR targets .
In Figure 3, we have plotted the minimum base station total power of OMA without information enhancement (OMA-No-En), NOMA without information enhancement (NOMA-No-En), and NOMA with information enhancement (NOMA-En) schemes versus the number of antennas as shown in Figure 3(a) and Figure 3(b) for a set of 100 channel realizations. All three models are solved based on Algorithm 1. As shown in Figure 3(a), the power of the three communication models decreases as the number of antennas increases. The reason is that as the antennas grow larger, the beamforming is more flexible to allocate information, thus reducing the total power. The total power of NOMA-No-En is less than that of OMA-No-En under the same parameters. For NOMA-En (as in problem (21a)–(21f)) and NOMA-No-En (as in problem (21a)–(21f)) without information enhances this part of the constraint, the constraint is enhanced by adding part of enhanced information in NOMA-En, which means that the constraint set of the whole optimization problem becomes larger, and the value of the objective function becomes smaller. Therefore, under the same parameters, the total power of NOMA-En is less than that of NOMA-No-En. It also proves that the method we proposed is effective. Figure 3(a) and Figure 3(b) are different in that their channel models and the other parameters are the same.

(a)

(b)

Example 2. Assume that the base station serves M = 2 users. It is assumed that channel 1 is used for all channels. Number of antennas in the base station K = 16. Other settings follow Example 1.
In Figure 4, we have plotted the minimum base station total power of OMA-No-En, NOMA-No-En, and NOMA-En schemes versus the distance between the center user and the edge user in Figure 4 for a set of 100 channel realizations. In OMA-No-En and NOMA-No-En, their optimization problem and the distance between the center user and the edge user are not unrelated, so the total power of their base stations does not change with the increase of the distance between the center user and the edge user. For NOMA-En, as the distance between the center user and the edge user increases, the channel becomes worse, and the information enhancement becomes smaller. For problem (21a)–(21f), its constraint set becomes smaller, causing the objective function to become larger. As this distance gets bigger, the constraint on the problem gets smaller, the target function gets bigger, and NOMA-En becomes closer to NOMA-No-En.

Example 3. Consider a scenario in which up to 4 users are served (as in problem (60a)–(60b)). Suppose that h_m and _m follow the channel model (3) with , and the SINR thresholds are the same, that is, . The distances between the base station and user m are (d₁,d₂,d₃,d₄)=(1.6, 0.8, 0.6, 0.4), and the distances between user M and user m are .
Figure 5 tests the differences in optimal values acquired by approximation SOCP. The figure displays the total transmission power of OMA without information enhancement and interference elimination (OMA-No-En-No-IF), NOMA without information enhancement and interference elimination (NOMA-No-En-No-IF), NOMA with information enhancement and without interference elimination (NOMA-En-No-IF), and NOMA with information enhancement and interference elimination (NOMA-En-IF) schemes versus the number of transmit antennas K. In Figure 5, we observe that the total transmission power of NOMA-No-En-No-IF is greater than the total transmission power of NOMA-No-En-IF, and the total transmission power of NOMA-No-En-IF is greater than the total transmission power of NOMA-En-IF, it is because that the set of constraints in NOMA-No-En-No-IF is less than the set of constraints in NOMA-No-En-IF, and the set of constraints in NOMA-No-En-IF is less than the set of constraints in NOMA-En-IF.

Example 4. Consider a scenario in which up to 3 users are served (as in problem (60)). Suppose that h_m and _m follow the channel model (3) with , and the SINR thresholds are the same, that is, . The distances between the base station and user m are (d₁, d₂, d₃) = (1.6, 0.8, 0.6), the distances between the user M and user m are , and the angle of departures (AoDs) in (3) are .
Figure 6 demonstrates the optimal total transmission power versus the number of transmit antennas under different path loss exponents. As can be seen, when the path loss exponent increases, the total power of the base station decreases.

Example 5. Suppose that the base station serves a set of four users u₁, u₂, u₃, u₄, which is an ordered set representing the decoding sequence with bigger index user having stronger channel. Number of antennas in the base station K = 20. Assume that h_m follows the channel model (3) with . The distances between the base station and user m are (d₁, d₂, d₃, d₄) = (1.6, 0.8, 0.6, 0.4), the distances between the user M and user m are , and the angle of departures in (3) are . The path loss exponent . In addition, we assume that . When we say that the base station serves three (two) users, it refers to the users .
In Figure 7, the relationship between the optimal total transmitted power and SINR is given. As can be seen, more transmission power is required when the base station has more value of the SINR. In order to satisfy the larger signal-to-noise ratio, the system needs larger transmission power. Furthermore, more transmission power is needed when the number of users is increasing. This is also sound since more transmission power is required in order to serve additional users.

6. Conclusion

Beamforming optimization problems in MISO NOMA communication network with information enhancement and internetwork interference elimination have been studied in this paper. First, for the two-user case in the communication system, we propose an iterative approximation algorithm based on SDP and SDR technology for the NOMA downlink beamforming optimization problem. Then, we propose an iterative approximation algorithm based on SOCP for the general multiusers case. Results verify that the optimal value of the sequence of SOCPs is not increasing and converges to a local optimal value. Finally, the performance of the proposed algorithms has been demonstrated by simulation results.

APPENDIX

A. Proof of Proposition 3.1

In order to present the proof of Proposition 3.1, let us first show the following lemma.

Lemma 7.1 (see [25]). Suppose that X is a N×N complex Hermitian positive semidefinite matrix of rank R, and A, B are two N×N given Hermitian matrices. Then, there is a rank-one decomposition such that and .

At this point, we are ready to prove Proposition 3.1.

Proof. Assume thatwhich are nonzero real numbers. Then, we haveConsidering Lemma 7.1, the optimal solution satisfies (A.4) (A.5), which means thatIt follows thatSuppose that . To satisfy condition (A.1), we have to renew the solution of rank-one decomposition asIt follows thatSimilarly, , have also rank-one decomposition by Lemma 7.1.
Therefore, the proof is complete.

Data Availability

The data in this paper used to support the findings are included in the article.

Conflicts of Interest

The authors declare that they have no conflicts of interest regarding the work reported in this paper.

Acknowledgments

The study was supported by the National Nature Science Foundation of China (No.62001115), the Project of Department of Education of Guangdong Province (No.2019KTSCX193), and GuangDong Basic and Applied Basic Research Foundation (No.2019A1515110136).

References

Z. Ding, Y. Liu, J. Choi et al., “Application of non-orthogonal multiple access in LTE and 5G networks,” IEEE Communications Magazine, vol. 55, no. 2, pp. 185–191, 2017.
View at: Publisher Site | Google Scholar
Z. Ding, X. Lei, G. K. Karagiannidis, R. Schober, J. Yuan, and V. K. Bhargava, “A survey on non-orthogonal multiple access for 5G networks: research challenges and future trends,” IEEE Journal on Selected Areas in Communications, vol. 35, no. 10, pp. 2181–2195, 2017.
View at: Publisher Site | Google Scholar
O. Ozel, K. Tutuncuoglu, J. Yang, S. Ulukus, and A. Yener, “Transmission with energy harvesting nodes in fading wireless channels: optimal policies,” IEEE Journal on Selected Areas in Communications, vol. 29, no. 8, pp. 1732–1743, 2011.
View at: Publisher Site | Google Scholar
Q. Sun, G. Zhu, C. Shen, X. Li, and Z. Zhong, “Joint beamforming design and time allocation for wireless powered communication networks,” IEEE Communications Letters, vol. 18, no. 10, pp. 1783–1786, 2014.
View at: Publisher Site | Google Scholar
L. R. Varshney, “Transporting information and energy simultaneously,” 2008 IEEE International Symposium on Information Theory, pp. 1612–1616, 2008.
View at: Publisher Site | Google Scholar
R. Zhang and C. K. Ho, “MIMO broadcasting for simultaneous wireless information and power transfer,” IEEE Transactions on Wireless Communications, vol. 12, no. 5, p. 1989, 2013.
View at: Publisher Site | Google Scholar
C. Shen, W.-C. Li, and T.-H. Chang, “Wireless information and energy transfer in multi-antenna interference channel,” IEEE Transactions on Signal Processing, vol. 62, no. 23, pp. 6249–6264, 2014.
View at: Publisher Site | Google Scholar
A. A. Nasir, X. Zhou, S. Durrani, and R. A. Kennedy, “Relaying protocols for wireless energy harvesting and information processing,” IEEE Transactions on Wireless Communications, vol. 12, no. 7, pp. 3622–3636, 2013.
View at: Publisher Site | Google Scholar
Z. Ding, I. Krikidis, B. Sharif, and H. V. Poor, “Wireless information and power transfer in cooperative networks with spatially random relays,” IEEE Transactions on Wireless Communications, vol. 13, no. 8, pp. 4440–4453, 2014.
View at: Publisher Site | Google Scholar
W. Yu and T. Lan, “Transmitter optimization for the multi-antenna downlink with per-antenna power constraints,” IEEE Transactions on Signal Processing, vol. 55, no. 6, pp. 2646–2660, 2007.
View at: Publisher Site | Google Scholar
J. Choi, “On generalized downlink beamforming with NOMA,” Journal of Communications and Networks, vol. 19, no. 4, pp. 319–328, 2017.
View at: Publisher Site | Google Scholar
Y. Liu, H. Xing, C. Pan, A. Nallanathan, M. Elkashlan, and L. Hanzo, “Multiple antenna assisted non-orthogonal multiple access,” IEEE Wireless Communications, vol. 25, no. 2, pp. 17–23, 2018.
View at: Publisher Site | Google Scholar
Z. Ding, M. Peng, and H. V. Poor, “Cooperative non-orthogonal multiple access in 5G systems,” IEEE Communications Letters, vol. 19, no. 8, pp. 1462–1465, 2015.
View at: Publisher Site | Google Scholar
Y. Xu, C. Shen, Z. Ding et al., “Joint beamforming and power-splitting control in downlink cooperative SWIPT NOMA systems,” IEEE Transactions on Signal Processing, vol. 65, no. 18, pp. 4874–4886, 2017.
View at: Publisher Site | Google Scholar
Y. Liu, Z. Ding, M. Elkashlan, and H. V. Poor, “Cooperative non-orthogonal multiple access with simultaneous wireless information and power transfer,” IEEE Journal on Selected Areas in Communications, vol. 34, no. 4, pp. 938–953, 2016.
View at: Publisher Site | Google Scholar
Y. Li, M. Jiang, Q. Zhang, Q. Li, and J. Qin, “Cooperative non-orthogonal multiple access in multiple-input-multiple-output channels,” IEEE Transactions on Wireless Communications, vol. 17, no. 3, pp. 2068–2079, 2018.
View at: Publisher Site | Google Scholar
M. Zeng, A. Yadav, O. A. Dobre, G. I. Tsiropoulos, and H. V. Poor, “Capacity comparison between MIMO-NOMA and MIMO-OMA with multiple users in a cluster,” IEEE Journal on Selected Areas in Communications, vol. 35, no. 10, pp. 2413–2424, 2017.
View at: Publisher Site | Google Scholar
X. Sun, N. Yang, S. Yan et al., “Joint beamforming and power allocation in downlink NOMA multiuser MIMO networks,” IEEE Transactions on Wireless Communications, vol. 17, no. 8, pp. 5367–5381, 2018.
View at: Publisher Site | Google Scholar
J. Y. Yang and W. J. Huang, “Joint beamforming and power allocation design in non-orthogonal multiple access systems,” in Proceedings of the 2016 International Computer Symposium (ICS), Chiayi, Taiwan, December 2016.
View at: Publisher Site | Google Scholar
Y. Huang and L. Zhou, “MISO NOMA downlink beam-forming optimization with per-antenna power,” Signal Processing, vol. 171, pp. 1–9, 2021.
View at: Google Scholar
Q. Shi, C. Peng, W. Xu, M. Hong, and Y. Cai, “Energy efficiency optimization for MISO SWIPT systems with zero-forcing beamforming,” IEEE Transactions on Signal Processing, vol. 64, no. 4, pp. 842–854, 2016.
View at: Publisher Site | Google Scholar
Z.-Q. Luo, W.-K. Ma, A. M. C. So, Y. Ye, and S. Zhang, “Semidefinite relaxation of quadratic optimization problems,” IEEE Signal Processing Magazine, vol. 27, no. 3, pp. 20–34, 2010.
View at: Publisher Site | Google Scholar
Y. Huang and D. P. Palomar, “Rank-constrained separable semidefinite programming with applications to optimal beamforming,” IEEE Transactions on Signal Processing, vol. 58, no. 2, pp. 664–678, 2010.
View at: Publisher Site | Google Scholar
Y. Huang and D. P. Palomar, “Randomized algorithms for optimal solutions of double-sided QCQP with applications in signal processing,” IEEE Transactions on Signal Processing, vol. 62, no. 5, pp. 1093–1108, 2014.
View at: Publisher Site | Google Scholar
Y. Huang and S. Zhang, “Complex matrix decomposition and quadratic programming,” Mathematics of Operations Research, vol. 32, no. 3, pp. 758–768, 2007.
View at: Publisher Site | Google Scholar

Copyright

Copyright © 2022 Qing Miao et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

PDF Download Citation

Download other formats

Order printed copies

Views

251

Downloads

329

Citations