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

Authors: Qing Miao; Longtao Zhou; Jiongqi Wang
Journal: Journal of Control Science and Engineering
 (2022)

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## 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.


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## Body

## 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 Figure1, 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.Figure 1
A downlink MISO transmission system.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 withK antennas transmits signals for M single-antenna users, and the signal transmitted by the base station can be expressed as follows:(1)x=∑m=1Msmwm,where wm is the beamforming vector for user m, and sm is the information symbol. The received signal of user m can be expressed as follows:(2)ym=hmHx+nm,where hm is the channel vector, nm and is the complex Gaussian white noise with mean 0 and variance σm2.To conduct successive interference cancellation at the users, a decoding sequence should be built. It depends on the users’ power level. Thus, we assume thathm, m = 1,…,M, follow the model of the Rician channel model [11, 20].(3)hm=βmζ1+ζaθm+11+ζum,where um is a Gaussian random vector with mean 0 and covariance 1/KI and aθ=1/K1;e−j2πd/λsinθ;…;e−jK−12πd/λsinθ is “the steering vector for a uniform linear array of half-wavelength spacing” [20]. Here, θm is the angle of departure to user m, and βm is calculated by 1/dmη [20], where dm is the distance from the base station to the user m, and η is the path loss exponent [20], which is a nonnegative number.In the paper, we usedS=u1,u2,…,um 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, um is able to decode the information for un, under quality-of-service conditions [20].(4)minn≤m<MSINRmn≥γn,1≤n<M,where SINRmn is expressed as(5)SINRmn=hmHwn2∑i=n+1MhmHwi2+σm2,n≤m,and γn is given by(6)γn=2Rn−1,where Rn is the target rate for user un.For userM, 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(7)SINRMn=1−βhMHwn2∑i=n+1M1−βhMHwi2+σM2,n≤M.Figure 2
The power splitting structure of group 1.The energy that is harvested by group 1 can be modeled as [21](8)E=β∑i=1MhMHwi2Γ,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, Γ=1/2. 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(9)PM=E1−Γ=β∑i=1MhMHwi2.


### 2.2. Cooperative Transmission Stage

In the second stage, the signal transmitted by userM is given by(10)y=∑m=1M−1smvm,where vm∈CK is the beamforming vector of group 1, and sm is the information transmitted to user m. The received signal by user m of group 2 is expressed as(11)Zm=gmHy+nm,and we get(12)Zm=gmHvmsm+∑i=1,i≠mM−1gmHvisi+nm,where gm is the channel vector between user M and user m(m∈{1…(M−1)}), and nm is the complex Gaussian white noise with mean 0 and variance σm2. For user m of group 2, the received SINR can be described as(13)SINRm2=gmHvn2∑i=1+i≠mM−1gmHvi2+σm2.At the end of the second stage, group 1 decodes the message {s1, s2, …, sM − 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(14)SINRm=SINRmm+SINRm2=hmHwm2∑i=m+1MhmHwi2+σm2+gmHvm2∑i=1,i≠mM−1gmHvi2+σm2.In order to enhance the information of userm, 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 Gm = [g1,…,gm-1,gm+1,…,gM-1] is defined, where Gm 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 vm as vm = Umqm, where Um is normalized and lies in the null space of Gm. Based on the definition of Um and using the singular value decomposition, we decompose Gm as(15)¯Gm=AmM−2,AmK−M+2DmBmH,where AmM−2 is the first M − 2 left eigenvectors of ¯Gm, which form an orthogonal basis of ¯Gm, and AmK−M+2 corresponding the zero eigenvalues represents the last K − M + 2 left eigenvectors of ¯Gm, which form an orthogonal basis of the null space of ¯Gm. Thus, Um is given by Um=AmK−M+2.Suppose that successive interference cancellation technology is employed for userm 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(16)Zm=gmHUmqmsm+nm.For userm of group 2, the SNR can be described as(17)SNRm2=gmHUmqmσm22.Therefore, the equivalent SINR at userm can be expressed as(18)SINRm=SINRmm+SNRm2=hmHwm2∑i=m+1MhmHwi2+σm2+gmHUmqm2σm2.In order to work normally forM users, the received power needs to be greater than or equal to the transmitted power(19)PM≥∑i=1M−1Uiqi2.The system design aims to minimize the total power of base andM 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(20a)minimizewm,qm,β∑m=1MwmHwm+∑m=1M−1UmqmHUmqm,(20b)subject tohmHwn2∑i=n+1MhmHwi2+σm2≥γn,n≤m<M,(20c)1−βhMHwn2∑i=n+1M1−βhMHwi2+σM2≥γn,n≤M,(20d)β∑i=1MhMHwi2≥∑i=1M−1Uiqi2,(20e)hmHwm2∑i=m+1MhmHwi+σm2+gmHUmqm2σm2≥γm,1≤m<M,(20f)0≤β≤1.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 sn needs to be no less than the target SINR γn. 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 γm. 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.


## 2.1. Direct Transmission Stage

For the communication network system in this paper, the base station withK antennas transmits signals for M single-antenna users, and the signal transmitted by the base station can be expressed as follows:(1)x=∑m=1Msmwm,where wm is the beamforming vector for user m, and sm is the information symbol. The received signal of user m can be expressed as follows:(2)ym=hmHx+nm,where hm is the channel vector, nm and is the complex Gaussian white noise with mean 0 and variance σm2.To conduct successive interference cancellation at the users, a decoding sequence should be built. It depends on the users’ power level. Thus, we assume thathm, m = 1,…,M, follow the model of the Rician channel model [11, 20].(3)hm=βmζ1+ζaθm+11+ζum,where um is a Gaussian random vector with mean 0 and covariance 1/KI and aθ=1/K1;e−j2πd/λsinθ;…;e−jK−12πd/λsinθ is “the steering vector for a uniform linear array of half-wavelength spacing” [20]. Here, θm is the angle of departure to user m, and βm is calculated by 1/dmη [20], where dm is the distance from the base station to the user m, and η is the path loss exponent [20], which is a nonnegative number.In the paper, we usedS=u1,u2,…,um 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, um is able to decode the information for un, under quality-of-service conditions [20].(4)minn≤m<MSINRmn≥γn,1≤n<M,where SINRmn is expressed as(5)SINRmn=hmHwn2∑i=n+1MhmHwi2+σm2,n≤m,and γn is given by(6)γn=2Rn−1,where Rn is the target rate for user un.For userM, 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(7)SINRMn=1−βhMHwn2∑i=n+1M1−βhMHwi2+σM2,n≤M.Figure 2
The power splitting structure of group 1.The energy that is harvested by group 1 can be modeled as [21](8)E=β∑i=1MhMHwi2Γ,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, Γ=1/2. 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(9)PM=E1−Γ=β∑i=1MhMHwi2.


## 2.2. Cooperative Transmission Stage

In the second stage, the signal transmitted by userM is given by(10)y=∑m=1M−1smvm,where vm∈CK is the beamforming vector of group 1, and sm is the information transmitted to user m. The received signal by user m of group 2 is expressed as(11)Zm=gmHy+nm,and we get(12)Zm=gmHvmsm+∑i=1,i≠mM−1gmHvisi+nm,where gm is the channel vector between user M and user m(m∈{1…(M−1)}), and nm is the complex Gaussian white noise with mean 0 and variance σm2. For user m of group 2, the received SINR can be described as(13)SINRm2=gmHvn2∑i=1+i≠mM−1gmHvi2+σm2.At the end of the second stage, group 1 decodes the message {s1, s2, …, sM − 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(14)SINRm=SINRmm+SINRm2=hmHwm2∑i=m+1MhmHwi2+σm2+gmHvm2∑i=1,i≠mM−1gmHvi2+σm2.In order to enhance the information of userm, 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 Gm = [g1,…,gm-1,gm+1,…,gM-1] is defined, where Gm 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 vm as vm = Umqm, where Um is normalized and lies in the null space of Gm. Based on the definition of Um and using the singular value decomposition, we decompose Gm as(15)¯Gm=AmM−2,AmK−M+2DmBmH,where AmM−2 is the first M − 2 left eigenvectors of ¯Gm, which form an orthogonal basis of ¯Gm, and AmK−M+2 corresponding the zero eigenvalues represents the last K − M + 2 left eigenvectors of ¯Gm, which form an orthogonal basis of the null space of ¯Gm. Thus, Um is given by Um=AmK−M+2.Suppose that successive interference cancellation technology is employed for userm 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(16)Zm=gmHUmqmsm+nm.For userm of group 2, the SNR can be described as(17)SNRm2=gmHUmqmσm22.Therefore, the equivalent SINR at userm can be expressed as(18)SINRm=SINRmm+SNRm2=hmHwm2∑i=m+1MhmHwi2+σm2+gmHUmqm2σm2.In order to work normally forM users, the received power needs to be greater than or equal to the transmitted power(19)PM≥∑i=1M−1Uiqi2.The system design aims to minimize the total power of base andM 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(20a)minimizewm,qm,β∑m=1MwmHwm+∑m=1M−1UmqmHUmqm,(20b)subject tohmHwn2∑i=n+1MhmHwi2+σm2≥γn,n≤m<M,(20c)1−βhMHwn2∑i=n+1M1−βhMHwi2+σM2≥γn,n≤M,(20d)β∑i=1MhMHwi2≥∑i=1M−1Uiqi2,(20e)hmHwm2∑i=m+1MhmHwi+σm2+gmHUmqm2σm2≥γm,1≤m<M,(20f)0≤β≤1.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 sn needs to be no less than the target SINR γn. 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 γm. 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(21a)minimizew1,w2,v1,βw1Hw1+w2Hw2,(21b)subject to1−βh2Hw121−βh2Hw22+σ22≥γ1,(21c)1−βh2Hw22σ22≥γ2,(21d)βh2Hw22+h2Hw12≥v12,(21e)h1Hw1h1Hw2+σ122+g1Hv12σ12≥γ1,(21f)0≤β≤1.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(22)V=v1v1H,Wi=wiwiH,i=1,2.Through adopting the SDR technology, problem (P2) can be relaxed as(23a)minimizeW1,W2,V1,βtrW1+W2,(23b)subject totrh2h2HW1−γ1W2≥γ1σ221−β,(23c)trh2h2HW2≥γ2σ221−β,(23d)trh2h2HW2+h2h2HW1≥trV1β,(23e)trh1h1HW1trh1h1HW2+σ12+trg1g1HV1σ12≥γ1,(23f)0≤β≤1,(23g)W1⪰0,W2⪰0,V1⪰0.In problem (P2), tr(·) denotes the trace of the square matrix argument. The curled inequality symbol⪰ (and its strict form ≻) denotes generalized inequality. A⪰B means that A − B is a Hermitian positive semidefinite matrix (A≻B for positive definiteness).Proposition 3.1.
Assume thatW1∗,W2∗,V1∗ is optimal for (23a)–(23g). Then, there always exist W1∗=w1∗w1∗H,W2∗=w2∗w2∗H,V1∗=v1∗v1∗H, and w1∗,w2∗,v1∗ 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 that(24a)trV1≤yβ,(24b)trh2h2HW2+h2h2HW1≥y.
We utilize the constrained concave-convex procedure to transform (24a) into a tractable approximation. It is observed that trV1≤1/4y+β2−1/4y−β2; we again observed that fy,β,y˜,β˜=1/4y˜+β˜2+1/2y˜+β˜2y+β−y˜−β˜ as the first-order Taylor expansion of the second quadratic term 1/4y+β2 around a given point y˜,β˜. Thus, (24a) can be approximated as(25)4trV1=y˜+β˜2+2y˜+β˜y+β−y˜−β˜−y−β2.
By introducing an auxiliary variablex≥0 into (23e), then constraint (23e) can be equivalently rewritten as(26a)x+trg1g1HV1≥γ1,(26b)trh1h1HW1≥xtrh1h1HW2+xσ12.
For 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 as(27)2xy≤ax2+ya2≤2z,so for condition (27), we can see that when a=y/x, the first equality can be obtained. So constraint (26b) can be approximately given by(28)anx2+trh1h1HW2an2≤2trh1h1HW1−2xσ12,where a(n) is the value of a at the n-th iteration, and it can be determined by(29)an=trh1h1HW2n−1xn−1.
Therefore, the problem in then-th iteration can be given by problem (P3)(30a)minimizeW1,W2,V1,x,y,βtrW1+W2+V1,(30b)subject to23b,23c,23f,24b,25,26a,28satisfied,(30c)W1⪰0,W2⪰0,V1⪰0,x≥0,y≥0.
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 W1∗,W2∗,V1∗ and satisfies(31)rank2W1∗+rank2W2∗+rank2V1∗≤4.
Since 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.Algorithm 1: Approximate SDP algorithm for (21a)–(21f).
Require:hm,gm,σm,γn,K,ζ,η,ξ,y0,β0,x0;Ensure: A solution w1∗,w2∗ andv1∗ for problem (21a)–(21f);(1)
Suppose thatW10,W20,V10 is an initial point; set l = 0 and r0=trW1+W2+V1 (a large number);(2)
do;(3)
solve SDP (30a)–(30c), obtaining optimal solution Wml+1,V1l+1,βl+1,xl+1,yl+1 and optimal value rl+1;(4)
solve(29), obtaining solution al+1;(5)
l:=l+1;(6)
UNTILrl−1−rl≤ξ;(7)
returnw∗,v∗.


## 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 {tmn,tMn}, (20a)–(20f) can be rewritten as(32a)minimizewm,qm,β∑m=1MwmHwm+∑m=1M−1UmqmHUmqm.(32b)subject to20d,20e,20fsatisfied,(32c)hmHwn≥tmn,(32d)hMHwn≥tMn,(32e)tmn≥γn∑i=n+1MhmHwi2+γnσm2,n≤m<M,(32f)tMn≥γn∑i=n+1MhMHwi2+γnσM21−β.It is nonconvex for (32c) and (32d) because of the absolute value, we propose a method of approximation [20] as follows:(33)hmHwn≥tmn,n≤m≤M,1≤n≤M.In order to design an algorithm for solving (20a)–(20f), we assume that w10,…,wm0 is an iteration starting point. Observe that(34)hmHwn≥wnHhmhmHwn0hmHwn0≥ℜwnHhmhmHwn0hmHwn0.It concludes that if(35)ℜwnHhmhmHwn0hmHwn0≥tmn,then one has(36)hmHwn≥tmn.Therefore, consider the second-order cone programming problem as follows:(37a)minimizewm,qm,β,tmn∑m=1MwmHwm+∑m=1M−1UmqmHUmqm,(37b)subject to20d,20e,20f,32e,32fsatisfied,(37c)ℜwnHhmhmHwn0hmHwn0≥tmn,1≤n≤M,n≤m≤M.For the following analysis, letl: = 1. Solve SOCP problem (37a)–(37c), finding a solution w1l,...,wMl;tmnl. Observe that hmHwn≥wnHhmhmHwnl/hmHwnl≥ℜwnHhmhmHwnl/hmHwnl, and that(38)ℜwnHhmhmHwnlhmHwnl≥tmnimplieshmHwn≥tmn.Then, construct another convex restriction of (20a)–(20f) in a similar way based on w1l,…,wMl(39a)minimizewm,tmn∑m=1MwmHwm+∑m=1M−1UmqmHUmqm,(39b)subject to20d,20e,20f,32e,32fsatisfied,(39c)ℜwnHhmhmHwnlhmHwnl≥tmn,1≤n≤M,n≤m≤M.Solve problem (39a)–(39c) to get w1l,…,wMl;tmnl. Set l: = l+1 and again deal with (39a)–(39c), and an iterative process is formulated in the way. Let(40)vl=∑m=1Mwml2,l=1,…be the optimal values. The property about to be demonstrated is that {vl} is a nonincreasing sequence.Proposition 4.1.
It holds thatvl≥vl+1 for l≥1.Proof.
The optimal solutionw1l,...,wMl;tmnl for problem (37a)–(37c) with wn0 replaced by wnl−1 is feasible for problem (39a)–(39c). In that case, the optimal solution w1l+1,...,wMl+1;tmnl+1 for (39a)–(39c) has the following property:(41)vl=∑m=1MwmlHwml≥∑m=1Mwml+1Hwml+1=vl+1.
We have an immediate check whether it is feasible for problem (39a)–(39c). Since replacing wn0 with wnl−1 is the optimal solution of (37a)–(37c), it is also feasible, so (37c) satisfies(42)ℜwnlHhmhmHwnl−1hmHwnl−1≥tmnl,which imply that(43)hmHwnl≥tmnl.
So we have(44)ℜwnlHhmhmHwnlhmHwnl=hmHwnl≥tmnl,and (39a) is also fulfilled(45)tmnl≥γn∑i=n+1MhmHwil2+γnσm2,for 1≤n≤M,n≤m≤M. It follows from (44), (45) that w1l,...,wMl;tmnl 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) as(46a)minimizewm∑m=1MwmHwm+∑m=1M−1UmqmHUmqm,(46b)subject to20d,20e,20fsatisfied,(46c)ℜwnHhmhmHwnlhmHwnl≥γn∑i=n+1MhmHwi2+γnσm2,n<m<M,1<n<M,(46d)ℜwnHhMhMHwnlhmHwnl≥γn∑i=n+1MhMHwi2+γnσm21−β.
For problem (46a)–(46d), by introducing an auxiliary variable ∂ to the constraint (20d), thus, (20d) can be rewritten as(47)β∂≥∑i=1M−1Uiqi2,(48)∑i=1MhMHwi2≥∂.
So for formula (47), we deal with it in the same way as (24a)(49)4∑i=1M−1trUiqi2≤∂˜+β˜2+2∂˜+β˜∂+β−∂˜−β˜−∂−β2.
To approximate the quadratic matrix inequality (48), we assume that an initial point wn0,qn0 is given, and that(50)wn−wn0wn−wn0H≈0,for n = 1,…M, and(51)qn−qn0qn−qn0H≈0.
This means thatwn0 is sufficiently close to wn for each n and qn0 is sufficiently close to qn. Then, we have(52)wnwnH≈wnwn0H+wn0wnH−wn0wn0H,∀n,(53)qnqnH≈qnqn0H+qn0qnH−qn0qn0H,and by this approximation, we can get(54)∑n=1MhMHwnwn0H+wn0wnH−wn0wn0HhM≥∂.
For problem (46a)–(46d), we can introduce an auxiliary variable xm, so the constraint (20e) can be written as(55)xm+gmHUmqm2σm2≥γm,1≤m<M,(56)hmHwm2∑i=m+1MhmHwi2+σm2≥xm,1≤m<M.
So for formula (55), we deal with it in the same way as (48)(57)xm+gmHUmqnqn0H+qn0qnH−qn0qn0HgmUmH≥γm,1≤m<M.
In order to approximate the inequality (56), by constructing a perfect plane, we can get(58)uv−u0v0v2≥0,|u|2v≥2ℜu0uv0−u02v02v.
For formula (56), it can be approximated by the perfect plane method, which can be obtained(59)2ℜhmHwmlwmHhmxml−hmHwmlwmlHhmxml2xm≥∑i=m+1MhmHwi2+σm2,1≤m<M.
Therefore, problem (20a)–(20f) is solved by solving the following optimization problem:(60a)minimizewm,qm,xm,β,∂∑m=1MwmHwm+∑m=1M−1UmqmHUmqm,(60b)subject to46c,46d,49,54,57,59,20fsatisfied.
Thus, an approximate SOCP algorithm is formulated as in Algorithm 2.Algorithm 2: An Approximate SOCP algorithm for (20a)–(20f).
Require:hm,σm,γn,Pk,M,K,wm0,qm0,∂˜,β˜,ξEnsure: A solution wm∗,qm∗ for problem (20a)–(20f);(1)
Suppose thatwm0,qm0 is an initial point; set l = 0 and vl=∑m=1MwmHwm+∑m=1M−1UmqmHUmqm (a large number);(2)
repeat;(3)
solve SOCP (60a)–(60b), obtaining optimal solution wml+1,qml+1;(4)
∂=∂,β=β,l:=l+1;(6)
untilvl−1−vl≤ξ.


## 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 userM, and the receiver noise power σm2=1 Watt for all m.Example 1.
Assume that the base station servesM = 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 hm follows Ν0,1/2dmηKI (i.e., (3) with ζ=0; similarly, suppose that gm also follows Ν0,1/2dmηKI; for channel 2, suppose that hm follows Ν0,1/2dmηKI+1/2dmηaθmMI (i.e., (3) with ζ=1); similarly, suppose that gm also follows Ν0,1/2dmηKI+1/2dmηaθmMI, where dm is the distance from the base station to user m, dmM is the distance from the user M to user m, θm is the angle of departure (AoD) between the base station and user m, θmM is the angle of departure between the user M and user m, and η denotes the path loss exponent. In the simulation, we set dM = d2 = 0.8 m (meter), d1 = 1.6 m, d12=1.2m, and η=1, and the AoDs in (3) are θ1,θ2,θ12=300,700,900. The SINR targets γ2=1,γ1=1.
In Figure3, 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.Figure 3
The optimal total transmission power versus the number of transmit antennas, withγ=1 and η=1.
(a)(b)Example 2.
Assume that the base station servesM = 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 Figure4, 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 g12 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.Figure 4
The optimal total transmission power versus the distance between the center user and the edge user, withγ=1,η=0 and K=16.Example 3.
Consider a scenario in which up to 4 users are served (as in problem (60a)–(60b)). Suppose that hm and gm follow the channel model (3) with ζ=0, and the SINR thresholds are the same, that is, γm=γ=1,∀m. 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 d14,d24,d34=1.4,0.9,0.5.
Figure5 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.Figure 5
The optimal total transmission power versus the number of transmit antennas, withγ=1 and ζ=0.Example 4.
Consider a scenario in which up to 3 users are served (as in problem (60)). Suppose thathm and gm follow the channel model (3) with ζ=0, and the SINR thresholds are the same, that is, γm=γ=1,∀m. 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 d13,d23=1.4,0.9, and the angle of departures (AoDs) in (3) are θ1,θ2,θ3,θ13,θ23=300,700,800,800,900.
Figure6 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.Figure 6
The optimal total transmission power versus the number of transmit antennas, withγ=1 and ζ=0.Example 5.
Suppose that the base station serves a set of four usersu1, 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 hm follows the channel model (3) with ζ=1. 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 d14,d24,d34=1.4,0.9,0.5, and the angle of departures in (3) are θ1,θ2,θ3,θ4,θ14,θ24,θ34=300,700,700,700,700,700,700. The path loss exponent η=1. In addition, we assume that γ1=γ2=γ3=γ4=γ5=γ6=γ. When we say that the base station serves three (two) users, it refers to the users u1,u2,u3u1,u2.
In Figure7, 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.Figure 7
The optimal total transmission power versus the value of SINR, withγ=1, K = 20, and η=1.


## 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.



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*Source: 1090582-2022-06-02.xml*