Transceiver Design for Cooperative Non-Orthogonal Multiple Access Systems with Wireless Energy Transfer

# Transceiver Design for Cooperative Non-Orthogonal Multiple Access Systems with Wireless Energy Transfer

## Abstract

In this paper, an energy harvesting (EH) based cooperative non-orthogonal multiple access (NOMA) system is considered, where node S simultaneously sends independent signals to a stronger node R and a weaker node D. We focus on the scenario that the direct link between S and D is too weak to meet the quality of service (QoS) of D. Based on the NOMA principle, node R, the stronger user, has prior knowledge about the information of the weaker user, node D. To satisfy the targeted rate of D, R also serves as an EH decode-and-forward (DF) relay to forward the traffic from S to D. In the sense of equivalent cognitive radio concept, node R viewed as a secondary user assists to boost D’s performance, in exchange for receiving its own information from S. Specifically, transmitter beamforming design, power splitting ratio optimization and receiver filter design to maximize node R’s rate are studied with the predefined QoS constraint of D and the power constraint of S. Since the problem is non-convex, we propose an iterative approach to solve it. Moreover, to reduce the computational complexity, a zero-forcing (ZF) based solution is also presented. Simulation results demonstrate that, both two proposed schemes have better performance than the direction transmission.

Cooperative non-orthogonal multiple access, energy harvesting, beamforming, convex optimization.

## I Introduction

Self-sustainability and high spectral efficiency are two important metrics for future wireless communication networks. As a promising solution to enabling self-sustainable communications, radio frequency energy harvesting (RF-EH) technology has recently rekindled considerable interest. The ambient electromagnetic radiation can be captured by the receiver antennas and converted into direct current (DC) voltage [1]. More importantly, RF-EH enables simultaneous wireless information and power transfer (SWIPT) [2]. To implement it, two practical receiver architectures called time switching (TS) and power splitting (PS) are proposed [3]. TS switches the receiver between information decoding (ID) and EH modes over time, while PS divides the received signal into two streams with one for ID and the other for EH. For relay-assisted networks with SWIPT, the energy-constrained relays are allowed to use the harvested RF energy broadcasted by sources to relay the sources’ information to destinations. The achievable throughput performance of decode-and-forward (DF) protocol is given in [4]. Furthermore, relay selection and energy cooperation strategy of multiple users are respectively studied in [5] and [6]. To enhance the EH efficiency, multiple antennas are introduced in relay systems. Joint beamforming design of source and relay node as well as power splitter ratio optimization is investigated in [7]. In addition, SWIPT has been extended to cooperative cognitive radio networks [8] and full duplex networks[9].

To improve the spectrum efficiency, non-orthogonal multiple access (NOMA) allows multiple users to be served in the same time and frequency resource by using power domain multiplexing. For user fairness, less powers are allocated to users who have better channel gains. Moreover, successive interference cancellation (SIC) is adopted by users with better channel conditions to subtract signals intended for other users before decoding their own. Based on the power allocation strategy, as proposed in [10], NOMA can be classified into two categories, i.e., fixed power allocation NOMA (F-NOMA), and cognitive radio inspired NOMA (CR-NOMA). F-NOMA means that user powers are strictly assigned according to the order of their channel conditions. The performance of downlink NOMA with randomly located user and the impact of user pairing are respectively characterized in [10] and [11] for F-NOMA. Despite that F-NOMA scheme has superior system performance, it does not work if multiple antennas are considered. This is owing to the fact that precoders would affect the channel conditions and hence it is challenging to order users.

As for the CR-NOMA scheme, users with better channel conditions are viewed as secondary users and opportunistically served by the source on the condition that the quality of service (QoS) of weaker users is satisfied. Based on this principle, the analytical outage probability of the stronger user is given in [10], since the weaker user’s QoS has already been guaranteed. For the multiple-antenna case, a zero-forcing (ZF) based beamforming design and user clustering strategy are investigated for the downlink multiuser NOMA systems [12]. In that paper, users within the same cluster share the same beamforming vector. To fully exploit the spatial multiplexing gain, two different beamforming vectors respectively for two users are optimized to maximize the system sum rate performance subject to the QoS constraint of the weaker user [13].

It is worth pointing out that the additional introduced secondary users (stronger users) deteriorate the performance of weaker users. In order to improve the reliability of weaker users, cooperative NOMA approach is proposed [14]. To be specific, stronger users serve as relays to forward the traffic from the source to weaker users. It is natural for stronger users to do this, since the messages intended for weaker users have been decoded and prior known by stronger users if the SIC is successful. In the sense of the equivalent cognitive radio concept, stronger users would like to relay messages intended for weaker users, in exchange for receiving their own. This cooperation is especially preferred when direct channels between the source and weaker users are too poor to guarantee their predefined QoS.

However, the QoS satisfaction for weaker users is brought by the stronger users’ extra transmission power consumption. The energy shortage at stronger users will break this cooperation strategy, even though the channel states between the source and stronger users are well enough for the information cooperation. This motivates us to introduce the wireless energy transfer to cooperative NOMA systems. That is, the source will transmit both the information and energy to stronger users, in return for stronger users to boost weaker users’ performance. Different from the user clustering approach and outage probability given in [15] with randomly deployed single-antenna users, in this paper, we focus on the beamforming design within one user cluster consisting of two paired users to further enhance the system performance.

In particular, we consider a RF-EH based cooperative NOMA system in which three nodes are included, i.e., M-antenna node S, N-antenna node R and single-antenna node D. Node R has a better connection to node S, while node D, whose service priority is higher, unfortunately has a worse channel condition. We particularly focus on the the case where the direct link between S and D is too weak to guarantee the required rate of D. It is a commonly seen situation when the direct link between S and D suffers from a deep fading or the required rate of S is too high. This motivates node R to simultaneously act as an EH relay to forward the traffic from S to D. Thus, the cooperative NOMA scheme is proposed. Multiple antennas at relay node are to enhance the spectral efficiency and energy transfer efficiency.

The main contributions of this work are summarized as follows:

1) In the proposed three nodes cooperative NOMA system, we focus on the transmitter beamforming design, power splitting ratio optimization and the receiver filter design to maximize the rate of R under constraints that the QoS of D is guaranteed and the transmission power of S is restricted.

2) Due to the coupling nature of variables, the considered problem is non-convex. Then, an iterative approach is presented. Specifically, with the fixed receiver filter, the optimal transmitter beamforming and power splitting ratio are obtained via semi-definite relaxation (SDR) and the dual method. With the fixed transmitter beamforming and power splitting ratio, the optimal receiver filter is also derived.

3) Moreover, to reduce the complexity, ZF-based solution is proposed to find a suboptimal transmitter beamforming and power splitting ratio with the fixed receiver filter.

4) Comparing these two schemes, the optimal transmitter beamforming scheme always outperforms ZF transmitter beamforming scheme in terms of node R’s rate. Yet, it has almost the same performance with ZF transmitter beamforming scheme in terms of the outage probability of node D. More importantly, both proposed schemes have better outage performance that the direct transmission.

The remainder of the paper is organized as follows. In Section II, system model and problem formulation are introduced. In Section III, we present an iterative solution to problem 1. In Section IV, we further state the ZF-based suboptimal solution to problem to reduce the complexity. The simulation results are presented and discussed in Section V. Finally, Section VI concludes the paper.

Notation: Bold lower and upper case letters are used to denote column vectors and matrices, respectively. The superscripts and is standard transpose and (Hermitian) conjugate transpose of , respectively. refers to the Euclidean norm of . and denote the rank and trace of matrix , respectively. means that matrix is positive semidefinite (negative semidefinite). is the orthogonal projection onto the column space of , while is the orthogonal projection onto the orthogonal complement of the column space of .

## Ii System Model and Problem Formulation

Considering a cooperative NOMA system, in which a M-antenna node S simultaneously communicates with a N-antenna node R and a single-antenna node D. Node R and node D are users with better and worse connections to S, respectively. We consider the scenario that the direct link between S and D is too weak to satisfy the rate demand of the node D. Therefore, the RF-EH based cooperative NOMA scheme needs to carry out. In particular, the energy-constrained node R also acts as a relay to first harvest the RF energy broadcasted by S and then uses all the harvested energy to forward the information from S to D. The PS approach to realize SWIPT is adopted at node R in this paper. Without loss of generality, we suppose that is normalized to be unity. All channels are assumed to be quasi-static, where the channel coefficients remain the same for each communication duration but vary randomly over different time slots. Note that our considered system model is readily applicable to the downlink transmission with receiver cooperation enhanced 5G systems. In 5G, the access-point (AP) will serve diverse devices with different capabilities, such as different number of antennas, different battery capacities, different data requirements, different priorities and so on.

### Ii-a Phase 1: Direct Transmission

During this phase, node S transmits two independent symbols1 and () with power 2 to nodes R and D respectively in the same frequency and time slot. The factor 2 is due to the fact that S only transmits signals during the first half duration. The transmitted signal at S can be written as

 x=√2PSw1x1+√2PSw2x2, (1)

where and denote the precoding vectors for R and D, respectively. The observations at D and R are respectively given by

 yD,1=√2PShHSDw1x1+√2PShHSDw2x2+nD,1, (2)
 yR,1=√2PSHHSRw1x1+√2PSHHSRw2x2+nR,1, (3)

where and denote the channel matrices from S to D and R, respectively. is additive Gaussian white noise (AWGNs) at D with variances , and is AWGNs vector at R, satisfying .

From (2), the received signal to interference plus noise ratio (SINR) at D to detect is given by

 γD,1=2PS∣∣hHSDw2∣∣22PS∣∣hHSDw1∣∣2+σ2D. (4)

Node R is assumed to be energy-limited and has the ability for RF-EH [16]. To decode information and harvest energy concurrently, the practical PS-based receiver architecture is applied at node R. The PS approach works as follows. The node R splits the received RF signal into two streams: one for decoding the information of R and D and the other for harvesting energy to power node R, with the relative power ratio of and , respectively. The stream flow for information decoding will be converted from the RF to the baseband, and consequently be written as

 yIDR,1=√ρyR,1+~nR,1=√ρ(√2PSHHSRw1x1+√2PSHHSRw2x2+nR,1)+~nR,1, (5)

where is the circuit noise vector caused by the signal frequency conversion from RF to baseband. After applying the receiver vector , the estimated signal at R can therefore be represented as

 xR,1=wHR[√ρ(√2PSHHSRw1x1+√2PSHHSRw2x2+nR,1)+~nR,1]. (6)

According to the NOMA protocol, SIC is carried out at node R. Specifically, R first decodes the information of D by treating the interference caused by as noise, and then removes this part from the received signal to decode its own information. Mathematically, the received SINRs at R to decode and can be respectively written as

 γD,1→R,1=2ρPS∣∣wHRHHSRw2∣∣22ρPS∣∣wHRHHSRw1∣∣2+ρσ2R∥wR∥2+~σ2R∥wR∥2, (7)
 (8)

which results in the rate of node R .

The signal flow for energy harvesting is

 yEHR,1=√1−ρyR,1=√1−ρ(√2PSHHSRw1x1+√2PSHHSRw2x2+nR,1). (9)

Let denote the energy harvesting efficiency, the harvested energy at R is

 E=η(1−ρ)(2PS(∥∥HHSRw1∥∥2+∥∥HHSRw2∥∥2))2. (10)

The noise power is ignored compared with the signal power.

We assume that the energy consumed for signal processing is negligible, as compared with the power for signal transmission. Moreover, the transmission period for two phases is equal. Accordingly, the total transmission power at R is

 PR=2ηPS(1−ρ)(∥∥HHSRw1∥∥2+∥∥HHSRw2∥∥2). (11)

### Ii-B Phase 2: Cooperative Transmission

In phase 2, node S keeps silent, and node R forwards the decoded signal to D with the transmission power . That is, DF protocol is used. The received signal at D is

 yD,2=√PRhHRDwDx2+nD,2, (12)

where and represent the channel vector from R to D and the AWGN at D, respectively; is R’s transmit beamforming. Intuitively, maximal ratio combining (MRC) is the best transmission choice, that is [17], since only a single data stream is considered here. Then, the received SNR is given by

 (13)

At the end of this phase, MRC strategy is applied to combine the signal of and . Consequently, the combined SINR at D is

 γMRCD,1,2=γD,1+γD,2=2PS∣∣hHSDw2∣∣22PS∣∣hHSDw1∣∣2+σ2D+2ηPS(1−ρ)(∥∥HHSRw1∥∥2+∥∥HHSRw2∥∥2)∥hRD∥2σ2D, (14)

which results in the achievable destination rate .

### Ii-C Problem Formulation

In accordance with the CR-NOMA proposed in [10], the node D, a user with weak channel condition, is viewed as a primary user who occupies the communication channel if orthogonal multiple access (OMA) is used. Based on the equivalent cognitive radio concept, node R is treated as the secondary user to co-work with node D under the underlay mode. Hence, it is of significant importance to meet the predefined QoS of the primary user D, especially when the direct link between S and D cannot satisfy the QoS of D. As a result, in this paper, we aim to maximize the rate of node R subject to the targeted rate constraint of node D and transmission power constraint of S. The optimization problem can be casted as

 P1: maxw1,w2,0≤ρ≤1,∥∥wR∥∥2=1 2ρPS∣∣wHRHHSRw1∣∣2ρσ2R∥wR∥2+~σ2R∥wR∥2 (15a) s. t. 2ρPS∣∣wHRHHSRw2∣∣22ρPS∣∣wHRhHSRw1∣∣2+ρσ2R∥wR∥2+~σ2R∥wR∥2≥γ′D, (15b) (15c) ∥w1∥2+∥w2∥2≤1, (15d)

where is the minimal SINR threshold at node D with the minimal rate requirement . What noteworthy is that the constraint (15b) is to ensure that node R can successfully detect node D’s information [11]. Different from the single-antenna case where the successful SIC decoding at R is guaranteed by its better channel gain, beamforming vectors at multiple-antenna S will change the SINRs of R and D. So it becomes necessary to add constraint (15b) [13]. Besides, (15c) and (15d) are the rate constraint of D and the transmission power constraint of S, respectively.

## Iii Optimization Solution

In this section, we propose an iterative approach to solve the non-convex problem .

### Iii-a Step one: Joint optimization of w1, w2 and ρ

With fixed , setting , the problem is simplified as

 P2: maxw1,w2,0≤ρ≤1 2ρPS∣∣~hHSRw1∣∣2ρσ2R+~σ2R (16a) s. t. 2ρPS∣∣~hHSRw2∣∣22ρPS∣∣~hHSRw1∣∣2+ρσ2R+~σ2R≥γ′D, (16b) (16c) ∥w1∥2+∥w2∥2≤1. (16d)

Obviously, problem 2 is non-convex, so the key idea to solve it lies in the reformulation of the problem. In order to solve problem 2 efficiently, we introduce a positive variable to rewrite the problem as the following 2.1:

 P2.1: maxw1,w2,0≤ρ≤1 2ρPS∣∣~hHSRw1∣∣2ρσ2R+~σ2R (17a) s. t. 2ρPS∣∣~hHSRw2∣∣22ρPS∣∣~hHSRw1∣∣2+ρσ2R+~σ2R≥γ′D, (17b) 2PS∣∣hHSDw2∣∣22PS∣∣hHSDw1∣∣2+σ2D≥Γ, (17c) 2η(1−ρ)PS(∣∣HHSRw1∣∣2+∣∣HHSRw2∣∣2)∥hRD∥2σ2D≥γ′D−Γ, (17d) ∥w1∥2+∥w2∥2≤1. (17e)

Clearly, there exists that makes the problem 2.1 identical to problem 2. In the following description, is treated as a constant.

We present the optimal solution to problem 2 by applying the celebrated technique of semidefinite relaxation (SDR). Define , , , and and ignore the rank-one constraint on and , the SDR of problem 2.1 can be expressed as

 P2.2: maxW1,W2,0≤ρ≤1 2PSTr(~HSRW1)σ2R+~σ2R/~σ2Rρρ (18a) s. t. (18b) 2PSTr(HSDW2)≥Γ(2PSTr(HSDW1)+σ2D), (18c) Tr(¯HSRW1)+Tr(¯HSRW2)≥(γ′D−Γ)σ2D2ηPS∥hRD∥2(1−ρ), (18d) Tr(W1)+Tr(W2)≤1. (18e)

Note that constraints (18b) and (18d) are convex owing to the fact that both and are convex functions with respect to with . However, Problem 2.2 is still nonconvex due to its objective function. Fortunately, this objective function is quasi-concave fractional. According to [18], a positive parameter can be introduced to formulate a new problem 2.3 which is closely related with 2.2.

 P2.3: maxW1,W2,0≤ρ≤1 2PSTr(~HSRW1)−t(σ2R+~σ2R/~σ2Rρρ) (19a) s. t. (19b) 2PSTr(HSDW2)≥Γ(2PSTr(HSDW1)+σ2D), (19c) Tr(¯HSRW1)+Tr(¯HSRW2)≥a1−ρ, (19d) Tr(W1)+Tr(W2)≤1, (19e)

where . Given and , Problem 2.3 is a convex semidefinite problem (SDP) and can be efficiently solved by off-the-shelf convex optimization solvers, e.g., CVX [19].

Remark 1: It is worth pointing out that problem 2.3 belongs to the so-called ¡°separate SDP¡± [20]. Let () be the optimal solution to problem 2.3. According to [20, Theorem 2.3], the optimal solution to problem 2.3 always satisfies , since the number of generalized constraints are 4. We consider the nontrivial case where , then and can be derived. So the SDR problem is tight.

Though the rank-one beamforming vectors can be directly achieved by solving problem 2.3, the computational complexity is high. To reduce the complexity, we resort to the Lagrangian dual problem of 2.3 for more insightful results.

Since problem 2.3 is convex and satisfies the Slater’s condition, its duality is zero. Let , , and denote the Lagrange multipliers respectively associated with four constraints of problem 2.3. Then, the Lagrangian function of problem 2.3 is given by

 L(W1,W2,ρ,λ1,λ2,λ3,λ4)=Tr(AW1)+Tr(BW2)−(t+λ1γ′D)~σ2Rρ−λ3a1−ρ  −tσ2R−λ1γ′Dσ2R−λ2Γσ2D+λ4, (20)

where

 A=2PS(1−λ1γ′D)~HSR+λ3¯HSR−2PSλ2ΓHSD−λ4I, (21)
 B=2PSλ1~HSR+λ3¯HSR+2PSλ2HSD−λ4I. (22)

With the Lagrangian function, the dual function of problem 2.3 is expressed as

 maxW1⪰0,W2⪰0,0≤ρ≤1 L(W1,W2,ρ,λ1,λ2,λ3,λ4) (23)

The optimal dual variables are represented as (), and hence the optimal and are denoted as and , respectively. To guarantee a bounded dual optimal value of (23), and must be negative semidefinite. As a result, we can obtain that and . In addition, according to (20) and (23), the optimal power splitter must be a solution of the following problem:

 P2.4: minρ (t+λ1γ′D)~σ2Rρ+λ3a1−ρ (24a) s. t. 0≤ρ≤1. (24b)

Proposition 1: The optimal solution to problem 2.4 is and the optimal value is , where .

Proof: See Appendix A.

Proposition 2: The optimal dual solution to problem 2.3 satisfies .

Proof: See Appendix B.

Define , then the Lagrangian dual problem of 2.3 is , which is expanded as (2.5)

 minλ1,λ2,λ3,λ4 −(t+λ1γ′D)~σ2R−λ3a−2√~σ2R(t+λ1γ′D)λ3a−tσ2R−λ1γ′Dσ2R−λ2Γσ2D+λ4 (25a) P2.5: s. t. A⪯0,B⪯0,λ1≥0,λ2≥0,λ3>0,λ4≥0. (25b)

The problem 2.5 is convex, since in (25a) is Geometric mean and thus concave [21]. Due to the zero dual gap, problem 2.5 has the same optimal value with problem 2.3.

With the optimal achieved by problem 2.5, based on Proposition 1, we can obtain . Moreover, the complementary slackness condition of problem 2.3 yields to and . Since and , we have and . Let and be the basis of the null space of and , respectively, and define and . Since , we have

 ⎧⎪ ⎪⎨⎪ ⎪⎩2PSτ21Tr(~HSRW′1)−t(σ2R+~σ2R/~σ2Rρ∗ρ∗)=d∗,τ21Tr(¯HSRW′1)+τ22Tr(¯HSRW′2)=a1−ρ∗, (26)

where is the optimal value of dual problem 2.5 and , are the power allocation coefficients for node R and D, respectively.

Thus, from (26), we have

 ⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩τ∗1=  ⎷d∗+t(σ2R+~σ2R/~σ2Rρ∗ρ∗)2PSTr(~HSRW′1),τ∗2= ⎷a1−ρ∗−τ∗21Tr(¯HSRW′1)Tr(¯HSRW′2). (27)

Then, optimal beamforming vectors are and with given and .

Remark 2: Note that complex variables and one real variable are to be optimized for problem 2.3, while only four real variables for problem 2.5. Obviously, problem 2.5 has a lower computational complexity than 2.3. Furthermore, the complexity reduction is remarkable as the number of antennas at S grows.

Now, we turn our attention to find the optimal and . Given , define the optimal value of problem 2.3 as and its dual function as . Using the zero dual gap, we have . It is easily checked that is a pointwise minimum of a family of affine function in terms of and as a result concave for . So the optimal can be found via the one-dimensional search. Based on (20), the gradient of is expressed as

 dϕ(Γ)dΓ=−2PSλ∗2Tr(HSDW∗1)−λ∗2σ2D + λ*3σ2Dη∥hRD∥2(1−ρ∗). (28)

According to the fractional programming [18], the optimal solution to problem 2.2 is the same with problem 2.3 when

 F(t∗)=maxW1,W2,ρ2PSTr(~HSRW1)−t∗(σ2R+~σ2R/~σ2Rρρ)=0. (29)

The optimal can be found by the Dinkelbach method [18]. Therefore, problem 2 is successfully solved. Detailed steps of proposed Algorithm 1 are summarized as below.

### Iii-B Step two: Optimization of wR

With fixed , and , define and , the optimization problem is formulated as

 P3: max∥wR∥2=1 ∣∣hH1wR∣∣2 (30a) s. t. 2ρPS∣∣hH2wR∣∣22ρPS∣∣hH1wR∣∣2+ρσ2R+~σ2R≥γ′D. (30b)

It is easy to observe that constraint (30b) is active at the optimum. That is,

 2ρPS∣∣hH2wR∣∣2=2ρPSγ′D∣∣hH1wR∣∣2+γ′Dσ2R+γ′D~σ2R . (31)

Since is only related to and , according to [22], the optimal can be parametrized as

 wR=√λ∏h2h1∥∥∏h2h1∥∥+√1−λ∏⊥h2h1∥∥∏⊥h2h1∥∥,0≤λ≤1. (32)

Then, we have

 ~f(λ)=∣∣hH1wR∣∣2=(√λ∥∥Πh2h1∥∥+√1−λ∥∥Π⊥h2h1∥∥)2 (33)
 and  ~g(λ)=∣∣hH2wR∣∣2=λ∥h2∥2