IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 17, NO. 8, AUGUST 2018 5367

Joint Beamforming and Power Allocation inDownlink NOMA Multiuser MIMO NetworksXiaofang Sun , Student Member, IEEE, Nan Yang , Member, IEEE, Shihao Yan , Member, IEEE,

Zhiguo Ding , Senior Member, IEEE, Derrick Wing Kwan Ng , Senior Member, IEEE,

Chao Shen , Member, IEEE, and Zhangdui Zhong, Senior Member, IEEE

Abstract— In this paper, a novel joint design of beamformingand power allocation is proposed for a multi-cell multiusermultiple-input multiple-output non-orthogonal multiple accessnetwork. In this network, base stations adopt coordinated multi-point for downlink transmission. We study a new scenario wherethe users are divided into two groups according to their quality-of-service requirements, rather than their channel qualities asinvestigated in the literature. Our proposed joint design aims tomaximize the sum rate of the users in one group with the besteffort while guaranteeing the minimum required target rates ofthe users in the other group. The joint design is formulated asa non-convex NP-hard problem. To make the problem tractable,a series of transformations is adopted to simplify the designproblem. Then, an iterative suboptimal resource allocation algo-

Manuscript received August 9, 2017; revised March 7, 2018; acceptedMay 16, 2018. Date of publication June 8, 2018; date of current ver-sion August 10, 2018. This work was supported in part by NSFC underGrants 61501024 and 61725101, in part by the Grant RCS2016ZZ004, byBeijing NSF under Grant L172020, in part by BJTU under Grant 2015RC087,in part by the China Railway Corporation under Grant 2016X003-L, in partby the Fundamental Research Funds for the Central Universities underGrant 2017JBM315, in part by the ARC Discovery Project under GrantDP180104062, in part by the U.K. EPSRC under Grant EP/N005597/1,in part by H2020-MSCA-RISE-2015 under Grant 690750, in part by theAustralian Research Council’s Discovery Early Career Researcher AwardFunding Scheme under Grant DE170100137, and in part by CSC. This paperwas presented in part at the IEEE Globecom Workshop, Washington, DC,USA, December 2016 [1]. The associate editor coordinating the review ofthis paper and approving it for publication was V. Cadambe. (Correspondingauthor: Chao Shen.)

X. Sun is with the State Key Laboratory of Rail Traffic Control andSafety, Beijing Jiaotong University, Beijing 100044, China, with the BeijingEngineering Research Center of High-Speed Railway Broadband MobileCommunications, Beijing Jiaotong University, Beijing 100044, China, andalso with the Research School of Engineering, The Australian National Uni-versity, Canberra, ACT 2601, Australia (e-mail: xiaofangsun@bjtu.edu.cn).

N. Yang is with the Research School of Engineering, TheAustralian National University, Canberra, ACT 2601, Australia (e-mail:nan.yang@anu.edu.au).

S. Yan is with the School of Engineering, Macquarie University, Sydney,NSW 2109, Australia (e-mail: shihao.yan@mq.edu.au).

Z. Ding is with the School of Electrical and Electronic Engineer-ing, The University of Manchester, Manchester M13 9PL, U.K. (e-mail:zhiguo.ding@gmail.com).

D. W. K. Ng is with the School of Electrical Engineering and Telecommu-nications, The University of New South Wales, Sydney, NSW 2052, Australia(e-mail: w.k.ng@unsw.edu.au).

C. Shen and Z. Zhong are with the State Key Laboratory of Rail TrafficControl and Safety, Beijing Jiaotong University, Beijing 100044, China, andalso with the Beijing Engineering Research Center of High-Speed RailwayBroadband Mobile Communications, Beijing Jiaotong University, Beijing100044, China (e-mail: chaoshen@bjtu.edu.cn; zhdzhong@bjtu.edu.cn).

Color versions of one or more of the figures in this paper are availableonline at http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TWC.2018.2842725

rithm based on successive convex approximation is proposed.In each iteration, a rank-constrained optimization problem issolved optimally via semidefinite program relaxation. Numericalresults reveal that the proposed scheme offers significant sum-rategains compared to the existing schemes and converges fast to asuboptimal solution.

Index Terms— NOMA, MIMO, CoMP, beamforming, powerallocation, successive convex approximation.

I. INTRODUCTION

NON-ORTHOGONAL multiple access (NOMA) hasrecently attracted much attention in both industry and

academia, as a promising technique for providing superiorspectral efficiency in 5G wireless networks [2]. Specifically,NOMA is a multiuser multiplexing scheme which enablessimultaneous multiple access in the power domain [3]. Thismakes it fundamentally different from conventional orthogo-nal multiple access (OMA) schemes, such as time divisionmultiple access, frequency division multiple access, and codedivision multiple access. Aided by NOMA, a base station (BS)is able to serve multiple users at the same time, frequency,and spreading code but at different power levels yielding ahigher flexibility and a more efficient use of spectrum andenergy. In order to unlock the potential benefit of NOMA, suc-cessive interference cancelation (SIC) is normally performedat some users such that they can remove the co-channelinterference incurred by NOMA and decode the desired signalssuccessively [4]. In practice, NOMA allocates more power tothe users with poor channel qualities to ensure the achievabletarget rates at these users, thus striking a balance betweennetwork throughput and user fairness [4], [5]. Moreover,NOMA has a definite superiority over OMA in terms of sumchannel capacity and ergodic sum capacity [6].

A. Related Studies and Motivations

Motivated by the fundamental works which establishedthe concepts of NOMA (see [7]–[10] and the refer-ences therein), Ding et al. [11] and Al-Imari et al. [12]systematically evaluated the performance of NOMA inthe downlink and the uplink, respectively. To maximizethe energy efficiency for a downlink multi-carrier NOMAsystem, Fang et al. [13] optimized subchannel assignmentand power allocation. Advocated by the unique benefitof multi-antenna systems, the application of multiple-input

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5368 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 17, NO. 8, AUGUST 2018

multiple-output (MIMO) techniques to NOMA was addressedin [6] and [14]–[24]. For instance, Zeng et al. [6] adopted anidentity matrix as the precoder and assumed that users areequipped with more antennas than the BS; thus users appliedzero-forcing (ZF) approach to eliminate intra-cluster interfer-ence. Hanif et al. [17] proposed a minorization-maximizationbased algorithm to maximize the downlink sum-rate, wherethe transmit signals of each user are processed by a sophisti-cated precoding vector. Considering a multiuser system whereusers transmiting multiple data streams, Choi [18] solved abeamforming power minimization problem by firstly obtainingthe optimal power allocation for given beamforming vectorsand then finding the optimal beamforming vectors iteratively.Ding et al. [19] considered the application of NOMA to amulti-user network with mixed multicast and unicast traf-fic. Sun et al. [20] aimed to maximize the system through-put and adopted an iteratively weighted minimum meansquare error approach to design the beamformer. Recently,the authors in [21] proposed a user clustering scheme andadopted the ZF beamforming approach for the maximiza-tion of the throughput in a single-cell scenario. Specifically,Xu et al. [22] introduced simultaneous wireless informationand power transfer into NOMA systems focusing on a two-usermulti-antenna single-cell scenario. As an enhanced version ofconventional MIMO, Ding and Poor [23] and Liu et al. [24]proposed massive-MIMO-NOMA downlink transmission pro-tocols. In particularly, the previous works relied on a keyassumption that users have significantly different channelgains. However, in some practical situations, e.g., the Internet-of-Things (IoT) scenarios [25], the locations of users are closeto each other and hence pairing users in terms of channelgains cannot fully exploit the promised performance gainfrom NOMA. To expand the limited application of NOMA,in this work, we study the joint beamforming and powerallocation design where users have similar channel gains tofurther unlock the potential benefit of NOMA.

We note that [6] and [14]–[24] focused on the applicationof NOMA in single-cell multi-antenna scenarios. Nevertheless,the spectral and energy efficiencies can be further improvedby introducing NOMA into multi-cell systems, which has alsobeen investigated in [26] and [27]. As shown in [26] and [27],interference is a key limiting factor in improving the capacityof multi-cell networks. To address this issue, multi-cell coop-eration has been proposed in [28]. Among various multi-cellcooperation techniques, coordinated multipoint (CoMP) is apromising and appealing one as it enables an adaptive coordi-nation among multiple BSs [29]. The ultimate goal of CoMPis to enhance the quality of useful signals and to mitigate theundesired interference for improving the network efficiencyand providing high quality-of-service (QoS) to users, espe-cially for the users, e.g. at the cell-edge, suffering from poorchannel qualities. To this end, multiple BSs can adopt eitherthe coordinated scheduling/beamforming scheme or the jointprocessing CoMP scheme to facilitate cooperation [30]. Forthe former scheme, the data of a user is required to be availableonly at its associated BS, but not to other BSs. Yet, userscheduling and beamforming decisions are made jointly viathe coordination among the BSs in the network. Differently,

for the latter scheme, user data is shared among multiple BSsof the network, thus requiring backhaul links with extremelyhigh capacity for information exchanging between all the BSs.In this paper, we focus on the former scheme to design thebeamforming, as coordinated beamforming offers promisingperformance gains via interference avoidance and is lesssophisticated compared to joint transmission [31].

B. Contributions

In this paper, we propose a new joint beamforming andpower allocation design for a generalized multi-cell MIMO-NOMA network. Notably, we study a new scenario where theusers are divided into two groups based on their QoS require-ments. Specifically, the users in Group 1 are expected to beserved with the best-effort, while the users in Group 2 imposestrict QoS requirements and need to be served with theirrequired target rates. Moreover, we consider a multi-cellnetwork in this paper, which is different from the single-cellsystem considered in our previous study [1]. In this multi-cellnetwork, our proposed design allows the multiple BSs to coop-erate with each other and jointly design beamforming vectorsand power allocation coefficients, which effectively suppressesthe inter-cell interference. In particular, the proposed beam-forming design takes into account the heterogeneous QoSrequirements of users and ensures a sufficient disparity of theeffective channel gains between any paired users. Therefore,the advantage of NOMA can be exploited.

The main contributions of this paper are summarized asfollows:

• To jointly design beamforming and power allocation inthe multi-cell multiuser MIMO-NOMA network, we firstformulate a joint design problem to maximize thesum-rate of users in Group 1, while ensuring the targetrates at the users in Group 2. Since this problem formu-lation is non-convex in general and hence challenging tosolve, we propose a series of transformations to simplifythe problem. We then propose an iterative suboptimalalgorithm based on the successive convex approxima-tion (SCA) to perform the coordination among BSs forjoint design of beamforming vectors and power allocationcoefficients.

• Unlike the problems studied in the literature which canbe directly solved by semidefinite relaxation (SDR),e.g. [32]–[34], the use of SDR in this work leadsto non-convex quadratic constraints and the proof ofrank-one solution is non-trivial. However, we analyticallyprove that the SDR is tight and verify our finding via sim-ulation. As such, the semidefinite program (SDP)-relaxedproblem is equivalent to the original joint design problem.

• Through numerical results, we demonstrate that our pro-posed design outperforms the existing NOMA and OMAschemes. Notably, we find that our proposed design canalways guarantee the QoS requirements of the usersin Group 2, while the existing NOMA scheme cannot.We also investigate the impact of network parameters,such as the maximum transmit power and the numbersof cells, on the performance of our proposed design

SUN et al.: JOINT BEAMFORMING AND POWER ALLOCATION 5369

Fig. 1. Illustration of a multi-cell multiuser MIMO network where NOMA isused in the downlink to serve two groups of users within the same frequencyresource simultaneously.

compared with existing schemes. We further find that ourproposed iterative resource allocation algorithm quicklyconverges to a suboptimal solution, i.e., in no morethan 10 iterations on average, and examine the impactof various network parameters on the convergence rate.Moreover, we have confirmed that our scheme exhibits acomparable performance to the optimal solution producedby the brute-force method in a small scale network, butincurring a much lower complexity.

The rest of the paper is organized as follows: Section IIpresents the system model and Section III formulates theresource allocation design as an optimization problem.In Section IV, the SDR approach and the SCA-based iterativealgorithm are proposed to solve the joint beamforming andpower allocation design problem. Simulation results are givenin Section V. Finally, Section VI concludes the paper.

Notation: Vectors and matrices are denoted by lower-caseand upper-case boldface symbols, respectively. (·)H denotesthe Hermitian transpose. Tr (·) denotes the trace operation.Rank (A) and Null (A) denote the rank and the null spaceof A, respectively, ‖ · ‖ denotes the Euclidean norm, and| · | denotes the absolute value. The distribution of a cir-cularly symmetric complex Gaussian (CSCG) variable withmean μ and covariance σ2 is denoted by CN (μ, σ2), and∼ means “distributed as.” E(·) denotes statistical expectation.∂F∂x denotes the first partial derivative of function F withrespect to variable x.

II. SYSTEM MODEL

In this paper, we consider an N -cell multiuser MIMOnetwork, as shown in Fig. 1. In each cell, a BS equippedwith M antennas communicates with 2K single-antenna users,where we assume that M ≥ K . The cell region of each BS ismodeled as a disk with radius Rc, where the BS is located atthe center of the disk. The 2K users in each cell are assumedto be divided into two groups, namely, Group 1 and Group 2.Without loss of generality, we assume that the number of

users in Group 1 is the same as that in Group 2 in eachcell, i.e., K users in Group 1 and K users in Group 2.In the considered network, NOMA is applied at the BSs suchthat they are able to simultaneously serve the two groups ofusers imposing different QoS requirements. We assume thatevery two users, one from Group 1 and one from Group 2,share the same resource (e.g. time slot, frequency, and space).These two users are considered as a cluster. The index of thek-th cluster served by the n-th BS is denoted by kn, wheren ∈ {1, 2, · · · , N} and k ∈ {1, 2, · · · , K}. The user inGroup 1 and the user in Group 2 in cluster kn are denoted byu1

knand u2

kn, respectively. We introduce the set Ln to represent

all the BSs in this network except for the n-th BS and denotethe set of the indices of the BSs in Ln by Nn, i.e., Nn ={1, 2, · · · , n − 1, n + 1, · · · , N}. Aiming at studying thebeamforming design for pairing users with heterogeneous QoSrequirements and taking fully advantage of CoMP, we assumethat perfect channel state information (CSI) of all the users inthe network, i.e., the full global CSI, is available at each BS viachannel reciprocity in time-division duplex systems or usersfeeding back in frequency-division duplex systems. The globalCSI can be obtained based on the channel estimation methodsstudied in the literature, e.g., [35]–[37].

In practical scenarios, the disparity between the channelconditions of two users is not necessarily significant, whichbrings difficulties to user pairing for implementing NOMA.Motivated by this, this work considers the scenario wherethe QoS requirements of the users in two groups are signifi-cantly different. For instance, the users in Group 1 requiringnon-delay sensitive applications are expected to be served withthe best-effort, whereas the users in Group 2 request delaysensitive applications requiring a constant data rate.

III. COOPERATIVE TRANSMISSION WITH NOMA

A. Zero Forcing Transmission at BSs

The information bearing vector adopted at BS n, denotedby sn, is given by sn = [s1n , · · · , sKn ]T , where skn =√

akns1kn

+√

bkns2kn

is the signal for cluster kn, s1kn

∼CN (0, 1) and s2

kn∼ CN (0, 1) are the signals intended to

u1kn

and u2kn

, respectively, and akn ≥ 0 and bkn ≥ 0are the portions of the transmit power allocated to u1

knand

u2kn

, respectively. Without loss of generality, we assume thatakn +bkn = 1 to guarantee the total transmit power constraint.

To facilitate the downlink multiuser transmission, an M×Kbeamforming matrix Pn is adopted at BS n. Mathematically,Pn can be expressed as Pn = [p1n , · · · ,pKn ], where pkn isthe beamforming vector designed for cluster kn. As observedin [3], a significant performance gain of NOMA over conven-tional OMA is achieved in the high signal-to-noise ratio (SNR)regime, particularly when the channel qualities of the twousers served using NOMA are significantly different from eachother. Motivated by this observation, to enlarge the disparitybetween the received SNRs of the two users in one cluster,in this work, we adopt ZF beamforming [38] to eliminatethe intra-cell interference at u1

knfrom other clusters within

the same cell. This is due to our assumption that u1kn

isexpected to be served with the best-effort, while u2

knonly

5370 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 17, NO. 8, AUGUST 2018

requires a constant target data rate. We note that the ZFbeamforming only requires local CSI at BS n, which reducesthe signaling overhead for cooperation. We also note thatthe QoS requirement at u2

knhas to be guaranteed. Thus,

it is important to ensure that the received interference atu2

knis carefully controlled. To this end, the beamforming

vector, pkn , needs to be designed such that u2kn

receivesa tolerable interference to achieve the target rate. To fulfillthe aforementioned requirements, we rewrite the beamformingvector pkn as pkn = Uknqkn , where Ukn is designed toeliminate the intra-cell interference for u1

knand qkn aims

to improve the data rate of u1kn

while guaranteeing the QoSrequirement of u2

kn.

For the sake of clarity, we define a set of new matri-ces Gkn ∈ C

M×(K−1), which contains the channel vec-tors from BS n to the remaining associated K − 1 usersin Group 1 except for u1

kn. As such, we have Gkn =[

gn1n , · · · ,gn(k−1)n,gn(k+1)n

, · · · ,gnKn

], where gnkn ∈

CM×1 represents the channel vector from BS n to u1

kn. Apply-

ing the singular value decomposition (SVD), we rewrite Gkn

as Gkn =[Akn Akn

]VknDkn , where Akn ∈ C

M×(K−1) isthe first K−1 left eigenvectors of Gkn forming an orthogonalbasis of Gkn and Akn ∈ C

M×(M−K+1) is the last M−K +1left eigenvectors of Gkn (corresponding to zero eigenvalues)forming an orthogonal basis of the null space of Gkn . SinceUkn is used to eliminate the intra-cell interference for u1

kn,

it lies in the null space of Gkn and can be written asUkn = Akn without loss of generality. Therefore, the signaltransmitted by BS n is given by

xn = Pnsn =K∑

k=1

Uknqkn

(√akns1

kn+√

bkns2kn

). (1)

B. SIC at Users in Group 1

The received signal at u1kn

is given by

y1kn

= gHnkn

xn +∑

i∈Nn

gHikn

xi + ωkn , (2)

where ωkn ∼ CN (0, σ2kn

)denotes the additive white

Gaussian noise (AWGN) at u1kn

with zero mean and vari-ance σ2

kn. Since ZF beamforming is adopted at BS n, we have

UHkn

Gkn = 0. Thus, substituting (1) into (2) yields

y1kn

=√

akngHnkn

Uknqkns1kn︸ ︷︷ ︸

desired signal

+√

bkngHnkn

Uknqkns2kn︸ ︷︷ ︸

intra-cluster interference

+∑

i∈Nn

K∑

j=1

gHikn

Ujiqjisji

︸ ︷︷ ︸inter-cell interference

+ωkn . (3)

In (3), the first term on the right-hand side is the desiredsignal for u1

kn, the second term is the intra-cluster interference,

i.e., the interference caused by the signal intended to u2kn

,and the third term is the inter-cell interference caused byBSs in Ln.

In the considered network, we assume that SIC is adoptedat u1

knto remove the interference caused by u2

kn, due to the

required best-effort service for u1kn

and a low target data raterequirement at u2

kn. Hence, u1

knhas to decode s2

knfirst. Based

on (3), the signal-to-interference-plus-noise ratio (SINR) fordecoding s2

knat u1

knis given by

SINR2k1

n

=bkn |gH

nknUknqkn |2

akn |gHnkn

Uknqkn |2 +∑

i∈Nn

∑Kj=1 |gH

iknUjiqji |2+σ2

kn

,

(4)

where we treat the interference as noise as commonly adoptedin the literature, e.g. [11]. Without loss of generality,we assume that the required target data rates at users inGroup 2 are the same, which is defined by R0. In order toguarantee the success of SIC, we need to ensure log2(1 +SINR2

k1n) ≥ R0. After performing SIC, the SINR for s1

knat

user u1kn

is given by

SINR1k1

n=

akn |gHnkn

Uknqkn |2∑

i∈Nn

∑Kj=1 |gH

iknUjiqji |2 + σ2

kn

. (5)

C. Direct Decoding at Users in Group 2

We next study the received signal at u2kn

, i.e., the userfrom Group 2 in cluster kn. We note that it is impossible tocompletely cancel the intra-cluster interference, the intra-cellinterference, and the inter-cell interference at u2

kn. As such,

the received signal at u2kn

is given by

y2kn

=√

bknhHnkn

Uknqkns2kn

+ Ikn + �kn , (6)

where hnkn represents the channel vector between BSn and u2

kn. Variable �kn ∼ CN (0, ς2

kn) is the AWGN at u2

kn.

In (6), the first term on the right-hand side is the desiredsignal for u2

knand Ikn represents the interference. Here, Ikn is

given by

Ikn =√

akn hHnkn

Uknqkns1kn︸ ︷︷ ︸

intra-cluster interference

+K∑

j=1,j �=k

hHnkn

Ujnqjnsjn

︸ ︷︷ ︸intra-cell interference

+∑

i∈Nn

K∑

j=1

hHikn

Ujiqjisji

︸ ︷︷ ︸inter-cell interference

, (7)

where the first term on the right-hand side is the intra-clusterinterference, the second term is the intra-cell interference, andthe third term is the inter-cell interference caused by the BSsin Ln. Following (6), the SINR for s2

knat u2

knis given by

SINR2k2

n=

bkn |hHnkn

Uknqkn |2E [|Ikn |2] + ς2

kn

, (8)

where the interference power is given by

E[|Ikn |2

]= akn |hH

nknUknqkn |2 +

K∑

j=1,j �=k

|hHnkn

Ujnqjn |2

+∑

i∈Nn

K∑

j=1

|hHikn

Ujiqji |2. (9)

SUN et al.: JOINT BEAMFORMING AND POWER ALLOCATION 5371

Note that in the considered network, we need to guaranteethe QoS requirement at u2

kn. As such, we need to ensure

log2

(1 + SINR2

k2n

)≥ R0 via a careful design of resource

allocation.

D. Problem Formulation

In the considered system, the BSs need to maximize thesum data rates of the users in Group 1 while guaranteeingthe target rates required at the users in Group 2. To this end,the BSs need to optimally design the beamforming vectors qkn

and the power allocation coefficients, i.e., akn and bkn . There-fore, the optimization problem for the BSs, denoted by P1,is formulated as

P1 : maximize{akn ,bkn ,qkn}

∀n,k

N∑

n=1

K∑

k=1

log2(1 + SINR1k1

n) (10a)

s. t. log2(1 + SINR2k1

n) ≥ R0, ∀n, k, (10b)

log2(1 + SINR2k2

n) ≥ R0, ∀n, k, (10c)

K∑

k=1

Tr(qknqHkn

) ≤ Pn, ∀n, (10d)

akn + bkn = 1, ∀n, k, (10e)

akn ≥ 0, bkn ≥ 0, ∀n, k, (10f)

where R0 is the data rate for users in Group 2. As per theShannon’s coding theorem, the data rates are required to beless than the channel capacities to correctly decode the databits at the receivers. Hence, constraint (10b) ensures the suc-cess of SIC decoding at u1

kn, constraint (10c) guarantees the

required target data rate at u2kn

, and constraint (10d) imposesthe maximum total power budget, Pn, ∀n ∈ {1, · · · , N},at each BS. We also note that the beamforming vectors,i.e., qkn , determine the power allocation among K clus-ters at each BS, and the power allocation constraints foru1

knand u2

knin each cluster are given by (10e) and (10f).

We further note that the objective function given by (10a)and the constraints given by (10b) and (10c) are non-convexwith respect to akn , bkn , and qkn , due to the couplingbetween optimization variables in the objective function andconstraints (10a), (10b), and (10c).

IV. JOINT DESIGN OF BEAMFORMING

AND POWER ALLOCATION

In this section, we aim to solve P1 in (10). To this end,a series of transformations are proposed to simplify this prob-lem. Then, SDR [32] and SCA [39] approaches are applied toperform the joint design of beamforming vectors and powerallocation coefficients.

A. Problem Reformulation

P1 is a non-convex problem which belongs to the classof NP-hard problems [32]. In general, a brute-force approachis needed to obtain a globally optimal solution. Thus,in this section, we first transform P1 into an equivalentrank-constrained SDP problem to facilitate the design of

a computationally efficient resource allocation. Specifically,we find that the beamforming variable qkn in (10) is in theform of qknqH

kn, ∀n, ∀k. Inspired by this, we introduce and

optimize the auxiliary optimization matrix Qkn = qknqHkn

,∀n, ∀k. It is noted that Qkn ∈ C(M−K+1)×(M−K+1) isrequired to be a rank-one positive semidefinite (PSD) matrix,i.e., Qkn 0 and Rank(Qkn) ≤ 1. Then, P1 can beequivalently written as P2 in terms of Qkn which is given by

P2 : maximize{akn ,Qkn}

∀n,∀k

N∑

n=1

K∑

k=1

log(

1 +akn Tr(GknknQkn)

wkn

)

(11a)

s. t. akn Tr(GknknQkn)

≤ Tr(GknknQkn)1 + γ

− γ

1 + γwkn , ∀n, k,

(11b)

akn Tr(HknknQkn)

≤ Tr(HknknQkn)1 + γ

− γ

1 + γrkn , ∀n, k,

(11c)K∑

k=1

Tr(Qkn) ≤ Pn, ∀n, (11d)

Qkn 0, ∀n, k, (11e)

0 ≤ akn ≤ 1, ∀n, k, (11f)

Rank(Qkn) ≤ 1, (11g)

where γ = 2R0 − 1,

Gjikn =UjigikngH

iknUji

σ2kn

, Hjikn =UjihiknhH

iknUji

ς2kn

,

wkn =∑

i∈Nn

K∑

j=1

Tr (GjiknQji) + 1,

rkn =K∑

j=1,j �=k

Tr(HjnknQjn)+∑

i∈Nn

K∑

j=1

Tr (HjiknQji)+1.

The proposed change of variables enables us to transform theconsidered problem with respect to qkn to a rank-constrainedSDP problem with respect to Qkn . We note that the optimiza-tion problem P2 is equivalent to the original problem P1 ifand only if Qkn is a rank-one PSD matrix. If the rank-oneconstraint is guaranteed, the vector solution to P1 can beretrieved from the matrix solution to P2. On the other hand,even if the rank-one constraint on Qkn is dropped, problem P2is still intractable due to the coupling between Qkn and akn .In the following, we will further transform and approximateproblem P2 to obtain a tractable formulation.

Now, we handle the coupling between the optimizationvariables in the objective function. We note that the logarithmfunction in the objective function is concave with respect to theinput argument. However, due to the received inter-cell inter-ference involved in the denominator and the joint design ofbeamforming vectors and power allocation in (11a), the objec-tive function is non-convex with respect to akn and Qkn .

5372 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 17, NO. 8, AUGUST 2018

As such, we first adopt the following transformation to theobjective function to circumvent its non-convexity.

We introduce a set of auxiliary variables ρkn to boundSINR1

k1n

from below, i.e., the achievable SINR at users inGroup 1. Specifically, ρkn is given by

ρkn ≤ akn Tr (GknknQkn)wkn

. (12)

Substituting ρkn into (11a), P2 is transformed into an equiv-alent optimization problem P2a, which is given by

P2a : maximize{akn ,Qkn ,ρkn}

∀n,k

N∑

n=1

K∑

k=1

log2 (1 + ρkn) (13a)

s. t. akn Tr (GknknQkn) ≥ ρknwkn , ∀n, k,

(13b)

ρkn ≥ 0, ∀n, k, (13c)

(11b) − (11g).

Due to the monotonically increasing property of logarithmfunctions, the value of the objective function (13a) increaseswith ρkn . Based on constraint (13b), the upper bound of ρkn

is akn Tr(GknknQkn)/wkn . Therefore, for maximizing theobjective function in P2a with (13b) and (13c), it is equivalentto maximize the objective function in P2.

We note that the functions on both sides of constraint (13b)and on the left-hand side of constraints (11b) and (11c) arebilinear functions with respect to ρkn , akn , and Qkn . There-fore, in the following subsection, we exploit the bilinearityof these optimization variables to design a tractable resourceallocation.

B. Successive Convex Approximation

Recall that the beamforming matrix Qkn and power alloca-tion coefficient akn are coupled together as bilinear functionsin constraints (11b), (11c), and (13b), e.g. akn Tr (GknknQkn)and akn Tr (HknknQkn). In fact, the Hessian matrix of a bilin-ear function is neither a positive nor a negative semidefinitematrix. Thus, bilinear functions are neither convex nor concavein general, which is an obstacle in designing a computationallyefficient resource allocation algorithm.

Now, we handle the bilinear terms in the following. We notethat the bilinear function akn Tr(GknknQkn) on the left-handside of (13b) is desired to be transformed into a concave func-tion. Whereas akn Tr(GknknQkn) and akn Tr(HknknQkn)on the left-hand side of (11b) and (11c), respectively, aredesired to be transformed into convex functions. In order toconvexify the considered constraints, we first adopt the Schurcomplement [40] to handle the bilinear constraint in (13b)which leads to the following equivalent constraints:

[akn tkn

tkn Tr(GknknQkn)

] 0, ∀n, k, (14)

t2kn

wkn

≥ ρkn , ∀n, k, (15)

where tkn is an auxiliary variable.However, (15) is still a non-convex constraint since it is

a difference-of-convex functions (DC) [41]. To address this

issue, we then tackle it via the SCA method based on thefirst-order Taylor expansion [42]. In particular, t2kn

/wkn onthe left-hand side of (15) is convex in both tkn and wkn , andthus can be tightly bounded from below with its first-orderapproximation. Specifically, for any fixed point (w̃kn , t̃kn)with w̃kn ≥ 1 and t̃kn ≥ 0, we have

t2kn

wkn

≥ 2t̃kn

w̃kn

tkn − t̃2kn

w̃2kn

wkn ≥ ρkn . (16)

By applying the concept of SCA [39], [42], we iterativelyupdate the fixed points w̃kn and t̃kn in the m-th iteration as

w̃(m)kn

= w(m−1)kn

, t̃(m)kn

= t(m−1)kn

. (17)

For handling the bilinear functions on the left-hand sideof (11b) and (11c), we adopt the SCA approach based onarithmetic-geometric mean (AGM) inequality such that theoriginal non-convex feasible set is sequentially upper boundedby a convex set. To this end, the non-convex bilinear functionsin (11b) and (11c) are replaced by their corresponding convexupper bounds which are given by

2akn Tr(GknknQkn) ≤ (aknckn)2 +(

Tr(GknknQkn)ckn

)2

,

(18)

2akn Tr(HknknQkn) ≤ (akndkn)2 +(

Tr(HknknQkn)dkn

)2

,

(19)

where ckn and dkn , ∀n, ∀k, are fixed feasible points. To tightenthe upper bounds, we iteratively update the fixed feasiblepoints ckn and dkn . The update equations in the m-th iterationare given by

c(m)kn

=

√√√√Tr(GknknQ(m−1)

kn)

a(m−1)kn

,

d(m)kn

=

√√√√Tr(HknknQ(m−1)

kn)

a(m−1)kn

, (20)

where the derivations are given in Appendix A. Then, the newconstraints in the m-th iteration are given by

(aknc

(m)kn

)2

+

(Tr(GknknQkn)

c(m)kn

)2

≤ 2 Tr(GknknQkn)1 + γ

− 2γ

1 + γwkn , ∀n, k, (21)

(aknd

(m)kn

)2

+

(Tr(HknknQkn)

d(m)kn

)2

≤ 2 Tr(HknknQkn)1 + γ

− 2γ

1 + γrkn , ∀n, k, (22)

Based on the aforementioned transformations and approx-imations, constraints given in (11b), (11c), and (13b) can beapproximated by some convex constraints. Now, the final diffi-culty to proceed arises from the rank-one constraint in (11g),which is combinatorial. To address this issue, we drop the

SUN et al.: JOINT BEAMFORMING AND POWER ALLOCATION 5373

constraint to obtain a relaxed version of P2a in (23), whichis denoted by P3 and as shown at the bottom of this page.

Now, the optimization problem P3 is convex for anygiven ckn , dkn , w̃kn , and t̃kn , and thus can be solved effi-ciently by off-the-shelf solvers for solving convex programs,e.g. CVX [43].

We note that the rank constraint is dropped in P3 and theobtained solution may not satisfy the rank constraint. We nextprove that the solution Qkn obtained in P3 can always satisfythe dropped rank constraint.

Theorem 1: The optimal solution Qkn obtained in P3 isalways a rank-one matrix, despite the relaxation of the rankconstraint.

Proof: Please refer to Appendix B.

Algorithm 1 Proposed Algorithm

1: Initialize ckn , dkn , w̃kn , t̃kn , ∀n, ∀k, R = 0, ε = 1,the maximum number of iterations Lmax, and iterationindex m = 1.

2: while ε ≥ 0.001 and Lmax ≤ m do3: Update {Q(m)

kn, a

(m)kn

, ρ(m)kn

, t(m)kn

} with fixed c(m)kn

, d(m)kn

,

w̃(m)kn

, t̃(m)kn

, by solving (23);4: Update the sum-rate of Group 1 users, R(m) by (23a);5: Update w̃

(m+1)kn

, t̃(m+1)kn

, c(m+1)kn

, and d(m+1)kn

basedon (17) and (20), respectively;

6: Update ε =∣∣R(m) − R(m−1)

∣∣ /R(m−1);

7: Update m = m + 1;8: end while9: Output

{Q(m)

kn, a

(m)kn

}

Then, we employ an iterative algorithm to tighten theobtained upper bounds, i.e., (21), (22), (16), as summarized inAlgorithm 1. In each iteration, the proposed iterative schemegenerates a sequence of feasible solutions {Qkn , akn} to theconvex optimization problem P3 successively.

C. Algorithm Convergence Analysis

In the above sections, we tackle the optimization problemP1 via transforming it into P2 and then approximate it by P3.We now discuss the connections among these optimizationproblems in the following lemma.

Lemma 1: Algorithm 1 converges to a stationary pointsatisfying the Karush-Kuhn-Tucker (KKT) conditions of P1.

Proof: Please refer to Appendix C.In other words, Algorithm 1 is able to achieve a sub-

optimal solution of P1 with polynomial-time computationalcomplexity.

V. NUMERICAL RESULTS

In this section, we numerically examine the performance ofour designed transmission scheme. In the simulation, we con-sider both small scale fading (i.e., Rayleigh fading) and pathloss in the channels. We model the channels as gnkn =g−α

nkng̃nkn and hnkn = h−α

nknh̃nkn , where gnkn and hnkn

represent the distances from BS n to u1kn

and u2kn

, respectively,g̃nkn and h̃nkn represent the Rayleigh fading coefficients fromBS n to u1

knand u2

kn, respectively, and α is the path loss

exponent. Here, the entries in g̃nkn and h̃nkn are modeled asCSCG random variables with zero mean and unit variance.We set the distance between every two neighboring BSs as1000 m. We assume that the locations of users in each cellare randomly and uniformly distributed in disks with radiusRc = 500 m centered at the location of the BS. The iterationerror tolerance, i.e., ε, in Algorithm 1 is 0.001. Without lossof generality, we also assume that the noise powers at all usersare the same with σ2

kn= ς2

kn= σ2 and the maximum transmit

powers at all BSs are identical with Pn = P, ∀n. For thesake of presentation, we define the average transmit SNR asρ = P/σ2 and denote the proposed joint beamforming andpower allocation design scheme as “NOMA-CoMP.” Besides,we set the sum-rate of users in Group 1 to zero if the

P3 :R � maximize{akn ,Qkn ,ρkn ,tkn}

∀n,k

N∑

n=1

K∑

k=1

log(1 + ρkn) (23a)

s. t. f1kn

�[

akn tkn

tkn Tr (GknknQkn)

] 0, ∀n, k, (23b)

f2kn

� 2t̃kn

w̃kn

tkn − t̃2kn

w̃2kn

wkn − ρkn ≥ 0, ∀n, k, (23c)

f3kn

� 21 + γ

(Tr (GknknQkn) − γwkn) −(

Tr (GknknQkn)ckn

)2

− (cknakn)2 ≥ 0, ∀n, k, (23d)

f4kn

� 21 + γ

(Tr (HknknQkn) − γrkn) −(

Tr (HknknQkn)dkn

)2

− (dknakn)2 ≥ 0, ∀n, k, (23e)

f5kn

� Qkn 0, ∀n, k, (23f)

f6n � Pn −

K∑

k=1

Tr (Qkn) ≥ 0, ∀n, (23g)

f7kn

� ρkn ≥ 0, f8kn

� 1 − akn ≥ 0, ∀n, k. (23h)

5374 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 17, NO. 8, AUGUST 2018

Fig. 2. Sum-rate of the users in Group 1 versus iteration index for differentnumbers of transmit antennas M and different numbers of users K withρ = 30 dB, γ = 0.2, α = 4, and N = 2.

Fig. 3. Sum-rate of the users in Group 1 versus iteration index for differenttarget rates R0 = log2 (1 + γ) and transmit SNR values ρ, with N = 2,M = 4, K = 3, and α = 4.

optimization problem in (23) is infeasible to account thepenalty of failure.

A. Convergence

We evaluate the convergence rate of the proposed NOMA-CoMP. In Fig. 2, we plot the sum-rate of users in Group 1 ver-sus the iteration index for different values of M and K .We observe that the convergence rate is faster when M = 4,K = 3, compared to M = 4, K = 2 and M = 6, K = 4.This is due to the fact that the solution space is spanned by2(M − K + 1)2 − (M − K + 1) independent variables ineach of the NK beamforming matrices to be optimized. Thus,the convergence rate increases when NK(2(M − K + 1)2 −(M − K + 1)) decreases.

In Fig. 3, we plot the sum-rate of users in Group 1 versusthe iteration index for different values of γ (or equivalently,different target rates given by R0 = log2 (1 + γ)) and theaverage transmit powers ρ. In this figure, we observe that the

TABLE I

Rλ WITH K = 2

TABLE II

Rλ WITH K = 3

sum-rate of the users in Group 1 decreases when γ increases.This is due to the fact that when the required target ratebecomes more stringent, more resources need to be allocatedto the users in Group 2. Hence, the BSs are less capable inmaximizing the sum-rate of the users in Group 1. Second,we observe that the slope of the sum-rate curves is similar toeach other, which indicates that the target data rate has almostnegligible impact on the convergence rate. Third, after at most15 iterations on average, our proposed algorithm converges toa stationary point. Thus, in the sequel, the maximum iterationnumber is set to be Lmax = 20 to illustrate the performanceof NOMA-CoMP in different scenarios.

B. Rank Results

In this section, we examine the rank of the optimal solu-tion, which allows us to verify the proof in Appendix B.In Tables I and II, we list Rλ for different numbers of transmitantennas, M , with K = 2 and K = 3, respectively. Here,Rλ is defined as the ratio between the largest eigenvalue andthe second largest eigenvalue of the optimal beamformingmatrix. From Tables I and II, we observe that the mini-mum value of Rλ is always sufficiently large, regardless ofM and K . This indicates that the solutions found by theproposed scheme are always rank-one.

C. Sum-Rate of Users

In Fig. 4, we plot the sum-rate of users in Group 1 versusthe average transmit SNR ρ for different schemes. Specifically,we compare the performance of the proposed NOMA-CoMPscheme with the performances of three schemes. The threeschemes are:

• The NOMA scheme with CoMP and fixed power alloca-tion is denoted by “NOMA-CoMP Fixed Power.” In thisscheme, the BSs design the beamforming vectors coordi-nately, but adopt fixed power allocation with akn = 0.4and bkn = 0.6.

• The uncoordinated NOMA scheme is denoted by“NOMA w/o CoMP.” In this scheme, each BS jointlydesigns the beamforming vectors and power allocation,but does not cooperate with others.

SUN et al.: JOINT BEAMFORMING AND POWER ALLOCATION 5375

Fig. 4. Sum-rate of the users in Group 1 versus ρ with N = 2, M = 6,K = 4, γ = 0.2, and α = 3.

• The OMA with CoMP scheme is denoted by“OMA-CoMP.” In this scheme, the BSs adopt OMA tocoordinately design the beamforming vectors followinga similar procedure to NOMA-CoMP.

We list the key observations from Fig. 4, as follows:1) Our proposed NOMA-CoMP always outperforms

NOMA-CoMP Fixed Power, which shows the poten-tial performance gain brought by the joint design ofbeamforming vectors and power allocation. In fact,the proposed scheme adaptively adjusts the transmitpower which enables a large disparity in the receivedsignal strengths at the paired users which is beneficialto NOMA.

2) Our proposed NOMA-CoMP outperforms NOMA w/oCoMP in the medium to the high SNR regime.Notably, the performance gap between NOMA-CoMPand NOMA w/o CoMP increases with ρ. This isbecause the inter-cell interference increases with ρ andNOMA w/o CoMP cannot suppress the interferencewhile NOMA-CoMP can efficiently do so.

3) Our proposed NOMA-CoMP slightly underperformsNOMA w/o CoMP in the low SNR regime due tothe following two reasons. First, each BS in NOMAw/o CoMP may use a higher transmit power thanin NOMA-CoMP, since each BS fully consumes themaximum transmit power in NOMA w/o CoMP, but notnecessary for the case in NOMA-CoMP. The proof offull power transmission in NOMA w/o CoMP is givenin Appendix D. Second, each BS may allocate a highertransmit power to the users in Group 1 in NOMA w/oCoMP than in NOMA-CoMP. This is caused by thefact that the BSs in NOMA-CoMP take into accountthe inter-cell interference in resource allocation whilethe BSs in NOMA w/o CoMP do not. This implies thatthe BSs in NOMA-CoMP use a higher transmit powerthan in NOMA w/o CoMP to guarantee the target ratesachieved by the users in Group 2.

4) Our proposed NOMA-CoMP outperforms OMA-CoMPin the medium to high SNR regime, while slightly

Fig. 5. Sum-rate of the users in Group 1 versus ρ with N = 2, M = 1,K = 1, γ = 0.2, and α = 3.

Fig. 6. Sum-rate of the users in Group 1 versus N with M = 4, K = 3,ρ = 50 dB, γ = 0.2, and α = 3.

underperforms OMA-CoMP in the low SNR regime.In fact, NOMA-CoMP does not have sufficient powerin the low SNR regime to ensure a significant channeldisparity between the paired users which limits thepotential gain brought by NOMA.

Fig. 5 depicts the sum-rate of users in Group 1 versusthe average transmit SNR ρ. We compare the performanceof the proposed suboptimal solution with the optimal one.To find the globally optimal solution, we adopt the brute-forcesearch algorithm. Notably, the cost of the brute-force algorithmincreases exponentially with the size of the problem. To illus-trate the performance gain between the proposed scheme andthe optimal scheme, we consider a simple two-cell scenario,where the BS is equipped with a single antenna and servestwo single-antenna NOMA users in each cell. We observethat our proposed algorithm can achieve almost the sameperformance as the optimal one, but only requires a muchlower computational complexity than the brute-force searchmethod, which is verified in Section V-A.

In Fig. 6, we plot the sum-rate of users in Group 1 versus thenumber of cells N . Specifically, we compare the performance

5376 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 17, NO. 8, AUGUST 2018

Fig. 7. Sum-rate of the users in Group 1 versus the path loss exponent αfor different ρ with N = 2, M = 6, K = 4, and γ = 0.2.

of the proposed NOMA-CoMP with that of NOMA-CoMPFixed Power and OMA-CoMP. We first observe that ourproposed NOMA-CoMP always outperforms NOMA-CoMPFixed Power. This is due to the fact that the efficiency of beam-forming depends highly on the power allocation in the con-sidered NOMA network. Indeed, our proposed NOMA-CoMPefficiently addresses this issue by jointly designing the powerallocation coefficients and beamforming vectors. We alsoobserve that there is a diminishing return in terms of per-formance gains when N is large. This is due to the fact thatwhen N increases, the inter-cell interference becomes moresevere. In particular, when N is sufficiently large, some of thedegrees of freedom offered by multiple antennas are exploitedto harness the interference received at users in Group 2,which reduces the capability of the BSs in focusing theenergy of information signals to the desired users. In addition,since the power allocation in the NOMA-CoMP Fixed Powerscheme is fixed, it has less flexibility than the NOMA-CoMPscheme in mitigating the interference via adaptive power allo-cation. This accounts for the increasing performance gap withN between NOMA-CoMP and NOMA-CoMP Fixed Power.Furthermore, we observe that the proposed NOMA-CoMP out-performs OMA-CoMP, which shows the advantage of NOMAover OMA for improving the spectral efficiency. Moreover,the sum-rate of OMA-CoMP decreases to zero when N islarge. This is because the BSs cannot guarantee the target ratesof users in Group 2 due to the exceedingly large interferencewhen there are more cells in the system.

In Fig. 7, we plot the sum-rate of users in Group 1 versuspath loss exponent α for different ρ, allowing us to examinethe impact of large scale fading on the network performance.We compare the performance of the proposed NOMA-CoMPscheme with that of NOMA w/o CoMP scheme. It is veryinteresting to find that the sum-rate of users in Group 1 firstincreases with the path loss exponent α and then decreaseswhen α is beyond certain value. This is attribute to the factthat when α increases, both the desired signal and interferencereceived by users decreases. However, the attenuation ofinterference occurs to be larger than that of desired signal

Fig. 8. Sum-rate of the users in Group 1 versus K for different numbers ofantennas at the BS with N = 2, γ = 0.2, α = 3, ρ = 50 dB.

because interference suffers from a more serious path loss.As α further increases, the operating regime of the systemis shifting from interference limited regime to noise limitedregime. In the noise limited regime, the desired signal strengthfurther decreases which decreases the sum-rate of users.Furthermore, we observe that the performance gain of theproposed NOMA-CoMP over NOMA w/o COMP increaseswhen α decreases. In this situation, the inter-cell interferencemanagement becomes more critical when the path loss expo-nent decreases and harnessing the inter-cell interference viaCoMP becomes necessary for improving system performance.

In Fig. 8, we plot the sum-rate of users in Group 1 versusthe number of clusters in each cell, i.e., K , for differentnumbers of antennas equipped at each BS M . We comparethe performance of the proposed NOMA-CoMP scheme withthat of NOMA-CoMP Fixed Power scheme. We observe thatthe sum-rate of users in Group 1 first increases with K andthen decreases when K is sufficiently large. Besides, the per-formance of the system always increases with M since thedegrees of freedom available for resource allocation increasewhen M increases or K decreases. In fact, there is a non-trivialtrade-off between the number of users in the network andthe system performance. When K is small, the BSs caneffectively exploit the multiuser diversity [7] in Group 1 forimproving the system performance. However, as the value ofK keeps increasing, the constraint on the minimum data raterequirements for the users in Group 2 becomes more stringent,cf. (10c). As a result, the BSs are forced to exploit the spatialdegrees of freedom to ensure the QoS to a larger number ofusers in Group 2, despite their potentially poor channel condi-tions. Eventually, the performance loss due to less flexibilityin resource allocation outweighs the performance gain broughtby multiuser diversity leading to the sum-rate degradation.Furthermore, the proposed NOMA-CoMP always outperformsNOMA-CoMP Fixed Power scheme due to the similar reasonas explained in Fig. 6.

VI. CONCLUSION

In this paper, we investigated the joint design of beam-forming and power allocation in the downlink of multi-cell

SUN et al.: JOINT BEAMFORMING AND POWER ALLOCATION 5377

multiuser MIMO-NOMA network. The CoMP technique wasapplied to the network to harness the interference and coor-dinate the information beams transmission among the cooper-ative BSs. In this network, the users were grouped based ontheir QoS requirements. We formulated the resource allocationdesign as an optimization problem to maximize the sum-rateof Group 1 users requiring the best-effort services. The pro-posed problem formulation took into account the minimumdata rate constraints imposed at the users in Group 2 withstrict QoS requirements. To solve the non-convex optimizationproblem, an iterative algorithm based on SCA was proposedto obtain a suboptimal solution. In each iteration, a rank-constrained optimization problem is solved optimally via SDR.Our results demonstrated that the proposed scheme can achievea superior sum-rate over the existing schemes and convergesfast to a stationary point. The beamforming design is studiedbased on the assumption of the perfect global CSI in thiswork. One promising future direction is to extend the currentwork to consider the impact of imperfect CSI via the robustoptimization approach, e.g., [30], [44], [45]. Correspondingly,the performance achieved by this work can serve as an upperbound for the NOMA scheme with imperfect CSI. Anotherone is to investigate user pairing to further improve the spectralefficiency.

APPENDIX AOPTIMAL ckn AND dkn

First, we define a function F of ckn as

F (ckn) = akn Tr(GknknQkn)

−12

(

(aknckn)2 +(

Tr(GknknQkn)ckn

)2)

. (24)

According to the properties of the AGM inequality, the opti-mal value of ckn , defined as c∗kn

, leads to F (c∗kn

)= 0.

From (24) we find that F (ckn) is a concave functionwith respect to ckn , since ∂2F (ckn) /∂c2

kn< 0. There-

fore, we obtain c∗knthat maximizes F (ckn) by solving

∂F (ckn) /∂ckn = 0, which leads to

c∗kn=

√Tr(GknknQkn)

akn

. (25)

The optimal value of dkn can be obtained by following thesimilar procedure to the derivation of c∗kn

.

APPENDIX BPROOF OF LEMMA 1

In this appendix, we prove that the optimal Qkn of P3 isalways a rank-one matrix. Specifically, the variable Qkn inthe imposed constraints (23c), (23d), and (23e) is in quadraticform. In addition, we focus on CoMP in a multi-cell scenario.These make this work challenging and fundamentally differentfrom the SDR problems in the literature, e.g. [34]. Thus,the rank-one property cannot be proved by using the existingresults in a straightforward manner. In the following, we provethe rank-one solutions specifically in the proposed problem P3.

We note that the power constraint given in (23g) maynot always hold with equality in P3, which is proved inAppendix D, where we define P̃n as the total transmit power atBS n with the optimal beamforming matrices, Q∗

kn, given by

P̃n �K∑

k=1

Tr(Q∗kn

). (26)

Accordingly, we have 0 < P̃n ≤ Pn. Replacing Pn with P̃n

in P3, we obtain a new optimization problem, defined as P4,which is given by

P4 : maximize{akn ,Qkn ,ρkn ,tkn}

∀n,k

N∑

n=1

K∑

k=1

log2(1 + ρkn) (27)

s. t. (23b), (23c), (23d), (23e), (23f), (23h),

f̃6n � P̃n −

K∑

k=1

Tr(Qkn) ≥ 0, ∀n. (28)

In fact, the solutions to P3 and P4 are the same. Thus,it is equivalent to prove the rank-one property of the optimalsolution Qkn obtained in P4. To this end, we focus on thedual problem of P4 and its corresponding KKT conditions.

We first denote the optimal dual variables associatedwith (23b) and (23f) by {λikn

0}, where i = 1, 5. We thendenote the optimal dual variables associated with (23c), (23d),(23e), and (23h) by {λikn

≥ 0}, where i = 2, 3, 4, 7,and 8. We further denote the optimal dual variables associatedwith (28) by λ6n . In addition, we express the entries in λ1kn

as

λ1kn=

[λ

(1)1kn

λ(2)1kn

λ(3)1kn

λ(4)1kn

]

. (29)

Then, the Lagrangian function for P4 is given in (30) as shownat the top of the next page.

According to the KKT conditions [46], [47] of P4, Q∗kn

and the optimal dual variables need to satisfy the followingequations:

λiknf i

kn= 0, i = 1, 5, ∀n, k, (31a)

λiknf i

kn= 0, i = 2, 3, 4, 7, 8, ∀n, k, (31b)

λ6n f̃6n = 0, ∀n, (31c)

∇Q∗knL = 0, ∀n, k. (31d)

Since the columns of Q∗kn

lie in the null space of λ5kn,

cf. (31a), we obtain Rank(Q∗

kn

)= M−K+1−Rank

(λ5kn

).

Thus, it is equivalent to explore the rank of λ5kn. By (31d),

we obtain

λ5kn= λ6nI − Xkn , (32)

where I is an (M − K + 1) × (M − K + 1) identity matrixand Xkn is in (33) as shown at the top of the next page.

We now define xmaxkn

as the largest eigenvalue of Xkn . Sinceλ∗

5knis positive semidefinite, λ6n and xmax

knneed to satisfy

λ6n ≥ xmaxkn

. (34)

Based on (26), the available power at each BS is fullyexhausted such that the corresponding optimal dual variableλ6n always satisfies λ6n > 0.

5378 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 17, NO. 8, AUGUST 2018

L =N∑

n=1

K∑

k=1

log (1 + ρkn) +N∑

n=1

K∑

k=1

Tr(

λ1kn

[akn tkn

tkn Tr (GknknQkn)

])

+N∑

n=1

K∑

k=1

λ2kn

(2t̃kn

w̃kn

tkn − t̃2kn

w̃2kn

wkn − ρkn

)

+N∑

n=1

K∑

k=1

λ3kn

(

2(

Tr (GknknQkn)1 + γ

− γ

1 + γwkn

)− (cknakn)2 −

(Tr (GknknQkn)

ckn

)2)

+N∑

n=1

K∑

k=1

λ4kn

(

2(

Tr (HknknQkn)1 + γ

− γ

1 + γrkn

)− (dknakn)2 −

(Tr (HknknQkn)

dkn

)2)

+N∑

n=1

K∑

k=1

Tr(λ5kn

Qkn

)+

N∑

n=1

λ6n

(

P̃n −K∑

k=1

Tr (Qkn)

)

+N∑

n=1

K∑

k=1

λ7knρkn +

N∑

n=1

K∑

k=1

λ8kn(1 − akn) . (30)

Xkn = λ(4)1kn

Gknkn +∑

i∈Nn

K∑

j=1

λ2jit̃2ji

w̃2ji

Gknji +2

1 + γ

(λ3kn

Gknkn + λ4knHknkn

)

− 2λ3kn

Tr(GknknQ∗

kn

)

c2kn

Gknkn − 2λ4kn

Tr(HknknQ∗

kn

)

d2kn

Hknkn − 2γ

1 + γ

∑

i∈Nn

K∑

j=1

λ3jiGknji

− 2γλ4ji

1 + γ

⎛

⎝K∑

j=1,j �=k

Hknjn +∑

i∈Nn

K∑

j=1

Hknji

⎞

⎠ . (33)

We next prove λ6n = xmaxkn

> 0 by contradiction.If xmax

kn< λ6n , we find that the smallest eigenvalue of

λ5knis λ6n − xmax

kn> 0, based on (32). Hence, λ5kn

isa full rank PSD matrix and the null space of λ5kn

is zero,i.e., Rank

(Q∗

kn

)= 0, which indicates that Q∗

knis a zero

matrix. However, this contradicts to∑K

k=1 Tr(Q∗

kn

)= P̃n for

P̃n > 0 and λ6n > 0. As such, xmaxkn

cannot be less than λ6n .We further note that if xmax

kn> λ6n , (34) is violated. Therefore,

the optimal solution must satisfy λ6n = xmaxkn

�= 0.Since λ6n = xmax

kn, the other eigenvalues of Xkn are less

than λ6n . As such, λ5knhas only one zero eigenvalue, which

is λ6n = xmaxkn

. Thus, we find that the rank of the optimalvalue of λ5kn

is M − K . From (31a), i.e., λ5knQ∗

kn= 0,

we demonstrate that the rank of Q∗kn

satisfies

Rank(Q∗

kn

)= Rank

(Null

(λ5kn

))

= (M − K + 1) − Rank(λ5kn

)= 1, (35)

which completes the proof.

APPENDIX CPROOF OF LEMMA 1

Recall that the optimization problems P2 and P2a areequivalent. To prove the convergence of Algorithm 1 to thestationary points of P2, it is equivalent to prove that of P2a.

Based on the constraint (13b) in P2a, we define a function

G(ρkn , wkn) = ρkn − akn Tr(GknknQkn)wkn

. (36)

It is noted that G(ρkn , wkn) ≤ 0 is always guaranteed.After a series of transformations and approximations basedon (14), (15), and (16), G(ρkn , wkn) is bounded above by

G̃(ρkn , wkn , w̃kn , tkn , t̃kn)

= ρkn −(

2t̃kn

w̃kn

tkn − t̃2kn

w̃2kn

wkn

)

≥ G(ρkn , wkn). (37)

In each iteration of the proposed Algorithm 1, G(ρkn , wkn)is replaced by G̃(ρkn , wkn , w̃kn , tkn , t̃kn), which is a differ-entiable convex function. Following the results from [48],the proposed algorithm in this work based on SCA convergesto a KKT point of problem P2a if the following conditionsare all satisfied:

1) G(ρkn , wkn) ≤ G̃(ρkn , wkn , w̃kn , tkn , t̃kn),2) G(ρ(m)

kn, w

(m)kn

) = G̃(ρ(m)kn

, w(m)kn

, w̃(m+1)kn

, t(m)kn

, t̃(m+1)kn

),

3)∂G(ρ

(m)kn

,w(m)kn

)

∂ρkn=

∂G̃(ρ(m)kn

,w(m)kn

,w̃(m+1)kn

,t(m)kn

,t̃(m+1)kn

)

∂ρkn,

4)∂G(ρ

(m)kn

,w(m)kn

)

∂wkn=

∂G̃(ρ(m)kn

,w(m)kn

,w̃(m+1)kn

,t(m)kn

,t̃(m+1)kn

)

∂wkn.

Since ∀ρkn ∈ S, we have that akn Tr(GknknQkn )ρkn

≥ w2kn

ρkn≥

2t̃kn

w̃kntkn − t̃2kn

w̃2kn

wkn based on (14) and (16), G(ρkn , wkn) and

G̃(ρkn , wkn , w̃kn , tkn , t̃kn) satisfy the first condition.

SUN et al.: JOINT BEAMFORMING AND POWER ALLOCATION 5379

Furthermore, based on the updating of the fixed points w̃kn ,t̃kn , i.e., (17), the second condition is satisfied.

Finally, we verify the third condition by deriving thefirst derivatives of G(ρkn , wkn) and G̃(ρkn , wkn , w̃kn , tkn , t̃kn)with respect to ρkn and wkn , respectively, which are given by

∂G(ρ(m)kn

)∂ρkn

=∂G̃(ρ(m)

kn, w̃

(m+1)kn

, t(m)kn

, t̃(m+1)kn

)∂ρkn

= 1,

(38)

∂G(ρ(m)kn

, w(m)kn

)∂wkn

=a(m)kn

Tr(GknknQ(m)kn

)

w(m)2kn

, (39)

∂G̃(ρ(m)kn

, w(m)kn

, w̃(m+1)kn

, t(m)kn

, t̃(m+1)kn

)∂wkn

=t̃(m+1)2kn

w̃(m+1)2kn

. (40)

Based on (14) and (17), i.e., a(m)kn

Tr(GknknQ(m)kn

) = t(m)2kn

=t̃(m+1)2kn

and w(m)kn

= w̃(m+1)kn

, the third condition is verifiedto be satisfied. Thus, the proposed algorithm converges to aKKT point of P2a.

On the other hand, as discussed in Section IV-A, P2 isequivalent to P1 as long as the rank-one property is guaran-teed. Based on Lemma 1, we can conclude that the proposedalgorithm achieves a stationary KKT point of P1.

APPENDIX DEQUAL POWER CONSTRAINT FOR

NOMA WITHOUT COMP

We now prove that the power constraint holds with equalityfor the NOMA system without CoMP. Since each BS designsits beamforming matrix and power allocation independently,the optimization problem for each BS using the NOMAwithout CoMP scheme is a special case of P3 with N = 1.Then, we prove by contradiction that the equality in thepower constraint in the single-cell case is active at the optimalsolution. Suppose that for BS n, the optimal beamformingmatrix, Q∗

kn, does not satisfy with equality in the power

constraint given by (11d), i.e.,∑K

k=1 Tr(Q∗

kn

)< Pn. We then

multiply a scaler � = Pn/∑K

k=1 Tr(Q∗

kn

)> 1 to the

optimal Q∗kn

. By doing so, we obtain a new solution, denotedby Q̄kn , and find that Q̄kn still satisfies the rate and powerconstraints. However, the value of the new objective functionwith Q̄kn is higher than that with Q∗

kn, which contradicts to

the claim of optimality. Therefore, the equality in the powerconstraint must hold for the NOMA without CoMP scheme.

We note that when N > 1, the power constraint forthe NOMA-CoMP scheme does not necessary active at theoptimal solution. If the transmit power increases at one BS,the performance of this cell increases while the performance ofother cells decreases. This is due to the fact that the inter-cellinterference increases with the transmit power at one BS.However, the sum-rate in the network may not increase. There-fore, the power constraint for the NOMA-CoMP scheme maynot satisfy with equality at the optimal solution. This explainswhy each BS fully consumes the maximum transmit power

for the NOMA without CoMP scheme, but not necessarily forthe NOMA-CoMP scheme, as indicated in Fig. 4.

REFERENCES

[1] X. Sun et al., “Joint beamforming and power allocation design in down-link non-orthogonal multiple access systems,” in Proc. IEEE GlobecomWorkshop, Washington, DC, USA, Dec. 2016, pp. 1–6.

[2] V. W. Wong, R. Schober, D. W. K. Ng, and L.-C. Wang, Key Technolo-gies for 5G Wireless Systems. Cambridge, U.K.: Cambridge Univ. Press,2017.

[3] Z. Ding et al., “Application of non-orthogonal multiple access in LTEand 5G networks,” IEEE Commun. Mag., vol. 55, no. 2, pp. 185–191,Feb. 2017.

[4] S. M. R. Islam, M. Zeng, and O. A. Dobre, “NOMA in 5G systems:Exciting possibilities for enhancing spectral efficiency,” IEEE 5G TechFocus, vol. 1, no. 2, Jun. 2017. Accessed: Jun. 7, 2018. [Online].Available: https://5g.ieee.org/tech-focus/june-2017/noma-in-5g-systems

[5] Z. Yang, Z. Ding, P. Fan, and N. Al-Dhahir, “A general power allo-cation scheme to guarantee quality of service in downlink and uplinkNOMA systems,” IEEE Trans. Wireless Commun., vol. 15, no. 11,pp. 7244–7257, Nov. 2016.

[6] M. Zeng, A. Yadav, O. A. Dobre, G. I. Tsiropoulos, and H. V. Poor,“Capacity comparison between MIMO-NOMA and MIMO-OMA withmultiple users in a cluster,” IEEE J. Sel. Areas Commun., vol. 35, no. 10,pp. 2413–2424, Oct. 2017.

[7] D. Tse and P. Viswanath, Fundamentals of Wireless Communication.Cambridge, U.K.: Cambridge Univ. Press, 2005.

[8] Y. Saito, Y. Kishiyama, A. Benjebbour, T. Nakamura, A. Li, andK. Higuchi, “Non-orthogonal multiple access (NOMA) for cellularfuture radio access,” in Proc. IEEE 77th VTC Spring, Dresden, Germany,Jun. 2013, pp. 1–5.

[9] A. Benjebbour, K. Saito, A. Li, Y. Kishiyama, and T. Nakamura, “Non-orthogonal multiple access (NOMA): Concept, performance evaluationand experimental trials,” in Proc. WINCOM. Marrakech, Morocco,Oct. 2015, pp. 1–6.

[10] A. Benjebbour, A. Li, K. Saito, Y. Saito, Y. Kishiyama, andT. Nakamura, “NOMA: From concept to standardization,” in Proc. IEEEConf. Standards Commun. Netw. (CSCN), Oct. 2015, pp. 18–23.

[11] Z. Ding, Z. Yang, P. Fan, and H. V. Poor, “On the performance ofnon-orthogonal multiple access in 5G systems with randomly deployedusers,” IEEE Signal Process. Lett., vol. 21, no. 12, pp. 1501–1505,Dec. 2014.

[12] M. Al-Imari, P. Xiao, M. A. Imran, and R. Tafazolli, “Uplink non-orthogonal multiple access for 5G wireless networks,” in Proc. 11th Int.Symp. Wireless Commun. Syst., Aug. 2014, pp. 781–785.

[13] F. Fang, H. Zhang, J. Cheng, and V. C. M. Leung, “Energy-efficientresource allocation for downlink non-orthogonal multiple access net-work,” IEEE Trans. Commun., vol. 64, no. 9, pp. 3722–3732, Sep. 2016.

[14] Q. Sun, S. Han, C.-L. I, and Z. Pan, “On the ergodic capacity ofMIMO NOMA systems,” IEEE Wireless Commun. Lett., vol. 4, no. 4,pp. 405–408, Aug. 2015.

[15] Z. Ding, F. Adachi, and H. V. Poor, “The application of MIMO to non-orthogonal multiple access,” IEEE Trans. Wireless Commun., vol. 15,no. 1, pp. 537–552, Jan. 2016.

[16] Z. Ding, R. Schober, and H. V. Poor, “A general MIMO frameworkfor NOMA downlink and uplink transmission based on signal align-ment,” IEEE Trans. Wireless Commun., vol. 15, no. 6, pp. 4438–4454,Jun. 2016.

[17] M. F. Hanif, Z. Ding, T. Ratnarajah, and G. K. Karagiannidis,“A minorization-maximization method for optimizing sum rate in thedownlink of non-orthogonal multiple access systems,” IEEE Trans.Signal Process., vol. 64, no. 1, pp. 76–88, Jan. 2016.

[18] J. Choi, “Minimum power multicast beamforming with superpositioncoding for multiresolution broadcast and application to NOMA systems,”IEEE Trans. Commun., vol. 63, no. 3, pp. 791–800, Mar. 2015.

[19] Z. Ding, Z. Zhao, M. Peng, and H. V. Poor, “On the spectral efficiencyand security enhancements of NOMA assisted multicast-unicast stream-ing,” IEEE Trans. Commun., vol. 65, no. 7, pp. 3151–3163, Jul. 2017.

[20] X. Sun, D. Duran-Herrmann, Z. Zhong, and Y. Yang, “Non-orthogonalmultiple access with weighted sum-rate optimization for downlinkbroadcast channel,” in Proc. IEEE MILCOM, Tampa, FL, USA,Oct. 2015, pp. 1176–1181.

[21] S. Ali, E. Hossain, and D. I. Kim, “Non-orthogonal multipleaccess (NOMA) for downlink multiuser MIMO systems: User clustering,beamforming, and power allocation,” IEEE Access, vol. 5, pp. 565–577,Mar. 2017.

5380 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 17, NO. 8, AUGUST 2018

[22] Y. Xu et al., “Joint beamforming and power-splitting control in downlinkcooperative SWIPT NOMA systems,” IEEE Trans. Signal Process.,vol. 65, no. 18, pp. 4874–4886, Sep. 2017.

[23] Z. Ding and H. V. Poor, “Design of massive-MIMO-NOMA with limitedfeedback,” IEEE Signal Process. Lett., vol. 23, no. 5, pp. 629–633,May 2016.

[24] Y. Liu, Z. Qin, M. Elkashlan, Y. Gao, and A. Nallanathan, “Non-orthogonal multiple access in massive MIMO aided heterogeneousnetworks,” in Proc. IEEE GLOBECOM, Washington, DC, USA,Dec. 2016, pp. 1–6.

[25] Z. Ding, L. Dai, and H. V. Poor, “MIMO-NOMA design for smallpacket transmission in the Internet of Things,” IEEE Access, vol. 4,pp. 1393–1405, 2016.

[26] H. Shuangfeng, C.-L. I, Z. Xu, and Q. Sun, “Energy efficiency andspectrum efficiency co-design: From NOMA to network NOMA,” IEEEMMTC E-Lett., vol. 9, no. 5, pp. 21–24, Sep. 2014.

[27] W. Shin, M. Vaezi, B. Lee, D. J. Love, J. Lee, and H. V. Poor, “Coordi-nated beamforming for multi-cell MIMO-NOMA,” IEEE Commun. Lett.,vol. 21, no. 1, pp. 84–87, Jan. 2017.

[28] D. Gesbert, S. Hanly, H. Huang, S. S. Shitz, O. Simeone, and W.Yu, “Multi-cell MIMO cooperative networks: A new look at interfer-ence,” IEEE J. Sel. Areas Commun., vol. 28, no. 9, pp. 1380–1408,Dec. 2010.

[29] R. Irmer et al., “Coordinated multipoint: Concepts, performance, andfield trial results,” IEEE Commun. Mag., vol. 49, no. 2, pp. 102–111,Feb. 2011.

[30] C. Shen et al. “Distributed robust multicell coordinated beamformingwith imperfect CSI: An ADMM approach,” IEEE Trans. Signal Process.,vol. 60, no. 6, pp. 2988–3003, Jun. 2012.

[31] I. Poole. 4G LTE CoMP, Coordinated Multipoint Tutorial.Accessed: Jun. 7, 2018. [Online]. Available: http://www.radio-electronics.com/info/cellulartelecomms/lte-long-term-evolution/4g-lte-advanced-comp-coordinated-multipoint.php

[32] Z.-Q. Luo, W.-K. Ma, A. M.-C. So, Y. Ye, and S. Zhang, “Semidefiniterelaxation of quadratic optimization problems,” IEEE Signal Process.Mag., vol. 27, no. 3, pp. 20–34, May 2010.

[33] N. D. Sidiropoulos, T. N. Davidson, and Z.-Q. Luo, “Transmit beam-forming for physical-layer multicasting,” IEEE Trans. Signal Process.,vol. 54, no. 6, pp. 2239–2251, Jun. 2006.

[34] E. Boshkovska, D. W. K. Ng, N. Zlatanov, A. Koelpin, and R. Schober,“Robust resource allocation for MIMO wireless powered communicationnetworks based on a non-linear EH model,” IEEE Trans. Commun.,vol. 65, no. 5, pp. 1984–1999, May 2017.

[35] N. Palleit and T. Weber, “Obtaining transmitter side channel stateinformation in MIMO FDD systems,” in Proc. IEEE PIMRC, Tokyo,Japan, Sep. 2009, pp. 2439–2443.

[36] J. Jose, A. Ashikhmin, P. Whiting, and S. Vishwanath, “Channelestimation and linear precoding in multiuser multiple-antenna TDDsystems,” IEEE Trans. Veh. Technol., vol. 60, no. 5, pp. 2102–2116,Jun. 2011.

[37] T. A. Thomas and F. W. Vook, “Method for obtaining full channel stateinformation for RF beamforming,” in Proc. IEEE Globecom, Austin,TX, USA, Dec. 2014, pp. 3496–3500.

[38] T. Yoo and A. Goldsmith, “On the optimality of multiantenna broad-cast scheduling using zero-forcing beamforming,” IEEE J. Sel. AreasCommun, vol. 24, no. 3, pp. 528–541, Mar. 2006.

[39] C. Shen, W.-C. Li, and T.-H. Chang, “Wireless information and energytransfer in multi-antenna interference channel,” IEEE Trans. SignalProcess., vol. 62, no. 23, pp. 6249–6264, Dec. 2014.

[40] F. Zhang, The Schur Complement and Its Applications. New York, NY,USA: Springer-Verlag, 2005.

[41] A. Khabbazibasmenj, F. Roemer, S. A. Vorobyov, and M. Haardt,“Sum-rate maximization in two-way AF MIMO relaying: Polynomialtime solutions to a class of DC programming problems,” IEEE Trans.Signal Process., vol. 60, no. 10, pp. 5478–5493, Oct. 2012.

[42] S. Boyd, Sequential Convex Programming, Standard EE364b, Stan-ford Univ., 2007. Accessed: Jun. 7, 2018. [Online]. Available:https://web.stanford.edu/class/ee364b/lectures/seq_slides.pdf

[43] M. Grant and S. Boyd. (Mar. 2014). CVX: MATLAB Software forDisciplined Convex Programming, Version 2.1. Accessed: Jun. 2018.[Online]. Available: http://cvxr.com/cvx

[44] Z. Wang, D. W. K. Ng, V. W. S. Wong, and R. Schober, “Robustbeamforming design in C-RAN with sigmoidal utility and capacity-limited backhaul,” IEEE Trans. Wireless Commun., vol. 16, no. 9,pp. 5583–5598, Sep. 2017.

[45] D. W. K. Ng, E. S. Lo, and R. Schober, “Robust beamforming forsecure communication in systems with wireless information and powertransfer,” IEEE Trans. Wireless Commun., vol. 13, no. 8, pp. 4599–4615,Aug. 2014.

[46] S. Boyd and L. Vandenberghe, Convex Optimization. Cambridge, U.K.:Cambridge Univ. Press, 2004.

[47] Q. Shi, L. Liu, W. Xu, and R. Zhang, “Joint transmit beamformingand receive power splitting for MISO SWIPT systems,” IEEE Trans.Wireless Commun., vol. 13, no. 6, pp. 3269–3280, Jun. 2014.

[48] B. R. Marks and G. P. Wright, “A general inner approximation algo-rithm for nonconvex mathematical programs,” Oper. Res., vol. 26,pp. 681–683, Jul. 1978.

Xiaofang Sun (S’14) received the B.S. degree fromBeijing Jiaotong University, Beijing, China, in 2013,where she is currently pursuing the Ph.D. degreewith the State Key Laboratory of Rail Traffic Controland Safety. From 2014 to 2015, she was a visitingstudent with the University of Nebraska–Lincoln,Omaha, NE, USA. From 2016 to 2017, she was avisiting student with The Australian National Uni-versity, Canberra, Australia. Her current researchinterests focus on resource allocation of wirelesscommunication networks and UAV-enabled wirelesscommunications.

Nan Yang (S’09–M’11) received the B.S. degreein electronics from China Agricultural Universityin 2005 and the M.S. and Ph.D. degrees in electronicengineering from the Beijing Institute of Technol-ogy in 2007 and 2011, respectively. He was aPost-Doctoral Research Fellow with The Universityof New South Wales from 2012 to 2014 and aPost-Doctoral Research Fellow with the Common-wealth Scientific and Industrial Research Organi-zation from 2010 to 2012. Since 2014, he hasbeen with the Research School of Engineering, The

Australian National University, where he is currently a Future EngineeringResearch Leadership Fellow and a Senior Lecturer. His general researchinterests include massive multi-antenna systems, millimeter-wave communica-tions, ultra-reliable low-latency communications, cyber-physical security, andmolecular communications. He was a co-recipient of the Best Paper Awardfrom the IEEE GlobeCOM 2016 and the IEEE VTC 2013-Spring. He receivedthe Top Editor Award from the Transactions on Emerging TelecommunicationsTechnologies in 2017, the Exemplary Reviewer Award from the IEEE TRANS-ACTIONS ON COMMUNICATIONS in 2016 and 2015, the Top Reviewer Awardfrom the IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY in 2015,the IEEE ComSoc Asia-Pacific Outstanding Young Researcher Award and theExemplary Reviewer Award from the IEEE WIRELESS COMMUNICATIONS

LETTERS in 2014, and the Exemplary Reviewer Award from the IEEECOMMUNICATIONS LETTERS in 2013 and 2012. He is currently servingon the Editorial Board of the IEEE TRANSACTIONS ON WIRELESS COM-MUNICATIONS, the IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY,and the Transactions on Emerging Telecommunications Technologies.

Shihao Yan (S’11–M’15) received the B.S. degreein communication engineering and the M.S. degreein communication and information systems fromShandong University, Jinan, China, in 2009 and2012, respectively, and the Ph.D. degree in electricalengineering from The University of New SouthWales, Sydney, Australia, in 2015. From 2015 to2017, he was a Post-Doctoral Research Fellow withthe Research School of Engineering, The AustraliaNational University, Canberra, Australia. He is cur-rently a University Research Fellow with the School

of Engineering, Macquarie University, Sydney. His current research interestsare in the areas of wireless communications and statistical signal process-ing, including physical-layer security, covert communications, and locationspoofing detection.

SUN et al.: JOINT BEAMFORMING AND POWER ALLOCATION 5381

Zhiguo Ding (S’03–M’05) received the B.Eng.degree in electrical engineering from the BeijingUniversity of Posts and Telecommunicationsin 2000 and the Ph.D. degree in electricalengineering from Imperial College London in 2005.From 2005 to 2018, he was with Queen’s UniversityBelfast, Imperial College, and Newcastle University.Since 2018, he has been a Chair Professor withThe University of Manchester. Since 2012, he hasalso been an Academic Visitor with PrincetonUniversity, Princeton, NJ, USA.

Dr. Ding’s research interests are 5G networks, game theory, cooperativeand energy harvesting networks, and statistical signal processing. He was arecipient of the IEEE Communication Letter Exemplary Reviewer Awardin 2012, the IEEE TVT Top Editor Award in 2017, and the EU Marie CurieFellowship from 2012 to 2014. He received the Best Paper Award fromthe IET Communication Conference on Wireless, Mobile and Computingin 2009. He is serving as an Editor for the IEEE TRANSACTIONS

ON COMMUNICATIONS, the IEEE TRANSACTIONS ON VEHICULAR

TECHNOLOGY, and the Journal of Wireless Communications and MobileComputing. He was an Editor of the IEEE WIRELESS COMMUNICATIONLETTERS and the IEEE COMMUNICATION LETTERS from 2013 to 2016.

Derrick Wing Kwan Ng (S’06–M’12–SM’17)received the bachelor’s degree (Hons.) and theM.Phil. degree in electronic engineering from TheHong Kong University of Science and Technol-ogy in 2006 and 2008, respectively, and the Ph.D.degree from The University of British Columbiain 2012. In 2011 and 2012, he was a VisitingScholar with the Centre Tecnològic de Telecomuni-cacions de Catalunya–Hong Kong. He was a SeniorPost-Doctoral Fellow with the Institute for Digi-tal Communications, Friedrich-Alexander-University

Erlangen-Nürnberg, Germany. He is currently a Senior Lecturer and anARC DECRA Research Fellow with The University of New South Wales,Sydney, Australia. His research interests include convex and non-convexoptimization, physical-layer security, wireless information and power transfer,and green (energy efficient) wireless communications. He was a recipient ofthe IEEE TCGCC Best Journal Paper Award in 2018. He received the BestPaper Award from the IEEE International Conference on Communicationsin 2018, the IEEE International Conference on Computing, Networking andCommunications in 2016, the IEEE Wireless Communications and Network-ing Conference in 2012, the IEEE Global Telecommunication Conferencein 2011, and the IEEE Third International Conference on Communications andNetworking in China in 2008. He has been serving as an Editorial Assistantfor the Editor-in-Chief of the IEEE TRANSACTIONS ON COMMUNICATIONS

since 2012. He is currently an Editor of the IEEE COMMUNICATIONS LET-TERS, the IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, and theIEEE TRANSACTIONS ON GREEN COMMUNICATIONS AND NETWORKING.He was honored as an Exemplary Reviewer for the IEEE TRANSACTIONS

ON COMMUNICATIONS in 2015 and 2017, the Top Reviewer for the IEEETRANSACTIONS ON VEHICULAR TECHNOLOGY in 2014 and 2016, and anExemplary Reviewer for the IEEE WIRELESS COMMUNICATIONS LETTERS

for 2012, 2014, and 2015 and the IEEE TRANSACTIONS ON WIRELESSCOMMUNICATIONS in 2017.

Chao Shen (S’12–M’13) received the B.S. degreein communication engineering and the Ph.D.degree in signal and information processing fromBeijing Jiaotong University (BJTU), Beijing, China,in 2003 and 2012, respectively. He was aPost-Doctoral Researcher with BJTU and also aVisiting Scholar with the University of Maryland,College Park, MD, USA, from 2014 to 2015, andalso with The Chinese University of Hong Kongfrom 2017 to 2018. Since 2015, he has been anAssociate Professor with the State Key Laboratory

of Rail Traffic Control and Safety, BJTU. His current research interests focuson the ultra-reliable and low-latency communications, UAV-enabled wirelesscommunications, and energy-efficient wireless communications for high-speedrails.

Zhangdui Zhong (SM’16) received the B.E. andM.S. degrees from Beijing Jiaotong University,Beijing, China, in 1983 and 1988, respectively.

He is currently a Professor and an Advisor ofPh.D. candidates with Beijing Jiaotong University,where he is also a Chief Scientist with the StateKey Laboratory of Rail Traffic Control and Safety.His research has been widely used in railwayengineering, such as the Qinghai–Xizang railway,the Datong–Qinhuangdao Heavy Haul railway, andmany high-speed railway lines in China. He has

authored or co-authored seven books and over 200 scientific research papersin his research area. He holds five invention patents. His interests includewireless communications for railways, control theory and techniques forrailways, and GSM-R systems. He is the Director of the Innovative ResearchTeam of the Ministry of Education, Beijing, and a Chief Scientist of theMinistry of Railways, Beijing. He is an Executive Council Member of theRadio Association of China, Beijing, and the Deputy Director of the RadioAssociation, Beijing.

Prof. Zhong was a recipient of the Mao Yisheng Scientific Award ofChina, the Zhan Tianyou Railway Honorary Award of China, and the Top10 Science/Technology Achievements Award of Chinese Universities.