Multi-Dimensional Resource Allocation for Uplink Throughput …repository.essex.ac.uk/23495/1/08435905.pdf · 2018-11-19 · J. Yang et al.: Multi-Dimensional Resource Allocation

SPECIAL SECTION ON WIRELESSLY POWERED NETWORKSALGORITHMS, APPLICATIONS AND TECHNOLOGIES

Received July 2, 2018, accepted August 8, 2018, date of publication August 14, 2018, date of current version September 21, 2018.

Digital Object Identifier 10.1109/ACCESS.2018.2865402

Multi-Dimensional Resource Allocation for UplinkThroughput Maximisation in Integrated Dataand Energy Communication NetworksJIANJUN YANG1, JIE HU 1, (Member, IEEE), KESI LV1, QIN YU1, (Member, IEEE),AND KUN YANG 1,2, (Senior Member, IEEE)1School of Information and Communication Engineering, University of Electronic Science and Technology of China, Chengdu 611731, China2School of Computer Science and Electronic Engineering, University of Essex, Colchester CO4 3SQ, U.K.

Corresponding author: Jie Hu ([email protected])

This work was supported in part by the National Natural Science Foundation of China (NSFC) under Grant 61601097, Grant U1705263,and Grant 61620106011, in part by the Fundamental Research Funds for the Central Universities under Grant ZYGX2016Z011,and in part by the Huawei Innovation Research Program (HIRP).

ABSTRACT The interdisciplinary research of the radio-frequency (RF) signal-based wireless powerand information transfer is expected to address the energy shortage issue in the massively deployedlow-power Internet of Things devices. Different from conventional wireless powered communicationnetworks (WPCNs), the hybrid base station (H-BS) adopts the simultaneous wireless information and powertransfer (SWIPT) for the sake of satisfying the downlink data and energy requests of the multiple userequipments (UEs). The energy harvested from the downlink transmissions can be depleted for supportingthe UEs’ uplink transmissions. Integrating SWIPT in the downlink transmission of the WPCN yields ageneric integrated data and energy communication network, where the H-BS is equipped with multipleantennas and both the downlink and uplink transmissions are slotted in the time-domain. Furthermore, boththe sum-throughput and the fair-throughput of the uplink transmissions are maximized by jointly optimizingthe transmit beamformer of the H-BS in the spatial-domain, the time-slot allocation in the time-domainand the signal splitting strategies of the UEs in the power domain, while satisfying the UEs’ minimumdownlink transmission requirements. Due to the non-convexity of the problem, a low-complexity successiveconvex approximation-based algorithm is relied upon for obtaining the optimal resource allocation schemein the time-domain, power-domain, and spatial-domain. The numerical results validate the efficiency of ourproposed resource allocation algorithm and they also demonstrate that supporting low-rate data servicesduring the downlink transmissions does not degrade the wireless power transfer and hence does not reducethe uplink throughput.

INDEX TERMS Simultaneous wireless information and power transfer (SWIPT), wireless powered com-munication network (WPCN), integrated data and energy communication network (IDEN), multiple anten-nas, multi-dimensional resource allocation, sum-throughput maximisation, fair-throughput maximisation,successive convex approximation (SCA).

NOMENCLATURECSI Channel State InformationDC Direct CurrentH-BS Hybrid Base StationIDEN Integrated Data and Energy communication NetworkIoT Internet of ThingsRF Radio FrequencySCA Successive Convex ApproximationSDR Semi-Definite Relaxation

SWIPT Simultaneous Wireless Information andPower Transfer

TDD Time Division DuplexTDMA Time Division Multiple AccessUE User EquipmentWPCN Wireless Powered Communication NetworkWIT Wireless Information TransferWPT Wireless Power Transfer

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J. Yang et al.: Multi-Dimensional Resource Allocation for Uplink Throughput Maximization in IDENs

I. INTRODUCTIONIn the upcoming era of the Internet of Things (IoT) [1],the low power IoT devices are massively deployed for thesake of data collecting, environment sensing and decisionmaking etc., which constitute the promising future of smartcities [2], industrial internet [3] and autonomous driving [4].However, the functions of the low-power IoT devices areheavily restricted by the energy supply, since they are onlypowered by the batteries having limited capacity. Since theIoT devices are normally deployed in unreachable places,frequently replacing batteries may result in an escalatingmaintenance expenditure of the operator [5].

Harvesting energy from the renewable sources, such assunlight, wind and tidal wave, is a feasible option for the sakeof increasing the lifetime of the low-power user equipments(UEs) in wireless communication networks [6]. Further-more, our environment is also full of electronic-magneticsignals emitted by various transmitters operating in the radio-frequency (RF) band [7]. The UEs can also harvest theenergy of these RF signals for charging their batteries byconverting the received RF signal to the direct current (DC).Muncuk et al. [8] designed an adjustable RF-DC converteroperating in various RF spectral bands ranging from 700MHzof LTE band and 850 MHz of GSM band to 900 MHz of ISMband. Their multi-band RF-DC converter can achieve at least45 % energy conversion efficiency. Alevizos and Bletsas [9]studied both the sensitivity and the non-linearity of RF-DCconverters and quantified their effect on the performance ofRF energy harvesting. However, energy harvesting is self-initiated by UEs, who are running out of energy. Moreover,UEs are only capable of passively gleaning energy from theenvironment, which results in low-efficiency, unpredictabil-ity and uncontrollability of the stochastic energy arrivals.In order to increase the UEs’ energy harvesting efficiency,RF signals can be exploited for enabling the ded-icated wireless power transfer (WPT). For example,Grover and Sahai [10] optimised the WPT performance ina single-input-single-output frequency selective channel.

The downlink RF signal based WPT is enabled for thesake of supporting the low-power UEs’ uplink transmission,which stimulates substantial research interests in the designof wireless powered communication networks (WPCNs)[11]–[21]. A hybrid base station (H-BS) plays an central rolein WPCNs, which both supports the downlink WPT and theuplink data reception. The ‘‘harvest-then-transfer’’ protocolhas been adopted by the UEs for dividing the downlink WPTand the uplink wireless information transfer (WIT) in thetime-domain [11].

However, the main challenge that deters WPCNs tobecome an reality is the lack of efficient resourceallocation schemes for striking a balanced performancebetween the downlink WPT and the uplink WIT.Therefore, Ju and Zhang [12] studied a time-division-multiple-access (TDMA) protocol enabled WPCN andthey optimised the time-domain resources for maximisingthe uplink sum-throughput, where the H-BS operates in

a half-duplex mode. Furthermore, Ju and Zhang [13] alsostudied the full-duplex aided WPCN. They proposed an opti-mal power and time allocation scheme for the sake of max-imising the weighted uplink sum-throughput by consideringboth the perfect and imperfect self-interference cancellation.Che et al. [14] analyzed the spatial throughput performancein a large-scale WPCN by exploiting the classic stochasticgeometry. In order to overcome the doubly near-far effect,which will be introduced at the beginning of Section IV,Chen et al. [15] proposed a pricing scheme to encourage theuser cooperation in a WPCN for the sake of improving theperformance of the uplink transmission.

Apart from the conventional uplink throughput maximi-sation [12]–[15], the energy efficiency of WPCNs havealso attracted substantial research interest. For instance,Wu et al. [16] proposed an optimal time and power allo-cation scheme for maximizing the achievable energy effi-ciency of the WPCN studied by also considering the initialbattery states of the UEs. Zewde and Gursoy [17] proposedan optimal resource allocation scheme for maximizing theeffective energy efficiency of the WPCN studied, wherethe UEs adopts the power-domain non-orthogonal-multiple-access protocol for supporting their uplink transmissions.

Furthermore, multiple antennas equipped at H-BSs mayprovide additional degree of freedom in the spatial domain,which thus substantially improve the performance ofthe WPCNs. For instance, Huang et al. [18] derived theclosed-form expression and the asymptotic expression inthe high signal-to-noise-ratio region of a single UE’s uplinkthroughput, which is powered by a H-BS equipped withmulti-antennas in a WPCN. Furthermore, Liu et al. [19]propose an optimal downlink energy beamforming forthe multi-antenna aided H-BS in order to maximise theminimum uplink throughput of the UEs in the WPCN.Ku and Lai [20] proposed a joint beamforming and timeallocation scheme for maximizing the sum-throughput ofthe wireless powered device-to-device communication pairs.Moreover, Yuan et al. [21] studied an cluster based cooper-ation in a WPCN, where the multi-antenna of the H-BS isexploited for efficiently beaming the energy to the clusterheads.

In the conventional WPCNs [11]–[21], the downlinktransmission of a H-BS completely contributes to thededicated WPT, which largely ignores the downlink datarequests of the UEs. However, coordinating the WPT andthe conventional WIT in the same RF band is challenging,since they may compete for the precious resources of the airinterface, which yields a large body of research concerningthe simultaneous wireless information and power transfer(SWIPT) [22]–[27]. Varshney et al. [22] is the pioneer ofexploring the inherent tradeoff between the WPT and theWIT from the information theoretical aspect by invokingan ideal receiver. In order to further advance the SWIPT tothe practical implementation, signal splitting based receivershave been widely investigated. Zhou et al. [23] proposedpractical receiver architectures for simultaneous reception

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of the information and energy, which rely upon the signalsplitting in either the power domain and the time-domain.Moreover, Xiong et al. [24] studied the rate-energy regionof both the power splitting and time switching aided SWIPTreceiver by taking the non-linearity of the RF-DC converterinto account. Lee et al. [25] jointly optimized the transmitcovariance matrices at the transmitter and the signal splittingratios in the time-domain1 at the receivers in order to achievethe boundary points of the rate-energy region and tominimizethe transmit energy consumption. Furthermore, Xu et al. [26]utilized the power splitting based SWIPT receiver at thelow-power UEs for simultaneous energy harvesting and thebeam-domain channel estimation, which is relied upon forscheduling the uplink transmissions of the UEs. Mishra andAlexandropoulos [27] proposed an optimal transmit pre-coding scheme of the multi-antenna aided transmitter andan optimal power splitter of the SWIPT receiver, whichis capable of maximizing the amount of energy harvestedby the receiver. It has been widely recognised that thepower splitting based receiver may reach a larger rate-energyregion than its time switching based counterpart, when linearRF-DC converters are adopted [23] and [24].

Furthermore, Lv et al. [28] firstly integrate the SWIPTin the downlink transmission of the conventional WPCN.However, they mainly focus on the time-domain resourceallocation for maximizing the uplink throughput performanceby assuming a single antenna at the H-BS. Against this back-ground, our novel contributions are summarized as below:• By integrating the SWIPT in the downlink transmissionof the WPCN, we establish a generic Integrated Dataand Energy communication Network (IDEN), where theUEs acquire both data and energy from the downlinktransmission of the H-BS and they upload their own datato the H-BS by exploiting the energy harvested duringtheir downlink transmissions.

• In order to maximise the sum-throughput and the fair-throughput of the UEs’ uplink transmissions, the corre-sponding multi-dimensional resource allocation schemeis obtained by jointly optimizing the transmit beam-former of the H-BS in the spatial domain, by optimizingthe time-slot allocation in the time-domain and by opti-mizing the UEs’ power splitting strategies in the powerdomain.

• The original non-convex optimization problem isdecomposed and transformed into several convex sub-problems, which have been iteratively solved by exploit-ing the successive convex approximation (SCA) basedalgorithm for obtaining the optimal resource alloca-tion scheme. The optimal initial transmit beamformeris found to significantly reduce the complexity of theSCA based algorithm.

• The numerical results demonstrate the efficiency of ourproposed multi-dimensional resource allocation scheme

1This is also widely regarded as the time switching between the energyharvesting mode and the information decoding mode.

and the advantage of our IDEN model over the conven-tional WPCN.

The rest of this paper is organized as follows. OurIDENmodel is introduced in Section II, which is followed bythe sum-throughput maximisation in Section III and the fair-throughput maximisation in Section IV, respectively. By pro-viding the numerical results in Section V, we finally concludethis paper in Section VI.

II. SYSTEM MODELThe IDEN system considered in our paper is illustratedin Fig.1, which consists of a single H-BS and a set of UEsU = {Ui|i = 1, 2, · · · ,K }. The total number of UEs isthus |U | = K . The H-BS equipped with Nt antennas has thefollowing pair of functions:• The H-BS transmits the modulated RF signals in thedownlink by simultaneously delivering both the data andenergy to the UEs;

• The H-BS receives the data uploaded by the UEs duringtheir uplink transmissions.

The H-BS operates in a time-division-duplex (TDD) mode inorder to avoid any potential interference between the down-link and uplink transmissions.

FIGURE 1. The (a) downlink and (b) uplink transmissions of the IDENconsisting of a single H-BS and K UEs.

The UEs in the IDEN system are batteryless but they are allequipped with super-capacitors. Their uplink transmissionscompletely rely on the energy harvested from the downlinktransmissions of the H-BS. The UEs equipped with a singleantenna has the following pair of functions:• The modulated RF signals received by the UEs duringthe downlink transmission of the H-BS are split in thepower domain in order to satisfy their data and energyrequests. Specifically, all the UEs in the system havediverse downlink throughput requirements, which aredenoted as D = {Di|i = 1, 2, · · · ,K }.

• Since super-capacitors have a very short dischargingcycle, they cannot store energy for a long period. There-fore, the UEs may deplete all the energy harvested fromthe H-BS for powering their own uplink transmissions.

The UEs also operates in a TDD mode.

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FIGURE 2. Frame structure of the (a) H-BS’s and (b) UE’s slottedoperation. EH: Energy Harvesting. ID: Information Decoding. Up:Uploading. St: Standby.

A. FRAME STRUCTUREIn our IDEN system, the operations of both the H-BS and theUEs are slotted in the time-domain by adopting the classicTDMA protocol in order to suppress the interference incurredby the transmissions of multiple UEs [29]. In general, a singletransmission frame consists of two phases, namely the controlphase having a duration of Tc and the transmission phasehaving a duration of Tt , as depicted in Fig.2. Therefore,the duration of a single transmission frame is T = Tc + Tt .The control signalling exchange is carried out during thecontrol phase in order to realize the following functions:• Channel Estimation. A training pilot can be sent bya UE to the H-BS. The received pilot can be furtherprocessed by the H-BS in order to extract the channelstate information (CSI) of the uplink between the UEand the H-BS. The acquired CSI of the uplink can alsobe regarded as that of the corresponding downlink byexploiting the channel reciprocity of the TDD system.This classic reverse-link training may significantly savethe energy of the batteryless UEs, since the UE cannotafford neither sophisticated signal processing nor feed-back transmission.

• Resource Allocation. Relying on the acquired CSI,the H-BS may carry out the resource allocation schemein order tomaximise the throughput performance. It theninforms the UEs of their signal splitting strategies andthe duration of their assigned time-slots.

• Synchronization. The H-BS and the UEs have to beaccurately synchronized by invoking the time-stamp-based synchronization approach [30]. The clock infor-mation of the H-BS is broadcasted to the UEs during thecontrol phase. The UEs may adjust their local clock tosynchronize themselves with the UEs. Synchronizationis particularly essential for the TDMA aided system.

The transmission phase is divided into a range of down-link time-slots denoted as tD = {tDj |j = 1, 2, · · · ,K } forsupporting the downlink transmissions of the H-BS anda range of uplink time-slots denoted as tU = {tUj |j =1, 2, · · · ,K } for supporting the uplink transmissions of the

multiple UEs. Therefore, we haveK∑i=1

(tDi + tUi ) ≤ Tt (1)

The detailed operations of the H-BS and the UEs during thetransmission phase are portrayed in Fig.2 and summarized asbelow:• For the downlink transmissions, the H-BSmodulates thedata requested by the UE Ui on the RF signal trans-mitted during the specific time-slot tDi , as illustratedin Fig.2(a). For the downlink reception, Ui split thereceived RF signal in the power-domain during tDi forsimultaneous energy harvesting and information decod-ing. Due to the broadcast nature of the wireless channels,Ui may also harvest energy from the modulatedRF signals requested by its peers during other time-slots,as illustrated in Fig.2(b).

• For the uplink transmissions, Ui may upload its data tothe H-BS during the specific time-slot tUi , as depictedin Fig.2(a), by exploiting the energy harvested fromall the time-slots tD. This is the classic ‘‘harvest-then-transfer’’ protocol [31]. During other time-slotsof the uplink transmissions, Ui may remain in thestandby mode without any other operation, as portrayedin Fig.2(b). The H-BS operates as a information receiverduring the entire uplink transmission process.

B. CHANNEL MODELThe far-field path-loss �i between the H-BS and the UEUi can be formulated as

�i = �0 ·

(did0

)α. (2)

In Eq.(2), d0 represents the near-field range of the trans-mit antenna, within which the power-law of the path-loss isinvalid, while �0 is the corresponding reference path-loss.The path-loss exponent is denoted as α in Eq.(2).The wireless channels between the H-BS and the UEs obey

the uncorrelated block fading [32]. The channel coefficientsremains unchanged during a single transmission frame butthey vary from a transmission frame to another. The downlinkchannel coefficient between the H-BS and Ui is representedby a 1 × Nt column vector hi. Since the channel reciprocityis assumed, the uplink channel coefficient is thus hTi , whichis the transpose of hi.

C. DOWNLINK TRANSMISSIONSWithout loss of generality, the H-BS is assumed to transmitthe modulated complex symbol xDi destined to the UEUi dur-ing the downlink time-slot tDi . We haveE[xDi x

D∗i ] = 1, where

xD∗i is the conjugate value of xDi . Considering an arbitrarytime-slot tDj , during which the symbol xDj is transmitted bythe H-BS, the RF signal received byUi’s antenna is expressedas

yDi,j =

√Pt�i

hiwtxDj + z0, (3)

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for ∀i, j = 1, · · · ,K , where Pt is the transmit power of theH-BS and the Nt × 1 vector wt is the H-BS’s normalisedtransmit beamformer, whereas z0 is the Gaussian distributednoise having a zero mean and a variance of σ 2

0 .As illustrated in Fig.2(b), Ui only decodes its requested

information during the time-slot tDi . At this time-slot,the received signal yDi,i is split in the power domain. A fraction√ρi of yDi,i is exploited for information decoding, which is

expressed as

yDi,i,id =√ρiyDi,i + zcov

=

√ρiPt�i

hiwtxDi +√ρiz0 + zcov, (4)

where zcov is the Gaussian distributed noise incurred bythe imperfect passband-to-baseband converter. The meanof zcov is zero and its variance is σ 2

cov. As a result, the cor-responding downlink throughput of Ui is formulated as

RDi = tDi · log

(1+

ρiPthiwtw∗t h∗i

�i(ρiσ 20 + σ

2cov)

)

≈ tDi · log(1+

ρiPthiwtw∗t h∗i

�iσ 2cov

), (5)

where w∗t and h∗i are the conjugate transpose of thevectors wt and hi. The approximation of Eq.(5) is obtainedbecause the power of the antenna noise is normally far lowerthan that of the passband-to-baseband converter noise [33].

The RF signal yDi,j received by Ui during the time-slot tDjfor j 6= i is fully exploited for energy harvesting. The receivedpower of yDi,j is P

Di,j = Pt�

−1i hiwtw∗t h

∗i . The energy harvested

by Ui during this time-slot is thus expressed as

EDi,j = ζ tDj P

Di,j = ζ t

Dj Pt�

−1i hiwtw∗t h

∗i , (6)

for i 6= j, where ζ is the RF-DC conversion efficiency.During the time-slot tDi , the fraction

√1− ρi of the received

signal yDi,i is exploited byUi for energy harvesting. Therefore,the amount of energy harvested by Ui during this time-slot isexpressed as

EDi,i = ζ tDi (1− ρi)Pt�

−1i hiwtw∗t h

∗i . (7)

As a result, the total energy harvested by Ui during all thedownlink time-slots tD can be derived as

EDi =K∑

j=1,j 6=i

EDi,j + EDi,i

= ζPt�−1i hiwtw∗t h

∗i

K∑j=1,j 6=i

tDj + (1− ρi)tDi

= γi

K∑j=1,j 6=i

tDj + (1− ρi)tDi

. (8)

where we have γi = ζPt�−1i hiwtw∗t h

∗i to simplify the

expression. Note that during the downlink transmission,the H-BS adopts the identical transmit beamformer wt for

all the downlink time-slots tD, which form a multicast beamfor the SWIPT. As a result, the choice of the transmit beam-former wt may satisfy the data and energy requests of themultiple UEs, which avoids the complexity of finding theoptimal point-to-point transmit beamformer for every singledownlink time-slot.

D. UPLINK TRANSMISSIONSAs illustrated in Fig.2(b), Ui only uploads its own data to theH-BS during the uplink time-slot tUi but remain silent duringothers. The transmit power of Ui is PUi = EDi /t

Ui , since all

the energy harvested by Ui during the downlink transmissionis now depleted for powering its own uplink transmission.As a result, the RF signals received by the H-BS’s multipleantennas during the uplink time-slot tUi is obtained as

yUi =

√EDitUi

hTi xUi + z0, (9)

where xUi having E[xUi xU∗i ] is the modulated complex sym-

bol uploaded by Ui and the Nt × 1 vector z0 representsthe Gaussian distributed noise received by the antennas ofthe H-BS. Every element in z0 has a zero mean and a vari-ance of σ 2

0 . The received signal yUi is then combined by the1× Nt receiver beamformer wi,r , which yields the signal forinformation decoding:

yUi,id = wi,ryUi + zcov

=

√EDitUi

wi,rhTi xUi + wi,rz0 + zcov, (10)

where zcov is the Gaussian distributed noise incurred bythe imperfect passband-to-baseband converter of the H-BS.We assume that the power of the converter noise atthe H-BS is equal to that of the UE. Since the maximum-ratio-combination (MRC) [34] has been proven to be theoptimal receiver beamformer, the corresponding wi,r can beexpressed as

wi,r =

√hih∗iNt

hi. (11)

As a result, the uplink throughput of Ui can be formulated as

RUi = tUi · log

(1+

EDitUi

wi,rhTi hT∗i w∗i,r

wi,rw∗i,rσ20 + σ

2cov

)

≈ tUi · log

(1+

EDitUi

wi,rhTi hT∗i w∗i,r

σ 2cov

)

= tUi · log

(1+

EDitUi

hi(h∗i hi)(h∗i hi)

Th∗iNtσ 2

cov

)

= tUi · log

(1+

EDitUi

giσ 2cov

)(12)

where EDi is given by Eq.(8) and the second approxima-tion is derived by ignoring the negligibly low antenna noise

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power σ 20 . The third equality of Eq.(12) is derived by sub-

stituting Eq.(11) into Eq.(12), while the fourth equality is

derived by letting gi =hi(h∗i hi)(h

∗i hi)

T h∗iNt

.

III. UPLINK SUM-THROUGHPUT MAXIMIZATIONIn this section, the time-, power- and spatial-domain resourceallocation scheme is designed for maximising the UEs’uplink sum-throughput. The original optimisation problemis decomposed into two sub-problems, which aim for opti-mising the time- and power-domain resource allocation andfor optimising the spatial-domain resource allocation, respec-tively. By invoking the SCA, an iterative algorithm is pro-posed for alternatively optimising this pair of sub-problemsin order to obtain the optimal solution.

The uplink sum-throughput maximisation problem is thenformulated as

(P1): maxwt ,tD,tU ,ρ

K∑i=1

RUi (wt , tD, tUi , ρi) (13)

s. t. RDi (wt , tDi , ρi) ≥ Di, ∀i = 1, · · · ,K , (13a)

0 ≤ ρi ≤ 1, ∀i = 1, · · · ,K , (13b)K∑i=1

(tDi + tUi ) ≤ Tt , (13c)

Tr(wtw∗t ) ≤ 1. (13d)

As expressed in Eq.(13), the objective of the optimisationproblem (P1) is to maximise the uplink sum-throughput ofall the UEs by optimising the transmit beamformer wt in thespatial-domain, by optimising the duration of the downlinktime-slots tD and that of the uplink time-slots tU in the time-domain as well as by optimising the power splitting ratiosρ = {ρi|i = 1, · · · ,K } in the power domain. The uplinkthroughput RUi of Ui in the objective (13) is a function of wt ,tD, tUi and ρi, according to Eq.(12). The downlink throughputRDi of Eq.(5) is a function of wt , tDi and ρi, which has tobe higher than Ui’s downlink throughput requirement Di,as expressed in the constraint (13a). The power splittingratios ρ are naturally constrained by (13b). As expressed inthe constraint (13c), the total duration of all the downlinkand uplink time-slots should be lower than the duration ofthe transmission phase Tt . The constraint (13d) representsthat the total power weights of the normalised transmitbeamformer should be lower than a unity, where Tr(·) rep-resents the trace of a matrix. By letting Wt = wtw∗t , theconstraint (13d) is equivalent to the following pair:Tr(Wt ) ≤ 1 and Rank(Wt ) = 1. The rank constraintimposes difficulty in solving the problem (P1). As a result,we relax the rank constraint on the semi-definite transmitbeamforming matrix Wt , which is well known as the semi-definite relaxation (SDR). Then, instead of wt , we will focuson the optimisation of the transmit beamforming matrix Wtwithout any rank constraint in the spatial-domain. By solvingthe relaxed problem (P1), we may obtain the optimal multi-dimensional resource allocation scheme {tD†, tU†, ρ†,W†

t }.

Unfortunately, due to the coupled parameters {tD, tU , ρ,Wt },(P1) is still a non-convex problem.

As a result, according to the basic principle of theSCA [35], (P1) can be decomposed into the following pairof sub-problems (P2) and (P3). Given a specific transmitbeamforming matrix Wt satisfying Tr(Wt ) ≤ 1, the sub-problem (P2) is formulated as

(P2): maxtD,tU ,ρ

K∑i=1

RUi (tD, tUi , ρi|Wt ) (14)

s. t. RDi (tDi , ρi|Wt ) ≥ Di, ∀i = 1, · · · ,K , (14a)

(13b) and (13c).

According to Eqs.(5) and (12), the downlink throughput RDiand the uplink throughput RUi are both functions of tD,tU and ρ. As expressed in the objective (14), given Wt ,(P2) aims for maximising the uplink sum-throughput byoptimising the time- and power-domain resource allocationscheme {tD‡, tU‡, ρ‡

|Wt }.Given a specific time- and power-domain resource alloca-

tion scheme {tD, tU , ρ}, the sub-problem (P3) is formulatedas

(P3): maxWt

K∑i=1

RUi (Wt |tD, tUi , ρi) (15)

s. t. RDi (Wt |tDi , ρi) ≥ Di, ∀i = 1, · · · ,K (15a)

Tr(Wt ) ≤ 1. (15b)

In (P3), RDi and RUi are both convex functions ofWt , accord-ing to Eqs.(5) and (12). As formulated in the objective (15),given {tD, tD, ρ}, the convex optimisation (P3) aims for max-imising the uplink sum-throughput by optimising the transmitbeamforming matrixW‡

t .In the rest of this section, we will elaborate on how we

jointly solve (P2) and (P3) in order to obtain the optimalresource allocation scheme {W†

t , tD†, tU†, ρ†}.

A. FEASIBILITY OF RESOURCE ALLOCATION.Let us first discuss the existence of a feasible resource alloca-tion scheme. We propose a minimisation problem of the totalduration of downlink time-slots tD by optimising the transmitbeamforming matrixWt , which is formulated as

(P1-1): minwt

K∑i=1

Di

log(1+

Pthiwtw∗t h∗i

�iσ 2cov

) , (16)

s. t. Tr(Wt ) ≤ 1. (16a)

The fractional expression of the objective (16) makes theproblem (P1-1) non-convex. Since a logarithmic function ismonotonously increasing with respect to its argument, takingthe logarithmic operation on the objective (16) may not alterthe optimality of (P1-1) but convert the non-convex objectiveto a convex one. As a result, the non-convex (P1-1) can be

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reformulated as the following convex problem:

(P1-2): minWt

K∑i=1

{logDi

− log[log

(1+

PtTr(h∗i hiWt )

�iσ 2cov

)]}, (17)

s. t. Tr(Wt ) ≤ 1, (17a)

where we have Tr(h∗i hiWt ) = hiwtw∗t h∗i . In order to effi-

ciently solve (P1-2), we design an algorithm by exploiting theclassic interior point method [36]. We then reformulate theconstrained problem (P1-2) as the following unconstrainedproblem:

(P1-3): minWt

f1(Wt )

=

K∑i=1

{logDi − log

[log

(1+

PtTr(h∗i hiWt )

�iσ 2cov

)]}−φ(1− Tr(Wt ))− φ(Det(Wt )), (18)

where Det(·) is the determinant operation on a matrix andφ(u) is the logarithmic barrier function [36], which is givenby

φ(u) =

log(u)m

, u > 0,

−∞, u ≤ 0,(19)

where the barrier parameter m should be higher than 0. Thegradient of f1(Wt ) can then be derived as

∇Wt f1(Wt )

=

K∑i=1

h∗i hi

(�iσ 2cov + Tr(h∗i hiWt ) log

(1+

PtTr(h∗i hiWt )�iσ 2cov

)+

1m

(I

1− Tr(Wt )−W−1t

), (20)

where I is a Nt × Nt identity matrix.Whenwe have a specific transmit beamformingmatrixWt ,

the searching direction is thus defined as1Wt = ∇Wt f1(Wt ).As a result, after the n-th iteration, the transmit beamformingmatrix is updated as W(n+1)

t = W(n)t − θ · 1Wt , where θ is

a scalar representing the searching step. In order to guaranteethe accuracy of the gradient descent method, the optimalsearching step is obtained as

θ = argminθf1(Wt − θ ·1Wt ). (21)

The gradient-descent method based algorithm for solving theproblem (P1-3) is thus detailed in Algorithm 1, which returnsus the optimal transmit beamforming matrixW†

t,0. Note that,the transmit beamforming matrix Wt in Algorithm 1 can beinitialised by the equal gain beamformer I/Nt .

If the resultant minimum total duration TD‡min of thedownlink time-slots is higher than Tt , the downlink through-put requirements D = {D1, · · · ,DK } cannot be sat-isfied. This tells us that no feasible resource allocation

Algorithm 1 The Gradient-Descent Method BasedAlgorithm for Solving Problems (P1-3), (P3-1) and (P6-2)Input: Path loss of {�1, · · · , �K }; Channel fading coeffi-

cients of {h1, · · · ,hK }; Transmit power of the H-BS Pt ;Initial transmit beamforming matrixWt ; Noise power ofthe converter σ 2

cov; Duration of the transmission phase Tt ;Resource allocation scheme in time- and power-domain{tD, tU , ρ}; Error tolerance ε.

Output: Optimal transmit beamforming matrixW†

t,0 or W†t ;

1: Obtain the searching direction gradient 1Wt ←

∇Wt f1(y) by Eq.(20), or 1Wt ← ∇Wt f2(y) byEq.(30), or 1Wt ← ∇Wt f3(y) by Eq.(44);

2: while ||1Wt || ≤ ε do3: Obtain the searching step θ by solving Eq.(21), or by

solving Eq.(32), or by Eq.(45);4: Update Wt ←Wt ∓ θ1Wt ;5: Obtain the searching direction gradient 1Wt ←

∇Wt f1(y) by Eq.(20), or 1Wt ← ∇Wt f2(y) byEq.(30), or 1Wt ← ∇Wt f3(y) by Eq.(44).

6: end while7: return W†

t,0 or W†t =Wt .

scheme {tD, tU , ρ,Wt } exist for maximising the uplink sum-throughput of the UEs. Hence, the H-BSmay reject the accessrequest of some UEs in order to find a feasible resourceallocation scheme.

Note that Algorithm 1 can also be exploited for solvingall the optimisation problem of the transmit beamformingmatrixWt in this paper, which includes the problems (P1-3),(P3-1) and (P6-2).

B. RESOURCE ALLOCATION IN TIME- ANDPOWER-DOMAINThe sub-problem (P2) is still non-convex, due to the cou-pled parameters tD and ρ induced by Eq.(8). By definingµ = {µi = tDi (1 − ρi)|, i = 1, · · · ,K }, (P2) is equiv-alently transformed to the following convex optimisationproblem (P2-1), which is formulated as

(P2-1): maxtD,tU ,µ

K∑i=1

RUi (tD, tUi , µi|Wt ) (22)

s. t. RDi (tDi , µi|Wt ) ≥ Di, ∀i = 1, · · · ,K , (22a)

0 ≤ µi ≤ tDi , ∀i = 1, · · · ,K , and (13c).

(22b)

By exploiting the classic Lagrange multiplier method,the corresponding Lagrange function of (P2-1) is derived as

L(tD, tU ,µ, λ, ξ )

=

K∑i=1

RUi (tD, tUi , µi|Wt )+ λ[Tc −

K∑i=1

(tDi + tUi )]

+

K∑i=1

ξi[RDi (tDi , µi|Wt )− Di], (23)

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where λ and ξ = {ξi|i = 1, · · · ,K } are the Lagrangemultipliers for the constraints (13c) and (22a), respectively.The optimal solution {tD‡, tU‡,µ‡

} of (P2-1) have to satisfythe following Karush-Kuhn-Tucker (KKT) conditions:

log(1+ pi)−pi

1+ pi= λ∑K

j=1 gjγj(1+ pi)σ 2

cov+ ξi[log(1+ qi)−

qi1+ qi

] = λ

gi1+ yi

= ξi1

1+ qi∑Ki=1(t

Di + t

Ui ) = Tt ,

tDi log(1+ qi) = Di,

(24)

for ∀i = 1, · · · ,K , where we have pi =γigi

∑Kj=1 t

Dj −µi

σ 2covand qi =

µiγitDi σ

2cov. Specifically, γi and gi are defined

in Eqs.(8) and (12), respectively. Given a specific valueof λ, the optimal solution can be derived as

tD‡i =Di

log(1+ q‡i ),

tU‡i = hiγi

∑Kj=1 t

D‡j − µ

‡i

p‡i σ2cov

,

µ‡i =

q‡i σ2cov

γitD‡i

,

(25)

where p‡i and q‡i can be obtained by solving the followingsystem of equations:

log(1+ pi)−pi

1+ pi= λ, (26)

(1+ qi) log(1+ qi)− qi =λ(1+ pi)−

∑Kj=1

gjγjσ 2

gi, (27)

The left-hand side of Eq.(26) is a monotonously increas-ing function of pi, while the left-hand side of Eq.(27) is amonotonously increasing function of qi. Observe from theirrespective right-hand side that both pi and qi monotonouslyincrease as we increase λ. Note that qi is positive, whichdetermines that the right-hand side of Eq.(27) should behigher than zero. Furthermore, increasing λ leads to theincrement of the right-hand side of Eq.(27). Therefore, if theright-hand side of Eq.(27) is lower than zero, we shouldincrease λ. The details of the binary search based algorithmis provided in Algorithm 2. Note that similar methodologyhas been used for obtaining the optimal time-domain resourceallocation scheme for maximising the sum-throughput in ourprevious work [28]. However, the binary searching basedAlgorithm 2 proposed in this paper substantially reduces thecomplexity of the time- and power-domain resource alloca-tion, when compared to the algorithm proposed in [28].

C. RESOURCE ALLOCATION IN SPATIAL-DOMAINAfter relaxing the rank constraint, the sub-problem (P3)is transformed into a convex optimisation problem.

Algorithm 2 A Binary Search Based Algorithm forSolving (P2-1)Input: Path loss of {�1, · · · , �K }; Channel fading coef-

ficients of {h1, · · · ,hK }; Transmit power of the H-BSPt ; Transmit beamforming matrix Wt ; Noise power ofthe converter σ 2

cov; Duration of the transmission phase Tt ;Error tolerance ε.

Output: Optimal resource allocation scheme {tD‡, tU‡, ρ‡}

in time- and power-domain.1: Initialise λmin← 0, λmax← A sufficiently large value;2: while λmax − λmin > ε do3: Update λmid ← 1

2 (λmax + λmin);4: Obtain p‡ ← {p‡1, · · · , p

‡K } by substituting λmid into

Eq.(26);5: if λmid (1+ p

‡i )−

∑Kj=1

gjγjσ 2cov

> 0,∀i then

6: Obtain q‡← {q‡1, · · · , q‡K } by substituting λmid and

p‡ into Eq.(27);7: Obtain {tD‡, tU‡,µ‡

} by substituting p‡ and q‡ intoEq.(25)

8: if∑K

i=1 tD‡i + t

U‡i ≤ Tt then

9: Update λmax← λmid ;10: else11: Update λmin← λmid ;12: end if13: else14: Update λmin← λmid ;15: end if16: end while17: Obtain ρ‡

← {ρ‡i =

µ‡i

tD‡i|i = 1, · · · ,K } ;

18: return {tD‡, tU‡, ρ‡}

Similarly to (P1-1), we also exploit the interior point methodfor efficiently solving this convex problem. Then, the originalconstrained problem (P3) is reformulated as the followingunconstrained problem:

(P3-1): maxWt

f2(Wt )

=

K∑i=1

[RUi (Wt |tD, tUi , ρ)+ φ(Tr(h

∗i hiWt )− ai)

]+φ(1− Tr(Wt ))+ φ(det(Wt )), (28)

where φ(u) is the logarithmic barrier function givenby Eq.(19). Furthermore, the parameter ai in (P3-1) isgiven by

ai =σ 2cov

ρi�i(e

DitDi − 1), (29)

for ∀i. The unconstrained problem (P3-1) can thusbe solved by exploiting the gradient descent method.Given a feasible Wt , the gradient of f2(Wt ) can be

47170 VOLUME 6, 2018


derived as

∇Wt f2(Wt )

=1m

(K∑i=1

h∗i hiTr(h∗i hiWt )− ai

−I

1− Tr(Wt )+W−1t

)

+

K∑i=1

tUi bih∗i hi

1+ biTr(h∗i hiWt ), (30)

where I is an Nt × Nt identity matrix and the parameter bi isexpressed as

bi =�igi(

∑Kj=1 t

Dj − t

Di ρi)

tUi σ2cov

, (31)

The optimal searching step is obtained as

θ = argminθf2(Wt + θ ·1Wt ), (32)

where 1Wt = ∇Wt f2(Wt ) is the searching direction. As aresult, after the n-th iteration, the transmit beamformingmatrix is updated as W(n+1)

t = W(n)t + θ · 1Wt . Therefore,

the optimal transmit beamforming matrix W‡t in the spatial

domain can be then obtained by exploiting Algorithm 1.

D. SCA BASED ALGORITHMThe optimal resource allocation scheme {tD†, tU†, ρ†,W†

t }

to (P1) can then be obtained by alternatively solving (P2-1)and (P3-1), which is detailed in Algorithm 3. Note that theexistence of a feasible resource allocation scheme to (P1) isvalidated by solving (P1-2) in Algorithm 1. This may sub-stantially increase the robustness and reduce the complexityof the SCA based Algorithm 3.

IV. UPLINK FAIR-THROUGHPUT MAXIMIZATIONThe UEs in the vicinity of the BS have good uplink chan-nels for satisfying their data uploading requirements. Theimprovement of their uplink throughput is substantial, whenmore resources are allocated to these UEs. However, the poorchannels connecting the distant UEs to the BS largely restrictstheir uplink throughputs. Even if more resources are allocatedto these UEs, the improvement of their uplink throughput islimited. As a result, in order to maximise the uplink sum-throughput, the BS may allocate more resources to the UEsin its vicinity, since it may results in substantial growth ofthe sum-throughput, while the data uploading requests of thedistant UEs may be largely ignored by the BS. This is theclassic ‘‘near-far effect’’ of the multi-user WIT system.

In the WPCN, the uplink transmission of the UEs is con-strained by the energy harvested from the downlink RF sig-nals transmitted by the H-BS, as expressed in Eq.(12). TheUEs in the vicinity of the H-BS may harvest more energyfrom the downlinks than their distant counterparts, whichfurther increase the gap between their uploading capabilities.Therefore, the H-BS is more willing to allocate resourcesto its nearby UEs, which results in the ‘‘doubly near-fareffect’’ [12].

Algorithm 3 SCA Based Algorithm for Solving (P1)Input: Path loss of {�1, · · · , �K }; Channel fading coef-

ficients of {h1, · · · ,hK }; Transmit power of the H-BSPt ; Noise power of the converter σ 2

cov; Duration of thetransmission phase Tt ; Error tolerance ε.

Output: Optimal resource allocation scheme{tD†, tU†, ρ†,W†

t } in time- power- and spatial-domain;Maximum uplink sum-throughput R†sum.

1: Initialise the transmit beamforming matrix Wt ← I/Ntin Algorithm 1;

2: Obtain a feasible transmit beamforming matrix W‡t,0

and the resultant minimum total duration TD‡min ofthe downlink time-slots by solving (P1-3) based onAlgorithm 1;

3: if TD‡min < Tt then4: Initialise an uplink sum-throughput recorderRsum,0←

0;5: Initialise another uplink sum-throughput recorder

Rsum,1← 2ε;6: Initialise the transmit beamforming matrix Wt ←

W‡t,0 in Algorithm 2 for solving (P2-1);

7: Initialise the transmit beamforming matrix Wt ←


8: while Rsum,1 − Rsum,0 > ε do9: Update Rsum,0← Rsum,1;10: Obtain the optimal solution {tD‡, tU‡, ρ‡

} to (P2-1)based on Algorithm 2;

11: Update {tD, tU , ρ} ← {tD‡, tU‡, ρ‡} in Algo-

rithm 1 for solving (P3-1);12: Obtain the optimal solution W‡

t to (P3-1) and theresultant uplink sum-throughput R‡sum based onAlgorithm 1;

13: Update Rsum,1← R‡sum;14: Update Wt ← W‡

t in Algorithm 1 for solving (P3-1) and in Algorithm 2 for solving (P2-1);

15: end while16: Update {tD†, tU†, ρ†,W†

t } ← {tD‡, tU‡, ρ‡,W‡t },

R†sum← R‡sum;17: else18: No feasible resource allocation scheme.19: end if20: return {tD†, tU†, ρ†,W†

t } and R†sum.

In order to overcome the unfairness incurred by the‘‘doubly near-far effect’’, we aim for maximising the uplinkfair-throughput in our IDEN, which is defined as the min-imum uplink throughput among the UEs. As a result,the uplink fair-throughput maximisation problem is formu-lated as

(P4): maxtD,tU ,ρ,wt

Rfair (33)

s. t. RUi (wt , tD, tUi , ρi) ≥ Rfair,∀i = 1, · · · ,K , (33a)

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RDi (wt , tDi , ρi) ≥ Di,∀i = 1, · · · ,K , (33b)

0 ≤ ρi ≤ 1,∀i = 1, · · · ,K , (33c)K∑i=1

(tDi + tUi ) ≤ T , (33d)

Tr(wtw∗t ) ≤ 1. (33e)

As expressed in the objective (33), (P4) aims for maximisingthe uplink fair-throughput Rfair by optimising the downlinktime-slots tD, the uplink time-slots tU and the power splittingratios ρ of the UEs as well as the transmit beamformer wtof the H-BS. The constraint (33a) defines the uplink fair-throughput Rfair, which is not higher than the uplink through-put of every UE. Other constraints are the same as our uplinksum-throughput maximisation problem (P1). Furthermore,based on the SDR method, we also relax the inherent rankconstraint of (33e) by letting Wt = wtw∗t and by substitut-ing (33e) with Tr(Wt ) ≤ 1.Unfortunately, the relaxed problem (P4) is still non-

convex, due to the coupled parameters {tD, tU , ρ,Wt }.By exploiting the SCA method, the relaxed problem (P4)can be decomposed into the following pair of sub-problems.Given a feasible transmit beamforming matrix Wt satisfyingTr(Wt ) ≤ 1, the first sub-problem aims for maximisingthe uplink fair-throughput by finding the optimal resourceallocation scheme {tD‡, tU‡, ρ‡

|Wt } in the time- and power-domain, which is formulated as

(P5): maxtD,tU ,ρ

Rfair (34)

s. t. RUi (tD, tUi , ρi|Wt ) ≥ Rfair,∀i = 1, · · · ,K , (34a)

RDi (tDi , ρi|Wt ) ≥ Di,∀i = 1, · · · ,K , (34b)

(33c) and (33d).

Given a feasible resource allocation scheme {tD, tU , ρ} inthe time- and power-domain satisfying the constraints (33c)and (33d), the second sub-problem aims for maximising theuplink fair-throughput by finding the optimal transmit beam-forming matrix {W‡

t |tD, tU , ρ} in the spatial-domain, whichis formulated as

(P6): maxWt

Rfair (35)

s. t. RUi (Wt |tD, tUi , ρi) ≥ Rfair,∀i = 1, · · · ,K , (35a)

RDi (Wt |tDi , ρi) ≥ Di,∀i = 1, · · · ,K , (35b)

Tr(Wt ) ≤ 1. (35c)

Moreover, the existence of the optimal resource allo-cation scheme for maximising the uplink fair-throughputRfair can also be validated by exploiting the methodologyintroduced in Section III-A. The resultant transmit beam-forming matrix W‡

t,0 can also be relied upon for initialisingthe iterative algorithm for solving (P4), which will be detailedin Section IV-C.

In the rest of this section, we will elaborate on how wejointly solve (P5) and (P6) in order to obtain the optimalresource allocation scheme {tD†, tU†, ρ†,W†

t }.

A. RESOURCE ALLOCATION IN TIME- ANDPOWER-DOMAINThe sub-problem (P5) is still non-convex, due to the cou-pled parameters tD and ρ. We then introduce a set of newparameters µ = {µi = tDi (1 − ρi)|i = 1, · · · ,K } andreformulate (P5) as

(P5-1): maxtD,tU ,µ

Rfair (36)

s. t. RUi (tD, tUi , µi|Wt ) ≥ Rfair,∀i = 1, · · · ,K , (36a)

RDi (tDi , µi|Wt ) ≥ Di,∀i = 1, · · · ,K , (36b)

0 ≤ µi ≤ tDi ,∀i = 1, · · · ,K , (36c)K∑i=1

(tDi + tUi ) ≤ Tt . (36d)

Since both the objective (36) and the constraints (36a)-(36d)are either convex or affine with respect to {tD, tU ,µ},(P5-1) is a convex optimisation problem. Note that thedownlink throughput RDi (t

Di , µi|Wt ) is a monotonously

increasing function of tDi , while the uplink throughputRUi (t

D, tUi , µi|Wt ) is also a monotonously increasing func-tion of tD and tUi . As a result, increasing the duration of theactual transmission phase T ′t =

∑Ki=1 (t

Di + t

Ui ) may increase

the fair throughput Rfair. In order to reduce the complexity ofdirectly solving (P5-1), given a specific Rfair, we formulatethe following convex optimisation problem:

(P5-2): mintD,tU ,µ

T ′t =K∑i=1

(tDi + tUi ), (37)

s. t. (36a), (36b) and (36c).

If the resultant minimum duration of the actual transmissionphase T ′t,min is higher than Tt , the current fair-throughput Rfairis not feasible and it has to be reduced. By contrast, if we haveT ′t,min < Tt , the current fair-throughput Rfair is feasible but ithas to be increased in order to achieve its maximum. Let usnow focus on the details of maximising Rfair.

By exploiting the Lagrangemultipliermethod, the Lagrangefunction of (P5-2) can be formulated as

L(tD, tU ,µ,λ, ξ )

=

K∑i=1

(tDi + tUi )+

K∑i=1

λi[Rfair − RUi (tD, tUi , µi|Wt )]

+

K∑i=1

ξi[Di − RDi (tDi , µi|Wt )], (38)

where λ = {λi|i = 1, · · · ,K } and ξ = {ξi|i = 1, · · · ,K } arethe Lagrange multipliers correspond to the constraints (36a)and (36b). Given a specific range of λ, the optimal solution{tD′

, tU′

,µ′} of (P5-2) has to satisfy the sameKKT conditionsgiven by Eq.(24) and thus they can be expressed as Eq.(25).

47172 VOLUME 6, 2018


However, the parameters pi and qi have different relationshipswith λ, which are expressed as

log(1+ pi)−pi

1+ pi=

1λi, (39)

(1+ qi) log(1+ qi)− qi =1+ piλigi

−

∑j 6=i λjγjgjλigi

. (40)

Therefore, the optimal pair of p‡i and q‡i can be derivedby solving Eqs.(39) and (40). Similar to Eqs.(26) and (27),the parameters pi and qi are also monotonously increasingwith respect to the corresponding λi. As a result, the optimalpair {p‡i , q

‡i } can be obtained by invoking the classic bisection

method on Eqs.(39) and (40), respectively.Furthermore, the sub-gradient of the Lagrange function

L(tD‡, tU‡,µ‡,λ, ξ ) with respect to λ is denoted as δ =

{δi|i = 1, · · · ,K }, whose i-th element is expressed as

δi =∂L∂λi= tU‡

i log(1+ p‡i )− Rfair. (41)

After the n-th iteration, the Lagrange multiplier set is updatedas λ(n+1)

= λ(n)− 1λδ, where 1λ is the searching step.

The iteration for obtaining the optimal Lagrange multiplierset λ‡ completes until we have |δ| ≤ εδ , where εδ representsthe absolute error tolerance on the sub-gradient.

We firstly initialise Rfair of (P5-2) with an arbitrary value.After solving (P5-2), we obtain the minimum duration of thetransmission phase T ′min,t . If we have T

′

min,t < Tt , we have toincrease Rfair and solve (P5-2) again. If we have T ′min,t > Tt ,we have to reduce Rfair and solve (P5-2) again. The binarysearching based algorithm is exploited for finding the opti-mal fair-throughput R‡fair. The details of the algorithm forsolving (P5-1) is provided in Algorithm 4. Note that similarmethodology has been used for obtaining the optimal time-domain resource allocation scheme for maximising the fair-throughput in our previous work [28]. However, the binarysearching based Algorithm 4 proposed in this paper sub-stantially reduces the complexity of the time- and power-domain resource allocation, when compared to the algorithmproposed in [28].

B. RESOURCE ALLOCATION IN SPATIAL-DOMAINGiven a specific resource allocation scheme {tD, tU , ρ},the fair-throughput maximisation problem (35) is convexwith respect to the transmit beamforming matrix Wt , afterthe SDR. Note that the uplink throughput RUi (Wt |tD, tUi , ρ)is a monotonously increasing function of Tr(Wt ) and so is thedownlink throughput RDi (Wt |tDi , ρi). As a result, increasingthe normalised power weights Tr(Wt ) of the transmit anten-nas is capable of increasing the uplink fair-throughput Rfair.Similarly to the formulation of (P5-2), given a specific Rfair,the following convex optimisation problem is formulated:

(P6-1): minWt

P′t,norm = Tr(Wt ) (42)

s. t. (35a) and (35b)

The problem (P6-1) can be solved by invoking the interior-point method, as we have done for solving (P1-2) and (P3-1).

Algorithm 4 The Binary Searching Based Algorithm forSolving (P5-1)Input: Path loss of {�1, · · · , �K }; Channel fading coeffi-

cients of {h1, · · · ,hK }; Transmit power of the H-BS Pt ;Feasible Transmit beamforming matrixWt ; Noise powerof the converter σ 2

cov; Duration of the transmission phaseTt ; Searching step for the Lagrange multipliers1λ; Errortolerance εδ and εr .

Output: Optimal resource allocation scheme {tD‡, tU‡, ρ‡}

in time- and power-domain; Maximum fair-throughputR‡fair.

1: Initialize Rmin ← 0 and Rmax ← A sufficiently highvalue;

2: while Rmax − Rmin > εr do3: Initialise the fair throughput Rfair←

Rmax+Rmin2 ;

4: Initialize λ← An arbitrary vector;5: Obtain {p‡i |i = 1, · · · ,K } and {q†i |i = 1, · · · ,K } by

solving Eqs.(39) and (40) with the aid of the bisectionmethod;

6: Obtain {tD‡, tU‡, ρ‡} by Eq.(25);

7: Obtain the sub-gradient δ = {δi|i = 1, · · · ,K } of theLagrange function L by Eq.(41);

8: while |δ| > εδ do9: Update λ← λ−1λδ;10: Update {p‡i |i = 1, · · · ,K } and {q†i |i = 1, · · · ,K }

by solving Eqs.(39) and (40);11: Update {tD‡, tU‡, ρ‡

} by Eq.(25);12: Update the sub-gradient δ = {δi|i = 1, · · · ,K } of

the Lagrange function L by Eq.(41);13: end while14: Obtain T ′min,t ←

∑i=1 K (tD‡i + t

U‡i );

15: if T ′min,t ≤ Tt then16: Rmax← Rfair;17: else18: Rmin← Rfair;19: end if20: end while21: Obtain ρ‡

← {ρ‡i =

µ‡i

tD‡i|i = 1, · · · ,K };

22: return {tD‡, tU‡, ρ‡} and R‡fair.

We then transform the original constrained problem (P6-1) tothe following unconstrained problem (P6-2):

(P6-2): minWt

f3(Wt )

= Tr(Wt )− φ(Det(Wt ))−K∑i=1

φ(Tr(h∗i hiWt )− ai

)−

K∑i=1

φ(Tr(h∗i hiWt )− ci

). (43)

In Eq.(43), we have ci = (eRfairtUi −1)bi and bi has been defined

in Eq.(31). The unconstrained problem (P6-2) can thus be

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Algorithm 5 The Binary Searching Based Method forSolving (P6)Input: Path loss of {�1, · · · , �K }; Channel fading coef-


cov; Duration of thetransmission phase Tt ; Environment settings; FeasibleTransmit beamforming matrix Wt ; Resource allocationscheme {tD, tU‡, ρ‡

} in time- and power-domain Errortolerance ε.

Output: Optimal transmit beamforming matrixW‡t ; Max-

imum fair-throughput R‡fair.1: Initialise Rmin ← 0 and Rmax ← a sufficiently high

value;2: Initialise W‡

t ←Wt ;3: while Rmax − Rmin > ε do4: Update R‡fair←

Rmax+Rmin2 .

5: Update W‡t by solving (P6-2) with the aid of Algo-

rithm 1;6: if Tr(W‡

t ) > 1 then7: Update Rmax← Rfair8: else9: Update Rmin← Rfair10: end if11: end while12: return W‡

t and R‡fair.

solved by exploiting the gradient descent method. Given afeasibleWt , the gradient of f3(Wt ) can be derived as

∇Wt f3(Wt ) = I−1m

[Wt +

K∑i=1

(h∗i hiWt

Tr(h∗i hiWt )− ai

+h∗i hiWt

Tr(h∗i hiWt )− ci

)]. (44)

The optimal searching step is obtained as

θ = argminθf3(Wt − θ ·1Wt ), (45)

where 1Wt = ∇Wt f3(Wt ) is the searching direction. As aresult, after the n-th iteration, the transmit beamformingmatrix is updated as W(n+1)

t = W(n)t − θ · 1Wt . Therefore,

the optimal transmit beamforming matrix W′‡t in the spatial

domain and the resultant P′min,t,norm = Tr(W′‡t ) of (P6-1) can

be then obtained by exploiting Algorithm 1.If we have P′min,t,norm < 1, we may increase Rfair

to better utilise the spatial-domain resource. If we haveP′min,t,norm > 1, we may reduce Rfair because the spatial-domain resource cannot satisfy this high fair-throughputrequirement. The binary searching based method can bethen exploited for finding the optimal transmit beamformingmatrix W‡

t and the maximum fair-throughput R‡fair, which isdetailed in Algorithm 5.

Algorithm 6 SCA Based Algorithm for Solving (P4)Input: Path loss of {�1, · · · , �K }; Channel fading coef-


cov; Duration of thetransmission phase Tt ; Error tolerance ε.

Output: Optimal resource allocation scheme{tD†, tU†, ρ†,W†

t } in time- power- and spatial-domain;Maximum uplink fair-throughput R†fair.

1: Initialise the transmit beamforming matrix Wt ← I/Ntin Algorithm 1;

2: Obtain a feasible transmit beamforming matrixW‡t,0 and

the resultant minimum total duration TD‡min of the down-link time-slots by solving (P1-3) based on Algorithm 1;

3: if TD‡min < Tt then4: Initialise an uplink fair-throughput recorder Rfair,0 ←

0;5: Initialise another uplink fair-throughput recorder

Rfair,1← 2ε;6: Initialise the transmit beamforming matrix Wt ←


7: Initialise the transmit beamforming matrix Wt ←

W‡t,0 in Algorithm 5 for solving (P6);

8: while Rfair,1 − Rfair,0 > ε do9: Update Rfair,0← Rfair,1;10: Obtain the optimal solution {tD‡, tU‡, ρ‡

} to (P5-1)based on Algorithm 4;

11: Update {tD, tU , ρ} ← {tD‡, tU‡, ρ‡} in Algo-

rithm 5 for solving (P6);12: Obtain the optimal solution W‡

t to (P6) and theresultant uplink fair-throughput R‡fair based on Algo-rithm 5;

13: Update Rfair,1← R‡fair;14: Update Wt ← W‡

t in Algorithm 4 for solving (P5-1) and in Algorithm 5 for solving (P6);

15: end while16: Update {tD†, tU†, ρ†,W†

t } ← {tD‡, tU‡, ρ‡,W‡t },

R†fair← R‡fair;17: else18: No feasible resource allocation scheme.19: end if20: return {tD†, tU†, ρ†,W†

t } and R†fair.

C. SCA BASED ALGORITHMThe optimal resource allocation scheme {tD†, tU†, ρ†,W†

t }

to (P4) can then be obtained by alternatively solving(P5-1) and (P6), which is detailed in Algorithm 6. Note thatthe existence of a feasible resource allocation scheme to (P4)is validated by solving (P1-2) in Algorithm 1. This may sub-stantially increase the robustness and reduce the complexityof the SCA based Algorithm 6.

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V. NUMERICAL RESULTSIn this section, numerical results are provided for character-ising the performance of our SCA based multi-dimensionalresource allocation algorithms. The transmit power of theH-BS is set to Pt = 30 dBm, which is not altered unlessspecific explanation. The power of the noise incurred by thepassband to baseband converter is set to σcov = −40 dBm.As shown in Eq.(2), the near-field reference distance of thetransmit antennas is set to d0 = 1 m. The RF signal propa-gating from the transmit antennas to this reference distancesuffers from the path-loss of �0 = 30 dB. The path-lossexponent is set to α = 2. The duration of the transmissionphase is Tt = 1 s. We consider a perfect RF-DC energyconversion by letting ζ = 1. Furthermore, uncorrelatedRayleigh block fading is assumed, which remains constantduring a single transmission frame T , as depicted in Fig.2,but varies from one transmission frame to another. Both thereal and imaginary parts of the channel fading coefficientobey a Gaussian distribution with a zero mean and a unityvariance. Without further explanation, the numerical resultsin this section present the average uplink throughput perfor-mance by averaging the randomness incurred by the Rayleighdistributed fading. The error tolerance of the SCA basedAlgorithms 3 and 6 is set to ε = 10−5. The main parametersettings are summarised in TABLE 1.

TABLE 1. Generic parameter settings.

A. CONVERGENCE OF THE SCA ALGORITHMWe firstly study the convergence of our proposed SCA basedresource allocation algorithms in Fig.3, which are adoptedby the H-BS for supporting the downlink/uplink transmis-sions of 5 UEs. The H-BS is equipped with 3 antennas. Thedistances from these UEs to the H-BS are {2, 2, 2, 3, 3} m,respectively. We assume that the UEs have identical down-link throughput requirement D ∈ {0, 0.5, 1.0, 2.0} bit/Hz.When we have D = 0 bit/Hz, our IDEN system degener-ates to a conventional WPCN. Other parameters are in linewith TABLE 1.

Then, we plot the number of iterations2 against the max-imum uplink sum-throughput and fair-throughput in Fig.3.Note that the expected uplink sum-throughput and fair-throughput in Fig.3 are evaluated by averaging the random-ness incurred by the Rayleigh distributed fading. As indicated

2We only quantify the number of iterations in the SCA based sum-throughput maximisation Algorithm 3 and fair-throughput maximisationAlgorithm 6.

in its legend, we study the SCA based resource allocationalgorithms with and without the optimal initialisation ofthe H-BS’s transmit beamformer W‡

t,0 by solving (P1-1).For the cases without the optimal transmit beam-former initialisation, the equal-gain transmit beamformerWt,0 = I/Nt is adopted. Furthermore, in the case of WPCNhaving D = 0, the equal-gain transmit beamformer is alsoadopted for initialising the SCA based resource allocationalgorithm.

Observe from Fig.3(a) that all the SCA based resourceallocation algorithms having different downlink through-put requirements quickly converge to the maximum uplinksum-throughput by a single iteration or two. Specifically,the algorithms initialised by the equal-gain transmit beam-former converges after two iterations. By contrast, afterbeing initialised by the optimal transmit beamformer W‡

t,0by solving (P1-1), the SCA based resource allocationAlgorithm 3 converges to the maximum uplink sum-throughput by only a single iteration.

FIGURE 3. The convergence of the SCA based multi-dimensional resourceallocation algorithm: (a) Uplink sum-throughput maximisation; (b) Uplinkfair-throughput maximisation.

We observe the similar trend from Fig.3(b) that all theSCA based resource allocation algorithms having differentdownlink throughput requirements converge to the maxi-mum uplink fair-throughput by a single iteration or two.Specifically, the algorithms initialised by the equal-gaintransmit beamformer converges after two iterations. By con-trast, after being initialised by the optimal transmitbeamformerW‡

t,0 by solving (P1-1), the SCA based resourceallocation Algorithm 6 converges to the maximum uplinkfair-throughput by only a single iteration.

The numerical results of Fig.3 validate the low-complexityof our SCA based multi-dimensional resource allocationalgorithm, which can be practically invoked by the H-BS forquickly responding to the dynamic information and energyrequests of the UEs in both the conventional WPCN and thegeneric IDEN. Furthermore, the low-complexity algorithmplays an important role in realising the low-latency WIT andWPT in the future IDEN.

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B. INDIVIDUAL UPLINK THROUGHPUTIn order to highlight the difference between the uplinksum-throughput and fair-throughput maximisation, we plotthe individual uplink throughput attained by 5 UEs in theIDEN during a specific transmission frame, when the sum-throughput and fair-throughput maximisation based resourceallocation schemes are invoked, respectively. The H-BS isequipped with 4 antennas. The UEs’ downlink throughputrequirements are all D = 1 bit/Hz. The distances betweenthese UEs and the H-BS are {2, 2, 2, 3, 3} m. During thistransmission frame, the complex downlink channel fadingcoefficients are given by the matrix H of Eq.(46), as shownat the bottom of the next page. By assuming the channelreciprocity, the uplink channel fading coefficients can berepresented by HT . Other parameter settings are in linewith TABLE 1.

Observe from Fig.4 that the attainable individual uplinkthroughput of the UEs varies, when the resource allocationaims for maximising the sum-throughput. For instance, moreresources in the time- power- and spatial-domain are allo-cated to U3, which may result in a fast growth of the uplinksum-throughput, since it is close to the H-BS and it hasa higher channel fading coefficients. However, as depictedin Fig.4, the uplink sum-throughput maximisation results inthe unfairness among the UEs in terms of their individualuplink throughput attained. For instance,U3 attains an uplinkthroughput as high as 2.96 bit/Hz but U5 can only attain anuplink throughput of 0.22 bit/Hz.

FIGURE 4. The individual uplink throughput of the UEs, when differentobjectives are invoked for the resource allocation.

The unfairness among the UEs may be addressed bymaximising the uplink fair-throughput. Observe from Fig.4that by maximising the uplink fair-throughput, the multi-dimensional resources aremore fairly allocated to all theUEs.As a result, every UEmay attain an identical individual uplinkthroughput of 0.84 bit/Hz. However, the performance of theuplink sum-throughput is inevitably sacrificed.

As a result, the numerical results of Fig.4 demonstratethat the uplink fair-throughput maximisation based resource

allocation scheme is capable of overcoming the doubly near-far effect.

C. MULTIPLE TRANSMIT ANTENNASWe also study the impact of the transmit antennas on theuplink throughput in Fig.5, where 3 UEs are supportedby the H-BS. The distances from these three UEs to theH-BS are {2, 2, 3} m, while their downlink throughputrequirements are all D = 1 bit/Hz. Other parameters are inline with TABLE 1.

FIGURE 5. The impact of the transmit antennas on the uplink throughputperformance: (a) uplink sum-throughput; (b) uplink fair-throughput.

We firstly characterise the uplink sum-throughput ofboth the sum-throughput maximisation oriented and thefair-throughput maximisation oriented resource allocationschemes in Fig.5(a). Obviously, increasing the number

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of transmit antennas of the H-BS increases the degreeof freedom in the spatial domain. Therefore, the uplinksum-throughput is substantially increased. Furthermore,the fair-throughput maximisation oriented resource alloca-tion scheme sacrifice its sum-throughput performance forachieving the optimal fairness. By letting D = 0 bit/Hz, ourIDEN degenerates to a conventional WPCN. Since the UEsin a WPCN do not require any downlink data, the downlinkresources can be fully exploited for delivering more energy tothe UEs. As a result, the uplink sum-throughput of theWPCNoutperforms its IDEN counterpart.

We then evaluate the uplink fair-throughput of boththe sum-throughput maximisation oriented and the fair-throughput maximisation oriented resource allocationschemes in Fig.5(b). Observe from Fig.5(b) that increas-ing the number of the H-BS’s transmit antennas iscapable of enhancing the fair-throughput. Furthermore,the fair-throughput of the sum-throughput maximisation ori-ented resource allocation scheme is very poor, due to thedoubly near-far effect.

The numerical results of Fig.5 demonstrate that the sum-throughput maximisation and the fair-throughput maximisa-tion are a pair of contradictory objectives, which should becarefully balanced in order to satisfy both the H-BS’s andthe UEs’ benefits. These numerical results also validate thatincreasing the spatial-domain degrees of freedom is capableof improving the throughput performance.

D. TRANSMIT POWER OF THE H-BSWe then plot the transmit power of the H-BS against thethroughput performance in Fig.6, when the resource allo-cation schemes having different objectives are invoked anddifferent number of antennas are equipped at the H-BS.There are 3 UEs supported by the H-BS. The distances fromthese UEs to the H-BS are {2, 2, 3} m, while their downlinkthroughput requirements are all D = 1 bit/Hz. Other param-eter settings are in line with TABLE 1.

The trend observed in Fig.6 is similar to that observedin Fig.5. Both the uplink sum-throughput and fair-throughputincrease as we increase the number of antennas equippedat the H-BS. Specifically, observe from Fig.6(a) that thesum-throughput of the sum-throughput maximisation ori-ented resource allocation scheme almost increases linearlyas we increase the transmit power of the H-BS. By contrast,the sum-throughput of the fair-throughput maximisation ori-ented resource allocation scheme gradually converge as wecontinuously increase the transmit power of the H-BS. Thisis because in order to guarantee the fairness, although the

FIGURE 6. The impact of the transmit power on the uplink throughputperformance: (a) uplink sum-throughput; (b) uplink fair-throughput.

H-BS has a higher transmit power, the transmit beam mayfocus more energy to the UEs having poor channels, whichlargely restricts the growth of the sum-throughput.

Furthermore, observe from Fig.6(b) that the fair-throughput of the fair-throughput maximisation orientedresource allocation scheme steadily increases as we increase

H =

h1h2h3h4h5

=−1.1480− 0.6003i, 0.1873− 2.1384i, 0.8404+ 0.1240i0.1049+ 0.4900i, −0.0825− 0.8396i, −0.8880+ 1.4367i0.7223+ 0.7394i, −1.9330+ 1.3546i, 0.1001− 1.9609i2.5855+ 1.7119i, −0.4390− 1.0722i, −0.5445− 0.1977i−0.6669− 0.1941i, −1.7947+ 0.9610i, 0.3035− 1.2078i

. (46)

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the transmit power of the H-BS. By contrast, the fair-throughput growth of the sum-throughput maximisation ori-ented resource allocation scheme is quite limited. This isbecause although the H-BS has a higher transmit power,the power may be beamed to the UEs having good channelquality in order to maximise the sum-throughput by somehowsacrificing the fairness among the UEs.

The numerical results of Fig.6 demonstrate that having ahigher transmit power, the H-BS may transfer more energyto the UEs during the downlink transmission, which largelyimprove the uplink throughput of the UEs.

E. DOWNLINK THROUGHPUT REQUIREMENTAt last, we study the impact of the downlink throughputrequirement on the throughput performance of our multi-dimensional resource allocation scheme. We have 3 UEs sup-ported by the H-BS equipped with 4 antennas. The distancesfrom these UEs to the H-BS are {2, 2, 3} m. All the 3 UEshave identical downlink throughput requirement D varyingfrom 0 bit/Hz to 4 bit/Hz. Other parameter settings are in linewith TABLE 1.

Observe from Fig.7 that as the downlink throughputrequirement increases, both the uplink sum-throughput andthe fair-throughput reduces. The reason is three-fold:

• In order to satisfy the demanding downlink throughputrequirement, more time-domain resources may be allo-cated to the downlink transmission, which thus reducesthe time-domain resources allocated to the UEs for theiruplink transmissions.

• More spatial-domain resources (e.g. the transmit beam)is allocated for satisfying the UEs’ downlink through-put requirement, which thus reduces the spatial-domain resources used for the WPT in the downlinktransmission.

FIGURE 7. The impact of the downlink throughput requirement on theuplink throughput.

• For the sake of satisfying the increasing downlinkthroughput requirement, the UEmay use a larger portionof the received RF signal for information reception,which thus reduce the amount of energy harvested forpowering their uplink transmissions.

Moreover, given the current parameter settings, the H-BS iscapable of satisfying the downlink throughput requirementas high as 4 bit/Hz but the uplink transmissions of the UEsare completely sacrificed. In this case, our IDEN degeneratesto the pure downlink WIT system. Furthermore, observefrom Fig.7 that when we have the downlink throughputrequirement lower than 0.5 bit/Hz, the uplink throughput onlyreduces a little, compared to the conventional WPCN.

The numerical results of Fig.7 characterise the trade-offbetween the downlink throughput requirement and the uplinkthroughput performance. Furthermore, they also demonstratethat the H-BS is capable of delivering low-rate downlink dataservice to the UEs without obviously degrading their uplinkthroughput. This characteristic can be exploited for deliveringthe low-rate commanding information to the IoT devices,while powering them for their uplink transmissions.

VI. CONCLUSIONIn a H-BS aided IDEN system studied in this paper, the UEsharvest energy and receive information from the downlinktransmission of the H-BS and their uplink transmissions arepowered by the energy harvested. The optimal resource allo-cation schemes in the time- power- and spatial domains areobtained by solving the uplink sum-throughput maximisa-tion and uplink fair-throughput maximisation problems. TheSCA based algorithms of low complexity are relied upon foriteratively approaching the optimal solutions. The numericalresults demonstrate that our SCAbased algorithmsmay effec-tively return us the optimal resource allocation solutions afteronly a single or two iterations, which is capable of realisingthe low-latencyWIT andWPT in the IDEN. Furthermore, ourIDEN is suitable for the IoT system having asymmetric down-link and uplink traffic load because the numerical resultsdemonstrate that having a downlink throughput requirementlower than 0.5 bit/Hz, the uplink throughput performancemay not be substantially degraded.

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JIANJUN YANG received the bachelor’s degreein information engineering and automation fromChongqing University, China, in 1994, and themaster’s degree in communication and informa-tion systems from the University of ElectronicScience and Technology of China in 2003. Heis currently a Lecturer with the Department ofNetwork Engineering, School of Information andCommunication Engineering, University of Elec-tronic Science and Technology of China. His

research interest is in the domain of resource allocation and optimization inenergy harvesting wireless networks and multitier and cooperative cellularnetworks.

JIE HU (S’11–M’16) received the B.Eng. andM.Sc. degrees from the Beijing University of Postsand Telecommunications, China, in 2008 and2011, respectively, and the Ph.D. degree fromthe Faculty of Physical Sciences and Engineer-ing, University of Southampton, U.K., in 2015.Since 2016, he has been with the School of Infor-mation and Communication Engineering, Univer-sity of Electronic Science and Technology ofChina, China, as an Associate Professor. He has a

broad range of interests in wireless communication and networking, suchas cognitive radio and cognitive networks, mobile social networks, inte-grated data and energy communication networks, and communication andcomputation convergence. His current research is funded by the NationalNatural Science Foundation of China. He is also in great partnership withindustry, such as Huawei and ZTE. He is also a regular TPC member ofseveral prestigious IEEE conferences, such as IEEE ICC and GLOBECOM.He has served for ZTE Communications as the guest editor of the specialissue Wireless Data and Energy Integrated Communication Networks.

KESI LV received the B.S degree from the Uni-versity of Electronic Science and Technology ofChina, China, in 2016, where he is currently pur-suing the master’s degree. His research interestsinclude integrated data and energy communicationnetworks and convex optimization.

VOLUME 6, 2018 47179


QIN YU received the B.S. degree in communi-cation engineering from the Chongqing Univer-sity of Posts and Telecommunications in 1996,and the M.S. and Ph.D. degrees in communicationand information engineering from the Universityof Electronic Science and Technology of China(UESTC) in 2002 and 2006, respectively. Shejoined the School of Information and Communica-tion Engineering, UESTC, in 2007. From 2007 to2009, she conducted post-doctoral research with

Prof. Z. Qin in information security at UESTC. Her current research inter-ests include data and energy integrated communication networks, wirelesscommunication and information security.

KUN YANG (SM’08) received the B.Sc. andM.Sc.degrees from the Computer Science Department,Jilin University, China, and the Ph.D. degree fromthe Department of Electronic and Electrical Engi-neering, University College London (UCL), U.K.He is currently a Chair Professor with the Schoolof Computer Science & Electronic Engineering,University of Essex, leading the Network Conver-gence Laboratory, U.K. He is also an affiliatedprofessor at UESTC, China. Before joining in Uni-

versity of Essex at 2003, he was at UCL on several European Union (EU)research projects for several years. He has published over 100 journal papers.His main research interests include wireless networks and communications,data and energy integrated networks, and computation-communication coop-eration. He manages research projects funded by various sources such asU.K. EPSRC, EU FP7/H2020, and industries. He serves on the editorialboards of both the IEEE and non-IEEE journals. He has been a fellow ofIET since 2009.

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