NOMA Throughput and Energy Efficiency in Energy HarvestingEnabled Networks

Nasir, A. A., Tuan, H. D., Duong, T. Q., & Debbah, M. (2019). NOMA Throughput and Energy Efficiency inEnergy Harvesting Enabled Networks. IEEE Transactions on Communications.https://doi.org/10.1109/TCOMM.2019.2919558

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NOMA Throughput and Energy Efficiency inEnergy Harvesting Enabled Networks

Ali A. Nasir, Hoang D. Tuan, Trung Q. Duong, and Merouane Debbah

Abstract—An energy harvesting (EH) enabled network iscapable of delivering energy to users, who are located suffi-ciently close to the base stations. However, the wireless energydelivery requires much more transmit power than what thenormal information delivery does. It is very challenging toprovide the quality of the wireless information and power de-livery simultaneously. It is of practical interest to employ non-orthogonal multiple access (NOMA) to improve the networkthroughput, while fulfilling the EH requirements. To realizeboth EH and information decoding, this paper considers atransmit time-switching (transmit-TS) protocol. Two impor-tant problems of users’ max-min throughput optimizationand energy efficiency maximization under power constraintand EH thresholds, which are non-convex in beamformingvectors, are addressed by efficient path-following algorithms.In addition, the conventional power splitting (PS)-based EHreceiver is also considered. The provided numerical resultsconfirm that the proposed transmit-TS based algorithmsclearly outperform the PS-based algorithms in terms of both,throughput and energy efficiency.

Index Terms—Wireless power delivery, energy harvest-ing, non-orthogonal multiple access (NOMA), transmit time-switching, nonconvex optimization, throughput, energy effi-ciency, quality-of-service (QoS).

I. INTRODUCTION

A. Motivation

Radio frequency (RF) energy harvesting (EH) [1]–[3]has emerged as a potential technology to energize the Inter-net of things. To realize both wireless EH and informationdecoding (ID), the users need to split the received signal forEH and ID either by power splitting (PS) or time switching(TS), where the latter is referred to as “receive-TS” [1], [4],[5]. PS has been shown mostly outperforming receive-TS

This work was supported in part by Vietnam National Foundation forScience and Technology Development (NAFOSTED) under Grant No.102.04-2017.301, in part by the KFUPM Research Project #SB171005,in part by the Australian Research Council’s Discovery Projects underProject DP190102501, and in part by the U.K. Royal Academy ofEngineering Research Fellowship under Grant RF1415\14\22.

Ali A. Nasir is with the Department of Electrical Engineering, KingFahd University of Petroleum and Minerals (KFUPM), Dhahran, SaudiArabia (email: anasir@kfupm.edu.sa)

Hoang D. Tuan is with the School of Electrical and Data Engineer-ing, University of Technology, Sydney, NSW 2007, Australia (email:tuan.hoang@uts.edu.au).

Trung Q. Duong is with the School of Electronics, Electrical Engineer-ing and Computer Science, Queen’s University Belfast, Belfast BT7 1NN,United Kingdom (e-mail: trung.q.duong@qub.ac.uk).

Merouane Debbah is with the Mathematical and Algorithmic SciencesLaboratory, France Research Center, Huawei Technologies, France (email:merouane.debbah@huawei.com).

but it is complicated and inefficient for practical imple-mentation. Recent findings in [4], [6] and [7] demonstratethe advantages of the new “transmit-TS” approach over thePS approach, where information and energy are transferredseparately so the users do not need any sophisticated devicefor EH.

For information transmission, one of the most criticaltasks is to provide the quality of service (QoS) in termsof throughput to those users, who are located far from theBS. Non-orthogonal multiple access (NOMA) technique(see e.g. [8] and [9]) is able to improve the throughputat these users by allowing the near-by users to access theinformation intended for them. NOMA is also capable ofsupporting numbers of users that are more than the numberof available orthogonal time-, frequency-, or code-domainresources, so it is considered as a potential candidate tosatisfy the radical spectral efficiency and massive connec-tivity requirements of 5G [10]. An efficient beamformingdesign for NOMA multicell systems was proposed in [11].

B. Literature Survey

A wireless powered communication network, wherethe BS charges users and enables them to transmit in-formation for uplink communication by using NOMA,was considered in [12] and [13]. A similar study for awireless-powered sensor network was conducted in [14].The trade-off among the energy efficiency, fairness, har-vested energy, and system sum rate of a NOMA basedheterogeneous network was investigated in [15]. Optimalresource allocation strategies for cognitive radio networkswith NOMA were designed in [16]. Wireless power transferand NOMA in machine-to-machine communication, wherethe machine-type communication device harvests energyin the downlink while transmitting information to the BSvia machine-type communication gateway in the uplink,were considered [17] and [18]. All these works consideredproblems for single antenna nodes, which either harvestenergy or decode information but not implement both EHand ID.

In EH based cooperative NOMA systems, the cell-centered or “nearly-located users” harvest energy from thewireless signals received from the BSs and act as a relayto forward the information to the “far-located users”. Mostof works used PS at the relay users. Outage expressionsfor far end users were derived in [19]–[22]. Different users

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pairing schemes for NOMA were proposed in [23]. Theproblem of maximizing the data rate of the strong userwhile satisfying the QoS requirement of the weak user wasconsidered in [24]. Antenna selection schemes at the BSwere proposed in [21].

A hybrid receive-TS/PS for EH at the relay users wasproposed in a recent work [25], while a simple TS modeat the relay receiver, where half of the time is dedicatedfor EH, was considered in [26].

Exploiting wireless EH in NOMA systems, the work[27] considered various energy harvesting protocols, e.g.,conventional receive-TS, PS and generalized (hybridTS/PS) under a setup of a single antenna BS and a singleuser pair. Wireless power transfer by PS and informationtransmission in a NOMA system was also studied in [28]under a simple setup with a single antenna BS. The work[29] considered an energy-constrained full-duplex informa-tion transmitter, which harvests energy from a dedicatedenergy transmitter while transmitting information to theinformation receivers by employing NOMA. A cognitiveradio based wireless information and power transfer net-work with NOMA was considered in [30].

C. Research Gap and Contribution

Recently, throughput optimization with multi-antennabeamforming and PS based EH in simple NOMA systemswith one BS serving one nearly-located user and one far-located user, was addressed in [24]. Like [29] and [30],semidefinite relaxation (SDR) is its tool for computation,which is quite inefficient because SDR has to first solve formatrix optimization of much higher dimension [6], [31]. Inaddition, it employed a PS-based approach for EH, whichis also not very practical due to the need of variable PS.Thus, an efficient solution for such important throughputmax-min optimization problem is still missing. Meanwhile,the energy efficiency (EE), which is defined as the sumthroughput per Joule of consumed energy (ratio of sumthroughput to power consumption), is a key performanceindicator for new wireless technologies [32]. Therefore, EEmaximization is an important target for EH enabled NOMAsystems. To the best of authors’ knowledge, the problemof EE maximization by multi-antenna beamforming in EHenabled NOMA network has not been addressed in the lit-erature. This problem is more computationally difficult thanthe throughput maximization problem due to the additionaloptimization variables appearing in the denominator of theEE function [33], [34].

This paper considers multi-antenna beamforming in anEH enabled NOMA network, which delivers informationand energy separately in fractional times, enabling the EHby simple devices. Thus, both throughput and harvestedenergy can be improved by separated information and en-ergy beamformers. We formulate two important problemsof throughput max-min optimization and EE maximizationunder power constraint and EH constraints at the nearly-

located users. As these problems are highly nonconvex,checking their feasibility is already computationally dif-ficult. Nevertheless, like [6], [7] for throughput max-minoptimization and EE in orthogonal multiple access (OMA)EH enabled networks, and [11] for throughput maximiza-tion in NOMA networks, we aim to develop efficient path-following algorithms for their computation. Compared tothese our previous works, the contribution of the paper istwo-fold:

• To address the problem of throughput max-min op-timization with new and much more complex non-convex constraints compared with those in [6], [7],[11], we propose completely new their inner convexapproximations. They lead to path-following algo-rithms of low-computational complexity, which con-verge rapidly to an optimal solution.

• To address the problem of EE maximization, wealso develop new path-following computational pro-cedures, which avoid approximations of the entire EEobjective functions proposed in [7]. As such, theyreveal that this problem is not more computationallycomplex than the problem of throughput maximiza-tion, contrary to the commonly accepted conception[33], [34]

For comparison purpose, we consider both PS-based andtransmit-TS based energy harvesting and propose novelalgorithms to solve above-mentioned novel beamformingproblems. Our numerical results confirm that the proposedtransmit-TS approach clearly outperforms the PS approachin terms of both, throughput and energy efficiency.

D. Organization and Notation

The paper is organized as follows. After a brief in-troduction of system model, Section II presents the for-mulation of throughput and EE maximization problemsand their computational solution for PS-based NOMAimplementation. Section III describes the formulation andsolution of such problems for transmit-TS based NOMAimplementation. Section IV evaluates the performance ofour proposed algorithms by numerical examples. Finally,Section V concludes the paper.

Notation. Bold-faced lower-case letters, e.g., x, are usedfor vectors and lower-case letters, e.g., x, are used for forscalars. xH , xT , and x∗ denote Hermitian transpose, nor-mal transpose, and conjugate of the vector x, respectively.‖ · ‖ stands for the vector’s Euclidean norm. C is the set ofall complex numbers, and ∅ is an empty set.

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Fig. 1. Downlink multiuser multicell interference scenario in a densenetwork consisting of K small cells. For clarity, the intercell interferencechannels are not shown, however, the interference occurs in all K cells.

II. SYSTEM MODEL AND PS-BASED NOMA

Consider a downlink system consisting of N cells underdense deployment, where the BS of each cell is equippedwith Nt antennas to serve 2K single-antenna-equippedusers (UEs) within its cell. In each cell, there are K nearUEs (cell-center UEs), which are located inside the innercircular area and K far UEs (cell-edge UEs), which arelocated in the ring area between inner circle and outerradius. A representative figure for three cells is shownin Fig. 1. The K far UEs in each cell are not only inpoorer channel conditions than other K near UEs but alsoare under more drastic inter-cell interference from adjacentcells.

Upon denoting I , {1, 2, · · · , N} and J ,{1, 2, · · · , 2K}, the j-th UE in the i-th cell is referred toas UE (i, j) ∈ S , I × J . The cell-center UEs are UE(i, j), j ∈ Jc , {1, · · · ,K} while the cell-edge UEs areUE (i, j), j ∈ Je , {K+1, · · · , 2K}. Thus the set of cell-center UEs and the set of cell-edge UEs are Sc , I × Jcand Se , I × Je, respectively. Due their proximity, thecell-center UE (i, j), j ∈ Jc is able to do both informationdecoding and energy harvesting. By exploiting the differen-tiated channel conditions between the cell-center and cell-edge UEs, each cell-center UE (i, j) ∈ Sc is randomlypaired with cell-edge UE (i, p(j)) ∈ Se of the same cell tocreate a virtual cluster to improve the network throughput.1

For notational convenience, the paired UE (i, p(j)) for UE(i, j) is chosen, such that p(j) = j +K.

1This is a practical user-pairing strategy. Using more sophisticateduser-pairing strategies may improve the performance of MIMO-NOMAnetworks (see e.g. [35]) but this is beyond the scope of this paper.

A. PS-based NOMA

For comparison point-of-view with transmit-TS basedNOMA, which will be presented in Section III, let usfirst develop the system model, problem formulation, andsolution approach for PS-based NOMA implementation.

The signal superpositions are precoded at the BSs priorto being transmitted to the UEs. Specifically, the messageintended for UE (i, j) is si,j ∈ C with E{|si,j |2} = 1,which is beamformed by vector wi,j ∈ CNt . The receivedsignals at UE (i, j) and UE (i, p(j)) are expressed as

yi,j =∑

(s,`)∈S

hs,i,jws,`ss,` + ni,j , (1)

and

yi,p(j) =∑

(s,`)∈S

hs,i,p(j)ws,`ss,` + ni,p(j), (2)

where hs,i,j ∈ C1×Nt is the MISO channel from the BSs ∈ I to UE (i, j) ∈ S and ni,j ∼ CN (0, σ) is theadditive noise. In this paper, we assume that full channelstate information is available by some means, e.g., throughcoordination among the BSs [36].

To implement simultaneous wireless information andpower transfer (SWIPT), the power splitter divides thereceived signal yi,j into two parts in the proportion ofαi,j : (1 − αi,j), where αi,j ∈ (0, 1) is termed as thePS ratio for UE (i, j). The first part √αi,jyi,j forms aninput to the ID receiver as:

√αi,jyi,j + z

ci,j =

√αi,j

∑(s,`)∈S

hs,i,jws,`ss,` + ni,j

+ zci,j , (3)where zci,j ∼ CN (0, σc) is the additional noise introducedby the ID receiver circuitry.

The energy of the second part√

1− αi,jyi,j of thereceived signal yi,j is harvested by the EH receiver of UE(i, j) as

Ei,j(w, αi,j) , ζi,j(1− αi,j) (pi,j(w)) . (4)

For notational convenience, we define w , [wi,j ](i,j)∈S ,which constitutes all possible beamforming vectors. Ascan be sensed from (3) and (4), the near-by UEs (i, j)harvest energy through wireless signals not only from theserving BSs but also from the neighboring BSs. Note thatthe harvested and stored energy Ei,j may be used laterfor different power constrained operations at cell-centerUEs (i, j), e.g., assisting uplink data transmission to theBS or performing downlink information processing. In (4),the constant ζi,j ∈ (0, 1) denotes the efficiency of energyconversion at the EH receiver, and

pi,j(w) =∑

(s,`)∈S

|hs,i,jws,`|2. (5)

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In NOMA, the message si,p(j) is decoded by UE (i, j)and (i, p(j)). The interference plus noise at UE (i, j) indecoding si,p(j) is

INci,p(j)(w) , αi,j(∑

(s,`)∈S\{(i,p(j))} |hs,i,jws,`|2

+σ) + σc= αi,j(‖Lci,p(j)(w)‖

2 + σ) + σc,

with

Lci,p(j)(w) , Col[hs,i,jws,`](s,`)∈S\{(i,p(j))}, (6)

which is a linear operator. Therefore, the signal-to-interference-plus-noise (SINR) in decoding si,p(j) at UE(i, j) is |hi,i,jwi,p(j)|2/

(‖Lci,p(j)(w)‖

2 + σ + σc/αi,j

).

Meanwhile, the edge-user UE (i, p(j) decodes its ownmessage si,p(j) only, so the interference plus noise at UE(i, p(j)) is INei,p(j)(w) , ‖Lei,p(j)(w)‖

2 +σ with the linearoperator

Lei,p(j)(w) , Col[hs,i,p(j)ws,`](s,`)∈S\{(i,p(j))}. (7)

In what follows, we use the general rate function ψ(x,y, ν)defined by

ψ(x,y, ν) = ln

(1 +

‖x‖2

‖y‖2 + σ + σcν

). (8)

Suppose that Ri,p(j) is the achievable rate in decodingsi,p(j). Since both near UE (i, j) and far UE (i, p(j)) willdecode the far user’s information si,p(j), thus, Ri,p(j) forUE (i, j) and UE (i, p(j)), respectively, is defined by

min{ψ(hi,i,jwi,p(j),Lci,p(j)(w), 1/αi,j),ψ(hi,i,jwi,p(j),Lei,p(j)(w), 0)}.

Thus Ri,p(j) satisfies the following constraints:

Ri,p(j) ≤ ψ(hi,i,jwi,p(j),Lci,p(j)(w), 1/αi,j), (9)Ri,p(j) ≤ ψ(hi,i,p(j)wi,p(j),Lei,p(j)(w), 0). (10)

UE (i, j) subtracts si,p(j) from the right hand side of (3) indecoding si,j . Then the achievable rate Ri,j by decodingsi,j is ψ(hi,i,jwi,j ,Lci,j(w), 1/αi,j) and is expressed bythe constraint:

Ri,j ≤ ψ(hi,i,jwi,j ,Lci,j(w), 1/αi,j), (11)

with

Lci,j(w) , Col[hs,i,jws,`](s,`)∈S\{(i,p(j)),(i,j)}. (12)

For convenience, we also use the notations

ααα , (αi,j)(i,j)∈SR , (Ri,j)(i,j)∈S .

We consider two basic problems:

1) Throughput max-min optimization

maxw,R,ααα

min(i,j)∈I×J

Ri,j s.t. (9), (10), (11), (13a)

ζi,j (pi,j(w)) ≥emini,j

1− αi,j, i ∈ I, j ∈ Jc, (13b)

0 < αi,j < 1, i ∈ I, j ∈ Jc, (13c)∑j∈J‖wi,j‖2 ≤ Pmaxi , i ∈ I, (13d)

where (13b) defines the EH constraint such that emini,j is theEH threshold and pi,j(w) is defined in (5), and Pmaxi in(13d) is the transmit power budget of BS i.

2) Energy-efficiency maximization under QoS constraints

maxw,R,ααα

F(w,R) ,∑

(i,j)∈I×J Ri,jξπ(w) + Pc

s.t. (9), (10), (11), (13b), (13c), (13d), (14a)Ri,j ≥ ri,j , i ∈ I, j ∈ Jc, (14b)

Ri,p(j) ≥ ri,p(j), i ∈ I, j ∈ Jc, (14c)

where π(w) =∑i∈I∑j∈J ‖wi,j‖2, ξ is the reciprocal

constant power amplifier efficiency, Pc , NtPA+Pcir, PAis the power dissipation at each transmit antenna, Pcir is thefixed circuit power consumption for base-band processing,and (14b) and (14c) are the QoS constraints, such that, ri,j ,i ∈ I, j ∈ J is the threshold rate to ensure a certain QoS.

B. Computational solutions for PS-based NOMA

Let us first address the throughput max-min optimizationproblem (13). We have to resolve the non-convex rateconstraints (9), (10), and (11) and the non-convex EHconstraint (13b). In order to deal with the non-convexity ofrate constraints (9), (10), and (11), we have to provide aconcave lower bounding function for ψ(x,y, 0) defined by(8), at a given point (x(κ),y(κ)) [5], [7]. In the AppendixA, we prove the following new and universal concavefunction bound:

ψ(x,y, µ) ≥ Λ(x,y, µ) (15)

over the trust region

2 0, (16)

with ψ(x(κ),y(κ), µ(κ)) = Λ(x(κ),y(κ), µ(κ)), where

Λ(x,y, µ) ,a(x(κ),y(κ), µ(κ))

− ‖x(κ)‖2

2

5

0 < b(x(κ),y(κ), µ(κ))

=‖y(κ)‖2 + µ̃(κ)

(‖x(κ)‖2 + ‖y(κ)‖2 + µ̃(κ))‖x(κ)‖2, (19)

0 < c(x(κ),y(κ), µ(κ))

=‖x(κ)‖2(

‖x(κ)‖2 + ‖y(κ)‖2 + µ̃(κ)) (‖y(κ)‖2 + µ̃(κ)

)(20)for µ̃(κ) = σ+σcµ(κ). Let (w(κ),R(κ),ααα(κ)) be a feasiblepoint for (13) that is found from the (κ − 1)th iteration,where ααα(κ) , (α(κ)i,j )(i,j)∈S and R(κ) , (R

(κ)i,j )(i,j)∈S .

Applying inequality (15) yields

ψ(hi,i,jwi,p(j),Lci,p(j)(w), αi,j)≥ Λ(hi,i,jwi,p(j),Lci,p(j)(w), 1/αi,j)

over the trust region

2 0, (21)

where the concave functionΛ(hi,i,jwi,p(j),Lci,p(j)(w), 1/αi,j) is defined from (17),(18)-(20) for x(κ) = hi,i,jw

(κ)i,p(j), y

(κ) = Lci,p(j)(w(κ)),

and µ(κ) = 1/α(κ)i,j . As such, the nonconvex constraint (9)is innerly approximated by the convex constraints (21)and

Ri,p(j) ≤ Λ(hi,i,jwi,p(j),Lci,p(j)(w), 1/αi,j). (22)

Analogously, the nonconvex constraints (10) and (11) areinnerly approximated by the convex constraints

Ri,p(j) ≤ Λ(hi,i,p(j)wi,p(j),Lei,p(j)(w), 0), (23)Ri,j ≤ Λ(hi,i,jwi,j ,Lci,j(w), αi,j), (24)

2 0, (25)

2 0, (26)

where the concave functionsΛ(hi,i,p(j)wi,p(j),Lei,p(j)(w), 0) is defined from (17)-(20)for

(x(κ),y(κ), µ̃(κ)) = (hi,i,p(j)w(κ)i,p(j),L

ei,p(j)(w

(κ)), σ),

while the concave functions Λ(hi,i,jwi,j ,Lci,j(w), 1/αi,j)is defined from (17)-(20) for

(x(κ),y(κ), µ̃(κ)) = (hi,i,jw(κ)i,j ,L

ci,j(w

(κ)), σ + 1/α(κ)i,j ).

Next, by using the following approximation [7]

|hHs,i,jx|2 ≥ −|hHs,i,jx|2 + 2<{(

x(κ))H

hs,i,jhHs,i,jx

},

an inner convex approximation for the nonconvex EHconstraint (13b) is given by

ζi,j∑

(s,`)∈S

(2 min(i,j)∈I×J R

(α)i,j because

(w(κ+1),R(κ+1),ααα(κ+1)) is the optimal solution of (28)while (w(κ),R(κ),ααα(κ)) is its feasible point. Therefore,Algorithm 1 is a path-following algorithm, which gener-ates a sequence {(w(κ),R(κ),ααα(κ))} of improved feasiblepoints for (13). By using similar arguments to that shownin [5], [6], we can easily show that it converges at leastto a locally optimal solution of (13), which satisfies theKarush-Kuhn-Tucker (KKT) optimality condition.

Since the EH constraint (13b) is nonconvex, locatingan initial feasible point (w(0),R(0),ααα(0)) for (13) is anonconvex problem, which is resolved via the iterations

maxw,R,ααα

min(i,j)∈I×Jc

{Ri,jr0− 1,

Ri,p(j)r0

− 1,

ζi,j∑

(s,`)∈S

(2 02 till reaching a value more than or equalto 0 and thus the feasibility for (13).

Algorithm 1 PS-based algorithm for throughput max-minoptimization problem (13)

1: Initialization: Set κ := 0 and initialize a feasible point(w(0),R(0),ααα(0)) for (13)

2: Repeat until convergence of the objective in (13):Solve the convex optimization problem (28) to generatethe feasible point (w(κ+1),R(κ+1),ααα(κ+1)) for (13);Reset κ := κ+ 1.

Next, we address the EE maximization problem (14),which is equivalent to the following minimization problem

minw,ααα,R

ξπ(w) + Pc∑(i,j)∈I×J Ri,j

s.t. (9), (10), (11), (13b), (13c), (13d), (14b), (14c). (30)

2r0 plays the role of good initial value of the max-min throughput

6

The advantage of this transformation is that the objectivefunction is already convex so there is no need of usingDinkelbach’s iteration or approximating it. Actually, theDinkelbach iteration for (14) still involves a nonconvexoptimization problem, which is as computationally difficultas (14) itself (see e.g., [34]). At the κ-th iteration, we solvethe following convex optimization problem of computa-tional complexity O

((2KN(Nt + 2))

3 (10KN +N))

togenerate its next feasible point (w(κ+1),R(κ+1),ααα(κ+1)):

minw,ααα,R

ξπ(w) + Pc∑(i,j)∈I×J Ri,j

s.t. (13c), (13d), (14b), (14c), (21)− (27). (31)

Algorithm 2 outlines a path-following computational proce-dure to solve the the energy-efficiency (EE) maximizationproblem (14). Like Algorithm 1, at least it converges to alocally optimal solution of (14), which satisfies the KKToptimality condition.

Locating a feasible point (w(0),R(0),ααα(0)) for the non-convex constraints (13b), (9), (10), and (11) to initializeAlgorithm 2 is resolved via the iterations

maxw,R,ααα

min(i,j)∈I×Jc

{Ri,jri,j− 1,

Ri,p(j)ri,p(j)

− 1,

ζi,j∑

(s,`)∈S

(2 0 [38, (78)]and (15), we can innerly approximate (34), (35) and (36)by

0.5Ri,p(j))2/R

(κ)i,p(j) +R

(κ)i,p(j)

1− ρ≤

Λ(hi,i,jwIi,p(j),L

ci,p(j)(w

I), 0), (38)

7

0.5(Ri,p(j))2/R

(κ)i,p(j) +R

(κ)i,p(j)

1− ρ≤

Λ(hi,i,p(j)wIi,p(j),L

ei,p(j)(w

I), 0), (39)

and

0.5(Ri,j)2/R(κ)i,j +R

(κ)i,j

1− ρ≤ Λ(hi,i,jwIi,j ,Lci,j(wI), 0)

(40)

over the trust regions

2 0, (41a)

2 0, (41b)

2 0, (41c)

with the concave functions Λ(hi,i,jwIi,p(j),Lci,p(j)(w

I), 0),Λ(hi,i,p(j)w

Ii,p(j),L

ei,p(j)(w

I), 0), andΛ(hi,i,jw

Ii,j ,Lci,j(wI), 0) defined from (17), (18)-(20) for

(x(κ),y(κ), µ̃(κ)) = (hi,i,jwI,(κ)i,p(j),L

ci,p(j)(w

I,(κ)), σ),

(x(κ),y(κ), µ̃(κ)) = (hi,i,p(j)wI,(κ)i,p(j),L

ei,p(j)(w

I,(κ)), σ),

and (x(κ),y(κ), µ̃(κ)) = (hi,i,jwI,(κ)i,j ,Lci,j(wI,(κ)), σ), re-

spectively.The EH constraint (37b) is innerly approximated by [7]∑

(s,`)∈Sc

[2<{

(wE,(κ)s,` )

HhHs,i,jhs,i,jwEs,`

}−∣∣∣hi,i,jwE,(κ)s,` ∣∣∣2] ≥ emini,jζi,jρ , i ∈ I, j ∈ Jc, (42)

while the power constraint (37d) is innerly approximatedby [7]

1

1− ρ∑j∈Jc

‖wEi,j‖2 +∑j∈J‖wIi,j‖2 ≤(

2

1− ρ(κ)− 1− ρ

(1− ρ(κ))2

)Pmaxi

+∑j∈J

(2<{

(wE,(κ)i,j )

HwEi,j

}− ‖wE,(κ)i,j ‖

2). (43)

In summary, at the κth iteration we solve the followingconvex optimization problem of computational complexityO((2KN(Nt +Nt/2 + 1) + 1)

3 (9KN +N + 1))

to generate the feasible point(wI,(κ+1),wE,(κ+1),R(κ), ρ(κ+1)) for (37):

maxwE ,wI ,R,ρ

min(i,j)∈I×J

Ri,j

s.t. (37c), (37e), (37f), (38)− (43). (44)

Algorithm 3 outlines a path-following computational pro-cedure to solve the throughput max-min optimization prob-lem (37). Like Algorithm 1, it converges at least to a

local optimal solution satisfying the KKT condition. Tofind an initial feasible point (wE,(0),wI,(0), ρ(0),R(0))of (37), we first fix ρ(0) and find (wE,(0),wI,(0)) byrandomly generating Nt × 1 complex vectors followed bytheir normalization to satisfy (37d), (37e), and (37f). Wethen find

(wE,(0),wI,(0),R(0)

)via the iterations

maxwE ,wI ,γ>0

γ s.t. (37e), (37f), (45a)

ζi,j∑

(s,`)∈Sc

(2

8

function in (46) is no longer convex, making equivalentlytransformed problem like (14) no longer useful. Never-theless, we now also develop another iterative procedure,where (46) is seen no more computationally difficult thanthe throughput optimization problem (37).

Let (wE,(κ),wI,(κ),R(κ), ρ(κ)) be a feasible point for(46) that is found from the (κ− 1)th iteration and

t(κ) ,∑(i,j)∈I×J R

(κ)i,j

ξ[ρ(κ)πE(wE,(κ)) + (1− ρ(κ))πI(wI,(κ))

]+ Pc

=∑(i,j)∈I×J R

(κ)i,j /(1− ρ(κ))

ξ

[πE(w

E,(κ))

(1− ρ(κ))− 1+ ξπI(w

I,(κ))

]+

Pc1− ρ(κ)

At the κth iteration we address the problem

maxwE ,wI ,R,ρ

∑(i,j)∈I×J Ri,j

1− ρ− t(κ)

[ξ

(1

1− ρ− 1)

×πE(wE) + ξπI(wI) +Pc

1− ρ

]s.t. (14b), (14c), (34)− (36), (37b)− (37f). (47)

Substituting x =∑

(i,j)∈I×J Ri,j , x̄ =∑

(i,j)∈I×J R(κ)i,j ,

and t = 1− ρ, t̄ = 1− ρ(κ) into the following inequality

x

t≥ 2√x̄

t̄

√x− x̄

t̄2t ∀ x > 0, x̄ > 0, t > 0, t̄ > 0, (48)

the proof of which is given by Appendix B, the first termin the objective of (47) is lower bounded by the concavefunction

f (κ)(R, ρ) , 2

√∑(i,j)∈I×J R

(κ)i,j

1− ρ(κ)

√ ∑(i,j)∈I×J

Ri,j

−∑

(i,j)∈I×J R(κ)i,j

(1− ρ(κ))2(1− ρ).

The second term g(w, ρ) , ξ(

11−ρ − 1

)πE(w

E) +

ξπI(wI) + Pc1−ρ in the objective of (47) is upper bounded

by the convex function

g(κ)(w, ρ) , 1(1−ρ)ξπE(wE) + ξπI(w

I) + Pc1−ρ

−ξ∑i∈I∑j∈Jc

(2

9

gain with Rician factor of 10 dB (for s = i while j ∈ Jc,i.e., channel between BS and its own cell-centered users).We set the path-loss exponents β = β1 = 3 for the formercase and β = β2 = 2 for the later case. Different values forpath-loss exponents have been proposed for different typeof users in the literature too [39], [40]. Considering theadopted path-loss model and practical simulation setup, thepower of the received signal at the cell-center UEs passesthe threshold minimum power requirement (−21 dBm with13 nm CMOS technology [1]) to carry out meaningful EH.For simplicity, set emini,j ≡ emin for the energy harvestingthresholds, ζi,j ≡ ζ, ∀i, j for the energy harvestingconversion, Pmaxi ≡ Pmax, ∀ i. Further, emin = −20 dBmfor the energy harvesting threshold, ζ = 0.5 for the energyconversion efficiency, and Pmax = 35 dBm are set (unlessstated otherwise). The bandwidth is set to B = 20 MHz,the carrier frequency is set to 2 GHz, and unless statedotehrwise, the power spectral density of additive whiteGaussian noise, σB or

σcB , is set to −174 dBm/Hz. For

energy efficiency maximization problems (14) and (46),the threshold rate ri,j = 0.5 bits/sec/Hz ∀ i, j (unlessstated otherwise), the power amplifier efficiency 1/ξ = 0.2,power dissipation at each transmit antenna PA = 0.6W(27.78 dBm), and circuit power consumption Pcir = 2.5W(33.97 dBm) are set [41], [42]. The convex solver CVX[43] is used.

A. Results for throughput max-min optimization problems(13) and (37):

On average (running 100 simulations and averagingover random channel realizations), the PS-based Algorithm1 requires 30.05 iterations, while TS-based Algorithm 3requires 18.25 iterations for convergence.

Fig. 3 plots the optimized max-min rate for varying num-ber of BS-antennas and fixed BS power budget Pmax = 35dBm by solving the power splitting (PS)-based problem(13) and the time switching (TS)-based problem (37). Asexpected, the rate increases by increasing the number ofantennas at the BS. Fig. 3 shows that the TS-based Al-gorithm 3 outperforms the PS-based Algorithm 1 in termsof achievable rate and the corresponding performance gapincreases by increasing the number of antennas mountedon the BS. In the TS-based model, the presence of moreantennas helps both information and energy beamformingvectors to scale their performance more progressively ascompared to the PS-based model with optimizing onlythe information beamforming vectors. Fig. 4 plots theoptimized worst user rate for varying values of BS transmitpower budget Pmax and fixed number of BS-antennasNt = 6, while solving the same power splitting (PS)-basedproblem (13) and time switching (TS)-based problem (37).Fig. 4 shows that increasing the transmit power budgetraises the level of achievable rate, however, the increasediminishes at higher values of transmit power budget, e.g.,there is a marginal improvement in the rate when the

transmit power budget is increased from Pmax = 43 dBmto 47 dBm. Similarly to Fig. 3, Fig. 4 shows that the TS-based Algorithm 3 outperforms the PS-based Algorithm 1in terms of achievable rate.

Fig. 5 plots the optimized worst user rate for varyingvalues of noise power spectral density σB (in dBm/Hz) andfixed value of transmit power budget Pmax = 35 dBmand BS antennas Nt = 6, while solving the same powersplitting (PS)-based problem (13) and time switching (TS)-based problem (37). Fig. 5 shows that, as expected, increas-ing the noise variance decreases the level of achievablerate. However, there is a minor decrease in the achievablerate for PS-based receiver. Therefore, though the TS-basedAlgorithm 3 outperforms the PS-based Algorithm 1 interms of achievable rate, the corresponding performancegap decreases by increasing the noise variance. Under asimilar simulation setup, Fig. 6 plots the optimized worstuser rate for varying values of path-loss exponent β1,showing that increasing the path-loss exponent increasesthe level of achievable rate. Indeed, it is an interestingresult as it goes against the usual expectation of decrease inthe achievable rate with the increase in path-loss exponent.The unexpected trend of increase in the achievable ratewith the increase in the path-loss exponent is due to theconsideration of multicell setup. The intercell interferencedecreases with the increase in the path-loss exponent,which boosts the achievable rate of the affected cell-edgeusers.

Remark 1: Figs. 3-6 show that the TS-based Algorithm 3outperforms the PS-based Algorithm 1 in terms of spectralefficiency. This is because for the TS-based Algorithm3, we separately optimize the beamforming vectors forinformation transmission and energy harvesting, whichresults in better design compared to the case of PS-basedAlgorithm 1, where same beamforming vector is used forboth information transmission and energy harvesting.

B. Results for energy efficiency maximization problems(14) and (46):

On average, the PS-based Algorithm 2 requires 15.05iterations, while the TS-based Algorithm 4 requires 15.7iterations before convergence.

Fig. 7 plots the optimized energy efficiency (EE) fora varying number of BS-antennas and fixed BS powerbudget Pmax = 35 dBm, while solving power splitting(PS)-based problem (14) and time switching (TS)-basedproblem (46). As expected, the EE increases by increasingthe number of antennas at the BS. We can also observefrom Fig. 7 that the TS-based Algorithm 4 outperforms thePS-based Algorithm 2 in terms of achievable EE and theperformance gap is more than 1 bit/Joule. Next, Fig. 8 plotsthe energy efficiency for varying values of BS transmitpower budget Pmax and fixed number of BS-antennasNt = 4, while solving the same power splitting (PS)-basedproblem (14) and time switching (TS)-based problem (46).

10

3 4 5 6 7Number of BS antennas, Nt

0

2

4

6

8

10

12w

ors

t user

rate

(bits/s

ec/H

z)

TS Algorithm 3PS Algorithm 1

Fig. 3. Optimized worst user rate for varying number of antennas andfixed BS power budget Pmax = 35 dBm, by solving the power splitting(PS)-based problem (13) and the time switching (TS)-based problem(37).

33 35 37 39 41 43 45 47BS transmit power budget, Pmax (dBm)

3

3.5

4

4.5

5

5.5

6

6.5

wors

t user

rate

(bits/s

ec/H

z)

TS Algorithm 3PS Algorithm 1

Fig. 4. Optimized worst user rate for varying values of BS transmitpower budget Pmax and fixed value of Nt = 6, by solving the powersplitting (PS)-based problem (13) and the time switching (TS)-basedproblem (37).

-175 -170 -165 -160 -155 -150noise power density, σ

B(dBm/Hz)

4

4.2

4.4

4.6

4.8

5

5.2

5.4

5.6

5.8

wors

t user

rate

(bits/s

ec/H

z)

TS Algorithm 3PS Algorithm 1

Fig. 5. Optimized worst user rate for varying values of noise powerspectral density σ

Band fixed value of Pmax = 35 dBm and BS antennas

Nt = 6 by solving the power splitting (PS)-based problem (13) andthe time switching (TS)-based problem (37).

2.8 2.9 3 3.1 3.2 3.3 3.4path loss exponent, β1

3.5

4

4.5

5

5.5

6w

ors

use

r ra

te (

bits/s

ec/H

z) TS Algorithm 3

PS Algorithm 1

Fig. 6. Optimized worst user rate for varying values of path-lossexponent β1 and fixed value of Pmax = 35 dBm and BS antennasNt = 6 by solving the power splitting (PS)-based problem (13) andthe time switching (TS)-based problem (37).

Fig. 8 shows that for TS-based implementation, increasingthe transmit power budget raises the level of achievableEE, however, for PS-based implementation, there is almostno improvement in EE when the transmit power budget isincreased from Pmax = 35 dBm to 45 dBm.

Fig. 9 plots the optimized energy efficiency for differentvalues of noise power spectral density σB (in dBm/Hz) andfixed value of transmit power budget Pmax = 35 dBm andBS antennas Nt = 4, while solving the power splitting(PS)-based problem (14) and time switching (TS)-basedproblem (46). Fig. 9 shows that, as expected, EE decreasesby increasing the noise variance. In addition, the TS-basedAlgorithm 4 outperforms the PS-based Algorithm 2 interms of EE. Nevertheless, this performance gap decreases

with the increase of noise variance.

C. Comparison with orthogonal multiple access (OMA):

As mention in Section I, the problem of EE maximiza-tion with multi-antenna beamforming in an EH-enabledNOMA network has not been addressed in the literaturebefore. In addition, the existing research did not consider“transmit-TS” approach for throughput max-min optimiza-tion of the EH-enabled NOMA network. Therefore, in thissubsection, we compare the performance of our proposedalgorithms with that obtained by OMA implementation.

Fig. 10 plots the optimized worst user rate for varyingvalues of BS transmit power budget Pmax and fixed numberof BS-antennas Nt = 6 by solving the same power splitting

11

4 5 6 7Number of BS antennas, Nt

3

4

5

6

7

8

9

10E

nerg

y E

ffic

iency (

bits/J

oule

)TS Algorithm 4PS Algorithm 2

Fig. 7. Optimized energy efficiency for varying number of antennas andfixed BS power budget Pmax = 35 dBm, while solving power splitting(PS)-based problem (14) and time switching (TS)-based problem (46).

33 35 37 39 41 43 45 47BS transmit power budget, Pmax (dBm)

5

5.5

6

6.5

7

7.5

Energ

y E

ffic

iency (

bits/J

oule

)

TS Algorithm 4PS Algorithm 2

Fig. 8. Optimized energy efficiency for varying values of noise powerspectral density σ

Band fixed BS transmit power budget Pmax = 35

and fixed value of Nt = 4, while solving power splitting (PS)-basedproblem (14) and time switching (TS)-based problem (46)

35 37 39 41 43 45BS transmit power budget, Pmax (dBm)

2.5

3

3.5

4

4.5

5

5.5

6

6.5

wors

t user

rate

(bits/s

ec/H

z)

TS Algorithm 3 (NOMA)PS Algorithm 1 (NOMA)TS (OMA)PS (OMA)

Fig. 10. Optimized worst user rate for varying values of BS transmitpower budget Pmax and fixed value of Nt = 6, while solving powersplitting (PS)-based NOMA problem (13), time switching (TS)-basedNOMA problem (37), and their OMA counterparts.

35 37 39 41 43 45BS transmit power budget, Pmax (dBm)

4

4.5

5

5.5

6

6.5

7

7.5E

nerg

y E

ffic

iency (

bits/J

oule

)

TS Algorithm 4 (NOMA)PS Algorithm 2 (NOMA)TS (OMA)PS (OMA)

Fig. 11. Optimized energy efficiency for varying values of BS transmitpower budget Pmax and fixed value of Nt = 4 and threshold rateri,j = 0.1 bits/sec/Hz (different from previous EE plots), while solvingpower splitting (PS)-based NOMA problem (14), time switching (TS)-based NOMA problem (46), and their OMA counterparts.

(PS)-based NOMA problem (13) and the time switching(TS)-based NOMA problem (37), and their orthogonalmultiple access (OMA) counterparts. In parallel, Fig. 11plots the EE for varying values of BS transmit powerbudget Pmax and fixed number of BS-antennas Nt = 4and threshold rate ri,j = 0.1 bits/sec/Hz (different fromprevious EE plots) by solving the power splitting (PS)-based NOMA problem (14), the time switching (TS)-based NOMA problem (46), and their OMA counterparts.We have to choose a smaller threshold rate ri,j = 0.1bits/sec/Hz because implementation with OMA schemefails to simultaneously satisfy both the higher threshold rateand the EH constraint for all simulations. We can clearly

observe from Figs. 10 and 11 that NOMA implementa-tion outperforms OMA implementation, in terms of both,throughput and energy efficiency, respectively.

V. CONCLUSIONS

This paper has considered an energy harvesting basedNOMA system, where transmit-TS approach is employedto realize both wireless energy harvesting and informationdecoding at the nearly-located users. We have formu-lated two important problems of worst-user throughputmaximization and energy efficiency maximization underpower constraint and energy harvesting constraints at thenearly-located users. For these problems, the optimization

12

-175 -170 -165 -160 -155 -150noise power desnity, σ

B(dBm/Hz)

3.5

4

4.5

5

5.5

6

6.5

7E

nerg

y E

ffic

iency (

bits/J

oule

)TS Algorithm 4PS Algorithm 2

Fig. 9. Optimized energy efficiency for varying values of of noisevariances σ and fixed value of Pmax = 35 dBm and BS antennas Nt = 4,while solving power splitting (PS)-based problem (14) and time switching(TS)-based problem (46)

objective and energy harvesting constraints are highly non-convex. To address this, we have developed efficient path-following algorithms to solve the two problems. We havealso proposed algorithms for the case if conventional PS-based approach is used for energy harvesting. Our nu-merical results confirmed that the proposed transmit-TSapproach clearly outperforms PS approach in terms ofboth, throughput and energy efficiency. One possible futureresearch direction is to consider many user-pairs per celland jointly optimize beamforming vectors and user-pairingstrategy.

APPENDIX A: PROOF FOR (15)

Define the function f(x, y) , ln(x−1 + y−1) whichis convex in x > 0 and y > 0 [44]. Then f(x, y) ≥f(x(κ), y(κ)) + 〈∇f(x(κ), y(κ)), (x, y) − (x(κ), y(κ))〉 forall x > 0, y > 0, x(κ) > 0, y(κ) > 0 [45], which meansthat

ln

(1

x+

1

y

)≥ ln

(1

x(κ)+

1

y(κ)

)+ 1

− 1x(κ) + y(κ)

(y(κ)

x(κ)x+

x(κ)

y(κ)y

). (50)

Substituting x = ‖x‖2, y = ‖y‖2 + σ + σcµ, x(κ) =‖x(κ)‖2, and y(κ) = ‖y(κ)‖2 + σ + σµ(κ), we obtain

ln(

(‖x‖2)−1 + (‖y‖2 + σ + σcµ)−1)≥

ln(

(‖x(κ)‖2)−1 + (‖y(κ)‖2 + µ̃(κ))−1)

+ 1

− 1‖x(κ)‖2 + ‖y(κ)‖2 + µ̃(κ)

(‖y(κ)‖2 + µ̃(κ)

‖x(κ)‖2‖x‖2

+‖x(κ)‖2

‖y(κ)‖2 + µ̃(κ)(‖y‖2 + µ̃(κ))

), (51)

where µ̃(κ) , σ + σcµ(κ). Next, as the functions ln(1/x)and ‖x‖2 are convex, it is true that

ln

(1

x

)≥ ln

(1

x(κ)

)− x− x

(κ)

x(κ), (52)

‖x‖2 ≥ 2 0,it is true that [45] x

2

t ≥ g(x̄, t̄) + 〈∇g(x̄, t̄), (x, t)− (x̄, t̄)〉= 2 x̄t̄ x −

x̄2

t̄2 t. Inequality (48) then follows by resettingx→

√x and x̄→

√x̄.

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Ali Arshad Nasir (S’09-M’13) is an AssistantProfessor in the Department of Electrical Engi-neering, King Fahd University of Petroleum andMinerals (KFUPM), Dhahran, KSA. Previously,he held the position of Assistant Professor inthe School of Electrical Engineering and Com-puter Science (SEECS) at National University ofSciences & Technology (NUST), Paksitan, from2015-2016. He received his Ph.D. in telecommu-nications engineering from the Australian Na-tional University (ANU), Australia in 2013 and

worked there as a Research Fellow from 2012-2015. His research interestsare in the area of signal processing in wireless communication systems.He is an Associate Editor for IEEE Canadian Journal of Electrical andComputer Engineering.

14

Hoang Duong Tuan received the Diploma(Hons.) and Ph.D. degrees in applied mathe-matics from Odessa State University, Ukraine,in 1987 and 1991, respectively. He spent nineacademic years in Japan as an Assistant Profes-sor in the Department of Electronic-MechanicalEngineering, Nagoya University, from 1994 to1999, and then as an Associate Professor in theDepartment of Electrical and Computer Engi-neering, Toyota Technological Institute, Nagoya,from 1999 to 2003. He was a Professor with the

School of Electrical Engineering and Telecommunications, University ofNew South Wales, from 2003 to 2011. He is currently a Professor withthe School of Electrical and Data Engineering, University of TechnologySydney. He has been involved in research with the areas of optimiza-tion, control, signal processing, wireless communication, and biomedicalengineering for more than 20 years.

Trung Q. Duong (S’05, M’12, SM’13) re-ceived his Ph.D. degree in TelecommunicationsSystems from Blekinge Institute of Technol-ogy (BTH), Sweden in 2012. Currently, he iswith Queen’s University Belfast (UK), wherehe was a Lecturer (Assistant Professor) from2013 to 2017 and a Reader (Associate Professor)from 2018. His current research interests includeInternet of Things (IoT), wireless communi-cations, molecular communications, and signalprocessing. He is the author or co-author of

330+ technical papers published in scientific journals (196 articles) andpresented at international conferences (125 papers).

Dr. Duong currently serves as an Editor for the IEEE TRANSACTIONSON WIRELESS COMMUNICATIONS, IEEE TRANSACTIONS ON COMMU-NICATIONS, IET COMMUNICATIONS, and a Lead Senior Editor for IEEECOMMUNICATIONS LETTERS. He was awarded the Best Paper Awardat the IEEE Vehicular Technology Conference (VTC-Spring) in 2013,IEEE International Conference on Communications (ICC) 2014, IEEEGlobal Communications Conference (GLOBECOM) 2016, and IEEEDigital Signal Processing Conference (DSP) 2017. He is the recipient ofprestigious Royal Academy of Engineering Research Fellowship (2016-2021) and has won a prestigious Newton Prize 2017.

Mérouane Debbah (S’01-M’04-SM’08-F’15)received the M.Sc. and Ph.D. degrees from theEcole Normale Suprieure Paris-Saclay, France.In 1996, he joined the Ecole Normale SuprieureParis-Saclay. He was with Motorola Labs,Saclay, France, from 1999 to 2002, and also withthe Vienna Research Center for Telecommunica-tions, Vienna, Austria, until 2003. From 2003to 2007, he was an Assistant Professor withthe Mobile Communications Department, Insti-tut Eurecom, Sophia Antipolis, France. From

2007 to 2014, he was the Director of the Alcatel-Lucent Chair on FlexibleRadio. Since 2007, he has been a Full Professor with CentraleSupelec,Gif-sur-Yvette, France. Since 2014, he has been a Vice-President of theHuawei France Research Center and the Director of the Mathematicaland Algorithmic Sciences Lab. He has managed 8 EU projects and morethan 24 national and international projects. His research interests lie infundamental mathematics, algorithms, statistics, information, and commu-nication sciences research. He was a recipient of the ERC Grant MORE(Advanced Mathematical Tools for Complex Network Engineering) from2012 to 2017. He received 20 best paper awards, among which the2015 IEEE Communications Society Leonard G. Abraham Prize, the2016 IEEE Communications Society Best Tutorial Paper Award, andthe 2018 IEEE Marconi Prize Paper Award. He is an Associate Editor-in-Chief of the journal Random Matrix: Theory and Applications. Hewas an Associate Area Editor and Senior Area Editor of the IEEETRANSACTIONS ON SIGNAL PROCESSING from 2011 to 2013 and from2013 to 2014, respectively.