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Wireless Information and Power Transfer: EnergyEfficiency Optimization in OFDMA Systems

Derrick Wing Kwan Ng,Member, IEEE,Ernest S. Lo,Member, IEEE,and Robert Schober,Fellow, IEEE

Abstract—This paper considers orthogonal frequency divisionmultiple access (OFDMA) systems with simultaneous wirelessinformation and power transfer. We study the resource allocationalgorithm design for maximization of the energy efficiency of datatransmission (bits/Joule delivered to the receivers). In particular,we focus on power splitting hybrid receivers which are able tosplit the received signals into two power streams for concurrentinformation decoding and energy harvesting. Two scenariosareinvestigated considering different power splitting abilities of thereceivers. In the first scenario, we assume receivers which cansplit the received power into a continuous set of power streamswith arbitrary power splitting ratios. In the second scenario, weexamine receivers which can split the received power only intoa discrete set of power streams with fixed power splitting ratios.For both scenarios, we formulate the corresponding algorithmdesign as a non-convex optimization problem which takes intoaccount the circuit power consumption, the minimum datarate requirements of delay constrained services, the minimumrequired system data rate, and the minimum amount of powerthat has to be delivered to the receivers. By exploiting frac-tional programming and dual decomposition, suboptimal iterativeresource allocation algorithms are developed to solve the non-convex problems. Simulation results illustrate that the proposediterative resource allocation algorithms approach the optimalsolution within a small number of iterations and unveil the trade-off between energy efficiency, system capacity, and wireless powertransfer: (1) wireless power transfer enhances the system energyefficiency by harvesting energy in the radio frequency, especiallyin the interference limited regime; (2) the presence of multiplereceivers is beneficial for the system capacity, but not necessarilyfor the system energy efficiency.

Index Terms—Energy efficiency, green communications, wire-less information and power transfer, non-convex optimization.

I. I NTRODUCTION

ORTHOGONAL frequency division multiple access(OFDMA) has been widely adopted as the air interface

in high speed wireless multiuser communication networks,due to its immunity to channel delay spread and flexibilityin resource allocation. In an OFDMA system, a widebandfrequency selective spectrum is converted into a number

Manuscript received March 16, 2013; revised July 31, 2013; acceptedOctober 1, 2013. The review of this paper was coordinated by Prof. A.Wyglinski. D. W. K. Ng and R. Schober are with the Institute for DigitalCommunications (IDC), Friedrich-Alexander-University Erlangen-Nurnberg(FAU), Germany (email:kwan, schober@lnt.de). Ernest S. Lo is with CentreTecnologic de Telecomunicacions de Catalunya - Hong Kong (CTTC-HK) (e-mail: ernest.lo@cttc.hk). This paper has been presented inpart at the IEEEWireless Communications and Networking Conference (WCNC)2013 [1]and the IEEE International Conference on Communications (ICC) 2013 [2].This work was supported in part by the AvH Professorship Program of theAlexander von Humboldt Foundation.

of orthogonal narrowband frequency flat subcarrier channelswhich facilitates the multiplexing of users’ data and theexploitation of multiuser diversity. On the other hand, nextgeneration communication systems are expected to supportmultiple users and to guarantee quality of service (QoS). Theincreasing demand for high data rate and ubiquitous serviceshas led to a high energy consumption in both transmitter(s)and receiver(s). Unfortunately, portable mobile devices areusually only equipped with limited energy supplies (batter-ies) which creates bottlenecks in perpetuating the lifetimeof networks. Besides, the battery capacity has improved ata very slow pace over the past decades and is unable tosatisfy the new energy requirements [3]. Consequently, energy-efficient mobile communication system design has becomea prominent approach for addressing this issue in energylimited networks [1]–[7]. Specifically, an enormous numberoftechnologies/methods such as energy harvesting and resourceallocation optimization have been proposed in the literature forimproving the energy efficiency (bits-per-Joule) of wirelesscommunication systems. Among the proposed technologies,energy harvesting from the environment is particularly ap-pealing as it constitutes a perpetual energy source. Moreimportantly, it provides self-sustainability to systems and isvirtually free of cost. In practice, numerous renewable energysources can be exploited for energy harvesting, including solar,tide, geothermal, and wind. However, these natural energysources are usually location, weather, or climate dependent andmay not always be available in enclosed/indoor environmentsor suitable for mobile devices. On the other hand, wirelesspower transfer technology, which enables the receivers to scav-enge energy from propagating electromagnetic waves (EM)in radio frequency (RF), has gained recent attention in bothindustry and academia [8]–[14]. Indeed, RF signals carry bothinformation and energy simultaneously. Thus, the RF energyradiated by the transmitter(s) can be recycled at the receiversfor prolonging the lifetime of networks. Yet, the utilizationof EM waves as a carrier for simultaneous information andpower transfer poses many new research challenges for bothresource allocation algorithm and receiver design. In [10]and[11], the fundamental trade-off between wireless informationand wireless power transfer was studied for flat fading andfrequency selective fading, respectively. Specifically, an idealreceiver was assumed in [10] and [11] such that informationdecoding and energy harvesting can be performed on the samereceived signal which is not possible in practice yet. As acompromise solution, three different types of receivers, namelypower splitting, separated, andtime-switchingreceivers, were

2

Power splitting unit

Battery

Receiver signal

processing core

Energy

harvesting unit

Transmitter

signal

processing

core

Power amplifier

A maximum power of

supplied by power grid

Antenna noise

Signal processing noiseInformation and power transfer

PGP

,

E

i kρ

Power splitting ratio

=

Power splitting ratio

=

,

2

i kI

σInterference power

,

I

i kρ

MobileMobile

Mobile

2az

σ

2sz

σ

Transmitter Mobile receivers

Transmitter Mobile receiver k

Information and power transfer

Fig. 1. An OFDMA downlink communication system withK = 3 mobile receivers. The upper half of the figure illustrates a block diagram of the transceivermodel for wireless information and power transfer.

proposed in [12], [13]. In particular, thepower splittingreceiver splits the received power into two power streams witha certain power splitting ratio for facilitating simultaneousenergy harvesting and information decoding in the receiver.The authors in [12] and [13] investigated the rate-energyregions for different types of receivers in two-user and point-to-point single carrier systems, respectively. However, theproblem formulations in [12] and [13] do not take into accountthe heterogeneous data rate requirements of users and thesolutions may not meet any data rate requirements and arenot applicable to multi-carrier systems with arbitrary numbersof users. On the other hand, the authors in [14] focused on theresource allocation algorithm design for a point-to-pointsingleuser system withpower splitting receiver in ergodic fadingchannels. Yet, the assumption of channel ergodicity may notbe well justified for delay sensitive services in practice sincethe transmitted data symbols of these services experience slowfading. Besides, the high power consumption in electroniccircuitries and RF transmission have been overlooked in [10]–[14] but play an important role in designing energy efficientcommunication systems [15], [16]. In other words, the energyefficiency of systems with energy harvesting receivers remainsunknown. In fact, if a portion of the transmitted RF energycan be harvested by the RF energy harvesting receivers, thesystem energy efficiency may improve. Yet, the system modelsadopted in [10]–[14] do not consider the energy recycling pro-cess from an energy efficiency point of view. Thus, it is unclearunder what conditions energy harvesting receivers improvethe system energy efficiency compared to traditional receiverswhich cannot harvest energy. By incorporating the circuitpower consumption and the RF energy harvesting ability ofthe receivers into the problem formulation, we presented in[1]and [2] two energy efficient resource allocation algorithmsfor

multicarrier systems employingseparatedreceivers andpowersplitting receivers, respectively. Nevertheless, [1] and [2] donot fully exploit the degrees of freedom in resource allocationsince data multiplexing of different users on different subcar-riers was not considered. Moreover, the algorithm proposedin [2] incurs a high computation complexity at the transmittersince the optimal power splitting ratio is found via full searchover a continuous variable. Furthermore, the power splittingratio may take only discrete levels in practice and the results in[2], which were obtained for continuous power splitting ratios,are not applicable in this case.

In this paper, we address the above issues. To this end, weformulate the resource allocation algorithm design for energyefficient communication in OFDMA systems with simultane-ous wireless information and power transfer as an optimizationproblem. In particular, we focus on the algorithm design forpower splitting receivers and consider both continuous anddiscrete power splitting ratios. Besides, data multiplexing ofusers on different subcarriers is incorporated into the problemformulation. The resulting non-convex optimization problemsare solved by iterative algorithms which combine nonlinearfractional programming and dual decomposition. Simulationresults illustrate an interesting trade-off between energy effi-ciency, wireless power transfer, and multiuser diversity.

II. SYSTEM MODEL

In this section, we present the adopted OFDMA signalmodel and the model for the hybrid information and energyharvesting receiver.

A. OFDMA Channel Model

We consider an OFDMA downlink system in which atransmitter servicesK mobile receivers. In particular, each

3

mobile receiver is able to decode information and harvestenergy from the received radio signals. All transceivers areequipped with a single antenna, cf. Figure 1. The total systembandwidth is B Hertz and there arenF subcarriers. Wefocus on quasi-static block fading channels and assume thatthe downlink channel gains can be accurately obtained byfeedback from the receivers. The downlink received symbolat receiverk ∈ 1, . . . , K on subcarrieri ∈ 1, . . . , nF isgiven by

Yi,k =√Pi,klkgkHi,kXi,k + Ii,k + Za

i,k, (1)

whereXi,k, Pi,k, andHi,k are the transmitted data symbol, thetransmitted power, and the multipath fading coefficient fromthe transmitter to receiverk on subcarrieri, respectively.lkandgk represent the path loss and shadowing attenuation fromthe transmitter to receiverk, respectively.Za

i,k is the additivewhite Gaussian noise (AWGN) originating from the antennaon subcarrieri of receiverk. It is modeled as Gaussian randomvariable with zero mean and varianceσ2

za , cf. Figure 1.Ii,k isthe received aggregate co-channel interference on subcarrieri of receiverk with zero mean and varianceσ2

Ii,k. Ii,k is

emitted by unintended transmitters sharing the same frequencychannel.

B. Hybrid Information and Energy Harvesting Receiver

In practice, the model of an energy harvesting receiverdepends on its specific implementation. For example, electro-magnetic induction and electromagnetic radiation are abletotransfer wireless power [11], [14]. Nevertheless, the associatedhardware circuitries in the receivers and the correspondingenergy harvesting efficiency can be significantly different. Be-sides, the signal used for decoding the modulated informationcannot be used for harvesting energy due to hardware limita-tions [14]. In order to isolate the resource allocation algorithmdesign from the specific hardware implementation details, wedo not assume a particular type of energy harvesting receiver.In this paper, we focus on receivers which consist of an energyharvesting unit and a conventional signal processing core unitfor concurrent energy harvesting and information decoding, cf.Figure 1. In particular, we adopt a receiver which splits thereceived signal into two power streams [14] in the RF frontend with power splitting ratios1 ρIi,k and ρEi,k. Subsequently,the two power streams with power splitting ratiosρEi,k andρIi,k are used for harvesting the energy and decoding theinformation contained in the signal, respectively. Indeed, byimposing power splitting ratios2 of ρIi,k = 1, ρEi,k = 0 andρIi,k = 0, ρEi,k = 1, the hybrid receiver reduces to a tradition in-formation receiver or energy harvesting receiver, respectively.Furthermore, we assume that the harvested energy is used toreplenish a rechargeable battery at the receiver. Besides,eachreceiver has a fixed power consumption ofPCR

Watts which

1Indeed,ρIi,k

and ρEi,k

represent the fractions of the received power ofuserk on subcarrieri used for information decoding and energy harvesting,respectively. Yet, we follow the convention in the literature [14] and adoptthe term “power splitting ratios” in the paper.

2 In this paper, a perfect passive power splitting unit is assumed; i.e., thepower splitting does not incur any power consumption and does not introduceany power loss or signal processing noise to the system.

is used for maintaining the routine operations in the receiverand is independent of the amount of harvested power. We notethat in practice the receivers may be powered by more thanone energy source and the harvested energy may be used asa supplement for supporting the energy consumption3 of thereceivers [17].

III. R ESOURCEALLOCATION - CONTINUOUS SET OF

POWER SPLITTING RATIOS

In this section, we consider the resource allocation algorithmdesign for maximizing the system energy efficiency for thecase of a continuous set of power splitting ratios. The derivedsolution provides not only a useful guideline for choosing asuitable number of discrete power splitting ratios in the powersplitting unit, but also serves as a performance benchmark forthe case of discrete power splitting ratios. Now, we definethe system energy efficiency by first introducing the weightedsystem capacity and the power dissipation of the system.

A. System Energy Efficiency

Assuming the availability of perfect channel state infor-mation (CSI) at the receiver, the channel capacity betweenthe transmitter and receiverk on subcarrieri with subcarrierbandwidthW = B/nF is given by4

Ci,k = W log2

(1 + Pi,kΓi,k

)and

Γi,k =ρIi,klkgk|Hi,k|

2

ρIi,k(σ2za + σ2

Ii,k) + σ2

zs

, (2)

whereσ2zs is the signal processing noise power at the receiver

and Pi,kΓi,k is the received signal-to-interference-plus-noiseratio (SINR) on subcarrieri at receiverk.

The weighted system capacityis defined as the aggregatenumber of bits delivered toK receivers and is given by

U(P ,S,ρ) =

nF∑

i=1

K∑

k=1

αksi,kCi,k [bits/s], (3)

whereP = Pi,k ≥ 0, ∀i, k is the power allocation policy,S = si,k ∈ 0, 1, ∀i, k is the subcarrier allocation pol-icy, and ρ = ρIi,k, ρ

Ei,k ≥ 0, ∀i, k is the power splitting

policy with variablesρIi,k and ρEi,k introduced in SectionII-B. αk ≥ 0, ∀k, is a non-negative weight which accountsfor the priorities of different receivers and is specified bythe application layer. In practice, proportional fairnessandmax-min fairness can be achieved by varying the valuesof αk over time5 [20]. On the other hand, for facilitatingthe resource allocation algorithm design, we incorporate thetotal power consumption of the system into the optimizationobjective function. In particular, the power consumption of the

3In this paper, the unit of Joule-per-second is used for energy consumption.Thus, the terms “power” and “energy” are interchangeable.

4The received interference signalIi,k on each subcarrier is treated asAWGN in order to simplify the algorithm design [13], [18].

5Optimizing the value ofαk for achieving different system objectives isbeyond the scope of this paper. Interested readers may referto [19] for furtherdetails.

4

considered system,UTP (P ,S,ρ), consists of five terms andcan be expressed as:

UTP (P ,S,ρ) = PCT+KPCR

+

K∑

k=1

nF∑

i=1

εPi,ksi,k

−

K∑

k=1

QDk−

K∑

k=1

QIk [Joule/s], (4)

where QDk=nF∑

i=1

( K∑

j=1

Pi,jsi,j

)lkgk|Hi,k|

2ηkρEi,k

︸ ︷︷ ︸Power harvested from information signal at receiverk

(5)

and QIk =

nF∑

i=1

(σ2za + σ2

Ii,k)ρEi,kηk

︸ ︷︷ ︸Power harvested from interference signal

and antenna noise at receiverk

. (6)

The first three terms in (4), i.e.,PCT+ KPCR

+∑Kk=1

∑nF

i=1 εPi,ksi,k, represent the power dissipation re-quired for supporting reliable communication.PCT

> 0 isthe constantsignal processing circuit power consumptioninthe transmitters, caused by filters, frequency synthesizer, etc.,and is independent of the power radiated by the transmitter.VariablesKPCR

and∑K

k=1

∑nF

i=1 εPi,ksi,k denote the totalcircuit power consumption in theK receivers and the powerdissipation in the power amplifier of the transmitter, respec-tively. To model the power inefficiency of the power amplifier,we introduce a multiplicative constant,ε ≥ 1, for the powerradiated by the transmitter in (4) which takes into accountthe joint effect of the drain efficiency and the power amplifieroutput backoff [15]. For example, ifε = 10, then10 Watts ofpower are consumed in the power amplifier for every 1 Watt ofpower radiated in the RF; the wasted power during the poweramplification is dissipated as heat. On the other hand, the lasttwo terms in (4), i.e.,−

∑Kk=1 QDk

−∑K

k=1 QIk , represent theharvested energy at theK receivers. The minus sign in front of∑K

k=1 QDkin (4) indicates that a portion of the power radiated

in the RF from the transmitter can be harvested by theK re-ceivers. Besides,0 ≤ ηk ≤ 1 in (5) and (6) is a constant whichdenotes the energy harvesting efficiency of mobile receiverk in converting the received radio signal to electrical energyfor storage. In fact, the termηklkgk|Hi,k|

2ρEi,k in (5) can beinterpreted as afrequency selective power transfer efficiencyfor transferring power from the transmitter to receiverk onsubcarrieri. Similarly, the minus sign in front of

∑K

k=1 QIk

in (4) accounts for the ability of the receivers to harvest energyfrom interference signals. The weightedenergy efficiencyofthe considered system is defined as the total average numberof bits successfully conveyed to theK receivers per Jouleconsumed energy and is given by

Ueff (P ,S,ρ) =U(P ,S,ρ)

UTP (P ,S,ρ)[bits/Joule]. (7)

It is interesting to note that the weighted energy efficiencyis a quasi-concave function with respect to the power al-location variables, cf. Appendix A. The quasi-concavity of

Ueff (P ,S,ρ) can be exploited to show the quasi-concavity ofa transformed version of the resource allocation problem, cf.Section III-B, Remark 2. Furthermore, strong interferencecanact as an energy source which supplies energy to the receiversand facilitates energy savings in the system. This effect isnotcaptured by other system models used in the literature [18],[21]. Therefore, unlike systems with non-energy harvestingreceivers, systems with energy harvesting receivers may ben-efit from interference as far as energy efficiency is concerned.The energy efficiency gain due to the introduction of energyharvesting receivers will be evaluated in the simulation section.On the other hand, we emphasize that the objective functionof the considered problem formulation does not capture howthe receivers utilize the harvested energy. For instance, if thereceivers use the harvested energy for extending their lifetime,an additional system capacity gain may be achieved sincethe receivers have more time to receive information. In othercases, the receivers may use the harvested energy for uplinktransmission which also results in a system capacity gain.

Remark 1: Mathematically, it is possible thatUTP (P ,S,ρ) takes a negative value. Yet,UTP (P ,S,ρ)> 0always holds for the considered system due to thefollowing reasons. First, it can be observed that∑nF

i=1

∑Kk=1 εPi,ksi,k ≥

∑nF

i=1

∑Kk=1 Pi,ksi,k >

∑Kk=1 QDk

,where the strict inequality is due to the second lawof thermodynamics from physics. In particular, thecommunication channel between the transmitter and theK receivers is a passive system which does not introduceextra energy. Besides, the energy of the desired signal receivedat the receiver is attenuated due to path loss and energyscavenging inefficiency. Second, we consider a system wherethe transmitter and the receivers consume non-negligibleamounts of power [16] for signal processing such thatPCT

+KPCR>

∑K

k=1 QIk .

B. Optimization Problem Formulation

The optimal power allocation policy,P∗, subcarrier alloca-tion policy,S∗, and power splitting policyρ∗, can be obtainedby solving the following optimization problem:

maxP,S,ρ

Ueff (P ,S,ρ) (8)

s.t. C1:QDk+QIk ≥ P req

mink, ∀k,

C2:nF∑

i=1

K∑

k=1

Pi,ksi,k ≤ Pmax,

C3: PCT+

nF∑

i=1

K∑

k=1

εPi,ksi,k ≤ PPG,

C4:nF∑

i=1

K∑

k=1

si,kCi,k ≥ Rmin,

C5:nF∑

i=1

si,kCi,k ≥ Rmink, ∀k ∈ D,

C6: Pi,k ≥ 0, ∀i, k, C7: si,k ∈ 0, 1, ∀i, k,

5

C8:K∑

k=1

si,k ≤ 1, ∀i, C9: ρEL ≤ ρEi,k ≤ ρEU , ∀i, k,

C10: ρIL ≤ ρIi,ksi,k ≤ ρIU , ∀i, k,

C11: ρIi,ksi,k + ρEi,k ≤ 1, ∀i, k, C12: ρEi,k = ρEj,k, ∀k, i 6= j.

Variable P reqmink

in C1 is a constant which specifies theminimum required power transfer to receiverk. The valueof Pmax in C2 puts an upper limit on the power radiatedby the transmitter. The value ofPmax is a constant whichdepends on the hardware limitations of the power amplifier.C3 limits the maximum power supplied by the power gridfor supporting the power consumption of the transmitter toPPG, cf. Figure 1.Rmin in C4 is the minimum required datarate of the system. AlthoughRmin is not an optimizationvariable in this paper, we can strike a balance between thesystem energy efficiency and the total system throughput byvarying its value. In particular, whenRmin is increasing,the resource allocator may increase the transmit power forsatisfying the higher data rate requirement by sacrificing thesystem energy efficiency. C5 is the minimum required datarate Rmink

for the delay constrainedservices of receiverk, and is specified by the application layer, andD denotesa set of receivers havingdelay constrainedservices. C6 isthe non-negative orthant constraint for the power allocationvariables. C7 and C8 indicate that each subcarrier can beallocated to at most one receiver exclusively in conveyinginformation; inter-user interference is avoided in the system.Besides, C8 indicates that some subcarriers can be excludedfrom the subcarrier selection process for energy efficiencymaximization. Boundary variablesρEU and ρEL in C9 denotethe constant upper and lower bounds of the power splittingratio for harvesting energy, respectively. The bounds are usedto account for the limited capability of the receivers in splittingthe received power. Similarly,ρIU and ρIL in C10 denote theconstant upper and lower bounds of the power splitting ratiofor information decoding whereρEU+ρIL = 1 andρEL+ρIU = 1.C11 is the power splitting constraint of the hybrid informationand energy harvesting receivers. Its physical meaning is thatthe power splitting unit, see Figure 1, is a passive device andno extra power gain can be achieved by the power splittingprocess. C12 constrains the power spitting ratio for energyharvesting,ρEi,k, such that it is identical for all subcarriersin each receiver. Theoretically,ρEi,k can be different acrossdifferent subcarriers. However, in this case, an analog adaptivepassive frequency selective power splitter is required whichresults in a high system complexity. Therefore, we considerthe more practical scenario whereρEi,k = ρEj,k, ∀k, ∀j 6= i.We do not explicitly imposeρIi,k = ρIj,k, ∀k, ∀j 6= i, in theproblem formulation. We note that it can be deduced that ifboth subcarrieri and subcarrierj are allocated to userk forinformation decoding,ρIi,k = ρIj,k will hold eventually dueto the constraintρEi,k = ρEj,k, ∀k, j 6= i. However, imposingρIi,k = ρIj,k, ∀k, j 6= i explicitly at the very beginning in theproblem formulation makes the development for an efficientsolution of the optimization problem more cumbersome.

C. Solution of the Optimization Problem

The key challenge in solving (8) is the lack of convexity ofthe problem formulation. In particular, although the objectivefunction in (8) is quasi-concave with respect to the powerallocation variables, the objective function is a jointly non-convex function with respect toP ,S,ρ. Besides, constraintsC1–C12 do not span a convex solution set due to the integerconstraint for subcarrier allocation in C7 and the coupledoptimization variables in C1. In general, there is no standardapproach for solving non-convex optimization problems. Inthe extreme case, an exhaustive search or branch-and-boundmethod is needed to obtain the global optimal solution whichiscomputationally infeasible even for smallK andnF . In orderto make the problem tractable, we transform the objectivefunction and approximate the transformed objective function inorder to simplify the problem. Subsequently, we use the con-straint relaxation approach for handling the integer constraintC7 to obtain a close-to-optimal resource allocation algorithm.Next, we introduce the objective function transformation via aparametric approach from nonlinear fractional programming.

D. Transformation of the Objective Function

For notational simplicity, we defineF as the set of feasiblesolutions6 of the optimization problem in (8) spanned byconstraints C1–C12. Without loss of generality, we assumethatP ,S,ρ ∈ F and we denoteq∗ as the maximum energyefficiency of the considered system which can be expressed as

q∗ =U(P∗,S∗,ρ∗)

UTP (P∗,S∗,ρ∗)= max

P,S,ρ

U(P ,S,ρ)

UTP (P ,S,ρ). (9)

Now, we introduce the following important Theorem whichis borrowed from nonlinear fractional programming [22] forsolving the optimization problem in (8).

Theorem 1:The resource allocation policy achieves themaximum energy efficiencyq∗ if and only if

maxP,S,ρ

U(P ,S,ρ)− q∗UTP (P ,S,ρ)

= U(P∗,S∗,ρ∗)− q∗UTP (P∗,S∗,ρ∗) = 0, (10)

for U(P ,S,ρ) ≥ 0 andUTP (P ,S,ρ) > 0.Proof: Please refer to [7, Appendix A] for a proof of

Theorem 1.Theorem 1 provides a necessary and sufficient condition in

describing the optimal resource allocation policy. In particu-lar, for an optimization problem with an objective functionin fractional form, there exists an equivalent optimizationproblem with an objective function in subtractive form, e.g.U(P ,S,ρ)−q∗UTP (P ,S,ρ) in the considered case, such thatboth problem formulations lead to the same optimal resourceallocation policy. Moreover, the optimal resource allocationpolicy will enforce the equality in (10) which provides anindicator for verifying the optimality of the solution. As aresult, we can focus on the equivalent objective function anddesign a resource allocation policy for satisfying Theorem1in the rest of the paper.

6We assume that the set is non-empty and compact.

6

TABLE IITERATIVE RESOURCEALLOCATION ALGORITHM.

Algorithm Iterative Resource Allocation Algorithm [Dinkel-bach method]1: Initialization: Lmax = the maximum number of iterations

and∆ = the maximum tolerance2: Setq = 0 and iteration indexj = 03: repeat Iteration Process: Main Loop4: For a givenq, obtain an intermediate resource allocation

policy P ′,S ′,ρ′ by solving the problem in (11)5: if U(P ′,S ′,ρ′)− qUTP (P

′,S ′,ρ′) < ∆ then6: Convergence= true7: return P∗,S∗,ρ∗ = P ′,S ′,ρ′ and q∗ =

U(P′,S′,ρ′)UTP (P′,S′,ρ′)

8: else9: Setq = U(P′,S′,ρ′)

UTP (P′,S′,ρ′) andj = j + 110: Convergence= false11: end if12: until Convergence= true or j = Lmax

E. Iterative Algorithm for Energy Efficiency Maximization

In this section, an iterative algorithm (known as the Dinkel-bach method7 [22]) is proposed for solving (8) with anequivalent objective function such that the obtained solutionsatisfies the conditions stated in Theorem 1. The proposedalgorithm is summarized in Table I (on the next page) and theconvergence to the optimal energy efficiency is guaranteed ifthe inner problem (11) is solved in each iteration.

Proof: Please refer to [7, Appendix B] for a proof ofconvergence.

As shown in Table I, we solve the following optimizationproblem for a given parameterq in each iteration in the mainloop, i.e., lines 3–12:

maxP,S,ρ

U(P ,S,ρ)− qUTP (P ,S,ρ)

s.t. C1 – C12. (11)

We note that for any value ofq generated by Algorithm Iin each iteration,U(P ,S,ρ) − qUTP (P ,S,ρ) ≥ 0 is alwaysvalid; negative energy efficiencies will not occur. Please referto [7, Proposition 3] for a proof. In fact, the transformedobjective function, i.e.,U(P ,S,ρ) − qUTP (P ,S,ρ), has aninteresting pricing interpretation from the field of economics.In particular,U(P ,S,ρ) indicates the system profit due toinformation transmission whileqUTP (P ,S,ρ) represents theassociated cost due to energy consumption. Besides, the termsQDk

andQIk in UTP (P ,S,ρ) are the corresponding rebateand discount to the energy cost via energy harvesting, respec-tively. The optimal value ofq represents a scaling factor forbalancing profit and cost.

1) Solution of the Main Loop Problem (11):The trans-formed problem has an objective function in subtractive form

7We note that the Dinkelbach method is an application of Newton’s methodfor root finding, please refer to [15], [23] for details.

and is parameterized by variableq. Unfortunately, there arestill two obstacles in tackling the problem. First, the powersplitting variables for information decoding and energy har-vesting, i.e.,ρIi,k and ρEi,k, are coupled with the power allo-cation variables in both the objective function and constraintC1 which complicates the solution. Second, the combinatorialconstraint C7 on the subcarrier allocation variables creates adisjoint feasible solution set which is a hurdle for solvingtheproblem via tools from convex optimization. In order to derivea tractable resource allocation algorithm, we approximatethetransformed objective function in the following. First, weapproximate theweighted system capacityas

U(P ,S,ρ) =

nF∑

i=1

K∑

k=1

αksi,kW log2

(1 + Pi,kΓi,k

)

≈ U(P ,S,ρ) (12)

whereU(P ,S,ρ) =

nF∑

i=1

K∑

k=1

αksi,kW log2

(Pi,kΓi,k

)(13)

which is a tight approximation for high SINR8, i.e.,Pi,kΓi,k ≫ 1. On the other hand, we adopt a lower bound onUTP (P ,S,ρ) in the transformed objective function in (11):

UTP (P ,S,ρ)

≥ UTP (P ,S,ρ) = PCT+KPCR

+

nF∑

i=1

K∑

k=1

εPi,ksi,k

−

K∑

k=1

nF∑

i=1

( K∑

j=1

Pi,jsi,j

)lkgk|Hi,k|

2ηk −

K∑

k=1

QIk . (14)

where UTP (P ,S,ρ) is obtained by settingρEi,k = 1 in

UTP (P ,S,ρ). Indeed,UTP (P ,S,ρ) can be interpreted as theuse of a theoretical receiver which is able to fully recycleand harvest the energy of the signal used for informationdecoding9. As a result, the transformed objective function canbe approximated by

U(P ,S,ρ)− qUTP (P ,S,ρ)

. U(P ,S,ρ)− qUTP (P ,S,ρ). (15)

On the other hand, by exploiting constraint C12, we canrewrite constraint C1 in the following form to remove theassociated non-convexity:

C1:nF∑

i=1

( K∑

j=1

Pi,jsi,j

)lkgk|Hi,k|

2ηk

+

nF∑

i=1

σ2Ii,k

ηk ≥P reqmink

ρE1,k, ∀k, (16)

8The high SINR assumption is used to bring the optimization probleminto convex form which makes the problem mathematically tractable. Indeed,in the proposed problem formulation, a minimum required system data rateRmin is set to guarantee a desired system data rate. Furthermore,we focuson a indoor line of sight communication environment, see Section V. Thus,it is unlikely that a selected receiver has low SINR.

9Note thatρEi,k

= 1 is only applied to the received energy harvested fromthe information signal, but not to the portion harvested from the interferencesignals.

7

where the right hand side of the inequality in (16) is due toconstraint C12:ρE1,k = ρE2,k = . . . = ρEi,k = . . . = ρEnF ,k.Besides, we can further simplify the algorithm design byreplacing C12 with the following equivalent constraint:

C13: ρE1,k = ρEr,k, ∀k, r ∈ 2, . . . , nF . (17)

Next, we handle the combinatorial constraint in C7 bytime-sharing relaxation [24], [25]. In particular, we relax thesubcarrier selection variablesi,k to be a real value betweenzero and one instead of a Boolean, i.e., C7:0 ≤ si,k ≤ 1. Asa result,si,k can be interpreted as a time-sharing factor in allo-cating subcarrieri toK receivers for delivering information. Inaddition, we introduce two new auxiliary variables and definethem asPi,k = Pi,ksi,k and ρIi,k = ρIi,ksi,k. They representthe actual transmitted power and the power splitting ratio forinformation decoding on subcarrieri for receiverk under thetime-sharing condition, respectively. On the other hand, we re-placeCi,k in C4 and C5 in (11) byCi,k = W log2

(Pi,k

si,kΓi,k

),

Γi,k = Γi,k|ρIi,k

=ρIi,k

si,k

whileCi,k ≥ Ci,k. We note that although

Ci,k ≥ Ci,k generally holds,Ci,k ≈ Ci,k is asymptoticallytight in the high SINR regime. Since the feasible solution setof the problem withCi,k in C4 and C5 is a subset of theoriginal problem10, the solution obtained with the Dinkelbachmethod for the approximated objective function can be usedas a suboptimal solution to the original optimization problemin (8). Nevertheless, it will be shown in the simulation sectionthat the proposed resource allocation algorithm achieves aclose-to-optimal performance in high SINR. Note that byconsidering the transformed optimization problem with theapproximated objective function and relaxed constraint C7ineach iteration of the Dinkelbach method, the proposed schemeconverges to the optimal solution of the problem with theapproximated objective function introduced in (14).

Remark 2: There is an alternative approach for solvingthe optimization problem in (8). Instead of usingU(P ,S,ρ)−qUTP (P ,S,ρ) . U(P ,S,ρ) − qUTP (P ,S,ρ) from (15) ineach iteration, we could exploit the fact that

U(P ,S,ρ)

UTP (P ,S,ρ).

U(P ,S,ρ)

UTP (P ,S,ρ).

Then, we can use a similar approach as in Appendix A toprove that U(P,S,ρ)

UTP (P,S,ρ)is jointly quasi-concave with respect

to (P ,S,ρ) under the time-sharing relaxation. As a result, aunique global optimal solution exist and can be obtained byusing the bisection method [26]. We note that both approachesachieve the same optimal value.

The problem with transformed objective function in (15) isjointly concave with respect to (w.r.t.) all optimization vari-ables (cf. Appendix B). Besides, it can be further verified thatthe primal problem satisfies Slater’s constraint qualification.

10In general, the constraint relaxation used in C7 may result in a supersetof the feasible solution set. Yet, it will be shown in the nextsection that theoptimal subcarrier allocation policy with respect to the approximated objectivefunction takes values of either zero or one on each subcarrier. In other words,the subcarrier allocation policy is Boolean even though it is allowed to takeany real value between zero and one; i.e., the size of the feasible solution setdoes not change with the constraint relaxation in C7.

As a result, strong duality holds and solving the dual problemis equivalent for solving the primal problem [26]. Motivatedby this fact, we solve the primal problem by solving its dualproblem in the following.

F. Dual Problem Formulation

In this subsection, the resource allocation policy is derivedvia solving the dual problem of (11) with approximated objec-tive function. For this purpose, we first need the Lagrangianfunction of the primal problem. The Lagrangian of (11) isgiven by

L(w, λ, γ, β, δ,υ,µ, ζ,P ,S,ρ) (18)

=

nF∑

i=1

K∑

k=1

(αk + γ + υk)si,kCi,k−λ( nF∑

i=1

K∑

k=1

Pi,k − Pmax

)

−q(PCT

+KPCR+

K∑

k=1

nF∑

i=1

εPi,k

−

K∑

k=1

nF∑

i=1

( K∑

j=1

Pi,j

)lkgk|Hi,k|

2ηk −

K∑

k=1

QIk

)− γRmin

−β(PPG − PCT

+

nF∑

i=1

K∑

k=1

εPi,k

)

−

nF∑

i=1

K∑

k=1

δi,k

(ρIi,k + ρEi,k − 1

)−

nF∑

i=1

ζi

( K∑

k=1

si,k − 1)

−

K∑

k=1

nF∑

i=2

µi,k

(ρE1,k − ρEi,k

)−

∑

k∈D

υkRmink

−K∑

k=1

wk

(P reqmink

ρE1,k−

nF∑

i=1

( K∑

j=1

Pi,j

)lkgk|Hi,k|

2ηk−

nF∑

i=1

σ2Ii,k

ηk

),

where Ci,k = W log2

(Pi,k

si,kΓi,k

)and Γi,k = Γi,k

∣∣∣ρIi,k

=ρIi,k

si,k

.

w has elementswk ≥ 0, k ∈ 1, . . . ,K, and is the Lagrangemultiplier vector corresponding to the individual minimumrequired power transfer constraint.λ ≥ 0 accounts for themaximum transmit power allowancePmax in constraint C2.β ≥ 0 is the Lagrange multiplier for C3 accounting for thepower usage from the power grid at the transmitter.γ is theLagrange multiplier associated with the minimum data raterequirement of the system in C4.υ denotes the Lagrange mul-tiplier vector for the minimum individual data rate requirementof receiverk in C5 and has elementsυk ≥ 0, k ∈ 1, . . . ,K.We note thatυk = 0, ∀k /∈ D, for the receivers requiringnon-delay constrained services.ζ is the Lagrange multipliervector accounting for the subcarrier assignment constraint C8and has elementsζi, i ∈ 1, . . . , nF . δ is the Lagrangemultiplier vector for the power splitting constraint C11 andhas elementsδi,k, i ∈ 1, . . . , nF , k ∈ 1, . . . ,K. µ is theLagrange multiplier vector for constraint C12 and has elementsµi,k, i ∈ 1, . . . , nF , k ∈ 1, . . . ,K. On the other hand, theboundary constraints C6, C7, C9, and C10 on the optimizationvariables are captured by the Karush-Kuhn-Tucker (KKT)conditions when deriving the resource allocation solutionin

8

the next section. Thus, the dual problem for the primal problem(11) is given by

minw,λ,γ,β,δ,υ,ζ≥0,µ

maxP,S,ρ

L(w, λ, γ, β, δ,υ,µ, ζ,P ,S,ρ). (19)

G. Dual Decomposition Solution

In this section, the optimal11 resource allocation policy isobtained via Lagrange dual decomposition [27]. Specifically,the dual problem in (19) is decomposed into a hierarchy oftwo levels. Level 1, the inner maximization in (19), consistsof nF subproblems with identical structure that can be solvedin parallel. Level 2, the outer minimization in (19), is themaster problem. The dual problem can be solved iteratively.Specifically, in each iteration, the transmitter solves thenF

subproblems by applying the KKT conditions for a fixed setof Lagrange multipliers. Then the solutions of the subproblemsare used for updating the Lagrange multiplier master problemvia the gradient method.

Level 1 (Subproblem Solution):Using standard convexoptimization techniques and the KKT conditions, for a givenq, in each iteration of the Dinkelbach method, the powerallocation policy and the power splitting policy on subcarrieri for receiverk are given by

P ∗i,k = si,kP

∗i,k = si,k

[W (αk + γ + υk)

ln(2)(Φi,k

)]+

, ∀i, k, (20)

Φi,k = qε+ βε+ λ−

K∑

k=1

(q + wk)lkgk|Hi,k|2ηk, (21)

ρE∗1,k =

[√[ wkPreqmink

δ1,k − q(σ2za + σ2

I1,k)ηk +

∑nF

j=2 µj,k

]+]ρE

U

ρEL

, (22)

∀k, and

ρE∗j,k =

[q(σ2

za + σ2Ij,k

)ηk − δj,k + µj,k

]ρEU

ρEL

, (23)

∀k, j ∈ 2, . . . , nF ,

ρI∗i,k = si,kρI∗i,k=si,k

[Ψi,k

2√ln(2)(δi,k)(σ2

za + σ2Ii,k

)

−σ2zs

2(σ2za + σ2

Ii,k)

]ρIU

ρIL

,∀i, k, (24)

Ψi,k =√4W (αk + γ + υk)(σ2

za + σ2Ii,k

) + (δi,k) ln(2)σ2zs

×√σ2zs (25)

and µ1,k = 0. Here, operators[x]+

and[x]cd

are defined

as[x]+

= max0, x and[x]cd

= c, if x > c,[x]cd

=

x, if d ≤ x ≤ c,[x]cd

= d, if d > x, respectively. Thepower allocation solution in (20) is known as multilevel water-filling. In particular, the water-level in allocating poweron

11In this section, an optimality refers to the optimality for the problemformulation using the time-sharing assumption and the approximated objectivefunction.

subcarrieri for receiver k, i.e., W (αk+γ+υk)ln(2)Φi,k

, is not onlydirectly proportional to the priority of receiverk via variableαk, but also depends on the channel gains of the otherK − 1receivers via the term

∑K

k=1(q + wk)lkgk|Hi,k|2ηk. Besides,

Lagrange multipliersγ, υk, andwk force the transmitter totransmit with a sufficiently high power to fulfill the system datarate requirement,Rmin, the individual data rate requirementsof the receivers having delay constrained services,Rmink

, andthe minimum power transfer requirement,P req

mink, for receiver

k, respectively. Moreover, as can be observed from (20), thepower allocation solutionP ∗

i,k is independent ofsi,k whichfacilitates a simple allocation design.

On the other hand, the power splitting ratio for informationdecoding,ρI∗i,k, is also in the form of water-filling and thewater-level depends on the priority of the receiver viaαk

in (24). Besides, Lagrange multiplierµi,k affects the powersplitting ratio solution for energy harvesting in (22) and (23)such thatρE1,k will eventually equalρEj,k, ∀j ∈ 2, . . . , nF ,as enforced by consensus constraint C12. Furthermore, whenσ2za + σ2

Ii,k≫ σ2

zs , the SINR on each subcarrier approachesPi,kρ

I∗i,klkgk|Hi,k|

2

ρI∗i,k

(σ2

za+σ2

Ii,k)+σ2

zs→

Pi,klkgk|Hi,k|2

σ2

za+σ2

Ii,k

and is independent of

ρI∗i,k. Besides,UTP (P ,S,ρ) is a monotonically decreasingfunction of ρE∗

i,k . Therefore,ρE∗i,k → ρEU and ρI∗i,k → ρIL be-

come the optimal power splitting policy for energy efficiencymaximization. Thus, ifσ2

za+σ2Ii,k

≫ σ2zs and the lower bound

of the power splitting ratio for information decoding is zero,i.e., ρIL = 0, then the solution suggests that in this case aninfinitesimally small portion of the received power is used atthe receivers for information decoding in order to achieve theoptimal performance; provided that the data rate constraintsC4 and C5 are satisfied. In other words, most of the receivedpower at the receivers should be used for energy harvesting.The above considerations indicate that in the interferencelimited regime, a hybrid information and energy harvestingreceiver must achieve a higher energy efficiency than thetraditional pure information receiver. This will be confirmedby simulation in Section V.

On the other hand, subcarrieri is assigned to receiverkwhen the following selection criterion is satisfied:

s∗i,k =

1 if k = argmax

aMi,a ,

0 otherwise(26)

where Mi,k (27)

= W(αk + γ + υk

)[log2

(P ∗i,klkgk|Hi,k|

2)

+ log2

( ρI∗i,k(ρI∗i,k)(σ

2za + σ2

Ii,k) + σ2

zs

)

−1

ln(2)−

σ2zs

ln(2)(ρI∗i,k(σ2za + σ2

Ii,k) + σ2

zs)

]− ζi

is the marginal benefit provided to the system when subcarrieri is assigned to serve receiverk. In other words, receiverk isselected for information transmission on subcarrieri if it canprovide the maximum marginal benefit to the system. Besides,if receiverk has a high priority or a stringent individual datarate requirement, it will have high values ofαk or υk and

9

the resource allocator at the transmitter will have a higherpreference to serve receiverk with subcarrieri. On the otherhand, it can be observed from (26) that although constraintrelaxation is used in constraint C7 for facilitating the designof the resource allocation algorithm, the subcarrier allocationpolicy on each subcarrier for the relaxed problem remainsBoolean; time sharing does not occur.

Remark 3: The above observation for the optimal powersplitting policy in the interference limited regime is valid forboth the problem with the original objective function and theproblem with the approximated objective function. Indeed,ifthe case of an interference limited regime is considered, theuse of the approximated objective function is not necessary.Instead, we can first setρE∗

i,k → ρEU and ρI∗i,k → ρIL in theproblem formulation.ρE∗

i,k and ρI∗i,k become constants and theassociated non-convexity vanishes. Then, we optimizeP andS by following a similar approach as used in (8)-(27).

Level 2 (Master Problem Solution):The Level 2 masterproblem in (19) can be solved by using the gradient methodwhich leads to the following Lagrange multiplier updateequations:

λ(u+ 1)=[λ(u)− ξ1(u)×

(Pmax −

nF∑

i=1

K∑

k=1

Pi,k

)]+, (28)

β(u+ 1)=[β(u)− ξ2(u)×

(PPG − PC (29)

−

nF∑

i=1

K∑

k=1

εPi,k)]+,

γ(u+ 1)=[γ(u)− ξ3(u)×

( nF∑

i=1

K∑

k=1

si,kCi,k

−Rmin

)]+, (30)

δi,k(u+ 1)=[δi,k(u)− ξ4(u)

×(1− ρIi,k − ρEi,k

)]+, ∀i, k, (31)

µr,k(u+ 1)=[µr,k(u)− ξ5(u)×

(ρEr,k − ρE1,k

)], (32)

∀r ∈ 2, . . . , nF , ∀k,

wk(u+ 1)=[wk(u)− ξ6(u)

×( nF∑

i=1

( K∑

j=1

Pi,j

)lkgk|Hi,k|

2ηk

+

nF∑

i=1

σ2Ii,k

ηk −P reqmink

ρE1,k,)]+

, ∀k, (33)

υk(u+ 1)=[υk(u)− ξ7(u)×

( nF∑

i=1

si,kCi,k

−Rmink

)]+, ∀k ∈ D, (34)

where index u ≥ 0 is the iteration index andξt(m),t ∈ 1, . . . , 7, are positive step sizes. Then, the updated12

Lagrange multipliers in (28)–(34) can be used for updating

12It can be observed that updatingζi is not necessary since it does notaffect the result of the subcarrier allocation solution in (26).

the resource allocation policy in (20)–(27) by solving thenF subproblems in (19). As the primal problem with theapproximated objective function is jointly concave w.r.t.theoptimization variables, it is guaranteed that the primal optimalsolution can be obtained by solving the problems in Level 1and Level 2 iteratively, provided that the chosen step sizes,ξt(m), are sufficiently small13. We note that other than theadopted gradient method, different iterative algorithms such asthe ellipsoid method can also be used for finding the optimalLagrange multipliers due to the convexity of the dual problem[26].

IV. RESOURCEALLOCATION DESIGN - DISCRETESET OF

POWER SPLITTING RATIOS

In practice, due to the high complexity associated with ahigh precision power splitting unit, the RF energy harvestingreceivers may only be capable of splitting the received powerinto two power streams based on a finite discrete set of powersplitting ratios. In this section, we design a resource allocationalgorithm for such receivers. In particular, we assume thatthere areN distinct power splitting ratios for energy harvestingand information decoding at each receiver. Thus, the powersplitting ratios for energy harvesting and information decodingon subcarrieri for mobile receiverk can be represented bythe following constraints:

C14: ρEi,k ∈ ρE1

k , ρE2

k , . . . , ρEn

k , . . . , ρEN

k ,

C15: ρIi,k ∈ ρI1k , ρI2k , . . . , ρInk , . . . , ρINk . (35)

ρEn

k andρInk , n ∈ 1, 2, . . . , N, are the possible power split-ting modes for energy harvesting and information decodingadopted in receiverk, respectively. The corresponding resourceallocation algorithm design can be formulated as the followingoptimization problem:

maxP,S,ρ

Ueff (P ,S,ρ)

s.t. C1–C8, C11, C12, C14, C15. (36)

Similar to the case of a continuous set of power splitting ratios,the objective function of the above problem formulation inher-its the non-convexity of the fractional form of the objectivefunction. Therefore, by using Theorem 1, we can transform theobjective function from the fractional form into a subtractiveform and solve the problem via the Dinkelbach method. Asshown in Table I, in each iteration of the main loop, i.e., lines3–12, we solve the following optimization problem for a givenparameterq:

maxP,S,ρ

U(P ,S,ρ)− qUTP (P ,S,ρ)

s.t. C1–C8, C11, C12, C14, C15. (37)

The additional difficulty in solving the above optimizationproblem compared to the problem formulation in (8) is thedisjoint/discrete nature of the optimization variablesρEi,k andρIi,k, cf. C14 and C15 in (35). In general, an exhaustive

13In the literature, different methods such as exact line search and back-tracking line search are commonly used for optimizing the step sizes [26],[28]. In this paper, we adopt a backtracking line search approach in optimizingthe step sizes for fast convergence.

10

search is required to obtain the global optimal solution andthesearch space grows in the order ofN2KnF , which may not becomputationally feasible for systems of moderate size. In thefollowing, we transform (37) into an optimization problemwith tractable solution by exploiting subcarrier time-sharingand the concept of power splitting mode selection. In particu-lar, different values ofρInk can be treated as different operatingmodes of receiverk with different equivalent SINRs. Then,we combine the subcarrier selection with the operating modeselection by augmenting the dimensions of the optimizationvariables. To this end, we define the channel capacity betweenthe transmitter and receiverk on subcarrieri with channelbandwidthW by using power splitting moden as

Cni,k = W log2

(1 + Pn

i,kΓni,k

)(38)

Γni,k =

ρInk lkgk|Hi,k|2

ρInk (σ2za + σ2

Ii,k) + σ2

zs

,

andPni,kΓ

ni,k is the received SINR on subcarrieri at receiver

k using power splitting moden for information decoding.The weighted system capacityis defined as the total averagenumber of bits successfully delivered to theK receivers viatheN power splitting modes and is given by

U(PN ,SN ) =

nF∑

i=1

K∑

k=1

N∑

n=1

αksni,kC

ni,k, (39)

where PN = Pni,k ≥ 0, ∀i, k, n is the power allocation

policy andSN = sni,k ∈ 0, 1, ∀i, k, n is the subcarrierallocation policy for the case of discrete power splittingratios. We note that the subcarrier allocation policy in (39)incorporates the power splitting mode selection for informationdecoding. On the other hand, the power consumption of thesystem can be written as

UTP (PN ,SN ,AN )

= PCT+KPCR

+

nF∑

i=1

K∑

k=1

N∑

n=1

εPni,ks

ni,k

−

K∑

k=1

N∑

n=1

QnDk

−

K∑

k=1

N∑

n=1

QnIk

(40)

whereQnDk=

nF∑

i=1

( K∑

j=1

N∑

m=1

Pmi,js

mi,j

)lkgk|Hi,k|

2ηkρEn

k ank

︸ ︷︷ ︸Power harvested from information signalat receiverk with power splitting moden

(41)

and QnIk=

nF∑

i=1

(σ2za + σ2

Ii,k)ρEn

k ankηk

︸ ︷︷ ︸Power harvested from interference and

antenna noise at receiverk with power splitting moden

. (42)

Here,AN = ank ∈ 0, 1, ∀k, n is the power splitting ratioselection policy for energy harvesting. In the above problemformulation,ank is the optimization variable which captures theselection of power splitting modeρEn

k for energy harvesting.Similar to the case of the continuous set of power splittingratios, we consider an approximation of the objective function

for facilitating a tractable resource allocation algorithm designin the following. First, the system capacity between thetransmitter and theK mobile receivers can be approximatedby

U(PN ,SN )(a)≈ U(PN ,SN )

=

nF∑

i=1

K∑

k=1

N∑

n=1

αksni,kC

ni,k,

Cni,k = W log2

(Pni,kΓ

ni,k

)(43)

and(a)≈ in (43) is due to the high SINR assumption. On the

other hand, a lower bound for the total power consumption ofthe system is given by

UTP (PN ,SN ,AN )

≥ UTP (PN ,SN ,AN )

= PCT+KPCR

+K∑

k=1

nF∑

i=1

N∑

n=1

εPni,ks

ni,k

−

nF∑

i=1

K∑

k=1

( K∑

j=1

N∑

m=1

Pmi,js

mi,j

)lkgk|Hi,k|

2ηk

−K∑

k=1

N∑

n=1

nF∑

i=1

QnIk, (44)

whereUTP (PN ,SN ,AN ) is obtained by settingρEn

k ank = 1.Then, the objective function for the optimization problem fordiscrete sets of power splitting factors is given by

U(PN ,SN )− qUTP (PN ,SN ,AN )

. U(PN ,SN )− qUTP (PN ,SN ,AN ). (45)

A. Optimization Problem Formulation

With a slight abuse of notation, we reformulate the op-timization problem for discrete power splitting ratios to besolved in each iteration as follows:

maxPN ,SN ,AN

U(PN ,SN )− qUTP (PN ,SN ,AN )

s.t. C1:N∑

n=1

QnDk

+QnIk

≥ P reqmink

, ∀k,

C2:nF∑

i=1

K∑

k=1

N∑

n=1

Pni,ks

ni,k ≤ Pmax,

C3: PC +

nF∑

i=1

K∑

k=1

N∑

n=1

εPni,ks

ni,k ≤ PPG,

C4:nF∑

i=1

K∑

k=1

N∑

n=1

sni,kCni,k ≥ Rmin,

C5:nF∑

i=1

K∑

k=1

N∑

n=1

sni,kCni,k ≥ Rmink

, ∀k ∈ D,

C6: Pni,k ≥ 0, ∀i, k, n, C7: sni,k ∈ 0, 1, ∀i, k, n,

C8:K∑

k=1

N∑

n=1

sni,k ≤ 1, ∀i, C9: ank ∈ 0, 1, ∀k, n,

11

C10:N∑

n=1

ank = 1, ∀k,

C11: ρInk sni,k +

N∑

m=1

ρEm

k amk ≤ 1, ∀i, k, n,

C12:N∑

m=1

ρImk smi,k + ρEn

k ank ≤ 1, ∀i, k, n. (46)

Constraints C1–C6 have the same physical meanings as in theproblem formulation in (8). Constraints C7–C12 are imposedto guarantee that in each receiver only one power splittingmode can be selected for information decoding and energyharvesting. Besides, C11 and C12 indicate that no extrapower gain can be achieved in the power splitting process.On the other hand, the non-convexity of the above problemformulation is caused by the coupled optimization variables inC1 and the combinatorial constraints in C7 and C9. In analogyto the techniques used for solving the optimization problemin(11), we solve problem (46) in the following two steps. In thefirst step, we relax constraints C7 and C9 such that variablessni,k andani,k can assume any value between zero and one, i.e.,C7: 0 ≤ sni,k ≤ 1, ∀i, k, n and C9:0 ≤ ani,k ≤ 1, ∀i, k, n. Thensni,k andani,k can be interpreted as the time sharing factors forreceiverk in utilizing subcarrieri with power splitting moden.We also define a new variablePn

i,k = sni,kPni,k for facilitating

the design of the resource allocation algorithm. In fact,Pni,k

represents the actual transmit power of the transmitter forreceiverk in subcarrieri if power splitting moden is usedunder the time sharing condition. In the second step, wereplace constraint C1 in (46) by

C1’:nF∑

i=1

( K∑

j=1

N∑

m=1

Pmi,j

)lkgk|Hi,k|

2ηk

+

nF∑

i=1

(σ2za + σ2

Ii,k)ηk ≥

ankPreqmink

ρEn

k

, ∀k, n. (47)

Although C1’ is equivalent to C1 in (46) only ifank takesa binary value, i.e.,ank ∈ 0, 1, and

∑K

k=1 ank = 1, it will

be shown in the next section that the optimal solution forankhas a binary form under constraint C1’, despite the adoptedconstraint relaxation. Consequently, the optimization problemwith the approximated objective function and the constraintrelaxation is now jointly concave w.r.t. all optimization vari-ables14. Besides, it satisfies Slater’s constraint qualification.Therefore, we can apply dual decomposition to solve theprimal problem via solving its dual problem. The Lagrangianfunction of the primal problem in (46) is given by

L(w, λ, γ, β, δ,υ,ϕ,κ, ζ,PN ,SN ,AN ) (48)

=

nF∑

i=1

K∑

k=1

N∑

n=1

(αk + γ + υk)sni,kC

ni,k

14The concavity of the above optimization problem can be proved byfollowing a similar approach as in Appendix B for the case of continuouspower splitting ratios.

− q(PCT

+KPCR+

K∑

k=1

N∑

n=1

nF∑

i=1

εPni,k

−K∑

k=1

nF∑

i=1

N∑

n=1

( K∑

j=1

Pni,j

)lkgk|Hi,k|

2ηk −K∑

k=1

N∑

n=1

QnIk

)

−γRmin −

nF∑

i=1

ζi

( K∑

k=1

N∑

n=1

sni,k − 1)

−λ( nF∑

i=1

K∑

k=1

N∑

n=1

Pni,k − Pmax

)

−

nF∑

i=1

K∑

k=1

N∑

n=1

δni,k

(ρInk sni,k +

N∑

m=1

ρEm

k amk − 1)

−

nF∑

i=1

K∑

k=1

N∑

n=1

κni,k

( N∑

m=1

ρImk smi,k + ρEn

k ank − 1)

−

K∑

k=1

N∑

n=1

wnk

(ankPreqmink

ρEn

k

−

nF∑

i=1

( K∑

j=1

N∑

m=1

Pmi,j

)lkgk|Hi,k|

2ηk

−

nF∑

i=1

(σ2za + σ2

Ii,k)ηk

)−

∑

k∈D

υkRmink

−

K∑

k=1

ϕk

( N∑

n=1

ank − 1)− β

(PPG − PCT

+

nF∑

i=1

K∑

k=1

N∑

n=1

εPni,k

),

where λ, β, and γ are the scalar Lagrange multipliers as-sociated to constraints C2, C3, and C4 in (46), respectively.w,υ,ϕ, δ, and κ are the Lagrange multiplier vectors forconstraints C1’, C5, C8, C10, C11, and C12 which haveelementswn

k ≥ 0, n ∈ 1, . . . , N, k ∈ 1, . . . ,K, υk ≥ 0,ζi ≥ 0, i ∈ 1, . . . , nF , ϕi, δni,k, andκn

i,k respectively. Wenote that there is no restriction on the value ofϕi since itis associated with equality constraint C10. Thus, the dualproblem is given by

minw,λ,γ,β,δ,υ,ζ,κ≥0,ϕ

(49)

maxPN ,SN ,AN

L(w, λ, γ, β, δ,υ,ϕ,κ, ζ,PN ,SN ,AN ).

B. Dual Decomposition Solution

By using dual decomposition and following a similar ap-proach as in Section III-G, the resource allocation policycan be obtained via an iterative procedure. For a given setof Lagrange multipliersw, λ, γ, β, δ,υ,ϕ,κ, ζ, the powerallocation policy, power splitting policy, and subcarrierallo-cation policy for receiverk using power splitting moden onsubcarrieri are given by

Pn∗i,k = sni,kP

n∗i,k = si,k

[W (αk + γ + υk)

ln(2)(Φi,k

)]+

, ∀i, k, (50)

an∗k =

1 if n = argmax

bT bk , ∀k,

0 otherwise(51)

12

T bk = ρEb

k

(qηk

nF∑

i=1

(σ2za + σ2

Ii,k)−

nF∑

i=1

N∑

m=1

δmi,k −

nF∑

i=1

κbi,k

)

−wb

kPreqmink

ρEb

k

− ϕk, (52)

sn∗i,k =

1 if n, k = argmax

c,bM c

i,b , ∀i,

0 otherwise(53)

Mni,k = W

(αk + γ + υk

)[log2

(Pn∗i,k lkgk|Hi,k|

2)

+ log2

( ρInkρInk (σ2

za + σ2Ii,k

) + σ2zs

)

−1

ln(2)−

σ2zs

ln(2)(ρIni,k(σ2za + σ2

Ii,k) + σ2

zs)

]

−δni,kρInk −

( N∑

m=1

κmi,k

)ρInk − ζi (54)

andΦi,k is defined in (25). The power allocation solution in(50) has a similar multi-level water filling interpretationas in(20). The difference between (20) and (50) is that the powerallocation in (50) is performed w.r.t. each power splittingmode. On the other hand, it can be observed from (51) and(53) that the optimal values ofank andsni,k are binary numbers,although time sharing relaxation is used for facilitating thealgorithm design.

Now, since the dual function is differentiable, we can updatethe set of Lagrange multipliers for a given set ofPN , SN , AN

by using the gradient method. The gradient update equationsare given by

λ(u+ 1)=[λ(u)− ξ1(u)×

(Pmax

−

nF∑

i=1

K∑

k=1

N∑

n=1

Pni,k

)]+, (55)

β(u+ 1)=[β(u)− ξ2(u)×

(PPG − PC

−

nF∑

i=1

K∑

k=1

N∑

n=1

εPni,k)

]+, (56)

γ(u+ 1)=[γ(u)− ξ3(u)×

(Rmin

−

nF∑

i=1

K∑

k=1

N∑

n=1

sni,kCni,k

)]+, (57)

δni,k(u+ 1)=[δni,k(u)− ξ4(u)

×(1− ρInk sni,k −

N∑

m=1

ρEm

k amk

)]+, ∀i, k, n,(58)

κni,k(u+ 1)=

[κni,k(u)− ξ5(u)

×(1−

N∑

m=1

ρImk smi,k − ρEk

k ank

)]+, ∀i, k, n, (59)

wnk (u+ 1)=

[wn

k (u)− ξ6(u) (60)

×( nF∑

i=1

( K∑

j=1

Pi,j

)lkgk|Hi,k|

2ηk

+

nF∑

i=1

(σ2za + σ2

Ii,k)ηk −

aknPreqmink

ρEn

k

,)]+

, ∀k, n,

υk(u+ 1)=[υk(u)− ξ7(u)×

(Rmink

−

nF∑

i=1

N∑

n=1

sni,kCni,k

)]+, ∀k ∈ D. (61)

Similar to the case of continuous power splitting ratios,updatingζi andϕi is not necessary since they will not affectthe power splitting mode selection and subcarrier allocation in(51) and (53), respectively.

V. RESULTS

In this section, simulation results are presented to demon-strate the energy efficiency and system capacity of the pro-posed resource allocation algorithms. We consider an indoorcommunication system withK receivers and the correspond-ing simulation parameters are provided in Table II. We notethat the 470 MHz frequency band will be used by IEEE802.11 for the next generation of Wi-Fi systems [31]. Besides,the wavelength of the carrier signal is0.6 meter which issmaller than the minimum distance between the transmitterand receivers. Thus, the far-field assumption of the channelmodel in [29] holds. Furthermore, in practice, the value ofcircuit power consumption depends on the specific hardwareimplementation and the application. On the other hand, weassume that all receivers have the same priorityαk = 1, ∀k,for illustrating the maximum achievable energy efficiency ofthe system. The shadowing of all communication links is setto gk = 1, ∀k, to account for the line-of-sight communicationsetting. Unless specified otherwise, we assume that there isonly one receiver requiring delay constrained service withaminimum data rate requirement ofRmink

= 10 Mbit/s. Inthe sequel, the total number of iterations is defined as thenumber of main loop iterations in the Dinkelbach method.For the case of continuous power splitting ratios, we setρEU = ρIU = 1 andρEL = ρIL = 0. Besides, there are five powersplitting ratios for the resource allocation with the discreteset of power splitting ratios:ρEi,k ∈ 1, 0.75, 0.5, 0.25, 0

and ρIi,k ∈ 0, 0.25, 0.5, 0.75, 1. Moreover, to ensure fastconvergence, the step sizes adopted in (28)–(34) and (55)–(61) are optimized via backtracking line search, cf. [26, page464] and [32, page 230]. Note that if the transmitter is unableto meet the minimum required system data rateRmin, theminimum required individual data rateRmink

, or the minimumrequired power transferP req

mink, we set the energy efficiency

and the system capacity for that channel realization to zeroto account for the corresponding failure. For the sake ofillustration, the performance curves of the proposed algorithmsfor the continuous and discrete sets of power splitting ratiosare labeled as “Proposed algorithm I” and “Proposed algorithmII” in Figures 2–7. The average energy efficiency of thesystem is computed according to (7) and averaged over 100000

13

TABLE IISYSTEM PARAMETERS

Maximum service distancedmax 10 mReference distanced0 1 mUsers distribution Uniformly distributed betweend0 anddmax

Multipath fading distribution Rician fading with Rician factor6 dBChannel path loss model TGn path loss model [29]Carrier center frequency 470 MHzNumber of subcarriersnF 128Total bandwidthB and subcarrier bandwidthW 20 MHz and156 kHzThermal noise power and antenna noise power per subcarrier −112 dBm and−115 dBmQuantization noise (12-bit uniform quantizer) per subcarrier −47 dBmMinimum system data rate requirementRmin 50 Mbit/sMinimum data rate requirement for delay constrained service Rmink

10 Mbit/sClass A/B power amplifier [30] with a power efficiency of16% ε = 6.25Maximum power supplyPPG 50 dBmMinimum required power transferP req

mink0 dBm

Energy harvesting efficiencyηk 0.8Circuit power consumptions:PCT

andPCR30 dBm and20 dBm

Effective antenna gain 12 dB

independent realizations of multipath fading and path lossattenuation.

A. Convergence and Optimality of Iterative Algorithm

Figure 2 depicts the average system energy efficiency of theproposed iterative algorithms for different levels of receivedinterference versus the number of iterations. Specifically, weare interested in the energy efficiency and convergence speedof the proposed algorithms. We plot the upper bound per-formance15 of the system after convergence to illustrate thesub-optimality of the proposed algorithms. The dashed linesrefer to the average energy efficiency upper bound for eachcase study. After only 5 iterations, the iterative algorithmsachieve over 95% of the upper bound value for all consideredscenarios. Besides, the convergence speed of the proposedalgorithms is invariant to the interference levels,σ2

Ii,k, which

is desirable for practical implementation.In the following case studies, the number of iterations

is set to 5 for illustrating the performance of the proposedalgorithms.

B. Energy Efficiency versus Maximum Allowed TransmitPower

Figure 3 shows the average system energy efficiency versusthe maximum transmit power allowance,Pmax, for differentreceived levels of interference,σ2

Ii,k. We first focus on the case

of a small number of receivers and moderate interference level.It can be observed that the average system energy efficiencyof the proposed algorithms is a monotonically non-decreasingfunction of Pmax. In particular, starting from a small valueof Pmax, the energy efficiency first quickly increases with anincreasingPmax and then saturates whenPmax > 18 dBm.This is due to the fact that the two proposed algorithms strikea balance between the system energy efficiency and the power

15The upper bound system performance is obtained by directly evaluatingthe upper bound objective function via replacing the energyconsumptionfunctionUTP (P,S,ρ) in (14) byU(P,S,ρ) for the continuous set of powersplitting ratios. We note that the approximation in (12) is asymptotically tightfor high SINR.

0 1 2 3 4 5 6 7 8 9 10

1

1.2

1.4

1.6

1.8

2

x 108

Number of iterations

Ave

rage

sys

tem

ene

rgy

effic

ienc

y (b

its/J

oule

)

Upper bound performance, σIi,k

2 = −50 dBm

Proposed algorithm I, σIi,k

2 = −50 dBm

Proposed algorithm II, σIi,k

2 = −50 dBm

Upper bound performance, σIi,k

2 = −40 dBm

Proposed algorithm I, σIi,k

2 = −40 dBm

Proposed algorithm II, σIi,k

2 = −40 dBm

Upper bound performance

σIi,k

2 = −50 dBm

σIi,k

2 = −40 dBm

Fig. 2. Average system energy efficiency (bits-per-Joule) versus number ofiterations with different levels of interference power,σ2

Ii,k, andPmax = 30

dBm. There areK = 3 receivers in the system. The dashed solid linesrepresent the upper bound of energy efficiency after algorithm convergence.

consumption of the system. In fact, once the maximum energyefficiency16 of the system is achieved, a further increase inthe transmit power would result in a degradation in energyefficiency. As expected, proposed algorithm I outperformsproposed algorithm II in all cases since the latter algorithm isdesigned based on discrete sets of power splitting ratios whichspan a smaller feasible solution set compared to algorithm I.Besides, Figure 3 reveals that although interference signalscan act as a viable energy source to the system, cf. (5) and(7), strong interference impairs the energy efficiency of thesystem; the energy harvesting gain due to strong interference17

is unable to compensate the corresponding capacity loss. For

16The maximum energy efficiency refers to the “maximum” w.r.t.thecorresponding problem formulation.

17Nevertheless, we would like to emphasize that the use of hybrid infor-mation and energy harvesting receivers provides a better energy efficiency tothe system compared to pure information receivers, as can beobserved frombaseline II.

14

5 10 15 20 25 30 350.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

x 108

Pmax

(dBm)

Ave

rage

sys

tem

ene

rgy

effic

ienc

y (b

its/J

oule

)

Proposed algorithm I, σIi,k

2 = −50 dBm

Proposed algorithm II, σIi,k

2 = −50 dBm

Baseline I, σIi,k

2 = −50 dBm

Baseline II, σIi,k

2 = −50 dBm

Proposed algorithm I, σIi,k

2 = −40 dBm

Proposed algorithm II, σIi,k

2 = −40 dBm

Baseline I, σIi,k

2 = −40 dBm

Baseline II, σIi,k

2 = −40 dBm

Proposed algorithm I, σIi,k

2 = −30 dBm

Proposed algorithm II, σIi,k

2 = −30 dBm

Baseline II, σIi,k

2 = −30 dBm

Baseline II, σIi,k

2 = −30 dBm

σIi,k

2 = − 30 dBm

σIi,k

2 = −50 dBm

σIi,k

2 = − 40 dBm

Fig. 3. Average system energy efficiency (bits-per-Joule) versus maximum transmit power allowance,Pmax, for K = 3 receivers and different levels ofinterference power,σ2

Ii,k.

comparison, Figure 3 also contains the energy efficiency oftwo baseline resource allocation schemes. For baseline I, wemaximize the weighted system capacity (bit/s) with respectto P ,S,ρ subject to constraints C1–C12 in (8), insteadof the energy efficiency. Specifically, baseline I focuses ona system with hybrid energy harvesting receivers which splitthe received signal into two power streams for a finite discreteset of power splitting ratios. On the other hand, baselineII maximizes the weighted energy efficiency of the systemwith respect toP ,S for receivers which do not haveenergy harvesting capability. Since the receivers of baselineII are unable to harvest energy, we do not impose constraintC1 for baseline II since a minimum power transfer to thereceivers is not required. Figure 3 reveals that in the lowtransmit power regime and for a small number of receivers,the system with power splitting receivers achieves a smallperformance gain compared to the system without energyharvesting receivers. This is because in the low transmit powerregime, the received power of the desired signal at the receiversmay not be sufficiently large for simultaneous informationdecoding and energy harvesting. As a result, power splittingoccurs for fulfilling constraint C1. On the other hand, theenergy efficiency gain achieved by the proposed algorithmattains its maximum in the high transmit power allowanceregime. It can be observed that approximately a 5% gainin energy efficiency can be achieved forK = 3 receiversand a moderate interference power level. We emphasize thatin the high transmit power allowance regime, the maximum

energy efficiency achieved by the system with hybrid energyharvesting receivers cannot be attained by the system withouthybrid energy harvesting receivers (baseline II) via increasingthe transmit power.

Now, we focus on the system with a moderate numberof receivers and moderate interference levels in the hightransmit power allowance regime. Figure 4 illustrates the av-erage system energy efficiency versus the number of receiversfor different interference power levels,σ2

Ii,k, and different

resource allocation algorithms. The maximum transmit powerallowance is set toPmax = 25 dBm. In particular, we comparethe proposed algorithm II, i.e., energy efficiency maximizationwith discrete power splitting ratio receivers, with baselineII. It can be observed from Figure 4 that the performancegain provided by energy harvesting depends on the number ofreceivers and the interference power level for a fixed maximumtransmit power allowancePmax. Indeed, when there is onlyone receiver, the performance gain due to energy harvestingis small since only a small portion of radiated power canbe harvested by the receiver. However, when there are morereceivers in the systems, more receivers participate in theenergy harvesting process and thus a larger amount of energycan be harvested from the RF signals. As a result, a largerperformance gain in terms of energy efficiency can be achievedby the proposed algorithm over baseline II when there aremore receivers in the system. Furthermore, it can be seen thatthe performance gain achieved by the proposed algorithm overbaseline II increases with the interference power level. The

15

1 2 3 4 5 6 7 82

3

4

5

6

7

8

9

10

11

12

x 107

Number of receivers

Ave

rage

sys

tem

ene

rgy

effic

ienc

y (b

its/J

oule

)

Proposed algorithm II, σIi,k

2 = −30 dBm

Baseline II, σIi,k

2 = −30 dBm

Proposed algorithm II, σIi,k

2 = −20 dBm

Baseline II, σIi,k

2 = −20 dBm

Proposed algorithm II, σIi,k

2 = −10 dBm

Baseline II, σIi,k

2 = −10 dBm

9% gain

10% gain

σIi,k

2 = −30 dBm

σIi,k

2 = −20 dBm

20% gain

σIi,k

2 = −10 dBm

Fig. 4. Average system energy efficiency (bits-per-Joule) versus number of receivers,K, for Pmax = 25 dBm and different levels of interference power,σ2Ii,k

.

reason behind this is twofold. First, the increase in interferencepower level provides some extra energy to the system forpotential energy harvesting. Second, the strong interferencetends to saturate the SINR on each subcarrier such that it isindependent ofρI∗i,k, i.e.,

Pi,kρI∗i,klkgk|Hi,k|

2

ρI∗i,k

(σ2

za+σ2

Ii,k)+σ2

zs⇒

Pi,klkgk|Hi,k|2

σ2

za+σ2

Ii,k

for σ2za+σ2

Ii,k≫ σ2

zs . Thus, using more of the received powerfor information decoding does not provide a significant gaininchannel capacity. Consequently, more received power is usedfor energy harvesting to reduce the total energy consumptionof the system (see also Figure 6), which enhances the systemenergy efficiency. On the other hand, we would like to empha-size that although the energy efficiency gain of algorithm IIcompared to baseline II depends on the system settings, energyharvesting may be necessary in some applications where otherpower sources are not available for the receivers regardless ofhow much can be gained in energy efficiency.

C. Average System Capacity versus Maximum Allowed Trans-mit Power

In Figure 5, we plot the average system capacity versus themaximum transmit power allowance,Pmax, for different levelsof interference power,σ2

Ii,k. We compare the two proposed

algorithms with the two aforementioned baseline schemes.For Pmax < 18 dBm, it can be observed that the averagesystem capacities of the two proposed algorithms scale withthe maximum transmit power allowancePmax. Yet, the systemcapacity gain due to a largerPmax begins to saturate in the

high transmit power allowance regime, i.e.,Pmax ≥ 18 dBm.Indeed, the proposed algorithms do not further increase thetransmit power in the RF if the system capacity gain due to ahigher transmit power cannot neutralize the associated energyconsumption increase required for boosting the transmit power.Baseline II has a similar behaviour as the proposed algorithmswith respect to the maximum transmit power allowance, sincebaseline II also focuses on energy efficiency maximization.On the other hand, as expected, baseline II achieves a systemcapacity gain over the other algorithms in the low transmitpower allowance regime. Indeed, the SINR of each subcarrieris monotonically increasing with respect toρIi,k. In otherwords, the SINR (capacity) on each subcarrier is maximizedwhen there is no power splitting, i.e.,ρIi,k = 1, ρEi,k = 0, whichis the setting for baseline II. We note that although baselineI focuses on system capacity maximization, power splittingoccurs for fulfilling constraint C1 which results in a lowersystem capacity compared to baseline II in the low transmitpower regime. However, the system capacity of baseline I ishigher than that of the other algorithms in the high transmitpower regime, e.g.,Pmax ≥ 18 dBm. This is attributed to thefact that in order to maximize the system capacity for baselineI, the transmitters always radiate all the available power.Yet,the higher system capacity of baseline I comes at the expenseof a low system energy efficiency, cf. Figure 3. In addition,we observe from Figure 5 that all the considered algorithmsare able to fulfill the system data rate requirement in C4 on

16

5 10 15 20 25 30 35

1

1.5

2

2.5

3

3.5

4

x 108

Pmax

(dBm)

Ave

rage

sys

tem

cap

acity

(bi

ts/s

)

Proposed algorithm I, σIi,k

2 = −50 dBm

Proposed algorithm II, σIi,k

2 = −50 dBm

Baseline I, σIi,k

2 = −50 dBm

Baseline II, σIi,k

2 = −50 dBm

Proposed algorithm I, σIi,k

2 = −40 dBm

Proposed algorithm I, σIi,k

2 = −40 dBm

Baseline I, σIi,k

2 = −40 dBm

Baseline II, σIi,k

2 = −40 dBm

Proposed algorithm I, σIi,k

2 = −30 dBm

Proposed algorithm II, σIi,k

2 = −30 dBm

Baseline I, σIi,k

2 = −30 dBm

Baseline II, σIi,k

2 = −30 dBm

14.9 15 15.1

2.64

2.66

2.68

2.7

x 108

σIi,k

2 = −50 dBm

σIi,k

2 = −40 dBm

σIi,k

2 = −30 dBm

Fig. 5. Average system capacity (bits-per-second) versus maximum transmit power allowance,Pmax, for different power levels of interference,σ2Ii,k

, andK = 3 receivers.

10 15 20 25 30 353

4

5

6

7

8

9

10

11

12

13

Pmax

(dBm)

Ave

rage

har

vest

ed p

ower

(dB

m)

Proposed algorithm II, σIi,k

2 = −30 dBm

Proposed algorithm II, σIi,k

2 = −40 dBm

Proposed algorithm II, σIi,k

2 = −50 dBm

σIi,k

2 = −30 dBm

σIi,k

2 = −40 dBm

σIi,k

2 = −50 dBm

Fig. 6. Average harvested power (dBm) of proposed algorithmII versus themaximum transmit power allowance,Pmax, for different interference powerlevels andK = 3 receivers. The double-sided arrow indicates the powerharvesting gain due to an increasing interference power level.

average.

D. Average Harvested Power versus Maximum Allowed Trans-mit Power

Figure 6 depicts the average harvested power of the pro-posed algorithm II versus the maximum allowed transmit

power,Pmax, for different levels of interference power,σ2Ii,k

.It can be seen that in the high transmit power regime, theamount of average harvested power in all considered scenariosis saturated. This is because for the proposed algorithm II,thetransmitter stops to increase the transmit power for energyefficiency maximization. Meanwhile, the average interferencepower level remains unchanged and no extra energy can beharvested by theK receivers. On the other hand, a higheramount of power is harvested by the receivers in the systemwhen the interference power level increases as the SINR isindependent ofρkIi,k on each subcarrier. Thus, splitting morereceived power for energy harvesting improves system energyefficiency and increases the average harvested power.

E. Average Energy Efficiency and System Capacity versusNumber of Receivers

Figure 7(a) and Figure 7(b) illustrate the average systemcapacity and the average system energy efficiency of theproposed algorithm II versus the number of mobile receiversfor different interference power levels,σ2

Ii,k, and different

receiver circuit power consumptions,PCR. In Figure 7(b), it

can be observed that the average system capacity increaseswith the number of receivers in the system since the proposedalgorithm is able to exploit multiuser diversity. Specifically,the transmitter has a higher chance of selecting a receiverwith good channel conditions when more receivers are in thesystem, which results in a system capacity gain. In addition,

17

1 2 3 4 5 6 7 8

1.2

1.4

1.6

1.8

2

2.2

2.4

x 108

Number of receivers

Ave

rage

sys

tem

ene

rgy

effic

ienc

y (b

its/J

oule

)

σIi,k

2 = −50 dBm

σIi,k

2 = −30 dBm

PC

R

= 10 dBm

PC

R

= 20 dBm

PC

R

= 15 dBm

PC

R

= 20 dBmP

CR

= 15 dBm

PC

R

= 10 dBm

(a) Average energy efficiency (bits-per-Joule) versusK.

1 2 3 4 5 6 7 81.4

1.6

1.8

2

2.2

2.4

2.6

2.8

3

3.2x 10

8

Number of receivers

Ave

rage

sys

tem

cap

acity

(bi

ts/s

)

PC

R

= 10 dBm, σIi,k

2 = −50 dBm

PC

R

= 15 dBm, σIi,k

2 = −50 dBm

PC

R

= 20 dBm, σIi,k

2 = −50 dBm

PC

R

= 10 dBm, σIi,k

2 = −30 dBm

PC

R

= 15 dBm, σIi,k

2 = −30 dBm

PC

R

= 20 dBm, σIi,k

2 = −30 dBm

σIi,k

2 = −50 dBm

σIi,k

2 = −30 dBm

(b) Average system capacity (bit-per-second) versusK.

Fig. 7. Average energy efficiency (bits-per-Joule) and average system capacity (bit-per-second) versus the number of receiversK for Pmax = 18 dBm,different receiver circuit power consumptions,PCR

, and different interference power levelsσ2Ii,k

.

although a higher interference power level impairs the averagesystem capacity, it does not decrease the performance gain dueto multiuser diversity as can be concluded from the slopes ofthe curves. Besides, a higher circuit power consumption in thereceivers does not have a large impact on the average systemcapacity. On the contrary, Figure 7(a) shows that the averagesystem energy efficiency does not necessarily monotonicallyincrease/decrease w.r.t. the number of receivers. In fact,fora moderate value of receiver circuit power consumption, e.g.PCR

≥ 15 dBm, for the considered system setting, the energyefficiency of the system first increases and then decreaseswith an increasing number of receivers. The enhancementof energy efficiency is mainly due to the multiuser diversitygain in the channel capacity when having multiple receivers.Besides, if more receivers participate in the energy harvestingprocess, a larger portion of energy can be harvested from theRF signals. Nevertheless, an extra circuit energy consumptionis incurred by each additional receiver. Indeed, when thenumber of receivers in the system orPCR

are large, thesystem performance gain due to multiuser diversity is unable tocompensate the total energy cost of the receivers sinceKPCR

increases linearly w.r.t. the number of receivers. Hence, theenergy efficiency of the system decreases with the number ofreceivers. As a matter of fact, the energy efficiency gain dueto the additional receivers depends on the trade-off betweenmultiuser diversity gain, the amount of harvested energy, andthe associated cost in having multiple receivers in the system.In the extreme case, whenPCR

→ 0, the energy efficiency willmonotonically increase w.r.t. the number of receivers, providedthat the optimization problems in (8) and (46) are feasible.

VI. CONCLUSIONS

In this paper, the resource allocation algorithm designfor simultaneous wireless information and power transfer inOFDMA systems was studied. We focused on power splittingreceivers which are able to split the received signals into

two power streams for concurrent information decoding andenergy harvesting. The algorithm design was formulated as anon-convex optimization problem which took into account aminimum system data rate requirement, minimum individualdata rate requirements of the receivers, a minimum requiredpower transfer, and the total system power dissipation. Wefirst focused on receivers with continuous sets of powersplitting ratios and proposed a resource allocation algorithm.The derived solution served as a building block for the designof a suboptimal resource allocation algorithm for receiverswith discrete sets of power splitting ratios. Simulation resultsshowed the excellent performance of the two proposed sub-optimal algorithms and also unveiled the trade-off betweenenergy efficiency, system capacity, and wireless power transfer.In particular, wireless power transfer enhances the systemenergy efficiency by harvesting energy in the radio frequency,especially in the interference limited regime.

APPENDIX

A. Proof of Quasi-Concavity of the Objective Function withrespect to the Power Allocation Variables

For facilitating the following presentation, we first providea definition for quasi-concave functions. Consider a functiong(·) with input vector x. g(x) is a strictly quasi-concavefunction if and only if its super-level set [26]

Lν = x ∈ Rn|g(x) ≥ ν, ν ∈ R, (62)

is concave, whereR denotes the set of real numbers. Supposeg(P ,S,ρ) = U(P,S,ρ)

UTP (P,S,ρ) . Now, we focus on the following set:

U(P ,S,ρ)

UTP (P ,S,ρ)≥ ν. (63)

When ν < 0, the set is empty sinceU(P ,S,ρ) > 0 andUTP (P ,S,ρ) > 0. As a result,Lα is an affine(/concave) set

18

[26] for ν < 0. Then, forν > 0, we can rewrite setLν as

U(P ,S,ρ)

UTP (P ,S,ρ)≥ ν

⇐⇒ U(P ,S,ρ)− νUTP (P ,S,ρ) ≥ 0. (64)

SinceU(P ,S,ρ) > 0 andUTP (P ,S,ρ) > 0 are concaveand affine functions with respect to the power allocationvariablesP , respectively, (64) is a concave function andLν

represents a concave set forν > 0. Therefore, the objectivefunction is a strictly quasi-concave function with respecttothe power allocation variables. In other words, the objectivefunction is either 1) first monotonically non-decreasing andthen monotonically non-increasing or 2) monotonically non-decreasing with respect to the power allocation variables.

In the following, we show that the objective function is firstmonotonically non-decreasing and then monotonically non-increasing with respect to the power allocation variables.Forfacilitating the following analysis, we assume that subcarrieri is allocated to receiverk, i.e., s∗i,k = 1, while ρE∗

i,k andρI∗i,kare given. Then, we take a partial derivative of the objectivefunction with respect toPi,k which yields

∂g(P ,S,ρ)

∂Pi,k

=Ai,k

Bi,k

(65)

whereAi,k = Γi,k

(Ωi,kPi,k + PCT

+KPCR−

K∑

k=1

QI,k

)

−Ωi,k(1 + Pi,kΓi,k) ln(1 + Pi,kΓi,k),

Bi,k = ln(2)(Pi,kΓi,k + 1)

×(Ωi,kPi,k + PCT

+KPCR−

K∑

k=1

QI,k

)2

, and

Ωi,k = ε−

K∑

k=1

lkgk|Hi,k|2ρE∗

i,k . (66)

We note thatΩi,k > 0 holds sincePi,k ≥ εPi,k >

Pi,k

∑Kk=1 lkgk|Hi,k|

2 where the strict inequality is due to thesecond law of thermodynamics from physics. Then, we have

∂g(P ,S,ρ)

∂Pi,k

∣∣∣Pi,k→0

= −WΓi,k∑K

k=1 QIk − PCT−KPCR

> 0 (67)

due to PCT+ KPCR

>∑K

k=1 QIk . In other words, theobjective function is increasing whenPi,k increases fromzero. On the other hand, whenPi,k is sufficiently large, e.g.,Pi,k → ∞, the first partial derivative of the objective functionwith respect toPi,k is given by

∂g(P ,S,ρ)

∂Pi,k

∣∣∣Pi,k→∞

= limPi,k→∞

Ai,k

Bi,k

(a)= lim

Pi,k→∞

−Ωi,kΓi,k log2(1 + Pi,kΓi,k)

(Γi,k(3Ωi,kPi,k+PCT+KPCR

−∑K

k=1 QIk)+2Ωi,k)

×1

(Ωi,kPi,k + PCT+KPCR

−∑K

k=1 QIk)

≤ 0, (68)

where (a) is due to the use of L’Hospital’s rule. Thus,the objective function is monotonic non-increasing for largeincreasingPi,k. By combining (67) and (68), we conclude

that the objective function is a quasi-concave function withrespect to the power allocation variables. Specifically, itis firstmonotonically non-decreasing and then monotonically non-increasing with respect toPi,k.

B. Proof of Concavity of the Transformed Problem with Ob-jective Function Approximation

The concavity of the optimization problem with approxi-mated objective function can be proved by the following fewsteps. First, we consider the concavity of functionU(P ,S,ρ)on a per subcarrier basis w.r.t. the optimization variablesPi,k, ρIi,k, and ρEi,k. For notational simplicity, we define avector xi,k = [Pi,k ρIi,k ρEi,k] and a functionfi,k(xi,k) =

Wαk log2

(Pi,kρ

Ii,klkgk|Hi,k|

2

ρIi,k

(σ2

za+σ2

Ii,k)+σ2

zs

)which takes vectorxi,k as

input. Then, we denote byH(fi,k(xi,k)), andτ1, τ2, andτ3the Hessian matrix of functionfi,k(xi,k) and the eigenvaluesof H(fi,k(xi,k)), respectively. The Hessian matrix of functionfi,k(xi,k) is given by

H(fi,k(xi,k))

=

−1(Pi,k)2 ln(2)

0 0

0−σ2

za

(σ2

za+2 ρIi,k σ2

Ii,k

)

(ρIi,k

)2 ln(2)(σ2

za+ρI

i,kσ2

Ii,k

)2 0

0 0 0

(69)

and the corresponding eigenvalues areτ1 = −1(Pi,k)2 ln(2)

, τ2 =

−σ2

za

(σ2

za+2 ρIi,k σ2

Ii,k

)

(ρIi,k

)2 ln(2)(σ2

za+ρI

i,kσ2

Ii,k

)2 , andτ3 = 0. Since τa ≤ 0, a ∈

1, 2, 3, H(fi,k(xi,k)) is a negative semi-definite matrix.In other words, functionfi,k(xi,k) is jointly concave w.r.t.Pi,k, ρIi,k, and ρEi,k. Then, we can perform the perspectivetransformation onfi,k(xi,k) which is given byui,k(xi,k) =si,kfi,k(xi,k/si,k). We note that the perspective transformationpreserves the concavity of the function [26] andui,k(xi,k) isjointly concave w.r.t.Pi,k, ρIi,k, ρEi,k, andsi,k. Subsequently,

U(P ,S,ρ) =∑nF

i=1

∑K

k=1 Wαk ui,k(xi,k) can be constructedas a non-negative weighted sum ofui,k(xi,k) which guaranteesthe concavity of the resulting function. Besides,UTP (P ,S,ρ)is an affine function of the optimization variables. Thus,U(P ,S,ρ) − qUTP (P ,S,ρ) is jointly concave w.r.t. theoptimization variables. On the other hand, constraints C1–C11(with relaxed constraint C7), C14, and C15 span a convexfeasible set. As a result, the transformed problem with theapproximated objective function is a concave maximizationproblem.

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