Wireless Power Meets Energy Harvesting: A Joint Energy Allocation Approach in OFDM-based System
aa r X i v : . [ c s . I T ] J a n Wireless Power Meets Energy Harvesting:A Joint Energy Allocation Approach inOFDM-based System
Xun Zhou, Chin Keong Ho,
Member, IEEE , and Rui Zhang,
Member, IEEE
Abstract —This paper investigates an orthogonal frequencydivision multiplexing (OFDM)-based wireless powered commu-nication system, where one user harvests energy from an energyaccess point (EAP) to power its information transmission to adata access point (DAP). The channels from the EAP to the user,i.e., the wireless energy transfer (WET) link, and from the userto the DAP, i.e., the wireless information transfer (WIT) link,vary over both time slots and sub-channels (SCs) in general.To avoid interference at DAP, WET and WIT are scheduledover orthogonal SCs at any slot. Our objective is to maximizethe achievable rate at the DAP by jointly optimizing the SCallocation over time and the power allocation over time andSCs for both WET and WIT links. Assuming availability offull channel state information (CSI), the structural results forthe optimal SC/power allocation are obtained and an offlinealgorithm is proposed to solve the problem. Furthermore, wepropose a low-complexity online algorithm when causal CSI isavailable.
Index Terms —Wireless power transfer, energy harvesting,wireless powered communication network (WPCN), orthogonalfrequency division multiplexing (OFDM).
I. I
NTRODUCTION
In conventional wireless networks, nodes are powered byfixed energy sources, e.g., batteries. Finite network lifetimedue to battery depletion becomes a fundamental bottleneckthat limits the performance of energy-constrained networks,e.g., wireless sensor networks. To prolong the operation timeof the networks, the batteries have to be replaced or replen-ished manually after depletion. However, in some applicationswireless nodes are deployed in conditions that replacement ofbatteries is inconvenient (e.g., for numerous sensors in large-scale sensor networks) or even infeasible (e.g., for implanteddevices in human body). Alternatively, harvesting energy fromrenewable energy sources to power wireless devices becomesappealing by providing perpetual energy without battery re-placement.Wireless communications with energy harvesting (EH)transmitter in fading channels have been considered in [1], [2],where the throughput is maximized by energy allocation over
This paper was presented in part at IEEE Global Conference on Signal andInformation Processing (GlobalSIP), December, 2014, Atlanta, USA.X. Zhou is with the Department of Electrical and Computer Engineering,National University of Singapore (e-mail:[email protected]).C. K. Ho is with the Institute for Infocomm Research, A*STAR, Singapore(e-mail:[email protected]).R. Zhang is with the Department of Electrical and Computer Engineering,National University of Singapore (e-mail:[email protected]). He is alsowith the Institute for Infocomm Research, A*STAR, Singapore.
Wireless information transfer (WIT)Wireless energy transfer (WET) Co-located EAP/DAP(HAP)DAPEAP useruser (b) Co-located EAP/DAP(a) Separated EAP/DAP
Fig. 1. Architecture of wireless powered communication network (WPCN):separated EAP/DAP and co-located EAP/DAP. time. In contrast to a conventional transmitter with fixed powersource, where the data transmission is adapted to the commu-nication channels, the EH transmitter adapts its transmissionboth to the communication channels and to the dynamics ofenergy arrivals. It is shown in [1], [2] that when the battery at atransmitter has infinite storage, the optimal transmission powerover slots follows a staircase water-filling (SWF) structure,where the water-levels (WLs) are nondecreasing over slots.This is in contrast with the case of total energy constraintat the transmitter, in which the optimal transmission power isgiven by conventional water-filling (WF), where all slots sharethe same WL. The works [1], [2] are extended to a two-hoprelay network in [3]–[6], where both source and relay nodesemploy EH to power their information transmission.The energy sources in [1]–[6] are not dedicated or control-lable, and are typically provided by the environment, such assolar energy, wind energy, thermal energy, and piezoelectricenergy. Thus, the amount of harvested energy greatly dependson the conditions of the environment. With advances of radiofrequency (RF) technologies, RF signals radiated from anaccess point (AP), referred to as wireless power , becomesa new viable source for EH. The radiation of RF signalsis controllable, hence can potentially supply continuous andstable energy for wireless nodes. Utilizing wireless poweras EH source to supply wireless nodes inspires research on wireless energy transfer (WET), the objective of which is tomaximize the harvested energy at wireless nodes (see, e.g.,[7]–[10]).Efficient WET by RF signals opens up the potential applica-tion of a new type of network, termed wireless powered com-munication network (WPCN) [11], where intended communi- cation users are powered by dedicated wireless power. Fig.1 illustrates the architecture of WPCN, where wireless userstransmit information to a data access points (DAP) using theenergy harvested from an energy access point (EAP). Hence,WET is performed at downlink from EAP to users, whilewireless information transfer (WIT) is performed at uplinkfrom users to DAP. For ease of practical implementation, eachuser is equipped with two antennas for EH and informationtransmission, respectively. In general, the DAP and EAPcan be separately located in the network, referred to as the separated EAP/DAP case (see Fig. 1(a)). A pair of DAP andEAP also can be co-located as a hybrid AP (HAP), providingthe dual function of energy transfer and data access, which isreferred to as the co-located EAP/DAP case (see Fig. 1(b)). Inboth cases, the channel state information (CSI) of WET/WITlinks is estimated at users and DAP, respectively, which arethen sent to a central controller (located at EAP or DAP forexample) for coordination of energy/information transmission.Separated EAP/DAP enjoys more flexibility in deploymentof EAP/DAP; however, additional coordination and synchro-nization between the EAP and DAP is necessary. Co-locatedEAP/DAP is advantageous in information sharing (e.g., thechannel estimation is simplified when uplink/downlink channelreciprocity applies) and hardware reuse (e.g., computationalunits). However, due to the same operation distance of WETand WIT for the co-located EAP/DAP case, the users faraway from the HAP achieve low throughput, since highertransmission power needs to be consumed at these users yetwith lower harvested energy, which is observed as a “doublynear-far” phenomenon in [12].For the case of co-located EAP/DAP, a harvest-then-transmitprotocol is proposed in [12] for a WPCN, where a HAPprovides downlink energy transmission and uplink data accessorthogonally in time to multiple EH users by the time divisionmultiple access (TDMA) scheme. The work [12] is extended toa full-duplex HAP setup in [13], where the downlink energytransmission and uplink information receiving is performedsimultaneously at the HAP. In [14], the authors extend thework in [12] by considering separated EAP/DAP, where theEAP is equipped with multiple antennas. In contrast to [12],which considers TDMA for uplink data transmission, spacedivision multiple access (SDMA) is considered in [15], [16] byemploying a multi-antenna HAP. Unlike [12], [13], [15] whichassume perfect CSI available at transmitters, [16] studies thecase of imperfect CSI by considering channel estimation.Furthermore, [17] investigates the limiting distribution of thestored energy, the average error rate, and outage probabilityat the user when on-off transmission policy is adopted at theuser assuming no CSIs for both WET/WIT links. The capacityof large-size WPCN with geographically distributed users isstudied in [18] and [19] for the separated EAP/DAP case, andin [20] for the co-located EAP/DAP case, based on the tool ofstochastic geometry. Cooperative communication for WPCNis studied in [21], where near users help relay the informationfrom far users to the HAP to overcome the doubly near-far problem.Another line of work on joint wireless energy and in-formation transmission has focused on the so-called SWIPT(simultaneous wireless information and power transfer) indownlink, which aims to characterize the achievable per-formance trade-off between harvesting energy and decodinginformation from the same signal waveform [22], [23]. SWIPTis studied under various setups, e.g., in [22], [24] for multiple-input multiple-output (MIMO) broadcast channels, in [25]–[27] for orthogonal frequency division multiplexing (OFDM)channels, and in [28] for relay channels.In this paper, we study an OFDM-based WPCN. Specif-ically, one user harvests energy from an EAP to powerits information transmission to a DAP, while the EAP andDAP can be either separated or co-located (see Fig. 2 forthe separated case). The total bandwidth of the system isequally divided into multiple orthogonal sub-channels (SCs).We consider energy/information transmission over finite timeslots, where the channels of the WET and WIT links in generalvary over both slots and SCs. Unlike [12] where energy andinformation transmissions are performed over different time,in this paper we consider they are scheduled over orthogonalSCs at any slot. As a result, the user can harvest energyand transmit information at the same time, provided that theenergy causality constraint [1], [2] is satisfied, i.e., the totalenergy consumed for information transmission until any giventime should be no greater than the total harvested energy sofar. Moreover, the SCs are allocated separately for WET andWIT. The frequency orthogonality for WET/WIT inevitablyintroduces a tradeoff between energy and information trans-missions. This additional design freedom via the SC allocationwill influence the power allocation for the WET/WIT links.Our objective is to maximize the achievable rate at the DAP byjointly optimizing the SC allocation over time and the powerallocation over time and SCs for both WET and WIT links.The problem is investigated under two types of CSI availabilityfor the WET and WIT links. The CSI can be either full CSI,which contains CSI of the past, present, and future slots, orcausal CSI, which contains CSI only of the past and presentslots.The main results of this paper are summarized as follows: • Given full CSI, we prove that the optimal solution sat-isfies the following structural properties. First, at anygiven slot WET occurs at most on one SC. Second,given SC allocation WET can only occur in any causallydominating slot, i.e., a slot that has a larger channel powergain on the allocated SC than any of its previous slots.Third, when the initial battery energy of the user is zero,the optimal power allocation at the WIT link performsSWF over slots. That is, at any given slot the powerallocated to different SCs has the same WL, while theWL increases after any causally dominating slot. • Given full CSI, we solve the problem by two stages. First,with given SC allocation, we obtain the optimal jointpower allocation over time and SCs for both WET and
WIT links. Second, we propose two heuristic schemes forthe SC allocation obtained by a dynamic and a static SCallocation. • Motivated by the optimal structural properties for the fullCSI case, we propose an online scheme for the causal CSIcase, namely, the scheme of dynamic SC with observe-then-transmit, which has low complexity. • For the full CSI case, our numerical results show thatthe performance by the proposed dynamic SC schemeis very close to the performance upper bound assumingnon-interfering simultaneous WIT and WET over anySC. Furthermore, the numerical results demonstrate thesuperiority of WPCN with dedicated EAP over conven-tional EH system with random energy arrivals. For thecausal CSI case, it is shown by numerical results that evenutilizing partial information of channels for the WET linkbrings significant benefits to the achievable rate.The rest of the paper is organized as follows. Section IIintroduces the system model. Section III presents the problemformulation. Section IV considers offline algorithm given fullCSI. Section V proposes online algorithm given causal CSI.Section VI provides numerical examples. Finally, Section VIIconcludes the paper.II. S
YSTEM M ODEL
We consider an OFDM-based wireless powered commu-nication system, where one user harvests energy from anEAP to power its information transmission to a DAP (seeFig. 2 for separated EAP/DAP). The EAP and DAP areeach equipped with one antenna, while the user is equippedwith two antennas. The EAP and DAP are connected tostable power supplies, whereas the user has no embeddedenergy sources. Consider energy/information transmission inone block, which is equally divided into K time slots, witheach slot being of duration T . Let T = 1 for convenience. Theslots are indexed in increasing order by k ∈ K , { , . . . , K } .The total bandwidth of the system is equally divided into N orthogonal sub-channels (SCs). The SC set is denoted by N = { , . . . , N } . The channel power gain from the EAP tothe user, i.e., the WET link, during slot k at SC n is denotedby h k,n > , k ∈ K , n ∈ N . The channel power gain from theuser to the DAP, i.e., the WIT link, during slot k at SC n isdenoted by g k,n > , k ∈ K , n ∈ N . It is assumed that h k,n ’sand g k,n ’s are constant within one slot and SC, but vary overslots and SCs. In practice, for the co-located EAP/DAP case, h k,n and g k,n are correlated; while for the separated EAP/DAPcase, h k,n and g k,n are independent. Our model is applicablefor both scenarios.Before the energy/information transmission during each slot k , the CSI of the WET and WIT links, i.e., h k,n , g k,n , n ∈ N ,is estimated. It is assumed that the channel estimation issufficiently accurate such that the performance degradationdue to the estimation error is negligible. Based on the CSI,energy/information transmission for the WET/WIT links isjointly scheduled. To avoid interference at the DAP from the Energy access point(EAP)
Wireless information transfer (WIT)Wireless energy transfer (WET)
Data access point(DAP)user
Energy receiver/Information transmitter ... ...
Fig. 2. A wireless powered communication system where one user harvestsenergy from the EAP to power its information transmission to the DAP. Theuser is equipped with two antennas for energy harvesting and informationtransmission, respectively, over orthogonal sub-channels (SCs). transmission signals by the EAP, WET and WIT are scheduledover orthogonal SCs . For notational simplicity, we define adummy SC n = 0 , where h k, = g k, = 0 for k ∈ K ,since there may be no SCs allocated for WET in slot k . Theextended SC set is denoted by N ′ = { } ∪ N . For each slot k ∈ K , the SC set N ′ is partitioned into two complementarydisjoint subsets for WET and WIT, denoted by N E k and N I k ,respectively, where N E k ⊆ N ′ , N I k ⊆ N ′ , and N I k = N ′ \N E k .The transmission power by the EAP during slot k on SC n is denoted by q k,n ≥ , k ∈ K , n ∈ N E k . The averagetransmission power at the EAP over K slots in each block isdenoted by Q , i.e., K K X k =1 X n ∈N E k q k,n ≤ Q. (1)The user harvests energy from the EAP by an energyreceiver, and the energy is then stored in an energy buffer topower the information transmitter. Assume the stored energyin the energy buffer at time instant k − , i.e., the time instantjust before slot k , is denoted by B k . The initial energy atthe buffer, i.e., B , is known. The transmission power by theinformation transmitter during slot k at SC n is denoted by p k,n ≥ , k ∈ K , n ∈ N I k . Assume the harvested energyduring slot k is ready for transmission at the end of slot k ,the transmission power constraint at the user is thus given by X n ∈N I k p k,n ≤ B k , k ∈ K . (2)We assume the storage of the energy buffer is sufficientlylarge compared to the harvested energy from the EAP, hence,no energy overflow at the energy buffer. We further assume In practice, strict orthogonality of SCs imposes high requirements forhardware design. Energy leakage from one SC to adjacent SC may resultin performance degradation, which is severe especially for the co-locatedEAP/DAP case as transmission power for the WET link is in general muchlarger than received power for the WIT link. For more detailed discussions,please refer to [29]. In practice, the transmission power at each slot and SC may also beconstrained by a peak power. In this paper, we assume the peak powerconstraint is sufficiently large compared to the transmission power, as WPCNin general operates at low power. except data transmission, other circuits at the user consumenegligible energy. This is justified when data transmissionconsumes much larger power than that by other circuits, whichis reasonable for most low-power (e.g., sensor) networks. Thestored energy in the energy buffer at time instant ( k + 1) − isthus given by B k +1 = B k + ζ X n ∈N E k h k,n q k,n − X n ∈N I k p k,n , k ∈ K (3)where ζ denotes the energy efficiency at the user , accountingfor both conversion and discharging losses. Combine (2) and(3), leading to the energy causality constraint i X k =1 X n ∈N I k p k,n ≤ ζ i − X k =1 X n ∈N E k h k,n q k,n + B , i ∈ K . (4)At the DAP, the receiver noise is modeled as a circularlysymmetric complex Gaussian (CSCG) random variable withzero mean and variance σ . Due to frequency orthogonaltransmission of energy and information, the energy signal fromEAP will not interfere with the information reception at DAP.Moreover, the gap for the achievable rate from the channelcapacity due to a practical modulation and coding scheme isdenoted by Γ ≥ . The average achievable rate at the DAP inbps/Hz is thus R = 1 KN K X k =1 X n ∈N I k log (cid:16) g k,n p k,n Γ σ (cid:17) . (5)III. P ROBLEM F ORMULATION
Our objective is to maximize the average rate at the DAP byjointly optimizing the SC allocation over time and the powerallocation over SCs and time for both WET and WIT links. Besides the radio signal from the EAP, the signal radiated by theinformation transmitter at the user can be a potential energy harvestingsource for the energy receiver [30]. However, the amount of the additionalenergy harvested from the information transmission is insignificant duringthe transmission block considered in this paper. This is because the powerfor information transmission is obtained from the harvested energy from theEAP, and the amount of additional energy harvested that can be recycled fromthe information transmission is insignificant, due to the pathloss between thetwo antennas at the user. For example, the pathloss can be 15dB, assuming anoperation frequency centered at 900MHz with the distance of the two antennasas λ/ , where λ denotes the wavelength of the transmission signal. In thiscase, the additional energy harvested from the information transmission isabout 3 % of the harvested energy from the EAP, which is negligible. Hence,in this paper, we assume the energy harvested from the RF signals radiatedby the information transmitter at the user is negligible as compared to energyharvested from the EAP. In practice, this amount of energy accumulated in along term still may be saved for transmission in future blocks. In this case,the accumulated amount of energy can be utilized as the initial battery energyfor the next transmission block. In practice, the energy efficiency may be a nonlinear function of thereceived power by the antenna, depending on specific EH circuits implemen-tation. In this paper, to simplify the system design, we assume the energyefficiency is constant and independent of the received power by the antenna.
This leads to the following optimization problem. max . {N E k } , { q k,n } , { p k,n } KN K X k =1 X n ∈N I k log (cid:16) g k,n p k,n Γ σ (cid:17) s . t . K K X k =1 X n ∈N E k q k,n ≤ Q, (6a) i X k =1 X n ∈N I k p k,n ≤ ζ i − X k =1 X n ∈N E k h k,n q k,n + B , i ∈ K . (6b)The SCs for WIT are not explicit optimization variablesbecause by definition a SC is used either for WIT or WETonly. The case where a SC n is neither used for WET norWIT is covered by assigning it to be used for WET with q k,n = 0 or WIT with p k,n = 0 . Since the energy harvestedduring the last slot K is not available for any informationtransmission, without loss of optimality there should be noenergy transmission at the last slot, i.e., N E K = { } , N I K = N ,as assumed henceforth. The optimal solutions are denoted by {N E k ∗ , k ∈ K} , { q ∗ k,n , k ∈ K , n ∈ N E k ∗ } , { p ∗ k,n , k ∈ K , n ∈N I k ∗ } , and the maximum average rate by R ∗ .Suppose that the SC allocation and power allocation for theWET are given, such that the constraint (6a) is satisfied. ThenProblem (6) is reduced to the conventional EH transmitter withenergy arrivals nP n ∈N E k h k,n q k,n , k ∈ K o [1], [2]. Hence,Problem (6) is more general with additional design freedomsavailable via the SC allocation and power allocation for theWET link, which will in turn influence the power allocationfor the WIT link.We first solve Problem (6) assuming full CSI available inSection IV. Based on the results for full CSI, we propose anonline algorithm for Problem (6) under causal CSI in SectionV. IV. O FFLINE A LGORITHM FOR J OINT SC AND P OWER A LLOCATION
In this section, we consider Problem (6) when full CSIis available, where all the h k,n ’s and g k,n ’s are a priori known by a central controller at the beginning of each blocktransmission. Our aim is to study the structural properties ofthe optimal transmission policy, which will provide importantinsights. Given SC allocation, by Propositions 4.1 and 4.2,we show that WET may occur only on the so-called causallydominating slots. Furthermore, Proposition 4.3 shows that thepower allocated for WET matches to the power consumedfor WIT during the intervals between the causally dominatingslots. The insights will be used for developing heuristic onlineschemes when only casual CSI is available.Given SC allocation N E k , k ∈ K , at slot k , the index ofthe best SC (i.e., the SC that has the largest channel powergain) for the WET link among SCs in N E k , is denoted by m ( k ) ∈ N ′ . Hence, m ( k ) = arg max { n ∈N E k } h k,n . (7) In the following proposition, we state that with given SCallocation, at each slot k WET may only occur on the SC m ( k ) . Proposition 4.1:
For Problem (6) with given SC allocation N E k , k ∈ K , we have q ∗ k,n = 0 for n = m ( k ) . Proof:
Please refer to Appendix A.The intuition of Proposition 4.1 is as follows. Given SCallocation N E k , k ∈ K , consider energy allocation for theWET link at any slot k with total energy P n ∈N E k q k,n .Since the harvested energy at the user increases linearly with q k,n , n ∈ N E k , the harvested energy at the user is maximizedby allocating all energy to q k,m ( k ) , which has the largest h k,n for n ∈ N E k .By Proposition 4.1, at each slot it is optimal to allocateat most one SC from the set N ′ to perform WET, as theremaining SCs can be utilized for potential WIT. We define aSC allocation function Π( k ) ∈ N ′ to denote the SC allocatedfor WET during slot k, k ∈ K . Hence, N E k = { Π( k ) } , N I k = N ′ \{ Π( k ) } , k ∈ K . Note that Π( k ) can be assigned to thedummy SC n = 0 in case there is no WET scheduled in slot k . Problem (6) is then reformulated by the following problem. max . { q k, Π( k ) } , { p k,n } , { Π( k ) } KN K X k =1 X n ∈N I k log (cid:16) g k,n p k,n Γ σ (cid:17) s . t . K K X k =1 q k, Π( k ) ≤ Q, (8a) i X k =1 X n ∈N I k p k,n ≤ ζ i − X k =1 h k, Π( k ) q k, Π( k ) + B , i ∈ K . (8b)Problem (8) is non-convex due to the integer SC alloca-tion function Π( k ) , k ∈ K . Hence, we solve Problem (8)by two stages: we first solve Problem (8) with given SCallocation Π( k ) , k ∈ K , where the joint power allocation forthe WET/WIT links is optimized; next, we propose heuristicschemes for the SC allocation. A. Joint Power Allocation
We first consider Problem (8) with given Π( k ) , k ∈ K ,where we focus on the joint power allocation design for theWET/WIT links. Given SC allocation Π( k ) , k ∈ K , then N E k , N I k , k ∈ K are known. For notational simplicity, when theSC allocation Π( k ) , k ∈ K is given in Problem (8), we dropthe subscript Π( k ) in q k, Π( k ) and h k, Π( k ) , i.e., q k , q k, Π( k ) , h k , h k, Π( k ) for k ∈ K . We note that h k = 0 when Π( k ) = 0 , k ∈ K .First, we investigate the properties for { q ∗ k } and { p ∗ k,n } forProblem (8) with given Π( k ) , k ∈ K . To this end, given SCallocation Π( k ) , k ∈ K , we define set D as follows D , { , if Π(1) ∈ N }∪ { k ∈ { , . . . , K − } : Π( k ) ∈ N , h k > h j , ∀ ≤ j < k } . (9) We note that for the slots in D , the channel power gain h k is increasing with the slot index k ; hence, the slots in D arecalled causally dominating slots. For convenience, we indexthe elements in set D = { d , d , . . . , d |D| } such that d i < d j for i < j . The complementary set of D is denoted by D c , i.e., D c = K\D .We partition the slot set K for the WIT link into subsets D i , { d i − + 1 , . . . , d i } , i = 1 , . . . , |D| + 1 , referred to asthe i th interval , where we set d = 0 and d |D| +1 = K fornotational simplicity. Thus, S i D i = K and D i ∩ D j = ∅ for i = j . In the following proposition, we show that WET onlyoccurs in the causally dominating slots in D . Proposition 4.2:
For Problem (8) with given Π( k ) , k ∈ K ,the optimal power allocation satisfies q ∗ k = 0 for k ∈ D c . Proof:
Please refer to Appendix B.
Remark 4.1:
Proposition 4.2 shows that WET occurs sparsely in time, i.e., WET occurs only when a slot dominatingall its previous slots. Intuitively, this is because instead ofallocating energy to any slot in D c , allocating the same amountof energy to an earlier slot in D which has larger channelpower gain at the WET link will result in a larger feasibleregion for { p k,n } .Further in Proposition 4.3, it is shown that if the energyat the user is used up after a particular casually dominatingslot, then the energy is used up after later causally dominatingslots. Proposition 4.3:
In Problem (8) with given Π( k ) , k ∈ K , if { q ∗ k } and { p ∗ k,n } satisfy d j X k =1 X n ∈N I k p ∗ k,n = ζ d j − X k =1 h k q ∗ k + B (10)where d ≤ d j ≤ d |D| , i.e., constraint (8b) holds with equalityat i = d j , then we have X k ∈D l +1 X n ∈N I k p ∗ k,n = ζh d l q ∗ d l , l = j, . . . , |D| . (11) Proof:
Please refer to Appendix C.Next, we discuss two cases for the initial battery energy B , i.e., the special case of B = 0 and the general case of B ≥ .
1) Zero Initial Battery Energy with B = 0 : We firstconsider the case B = 0 . With B = 0 , from Proposition4.2 and (8b), we have d X k =1 X n ∈N I k p k,n = 0 . (12)Thus, p ∗ k,n = 0 , k ∈ D , n ∈ N I k (13)From Proposition 4.2, q ∗ k = 0 , k ∈ D c . Henceforth, weconsider optimization for { p k,n , k = d + 1 , . . . , K, n ∈ N I k } and { q k , k ∈ D} . From (12), constraint (8b) holds with equality at i = d ,from Proposition 4.3 we have X k ∈D l +1 X n ∈N I k p ∗ k,n = ζh d l q ∗ d l , l = 1 , . . . , |D| . (14)Define the effective channel power gain as g ′ k,n = h d i g k,n , k ∈ D i +1 , i = 1 , . . . , |D| , n ∈ N I k (15)Define { p ′ k,n } as p ′ k,n = p k,n /h d i , k ∈ D i +1 , i = 1 , . . . , |D| , n ∈ N I k . (16)From (13)-(16), Problem (8) with given Π( k ) , k ∈ K and B = 0 is equivalent to the following problem. max . { q di } , { p ′ k,n } KN K X k = d +1 X n ∈N I k log (cid:18) g ′ k,n p ′ k,n Γ σ (cid:19) s . t . K X k = d +1 X n ∈N I k p ′ k,n = ζKQ, (17a) X k ∈D i +1 X n ∈N I k p ′ k,n = ζq d i , i = 1 , . . . , |D| . (17b)We recognize that the optimization over { p ′ k,n , k = d +1 , . . . , K, n ∈ N I k } is then a water-filling (WF) problem overtime slots k ∈ { d + 1 , . . . , K } and SCs n ∈ N I k , because { q d i } can be arbitrarily chosen and thus the last constraintbecomes redundant. The optimal { p ′ k,n , k = d +1 , . . . , K, n ∈N I k } is then obtained by the so-called WF power allocationover slots/SCs, given by p ′ k,n = λKN ln 2 − Γ σ g ′ k,n ! + , k = d + 1 , . . . , K (18)where ( a ) + , max(0 , a ) , and λ satisfies P Ki = d +1 P n ∈N I k p ′ k,n = ζKQ . The water-level (WL)is given by ( λKN ln 2) − . From (13), (16), and (18), theoptimal { p ∗ k,n , k ∈ K , n ∈ N I k } is given by p ∗ k,n = , k ∈ D , (cid:16) h di λKN ln 2 − Γ σ g k,n (cid:17) + , k ∈ D i +1 , i = 1 , . . . , |D| . (19)From (17b) and Proposition 4.2, the optimal { q ∗ k , k ∈ K} isgiven by q j = ζ P k ∈D i +1 P n ∈N I k p ′ k,n , j = d i , i = 1 , . . . , |D| , , otherwise . (20) Remark 4.2:
From (19), the optimal power allocation forthe WIT link is adaptive to channels for both WET and WITlinks. Moreover, the WLs are the same for slots and SCs in thesame interval, while the WL for interval D i +1 is increasingover index i . Thus, the power allocation for the WIT linkperforms staircase water-filling (SWF) over slots.
2) Arbitrary Initial Battery Energy with B ≥ : Now weconsider the case with arbitrary initial battery energy at theuser, i.e., B ≥ .We note that in Problem (8) with given Π( k ) , k ∈ K , { q ∗ k } and { p ∗ k,n } satisfy constraint (8b) with equality at the lastslot K = d |D| +1 ; otherwise, the objective function can beincreased by increasing some p K,n . Let d x , ≤ x ≤ |D| + 1 ,denote the first slot index in D ∪ { K } such that { q ∗ k } and { p ∗ k,n } satisfy constraint (8b) with equality. Hence, d i X k =1 X n ∈N I k p ∗ k,n < ζ i − X k =1 h d k q ∗ d k + B , i = 1 , . . . , x − , (21) d x X k =1 X n ∈N I k p ∗ k,n = ζ x − X k =1 h d k q ∗ d k + B (22)where we define h , and q , . Lemma 4.1:
For Problem (8) with given Π( k ) , k ∈ K , q ∗ d k = 0 for k < x − . The optimal { q ∗ k } and { p ∗ k,n } satisfy d x X k =1 X n ∈N I k p ∗ k,n = ζh d x − q ∗ d x − + B . (23) Proof:
Please refer to Appendix D.Similar to the case of B = 0 , we define the effectivechannel power gain as g ′ k,n = ( h d x − g k,n , k = 1 , . . . , d x , n ∈ N I k ,h d i g k,n , k ∈ D i +1 , i = x, . . . , |D| , n ∈ N I k . (24)Define { p ′ k,n } as p ′ k,n = ( p k,n /h d x − , k = 1 , . . . , d x , n ∈ N I k ,p k,n /h d i , k ∈ D i +1 , i = x, . . . , |D| , n ∈ N I k . (25)In the following lemma, we show that Problem (8) withgiven Π( k ) , k ∈ K is equivalent to a problem with optimizingvariables { p ′ k,n } . Lemma 4.2:
Problem (8) with given Π( k ) , k ∈ K is equiv-alent to the following problem. max . { p ′ k,n } KN K X k =1 X n ∈N I k log (cid:18) g ′ k,n p ′ k,n Γ σ (cid:19) s . t . K X k =1 X n ∈N I k p ′ k,n ≤ ζKQ + B h d x − , (26a) d x − X k =1 X n ∈N I k p ′ k,n ≤ B h d x − , (26b) d x X k =1 X n ∈N I k p ′ k,n ≥ B h d x − . (26c) The optimal { p ∗ k,n } is obtained by (25); the optimal { q ∗ k } isobtained by q j = ζ d x P k =1 P n ∈N I k p ′ k,n − B h dx − ! , j = d x − , ζ P k ∈D i +1 P n ∈N I k p ′ k,n , j = d i , i = x, . . . , |D| , , otherwise . (27) Proof:
Please refer to Appendix E.Problem (26) is solved by the following proposition.
Proposition 4.4:
For Problem (26), the optimal { p ′ k,n , k ∈K , n ∈ N I k } is either given by p ′ k,n = λKN ln 2 − Γ σ g ′ k,n ! + (28)where λ satisfies P Kk =1 P n ∈N I k p ′ k,n = ζKQ + B /h d x − ; orgiven by p ′ k,n = (cid:16) λ − µ ) KN ln 2 − Γ σ g ′ k,n (cid:17) + , k = 1 , . . . , d x , (cid:16) λKN ln 2 − Γ σ g ′ k,n (cid:17) + , k = d x + 1 , . . . , K. (29)where λ and µ satisfy P d x k =1 P n ∈N I k p ′ k,n = B /h d x − and P Kk = d x +1 P n ∈N I k p ′ k,n = ζKQ . Proof:
Please refer to Appendix F.To summarize, Problem (8) given Π( k ) , k ∈ K can besolved as follows: for each d x , ≤ x ≤ |D| +1 , solve Problem(26) to obtain { q k } , { p k,n } , and the objective value, denotedby R ( d x ) . The optimal d x is then obtained by the d x whichachieves the largest rate R ( d x ) and the corresponding powerallocation { q k } and { p k,n } satisfy the constraints (8a) and(8b). We propose Algorithm 1 to solve Problem (8) with given Π( k ) , k ∈ K . B. SC Allocation
Next, we consider the SC allocation design for Problem (8),i.e., the integer function Π( k ) , k ∈ K . The optimization on theinteger function Π( k ) , k ∈ K is non-convex. In general, thecomplexity of exhaustive search over all possible Π( k ) , k ∈ K is O ( N K ) . Hence, we propose heuristic schemes for the SCallocation, which are easy to implement in practice, namelythe dynamic SC scheme and the static SC scheme.Define a SC allocation function ˜Π( k ) , which allocates thebest SC for the WET link among all SCs N ′ at each slot k for WET, i.e., ˜Π( k ) = arg max n ∈N ′ h k,n , k ∈ K . (30)Let ˜ D denote the causally dominating slot set obtained by (9)given SC allocation ˜Π( k ) . From Proposition 4.2, WET shouldoccur only at causally dominating slots, hence, we let Π( k ) =0 for k ∈ ˜ D c such that potential information transmission Algorithm 1:
Algorithm for solving Problem (8) withgiven Π( k ) , k ∈ K . Input : number of slots K ; number of SCs N ; initialbattery energy B ; channel power gain for theWET/WIT links { h k,n } and { g k,n } ; SC allocation Π( k ) , k ∈ K Output : optimal value R ( d ∗ x ) ; optimal power allocation { q ∗ k } and { p ∗ k,n } for each x = 1 , . . . , |D| + 1 do Set effective channel power gain { g ′ k,n } by (24); Obtain p ′ k,n , k ∈ K , n ∈ N I k by WF algorithm withtotal power ζKQ + B /h d x − ; if d x P k =1 P n ∈N I k p ′ k,n < B /h d x − then Obtain p ′ k,n , k = 1 , . . . , d x , n ∈ N I k by WFalgorithm with total power B /h d x − ; Obtain p ′ k,n , k = d x + 1 , . . . , K, n ∈ N I k by WFalgorithm with total power ζKQ ; end Obtain { p k,n } and { q k } by (25) and (27),respectively, and obtain the corresponding rate R ( d x ) ; if { q k } and { p k,n } do not satisfy (8a) or (8b) then Set R ( d x ) as zero; end end Set d ∗ x = arg max d x R ( d x ) ; The achievable rate for Problem (8) with given Π( k ) , k ∈ K is given by R ( d ∗ x ) . The optimal powerallocation { q ∗ k } and { p ∗ k,n } are obtained by step 8correspond to the x ∗ th iteration.can be performed at SCs ˜Π( k ) , k ∈ ˜ D c . In the dynamic SCscheme , the SC allocation is then given by Π( k ) = ( ˜Π( k ) , k ∈ ˜ D , , otherwise . (31) Remark 4.3:
In Problem (8), a performance upper boundfor any SC allocation is obtained by allowing energy andinformation to transmit simultaneously using the same SC,while employing perfect interference cancellation at the DAP.Mathematically, this is equivalent to letting N E k = N I k = N ′ , k ∈ K in Problem (6), which is then solved by thefollowing lemma. Lemma 4.3:
Problem (6) with N E k = N I k = N ′ , k ∈ K achieves same rate as Problem (8) with Π( k ) given in (31)and N I k = N ′ , k ∈ K . Proof:
Please refer to Appendix G.In the static SC scheme , one SC is selected and fixed forWET throughout the whole transmission block, i.e., Π( k ) = n, k ∈ K , where the optimal choice of n is obtained byexhaustive search over the SC set N ′ and selecting the onewhich achieves the largest rate. Therefore, the complexity ofexhaustive search over all possible Π( k ) = n, k ∈ K is O ( N ) . V. O
NLINE A LGORITHM FOR
SC A
LLOCATION
In this section, we consider online algorithms when causalCSI is available. In general, online algorithms can be designedoptimally based on dynamic programming (DP) [1]. However,the DP approach usually involves recursive computation withhigh computing complexity, which may be complicated forpractical implementation. Furthermore, DP requires knowl-edge of channel statistics, e.g., the joint probability densityfunction of the channel power gains for the WET/WIT links,which may be non-stationary or not available. Therefore,we aim to design online algorithm that has low complexityand requires only the past and present channel observations.Motivated by the results for the full CSI case, our onlinealgorithm partitions the transmission block into subsets, andperform WET on the expected best SC in each subset. Inparticular, a simple scheme is proposed for the SC selection,which requires channel observations only of the past andpresent slots for the WET link.For the full CSI case (assuming zero initial battery energy),the transmission block is partitioned as intervals according tothe channels for the WET link, and the information trans-mission during each interval D i +1 , i = 1 , . . . , |D| is poweredby the harvested energy during its prior slot d i (c.f. Propo-sition 4.3). The required amount of energy for informationtransmission is harvested at an earlier slot to ensure thatthere is always sufficient energy for WIT, i.e., no energyoutage at the energy buffer. Motivated by this observation,we partition the transmission block K into subsets, referredto as windows , denoted by W i , i = 1 , . . . , W , where W denotes the number of windows. WET is performed in eachwindow W i , i = 1 , . . . , W − , and the harvested energyduring W i is utilized to power information transmission duringthe next window W i +1 , which ensures no energy outage forWIT during the block (except the first window). No WET isperformed in the last window. In particular, the first windowconsists of the first slot, while the remaining K − slots arepartitioned into W − windows, each window consists of L slots, where L denotes the window size, with ≤ L ≤ K − .For simplicity, we assume K − is divisible by L ; hence, W = ( K − /L + 1 . Notice that the partitioned windowsfor the causal CSI case are fixed, which is independent of thechannels for the WET link.In each window W i , i = 1 , . . . , W − , one SC is selectedto perform WET. We assume the transmission energy at EAPis equally scheduled to the windows W i , i = 1 , . . . , W − ,hence, the EAP transmit with power KQ/ ( W − at theselected SC in each window. For information transmissionat the user, two energy sources are available, i.e., the initialbattery energy B and the energy harvested from EAP. Sinceonly causal CSI is available, we assume B is equally sched-uled for information transmission over all K slots, hence eachslot is scheduled with transmission power B /K . The energyharvested during window W i , i = 1 , . . . , W − , is utilizedfor information transmission during the next window W i +1 ,where each slot is scheduled with equal transmission power. At each slot k , { p k,n , n ∈ N I k } is obtained by the WF powerallocation over SCs n ∈ N I k .Next, we investigate the SC selection in each window W i , i = 1 , . . . , W − . As revealed by the full CSI case,WET is performed on one SC to power its subsequent interval,hence, we aim to select one SC that is expected to have thelargest channel power gain for the WET link among all SCsin each window to perform WET. It is necessary for a SCto be best among all SCs in a window that it is the bestSC at its current slot, hence, the SC is selected from theset { ˜Π( k ) , k ∈ W i } . For the first window W = { } , thebest SC ˜Π(1) is selected to perform WET. Consider otherwindows W i , i = 2 , . . . , W − . Assume the channel powergain at the best SC at the k th slot in the window is denoted by h [ k ] , where k = 1 , . . . , L . The SC selection problem is thenformulated as a stopping problem described as follows. Givena sequentially occurring random sequence h [1] , h [2] , . . . , h [ L ] ,the permutations of which are equally likely, our objective isto select a slot to stop, the index of which is denoted by s ,such that the probability of h [ s ] > h [ j ] , ∀ j = 1 , . . . , L, j = s ,denoted by P r , is maximized. The challenge is that at anyslot k = 1 , . . . , L , the decision of whether to stop at currentslot (i.e., s = k ) or stop at latter slots (i.e., s = k ) needsto be made immediately, based on causal information, i.e., { h [ j ] , ≤ j ≤ k } . The decision of s = k suffers apotential loss when better channels occur in subsequent slotsin the window; whereas the decision of s = k risks theprobability that a better channel never occurs subsequently.The stopping problem can be viewed as a classic SecretaryProblem [31]. A necessary condition for stopping at slot s isthat h [ s ] > h [ j ] , ∀ j = 1 , . . . , L, j < s , i.e., slot s causallydominates all its previous slots in the window; otherwise, theprobability P r becomes zero. Hence, the optimal stopping rulelies in a class of policies, which are described as follows:Define the cutoff slot f ( L ) , which is a parameter that can beoptimized, and ≤ f ( L ) ≤ L . The first f ( L ) − slots arefor observation. During the remaining L − f ( L ) + 1 slots, thefirst slot (if any) that causally dominates all its previous slots,is selected as s ; if no slot is selected until the last slot, then s = L . The probability P r is given by [31] P r = L , f ( L ) = 1 , f ( L ) − L L P l = f ( L ) 1 l − , < f ( L ) ≤ L. (32)The optimal cutoff slot f ∗ ( L ) that maximizes P r is thusobtained as f ∗ ( L ) = arg max ≤ f ( L ) ≤ L P r . The above SC selectionscheme is referred to as dynamic SC with observe-then-transmit (OTT) .An example of the scheme of dynamic SC with OTT isillustrated in Fig. 3, where the total number of slots is K = 16 , In practice, the user may be imposed on a peak power constraint on itstransmission power on each SC n during each slot k , i.e., p k,n ≤ P peak . Inthis case, at each slot k , the power allocation at the user { p k,n , n ∈ N I k } is then obtained by the (revised) WF power allocation with additional peakpower constraint p k,n ≤ P peak . frequency no transmissiontimeSC 1SC 4 ... WET WITwindow 2 window 3 window 4(a) Energy utilization at EAPfrequency timeSC 1SC 4 ... window 2 window 3 window 4(b) Energy utilization at userobserve
Fig. 3. Energy utilization for the scheme of dynamic SC with OTT, where K = 16 , N = 4 and L = 5 . and the window size is L = 5 . The windows are obtainedas W = { } , W = { , . . . , } , W = { , . . . , } , W = { , . . . , } . In W , the best SC (SC 3) at WET link in slotone is selected to transmit energy. From (32), the cutoff slotis obtained as f ∗ ( L ) = 3 . Hence, for W and W , in eachwindow the first two slots are for observing h k,n , n ∈ N ,and the first slot (if any) during the remaining three slotsthat causally dominates all its previous slots in the windowis selected for WET; otherwise, the last slot is scheduled forWET. In W , there is no energy transmission from EAP. InFig. 3(a), WET is performed at SC 3 during slot 1, SC 2 duringslot 5, and SC 1 during slot 11; hence, in Fig. 3(b), the usertransmits information at the remaining SCs.VI. N UMERICAL E XAMPLE
In this section, we provide numerical examples. We focuson the separated EAP/DAP case, where the distances fromEAP to the user and from the user to DAP are assumed to be3meter (m) and 7m, respectively. The total number of slotsis set to be K = 61 . The total bandwidth of the systemis assumed to be MHz, centered at
MHz, which isequally divided into N = 16 SCs, each with bandwidth MHz.The frequency-selective channels are generated by a multi-path power delay profile with exponential distribution A ( τ ) = σ rms e − τ/σ rms , τ > , where σ rms denotes the root mean square(rms) delay spread. Assume σ rms = 0 . µ s, the coherencebandwidth is therefore given by B c = πσ rms ≈ MHz. Thechannels over slots are generated independently. In later simu-lations, all achievable rates are averaged over independentchannel realizations. Assuming the path-loss exponent is three,the signal power attenuation at transmission distance d (inmeter) is then approximately ( − . −
30 log d ) dB [32]. Thereceiver noise power spectrum density at DAP is assumed tobe − dBm/Hz, and Γ = 9 dB. The initial batter energy is
20 30 40 50 60 70 80 90 10000.20.40.60.811.21.41.61.82 Transmission power at EAP (mW) R a t e ( bp s / H z ) upper bounddynamic SC, joint WET/WITdynamic SC, constant WETstatic SC, joint WET/WITstatic SC, constant WETrandom energy arrivals Fig. 4. Performance comparison for offline algorithms when full CSI isavailable. set to be B = 0 . The energy efficiency ζ is assumed to be0.2. A. Offline Algorithms Under Full CSI
First, consider the full CSI case, in which we compare theperformance by different offline schemes. As benchmark, weconsider the system in [1], [2] with random energy arrivalsat the EH user. In particular, the EAP in Fig. 2 is replacedby an ambient RF transmitter which is oblivious of the WETlink, and hence its transmit power over time is random to theuser, since it is adapted to its own information transmissionlink (to another receiver). Throughout the whole transmissionblock, the ambient transmitter transmits over a fixed SC (e.g.,the first SC), and the remaining ( N − SCs are for theinformation transmission at the EH user. In simulation, thetransmit power at the ambient transmitter q k, , k ∈ K arerandomly generated by the uniform distribution over [0 , ,and then are normalized such that /K P Kk =1 q k, = Q .Hence, during each slot k, k ∈ K , a random energy h k, q k, arrives at the user. Given { h k, q k, , k ∈ K} , the achievablerates are obtained by optimizing { p k,n , k ∈ K , n ∈ N \{ }} according to [1], [2]. The performance of this system isobtained by averaging the results from realizations ofrandom transmission power { q k, , k ∈ K} . In addition, theperformance upper bound obtained by the ideal DAP withperfect interference cancellation (refer to Remark 4.3) is alsoconsidered as benchmark. Besides the optimal joint WET/WITtransmission, for comparison we also consider a sub-optimalWET scheme referred to as constant WET, where the EAPtransmits constant power Q each slot at given SC (by dy-namic/static SC schemes).Fig. 4 shows the achievable rates at DAP versus trans-mission power at EAP by different offline schemes. In Fig.4, it is observed that the achievable rates by the proposeddynamic SC scheme with joint WET/WIT transmission arevery close to that by the upper bound. Comparing the joint
20 30 40 50 60 70 80 90 10000.20.40.60.811.21.41.61.82 Transmission power at EAP (mW) R a t e ( bp s / H z ) offlinedynamic SC, OTTdynamic SC, no observestatic SC, OTTstatic SC, no observe Fig. 5. Performance comparison for online algorithms when casual CSI isavailable, where L = 15 . WET/WIT with constant WET transmission schemes, it isobserved that for both dynamic and static SC, the jointWET/WIT schemes achieve much larger rates than that by theconstant WET schemes, which demonstrates the importanceof optimal energy allocation over time for the WET link.Comparing the dynamic SC and the static SC schemes, itis observed that for either joint WET/WIT or constant WETtransmission, the dynamic SC scheme is superior than thestatic SC scheme, and the performance gap is larger whenEAP performs constant WET transmission. This is becausethat the dynamic SC scheme exploits more frequency diversityfor WET; in contrast, the available channels for WET areconstrained on one SC over the whole transmission block.Hence, in general more energy can be harvested to supporthigher data rate by the dynamic SC scheme than by the staticSC scheme. It implies that optimizing SC allocation is impor-tant to the performance, especially when EAP performs sub-optimal constant-power WET. Last, comparing the achievablerates by the wireless powered communication system withdynamic SC, joint WET/WIT and that by the system withrandom energy arrivals, a remarkable performance improve-ment is observed by the wireless powered system, whichdemonstrates the superiority of WPCN with dedicated EAPover conventional EH system with random energy arrivals.
B. Online Algorithms Under Causal CSI
Next, consider the causal CSI case, in which we comparethe performance by different online schemes. Besides thescheme of dynamic SC with OTT proposed in Section V,for comparison we also consider other window-based onlineschemes, where a static SC (e.g., the first SC) is fixed forWET, or the EAP selects the first slot in each window (i.e.,no channel observation) to perform WET. In addition, theperformance by the offline dynamic SC with joint WET/WITscheme is considered as a benchmark. R a t e ( bp s / H z ) offlinedynamic SC, OTTdynamic SC, no observestatic SC, OTTstatic SC, no observe Fig. 6. Performance comparison for online algorithms when casual CSI isavailable, where Q = 60 mW. Fig. 5 shows the achievable rates by different window-based online schemes versus the transmission power at EAP Q . In Fig. 5, the window size L is set to be L = 15 withoptimal cutoff slot f ∗ ( L ) = 6 . It is observed in Fig. 5 thatthe achievable rates by online schemes are smaller than thatby the offline scheme, due to lack of future information ofchannels for the WET/WIT links. In Fig. 5, comparing theperformance by the dynamic and static SC schemes, it isobserved that the dynamic SC schemes achieve larger rates.Comparing the performance by the OTT schemes and thatby the no-observation schemes, it is observed that the OTTschemes are superior, as observation for the WET link helpsto employ more efficient energy transmission by transmittingon SC that is expected to have large channel power gain.Fig. 6 further shows the achievable rates by differentwindow-based online schemes versus the window size L . InFig. 6, the transmission power at EAP is set to be Q = 60 mW.Similar as in Fig. 5, it is observed in Fig. 6 that the dynamicSC scheme is superior than the static SC scheme, and the OTTscheme is better than the no-observation scheme. We noticethat in Fig. 6 the performance by the OTT schemes degrade tothat by the no-observation schemes when L = 1 , , , as noobservation is performed for these special cases. Furthermore,in Fig. 6, it is observed that as the window size increases, theachievable rates by the no-observation schemes are indepen-dent of the window size; whereas the achievable rates by theOTT schemes first increase and then decrease. Intuitively, thismay be because that with larger window size more observationslots help to select SCs with large channel power gain toperform WET. However, smaller window size results in morenumber of selected SCs, which helps to compensate the lossof selecting poor SCs (in the last slot of each window).VII. C ONCLUSION
This paper studied an OFDM-based wireless powered com-munication system, where a user harvests energy from the EAP to power its information transmission to the DAP. The energytransmission by the EAP and the information transmission bythe user is performed over orthogonal SCs. The achievablerate at the DAP is maximized by jointly optimizing the SCallocation over time and power allocation over time and SCsfor both WET and WIT links. Numerical results demonstratethat by dynamic SC allocation and joint power allocation,the performance is improved remarkably as compared to aconventional EH system where the information transmitteris powered by random energy arrivals. Furthermore, bothcases of full CSI and causal CSI are considered. Numericalresults show that for the case of causal CSI utilizing partialinformation of the channels for the WET link can be beneficialto the achievable rate. In general, there is still performancegap between the online and offline algorithms. Therefore, howto further reduce this gap is worthy of future investigation.In this paper we consider the single-user scenario for thepurpose of exposition. As an extension, it is interesting toinvestigate the more general scenario of multiple users co-existing in the system. For the multiuser case, the broadcastsignals by the EAP may provide wireless power to all userssimultaneously. The information transmission from differentusers to the DAP may be coordinated by the orthogonalfrequency division multiple access (OFDMA) scheme. Hence,at each slot, besides the SC allocation between WET and WITlinks, the SCs for the WIT links need to be further allocatedamong multiple users. The complicated twofold SC allocationinevitably results in a non-convex optimization problem, whichin general is challenging to solve optimally.A PPENDIX AP ROOF OF P ROPOSITION K , we proveProposition 4.1 for ≤ k ≤ K − . For optimal q ∗ k,n , k ∈K , n ∈ N E k , assume there exists a slot j, ≤ j ≤ K − andSC l ∈ N E j , l = m ( j ) , such that q ∗ j,l > . We construct adifferent power allocation for the WET link as follows: ˆ q k,n = P u ∈N E k q ∗ j,u , k = j, n = m ( j ) , , k = j, n = m ( j ) ,q ∗ k,n , k = j, n ∈ N E k . (33)From (33), ˆ q k,n , k ∈ K , n ∈ N E k satisfies (6a). Since q ∗ j,l > ,we have X n ∈N E j h j,n (ˆ q j,n − q ∗ j,n ) = X n ∈N E j (cid:0) h j,m ( j ) − h j,n (cid:1) q ∗ j,n > . (34)From (34), by ˆ q k,n , k ∈ K , n ∈ N E k , a larger feasibleregion for p k,n , k ∈ K , n ∈ N I k is obtained than that by q ∗ k,n , k ∈ K , n ∈ N E k , thus a larger achievable rate can beobtained by increasing some p K,n , n ∈ N , which contradictsthe assumption that q ∗ k,n , k ∈ K , n ∈ N E k is optimal. Hence, q ∗ k,n = 0 for n = m ( k ) . Proposition 4.1 is thus proved. A PPENDIX BP ROOF OF P ROPOSITION Π( k ) , k ∈ K , there are two possiblecases for slots in set D c , i.e., Π( k ) = 0 or Π( k ) ∈ N . For k ∈ D c , Π( k ) = 0 , we have q ∗ k = 0 , since no SC is availablefor WET during the slot k . Next, we prove that q ∗ k = 0 for k ∈ D c , Π( k ) ∈ N . For any power allocation { q k } , { p k,n } thatsatisfy the constraints (8a) and (8b), assume there exists a slot i ∈ D c with Π( i ) ∈ N and q i > . By the definition of set D ,there exists a slot ≤ j < i such that h j > h i > , Π( j ) ∈ N .We construct a power allocation strategy { ˆ q k } , { ˆ p k,n } givenby ˆ q k = q j + q i , k = j, , k = i,q k , otherwise . ˆ p k,n = ( p k,n + ζ ( h j − h i ) q i N , k = i, n ∈ N I k ,p k,n , otherwise . It can be verified that { ˆ q k } and { ˆ p k,n } satisfy the constraints(8a) and (8b). Since h j > h i and q i > , the achievable rateby { ˆ q k } , { ˆ p k,n } is larger than that by { q k } , { p k,n } , i.e., { q k } , { p k,n } is not optimal. Hence, the optimal solution satisfiesthat q ∗ k = 0 for k ∈ D c . The proof of Proposition 4.2 is thuscompleted. A PPENDIX CP ROOF OF P ROPOSITION { q ∗ k } and { p ∗ k,n } satisfy (8b) with equalityat i = d j , where ≤ j ≤ |D| , then they satisfy (8b) withequality at i = d j +1 , i.e., d j +1 X k =1 X n ∈N I k p ∗ k,n = ζ d j +1 − X k =1 h k q ∗ k + B . (35)Note that (35) is satisfied for j = |D| ; otherwise, the objectivefunction in Problem (8) can be increased by increasing some p K,n .Next, we prove (35) for the case ≤ j ≤ |D|− by contra-diction. The optimal solutions { q ∗ k } and { p ∗ k,n } satisfy the con-straints (8a) and (8b). Assume { q ∗ k } and { p ∗ k,n } do not satisfy(35), i.e., ∆ , ζ P d j +1 − k =1 h k q ∗ k + B − P d j +1 k =1 P n ∈N I k p ∗ k,n > . From (10) and Proposition 4.2, we have ∆ = ζh d j q ∗ d j − d j +1 X k = d j +1 X n ∈N I k p ∗ k,n . (36)Now, we construct a power allocation strategy { ˆ q k , k ∈ K} , { ˆ p k,n , k ∈ K , n ∈ N I k } given by ˆ q k = q ∗ d j − ∆ ζh dj , k = d j ,q ∗ d j +1 + ∆ ζh dj , k = d j +1 ,q ∗ k , otherwise . (37) ˆ p k,n = p ∗ k,n , k = 1 , . . . , d j +1 ,p ∗ k,n + ( h dj +1 − h dj ) ∆ h dj ( K − d j +1 ) N , k = d j +1 + 1 , . . . , K. (38)It can be verified that { ˆ q k } and { ˆ p k,n } satisfy the constraints(8a) and (8b). Since ∆ > and h d j +1 > h d j , the powerallocation { ˆ q k } and { ˆ p k,n } achieve larger rate than { q ∗ k } and { p ∗ k,n } , which contradicts the assumption that { q ∗ k } and { p ∗ k,n } are optimal for (8). Therefore, { q ∗ k } and { p ∗ k,n } satisfy (35).By induction, d l +1 X k =1 X n ∈N I k p ∗ k,n = ζ d l +1 − X k =1 h k q ∗ k + B , l = j, . . . , |D| . (39)It follows from (39) that, for l = j, . . . , |D| , X k ∈D l +1 X n ∈N I k p ∗ k,n = ζ d l +1 − X k = d l h k q ∗ k = h d l q ∗ d l (40)which completes the proof of Proposition 4.3.A PPENDIX DP ROOF OF L EMMA ≤ x ≤ , from (22), (23) is satis-fied. For the case < x ≤ |D| + 1 , we first prove q ∗ d k = 0 for ≤ k ≤ x − by contradiction. As-sume there exists q ∗ d j > for ≤ j ≤ x − . Define ∆ , min i = j +1 ,...,x − ζ i − P k =1 h d k q d k + B − d i P k =1 P n ∈N I k p k,n ! .From (21), we have ∆ > . We construct a power allocationstrategy { ˆ q k , k ∈ K} and { ˆ p k,n , k ∈ K , n ∈ N I k } given by ˆ q k = q ∗ d j − min (cid:16) q ∗ d j , ∆ ζh dj (cid:17) , k = d j ,q ∗ d x − + min (cid:16) q ∗ d j , ∆ ζh dj (cid:17) , k = d x − ,q ∗ k , otherwise . ˆ p k,n = p ∗ k,n + ζ ( h dx − − h dj ) N min (cid:16) q ∗ d j , ∆ ζh dj (cid:17) , k = d x ,p ∗ k,n , otherwise . It can be verified that { ˆ q k } and { ˆ p k,n } satisfy the constraints(8a) and (8b). Since h d x − > h d j , q ∗ d j > , and ∆ > , thepower allocation { ˆ q k } and { ˆ p k,n } achieve larger rate than { q ∗ k } and { p ∗ k,n } , which contradicts the assumption that { q ∗ k } and { p ∗ k,n } are optimal for (8). Therefore, q ∗ d k = 0 for ≤ k ≤ x − . Then (23) follows from (22). The proof of Lemma 4.1then completes. A PPENDIX EP ROOF OF L EMMA { Π( k ) } , we prove the equivalence between Problems(8) and (26). It is sufficient for us to prove that given optimalsolution { q k,n } , { p k,n } for Problem (8), { p ′ k,n } obtained by(25) is optimal for Problem (26); given optimal solution { p ′ k,n } for Problem (26), { q k } , { p k,n } obtained by (27) and (25) is optimal for Problem (8). For convenience, the optimal value ofProblems (8) and (26) are denoted by R ∗ and R ′ , respectively.Given optimal solution { q k,n } , { p k,n } for Problem (8), then { q k,n } , { p k,n } satisfy constraints (8a) and (8b). We obtain { p ′ k,n } by (25). Since g k,n p k,n = g ′ k,n p ′ k,n , the average rateachieved by { p ′ k,n } equals to R ∗ . Next, we prove that { p ′ k,n } is a feasible solution for Problem (26). From Lemma 4.1, (25),and (21), { p ′ k,n } satisfy constraints (26b) and (26c). FromProposition (22), and 4.3, { q k,n } , { p k,n } satisfy X k ∈D l +1 X n ∈N I k p ∗ k,n = ζh d l q ∗ d l , l = x, . . . , |D| . (41)From Proposition 4.2, (23), and (25), it follows that K X k =1 X n ∈N I k p ′ k,n ≤ ζ |D| X i = x − q d i + B h d x − ≤ ζKQ + B h d x − . (42)It follows that { p ′ k,n } satisfy constraint (26a); thus, { p ′ k,n } isa feasible solution for Problem (8). Therefore, the average rateachieved by { p ′ k,n } is no larger than R ′ ; i.e., R ∗ ≤ R ′ , wherethe equality holds if and only if { p ′ k,n } is optimal for Problem(26).Given optimal solution { p ′ k,n } for Problem (26), then { p ′ k,n } satisfy constraints (26a), (26b), and (26c). We obtain q k and p k,n by (27) and (25). Since g k,n p k,n = g ′ k,n p ′ k,n , the averagerate achieved by { q k } , { p k,n } equals to R ′ . Next, we provethat { q k } , { p k,n } is a feasible solution for Problem (8). From(26a) and (27), { q k } satisfy constraint (8a). From (25), (26b),and (27), { q k } and { p k,n } satisfy constraints (8a) and (8a).Therefore, { q k } , { p k,n } is a feasible solution for Problem (8).It follows that the average rate achieved by { q k } , { p k,n } isno larger than R ∗ ; thus, R ′ ≤ R ∗ , where the equality holds ifand only if { q k } , { p k,n } is optimal for Problem (8).From R ∗ ≤ R ′ and R ′ ≤ R ∗ , we have R ∗ = R ′ . Therefore,given optimal solution { q k,n } , { p k,n } for Problem (8), { p ′ k,n } obtained by (25) is optimal for Problem (26); given optimalsolution { p ′ k,n } for Problem (26), { q k } , { p k,n } obtained by(27) and (25) is optimal for Problem (8). The proof of Lemma4.2 completes. A PPENDIX FP ROOF OF P ROPOSITION The Lagrangian of Problem (26) is given by L (cid:0) { p ′ k,n } , λ, δ, µ (cid:1) = 1 KN K X k =1 X n ∈N I k log (cid:18) g ′ k,n p ′ k,n Γ σ (cid:19) + λ ζKQ + B h d x − − K X k =1 X n ∈N I k p ′ k,n + δ B h d x − − d x − X k =1 X n ∈N I k p ′ k,n + µ d x X k =1 X n ∈N I k p ′ k,n − B h d x − (43)where λ, δ , and µ are the non-negative dual variables associ-ated with the corresponding constraints in Problem (26). Thenecessary and sufficient conditions for { p ′ k,n } and λ, δ, µ tobe both primal and dual optimal are given by the Karush-Kuhn-Tucker (KKT) optimality conditions: { p ′ k,n } satisfy allthe constraints in Problem (26), and λ ζKQ + B h d x − − K X k =1 X n ∈N I k p ′ k,n = 0 , (44a) δ B h d x − − d x − X k =1 X n ∈N I k p ′ k,n = 0 , (44b) µ d x X k =1 X n ∈N I k p ′ k,n − B h d x − = 0 , (44c) ∂ L (cid:16) { p ′ k,n } , λ, δ, µ (cid:17) ∂p ′ k,n = 0 . (44d)From (21) and Lemma 4.1, P d x − k =1 P n ∈N I k p ∗ k,n < B ;therefore, the optimal { p ′ k,n } satisfies P d x − k =1 P n ∈N I k p ′ k,n < B /h d x − . It follows that the optimal δ = 0 by (44b).From (44d) and δ = 0 , the optimal p ′ k,n , k ∈ K , n ∈ N I k isgiven by p ′ k,n = (cid:16) λ − µ ) KN ln 2 − Γ σ g ′ k,n (cid:17) + , k = 1 , . . . , d x , (cid:16) λKN ln 2 − Γ σ g ′ k,n (cid:17) + , k = d x + 1 , . . . , K. (45)If the optimal µ > , from (44a) and (44c), we have P d x k =1 P n ∈N I k p ′ k,n = B /h d x − and P Kk = d x +1 P n ∈N I k p ′ k,n = ζKQ . If the optimal µ = 0 , then p ′ k,n = λKN ln 2 − Γ σ g ′ k,n ! + , k ∈ K , n ∈ N I k (46)where λ satisfies P Kk =1 P n ∈N I k p ′ k,n = ζKQ + B /h d x − by(44a). Proposition 4.4 is thus proved. A PPENDIX GP ROOF OF L EMMA N E k = N I k = N ′ , k ∈ K . Since N E k = N ′ , from (7) and (30), m ( k ) = ˜Π( k ) , k ∈ K . ByProposition 4.1, we have q ∗ k,n = 0 , n = ˜Π( k ) , k ∈ K forProblem (6) with N E k = N I k = N ′ , k ∈ K . It follows thatProblem (6) with N E k = N I k = N ′ , k ∈ K achieves samerate as Problem (8) with Π( k ) = ˜Π( k ) , N I k = N ′ , k ∈ K .From Proposition 4.2, q ∗ k = 0 , k ∈ ˜ D c for Problem (8) with Π( k ) = ˜Π( k ) , N I k = N ′ , k ∈ K . It follows that Problem (8)with Π( k ) = ˜Π( k ) , N I k = N ′ , k ∈ K achieves same rate asProblem (8) with Π( k ) given in (31) and N I k = N ′ , k ∈ K .Therefore, Problem (6) with N E k = N I k = N ′ , k ∈ K achievessame rate as Problem (8) with Π( k ) given in (31) and N I k = N ′ , k ∈ K . This thus completes the proof of Lemma 4.3.R EFERENCES[1] C. K. Ho and R. Zhang, “Optimal energy allocation for wirelesscommunications with energy harvesting constraints,”
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