TRADITIONALLY, energy-constrained wireless networks, Throughput Maximization in Wireless Powered Communication Networks

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1 48 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 3, NO., JANUARY 4 Throughpu Maxmzaon n Wreless Powered Communcaon Neworks Hyungsk Ju, Member, IEEE, and Ru Zhang, Member, IEEE Absrac Ths paper sudes he newly emergng wreless powered communcaon nework n whch one hybrd access pon H-AP wh consan power supply coordnaes he wreless energy/nformaon ransmssons o/from a se of dsrbued users ha do no have oher energy sources. A harves-henransm proocol s proposed where all users frs harves he wreless energy broadcas by he H-AP n he downlnk DL and hen send her ndependen nformaon o he H- AP n he uplnk UL by me-dvson-mulple-access TDMA. Frs, we sudy he sum-hroughpu maxmzaon of all users by jonly opmzng he me allocaon for he DL wreless power ransfer versus he users UL nformaon ransmssons gven a oal me consran based on he users DL and UL channels as well as her average harvesed energy values. By applyng convex opmzaon echnques, we oban he closedform expressons for he opmal me allocaons o maxmze he sum-hroughpu. Our soluon reveals an neresng doubly near-far phenomenon due o boh he DL and UL dsancedependen sgnal aenuaon, where a far user from he H- AP, whch receves less wreless energy han a nearer user n he DL, has o ransm wh more power n he UL for relable nformaon ransmsson. As a resul, he maxmum sumhroughpu s shown o be acheved by allocang subsanally more me o he near users han he far users, hus resulng n unfar rae allocaon among dfferen users. To overcome hs problem, we furhermore propose a new performance merc socalled common-hroughpu wh he addonal consran ha all users should be allocaed wh an equal rae regardless of her dsances o he H-AP. We presen an effcen algorhm o solve he common-hroughpu maxmzaon problem. Smulaon resuls demonsrae he effecveness of he common-hroughpu approach for solvng he new doubly near-far problem n wreless powered communcaon neworks. Index Terms Wreless power, energy harvesng, hroughpu maxmzaon, doubly near-far problem, TDMA, convex opmzaon. I. INTRODUCTION TRADITIONALLY, energy-consraned wreless neworks, such as sensor neworks, are powered by fxed energy sources, e.g. baeres, whch have lmed operaon me. Alhough he lfeme of he nework can be exended Manuscrp receved Aprl 3, 3; revsed Augus 7, 3; acceped Ocober 8, 3. The assocae edor coordnang he revew of hs paper and approvng for publcaon was H. Hassanen. Ths paper wll be presened n par a he IEEE Global Communcaons Conference Globecom, December 9-3, 3, Alana, USA. H. Ju s wh he Deparmen of Elecrcal and Compuer Engneerng, Naonal Unversy of Sngapore e-mal: elejhs@nus.edu.sg. R. Zhang s wh he Deparmen of Elecrcal and Compuer Engneerng, Naonal Unversy of Sngapore e-mal: elezhang@nus.edu.sg. He s also wh he Insue for Infocomm Research, A*STAR, Sngapore. Ths work was suppored n par by he Naonal Unversy of Sngapore under research gran R Dgal Objec Idenfer.9/TWC /4$3. c 4 IEEE by replacng or rechargng he baeres, may be nconvenen, cosly, dangerous e.g., n a oxc envronmen or even mpossble e.g., for sensors mplaned n human bodes. As an alernave soluon o prolong he nework s lfeme, energy harvesng has recenly drawn sgnfcan neress snce poenally provdes unlmed power supples o wreless neworks by scavengng energy from he envronmen. In parcular, rado sgnals radaed by amben ransmers become a vable new source for wreless energy harvesng. I has been repored ha 3.5mW and uw of wreless power can be harvesed from rado-frequency RF sgnals a dsances of.6 and meers, respecvely, usng Powercas RF energyharveser operang a 95MHz []. Furhermore, recen advance n desgnng hghly effcen recfyng anennas wll enable more effcen wreless energy harvesng from RF sgnals n he near fuure []. I s worh nong ha here has been recenly a growng neres n sudyng wreless powered communcaon neworks WPCNs, where energy harvesed from amben RF sgnals s used o power wreless ermnals n he nework, e.g., [3]-[5]. In [3], a wreless powered sensor nework was nvesgaed, where a moble chargng vehcle movng n he nework s employed as he energy ransmer o wrelessly power he sensor nodes. In [4], he wreless powered cellular nework was suded n whch dedcaed power-beacons are deployed n he cellular nework o charge moble ermnals. Moreover, he wreless powered cognve rado nework has been consdered n [5], where acve prmary users are ulzed as energy ransmers for chargng her nearby secondary users ha are no allowed o ransm over he same channel due o srong nerference. Furhermore, snce rado sgnals carry energy as well as nformaon a he same me, a jon nvesgaon of smulaneous wreless nformaon and power ransfer SWIPT has recenly drawn a sgnfcan aenon see e.g. [6]-[] and he references heren. In hs paper, we sudy a new ype of WPCN as shown n Fg., n whch one hybrd access pon H-AP wh consan power supply e.g. baery coordnaes he wreless energy/nformaon ransmssons o/from a se of dsrbued users ha are assumed o have no oher energy sources. All users are each equpped wh a rechargeable baery and hus can harves and sore he wreless energy broadcas by he H- AP. Unlke pror works on SWIPT [6]-[], whch focused on he smulaneous energy and nformaon ransmssons o users n he downlnk DL, n hs paper we consder a dfferen seup where he H-AP broadcass only wreless energy o all users n he DL whle he users ransm her ndependen nformaon usng her ndvdually harvesed

2 JU and ZHANG: THROUGHPUT MAXIMIZATION IN WIRELESS POWERED COMMUNICATION NETWORKS 49 DL for WET UL for WIT U h g H-AP U U U K h g U T T T K T Hybrd AP g K h K U K Energy ransfer Informaon ransfer Fg.. A wreless powered communcaon nework WPCN wh wreless energy ransfer WET n he downlnk DL and wreless nformaon ransmssons WITs n he uplnk UL. energy o he H-AP n he uplnk UL. We are neresed n maxmzng he UL hroughpu of he aforemenoned WPCN by opmally allocang he me for he DL wreless energy ransfer WET by he H-AP and he UL wreless nformaon ransmssons WITs by dfferen users. The man conrbuons of hs paper are summarzed as follows: We propose a proocol ermed harves-hen-ransm for he WPCN depced n Fg., where he H-AP frs broadcass wreless energy o all users n he DL, and hen he users ransm her ndependen nformaon o he H-AP n he UL usng her ndvdually harvesed energy by me-dvson-mulple-access TDMA. Wh he proposed proocol, we frs maxmze he sumhroughpu of he WPCN by jonly opmzng he me allocaed o he DL WET and he UL WITs gven a oal me consran, based on he users DL and UL channels as well as her average harvesed energy amoun. I s shown ha he sum-hroughpu maxmzaon problem s convex, and herefore we derve closed-form expressons for he opmal me allocaons by applyng convex opmzaon echnques []. Our soluon reveals an neresng new doubly near-far phenomenon n he WPCN, when a far user from he H- AP receves less amoun of wreless energy han a nearer user n he DL, bu has o ransm wh more power n he UL for achevng he same nformaon rae due o he doubly dsance-dependen sgnal aenuaon n boh he DL WET and UL WIT. Consequenly, he sumhroughpu maxmzaon soluon s shown o allocae subsanally more me o he near users han he far users, hus resulng n unfar achevable raes among dfferen users. To overcome he doubly near-far problem, we furhermore propose a new performance merc referred o as common-hroughpu wh he addonal consran ha all users should be allocaed wh an equal rae n her UL WITs regardless of her dsances o he H-AP. We propose an effcen algorhm o maxmze he commonhroughpu of he WPCN by re-opmzng he me Fg.. The harves-hen-ransm proocol. allocaed for he DL WET and UL WITs. By comparng he maxmum sum- versus common-hroughpu, we characerze he fundamenal hroughpu-farness rade-offs n awpcn. The res of hs paper s organzed as follows. Secon II presens he WPCN model and he proposed harveshen-ransm proocol. Secon III sudes he sum-hroughpu maxmzaon problem, and characerzes he doubly near-far phenomenon. Secon IV formulaes he common-hroughpu maxmzaon problem and presens an effcen algorhm o solve. Secon V presens smulaon resuls on he sumhroughpu versus common-hroughpu comparson. Fnally, Secon VI concludes he paper. II. SYSTEM MODEL As shown n Fg., hs paper consders a WPCN wh WET n he DL and WITs n he UL. The nework consss of one H- AP and K users e.g., sensors denoed by U, =,, K. I s assumed ha he H-AP and all user ermnals are equpped wh one sngle anenna each. I s furher assumed ha he H-AP and all he users operae over he same frequency band. In addon, all user ermnals are assumed o have no oher embedded energy sources; hus, he users need o harves energy from he receved sgnals broadcas by he H-AP n he DL, whch s sored n a rechargeable baery and hen used o power operang crcus and ransm nformaon n he UL. The DL channel from he H-AP o user U and he correspondng reversed UL channel are denoed by complex random varables h and g, respecvely, wh channel power gans h = h and g = g. I s assumed ha boh he DL and UL channels are quas-sac fla-fadng, where h s and g s reman consan durng each block ransmsson me, denoed by T, bu can vary from one block o anoher. I s furher assumed ha he H-AP knows boh h and g, =,, K, perfecly a he begnnng of each block. The nework adops a harves-hen-ransm proocol as shown n Fg.. In each block, he frs T amoun of me, < <, s assgned o he DL for he H-AP o broadcas wreless energy o all users, whle he remanng me n he same block s assgned o he UL for nformaon ransmssons, durng whch users ransm her ndependen nformaon o he H-AP by TDMA. The amoun of me assgned o user U n he UL s denoed by T, <, =, K. Snce,,, K represen he me porons n each block allocaed o he H-AP and users U,, U K for UL WET and DL WITs, respecvely, we have. =

3 4 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 3, NO., JANUARY 4 For convenence, we assume a normalzed un block me T =n he sequel whou loss of generaly; hence, we can use boh he erms of energy and power nerchangeably. Durng he DL phase, he ransmed baseband sgnal of he H-AP n one block of neres s denoed by x A. We assume ha x A s an arbrary complex random sgnal sasfyng E[ x A ]=P A,whereP A denoes he ransm power a he H-AP. The receved sgnal a U s hen expressed as y = h x A + z, =,, K, where y and z denoe he receved sgnal and nose a U, respecvely. I s assumed ha P A s suffcenly large such ha he energy harvesed due o he recever nose s neglgble. Thus, he amoun of energy harvesed by each user n he DL can be expressed as assumng un block me,.e., T = E = ζ P A h, =,, K, 3 where <ζ <, =,K, s he energy harvesng effcency a each recever. For convenence, s assumed ha ζ = = ζ K = ζ n he sequel of hs paper. Afer he users replensh her energy durng he DL phase, n he subsequen UL phase hey ransm ndependen nformaon o he H-AP n her allocaed me slos. I s assumed ha a each user ermnal, a fxed poron of he harvesed energy gven by 3 s used for s nformaon ransmsson n he UL, denoed by η for U, <η, =,, K. Whn amoun of me assgned o U, we denoe x as he complex baseband sgnal ransmed by U, =,, K. We assume Gaussan npus,.e., x CN,P,whereCNμ, σ sands for a crcularly symmerc complex Gaussan CSCG random varable wh mean μ and varance σ,andp denoes he average ransm power a U, whch s gven by P = η E, =,,K. 4 For he purpose of exposon, we assume η =,, nhe sequel,.e., all he energy harvesed a each user s used for s UL nformaon ransmsson. The receved sgnal a he H-AP n he h UL slo s hen expressed as y A, = g x + z A,, =,,K, 5 where y A, and z A, denoe he receved sgnal and nose a he H-AP, respecvely, durng slo. I s assumed ha z A, CN,σ,. From 3-5, he achevable UL hroughpu of U n bs/second/hz bps/hz can be expressed as R = log + g P Γσ = log +γ, =,, K, 6 where = [ K ], and Γ represens he sgnal-onose rao SNR gap from he addve whe Gaussan nose Noe ha x A can also be used o send DL nformaon a he same me; however, hs usage wll no be consdered n hs paper. Ineresed readers may refer o recen works on SWIPT [6]-[]. Throughpu bps/hz Opmal me allocaon o DL WET * Opmal me allocaon o UL WIT Fg. 3. Throughpu versus me allocaed o DL WET n a sngle-user WPCN wh γ =db. AWGN channel capacy due o a praccal modulaon and codng scheme MCS used. In addon, γ s gven by γ = ζh g P A Γσ, =,, K. 7 From 6, s observed ha R ncreases wh for a gven. In addon, can also be shown ha R ncreases wh for a gven.however, and s canno be ncreased a he same me gven her oal me consran n. Fg. 3 shows he hroughpu gven n 6 for he specal case of one sngle user n he nework,.e., K =,versus he me allocaed o he DL WET,, wh γ = db, assumng ha holds wh equaly,.e., for he UL WIT =. I s observed ha he hroughpu s zero when =,.e., no me s assgned for WET o he user n he DL and hus no energy s avalable for WIT n he UL, as well as when =or = =,.e., no me s assgned o he user for WIT n he UL. I s also observed ha he hroughpu frs ncreases wh when < =.4, bu decreases wh ncreasng when >,where s he opmal me allocaon o maxmze he hroughpu. Ths can be explaned as follows. Wh small, he amoun of energy harvesed by U n he DL s small. In hs regme, as U harvess more energy wh ncreasng,.e., more energy s avalable for he nformaon ransmsson n he UL, he hroughpu ncreases wh.however,as becomes larger han, he hroughpu s decreased more sgnfcanly due o he reducon n he allocaed UL ransmsson me, ;as a resul, he hroughpu sars o decrease wh ncreasng. Therefore, here exss a unque opmal o maxmze he hroughpu. III. SUM-THROUGHPUT MAXIMIZATION In hs secon, we characerze he maxmum sumhroughpu of he WPCN presened n Secon II wh arbrary number of users, K. From 6, he sum-hroughpu of all users s gven by R sum = K R, whch s a funcon of he

4 JU and ZHANG: THROUGHPUT MAXIMIZATION IN WIRELESS POWERED COMMUNICATION NETWORKS fz A > < A < U.6 o d e a c l lo a Increasng Sum-Throughpu = * *, z allocaed o U Fg. 4. Plo of f z gven n 9 versus z. Fg. 5. Sum-hroughpu n bps/hz versus me allocaon. DL and UL me allocaon. Therefore, from he sumhroughpu maxmzaon problem s formulaed as P : max s.. R sum, =, =,, K. 8 Lemma 3.: R s a concave funcon of for any gven {,, K}. Proof: Please refer o Appendx A. From Lemma 3., follows ha R sum s also a concave funcon of snce s he summaon of R s. Therefore, P s a convex opmzaon problem, and hus can be solved by convex opmzaon echnques. To solve P, we frs have he followng lemma. Lemma 3.: Gven A>, here exss a unque z > ha s he soluon of f z =A, where f z Δ = z ln z z +, z. 9 Proof: Please refer o Appendx B. Fg. 4 shows f z gven n 9 wh z. I s observed ha f z s a convex funcon over z where he mnmum s aaned a z =wh f =. Therefore, gven <A, here are wo dfferen soluons for f z = A, among whch one s smaller han and he oher s larger han,.e., z >. On he oher hand, f A>, here s only one soluon for f z =A, whch s larger han,.e., z >. The above observaons are hus n accordance wh Lemma 3.. Proposon 3.: The opmal me allocaon soluon for P, denoed by =[ K ],sgvenby = { z A+z γ A+z, =, =,,K where A = Δ K γ > and z > s he correspondng soluon of f z =A as gven by Lemma 3.. Proof: Please refer o Appendx C. I s worh nong ha A> always holds snce from 7 we have γ >, =,,,K, provded ha h and g. Hence, gven a se of srcly posve γ s, accordng o Lemma 3. z > s unquely deermned wh he presumed A, hus resulng n a unque soluon for P, wh >, =,,,K,.e., he me allocaed o he DL WET s always greaer han zero, and so s he me allocaed o each user n he UL WIT, provded ha γ >,. Furhermore, from Proposon 3. we have he followng corollary. Corollary 3.: In he opmal me allocaon soluon of P, s a monooncally decreasng funcon of A>. Proof: Please refer o Appendx D. From Corollary 3., s nferred ha he me allocaed o he DL WET decreases wh ncreasng γ s, or channel power gans h s and/or g s, snce A = K γ and γ h g, =,,K, as shown n 7. As a resul, s, =,,K, ncrease wh A,.e., he me allocaed o he UL WIT ncreases wh γ s. Ths s an neresng observaon mplyng ha when he channel power gans, h s and g s, become larger, we should allocae more me o he UL WITs nsead of he DL WET o maxmze he sum-hroughpu. Ths s because wh larger γ s, he requred energy for UL WITs becomes smaller gven any ransmsson rae; hus, each user can harves suffcen amoun of wreless energy from he H- AP even wh a smaller me allocaed o he DL WET. Noe ha he sum-hroughpu maxmzaon soluon gven n allocaes more me o a near user o he H-AP han a far user, snce n pracce h D α d, g D αu,and γ h g accordng o 7, where α d and α u denoe he channel pahloss exponens n he DL and UL, respecvely, and D denoes he dsance beween he H-AP and U. Thus, from follows ha D α d+α u, =,,K, whch resuls n an unfar me and hroughpu allocaon among users n he WPCN, a phenomenon ermed doubly near-far problem. To furher llusrae hs ssue, Fg. 5 shows he sum-hroughpu R sum versus UL WIT me allocaon and for a wo-user nework wh K =and D = D. I s assumed ha he channel recprocy holds for he DL and UL and hus h = g, =,, wh α d = α u =.

5 4 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 3, NO., JANUARY 4 Convenonal TDMA nework WPCN he hroughpu rao of he wo users n he WPNC case decreases wce faser han ha n he convenonal TDMA n he logarhm scale due o he more severe doubly nearfar problem Pahloss Exponen, α d = α u = α Fg. 6. Throughpu rao n a wo-user WPCN versus convenonal TDMA nework wh equal energy supply. Accordngly, we se γ = db and γ = db, wh γ /γ = D /D α d+α u. I s observed from Fg. 5 ha R sum =when = =, = + =,or + =, =, snce no me s allocaed o he users for he UL WITs n he former case, whle no me s allocaed o he DL WET n he laer case. The numercal resul of sum-hroughpu clearly shows ha R sum s srcly posve when < + <. In addon, s observed ha he opmal DL and UL me allocaon o maxmze he sum-hroughpu s =[.44,.74,.445] where =6,.e., =D /D α d+α u, whch s conssen wh. Furhermore, a he opmal, R sum =4.58 bps/hz, wh R = 4.3 bps/hz and R =.45 bps/hz, whch demonsraes he very unfar hroughpu allocaon beween he wo users due o he doubly near-far problem. For comparson, we consder UL ransmssons n a convenonal TDMA-based wreless nework wh WIT only [3]- [6], where each user s equpped wh a consan energy supply, and hus has an equal energy consumpon a each block denoed by Ē. I hen follows from 7 ha γ D αu and hus from, he opmal me allocaon o maxmze he sum-hroughpu of such a convenonal TDMA nework should sasfy D αu, =,, K,and =snce no DL WET s needed. Clearly, he WPCN suffers from a more severe near-far problem han he convenonal TDMA nework. Wh he same seup as for Fg. 5, n Fg. 6 we show he opmal hroughpu of U, R, normalzed by ha of U, R, n a WPCN versus ha n a convenonal TDMA nework for dfferen values of he pahloss exponen α, wh α d = α u = α. For he WPCN, γ s se o be fxed as γ = db for U, whle γ =, 7, 4,, and db for α =,.5, 3, 3.5, and 4, respecvely, snce γ /γ =D /D α. For he convenonal TDMA nework, Ē for boh U and U are assumed o be Ē = E + E wh E and E denong he average harvesed energy a U and U n he WPCN under comparson, respecvely; hen follows ha for he convenonal TDMA nework, γ =3dB and γ = 7., 5.5, 4.,.4, and.9db for α =,.5, 3, 3.5, and4, respecvely. From Fg. 6, s observed ha IV. COMMON-THROUGHPUT MAXIMIZATION In hs secon, we ackle he doubly near-far problem n he WPNC by applyng he common-hroughpu maxmzaon approach, whch guaranees equal hroughpu allocaons o all users and ye maxmze her sum-hroughpu. From and 6, he common-hroughpu maxmzaon problem s formulaed as P : max R, R s.. R R, = K, D, where R denoes he common-hroughpu and D s he feasble se of specfed by and 8. Remark 4.: Problem P s desgned o guaranee he hroughpu of he user wh he wors channel condon, e.g., of he larges dsance from he H-AP. Snce R gven by 6 s a monooncally ncreasng funcon of boh and, can be easly shown ha he opmal me allocaon soluon for P should allocae he same opmal hroughpu o all he users, denoed by R = R = = R K, wh = =, when he mnmum user hroughpu n he nework s maxmzed. In addon, allocang equal hroughpu o all users can be relevan n pracce, snce one ypcal applcaon of he WPCN s sensor nework, where all he sensors may need o perodcally send her sensng daa o a funcon cener modelled as he H-AP n our seup wh he same rae. The maxmum common-hroughpu R s he maxmum of all he feasble common-hroughpu R ha sasfes he rae nequales n of P. To solve P, gven any R >, we frs consder he followng feasbly problem: Fnd s.. R R, =,,K, D. Snce he problem n s convex, we consder s Lagrangan gven by L, λ = λ R R, 3 where λ =[λ,,λ K ] denoes he componenwse nequaly consss of he Lagrange mulplers assocaed wh he K user hroughpu consrans n problem. The dual funcon of problem s hen gven by G λ =mnl, λ. 4 D The dual funcon G λ can be used o deermne wheher problem s feasble, as provded n he followng lemma. Lemma 4.: For a gven R >, problem s nfeasble f and only f here exss an λ such ha Gλ >.

6 JU and ZHANG: THROUGHPUT MAXIMIZATION IN WIRELESS POWERED COMMUNICATION NETWORKS 43 Proof: Please refer o Appendx E. Nex, we oban Gλ n 4 for a gven λ by solvng he followng weghed sum-hroughpu maxmzaon problem, whch follows from 3. max λ R s.. D. 5 Lke P, he problem n 5 s convex and hus can be solved by convex opmzaon echnques. Smlar o Proposon 3. for he sum-hroughpu maxmzaon case wh λ =,, we oban he opmal me allocaon soluon for he weghed sum-hroughpu maxmzaon problem n 5, gven n he followng proposon. Proposon 4.: Gven λ, he opmal me allocaon soluon for 5, denoed by =[,,, K ],s = + K, 6 γj /zj j= j= γ /z =, =,, K, 7 + K γj /zj where z, =,,K, s he soluon of he followng equaons: ln + z z = μ ln, 8 +z λ λ γ +z = μ ln, 9 wh μ > beng a consan. Proof: Please refer o Appendx F. Wh Proposon 4., we can compue effcenly as follows. Denoe he lef-hand sdes LHSs of 8 and 9 as Q z and S z wh z =[z,z,, z K ], respecvely. Noe ha Q z s an ncreasng funcon of z, =,,K see Appendx C, whereas S z s a decreasng funcon wh respec o each ndvdual z. Gven any μ>, suppose ha z s he soluon of Q z = μ λ ln, =,,K, n 8. Wh hese z s, here are wo possble cases o consder nex. If n 9 he resulng S z > μln, we should ncrease μ snce z s sasfyng Q z = μ λ ln, =,,K, wll ncrease wh μ gven ha Q z,, s an ncreasng funcon of z ; as a resul, S z wll decrease snce s a decreasng funcon of each ndvdual z.oherwse,μ should be decreased o sasfy 9 f S z <μln. Therefore, z s and μ can be obaned by eravely updang z s and μ as above unl convergence s reached. Then, can be compued from 6 and 7 accordngly. Gven R, λ, and he obaned by solvng problem 5 wh Proposon 4., we can compue he correspondng R, =,,K, and hus G λ n 4 usng 3. If G λ >, follows from Lemma 4. ha problem s nfeasble,.e., R > R. Therefore, we should decrease R and solve he feasbly problem n agan. On he oher hand, f G λ, we can updae λ usng sub-graden based TABLE I ALGORITHM TO SOLVE P. Inalze R mn =, R max > R. 3 Repea. R = R mn + R max.. Inalze λ. 3. Gven λ, solve he problem n 5 by Proposon Compue G λ usng If G λ >, R s nfeasble, se R max R, goosep. Oherwse, updae λ usng he ellpsod mehod and he subgraden of G λ gven by. If he soppng crera of he ellpsod mehod s no me, go o sep Se R mn R. 3 Unl R max R mn <δ,whereδ> s a gven error olerance. U.6 o d e a c l lo a Fg * *,. Increasng Common-Throughpu + = allocaed o U.6. Common-hroughpu n bps/hz versus me allocaon. algorhms, e.g. he ellpsod mehod [7], wh he subgraden of G λ, denoed by υ =[υ υ υ K ] T,gvenby υ = log + γ R, K, unl λ converges o λ wh λ denong he maxmzer of G λ or he opmal dual soluon for problem. If G λ, hen follows ha problem s feasble and hus R R. In hs case, R should be ncreased for solvng he feasbly problem n agan. Consequenly, R can be obaned numercally by eravely updang R by a smple bsecon search []. To summarze, one algorhm o solve P sgvenntablei. Fg. 7 shows he common-hroughpu n bps/hz versus and for he same wo-user channel seup as for Fg. 5. I s observed ha he opmal me allocaon for P s gven by =[.3683,.386,.493], whch resuls n R = R =R =.46bps/Hz. Comparng o Fg. 5 where he sum-hroughpu s maxmzed, he me poron allocaed o he near user, U, s decreased subsanally from.74 o.737, whle ha o he far user, U, s grealy ncreased from.445 o Consequenly, he The compuaonal complexy of he proposed algorhm n Table I can be shown o be OK 3 snce a each eraon performs K one-dmenson searches each wh he complexy of O o fnd, and he ellpsod mehod has he complexy of OK [7] o converge. 3 The nal value of R max can be chosen as any arbrary large number such ha sasfes R max > R.

7 44 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 3, NO., JANUARY 4 P P z H / 3 s p b.5 U f o u p h g u o r h T.5 b Soluon of P Soluon of P Pahloss Exponen α.5 a Throughpu of U bps/hz Fg. 8. Comparson of he rao of me allocaed o U and U n P versus P. Fg. 9. Throughpu regon of a wo-user WPCN wh a correspondng o he maxmum sum-hroughpu; and b correspondng o he maxmum common-hroughpu. hroughpu of U s ncreased from.45bps/hz o.46bps/hz, whle ha of U s decreased from 4.3bps/Hz o.46bps/hz, he same hroughpu as U. Ths resul shows he effecveness of he proposed common-hroughpu approach for acklng he doubly near-far problem n a WPCN. Fg. 8 shows he rao of he opmal me allocaed o he far user U over ha o he near user U,.e., /,np versus P, wh dfferen values of he common pahloss exponen α n boh DL and UL, where he same wo-user channel seup as for Fg. 5 s consdered. I s assumed ha γ for he near user U s fxed as γ =db and γ for he far user U s se he same as for Fg. 6. I s observed ha he me rao of P n he logarhm scale decreases lnearly wh α o maxmze he sum-hroughpu, whch can also be nferred from, due o he doubly near-far problem. On he conrary, / s observed o ncrease wh α o maxmze he common-hroughpu n P, snce more me s allocaed o he far user U nsead of he near user U as he rao beween γ and γ,.e., γ /γ, ncreases wh α. Noce ha P and P deal wh wo exreme cases of hroughpu allocaon o he users n a WPNC where he farness s compleely gnored and a src equal farness s mposed, respecvely. More generally, Fg. 9 shows he achevable hroughpu regon of a wo-user WPCN by solvng he weghed sum-hroughpu maxmzaon problem n 5 wh dfferen hroughpu weghs for he near and far users, under he same channel seup as for Fg. 5. I s observed ha he boundary of he hroughpu regon characerzes all he opmal hroughpu-farness rade-offs n hs wo-user WPCN, whch nclude he hroughpu pars obaned by solvng P for he maxmum sum-hroughpu and by solvng P forhe maxmum common-hroughpu, shown as pons a andb n he fgure, respecvely. Remark 4.: I s worh nong ha he common-hroughpu approach for characerzng he achevable rae regon of muluser communcaon sysems under src farness consrans can be consdered as one specal case of he rae-profle mehod proposed n [8]. Hence, he common-hroughpu approach nvesgaed n hs paper can be easly exended o he general case where he requred hroughpu of each user [ s dfferen ] usng he rae-profle mehod. Gven R = R R RK wh R denong he requred hroughpu of user, =,,K, he correspondng rae profle vecor s defned as β =[β β β K ] where β = R / K R j j= Noe ha he common-hroughpu maxmzaon problem P s hus for a specal case wh β = /K,. The opmal me allocaon soluon o maxmze he sysem sumhroughpu subjec o he rae farness consran wh any gven β can be obaned usng he same algorhm proposed for P n hs paper, wh he hroughpu consran n replaced by R β R, =,, K,where R here denoes he sum-hroughpu of all users. V. SIMULATION RESULT In hs secon, we compare he maxmum sum-hroughpu by P versus he maxmum common-hroughpu by P n an example WPCN. The bandwdh s se as MHz. I s assumed ha he channel recprocy holds for he DL and UL and hus h = g, =,,K, wh he same pahloss exponen α d = α u = α. Accordngly, boh he DL and UL channel power gans are modeled as h = g = 3 ρ D α, =,,K,whereρ represens he addonal channel shor-erm fadng whch s assumed o be Raylegh dsrbued, and hus ρ s an exponenally dsrbued random varable wh un mean. Noe ha n he above channel model, a 3dB average sgnal power aenuaon s assumed a a reference dsance of m. The AWGN a he H-AP recever s assumed o have a whe power specral densy of 6dBm/Hz. For each user, he energy harvesng effcency for WET s assumed o be ζ =.5. Fnally, we se Γ=9.8dB assumng ha an uncoded quadraure amplude modulaon QAM s employed [9]. Fg. shows he maxmum sum-hroughpu versus he maxmum common-hroughpu n he same WPCN wh K =, D = 5m, and D = m for dfferen values of ransm power a H-AP, P A, n dbm, by averagng over

8 JU and ZHANG: THROUGHPUT MAXIMIZATION IN WIRELESS POWERED COMMUNICATION NETWORKS s p b 4 M u p h g u 3 r o h T e g r a e v A Max. Common-Throughpu for P Max. Sum-Throughpu for P Normalzed Max. Sum-Throughpu Throughpu of U n P Throughpu of U n P Average Throughpu Mbps Max. Common-Throughpu for P Max. Sum-Throughpu for P Normalzed Max. Sum-Throughpu Throughpu of U n P Throughpu of U n P P A dbm Pahloss Exponen α Fg.. Sum-hroughpu vs. common-hroughpu. Fg.. Throughpu vs. pahloss exponen. randomly generaed fadng channel realzaons, wh fxed α =. As shown n Fg., when he sum-hroughpu s maxmzed, he hroughpu of U domnaes over ha of U due o he doubly near-far problem, whch resuls n noably unfar rae allocaon beween he near user U andfaruser U n hs example. I s also observed ha he maxmum common-hroughpu for he wo users s smaller han he normalzed maxmum sum-hroughpu by he number of users,.e., R sum /K K =n hs example, whch s a cos o pay n order o ensure a srcly far rae allocaon o he wo users regardless of her dsances from he H-AP. Nex, by fxng P A =dbm, Fg. shows he hroughpu comparson for dfferen values of he common pahloss exponen α n boh he DL and UL n he same WPCN as for Fg.. I s observed ha when he sum-hroughpu s maxmzed, he hroughpu of he near user U converges o he maxmum sum-hroughpu as α ncreases, whereas ha of he far user U converges o zero, whch ndcaes ha he WPCN suffers from a more severe unfar rae allocaon beween he near and far users as he pahloss exponen ncreases, due o he doubly near-far problem. In addon, he maxmum commonhroughpu for he wo users s observed o decrease faser wh ncreasng α han he normalzed maxmum sum-hroughpu. Ths s because as α ncreases, P allocaes more me o he far user U nsead of near user U n order o ensure he equal hroughpu allocaon among users snce he rao γ /γ ncreases wh α, whereas P allocaes more me o U nsead of U as α ncreases. A las, Fg. shows he hroughpu over number of users, K. I s assumed ha K users n he nework are equally separaed from he H-AP accordng o D = DK K, =, K, where D K = m. The ransm power a he H-AP and he pahloss exponen are se o be fxed as P A = dbm and α =, respecvely. In addon, we compare wh he hroughpu achevable by equal me allocaon ETA,.e., = K+, =,,K,asalowcomplexy me allocaon scheme. I s observed ha boh he normalzed maxmum sum-hroughpu by solvng P and he maxmum common-hroughpu by solvng P decreases wh ncreasng K, and hey ouperform he sum-hroughpu Average Throughpu Mbps/Hz Fg Normalzed Max. Sum-Throughpu Normalzed Sum-Throughpu by ETA Max. Common-Throughpu Mn. Throughpu by ETA Number of users K Throughpu vs. number of users K. and he mnmum hroughpu over all users by he heursc ETA scheme, respecvely. VI. CONCLUSION Ths paper has suded a new ype of wreless RF rado frequency powered communcaon nework wh a harveshen-ransm proocol, where he H-AP frs broadcass wreless energy o dsrbued users n he downlnk and hen he users ransm her ndependen nformaon o he H-AP n he uplnk by TDMA. Our resuls reveal an neresng new phenomenon n such hybrd energy-nformaon ransmsson neworks, so-called doubly near-far problem, whch s due o he folded sgnal aenuaon n boh he downlnk WET and uplnk WIT. As a resul, noably unfar me and hroughpu allocaon among he users occurs when he convenonal merc of nework sum-hroughpu s maxmzed. To overcome hs problem, we propose a new common-hroughpu maxmzaon approach o allocae equal raes o all users regardless of her dsances from he H-AP by allocang he ransmsson me o users nversely proporonal o her dsances o he H-AP. Smulaon resuls showed ha hs approach s effecve n

9 46 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 3, NO., JANUARY 4 solvng he doubly near-far problem n he WPCN, bu a a cos of sum-hroughpu degradaon. APPENDIX A PROOF OF LEMMA 3. Denoe he Hessan of R defned n 6 as [ ] R = d j,m, j, m K, where d j,m denoes he elemen of R a he jh row and mh column. From 6, he dagonal enres of R,.e., j = m, can be expressed as d j,j = ln γ ln γ 3 β β, j =, j =, oherwse, where β =+ γ. In addon, he off-dagonal enres of R can be expressed as d j,m = d m,j = { ln γ β, j = and m =, oherwse. Gven an arbrary real vecor v = [v,v,,v K ] T, snce, can be shown from and ha v T R v = ln, β γ 3 v v.e., R,, s a negave semdefne marx. Therefore, R s a concave funcon of =[ N ] T []. Ths complees he proof of Lemma 3.. From 9, we have APPENDIX B PROOF OF LEMMA 3. lm f z =, 3 z f z z =lnz, 4 f z z = z. 5 Thus follows from 4 and 5 ha f z s a convex funcon over z and s mnmum s aaned a z = wh f =. Thsmpleshaf z wh z and s monooncally ncreasng wh z n hs regme. Therefore, gven A>, f z =A has a unque soluon z >. Ths complees he proof of Lemma 3.. APPENDIX C PROOF OF PROPOSITION 3. The Lagrangan of P s gven by K L sum,ν=r sum ν, 6 where ν denoes he Lagrange mulpler assocaed wh he consran n. The dual funcon of P s hus gven by G ν =mn L sum,ν, 7 D where D s he feasble se of specfedbyand8. I can be shown from and 8 ha here exss an D wh >, =,, K, sasfyng K <, and hus = srong dualy holds for hs problem hanks o he Slaer s condon []. Snce P s a convex opmzaon problem for whch he srong dualy holds, he Karush-Kuhn-Tucker KKT condons are boh necessary and suffcen for he global opmaly of P, whch are gven by =, 8 = K ν =, 9 = R sum ν =, =,,K, 3 where s and ν denoe he opmal prmal and dual soluons of P, respecvely. I can be easly verfed ha = mus hold for P and hus from 9 whou = loss of generaly, we assume ν >. From 3, follows ha γ = ν ln, 3 +γ γ = ν ln, K, 3 where x s defned as x =ln+x Δ x, +x x. 33 Gven, j K, from 3 we have γ = γ j j, j. 34 I can be easly shown ha x s a monooncally ncreasng funcon of x snce d x /dx = x + x for x. Therefore, equaly n 34 holds f and only f γ = γ j,, j K,.e., j γ = γ = γk K = C. 35

10 JU and ZHANG: THROUGHPUT MAXIMIZATION IN WIRELESS POWERED COMMUNICATION NETWORKS 47 Noe ha = K j j= respecvely. Therefore, can be expressed as = γ = γ j and j = γj γ from 8 and 35, j= γ A, 36 where A = K γ j. In addon, follows from 3, 35, and j= 36 ha ln + C C A +C = +C where C s defned n 35. Snce C = can modfy 37 as, K, 37 A from 36, we z ln z z A +=, 38 where z =+ A. I s observed ha z > f A> and < <. From Lemma 3., here exss a unque z > ha s he soluon of 38. Therefore, he opmal me allocaon o he DL WET s gven by = z A + z. 39 In addon, from 36 and 39, he opmal me allocaon o he UL WITs,, K, sgvenby = γ A + z. 4 Ths hus proves Proposon 3.. APPENDIX D PROOF OF COROLLARY 3. I can be easly shown from ha =wh A =. From 38 and 39, can be alernavely expressed as = z z ln z. 4 Gven A and hus z, boh z ln z and z n 4 ncrease wh A snce z ncreases wh A as shown n he proof of Lemma 3. gven n Appendx B. Furhermore, snce d dz z ln z =+lnz and d dz z =, follows d ha dz z ln z > d dz z wh z >,.e., z ln z ncreases faser wh z han z. Therefore, can be verfed ha z ln z ncreases faser wh A han z, and hus decreases monooncally wh ncreasng A. Fnally, can be shown ha as A from he fac ha z ncreases wh A. Ths hus complees he proof of Corollary 3.. APPENDIX E PROOF OF LEMMA 4. We frs prove he f par of Lemma 4.. If Ds a feasble soluon for gven R >,.e., R R, =,,K, hen for any λ follows from 3 ha and hus max λ G λ L, λ, G λ, whch conradcs wh he gven assumpon ha here exss an λ such ha G λ >. The f par s hus proved. Nex, we prove he only f par of Lemma 4. by showng ha s ransposon s rue,.e, he problem n s feasble f G λ, λ, by conradcon. Suppose ha problem s feasble and here exs an λ where G λ >. However, snce s assumed o be feasble, here exss an Dsuch ha R R,, resulng n λ R R snce λ. From 3 and 4, we hus have G λ K λ R R. Ths conradcs G λ >, and hus problem s feasble f G λ, λ. The only f par s hus proved. Combnng he above proofs of boh f and only f pars, Lemma 4. hus follows. APPENDIX F PROOF OF PROPOSITION 4. Gven λ, he Lagrangan of problem 5 s gven by K L WSR,μ= λ R μ, 4 where μ denoes he Lagrange mulpler assocaed wh he consran n. The dual funcon of problem 5 s hus gven by G WSR μ =mn L WSR,μ. 43 D Smlar o P, can be easly shown ha he problem n 5 s convex wh zero dualy gap. Therefore, he followng KKT condons mus be sasfed by he opmal prmal and dual soluons of problem 5: ln +γ γ +γ = = μ λ ln, =,,K, 44 λ γ +γ = μ ln, 45 =, 46 = where μ > s he opmal dual soluon. We hen oban 8 and 9 by changng varables as z = γ, =,,K, 47 n 44 and 45, respecvely. I s worh nong ha z z K and μ sasfyng boh 8 and 9 are unquely deermned snce K + varables are soluons of K + ndependen equaons and ln + z z +z s a monooncally ncreasng funcon of z. In addon, snce = K from 46 and = γ z from 47, follows ha γ + =, 48 z from whch we oban 6. Fnally, we oban 7 from 47 and 48. Ths hus complees he proof of Proposon 4..

11 48 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 3, NO., JANUARY 4 REFERENCES [] A. M. Zungeru, L. M. Ang, S. Prabaharan, and K. P. Seng, Rado frequency energy harvesng and managemen for wreless sensor neworks, Green Moble Devces New.: Energy Op. Scav. Tech., CRC Press, pp ,. [] R.J.M.Vullers,R.V.Schajk,I.Doms,C.V.Hoof,andR.Merens, Mcropower energy harvesng, Elsever Sold-Sae Crcus, vol. 53, no. 7, pp , July 9. [3] Y. Sh, L. Xe, Y. T. Hou, and H. D. Sheral, On renewable sensor neworks wh wreless energy ransfer, n Proc. IEEE INFOCOM, pp [4] K. Huang and V. K. N. Lau, Enablng wreless power ransfer n cellular neworks: archecure, modelng and deploymen, submed for publcaon. Avalable: arxv: [5] S. H. Lee, R. Zhang, and K. B. Huang, Opporunsc wreless energy harvesng n cognve rado neworks, IEEE Trans. Wreless Commun., vol., no. 9, pp , Sep. 3. [6] L. R. Varshney, Transporng nformaon and energy smulaneously, n Proc. 8 IEEE In. Symp. Inf. Theory, pp [7] P. Grover and A. Saha, Shannon mees Tesla: wreless nformaon and power ransfer, n Proc. IEEE In. Symp. Inf. Theory, pp [8] R. Zhang and C. K. Ho, MIMO broadcasng for smulaneous wreless nformaon and power ransfer, IEEE Trans. Wreless Commun., vol., no. 5, pp. 989, May 3. [9] L. Lu, R. Zhang, and K. C. Chua, Wreless nformaon ransfer wh opporunsc energy harvesng, IEEE Trans. Wreless Commun., vol., no., pp. 88 3, Jan. 3. [] X. Zhou, R. Zhang, and C. K. Ho, Wreless nformaon and power ransfer: Archecure desgn and rae-energy radeoff, o appear n IEEE Trans. Commun.. Avalable: arxv:5.68. [] A. M. Fouladgar and O. Smeone, On he ransfer of nformaon and energy n mul-user sysems, IEEE Commun. Le., vol. 6, no., pp , Nov.. [] S. Boyd and L. Vandenberghe, Convex Opmzaon. Cambrdge Unversy Press, 4. [3] R. Knopp and P. A. Humble, Informaon capacy and power conrol n sngle-cell mul-user communcaons, n Proc. 995 IEEE In. Conf. Commun., pp [4] D. N. C. Tse and S. V. Hanly, Mulaccess fadng channels par I: polymarod srucure, opmal resource allocaon and hroughpu capaces, IEEE Trans. Inf. Theory, vol. 44, no. 7, pp , Nov [5] L. L and A. J. Goldsmh, Capacy and opmal resource allocaon for fadng broadcas channels par I: ergodc capacy, IEEE Trans. Inf. Theory, vol. 47, no. 3, pp. 83, Mar.. [6] R. Zhang, S. Cu, and Y. -C. Lang, On ergodc capacy of fadng cognve mulple-access and broadcas channels, IEEE Trans. Inf. Theory, vol. 55, no., pp , Nov. 9. [7] S. Boyd, EE364b Lecure Noes. Sanford, CA: Sanford Unv. Avalable: hp:// mehod sldes.pdf. [8] M. Mohsen, R. Zhang, and J. M. Coff, Opmzed ransmsson for fadng mulple-access and broadcas channels wh mulple anennas, IEEE J. Sel. Areas Commun., vol. 4, no. 8, pp , Aug. 6. [9] A. Goldsmh, Wreless Communcaons. Cambrdge Unversy Press, 5. Hyungsk Ju S 8-M receved he B.S. and Ph.D. degrees n Elecrcal and Elecronc Engneerng from Yonse Unversy, Seoul, Korea, n 5 and, respecvely. From Sep. o Mar., he worked as a Posdocoral Researcher n he Informaon and Telecommuncaon Laboraory ITL a Yonse Unversy. Snce Mar., he has joned he Deparmen of Elecrcal and Compuer Engneerng of he Naonal Unversy of Sngapore as a research fellow. Hs curren research neress nclude wreless nformaon and power ransfer, wreless powered neworks, fullduplex wreless communcaon, relay-based mul-hop communcaon and full-duplex relay sysems. Dr. Ju receved he 4h Humanech Paper Award, Samsung Elecroncs, Ld., n Feb. 8. Ru Zhang S -M 7 receved he B.Eng. Frs- Class Hons. and M.Eng. degrees from he Naonal Unversy of Sngapore n and, respecvely, and he Ph.D. degree from he Sanford Unversy, Sanford, CA USA, n 7, all n elecrcal engneerng. Snce 7, he has worked wh he Insue for Infocomm Research, A-STAR, Sngapore, where he s now a Senor Research Scens. Snce, he has joned he Deparmen of Elecrcal and Compuer Engneerng of he Naonal Unversy of Sngapore as an Asssan Professor. Hs curren research neress nclude muluser MIMO, cognve rado, cooperave communcaon, energy effcen and energy harvesng wreless communcaon, wreless nformaon and power ransfer, smar grd, and opmzaon heory. Dr. Zhang has auhored/coauhored over 5 nernaonally refereed journal and conference papers. He was he co-recpen of he Bes Paper Award from he IEEE PIMRC n 5. He was he recpen of he 6h IEEE ComSoc Asa-Pacfc Bes Young Researcher Award n, and he Young Invesgaor Award of he Naonal Unversy of Sngapore n. He s now an eleced member of IEEE Sgnal Processng Socey SPCOM and SAM Techncal Commees, and an edor for he IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS and he IEEE TRANSACTIONS ON SIGNAL PROCESSING.