Rate Maximization in Wireless Powered Communication Systems with Non-Ideal Circuit Power Consumption

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1 krotalasna revja Decembar 06 Rate axmzaton n reless Powered Communcaton Systems wth Non-Ideal Crcut Power Consumpton Ivana Nkoloska, Zoran Hadz-Velkov, Hrstna Chngoska Abstract The key concern n battery-powered communcaton s the lmted power avalablty and the need for battery replacement. Energy harvestng EH wreless communcaton has promsed to tackle these ssues successfully by devsng optmal polces that take nto account the harvested energy causalty. Ths task s partcularly challengng f the processng energy cost, caused by the non-deal crcut consumpton of the EH devces, s ncorporated nto these polces, together wth the RF transmt power at the output of these devces. In ths paper, we consder a pont-to-pont communcatons system wth RF energy harvestng, whch conssts of a base staton BS that broadcasts RF energy, and an EH user EHU that harvests ths energy and transmts nformaton to the BS. The nformaton transmtter employs harvest-then-transmt protocol, such that the EHU n each nformaton transmsson spends the total amount of the energy harvested from the BS durng the precedng RF broadcast. For such system, we propose a novel transmsson scheme that optmzes the BS transmt power and the tme sharng between the BS and EHU transmssons. The numercal results det the achevable average rates of the proposed scheme. Increasng the EHU s processng cost sgnfcantly deterorates the achevable rate, and, therefore, should certanly be consdered n the protocol desgn. Index Terms Energy harvestng, reless power transfer, Processng cost, Pont-to-pont lnks, Power and tme allocaton. I. INTRODUCTION Recent advances n ultra-low power wreless communcatons and energy-harvestng technologes have made selfsustanable devces feasble. Tyally, the major concern for these devces s battery lfe and replacement. Applyng energy harvestng technques to these devces can sgnfcantly extend battery lfe and sometmes even entrely elmnate the need for a battery []-[]. To maxmze the system performance, the EH transmtters should adapt ther output powers based on the energy harvested up to the tme of transmsson causalty Ivana Nkoloska s wth the Faculty of Electrcal Engneerng and Informaton Technologes, Ss. Cyrl and ethodus Unversty, Rugjer Boshkovk bb, 000 Skopje, R. acedona, E-mal: vanan@fet.ukm.edu.mk Zoran Hadz-Velkov s wth the Faculty of Electrcal Engneerng and Informaton Technologes, Ss. Cyrl and ethodus Unversty, Rugjer Boshkovk bb, 000 Skopje, R. acedona, E-mal: zoranhv@fet.ukm.edu.mk Hrstna Chngoska s wth the Faculty of Electrcal Engneerng and Informaton Technologes, Ss. Cyrl and ethodus Unversty, Rugjer Boshkovk bb, 000 Skopje, R. acedona, E-mal: cngoska@fet.ukm.edu.mk Ths s an extended verson of the paper Performance of Communcaton Systems wth RF Energy-Harvestng and Processng Cost presented at th Internatonal Conference on Telecommuncatons n odern Satellte, Cable and Broadcastng Servces - TELSIKS 05, held n October 05 n Ns, Serba. constrant [3]-[4]. Ambent energy n the form of manmade electromagnetc radaton s abundant regardless of the system s locaton.e. ndoor or outdoor systems or tme of day. The rado frequency energy can be easly found n surroundngs as t s used wdely by many applcatons lke televson broadcastng, telecommuncaton, mcrowave etc. It s ubqutous and free and hghly effcent. Harvestng the RF radaton s an approach known as wreless power transfer PT [5]-[6]. The PT can be realzed n the form of a smultaneous wreless nformaton and power transfer SIPT from the same transmtter [7], or, alternatvely, the energy and nformaton can be transmtted over orthogonal ether n tme or frequency channels. The latter opton gves rse to the so called wreless powered communcatons networks PCNs [8], whch tyally rely on tme-dvson multple access TDA. In order to develop PCNs, the network nodes must rely on new ultra-low power communcaton schemes that allow them to make the best use of the small amounts of ambent energy they have at ther dsposal. In [8], the authors determne the optmal TDA scheme among the half-duplex nodes ether BS or EHUs, dependng on the channel fadng states. The optmal tme-sharng n PCNs where one BS operatng n full-duplex broadcasts wreless energy to a set of dstrbuted users n downlnk and at the same tme receves ndependent nformaton from the users va tme-dvson-multple-access TDA n uplnk, was studed n [9]. The paper [0] studes the PCN wth full-duplex nodes, where the BS s equpped wth two antennas. The authors n [], consder PCNs, where the nodes choose between two power levels, a constant desred power, or lower power when ts EH battery has stored nsuffcent energy. The paper [], studes relayng systems wth RF energy harvestng. In ths paper, we propose a novel transmsson scheme, whch optmally adapt the BS transmt power and the duraton of the EH and IT phases n each TDA epoch. The scheme ncorporates the constant processng energy cost, denoted by p c, such that the total power consumpton by the EHU ncludes both the processng and RF transmt powers. The proposed scheme drastcally outperforms schemes whch only employ optmal tme allocaton among the network nodes. II. SYSTE ODEL As deted n Fg. a, we consder a pont-to-pont lnk consstng of a half-duplex BS and a half-duplex EHU, whch 6

2 December, 06 crowave Revew where P S s the EHU s transmt power. The amount of harvested power by the EHU s E = N 0 p τ T whch s completely spent n the successve IT of duraton τ T as E = P S + p c τ T 3 Thus, the EHU s transmt power n epoch s gven by Fg. : a Pont-to-pont EHU-BS, lnk b TDA epoch/frame structure operate n a fadng envronment. The BS broadcasts RF energy to the EHU, whereas the EHU transmts nformaton back to the BS. The EHU s equpped wth rechargeable EH batteres that harvest the RF energy broadcasted from the BS. The IT from the EHU to the BS IT phase, and the PT from the BS to the EHU EH phase are realzed as successve sgnal transmssons usng TDA over a common channel, where each TDA frame/epoch s of duraton T. Each TDA epoch conssts of an EH phase and an IT phase Fg. b. The channel between the BS and the EHU s a quas-statc block fadng channel, whch s constant durng a sngle block but changes ndependently from one block to the next. The duraton of one fadng block s assumed equal to T, and one block concdes wth a sngle epoch. In epoch, the fadng power gans of the BS-EHU channel s denoted by x. For convenence, these gans are normalzed by the addtve whte gaussan nose AGN power, yeldng = x /N 0 wth an average value of Ω = E[x ]/N 0, where E[ ] denotes expectaton. The channels are assume to be recprocal,.e., n epoch, the gans of BS-EHU and EHU-BS channels are the same. In epoch, the BS transmt power s denoted by p, the duraton of the EH phase s τ T and the duraton of the IT phase s τ T. e assume that p satsfes an average power constrant P avg.e., E[p τ ] = P avg, and a maxmum power constrant P ma.e., 0 p P max. A. Processng Cost In practcal EH transmtters, besdes ther transmt power, an addtonal power s also consumed by ts non-deal electrc crcutry e.g., AC/DC converter, analog RF amplfer, processor etc., denoted as the processng energy cost [3]. In short range communcatons, as n most wreless sensor networks, where nter-node dstances are less than 0m, processng energy consumpton can be comparable to the transmsson energy [4]. e consder the followng practcal model for the total power consumpton of the EHU: P S + p c, P S > 0 p t = 0, P S 0, P S = N 0p τ τ p c, 4 and the achevable nformaton rate n ths epoch s gven by R S = τ log + P S. e am at maxmzng the EHU s average throughput, gven by R = lm R S. 5 = III. JOINT POER AND TIE ALLOCATION e am at maxmzng the achevable rate by optmzng P S and τ. Therefore, we defne the followng optmzaton problem: max P S,τ, s.t. C : C : τ log + P S = = τ P S + p c N 0 P avg τ P S + p c N 0 P max τ C3 : P S 0 C4 : 0 < τ. 6 Due to the products and ratos of the optmzaton varables, 6 s not a convex optmzaton problem. After ntroducng the change of varables, and = τ 7 α = τ P S / 8 problem 6 s transformed nto a convex optmzaton problem, s.t. C : max α,, = = log + α x α + p c N 0 P avg C : α + + N 0 P max N 0 P max C3 : α 0 C4 : 0 <. 9 7

3 krotalasna revja Decembar 06 The soluton of the optmzaton problem 6 s gven by the followng Theorem: Theorem. The optmal BS transmt power s gven by Pmax p, 0 < τ N 0 P max τ = p c < λ x 0, otherwse. The optmal duraton of the EH phase, τ, s gven by, τ = N 0x Pmax N 0x Pmax+x + N0 x Pmax+ e λpmax 0 where denotes the Lambert functon. The constant λ s found from / = p τ = P avg. Proof: Please refer to Appendx A. Note, when settng p c = 0, the optmal soluton for ths optmzaton problem becomes: The optmal BS transmt power s gven by p P max, x = > λ 0, otherwse. The optmal duraton of the EH phase, τ, s gven by, τ = N 0x P max N 0 x P +, max N0 x P max e λp max 3. The constant λ s found from / = p τ = P avg. Interestngly, the condton n 0, collapses down to a much smpler form, gven n, as shown n Appendx B. IV. PRACTICAL IPLEENTATION ALGORITH The polcy proposed n Theorem, s dffcult to mplement n practce, snce the values of the Lagrangan multplers are not known n advance when the PDFs of the fadng channels are unknown. In ths secton, we propose an onlne polcy n whch the actual value of λ s replaced by ts estmate n epoch, ˆλ. Therefore Eqs. 0- become: ˆp = ˆτ = Pmax, 0 < τn0pmax τ 0, otherwse. N 0x Pmax N 0x Pmax+x p c < ˆλ x + N0 x Pmax+ e ˆλPmax, 4 5 Now, the optmal values of the BS output power ˆp and the duraton of the EH phase ˆτ are estmates of p and τ, snce they are based on the estmate of the value of the Lagrangan multpler. The estmate of λ s calculated by applyng the stochastc gradent descent method accordng to [5, Secton III.C], as ˆλ 0 = ˆλ 0 + β 0 ˆp nˆτ 0n P avg, 6 n= 8, Achevable sum rate bts per symbol.5.5 Pavg = 0, 9, 8, 7, 6, 5, 4, 3,, Processng energy costm Fg. : axmum achevable throughput for dfferent values of the processng energy cost where β 0 s an arbtrary small step sze, and the average s estmated from the prevous epochs, as n=. hat makes ths algorthm attractve s ts capablty to adapt to sudden changes n channel statstc. The algorthm rapdly learns ths changes and s able to converge to a new optmum. V. NUERICAL RESULTS e llustllustrates the achevable rate n a pont to pont lnk for dfferent values of the processng cost. As expected the rate decreases wth the ncrease of the processng energy cost. Even more, for smaller values of the average BS output power, ncreasng the p c decreases the rate up to one quarter of ts value at 00n. Ths llustrates the crpplng effect processng energy cost has on ths knd on communcaton. However, the performance of of the proposed practcal algorthm, can not be surpassed n such scenaros, when p c s taken nto account. Fg 3. presents the achevable rate for dfferent values of the BS output power. The ncrease n rate s agan notceable for as p c becomes smaller. As prevously stated, when p c = 0, the optmal soluton collapses down to Eqs. -3 and ths s the maxmum achevable rate n an energy harvestng pontto-pont system. As a benchmark we apply the optmal tme allocaton scheme, accordng to [8], whch maxmzes the uplnk sumrate of the PCN by optmzng the duratons of the EH and IT phases, for fxed BS output power n all epochs.e., p = P avg. Eqs. -3 can be seen as an extenson of ths benchmark. Note, that, the optmal tme allocaton scheme does not take nto account the processng energy cost. Stll, when we set p c = 0, we can llustrate the superor performances of jont power and tme allocaton Fg. 4 compared to the optmal tme allocaton gven n [8]. VI. CONCLUSION In ths paper, we optmze a pont-to-pont wreless powered communcaton system wth non-deal crcut power consump-

4 December, 06 Achevable sum rate bts per symbol Achevable rate bts per symbol Pc = 000n Pc = 700n Pc = 500n Pc = Average transmt power Fg. 3: axmum achevable throughput vs. BS output power Jont optmal power/tme allocaton Optmal tme allocaton Average BS transmt power Fg. 4: axmum achevable throughput vs. BS output power ton. e proposed a transmsson scheme that jontly optmzes the BS transmt power allocaton and the tme-sharng n the dynamc TDA frame. Ths scheme can be also effcently used for practcal scenaros, when the processng cost s consdered. The proposed algorthm for practcal mplementaton s capable of teratvely fndng the optmal resource allocaton even when the fadng channel dstrbuton s unknown a pror. Numercal results show that the proposed scheme outperforms the exstng schemes n the lterature. In the future, we am to generalze ths scheme for wreless powered communcaton networks wth an arbtrary number of users. APPENDIX A PROOF OF THEORE The Lagrangan of problem 9 can be wrtten as L α,, λ = log + α x p c λ α + x P N 0 + µ α q α + + P max N 0 P max N 0 +υ γ 7 where λ s assocated wth C, µ wth C, q wth C3, and υ and γ wth C4 n problem 9. By dfferentatng Eq.7 wth respect to α and, we get L L α = λ p c q x crowave Revew + αx λ + µ q = 0, 8 = log + α x + P max N 0 α x + α x + υ γ = 0 9 Accordng to KKT condtons, the complementary slackness should be satsfed, : µ α = q α + + P max N 0 P max N 0 = υ = γ = 0 0 where µ 0, q 0, υ 0 and γ 0. If we consder the prevous expressons together wth the complementary slackness condtons we can conclude that: Case : hen α = 0, from the defnton of α, s also 0. Naturally, no power s allocated n epoch,.e. p = 0. Case : hen 0 < α < N 0 P max p c, then the slackness condtons requre µ = 0 and q = 0. From Eq.8 follows that α = λ x and from Eq.9 follows that x log + λ λ x, λ p c x = 0 whch can not hold for arbtrary. Therefore we can conclude that the optmal p, can not be n the nterval 0, P max. Case 3: hen α = N 0 P max p c, then 0 < <, q > 0 and µ = 0. Therefore, from Eq.8 we get q = x p c + P max N 0 x λ > 0. 3 By usng the expresson for q, we can compute the condton under whch p = P max. Then, by usng Eq.9 and replacng α wth N 0 P max p c, we can fnally obtan the followng transcendental equaton, log p c + N 0P max x τ + N 0 λp max τ N 0 P max x = τ p c + N 0 P max x τ, 4 whose closed form soluton can be found by applyng the defnton of the Lambert functon and s gven by. 9

5 krotalasna revja Decembar 06 A. Lambert functon The Lambert functon, also called the omega functon or product logarthm, s a set of functons, namely the branches of the nverse relaton of fz = ze z where e z s the exponental functon and z s any complex number [6],.e. z = f ze z = ze z. 5 By substtutng the above equaton n z = ze z, we get the defnng equaton for the functon and for the relaton n general as z = z e z 6 for any complex number z. Settng y = p c + N 0P max x τ τ, 7 4 can be rewrtten n the form of 6, fnally yeldng. The Lambert functon s ntegrated as a specal functon n numerous software packages, lke atlab or aple. It has many applcatons, rangng from bochemstry, to modelng of fadng channels [7] and cooperatve methods of power allocaton [8]. APPENDIX B SUFFICIENT CONDITION FOR The condton n s obtaned from the followng nequalty condton: or, equvalently or, equvalently τ τ 0 < τ N 0 P max < τ λ x, 8 λ x > N 0 P max > 0, 9 λ x 0 < τ < N 0 P max + λ. 30 x In order to satsfy the constrant 0 < τ <, 30 suggest the followng addtonal condton: λ < x. 3 If 3 s satsfed, τ, always satsfes the condtons In order to prove ths, we defne the an auxlary functon, as fτ = N 0 P max x τ + N 0 P max x τ N 0 λp max log + N 0P max x τ. 3 τ Note that 3, orgnates from the transcendental equaton, whose closed form s gven by 3. Let us assume that 3 s satsfed. If we now set set τ to the lower bound of 30, we have Then, f we set τ to the upper bound of 30, we obtan λ f x N 0 P max + λ = λ x x log < x λ Snce fτ s decreasng n τ and changes the sgn at the lower and upper bounds of 30, we conclude that fτ = 0, s always between these bounds. Therefore, assumng 3, the condtons 8-30 are always satsfed by the soluton of 3. REFERENCES [] D. Gunduz, K. Stamatou, N. chelus, and. Zorz, Desgnng Intellgent Energy Harvestng Communcaton Systems, IEEE Commun. agazne, vol. 5, no., pp. 0-6, January 04. [] C. K. Ho, and R. Zhang, Optmal Energy Allocaton for reless Communcatons wth Energy Harvestng Constrants, IEEE Trans. Sgnal Proccessng, vol. 60, no. 9, pp , ay 0. [3] I. Krkds, T. Charalambous, and J. S. Thompson, Stablty Analyss and Power Optmzaton for Energy Harvestng Cooperatve Networks, IEEE Sgnal Processng Letters, vol. 9, no., pp. 0-3, November 0. [4] Z. Dng, and H. V. Poor, Cooperatve Energy Harvestng Networks wth Spatally Random Users, IEEE Sgnal Processng Letters, vol. 0, no., pp. -4, October 03. [5] L. R. Varshney, Transportng Informaton and Energy Smultaneousely, Proc. IEEE ISIT, pp. 6-66, July 008. [6] P. Grover, and A. Saha, Shannon eets Tesla: reless Informaton and Power Transfer, Proc. IEEE ISIT, pp , June 00. [7] R. Zhang, and C. K. Ho, IO Broadcastng for Smultaneous reless Informaton and Power Transfer, IEEE Trans. reless Commun., vol., no. 5, pp , ay 03. [8] H. Yu, and R. Zhang, Througnput axmzaton n reless Powered Communcaton Networks, IEEE Trans. reless Commun., vol 3, no., pp , January 04. [9] H. Ju, and R Zhang, Optmal Resource Allocaton n Full-Duplex reless-powered Communcaton Network, IEEE Trans. Commun., vol. 6, no. 0, pp , October 04. [0] X. Kang, C. Keong Ho, and S. Sun, Optmal Tme Allocaton for Dynamc-TDA-Based reless Powered Communcaton Networks, Proc. IEEE Globecom 04, Sgnal Processng for Communcatons Symposum, December 04. [] He Henry Chen, et al., Harvest-then-Cooperate: reless-powered Cooperatve Communcatons, arxv: v, arch 05. [] R. ors, D. S. chalopoulos, and R. Schober, Performance Analyss of reless Powered Communcaton wth Fnte/Infnte Energy Storage, arxv: v, October 04. [3] O. Orhan, D. Gunduz, and E. Erkp, Throughput axmzaton for an Energy Harvestng Communcaton System wth Processng Cost, Informaton Theory orkshop IT pp , 03 September 0. [4] S. Cu, A. J. Goldsmth, and A. Baha, Energy-Constraned odulaton Optmzaton, IEEE Trans. reless Commun., vol. 4, no. 5, pp , September 005. [5] X. ang, and N. Gao, Stochastc Resource Allocaton over Fadng ultple Access and Broadcast Channels, IEEE Trans. Info. Theory, vol. 56, no. 5, pp , ay 00. [6] R.. Corless, G. H. Gonnet, D. E.G. Hare, D. J. Jeffrey, and D. E. Knuth, On the Lambert Functon, Adv. Computatonal ath., vol. 5, no., pp , January 996. [7] C. Tellambura and D. Senaratne, Accurate Computaton of the GF of the Lognormal Dstrbuton and ts Applcaton to sum of Lognormals, IEEE Trans. Commun., vol. 58, no. 5, pp , ay 00. [8] D. u, Y. Ca, L. Zhou, and J. ang, A Cooperatve Communcaton Scheme Based on Coalton Formaton Game n Clustered reless Sensor Networks, IEEE Trans. reless Commun., vol., no. 3, pp , arch 0. f0 = N 0 P max x λ >