Information-Theoretic Analysis of an Energy Harvesting Communication System

Transcription

1 Iformatio-Theoretic Aalysis of a Eergy Harvestig Commuicatio System Omur Ozel Seur Ulukus Departmet of Electrical ad Computer Egieerig Uiversity of Marylad, College Park, MD 074 omur@umd.edu ulukus@umd.edu Abstract I eergy harvestig commuicatio systems, a exogeous recharge process supplies eergy for the data trasmissio ad arrivig eergy ca be buffered i a battery before cosumptio. Trasmissio is iterrupted if there is ot sufficiet eergy. We address commuicatio with such radom eergy arrivals i a iformatio-theoretic settig. Based o the classical additive white Gaussia oise (AWGN) chael model, we study the codig problem with radom eergy arrivals at the trasmitter. We show that the capacity of the AWGN chael with stochastic eergy arrivals is equal to the capacity with a average power costrait equal to the average recharge rate. We provide two differet capacity achievig schemes: save-adtrasmit ad best-effort-trasmit. Next, we cosider the case where eergy arrivals have time-varyig average i a larger time scale. We derive the optimal offlie power allocatio for maximum average throughput ad provide a algorithm that fids the optimal power allocatio. I. INTRODUCTION I this paper, we aalyze poit-to-poit commuicatio of eergy harvestig odes from a iformatio-theoretic perspective. We focus o wireless etworkig applicatios where odes (e.g., sesors odes) ca harvest eergy from ature through various differet sources, such as solar cells, vibratio absorptio devices, water mills, thermoelectric geerators, microbial fuel cells, etc. I such systems, eergy that becomes available for data trasmissio ca be modeled as a exogeous recharge process. Therefore, ulike traditioal battery-powered systems, i these systems, eergy is ot a determiistic quatity, but is a radom process which varies stochastically i time at a scale o the order of symbol duratio. The trasmissio ca be iterrupted due to lack of eergy i the battery. O the other had, excess eergy ca be buffered i the battery before cosumptio for trasmissio. This model requires a major shift i terms of the power costrait imposed o the chael iput compared to those i the existig literature. To illustrate, i iformatio-theoretic approaches, there are two widely used iput costraits o the chael iputs of cotiuous-alphabet chaels: average power costrait ad amplitude costrait. If the iput is average power costraied, the ay codeword should be such that while each symbol ca take ay real value, the average power of the etire codeword should be o more tha the power costrait. O This work was supported by NSF Grats CCF , CCF , CNS 07-63, CCF ad CNS the other had, if the iput is amplitude costraied, the every code symbol should be less tha the costrait i amplitude. It is clear that i a eergy harvestig model, the costrait imposed o the chael iput is differet tha these costraits, i that, while code symbols are istataeously amplitude costraied, eergy ca be saved i the battery for later use. This amouts to a uprecedeted iput costrait o the chael iput. I this cotext, the mai goal of this paper is to ivestigate the effect of stochastic eergy arrivals o the commuicatio i a iformatio-theoretic framework. I particular, we augmet a eergy buffer to the classical AWGN system ad study iformatio-theoretically achievable rates. First, we cosider the settig where eergy arrives at the trasmitter as a stochastic process, which varies o the order of symbol duratio. The problem is posed as desig of a codebook that complies with all of the iput costraits. The iput costrait imposed by stochastic eergy arrivals is a radom oe ad i this sese it geeralizes classical determiistic average or amplitude power costraits. The recharge process together with past code symbols determie the allowable rage of iputs i each chael use. We start with showig that the capacity of the AWGN chael with a average power costrait equal to the average recharge rate is a upper boud for the capacity i the eergy harvestig system. The, we develop a scheme called save-ad-trasmit scheme that achieves this upper boud ad hece the capacity. The save-ad-trasmit scheme relies o sedig zero code symbols i a portio of the total block legth, which becomes egligible as the block legth gets larger. Next, based o feasibility to sed the code symbol i a chael use, we provide a alterative scheme called the best-effort-trasmit scheme for achievig the capacity. Wheever the available eergy is sufficiet to sed the code symbol, it is put to the chael, while a zero is put to the chael if there is ot eough eergy i the battery. We show that this scheme achieves rates arbitrarily close to the capacity. Secodly, we address a typical behavior i certai eergy harvestig sesors, such as solar-powered sesors, where the recharge process is ot ergodic or statioary. I this case, we assume that the average rechargig rate is ot a costat i time, but rather fosters time variatio i a scale much larger tha the time scale that commuicatio takes place. Accordigly, we exted the formulatio to the case i which recharge

2 process has a mea value that is varyig i sufficietly log time ad we call each such sufficietly log time a slot. We derive the optimal power cotrol for maximum average throughput i this case, ad provide a geometric iterpretatio for the resultig power allocatio. We illustrate the advatage of the optimal power allocatio i a umerical study. A. Relevat Literature To the best of our kowledge, this work is the first attempt to aalyze a eergy harvestig commuicatio system from a iformatio-theoretic perspective. There are may motivatig works i the etworkig literature. I [], Lei et. al. address repleishmet i oe hop trasmissio. Formulatig trasmissio strategy as a Markov decisio process, [] uses dyamic programmig techiques for optimizatio of trasmissio policy uder repleishmet. I [], Gatziaas et. al. exted classical wireless etwork schedulig results to a etwork with users havig rechargeable batteries. Each battery is cosidered as a eergy queue, ad data ad eergy queues are updated simultaeously i a iteractio determied by rate versus power relatioship. Stability of data queues is studied usig Lyapuov techiques. I [3], [4], i a similar eergy harvestig settig, a dyamic power maagemet policy is proposed ad is show to stabilize data queues. I each slot, eergy spet is equal to the average recharge rate. Moreover, uder a liear approximatio, some delay-optimal schemes are proposed. The capacity of scalar AWGN chael has bee extesively studied i the literature uder differet costraits o the iput sigal. Average power (or the secod momet) costrait o the iput yields the well-kow result that the capacity achievig iput distributio is Gaussia with variace equal to the power costrait. Smith [5], [6] cosiders amplitude costraits i additio to average power costraits ad cocludes that the capacity achievig iput distributio fuctio is a step fuctio with fiite umber of icrease poits. Moreover, Shamai ad Bar-David [7] exted Smith s result to amplitude costraied quadrature Gaussia chael for which the optimal iput distributio is cocetrated to fiite umber of uiform phase circles withi the amplitude costrait. II. AWGN CHANNEL WITH RANDOM ENERGY ARRIVALS We cosider scalar AWGN chael characterized by the iput X, output Y, additive oise N with uit ormal distributio N(0,) ad a battery (see Fig. ). Iput ad output alphabets are take as real umbers. Eergy eters the system from a power source that supplies E i uits of eergy i the ith chael use where E i 0. {E,E,...,E } is the time sequece of supplied eergy i chael uses. E i is a i.i.d. sequece with average value P, i.e., E [E i ] = P, for all i. We assume that the eergy stored ad depleted from the battery are for oly commuicatio purposes (for example, the eergy required for processig is out of the model). Existig eergy i the battery ca be retrieved without ay loss ad the battery capacity is large eough so that every quata of icomig eergy ca be stored i the battery. This assumptio W E i X i N i Ecoder Decoder Ŵ Fig.. AWGN chael with radom eergy arrivals. is especially valid for the curret state of the techology i which batteries have very large capacities compared to the rate of harvested eergy flow [3]: E max P. The battery is iitially empty ad eergy eeded for commuicatio of a message is obtaied from the arrivig eergy durig trasmissio of the correspodig codeword subject to causality. I particular, E i uits of eergy is added to the battery ad Xi uits of eergy is depleted from the battery i the ith chael use. This brigs us to the followig cumulative power costraits o the iput based o causality: k Xi Y i k E i, k =,..., () This is illustrated i Fig., where i each chael use, E i amout of eergy arrives ad Xi amout of eergy is used. Note that the costraits i () are upo the support set of the radom variables X i. The first costrait restricts the support set of X to [ E, E ]. The secod costrait is X + X E + E. I geeral, lettig S i deote [ i j= (E j Xj )]+, i chael use i, the symbol X i is subject to the costrait Xi E i + S i. The iput costraits i () are challegig because E i are radom ad these costraits itroduce memory (i time) i the chael iputs. Radomess i E i makes the problem similar to fadig chaels i that state of the recharge process (i.e., low or high E i ) affects istataeous quality of commuicatio. Moreover, this time variatio i recharge process allows opportuistic cotrol of eergy as i fadig chaels. However, recharge process ca be collected i battery ulike a fadig state. I fact, we will see that, this ature of eergy arrivals makes it more advatageous to save eergy i the battery for future use whe a peak occurs i the recharge process, as opposed to opportuistically ride the peaks. III. THE CAPACITY We will ivoke the geeral capacity formula of Verdu ad Ha [8]. For fixed, let f be the joit desity fuctio of radom variables {X i } ad let F be the set of variable joit desity fuctios that satisfy the costraits i (). Sice AWGN is a iformatio-stable chael [8], the capacity of the chael i Fig. with costraits i () is: C = lim max f F I(X ;Y ) ()

3 +E +E +E X X X X..... X X Fig.. For time i, E i deotes the eergy arrivig, ad X i deotes the chael iput. Therefore, Xi is the eergy used at time i. I geeral, capacity achievig iput distributio is i the form of product of margial distributios (idepedet distributio) [8]. However, ote that the power costraits create depedece amog the radom variables. The costrait o X i+ is depedet o give value of X j, j i. Though idepedet processes achieve higher mutual iformatio tha the oes with the same margial distributio but with correlatio [8], the capacity that we seek i this problem does ot let the process be idepedet. This problem falls i the family of problems of fidig capacity uder depedece costraits o code symbols which is by itself iterestig ad less studied. A upper boud for C is the correspodig AWGN capacity with average power P, as X i E i ad by the i.i.d. ature of E i, ivokig strog law of large umbers [9], E i P with probability. Therefore, each codeword satisfyig costraits i () automatically satisfies lim X i P with probability. However, if a codeword satisfies average power costrait, it does ot ecessarily satisfy the costraits i (). Hece, we get the followig boud: C log ( + P) (3) Next, we state ad prove that the above upper boud ca be achieved. Theorem The capacity of the AWGN chael uder i.i.d. radom eergy arrivals E i, where E[E i ] = P, is idepedet of the realizatios of E i ad equal to the chael capacity uder average power costrait P : C = log ( + P) (4) To prove the theorem, we eed a achievable scheme. Achievig the capacity requires desig of a (, R,ǫ ) codebook with ecodig fuctio {fk (.)} k= ad decodig fuctio g (.) where is the code legth, R is the code size ad ǫ is the probability of error such that ǫ 0 ad R C. There are two separate causes of error. The first oe is that ay codeword does ot satisfy the iput costraits. I this case, the ecoder does ot sed that codeword but seds a all zero codeword. Hece, all zero codeword is assumed to Note that zero is costless (requires o eergy) ad therefore all zero codeword satisfies the iput costraits for all realizatios of the eergy arrival process. be i the codebook ad it is decoded to message 0 at the receiver. If the decoder maps received sigal to message 0, it is accepted as a error. Secod cause of error is the actual decodig error at the receiver. If the received sigal is decoded to a message that is differet from the message set, the a error occurs. Accordigly, the error evet is defied as uio of these two evets. While desigig the codebook ad the ecodig/decodig rule, a first approach ca be to optimize the codebook desig subject to the iput costraits. Therefore, the occurrece of the first type of error (i.e., isufficiet eergy) is elimiated from the begiig. However, we will show that, it is ot ecessary to solve this difficult optimizatio problem. Istead, the asymptotic behavior of the costraits allows us to desig codes that do ot obey the iput costraits for a particular with a small probability but the requiremet is that small probability goes to zero asymptotically. For the error probability to go to zero, we eed to average out the error due to radomess itroduced by the chael ad due to the radomess i the eergy arrivals. To do this, we propose a scheme that implemets a save-ad-trasmit priciple i that error due to radomess i eergy is averaged out first ad the the chael codig is performed. I additio, it is possible to maitai error-free commuicatio eve if some code symbols caot be put to the chael correctly. Therefore, we also ivestigate achievable rates usig a best-effort-trasmit scheme, which ca iterestigly approach ǫ eighborhood of the capacity for ay ǫ > 0. IV. THE SAVE-AND-TRANSMIT SCHEME We ow preset a achievable scheme. Let h() o() such that lim h() =. Here, o() deotes the class y() of fuctios y() such that lim = 0. Cosider the sequece of codes with code legth ad rate R such that the first h() symbols of each codeword is zero ad the remaiig h() codewords are chose as idepedet radom variables from the (capacity achievig) Gaussia distributio with variace P where P is the average recharge rate. That is, for k =,,...,h(), we have, the ecodig fuctio, fk (m) = 0 for all m {,,..., R }. For k = h() +,...,, fk (m) comes from realizatios of a Gaussia distributed radom variable of variace P for all m {,,..., R }. As grows large, by strog law of large umbers, at time idex h(), about h()p amout of eergy is collected i the battery with very high probability. Sice o eergy is cosumed up to this time, the codebook is feasible util time h(). After time h(), the probability that a code symbol requires more eergy tha the eergy available i the battery goes to zero ad eergy cosumed is accouted for the eergy arrived. I the ext h() chael uses, h()p uits of eergy is collected i the battery ad aother h()p is cosumed for data trasmissio. We repeat this procedure for a total of /h() times, where i each iterval of h() symbols, we collect about h()p uits of eergy ad use about h()p uits of eergy collected i the previous block of h() symbols. The first collected amout guaratees that the codewords are

4 ..... P [ h()]p h()p eergy arrival eergy expediture X h()+ X h()+ X X h() Fig. 3. The codewords i the save-ad-trasmit scheme. First h() code symbols are idetically zero for all codewords. Remaiig h() code symbols are selected as i.i.d. Gaussia distributed. Evolutio of a sample eergy arrival ad eergy expediture for a particular codeword is illustrated. Collected eergy i the first h() chael uses makes it asymptotically impossible to deem ay codeword ifeasible. always feasible (see Fig. 3). More specifically, as the legth of the codewords gets large, by strog law of large umbers, we have u i=h() X i u E i almost surely for all u > h(). For a precise proof of these argumets, we eed to make use of Marcikiewicz-Zygmud type strog laws for sums of i.i.d. radom variables. Please see [0] for a complete proof. The achievable rate for this scheme is I(X ;Y ) = j=h() I(X j ;Y j ) = h() log ( + P) log( + P) (5) I other words, commuicatio ca start with h() amout of delay ad the capacity of a correspodig average power costraied AWGN chael ca be achieved. Sice, this scheme is asymptotically feasible, this proves the achievability part ad completes the proof of Theorem. It should be emphasized that the above achievable scheme is obtaied without usig causal or ocausal iformatio of eergy arrivals. For ay realizatio of E i, we itroduce a reasoable delay h() to save eergy ad afterwards trasmit with average power P. This scheme obeys the iput costrait with probability ad achieves the upper boud i (3). The ability to collect eergy i the battery allows desiger to apply a save-ad-trasmit strategy so that the ucertaity i the eergy arrivals is elimiated first ad the the ucertaity i the chael is dealt with by meas of appropriate chael codig. V. THE BEST-EFFORT-TRANSMIT SCHEME Let X = (X,X,...,X ) be a codeword of legth where X i is the code symbol to be set i chael use i ad the codebook be C. The codebook that two parties agree upo is determied by geeratig idepedet Gaussia distributed radom variables with mea zero ad variace P ǫ for each code symbol. Let S(i) be the battery eergy just before the ith chael use starts. I the best-effort-trasmit scheme, the code symbol X i ca be put to the chael if S(i) X i. Otherwise, trasmitter puts a code symbol 0 to the chael as battery does ot have sufficiet eergy to sed symbol X i. Hece, the iput to the chael is X i (S(i) Xi ) ad therefore there is a possible mismatch. Yet, we will show i the followig that the mismatch does ot affect the correct decodig of the message at the receiver. The battery eergy is updated as follows S(i + ) = S(i) + E i X i (S(i) X i ) (6) The eergy updates i (6) are aalogous to the queue updates i classical slotted systems [] where the otio of a slot is replaced with chael use. Therefore, battery acts like a eergy queue. Ulike i the classical systems with packet queues, our aim is to keep the eergy queue ustable so that the availability of the eergy to sed a code symbol is asymptotically guarateed. The key to the updates i (6) is to put 0 symbol o the chael as it guaratees that ifeasible code symbols are observed oly fiitely may times, which is stated i the followig theorem, ad detailed proof is agai skipped for brevity ad ca be foud i [0]. Theorem Let X be a codeword such that {X i } is a i.i.d. real radom sequece with mea zero ad variace P ǫ for some ǫ > 0 sufficietly small. Let E i be the eergy arrivals ad S(i) be the battery eergy which is updated as i (6). The, oly fiitely may code symbols are ifeasible. log(+(p ǫ)) It is well kow that we ca pack about codewords with block legth, each havig average power less tha or equal to P ǫ, i the R space such that, wheever oe of them is set by the trasmitter, a joit typicality decoder [9] which checks whether a codeword is joitly typical with the received sigal vector, ca recostruct the codeword at the receiver with probability of error approachig zero. I the besteffort-trasmit scheme, the received sigal vector will still be joitly typical with the set codeword, sice the umber of ifeasible code symbols is oly fiitely may due to Theorem. Hece, the same decoder ca recover the codeword if the chael iput is X i (S(i) Xi ) i each chael use. Therefore, the code rate R = log ( + (P ǫ)) is achievable, ad hece by cotiuity of log(.), R < log ( + P) ca be achieved. VI. OPTIMAL POWER CONTROL IN A LARGE TIME SCALE We have see that classical AWGN capacity with average power costrait ca be achieved with o() delay for fuelig the battery with eergy if the recharge process is i.i.d. However, the recharge process ca deviate from its i.i.d. characteristic i a large time scale. I particular, the mea value of the recharge process may vary after a log time. I the classical example of sesor odes fueled with solar power, mea recharge rate chages depedig o the time of the day. As a example, the mea recharge rate may vary i oe-hour slots ad the sesor may be o for twelve hours a day, i which case, a careful maagemet of eergy expediture i each slot

5 will be required to optimize the average performace durig the day. We geeralize the system model for L large time slots (see Fig. 4). We assume that the duratio of each slot is T s (large eough). For each slot i =,,...,L, average recharge rate is P i (i) ad P tr (i) uits of power is allocated for data trasmissio. I slot i, P i (i)t s uits of eergy eters the battery ad P tr (i)t s uits of eergy is spet for commuicatio. If P i (i) > P tr (i), (P i (i) P tr (i))t s uits of eergy is saved, or otherwise (P tr (i) P i (i))t s uits of eergy is depleted from the battery. Assumig zero iitial eergy i the battery ad large eough battery capacity, every uit of icomig eergy is saved i the battery. The causality of eergy arrivals requires P tr (i) P i (i), l =,...,L (7) Aroud log ( + P tr(i)) T s bits of data are set i slot i. Before the commuicatio starts, suppose the desiger kows the mea recharge rates P i (i) for all i, calculates P tr (i) ad adjusts the average power of codewords i slot i to P tr (i) durig trasmissio. We allocate trasmit power to each slot subject to causality costrait so that average throughput i L slots is optimized: max s.t. L L log ( + P tr(i)) P tr (i) P i (i), l =,...,L (8) We will deote the solutio of the above optimizatio problem as P tr = [P tr(),p tr(),...,p tr(l)]. Note that the slot duratio T s does ot appear i the optimizatio problem. The rest of this sectio is devoted to characterizig P tr. After developig ecessary cocepts, we preset a algorithm to fid P tr ad illustrate it for several cases. A. Solutio of the Optimizatio Problem I order to uderstad how the optimal solutio may look like, we will first pose the simplest versio of the problem. Suppose there is clt S amout of eergy available i the battery where c > 0 is a costat ad the recharge process is zero. The, the optimal power allocatio strategy P tr is L the oe that maximizes L log ( + P tr(i)) subject to L P tr(i) cl. By Jese s iequality [9], optimal strategy is Ptr(i) = c. That is, if P tr (i) P tr (j) for i j, the we ca always achieve higher average throughput by settig P tr(i) = P tr(j) = Ptr(i)+Ptr(j). Returig to the origial problem, above observatio traslates to a geeral optimality criterio for the problem, stated i the followig theorem. Please see [0] for a proof of this theorem. Chagig the average power of codewords requires usig differet codebooks i each slot. However, scalig a commo codebook by slot power P tr(i) works as well. This ca also be iterpreted as a codebook with dyamic power allocatio [] i slow time variatio. E(0) slot E() slot E() E(L ) slot L E(L) P i () P i () P i (L) Ts Ts (L )Ts LTs P tr () P tr () P tr (L) Fig. 4. L large time slots. I each slot, sufficietly large time passes to achieve AWGN capacity with average power costraied to allocated power i that slot. Theorem 3 Let P tr = [P tr (),...,P tr (L)] be ay power allocatio such that l P tr(i) l P i(i), for l =,...,L. Assume P tr P tr be aother power allocatio such that l P tr(i) l P i(i), l =,...,L ad for some e,s {,,..,L} with e < s { P tr(i) c, i {e,e +,...,s} = P tr (i), i / {e,e +,...,s}. The, L log ( + P tr(i)) > L log ( + P tr(i)). This theorem proposes a method to optimize the power allocatio. It should be performed such that variatio i power from slot to slot is avoided as much as possible ad the eergy should be cosumed i as smooth a way as possible. We will ow make this more precise by describig the method of fidig the solutio P tr. It is iferred that P tr must take a costat value (ot ecessarily the same value) i each slot. I additio, we ca partitio the whole iterval ito disjoit itervals over which P tr(i) is costat as follows: P tr(i) = eergy recharged i [ kt s, k+ T s ] ( k+ k )T s (9) for all i i { k +,..., k+ } for some subset { k } of {,,..,L} with k+ > k. Thus, the problem ca also be cast as fidig the right subset { k } that optimizes the average throughput. Note that 0 ad L are always i this subset. Hece, = 0 ad the elemet with last order is L. Defie the cumulative eergy arrival rate as e(i) = i j= P i(j) for all i { : L} ad reset e(0) = 0. I each poit, we have to use the kowledge of the amout of eergy that is goig to arrive i the remaiig time iterval ad determie feasible costat power strategies ad fid the optimal oe amog them. We have = 0 ad ext i are determied for optimal strategy as follows [0]: e(k) e( i i = arg mi ) k { i +:L} k i The, the optimal power allocatio is as follows: (0) Ptr(i) = e( i ) e( i ) i, for all i { i : i } () i A illustratio of the operatio of the algorithm is preseted i Fig. 5. The operatio of the algorithm has the followig geometric structure. At each poit i, lies are draw from

6 P i (i) 3 4 slot i Cumulative: i Pi(j) selected lies 3 4 slot i Avg. Throughput (bits / ch. use) Optimal power maagemet No power maagemet Upper boud o the avg. throughput Fig. 5. Operatio of the optimal power allocatio algorithm. O the left figure, the power impulses are show ad o the right figure, cumulative eergy arrival rate is show ad costructio of optimal power allocatio as miimum feasible piecewise liear curve that is uder the cumulative arrivals is illustrated. Ptr (i) is the slope of the selected lie i slot i. cumulative eergy poit (i, e(i)) to future poits (k, e(k)) for all k > i ad the oe with miimum slope is chose, say k. That slope is the allocated power for all slots betwee i ad k. B. A Numerical Study of the Algorithm The optimal power maagemet algorithm takes the arrival power vector [P i (),...,P i (L)] ad outputs the trasmit power vector [P tr(),...,p tr(l)]. We let the arrival rates of eergy i all slots, P i (i), follow a i.i.d. expoetial distributio. A bechmark algorithm is simply o power maagemet algorithm, i.e., P tr (i) = P i (i). I this simple scheme, the eergy arrival rate i each slot is take as the commuicatio power i that slot. This scheme yields a average throughput T lb = L L log ( + P i(i)) () which is a lower boud. However, if the desiger has the iformatio of arrival rates i future slots, the optimal power maagemet algorithm ca improve the average throughput. It is clear that a upper boud for the average throughput is ( ) T ub = log + L P i (i) (3) L The compariso of performaces of optimal power maagemet with the upper boud T ub ad the lower boud T lb (o power maagemet) is give i Fig. 6 for a L = 0 slot system. We observe that as the variace of the arrival rates icreases, the advatage of optimal power maagemet becomes more apparet with respect to o power maagemet. Because, the power is more peaky as the variace is icreased yet for better throughput, trasmitter should ot ride the peaks but rather save the eergy for future use. Aother observatio is that the differece betwee upper boud ad average throughput with optimal power maagemet also icreases as the stadard deviatio of the arrival rate is icreased. Hece, the causality costrait becomes more restrictive as the variatio i the arrival rate is icreased Mea/Std. Deviatio of Arrival Rate Fig. 6. Average throughput i L = 0 slots is plotted for differet values of mea/stadard deviatio of the arrival rate. As the variatio i the arrival rate icreases, optimal power maagemet algorithm yields higher advatage over o power maagemet. VII. CONCLUSION We studied commuicatio i a AWGN chael uder radom eergy arrivals usig a iformatio-theoretic framework. We showed that the capacity of AWGN uder eergy harvestig is the same as the capacity of the AWGN chael, where the average power is costraied to the average recharge rate. This upper boud ca be achieved by save-ad-trasmit ad best-effort-trasmit schemes. Next, we addressed time varyig recharge rates i large time scales. We obtaied a algorithm to fid the optimal power maagemet for maximum average throughput. We illustrated its operatio ad provided a umerical study to visualize the advatage of the algorithm. REFERENCES [] J. Lei, R. Yates, ad L. Greestei, A geeric framework for optimizig sigle-hop trasmissio policy of repleishable sesors, IEEE Trasactios o Wireless Commuicatios, vol. 8, pp , Feb [] M. Gatziaas, L. Georgiadis, ad L. Tassiulas, Cotrol of wireless etworks with rechargeable batteries, IEEE Trasactios o Wireless Commuicatios, vol. 9, pp , February 00. [3] V. Sharma, U. Mukherji, V. Joseph, ad S. Gupta, Optimal eergy maagemet policies for eergy harvestig sesor odes, IEEE Trasactios o Wireless Commuicatios, vol. 9, pp , April 00. [4] V. Sharma, U. Mukherji, ad V. Joseph, Efficiet eergy maagemet policies for etworks with eergy harvestig sesor odes, i Proc. Allerto Cof. Commu. Cotrol ad Comput., October 008. [5] J. G. Smith, The iformatio capacity of amplitude ad variacecostraied scalar Gaussia chaels, Iformatio ad Cotrol, vol. 8, pp. 03 9, 97. [6] J. G. Smith, O the iformatio capacity of peak ad average power costraied Gaussia chaels. PhD thesis, Uiv. Califoria, Berkeley, CA, USA, 969. [7] S. Shamai ad I. Bar-David, The capacity of average ad peakpower-limited quadrature Gaussia chaels, IEEE Trasactios o Iformatio Theory, vol. 4, pp , July 995. [8] S. Verdu ad T. S. Ha, A geeral formula for chael capacity, IEEE Trasactios o Iformatio Theory, vol. 40, pp , July 994. [9] T. M. Cover ad J. Thomas, Elemets of Iformatio Theory. New York: Joh Wiley ad Sos Ic., 006. [0] O. Ozel ad S. Ulukus, Achievig AWGN capacity uder stochastic eergy harvestig, to be submitted. [] G. Caire ad S. S. (Shitz), O the capacity of some chaels with chael state iformatio, IEEE Trasactios o Iformatio Theory, vol. 45, pp , September 999.