On the Capacity of Energy Harvesting Communication Link Mehdi Ashraphijuo, Vaneet Aggarwal, Senior Member, IEEE, and Xiaodong Wang, Fellow, IEEE

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1 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 33, NO., DECEMBER On the Caacity of Energy Harvesting Communication Lin Mehdi Ashrahijuo, Vaneet Aggarwal, Senior Member, IEEE, and Xiaodong Wang, Fellow, IEEE Abstract We consider an energy harvesting oint-to-oint communication system where the transmitter is owered by an energy arrival rocess and is equied with a battery of finite caacity B max, which could be used for saving energy for future use. We assume a discrete i.i.d. energy arrival rocess where at each time ste, energy of amount A i is harvested with robability i i {,,...,K} indeendent of the other time stes. We rovide uer and lower bounds on the caacity of this channel. These bounds are shown to be within a constant ga for K 3for all arameters, and for K > 3 when the battery caacity B max is small or large enough, where this constant does not deend on any energy or battery arameters. Index Terms Energy harvesting, channel caacity, achievability, uer bound, energy arrival rocess. I. INTRODUCTION W IRELESS communication networs that are comosed of devices that can harvest energy from nature, reresent the green future of wireless systems. The simlest model that catures the communication scenario is a discrete-time AWGN channel. In most communication systems, the transmission ower is a major cost [] [4]. This cost can be artially alleviated by using energy harvesting devices [5] [8]. For transmitters that are owered by energy harvesters, energy is tyically relenished by the energy harvester and exended for communications or other rocesses; any unused energy is then stored in an energy storage, such as a rechargeable battery [9], [0]. The i.i.d. arrival rocess of the incoming energy in energy harvesting is a good starting oint for general results and is a common assumtion in literature [5] [7], [0], []. However, unlie conventional communications where devices are only subject to a ower constraint or a total energy constraint, transmitters with energy harvesting caabilities are subject to additional energy harvesting constraints [] [4]. It is noteworthy to mention other directions in the literature on characterizing the energy harvesting communication system caacity such as Manuscrit received March 8, 05; revised June 4, 05; acceted Setember 3, 05. Date of ublication Setember 5, 05; date of current version November 6, 05. The material in this aer was resented in art at the Annual Allerton Conference on Communication, Control, and Comuting, Chamaign, IL, USA, October 05. This wor was suorted in art by the U.S. National Science Foundation under grant CCF-565. M. Ashrahijuo is with the Electrical Engineering Deartment, Columbia University, New Yor, NY 007 USA mehdi@ee.columbia.edu. V. Aggarwal is with the Industrial Engineering Deartment, Purdue University, West Lafayette, IN USA vaneet@urdue.edu. X. Wang is with the Electrical Engineering Deartment, Columbia University, New Yor, NY 007 USA wangx@ee.columbia.edu. Color versions of one or more of the figures in this aer are available online at htt://ieeexlore.ieee.org. Digital Object Identifier 0.09/JSAC acet scheduling otimality [5], [6] and throughut analysis with unreliable energy sources [7], [8]. Secifically, in every time ste, the transmitter is constrained to use at most the amount of stored energy that is currently available, although more energy may become available in the future slots. Thus, a causality constraint is imosed on the use of the harvested energy. This raises many interesting issues in the design of efficient energy harvesting communication schemes. In this aer, we study the bounds on the caacity of AWGN channel where the transmit node is owered by stochastic energy arrivals, and is equied with the caability of storing and drawing energy from a finite caacity battery. The caacity of a single-user channel with infinite battery caacity is obtained in [9], where it is shown that the caacity equals to that of a classical AWGN channel with an average ower constraint equal to the average energy harvesting rate. Desite the recent significant efforts to characterize the caacity of the energy harvesting channel in the finite battery case [0] [], there is still a lac of comlete understanding. For examle, [] rovides a formulation of the caacity and derives a lower bound on the caacity that is only numerically comutable. However, it is difficult to obtain useful insights from numerical evaluations. Even in the case of no battery, where [3] rovides an exact single-letter characterization of the caacity in terms of an otimization roblem, the corresonding otimization is difficult to solve and requires numerical evaluations. The authors of [4] investigated the case of constant inut energy with a limited battery. Recently, the aroximate caacity of a single-user energy harvesting system with discrete energy arrivals has been characterized in [8], where it is assumed that all the incoming energy is stored in the battery and the transmission energy is taen from the battery. We note that [8] considers a similar energy harvesting model with the difference that the incoming energy cannot be used directly when it arrives at the transmitter and consequently if there is not enough sace in the battery it will be wasted. In contrast, we assume that the incoming energy can be used for transmission and consequently increases the flexibility of the transmitter, which rovides the ossibility of achieving better rates. The results in [8] have been further extended to fading channels in [5], [6]. We assume that the energy arrival is an i.i.d. discrete random rocess that taes K values of A,...,A K, where energy A i arrives with robability i. The incoming energy in art is used for transmission and in art is stored into the battery, which has a caacity B max. We rovide uer and lower bounds on the caacity of such channel when the receiver does not IEEE. Personal use is ermitted, but reublication/redistribution requires IEEE ermission. See htt:// for more information.

2 67 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 33, NO., DECEMBER 05 now the energy arrival rocess. These bounds are within a constant ga for K 3.04 bits for K,.884 bits for K and 4.46 bits for K 3, where the constant does not deend on any energy or battery arameters. For K > 3, the ga between the bounds is shown to be a constant that is indeendent of all arameters in the cases of small enough B max A A and large enough B max A K K i i A i battery sizes. Our aroximate caacity characterizations rovide imortant insights on the otimal design of energy harvesting communication systems. For lower bound, we introduce multile strategies and choose the best for each set of system model arameters which maes our results different from that in [8] where the lower bound is based on a single strategy. For each strategy, we derive a unique energy allocation olicy that is time invariant and the consumtion of each arriving energy acage is decreasing over time with a geometric arameter across different eochs. The roosed uer bound accounts for the flexibility of incoming harvested energy, and thus needs to decide how much incoming energy to store in battery, and how much to utilize for transmission in each time instant which maes our results more comlex than that in [8] where all the incoming energy can only be stored in the battery. The remainder of this aer is organized as follows. Section II introduces the model for a oint-to-oint communication system, equied with an energy harvesting device. Sections III and IV give the results on the uer and lower bounds of the caacity, resectively. These bounds are shown to be within a constant ga in several cases in Section V. Finally, Section VI concludes the aer. II. SYSTEM MODEL We consider a oint-to-oint channel with a single transmitter, equied with an energy harvesting device that has a battery with a caacity of B max. If the battery is not full, the harvested energy is in art stored in the battery and in art used for transmission; if the battery is full, the transmitter can directly use all the harvested energy. In both cases, some additional amount of energy that is stored in the battery can also be used for transmission. Let X t denote the scalar real inut to the channel at time t. We consider a discrete-time AWGN channel, where the outut of the channel is given by Y t X t N t, where N t N0, is the additive white Gaussian noise. At each time t, the system harvests E t units of energy that is causally nown at the transmitter i.e., at time t the transmitter nows E t, E t,...butis not nown at the receiver. Let B t be the available energy in the battery at time t. We assume that at each time, the system first harvests energy and then transmits the signal X t. The square of X t is constrained by the available energy B t lus the harvested energy E t, i.e., X t B t E t. And the available battery energy B t is udated as B t min{b max, B t E t X t }. Note that in order to not waste the harvested energy, we should choose X t such that B t E t X t B max, which leads to the lower bound on X t, i.e., B t E t B max X t. 3 Here, we consider the case that the harvested energy E t is a K -level i.i.d. rocess as E t A with robability,,...,k, 4 where 0 A <...< A K, K and,..., K > 0. Definition : The encoding functions f t, t,, n and the decoding function g are defined as f t : M E t X, t,, n, 5 g : Y n M, 6 where X Y R, E {A,, A K } and M {,, M} is the set of messages to be transmitted. To transmit message w M, attimet,, n, the transmitter sends X t f t w, {E i } t i0. The battery state B t is a deterministic function of {X i } t i0, {E i} t i0, therefore also of w, {E i} t i0.the functions f t must satisfy the energy constraints and 3: B t w, {Ei } t i0 Et B max f t w, {Ei } t i0 Bt w, {Ei } t i0 Et. 7 The receiver estimates ŵ g {Y i } n i0. The robability of error is P n e M M P ŵ w w was transmitted. 8 w The rate of an M, n code is M n. We say rate R is achievable if for every δ>0 there exists, for all sufficiently large n, anm, n code with rate M n > R δ, and error P e n 0. The caacity C of the above system with arameters A,...,A K,,..., K and B max is defined as the suremum of all achievable rates. Remar : For the secial case of B max, the caacity of the above system has been characterized in [9]. It is shown that in the otimal transmission scheme, nothing is transmitted for the first few time slots so that enough energy is accumulated. This is followed by transmission using the average harvested energy in every time ste. Hence, the caacity is characterized by the average energy arrival rate. [3] also characterized the caacity with infinite buffer size using a different aroach. Remar : In [8], for the Bernoulli energy arrival, i.e., K, A 0, the aroximate caacity is characterized under a different battery model where the transmission energy can only be taen from the battery and thus even if A > B max,theextra energy, A B max, cannot be utilized. Remar 3: In the remainder of the aer, we assume K > since for K i.e., constant inut energy, A is an uer bound on the caacity since A is the average ower. Further, a lower bound on the caacity is A.04,

3 ASHRAPHIJUO et al.: ON THE CAPACITY OF ENERGY HARVESTING COMMUNICATION LINK 673 which can be achieved by the amlitude-constrained AWGN channel [8, Lemma ], leading to a maximal ga of.04 bits between the uer and lower bounds. The authors of [4] investigated the case of constant inut energy with a finite-caacity battery and obtained better bounds on caacity, where the lower bound and uer bound are quite close. III. UPPER BOUND An uer bound on the caacity of the discrete-time AWGN channel with an energy harvesting transmitter is given as follows. Theorem : For K, 0 A <...<A K, K,,..., K > 0 and B max 0, the caacity is uer bounded by C ub,k which is the solution to the following otimization roblem: C ub,k max S K z,...,z K z,...,z K subject to z i min{b max, A i }, i,...,k, K i z i 0, 9 i where S K z,...,z K K i i A i z i Ki A i z i. Proof: The claimed uer bound holds intuitively, because at times of energy arrival A i, i >, if the average energy that is ut in the battery is z i, then the caacity in these time slots is uer bounded by A i z i. The energy z i taen out from the incoming energy and stored in the battery can be utilized in the time slots with energy arrival of A. And consequently, an average ower constraint forms the uer bound. Define gt Xt as the ower allocation strategy that maximizes the long-term average transmission rate over the class of feasible online olicies and and also define g i t gt Et A i. Then, the caacity is uer bounded as C lim inf N E a lim inf N E [ N N t [ K N i [ b lim inf E K N c K i i gt N t ] ] g it ] N i N N g i t, N i i t E [ g i t ], 0 where N i is the number of occurrences of E t A i for t {,...,N}, and as N, N i N i. In the above, a follows by searating the incoming energies by their levels, b follows from the concavity of the function and c follows from the law of large numbers. Suose that an average of x i energy is stored into the battery and y i i energy is drawn from the battery when the energy arrival is A i for i >, then Eg i t i A i x i y i for i > and Eg t A K i i x i y i.asn,usingthe law of large numbers, it can be concluded that the caacity is uer bounded by C max 0x i,y i,x i y i B max, 0 K i i x i y i { K i i A i x i y i } Ki i x i y i A. By defining z i x i y i we get the result stated in the theorem. The following two results follow from Theorem. Corollary : For the case of K, the uer bound can be reduced to { C ub, max 0xB max A x A x }. The otimal value of x in is x min{b max, A A }. The otimum solution to 9 for general K is given exlicitely by the following theorem. The roof is given in Aendix A. Theorem : The exlicit uer bound is given as follows for different ranges of B max. A For B max A A, K l C ub,k A l B max l A B Kj max j. 3 s B For A s r r s i i A i B max A s sr r si i A i, s K, K l Kis C ub,k A i l B max ls si i A i B Kjs max j. 4 K ms m C For A K K i i A i B max, C ub,k K i A i. 5 i IV. THE LOWER BOUND The insights develoed in derivation of the uer bound can be used to give an achievability scheme, which achieves the rate given in the following theorem.

4 674 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 33, NO., DECEMBER 05 Fig.. Illustration of the energy allocation olicy for C 3 lb,4 x with q q 3 and r q 3. Note that the energy control olicy is reset each time a acet A 3 or A 4 arrives. Theorem 3: For K, 0 A <...<A K, K,,..., K > 0, and B max 0 define q K i i. Then, therateof max max,...,k 0x min{b max,a } C lb,k x can be achieved, where K Clb,K x j A j x j h A h xq q.884 K K >. 6 The rest of this section roves this theorem. We show the achievability of Clb,K x, for any {,...,K } and x [0, min{b max, A }]. Let the time slots of two consecutive energy arrivals of at least A be T and T, i.e., E T, E T A and E t < A for T < t < T. Then, the energy of E T x is used at time T and the energy of x goes into the battery. At any time t between T and T, the energy of q B t is extracted from the battery, and thus the energy of E t q B t can be used for transmission, where E t < A, and the residual energy at time t in the battery is B t q B t. Thus, the energy usage from the battery at time t {T,...,T } is q q tt x. We note that this is a geometric random variable with arameter q, and thus has the mean q. We ignore the battery energy residue at T, and consider the next interval starting at T to find the energy usage after T. This olicy can be evaluated to give the desired bound. The energy utilization strategy g t roosed above is of the form g t φ j, where j t max{τ : E τ A i, i, τ t}, i.e., the strategy is invariant across time and the allocated energy deends on the number of time stes since the last energy arrival A i, i. The random variable φ j is defined as q q j x A, w.. q, φ j., q q j x A, w.. q, for all j, 7 and A x, w.. q, φ A K x, w.. K q, Fig. deicts an examle of energy allocation olicy for Clb,4 3 x. The idea for the achievability scheme is that if both the transmitter and receiver now at each time arrival what energy acet A j arrives, they can agree on an energy allocation strategy ahead of time. Communication roceeds as follows: At each time ste t,the transmitter sees the realization of the energy rocess E t,let j t max{t t : E t A }, i.e., the number of time stes since the last time battery was recharged with energy acet arrival A j, j. Letφ i denote the amount of energy allocated to transmission, i channel uses after the last time the battery was recharged via acet arrivals A j, j, i 0,,. We concentrate on an energy allocation olicy φ i that is invariant across different eochs the eriod of time between two adjacent acet arrivals A j and A j, j, j. In other words, if energy A j, j arrives at the current channel use, we allocate φ 0 amount of energy for transmission; if energy A j, j arrived in the revious channel use but not the current channel use, then we allocate φ amount of energy for transmission, and so on till the next arrival of energy A j, j. Consider n consecutive time slots of energy arrivals where n is large enough. Denote n i, i 0 as the number of times slots t that their corresonding j t max{t t : E t A } is equal to i.ifn goes to infinity it can be shown that n i, i 0 goes to infinity, as well. The transmitter and the receiver agree on a sequence of M codeboos: C 0, C, C,, CM with large enough M, each codeboo C i consisting of ni R i codewords where R i is the rate of the codeboo and codeboo C i is amlitude-constrained to φ i, i.e., the symbols of each codeword in C i are such that X t φ i if i t max{t t : E t A }. This ensures that the symbol transmitted at the corresonding time will not exceed the energy

5 ASHRAPHIJUO et al.: ON THE CAPACITY OF ENERGY HARVESTING COMMUNICATION LINK 675 constraint φ i. The transmitter chooses a codeword c,i C i, i {0,,, M} to communicate to the receiver. More secifically, in the l th occurrence of i, the transmitter sends the l th symbol of codeword c,i, i.e., uon the arrival of the first energy acet A j, j, the transmitter sends the first symbol of c,0 C 0 ; if there is no energy acet arrival A j, j in the next channel use, it transmits the first symbol of c, C in the next channel use, etc. Once the second energy acet A j, j arrives, the transmitter transmits the second symbol of c,0, then the second symbol of c,,etc.if j > M, the transmitter transmits zero symbol. Communication ends when the transmitter observes the arrival of the n 0 th energy acet of energy A j, j. We assume that communication starts with the arrival of the first energy acet of energy A j, j. We assume that HE t bits is used to communicate the incoming energy level E t to the receiver. The receiver can trac the codeboo used by the transmitter and decode each codeword searately by nowing the energy arrival E t in the transmitter. Let {S l} l L be the inter-arrival times between the lth and l th energy arrivals of E t A, where L is the total number of energy arrivals of E t A between t and t N, i.e., L l S l N < L l S l. Notice that S l s are i.i.d. geometric random variables with arameter q, and thus has the mean q. We can lower bound the rate achieved by g t in terms of these new variables as lim inf N N N t lim inf L a E b q E g t L S l l j0 E[ φ j] L l S l [ S [ j0 E φ j i i E[S ] i PS i j0 j0 ]] E φ j i q q q i E φ j j0 i j q q i E q q j E j0 φ j φ j, 9 where a follows from the law of large numbers and b follows from E[S ] q. Using Eqn. 9 and the discussed energy allocation strategy before it, the rate in Lemma below could be achieved, which will be further lower bounded to get the bound in the statement of the theorem. First, we give the following lemma which will be used in the roof of Lemma. Lemma : [8, Lemma ] Lower bound on amlitudeconstrained AWGN caacity max I X; Y A x:x A Lemma : The caacity C of a system with i.i.d. energy arrival rocess that is only causally nown at the transmitter but not at the receiver is lower bounded by C q q j E φ j j0.04 H,..., K. Proof: We showed that 9 is achievable for an AWGN channel with channel state information i.e., energy arrival rocess at the receiver. For the model in this aer, the ga of.04 is due to amlitude-constrained AWGN given in Lemma, and H,..., K is due to having no channel state information at the receiver since the energy level can be communicated to the receiver in H,..., K bits. The next result further lower bounds the achievable rate in Lemma. The roof is given in Aendix B. Lemma 3: The following inequality holds for all 0 x min{b max, A }: RHS of K j h j A j x q A h q.884 K K >. Using Lemma and Lemma 3, Theorem 3 then follows. V. THE GAP BETWEEN THE BOUNDS In this section, we show that the ga between the uer and lower bounds is bounded by some constant in several cases, where the constant does not deend on any energy or battery arameters. A. Case of K From Theorems and 3, the following corollary follows. Corollary : For K, the uer and lower bounds on the caacity are within a constant ga. More formally, C ub, Clb, x.884 bits 3 for,, A, A and B max 0. Proof: From Theorem 3, we get the lower bound A x.884, 4 C lb, x A x for all 0 x min{b max, A }. The uer bound is given by Corollary, i.e., Eq. that together with 4 gives 3.

6 676 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 33, NO., DECEMBER 05 Fig.. Comarison of the bounds of the roosed model with those in [8] for K. We can further consider a more general case where the usage of the battery is not free. In this model, the cost of storing energy into the battery is denoted through the coefficient 0 < r in. Similarly, the cost of taing energy out the battery is denoted through the coefficient 0 < r out. Then Eqns. and become X t r out B t E t and B t B t r in E t X t r out X t E t, resectively. And the uer bound in Corollary and the constant ga result in Corollary can be generalized as follows. The roof is given in Aendix C. Theorem 4: For the case of K with the battery efficiency arameters r in and r out, we have the following uer bound on the caacity C ub, max 0xB max { A x A r out r in x }. 5 { where x min B max, A A r outr in r out r in }.Wealso have C ub,.884 C C ub,, 6 For K, assuming, A 0 and A 0 5, in Fig. we lot the uer and lower bounds for r in r out and r in r out 0.8, resectively, as well as the bounds in [8] for varying values of B max. The bounds decrease with a decrease in efficiencies r in and r out. Recall that [8] assumes a different system model from the one used in this aer, where the harvested energy cannot be directly used without being stored in the battery first. Since the achievability scheme in [8] is also a feasible strategy for our model, the ga can be imroved by choosing the maximum of the achievable rate in this aer and that in [8]. Fig. 3. Comarison of uer and lower bounds of the roosed model for K 3. B. Case of K 3 The next result gives the ga between the lower and uer bounds for K 3. Theorem 5: For K 3, the ga between the uer and the lower bounds is bounded by 4.46 bits. More formally } C ub,3 max {Clb,3 x, C3 lb,3 x bits 7 for,, 3, A, A, A 3 and B max 0. Proof: We note that for K 3 there is a constant of.884 K K > 3.96 in 6, and we can show that the ga between the uer bound C ub,3 given in Theorem and the maximum of the lower bounds Clb,3 x and Clb,3 3 x 3 given in 6 is at most 0.5 lus the above 3.96 bits. In order to show this ga of 0.5 bit, we consider 5 cases deending on the values of B max, A, A and A 3. The detailed ga evaluation for these five cases is given in Aendix D. For K 3, assuming 3 3, A 0, A 0 3 and A 3 0 6, in Fig. 3 we lot the uer and lower bounds as a function of B max. It is seen that both bounds increase with the battery caacity; the uer bound saturates at 3 i i A i 9.74 bit/s and the lower bound saturates at bit/s. C. Case of K > 3 The roosed achievability scheme can achieve a rate that is within a constant ga to the uer bound of the caacity for general K for arts A and C of Theorem as shown in the following theorem. Theorem 6: The following constant ga results for general K hold: For B max A A,wehave C ub,k C lb,k B max.34 K. 8

7 ASHRAPHIJUO et al.: ON THE CAPACITY OF ENERGY HARVESTING COMMUNICATION LINK 677 For A K K i i A i B max,wehave u C ub,k Clb,K u A u j.34 3 K, 9 j where u arg max i A i Kji j. Proof: Part : Follows from 3 and 6 that C ub,k C lb,k B max.34 K. 30 Part : Using 5 we have C ub,k K i A i i K a i A i KA u A u K j ji K j ju K j ju K, 3 where a follows from the fact that u arg max i A Kji i j. Moreover, using 6 we have u A u C u lb,k u j lu j j j A u Kmu u n n m A j u }{{} Au u Kmu n n m u K l u A l A u j.34 K j }{{} A u A u A u u j j K j ju.34 K. 3 For K 5, assuming, 5 8, A 0, A i 0.5i0 5, i {,...,5}, in Fig. 4, we lot the uer and lower bounds for a range of values of B max.forb max A A 5 0 4, an uer bound is given in Case A of Theorem and its ga to the achievable rate is shown to be within bits in Part of Theorem 6. Also, for B max A K K i i A i , an uer bound is H A z Kl A l z l 3 Kl A l z l. K Kl A l z l 3 A 3 A 3 z 3 Kl l z l 3 Kl A l z l. K 3 A Kl l z l K Kl A l z l 3 K Kl A l z l K A K z K K Kl A l z l a Kl A l z l 3 Kl A l z l. K Kl A l z l 3 Kl A l z l 3 Kl A l z l. K 3 Kl A l z l K Kl A l z l 3 K Kl A l z l K Kl A l z l A Kl [,..., K ] T [,..., K ] 0, 33 l z l

8 678 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 33, NO., DECEMBER 05 We also define μ gi,μ hi, i {,...,K },μ u as the corresonding Lagrangian multiliers of these constraints, resectively. A solution z,...,z K is otimum if it has the following roerties: K S K z,...,z K μ gi g i z,...,z K K i i μ hi h i z,...,z K μ u uz,...,z K, 37 μ u 0,μ gi 0,μ hi 0, for all i,...,k, μ gi g i z,...,z K 0, for all i,...,k, 38 μ hi h i z,...,z K 0, for all i,...,k, μ u uz,...,z K Fig. 4. Comarison of uer and lower bounds of the roosed model for K 5. given in Case C of Theorem and its ga to the achievable rate is shown to be within bits in Part of Theorem 6. In addition, for A A < B max < A K Ki i A i, an uer bound is given in Case B of Theorem and Clb,5 3 min{a j 3 j, B max } is considered as the lower bound. Further, it can be seen that both bounds increase with the battery caacity, and the uer bound saturates at 5 i i A i VI. CONCLUSIONS We have considered an energy-harvesting communication system where a transmitter owered by an exogenous energy arrival rocess, modeled as a discrete random rocess, and equied with a battery of finite caacity, communicates over a discrete-time AWGN channel. We have develoed uer and lower bounds on the caacity of such systems, which are shown to be within a constant ga for K 3, and some cases of K > 3. Extension to fading channels is a future wor. APPENDIX A PROOF OF THEOREM We first show that S K z,...,z K is concave by showing its Hessian matrix is negative-semidefinite. The Hessian is given by 33, as shown at the bottom of the revious age, where a follows from the fact that i < 0, for all i,...,k. A i z i Define the following functions corresonding to the constraints g i z,...,z K z i B max 0, i,...,k, 34 h i z,...,z K z i A i 0, i,...,k, 35 uz,...,z K z... K z K Now we rove the theorem for the given 3 cases: A For B max A A we show the otimality of z l B max, l {,...,K } which by substituting in the uer bound 9 results in 3. We have uz,...,z K B max < 0 and also for all i,...,k h i z,...,z K B max A i A A A i a < 0, 40 where a follows from the fact that A < A < A 3 <...< A K. So, we should have μ hi 0, i {,...,K },μ u 0. Also, g i z,...,z K 0, i {,...,K }. To show the otimality, it is sufficient to show that μ gi 0, for all i,...,k. Using these and 37, we get μ gi i A B max, A i B max for all i,...,k. Moreover, we have μ gi 0, for all i,...,k, since μ gi i A B A max i B max a A i A B max A B max A i B max A A B max A B max A i B max b 0, where a follows from A < A 3 <...<A K and b follows from B max A A. s B For A s r r s i i A i < B max A s sr r si i A i we show the otimality of sm zl A l m A m K ns n B max sl, l {,...,s} and l zm B max, l {s,...,k } which by substituting in

9 ASHRAPHIJUO et al.: ON THE CAPACITY OF ENERGY HARVESTING COMMUNICATION LINK 679 the uer bound 9 results in 4. For all i,...,s we have g i z,...,z K z i B max sm m A m K A i ns n B max sl l B max a < 0, where a follows from A s A s s r r r r s i i A i <...< s i i A i < B max.wealsohave h i z,...,z K z i A i sm m A m K ns n B max sl < 0, 4 l for all i,...,s. It can be also seen that for all i s,...,k h i z,...,z K z i A i B max A i s s A s r i A i A s r i K s i A i < 0. 4 In addition, we have A s K uz,...,z K j z j j r rs B max K s K i js j <0. 43 So, we should have μ g j 0, j {,...,s},μ hi 0, i {,...,K}, and μ u 0. Also, g i z,...,z K 0, i {s,...,k }. It is sufficient to show that μ gi 0, for all i s,...,k. Using these and 37, we get that the first s elements of S K z,...,z are 0 and also μ gi i sm m A m K ns n B max sl l, A i B max for all i s,...,k. Moreover, we have μ gi 0, for all i s,...,k, since 44, shown at the bottom of the age, holds where a follows from A s < A s <...< A K and b follows from B max A sr s r si i A i. C For A K K i i A i B max we show the otimality of zl A l K i i A i, l {,...,K } which by substituting in the uer bound 9 results 5. For this set of zi s, we get S K z,...,z K 0 which shows the otimality. APPENDIX B PROOF OF LEMMA 3 We divide the roof into two arts; K > and K. A. Proof for K > For the case of K >, we divide the roof into three cases. The first case is when q x q A > A, the second one is when q x q A A, and the third one is when q x q A q A q x q A q,for< q < and A.4. Let choice of A is a result of erforming a minimization of a function that will be mentioned in Remar 4. I For the case q x q A q A q x q A q,for< q <. We have K j j A j x q A h q K j hq h j A j x q j A j q A h q, 45 μ gi i sm m A m K ns n B max sl l A i B max sm m A m K ns n B max sl l A i B max sm m A m K ns n B max sl A i B max l A i s l l s m m A m B max sm s l l m A m K ns n B max sl A i B max l a A s s l l s m m A m B max b sm s l l m A m 0, 44 K ns n B max sl A i B max l

10 680 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 33, NO., DECEMBER 05 and j0 q q j E φ j a q h K n n q A nx q q q j j } {{ } j K n q q j j q q j x A h }{{} > q q jx Ah q q jx Ah q j q j q x q A h hq jq q j q x A h q hq q 46 n A nx, 47 where a follows from 7 and 8. Using 45 and 46 we get K j j A j x q A h q h q q j E φ j q j j0 hq q j j j a q j 0.7 j A hq q x A h q jq q j hq j A hq jq q j hq j A hq q A h q q q q x q A h q x q A h q j q j b q j q j A ln hq j A ln hq j.4 ln hq q x q A h A j , 49 where a follows since j jq q j q c q d q q 0.7, and b follows from A.4. To show c, we use the identity j0 jq q j q E[X] q q, where X geometricq, and j0 q q j and d follows from the fact that Gq q q q is a continuous bounded function of q 0,. Furthermore, it is monotonically decreasing and is uer bounded by lim q 0 Gq ln 0.7. Remar 4: The choice of A.4 is made since it minimizes the RHS of 48. In other words, for q { { }} j j, arg min A max 0 A A ln.4, and thus choosing A.4 results in the best bound using 48. II For the case q x q A >.4, we have: j0 q q j E φ j K j A j x q A h q j h a q q x A h A h q q h }{{} jq q j q j q x h q A h jq q j q j q j q x q A q x q A h jq q j q 50

11 ASHRAPHIJUO et al.: ON THE CAPACITY OF ENERGY HARVESTING COMMUNICATION LINK 68 b q q x 0.7 q A q q ln x 0.7 q A , 5 ln.4 where a follows from 46 and b follows from Ste a used in 48. III For the case q x q A.4 we have: K j j A j x q A h q j0 a h h q q j E φ j q A h q q q j j h q q j x A h h h q A h q , 53 where a follows from 46. So, we get K j j A j x q A h q j0 h q q j E φ j.30, 54 which together with H,, K K comletes the roof for K >, i.e., by comaring this to the statement of the lemma and the fact that.884 K H,..., K.30 K H,..., K.30, the result is roven. when x A A, where A.4. The choice of constant.4 is exlained in Remar 5. I For the case x A > A we have: K j j A j x h j0 q A h q q q j E φ j H, a q q x q A q q q x A H, ln x A H, H, ln A H, ln A H, ln A b.5.844, 55 ln A B. Proof for K For the case of K, we divide the roof into two cases. The first case is when x A > A and the second one is where a follows from 50, b is due to the fact that the exression is a continuous bounded function of 0,. Furthermore, it is concave and attains a maximum value of ln A.5 at 0.43.

12 68 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 33, NO., DECEMBER 05 II For the case x A A we have: K j A j x j q A h q h q q j E φ j h j0 a b q A h H, q x A H, H, A H A.844, 56 where a follows from 5 and b follows from the fact that K. Remar 5: If the ga result in the two cases is et as function of A rather than substituting the value as in the roof above, ln A.5 from 55 is a strictly decreasing function of A and A from 56 is a strictly increasing function of A. Since the ga is the maximum of two gas, otimizing the maximum of the gas gives A.4 using which we get both the RHS of 55 and the RHS of to be equal. So, for K we get K j j A j x q A h q j0 h q q j E φ j H,.844, 57 which together with H comletes the roof for K, i.e., by comaring this to the statement of the lemma and the fact that , the result follows. APPENDIX C PROOF OF THEOREM 4 We divide the roof of Theorem 4 into two arts, corresonding to the uer and lower bounds, resectively. C. Uer Bound Similar to the roof of Theorem, suose that an average of x energy is stored into the battery and y energy is drawn from the battery when the energy arrival is A, then Eg t A x r out y and Eg t A r out r in x r in y.asn, using the law of large numbers, we see that the caacity is uer bounded by { C max 0x B max, A x r out y 0 r out r in x r y in A r out r in x r } in y. 58 Now, we relace the variable x by x x min{x, y r out r in } and also the variable y by y y r out min{x, y r out r in }. We see that A x r outy A r out r in x y r in A x r out y A r out r in x y r in r out r in min{x, y r out r in } r out r in A x r out y A r out r in x y. 59 r in Thus, C ub, with x and y is less than or equal to C ub, with x and y if min{x, y r out r in } > 0. So, for otimal x and y,min{x, y r out r in }0. Since r out r in x r in y 0, we get y 0 and thus the uer bound in 5 follows. D. Lower Bound To show the achievability given in 6, we show that the rate of max x can be achieved, where 0xmin{B max,a } C lb, Clb, x A x A r inr out x Let the time slots of two consecutive energy arrivals of A be T and T, i.e., E T, E T A and E t A for T < t < T. Then, the energy of E T x is used at time T and the energy of r in x goes into the battery is. At any time t between T and T, the energy of B t is extracted from the battery, and thus the energy of E t r out B t can be used for transmission, where E t A, and the residual energy at time t in the battery is B t B t. Thus, the energy usage from the battery at time t {T,...,T } is r in r out tt x. We note that this is a geometric random variable with arameter, and thus has the mean. We ignore the battery energy residue at T, and consider the next interval starting at T to find the energy usage after T.

13 ASHRAPHIJUO et al.: ON THE CAPACITY OF ENERGY HARVESTING COMMUNICATION LINK 683 The energy utilization strategy gt roosed above is of the form gt φj, where j t max{τ : E τ A, τ t}, i.e., the strategy is invariant across time and the allocated energy deends on the number of time stes since the last energy arrival of A. The random variable φj is defined as φj r in r out j x A, for all j, and φ0 A x. 6 The rest of the roof is straightforward and in the same lines as in the case of r in r out as in Section IV, with the only difference that in the roof of Lemma 3 in Aendix VI the choice of two cases change as follows. The first case is when r in r out x A >.4 and the second one is when r in r out x A.4. and Clb,3 3 A 3 A 3 A 3 A 3 A A 3 A 3 A A 3 A 3 A 3.96 A 3 A 3 A So, the maximum of the rates in 64 and 65 is achievable because of B max A 3 A A A and it is seen from 63 that their maximum is within a constant ga of 4.46 bits to the uer bound. APPENDIX D PROOF OF THEOREM 5 Case : B max A A : The uer bound in 3 can be achieved within 3.96 bits using the achievability scheme Clb,3 B max in Theorem 3. Case : B max A 3 A : We have Then using 5 we get A A A 3 3 A A 3 A 3 3 A A 3 A A 3 A 3 A max{a A 3 A,A 3 A 3 A }. 6 C ub,3 A A A 3 3 max{a A 3 A, A 3 A 3 A }. 63 Also, we have the following two achievable rates Clb,3 A A 3 A 3 A A A A A A 3 A A 3.96 A A 3 A 3.96, 64 3 Case 3: A A <B max < A 3 A and A A 3 A A 3 A 3 A : For this case, similar to the revious case, we get C ub,3 A A A 3 3 max{a A 3 A, A 3 A 3 A } A A 3 A, 66 and the rate in 64 is achievable because of B max A A and from 66 the maximum is within a constant ga of 4.46 bits to the uer bound. 4 Case 4: A A <B max < A 3 3 i i A i and A A 3 A <A 3 A 3 A : In this case for the uer bound in Theorem, we have z3 B max. So, C ub,3 max 0x B max { A x 3 A 3 B max A x 3 B max 3 A 3 B max A A A 3 B max }. 67 Now, divide the achievability scheme into two arts: A A 3 B max : The scheme Clb,3 3 x with x min{b max, A 3 }B max gives the caacity within 4.46 bits to the uer bound 67 because Clb,3 3 B max equates shown at the bottom of the next age. B A A > 3 B max : For this set of of arameters, we have > 3. This is because if 3,itresults

14 684 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 33, NO., DECEMBER 05 A A A A 3 > B max which is a contradiction to the initial assumtion of A A B max. Use the strategy Clb,3 3 A A for the achievability. Then, we get C ub,3 Clb,3 3 A A 3.96 A A {}}{ A A A 3 B max A A A A A A A A A 3 A A A A A A 3 68 a. 69 where a follows since 3 which can be easily shown. 5 Case 5: 3 i i A i < B max < A 3 A and A A 3 A <A 3 A 3 A : Tae the uer bound C ub,3 A A 3 A 3. For achievability consider Clb,3 3 A 3 3 i i A i. We could see that 3 x 3 i i A i A A, so we will have 3 A 3 i A i 3.96 C 3 lb,3 i 3 3 i A i i 3 i A i i 3 i A i i A A A A, and we get C ub Clb,3 3 A 3 c 3 i A i 3.96 i A A 3 i A i i A A 3 i A i a b lim A lim A max 3 3 e d { { { i A A 3 A 3 A A 3 A 3 A 70 A 3 A 3 } A A 3 A 3 A A 3 A 3 } A 7 A 3 A 3 3 e } 3 e 3 3 7, 73 where a follows from the fact that the function gx x x is increasing and also A A A 3 i i A i A 3 A 3 and b follows from the fact that 7 has a ositive derivative resect to A, c follows from 3 3 A 3 A and d follows from the fact that 7 has a negative derivative resect to e and e 3 3 in 73. Clb,3 3 B max A 3 B max A 3 B max A 3 B max. }{{ } A 3 B max A 3 B max A A A 3 B max

15 ASHRAPHIJUO et al.: ON THE CAPACITY OF ENERGY HARVESTING COMMUNICATION LINK 685 It remains to show that 73 /. We divide the roof into two arts: A : Let tae m n, m and 3 m n. Then, we get m mn 3. 3 m mnn 3 So, we have 3 3. B > :Lettae m, m n and 3 m n. Then, we get m m n nm n m m nm n nm n m nm n m nm n m m n max n { m m nm n m n m nm n }. 74 We see that otimal n, n m m m. Then, RHS of 74 is equal to m mm mmmm m, where 0 < m mm. It can be seen that m is an increasing function with resect mm to m, and is thus. ACKNOWLEDGMENT We would lie to than the anonymous reviewers for their comments and suggestions which significantly imroved the aer. REFERENCES [] S. Lindsey and C. Raghavendra, PEGASIS: Power-efficient gathering in sensor information systems, in Proc. IEEE Aeros. Conf., 00, vol. 3, [] W. Heinzelman, A. Chandraasan, and H. Balarishnan, Energyefficient communication rotocol for wireless microsensor networs, in Proc. 33rd Annu. Hawaii Int. Conf. Syst. Sci., Jan. 000,. 0. [3] R. 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16 686 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 33, NO., DECEMBER 05 Mehdi Ashrahijuo received the B.Sc. and M.Sc. degrees in electrical engineering from Sharif University of Technoy, Tehran, Iran, in 00 and 0, resectively. He is currently ursuing the Ph.D. degree in electrical engineering from Columbia University, New Yor, NY, USA. His research interests include networ information theory and statistical signal rocessing with alications in wireless communication, digital communications, underwater sensor networs, and alications of the queuing theory in wireless sensor networs, develoing fundamental rinciles for communication networ design, with emhasis on develoing interference management and understanding the role of feedbac and cooeration. He was the reciient of the Qualcomm Innovation Fellowshi in 04. Vaneet Aggarwal S 08 M SM 5 received the B.Tech. degree from the Indian Institute of Technoy, Kanur, India, and the M.A. and Ph.D. degrees from the Princeton University, Princeton, NJ, USA, all in electrical engineering, in 005, 007, and 00, resectively. He is currently an Assistant Professor with the Purdue University, West Lafayette, IN, USA. Prior to this, he was a Senior Member of Technical Staff Research with AT&T Labs-Research, Middletown, NJ, USA, and an Adjunct Assistant Professor with the Columbia University, New Yor, NY, USA. His research interests include alications of information and coding theory to wireless systems and distributed storage systems. He was the reciient of the Princeton University s Porter Ogden Jacobus Honorific Fellowshi in 009. Xiaodong Wang S 98 M 98 SM 04 F 08 received the Ph.D. degree in electrical engineering from the Princeton University, Princeton, NJ, USA. He is a Professor of electrical engineering with Columbia University, New Yor, NY, USA. Among his ublications is a boo entitled Wireless Communication Systems: Advanced Techniques for Signal Recetion Prentice Hall in 003. His research interests include comuting, signal rocessing and communications, and he has ublished extensively in these areas, and wireless communications, statistical signal rocessing, and genomic signal rocessing. He has served as an Associate Editor for the IEEE TRANSACTIONS ON COMMUNICATIONS, the IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, the IEEE TRANSACTIONS ON SIGNAL PROCESSING, and the IEEE TRANSACTIONS ON INFORMATION THEORY. He was the reciient of the 999 NSF CAREER Award, the 00 IEEE Communications Society and Information Theory Society Joint Paer Award, and the 0 IEEE Communication Society Award for Outstanding Paer on New Communication Toics. He is listed as an ISI Highly-cited Author.