A Non-Asymptotic Achievable Rate for the AWGN Energy-Harvesting Channel using Save-and-Transmit

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1 016 IEEE Iteratioal Symposium o Iformatio Theory A No-Asymptotic Achievable Rate for the AWGN Eergy-Harvestig Chael usig Save-ad-Trasmit Silas L. Fog, Vicet Y. F. Ta, ad Jig Yag Departmet of Electrical ad Computer Egieerig, Natioal Uiversity of Sigapore, Sigapore silas_fog, vta@us.edu.sg Departmet of Electrical Egieerig, Uiversity of Arkasas, AR, USA jigyag@uark.edu Abstract This paper ivestigates the iformatio-theoretic limits of the additive white Gaussia oise (AWGN) eergyharvestig (EH) chael i the fiite blocklegth regime. The EH process is characterized by a sequece of i.i.d. radom variables with fiite variaces. We use the save-ad-trasmit strategy proposed by Ozel ad Ulukus (01) together with Shao s o-asymptotic achievability boud to obtai a lower boud o the achievable rate for the AWGN EH chael. The first-order term of the lower boud o the achievable rate is equal to C ad the secod-order (backoff from capacity) term is proportioal to log, deotes the blocklegth ad C deotes the capacity of the EH chael, which is the same as the capacity without the EH costraits. The costat of proportioality of the backoff term is foud ad qualitative iterpretatios are provided. I. INTRODUCTION The eergy-harvestig (EH) chael cosists of oe source equipped with a eergy buffer, ad oe destiatio. For simplicity, i this paper, we assume that the buffer has ifiite capacity. At each discrete time k 1,,..., a radom amout of eergy E k [0, ) arrives at the buffer ad the source trasmits a symbol X k (, ) such that k X l k E l almost surely. (1) I other words, the total harvested eergy k E l must be o smaller tha the eergy of the codeword k X l at every discrete time k for trasmissio to take place successfully. We assume that E l are idepedet ad idetically distributed (i.i.d.) o-egative radom variables, E[E 1 ] = ad E[E1] < +. The destiatio receives Y k = X k + Z k at time slot k for each k N Z k are i.i.d. stadard ormal radom variables. We refer to the above EH chael as the additive white Gaussia oise (AWGN) EH chael. It was show by Ozel ad Ulukus [1] that the capacity of the AWGN EH chael is C 1 log(1 + ), () = E[E 1 ] is the expectatio of the amout of harvested eergy for each eergy arrival. The AWGN EH chael models real-world, practical situatios eergy may ot be fully available at the time of trasmissio ad its uavailability may result i the trasmitter ot beig able to put out the desired codeword. This model is applicable i large-scale sesor etworks each ode is equipped with a EH device that collects a stochastic amout of eergy. See [] for a comprehesive review of recet advaces i EH wireless commuicatios. Although the capacity i () is uchaged vis-à-vis the AWGN chael without the EH costraits, we show i this work that if oe uses the save-ad-trasmit strategy [1] to take the EH costraits ito accout, the there ca potetially be a sigificat backoff from capacity at moderate blocklegths compared to the case whe EH costraits are abset (cf. [3]). A. Mai Cotributio We prove a achievable fiite blocklegth boud for the AWGN EH chael uder the EH costraits i (1) based o the save-ad-trasmit strategy of [1]. Durig the savig phase of the save-ad-trasmit strategy, we save eergy for a certai umber of time slots ad o iformatio is trasmitted. Subsequetly, durig the trasmissio phase, we use the remaiig time slots to sed iformatio. By carefully developig various cocetratio bouds to cotrol the probability that the available eergy is isufficiet to support the trasmitted codeword durig the trasmissio phase (i.e., that k E l < k X l ), we show that the backoff from capacity C at a blocklegth is o larger tha O( 1 log ). Furthermore, by scrutiizig the aalysis of Ozel ad Ulukus [1], oe ca also deduce that the backoff from capacity is o larger tha O( 1/ log ). Thus, our aalysis results i a slightly smaller (tighter) backoff tha what was implied by the authors i [1]. I additio, our aalysis oly requires miimal statistical assumptios o the EH process E l. Ideed, apart from assumig that the process is i.i.d., we oly assume that the secod momet of the EH radom variable E 1 is bouded, i.e., E[E1] <. I previous results such as [1, Lemmas 1 & ], more restrictive assumptios o E 1 were made (e.g., E [ ] e Eγ 1 is bouded for some γ (0, 1)), which may be hard to verify i practice. B. Related Work Iformatio-theoretic characterizatios of EH commuicatio chaels have bee ivestigated recetly. As eergy arrives radomly to the trasmitter, codewords must satisfy the cumulative stochastic eergy costraits. The impact of the stochastic eergy supply o the chael capacity was /16/$ IEEE 455

2 016 IEEE Iteratioal Symposium o Iformatio Theory characterized for the AWGN chael with a i.i.d. EH process i [1] ad with a statioary ergodic EH process i [4]. The aforemetioed studies showed that with a ulimited battery, the capacity of the AWGN chael with stochastic eergy costraits is equal to the capacity of the same chael uder a average power costrait, as log as the average power equals the average recharge rate of the battery. The study of fiite blocklegth fudametal limits i Shao-theoretic problems was udertake by olyaskiy, oor ad Verdú [3]. Such a study is useful as it provides guidelies regardig the required backoff from the asymptotic fudametal limit (capacity) whe oe operates at fiite blocklegths. For a survey, please see [5]. C. aper Outlie This paper is orgaized as follows. The otatio used i this paper is described i the ext subsectio. Sectio II states the formulatio of the AWGN EH chael ad presets our mai theorem. Numerical results are also provided. Sectio III describes the save-ad-trasmit strategy ad presets a proof sketch of our mai theorem. D. Notatio We use X to deote the radom tuple (X 1, X,..., X ). We let p Y X deote the coditioal probability distributio of Y give X, ad let p X,Y = p X p Y X deote the probability distributio of (X, Y ). To make the depedece o the distributio explicit, we let r px g(x) A deote x X p X(x)1g(x) A dx for ay set A R ad ay realvalued fuctio g with domai X. The expectatio ad the variace of g(x) are deoted as E px [g(x)] ad Var px [g(x)] respectively. We let N (z; µ, σ 1 ) exp ( ) πσ (z µ) σ deote the probability desity fuctio of the Gaussia radom variable Z whose mea ad variace are µ ad σ respectively. We will take all logarithms to base e throughout this paper. II. AWGN ENERGY-HARVESTING CHANNEL A. roblem Formulatio ad Mai Result The AWGN EH chael cosists of oe source ad oe destiatio, deoted by s ad d respectively. Node s trasmits iformatio to ode d i time slots as follows. Node s chooses message W ad seds W to ode d, W is uiform o its alphabet. The for each k 1,,...,, ode s trasmits X k R ad ode d receives Y k R i time slot k. Let E 1, E,..., E be i.i.d. radom variables that satisfy re 1 < 0 = 0, E[E 1 ] = ad E[E 1] <. We assume the followig for each k 1,,..., : (i) E k ad (W, E k 1, X k 1, Y k 1 ) are idepedet, i.e., p W,E k,x k 1,Y k 1 = p E k p W,E k 1,X k 1,Y k 1 (ii) Every codeword X trasmitted by s should satisfy r k X l k E l = 1. (3) After time slots, ode d declares Ŵ to be the trasmitted W based o Y. Formally, we defie a code as follows: Defiitio 1: A (, M)-code cosists of a message set W 1,,..., M at ode s W is uiform o W, a sequece of ecodig fuctios f k : W R k + R for each k 1,,..., such that X k = f k (W, E k ) ad (3) holds, ad a decodig fuctio ϕ : R W at ode d which produces Ŵ = ϕ(y ). Defiitio : The AWGN EH chael is characterized by q Y X such that the followig holds for ay (, M)-code: For each k 1,,...,, p W,Ek,X k,y k = p W,E k,x k,y k 1p Y k X k p Yk X k (y k x k ) = q Y X (y k x k ) = N (y k x k ; 0, 1). (4) We call a (, M)-code with average probability of decodig error r Ŵ W o larger tha a (, M, )- code. For ay [0, 1), a rate R is said to be -achievable if there exists a sequece of (, M, )-codes such that 1 lim if log M R ad lim sup. The - capacity for the AWGN EH chael, deoted by C, is defied to be C supr : R is -achievable. The followig theorem is the mai result i this paper. The proof sketch is provided i Sectio III. Theorem 1: Let (0, 1), ad defie a max E pe1 [E1], 1. (5) Suppose 3 is a sufficietly large iteger such that log max E pe1 [E1] /, 1, (6) ad ( log( + ) log( ) ) 4 (7) log e ( + )/. (8) The, there exists a ( + m, M, )-code such that log M log(1 + ) ( + ) ( + 1) (9) m 6 a log / (10) deotes the legth of the iitial savig period before ay trasmissio occurs ad deotes the legth of the actual trasmissio period. I particular, there exists a (, M, )- code with + m such that a log log M 3 log(1 + ) log(1 + ) ( + ) ( + 1) ( ) log(1 + ) 1. (11) Remark 1: Sice 1 E k coverges to E pe1 [E 1 ] = with probability oe by the strog law of large umbers, it follows from the power costrait (3) ad the strog coverse theorem for the AWGN chael [6,7] that the -capacity of 456

3 016 IEEE Iteratioal Symposium o Iformatio Theory the AWGN EH chael is upper bouded by 1 log(1 + ). Therefore, by ormalizig both side of (11) by ad takig the limit, we see that Theorem 1 implies that the -capacity is C = C = 1 log(1 + ), [0, 1). Remark : The ivestigatio of the save-ad-trasmit scheme by Ozel ad Ulukus i [1, Lemma ] implies that log(1 + ) O( (log ) α ) ats is achievable over chael uses for ay α > 1 ad for. Theorem 1 improves the lower boud of the secod-order term because the backoff term improves from O( (log ) α ) to O( log ). Remark 3: It follows from (5) ad (11) that the coefficiet of the secod-order term achieved by the save-ad-trasmit strategy is at least ν 3 log(1 + ) max E pe1 [E 1 ], 1. By ispectig the equatios from (33) to (35) i the proof sketch of the theorem, we ca see that (11) is a direct cosequece of (9) ad the secod-order term i (11) is due to the savig period oly, which meas ν is affected by the legth of the savig period m aloe (but ot ). As icreases, the magitude of ν icreases ad hece a loger savig period is required to guaratee a certai probability of outage, amely that the trasmitted codeword does ot satisfy all the EH costraits. This corroborates the fact that as icreases, the variace of each Gaussia codeword icreases ad hece a loger savig period is required to maitai a certai outage probability. Similarly, as E pe1 [E 1] icreases while is fixed, the variace of the eergy arrival process is larger ad hece a loger savig period is required to maitai a certai outage probability. B. Numerical Results I this sectio, we illustrate achievable rates as a fuctio of per Theorem 1. More specifically, we defie R (EH), log(1 + ) (+) ( +1) (1) + m to be the o-asymptotic rate achievable by save-ad-trasmit accordig to (9) ad (10). We plot R, agaist i Figure 1 for E[E 1 ] = = 3dB ad various values of ad Var[E 1 ] = E[E1], correspodig to the lies idicated as (EH) respectively. I order to demostrate how much the EH costraits (3) degrade the o-asymptotic achievable rates compared to the peak power costrait r X k = 1, (13) i Figure 1 we also plot the optimal trasmissio rate R (No-EH), uder the peak power costrait (13), correspodig to the lies idicated by (No-EH). Due to olyaskiy-oor-verdú [3, Th. 54, Eq. (94)] ad Ta-Tomamichel [8, Th. 1], V( ) R, (No-EH) = C + Φ 1 () + log ( ) 1 + O (14) ( +)(log e) V( ) ( +1) is kow as the Gaussia dispersio fuctio ad Φ 1 is the iverse of the cumulative distributio fuctio for the stadard Gaussia distributio. We igore the fial correctio term i (14) whe we plot R, (No-EH) because it is egligible compared with the first three terms. I Figure 1(a), we see that as icreases, the backoff of both R, (EH) ad R, (No-EH) from the capacity decreases, which is due to the icrease of the magitude of the secod term i (1). I Figure 1(b), we see that as Var[E 1 ] icreases for a fixed E[E 1 ], the backoff from the capacity icreases, which is due to the explaatio i Remark 3 that a loger savig period is required as E[E1] icreases. As we ca see from Figures 1(a) ad 1(b), the performace degradatio due to the EH costraits ad the save-ad-trasmit strategy compared to the peak power costrait is sigificat. III. SAVE-AND-TRANSMIT STRATEGY I this sectio, we ivestigate the save-ad-trasmit scheme proposed i [1, Sec. IV] i the fiite blocklegth regime ad use this achievability scheme to prove Theorem 1. Due to space limitatio, oly a proof sketch of Theorem 1 is provided. The complete proof ca be foud i the log versio of this paper [9]. The proof of Theorem 1 relies o the followig result which is useful for obtaiig a lower boud o the legth of the eergy-savig phase. Lemma 1 ( [9, Sec. III-A]): Suppose X ad E m+ are two radom tuples, each cosistig of i.i.d. radom variables such that X 1 N (x 1 ; 0, ), r pe1 E 1 < 0 = 0, ad E pe1 [E 1 ] = E px1 [X1 ] =. Defie a as i (5). The, k r px p E m+ Xl E l ( ) e e log m log log a for all sufficietly large that satisfies (6). roof sketch of Theorem 1: Fix a (0, 1). Defie a as i (5). Fix a sufficietly large 3 such that (6), (7) ad (8) hold. Defie m as i (10) to be the umber of time slots used for savig eergy. Cosider the radom code that uses the chael m + times as follows: Save-ad-Trasmit Radom Codebook Costructio Let 0 m deote the legth-m zero tuple. Defie p X (x) N (x; 0, ) (15) to be the distributio of a zero-mea Gaussia radom variable X with variace. I additio, let p X be the product distributio of idepedet copies of X. Costruct M i.i.d. radom tuples deoted by X (1), X (),..., X (M) such that X (1) is distributed accordig to p X, M will 457

4 016 IEEE Iteratioal Symposium o Iformatio Theory Achievable Rate 0.3 = (EH) 0. = 0.01 (EH) = 0.05 (EH) = (No-EH) 0.1 = 0.01 (No-EH) = 0.05 (No-EH) Capacity Blocklegth (a) E[E 1 ] = = 3 db ad Var[E 1 ] = 100 Achievable Rate Var[E 1 ] = 0 (EH) Var[E 1 ] = 100 (EH) Var[E 1 ] = 00 (EH) No-EH Capacity Blocklegth (b) E[E 1 ] = = 3 db ad = 0.01 Fig. 1. Achievable rates for the save-ad-trasmit scheme i (1) whe the error probability is varied (left) ad whe the variace of the EH process is varied (right). O the plot o the right (Figure 1(b)) for the No-EH lie, the peak power (cf. (13)) is kept at = 3 db be carefully chose later whe we evaluate the probability of decodig error. Defie X m+ (i) (0 m, X (i)) (16) for each i 1,,..., M ad costruct the radom codebook Xm+ (i) i 1,,..., M. (17) The codebook is revealed to both the ecoder ad the decoder. To facilitate discussio, we let X k (i) ad X k (i) deote the k th symbols i X (i) ad X m+ (i) respectively for each i. Sice the first m symbols of each radom codeword X m+ (i) are zeros by (16), the source will just trasmit 0 with probability oe util time slot m+1 whe the amout of eergy m+1 E k is available for ecodig X m+1 (W ) (16) = X 1 (W ). Ecodig uder the EH Costraits The source s has the kowledge of E k before trasmittig its symbol i time slot k for each k 1,,..., m +. For each i 1,,..., M, recallig that Xk (i) is the k th elemet of Xm+ (i) (16) = (0 m, X (i)), we costruct recursively for k = 1,,..., m + the radom variable ˆX k (i, E k ) X k (i) if ( X k (i)) k E l k 1 ( ˆX l (i, E l )), (18) 0 otherwise. To sed message W which is uiformly distributed o 1,,..., M, ode s trasmits ˆXk (W, E k ) i time slot k for each k 1,,..., m +. Sice ode s trasmits 0 with probability oe i the first m times slots by (16) ad (18), the trasmitted codeword ( ˆX 1 (W, E 1 ), ˆX (W, E ),..., ˆX m+ (W, E m+ )) satisfies the EH costraits (3) by (18). Threshold Decodig Upo receivig Ŷ m+ = ˆX m+ (W, E m+ ) + Z m+ (19) ˆX m+ (W, E m+ ) ( ˆX 1 (W, E 1 ), ˆX (W, E ),..., ˆX m+ (W, E m+ )) (0) deotes the trasmitted tuple specified i (18) ad Z m+ N (z m+ ; 0, 1) by the chael law (cf. (4)), ode d costructs its subtuple deoted by Ȳ by keepig oly the last symbols of Ŷ m+. Recallig that q Y X deotes the chael law ad p X was defied i (15), we defie the joit distributio ad defie p X,Y as p X,Y (x, y ) p X,Y p X q Y X, (1) p X,Y (x k, y k ) for all (x, y ) R. The, ode d declares ϕ(ȳ ) 1,,..., M (with a slight abuse of otatio, we write ϕ(ȳ ) istead of ϕ(ŷ m+ )) to be the trasmitted message ϕ(ȳ ) is the decodig fuctio defied as follows: If there exists a uique idex j such that ( py X log (Ȳ X ) (j)) > log M + 1 p Y (Ȳ 4, () ) the ϕ(ȳ ) is assiged the value j. Otherwise, ϕ(ȳ ) is assiged a radom value uiformly distributed o 1,,..., M. robability of Violatig the EH Costraits Defiig X (W, E m+ ) to be the tuple cotaiig the last symbols of ˆXm+ (W, E m+ ), we obtai from (0), (18) 458

5 016 IEEE Iteratioal Symposium o Iformatio Theory ad (16) that X (W, E r m+ ) k = X (X (W ) l (W )) E l = 1. (3) Combiig Lemma 1, (5) ad (6) ad otig that E m+ ad (W, X (W )) are idepedet by costructio, we obtai k r (X l (W )) E l ( ) e 1 e log m log log a. (4) Combiig (3) ad (4) ad usig the boud o m i (10) ad the boud o log i (8), we ca obtai r X (W, E m+ ) = X (W ) 1 +. (5) Calculatig the robability of Decodig Error Defiig Z to be the tuple cotaiig the last symbols of Z m+ ad recallig X (W, E m+ ) ad Ȳ are the tuples cotaiig the last symbols of ˆX m+ (W, E m+ ) ad Ŷ m+ respectively, we obtai from (19) ad (5) that r Ȳ = X (W ) + Z 1 +, (6) X (W ) ad Z are idepedet. Let E i w deote ( py X log (X (w) + Z X ) (i)) p Y (X (w) + Z log M (7) ) Usig the Shao s boud [10] ad (7) ad recallig the symmetry of the radom codebook, after some calculatios we ca obtai for each w r pw ( M i=1 p X (i))p Z Ew w j 1,,...,M\w E j w c W = w r p X k (1)p Zk E (8) I order to obtai a simple upper boud o the first term i (8), we choose M to be the uique iteger that satisfies log(m + 1) ( )] py X (Y X) E px,y [log p Y (Y ) ( )] ( + ) py X (Y X) Var px,y [log 1 4 p Y (Y ) > log M. (9) Usig (7), (9) ad the Chebyshev s iequality, we have r p X k (1)p E1 1 Zk +. (30) The decodig error probability ca ow be bouded above as r pw,x (W ) p Z p Ȳ W,X (W ), ϕ( Ȳ ) W Z (6) r ϕ(ȳ ) W Ȳ = X (W ) + Z + + r ϕ(x (W ) + Z ) W + + (31) (a) (3) (a) follows from the threshold decodig rule (cf. () ad (7)), (8) ad (30). Usig (10), (17), (9) ad (3), we coclude that the costructed code is a ( + m, M, )-code that satisfies (9), which implies from (1), (15) ad (4) that log M + 1 log(1 + ) ( + ) ( + 1) 1 4 (33) m log(1 + ) ( + ) ( + 1) ( ) 1 4 (34) +m. Equatio (9) the follows from (33). Sice m 0 ad m 6 a log + 1 by (10), it follows that 6 a log a log + 1 m. (35) Equatio (11) the follows from (34) ad (35). Ackowledgemets Silas Fog ad Vicet Ta gratefully ackowledge fiacial support from the Natioal Uiversity of Sigapore (NUS) uder grat R A98-750/133 ad NUS Youg Ivestigator Award R B Jig Yag is supported by Uited States Natioal Sciece Foudatio (NSF) uder grats ECCS ad ECCS REFERENCES [1] O. Ozel ad S. Ulukus, Achievig AWGN capacity uder stochastic eergy harvestig, IEEE Tras. If. Theory, vol. 58, o. 10, pp , 01. [] S. Ulukus, A. Yeer, E. Erkip, O. Simeoe, M. Zorzi,. Grover, ad K. Huag, Eergy harvestig wireless commuicatios: A review of recet advaces, IEEE J. Sel. Areas Commu., vol. 33, o. 3, pp , 015. [3] Y. olyaskiy, H. V. oor, ad S. Verdú, Chael codig rate i the fiite blocklegth regime, IEEE Tras. If. Theory, vol. 56, o. 5, pp , 010. [4] R. Rajesh, V. Sharma, ad. Viswaath, Capacity of Gaussia chaels with eergy harvestig ad processig cost, IEEE Tras. If. Theory, vol. 60, o. 5, pp , 014. [5] V. Y. F. Ta, Asymptotic estimates i iformatio theory with ovaishig error probabilities, Foudatios ad Treds i Commuicatios ad Iformatio Theory, vol. 11, o. 1-, pp , 014. [6] K. Yoshihara, Simple proofs for the strog coverse theorems i some chaels, Kodai Mathematical Joural, vol. 16, o. 4, pp. 13, [7] V. Kostia ad S. Verdú, Chaels with cost costraits: Strog coverse ad dispersio, IEEE Tras. If. Theory, vol. 61, o. 5, pp , 015. [8] V. Y. F. Ta ad M. Tomamichel, The third-order term i the ormal approximatio for the AWGN chael, IEEE Tras. If. Theory, vol. 61, o. 5, pp , 015. [9] S. L. Fog, V. Y. F. Ta, ad J. Yag, No-asymptotic achievable rates for eergy-harvestig chaels usig save-ad-trasmit, submitted to IEEE J. Sel. Areas Commu., Jul. 015, [10] C. E. Shao, Certai results i codig theory for oisy chaels, Iformatio ad Cotrol, vol. 1, pp. 6 5,