Non-Asymptotic Achievable Rates for Gaussian Energy-Harvesting Channels: Best-Effort and Save-and-Transmit

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1 No-Asymptotic Achievable Rates for Gaussia Eergy-Harvestig Chaels: Best-Effort ad Save-ad-Trasmit Silas L. Fog, Jig Yag, ad Ayli Yeer arxiv: v [cs.it] 30 May 08 Abstract A additive white Gaussia oise (AWGN eergy-harvestig (EH chael is cosidered where the trasmitter is equipped with a ifiite-sized battery. The eergy arrival process is modeled as a sequece of idepedet ad idetically distributed (i.i.d. radom variables. The capacity of this chael is kow ad is achievable by the so-called best-effort ad save-ad-trasmit schemes. This paper ivestigates the best-effort scheme i the fiite blocklegth regime, ad establishes the first o-asymptotic achievable rate for it. The first-order term of the oasymptotic achievable rate equals the capacity, ad the secod-order term is proportioal to where is the blocklegth. The proof techique ivolves aalyzig the escape probability of a Markov process. I additio, we use this ew proof techique to aalyze the save-ad-trasmit ad derive a ew o-asymptotic achievable rate ( for it, whose first-order ad secod-order terms achieve the capacity ad the scalig O respectively. For all sufficietly large sigal-to-oise ratios (SNRs, this ew achievable rate outperforms the existig oes. The results are also exteded to the block eergy arrival model where the legth of each eergy block L grows subliearly i. It is ( ( show that best-effort ad save-ad-trasmit achieve the secod-order scaligs O ad O respectively. I. INTRODUCTION maxlog,l I this paper, we cosider commuicatio over a eergy-harvestig (EH chael which has a iput alphabet X, a output alphabet Y ad a ifiite-sized battery that stores eergy harvested from the eviromet. The chael law of the EH chael is characterized by a coditioal distributio q Y X where X X ad Y Y deote the chael iput ad output respectively. A source ode wats to trasmit a message to a destiatio ode through the EH chael. Let c : X [0, be a cost fuctio associated with the EH chael, where c(x represets the log L This work was sposored by NSF uder Grat CCF 4347 ad Grat CNS This paper will be preseted i part at 08 IEEE Iteratioal Symposium o Iformatio Theory. S. L. Fog is with the Departmet of Electrical ad Computer Egieerig, Uiversity of Toroto, Toroto, ON M5S 3G4, Caada ( silas.fog@utoroto.ca. J. Yag ad A. Yeer are with the Departmet of Electrical Egieerig, The Pesylvaia State Uiversity, Uiversity Park, PA 680, USA ( yagjig,yeer@egr.psu.edu. May 3, 08

2 amout of eergy used for trasmittig x X. At each discrete time k,,..., a radom amout of eergy E k arrives at the battery buffer ad the source trasmits a symbol X k X such that k c(x i k i= i= E i almost surely. This implies that the total harvested eergy k i= E i must be o smaller tha the eergy of the codeword k i= c(x i at every discrete time k for trasmissio to take place successfully. The destiatio receives Y k from the chael output i time slot k for each k,,..., where (X k, Y k is distributed accordig to the chael law such that p Yk X k (y k x k = q Y X (y k x k for all (x k, y k X Y. We assume that E i i= are idepedet ad idetically distributed (i.i.d., where E is a o-egative radom variable. To simplify otatio, we write E E if there is o ambiguity. Throughout the paper, we let P E[E], the expected value of E, be the sigal-to-oise ratio (SBR of the chael. This paper focuses o the additive white Gaussia oise (AWGN model where X = Y = R, q Y X (y x π e (y x ad c(x x. Uder the AWGN model, Y k = X k +Z k for each time k where Z k is a stadard ormal radom variable which is idepedet of X k ad the Gaussia oises Z k are idepedet. Referece [] showed that the capacity of this chael is log( + P ad proposed two capacity-achievig schemes, amely save-ad-trasmit ad best-effort. The save-ad-trasmit scheme cosists of a iitial savig phase ad a subsequet trasmissio phase. The trasmitter remais silet i the savig phase so that eergy accumulates i the battery. I the trasmissio phase, the trasmitter seds the symbols of a radom Gaussia codeword with variace P as log as the battery has sufficiet eergy where 0 < P deotes some small offset from P. The best-effort scheme by cotrast does ot have a iitial savig phase. Therefore, iformatio is set right away ad the trasmitter uses every opportuity (as log as it has sufficiet eergy to output the symbols of a radom Gaussia codeword with variace P for some 0 < P. Followig referece [], a umber of o-asymptotic achievable rates for the save-ad-trasmit scheme have bee preseted i refereces [] [4]. By cotrast, o o-asymptotic achievable rate exists for the best-effort scheme except for a special discrete memoryless EH chael with ifiite battery studied i [5] ad a special discrete memoryless EH chael with o battery studied i [6]. A mai goal of this paper is to provide a o-asymptotic achievable rate for best-effort over the AWGN EH chael. Note that the results i this paper cease to hold if the size of the battery is fiite. The chael capacity for the fiite battery case is the subject of recet iterest, see [7] [9]. A. Related Work The chael capacity of the AWGN EH chael was characterized i [], which showed that the capacity of the AWGN chael with a ifiite-sized battery subject to EH costraits is equal to the capacity of the same chael uder a average power costrait where the average power equals the average recharge rate of the battery. I particular, save-ad-trasmit [, Sec. IV] ad best-effort [, Sec. V] were proposed as capacity-achievig strategies. May 3, 08

3 3 For a fixed tolerable error probability ε, Fog et al. [] performed a fiite blocklegth aalysis of save-adtrasmit schemes proposed i [] ad obtaied a o-asymptotic achievable rate for the AWGN EH chael. The first-, secod- ad third-order terms of the o-asymptotic achievable rate preseted i [, Th. ] are equal to the log +ε capacity, c ad c ε respectively where c ad c are some positive costats that do ot deped o ad ε. Subsequetly, referece [3] refied the aalysis i [] ad improved the secod-order term to c3 ε where c 3 is some positive costat that does ot deped o ad ε. Referece [4] further improved the secod-order log(/ε term to sup c 4 + P (P + Φ (ε where c 4 is some positive costat that does ot deped o ad ε. ε 0,ε 0, ε +ε =ε B. Mai Cotributios We use O(, Θ(, ω( ad o( to deote stadard asymptotic Bachma-Ladau otatios except our covetio that they must be o-egative. I the first part of this paper, we propose a best-effort scheme for the AWGN EH chael ad derive a o-asymptotic achievable rate for the best-effort scheme. The derivatio ivolves ( carefully desigig the trasmitted power to be P O so that we ca effectively boud the umber of mismatched positios betwee the desired trasmitted codeword ad the actual trasmitted codeword subject to a fixed blocklegth. I the secod part of this paper, we propose a save-ad-trasmit scheme with a similar trasmitted power P O ( log / ad obtai a ew o-asymptotic achievable rate which outperforms the best existig oe for all sufficietly large SNRs P = E[X] > 0. The aforemetioed results are exteded to the block eergy arrival model where the legth of each eergy block L grows subliearly i [4,0,]. We show that best-effort ad ( ( maxlog,l L save-ad-trasmit achieve the secod-order scaligs O ad O respectively. I particular, ( if L = ω(log, the both of our proposed schemes achieve the optimal secod-order scalig O. This work provides the first fiite blocklegth aalysis of the best-effort scheme over the AWGN EH chael ad presets a o-asymptotic achievable rate. It shows that the first- ad secod-order terms of the asymptotic ( log achievable rate are equal to the capacity ad c ε log log 5 = O respectively where c 5 is some ( positive costat that does ot deped o ad ε. This secod-order scalig O greatly improves the state-of-the-art result o( i []. I additio, this work obtais a ew o-asymptotic achievable rate for savead-trasmit schemes which outperforms the best existig oe derived i [4] for all sufficietly large sigal-to-oise ratios (SNRs. More specifically, the first- ad secod-order terms of the ew o-asymptotic achievable rate are log(/ε equal to the capacity ad sup c 6 + P (P + Φ (ε respectively where c 6 does ot deped ε 0,ε 0, ε +ε =ε o ad ε ad satisfies 0 < c 6 < c 4 for all sufficietly large P > 0 (c 4 refers to the quatity i the previous subsectio. log log L C. Paper Outlie This paper is orgaized as follows. The otatio of this paper is preseted i the ext subsectio. Sectio II presets the model of the AWGN EH chael. Sectio III describes the best-effort scheme, states prelimiary results, ad presets the first mai result a o-asymptotic achievable rate for the best-effort scheme. Sectio IV May 3, 08

4 4 trasmitter AWGN receiver Fig.. The AWGN EH chael discusses the save-ad-trasmit scheme ad presets the secod mai result a ew o-asymptotic achievable rate for the save-ad-trasmit scheme. Sectios V ad VI cotai the proof of the first ad secod mai results respectively. Sectio VII geeralizes the results i Sectios III ad IV to the block eergy arrival model, whose proofs are provided i Sectios VIII ad IX respectively. Sectio X cotais umerical results which demostrate the performace advatage of our proposed schemes over the state-of-the-art schemes for both i.i.d. ad block eergy arrivals. Sectio XI cocludes the paper. D. Notatio The sets of atural umbers, real umbers ad o-egative real umbers are deoted by N, R ad R + respectively. The Euclidea orm of a vector a R L is deoted by a def = throughout the paper. L l= a l. All logarithms are take to base e We use PE to represet the probability of a evet E, ad we let E be the idicator fuctio of E. Radom variables are deoted by capital letters (e.g., X, ad the realizatio ad the alphabet of a radom variable are deoted by the correspodig small letter (e.g., x ad calligraphic fot (e.g., X respectively. We use X to deote a radom tuple (X, X,..., X, where all of the elemets X k have the same alphabet X. We let p X ad p Y X deote the probability distributio of X ad the coditioal probability distributio of Y give X respectively for radom variables X ad Y. We let p X p Y X deote the joit distributio of (X, Y, i.e., p X p Y X (x, y = p X (xp Y X (y x for all x ad y. For radom variable X p X ad ay real-valued fuctio g whose domai icludes X, we let P px g(x ξ deote X p X(x(g(x ξ dx for ay real costat ξ. For ay fuctio f whose domai cotais X, we use E px [f(x] to deote the expectatio of f(x where X is distributed accordig to p X. For simplicity, we omit the subscript of a otatio whe there is o ambiguity. The distributio of a Gaussia radom variable Z whose mea ad variace are µ ad σ respectively is deoted by N (z; µ, σ (z µ e σ πσ. II. THE AWGN EH CHANNEL A. Problem Formulatio The AWGN EH chael, as illustrated i Figure, cosists of oe trasmitter ad oe receiver. Eergy harvestig ad commuicatio occur i time slots, i.e., chael uses. I each time slot, a radom amout of eergy E, May 3, 08

5 5 E = R +, is harvested where E[E] = P > 0; E[E ] <. ( The eergy-harvestig process is characterized by idepedet copies of E deoted by E, E,..., E. Before the time slots, the trasmitter chooses a message W. For each k,,...,, the trasmitter cosumes Xk uits of eergy to trasmit X k R based o (W, E k ad the receiver observes Y k R i time slot k. The eergy state iformatio E k is kow by the trasmitter at time k before ecodig X k, but the receiver has o access to E k. For each k,,...,, we have: (i E k ad (W, E k, X k, Y k are idepedet, i.e., p W,E k,x k,y = p k E k p W,E k,x k,y k. ( (ii For w W ad every e R +, a trasmitted codeword X should satisfy k k P Xi e i W = w, E = e = (3 i= i= for each k,,...,. After time slots, the receiver declares Ŵ to be the trasmitted W based o Y. B. Stadard Defiitios Formally, we defie a code as follows: Defiitio : A (, M-code cosists of the followig: A message set W,,..., M, where W is uiform o W. A sequece of ecodig fuctios f k : W R k + R for each k,,...,, where f k is used by the trasmitter at time slot k for ecodig X k accordig to X k = f k (W, E k. 3 A decodig fuctio ϕ : R W, for decodig W at the receiver, i.e., Ŵ = ϕ(y. If the sequece of ecodig fuctios f i satisfies (3, the code is also called a (, M-EH code. If a (, M-code does ot satisfy the EH costraits (3 durig the ecodig process (i.e., X is a fuctio of W aloe, the the (, M-EH code ca be viewed as a (, M-code for the usual AWGN chael without ay cost costrait [,3]. Defiitio : The AWGN EH chael is characterized by a coditioal probability distributio q Y X (y x N (y; x, such that the followig holds for ay (, M-code: For each k,,...,, p W,Ek,X k,y k = p W,E k,x k,y k p Y k X k where p Yk X k (y k x k = q Y X (y k x k = π e (y k x k (4 for all x k X ad y k Y. May 3, 08

6 6 For ay (, M-code defied o the AWGN EH chael, let p W,E,X,Y,Ŵ by the code. We ca factorize p W,E,X,Y,Ŵ as be the joit distributio iduced ( p W,E,X,Y,Ŵ = p W p Ek p Xk W,E kp Y k X k pŵ Y, (5 which follows from the i.i.d. assumptio of the EH process E i (, the fact by Defiitio that X i is a fuctio of (W, E i ad the memoryless property of the chael q Y X described i Defiitio. Defiitio 3: For a (, M-code defied o the AWGN EH chael, we ca calculate accordig to (5 the average probability of decodig error defied as P Ŵ W. We call a (, M-EH code with average probability of decodig error o larger tha ε a (, M, ε-eh code. Defiitio 4: Let ε (0, be a real umber. A rate R is said to be ε-achievable for the EH chael if there exists a sequece of (, M, ε-eh codes such that lim if log M R. Defiitio 5: The ε-capacity of the AWGN EH chael, deoted by C ε, is defied to be C ε supr : R is ε-achievable for the EH chael. The capacity of the AWGN EH chael is C if ε>0 C ε. Defie the capacity fuctio C(x log( + x (6 for all x 0. It was show i [] that C ε = C = C(P for all ε (0, where P E[E] (7 ca be viewed as the sigal-to-oise ratio (SNR of the AWGN EH chael. A. Best-Effort Scheme III. AN ACHIEVABLE RATE FOR BEST-EFFORT Fix a blocklegth. Choose a positive real umber S < P = E[E] ad let p X (x N (x; 0, S (8 such that S = E px [X ]. (9 The codebook cosists of M mutually idepedet radom codewords, which are costructed as follows. For each message w W, a legth- codeword X (w (X (w, X (w,... X (w cosistig of i.i.d. symbols is May 3, 08

7 7 costructed where X (w p X. Suppose W = w ad E = e, i.e., the trasmitter chooses message w W ad the realizatio of E is e R +. The, the trasmitter uses the followig best-effort (, M-EH code with ecodig fuctios f best k ad decodig fuctio ϕbest. Defie f best, f best,..., f best i a recursive maer where X k (w if ( X k (w ek + k ( e i ( fi best (w, e i, i= (0 0 otherwise. fk best (w, e k For each k,,...,, the trasmitter seds X k (W f best k (W, E k. ( By costructio, k ( P Xi (W k E i =. ( i= i= Upo receivig Ỹ (W (Ỹ(W, Ỹ(W,..., Ỹ(W where Ỹk(W is geerated accordig to PỸk(W = b X k (W = a q Y X (b a, (3 the receiver declares that ϕ best (Ỹ (W = j if j is the uique iteger i W that satisfies where p Y log q Y X(Ỹk(W X k (j p Y (Ỹk(W log ξ, (4 is the margial distributio of p X q Y X ad log ξ is a arbitrary threshold which will be carefully chose later i (67. Otherwise, the receiver chooses ϕ best (Ỹ (W W accordig to the uiform distributio. The scheme described above is said to be best-effort because accordig to (0 ad (34, the trasmitter tries its best to output the desired symbol X k (W wheever the battery cotais eough eergy for trasmittig X k (W. B. Prelimiaries A importat quatity that determies the performace of the best-effort (, M-EH code is Q ( k,,..., X k (W X k (W, (5 which is a radom set that specifies the mismatched positios betwee X (W ad X (W. The followig lemma cocers the umber of mismatched positios betwee X (W ad X (W. The proof, which is based o aalyzig the escape probability of a Markov process is provided i Appedix A. Lemma : Fix ay ad ay ρ (0, such that 4ρ ρ <, (6 May 3, 08

8 8 ad fix a best-effort (, M-EH code with S P ( ρ. (7 Defie ρ P α E[E ] + 3S (8 ad For ay γ R +, we have β P Q ( γ + α + 63α S. (9 e γ (P β + α E[E ]. (0 Remark : I the proof of Lemma i Appedix A, a importat step is aalyzig the escape probability (46 of the Markov process E + k ( i= Ei Xi τ where τ is the stoppig time whe the value of the Markov process hits ay egative umber a < 0. k= The followig lemma [4] is stadard for provig achievability results i the fiite blocklegth regime ad its proof ca be foud i [5, Th. 3.8.]. Lemma (Implied by Shao s boud [4]: Let p X,Y be the probability distributio of a pair of radom variables (X, Y. Suppose (X (, Y ( p X,Y ad (X (, Y ( p X,Y are idepedet. The for each δ > 0 ad each M N, we have P log p Y X (Y ( X ( p Y (Y > log M + δ ( M e δ. ( The followig lemma is a modificatio of the Shao s boud stated i the previous lemma, ad its proof is provided i Appedix B. Lemma 3: Suppose we are give a best-effort (, M-EH code as described i Sectio III-A. The for each γ R +, each δ > 0 ad each M N, we have P log p Y X (Ỹ ( X ( > log M + δ p Y (Ỹ ( Q ( < γ + M e δ (S + γ+. ( C. A Achievable Rate for the Best-Effort Scheme The followig theorem is the first mai result of this paper. The proof relies o Lemma ad Lemma 3, ad it will be preseted i Sectio V. Theorem : Fix a ε (0,, fix ay, ad fix a ρ (0, that satisfies (6. Defie S, α ad β as i (7, (8 ad (9 respectively, ad defie log ε γ. (3 P β + α E[E ] May 3, 08

9 9 Let p X = N (x; 0, S ad let p Y deote the variace ad the third absolute momet of log q Y X (Y X p Y (Y that ε + ε ε, if is sufficietly large such that ε code which satisfies = N (y; 0, S + be the margial distributio of p X q Y X, ad let σ ad T respectively. For ay ε > 0 ad ε > 0 such T σ 3 3 > 0, the there exists a best-effort (, M-EH log M log( + S + ( σ Φ ε T σ 3 3 S (γ + log log (4 ad P ϕ best( Ỹ (W W ε. (5 I particular, the probability of seeig more tha γ + mismatch evets ca be bouded as P Q ( γ + ε. (6 The followig corollary is a direct cosequece of Theorem, ad it states that the best-effort scheme achieves ( a o-asymptotic rate whose secod-order term scales as O. The proof of Corollary 4 is provided i Appedix C. Corollary 4: Fix a ε (0,, ad fix ay ε > 0 ad ε > 0 such that ε + ε ε. There exists a costat κ > 0 which does ot deped o such that for all sufficietly large, we ca costruct a best-effort (, M, ε-eh code with ρ (P + (E[E ] + 3P log ε P log ( log log = Θ (7 ad S = P ( ρ = P Θ ( log (8 which satisfies log M log( + P (E[E ] + 3P log ε P + log P (P + Φ (ε κ log. (9 I particular, the probability of seeig more tha Θ( log mismatch evets ca be bouded as where γ = Θ( log accordig to (3, (8, (9 ad (7. P Q ( γ + ε (30 IV. AN ACHIEVABLE RATE FOR SAVE-AND-TRANSMIT I Sectio III-A, we have described the costructio of the codebook of a best-effort scheme. I this sectio, we would like to ivestigate a save-ad-trasmit strategy that uses a similar codebook. May 3, 08

10 0 A. Save-ad-Trasmit Scheme Fix a blocklegth. Choose a positive real umber S < P = E[E] ad let p X (x N (x; 0, S (3 such that S = E px [X ]. (3 The codebook cosists of M mutually idepedet radom codewords deoted by X (w w W, which are costructed as described i Sectio III-A. Suppose W = w ad E = e, i.e., the trasmitter chooses message w W ad the realizatio of E is e R +. The, the trasmitter uses the followig save-ad-trasmit (, M-EH code with ecodig fuctios fk save ad decodig fuctio ϕsave. The save-ad-trasmit code cosists of a iitial savig phase ad a subsequet trasmissio phase. Defie γ to be the umber of time slots i the iitial savig phase durig which o eergy is cosumed ad hece o iformatio is coveyed. Defie f save, f save,..., f save i a recursive maer where fk save (w, e k X k (w 0 otherwise. For each k,,...,, the trasmitter seds if k > γ ad ( X k (w ek + k i= ( e i ( f save i (w, e i, (33 X k (W f save k (W, E k. (34 By costructio, the EH costraits ( hold. Upo receivig Ỹ (W (Ỹ(W, Ỹ(W,..., Ỹ(W where Ỹ k (W is geerated accordig to (3, the receiver declares that ϕ save (Ỹ (W = j if j is the uique iteger i W that satisfies where p Y k=γ+ log q Y X(Ỹk(W X k (j p Y (Ỹk(W log ξ, (35 is the margial distributio of p X q Y X ad log ξ is a arbitrary threshold which will be carefully chose later i (76. Otherwise, the receiver chooses ϕ save (Ỹ (W W accordig to the uiform distributio. The followig lemma states a upper boud o the probability of a mismatch evet occurrig i the trasmissio phase give that the savig phase lasts for γ time slots. The proof of Lemma 5 is quite similar to the proof of Lemma established for the best-effort scheme, ad it is provided i Appedix E. Lemma 5: Fix ay ad ay ρ (0, such that (6 holds, ad fix a save-ad-trasmit (, M-EH code with S, α ad β beig defied as i (7, (8 ad (9 respectively. For ay γ N, we have k k ( P E i < Xi e γ P β + α E[E ]. (36 k=γ+ i= i=γ+ May 3, 08

11 B. A Achievable Rate for the Save-ad-Trasmit Scheme The followig theorem is the secod mai result of this paper. The proof relies o Lemma 5 ad Lemma, ad it will be provided i Sectio VI. Theorem : Fix a ε (0,, ad fix ay ad fix a ρ (0, that satisfies (6. Defie S, α ad β as i (7, (8 ad (9 respectively, ad defie log ε γ P β + α E[E ] Let p X = N (x; 0, S ad let p Y deote the variace ad the third absolute momet of log q Y X (Y X p Y (Y = N (y; 0, S + be the margial distributio of p X q Y X, ad let σ ad T that ε + ε ε, if is sufficietly large such that γ > 0 ad ε save-ad-trasmit (, M-EH code which satisfies log M γ log( + S + ( γ σ Φ (ε (37 respectively. For ay ε > 0 ad ε > 0 such T σ 3 γ > 0, the there exists a T σ 3 log (38 γ ad P ϕ save( Ỹ (W W ε. (39 I particular, the probability of seeig a mismatch evet i the trasmissio phase ca be bouded as k k P E i < Xi ε. (40 k=γ + i= i=γ + The followig corollary is a direct cosequece of Theorem, ad it states a o-asymptotic rate for the save-adtrasmit scheme whose secod-order term scales as O (. The proof of Corollary 6 is provided i Appedix F. Corollary 6: Fix a ε (0,, ad fix ay ε > 0 ad ε > 0 such that ε + ε ε. There exists a costat κ > 0 which does ot deped o such that for all sufficietly large, we ca costruct a save-ad-trasmit (, M, ε-eh code with ρ ad S = P ( ρ which satisfies log M log( + P (P + (E[E ] + 3P log( + P log ε P P (E[E ] + 3P log( + P log ε P (P + ( = Θ (4 P + (P + Φ (ε κ. (4 3/4 I particular, the probability of seeig a mismatch evet i the trasmissio phase ca be bouded as k k P E i < Xi ε (43 k=γ + i= where γ = Θ( accordig to (37, (8, (9 ad (4. i=γ + May 3, 08

12 Remark : The parameters ρ ad γ i Corollary 6 have bee optimized to achieve the secod-order scalig O(/. Fix ay ε > 0. The best existig lower boud o the secod-order term of log M was derived i [4, Th. ], which states that there exists a save-ad-trasmit (, M, ε-eh code that satisfies ( lim if log M log( + P log( + P (E[E P ] + P log P + ε P + Φ (ε (44 for ay ε > 0 ad ε > 0 such that ε + ε ε. Note that the secod-order term of the best existig lower boud as stated o the RHS of (44 decays as log( + P ( + E[E ] P log ε + Φ (ε as P teds to. O the other had, it follows from (4 i Corollary 6 that the secod-order term of our lower boud decays as (3 + E[E ] P log( + P log ε + Φ (ε as P teds to. Cosequetly, the secod-order term achievable by the save-ad-trasmit scheme guarateed by Corollary 6 is strictly larger (less egative tha the best existig boud for all sufficietly large P > 0. V. PROOF OF THEOREM Fix a ε (0, ad ay ε > 0 ad ε > 0 such that ε + ε = ε. Fix a ad a ρ (0, that satisfies (6. Cosider a best-effort (, M-code described i Sectio III-A where the correspodig S ad p X are defied accordig to (7 ad (8 respectively, i.e., S P ( ρ ad p X (x N (x; 0, S. I additio, let p Y = N (y; 0, S + be the margial distributio of p X q Y X, ad defie α, β ad γ as i (8, (9 ad (3 respectively. Cosider the probability of decodig error ϕ best P ( Ỹ (W W P ϕ best ( Ỹ (W W Q ( < γ + + ε (45 which follows from the fact that P Q ( γ + e log ε ε (46 due to the uio boud, Lemma ad the defiitio of γ i (3. With the give code costructio, it follows that P max (X k( 8S log k,,..., whose derivatio is relegated to Appedix D. To simplify otatio, defie the evets P px X 8S log, (47 E Q ( < γ + (48 ad E max (X k( < 8S log. (49 k,,..., I additio, defie ı(a; b log q Y X(b a p Y (b = log( + S + S (b a + a(b a + a (S + (50 May 3, 08

13 3 for all (a, b R + where ı(a; b is used i the decodig rule (4. Followig (45 ad lettig ξ > 0 be a arbitrary positive umber to be determied later i (67, we obtai from the symmetry of the codebook, the ecodig rule (0, the decodig rule (4, the uio boud ad (47 that ϕ best P ( Ỹ (W W E ( M P ı(x k (; Ỹk( < log ξ ı(x k (i; Ỹk(i log ξ E i= P ı(x k (; Ỹk( < log ξ E E i= M + P ı(x k (i; Ỹk( log ξ E E +. (53 I order to boud the first term i (53, we cosider P ı(x k (; Ỹk( < log ξ E E = P ı(x k (; Y k ( + k Q ( Followig (54, we let p Z (z N (z; 0, ad cosider for each k Q ( (5 (5 ( ı(x k (; Ỹk( ı(x k (; Y k ( < log ξ E E. (54 P ı(x k (; Ỹk( ı(x k (; Y k ( 0S log E S XZ (S + X = P px p Z 0S log X < 8S log (55 (S + where the equality follows from the followig fact due to the defiitio of ı( ; i (50 ad the facts that X k ( N (x k (; 0, S ad Ỹk( = Y k ( X k ( N (y k ( x k (; 0, : ı(x k (; Ỹk( ı(x k (; Y k ( = S X k (Z k (S + (X k ( (S + (56 with the defiitio Z k Y k ( X k (. Followig (46, we cosider the chai of iequalities below for each k Q ( : P px p Z S XZ (S + X (S + 0S log X < 8S log S xz (S + x sup P pz 0S log x : x <8S log (S + = sup P pz e xz 0(S+ e (S+x S x : x <8S log sup [ E pz e xz ] 0(S+ e (S+x S (59 x : x <8S log = sup 0(S+ e (S+x S x : x <8S log (57 (58 + x (60 < 0(S+ e 8(S+ log (6 May 3, 08

14 4 < (6 where (59 is due to Markov s iequality. Combiig (54, (55 ad (6 ad usig the uio boud, we have P ı(x k (; Ỹk( < log ξ E E P ı(x k (; Y k ( 0S Q ( log < log ξ E E + P ı(x k (; Y k ( < log ξ + 0S (γ + log + where the last iequality follows from the defiitio of E i (48. I order to boud the secod term i (53, we use Lemma 3 to obtai P ı(x k (; Ỹk( log ξ E E M e (log ξ log M (S + γ+. (65 Cosequetly, it follows from (45, (53 ad (64 that P ϕ best( Ỹ (W W ( P ı(x k (; Ỹk( < log ξ E E + e log ξ log M γ+ log(s + + ε + + (66 The remaider of the proof follows from stadard steps, outlied below for the sake of completeess. Let µ = log( + S, σ = S S + respectively. Choose ad T < deote the mea, the variace ad the third absolute momet of ı(x; Y log ξ µ + ( σ Φ ε T σ 3 3 0S (γ + log. (67 (63 (64 It the follows from Berry-Essée theorem [6], i.e., P V k µ σ a Φ(a T σ 3 (68 for all a R, that P ı(x k (; Y k ( < log ξ + 0S (γ + log I order to boud the secod term i (53, we choose log M log ξ γ + log(s + log ε 3. (69 (70 log ξ S (γ + log log. (7 Cosequetly, (5 follows from (66, (69 ad (70, ad (4 follows from (67 ad (7. I additio, (6 follows from (46. May 3, 08

15 5 VI. PROOF OF THEOREM Fix a ε (0, ad let ε > 0 ad ε > 0 such that ε + ε = ε. Fix a ad a ρ (0, that satisfies (6. Cosider a save-ad-trasmit (, M-code described i Sectio IV-A with S P ( ρ (7 where the correspodig p X is defied accordig to (3, i.e., p X (x N (x; 0, S. I additio, let p Y = N (y; 0, S + be the margial distributio of p X q Y X, ad defie α, β ad γ as i (8, (9 ad (37 respectively. We use Lemma 5 ad the defiitio of γ i (37 to boud the probability of decodig error as P ϕ save( Ỹ (W W P ϕ save ( Ỹ (W W k=γ + k E i i= k i=γ + X i + ε. (73 By the symmetry of the radom codebook X (w w W, the ecodig rule i (33, the decodig rule i (35 ad the uio boud, we obtai P ϕ save ( Ỹ (W W k k E i Xi k=γ + i= i=γ + P ı(x k (; Y k ( < log ξ + (M P k=γ + k=γ + ı(x k (; Y k ( log ξ where log ξ is a arbitrary threshold to be determied later i (76. Combiig (73 ad (74 ad usig Lemma, we obtai P ϕ save( Ỹ (W W P k=γ + (74 ı(x k (; Y k ( < log ξ + e (log ξ log M. (75 The remaider of the proof follows from stadard steps, outlied below for the sake of completeess. Let µ = log( + S, σ = S S + respectively. Choose ad T < deote the mea, the variace ad the third absolute momet of ı(x; Y log ξ ( γ µ + ( γ σ Φ (ε T σ 3 ( γ. (76 It the follows from Berry-Essée theorem [6] as stated i (68 that P ı(x k (; Y k ( < log ξ ε. (77 k=γ + I order to boud the secod term i (75, we choose log M log ξ log (78 log ξ log. (79 May 3, 08

16 6 Cosequetly, (39 follows from (75, (77 ad (78, ad (38 follows from (76 ad (79. VII. THE BLOCK ENERGY ARRIVAL MODEL I this sectio, we geeralize our achievable rates for save-ad-trasmit ad best-effort to the block eergy arrival model [4,0,], which is useful for modelig practical scearios whe the eergy-arrival process (e.g., solar eergy, wid eergy, ambiet radio-frequecy (RF eergy, etc. evolves at a slower timescale compared to the trasmissio process. A. Block Eergy Arrival We follow the formulatio i [4], which assumes that E i i= arrive at the buffer i a block-by-block maer as follows: For each l N, let b l (l L (80 such that b l + is the idex of the first chael use withi the l th block of eergy arrival, where L deotes the legth of each block. The EH radom variables that mark the startig poits of the blocks (i.e., E bl + l= are assumed to be idepedet ad idetically distributed (i.i.d. radom variables where E = E satisfies (. I additio, we assume E bl + = E bl + =... = E bl +L (8 for all l N. I other words, the harvested eergy i each chael use withi a block remais costat while the harvested eergy across differet blocks is characterized by a sequece of i.i.d. radom variables with mea equal to P. By costructio, we have the followig for each k,,..., ad all e k R k +, p Ek E k (e p k e k E (e k if k = b l + for some l N, = e k = e k otherwise. The legth of each eergy-arrival block L is assumed to remai costat or grow subliearly i. (8 B. Blockwise Best-Effort Fix a blocklegth. Choose a atural umber L ad defie = /L. Choose a positive real umber S < P = E[E] ad let p X (x N (x; 0, S (83 such that S = E px [X ]. (84 The codebook cosists of M mutually idepedet radom codewords deoted by X (w w W, which are costructed as described i Sectio III-A. Suppose W = w ad E = e, i.e., the trasmitter chooses message w May 3, 08

17 7 W ad the realizatio of E is e R +. The, the trasmitter uses the followig blockwise best-effort (, M- EH code with block ecodig fuctios fl best l= ad decodig fuctio ϕbest. Defie f best, f best,..., f best recursive maer where fl best (w, e b l (X bl +(w, X bl +(w,..., X bl +L(w if l < ad L ( Xbl +j(w l L j= k=b + i= j= (X b +(w, X b +(w,..., X (w if l = ad ( Xk (w l L (0, 0,..., 0 mil, b l times otherwise. i= j= e bi+j l i= e bi+j l i= f best i a i (w, e i, f best i (w, e i, I other words, the trasmitter outputs a block of L symbols (X bl +(w, X bl +(w,..., X bl +L(w durig time b l + to b l + L if the eergy i the battery at time b l (i.e., l L i= j= e b i+j l i= f best i (w, e i ca support the trasmissio of the whole block of symbols. Defie ( X bl +(W, X bl +(W,..., X bl +L(W fl best (W, E b l ( for each l,,...,, ad defie Xb +(W, X b +(W,..., X (W f best (W, E b. For each time k,,...,, Xk (W is trasmitted. Upo receivig Ỹ (W (Ỹ(W, Ỹ(W,..., Ỹ(W where Ỹk(W is geerated accordig to (3, the receiver declares that ϕ best (Ỹ (W = j if j is the uique iteger i W that satisfies (4 where p Y is the margial distributio of p X q Y X ad log ξ is a arbitrary threshold which will be carefully chose later i (37. Otherwise, the receiver chooses ϕ best (Ỹ (W W accordig to the uiform distributio. Defie Q ( as i (5, which is a radom set that specifies the mismatched positios betwee X (W ad X (W. The followig lemma is a straightforward extesio of Lemma, ad its proof is cotaied i Appedix G for the sake of completeess. Lemma 7: Fix ay, ay L < ad ay ρ (0, such that (85 4ρ ρ <, (86 ad fix a blockwise best-effort (, M-EH code with S P ( ρ. (87 Defie ρ P α,l LE[E ] + 3S (88 ad For ay γ R +, we have β,l P Q ( L(γ + α,l + 63α,L S. (89 e Lγ (P β,l + Lα,L E[E ]. (90 May 3, 08

18 8 The followig theorem is the first mai result uder the block eergy arrival model. The proof relies o Lemma 7 ad Lemma 3, ad it will be preseted i Sectio VIII. Theorem 3: Fix a ε (0,. Fix ay N, fix ay L ad fix a ρ (0, that satisfies (86. Defie S, α,l ad β,l as i (87, (88 ad (89 respectively, ad defie Let p X = N (x; 0, S ad let p Y deote the variace ad the third absolute momet of log q Y X (Y X p Y (Y that ε + ε ε, if is sufficietly large such that ε (, M-EH code which satisfies log ε γ,l. (9 LP β,l + L α,l E[E ] = N (y; 0, S + be the margial distributio of p X q Y X, ad let σ ad T log M log( + S + ( σ Φ ε T σ 3 3 respectively. For ay ε > 0 ad ε > 0 such T σ 3 3 > 0, the there exists a blockwise best-effort S (L log + 8 log + (γ,l + log P (9 ad P ϕ best( Ỹ (W W ε. (93 I particular, the probability of seeig more tha L(γ,L + mismatch evets ca be bouded as P Q ( L(γ,L + ε. (94 The followig corollary is a direct cosequece of Theorem 3, ad it states that the best-effort scheme achieves a ( o-asymptotic rate whose secod-order term scales as O. The proof of Corollary 8 is provided i Appedix H. maxlog,l Corollary 8: Fix a ε (0,, ad fix ay ε > 0 ad ε > 0 such that ε +ε ε. There exists a costat κ > 0 which does ot deped o such that for all sufficietly large ad ay L, we ca costruct a blockwise best-effort (, M, ε-eh code with (L log + 8 log + (P + (LE[E ] + 3P log ( ε maxlog, L ρ P = Θ L (95 ad S = P ( ρ = P Θ ( maxlog, L (96 which satisfies log M log( + P (L log + 8 log + (LE[E ] + 3P log ε L(P + P (P + Φ (ε κ maxlog, L. (97 May 3, 08

19 9 ( I particular, the probability of seeig more tha Θ L ( where γ,l = Θ maxlog,l maxlog,l mismatch evets ca be bouded as P Q ( L(γ,L + ε (98 accordig to (9, (88, (89 ad (95. Remark 3: The parameters ρ ad γ,l i Corollary 8 have bee optimized to achieve the secod-order scalig ( O. maxlog,l The followig result is a direct cosequece of Corollary 8. Theorem 4: Fix ay ε (0,. Suppose L = ω(log o(, i.e., lim log L = lim L = 0. The for all sufficietly large, there exists a blockwise best-effort (, M, ε-eh code such that log M ( log( + P (log E[E ] log ε L L P + o. (99 Remark 4: Let M,ε supm N There exists a (, M, ε-eh code be the maximum alphabet size of the message we ca trasmit usig a (, M, ε-eh code. For the case L = ( ω(log o(, it has bee show i [4] that log M,ε log(+p +O L. Therefore, Theorem 4 implies ( that the blockwise best-effort scheme achieve the optimal secod-order scalig O if L = ω(log o(. L C. Blockwise Save-ad-Trasmit Fix a blocklegth. Choose L ad defie = /L. Choose a positive real umber S < P = E[E] ad let p X (x N (x; 0, S (00 such that S = E px [X ]. (0 The codebook cosists of M mutually idepedet radom codewords deoted by X (w w W, which are costructed as described i Sectio III-A. Suppose W = w ad E = e. The, the trasmitter uses the followig blockwise save-ad-trasmit (, M-EH code with ecodig fuctios f save k ad decodig fuctio ϕ save. Defie γ to be the umber of blocks of L cosecutive time slots i the iitial savig phase. Defie f save, f save,..., f save i a recursive maer where X k (w fk save (w, e k 0 otherwise. if k > Lγ ad ( X k (w ek + k i= ( e i ( f save i (w, e i, (0 May 3, 08

20 0 For each k,,...,, the trasmitter seds X k (W f save k (W, E k. (03 Upo receivig Ỹ (W (Ỹ(W, Ỹ(W,..., Ỹ(W where Ỹk(W is geerated accordig to (3, the receiver declares that ϕ save (Ỹ (W = j if j is the uique iteger i W that satisfies where p Y k=lγ+ log q Y X(Ỹk(W X k (j p Y (Ỹk(W log ξ, (04 is the margial distributio of p X q Y X ad log ξ is a arbitrary threshold which will be carefully chose later i (4. Otherwise, the receiver chooses ϕ save (Ỹ (W W accordig to the uiform distributio. The followig lemma is a straightforward extesio of Lemma 5 which states a upper boud o the probability of a mismatch evet occurrig i the trasmissio phase give that the savig phase lasts for γ blocks of L time slots. The proof of Lemma 9 is cotaied i Appedix I for the sake of completeess. Lemma 9: Fix ay, ay L ad ay ρ (0, such that (6 holds, ad fix a blockwise save-ad-trasmit (, M-EH code with S, α,l ad β,l beig defied as i (87, (88 ad (89 respectively. For ay γ N, we have P k=lγ+ k E i < i= k i=lγ+ X i ( e L(γ P β,l + Lα,L E[E ]. (05 The followig theorem is the secod mai result uder the block eergy arrival model. The proof relies o Lemma 9 ad Lemma, ad it will be provided i Sectio IX. Theorem 5: Fix a ε (0,. Fix ay N, fix ay L ad fix a ρ (0, that satisfies (86. Defie S, α,l ad β,l as i (87, (88 ad (89 respectively, ad defie log γ,l ε LP β,l + L α,l E[E ] +. (06 Let p X = N (x; 0, S ad let p Y deote the variace ad the third absolute momet of log q Y X (Y X p Y (Y = N (y; 0, S + be the margial distributio of p X q Y X, ad let σ ad T respectively. For ay ε > 0 ad ε > 0 such T that ε +ε ε, if Lγ,L > 0 ad ε σ 3 Lγ,L > 0, the there exists a blockwise save-ad-trasmit (, M-EH code which satisfies log M Lγ,L log( + S + ( Lγ,L σ Φ (ε T σ 3 log (07 Lγ,L ad P ϕ save( Ỹ (W W ε. (08 May 3, 08

21 I particular, the probability of seeig a mismatch evet i the trasmissio phase ca be bouded as k k P E i < Xi ε. (09 k=lγ,l + i= i=lγ,l + The followig corollary is a direct cosequece of Theorem 5, ad it states a o-asymptotic rate for the blockwise ( save-ad-trasmit scheme whose secod-order term scales as O. The proof of Corollary 0 is provided i Appedix J. Corollary 0: Fix a ε (0,, ad fix ay ε > 0 ad ε > 0 such that ε + ε ε. Suppose L = o(. There exists a costat κ > 0 which does ot deped o such that for ay sufficietly large, we ca costruct a blockwise save-ad-trasmit (, M, ε-eh code with (P + (LE[E ] + 3P log( + P log ( ε L ρ P = Θ P ad S = P ( ρ which satisfies log M log( + P (LE[E ] + 3P log( + P log ε P (P + L (0 P + (P + Φ (ε κl/4. ( 3/4 I particular, the probability of seeig a mismatch evet i the trasmissio phase whe the savig phase is Θ( L ca be bouded as (09 where γ,l = Θ( /L accordig to (06, (88, (89 ad (0. Remark 5: The parameters ρ ad γ,l i Corollary 0 have bee optimized to achieve the secod-order scalig ( O. L The followig result is a direct cosequece of Corollary 0. Theorem 6: Fix ay ε (0,. Suppose L = ω( o(, i.e., lim L = lim L = 0. The for all sufficietly large, there exists a blockwise save-ad-trasmit (, M, ε-eh code such that log M ( log( + P E[E ] log( + P log ε L L P (P + o. ( of Remark 6: Fix ay ε > 0. Fix ay L = ω( o(. The best existig lower boud o the secod-order term satisfies log M was derived i [4, Th. ], which states that there exists a save-ad-trasmit (, M, ε-eh code that lim if ( log M L log( + P log( + P (E[E P ] + P log ε. (3 Note that the secod-order term of the best existig lower boud as stated o the RHS of (3 decays as log( + P ( + E[E ] P log ε as P teds to. O the other had, it follows from ( i Theorem 6 that E[E the secod-order term of our lower boud decays as ] P log( + P log ε as P teds to. Cosequetly, the secod-order term achievable by the save-ad-trasmit scheme guarateed by Theorem 6 is strictly larger (less egative tha the best existig boud for all sufficietly large P > 0. May 3, 08

22 VIII. PROOF OF THEOREM 3 Fix a ε (0, ad ay ε > 0 ad ε > 0 such that ε +ε = ε. Fix a N, a L < ad a ρ (0, that satisfies (86. Cosider a blockwise best-effort (, M-code described i Sectio VII-B where the correspodig S ad p X are defied accordig to (87 ad (83 respectively. I additio, let p Y = N (y; 0, S + be the margial distributio of p X q Y X, ad defie α,l, β,l ad γ,l as i (88, (89 ad (9 respectively. Cosider the probability of decodig error ϕ best P ( Ỹ (W W P ϕ best ( Ỹ (W W Q ( < L(γ,L + + ε (4 which follows from the fact that P Q ( L(γ,L + e log ε ε (5 due to the uio boud, Lemma 7 ad the defiitio of γ,l i (9. Usig the covetio X k ( = 0 determiistically for all k >, it follows that P max (Xbl +(, X bl +(,..., X bl +L( S (L log + 3 log l,,..., /L L L P p X Xk S (L log + 3 log, where the derivatio of the last iequality is relegated to Appedix K. To simplify otatio, defie (6 (7,L L log + 3 log (8 ad +,L L log + 4 log, (9 ad defie the evets E Q ( < L(γ,L + (0 ad E max l,,..., /L j= L (X bl +j( < S,L. ( I additio, defie ı(a; b as i (50 where ı(a; b is used i the decodig rule specified by (4. Followig (4 ad lettig ξ > 0 be a arbitrary positive umber to be determied later i (37, we obtai from the symmetry of May 3, 08

23 3 the codebook, the ecodig rule (0, the decodig rule (4, the uio boud ad (7 that ϕ best P ( Ỹ (W W E ( M P ı(x k (; Ỹk( < log ξ ı(x k (i; Ỹk(i log ξ E i= P ı(x k (; Ỹk( < log ξ E E i= ( (3 M + P ı(x k (i; Ỹk( log ξ E E +. (4 I order to boud the first term i (4, we cosider P ı(x k (; Ỹk( < log ξ E E = P ı(x k (; Y k ( + k Q ( ( ı(x k (; Ỹk( ı(x k (; Y k ( < log ξ E E. (5 Followig (5, we let p Z (z N (z k; 0, ad cosider the chai of iequalities below for each block of L cosecutive mismatched positios i Q ( deoted by b l +, b l +,..., b l + L: bl +L P ı(x k (; Ỹk( ı(x k (; Y k ( S +,L E k=b l + = P px p Z bl +L k=b l + sup x L : x L <S,L P pz = sup x L : x L <S,L P p Z sup x L : x L <S,L E pz S X k Z k (S + Xk S +,L E (S + L S x k Z k (S + x k S +,L (S + = sup e (S+ + L,L e x L : x L <S,L e L x kz k e (S+ +,L e L (S+x k S (6 (7 (8 [e ] L x L kz k e (S+ + (S+x k,l e S (9 (S+x k + x k S (30 < e (S+ +,L e (S +,L (3 < (3 where (6 is due to (56 with the defiitio Z k Y k ( X k (; (9 is due to Markov s iequality; ad (3 is due to the defiitios of,l ad +,L i (8 ad (9 respectively. May 3, 08

24 4 Combiig (5 ad (3 ad usig the uio boud, we have P ı(x k (; Ỹk( < log ξ E E Q P ı(x k (; Y k ( S + (,L < log ξ E E + L P ı(x k (; Y k ( < log ξ + S +,L ( γ,l + + where (33 is due to the uio boud, the fact that Q ( has at most (33 (34 Q ( L blocks of cosecutive mismatched positios (oly the last block may have legth other tha L, ad the fact that (3 holds eve if L is replaced by ay atural umber L < L; ad (34 follows from the defiitio of E i (0. I order to boud the secod term i (4, we use Lemma 3 to obtai P ı(x k (; Ỹk( log ξ E E M e (log ξ log M (S + γ,l +. (35 Cosequetly, it follows from (4, (4 ad (34 that P ϕ best( Ỹ (W W ( P ı(x k (; Ỹk( < log ξ E E + e log ξ log M γ,l + log(s + + ε + + Let µ = log( + S, σ = ı(x; Y respectively. Choose S S + (36 ad T < deote the mea, the variace ad the third absolute momet of log ξ µ + ( σ Φ ε T σ 3 3 S +,L ( γ,l +. (37 It the follows from Berry-Essée theorem stated i (68 that P ı(x k (; Y k ( < log ξ + S +,L ( γ,l + I order to boud the secod term i (4, we choose log M log ξ γ,l + log(s + log ε 3. (38 (39 log ξ S ( γ,l + log P. (40 Cosequetly, (93 follows from (36, (38 ad (39, ad (9 follows from (37 ad (40. I additio, (94 follows from (5. May 3, 08

25 5 IX. PROOF OF THEOREM 5 Fix a ε (0, ad ay ε > 0 ad ε > 0 such that ε + ε = ε. Fix a N, a L < ad a ρ (0, that satisfies (6. Cosider a blockwise save-ad-trasmit (, M-code described i Sectio VII-C where the correspodig S ad p X are defied accordig to (87 ad (83 respectively. I additio, let p Y = N (y; 0, S + be the margial distributio of p X q Y X, ad defie α,l, β,l ad γ,l as i (88, (89 ad (06 respectively. We use Lemma 9 ad the defiitio of γ,l i (06 to boud the probability of decodig error as P ϕ save( Ỹ (W W P ϕ save ( Ỹ (W W k=lγ,l + k E i i= k i=lγ,l + X i + ε. (4 Let ad log ξ ( Lγ,L µ + ( Lγ,L σ Φ (ε T σ 3 ( Lγ,L (4 log M log ξ log (43 log ξ log. (44 Followig similar procedures startig from (73 i the proof of Theorem i Sectio VI, we ca show that (08 ad (07 hold. X. NUMERICAL RESULTS I this sectio, we umerically compare the performace of our proposed save-ad-trasmit scheme with the state-of-the-art scheme uder the followig two cases: The i.i.d. eergy arrival case with L =, ad the block eergy arrival case with L =. Sice the proposed best-effort scheme does ot achieve a positive rate for reasoable blocklegths, we omit best-effort i our umerical results. A. Case L = Figure (a plots the achievable rate up to the Θ(/ term of our proposed save-ad-trasmit scheme ad the best existig scheme accordig to (4 ad (44 respectively for the high SNR regime P = 5 db ad E[X ] = 3P ad for two values of ε = ε = ε/. I additio, we compare i Figure (b the two schemes for the low SNR regime P = db. For the high SNR regime, Figure (a shows that our proposed save-ad-trasmit outperforms the state-of-the-art at reasoable values of the blocklegth. O the other had, the state-of-the-art outperforms our proposed save-ad-trasmit for the low SNR regime as show i Figure (b. The two plots i Figure agree with Remark. May 3, 08

26 (a SNR = 5 db (b SNR = db Fig.. Achievable rates for the proposed save-ad-trasmit ad the state-of-the-art for L = where ε = ε = ε/ (a SNR = 5 db (b SNR = db Fig. 3. Achievable rates for the proposed save-ad-trasmit ad the state-of-the-art for L =. B. Case L = Figure 3(a plots the achievable rate up to the Θ( L/ term of our proposed save-ad-trasmit scheme ad the best existig scheme accordig to (99, ( ad (3 respectively for the high SNR regime P = 5 db ad E[X ] = 3P ad for two values of ε. I additio, we compare i Figure 3(b the two schemes for the low SNR regime P = db. For the high SNR regime, Figure 3(a shows that our proposed save-ad-trasmit outperforms the state-of-the-art at reasoable values of the blocklegth. O the other had, the state-of-the-art outperforms our proposed save-ad-trasmit for the low SNR regime as show i Figure 3(b. The two plots i Figure 3 agree with Remark 6. May 3, 08

27 7 XI. CONCLUDING REMARKS I this paper, we obtai the first o-asymptotic achievable rate for the best-effort scheme over the AWGN EH ( chael. The secod-order scalig of the o-asymptotic rate for the best-effort scheme is O. The proof techique ivolves aalyzig the escape probability of a Markov process. The, we use a similar proof techique ad obtai a ew o-asymptotic achievable rate for save-ad-trasmit over the same chael. The secod-order scalig of the o-asymptotic rate for save-ad-trasmit is O (. The achievable rates for best-effort ad save-ad-trasmit are exteded to the block eergy arrival model where L = o(, ad we show that best-effort ( ( maxlog,l L ad save-ad-trasmit achieve the secod-order scaligs O ad O. I particular, both ( L schemes achieve the optimal scalig O for the case L = ω(log. ( For the simplest case L =, the best-effort scheme does ot achieve the optimal scalig O. A straightforward verificatio by MATLAB reveals that the average umber of mismatched positios for a best-effort scheme is of the order o( while the average umber of mismatched positios for a save-ad-trasmit scheme is of the order Θ( (. A future directio may ivolve improvig the secod-order scalig O for L = for best-effort schemes by possibly provig a sharper probability boud tha (0 i Lemma. Aother iterestig directio is to explore the fiite-battery case ad complemet existig results [8,9] by derivig ew o-asymptotic achievable rates for best-effort ad save-ad-trasmit schemes with fiite battery. log log APPENDIX A PROOF OF LEMMA Fix a N ad a ρ (0, that satisfies (6, ad fix a best-effort (, M-EH code as described i Sectio III-A. Let p X be as defied i (8 where S is as defied i (7, ad let p E be the distributio of a EH radom variable that satisfies (. I this proof, all the probability, expectatio ad variace terms are evaluated accordig to p X p E where p X ad p E deote the ifiite product distributios of p X ad p E respectively. Cosider the Markov process E + k ( i= Ei Xi τ where τ is the stoppig time whe the value of the k= Markov process hits ay egative umber a < 0. By the defiitio of τ, we have τ ( P E + Ei Xi < 0 τ < = (45 ad i= P τ = = P E + k= k i= ( Ei Xi 0, (46 where P τ = deotes the escape probability. I order to show (0, we first fix a γ N, let τ, τ,..., τ γ be γ idepedet copies of τ. Due to the costructio of the best-effort scheme, eergy is saved but ot cosumed durig each mismatch evet, which implies that τ serves as a lower boud o the umber of time slots betwee May 3, 08

28 8 two mismatch evets, which the implies that γ P Q ( γ + P τ k < (47 = (Pτ < γ. (48 I order to obtai a upper boud o P τ <, we first costruct the followig sequece deoted by B k. For each k N, defie B k recursively as By ispectig (49, we have E if k =, B k B k + E k Xk if k ad B k 0, B k if k ad B k < 0. B < 0 = Combiig (48, (46 ad (50, we obtai E + k= (49 k (E i Xi < 0. (50 i= P Q ( γ + (PB < 0 γ. (5 It remais to obtai a upper boud o PB < 0. To this ed, we first defie for each k N B if k =, U k B k B k otherwise E if k =, = E k Xk if k, 3,... ad B k 0, 0 if k, 3,... ad B k < 0 (5 (53 where (53 follows from (49. It the follows from (50 ad (53 that B < 0 = U k < 0, hece PB < 0 = P U k < 0. (54 Followig (54, we cosider the chai of iequalities below for ay t > 0: P U k < 0 = P e t U k > E [e t ] U k (55 (56 May 3, 08

29 9 where the iequality follows from Markov s iequality. I order to simplify the RHS of (56, we cosider the followig chai of iequalities for each i, 3,...: where (57 is due to (53. E [e t ] i U k [ = E E [e t i U k U i ]] [ = E E [E [ e tui U i ] e t i [ [ E E max = max E [ E U k U i ]] ] e t(ei X i U i, e t i ] [ e t(ei X i ], E [e t i U k U k U i ]] (57 (58 follows from the idepedece betwee (E i, X i ad U i due to the idepedece betwee (E i, X i ad (E i, X i. Combiig (54, (56 ad (58, we have (58 P B < 0 E [ e tu] ( max E [e ] t(e X, (59 = E [ e te] ( max E [e ] t(e X,. (60 I order to simplify the RHS of (60, we use the followig two facts, whose proofs ca be foud i [, Appedix]: For ay y 0, + y e y + y + y e y (6 ad y e y y + y. (6 Let t > 0 be the positive solutio of the quadratic equatio Straightforward calculatios reveal that t = (P S E[E ] + 3S ( + 63S t. (63 t = (E[E ] + 3S + (E[E ] + 3S + 5S(P 3 S 378S 3 4(P S 63 S 3 4ρ P = 63 S 3 < 6S (64 (65 (66 (67 where May 3, 08

30 30 (65 is due to the fact that a + b a + b for all (a, b R +. (67 is due to (6 ad (7. Usig the defiitio of p X i (8 ad (67, we have I additio, usig (67 ad straightforward algebra, we obtai ad [ E X 4 e tx] 3S = <. (68 ( S t 5/ ( S t + 3S t (69 ( + 3St 5/ ( + 3S t 3 (70 + 9S t + 7S t + 7S 3 t 3 (7 + 63S t. (7 Followig (60, we use (6 ad (6 to obtai E [ e te] tp + t E[E ] e tp + t E[E ] (73 ad [ E e tx] + ts + t E[X 4 e tx ] e ts+ t E[X 4 e tx ], (74 which implies that E [ e te] E [e tx] e t t(p S+ ( E[E ]+E[X 4 e tx ] (75 (76 where (76 follows from the fact due to (63, (68, (69 ad (7 that Usig (60, (73 ad (76, we obtai t (P S E[E ] + E [ X 4 etx]. (77 P B < 0 e tp + t E[E ]. (78 Combiig (5 ad (78, we have P Q ( γ + e γ (tp t E[E ]. (79 May 3, 08