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

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1 08 IEEE Iteratioal Symposium o Iformatio Theory ISIT No-Asymptotic Achievable Rates for Gaussia Eergy-Harvestig Chaels: Best-Effort ad Save-ad-Trasmit Silas L Fog Departmet of Electrical ad Computer Egieerig Uiversity of Toroto Toroto, ON M5S 3G4, Caada silasfog@utorotoca Jig Yag ad Ayli Yeer Departmet of Electrical Egieerig The esylvaia State Uiversity Uiversity ark, A 680, USA yagjig, yeer}@egrpsuedu Abstract A additive white Gaussia oise AWGN eergyharvestig EH chael is cosidered the trasmitter is equipped with a ifiite-sized battery which stores eergy harvested from the eviromet The eergy arrival process is modeled as a sequece of idepedet ad idetically distributed iid radom variables The capacity of this chael is kow ad is achievable by the so-called best-effort ad save-adtrasmit schemes This paper ivestigates the best-effort scheme i the fiite blocklegth regime ad establishes the first oasymptotic achievable rate for it The first-order term of the oasymptotic achievable rate equals the capacity, ad the secodorder term is proportioal to log / deotes 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 obtai a ew o-asymptotic achievable rate for it, whose first-order ad secod-order terms achieve the capacity ad the scalig / respectively For all sufficietly large sigal-to-oise ratios SNRs, our ew achievable rate outperforms the existig oes I INTRODUCTION I this paper, we cosider commuicatio over a eergyharvestig EH chael betwee a trasmitted equipped with a ifiite-capacity battery ad a receiver as illustrated i Figure The trasmitter wats to trasmit a message to the receiver through the EH chael, ad the chael oise is modeled as a additive white Gaussia oise AWGN At each discrete time k,, }, a radom amout of eergy E k arrives at the battery ad the trasmitter seds a symbol X k X such that Xl almost surely E l Xl deote the eergy cosumed by trasmittig X l This implies that 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 The receiver observes Y k = X k +Z k at each time k Z k is a stadard ormal radom variable which is idepedet of X k ad Z k } k= are idepedet We assume that E l} are idepedet ad idetically distributed iid, E is a o-egative radom variable To simplify otatio, we write E E if there is o ambiguity Throughout the paper, we let E[E], the expected value of E, be the sigal-to-oise ratio SNR of the chael For the AWGN EH chael described above, referece [] showed that the capacity equals log + 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 will be accumulated withi the battery I the trasmissio phase the trasmitter seds the symbols of a radom Gaussia codeword with variace as log as the battery has sufficiet eergy 0 < deotes some small offset from The best-effort scheme ca be viewed as a save-ad-trasmit scheme without 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 for some 0 < Followig referece [], a umber of o-asymptotic achievable rates for save-ad-trasmit schemes have bee preseted [] [4] By cotrast, o o-asymptotic achievable rate exists for the best-effort scheme except for a special discrete memoryless EH chael with fiite battery studied i [5] ad a special discrete memoryless EH chael with o battery studied i [6] Therefore, we are motivated to prove the first o-asymptotic achievable rates for best-effort schemes over the AWGN EH chael This paper cotais two mai results First, we derive the first o-asymptotic achievable rate for the best-effort scheme The derivatio ivolves carefully desigig the trasmitted power to be O log / so that we ca effectively boud the umber of mismatched positios betwee the desired trasmitted codeword ad the actual trasmitted codeword for a fixed blocklegth Secod, we propose a savead-trasmit scheme with a similar trasmitted power /8/$ IEEE 87

2 08 IEEE Iteratioal Symposium o Iformatio Theory ISIT trasmitter AWGN receiver Fig The AWGN EH chael O log / ad obtai a ew o-asymptotic achievable rate which outperforms the best existig oe for all sufficietly large SNRs = E[X] > 0 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] The rest of the paper is orgaized as follows Sectio II describes the otatio used i this paper Sectio III presets the formulatio of the AWGN EH chael Sectio IV describes the best-effort scheme ad presets the first mai result Sectio V discusses the save-ad-trasmit scheme ad presets the secod mai result Sectio VI cocludes this paper ad discusses the extesio of the results to the block eergy arrival model [4], [0] II NOTATION We use O, Θ, ω ad o to deote stadard asymptotic Bachma-Ladau otatios except our covetio that they must be o-egative The sets of atural umbers, real umbers ad o-egative real umbers are deoted by N, R ad R + respectively All logarithms are take to base e throughout the paper Radom variables are deoted by capital letters eg, X, ad the realizatio ad the alphabet of a radom variable are deoted by the correspodig small letter eg, x ad calligraphic fot eg, X respectively We use X to deote a radom tuple X, X,, X, all 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 For ay fuctio f whose domai cotais X, we use E px [fx] to deote the expectatio of fx 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 zµ e σ πσ A roblem Formulatio III THE AWGN EH CHANNEL The AWGN EH chael, as illustrated i Figure, cosists of oe trasmitter ad oe receiver Eergy harvestig ad commuicatio occur i time slots, ie, chael uses I each time slot, a radom amout of eergy E, E = R +, is harvested E[E] > 0 ad E[E ] < The eergyharvestig 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, ie, p W,Ek,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 } Xl e l W = w, E = e = for each k,,, } After time slots, the receiver declares Ŵ based o Y to be the trasmitted W B Stadard Defiitios Formally, we defie a code as follows: Defiitio : A, M-code cosists of the followig: A message set W,,, M}, W is uiform o W A sequece of ecodig fuctios f k : W R k + R for each k,,, }, 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, ie, Ŵ = ϕy If the sequece of ecodig fuctios f i satisfies, the code is also called a, M-EH code 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 kp Y k X k p Yk X k y k x k = q Y X y k x k = π e y k x k 3 for all x k X ad y k Y For ay, M-code defied o the AWGN EH chael, let p W,E,X,Y,Ŵ be the joit distributio iduced by the code We ca factorize p W,E,X,Y,Ŵ as p W,E,X,Y,Ŵ = p W p Ek p Xk W,E kp Y k X k pŵ Y, k= which follows from the iid 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 4 87

3 08 IEEE Iteratioal Symposium o Iformatio Theory ISIT Defiitio 3: For a, M-code defied o the AWGN EH chael, we ca calculate accordig to 4 the average probability of decodig error defied as 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 Cx log + x for all x 0 ad defie E[E] It was show i [] that C ε = C = C for all ε 0, = E[X] ca be viewed as the sigal-to-oise ratio SNR of the AWGN EH chael IV AN ACHIEVABLE RATE FOR BEST-EFFORT A Best-Effort Scheme Fix a blocklegth Choose a positive real umber S < = E[E] ad let such that p X x N x; 0, S 5 S = E px [X ] 6 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 iid symbols is costructed X w p X Suppose W = w ad E = e, ie, 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 fk best } k= ad decodig fuctio ϕbest Defie f best, f best,, f best i a recursive maer fk best w, e k X k w 0 otherwise if X k w ek + k For each k,,, }, the trasmitter seds e l f best l w, e l, 7 X k W f best k W, E k 8 By costructio, Xl W } E l = 9 Upo receivig Ỹ W ỸW, ỸW,, ỸW ỸkW is geerated accordig to ỸkW = b X k W = a} q Y X b a, 0 the receiver declares that ϕ best Ỹ W = j if j is the uique iteger i W that satisfies log q Y XỸkW X k j log ξ, p Y ỸkW k= p Y is the margial distributio of p X q Y X ad log ξ is a arbitrary threshold Otherwise, the receiver chooses ϕ best Ỹ W W accordig to the uiform distributio The code described above is said to be best-effort because accordig to 7 ad 8, 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 relimiaries A importat quatity that determies the performace of the best-effort, M-EH code is Q k,,, } X } k W X k W, 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 Lemma : Fix ay ad ay ρ 0, such that 4ρ ρ <, 3 ad fix a best-effort, M-EH code with Defie ad S ρ 4 ρ α E[E ] + 3S β 5 α + 63α S 6 For ay γ R +, we have } Q γ + e γ β + α E[E ] 7 Remark : The proof of Lemma ca be foud i [, Appedix A] A importat step i the proof is aalyzig the escape probability τ = } of the Markov process E + k } l= El Xl τ τ is the stoppig time k= whe the value of the Markov process hits ay egative umber a < 0 I particular, τ = } e β + α E[E ] The followig lemma [] is stadard for provig achievability results i the fiite blocklegth regime ad its proof ca be foud i [3, Th 38] Lemma Implied by Shao s boud []: Let p X,Y be the probability distributio of a pair of radom variables X, Y Suppose X, Y p X,Y ad 873

4 08 IEEE Iteratioal Symposium o Iformatio Theory ISIT X, Y p X,Y are idepedet The for each δ > 0 ad each M N, we have log p Y X Y X } p Y Y > log M + δ M eδ The followig lemma is a modificatio of the Shao s boud stated i the previous lemma, ad its proof ca be foud i [, Appedix B] Lemma 3: Suppose we are give a best-effort, M-EH code as described i Sectio IV-A The for each γ R +, each δ > 0 ad each M N, we have Q < γ + } log p Y X Ỹ X p Y Ỹ } >logme δ S γ+ + Me δ 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 is cotaied i [, Sec IV ad Appedix B] Theorem : 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, there exists a best-effort, M, ε-eh code with ρ ad some choice of + E[E ] + 3 log ε log S = ρ = Θ log ξ = log M + O log log, log M log + E[E ] + 3 log ε log + + Φ ε κ log I particular, the probability of seeig more tha Θ / log mismatch evets ca be bouded as γ Q γ + } ε log ε β + α E[E ] = Θ log V AN ACHIEVABLE RATE FOR SAVE-AND-TRANSMIT I Sectio IV-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 A Save-ad-Trasmit Scheme Fix a blocklegth Choose a positive real umber S < = E[E] ad let p X ad S as defied i 5 ad 6 respectively The codebook cosists of M mutually idepedet radom codewords deoted by X w w W}, which are costructed as described i Sectio IV-A Suppose W = w ad E = e, ie, 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 } k= ad decodig fuctio ϕsave The savead-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 fk save w, e k X k w if k > γ ad Xk w ek + k e l fl save w, e l, 0 otherwise For each k,,, }, the trasmitter seds X k W fk save W, E k 8 By costructio, Xl W } E l = 9 Upo receivig Ỹ W ỸW, ỸW,, ỸW ỸkW is geerated accordig to 0, the receiver declares that ϕ save Ỹ W = j if j is the uique iteger i W that satisfies log q Y XỸkW X k j log ξ, p k=γ+ Y ỸkW 0 p Y is the margial distributio of p X q Y X ad log ξ is a arbitrary threshold 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 4 is quite similar to the proof of Lemma established for the best-effort scheme, ad it ca be foud i [, Appedix E] Lemma 4: Fix ay ad ay ρ 0, such that 3 holds, ad fix a save-ad-trasmit, M-EH code with S, α ad β beig defied as i 4, 5 ad 6 respectively For ay γ N, we have k=γ+ E i < i= i=γ+ X i eγ β + α E[E ] 874

5 08 IEEE Iteratioal Symposium o Iformatio Theory ISIT 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 4 ad Lemma, ad it is cotaied i [, Sec VI ad Appedix F] Theorem : 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, there exists a save-ad-trasmit, M, ε-eh code with + E[E ] + 3 log + log ε ρ, ad some choice of S = ρ = Θ/ log ξ = log M + Olog log M log + E[E ] + 3 log + log ε Φ ε κ 3/4 I particular, the probability of seeig a mismatch evet i the trasmissio phase ca be bouded as E i < Xi ε k=γ + i= i=γ + log ε γ = Θ β + α E[E ] The parameters ρ ad γ i Theorem have bee optimized to achieve the secod-order scalig O/ Fix ay ε > 0 The best existig lower boud o the secod-order term log M was derived i [4, Th ], which states that there of exists a save-ad-trasmit, M, ε-eh code that satisfies log M log + lim if log + E[E ] + log + ε + Φ ε 3 for ay ε > 0 ad ε > 0 such that ε + ε ε Note that the secod-order term of the best existig lower boudas stated o the RHS of 3 decays as log + + E[E ] log ε + Φ ε as teds to O the other had, it follows from i Theorem that the secod-order term of our lower boud decays as 3 + E[E ] log + log ε + Φ ε as teds to Cosequetly, the secod-order term achievable by the save-ad-trasmit scheme guarateed by Theorem is strictly larger less egative tha the best existig boud for all sufficietly large > 0 VI CONCLUDING REMARKS I this work, we prove 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 besteffort is O log / 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 savead-trasmit have bee exteded to the block eergy arrival model i the log versio of this paper [] the eergy arrives i a block iid fashio [4], [0] If the legth of each eergy block L grows subliearly i, ie, L = o, we show i [] that best-effort ad save-ad-trasmit achieve the secod-order scaligs O maxlog, L}/ ad O L/ A future directio may improve the secodorder scalig O log / for L = for best-effort schemes by possibly provig a sharper probability boud tha 7 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 with fiite battery REFERENCES [] O Ozel ad S Ulukus, Achievig AWGN capacity uder stochastic eergy harvestig, IEEE Tras If Theory, vol 58, o 0, pp , 0 [] S L Fog, V Y F Ta, ad J Yag, No-asymptotic achievable rates for eergy-harvestig chaels usig save-ad-trasmit, IEEE J Sel Areas Commu, vol 34, o, pp , 06 [3] K G Sheoy ad V Sharma, Fiite blocklegth achievable rates for eergy harvestig AWGN chaels with ifiite buffer, i roc IEEE Itl Symp If Theory, Barcelo, Spai, Jul 06, pp [4] S L Fog, V Y F Ta, ad A Özgür, O achievable rates of AWGN eergy-harvestig chaels with block eergy arrival ad o-vaishig error probabilities, IEEE Tras If Theory, vol 64, o 3, pp , 08 [5] J Yag, Achievable rate for eergy harvestig chael with fiite blocklegth, i roc IEEE Itl Symp If Theory, Hoolulu, HI, USA, Ju 04, pp 8 85 [6] E MolaviaJazi ad A Yeer, Low-latecy commuicatios over zerobattery eergy harvestig chaels, i roc IEEE Global Commuicatios Coferece Globecom 5, Sa Diego, CA, Dec 05 [7] K Tutucuoglu, O Ozel, A Yeer, ad S Ulukus, The biary eergy harvestig chael with a uit-sized battery, IEEE Tras If Theory, vol 63, o 7, pp , 07 [8] D Shaviv, -M Nguye, ad A Özgür, Capacity of the eergy harvestig chael with a fiite battery, IEEE Tras If Theory, vol 6, o, pp , 06 [9] W Mao ad B Hassibi, Capacity aalysis of discrete eergy harvestig chaels, IEEE Tras If Theory, vol 63, o 9, pp , 07 [0] F Zhag ad V K N Lau, Closed-form delay-optimal power cotrol for eergy harvestig wireless system with fiite eergy storage, IEEE Tras Sigal rocess, vol 6, o, pp , 04 [] S L Fog, J Yag, ad A Yeer, No-asymptotic achievable rates for Gaussia eergy-harvestig chaels: Best-effort ad save-adtrasmit, arxiv: [csit] [] C E Shao, Certai results i codig theory for oisy chaels, Iformatio ad Cotrol, vol, pp 6 5, 957 [3] T S Ha, Iformatio-Spectrum Methods i Iformatio Theory Berli, Germay: Spriger, Feb