State Amplification and State Masking for the Binary Energy Harvesting Channel

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1 State Amplfcaton and State Maskng for the Bnary Energy Harvestng Channel Kaya Tutuncuoglu, Omur Ozel 2, Ayln Yener, and Sennur Ulukus 2 Department of Electrcal Engneerng, The Pennsylvana State Unversty, Unversty Park, PA Department of Electrcal and Computer Engneerng, Unversty of Maryland, College Park, MD Abstract In ths paper, we consder a bnary energy harvestng transmtter that wshes to control the amount of sde nformaton the recever can obtan about ts energy harvests. Specfcally, we study state amplfcaton and state maskng, whch defne the maxmum and mnmum amount of state nformaton conveyed to the recever for a gven message rate, respectvely. For an ndependent and dentcally dstrbuted energy harvestng process, we frst fnd the amplfcaton and maskng regons for a transmtter wthout a battery and a transmtter wth an nfnte battery. Next, we fnd nner bounds for these regons for a unt-szed battery at the transmtter usng two dfferent encodng schemes, usng nstantaneous Shannon strateges and usng a scheme based on the equvalent tmng channel ntroduced n our prevous work. We observe that the former provdes better state amplfcaton, whle the latter provdes better state maskng. I. INTRODUCTION In the energy harvestng channel, the harvested energy s a random process that s revealed to the transmtter causally throughout the transmsson. Snce the transmtted symbols are constraned by the energy avalable at the encoder, the decoder obtans some nformaton about the energy harvestng process n addton to the ntended message. Dependng on the applcaton, t may be desrable to maxmze or mnmze ths sde nformaton about the harvested energy, e.g., t may facltate smart schedulng based on energy harvestng rates, but may also reveal the poston or energy source of a wreless node. These scenaros result n the problems of state amplfcaton [], [2] and state maskng [3], respectvely. The capacty of an energy harvestng channel s determned n prevous work for two extreme cases for the AWGN channel. When the battery-sze s unlmted, [4] shows that the capacty s equal to the capacty of the same system wth an average power constrant equal to the average recharge rate. At the other extreme, when the battery sze s zero, the system becomes a stochastc ampltude-constraned channel and the capacty n ths case s acheved [5] by usng Shannon strateges [6]. The capacty for the case of a fnte-szed battery s open n general. In the specal case of a noseless bnary channel and a unt-szed battery, [7] shows that the channel s equvalent to an addtve geometrc-nose tmng channel, for whch upper bounds and achevable rates usng tmng channel based encodng schemes are proposed. Concurrently, n [8], encodng schemes that utlze Shannon strateges are studed Ths work was supported by NSF Grants CNS /CNS , and CCF /CCF n a more general settng, and the correspondng achevable rates are found, where only the nstantaneous battery states are used to determne channel nput at each channel use. For the channel model n [7], [9] ntroduces a tghter upper bound and a better encodng scheme, outperformng..d. and frst order Markov Shannon strateges that utlze current battery state only. Reference [0] extends the model of [9] to a nosy channel wth arbtrary battery sze, and obtans upper bounds by provdng battery state nformaton to the recever. The fnte-szed battery model s also of nterest n an AWGN channel. In [], an approxmate capacty for ths channel model s found wth a constant gap usng ampltude constraned codebooks. For determnstc energy harvestng n an AWGN channel, [2] evaluates a lower bound on the capacty va the volume of the feasble nput set. The state amplfcaton problem,.e., sendng nformaton about system state along wth the message, s frst studed n []. Ths reference quantfes the sde nformaton revealed to the recever as the mutual nformaton between the output sequence and the state sequence. Ths problem s later consdered n terms of the dstorton n channel state estmaton n [2]. The state maskng counterpart,.e., concealng channel state as much as possble whle sendng a message, s ntroduced n [3] for both causal and non-causal state nformaton at the encoder. In an energy harvestng settng, the state amplfcaton problem s frst studed n [3] for an AWGN channel wth nfnte and no battery at the encoder. In ths paper, we evaluate state amplfcaton and state maskng regons, whch outlne the set of achevable message rates and state revealng rates for a bnary energy harvestng channel. Our goal s to nvestgate the trade-off between the energy arrval nformaton that the recever can extract from the communcaton and the message transmsson rate. We consder bnary energy arrvals as the state of the channel, where each energy unt corresponds to the energy requred to send a through the bnary channel once. We consder the no battery case and the nfnte battery case, for whch the channel capactes are known, and the unt battery case, for whch we derve nner bounds usng the achevable rates n [7], [8]. II. CHANNEL MODEL AND PROBLEM FORMULATION We consder a bnary symmetrc channel (BSC) wth an energy harvestng transmtter, as shown n Fg.. The crossover probablty of the bnary symmetrc channel s denoted by

2 W Fg.. E Encoder X Channel Y Decoder The energy harvestng channel wth battery sze. p e 2. In the th channel use, the encoder harvests an energy of E 0, },.e., harvests a unt of energy or not, and sends a bnary symbol X 0, } through the channel. The energy cost of sendng a s unt, whle sendng a zero does not requre any energy. The energy arrvals E are ndependent and dentcally dstrbuted (..d.) wth the probablty of arrval P r[e = ] = q. E values are revealed to the encoder causally, whle not avalable to the decoder. We consder two cases for the energy storage and consumpton model. If the encoder has no battery, denoted by = 0, then the channel nput s constraned by the energy harvested wthn the same channel use,.e., X E. For > 0, we consder a transmt-frst model. Denotng the state of the battery at the begnnng of the th channel use as B, the encoder frst transmts X B, and then stores the harvested energy E n the battery, provded that the battery capacty permts storage. Hence, the battery state evolves as B + = mnb X + E, }. () In ths work, we are nterested n how much the decoder can learn about the energy arrval process E n. From the decoder s perspectve, there are 2 H(En) possble energy arrval sequences, snce 2 H(En) s the sze of the typcal set for E n. Upon recevng Y n, the decoder can reduce the sze of ths lst to those that are possble gven Y n, whch has a sze of 2 H(En Y n). Hence, the reducton n the entropy of E n for the decoder can be expressed as = n (H(En ) H(E n Y n )) = n I(En ; Y n ). (2) Note that the value of s related strongly to the encodng scheme adopted by the encoder. For example, the encoder can choose to send X = 0 for all, thus achevng no message rate, but obtanng = 0. On the other extreme, the encoder can choose X = E, once agan achevng no message rate, but obtanng = H(E n ). For a non-zero rate, dfferent encodng schemes achevng the same rate may yeld dfferent values for. A. State Amplfcaton Problem In the state amplfcaton problem, the encoder wshes the decoder to obtan as much nformaton as possble about the energy harvestng process E n,.e., maxmze, whle relably conveyng a message wth some rate R. Ths problem s frst consdered n [], where the achevable message rates and state Ŵ amplfcaton rates are shown to satsfy R I(U; Y ), (3) H(S), (4) R + I(X, S; Y ), (5) for a memoryless channel wth state S known causally at the transmtter. Here, U s an auxlary random varable yeldng the jont dstrbuton p(s)p(u)p(x u, s)p(y x, s). B. State Maskng Problem The state maskng problem, studed frst n [3], fnds a lower bound on for any gven message rate R. Hence, t ndcates the mnmum amount of nformaton that must be revealed to the decoder about the state n order to acheve some rate R. The achevable (R, ) regons are obtaned by the unon of the regons R I(U; Y ), (6) I(S; Y U), (7) for causally avalable state at the encoder [3]. Note that (3) and (6) are dentcal, and (7) provdes a lower bound on whle (4) and (5) provde upper bounds. In the remander of ths paper, we consder the state amplfcaton and state maskng problems ndvdually for the cases = 0, =, and =. III. NO BATTERY CASE: = 0 For = 0, the energy avalable at channel use s E, whch s..d., and therefore the state of the channel s memoryless. Hence, the results of [] extend drectly to ths case. Gven the two states, E 0, }, the two nputs X 0, }, and the restrcton X E, there are two feasble mappngs from E to X. We refer to these mappngs as strateges, and denote them as U = (X, X), where X s the channel nput when E = 0 and X s the channel nput when E =. The two feasble strateges are (0, 0) and (0, ), correspondng to always transmttng a zero and attemptng to send a, respectvely. For an encodng strategy wth P r[u = (0, )] = p, the exact (R, ) regon for state amplfcaton s obtaned as R H(pq p e ) ph(q p e ) ( p)h(p e ), (8) H(q), (9) R + H(pq p e ) H(p e ), (0) where H(a) denotes the bnary entropy functon, and p q = p( q) + ( p)q. Next, we utlze the results of [3], and characterze the exact (R, ) regon for state maskng as (8) and ph(q p e ) ph(p e ). () We remark that for a noseless bnary channel,.e., p e = 0, the bounds for n (8)-(0) and () match only at R = H(pq) ph(q), = ph(q). (2)

3 IV. INFINITE BATTERY CASE: = We next consder the nfnte battery case,.e., =. Wth an nfnte battery, [4] showed that a save-and-transmt scheme acheves the AWGN capacty wth average transmt power. Ths scheme frst saves energy for a neglgble duraton of the transmsson, and then encodes as f constraned by an average power constrant only. The save-and-transmt scheme can be extended to the bnary channel, yeldng the capacty H(q p e ) H(p e ), q C BSC = 2, H(p e ), q > 2, (3) wth the channel nput dstrbuton P r[x = ] = mn ( q, 2). Note that ths s also the capacty of a BSC wth an nput constrant E[X] q, as s the case n [3] for the AWGN channel. Wth ths observaton, we present the state amplfcaton regon for ths channel n the followng lemma. Lemma The exact (R, ) regon for the bnary EH channel wth an nfnte-szed battery at the transmtter satsfes R + C BSC, 0 H(q). (4) Proof: We frst show the achevablty of these (R, ) pars. Clearly, the rate R = C BSC s achevable wth the saveand-transmt scheme, whch we assume to yeld = 0. Furthermore, by compressng the E n sequence and sendng t as a part of the message, the encoder can trade any porton of the message rate R wth, provded that ths porton does not exceed H(q) for q 0.5 and for q > 0.5. Due to the causal avalablty of E, ths s performed n a block Markov fashon. For the converse, we wrte I(X n ;Y n ) = I(X n, E n, W ; Y n ) (5) I(E n, W ; Y n ) (6) = I(E n ; Y n ) + H(W ) H(W Y n, E n ) (7) I(E n ; Y n ) + H(W ) H(ɛ) ɛ log(nr) (8) = n + nr H(ɛ) ɛ log(nr) (9) where W s the message and ɛ s the decodng error probablty. Here, (5) follows from the Markov chan (W, E n ) X n Y n, (7) s due to the ndependence of W and E n, and (8) follows from Fano s nequalty. Hence, whenever the decodng error probablty ɛ goes to zero as n, we have + R lm n I(Xn ; Y n ) C BSC, (20) whch concludes the converse. For the maskng problem, as emphaszed n [3], snce (R, ) = (C BSC, 0) s achevable, perfect maskng of the state E n s possble usng the save-and-transmt scheme. Hence, we have 0 as the maskng lower bound. V. UNIT BATTERY CASE: = In precedng sectons, we consdered cases for whch the channel capacty s known. Snce ths s not the case for =, we are not able to determne the entre (R, ) regon. In ths secton, we utlze two encodng schemes, proposed by [8] and [7], to fnd nner bounds on the (R, ) regon for the bnary noseless channel wth p e = 0. Although the channel s noseless, ths s a non-trval model due to ts memory and the state s dependence on the channel nput, for whch the capacty s an open problem. A. Instantaneous Shannon Strateges In [8], the Shannon strateges of [6], whch are capacty achevng for a memoryless channel, are used to fnd achevable rates for the energy harvestng channel. These strateges are mappngs from the current battery state B to the channel nput X. Gven two battery states, B 0, }, two nput symbols X 0, }, and the restrcton X S n each channel use, there are two feasble mappngs from B to X. As n Secton III, we denote these strateges as U (0, 0), (0, )}, where the former gves X = 0 for all B and the latter gves X = 0 for B = 0 and X = for B =. We consder a codebook consstng of strategy codewords U n, generated..d. wth P r[u = (0, )] = p. Upon selectng the codeword correspondng to the message, the encoder chooses X based on U and B n the th channel use. The acheved message rate for ths encodng scheme s gven by R IID = lm n I(U n ; Y n ). (2) To fnd the correspondng for ths encodng scheme, we frst defne the random varable ψ j = 0, E k = 0, k < j, (22), otherwse, whch s an ndcator of whether a unt of energy has arrved or not between the th and jth channel uses. We then defne the set Ψ(u n ) = ψ 2, ψ 3 2,... } as the collecton of mutually exclusve ndcators ψ k+ k, where =, and k, k = 2, 3,... are the channel ndces that satsfy u k = (0, ). In other words, Ψ(u n ) s the set of ndcators that show whether energy s avalable or not for each attempt of sendng a gven the strategy sequence u n. We then wrte H(E n Y n ) H(E n Y n, U n ) = I(E n ; U n Y n ) (23) H(U n Y n ) (24) H(W Y n ) (25) H(ɛ) ɛ log(nr) (26) where ɛ s the probablty of decodng error. Here, (25) s due to u n beng a functon of message w, and (26) s due to Fano s nequalty. Hence, whenever the error probablty ɛ goes to zero as n, we have lm n H(En Y n ) = lm Wth ths observaton, we wrte as lm n H(En Y n, U n ). (27) n I(En ; Y n ) = lm n (H(En ) H(E n Y n, U n )) (28)

4 = lm n (H(En ) H(E n Y n, U n, Ψ(U n ))) (29) = lm = lm n (H(En ) H(E n Ψ(U n ), U n )) (30) n H(Ψ(U n ) U n ) (3) Here, (28) follows from (27), and (29) holds as Ψ(U n ) can be obtaned from Y n and U n. Smlarly, (30) follows snce Y n can be obtaned from U n and Ψ(U n ). Fnally, (3) holds snce H(Ψ(U n ) U n, E n ) = 0, and U n s ndependent of E n. What the seres of equaltes n (28)-(3) mply s that, observng Y n, the decoder learns the ndcators Ψ(U n ) about E n, and nothng more. Note that the ntervals ( k, k+ ) are dsjont, and therefore the elements of the set Ψ(u n ) are ndependent. Snce P r[ψ j = 0] = ( q) j and u are generated..d. wth probablty P r[u = (0, )] = p, (3) can be further smplfed as = lm n ( p(u n ) u n k H ( ( q) δ k) ) (32) where δ k = k+ k denotes the number of channel uses between the kth and k+st u = (0, ) n u n, and s dstrbuted..d. geometrc wth parameter p. As n, due to the law of large numbers, the set Ψ(u n ) has np elements, yeldng = lm n I(En ; Y n ) = p 2 ( p) δ H ( ( q) δ). δ= (33) Lemma 2 For the..d. encodng scheme wth Shannon strateges, the decrease n entropy of E n upon observng Y n,.e.,, s equal to (33) for the noseless channel. B. Tmng-Based Encodng For the bnary noseless channel, [7] ntroduced an alternatve scheme where encodng s performed on an equvalent tmng channel. In the tmng channel, the number of channel uses spent watng for an energy are denoted by Z, and the number of channel uses between the energy arrval and the departure of a are denoted by V. The decoder observes the number of channel uses between consecutve s, gven by T = V + Z. (34) The tmng representaton s depcted n Fg. 2. Note that one use of the tmng channel corresponds to T uses of the bnary channel. Snce energy arrvals are..d., Z values are..d. and geometrc dstrbuted. The encodng scheme of [7] for frame length N uses Shannon strateges U 0,,..., N } to generate an..d. codebook. For codeword U m and causally revealed Z m, the encoder nserts V = (U Z mod N) + (35) to the tmng channel. Recevng T, the decoder can obtan U = T mod N wthout error. Hence, the rate acheved Z T V Z 2 V 2 V 3 T 2 T 3 Fg. 2. Graphcal representaton of the varables n the equvalent tmng channel T n, V n and Z n. Here, crcles represent energy arrvals and trangles represent transmsson of a symbol. wth ths scheme per use of the bnary channel s R (N) A = max p(u) H(U) E[T ]... bts/ch. use, (36) where E[T ] s the average length of each use of the tmng channel. For ths encodng scheme, we next calculate as a functon of the strategy dstrbuton p(u). For a gven output sequence T m = t m, we defne a = j= t j. Then, the th use of the tmng channel les on the a + st to a + t th uses of the bnary channel. For ths nterval, the decoder can nfer the followng: For t N, we have z t from (34). Usng the defnton n (22), ths mples ψa a+t =. Otherwse, for t > N, we have z t N snce v N by defnton. Ths mples that ψa a+t N = 0 and ψ a+t a +t N =. Based on these two cases, we defne the sets Ψ 0 (t m ) = ψa a+t N, (37) t >N Ψ (t m ) = and wrte t N ψ a+t a ( t >N ψ a+t a +t N ), (38) H(E n T m ) = t m p(t m )H(E n T m = t m ) (39) = t m p(t m )H(E n T m = t m, U m = u m, Ψ 0 (t m ) = 0, Ψ (t m ) = ) (40) = t m p(t m )H(E n Ψ 0 (t m ) = 0, Ψ (t m ) = ) (4) where Ψ 0 (t m ) = 0 denotes element-wse equalty for all elements of the set Ψ0 (t m ). Here, (40) holds snce u m can be obtaned from t m, whch also reveals that Ψ 0 (t m ) = 0 and Ψ (t m ) =. Note that (4) s the entropy of E n gven that parts of E n are zero and parts nclude at least one non-zero arrval. Calculatng and averagng over t m, we get n I(En ; Y n ) = H(q) E[T ] E [ δh(q) H(( q) δ ) ( q) δ ], (42) where δ s a random varable denotng the length of ψ terms n Ψ (t m ), and s dstrbuted as k, k < N, w.p. p U (k ) ( ( q) k), δ = N N, w.p. u=0 p U (u)( q) u+. (43)

5 =(/n)i(e n ;Y n ) (State entropy reducton rate) E = max =0 E = wth d Shannon enc. max = wth tmng enc. =(/n)i(e n ;Y n ) (State entropy reducton rate) E = max =0 E = wth d Shannon enc. max = wth tmng enc R (Message rate) R (Message rate) Fg. 3. The maxmum values wth respect to message rate R,.e., state amplfcaton boundares, for q = 0.5 and p e = 0. Fg. 4. The mnmum values wth respect to message rate R,.e., state maskng boundares, for q = 0.5 and p e = 0. Lemma 3 For the encodng scheme n [7], the decrease n entropy of E n upon observng Y n,.e.,, for a specfc auxlary dstrbuton p U (u) s equal to (42). Fnally, we obtan (R, ) pars for ths encodng scheme by exhaustvely searchng p U (u) and usng (36) and (42). VI. NUMERICAL RESULTS For comparson, we evaluate the state amplfcaton and state maskng regons for the = 0 and = cases, and the maxmum and mnmum achevable values of wth respect to some message rate R for the = case, n a noseless channel wth q = 0.5. The state amplfcaton results are plotted n Fg. 3, and the state maskng results are plotted n Fg. 4. In Fg. 3, we observe that the nstantaneous Shannon encodng strategy n Secton V-A performs state amplfcaton almost as good as the deal case of Secton IV for low message rates. As message rate approaches the best achevable rate, state amplfcaton s sacrfced. Moreover, we note that for the most part, nstantaneous Shannon encodng provdes more state nformaton than the tmng channel based encodng strategy n Secton V-B. For the state maskng problem n Fg. 4, we observe that for low rates, = provdes sgnfcantly better state maskng compared to the = 0 case. Smlar to the state amplfcaton case, tmng-based encodng delvers less state nformaton than nstantaneous Shannon encodng, although ths s desrable for the state maskng case. Hence, we conclude that tmng-based encodng outperforms nstantaneous Shannon encodng n state maskng, whle the reverse s true n state amplfcaton. VII. CONCLUSION In ths paper, we consdered the problems of state amplfcaton and state maskng n an energy harvestng bnary symmetrc channel. We focused on the no battery, nfnte battery, and unt battery cases. For the no battery case, we obtaned the regons b prevous results. For the nfnte battery case, we found that perfect state amplfcaton and perfect state maskng are possble, n the sense that message and state nformaton rates add up to the capacty of the channel n the case of state amplfcaton. For the unt battery case, we compared the nstantaneous Shannon strategy encodng scheme and the tmng channel based encodng scheme n the noseless case, and observed that the former provdes better state amplfcaton whle the latter provdes better state maskng. REFERENCES [] Y. H. Km, A. Sutvong, and T. M. Cover. State amplfcaton. IEEE Trans. on Inform. Theory, 54(5): , May [2] C. Choudhur, Y. H. Km, and U. Mtra. Causal state amplfcaton. In IEEE ISIT, pages 20 24, August 20. [3] N. Merhav and S. Shama. Informaton rates subject to state maskng. IEEE Trans. on Inform. Theory, 53(6): , June [4] O. Ozel and S. Ulukus. Achevng AWGN capacty under stochastc energy harvestng. IEEE Trans. on Informaton Theory, 58(0): , October 202. [5] O. Ozel and S. Ulukus. AWGN channel under tme-varyng ampltude constrants wth causal nformaton at the transmtter. In Aslomar Conference, November 20. [6] C. E. Shannon. Channels wth sde nformaton at the transmtter. IBM Jour. of Research and Development, 2(4): , 958. [7] K. Tutuncuoglu, O. Ozel, A. Yener, and S. Ulukus. Bnary energy harvestng channel wth fnte energy storage. In IEEE ISIT, July 203. [8] W. Mao and B. Hassb. On the capacty of a communcaton system wth energy harvestng and a lmted battery. In IEEE ISIT, July 203. [9] K. Tutuncuoglu, O. Ozel, A. Yener, and S. Ulukus. Improved capacty bounds for the bnary energy harvestng channel. In IEEE ISIT, June 204. [0] O. Ozel, K. Tutuncuoglu, S. Ulukus, and A. Yener. Capacty of the dscrete memoryless energy harvestng channel wth sde nformaton. In IEEE ISIT, June 204. [] Y. Dong and A. Özgür. Approxmate capacty of energy harvestng communcaton wth fnte battery. In IEEE ISIT, June 204. [2] V. Jog and V. Anantharam. An energy harvestng AWGN channel wth a fnte battery. In IEEE ISIT, June 204. [3] O. Ozel and S. Ulukus. Energy state amplfcaton n an energy harvestng communcaton system. In IEEE ISIT, July 202.