1 Sae Esimaion in Energy Harvesing Sysems Omur Ozel Venka Ananharam Deparmen of Elecrical Engineering and Compuer Sciences Universiy of California, Berkeley, CA 94720 ozel@berkeley.edu ananh@berkeley.edu Absrac We consider a discree ime scalar Kalman filering problem over an erasure channel wih an energy harvesing ransmier boh wih and wihou side informaion abou he erasure sae of he channel available a he ransmier. The cos of ransmiing an observaion is one uni of energy and he ransmier conrols is energy expendiure by leveraging he available energy sorage and side informaion. In his seing, he observaions are inermienly available a he receiver and his inermiency is conrolled by he ransmier. We sudy he hreshold for he growh rae of he dynamics ha guaranees he boundedness of he asympoic expeced sae esimaion error covariance under differen sysem seings. I. INTRODUCTION Wih recen developmens in MEMS, RF, and solar echnologies, energy harvesing is becoming an inegral par of many wireless sensors. In applicaions wih energy harvesing sensors, energy o run he circuiry of he sensor o sense, process, and ransmi he measured daa is ypically only inermienly available. Energy sorage is ypically available o save a porion of he harvesed energy for laer use. This enables he deploymen of an energy managemen policy o conrol he effecs of he inermiency in he arrival of energy. In his paper, we address he implicaions of his abiliy o manage he energy inermiency in he conex of sae esimaion of a scalar dynamical sysem a a remoe locaion conneced o he source of he dynamics over a noisy channel. Energy managemen for energy harvesing applicaions has recenly been an acive opic of research. In [1], [2], his problem was sudied in an informaion heoreic seing. [1] considers he channel capaciy of a poin-opoin addiive Gaussian channel wih an energy harvesing ransmier and unlimied energy sorage. I is shown ha he energy inermiency is smoohed ou by unlimied energy sorage in deermining he channel capaciy and he ransmier is effecively average power consrained. Laer, in [2], his problem is invesigaed in he finie baery regime. This work reveals ha sudying he growh rae wih block lengh of he volume of he se of energy feasible codewords enables quie accurae approximae capaciy expressions o be wrien in he finie baery regime. [3] considers he channel capaciy when i is he receiver ha is energy harvesing, focusing on he rade-off a he receiver beween he energy cos of sampling and he energy cos of decoding. In [4], asympoic opimaliy for scheduling sensing epochs is addressed in he finie and infinie baery regimes. In conras o hese works, our work is largely moivaed by he problem of Kalman filering wih inermien observaions ha was pioneered by he seminal work [5]. This work invesigaes he sae esimaion error covariance in a Gaussian Kalman filering problem wih vecor dynamics where he observaions are randomly erased by an i.i.d. (independen and idenically disribued) erasure process. The esimaion error covariance goes from bounded o asympoically unbounded a a hreshold for he erasure probabiliy, and [5] deermines his hreshold in cerain cases. This problem has laer been considered in various differen seings [6] [13]. Noable among hese is he work of Park and Sahai [12], [13], which deermines he exac value of he hreshold in he mos general seing wih i.i.d. erasures. Addiionally, he role of sampling in he deerminaion of his hreshold is considered in [12], [13]. Laer, in [14], he value of side informaion in he conex of his Kalman filering problem is addressed. Closer o he spiri of our work, in he conex of energy harvesing, [15] considers a remoe esimaion problem wih an energy harvesing sensor and a remoe esimaor. The objecive is o joinly minimize he communicaion coss and disorion by a communicaion sraegy a he ransmier and an esimaion sraegy a he receiver. Using dynamic programming, [15] shows ha joinly opimal sraegies are characerized by hresholds and he opimal esimae is a funcion of he mos recen observaion. This work, however, does no consider channel erasures. Our work is mos closely relaed o he work of [16], which considers a problem formulaion very similar o and in many ways more general han he one presened below. Our work is disinguished from [16] by he focus on he explici deerminaion of he hreshold on he growh rae of he dynamics ha allows for asympoically bounded esimaion error covariance. Throughou he paper, we use he expressions ransmission policy and ransmission sraegy inerchangeably. We wrie := for equaliy by definiion. For any sequence (α, 0), we wrie α for (α 0,..., α ). II. THE SETUP AND MODEL We consider he Kalman filering problem wih an energy harvesing ransmier. The seing corresponding o his problem is shown in Figure 1. The sae of he
2 sysem x evolves according o he following sochasic rule: x +1 = ax + ω, (1) where a is a real number wih a > 1, x 0 N (0, Π 0 ) wih Π 0 > 0, and ω N (0, κ) are i.i.d. wih (ω, 0) and x 0 being muually independen. The observaions are inermienly available o he receiver. The inermiency of he observaions is due o boh he channel erasures and decision of he ransmier o no ransmi he observaion, which in urn is moivaed by he need for energy managemen. The channel is of erasure ype wih wo possible saes 0 and 1 and erasure probabiliy p e. This is represened by a channel erasure process (γ, 0), which is i.i.d. and independen of (x 0, (ω, 0)), wih P (γ = 0) = p e. The energy arrival process is an i.i.d. binary process (E, 0) ha akes values 0 or 1 wih P (E = 1) = p a, and is independen of (x 0, ((ω, γ ), 0)). A each ime 0 he ransmier decides o ransmi or no o ransmi he observaion y = cx +ν, where c 0 and (ν, 0) is an i.i.d. observaion noise process independen of (x 0, ((ω, γ, E ), 0)), wih ν N (0, 1). This ransmission policy is denoed as S {0, 1} where S = 1 means he ransmier decides o send y and incurs a depleion of one uni of energy from he baery. There is no cos associaed o no ransmiing. Arriving energy is immediaely available for daa ransmission. Accordingly, S = 1 is possible only when he baery is non-empy or E = 1. The number of unis of energy in he baery a ime, B, is updaed as B +1 = min{b + E S, B max }, (2) where S = 1 is possible only when B + E > 0. Here, B max is he baery sorage capaciy and B 0 is iniial number of unis of energy in he baery, which is assumed o be deerminisic, i.e. a fixed ineger. This is only for noaional convenience, as discussed in Remark 2. We assume B 0 is finie and B 0 {0, 1,..., B max }. We consider several scenarios for he informaion available o he ransmier when deermining S. Access o he realizaion of he erasure environmen ha will be faced on he curren ransmission is called side informaion in his documen. Furher, we also make a disincion beween he causal and he anicipaive cases - he laer is mean o capure he scenario where he ransmier migh have lookahead informaion abou he fuure of he energy arrival process and/or he erasure environmen of he channel. Anicipaive scenarios of his kind migh be of ineres in pracice if he energy arrival process and/or he erasure environmen is influenced by some feaures exernal o he model ha he ransmier has access o. More precisely, he following scenarios for he informaion available o he ransmier will show up in his documen. In all hese scenarios, (ζ, 0) is a sequence of i.i.d. random variables, each uniformly disribued on [0, 1], and independen of (x 0, ((ω, ν, E, γ ), 0)), which are used by he ransmier for randomizaion of B max E {0,1} Sysem x +1 = ax + ω y = cx + ν y B Transmier y if S = 1, 0 else Erasure Channel γ Possible Side γ {0,1} Informaion ỹ Esimaor Fig. 1. Sysem model. The side informaion may or may no be presen. he ransmission sraegy, if desired. S = S (B 0, E, ζ ), which we call he causal case wih no side informaion. S = S (B 0, E, γ, ζ ), which we call he causal case wih side informaion. S = S (B 0, Ē, ζ ), which we call he anicipaive case wih no side informaion. S = S (B 0, Ē, γ, ζ ), which we call he energy anicipaive case wih side informaion. S = S (B 0, Ē, γ, ζ ) which we call he anicipaive case wih side informaion. The reason for explicily including B 0 in hese bulles is o accommodae he siuaion where B 0 migh be random, as described in Remark 2. I is imporan o noe ha in all hese cases he decision of he ransmier as o when o ransmi he observaion does no depend on he observaions. A consequence of his is ha ((E, γ, S, ζ ), 0) is independen of ((x, y, ω, ν ), 0). This independence is crucial o suppor he resuls claimed in his paper. We urn now o discuss he receiver/esimaor. Le γ := γ S, so ( γ, 0) is he effecive erasure sequence. We are ineresed in sudying he evoluion of he predicion error covariance as he receiver ries o predic he sae a ime, given all he observaions available o i prior o ime, 0. This informaion is ( γ 1, η 1 ), where η := y 1( γ = 1) for 0. The MMSE (minimum mean square error) predicor of he sae is hen given by ˆx 1 := E [ x γ 1, η 1], 0, (3) which reads ˆx 0 1 = E[x 0 ] a = 0, by convenion, and he predicion error covariance is Π := E [ (x ˆx 1 ) 2 γ 1, η 1], (4) which reads Π 0 = E[ ( x 0 ˆx 0 1 ) 2] a = 0, by convenion, and is consisen wih he prior use of he noaion Π 0, because ˆx 0 1 = 0. Regarding he evoluion of he predicion error covariance, one can use radiional Kalman filering heory o prove he following lemma. Lemma 1 The predicion error covariance evolves according o he following random Riccai equaion, iniial-
3 ized a Π 0 : Π +1 = a 2 Π + κ γ a 2 c 2 Π 2 c 2 Π + 1. (5) Proof: The correcness of he iniializaion is seen from ˆx 0 1 = 0. In order o prove he lemma, we consider, for σ 2 > 0, he scenario where he observaions ( γ 1, η 1 ) are augmened o ( γ 1, η 1 ), where { y, if γ η := = 1, (6) y + β, if γ = 0, where (β, 0) is an i.i.d. sequence independen of (x 0, ((ω, γ, E, ν, ζ ), 0)), wih β N (0, σ 2 ). In his augmened scenario, because ( γ, 0) is independen of ((x, y, ω, ν ), 0), radiional Kalman filering heory applies, and we can wrie an evoluion equaion for he corresponding predicion error covariance. We hen focus on he asympoic scenario σ 2. The updae equaions for he predicor error covariance become equaion (5) in his limi. Furher, in his asympoic limi, if he channel incurs erasure, i.e. γ = 0, or he ransmier decides o no ransmi he observaion, i.e. S = 0, i is as if he esimaor observes noise of infinie variance in he oupu, i.e., he measuremen is los. The correcness of his inerpreaion of he asympoic limi can be jusified as done in [5]. Remark 1 Even hough he evoluion equaion for he predicion error covariance is idenical for all he ransmission informaion srucures we consider, he acual predicion error covariance processes are differen, because hey are driven by differen ( γ, 0). Remark 2 The assumpion ha B 0 is deerminisic may give he impression ha B 0 is assumed o be known o he receiver. However, all he resuls in his documen hold when B 0 is assumed o be a random finie ineger aking values in {0, 1,..., B max }, independen of (x 0, ((ω, γ, E, ζ ), 0)), and no known o he receiver. This is because (B 0, (E, γ, S, ζ ), 0) would hen be independen of ((x, y, ω, ν ), 0) under our assumpion ha he ransmier does no have access o he observaions in deciding he imes a which o ransmi. This would imply ha he evoluion of he predicion error covariance a he receiver coninues o obey equaion (5). Corollary 1 For 0, we have κ Π +1 κ + a2 c 2 if γ = 1. (7) Proof: From equaion (5), we have Π +1 = a 2 Π + κ a2 c 2 Π 2 c 2 Π + 1 if γ = 1. Since a2 c 2 Π 2 c 2 Π +1 a2 Π, we have he claimed lower bound. For he upper bound, observe ha a 2 u a2 c 2 u 2 c 2 u + 1 a2 for all u 0. c2 Corollary 2 For each 0, le denoe he ime elapsed since he mos recen ransmission if here has been a prior ransmission, and = oherwise. Noe ha = does no disinguish beween wheher or no here was a ransmission a ime 0. Then we have Π +1 min(π 0, κ)a 2, for all 0. (8) In paricular, his implies ha Π min(π 0, κ), for all 0. (9) Proof: From equaion (5), Π 1 κ if γ 0 = 0, and his is also rue if γ 0 = 1, by equaion (7), so since 0 = 0 he claim in equaion (8) is rue for = 0, i.e. for Π 1. Assume i is rue for Π for some 1. If S = 0, hen γ = 0, so from equaion (5) we have Π +1 a 2 Π, and since = 1 + 1 in his case, he claim is rue. If S = 1 and γ = 1, hen from equaion (7) we have Π +1 κ, and since = 0 in his case, he claim is rue. If S = 1 and γ = 0, we have = 0 and Π +1 κ from equaion (5), so he claim is once again rue. This proves equaion (8) by inducion. The ruh of equaion (9) follows immediaely, since i is also rue a = 0. Corollary 3 For each 0, le denoe he ime elapsed since he mos recen successful ransmission if here has been a successful prior ransmission, and = oherwise. Noe ha = does no disinguish beween wheher or no here was a successful ransmission a ime 0. Then we have where Ma 2 Π +1, for all 0, (10) M := a 2 (max(π 0, 1 c 2 ) + κ a 2 1 ). (11) Proof: To prove equaion (10), we will acually prove a 2 max(π 0, 1 c 2 )a2 + κ a 2 1 (a2( +1) 1) Π +1, for all 0. Since 0 = 0, his is rue a = 0, by equaions (5) and (7). Assume i is rue for Π for some 1. If γ = 0, hen = 1 + 1, and since a 2 κ a 2 1 (a2( 1+1) 1) + κ = κ a 2 1 (a2( +1) 1). he claim holds for Π +1. If γ = 1, hen = 0, and he claim holds for Π +1 by equaion (7), compleing he proof.
4 Lemma 2 Le T 1 and T 2 be independen random variables wih P (T 1 = k) = (1 α) k 1 α and P (T 2 = k) = (1 β) k 1 β, k 1, where 0 < α, β < 1. Le T := T 1 +T 2. Noe ha T 2. We have lim k P (T = k+1 T > k) = min(α, β) Proof: The saemen follows by a direc calculaion. We nex urn o presen our main resuls. Our ineres is in he boundedness of he ime asympoe of he expeced value of he predicion error covariance, i.e. we are ineresed in sudying (E[Π ], 0). 1 A. p a = 0 III. EXTREME CASES The case p a = 0 corresponds o no arrival of energy. In his case γ = 0 for all 0, excep for a mos B 0 ime insances, so we always have lim E[Π ] =, from equaion (5). B. p e = 1 In he case p e = 1 all ransmissions are erased. Thus we have γ = 0 for all 0, so we always have lim E[Π ] = in his case, from equaion (5). C. p a = 1 The case p a = 1 corresponds o arrival of energy in every ime slo. We may assume ha 0 p e < 1 since he case p e = 1 has already been covered. I is inuiively apparen ha he bes ransmission sraegy o minimize he growh of he expeced predicor error covariance is o ransmi he observaion in each ime slo. This is formally jusified by he following resul, which covers all five ransmier informaion srucures of ineres and also idenifies he hreshold a for he asympoic boundedness of he expeced predicor error covariance. Theorem 1 When p a = 1 and 0 p e < 1, if a 2 p e < 1 he causal deerminisic sraegy wih no side informaion of ransmiing a every energy arrival ime has lim sup E[Π ] <. Conversely, when a 2 p e > 1, for any randomized anicipaive policy wih side informaion we have lim E[Π ] =. Proof: For he firs saemen, firs wrie equaion (5) as Π +1 = (a 2 Π + κ)1( γ = 0) + Π +1 1( γ = 1), (12) From equaion (7) we ge Π +1 a 2 Π 1( γ = 0) + a2 c 2 1( γ = 1) + κ. (13) Now, for he policy under consideraion, we have γ = γ for all 0. This gives, on aking expecaions and observing ha γ is independen of Π, E[Π +1 ] a 2 p e E[Π ] + a2 c 2 + κ, 1 When B 0 is random, we are in effec sudying (E[Π B 0 ], 0). which proves he saemen. For he second saemen, from equaion (5) we have which can be ieraed o give and hence Π +1 a 2 Π 1( γ = 0), (14) Π +1 a 2(+1) Π 0 1( γ 0 = 0,..., γ = 0), (15) Π +1 a 2(+1) Π 0 1(γ 0 = 0,..., γ = 0). Taking expecaions gives E[Π +1 ] (a 2 p e ) +1 Π 0, which proves he saemen. D. B max = 0 We may assume ha 0 < p a < 1 and 0 p e < 1, since he oher cases have already been covered. Noe ha we mus necessarily have B 0 = 0 when B max = 0. I is inuiively apparen ha he bes ransmission sraegy o minimize he growh of he expeced predicor error covariance is o ransmi he observaion in each ime slo. This is formally jusified by he following resul, which covers all five ransmier informaion srucures of ineres and also idenifies he hreshold a for he asympoic boundedness of he expeced predicor error covariance. Theorem 2 When B max = 0, 0 < p a < 1, and 0 p e < 1, if a 2 (1 p a (1 p e )) < 1 he causal deerminisic sraegy wih no side informaion of ransmiing a every energy arrival ime has lim sup E[Π ] <. Conversely, when a 2 (1 p a (1 p e )) > 1, for any randomized anicipaive policy wih side informaion we have lim E[Π ] =. Proof: The proof is idenical in srucure o he proof of Theorem 1. For he firs saemen, we wrie equaion (5) in he form of equaion (12) and hen use equaion (7) o ge he bound in equaion (13). We observe ha 1( γ = 0) = 1 1(γ = 1, E = 1) for he policy under consideraion. We hen ake expecaions, observing ha (γ, E ) is independen of Π, o ge E[Π +1 ] a 2 (1 p a (1 p e ))E[Π ] + a2 c 2 + κ, which proves he saemen. For he second saemen, we sar wih he lower bound in equaion (14) as in he proof of Theorem 1 and ierae as here o ge he lower bound in equaion (15). We hen observe ha 1( γ s = 0) 1 1(γ s = 1, E s = 1) for all s 0 o ge Π +1 a 2(+1) Π 0 (1 1(γ s = 1, E s = 1)). s=0 Taking expecaions gives E[Π +1 ] (a 2 (1 p a (1 p e ))) +1 Π 0, which proves he saemen.
5 IV. THE CASE OF p e = 0 We may assume ha 1 B max and 0 < p a < 1, since he oher cases have already been covered. When he channel is erasure-free, i.e., p e = 0, no disincion needs o be made beween he case wih side informaion and he case wihou side informaion, so ha here are in effec only wo ransmier informaion srucures of ineres, namely he causal case wih no side informaion, and he anicipaive case wih no side informaion. A. The causal case wih no side informaion The hreshold a for asympoic boundedness of he expeced predicion error covariance is idenified in he following heorem. Theorem 3 Le p e = 0, 0 < p a < 1, and 1 B max. Le he iniial baery level be any finie ineger B 0 {0, 1,..., B max }. If a 2 (1 p a ) < 1 he causal deerminisic sraegy wih no side informaion of ransmiing a every energy arrival ime has lim sup E[Π ] <. Conversely, if a 2 (1 p a ) > 1 hen for every causal randomized sraegy wih no side informaion we have lim E[Π ] =. Proof: The proof of he firs saemen follows he paern of he corresponding par of he proof of Theorem 1. Wriing equaion (5) for his policy as in equaion (12), we hen use equaion (7) o ge he inequaliy in equaion (13). We hen observe ha 1( γ = 0) equals 1(E = 0) for he ransmission sraegy under consideraion. We ake expecaions in equaion (13), observing ha E is independen of Π, o ge E[Π +1 ] a 2 (1 p a )E[Π ] + a2 c 2 + κ, which proves he saemen. To prove he second saemen, pick ɛ > 0 such ha a 2(1 ɛ) (1 p a ) > 1. Fix any causal randomized sraegy wih no side informaion. Noe ha 1( γ = 0) = 1(S = 0), because p e = 0. We claim ha lim P (S ɛ = 0,..., S = 0 E 0 = 0,..., E = 0) = 1. To see his, for noaional convenience, le us denoe ψ := P (S ɛ = 0,..., S = 0 E 0 = 0,..., E = 0). Firs noe ha (16) ψ = P (S ɛ = 0,..., S = 0 E 0 = 0,..., E u = 0), (17) for all u. This is because (E +1,..., E u ) is independen of (S ɛ,..., S, E ), because he sraegy is causal. We will show ha, for all δ > 0, here can be only finiely many values of for which ψ 1 δ, which suffices o esablish he claim of equaion (16). To see his, given δ > 0, suppose here is a value of, call i 1, for which ψ 1 δ. Pick s 1 so large ha 1 < ɛs 1 and ask if here is value of, > s 1, for which ψ 1 δ. If here is such a value, call i 2, hen pick s 2 so large ha 2 < ɛs 2, and ask if here is a value of, > s 2, for which ψ 1 δ, and so on. There can be a mos B0 δ imes where he kind of choice of desired can be found. To see his, suppose here are imes 1 < 2 <... < N of he kind desired where N > B0 δ. In view of equaion (17), we have, for each 1 n N, P ((S ɛn = 0,..., S n = 0) c E 0 = 0,..., E N = 0) δ. Since he ime inervals { ɛ n,..., n } for 1 n N are disjoin, adding hese inequaliies shows ha he expeced number of ransmissions over he ime inerval {0,..., N }, condiioned on he even {E 0 = 0,..., E N = 0}, is sricly bigger han B 0, bu his is impossible. Having esablished he correcness of he claim in equaion (16), we hen sar wih equaion (5) and ge he inequaliy in equaion (14), as in he proof of Theorem 1. Raher han ieraing all he way o ge he inequaliy in equaion (15), we ierae parially o ge Π +1 a 2(+1 ɛ ) Π ɛ 1( γ ɛ = 0,..., γ = 0). From he lower bound in equaion (9), his gives Π +1 a 2(+1 ɛ ) min(π 0, κ)1( γ ɛ = 0,..., γ = 0), = a 2(+1 ɛ ) min(π 0, κ)1(s ɛ = 0,..., S = 0). Taking expecaions, we have E[Π +1 ] a 2(+1 ɛ ) min(π 0, κ)p (S ɛ = 0,..., S = 0), a 2(+1 ɛ ) min(π 0, κ)ψ (1 p a ) +1, (a 2(1 ɛ) (1 p a )) min(π 0, κ)ψ (1 p a ), which, in view of equaion (16), proves he second saemen of he heorem. B. The anicipaive case wih no side informaion The hreshold a for asympoic boundedness of he expeced predicion error covariance is idenified, in a weak form, in he following heorem. Noe ha he converse saemen is weaker han desired. Wha we would really like o prove is ha lim inf E[Π ] =. Theorem 4 Le p e = 0, 0 < p a < 1, and 1 B max. Le he iniial baery level be any finie ineger B 0 {0, 1,..., B max }. If a 2 (1 p a ) < 1 here is an anicipaive randomized sraegy wih no side informaion, which has lim sup E[Π ] <. Conversely, if a 2 (1 p a ) > 1 hen for every anicipaive randomized sraegy wih no side informaion we have 1 E[Π u] =. lim 1
6 Proof: We consruc he sraegy promised in he firs saemen of he heorem. We may assume ha B 0 1, since he case B 0 = 0 is already covered by Theorem 3, in fac wih a causal deerminisic sraegy. If here is an energy arrival a ime 0 we ignore i, i.e. we keep he one uni of energy i brings in he baery if possible ill i is los due o baery overflow (which migh never happen), wihou ever using i. The successive energy arrival imes from ime 1 onwards are denoed L 1, L 1 + L 2, L 1 +L 2 +L 3,.... Thus he L k for k 1 are i.i.d. wih P (L k = s) = (1 p a ) s 1 p a for s 1. The ransmier generaes B 0 1 addiional i.i.d. random variables, denoed L 0, L 1, L 2,..., L B0+2, based on is random seed ζ 0. We also have P (L k = s) = (1 p a ) s 1 p a for s 1 for each B 0 + 2 k 0. Concepually, we hink of hese arificially generaed random variables as represening arrivals a imes L 0, L 0 L 1,..., L 0 L 1... L B0+2. We also hink of one of he unis of energy in he baery a ime 0 (recall we have assumed ha B 0 1) as having arrived a ime 0. Wih his viewpoin in mind, le us wrie A k for L k+1... L 0, for k =,..., 1, wrie A 0 for 0, and wrie A k for L 1 +... + L k for k 1. We also define B0 W k := L k+1 + B 0 1 L k+2 +... + 1 L k+b0 B 0 (18) for k. Our sraegy is o ransmi a each of he imes of he ype A k + W k, k, ha is nonnegaive. To show ha his policy works, we firs need o verify ha each of he proposed ransmission imes is disinc. This can be seen by checking ha, for each k, A k+1 + W k+1 (A k + W k ) = L k+1 + W k+1 W k = L k+1 + B0 L k+1 +... + L k+1+b0 j=1 jl k+b 0 j+2 1. B0 Noe ha his calculaion also shows ha 1+ L k+1 +... + L k+1+b0 j=1 jl k+b 0 j+1 A k+1 +W k+1 (A k +W k ). We also need o verify ha his policy is energy-feasible, i.e. ha i does no require a ransmission o occur a a ime unless here is energy in he baery a ha ime or here is an energy arrival a ha ime. To see his, i suffices o verify ha for each k we have A k + W k A k, which is obvious, because W k 0, and also A k + W k < A k+b0, which follows from B0 j=1 jl k+b 0 j+1 < L k+1 +... + L k+b0, where he sric inequaliy is because each L i is sricly posiive. Finally, we claim ha lim sup E[Π ] < under his policy. To show his, consider any ime > 0. By Corollary 3, if he mos recen ransmission occurred a ime 0 k, we have Π +1 Ma 2k, while if here has been no ransmission a any ime in {0, 1,..., }, we have Π +1 Ma 2. Here M is given by equaion (11). We have E[Π +1 ] Ma 2 P (A 0 + W 0 > ) + Ma 2k P (A j + W j = k, j=0 A j+1 + W j+1 > ), Ma 2k (k + ) P (L 1 +... + L B0+1 ()(k 1)). L 1 +... + L B0+1 is negaive binomial wih success probabiliy p a and as a 2 (1 p a ) B0+1 < 1, we have k=1 a2k (k + )P (L 1 +... + L B0+1 (B 0 + 1)(k 1)) <. Therefore, he firs saemen in he claim is proved. To prove he second saemen, pick ɛ > 0 such ha a 2(1 2ɛ) (1 p a ) > 1. Le T 1 denoe he ime of he firs energy arrival, so P (T 1 = s) = (1 p a ) s p a, s 0. For each 0, as in Corollary 2, le denoe he ime elapsed since he mos recen ransmission under he policy under consideraion, if here has been a prior ransmission, and = oherwise. From equaion (8), we have Π, u+1 min(π 0, κ)a 2 u 1(T 1 ), for all 0 u 1. Hence 1 1 1 1 Π u+1 c 1 a 2 u 1(T 1 ), where c 1 := min(π 0, κ). Taking expecaions gives 1 1 c 1 s= E[Π u+1 ] (1 p a ) s p a E[ 1 1 a 2 u T 1 = s]. (19) Condiioned on T 1 = s, here can be a mos B 0 ransmission aemps in he inerval {0, 1,..., 1}. If u 1,..., u B0, u B0+1 are he lenghs of duraions ino which hese ransmissions spli he duraion {0, 1,..., 1}, so B0+1 k=1 u k = (and some of he u k are allowed o be 0
7 if here are fewer han B 0 ransmissions), hen we have 1 a 2 u = B 0+1 k=1 a 2u k 1 a 2 1 where c 2 := B0+1 a 2 1, by Jensen s inequaliy. Puing hese ogeher gives c 2(a 2 1), 1 1 1 E[Π u+1 ] c 3 (1 p a) a 2, for a suiable consan c 3 > 0, for all sufficienly large, which complees he proof of he second saemen of he heorem. V. 0 < p e < 1 WITH NO SIDE INFORMATION Suppose 0 < p e < 1. We may assume ha 1 B max and 0 < p a < 1, since he oher cases have already been covered. If here is no side informaion here are wo informaion srucures of ineres for he ransmier, and he following heorem idenifies he hreshold a for asympoic boundedness of he expeced predicion error covariance in boh hese cases. Theorem 5 Suppose 0 < p e < 1, 0 < p a < 1, 1 B max, and le he iniial baery level be any finie ineger B 0 {0, 1,..., B max }. If a 2 (1 p a (1 p e )) < 1 he causal deerminisic policy wih no side informaion of ransmiing a every energy arrival ime has lim sup E[Π ] <. Conversely, if a 2 (1 p a (1 p e )) > 1 for every anicipaive randomized policy wih no side informaion we have lim E[Π ] =. Proof: The proof of he firs saemen is idenical o he proof of he firs saemen in Theorem 2 (he condiion ha B max = 0 in he hypohesis of ha heorem was no used in he proof of is firs saemen). To prove he second saemen, noe ha for each 0 he probabiliy of having k arrivals of energy in he ime inerval {0,..., } is ( ) +1 k p k a (1 p a ) +1 k, 0 k + 1. The oal number of ransmission aemps of he policy during his ime inerval can be a mos B 0 +k if here are k arrivals of energy during he inerval. Since (B 0, ((E, ζ ), 0)) is independen of (γ, 0), each ransmission aemp will be independenly erased wih probabiliy p e, irrespecive of he paern of he ransmission aemps. We conclude ha P ( γ = 0 ) +1 p B0+k e ( + 1 k ) p k a(1 p a ) +1 k, = p B0 e (1 p a + p a p e ) +1, (20) where 0 denoes he sring of + 1 zeros, (0,..., 0). We hen proceed as in he proof of he second saemen of Theorem 1. From equaion (5) we have he lower bound in equaion (14). Ieraing his gives he lower bound in equaion (15). Taking expecaions and using equaion (20) gives E[Π +1 ] a 2(+1) Π 0 p B0 e (1 p a + p a p e ) +1, which proves he saemen. VI. 0 < p e < 1 WITH SIDE INFORMATION Suppose 0 < p e < 1. We may assume ha 1 B max and 0 < p a < 1, since he oher cases have already been covered. If here is side informaion here are hree informaion srucures of ineres for he ransmier. The following heorem idenifies he hreshold a for asympoic boundedness of he expeced predicion error covariance in he causal case wih side informaion. Theorem 6 Suppose 0 < p e < 1, 0 < p a < 1, 1 B max, and le he iniial baery level be any finie ineger B 0 {0, 1,..., B max }. If a 2 (1 p a ) < 1 and a 2 p e < 1, here is a deerminisic causal ransmission sraegy wih side informaion such ha lim sup E[Π ] <. Conversely, if eiher a 2 (1 p a ) > 1 or a 2 p e > 1, for every causal randomized policy wih side informaion we have lim E[Π ] =. Proof: To prove he firs saemen we consider he following sraegy: wai for a fresh uni of energy o arrive a ime 1 or laer. Once i has arrived, he baery will have a leas one uni of energy. Then wai ill he firs ime ha here is side informaion ha he channel is erasurefree, and ransmi a ha ime, call i V 1. Wai for he firs arrival of energy a one of he imes from V 1 + 1 onwards, hen, once i has arrived, wai for he nex side informaion ha he channel is erasure-free and ransmi a ha ime, ec. Wih his sraegy, ransmissions occur a he ime of a renewal process wih iner-ransmission imes i.i.d. and having he disribuion of V, which is he sum of wo independen random variables T 1 and T 2, wih P (T 1 = k) = (1 p a ) k 1 p a for k 1 and P (T 2 = k) = p k 1 e (1 p e ) for k 1. Since all ransmissions are guaraneed o be successful, = for all 0, and (, 0) evolves as a Markov chain wih P ( +1 = 1 = 0) = 1, P ( +1 = k + 1 = k) = P (V > k + 1 V > k), P ( +1 = 0 = k) = P (V = k + 1 V > k). By equaion (10), we have E[Π +1 ] ME[a 2 ] = a 2k P ( = k),
8 wih M given as in equaion (11). We have +1 a 2k P ( +1 = k) = P ( +1 = 0) + a 2 P ( +1 = 1) +1 + a 2k P ( +1 = k), k=2 = P ( +1 = 0) + a 2 P ( = 0) + a 2 ( a 2k P ( = k))p (V > k + 1 V > k). k=1 According o Lemma 2, lim k P (V > k + 1 V > k) = max((1 p a ), p e ), from which i follows ha if we have boh a 2 (1 p a ) < 1 and a 2 p e < 1, hen, for all ɛ > 0, we have +1 a 2k P ( +1 = k) C + a 2 (max((1 p a ), p e ) + ɛ)( a 2k P ( = k)) for some finie consan C, for all sufficienly large. This implies ha lim sup E[Π +1 ] <. To prove he second saemen, fix 1 B max and he iniial baery level B 0 {1,..., B max }, a finie ineger. Fix 0 < p a < 1, and consider any 0 < p e < 1. Suppose here exiss a causal randomized policy wih side informaion for which lim inf E[Π ] <. Consider he scenario where p e = 0. The ransmier can arificially creae an i.i.d. sequence of erasure wih probabiliy p e and implemen his sraegy, because, having access o he realizaion of he erasures, i has he requisie side informaion. Bu we already know from Theorem 3 ha if a 2 (1 p a ) > 1 we have lim E[Π ] =. This means ha for every 0 < p e < 1, for every causal randomized policy wih side informaion we mus have lim E[Π ] =. Similarly, fix 0 < p e < 1 and consider any 0 < p a < 1. Suppose here exiss a causal randomized policy wih side informaion for which lim inf E[Π ] <. Consider he scenario where p a = 1. The ransmier can arifically drop energy arrivals i.i.d. wih drop probabiliy 1 p a, and hen implemen he given sraegy. Bu we already know from Theorem 1 ha if a 2 p e > 1 for every randomized causal policy wih side informaion we have lim E[Π ] = (in fac we know his for randomized anicipaive policies wih side informaion). Thus i mus be he case ha for all 0 < p a < 1 for every causal randomized policy wih side informaion we have lim E[Π ] =. This complees he proof of he second saemen. For he wo anicipaive cases wih side informaion, a he ime of wriing we have only he following weak parial resul. Theorem 7 Suppose 0 < p e < 1, 0 < p a < 1, B max =, and le he iniial baery level be any finie ineger B 0 {0, 1,...}. If a 2 (1 p a ) < 1 and a 2 p e < 1, here is a randomized energy anicipaive ransmission sraegy wih side informaion such ha lim sup E[Π ] <. On he converse side, suppose 0 < p e < 1, 0 < p a < 1, 1 B max, and le he iniial baery level be any finie ineger B 0 {0, 1,..., B max }. If eiher a 2 (1 p a ) > 1 or a 2 p e > 1, for every randomized anicipaive policy wih side informaion we 1 1 have lim E[Π u] =. Proof: For he proof of he firs saemen, we se up he imes A k and he inervals W k, k exacly as in he proof of he firs saemen of Theorem 4, and view hose of he imes A k + W k, k ha are nonnegaive as poenial ransmission imes. Le A J + W J be he firs such nonnegaive ime (so J 0, and J is also a random variable). Consider A J + W J. We wai ill he firs ime U J > A J + W J a which we have he side informaion ha he channel is erasure-free and ransmi a ha ime. Le T J := U J (A J + W J ) We adjus each of he poenial ransmission imes A k + W k, k J +1 o A k +W k +T J. When we ge o ime A J+1 + W J+1 + T J, we wai ill he firs ime U J+1 > A J+1 + W J+1 + T J a which we ge he side informaion ha he channel is erasure-free. Le T J+1 := U J+1 (A J+1 + W J+1 + T J ). We adjus each of he poenial ransmission imes A k +W k +T J, k J +2 o A k +W k +T J +T J+1, and so on. Noe ha, since he erasure sequence is independen of he sequence of energy arrivals and he randomizaion variables, he sequence of random variables (T J, T J+1,...) is he porion, defined by J, of an i.i.d sequence of random variables (T k, k ), where P (T k = l) = p l 1 e (1 p e ) for l 1. The imes a which ransmissions ake place under his energy anicipaive policy are disinc, since each ransmission akes place a leas one uni of ime afer he preceding one. The policy is energy feasible because he policy of he firs par of he proof of Theorem 4 was energy feasible. Indeed, in ha policy he energy ha arrived a A k can be hough of as desined o be used a ime A k + W k. Here his energy sis in he baery longer, and is used a he ime U k (his is where we need he assumpion B max = ). The proof ha lim sup E[Π ] < for his policy also parallels he proof of he corresponding saemen in he proof of Theorem 4. Le T be independen of (L 1, L 2,..., L B0+1) wih he disribuion P (T = l) = p l 1 e (1 p e ), l 1. I suffices o replace he occurence of L 1 +... + L B0+1 in he proof of Theorem 4 by L 1 +... + L B0+1 + T. Since L 1 +... + L B0+1 is negaive binomial wih success probabiliy p a and as we have assumed boh a 2 (1 p a ) B0+1 < 1 and a 2 p e < 1, we have k=1 a2k (k +)P (L 1 +...+L B0+1 +()T
9 ()(k 1)) <. Therefore, he firs saemen of he heorem is proved. For proof of he second saemen, fix 1 B max, and a finie ineger B 0 {1,..., B max } as he iniial level of he baery. Firs fix 0 < p a < 1, and consider any 0 < p e < 1. If here were a randomized anicipaive sraegy wih side informaion for which 1 E[Π u] <, he ransmier could have 1 lim ensured his propery in he scenario when p e = 0 by simply arifically creaing i.i.d. erasures wih erasure probabiliy p e and implemening he given sraegy. Since he erasures are creaed by he ransmier, i has access o hem, so i is possible o implemen his sraegy. Bu we already know from Theorem 4 ha for p e = 0, if a 2 (1 p a ) > 1, hen for every randomized energy anicipaive sraegy we have lim E[Π u] =. 1 1 Thus i mus be he case ha for each 0 < p e < 1, for every randomized anicipaive sraegy wih side informaion, whenever a 2 (1 p a ) > 1 we also have 1 1 lim E[Π u] =. Similarly, fix 0 < p e < 1 and consider any 0 < p a < 1. Suppose here were a randomized anicipaive sraegy 1 1 wih side informaion for which lim E[Π u] <. The ransmier could implemen his sraegy in he scenario when p a = 1 by simply dropping energy arrivals in an i.i.d. fashion wih drop probabiliy 1 p a. Since he ransmier knows when he surviving energy arrivals occur, i is possible for i o implemen his policy. Bu we already know from Theorem 1 ha when p a = 1, if a 2 p e > 1, we have lim E[Π ] = for every randomized anicipaive policy wih side informaion. Thus i mus be he case ha for all 0 < p a < 1, if a 2 p e > 1 we also have lim E[Π ] = for every randomized anicipaive policy wih side informaion, which complees he proof of he second saemen. VII. CONCLUSION We considered a scalar inermien Kalman filering problem over an erasure channel wih an energy harvesing ransmier. We focused on he hreshold on he growh rae of he sysem dynamics ha admis an asympoically bounded sae esimaion error covariance a he receiver. We showed ha he presence of anicipaive knowledge abou he fuure energy arrivals as well as unlimied energy sorage does no aler he hreshold if he ransmier has no side informaion abou he curren erasure sae of he channel. On he conrary, we showed ha side informaion regarding he curren erasure sae of he channel could lead o an improvemen in his hreshold when combined wih energy sorage and anicipaive knowledge abou he energy arrivals. These resuls sugges new direcions of research o reveal he synergies beween channel side informaion, energy sorage capabiliy, and he poenial o provide anicipaive informaion abou fuure energy arrivals, in neworks of energy harvesing nodes ha inerac wih dynamical sysems. ACKNOWLEDGEMENT Research suppored by he NSF Science and Technology Cener gran CCF-0939370, Science of Informaion, and he NSF grans ECCS-1343398 and CNS-1527846. REFERENCES [1] O. Ozel and S. 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