Minimum Energy Coding in CDMA Wireless Sensor Networks

Transcription

1 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 8, NO., FEBRUARY Mnmum Energy Codng n CDMA Wreless Sensor Networks C. Fschone, Member, IEEE, K. H. Johansson, Member, IEEE, A. Sangovann-Vncentell, Fellow, IEEE, and B. Zurta Ares Abstract A theoretcal framework s proposed for accurate comparson of mnmum energy codng n Coded Dvson Multple Access CDMA Wreless Sensor Networks WSNs. Energy consumpton and relablty are analyzed for two codng schemes: Mnmum Energy codng ME, and Modfed Mnmum Energy codng MME. A detaled model of consumed energy s descrbed as functon of the codng, rado transmt power, the characterstcs of the transcevers, and the dynamcs of the wreless channel. Snce CDMA s strongly lmted by mult-access nterference, the system model ncludes all the relevant characterstcs of wreless propagaton. A dstrbuted and asynchronous algorthm, whch mnmzes the total energy consumpton by controllng the rado power, s developed. Numercal results are presented to valdate the theoretcal analyss and show under whch condtons MME outperforms ME wth respect to energy consumpton and bt error rate. It s concluded that MME s more energy effcent than ME only for short codewords. Index Terms Wreless sensor network WSNs, mnmum energy codng, CDMA, OOK, power control, outages, parallel and dstrbuted computaton. I. INTRODUCTION WIRELESS sensor networks have the potental of dwarfng the revoluton that the Internet has brought to the world of computng, entertanment, work and human nteracton. They wll make t possble to connect computng wth the physcal envronment yeldng the so called Physcal Internet. Gven the small dmensons of the sensng devces and ther naccessblty when deployed n the envronment, ther computng power and the energy resources are necessarly lmted. Wreless communcaton consumes a substantal part of energy, whch s related to the need of transmttng wth few errors. An optmzaton problem arses where the energy spent n transmttng and recevng s mnmzed whle a gven probablty of correct transmsson s guaranteed. Source codng, rado power control and sleep dscplnes are Manuscrpt receved February 0, 008; revsed June 0, 008; accepted August 13, 008. The assocate edtor coordnatng the revew of ths paper and approvng t for publcaton was K. B. Lee. C. Fschone and K. H. Johansson are wth the ACCESS Lnnaeus Center, Electrcal Engneerng, Royal Insttute of Technology, Stockholm, Sweden e-mal: {carlof, kallej}@ee.kth.se. A. Sangovann-Vncentell s wth the Department of Electrcal Engneerng and Computer Scences, Unversty of Calforna at Berkeley e-mal: alberto@eecs.berkeley.edu. B. Zurta Ares was wth the Royal Insttute of Technology, Stockholm, Sweden, when contrbutng to ths work e-mal: bengno.zurta@gmal.com. A. Sangovann Vncentell and C. Fschone wsh to acknowledge the support of the NSF ITR CHESS and the GSRC. The work was partally funded also by the Swedsh Foundaton for Strategc Research and the Swedsh Research Councl and VINNOVA. Part of the topcs of ths paper were presented at EWSN 07. Dgtal Object Identfer /TWC /09$5.00 c 009 IEEE all technques that have been used to mnmze the energy consumpton of wreless sensor networks. In ths paper we focus on mnmzng the energy consumpton of wreless sensor networks by usng low energy codng schemes. The two codng technques analyzed here are Mnmum Energy ME codng used n conjuncton wth On-Off Keyng OOK modulaton [1], [], and the Modfed Mnmum Energy MME codng, a varant of the ME codng wth sleep dscplnes at the recever [3], [4]. In ME and MME codng, nformaton s coded n dgtal form so that zero-one patterns determne n large part energy consumpton by the transmtter and the recever. If we assume that the ones are the nformaton that s actually sent whle zeros correspond to no transmsson, then ME optmzes energy by mnmzng the number of ones present n the coded message to be transmtted. In partcular, ME encodes hgh probablty source codewords wth coded codewords havng a small number of ones. Ths smple yet powerful dea, proposed by Ern and Asada, has been extended by Prakash and Gupta [5], who have proposed ME codng along wth channel codng for the case of sources wth unknown statstcs. Tang et al. [6] have nvestgated the bt error rate of ME codng and OOK modulaton for both coherent and non coherent recevers n AWGN channels. Lu and Asada [7] have appled ME codng to CDMA wreless systems and reported that Mult-Access Interference MAI s reduced when usng ME codng. Snce small MAI mples fewer bt errors, ME has good relablty. ME and CDMA were consdered also by Km and Andrews [4], when they proposed the MME scheme. Snce rado power plays an mportant role n the energy balance for transmsson, t s of paramount mportance to consder the jont effect of rado power control and mnmum energy codng on the overall energy consumpton. Zurta et al. [8] presented an analyss of ME and MME for WSNs n a slow fadng propagaton envronment usng heurstc power control and detecton algorthms. In ths paper we offer a complete framework to compare and optmze the ME and MME approaches wth respect to energy consumpton whle satsfyng constrants on communcaton errors. To do so, we develop an accurate characterzaton of the energy spent for codng, transmttng and recevng takng nto account rado power, and average number of rado module startups ncludng a detaled model of the MAI and wreless channel wth path loss, slow and fast fadng presented n Secton III; a decentralzed rado power control algorthm, whch Authorzed lcensed use lmted to: KTH THE ROYAL INSTITUTE OF TECHNOLOGY. Downloaded on Aprl 9, 009 at 07:31 from IEEE Xplore. Restrctons apply.

2 986 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 8, NO., FEBRUARY 009 transmtter-recever par Rx 1 Tx Tx Rx Tx 3 Fg. 1. System scenaro: K asynchronous pars of nodes are smultaneously transmttng. L0=Orgnal Codeword Length ME Codng Rx Tx K Rx Redundant Bts L0 = Orgnal Codeword ME Codng Ls=Subframe Length Subframe # Subframe #3 Indcator bts 1: no hgh bts n ths subframe, 0: there are Ns = number of subframes Redundant Bts LMME = New Codeword Length more zeros Fg. 3. MME codeword. The orgnal codeword s mapped nto a MME codeword parttoned n sub-frames. Each sub-frame starts wth an ndcator bt from [4]. LME = New Codeword Length more zeros Fg.. Mappng of the source codewords nto the codewords of the Mnmum Energy Codng from [7]. mnmzes the total energy spent for transmsson and recepton usng the models above Secton IV; an optmal decson threshold for OOK detecton, as functon of the ME and MME codng schemes and rado powers Secton V; a characterzaton of the bt error probabltes of ME and MME codng when the rado power control algorthm s deployed Secton VI. Comparng our approach to exstng contrbutons [1], [] and [4] [7], we are able to provde a characterzaton of total energy consumpton wth respect to all the parameters of a complete system scenaro, namely: the energy due to ME, MME, CDMA, the wreless channel, and the transcever. As a consequence of ths more detaled study, our analyss leads to a dfferent result from the lterature when comparng ME and MME as shown n the numercal results presented n Secton VII. II. SYSTEM DESCRIPTION Consder a scenaro where there are K transmtter-recever pars of nodes see Fg. 1. Data sensed by a node s at frst coded ether wth ME or wth MME codng. Wth ME codng [1], [], each codeword havng large probablty s mapped nto a new codeword havng less number of one or hgh bts. Denote wth L 0 the length of the source codeword, and wth L ME the length of the ME codeword, where L 0 L ME. The extra bts added are the redundant bts needed for channel codng see Fg.. Let α ME be the probablty of havng hgh bts n an ME codeword. MME codng [4] explots a structure of the codeword that allows the recever to go n a sleep state, where the rado electronc crcutry s swtched off [3]. Wth MME, the ME codewords are parttoned nto N s sub-frames of length L s, where each sub-frame starts wth an ndcator bt b nd see Fg. 3. When b nd s a one hgh bt, t ndcates that there are no hgh bts n that sub-frame, so there s no need for decodng, and the recever can go to the sleep state. Conversely, f b nd s a zero low bt, t ndcates that there are hgh bts n the sub-frame, so the decodng operaton must be performed, and the recever cannot go to sleep. The ME or MME coded bts are then handled by an OOK modulator: only bts havng value one hgh bts are DS-CDMA processed and transmtted. An asynchronous DS- CDMA wreless access scheme s consdered, where the same fxed bandwdth W s allocated to each transmtter-recever par. The processng gan s denoted wth G = T b /T c, where T b s the bt nterval, and T c s the chp nterval. The transmtted sgnal, after beng attenuated by the wreless channel, s receved corrupted by an addtve Gaussan nose and MAI caused by other transmttng nodes. The output of the coherent correlaton recever of lnk can be expressed as [9] Z t =D t+i t+n g t, 1 where D t s the sgnal bearng the nformaton for the par, I t s the nterference due to the presence of multple transmttng nodes causng MAI and N g t s the AWGN nose, whch s modeled as a Gaussan random varable wth zero mean and varance N 0 T b /4. In partcular, t can be proved that [9] h, tp D t =ν t T b ν tμ Z t, and that the varance of the MAI plus the AWGN, on a bt tme scale and condtoned to the dstrbuton of the hgh bts and the wreless channel, s T b 6G K j=1 j ν j th j tp j + T b 4 N 0 σ Z t. 3 Authorzed lcensed use lmted to: KTH THE ROYAL INSTITUTE OF TECHNOLOGY. Downloaded on Aprl 9, 009 at 07:31 from IEEE Xplore. Restrctons apply.

3 FISCHIONE et al.: MINIMUM ENERGY CODING IN CDMA WIRELESS SENSOR NETWORKS 987 Fg. 4. Relaton among the system parameters wreless channel, codng, transmt rado power and the bt error rate BER and the energy consumpton of ME and MME. where E tx and E rx are the average energy consumpton of a node whle transmttng and recevng, respectvely. The power consumpton of the electronc crcuts, whle transmttng and processng a codeword, s denoted wth P tx,ckt,and whle recevng s denoted wth P rx,ckt. Note that P tx,ckt and P rx,ckt do not nclude the rado power, whch s P. The functon fp accounts for the energy spent by the rado module to transmt at rado power P. fp s an ncreasng functon of P, snce the hgher s the rado power transmtted, the hgher s the energy draned by the transmt antenna. T tx,me s the transmtter actvty tme per ME codeword, and T rx,me s the recever actvty tme per codeword; fnally, T s s the start up tme of the rado transcever,.e., the tme employed to go from the sleep state to the actve state. In and 3, P, =1,...,K, denotes the rado power of the sender node n lnk. We ntroduce the vector P = [P 1,...,P,...,P K ] T. The term ν j t s a bnary random varable abstractng the transmsson of a hgh bt ν j t =1 or low bts ν j t =0, wth probablty mass functon Pr[ν j t = 1] = a and Pr[ν j t = 0] = 1 a, respectvely. In partcular, a = α ME for ME codng, whereas a = α MME + N s /N s L s =α MME +1/L s for MME codng, because ndcator bts have been added to the MME codeword wth respect to the ME codeword recall Fg. and Fg. 3. Throughout the paper, we assume that α ME = α MME.Let vector νt =[ν 1 t,...,ν t,...ν K t] T denote the nodes codng actvty. In and 3, the wreless channel coeffcent assocated to the path from the transmtter of lnk j to the recever of lnk s h j t = l j r j texpξ j t where l j s the path loss, whch s dependent on the dstance and propagaton envronment [10]. We consder the Nakagam dstrbuton for the fast fadng, wth correlaton 1 and parameter m, so r j t has a gamma dstrbuton havng average μ r and correlaton ρ r [10]. The term exp ξ j t s the shadow fadng component, wth ξ j t beng a Gaussan random varable havng zero average and standard devaton σ ξj. We ntroduce the vector h t =[h 1, t,...,h, t,...h K, t] T to denote the wreless channel coeffcents seen by the recever of the par. III. ENERGY MODEL Here we present an orgnal characterzaton of the energy spent to transmt nformaton usng ME and MME as functon of the codng actvty, the transmt rado powers, the wreless channel, and the hardware platform. The relaton among these system parameters wth the bt error rate and the energy consumpton of ME and MME s summarzed n Fg. 4. A. ME Codng Consder lnk between a par of transmtter and recever nodes. The energy consumpton per ME codeword spent over the lnk can be expressed as follows: E ME =E tx + E rx =P tx,ckt [ T tx,me + T s ] + α ME fp T tx,me + P rx,ckt [ T rx,me + T s ], 4 B. MME Codng By adoptng the same parameter defnton used n 4, we model the energy consumpton per MME codeword as follows: E MME =E tx + E rx [ ] =P tx,ckt T tx,mme + T s + α MME fp T tx,mme + P rx,ckt T rx,mme [ + P rx,ckt + P rx,ckt,st] N +1T s. 5 In 5, T tx,mme s the transmtter actvty tme per MME codeword, and T rx,mme s the recever actvty tme per codeword. We wll see later that T rx,mme depends on the rado powers. P rx,ckt,st s the extra energy overhead spent n the start-up phase. In 5, we have ntroduced the average number of tmes, denoted wth N, that the recever has to awake from the sleep state. Ths term, as T rx,mme, depends on the bt error rate, and hence on the rado powers. It plays a fundamental role when evaluatng MME energy consumpton: each tme the rado recever module s turned on, t spends an amount of energy gven by P rx,ckt T s.the term [ P rx,ckt + P rx,ckt,st] N +1T s may be an mportant pece of energy, because P rx,ckt s the largest term among the power components at the recever t s comparable wth the largest value of fp T tx,mme n off-the-shelf nodes [11] [1], T s s not neglgble, and nether s N. fp and N have not been ncluded n the energy model proposed n [4]. The consequence of takng all these parameters nto account wll lead to the concluson that MME offers adequate performance mprovement wth respect to ME only n some but not all cases, as we shall see later. In the followng sectons, we nvestgate the components that concur to the energy consumpton of ME and MME codng, namely, the energy needed for rado transmsson, and the bt error rate, whch determnes the number of starts-up of the rado module. IV. OPTIMAL TRANSMIT RADIO POWER In ths secton, we propose an algorthm for mnmzng the total energy consumpton by optmally selectng the rado powers P. Rado powers must be allocated to ensure a gven qualty of the receved sgnal. The Sgnal to Interference plus Nose Rato SINR s a typcal qualty measure. Consder lnk Authorzed lcensed use lmted to: KTH THE ROYAL INSTITUTE OF TECHNOLOGY. Downloaded on Aprl 9, 009 at 07:31 from IEEE Xplore. Restrctons apply.

4 988 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 8, NO., FEBRUARY 009 between a par of transmtter and a recever nodes. The SINR s defned as SINR t = ν tμ Z t σz. 6 t Note that the SINR s a random varable, snce t depends on the wreless channel coeffcents h t, aswellasonthe codng ν t. Moreover, t s functon of the transmt powers P. Whenν t =0, the SINR s zero. To mnmze the total energy consumpton of the overall system, we propose an optmzaton problem whose objectve functon s the sum of the energy to transmt and receve, whle the constrants are expressed n terms of outage probablty of the SINR: K P : mn E P 7 P =1 s.t. Pr [SINR t ν t =1] P out, =1,...,K. In Problem P, E P s the total energy consumpton, and t s gven by 4 for ME codng, and by 5 for MME codng. s the SINR threshold under whch outages occur. Solvng the optmzaton problem ensures that the outage probabltes reman below P out. We assume that Problem P s feasble throughout the paper. From a physcal pont of vew, feasblty means that all nodes can transmt wth a certan rado power whle satsfyng the outage constrant. To solve Problem P, the constrants related to the outage have to be modeled. Note that the outage probablty depends on the dstrbuton of the SINR, whch, n turn, depends on the wreless channel h t and the dstrbuton of hgh bts νt.e., the codng. Snce the dstrbuton of the SINR s unknown, we make use of an approxmaton. We note that μ Z t n the numerator of the SINR 6 s a product of a gamma random varable, r t, and a log-normal one, exp ξ t. Then, as proposed n [10], ths product can be well approxmated wth a log-normal random varable: μ Z t = T b P l r texpξ t exp X t. 8 where X t s a Gaussan random varable havng average and standard devaton, respectvely μ X =lnp l Tb ln + ψm ln m and σx = ζ,m+σξ,whereψm s Euler s ps functon, and ζ,m s Remann s zeta functon. The denomnator of the SINR s sum of log-normal random varables weghted by gamma and bnary random varables. Therefore, followng the same approach as n [13], the Wlknson s moment-matchng method can be appled and the denomnator s well approxmated wth a log-normal random varable σz t expy t, wherey t s a Gaussan random varable havng average and standard devaton obtaned by matchng the frst and second order moments of σz t: μ Y =lne νt,ht[σ Z t] 1 ln E νt,h t[σ 4 Z t], σ Y =lne νt,ht[σ 4 Z t] ln E νt,ht[σ Z t]. The expressons of E νt,ht[σ Z t] and E νt,ht[σ 4 Z t] can be easly derved by applyng the lnear and dstrbutve propertes of the statstcal expectaton, and rememberng that the vectors h t and νt are statstcally ndependent [8]. Usng the prevous approxmatons, the outage probablty can be computed whle takng nto account the wreless coeffcents, the transmsson powers, and the dstrbuton of hgh bts. It follows that Pr [SINR t ν t =1] Pr [expx t Y t ] =1 Q ln μ X + μ Y, 9 σ + X σ Y where Qx = 1/ π / x e t dt s the complementary standard Gaussan dstrbuton. Expresson 9 can be used to rewrte the constrants n Problem P, whch becomes: P : mn P K E P =1 s.t. P g P, =1,...,K where g P = exp μ X +lnp + μ Y q σ X + σ Y and q = Q 1 1 P out. Problem P s a centralzed problem, n the sense that a central entty needs to collect nformaton related to all rado lnk coeffcents, compute the soluton, and fnally dspatch the soluton to all other transmttng nodes. A centralzed mplementaton has obvous dsadvantages n terms of communcaton resources, delays and robustness. Nevertheless, by followng the method proposed n [14], Problem P can be solved usng a dstrbuted strategy. Frst, t s possble to show that g P does not depend on P, because nether μ X ln P, σ X, μ Y, nor σ Y depend on P.Furthermore, g P s monotoncally non-decreasng wth P. Fromthese propertes, we have the followng result: Theorem 1: Let E P : R K R be an ncreasng functon of P. Then, Problem P admts a unque optmal soluton P, such that P = g P. Proof: The proof s smlar to the proof of Theorem 1 n [14]. Ths theorem suggests solvng Problem P just by lookng at the soluton of the system of non-lnear equatons gven by the constrants. Snce g P s a non-decreasng functon, t s also a contracton mappngs n P. The component soluton method, as defned n [15, Pag ], can be appled to solve Problem P n a fully dstrbuted and asynchronous manner usng the followng algorthm: P n :=g Pn 1, 10 where n s a local teraton tme. The algorthm s stopped when P n P n 1 ε, whereε denotes the precson of the soluton. The algorthm s fully dstrbuted, because at recever node of lnk we need not know the transmt powers of all nodes, but only the statstcal moments μ X, σ X, μ Y,andσ Y, whch can be easly estmated locally through sample averages. Algorthm 10 converges exponentally to the optmal soluton [15]. The numercal results of Secton VII show that convergence s fast; n the example descrbed n Secton VII, convergence was acheved n less than 5 teratons. Authorzed lcensed use lmted to: KTH THE ROYAL INSTITUTE OF TECHNOLOGY. Downloaded on Aprl 9, 009 at 07:31 from IEEE Xplore. Restrctons apply.

5 FISCHIONE et al.: MINIMUM ENERGY CODING IN CDMA WIRELESS SENSOR NETWORKS 989 In the sequel, we assume that the rado powers are computed by solvng Problem P. We nvestgate the bt error probabltes of ME and MME, and how these probabltes relate to the total energy consumpton. V. BIT ERROR PROBABILITY The dervaton of the BER usng the decson varable 1 dstngushes two cases of error: the decson varable s decoded as a low bt, when a hgh bt was transmtted; or the decson varable s decoded as a hgh bt when a low bt was transmtted. We denote these probabltes wth e 0,ht,νt and e 1,ht,νt, respectvely, where: e 0,ht,νt = Pr[Z t δ ν t =0, h t, νt] δ = Q, 11 σ Z t e 1,ht,νt = Pr[Z t <δ ν t =1, h t, νt] μz t δ = Q, 1 σ Z t where δ s the decson threshold for the varable Z t and μ Z t and σz t have been defned n 8 and 3, respectvely. The probabltes n 11 and 1 are computed under the obvous assumpton that the decson threshold δ must le n the nterval [0,μ Z t], and adoptng the usual standard Gaussan approxmaton [10], where Z t s modelled as a Gaussan random varable condtoned to the dstrbuton of the channel coeffcents and codng. Specfcally, t s assumed that Z t Nν tμ Z t,σ Z t. The bt error probablty, condtoned to the channel coeffcents and codng, s Φ h t,νtδ =Pr[ν t =0]e 0,h t,ν tδ +Pr[ν t =1]e 1,h t,ν tδ 13 δ μz t δ =1 a Q + aq. σ Z t σ Z t Ths expresson could be mnmzed wth respect to δ. Indeed, 13 s convex n δ, because Qx s convex for x 0, whereas the arguments of the frst and second Q are concave, and a convex functon of a concave functon s convex [16]. We can analytcally solve for the optmal soluton: δ = μ Z t σ Z t μ Z t ln a 1 a, 14 where the optmal threshold s dependent on the nstantaneous values of the channel coeffcents h t va μ Z t and σ Z t. However, there are at least two reasons that prevent usng 14: frst, for every bt tme, each node should be able to detect f other nodes are transmttng.e. nstantaneous global codng actvty knowledge s requred; second, each recever node should be equpped wth a channel estmator that provdes the vector h t at each bt tme nstant.e. global nstantaneous wreless channel estmaton for each nterferng node s requred. Implementng these tasks on local nodes s prohbtve, snce they have reduced computng resources. An alternatve approach s based on takng the average of the BER wth respect to the channel coeffcent and codng, and then mnmzng the resultng expresson. Wth ths approach, the optmal threshold depends only on the averages of the MAI, whch s smple to compute, as we dscuss below. By averagng 13 wth respect to h t and νt we obtan [ δ Φ δ =E νt,ht 1 a Q σ Z t ] μz t δ +aq. 15 σ Z t Mnmzng 15 wth respect to δ s dffcult, because the functon s non lnear and no closed-form s avalable for Φ δ. Nevertheless, numercal technques can be appled. Frst, we use the Strlng approxmaton to compute the average of a Q functon: gven a log-normal random varable ζ b, havng average μ ζb and varance σζ b, the followng approxmaton follows [10]: E [Q ζ b ] 3 Q μ ζ b Q μ ζb + 3σ ζb Q μ ζb 3σ ζb F ζ b. 16 Now, defne the random varables ζ 0δ = δ δ exp Yt, σ Z t [ ζ 1δ = μz t δ Xt exp σ Z t δ ] exp Yt. Computng the average and standard devaton of a log-normal random varable from the natural logarthm of the varable, we have: μ ζ0 δ =δ exp 1 μ Y + 18 σy, σζ 0 δ =δ exp μ Y + 1 σ Y [1 exp 14 ] σy, and [ 1 μ ζ1 δ = exp μ X + 1 ] 8 σ X δ exp 1 μ Y + 18 σy, [ σζ 1 δ = exp μ X + 1 σx + δ 1 δ exp μx + 1 ] 8 σ X exp μ Y + 1 σy [ exp μ X + 14 σx + δ 1 δ exp μx + 1 ] 8 σ X exp μ Y + 14 σy. Usng these expressons, together wth 15 and 16, we obtan Φ δ 3 [1 aq μ ζ 0 δ + aq μ ζ1 δ ] + 1 [ 1 aq μ ζ0 δ + 3σ ζ0 δ 6 + aq μ ζ1 δ + 3σ ζ1 δ +1 aq μ ζ0 δ 3σ ζ0 δ + aq μ ζ1 δ ] 3σ ζ1 δ. 17 Smple optmzaton algorthms such as the steepest descent or the bsecton algorthms [16] can be appled to compute the Authorzed lcensed use lmted to: KTH THE ROYAL INSTITUTE OF TECHNOLOGY. Downloaded on Aprl 9, 009 at 07:31 from IEEE Xplore. Restrctons apply.

6 990 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 8, NO., FEBRUARY 009 mnmum of Φ δ.letδ be the value of δ that mnmzes Φ δ. Fnally, denote the average values of 11 and 1 computed n δ as e 0 = E e 0,ht,νt = F ζ 0 δ, e 1 = E e 1,ht,νt = F ζ 1 δ. These expressons are used next to derve the bt error probablty for the ME and MME codng. A. Bt Error Probablty n ME Codng The BER n the ME case, denoted wth Φ ME, can be easly computed by usng 17 and δ,whereα takes the value α ME. B. Bt Error Probablty n MME Codng An analyss on the MME performance regardng ts BER demands a careful study whch takes nto account the specal nature of the MME codeword, as we see next. We compute the average BER of the MME codng as the rato between the average number of erroneous bts per MME codeword and the codeword length: Φ MME = n,sf N s L MME, 18 where N s s the number of sub-frames per codeword and n,sf s the average number of erroneous bts n a sub-frame transmtted over lnk n,sf = L s 1 n=1 nψ n, 19 where Ψ n stands for the probablty of havng n errors n the sub-frame. To characterze Ψ n, we need the followng defntons: the event A n happens when the ndcator bt b nd was transmtted and there are n decodng errors n the followng sub-frame; the event B n happens when there are n hgh bts n the sub-frame. Then Pr[A n b nd ]= and Also, Pr[B n] = Ls 1 n Ls 1 n e bnd n 1 e bnd Ls 1 n, α n MME 1 α MME Ls 1 n. Pr [b nd =0]=1 1 α MME Ls 1, Pr [b nd =1]=1 α MME Ls 1. Then, we have the followng result Proposton 1: The probablty of detectng n errors n a sub-frame s Ψ n =Pr[b nd =1]e 1 Pr[A n b nd =1] +Pr[b nd =0] { 1 e 0 Pr[A n b nd =0] +e 0 Pr[B n] }, 0 Proof: Recallng that each sub-frame starts wth an ndcator bt, Ψ n can be computed consderng three cases: 1 The ndcator bt s a one and the recever erroneously decodes the ndcator bt as a zero, wth catastrophc consequences on the decodng of the sub-frame. The recever decodes the bts of the sub-frame fndng n errors wth probablty Pr[A n b nd =1]. The ndcator bt s zero and the recever performs decodng correctly. Then the recever detects the bts of the sub-frame, fndng n bts n error wth probablty Pr[A n b nd =0]. 3 The ndcator bt s zero and the recever erroneously nterprets t as a one. Then the recever catastrophcally consders all the followng bts as f t were zero, makng n errors wth probablty Pr[B n]. By summng up the probabltes of all the cases above, we obtan 0, whch concludes the proof. Fnally, the BER of MME s gven by 18, 19, and 0. VI. ENERGY CONSUMPTION Here we put together the analyss presented n Sectons IV and V to characterze the energy consumpton of the ME and MME schema accurately. A. ME Codng The energy consumpton of the ME codng scheme s defned as the average of the energy consumpton of all the sensor nodes: E ME = 1 K E ME, 1 K where the term E ME s defned n 4 usng the rado power levels as obtaned by Algorthm 10. The total energy consumpton 1 s non-decreasng wth P, snce so does f P n 4. Consderng the energy model for a system usng ME codng 4, settng α ME =1, and computng the rado powers wth such an α, we obtan the energy consumpton of the BPSK case. The energy gan of the ME codng wth respect to BPSK s defned as the rato n db of the energy used n a BPSK system and 1 =1 ρ ME =10log EBPSK E ME. B. MME Codng To compute the energy needed by MME codng, t s necessary to characterze T rx,mme and N n 5. The older s gven by the average tme the recever s n the actve state per MME codeword tmes the bt tme: T rx,mme = n T b. 3 We have the followng results Proposton : The average tme the recever s n the actve state per MME codeword s { n = N s 1+Pr[bnd =0]1 e 0 L s 1 +Pr[b nd =1]e 1 L s 1 }. Proof: Snce the sub-frame structure of an MME codeword, the recever wakes up N s tmes to check the ndcator bt. If the ndcator bt s low and t s decoded as such, the recever stays awake for the overall sub-frame,.e., for Authorzed lcensed use lmted to: KTH THE ROYAL INSTITUTE OF TECHNOLOGY. Downloaded on Aprl 9, 009 at 07:31 from IEEE Xplore. Restrctons apply.

7 FISCHIONE et al.: MINIMUM ENERGY CODING IN CDMA WIRELESS SENSOR NETWORKS 991 L s 1 bt. If the ndcator bt s hgh, but t s erroneously decoded as a low one, agan the recever stays awake for the overall sub-frame. Addng up all the contrbutons, we obtan the expresson. Proposton 3: The average number of tmes that the rado module of the recever goes from off to on s N = N s Pr[bnd =0]e 0 +Pr[b nd =1]1 e 1. 4 Proof: The number of tmes that a recever awakes can be computed from the number of tmes the recever goes from the on to the off state. Indeed, each tme the recever turns off, t must turn on to decode the followng ndcator bt. The recever s turned off each tme an ndcator bt s detected as hgh. For each of the N s sub-frames, ths happens wth probablty Pr[b nd =0]e 0 +Pr[b nd =1]1 e 1, from whch 4 s obtaned. Eq. 3 and Eq. 4 can be used wth 5 to obtan the energy consumpton for the generc lnk. Thus, averagng over all the lnks, the MME energy s: E MME = 1 K K =1 E MME. 5 The total energy consumpton 5 s non-decreasng wth P, snce so do f P and T rx,mme +N +1T s. Fnally, the MME energy gan s defned as follows: ρ MME = 10log EME. 6 E MME VII. NUMERICAL RESULTS In ths secton, we provde numercal evaluaton of the bt error probablty and the total energy consumpton of ME and MME codng. Analytcal results, as obtaned from the analyss carred out n the prevous sectons, are compared wth those obtaned from smulatons. The results have been carred out takng as reference the Tmote Sky wreless sensor nodes [11], whch features the CC40 rado transcever module by Chpcon [1]. These sensors make use of DS-CDMA wth a bt rate of 1/T b = 50 Kbps, and a processng gan G = 8. The smulaton parameters, whch are ntroduced n the sequel, are therefore consstent wth ths hardware platform. A system scenaro wth K =8pars of nodes s analyzed. Nodes are deployed over an area where the maxmum dstance between a source node and a destnaton node of a par s randomly chosen between m and 15 m, and the maxmum dstance between the transmtter of an nterferng par and a recever s 15m. Ths choce s the most general and nterestng, because each par has a dfferent dstance source-destnaton, and each recever wll experence dfferent nterference, so that all the cases are accounted for. No larger dstances are allowed, snce ths would requre levels of rado power larger than 0 dbm, whch s not possble wth the Tmote Sky wreless sensor nodes. For each par, j =1...K, the standard devaton of the shadowng σ ξj s randomly selected between db and 4dB. We assumed Ralegh fadng,.e., m =1. The power of the nose s set to N 0 = 130 dbm. The values of the hardware energy consumpton are P tx,ckt = W, P rx,ckt = W, and P rx,ckt,st =0W. Furthermore, fp =VCP,where dbm par 1 par par 3 par 4 par 5 par teratons Fg. 5. Convergence of the power mnmzaton algorthm, α = 0.19, P out =0.01 =1. On the x-axs the number of teratons s reported. V =1.8V s the voltage consumpton and CP s the current consumpton of the electronc crcut needed to transmt wth rado power P. The followng relaton can be derved [17]: CP = log 3 P log P [ log 10 3 P 10 3 ] We set the start-up tme to T s =10T b, a reasonable fgure. Values of T s 10T b provde the same results, as those shown below. Note that the start-up tme of the Chpcon rado module [1] s s. Ths case s however not that nterestng to study because ths long start-up tme s not realstc for ME and MME codng. If numercal results were obtaned wth ths fgure, no advantage of MME wth respect to ME would be acheved no matter what system parameters fadng, number of nodes, codng actvty, codeword length are, because ths tme s very long when compared to the bt tme. Furthermore, such a hgh wake up tme would put a severe lmtaton to packet transmsson, snce no bt recepton s possble durng that long tme. In Fg. 5, the convergence trace of the power control Algorthm 10 s reported for the case of Pout = 0.01, α ME =0.19, and =1. Each curve refers to the rado power of a par. Convergence s fast, wth less than 5 teratons. Ths behavor remans the same for other choces of the system parameters. In the sequel, we present numercal results for two representatve cases: short codewords N s =3, L s =0,Fgs.6 8, and long codewords N s =0, L s =0, Fgs These parameters were chosen consderng that the longest codeword allowed for the payload wth the Tmote Sky sensor nodes s 760 bt. In Fg. 6, the ME energy gan as obtaned by n the case of N s =3and L s =0s plotted versus the SINR threshold. Each curve refers to a dfferent value of the codng α ME, as obtaned after the convergence of the rado power control Algorthm 10. It s evdent that the ME codng ntroduces sgnfcant energy savngs, n partcular for low values of α ME. Surprsngly, as tends to 1, the gan quckly decreases. Ths s due to the fact that large values of requre hgher rado Authorzed lcensed use lmted to: KTH THE ROYAL INSTITUTE OF TECHNOLOGY. Downloaded on Aprl 9, 009 at 07:31 from IEEE Xplore. Restrctons apply.

8 99 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 8, NO., FEBRUARY α= 0.09 α= 0.9 ME codng gan Fg. 6. ME Energy Gan as functon of the SINR threshold for dfferent values of α ME = α, Pout =0.01, N s =3and L s =0. BER 10 1 BPSK BPSK sm ME ME sm MME MME sm α = 0.9 BPSK Fg. 8. BER for BPSK, ME and MME as functon of the SINR threshold for dfferent values of α ME = α MME = α, Pout =0.01, N s =3and L s = sm 1 MME Codng Gan α = 0.9 sm α = 0.9 sm ME codng gan α = Fg. 7. MME Energy Gan as functon of the SINR threshold for dfferent values of α MME = α, Pout =0.01, N s =3and L s =0. 0 Fg. 9. ME Energy Gan as functon of the SINR threshold for dfferent values of α ME = α, Pout =0.01, N s =0and L s =0. powers, wth the consequence that the energy spent for rado power transmsson α ME fp assumes large values when compared to the energy spent by the electronc crcuts whle transmttng and processng a codeword. No value of larger than 1 s admtted, because the outage constrant cannot be guaranteed,.e., for > 1 there may be at least a par n outage. In Fg. 7, the MME energy gan as obtaned by 6 n the case of N s = 3 and L s = 0 s plotted versus the SINR threshold. Each curve refers to a dfferent value of the codng α MME, as obtaned after the convergence of the Algorthm 10. MME codng does not show any advantage wth respect to ME when <0.7. Indeed, low values of mply large value of the MME bt error probablty, whch causes the recever to commt frequent errors when decodng of the ndcator bt. The consequence s that the recever s awake even though t was not necessary. Furthermore, recallng that the MME energy balance 5 ncludes the extra term N wth respect to the ME energy consumpton 4, ths explans the poor performance of MME when 0.7. On the contrary, when 0.7, the ME bt error probablty decreases, so that the MME recever goes often to sleep, and N s also reduced. The MME gan ncreases as the the codng actvty ncreases. Indeed, the ME codng gan worsens n such crcumstances, whle the MME s able to save energy by gong to sleep. Ths s an nterestng result, snce t means that for large t s better to use short codes.e., codes wth larger actvty, rather then longer codes. In Fg. 8, the bt error probablty n the case of N s =3,and L s =0s reported for BPSK, ME and MME codng versus the SINR threshold as obtaned n secton V-A and V-B. Each curve s assocated to a dfferent value of the codng actvty, as obtaned after the convergence of the rado power control Algorthm 10. There s no common pattern n the behavor of the bt error rate wth respect to ME and MME Authorzed lcensed use lmted to: KTH THE ROYAL INSTITUTE OF TECHNOLOGY. Downloaded on Aprl 9, 009 at 07:31 from IEEE Xplore. Restrctons apply.

9 FISCHIONE et al.: MINIMUM ENERGY CODING IN CDMA WIRELESS SENSOR NETWORKS MME Codng Gan sm α = 0.9 sm α = 0.9 sm 0.15 BER BPSK BPSK sm α = ME ME sm MME MME sm BPSK Fg. 10. MME Energy Gan as functon of the SINR threshold for dfferent values of α MME = α, Pout =0.01, N s =0and L s =0. Fg. 1. BER for BPSK, ME and MME as functon of the SINR threshold for dfferent values of α ME = α MME = α, Pout =0.01, N s =0and L s =0. MME Codng Gan at the recever sm α = 0.9 sm α = 0.9 sm 0. Fg. 11. MME Energy Gan at the recever as functon of the SINR threshold for dfferent values of α MME = α, Pout =0.01, N s =0and L s =0. codng. The BER of BPSK s lower then the BER of ME and MME for 0.6. Ths s an nterestng result, snce one would expect that the OOK modulaton ncreases the BER. However, ME and MME cause less nterference, whch explans ther better performance when <0.5. Bythesame argument, ncreasng the codng actvty leads often to an ncrease of the BER of ME and MME. However, notce that small and large α MME yeld the worst case of the BER recall that ths choce of parameters gves the best results n terms of energy. Fg. 9 presents the ME energy gan as obtaned by n the case of N s =0and L s =0versus the SINR threshold. The man dfference wth respect to the short codeword case n Fg. 6 s that the gan does not decrease substantally when ncreases. Ths s because a longer codeword rses the mportance of the terms P tx,ckt and P rx,ckt n and 6, respectvely, so that the energy for rado power s less relevant wth respect to the case of short codewords. In Fg. 10, the MME energy gan as obtaned by 6 n the case of N s = 0 and L s = 0 s plotted versus the SINR threshold. Each curve refers to a dfferent value of the codng α MME. Observe that MME codng does not show any advantage wth respect to ME for any value of. Choosng large codewords means to ncrease substantally the term T tx,mme = N s L s T b n 6, whereas notce that T tx,me = N s L s 1T b. The tme spent to transmt s responsble also for the energy spent for rado power transmsson. Ths explans the poor performance of MME. However, MME s stll able to save energy at the recever node, asshownnfg.11.thsgansdefned smlarly to 6, but usng only the energy E rx n 4 for ME, and E rx n 5 for MME. The recever MME energy gan decreases wth, snce so does the number of starts-up N.Themajor consequence that can be nferred from Fg. 7 and Fg. 10 s that MME does not allow for global energy savngs when usng large codewords. Fnally, In Fg. 1, the bt error probablty n the case of N s =0and L s =0s reported for BPSK, ME, and MME codng versus the SINR threshold. The same consderatons made for Fg. 8 stll hold. The man dfference s that there s just a slght ncrease of the BER, as obvous consequence of longer codewords. VIII. CONCLUSIONS We presented a general framework for the comparson of ME and MME codng n CDMA-based wreless sensor networks. The analyss took nto account rado power consumpton, energy consumpton of the electronc crcut transcevers, and bt error probablty. A dstrbuted mnmzaton of the total energy consumpton was proposed, and a novel detecton threshold method for OOK modulaton was suggested. Numercal results show that ME codng always outperforms BPSK n terms of energy consumpton, and, for certan regons of outage and codng actvty, also n terms of bt error rate. MME codng outperforms ME only for small-sze codewords, Authorzed lcensed use lmted to: KTH THE ROYAL INSTITUTE OF TECHNOLOGY. Downloaded on Aprl 9, 009 at 07:31 from IEEE Xplore. Restrctons apply.

10 994 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 8, NO., FEBRUARY 009 whle t shows poor performance for large codewords, because of the sgnfcant energy spent for start-up. However, as technology evolves and smaller start-up tmes and energy are possble, MME may be favorable also wth longer codewords. REFERENCES [1] C. Ern and H. H. Asada, Energy optmal codes for wreless communcatons," n Proc. 38th IEEE Conf. of Decson and Control, [] C. Ern, H. H. Asada, and K.-Y. Su, Mnmum energy codng for RF transmsson," n Proc. IEEE Wreless Commun. Networkng Conf., [3] C. Fschone, A. Bonvento, A. Sangovann-Vncentell, F. Santucc, and K. H. Johansson, Performance analyss of collaboratve spato-temporal processng for wreless sensor networks," n Proc. IEEE Consumer Commun. Conf. 006, Jan [4] J. Km and J. G. Andrews, An energy effcent source codng and modulaton scheme for wreless sensor networks," n Proc. 6th IEEE Workshop Sgnal Processng Advances Wreless Commun., 005. [5] Y. Prakash and S. K. Gupta, Energy effcent source codng and modulaton for wreless applcatons," n Proc. IEEE Wreless Commun. Networkng Conf., 003. [6] Q. Tang, S. Gupta, and L. Schwebert, BER performance analyss of an on-off keyng based mnmum energy codng for energy constraned wreless sensor applcaton," n Proc. IEEE Internatonal Conf. Commun., 005. [7] C. H. Lu and H. H. Asada, A source codng and modulaton method for power savng and nterference reducton n DS-CDMA sensor networks systems," n Proc. Amercal Control Conf., 00. [8] B. Zurta Ares, C. Fschone, and K. H. Johansson, Energy consumpton of mnmum energy codng n CDMA wreless sensor networks," n Proc. 4th European Conf. Sensor Networks 007, Jan [9] F. Santucc, G. Durastante, F. Grazos, and C. Fschone, Power allocaton and control n multmeda CDMA wreless systems," Kluwer Telecommun. Syst., vol. 3, pp , May/June 003. [10] G. L. Stüber, Prncples of Moble Communcaton. Kluwer Academc Publshers, [11] Motev," San Francsco, CA, Tmote Sky Data Sheet, 006. [1] Chpcon Products,.4 GHz IEEE zgbee-ready RF transcever," techncal report, Texas Instruments, 006. [13] C. Fschone, F. Grazos, and F. Santucc, Approxmaton for a sum of on-off log-normal processes wth wreless applcatons," IEEE Trans. Commun., vol. 55, no. 10, pp , Oct [14] C. Fschone, M. Butuss, K. H. Johansson, and M. D Angelo, Power and rate control wth outage constrants n CDMA wreless networks," IEEE Trans. Commun., 008, to appear. [15] D. P. Bertsekas and J. N. Tstskls, Parallel and Dstrbuted Computaton: Numercal Methods. Athena Scentfc, [16] S. Boyd and L. Vandenberghe, Convex Optmzaton. Cambrdge Unversty Press, 004. [17] D. Lymberopoulos and A. Savvdes. XYZ: a moton-enabled, power aware sensor node platform for dstrbuted sensor network applcatons," n Proc. Informaton Processng Sensor Networks IPSN 005. Carlo Fschone receved the Ph.D. degree n Electrcal and Informaton Engneerng and the Laurea degree n Electronc Engneerng summa cum Laude n May 005 and Aprl 001, respectvely, both from the Unversty of L Aqula, Italy. Snce May 008 he s wth the Royal Insttute of Technology, ACCESS Lnnaeus Centre, Electrcal Engneerng, Stockholm, Sweden, as an Assstant Professor. He was a postdoctoral research assocate at the Unversty of Calforna at Berkeley, Department of Electrcal Engneerng and Computer Scences May September 008, and a postdoctoral research assocate at the Automatc Control Group, Royal Insttute of Technology May May 007. He receved the best paper award for the IEEE TRANSACTIONS ON INDUSTRIAL INFORMATICS durng the year 007, a best paper award of the IEEE Internatonal Conference on Moble Ad-hoc and Sensor System 05 IEEE MASS 05, the Ferdnando Flauro award from the Unversty of L Aqula, Italy, and the Alta Formazone award of the Abruzzo Regon Government, Italy. Hs research nterests are n the area of wreless networks and networked embedded systems, wth partcular reference to wreless communcaton theory, control of wreless networks, and optmzaton. He s a member of the IEEE Communcatons Socety and serves as revewer for techncal journals and conferences. Karl H. Johansson receved an M.Sc. and a Ph.D. n Electrcal Engneerng n 199 and 1997, respectvely, both from Lund Unversty n Sweden. He s currently Professor and Co-drector of the ACCESS Lnnaeus Centre at the School of Electrcal Engneerng, Royal Insttute of Technology, Sweden. He holds a Senor Researcher Poston at the Swedsh Research Councl. He has held vstng postons at UC Berkeley and Calforna Insttute of Technology Hs research nterests are n networked control systems, hybrd and embedded control, and control applcatons n automotve, automaton and communcaton systems. He s the Char of the Internatonal Federaton of Automatc Control IFAC Techncal Commttee on Networked Systems snce 008. He has served on the Executve Commttees of the European research projects HYCON and RUNES, both on networked embedded systems. He s on the edtoral boards of IEEE TRANSACTIONS ON AUTOMATIC CONTROL and IET CONTROL THEORY &APPLICATIONS, and prevously of AUTOMATICA. He was awarded an Indvdual Grant for the Advancement of Research Leaders from the Swedsh Foundaton for Strategc Research n 005. He receved the trennal Young Author Prze from IFAC n 1996 and the Pecce Award from the Internatonal Insttute of System Analyss, Austra, n He receved Young Researcher Awards from Scana n 1996 and from Ercsson n 1998 and Bengno Zurta Ares receved the M.Sc. n Telecommuncatons Engneerng from Unversty of Sevlle, Span, n 006. He developed hs M.Sc. Thess at the Royal Insttute of Technology, Stockholm, Sweden n 006, where he was an assstant researcher at the Automatc Control Group, School of Electrcal Engneerng. Hs research nterests are n the area of networked control systems, embedded control and communcaton systems, focusng manly on wreless sensor networks. Snce 007, he s workng n an aerospace and mltary company, developng boarded systems. Alberto Sangovann-Vncentell holds the Edgar L. and Harold H. Buttner Char of Electrcal Engneerng and Computer Scences at the Unversty of Calforna at Berkeley where he has been on the Faculty snce In , he spent a year as a Vstng Scentst at the Mathematcal Scences Department of the IBM T.J. Watson Research Center. In 1987, he was Vstng Professor at MIT. He was a co-founder of Cadence and Synopsys, the two leadng companes n the area of Electronc Desgn Automaton. He s the Chef Technology Advsor of Cadence Desgn System. He s a member of the Board of Drectors of Cadence, Soncs Inc., UPEK, Value Partners and Accent. He was a member of the HP Strategc Technology Advsory Board. He s a member of the GM Scence and Technology Advsory Board, and the founder and Scentfc Drector of PARADES, a European Group of Economc Interest supported by Cadence and ST Mcroelectroncs. He s a member of the Hgh-Level Group and of the Steerng Commttee of the EU Artems Technology Platform. In 1981 he receved the Dstngushed Teachng Award of the Unversty of Calforna. He receved the worldwde 1995 Graduate Teachng Award of the IEEE for nspratonal teachng of graduate students." He has receved numerous best paper awards ncludng the Gullemn-Cauer Award and the Darlngton Award In 001, he was gven the prestgous Kaufman Award of the Electronc Desgn Automaton Councl for poneerng contrbutons to EDA. In 00, he was the recpent of the Arstotle Award of the Semconductor Research Corporaton. He was elected Fellow of the IEEE n 198 and to the Natonal Academy of Engneerng n He s an author of over 800 papers and 15 books n the area of desgn tools and methodologes, large-scale systems, embedded controllers, hybrd systems and nnovaton. Authorzed lcensed use lmted to: KTH THE ROYAL INSTITUTE OF TECHNOLOGY. Downloaded on Aprl 9, 009 at 07:31 from IEEE Xplore. Restrctons apply.