Research Article Achieving Distributed Consensus in UWB Sensor Networks: A Low Sampling Rate Scheme with Quantized Measurements

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1 Hndaw Publshng Corporaon Inernaonal Journal of Dsrbued Sensor Newors Volume 214, Arcle ID 75475, 12 pages hp://dx.do.org/1.1155/214/75475 Research Arcle Achevng Dsrbued Consensus n UWB Sensor Newors: A Low Samplng Rae Scheme wh Quanzed Measuremens Lorenzo Taponecco 1 and Luca Sangune 1,2 1 Deparmen of Informaon Engneerng, Unversy of Psa, Va Caruso, Psa, Ialy 2 Alcael-Lucen Char, École Supéreure d Élecrcé(Supélec), 3 rue Jolo Cure, Plaeau de Moulon, Gf-Sur-Yvee, France Correspondence should be addressed o Lorenzo Taponecco; lorenzo.aponecco@e.unp. Receved 14 January 214; Acceped 27 February 214; Publshed 8 Aprl 214 Academc Edor: Javer Bajo Copyrgh 214 L. Taponecco and L. Sangune. Ths s an open access arcle dsrbued under he Creave Commons Arbuon Lcense, whch perms unresrced use, dsrbuon, and reproducon n any medum, provded he orgnal wor s properly ced. Dsrbued consensus n sensor newors has receved grea aenon n he las few years. Mos of he research acvy has been devoed o sudy he sensor neracons ha allow he convergence of dsrbued consensus algorhms oward a globally opmal decson. On he oher hand, he problem of desgnng an approprae rado nerface enablng such neracons has receved lle aenon n he leraure. Movaed by he above consderaon, n hs wor an ulrawdeband sensor newor s consdered and a physcal layer scheme s desgned, whch allows he acve sensors o acheve consensus n a dsrbued manner whou he need of any admsson proocol. We focus on he class of he so-called quanzed dsrbued consensus algorhms n whch he local measuremens or curren saes of each sensor belong o a fne se. Parcular aenon s devoed o address he praccal mplemenaon ssues as well as o he developmen of a recever archecure wh he same performance of exsng alernaves based on an all-dgal mplemenaon bu wh a much lower samplng frequency on he order of MHz nsead of GHz. 1. Inroducon Asensornewordenoesacolleconofspaallydsrbued rado ranscevers equpped wh sensors ha communcae hrough wreless lns whou he need of any fxed nfrasrucure (see [1, 2] andreferencesheren).theabsenceof any cenralzed conrol mechansm maes sensor newors parcularly sued for a large number of cvl and ndusral applcaons ncludng survellance, healhcare, facoryauomaon, and n-vehcle sensng [3]. Alhough her poenal benefs n all he above applcaons are wdely recognzed, he mplemenaon of sensor newors poses several echncal challenges, whch are subsanally dfferen from hose of convenonal communcaon sysems. One of he prmary neress s represened by he need o conjugae he relave unrelably of a sngle sensor (due o s lmed complexy and energy avalably) wh he hgh relably requred o he whole newor. For hs reason, an nense research acvy has been recenly devoed o desgnng algorhms by whch all he acve sensors may reach an agreemen on ceran quanes of neres n a dsrbued manner. Ths problem s nown as consensus n he leraure and has a long sandng radon n compuer scence (see, e.g., [4 6]andreferences herenfora comprehensve overvew of he problem). In sensor newors, parcular aenon has been gven o he characerzaon of he condons for achevng consensus (see, e.g., [7, 8]) whle only few wors deal wh he problem of desgnng approprae rado nerfaces ha allow achevng consensus. Movaed by he above consderaon, n hs wor, we concenrae on he physcal layer and assume ha he acve sensors nerac drecly n a peer-o-peer fashon whou employng any admsson proocol and usng ulrawdeband wh mpulse rado (UWB) as a common ar nerface [9]. Ths echnology s nowadays consdered as one of he mos promsng canddae for supporng emergng sensor newor applcaons [1]. In UWB sysems he nformaon s conveyed by low-power ulrashor pulses whose bandwdh s on he order of a few GHz. The low ransm power faclaes he coexsence of UWB devces wh dfferen

2 2 Inernaonal Journal of Dsrbued Sensor Newors ypes of wreless servces (a suaon ha s lely o occur wh sensor newors) whle he shor duraon provdes he sysem wh hgh-me resoluon, whch mproves rangng accuracy (parcularly useful for a large varey of sensor newor applcaons such as conrol and monorng). In addon, UWB sysems provde he possbly o realze ranscever archecures n a low-energy consumpon and negraed fashon as s desrable n sensor newors n order o reduce he sze and ncrease he baery lfe of he wreless devces. All hese feaures mae he UWB echnology parcularly appealng for he aforemenoned applcaons and jusfy s adopon as a physcal layer echnque n he IEEE a sandard [11]. Afrsaempodesgnaradonerfaceforhepraccal mplemenaon of a dsrbued consensus algorhm was orgnally presened by Aawa e al. n [12] fornerbase saon synchronzaon n mcrocellular sysems and laer exended o nervehcle communcaons by Sourour and Naagawa n [13]. Such a scheme maes use of a pulseposon modulaon (PPM) echnque o ransm he curren sae of each vehcle. Ths nformaon s hen recovered a he recever measurng he ampludes and relave me of arrvals of he sgnal peas exceedng a properly desgned hreshold. Unforunaely, he above soluon provdes good performance only n hose applcaons n whch he channel can be modeled as Gaussan. Obvously, hs model does no hold rue n boh ndoor and oudoor UWB ransmssons n whch he sensor newor covers a nonneglgble area and each ransmed pulse propagaes hrough a large number of dsnc pahs and s seen by he recever as he superposon of mulple echoes, each characerzed by a dfferen shape and a random poson n he me scale. A scheme ha s robus o mulpah propagaon was dscussed by Smeone and Spagnoln n [14]. Here, he auhors propose abarycener-basedme-deecorwhoseamsoesmaea convex combnaon of he arrval mes of all he receved pulses raher han explcly esmang only hose assocaed wh he larges peas. A smlar approach s followed by Pescosoldo e al. n [15] whle oher soluons can be found n [15, 16]. Dfferenly from [14], however, he resul n [15] s acheved by applcaon of an alernave soluon based on a double negraon of he receved sgnal. As dscussed laer, he man drawbac of hese wo schemes s ha hey requre all-dgal recevers. Albe possble n prncple, he realzaon of an all-dgal recever n UWB sensor newors s challengng. The man problem s represened by he need of usng analog-o-dgal converers (ADCs) operang n he mul-ghz range and characerzed by boh low-power consumpon and hgh-moderae b resoluon. Unforunaely, all hese requremens canno be easly sasfed wh curren echnology. For example, he amoun of power requred by acommonflashadcncreaseslnearlywhhesamplng raeandexponenallywhhebresoluon [17]. Vce versa, he so called successve approxmaon regser ADCs can be employed only n hose applcaons characerzed by low samplng rae and hgh b resoluon [18]. In summary, an all-dgal mplemenaon of he UWB recever n sensor newors appears unfeasble. Movaed by he above dscusson, n hs wor, a smple erave dsrbued algorhm s consdered and a physcal layer scheme s proposed ha allows achevng consensus on a common parameer of neres wh affordable complexy. In parcular, we consder a collecon of sensors conneced by wreless lns operang n a half-duplex manner and composed of he followng basc componens. A ransducer ha s used o monor he physcal parameer of neres and a dynamcal sysem whose sae evolves n me accordng o he local measuremens and he saes of nearby nodes. Fnally, a rado nerface ha s used o ransm he sae of he dynamcal sysem and receve hose of nearby sensors. We consder he class of he so-called quanzed dsrbued consensus algorhms n whch he local measuremens or curren saes of each sensor belong o a fne se (see, e.g., [19 21] andreferencesheren).inparcular,heeffecve local measuremen of each sensor s frs quanzed on aspecfednumberoflevelsandhenlaunchedoverhe physcal channel usng an UWB sgnal wh PPM. Each sensor updaes s curren sae accordng o he erave algorhm on he bass of he sgnals receved from s neghbors. In parcular, he sae ncremen s compued resorng o he barycener-based scheme llusraed n [14]. The laer s frs revsed o be appled o he sysem under nvesgaon and hen exended as follows. Frs, we dscuss he condons under whch s operaon s guaraneed n UWB ransmssons operang over frequency-selecve channels. Second, we propose a smple alernave soluon, whch operaes a a much lower samplng frequency on he order of MHz. Ths ranslaes no a recever archecure of reduced complexy and low-energy consumpon n accordance wh he mplemenaon consrans posed by UWB sensor newors. Thrd, we assume ha he analog esmae of he sae ncremen obaned wh such a reduced complexy soluon s processed by an N b -b quanzer before beng passed o he erave algorhm. Then, we dscuss he crucal ssue of he sysem parameer seng, whch s requred o ensure he convergence of he consensus algorhm. For compleeness, we reurn also o he mehod proposed n [15] and show ha s mahemacally equvalen o [14] excep for a normalzaon facor, whch depends on he energy of he receved sgnal. Numercal resuls are used o assess he effecveness of he proposed scheme and o mae comparsons wh he exsng alernaves. To hs end, an UWB scenaro nspred by he IEEE a sandard s adoped. These resuls gve a useful gudance on how o mprove he performance of he nvesgaed soluon n praccal applcaons and on how all he consdered aspecs nerac o each oher. To he bes of our nowledge, hs s he frs me ha such an analyss s carred ou n a praccal smulaon seup for UWB sensor newors. In summary, he major conrbuons of hs wor can be summarsed as follows. (a) I proposes a smple recever archecure whch has a much lower complexy compared o exsng alernaves. (b) I provdes a suffcenly dealed analyss of he communcaon and mplemenaon problems arsng n

3 Inernaonal Journal of Dsrbued Sensor Newors 3 sensor newors usng UWB echnologes. Ths may represen n our opnon a useful gudance for sysem developers ha am a seng he sysem parameers of praccal applcaons. (c) Some crcal mplemenaon ssues arsng n praccal applcaons are dscussed such as () he developmen of a recever archecure of reduced complexy and low-energy consumpon n accordance wh he mplemenaon consrans posed by UWB sensors; () he desgn of a suable hreshold able o remove he exra nose wh ensung enhancemen of he sysem performance; () he sudy of he mpac of he number of bs used by he quanzer on he sysem performance. (d) The performance of he proposed physcal layer scheme s assessed and valdaed usng dfferen channel models nspred o he IEEE a sandard. All hs may serve as an ncenve o he research communy for furher nvesgaons snce mos of he problems addressed n hs wor have been compleely negleced so far. The res of he paper s organzed as follows. Nex secon formulaes he problem and descrbes he sysem under nvesgaon. In Secon 3 he barycener-based scheme s frs revsed and hen modfed o enable s physcal mplemenaon. Some useful remars are dscussed n Secon 4. Numercal resuls are presened n Secon5 whle some conclusons are drawn n Secon Problem Saemen and Sysem Descrpon 2.1. Problem Saemen. We consder a collecon of M sensors and we call θ Rhe physcal parameer o be esmaed. Thus, denong by χ (1) he nal esmae of he h sensor, we may wre χ (1) =θ+ς for =1,2,...,M, (1) where ς s a dsurbance erm modeled as a zero mean random varable. As menoned before, we consder he class of quanzed consensus algorhms n whch he nal esmae χ (1) aes values n he fne range (χ mn,χ max ) and s mapped no he fne se Z of cardnaly Z accordng o he followng rule: z (1) = χ(1) z I mn, (2) z where c ndcaes he neger closes o c and I z s a desgn parameer whose dmenson s dcaed by he ype of physcal parameer o be esmaed whle z mn s gven by z mn = χ mn /I z (for smplcy, we assume ha χ mn s amulpleofi z ). I s worh observng ha I z plays a ey role on he esmaon accuracy of he consensus algorhm as conrols he quanzaon error affecng he exchanged nformaon. Albe s praccal neres, s no consdered n hs wor snce depends on he specfc applcaon for whch he sensor newor s employed. Only a commen wll be provded n Secon 5 o show how s effecs can be aen no accoun. The nformaon assocaed wh he nal saes s exchanged among he acve sensors and used by he dsrbued algorhm o evenually drve he newor oward a consensus from whch an esmae of θ s evenually compued. Snce we consder low mobly applcaons n whch he propagaon channel changes slowly compared o he convergence me of he erave algorhm (as shown n Secon 5, he convergence me of he proposed algorhm s relavely small,.e., on he order of few mcroseconds), he connecon lns are assumed o manan consan over he me nerval requred by he consensus algorhm o converge. In he above crcumsances, he dynamc of he erave algorhm can be mahemacally descrbed by he followng recurson: z (n+1) =z (n) + μ (n) e (n) for n=1,2,..., (3) where z (n) denoes he esmae or sae of he h sensor a he nh eraon and μ(n) s a nonnegave desgn parameer (nown as sep sze)haconrolsheconvergenceproperes of he erave algorhm whle e (n) = ρ, (z (n) z (n) ) (4) M s he correspondng sae ncremen. In he above equaon, M denoes he se ha conans he ndexes of he neghbors communcang wh he h sensor whle he quanes {ρ, } are he real-valued nonnegave couplng coeffcens accounng for he neracons among he conneced sensors. In Secon 5, wewllshowhaanappropraesengof he sysem parameers allows he erave procedure n (3)o converge o he followng asympoc resul: lm n z(n) = 1 M M =1 z (1). (5) From (3) and(4), we see ha he curren sae z (n) can be updaed only f we assume ha he generc h sensor has nowledge of he saes {z (n) ;=} of s neghbors. Obvously, hs assumpon s no vald n praccal applcaons. For hs reason, an UWB-based rado nerface o exchange hs nformaon s descrbed n he nex Sysem Descrpon. Smlar o [14, 15], we consder a frequency-synchronzed newor n whch all he acve sensors have a cloc ha evolves n me wh a common perod of duraon T. Asn[14, 15], we also assume ha he consensus phase s preceded by an acquson procedure ha allows he sensors o algn he me reference scales. Each sensor ransms s own sae usng an UWB sgnal wh PPM. For hs purpose, he sae varable z (n) for any = 1, 2,..., M and n=,1,...s mapped pror o ransmsson ono s correspondng mng delay as follows: x (n) =Δz (n), (6)

4 4 Inernaonal Journal of Dsrbued Sensor Newors where Δ s a sysem parameer desgned laer. The quany x (n) s hen used o modulae he poson of ulrashor pulses whose duraon s on he order of a few nanoseconds. The sgnal ransmed by he h sensor durng he nh symbol aes he form llusraed n Fgure1 and s mahemacally descrbed by s (n) () =p( x (n) nt), (7) where p() denoes an ulrashor pulse. The ransmed sgnals propagae hrough dfferen channels and undergo mulpah propagaon. A each sensor, he ncomng waveforms are mplcly recombned by he receve anenna and fed o a receve fler, whch has a recangular ransfer funcon wh bandwdh B suffcenly large o pass he sgnal pulses undsored. As dscussed laer, f he symbol duraon T s properly chosen, he sgnal receved a he h sensor durng he nh symbol can be mahemacally wren as () ={ y(n) () +w (n) () f s (n) f s (n) () = () =, where we have aen no accoun ha each sensor operaes n an half-duplex manner meanng ha canno receve he sgnal durng he me n whch s ransmng. In he above equaon, w (n) () s he hermal nose modeled as agaussanrandomprocesswhzeromeanandwo-sded power specral densy N /2 whle y (n) () aes he form (8) y (n) () = h, ( x (n) nt). M In addon, h, () denoes he overall channel mpulse response beween he h and hsensor and s gven by h, () = L, 1 l= (9) γ, (l) g( τ, (l) ], ), (1) where L, denoes he number of dsnc pahs whle g() s obaned as he convoluon of he ransm and he receve flers. Fnally, γ, (l) s he gan of he lh pah and τ, (l) s s correspondng delay whle he quany ], sands for a possble msalgnmen, due o he mng errors of he nal synchronzaon phase, beween he me reference scales of user and (see Fgure 2). We are now lef wh he problem of desgnng he symbol duraon T. The laer mus be chosen long enough so as o accommodae he ransmsson nervals, he channel delay spreads, and he possble mng msalgnmens among he acve sensors. Mahemacally, hs amouns o seng T=T lef + Z Δ+T rgh, (11) where T lef and T rgh are desgn parameers ha depend on he channel characerscs and he resdual mng offses. In parcular, he quany T lef accouns for he maxmum lefward mng shf among he sgnals receved from all he sensors and s gven by T lef = ] max ] mn τ mn, (12) s (n) () nt x (n) T nt + T Fgure 1: Illusraon of he UWB sgnal wh PPM. s (n) () T () nt, nt x (n), +x (n) +τ, () nt + T nt + T Fgure 2: Illusraon of he relaonshp beween he sgnal ransmed by sensor and he correspondng sgnal receved by sensor. where τ mn = mn,,l {τ, (l)} whle ] max and ] mn are, respecvely, defned as ] max = max, {], } and ] mn = mn, {], }.Onheoherhand,T rgh denoes he maxmum rghward mng shf nduced by he channels of he acve sensors and aes he form wh τ max = max,,l {τ, (l)}. T rgh = ] max ] mn +τ max (13) 3. A Praccal Sae Incremen Esmaor In hs secon, we frs adap he mehod llusraed n [14] o he sysem under nvesgaon and hen we dscuss s mplemenaon ssues Barycener-Based Esmaon. Whou loss of generaly, we concenrae on he sgnal receved by he h sensor durng henh nervaland assume ha =denoes he sar of he consdered nerval n he h me scale. We sar shfng lefward he receved sgnal by a quany.nex,wecompuehebarycenerofheresulngsgnal ( + x(n) ) over he observaon wndow of duraon T and gven by T (n) n =[nt T lef,nt+ Z Δ + T rgh ].Thsproduces x (n) where E (n) r B (n) = 1 E (n) r 2 T (n) ( + x (n) )d, (14) n s he energy of ();has,e(n) r = T (n) 2 ()d. n We begn by observng ha n UWB sysems he me nerval requred o ransm he nformaon sgnal s (n) ()

5 Inernaonal Journal of Dsrbued Sensor Newors 5 () nt,2 +x (n) 2 +τ,2 (),1 +x (n) 1 +τ,1 (),3 +x (n) 3 +τ,3 () nt + T Fgure 3: Illusraon of he h receved sgnal. s usually a small fracon of he symbol duraon T. Then, o ease he noaon and o faclae he mahemacal compuaons, we may reasonably approxmae (8) as follows: () =y (n) () +w (n) (). (15) To proceed furher, we assume ha he sgnal-o-nose rao (SNR) s relavely large so ha he nose conrbuon can be negleced. Is effecs wll be aen no accoun laer. Moreover, we assume ha he sgnals receved from dfferen sensors do no overlap n me. Ths suaon s seched n Fgure 3 and amouns o sayng ha he overall channel mpulse responses sasfy he followng condon: h, ( τ nt) h,m ( ψ nt) d = T (n) n for any n, =m, τ=ψ. (16) Ths assumpon s no exacly sasfed n praccal applcaons and s adoped n hs wor only o smplfy he analyss. Alhough no exacly sasfed and nroduced only for analycal purposes, hs assumpon s que reasonable n UWB sysems hans o he ulrashor naure of pulses and he heavy mulpah behavour of propagaon channels ha mae he receved sgnals hghly uncorrelaed [22]. Ths means ha he proposed soluons wll operae n a msmached mode whose mpac on he sysem performance wll be evaluaed by means of numercal resuls. In all he above crcumsances (all of hem wll be removed n Secon 5 where he performance of he proposed algorhm s evaluaed numercally), we have ha sgnal ( + x (n) ) n (14)aesheform 2 2 whle E (n) r ( + x (n) )= h 2, ( x(n) +x (n) nt) (17) M s gven by E r = M E h,, (18) where he funconal dependence from n has been omed due o he above assumpons and we have defned E h, he energy of h, ().Subsung(17)no(14)produces B (n) 1 = h E M r T 2 (n), ( x(n) +x (n) nt)d (19) n whch can be rewren n he followng equvalen form B (n) where λ, s gven by = H, + λ, (x (n) x (n) ), (2) M M λ, = E h, E r (21) whle H, s defned as H, = (1/E r ) T (n) h 2, ( nt)d. n Dvdngbohsdes of (2) byδ,usng(6), we oban 1 Δ B(n) = 1 Δ H, + λ, (z (n) z (n) ) (22) M M from whch follows ha B (n) /Δ s a based esmae of he sae ncremen gven by (4) afer replacng ρ, wh λ,.to ge rd of he bas erm (1/Δ) M H,, we follow he same lne of reasonng llusraed n [15] and adop a soluon ha was orgnally presened n [23]. In parcular, reles on he assumpon ha he ransmsson s preceded by a plo symbol durng whch each sensor ses s own sae o zero (.e., x () =) and ransms he followng sgnal s () () = p() for =1,2,...,M.Inabsenceofnose,heh receved sgnal aes he form r () () = M h, () and s used o compue B () = 1 r E r T ()2 () () d. (23) n Parallelng he same seps as before, we oban B () = M H,. (24) Once B () s obaned, can be exploed o compue he followng quany: e (n) = 1 Δ (B(n) B () )= λ, (z (n) M z (n) ) (25) whch s used o updae he sae of he h sensor accordng o (3). The mng offse x (n+1) s hen obaned from z (n+1) as follows: x (n+1) =Δz (n+1). (26) The laer s evenually used o modulae he poson of he UWB sgnal durng he n+1me nerval. In he sequel, he scheme based on (3) and(25)-(26) sreferredoashe barycener-based sae ncremen esmaor (BIE) Implemenaon Issues. We begn by observng ha he mplemenaon of BIE needs he compuaon of (14) and (23). As depced n Fgure 4, evaluanghesequanes requres o delay he sgnal a he oupu of he LPF by a quany ha n (14) depends on he ndex n. The resulng sgnal s passed hrough a square law devce whose oupu s

6 6 Inernaonal Journal of Dsrbued Sensor Newors hen mulpled by he connuous sgnal and fnally fed o an negrae and dump crcu wh rae 1/T. Alhoughaafrs glance all he above operaons seem easly mplemenable, an accurae nspecon reveals ha among hem here s one ha may preven he praccal realzaon of BIE. Ths s represened by he unable delay lne, whch mus handle awdebandanalogsgnalandmusalsobeveryaccurae o guaranee he convergence of he erave algorhm. Such an accurae and conrolled devce canno be deployed wh analog archecures unless expensve and cumbersome equpmens are used. However, hs s n sharp conras o he low-cos and low-energy consumpon requremens of UWB sensor newors [24]. A possble way ou s o mae use of an all-dgal recever [14], whch operaes drecly on he samples of he sgnal a he LPF oupu aen a Nyqus rae. As menoned prevously, also hs approach s no sued for sensor newors as would requre ADCs wh samplng frequency of some GHz, whch are oo expensve and exremely energy consumng [25]. Movaed by he above argumens, n he sequel we propose an alernave soluon, whch dspenses from all he above mparmens. We sar compung he barycener of he receved sgnals ().Thsproduces B (n) = 1 E (n) r 2 T (n) () d (27) n from whch usng he same argumens of he prevous secon follows ha B (n) = H, + λ, x (n). M M Then, we compue he followng quany: e (n) (28) = 1 Δ (B (n) B () x (n) ). (29) Subsung (24) and (28) no he above equaon and observng ha M λ, =1,weoban e (n) = λ, (z (n) z (n) ) (3) M whch represens an esmae of he sae ncremen n he form gven by (25). From (24) and(27), we see ha he compuaon of (29) does no requre any delay lne as operaes drecly on he receved sgnals. Indeed, he recever bloc dagram of he proposed algorhm s ha seched n Fgure 4 whou he unable delay lne. The recever archecure s now composed by wo negraed and dump crcus, whch operae a symbol rae of 1/T and by a quanzer. Ineresngly, 1/T s much lower han he Nyqus rae snce n he nvesgaed sysem he Nyqus rae s on he order of some GHz whle he symbol rae s on he order of some MHz. We now observe ha he sae ncremen n (29)depends on B (n) and B ().Thelaerareanalogquaneshamay ae any value n he nerval (, T). In praccal applcaons, however, only a quanzed verson of hese varables can be passed o he consensus algorhm. For hs reason, we assume ha he recever s equpped wh a unform N b -b quanzer whch produces B (n) B () = B (n) T T res, res = B() T T res, res (31) where T res = T/2 N b denoes he me-resoluon of he quanzer. Replacng B (n) and B () (n) () wh B and B no n (29)yelds e (n) e (n) = 1 Δ (n) () ( B B x (n) ) (32) whch s evenually employed o updae he sae varable accordng o (3). The above scheme s called reducedcomplexy BIE () n he sequel. 4. Remars () Removng E (n) r from he rgh-hand sde of (14)produces P (n) = r T (n)2 (n) (+x (n) ) d (33) n from whch usng he rule of negraon by pars we oban P (n) Leng (n) yelds wh =T E (n) r T (n) n = x (n) u (n) P (n) () = 2 ( +x (n) )d d. (34) and usng sandard manpulaons =T E (n) r + u T (n) (n) () d (35) n x (n) + x (n) 2 ( (n) )d (n) (36) whch are equvalen o he soluon llusraed n [15]. As ancpaed before, hs means ha he schemes n [14, 15] are subsanally he same excep for he normalzaon facor.smlaro[14], from (35) and(36), follows ha he mehod n [15] has a praccal dsadvanage: requres o negrae he receved sgnal over a sldng wndow of amplude [ x (n), x(n) +].Thsoperaoncanonlybecompued n dgal form, hereby mang unsued for praccal applcaons. Followng he same lne of reasonng employed n he prevous secon for [14], a reduced complexy soluon can be easly derved. () I s worh observng ha he scheme presened n [15] does no employ any square devce. Ths s due o he fac ha he auhors mae use of pulses wh un area (hs amouns E (n) r

7 Inernaonal Journal of Dsrbued Sensor Newors 7 () z Δ+T rgh T lef Delay LPF Quanzer BIE lne () 2 nt T T lef z (n+1) z Δ+T rgh T lef We have denoed T :=mod T Fgure 4: Bloc dagram of he recever employng BIE. o sayng ha τ g g()d = 1 wh τ g beng he duraon of g()) raherhanzeroasassumednhswor.underhs assumpon, he esmae of he error sae ncremen can be compued as 2 () SNR = 1 db e (n) = 1 Δ (P (n) P () E r x (n) ), (37) where P (n) and E r are now gven by P (n) = T (n) n E r = T (n) n () d, () d. Parallelng he same seps of before yelds e (n) where λ, are now defned as follows: (38) = λ, (z (n) z (n) ), (39) M λ, = h, ( nt) d. (4) T (n) n Ineresngly, he above resuls can be derved whou mposng he condon (16)on he channel mpulse responses of he acve sensors. On he conrary, when a square law devce s adoped he no nerference condon mus be fulflled. Ths fac s parcularly appealng as would mae he scheme based on (37)-(38) nsensveohenerferenceamonghe dfferen channel mpulse responses. The man mparmen of such a soluon s ha would requre bandpass communcaons snce he drec curren of asgnalcannobephyscallyransmednbaseband.albe feasble, hs approach seems o be unsued for praccal UWB sensor newors as would requre he use of expensve and power consumpon devces for he demodulaon procedure. () As menoned prevously, n UWB sysems, he sgnals ransmed by he acve sensors propagae hrough dfferen pahs whose number s on he order of hundreds. Ths ranslaes no a severe dsperson of he sgnal power a he recevng ermnal. Then, may happen ha even for 2 () T SNR = 2 db Fgure 5: Illusraon of he sgnal 2 () for dfferen values of SNR when he number of acve sensors s M=4. moderae values of SNRs he receved sgnal s overwhelmed by he hermal nose, hereby prevenng he applcaon of all he nvesgaed soluons. Ths suaon s depced n Fgure 5, wherehesquareofherecevedsgnalsshown for wo dfferen values of SNR assumng ha he number of acve sensors s M = 4.Assseen,whenheSNRs fxed o 1 db he useful componen canno be dsngushed from he receved sgnal as s compleely overwhelmed by he hermal nose. A beer suaon s observed for an SNR of 2 db. However, even for hs favorable case has been proven by compuer smulaons ha all he nvesgaed soluons fal. Ths fac can be explaned observng ha he observaon wndow s equal o T (n) n.asdepcednfgure 5, s much larger han he suppor of he useful componen as has been chosen so as o accommodae he channel delay spreads and all possble mng msalgnmens. Then, may happen ha, even for consderable values of SNR, he oal amoun of colleced nose s oo large o acheve accepable performance. I s worh observng ha hese ssues have been also parally dscussed n [14] buonlyforagaussan T (n) n

8 8 Inernaonal Journal of Dsrbued Sensor Newors channel n whch hey are less harmful for he followng wo reasons. Frs, all he ransmed power s concenraed n a sngle pulse ha sands more easly over he nose. Second, n a Gaussan channel he observaon wndow can be chosen shorer snce he delay spread s lmed o he duraon of he ransmed pulses. Apossblesoluonoconrasallheaboveproblemss o mae use of a properly desgned hreshold λ,whchallows removng he exra nose. Omng he me nerval n for noaonal smplcy, hs amouns o replacng r 2 () wh r 2 () ={r2 () f r2 () λ f r 2 () < λ. (41) To desgn λ, we compue he probably ha he nose conrbuon s greaer han λ whn heobservaon wndow. Usng (15)yelds r 2 () =y2 () +2y () w () +w 2 () (42) from whch follows ha such a probably s gven by P h (λ) =2 Q( λ ), (43) σ2 where we have used he fac ha w () s a Gaussan random process wh zero mean and varance σ 2 =N B.Numercal resuls ndcaes ha a good choce for λ s such ha P h (λ) s equal o 1 5.Unlessoherwsespecfed,hsvaluesadoped for he nvesgaed schemes n all subsequen smulaons. 5. Smulaon Resuls Mone Carlo smulaons have been run o assess he performance of he proposed soluon. The sysem parameers are summarzed as follows Sysem Parameers. The monocycle p() s shaped as he second dervave of a Gaussan funcon wh duraon T p equal o 1 ns whle he bandwdh B of he recever fler s chosen equal o 4 GHz. Unless oherwse specfed, he channel sascs are generaed as specfed by he model CM1 llusraed n [22]andheresulsareobanedaveragngover 1 3 ndependen channel realzaons. The nal observaons {z (1) } are assumed o be unformly dsrbued whn he se Z wh Z = 2 whle he parameers T lef and T rgh are obaned from (12)and(13) seng τ max =22ns, τ mn =1ns, ] max =15ns, and ] mn = 15 ns. The mng msalgnmens are randomly chosen from he nerval [] mn, ] max ].Then,wehavehaT lef = 2ns and T rgh = 25ns. In order o mprove he seady sae performance of he quanzed consensus algorhm, he sep sze μ(n) n (3) schosenequaloμ(n) = μ /n, whereμ s a nonnegave desgn parameer [26]. Ths provdes he sysem wh a ceran robusness agans he dermenal effecs of he quanzaon nose nroduced by he recurson n (3). Unless oherwse specfed, he performance of he nvesgaed soluons has been assessed by measurng he roomean-square error (RMSE) of he esmaes a he seady sae: where η s gven by = E{ 1 M η= 1 M M [z (N s) =1 M z (1) =1 η] 2 }, (44) (45) whle N s denoes he eraon ndex for whch he algorhm acheves s seady sae. In he sequel, such a sae s acheved a he eraon ndex from whch he varaons of he are lmed n he range ±1 2.Ineresngly,wehavefound hrough numercal resuls ha he proposed algorhm s vrually unbased, so he RMSE and he sandard devaon of he esmaon error are praccally he same. We now show how he defned above can be used o compue he RMSE of he unnown parameer θ: RMSE θ = E{ 1 M M =1 [ θ θ] 2 }, (46) where θ denoes he esmae of θ a he h sensor durng he seady sae. Recallng (1)and(2), we have ha θ s gven by θ =I z z (N s). (47) We now le ε =z (N s) ηand use (45) orewreherghhand sde of (47) n he followng equvalen form: θ =I z ε + 1 M M =1 I z z (1). (48) From (1) and(2), we have ha I z z (1) =θ+a +ς,where a s a random varable dependng on he quanzaon error, whch s assumed o be unformly dsrbued whn he range ±I z /2. Collecng all he above facs ogeher and assumng ha ε, a,andς are sascally ndependen, usng sandard compuaons, follows ha RMSE θ aes he form RMSE θ = σ2 ς M 2 +I2 z (MSE 1 z + 12 M 2 ), (49) where σ ς denoes he sandard devaon of he nose erm ς.theaboveresulsusefuloevaluaeherelaonshp beween he esmaon accuracy provded by each sensor worng ndvdually (represened by σ ς )whhaensured by consensus (gven by RMSE θ ). Assume, for example, ha θ s a emperaure and M = 2.Ifσ ς = 3,MSE z =.5 (see, e.g., Fgure 8) andi z = 1,from(49), follows ha RMSE θ =.72.Thsmeanshaconsensusmayensure esmaon accuracy four mes beer han ha of a sngle sensor.

9 Inernaonal Journal of Dsrbued Sensor Newors M= 2, SNR = 25 db 6 5 M= 2, SNR = 25 db, Δ = 25 ns, and μ = Δ (ns) SNR (db) 4 μ =3 μ =4 μ =5 N b =4 N b =5 N b =6 N b =7 N b =8 No quanzaon Fgure 6: of a he seady sae as a funcon of μ for dfferen values of μ wh SNR = 25 db, N b =8,andM=2. Fgure 8: of a he seady sae as a funcon of SNR for dfferen values of N b wh Δ =25ns,μ =4,andM= μ =3 μ =4 μ =5 M= 2, SNR =25dB, and Δ=25ns nt (ns) Fgure 7: (n) of versus nt for dfferen values of μ wh SNR = 25 db, N b =8,andM= Performance Evaluaon. We begn by assessng he mpac of he sysem parameers Δ and μ on he performance of. Fgure 6 llusraes he of as a funcon of Δ for dfferen values of μ when he SNR s fxed o 25 db, M = 2,andN b = 8.Weseehahe bes performance s obaned wh μ equal o 4 and ha he eeps praccally consan for values of Δ larger han 25 ns, whle an ncrease s observed for Δ<25ns. Such abehavorsuggessochooseavalueofδ greaer han or equal o 25 ns. From (11), s seen ha Δ canno be chosen arbrarly large as would ncrease he symbol duraon and consequenly he requred converge me. To sasfy hese wo conflcng requremens, n all he subsequen smulaons, we adop Δ=25,whchcorrespondsoasymbolduraonT of.77 μs. A close nspecon of he resuls of Fgure 6 ndcaes ha he bes performance a he seady sae s acheved for μ = 4. However, hs fac s no enough o fx μ = 4 snce he sep sze mus be chosen so as o acheve a reasonable radeoff beween seady sae performance and convergence capables. For hs reason, n Fgure7, we llusrae how he evolves n me wh he eraon ndex for dfferen values of μ when he sysem parameers are he same of Fgure 6.Forhspurpose,heperformances assessed by evaluang he a he nh eraon, whch s mahemacally gven by (n) = E{ 1 M M [z (n) =1 η] 2 }. (5) From he resuls of Fgure 7, weobservehahasa very shor convergence me as acheves he equlbrum n only 3 μswhenμ =3whle 4 μsareneededforμ =4.Then, we may reasonably se μ =4as ndcaed by he resuls of Fgure 6. Fgure 8 shows he of as a funcon of SNR for dfferen values of N b wh Δ =25ns,μ =4,andM = 2. The curve labelled No quanzaon corresponds o he performance of when no quanzaon s employed a he recever and serves a benchmar. As seen, reducng N b up o 6 does no affec apprecably he performance of for SNR values of praccal neres. Snce a small number of bs resuls no a quanzer of reduced complexy

10 1 Inernaonal Journal of Dsrbued Sensor Newors SNR = 25 db, Δ = 25 ns,μ = 4, and Nb =6 8 M = 2, Δ = 25 ns, and μ = SNR (db) 4 2 SNR (db) 4 M = 1 M = 2 M=3 Fgure 9: of versus SNR n db for dfferen values of M wh Δ =25ns,μ =4,andN b = P h (λ) = 1 3 P h (λ) = M=2,Δ=25ns,μ = 4, and Nb =6 SNR (db) 4 P h (λ) = 1 7 P h (λ) = 1 Fgure 1: of versus SNR n db for dfferen values of P h (λ) wh Δ =25ns,μ =4, N b =6,andM=2. and lower energy consumpon, we choose N b equal o 6 n all subsequen smulaons. Fgure 9 depcs he versus SNR when he number of acve sensors s M = 1, 2, and 3. As expeced, he reduces as he SNR ncreases. Ineresngly, we observe ha n he low SNR regme ncreasng M mproves he sysem performance whle only margnal dfferences are observed for all he oher SNR values of praccal neres. We now reurn o he need of usng a properly desgned hreshold o remove he nose conrbuon a he oupu of he square devce. For hs purpose, n Fgure 1 we plo BIE Fgure 11: of and BIE versus SNR n db wh Δ = 25 ns, μ =4,andM=2. he versus SNR for dfferen values of λ.forcomparsons, we repor also he performance of a sysem n whch he hreshold s se o zero. From he resuls of Fgure 1 follows ha our argumens n Secon 4 were correc. In fac, a gan greaer han 15 db s acheved n all nvesgaed scenaros n whch he hreshold s employed. As ancpaed, he bes resuls are obaned for λ such ha P h (λ) = 1 5. In Fgure 11,wecomparewhBIE.Comparsons wh BIE are made under a common smulaon seup, whch ncludes he same Δ=25ns and μ =4as well as he same hreshold λ.fromhe resuls of Fgure 11,we see ha has vrually he same of BIE for all he nvesgaed SNRs. Ths resul s acheved wh reduced complexy snce he proposed soluon operaes wh a much lower samplng frequency. All he above resuls are obaned consderng he CM1 scenaro. We now assess how he propagaon channel nfluences he sysem performance. For hs purpose, Fgure 12 llusraes he versus SNR for he followng channel propagaon models: CM1, CM2, CM3, and CM4. The laer are four of he mos represenave channel models defned for he IEEE a sandard (see [22] for more deals). In parcular, CM1 and CM2 apples o a lne-of-sgh (LOS) and a non-lne-of-sgh (NLOS) propagaon n resdenal envronmens, respecvely, whle CM3 and CM4 reflec a LOS and a NLOS scenaro n offce envronmens. In parcular, he buldng srucures of resdenal envronmens are characerzed by small uns, wh ndoor walls of reasonable hcness and cover a range from 7 o 2 m. For offce envronmens, some of he rooms are comparable n sze o resdenal, bu oher rooms (especally cubcle areas, laboraores, ec.) are consderably larger. Areas wh many small offces are ypcally lned by long corrdors. Each of he offces ypcally conans furnure, booshelves on he walls, and so forh, whch adds o he aenuaon gven by he (ypcally hn) offce paronng. Offce envronmens cover

11 Inernaonal Journal of Dsrbued Sensor Newors M = 2, Δ = 25 ns,μ =4, and N b =6 such problems. In parcular, a hardware mplemenaon for esng he performance of he proposed soluon under real world condons would be a very neresng opc for fuure research. 6 4 Conflc of Ineress The auhors declare ha here s no conflc of neress regardng he publcaon of hs paper. 2 CM1 CM2 2 SNR (db) CM3 CM4 Fgure 12: of versus SNR n db for dfferen channel models wh Δ =25ns,μ =4, N b =6,andM=2. arangefrom3o28m.weobservehahebesperformance s acheved for CM1. However, changng he propagaon channel ncurs n a loss of less han 4 db. 6. Conclusons and Dscussons We have presened and nvesgaed he behavours of a physcal layer scheme for achevng consensus n UWB sensor newors. The praccal mplemenaon ssues of he proposed soluon have been nvesgaed and s performance has been deeply analyzed. Parcular aenon has been devoed o he sudy of he mpac of he number of bs used by he quanzer on he sysem performance and o a proper desgn of he sysem parameers n order o acheve fas convergence me wh hgh esmaon accuracy. We have found ha wh a 6-b resoluon he loss, compared o an deal recever wh no quanzaon, s no apprecably and ha a seady sae wh neglgble esmaon errors can be acheved n less han 4 μs. Theperformancehas been evaluaed under a praccal smulaon seup nspred o he IEEE a sandards. On he bass of such an analyss, urns ou ha he proposed scheme provdes he same performance of an exsng alernave based on an all-dgal mplemenaon. However, hs resul s obaned wh a reduced complexy and low-energy consumpon as allows he recever o operae a a much lower samplng frequency on he order of MHz nsead of GHz. Ths maes parcularly sued for praccal applcaons. To he bes of our nowledge, hs s he frs me ha an analyss on desgnng approprae physcal layer schemes o acheve consensus n praccal applcaons s carred ou. Indeed, mos of he exsng wors n hs feld are largely focused essenally on he mahemacal characerzaon of he condons for achevng consensus. We hope ha hs wor may ac as an ncenve o he research communy for furher nvesgang 4 References [1] I. F. Ayldz, W. Su, Y. Sanarasubramanam, and E. Cayrc, A survey on sensor newors, IEEE Communcaons Magazne, vol. 4, no. 8, pp , 22. [2] V. Raghunahan, S. Ganerwal, and M. Srvasava, Emergng echnques for long lved wreless sensor newors, IEEE Communcaons Magazne,vol.44,no.4,pp ,26. [3] K. Römer and F. Maern, The desgn space of wreless sensor newors, IEEE Wreless Communcaons,vol.11,no.6,pp.54 61, 24. [4] R. Olfa-Saber, J. A. Fax, and R. M. Murray, Consensus and cooperaon n newored mul-agen sysems, Proceedngs of he IEEE,vol.95,no.1,pp ,27. [5] T. C. Aysal, M. E. Yldz, A. D. Sarwae, and A. Scaglone, Broadcas gossp algorhms for consensus, IEEE Transacons on Sgnal Processng,vol.57,no.7,pp ,29. [6] S. Kr, A. Scaglone, and R. J. Thomas, A scalable wreless communcaon archecure for average consensus, n Proceedngs of he 46h IEEE Conference on Decson and Conrol (CDC 7),pp.32 37,December27. [7] G. Scuar, S. Barbarossa, and L. Pescosoldo, Dsrbued decson hrough self-synchronzng sensor newors n he presence of propagaon delays and asymmerc channels, IEEE Transacons on Sgnal Processng, vol.56,no.4,pp , 28. [8] G. Scuar and S. Barbarossa, Dsrbued consensus over wreless sensor newors affeced by mulpah fadng, IEEE Transacons on Sgnal Processng,vol.56,no.8,pp , 28. [9] L. Yang and G. B. Gannas, Ulra-wdeband communcaons, IEEE Sgnal Processng Magazne, vol. 21, no. 6, pp , 24. [1] J. Zhang, P. V. Orl, Z. Sahnoglu, A. F. Molsch, and P. Knney, UWB sysems for wreless sensor newors, Proceedngs of he IEEE,vol.97,no.2,pp ,29. [11] Par 15.4: wreless medum access conrol (MAC) and physcal layer (PHY) specfcaons for low-rae wreless personal area newors (LRWPANs), Tech. Rep. IEEE P a/D4, July 26, Amendmen of IEEE Sd [12] Y. Aawa, H. Andoh, and T. Kohama, Auonomous decenralzed ner-base-saon synchronzaon for TDMA mcrocellular sysems, n Proceedngs of he 41s IEEE Vehcular Technology Conference,pp ,May1991. [13] E. Sourour and M. Naagawa, Muual decenralzed synchronzaon for nervehcle communcaons, IEEE Transacons on Vehcular Technology,vol.48,no.6,pp ,1999. [14] O. Smeone and U. Spagnoln, Dsrbued me synchronzaon n wreless sensor newors wh coupled dscree-me oscllaors, Eurasp Journal on Wreless Communcaons and Neworng, vol. 27, Arcle ID 5754, 27.

12 12 Inernaonal Journal of Dsrbued Sensor Newors [15] L. Pescosoldo, S. Barbarossa, and G. Scuar, Average consensus algorhms robus agans channel nose, n Proceedngs of he 9h IEEE Worshop on Sgnal Processng Advances n Wreless Communcaons (SPAWC 8),pp ,July28. [16] C. Seffens and M. Pesaveno, A physcal layer average consensus algorhm for wreless sensor newors, n Proceedngs of he 16h Inernaonal ITG Worshop on Smar Anennas (WSA 12),pp.7 77,March212. [17] Y. Vanderperren, G. Leus, and W. Dehaene, An approach for specfyng he ADC and AGC requremens for UWB dgal recevers, n Proceedngs of he IET Semnar on Ulra Wdeband Sysems, Technologes and Applcaons,pp.196 2,Aprl26. [18] Y. N. Sheung, B. Jalal, Z. Pengbe, J. Wlson, and M. Ismal, A low-volage CMOS 5-b 6MHz 3mW SAR ADC for UWB wreless recevers, n Proceedngs of he 48h Mdwes Symposum on Crcus and Sysems (MWSCAS 5), vol.1,pp , Augus 25. [19] A. Kashyap, T. Başar, and R. Sran, Quanzed consensus, n Proceedngs of he IEEE Inernaonal Symposum on Informaon Theory (ISIT 6), pp , Seale, Wash, USA, July 26. [2] S. Kar and J. M. F. Moura, Dsrbued average consensus n sensor newors wh quanzed ner-sensor communcaon, n Proceedngs of he IEEE Inernaonal Conference on Acouscs, Speech and Sgnal Processng (ICASSP 8), pp ,Las Vegas, Nev, USA, Aprl 28. [21] J. Fang and H. L, An adapve quanzaon scheme for dsrbued consensus, n Proceedngs of he IEEE Inernaonal Conference on Acouscs, Speech, and Sgnal Processng (ICASSP 9), pp , Tape, Tawan, Aprl 29. [22] A. F. Molsch, K. Balarshnan, C. C. Chong e al., IEEE a channel model fnal repor, Tech. Rep., February 25. [23] F. Tong and Y. Aawa, Theorecal analyss of nerbase-saon synchronzaon sysems, IEEE Transacons on Communcaons,vol.46,no.5,pp ,1998. [24] V. Raghunahan, C. Schurgers, S. Par, and M. B. Srvasava, Energy-aware wreless mcrosensor newors, IEEE Sgnal Processng Magazne,vol.19,no.2,pp.4 5,22. [25] R. H. Walden, Analog-o-dgal converer survey and analyss, IEEE Journal on Seleced Areas n Communcaons, vol.17,no. 4, pp , [26] L. Xao and S. Boyd, Fas lnear eraons for dsrbued averagng, n Proceedngs of he 42nd IEEE Conference on Decson and Conrol,vol.5,pp ,December23.