1754 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 6, NO. 5, MAY 2007

1754 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 6, NO. 5, MAY 007 On th Fasibility of Distributd Bamforming in Wirlss Ntworks R. Mudumbai, Studnt Mmbr, IEEE, G. Barriac, Mmbr, IEEE, and U. Madhow, Fllow, IEEE Abstract Enrgy fficint communication is a fundamntal problm in wirlss ad-hoc and snsor ntworks. In this papr, w xplor th fasibility of a distributd bamforming approach to this problm, with a clustr of distributd transmittrs mulating a cntralizd antnna array so as to transmit a common mssag signal cohrntly to a distant Bas Station. Th potntial SNR gains from bamforming ar wll-known. Howvr, ralizing ths gains rquirs synchronization of th individual carrir signals in phas and frquncy. In this papr w show that a larg fraction of th bamforming gains can b ralisd vn with imprfct synchronization corrsponding to phas rrors with modratly larg varianc. W prsnt a mastr-slav architctur whr a dsignatd mastr transmittr coordinats th synchronization of othr (slav) transmittrs for bamforming. W obsrv that th transmittrs can achiv distributd bamforming with minimal coordination with th Bas Station using channl rciprocity. Thus, inxpnsiv local coordination with a mastr transmittr maks th xpnsiv communication with a distant Bas Station rcivr mor fficint. Howvr, th duplxing constraints of th wirlss channl plac a fundamntal limitation on th achivabl accuracy of synchronization. W prsnt a stochastic analysis that dmonstrats th robustnss of bamforming gains with imprfct synchronization, and dmonstrat a tradoff btwn synchronization ovrhad and bamforming gains. W also prsnt simulation rsults for th phas rrors that validat th analysis. Indx Trms Distributd bamforming, synchronization, wirlss ntworks, snsor ntworks, spac-tim communication. I. INTRODUCTION DISTRIBUTED bamforming has th potntial of gratly nhancing nrgy fficincy in wirlss ntworks. A group of cooprativ transmittrs can mulat an antnna array by transmitting a common mssag signal in such a way that th transmission is focusd in th dirction of th intndd Bas Station rcivr (BS). For a narrowband mssag signal this can b arrangd by adjusting th carrir phas of ach transmittr in such a way that th individual transmissions combin cohrntly at th rcivr. Th nrgy fficincy gains from bamforming ar wll-known in th litratur; if a singl lmnt antnna transmitting with powr P T achivs a rcivd signal to nois ratio (SNR) of ρ 1, an array of N Manuscript rcivd August 6, 005; rvisd May, 006 and Sptmbr 1, 006; accptd Octobr 19, 006. Th associat ditor coordinating th rviw of this papr and approving it for publication was A. Swami. This work was supportd by th National Scinc Foundation undr grants CCF- 043105, CNS-050335 and ANI-00118, by th Offic of Naval Rsarch undr grants N00014-06-1-0066 and N00014-03-1-0090, and by th Institut for Collaborativ Biotchnologis through grant DAAD19-03-D-0004 from th U.S. Army Rsarch Offic. R. Mudumbai and U. Madhow ar with th Dpartmnt of Elctrical and Computr Enginring, Univrsity of California Santa Barbara, CA 93106 USA (-mail: {raghu, madhow}@c.ucsb.du). G. Barriac is with Qualcomm Inc., 9950 Barns Canyon Rd, San Digo, CA 911 USA (-mail: gbarriac@qualcomm.com). Digital Objct Idntifir 10.1109/TWC.007.05610. Fig. 1. 1536-176/07$5.00 c 007 IEEE Communication modl for a snsor ntwork. idntical antnnas can us bamforming to achiv a SNR of ρ N = Nρ 1 with th sam total transmit powr P T,i.. with ach antnna transmitting with powr PT N. Physically, this SNR incras ariss from th incrasd dirctivity of th transmission from th antnna array; bamforming focuss N tims mor of th transmittd lctromagntic nrgy in th dirction of th rcivr. Th biggst challng in ralizing ths gains is th rquirmnt of phas and frquncy synchronization of th high-frquncy carrir signals. In this papr, w prsnt a mastr-slav protocol for synchronization, xamin its prformanc using thory and simulation, and invstigat th fasibility of distributd bamforming with imprfct synchronization. Whil th idas hr may b of gnral applicability to diffrnt kinds of wirlss ntworks, w focus on distributd bamforming in th contxt of a clustr of nrgy-constraind wirlss snsor nods communicating with a distant Bas Station rcivr (BS), as illustratd in Fig. 1. Th main assumption is that local communication among th cooprating snsors is inxpnsiv compard to transmitting to th Bas Station. Accordingly, w considr a mastr-slav architctur, whr a dsignatd mastr snsor coordinats th calibration and synchronization of th carrir signals of th othr slav snsors. In this way, th snsors us chap local communication btwn th mastr and th slav snsors to mulat a cntralizd antnna array, and to avoid th nd for coordinating with th distant BS. In a traditional (cntralizd) multi-antnna transmittr, on way to prform bamforming is by xploiting rciprocity to stimat th complx channl gains to ach antnna lmnt. Ths channl gains ar computd in a cntralizd mannr with rfrnc to a RF carrir signal supplid by a local oscillator. Howvr, in a distributd stting, ach snsor has sparat RF carrir signals supplid by sparat local oscillator circuits. Ths carrir signals ar not synchronizd apriori. In th absnc of carrir synchronization, it is not possibl to stimat and pr-compnsat th channl phas rsponss so as to assur phas cohrnc of all signals at th rcivr.

MUDUMBAI t al.: ON THE FEASIBILITY OF DISTRIBUTED BEAMFORMING IN WIRELESS NETWORKS 1755 It is ncouraging to not that th achivabl bamforming gains from imprfct phas synchronization ar substantial. Considr th simpl xampl of two qual amplitud signals from two transmittrs combining at th BS with rlativ phas rror of δ. Th rsulting signal amplitud is givn by 1+ jδ =cos ( δ ). Evn a significant phas rror of δ =30 givs a signal amplitud of 1.93, whichis96% of th maximum possibl amplitud of.0 corrsponding to th zro phas rror cas. Mor gnrally, w show that it is possibl to achiv SNR of up to 70% of th maximum with modratly larg phas rrors 1 on th ordr of 60. Th fasibility of distributd bamforming thn dpnds on bing abl to kp th synchronization rrors sufficintly small. W xamin th diffrnt possibl sourcs of phas rror in dtail. W obsrv that phas nois in practical oscillators causs thm to drift out of synchronization, thrfor, it is ncssary for th mastr snsor to rsynchroniz th slavs priodically. This, combind with th duplxing constraints of th wirlss channl (i.. it is not possibl to transmit and rciv on th sam frquncy simultanously), rvals a fundamntal tradoff btwn synchronization ovrhad and bamforming gain. W quantify this tradoff using a stochastic modl for th intrnal phas nois of oscillators. Thr is now a growing body of rsarch about cooprativ transmission systms, including studis of distributd coding tchniqus for spac-tim divrsity gains [1]. Divrsity schms do not offr avrag SNR gains, but rathr rduc th probability of an outag vnt. Distributd divrsity schms ar, thrfor, of intrst only in fading channls, and do not rquir cohrnt combining of signals. Howvr, bcaus typical divrsity coding schms [] rquir a basband channl that is constant at last ovr th lngth of a codword, thr is an implicit assumption of carrir frquncy synchronization among th cooprating transmittrs. In contrast, bamforming offrs SNR gains in both dtrministic and fading channls, and in addition for fading channls rducs th probability of outag. Howvr, distributd bamforming rquirs a globally consistnt phas rfrnc in addition to carrir frquncy synchronization. Othr authors hav also indpndntly considrd th ida of using distributd transmittrs as a virtual antnna array [3]. Th prformanc of distributd bamforming has bn prviously studid from an information thortic prspctiv [4], [5]. In [6], th authors studid th scalability of adhoc ntworks using distributd bamforming with a Listn and Transmit protocol. Furthr, thy considrd th ffcts of imprfct synchronization and showd that th scalability rsults still hold in th prsnc of synchronization rrors. In [7], th authors propos a synchronization protocol that is suitabl for cohrnt transmission. Howvr, this rquirs significant coordination with th distant BS, and dos not scal for a larg numbr of transmittrs. In rcnt work, th authors in [8] propos a distributd phas synchronization protocol for two transmittrs. Most work in th litratur on clock synchronization, (.g. [9]) focuss on ntwork synchronization, and is unsuitabl for distributd bamforming. 1 Th phas rrors ar th rsult of random nois, and ar, thrfor, random variabls. In this papr w us th trm larg phas rror to indicat a phas rror distribution with a larg root-man squard rror. Th authors in [10] also studid th prformanc of distributd bamforming in snsor ntworks. Thy modl th snsor locations as random and valuat th ffct of location (and phas) uncrtainty on th avrag bampattrn. Ths rsults ar consistnt with and complmntary to th rsults in this papr. Whil [10] xamins th bampattrn avragd ovr all possibl snsor placmnts, w considr som fixd placmnt and xamin th synchronization procss in dtail. In summary, most prior work on cooprativ communication [10], [4], [1], [3] taks synchronization as a givn. To th bst of our knowldg, th protocol prsntd in this papr (which minimizs coordination with th BS), and th complmntary protocol in our rlatd work [11] (which utilizs fdback from th BS to minimiz coordination among th transmittrs), ar th first dtaild studis of th synchronization mchanisms rquird for larg-scal cooprativ communication. Th rst of this papr is organizd as follows. Sction II prsnts a simpl analysis of th ffct of imprfct synchronization on bamforming gain. This shows th high tolranc for phas rrors. W prsnt a mastr-slav architctur for synchronization in Sction III, and driv a distributd protocol for bamforming basd on this architctur. Th main contributor to th rsidual phas rror is oscillator nois, and Sction IV offrs a stochastic analysis of this nois, and idntifis a tradoff btwn th synchronization ovrhad and bamforming gain. Sction V concluds. II. ANALYSIS OF BEAMFORMING GAIN W considr a clustr of N snsors, communicating a common (basband) mssag signal m(t) to a distant Bas Station rcivr, by modulating m(t) with a carrir signal at frquncy f c. Each snsor drivs its carrir signal from a sparat local oscillator, thrfor, th carrir signals of th diffrnt snsors ar not initially synchronizd to ach othr. Thrfor, an xplicit synchronization procss is ncssary. Bfor w prsnt our algorithm for carrir synchronization, w show using a simpl analysis that bamforming gains ar robust to modratly larg phas rrors. For this sction w assum that th synchronization algorithm allows ach snsor to obtain synchronizd carrir signals at frquncy f c and an stimat of thir own channl gain to th BS. Using this th snsors can cooprativly transmit th mssag m(t) by bamforming, just lik a cntralizd antnna array. Th rsulting rcivd signal r(t) is th suprposition of th channl-attnuatd transmissions of all th snsors and additiv nois n(t): ( r(t) =R m(t) jπfct ) gi h i jφ i(t) + n(t) (1) i whr g i is th pr-amplification and φ i (t) is th cumulativ phas rror from th synchronization procss for slav i. Undr a constraint on th total transmit powr, th optimum g i h i. Th phas rrors hav two ffcts on th rcivd signal: a rduction in th avrag SNR, and a tim-dpndnt fluctuation of th rcivd phas. Th lattr ffct may caus limitations in th cohrnt dmodulation of digital signals. Howvr, thr ar svral mthods,.g. diffrntial modulation, availabl to dal with ths fluctuations providd th

1756 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 6, NO. 5, MAY 007 tim-variations ar not too rapid. In this papr, w concntrat on th first ffct i.. th rduction in avrag SNR. This is appropriat for powr-limitd snsor ntworks, whr th fasibl communication rang is limitd by SNR. For simplicity of notation, w supprss th tim-dpndnc of φ i (t) in this sction. W modl th channl cofficints h i, i = 1...N, as indpndnt circularly symmtric complx normal random variabls with zro man and unit varianc, as dnotd by h i CN(0, 1). This is an appropriat modl for a non-los wirlss channl. This allows us to valuat th variation of bamforming gain in fading channls i.. outag mitigation. W also assum that th phas rrors φ i ar indpndnt and idntically distributd random variabls for all th snsors i. Equation (1) motivats as our figur of mrit, th bamforming gain dfind as th normalizd rcivd powr P R, givn that th total transmit powr is P T =1: P R = 1 N h i jφ i () N 1 Proposition 1: N P R ( ) β φ a.s. as N,whr β φ = E[cos φ i ] and a.s. dnots almost sur convrgnc. In othr words, whn th total transmit powr is kpt a constant, th rcivd signal powr incrass linarly with N as N. Not that whn thr ar no phas rrors, i.. f φ (φ i )=δ(0), thn 1 N P R 1 a.s. Proposition : For finit N, E[P R ]=1+(N 1) ( β φ ). Thus, vn for finit N, th xpctd valu of th rcivd signal powr incrass linarly with N. (β φ is dfind as in Proposition 1, i.. β φ = E[cos φ i ].) In th absnc of phas rrors, Proposition givs that E[P R ]=N. Proposition 3: Whn N is larg nough for th cntral limit thorm to apply, P R X c + X s (3) whr X c N(m c,σc ), X s N(0,σs), and th paramtrs m c, σc,andσ s, ar givn as follows: m c = NE[cos(φ i )] σc = E[cos (φ i )] E[cos(φ i )] σ s = E[sin (φ i )] (4) Th varianc of th rcivd signal powr is thn var[p R ]=4σ c m c +σ4 c +σ4 s (5) which incrass linarly with N. Whn thr ar no phas rrors, (5) rducs to var[p R ]= 4N. (Rfr to th Appndix for a proof of ths rsults.) Proposition implis that as long as th distribution of phas rrors is such that β φ E ( cos φ i ) is clos to 1, larg gains can still b ralizd using distributd bamforming. Fig.. E[P R ]/N vs N, mpirical and analytical rsults. Th four sts of curvs ar for (top to bottom), Δ=0.1 :0.1 :0.4. W now prsnt som numrical rsults comparing th abov analytical modl with Mont-Carlo simulations prformd using SIMULINK. W assum that th snsors transmit a binary puls train modulatd by BPSK, with a bit-rat small compard to th carrir frquncy: m(t) = k p(t kt)s k (6) whr {s k } is th BPSK symbol stram, and p(t) is th transmittd puls. Th avrag powr of th puls p(t),t = 0..T is normalizd to 1 N and E[ s k ]=1so that th total powr transmittd by all th snsors is P T =1.Furthrw assum that th phass φ i ar distributd uniformly in th rang ( Δπ, Δπ). Figur shows th variation of avrag bamforming gain E(P normalizd to th maximum possibl: i.. R) N against th phas rror paramtr Δ. W find that bamforming gains of mor than 70% of th maximum ar possibl with phas rrors as larg as of 60. In othr words, th trm β φ dcrass vry slowly with th paramtr Δ, which lads to th ky conclusion that th bamforming gains ar robust to modratly larg phas rrors. Whil Fig. shows th avrag bamforming gain, th actual bamforming gain is a random variabl. W now look at th variation of th SNR with th phas rrors uniformly distributd as abov. Histograms of P R, calculatd using th Normal approximations as in Proposition 3, ar shown in Fig. 3whrΔ=0.1 and N =10:10:40. Th histograms in Fig. 3 show incrasd avraging for largr numbrs of transmittrs. This is xprssd quantitativly in Proposition 3, which shows that whil th man of P R is proportional to N, th standard dviation is proportional to var(pr) N i.. th fractional dviation P R dcrass with incrasing N. This mans that th probability of an outag vnt.g. whr th rcivd SNR is smallr than 70% of its man, dcrass with incrasing N, showing that bamforming has th ffct of mitigating fading. This is tru for prfct and imprfct synchronization. Of cours, th xistnc of phas rrors can only incras th varianc ovr that of an idal, rror fr systm.

MUDUMBAI t al.: ON THE FEASIBILITY OF DISTRIBUTED BEAMFORMING IN WIRELESS NETWORKS 1757 Fig. 3. Histograms of P R, Δ=0.1. Fig. 4. Mastr-Slav architctur for carrir synchronization. III. A MASTER-SLAVE ARCHITECTURE FOR BEAMFORMING In this sction, w prsnt a protocol for achiving carrir phas synchronization basd on a mastr-slav architctur. This is a multi-stp procss, and ach stp contributs to th ovrall phas rror φ i (t) that limits th bamforming gain. W now look at ach stp of th synchronization in dtail. Th ida bhind th protocol is illustratd in Fig. 4. Th mastr snsor has a local oscillator which gnrats a sinusoid c 0 (t): c 0 (t) =R ( c 0 (t) ),whr c 0 (t) = j(πfct+γ0) (7) that srvs as th rfrnc signal for th ntwork. Th mastr snsor broadcasts c 0 (t) to all th slavs. W assum that th local communication channl btwn mastr and slav snsors has a larg SNR and ignor th rcivr nois in this channl. Aftr rcption and amplification, th slav snsor i rcivs th signal broadcast by rcption and amplification, th slav snsor i rcivs th signal broadcast by th mastr as: c i,0 (t) =R ( c i,0 (t) ),whr c i,0 (t) =A i,0 j(πfct+γ0 γi) (8) whr γ i is th phas shift btwn th mastr and slav. A i,0 is th amplitud of th rcivd signal, its prcis valu is unimportant to th phas synchronization procss (as th PLL is only snsitiv to its phas). W simply st th trm A i,0 to unity, and th constant γ 0 to zro for simplicity. Th snsor i uss this signal c i,0 (t) from (8) as input to a scond-ordr phas lockd-loop, drivn by a VCO with a quiscnt frquncy clos to f c. From PLL thory [1], w can show that th stady-stat phas rror btwn VCO output and c i,0 (t) is zro, and thrfor, th stady-stat VCO output can b usd as a carrir signal consistnt across all snsors - providd that th offst γ i can b corrctd for. Th phas offst γ i is th total phas shift btwn th mastr snsors rfrnc oscillator signal c 0 (t), and th input signal at th slav snsors PLL to which th slav VCO is synchronizd in stady-stat. On contribution to γ i is from th phas rspons of th RF amplifirs at th mastr and slav snsor. Ths offsts ar fixd and prcisly known, and thrfor, can b corrctd for. Howvr, th propagation dlay of th wirlss channl btwn mastr and slav also contributs to γ i. This contribution can b charactrizd by an ffctiv channl lngth d i as γ i = πfcdi c. Unfortunatly, for th high-frquncy RF carrirs typical of wirlss ntworks, vn a small uncrtainty in channl lngth d i causs substantial phas uncrtainty.g. at f c =1.0 GHz, th wavlngth of th transmission is 30 cm, and an uncrtainty of 15 cm in th channl lngth causs an uncrtainty of 180 in γ i. If lft uncorrctd this is disastrous for distributd bamforming, bcaus a 180 offst would chang constructiv intrfrnc btwn transmittrs into dstructiv intrfrnc. In cntralizd antnna arrays, th array lmnts ar arrangd in a known gomtry, and thrfor, th offst for ach lmnt can b prcisly computd. This is not a rasonabl assumption for ad-hoc and snsor ntworks considrd in this papr. Thus it is ncssary to dvlop mthods to xplicitly masur and corrct for this unknown offst. Fortunatly, if th snsors ar not moving rlativ to ach othr, this offst stays roughly constant for significant tim intrvals, and thrfor, frqunt rcalibration is not rquird. In Sction III-A, w dscrib a protocol for prforming this calibration, basd on ach slav snsor transmitting thir frquncy-lockd carrir signal c i,0 (t) back to th mastr snsor. W now sktch th procss of channl stimation, and th algorithm for distributd bamforming assuming that slav i has an stimat ˆγ i = γ i + φ i of its phas offst, whr φ i is th stimation rror in th phas calibration. Slav i thn has th calibratd carrir signal c i (t), which it uss to prform channl stimation and bamforming: c i (t) =R ( c i (t) ) whr c i (t) = c i,0 (t) jˆγi = jπfct+jφ i (9) So far th synchronization procss has bn coordinatd within th snsor ntwork by th mastr snsor without

1758 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 6, NO. 5, MAY 007 Fig. 5. Th Tim-Division Duplxing constraint. τ i, w gt an ffctiv phas nois: φ h i =πδfτ i. Thrfor, w rwrit (11) as: ĥ i = C h i j( φ i +φh i ) (1) Fig. 6. Schmatic of a slav snsor. rquiring any intraction with th BS. In ordr for th snsors to bamform towards th BS, som information about th dirction of th BS, or mor prcisly th channl rspons to th BS is rquird. Using channl rciprocity allows us to achiv this with only a minimum intraction with th BS. Spcifically th BS broadcasts an unmodulatd carrir signal g(t): g(t) =R( g(t)) = R ( jπfc,0t+φ0) (10) Each snsor indpndntly dmodulats its rcivd signal g i (t) =R ( h i g(t) ) using c i (t) to obtain an stimat ĥi of its own complx channl gain h i to th rcivr (for a narrowband mssag signal, th linar tim-invariant channl to BS is rprsntd as a scalar complx gain). Mor prcisly, th channl stimat ĥi is obtaind by th snsor i by dmodulating th rcivd carrir signal g i (t) using c i (t), and sampling th rsult at som fixd tim t h. Not that whil th snsor nods hav a mutually consistnt carrir signal, th BS s carrir has not bn xplicitly synchronizd to th mastr snsor s rfrnc carrir, and thrfor, would not b at th sam frquncy as th snsors. Ltting f c,0 = f c +Δf w hav: ĥ i = h i j(φ0 φ i +φ h) (11) whr φ h =πδf t h. W obsrv that th trm φ 0 is just a constant scaling trm and adds no rlativ phas rrors btwn snsors. Similarly th trm φ h adds no rlativ phas rror so long as th sampling trm t h is idntical for all snsors. If th sampling tims ar off du to timing rrors whr C is a (complx) scaling constant that has no impact on th bamforming procss. For simplicity, w tak C =1. Th snsors now us th synchronizd carrir signal c i (t), and th channl stimat ĥi to modulat th mssag signal for bamforming. Th slav snsors obtain thir carrir signal from th VCO that is synchronizd to th rfrnc signal from th mastr snsor, howvr, it is not possibl for th slav snsors to rciv a synchronization signal from th mastr snsor, whil thy ar transmitting. Thrfor, th VCOs of th slav snsors nd to oprat in an opn-loop mod as shown in Fig. 6, whil th slav snsors ar transmitting. Whil in th opn-loop mod, th slav s carrir signals obtaind from th VCO undrgos uncompnsatd phas drift bcaus of intrnal oscillator nois, and ovr tim, th diffrnt slav carrirs drift out of phas. This motivats th timdivision duplxd mod of opration shown in Fig. 5, whr th mastr snsor priodically transmits a rfrnc carrir signal to rsynchroniz th slav carrirs, to kp th total phas rror boundd. Th phas nois can b considrd as a cyclostationary random procss with priod T = T 1 + T,and w analyz it in dtail in Sction IV. Th noisy carrir signal usd by th slav snsor i for modulation can b writtn as: c o i (t) =R ( c o i (t) ) whr c o i (t) = c i (t) jφd i (t) = jπfct+jφ i +jφd i (t) (13) φ d i (t) rprsnts th uncompnsatd VCO drift whn slav i is transmitting. Aftr modulation by th carrir signal c o i (t),slav snsor i applis a complx amplification ĥ i to compnsat for th channl, and transmits th signal: s i (t) =R ( s i (t) ) whr s i (t) =ĥ i m(t) c o i (t) (14)

MUDUMBAI t al.: ON THE FEASIBILITY OF DISTRIBUTED BEAMFORMING IN WIRELESS NETWORKS 1759 Th rcivd signal at th BS is thn givn by r(t) =R ( h i s i (t)+n(t) ) i = R ( m(t) h i ĥ i co i (t)) i ( = R m(t) hi jπf ct jφ h i +jφ i +jφd(t)) i. (15) i Comparing (15) with (1), w hav for th total carrir phas rror φ i (t) = φ h i +φ i + φ d i (t) (16) Equation (16) shows th diffrnt contributions to th total phas rror in th rcivd signal at th BS. In Sction IV w look at th phas rror in dtail; w argu that th dominant componnt is th drift trm φ d i (t), and show quantitativly how it affcts th total bamforming gains. A. Closd-Loop Mthod for Carrir Phas Calibration In this sction, w propos a flxibl mthod for carrir phas calibration, whr th mastr snsor masurs th roundtrip phas offst, and uss it to stimat th unknown phas offst γ i from (8) for ach slav, assuming symmtry in th forward and rvrs channls to th slav nods. Th flxibility of this mthod coms at th pric of complxity, and th ncssity of synchronizing ach of th slavs individually. Howvr, th calibration procss has to b rpatd only whn th RF channl btwn th mastr and slav snsor changs, thrfor, th ovrhad from this procss is small. Rmark: In th idal cas whr th rlativ positions of th mastr and slav snsors as wll as any multi-path scattrrs do not chang, th calibration procss has to b prformd only onc (at startup tim). In practic, wirlss channls ar not prfctly static: mobil scattrrs and physical changs in th mdium may chang th channl phas rspons vn whn th snsors ar stationary. Thrfor, it maks sns to rcalibrat th slav snsors priodically to track th channl changs. Fortunatly, th channl variations ar slow compard to th channl transmission tims, and th robustnss bnfits of this priodic rcalibration (.g. vry 100 sconds) outwigh th small xtra ovrhad. Fig. 7 illustrats th procss of round-trip phas offst stimation. Th basic ida is for th slav snsor i to transmit back to th mastr snsor th (uncompnsatd) VCO signal c i,0 (t) rprsntd in (8). Th symmtry of th forward and rvrs mastr-slav channls imply that th signal c i,1 at th mastr snsor can b writtn as: c i,1 (t) =A ( i,1 R j(πf lt+γ ) 0 γ i) (17) whr A i,1 is th rcivd signal amplitud at th mastr snsor (A i,1 is qual to A i,0 by symmtry, but th actual valu is not rlvant to th phas nois, thrfor, A i,1 is st to unity for th discussion). Estimating th phas diffrnc btwn c i,1 (t) from (17) and c 0 (t) from (7) givs: Δφ i = ( γ i mod π ) (18) Givn a masurd valu of Δφ, w hav th stimatd valu of th offst γ i : ˆγ i = Δφ (19) Fig. 7. Round-trip phas calibration. Rmark: Thr is on subtlty that nds to b notd hr: th round-trip masurmnt of phas offst as in (19) lavs a 180 ambiguity in γ i. In othr words, by masuring Δφ i w cannot distinguish btwn γ i and γ i + 180. Whil it is possibl to rsolv this ambiguity by xchanging anothr st of mssags btwn mastr and slav i, it turns out that a 180 phas diffrnc dos not affct th bamforming procss. Th rason is that th sam carrir c i (t) is usd by slav i for both channl stimation and distributd bamforming, and as (16) shows, th two ambiguitis cancl ach othr. B. Discussion Th tim-division duplxing rquirmnt for th mastrslav link is th most important constraint of th synchronization protocol of Sction III. Othr authors [8] hav considrd using two frquncis to avoid this problm, with on frquncy f 1 rsrvd for th mastr-slav link and th slav snsors bamforming to th BS on a compltly diffrnt frquncy f. In such schms, th slav snsors us a frquncy dividing PLL to obtain a carrir signal at frquncy f as f = m n f 1,whrm and n ar intgrs. Undr this schm th slav PLLs do not nd to b opn-loop whil transmitting, thrfor, an intrsting qustion is whthr such a frquncy division duplxd (FDD) architctur can liminat th problm of uncompnsatd carrir drift. Unfortunatly, th frquncy dividr introducs a phas ambiguity of intgr multipls of π n in th drivd carrir signal. Whil it may appar that a constant phas ambiguity can b stimatd and corrctd for in a on-tim calibration procss, closr analysis shows that such phas ambiguitis may also occur during th dynamical opration of th PLL,.g. du to cycl slips [1]. Thrfor, priodic rcalibration is still ncssary vn with a FDD architctur, and w conjctur that a tradoff btwn th synchronization ovrhad and th achivabl bamforming gain still applis in this cas.

1760 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 6, NO. 5, MAY 007 IV. ANALYSIS OF PHASE ERROR So far, w hav dscribd a protocol for carrir synchronization and bamforming, whil numrating th diffrnt sourcs of phas rrors φ i, φh i and φd i (t). Of th thr diffrnt sourcs of rror, φ i and φ h i ar constant calibration rrors, whras φ d i (t) is a tim-varying nois trm that ariss from oscillator drift. Thortically, w could prform carrir phas calibration and channl stimation svral tims indpndntly and rduc th rror trms φ i and φh i to arbitrarily small lvls. Howvr, th drift trm φ d i (t) rprsnts an irrducibl phas rror. Thrfor, w considr this as th dominant caus of prformanc dgradation and w now dvlop a stochastic modl to charactriz it. Th prvious discussion in Sction III motivatd th timdivision duplxd (TDD) mod of opration as shown in Fig. 5, whr th slav snsors altrnat btwn sync and transmit timslots. Th timslots T 1 whr th slav snsor synchroniz to th mastr is a synchronization ovrhad, thrfor, it is dsirabl to kp it small rlativ to th usful timslots T. T 1 is dtrmind by th sttling tim of th slav PLL, and T is dtrmind by th maximum admissibl phas rror, and th statistics of oscillator phas nois. By tolrating largr phas rror, w ar abl to mak T highr and thrby rduc th synchronization ovrhad. W show in Sction II that th SNR gains from bamforming ar robust to modratly larg phas rrors. In th rmaindr of this sction, w offr a quantitativ analysis of this tradoff using a stochastic modl for oscillator phas nois. Considr th PLL of th slav snsor as shown in Fig. 6. W us a loop-filtr with on pol to obtain a scond-ordr PLL with th closd-loop transfr function[1]: H(s) = s +ξω n s + ωn (0) whr ω n is th natural frquncy and ξ is th damping ratio of th loop. By standard PLL thory, th stady stat phas rror of a scond-ordr PLL is zro, and if w rquir a 90 phas margin, thn w nd a damping ratio of at last ξ = 1.0, and th sttling thn w nd a damping ratio of at last ξ = 1.0, and th sttling tim (dfind as th tim rquird for th phas rror to dcras to lss than a givn small fraction, say ρ =1% of th initial rror) is T s 4 ω n. Sinc th synchronization timslot T 1 T s,in ordr to minimiz ovrhad w want to mak T s as small as possibl. Howvr, w obsrv that th loop has a lowpass frquncy rspons with approximat bandwidth of ω n, thrfor, incrasing ω n also incrass th phas nois. At th nd of th sync timslot, th loop-filtr output is sampld and th VCO input is hld to this valu for th duration of th transmit timslot. Th phas rror procss in th transmit timslot dtrmins th achivabl bamforming gain. A. Stochastic modl for th phas nois procss In ordr to study this mor quantitativly, w assum that th PLL input signal from th mastr snsor (in th synchronization timslot) is noislss, and th only sourc of phas rror is intrnal phas nois φ d i (t) in th slav snsors s Fig. 8. Simulation of oscillator phas drift. local oscillator signal: c i (t) =R ( jπfct+jφd i (t)) (1) (W also assum that th PLL phas drift is always small nough to allow th us of a linarizd modl.) Th traditional way to masur phas nois is by spcifying its root-man squard frquncy dviation and Allan varianc [1]. Howvr, ths masurs ar most usful if th nois procss is stationary in tim. In our cas th drift procss φ d i (t) is not stationary; in th transmit timslot, th dominant phas nois contribution is from a random rsidual frquncy offst that causs th phas rror to incras linarly in tim until th nxt sync timslot (s Fig. 8). Thrfor, th statistics ar mor appropriatly modld as cyclostationary with th priod T = T 1 + T. W us a mor fundamntal approach to modl this procss. In our modl, th phas rror in th oscillator in closdloop (i.. in th sync timslot) consists of two componnts: a dcaying transint of th initial phas offst, and a phas nois intrnal to th oscillator. Th phas rror in th frrunning oscillator (i.. in th transmit timslot) has thos two componnts and an additional linar phas drift. Th linar drift ariss bcaus th VCO frquncy st by th sampl-andhold (s Fig. 6) may hav a small but non-zro offst from th rfrnc frquncy f l. Th oscillator intrnal phas nois is modld as a widband (whit) Gaussian nois procss with spctral dnsity N p. Whil phas nois in practical oscillators may also hav othr typs of spctral dnsitis.g. flickr nois and random-walk nois, whit Gaussian phas nois rprsnts a worst cas in trms of larg instantanous frquncy dviations, bcaus of th powr in th high frquncis. Lt N p b th normalizd spctral dnsity dfind such that th total powr of th phas nois is N p ω n. In othr words, a whit Gaussian phas nois with spctral dnsity N p will hav th sam powr in a systm of bandwidth ω n as th oscillator s total intrnal phas nois. Sinc th phas rror procss φ d i (t) is a zro-man Gaussian procss at all tims, thrfor, w charactriz its statistics by computing its varianc at th ky tim instants lablld A, B

MUDUMBAI t al.: ON THE FEASIBILITY OF DISTRIBUTED BEAMFORMING IN WIRELESS NETWORKS 1761 and C in Fig. 5. Lt th random phas valus at ths instants b dnotd as φ A, φ B and φ C, and thir standard dviations as σ A, σ B and σ C rspctivly. By th cyclostationarity of φ d i (t), σ C σ A. Using th linarity of th PLL s phas rspons, w can writ th phas at tim B as th suprposition of th dtrministic dcay of th initial rror φ A, to a small fraction ρ of its starting valu (ial rror φ A, to a small fraction ρ of its starting valu (.g. ρ =1%), and a nois trm: φ B = ρφ A + ψ 1 () φ B is small by dsign, and its varianc can b writtn as: σ B = ρ σ A + N p ω n (3) In addition to th small phas rror, at tim instant B, th VCO input is sampld to st th VCO frquncy for th transmit timslot. Th sampld valu has a random offst Δf from th rfrnc carrir frquncy, and this offst consists of a transint trm and a nois trm: Δf = ρω n φ A + ω n ψ 3 Thrfor σ f = ρ ω n σ A + ω3 n N p (4) W hav for th volution of th phas btwn tim instants BandC: φ C =ΔfT + φ B + ψ (5) Of th thr trms in (5), th frquncy offst is th dominant trm bcaus it causs a phas drift that grows with tim. Th phas φ B is small by dsign, and ψ rprsnts a stationary trm, and w can safly nglct both trms compard to th linar drift. This is also illustratd in th simulation shown in Fig. 8. Thus w hav: Combining (4), (3) and (6), w gt: σ C σ A = σ f T (6) σa = N pωn 3T 1 ρ ωn T (7) Fig. 8 shows a simulation of th phas rror ovr tim with T 1 = 150μsc, T = 0.85 ms, ω n = 100 khz, ρ = 1% and N p =7 10 11 Hz 1 or 101 dbc/hz. Th VCO in this simulation has a quiscnt frquncy that is 1 khz offst from th rfrnc carrir signal. Th spctral dnsity of phas nois is chosn consrvativly compard to typical numbrs rportd.g. 110 dbc/hz in [13]. For ths numbrs, w gt σ A 4 from (7). Sinc th phas rror is a Gaussian variabl with standard dviation smallr than 4 at all tims, β φ = E(cos φ i ) 0.91, and by Proposition 1, w can s that avrag bamforming gains of at last 91% ar achivabl. This is an avrag numbr and occasionally, phas rrors largr than this can occur as sn in Fig. 8, whr phas rror bcoms almost 35 at on point. Evn with this larg phas rror, th rsulting bamforming gain is 81% of th maximum. This confirms th rsults of Sction II, that bamforming gain is robust to phas rrors and dmonstrats th basic fasibility of th distributd bamforming algorithm. B. Comparison with fundamntal Cramr-Rao bounds So far in this analysis w hav limitd ourslvs arbitrarily to a scond-ordr PLL bcaus it is th most commonly usd dvic in practic. Howvr, w can also driv fundamntal limits on th siz of th frquncy and phas offsts, by viwing th PLL as a frquncy and phas stimator. Th PLL uss th (noisy) oscillator signal in th sync timslot to form stimats ˆf l and ˆφ. It uss th stimat ˆφ to driv th phas diffrnc with th PLL input to zro, and ˆf l to tun th VCO s input to th frquncy of th rfrnc, and th sampland-hold lmnt kps th VCO tund to that stimat in th transmit timslot. Th Cramr-Rao lowr bound for th varianc of ths offsts has bn computd in prvious work on frquncy stimation [14]: ˆσ f. = var( ˆf l )= 3N p π T1 3 ˆσ φ. = var( ˆφ) = N p (8) T 1 Using th sam valus usd in Fig. 8, w find ˆσ f =.5 Hz, and ˆσ φ 1.Sincˆσ f =.5 Hz, and ˆσ φ 1.Sinc ˆσ f is substantially smallr than th PLL s root-man squard frquncy offst σ f = 418 Hz, w conclud that thr is significant suboptimality in using an analog PLL, thrfor, prformanc can b furthr improvd by using optimal digital procssing. V. CONCLUSION W hav invstigatd a mastr-slav architctur for achiving th carrir synchronization ncssary for distributd transmit bamforming. Thr ar svral sourcs of synchronization rror in this procdur, and th aggrgat phas rrors limit th achivabl SNR gains from bamforming. W idntifid th dominant sourc of rrors as VCO drift arising from tim-division duplxd opration of th synchronization protocol. W xamind th phas nois procss of a scond ordr analog PLL by simulation and analysis, and calculatd th rsulting bamforming gains. W also compard th prformanc of th PLL with th fundamntal Cramr- Rao stimation bounds. Our rsults show that vn with th suboptimal analog PLL, and with phas rrors on th ordr of 60, it is possibl to achiv SNR gains of 70% of th maximum. In summary, our invstigation indicats that implmntation of distributd bamforming at RF frquncis is challnging but potntially fasibl. On way to improv th bamforming prformanc is to us an optimal frquncy and phas stimator instad of an analog PLL. This would rquir a mor sophisticatd digital implmntation, and a dtaild prformanc study of such a schm is a topic for futur work. In this papr, w studid a rciprocity basd approach to channl stimation dsignd to minimiz coordination with th BS. An altrnativ mthodology for distributd bamforming is to us fdback from th rcivr (i.. th BS) for distributd phas synchronization. W xplord this approach in [11], [15] and our rsults ar promising. It is also possibl to combin ths idas with multi-hop routing schms for wirlss ntworks. Ths issus ar not considrd in this prsnt work and ar intrsting aras for furthr inquiry.

176 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 6, NO. 5, MAY 007 APPENDIX I PROOFS: Proof of Proposition 1: W can rwrit () as follows: 1 N P R = N hi jφ i (9) N Invoking th law of larg numbrs, and th fact that th { hi }, ar i.i.d. xponntial random variabls which ar indpndnt from th i.i.d {φ i },whav 1 N [ h i jφ ] i E h i (cos φi + j sin φ i ) a.s. (30) N Th xpctation on th RHS of (30) simplifis as follows [ ] E h i (cos φi + j sin φ i ) = E[ h i ]E[cos φi ] = E[cos φ i ] (31) W hav assumd that φ i is symmtrically distributd around 0, and hnc E[sin φ i ] = 0. Equation (31) rsults bcaus h i CN(0, 1) and hnc hi is xponntial with unit man. W thus hav that 1 N h i jφ i (E[cos(φ i )]) a.s. (3) N sinc continuous functions of variabls which ar convrging almost surly also convrg almost surly, and th dsird rsult follows. Proof of Proposition : Th xpctd valu of P R can b writtn as [ E[P R ] = 1 N ] N E hi N jφ i hl jφ l l=1 = 1 N(N 1) (N + E[ h1 h N.R( j(φ1 φ) )] (33) = 1 ( ) N(N 1) N + E[cos(φ 1 φ )] N = 1+(N 1)E[cos(φ 1 φ )] = 1+(N 1)E[cos(φ 1 )cos(φ ) sin(φ 1 )sin(φ )] = 1+(N 1)E[cos(φ i )] (34) whr w hav usd th fact that th {h i }, {φ i } ar i.i.d. and that th {φ i } ar symmtrically distributd around 0. Proof of Proposition 3: W onc again bgin with th dfinition for P R. 1 N P R = hi cos(φi )+j hi sin(φi ) N 1 N = ( hi cos(φi ) α) N +j 1 N hi sin(φi ) + Nα (35) N whr α = E[ h i cos(φi )] = E[cos(φ i )]. Invoking th cntral limit thorm, as N gts larg, th first trm in (35) tnds to a Gaussian random variabl with man 0 and varianc σc var[ h i cos(φi )]. Similarly, th scond trm tnds to a Gaussian random variabl with man 0 and varianc σs var[ hi sin(φi )]. Sinc th last trm in (35) is ral and constant, it only shifts th man of th first Gaussian random variabl, so w can writ P R Xc + jx s (36) whr X c N( Nα,σc ), and X s N(0,σs ). Making us of th fact that hi is a unit man xponntial random variabl, σc = var[ hi cos(φi )] = E[ h i 4 cos (φ i )] E[ h i cos(φi )] =E[cos (φ i )] E[cos(φ i )] and similarly, σ s =E[sin (φ i )] (37) Ltting m c Nα,whavthatP R = Xc + X s,asgivn. Th varianc of P R follows from standard calculations for momnts of Gaussian random variabls. REFERENCES [1] J. Lanman and G. Wornll, Distributd spac-tim-codd protocols for xploiting cooprativ divrsity in wirlss ntworks, IEEE Trans. Inf. Thory, vol. 49, no. 10, pp. 415 45, Oct. 003. [] S. Alamouti, A simpl transmit divrsity tchniqu for wirlss communications, IEEE J. Sl. Aras Commun., vol. 16, no. 8, pp. 1451 1458, Oct. 1998. [3] M. Dohlr, J. Dominguz, and H. Aghvami, Link capacity analysis for virtual antnna arrays, in Proc. 56th IEEE Vhicular Tchnology Confrnc 00, vol. 1, pp. 440 443. [4] A. swol Hu and S. Srvtto, Optimal dtction for a distributd transmission array, in Proc. IEEE Intrnational Symposium on Information Thory 003, pp. 00 00. [5] O. Oyman, R. Nabar, H. Bolcski, and A. Paulraj, Charactrizing th statistical proprtis of mutual information in MIMO channls, IEEE Trans. Signal Procssing, vol. 51, no. 11, pp. 784 795, Nov. 003. [6] B. Hassibi and A. Dana, On th powr fficincy of snsory and adhoc wirlss ntworks, in Proc. IEEE Intrnational Symposium on Information Thory 003, pp. 41 41. [7] Y.-S. Tu and G. Potti, Cohrnt cooprativ transmission from multipl adjacnt antnnas to a distant stationary antnna through AWGN channls, in Proc. 55th IEEE Vhicular Tchnology Confrnc Spring 00, vol. 1, pp. 130 134. [8] D. Brown, A mthod for carrir frquncy and phas synchronization of two autonomous cooprativ transmittrs. [Onlin]. Availabl: http://spinlab.wpi.du/publications/confrncs/brown SPAWC 005.pdf [9] J. Elson, L. Girod, and D. Estrin, Fin-graind ntwork tim synchronization using rfrnc broadcasts, SIGOPS Opr. Syst. Rv., vol. 36, no. SI, pp. 147 163, 00. [10] H. Ochiai, P. Mitran, H. Poor, and V. Tarokh, Collaborativ bamforming for distributd wirlss ad hoc snsor ntworks, IEEE Trans. Signal Procss. (S also IEEE Trans. Acoustics, Spch, Signal Procssing, vol. 53, no. 1053-1058, pp. 4110 414, 005.) [11] R. Mudumbai, J. Hspanha, U. Madhow, and G. Barriac, Scalabl fdback control for distributd bamforming in snsor ntworks, in Proc. IEEE Intl. Symp. on Inform. Thory (ISIT) 005. [1] H. Myr and G. Aschid, Synchronization in Digital Communications. Nw York: John Wily and Sons, 1990. [13] B. Razavi, A study of phas nois in CMOS oscillators, IEEE J. Solid- Stat Circuits, vol. 31, no. 3, pp. 331 343, 1996. [14] D. Rif and R. Boorstyn, Singl ton paramtr stimation from discrt-tim obsrvations, IEEE Trans. Inf. Thory, vol. 0, no. 5, pp. 591 598, 1974.

MUDUMBAI t al.: ON THE FEASIBILITY OF DISTRIBUTED BEAMFORMING IN WIRELESS NETWORKS 1763 [15] R. Mudumbai, B. Wild, U. Madhow, and K. Ramchandran, Distributd bamforming using 1 bit fdback: from concpt to ralization, in Proc. 44th Allrton Confrnc on Communication Control and Computing, Spt. 006. Raghuraman Mudumbai rcivd th B. Tch dgr in lctrical nginring from th Indian Institut of Tchnology, Madras, India in 1998, and th MS dgr in lctrical nginring from Polytchnic Univrsity, Brooklyn, USA in 000. H is currntly working for his Ph.D at th Univrsity of California Santa Barbara. Gwn Barriac rcivd th B.S. dgr in lctrical nginring from Princton Univrsity, Princton, NJ, in 1999, and th M.S. and Ph.D. dgrs in lctrical nginring from th Univrsity of California at Santa Barbara in 004. Sh is currntly with Qualcomm, San Digo, CA. Upamanyu Madhow (S86M90SM96F05) rcivd th bachlor s dgr in lctrical nginring from th Indian Institut of Tchnology, Kanpur, India, in 1985, and th M.S. and Ph.D. dgrs in lctrical nginring from th Univrsity of Illinois, Urbana-Champaign, in 1987 and 1990, rspctivly. From 1990 to 1991, h was a Visiting Assistant Profssor at th Univrsity of Illinois. From 1991 to 1994, h was a Rsarch Scintist with Bll Communications Rsarch, Morristown, NJ. From 1994 to 1999, h was on th faculty of th Dpartmnt of Elctrical and Computr Enginring, Univrsity of Illinois, Urbana-Champaign. Sinc Dcmbr 1999, h has bn with th Dpartmnt of Elctrical and Computr Enginring, Univrsity of California, Santa Barbara, whr h is currntly a Profssor. His rsarch intrsts ar in communication systms and ntworking, with currnt mphasis on wirlss communication, snsor ntworks, and data hiding. Dr. Madhow is a rcipint of th NSF CAREER award. H has srvd as Associat Editor for Sprad Spctrum for th IEEE Transactions on Communications, and as Associat Editor for Dtction and Estimation for th IEEE Transactions on Information Thory.