Retrodrectve Dstrbuted Transmt Beamformng wth Two-Way Source Synchronzaton Robert D. Preuss and D. Rchard Brown III Abstract Dstrbuted transmt beamformng has recenty been proposed as a technque n whch severa snge-antenna sources cooperate to form a vrtua antenna array and smutaneousy transmt wth phase-agned carrers such that the passband sgnas coherenty combne at an ntended destnaton. The power gans of dstrbuted transmt beamformng can provde ncreased range, rate, energy effcency, and/or securty, as we as reduce nterference. Dstrbuted transmt beamformng, however, typcay requres precse synchronzaton between the sources wth tmng errors on the order of pcoseconds. In ths paper, a new two-way synchronzaton protoco s deveoped to factate precse source synchronzaton and retrodrectve dstrbuted transmt beamformng. The two-way synchronzaton protoco s deveoped under the assumpton that a processng at each source node s performed wth oca observatons n oca tme. An anayss of the statstca propertes of the phase and frequency estmaton errors n the two-way synchronzaton protoco and the resutng power gan of a dstrbuted transmt beamformer usng ths protoco s provded. Numerca exampes are aso presented characterzng the performance of dstrbuted transmt beamformng n a system usng two-way source synchronzaton. The numerca resuts demonstrate that near-dea beamformng performance can be acheved wth ow synchronzaton overhead. I. INTRODUCTION Dstrbuted transmt beamformng s a technque n whch mutpe ndvdua snge-antenna sources smutaneousy transmt a common message and contro the phase and frequency of ther carrers so that ther bandpass sgnas constructvey combne at an ntended destnaton. The transmtters n a dstrbuted transmt beamformer form a vrtua antenna array and, n prncpe, can acheve a of the gans of a conventona antenna array, e.g. ncreased range, rate, and/or energy effcency, wthout the sze, cost, and compexty of a conventona antenna array. Dstrbuted transmt beamformng can aso provde benefts n terms of securty and nterference reducton snce ess transmt power s scattered n unntended drectons. A common assumpton n the terature s that dstrbuted transmt beamformng can be performed by usng tmedvson-dupexng (TDD) and a phase conjugaton technque smar to the retrodrectve Pon array [1] technque deveoped for conventona antenna arrays. The approach s as foows: () the destnaton node frst broadcasts a sgna R.D. Preuss s a Senor Member of the IEEE vng n Arngton, MA 02476 USA. e-ma: r.preuss@eee.org D.R. Brown III s an Assocate Professor wth the Eectrca and Computer Engneerng Department, Worcester Poytechnc Insttute, Worcester, MA 01609 USA. e-ma: drb@ece.wp.edu. Ths work was supported by NSF award CCF-0447743. receved by each source node and () each source node then transmts back to the destnaton at the same frequency but wth conjugate phase. In prncpe, the phase conjugaton at each transmtter cances the phase shft of the channe. Ths causes the carrers to arrve n phase and coherenty combne at the destnaton. Whe ths retrodrectve transmsson technque s known to be effectve for conventona antenna arrays where each antenna eement s connected to a common oca oscator, t s actuay key to be neffectve n dstrbuted transmsson systems n whch each source node has ts own ndependent oca oscator f the source nodes are not pre-synchronzed. To demonstrate the crtca roe of synchronzaton n dstrbuted transmt beamformng, consder a system wth two unsynchronzed source nodes, denoted as S 1 and S 2, and one destnaton node. Denote the tme at the destnaton node as t and the tme at the source nodes as t 1 and t 2. For purposes of ustraton, we assume that S has an unknown fxed oca tme offset wth respect to the destnaton node s tme such that t = t. Fgure 1 shows a TDD tmene n whch the unknown oca cock offsets 1 and 2 are dfferent,.e. the source nodes are not pre-synchronzed. In the frst step of TDD operaton, the destnaton node broadcasts the sgna x 0 (t) = exp{jω 0 t}i t [0,T) to the source nodes, where T s the sgna duraton and the ndcator functon I t A = 1 when t A, and s otherwse equa to zero. Ths sgna s represented as the sod-ne sgna n Fgure 1. Assumng snge-path unt-gan channes and gnorng nose, the sgna receved by S can then be wrtten as y (t) = exp{jω 0 (t τ 0, )}I t [τ0,,τ 0,T) (1) for = 1, 2 where τ 0, s the unknown propagaton deay of the channe from the destnaton node to S. These sgnas are ustrated as the sod-ne sgnas on the source node s tmenes n Fgure 1. Note that (1) s wrtten n the destnaton node s oca tme. In the source node s oca tme, y (t ) = exp{jω 0 (t τ 0, )}I t [ τ 0,, τ 0,T) for = 1, 2. The phase estmate at S s then cacuated as the phase at t = 0,.e. ω 0 ( τ 0, ). In the second step of TDD operaton, both source nodes transmt wth conjugate phase back to the destnaton. The carrer transmtted by S can be wrtten as x (t ) = exp{jω 0 (t τ 0, )}I t [s,s T ) (2) where T s the transmsson duraton and s s the startng tme of the transmsson for source node. These sgnas
source node 1 T destnaton node 2 1 source node 2 τ 0,1 τ 0,2 x 0 (t) y 1 (t) y 2 (t) x 2 (t) T x 1 (t) y 0 (t) Fg. 1. An exampe of tme-dvson-dupexng (TDD) n a system wth two unsynchronzed source nodes. The carrers transmtted by the source nodes n ths exampe fuy cance each other at the destnaton node. are shown as the dotted and dash-dotted sgnas for S 1 and S 2, respectvey, on the source node s tmenes n Fgure 1. Convertng (2) to the destnaton node s oca tme, we have x (t) = exp{jω 0 (t 2 τ 0, )}I t [s,s T ). The aggregate sgna receved by the destnaton node after propagaton from each source to the destnaton s then y 0 (t) = 2 exp{jω 0 (t 2 )}I t [s,s T ) (3) =1 where s := s τ 0,. Ths ast expresson exposes the two eements of synchronzaton necessary to ensure coherent combnng of the sgnas at the destnaton node. Frst, n order for the carrers to constructvey combne, (3) requres that 2ω 0 1 2ω 0 2 (mod 2π). Ths condton s necessary and suffcent to acheve carrer coherence. The second synchronzaton eement requred to ensure the sgnas coherenty combne at the destnaton node reates to the start tme of transmsson at each source node. In order for source nodes sgnas to arrve at the same tme at the destnaton, the transmsson start tmes must be staggered such that s 1 = s 2. Ths condton s necessary and suffcent to acheve message coherence. The focus of ths paper s prmary on the probem of achevng carrer coherence snce, as shown n ths exampe, the effects of carrer offset can be crtca. Moreover, carrer coherence s usuay consdered the more dffcut probem because the synchronzaton accuracy requred for carrer coherence s typcay on the order of pcoseconds. The probem of message coherence s aso mportant and has been consdered n [2], but the tmng accuracy requrements are ess strngent and the effects of message offset,.e. ntersymbo nterference, are usuay ess crtca. t 1 t t 2 Severa carrer synchronzaton technques have recenty been proposed for dstrbuted transmt beamformng ncudng fu-feedback cosed-oop [3], one-bt cosed-oop [4] [6], master-save open-oop [7], and round-trp open-oop carrer synchronzaton [8], [9]. Each of these technques has advantages and dsadvantages n partcuar appcatons, as dscussed n the survey artce [10]. In ths paper, we descrbe a new synchronzaton technque caed two-way synchronzaton [11] and demonstrate ts effcacy n nose-free and nosy channes. Two-way synchronzaton s smar n some aspects to round-trp synchronzaton, but, unke the round-trp carrer synchronzaton technques descrbed n [8], [9], two-way synchronzaton s performed among the source nodes pror to the transmsson of a beacon from the ntended destnaton. The man contrbutons of ths paper are a descrpton of the two-way carrer synchronzaton technque n a system where each source node has an ndependent oca oscator. We aso show how approprate transmsson phases can be generated to enabe beamformng to an ntended destnaton. We then anayze the statstca propertes of the two-way synchronzaton protoco n terms of the estmaton errors and oscator phase nose. We concude wth numerca exampes that show that the two-way synchronzaton overhead can be sma wth respect to the expected usefu beamformng tme. II. SYSTEM MODEL We consder the system ustrated n Fgure 2 one destnaton node, denoted as node 0, and M source nodes, denoted as nodes S 1,..., S M. A nodes are assumed to possess a snge sotropc antenna. The channe between the destnaton node and S m s modeed as a causa near tme-nvarant (LTI) system wth mpuse response g m (t). The channe between S m and S n s aso modeed as a causa near tme-nvarant (LTI) system wth mpuse response h m,n (t) The nose n each channe s addtve, whte, and Gaussan and the mpuse response of each channe n the system s assumed to be recproca,.e. h m,n (t) = h n,m (t). h 1,2 (t) S 2 Fg. 2. source nodes S 1 h 2,3 (t) h 1,3 (t) S 3 g 2 (t) g 1 (t) g 3 (t) destnaton node A system wth M = 3 source nodes and one destnaton node. We assume the oca tme at S has an unknown rate offset β and an unknown tme offset wth respect to a reference tme t such that t = β (t ). (4) 0
Ths mode does not ncude the effect of oscator phase nose but s reasonabe over short duratons, e.g. durng synchronzaton. The effects of oscator phase nose durng beamformng are consdered n Secton V. III. TWO-WAY SOURCE SYNCHRONIZATION PROTOCOL The two-way source synchronzaton protoco s ntated by S 1 transmttng a snusoda beacon to S 2. Ths snusoda beacon s retransmtted through ncreasng ndces S 2 S 3 S S M ( forward propagaton ), where each retransmsson s a perodc extenson of the beacon receved n the prevous tmesot. A second snusoda beacon, ntated by S M, s smary transmtted through the decreasng ndces S M S S 2 S 1 ( backward propagaton ). Assumng approxmatey the same frequency s used for the forward and backward propagated beacons, 2M 2 nonoverappng tme sots (enumerated as TS (1),...,TS (2M 2) ) are used to ensure there s no mutua nterference among the 2M 2 ndvdua transmssons n the two-way source synchronzaton protoco. The sgnas exchanged and estmates generated n each tmesot are expcty descrbed for the forward propagaton stage as foows. In TS (1), S 1 transmts a snusoda beacon x (1) 1 (t 1) = exp{j(ω 1 t 1 φ 1 )}I t1 T (1) 1 to S 2 where T (1) 1 s the transmsson nterva of S 1 n TS (1). Note that x (1) (t 1 ) s expressed n oca tme for S 1. Ths beacon propagates through the channe to S 2 and s receved n oca tme at S 2 as y (1) 2 (t 2) = a 1,2 exp {j (f 1,2 (t 2 ) φ 1 )} I t2 T w (1) (1) 2 (t 2) 2 where T (1) 2 s the recepton nterva of S 2 n TS (1), w (1) 2 (t 2) s the( nose n the sgna ) receved by S 2 n TS (1), f 1,2 (t 2 ) := β 1 ω t2 1 β 2 1 2 ψ 1,2, and a 1,2 = H 1,2 (β 1 ω 1 ) and ψ 1,2 = H 1,2 (β 1 ω 1 ) are the amptude and phase shft, respectvey, of the LTI channe between S 1 and S 2 at the true frequency β 1 ω 1. Ths observaton s then used by S 2 to generate frequency and phase estmates ˆω (1) 2 = β 1ω 1 ω (1) 2, and β 2 (5) ˆφ (1) (1) 2 = β 1 ω 1 ( 1 2 ) ψ 1,2 φ 1 φ 2 (6) where ω (1) (1) 2 and φ 2 are the frequency and phase estmaton error, respectvey, at S 2 n TS (1). Ths process s repeated through ncreasng source node ndces. In each tmesot, a source node transmts a perodc extenson of the beacon t receved n the pror tmesot to the next source node. The sgna transmtted by S 1 to S n TS ( 1) s x ( 1) 1 (t 1) = exp{j(ˆω ( 2) 1 t 1 ˆφ ( 2). After propagaton through the LTI chan- 1 )}I t 1 T ( 1) 1 ne to S, the sgna s receved as y ( 1) (t ) = a 1, exp { j w ( 1) (t ) ( f 1, (t ) )} ( 2) ˆφ 1 I t T ( 1) where f 1, := β 1ˆω ( 2) 1 ( t β 1 )ψ 1,. Ths observaton s then used by S to generate frequency and phase estmates ˆω ( 1) ˆφ ( 1) = β 1ˆω ( 2) 1 ω ( 1), and (7) β = β 1ˆω ( 2) 1 ( 1 ) ψ 1, for = 3,...,M, where ω ( 1) φ ( 1) ( 2) ( 1) ˆφ 1 φ (8) and are the frequency and phase estmaton error, respectvey, at S n TS ( 1). The forward propagaton stage concudes at the end of TS (). Backward propagaton s the same as forward propagaton except S M ntates the process by transmttng a snusoda beacon x (M) M (t M) = exp{j(ω M t M φ M )}I tm to S T (M). M The beacons are retransmtted through decreasng ndces = M 1,...,1 and the backward propagaton stage concudes after S 1 receves the fna beacon n TS (2M 2). At the end of the two-way source synchronzaton protoco, each source except S 1 and S M has two sets of phase and frequency estmates. Sources S 1 and S M use ther nta beacon phase and frequency (ω 1 and φ 1 or ω M and φ M ) as ther other estmates. Note that these estmates have no estmaton error. For notatona convenence, we denote the four estmates obtaned by S as ˆω,1, ˆω,2, ˆφ,1, and ˆφ,2. IV. SYNCHRONIZATION AND BEAMFORMING After the exchange of beacons, each source adds ts frst and second estmates to synthesze a synchronzed oca oscator (SLO) wth frequency ˆω = ˆω,1 ˆω,2 and nta phase ˆφ = ˆφ,1 ˆφ,2. If we temporary assume that each source node s phase and frequency estmates are perfect 1 n the sense that there s no estmaton error n each tmesot, t s not dffcut to show that the SLO phase ξ := ˆω t ˆφ s dentca at a source nodes (moduo 2π). To see ths, we can use (7) n the forward propagaton stage to wrte the frst frequency estmate at S as ˆω,1 = β 1 ˆω 1,1 = β 1 ω 1 β β for = 2,...,M. The second equaty resuts from a recursve appcaton of the frst equaty and the fact that ˆω 1,1 := ω 1. Aong the same nes, we can use (7) n the backward propagaton stage to wrte the second frequency estmate at S as ˆω,2 = β 1 β ˆω 1,2 = β M β ω M for = M 1,...,1 where ˆω M,2 := ω M. The resutng frequency at S s then ˆω = ˆω,1 ˆω,2 = β 1ω 1 β M ω M β. (9) The frst phase estmate at S can be cacuated from (7) and (8) n the forward propagaton stage as ˆφ,1 = β 1 ω 1 ( 1 ) ψ 1, ˆφ 1,1 1 = β 1 ω 1 ( 1 ) ψ,1 φ 1 =1 1 Imperfect estmates are consdered n Secton V.
for = 2,...,M where we have used β 1ˆω ( 2) 1 = β 1ˆω 1,1 = β 1 ω 1 and where the second equaty resuts from a recursve appcaton of the frst equaty. Aong the same nes, we can use (7) and (8) n the backward propagaton stage to wrte the second phase estmate at S as ˆφ,2 = β M ω M ( M ) ψ 1, φ M for = M 1,...,1. Snce ψ 1, = ψ,1,.e. the channes have recproca phase shfts, the resutng phase at S can be wrtten as = ˆφ = β 1 ω 1 ( 1 )β M ω M ( M ) ψφ 1 φ M (10) where we have defned ψ := =1 ψ,1. Puttng t a together, the SLO phase at S s then ξ = β 1ω 1 β M ω M β t β 1 ω 1 ( 1 ) φ 1 β M ω M ( M ) φ M ψ = (β 1 ω 1 β M ω M )t γ 1 γ M ψ where the second equaty resuts from (4) and γ m := β m ω m m φ m. Hence, even though each source node possesses ts own oca noton of tme and operates ony on ts own oca estmates, each source node s abe to synthesze a synchronzed oca oscator after two-way synchronzaton. After the formaton of the SLOs, retrodrectve dstrbuted transmt beamformng can be performed usng TDD technques such as those descrbed n [1]. For notatona smpcty, assume that the destnaton s noton of tme s reference tme so that t 0 = t. After recevng the transmsson from the destnaton at frequency ω 0, each source, for exampe, estmates the frequency and phase of ths transmsson and subtract these ˆφ estmates, denoted as ˆω and, respectvey, from the SLO frequency and phase to generate the beamformng carrer { ( x (bf) (t ) = exp j (ˆω ˆω )t ˆφ )} ˆφ. (11) Assumng agan that the estmates are perfect, the sum of these carrers after propagaton to the destnaton can be wrtten as y (bf) 0 (t) = M =1 a 0, exp { j ( ωt γ ψ )} I t T (bf),0 w (bf) 0 (t) where ω := β 1 ω 1 β M ω M ω 0 and γ := γ 1 γ M γ 0. The receved power of the aggregate unmoduated carrers at the destnaton node n ths case s y (bf) 0 (t) 2 = ( a 0,) 2. Ths corresponds to the power of an dea transmt beamformer, when each source node transmts wth unt carrer amptude. V. PERFORMANCE ANALYSIS WITH ESTIMATION ERROR Estmaton errors ncurred durng two-way synchronzaton and source-destnaton channe phase estmaton as we as phase nose at each source node a ead to some oss of performance wth respect to the dea transmt beamformer. At tme t, the power of the aggregate carrers from the M source nodes receved at the destnaton can be expressed as y (bf) 0 (t) 2 = MX MX X a 2 0,m a 0,ma 0,n cos (δ m,n(t)) (12) m=1 m=1 n m where the non-dea nature of the dstrbuted beamformer s captured n the carrer offset terms between S m and S n δ m,n (t) := (ˆω m ˆω m )β m(t m ) (ˆω n ˆω n )β n (t n ) ˆφ (ˆφ m m ψ m,0) (ˆφ n ˆφ n ψ n,0 ) χ m (t) χ n (t) (13) where χ m (t) χ n (t) represents the dfference n the phase nose processes of the SLOs between S m and S n. Note that (13) s composed of three components: carrer frequency offset, nta carrer phase offset at t = 0, and phase nose. We can rewrte (13) n these terms as δ m,n (t) = ω m,n t φ m,n χ m,n (t). (14) The frequency and phase estmates n (13) can be wrtten as ˆω m = β 1ω 1 β M ω M ω m β m, (15) ˆω m = ω 0 ω m, β m (16) ˆφ m = β 1 ω 1 ( 1 m ) β M ω M ( M m ) φ 1 φ M ψ φ m (17) ˆφ m = ω 0 ( 0 m ) ψ 0,m φ 0 φ m. (18) Substtutng these expressons nto (13) aows us to wrte the frequency and phase offsets n (14) n terms of the ndvdua estmaton errors as ω m,n = ( ω m ω m ) ( ω n ω n ) (19) φ m,n = ( φ m φ m ) ( φ n φ n ) m ( ω m ω m ) n ( ω n ω n ). (20) The carrer frequency and phase offsets between S m and S n are anayzed n terms of the consttuent estmaton errors n the foowng sectons. The statstca propertes of the phase nose processes are dscussed n Secton V-D. A. Frequency and Phase Estmaton Error Statstcs To factate anayss, we assume a of the estmates are unbased and that the estmaton errors are jonty Gaussan dstrbuted. It can be shown that the covarances E{ˆω mˆω n }, E{ˆφ m ˆφn }, E{ˆω m ˆω n }, and E{ˆφ m ˆφ n } are a zero except when m = n snce observatons n dfferent tmesots are affected by ndependent nose reazatons and observatons at dfferent source nodes are aso affected by ndependent nose reazatons. It can aso be shown that a of the other ˆφ covarances are zero except E{ˆω m ˆφm } and E{ˆω m m } snce frequency and phase estmates obtaned from the same observaton at a partcuar source node are not ndependent.
It s possbe to bound the non-zero covarances wth the Cramer-Rao bound (CRB) [12]. Gven an N s -sampe observaton of a compex exponenta of amptude a, the CRB for the covarance of the frequency and phase estmates s [13] cov {[ω, φ] } σ2 a 2 [ 1 Ts 2Ns(Q P2 ) (n 0P) T sn s(q P 2 ) (n 0P) T sn s(q P 2 ) n 2 0 2n0PQ N s(q P 2 ) ] (21) where σ 2 s the varance of the uncorreated rea and magnary components of the ndependent, dentcay dstrbuted, zeromean, compex Gaussan nose sampes, T s s the sampng perod, n 0 s the ndex of the frst sampe of the observaton n the observer s oca tme, P := (N s 1)/2, Q := (N s 1)(2N s 1)/6, and A B means that A B s postve semdefnte. These resuts can be used as a reasonabe approxmaton for the non-zero covarances when each source node uses an unbased and effcent estmator, e.g. the maxmum kehood estmator for arge N s [12], to generate the oca phase and frequency estmates. B. Carrer Frequency Offset In the forward propagaton stage of the two-way synchronzaton protoco, the estmaton error ω ( 1) n (7) s defned wth respect to the true frequency of the sgna transmtted by S 1 n TS ( 1). In TS (1), the true frequency of transmsson s β 1 ω 1. In TS ( 1) for = 3,...,M, the true frequency of transmsson s β 1ˆω ( 2) 1. The sera nature of the transmssons n the two-way synchronzaton protoco mpes that the frequency error at S wth respect to the nta true beacon frequency β 1 ω 1 s an accumuaton of the ndvdua frequency estmaton errors,.e. ω (1) 2 ω ( 1). The same s true for the backward propagaton stage except the true frequency of the nta beacon s β M ω M. The frequency error of the SLO at S m can thus be computed from recursve appcaton of (7) for the forward and backward propagaton stages as ω m = m =2 ω ( 1) =m ω (2M 1) (22) where the frst and second sums correspond to the accumuated estmaton error at S n the forward and backward propagaton stages, respectvey. Based on (19) and the assumptons n Secton V-A, ths resut shows that the carrer frequency offsets between S m and S n are zero-mean and jonty Gaussan dstrbuted wth covarances that can be straghtforwardy computed n terms of the consttuent estmaton error covarances. C. Carrer Phase Offset Smar to the frequency estmaton errors, the phase estmaton errors n the forward and backward propagaton stages of the two-way synchronzaton protoco accumuate as the sgnas propagate through ncreasng and decreasng source node ndces. The accumuaton of phase error at S, however, s due to both consttuent phase and frequency estmaton errors. In the forward propagaton stage of the two-way synchronzaton protoco, we can recursvey appy (7) and (8) to wrte the frst oca phase estmate at S m as m ˆφ ( 1) m = β 1 ω 1 ( 1 m ) φ 1 ψ 1, m =2 m 1 φ ( 1) =2 =2 ω ( 1) ( m ) for m = 2,...,M. Smary, the second oca phase estmate obtaned durng backward propagaton at S m s X ˆφ (2M m 1) m = β Mω M( M m) φ M X =m φ (2M 1) X =m1 =m ψ,1 ω (2M 1) ( m) for m = 1,...,M 1. These estmates are summed at S m to generate the SLO phase. The resutng phase error s then φ = 1 φ ( 1) =2 =2 = ω ( 1) ( ) φ (2M 1) =1 ω (2M 1) ( ). Based on (20), (22), and the assumptons n Secton V-A, ths resut shows that the carrer phase offsets between S m and S n are zero-mean and jonty Gaussan dstrbuted wth covarances that can be straghtforwardy computed n terms of the consttuent estmaton error covarances. D. Phase Nose Phase nose causes the phase of the SLO at each source node to randomy wander from the phase obtaned at the end of the two-way synchronzaton protoco. As shown n [9], ths can estabsh a ceng on the reabe beamformng tme even n the absence of estmaton error. The phase nose χ (t) at S can be modeed as a zero-mean non-statonary Gaussan random process, ndependent of the estmaton errors, wth varance ncreasng neary wth tme,.e. σχ 2 (t) = r(t T (sync) ) for t T (sync), where T (sync) s the tme at whch S generates estmates ˆω and ˆφ. The varance parameter r s a functon of the physca propertes of the oscator ncudng ts natura frequency and physca type [14]. We assume that a source nodes share the same vaue of r but have ndependent phase nose processes. VI. NUMERICAL RESULTS Ths secton presents numerca exampes of the performance retrodrectve dstrbuted transmt beamformng n a system usng two-way source synchronzaton. To provde a far comparson wth snge-source transmsson, we normaze the transmt power of each source node by M so that the tota transmt power s fxed. We compute the mean beamformng gan wth respect to snge-source transmsson. The scenaro consdered n ths secton assumes 1 ms observatons durng the forward and backward propagaton
stages of the two-way synchronzaton protoco. A channes are assumed to have unt gan and a sgnas are assumed to be receved at a sgna to nose rato of 10dB. At the concuson of the fna synchronzaton tmesot, the source nodes form ther SLOs and the destnaton mmedatey broadcasts a 1 ms beacon. The CRB resuts n (21) are used to generate the jonty Gaussan consttuent estmaton errors wth approprate covarances. The beamformng power at the destnaton for each reazaton of the estmaton errors and phase nose processes s computed usng the resuts n Secton V. Fgure 3 shows the beamformng gan as a functon of tme for dfferent numbers of source nodes (M) and dfferent eves of oca oscator phase nose (r). The r = 0 resuts correspond to the case wth no phase nose and soate the effect of carrer phase and frequency offsets on the mean beamformng gan. The r = 1 resuts correspond to the case when the each source node has an ndependent phase nose process typca of a ow-cost oscator. In ths case, as expected, the mean beamformng gan degrades more qucky. In both cases, perodc resynchronzaton s necessary to prevent the nodes from sppng out of synchroncty and transmttng ncoherenty. The overhead requred for perodc resynchronzaton, however, can be ow wth respect to the amount of beamformng tme. For exampe, n the case wth M = 8 source nodes, the mean beamformng gan of source nodes wth ow-cost oscators s wthn 1dB of dea for approxmatey 240 ms. The synchronzaton tme n ths case s 14 ms, correspondng to an overhead of approxmatey 5%. Even ower overheads can be acheved by usng oscators wth better phase nose characterstcs, e.g. temperature controed oscators. mean beamformng gan (db) 16 14 12 10 8 6 4 2 ncoherent 0 10 2 10 1 10 0 10 1 beamformng tme (seconds) dea dea dea M=32, r=1 M=32, r=0 M=8, r=1 M=8, r=0 M=2, r=1 M=2, r=0 Fg. 3. Mean receved beamformng gan as a functon of beamformng tme, number of source nodes, and oca oscator phase nose parameter. 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Boorstyn, Snge-tone parameter estmaton from dscrete-tme observatons, IEEE Trans. on Informaton Theory, vo. 20, no. 5, pp. 591 598, September 1974. [14] A. Demr, A. Mehrotra, and J. Roychowdhury, Phase nose n oscators: A unfyng theory and numerca methods and characterzaton, IEEE Trans. on Crcuts and Systems I: Fund. Theory and App., vo. 47, no. 5, pp. 655 674, May 2000. The resuts n Fgure 3 aso show that ncreasng the number of source nodes partcpatng n the dstrbuted transmt beamformer ncreases the mean receved power at the destnaton, up to the pont n tme when ncoherent transmsson begns.