Distributed Transmit Beamforming using Feedback Control

Transcription

1 1 Dstrbuted Transmt Beamformng usng Feedback Control R. Mudumba, Student Member, IEEE, J. Hespanha, Senor Member, IEEE, U. Madhow, Fellow, IEEE and G. Barrac, Member, IEEE arxv:cs/6372v1 [cs.it] 18 Mar 26 Abstract A smple feedback control algorthm s presented for dstrbuted beamformng n a wreless network. A network of wreless sensors that seek to cooperatvely transmt a common message sgnal to a Base Staton BS s consdered. In ths case, t s well-known that substantal energy effcences are possble by usng dstrbuted beamformng. The feedback algorthm s shown to acheve the carrer phase coherence requred for beamformng n a scalable and dstrbuted manner. In the proposed algorthm, each sensor ndependently makes a random adjustment to ts carrer phase. Assumng that the BS s able to broadcast one bt of feedback each tmeslot about the change n receved sgnal to nose rato SNR, the sensors are able to keep the favorable phase adjustments and dscard the unfavorable ones, asymptotcally achevng perfect phase coherence. A novel analytcal model s derved that accurately predcts the convergence rate. The analytcal model s used to optmze the algorthm for fast convergence and to establsh the scalablty of the algorthm. Index Terms Dstrbuted beamformng, synchronza- Ths work was supported by the Natonal Scence Foundaton under grants CCF-43125, ANI and EIA-8134, by the Offce of Naval Research under grant N , and by the Insttute for Collaboratve Botechnologes through grant DAAD19-3-D-4 from the U.S. Army Research Offce. R. Mudumba, J. Hespanha and U. Madhow are wth the Department of Electrcal and Computer Engneerng, Unversty of Calforna Santa Barbara G. Barrac s wth Qualcomm Inc., San Dego, CA ton, wreless networks, sensor networks, space-tme communcaton. I. INTRODUCTION Energy effcent communcaton s mportant for wreless ad-hoc and sensor networks. We consder the problem of cooperatve communcaton n a sensor network, where there are multple transmtters e.g., sensor nodes seekng to transmt a common message sgnal to a dstant Base Staton recever BS. In partcular, we nvestgate dstrbuted beamformng, where multple transmtters coordnate ther transmssons to combne coherently at the ntended recever. Wth beamformng, the sgnals transmtted from each antenna undergo constructve nterference at the recever, the multple transmtters actng as a vrtual antenna array. Thus, the receved sgnal magntude ncreases n proporton to number of transmtters N, and the SNR ncreases proportonal to N 2, whereas the total transmt power only ncreases proportonal to N. Ths N-fold ncrease n energy effcency, however, requres precse control of the carrer phases at each transmtter n order that the transmtted sgnals arrve n phase at the recever. In ths paper, we propose a feedback control protocol for achevng such phase coherence. The protocol s based on a fully dstrbuted teratve algorthm, n whch each transmtter

2 2 ndependently adjusts ts phase by a small amount n a manner dependng on a sngle bt of feedback from the BS. The algorthm s scalable, n that convergence to phase coherence occurs n a tme that s lnear n the number of cooperatng transmtters. Pror work on cooperatve communcaton manly focuses on explotng spatal dversty for several wreless relayng and networkng problems [1], [2]. Such dstrbuted dversty methods requre dfferent transmtters to transmt nformaton on orthogonal channels, whch are then combned at the recever. The resultng dversty gans could be substantal n terms of smoothng out statstcal fluctuatons n receved power due to fadng and shadowng envronments. However, unlke dstrbuted beamformng, dstrbuted dversty does not provde a gan n energy effcency n terms of average receved power, whch smply scales wth the transmtted power. On the other hand, the coherent combnng of sgnals at the recever due to dstrbuted beamformng also provdes dversty gans. Recent papers dscussng potental gans from dstrbuted beamformng nclude [3], whch nvestgates the use of beamformng for relay under deal coherence at the recever, and [4], whch shows that even partal phase synchronzaton leads to sgnfcant ncrease n power effcency n wreless ad hoc networks. The beam patterns resultng from dstrbuted beamformng usng randomly placed nodes are nvestgated n [5]. However, the techncal feasblty of dstrbuted beamformng s not nvestgated n the precedng papers. In our pror work [6], [7], we recognzed that the key techncal bottleneck n dstrbuted beamformng s carrer phase synchronzaton across cooperatng nodes. We presented a protocol n whch the nodes frst establsh a common carrer phase reference usng a master-slave archtecture, thus provdng a drect emulaton of a centralzed multantenna system. Ths s a challengng problem, because even small tmng errors lead to large phase errors at the carrer frequences of nterest. Once phase synchronzaton s acheved, recprocty was proposed as a means of measurng the channel phase response to the BS. In ths paper, we present an alternatve method of achevng coherent transmsson teratvely usng a smple feedback control algorthm, whch removes the need for explct estmaton of the channel to the BS, and greatly reduces the level of coordnaton requred among the sensors. Other related work on synchronzaton n sensor networks s based on pulse-coupled oscllator networks [8] and bologcally nspred frefly synchronzaton [9] methods. These methods are elegant, robust and sutable for dstrbuted mplementaton, however they are lmted by assumptons of zero propagaton delay and the requrement of mesh-connectvty, and are not sutable for carrer phase synchronzaton. We consder the followng model to llustrate our deas. The protocol s ntalzed by each sensor transmttng a common message sgnal modulated by a carrer wth an arbtary phase offset. Ths phase offset s a result of unknown tmng synchronzaton errors, and s therefore unknown. When the sensors wreless channel s lnear and tme-nvarant, the receved sgnal s the message sgnal modulated by an effectve carrer sgnal that s the phasor sum of the channel-attenuated carrer sgnals of the ndvdual sensors. At perodc ntervals, the BS broadcasts a sngle bt of feedback to the sensors ndcatng whether or not the receved SNR level ncreased n the precedng nterval. Each sensor ntroduces an ndependent random perturbaton of ther transmtted phase offset. When ths results n ncreased total SNR compared to the prevous tme ntervals as ndcated by feedback from the BS, the new phase offset s set equal to the perturbed phase by

3 3 each sensor; otherwse, the new phase offset s set equal to the phase pror to the perturbaton. Each sensor then ntroduces a new random perturbaton, and the process contnues. We show that ths procedure asymptotcally converges to perfect coherence of the receved sgnals, and provde a novel analyss that accurately predcts the rate of convergence. We verfy the analytcal model usng Monte-Carlo smulatons, and use t to optmze the convergence rate of the algorthm. The rest of ths paper s organzed as follows. Secton II descrbes our communcaton model for the sensor network. A feedback control protocol for dstrbuted beamformng s descrbed n Secton III-A and ts asymptotc convergence s shown n Secton III-B. Secton IV descrbes an analytcal model to characterze the convergence behavor of the protocol. Some analytcal and smulaton results are presented n Secton V. Secton V-A presents an optmzed verson of the feedback control protocol. Sectons V-B and V-C present some results on scalablty, and the effect of tme-varyng channels respectvely. Secton VI concludes the paper wth a short dscusson of open ssues. II. COMMUNICATION MODEL FOR A SENSOR NETWORK We consder a system of N sensors transmttng a common message sgnal mt to a recever. Each sensor s power constraned to a maxmum transmt power of P. The message mt could represent raw measurement data, or t could be a waveform encoded wth dgtal data. We now lst the assumptons n ths model. 1 The sensors communcate wth the recever over a narrowband wreless channel at some carrer frequency, f c. In partcular, the message bandwdth B < W c, where B s the bandwdth of mt and W c s the coherence bandwdth of each sensor s channel. Ths means that each sensor has a flatfadng channel to the recever. Therefore the sensor s channel can be represented by a complex scalar gan h. 2 Each sensor has a local oscllator synchronzed to the carrer frequency f c.e. carrer drft s small. One way to ensure ths s to use Phase-Locked Loops PLLs to synchronze to a reference tone transmtted by a desgnated master sensor as n [6]. In ths paper, we use complex-baseband notaton for all the transmtted sgnals referred to the common carrer frequency f c. 3 The local carrer of each sensor has an unknown phase offset, γ relatve to the recever s phase reference. Note that synchronzaton usng PLLs stll results n ndependent random phase offsets γ = 2πf c τ mod 2π, because of tmng synchronzaton errors τ that are fundamentally lmted by propagaton delay effects. 4 The sensors communcaton channel s tmeslotted wth slot length T. The sensors only transmt at the begnnng of a slot. Ths requres some coarse tmng synchronzaton: τ T where τ s the tmng error of sensor. 5 Tmng errors among sensors are small compared to a symbol nterval a symbol nterval T s s nomnally defned as nverse bandwdth: T s = 1 B. For a dgtally modulated message sgnal mt, ths means that Inter Symbol Interference ISI can be neglected. 6 The channels h are assumed to exhbt slowfadng,.e. the channel gans stay roughly constant for several tme-slots. In other words T s T T c, where T c s the coherence tme of the sensor channels.

4 4 Dstrbuted transmsson model: The communcaton process begns wth the recever broadcastng a sgnal to the sensors to transmt ther measured data. The sensors then transmt the message sgnal at the next tme-slot. Specfcally, each sensor transmts: s t = A g mt τ, where τ s the tmng error of sensor, A P s the ampltude of the transmsson, and g s a complex amplfcaton performed by sensor. Our objectve s to choose g to acheve optmum receved SNR, gven transmt power constrant of P on each sensor. For smplcty, we wrte h = a e jψ and g = b e jθ, where b 1 to satsfy the power constrant. Then the receved sgnal s: rt = N h s te jγ + nt 1 =1 = A = A N h g e jγ mt τ + nt =1 N a b e jγ+θ+ψ mt τ + nt. 2 =1 In the frequency doman, ths becomes: Rf = A N a b e jγ+θ+ψ Mfe jfτ + Nf =1 A Mf N a b e jγ+θ+ψ + Nf, 3 =1 where nt s the addtve nose at the recever and Nf s ts Fourer transform over the frequency range f B 2 İn 1, the phase term γ accounts for the phase offset n sensor. In 3, we set e jfτ 1 because fτ Bτ τ T s 1. Equaton 3 motvates a fgure of mert for the beamformng gan: G = N a b e jγ+θ+ψ 4 =1 whch s proportonal to the square-root of receved SNR. Note that b 1, n order to satsfy the power constrant on sensor. From the Cauchy-Schwartz Inequalty, we can see that to maxmze G, t s necessary. that the receved carrer phases Φ = γ + θ + ψ, are all equal: G = N a b e jφ =1 N a e jφ =1 = N a a e jφ 5 =1 G opt N a, wth equalty f and only f Φ = Φ j and b = 1 =1 6 However sensor s unable to estmate ether γ or ψ because of the lack of a common carrer phase reference. In the rest of ths paper, we propose and analyze a feedback control technque for sensor to dynamcally compute the optmal value of θ so as to acheve the condton for equalty n 6. III. FEEDBACK CONTROL PROTOCOL Fg. 1. Phase synchronzaton usng recever feedback Fg. 1 llustrates the process of phase synchronzaton usng feedback control. In ths secton, we descrbe the feedback control algorthm, and prove ts asymptotc convergence. A. Descrpton of Algorthm The protocol for dstrbuted beamformng works as follows: each sensor starts wth an arbtrary unsynchronzed phase offset γ. In each tme-slot, the sensor

5 5 apples a random perturbaton to θ and observes the resultng receved sgnal strength y[n] through feedback. The objectve s to adjust ts phase to maxmze y[n] through coherent combnng at the recever. Each phase perturbaton s a guess by each sensor about the correct phase adjustment requred to ncrease the overall receved sgnal strength. If the receved SNR s found to ncrease as a result of ths perturbaton, the sensor adds the approprate phase offset, and repeats the process. Ths works lke a dstrbuted, randomzed gradent search procedure, and eventually converges to the correct phase offsets for each sensor to acheve dstrbuted beamformng. Fg. 2 shows the convergence to beamformng wth N = 1 sensors. teratons 1 teratons teratons 5 teratons Fg. 2. Convergence of feedback control algorthm Let n denote the tme-slot ndex and Y[n] the ampltude of the receved sgnal n tme-slot n. From 3, we have: Y[n] a e jφ[n] where Φ [n] s the receved sgnal phase correspondng to sensor. We set the proportonalty constant to unty for smplcty of analyss. At each tme nstant n, let θ [n] be the best known carrer phase at sensor for maxmum receved SNR. Each sensor uses the dstrbuted feedback algorthm to dynamcally adjust θ [n] to satsfy 6 asymptotcally. The algorthm works as follows. Intally the phases are set to zero: θ [1] =. At each tme-slot n, each sensor apples a random phase perturbaton δ [n] to θ [n] for ts transmsson. As a result, the receved phase s gven by: Φ [n] = γ +θ [n]+δ [n]+ψ. The BS measures Y[n] and keeps a record of the hghest observed sgnal strength Y best [n] = max k<n Y[k] n all prevous tmeslots. At the end of each tmeslot, the BS broadcasts a one-bt feedback message that ndcates whether the receved sgnal strength of the precedng tmeslot was hgher than the prevous hghest sgnal strength. Dependng on the feedback message, each sensor updates ts phase accordng to: θ [n] + δ [n] Y[n] > Y best [n] θ [n + 1] = θ [n] otherwse. 7 Smultaneously, The BS also updates ts hghest receved sgnal strength: Y best [n + 1] = max Y best [n], Y[n] 8 Ths has the effect of retanng the phase perturbatons that ncrease SNR and dscardng the unfavorable ones that do not ncrease SNR. The sensors and the BS repeat the same procedure n the next tmeslot. The random perturbaton δ [n] s chosen ndependently across sensors from a probablty dstrbuton δ [n] f δ δ, where the densty functon f δ δ s a parameter of the protocol. In ths paper, we consder prmarly two smple dstrbutons for f δ δ : the two valued dstrbuton where δ = ±δ wth probablty.5, and the unform dstrbuton where δ unform[ δ, δ ]. We allow for the possblty that the dstrbuton f δ δ dynamcally changes n tme. It follows from 7 that f the algorthm were to be termnated at tmeslot n, the best achevable sgnal strength usng the feedback nformaton receved so

6 6 far, s equal to Y best [n], whch correspond to sensor transmttng wth the phase θ [n]. Y best [n] a e Φ[n] where Φ [n] = γ + θ [n] + ψ 9 B. Asymptotc Coherence We now show that the feedback control protocol outlned n Secton III-A asymptotcally acheves phase coherence for any ntal values of the phases Φ. Let Φ denote the vector of the receved phase angles Φ. We defne the functon Mag Φ to be the receved sgnal strength correspondng to receved phase Φ: Mag Φ =. a e jφ 1 Phase coherence means Φ = Φ j = Φ const, where Φ const s an arbtrary phase constant. In order to remove ths ambguty, t s convenent to work wth the rotated phase values φ = Φ Φ, where Φ s a constant chosen such that the phase of the total receved sgnal s zero. Ths s just a convenent shft of the recever s phase reference and as 1 shows, such a shft has no mpact on the receved sgnal strength: Bound [1]. In general the lmt G would depend on the startng phase angles φ. We now provde an argument that shows under mld condtons on the probablty densty functon f δ δ, that n fact {Y best [n]} converges to the constant G opt wth probablty 1 for arbtrary startng phases φ. The followng proposton wll be needed to establsh the convergence. Proposton 1: Consder a dstrbuton f δ δ that has non-zero support n an nterval δ, δ. Gven any φ, and Mag φ < G opt ǫ, where ǫ > s arbtrary, there exst constants ǫ 1 > and ρ > such that ProbMag φ + δ Mag φ > ǫ 1 > ρ. Proof. For the class of dstrbutons f δ δ that we consder, the probablty of choosng δ n any fnte nterval I δ, δ s non-zero. One example of such a class of dstrbutons s f δ δ unform[ δ, δ ]. Recall that the phase reference s chosen such that the total receved sgnal a e jφ has zero phase. Frst we sort all the phases φ n the vector φ n the descendng order of φ to get the sorted phases φ satsfyng φ 1 > φ 2 >... > φ N, and the correspondng sorted channel gans a. We use the condton Mag φ < G opt ǫ to get: Mag φ Mag Φ 11 cosφ 1 a < a cosφ Gopt ǫ We nterpret the feedback control algorthm as a dscrete-tme random process Y best [n] n the state-space of φ, the state-space beng the N-dmensonal space of the phases φ constraned by the condton that the phase of the receved sgnal s zero. We observe that the sequence {Y best [n]} s monotoncally non-decreasng, and s upperbounded by G opt as shown n 5. Therefore each realzaton of {Y best [n]} s always guaranteed to converge to some lmt G G opt. Furthermore G = lm n Y best [n] sup n Y best [n].e. the lmt G of the sequence {Y best [n]} s the same as ts Least Upper φ 1 > φ. ǫ = cos 1 G opt ǫ a 12 Now we choose a phase perturbaton δ 1 that decreases φ 1. Ths makes the most ms-algned phase n φ closer to the receved sgnal phase, and thus ncreases the magntude of the receved sgnal. If φ 1 >, then we need to choose a δ 1 <, whereas f φ 1 <, we need δ 1 >. In the followng, we assume that φ 1 > and φ ǫ > δ. The argument below does not depend on these assumptons, and can be easly modfed for the other cases. Consder δ 1 δ, δ 2. Ths s an nterval n

7 7 whch f δ δ 1 s non-zero, therefore there s a non-zero probablty ρ 1 > of choosng such a δ 1. We have: a 1 cosφ 1 + δ 1 a 1 cosφ 1 > 2ǫ 1 Theorem 1: For the class of dstrbutons f δ δ consdered n Proposton 1, startng from an arbtrary φ, the feedback algorthm converges to perfect coherence of the receved sgnals almost surely,.e. Y best [n] G opt. a where ǫ 1 = 1 snφ ǫ δ 2 δ 4 13 or equvalently φ[n].e. φ [n], wth We observe that ǫ 1 and ρ 1 do not dependent on φ. probablty 1. The perturbaton δ 1 by tself wll acheve a non-zero Proof: We wsh to show that the sequence Y best [n] = ncrease n total receved sgnal, provded that the other Mag φ[n] G opt gven an arbtrary φ[1] = φ. phases φ Consder an arbtrarly small ǫ > and defne T ǫ φ do not get too ms-algned by ther respectve as the frst tmeslot when the receved sgnal exceeds δ : Mag φ + δ Mag φ = G opt ǫ. a cosφ + δ cosφ By defnton f n < T ǫ φ, then Y best [n] = = a 1 cosφ 1 + δ 1 cosφ 1 + Mag φ[n] a < G opt ǫ, cosφ + δ cosφ and by Proposton 1, >1 ProbY best [n + 1] Y best [n] > ǫ 1 > ρ for some > 2ǫ 1 + a cosφ + δ cosφ constants ǫ 1 > and ρ >. We have: >1 14 E Y best [n + 1] Y best [n] > ǫ 1 ρ, n < T ǫ φ 17 We note that snce Mag φ s contnuous n each of the phases φ, we can always fnd a ǫ > to satsfy: a cosφ +δ cosφ ǫ 1 < N 1, δ < ǫ 15. In partcular the choce ǫ = ǫ 1 a N 1, satsfes 15, and ths choce of ǫ s ndependent of φ. Wth the δ s chosen to satsfy 15, we have: ǫ 1 < >1 a cosφ + δ cosφ < ǫ 1 16 Snce f δ δ has non-zero support n each of the nonzero ntervals ǫ, ǫ, the probablty ρ of choosng δ to satsfy 15 s non-zero,.e. ρ >, whch s ndependent of φ. Fnally, we recall that each of the δ are chosen ndependently, and therefore wth probablty ρ = ρ >, t s possble to fnd δ 1 to satsfy 13 and δ, > 1 to satsfy 15. For δ chosen as above, Mag φ + δ Mag φ > ǫ 1, and therefore Proposton 1 follows. Usng 17 we have: G opt Y best [n + 1] n = Y best [1] + Y best [k + 1] Y best [k] > k=1 n Y best [k + 1] Y best [k] k=1 Takng expectaton we have: n G opt > E Ybest [k + 1] Y best [k] k=1 18 n > Prob T ǫ φ > n E Ybest [k + 1] Y best [k] Tǫ φ > n k=1 > Prob T ǫ φ > n nǫ 1 ρ 19 where we obtaned 19 by usng 17. Therefore we have ProbT ǫ φ > n < 1 G opt n ǫ 1ρ, as n. Snce ths s true for an arbtrarly small ǫ, we have shown that Y best [n] G opt and φ[n] almost surely.

8 8 IV. ANALYTICAL MODEL FOR CONVERGENCE The analyss n Secton III-B shows that the feedback control algorthm of Secton III-A asymptotcally converges for a large class of dstrbutons f δ δ ; however t provdes no nsght nto the rates of convergence. We now derve an analytcal model based on smple, ntutve deas that predcts the convergence behavor of the protocol accurately. We then use ths analytcal model, to optmze f δ δ for fast convergence. Fg. 3. receved sgnal strength: y[n] tmeslots, n Motvatng the Analytcal Model: two smulated nstances wth N = 1, f δ δ unform[ π 2, π 2 ]. A. Dervaton of Analytcal Model The basc dea behnd our analytcal model s that the convergence rate of typcal realzatons of Y best [n] s well-modeled by computng the expected ncrease n sgnal strength at each tme-nterval gven a dstrbuton f δ δ. Ths s llustrated n Fg. 3, where we show two separate realzatons of Y best [n] from a Monte-Carlo smulaton of the feedback algorthm. We defne the averaged sequence y[n] recursvely as the condtonal value of Y best [n + 1] gven Y best [n]: y[1] = E Mag φ[1] 2 y[n + 1] = E δ[n] Y best [n + 1] Y best [n] = y[n] 21 The ntal value y[1] n 2 s set under the assumpton that the receved phases φ[1] are randomly dstrbuted n [, 2π. For subsequent tmeslots, Y best [n + 1] n 21 s condtoned on Y best [n] but the phase vector φ[n] s not known. Some remarks are n order regardng ths defnton, partcularly the relatonshp of y[n] wth the uncondtonally averaged Y best [n]. Let y[n+1] = F y[n], where Fy. = E Y best [n+1] Y best [n] = y Consder: 22 E Y best [n + 1] = E Ybest [n] E Y best [n + 1] Y best [n] = E Ybest [n] F Y best [n] F E Y best [n] y[n + 1] 23 In most cases, the functon Fy s concave, and therefore by Jensen s Inequalty the approxmaton n 23 represents an overestmate of the uncondtonal average of Y best [n + 1]. Also n dfferent nstances of the algorthm, we would expect to see dfferent random evolutons of φ[n] and Y best [n] wth tme, and an averaged quantty only provdes partal nformaton about the convergence rate. Fortunately, as Fg. 3 shows, even over multple nstances of the algorthm, the convergence rate remans hghly predctable, and the average characterzes the actual convergence reasonably well. Snce the varaton of the random Y best [n] around ts average value s small, the approxmaton n 23 also works well. Our goal s to compute Fy as defned n 22. Note that whle 22 s condtoned on Y best [n] beng known, the phase vector φ[n] s unknown. As Y best [n] ncreases, the phases φ [n] become ncreasngly clustered together, however ther exact values are determned by ther ntal values, and the random perturbatons from prevous tme-slots. In order to compute the expectaton n 22, we need some nformaton about φ[n].

9 9 We show n Secton IV-B that the phases φ [n] can be accurately modeled as clustered together accordng to a statstcal dstrbuton that s determned parametrcally as a functon of Y best [n] alone. Ths s analogous to the technque n equlbrum statstcal mechancs, where the ndvdual postons and veloctes of partcles n an ensemble s unknown, but accurate macroscopc results are obtaned by modelng the knetc energes as followng the Boltzmann dstrbuton, whch s fully determned by a sngle parameter the average knetc energy or the temperature. In our case, φ [n] are modeled as ndependent and dentcally dstrbuted for all accordng to a dstrbuton satsfyng the constrant: y[n] = a cosφ NE a Eφ cosφ 24 CLT. Mag φ + δ = = = a e jφ+jδ 27, C δ y[n] + x 1 + jx2 28 a cosφ cosδ sn φ sn δ + j a cosφ sn δ where C δ = E δ cosδ, 29 x 1 = a cosφ cosδ C δ snφ sn δ, x 2 = a cosφ sn δ + sn φ cosδ The random varables x 1, x 2 are llustrated n Fg Therefore, even though the ndvdual φ are unknown, we can compute all aggregate functons of φ usng ths dstrbuton, as f the φ are known. Ths s an extremely powerful tool, and we now use t to compute Fy treatng φ[n] as a gven. Secton IV-B completes the computaton by dervng the dstrbuton used to specfy φ[n] gven Y best [n]. From the condton Y best [n] = y[n], we have: y[n] = a e jφ = a cosφ 25 where we used the fact that the magnary part of the receved sgnal s zero for our choce of phase reference. We have the followng expressons omttng the tmendex on φ[n] and δ[n] for convenence: Mag φ + δ f Mag φ + δ > y[n] Y best [n + 1] = y[n] otherwse. 26 We now express Mag φ + δ as a sum of..d. terms from each sensor, and nvoke the Central Lmt Theorem Fg. 4. Perturbaton n the total receved sgnal. Both x 1 and x 2 as defned n 28 are lnear combnatons of d random varables, snδ and cosδ. Therefore as the number of sensors N ncreases, these random varables can be well-modeled as Gaussan, as per the CLT [11]. Futhermore, x 1, x 2 are zero-mean random varables, and ther respectve varances σ 2 1, σ2 2 are related by: σ1 2 = 1 2 σ2 2 = 1 2 a 2 a 2 1 Cδ 2 cos2φ Cδ 2 C 2δ 1 cos2φ C 2δ where C 2δ = E δ cos2δ 32 Wth these smplfcatons, the statstcs of y[n + 1] only depends on the densty functon f δ δ through C δ and C 2δ. We have the followng proposton.

10 1 Proposton 2: Assumng that the CLT apples for random varable x 1, the expected value of the receved sgnal strength s gven by: y[n + 1] y[n] 1 p 1 C δ + σ 1 e 2π y[n]1 Cδ where p = Q σ 1 y[n]1 C 2 δ 2σ Proof. Frst we observe that the small magnary component x 2 of the perturbaton mostly rotates the receved sgnal, wth most of the ncrease n y[n + 1] comng from x 1 see Fg. 4. Mag φ + δ = C δ y[n] + x 1 + jx 2 C δ y[n] + x 1 35 Defnng p as the probablty that Y best [n + 1] > y[n], 33, 34 readly follow from 35, 26 usng Gaussan statstcs. We can rewrte 33 as: y[n + 1] = F y[n] = y[n] + f y[n] where fy =. y1 Cδ σ 1 g σ 1 and gx. = 1 2π e x2 2 xqx 36 Proposton 2 does not yet allow us to compute the y[n] because t nvolves the varance σ 1 that depends on the phases φ of the ndvdual sensors. In the next secton we present a statstcal dstrbuton for φ that allows us to calculate aggregate quanttes such as σ 1 wthout knowledge of the ndvdual φ. B. Statstcal Characterzaton of Sensor Channels The statstcal model s based on the assumpton that each sensor has a channel to the BS of smlar qualty, and unknown phase. Ths means that the a s are all approxmately equal, and that the ntal values of the phases φ are dstrbuted ndependently 1 and unformly n [, 2π. In partcular, we set a = 1 for all sensors, whch gves G opt = N. As the algorthm progresses towards convergence, the values of φ are dstrbuted over a smaller and smaller range. In general, we expect that the dstrbuton f φ φ of φ [n] depends on the number of sensors N, the teraton ndex n, and the dstrbuton of the perturbatons f δ δ. In the sprt of the statstcal model, we consder large N, and look for a class of dstrbutons that approxmate f φ φ. Probablty Densty Fg Phase angle radans Comparng a Laplacan Dstrbuton wth a Hstogram of Emprcally Observed Phase Angles We fnd that the Laplacan probablty dstrbuton gves the best results 2 n terms of accurately predctng the convergence behavor of the algorthm of Secton III-A. Fg. 5 shows an emprcally derved hstogram 1 It s mportant to note that the φ are not random varables, however we statstcally parametrze them usng a probablty dstrbuton for the sake of compactness. 2 The Laplacan dstrbuton for φ s emprcally found to work well, when compared wth other famles of dstrbutons lke the unform and trangular dstrbutons.

11 11 from a Monte-Carlo smulaton of the feedback control algorthm. A Laplacan approxmaton s also plotted alongsde the hstogram. We now explan the detals of the approxmaton. The Laplacan densty functon s gven by [11]: f φ φ = 1 2φ e φ φ 37 For φ dstrbuted accordng to 37, we also have: E 1 cosφ = 1 + φ 2 E 1 cos2φ = 1 + 4φ Therefore gven that at teraton n of the feedback algorthm, the phase angles are φ[n] = [φ 1 φ 2...φ N ], we have: y[n] = Mag φ[n] = a e jφ cosφ 4 where we used a = 1 n 1. Now f we parametrze all the φ usng a Laplacan dstrbuton, we can set φ such that cosφ NE φ cosφ. Thus we use 38 to rewrte 4 as: y[n] = N 1 + φ 2 We are now able to determne σ 1 gven y[n]. Proposton 3: The varance σ 2 1 of x 1 s gven by: σ 2 1 = N 2 1 C 2 δ y[n] N 4 3 y[n] N Cδ 2 C 2δ Proof. Equaton 42 follows usng 32, and the value of the Laplacan parameter from 41 along wth the observaton that cos2φ = NE φ cos2φ = N. 1+4φ 2 Usng Propostons 2 and 3, we are able to analytcally derve the average convergence behavor of the feedback control algorthm. In partcular, we recursvely calculate y[n] by substtutng the varance σ 2 1 from 42 nto 33. C. Summary of Analytcal Model We now summarze the analytcal model derved n Sectons IV-A and IV-B. Our objectve s to model the ncrease over tme of the receved sgnal strength by averagng over all possble values of the random perturbatons. As mentoned before, we set the channel attenuatons for each sensor to unty.e. a = 1. 1 Intally we set the receved sgnal strength as y[1] = N. Ths s the expected value of the sgnal strength f the ntal phase angles are all chosen ndependently n [, 2π. 2 At each tme-nterval teraton n > 1, gven the probablty dstrbuton of the perturbatons f δ δ and the value of y[n], we compute the Laplacan parameter φ usng 41, and then compute the Gaussan varance σ 2 1 usng 42 and fnally y[n+ 1] usng the Gaussan statstcs n 35 and 26. V. PERFORMANCE ANALYSIS OF FEEDBACK CONTROL PROTOCOL We now present some results obtaned from the analytcal model of Secton IV. Fg. 6 shows the evoluton of y[n] derved from the analytcal model and also from a Monte-Carlo smulaton wth N = 1, for two dfferent choces of the dstrbuton f δ δ : a unform dstrbuton n [ π 3, π 3 ] and a dstrbuton choosng ± π 3 wth equal probablty. The close match observed between the analytcal model and the smulaton data provdes valdaton for the analytcal model. We observe from Fg. 6, that the receved sgnal grows rapdly n the begnnng, but after y[n] gets to wthn about 25% of G opt, the rate of convergence becomes slower. Also whle the smple two-valued probablty dstrbuton appears to gve good results, t does not satsfy the condton for asymptotc coherence derved n Secton III-B.

12 receved sgnal strength, y[n] smulated analytcal receved sgnal strength, y[n] smulated analytcal tmeslot ndex, n tmeslot ndex, n a f δ δ unform π 3, π 3 b f δ δ ± π 3 Fg. 6. Comparson of Analytcal Model wth Monte-Carlo Smulaton of Feedback Control Algorthm A. Optmzng the Random Perturbatons In Fg. 6, we used the same dstrbuton for the perturbatons for all teratons of the algorthm. However ths choce s not optmal: ntuton suggests that t s best to choose larger perturbatons ntally to speed up the convergence and make the dstrbuton narrower when the phase angles are closer to coherence. We now use the analytcal model to dynamcally choose the dstrbuton f δ δ as a functon of y[n]. The general problem of choosng a dstrbuton s a problem n calculus of varatons. Fortunately, t s possble to restrct ourselves to a famly of dstrbutons wthout losng optmalty, because the analytcal model only depends on the dstrbuton through the two parameters C δ, C 2δ. Furthermore the parameters C δ, C 2δ are hghly correlated. To see ths recall from 31 and 32 the defntons of C δ and C 2δ as the expected values of cosδ and cos2δ respectvely. Usng the dentty cos2δ = 2 cos 2 δ 1 and Jensen s Inequalty, we can show that C 2δ s constraned by the value of C δ : 2Cδ 2 1 C 2δ 2C δ 1 43 We are nterested n δ correspondng to small random perturbatons.e. δ π 2. For such small values of δ, 43 allows only a small range of possble values of C 2δ. Indeed we observe that cosδ and cos2δ are very well approxmated by the frst two terms of the Taylor seres: cosδ 1 δ2 2, f δ π 2 44 Equaton 44 ndcates that both C δ and C 2δ are essentally determned by the second moment of δ, and therefore even a one-parameter famly of dstrbutons f δ δ s suffcent to acheve optmalty of the convergence rate. Fg. 7 shows plots of the optmal choces of the C δ, C 2δ par wth N = 2 over 1 tmeslots for two famles of dstrbutons: the 3-pont dstrbutons P±δ = p, P = 1 2p parametersed by the par δ, p, and the dstrbutons unform[ δ, δ ] parametrsed by δ. At each teraton of the protocol, we used the analytcal model to compute the value of the parameters.e. the par δ, p n case and δ n case that maxmzes the y[n + 1] gven y[n]; the optmal parameters n each case were determned numercally usng a smple search procedure. The two

13 13 curves n Fg. 7 were obtaned by plottng C δ, C 2δ par correspondng to the optmal parameters for cases and at each tmeslot. The 3-pont dstrbuton s flexble enough to permt any C δ, C 2δ n the feasble regon of 43. For the example of Fg. 7, t s clear that the unform dstrbuton acheves values of C δ, C 2δ that s close to optmal, thereby confrmng the ntuton of optmzed over unform dstrbutons.8.6 optmzed over general dstrbutons.4 receved sgnal y[n] optmum δ n degrees optmzed tmeslots, n C 2δ Feasble Regon Fg. 8. Optmzed algorthm compared to fxed f δ δ unform[ δ, δ ] for dfferent δ and N = 2 B. Scalablty Results Fg. 7. C δ Near-Optmalty of a One-Parameter Dstrbuton We now use the famly of dstrbutons f δ δ unform[ δ, δ ] to obtan nsght nto the optmal convergence rate. Fg. 8 shows y[n] as a functon of n for fxed values of δ as well as for the optmzed algorthm. We observe that the convergence rate decreases wth tme n all cases, and the optmzed algorthm converges sgnfcantly faster than any fxed nstance. Fg. 8 also shows the varaton of optmal δ wth tme. Ths confrms our ntuton that at the ntal stages of the algorthm, t s preferable to use larger perturbatons correspondng to large δ, and when y[n] gets closer to G opt, t s optmum to use narrower dstrbutons correspondng to smaller δ. We now turn to the analytcal model to study the scalablty of the feedback algorthm wth the number of transmttng sensors N. We show the followng scalablty results: - The expected receved sgnal strength at any tme, always ncreases when more transmtters are added. - The number of tmeslots requred for the expected sgnal strength to reach wthn a certan fracton of convergence always ncreases wth more transmtters, but ncreases no faster than lnearly n the number of transmtters. Theorem 2: Let y 1 [n] and y 2 [n] be the expected receved sgnal magntude at tmeslot n when the number of transmttng sensors s N 1 and N 2 respectvely. If the sensors use the same dstrbutons f δ δ for all tmeslots n, and N 2 > N 1, then the followng holds for all n: and y 1[n] N 1 y 2 [n] y 1 [n] 45 y 2[n] N 2 46

14 14 Proof. We offer a proof by nducton. From Secton IV-C, we know that y 2 [1] > y 1 [1] and y1[1] N 1 > y2[1] N 2. To prove 45, we need to show that y 2 [n + 1] > y 1 [n + 1] gven y 2 [n] > y 1 [n]. We wrte y 1 [n+1] = F 1 y 1 [n], y 2 [n+1] = F 2 y 2 [n] where F 1 y and F 2 y are defned as n 36. Note that F 1 y 1 [n] and F 2 y 2 [n] depends on the tme ndex n not only through y 1 [n] y 2 [n], but also through the dstrbuton f δ δ. We have suppressed ths addtonal tme-dependence to keep the notaton smple. The functons F 1 y and F 2 y satsfy the followng propertes: F 2 y > F 1 y, y 47 F 1 y + > F 1 y, and F 2 y + > F 2 y f y + > y 48 To see ths we observe from 42 that for the same value of y, σ 1 s larger for larger N, and snce fy n 36 ncreases wth σ 1, 47 follows. To show 48, t s suffcent to show that F 1 y and F 2 y have a postve dervatve wth respect to y. Ths can be shown readly by dfferentatng the expresson n 36: df 1 y dy = d y1 Cδ varance σ1 y+fy = 1 1 Cδ Q 2. Wth y[n] = f Gopt = f N we have: > C δ > dy σ C δ 2 Cδ 2 C 2δ 1 Cδ 2 52 We are now ready to complete the proof of 45 by nducton. Gven that y 2 [n] > y 1 [n], we have: y 2 [n+1] F 2 y2 [n] > F 1 y2 [n] > F 1 y1 [n] y 1 [n+1] where we used 47 and 48 for the two nequaltes. 5 Ths completes the proof of 45. The proof of 46 by nducton s smlar and s omtted. dstrbutons. We apply Theorem 2 to the case where we use the dstrbuton f δ δ optmzed for N 1 sensors n both cases. By defnton ỹ 2 [n] y 2 [n], and ỹ 1 [n] = y 1 [n], therefore ỹ 2 [n] ỹ 1 [n], n. Ths proves 45. Usng the same argument for the dstrbuton f δ δ optmzed for N 2 sensors, we can prove 46. Another mportant crteron for scalablty s the number of tmeslots T f N requred for the algorthm to converge to a fxed fracton, say f =.75 or 75% of the maxmum for N transmttng sensors. Theorem 2 shows that T f N s an ncreasng functon of N. Next we show that when the feedback algorthm s approprately optmzed, T f N ncreases wth N no faster than lnearly. Theorem 3: The number of tmeslots to convergence satsfes the followng: T f N lm N N t f, where t f s some constant. 51 Proof. Frst we use 43 to get a lower-bound for the Usng the upper bound from 52 n 42, we have: σ1 2 > N 1 C2 δ 4N 4y[n] 2 4N 3y[n] 1 f > 2N1 C δ 4 3f We now use a bound for the Gaussan Q-Functon: Qx > 1 e 1 x2 2 2π x 1 x x Corollary: The scalablty relatons 45 and 46 hold when the sensors use optmzed dstrbutons f δ δ n both cases. Proof. Let ỹ 1 [n] and ỹ 2 [n] be the expected receved sgnal magntudes usng the respectve optmzed Usng 54, we rewrte 33 to get: y[n] =. y[n + 1] y[n] > σ 1 e x2 2 2π 1 x 2 3 x 4 55 where x = y[n]1 C δ σ 1 56

15 15 The bound n 55 has a maxmum at x 3.6; choosng a C δ such that x s close to x, does not necessarly optmze the RHS n 55, because σ 1 also depends on C δ. However such a choce for C δ does provde a meanngful lower bound on the optmal y [n]. σ 1 2x 1 f 57 f 4 3f y [n] > 2 1 f 1 e x f 4 3f 2π x x 3 58 where 57 s obtaned by backsubsttutng 56 nto 53. Let us denote the RHS of 58 by Kf. We observe that the lower bound n 58 only depends on the fracton f = y[n] N. Let T f, fn be the number of tmeslots requred for the feedback algorthm to ncrease y[n] from a fracton f f to a fracton f of convergence. If f s small enough, we can use 58 to wrte: C. Trackng Tme-varyng channels So far we have focused on the smple case of tmenvarant wreless channels from each sensor to the BS. In practce, the channel phase response vares because of Doppler effects arsng from the moton of the sensors or scatterng elements relatve to the BS. In the dstrbuted beamformng scenaro, Doppler effects also arse because of drfts n carrer frequency between the local oscllators of multple sensors. Therefore an mportant performance metrc for the feedback control algorthm s ts ablty to track tme-varyng channels. Intutvely we expect that the algorthm should track well as long as the tme-scale of the channel varatons s smaller than the convergence tme of the algorthm. In lght of the scalablty results n Secton V-B, the algorthm performs better for smaller N because the correspondng convergence tme s smaller. f N = y[n] y[n T f, f N] = T f, f N t=1 Therefore T f, f N < f N Kf y[n t] y[n] T f, f N > KfT f, f N 59 6 Snce T f s just a sum of terms lke T f, f, 51 mmedately follows. Theorem 3 s llustrated by the results n Fg. 9, where the number of tmeslots requred to get wthn a certan fracton of convergence s plotted aganst number of transmtters N for a fxed dstrbuton Fg. 9a as well as optmzed dstrbutons Fg. 9b. These results show that the feedback algorthm s hghly scalable wth number of transmtters. receved sgnal strength, y[n] δ=unform[ π/3,π/3] δ=unform[ π/6,π/6] δ=unform[ π/1,π/1] tmeslots ndex, n Fg. 1. Trackng Performance for Tme Varyng Channels: N = 1, Doppler rate= π 2 radans/tmeslot A smulaton of y[n] wth tme n the presence of channel tme-varatons s shown n Fg. 1. Ths plot uses a fxed dstrbuton for the phase perturbatons, as

16 tmeslots to convergence % 6% 5% tme slots to convergence 1 5 8% 75% 7% number of sensors, N number of sensors, N a f δ δ unform π 9, π 9 b Optmzed Dstrbuton Fg. 9. Scalablty of Feedback Control Algorthm wth Number of Sensors the analytcal model for optmzaton s not applcable to the tme-varyng case. A more detaled study of the trackng performance of the feedback control algorthm s beyond the scope of the present work. VI. CONCLUSION In ths paper, we presented a smple algorthm for dstrbuted beamformng n sensor networks, that s based on the dea of usng SNR feedback from the recever to perform phase synchronzaton n an teratve manner. Ths algorthm can be easly mplemented n a decentralzed manner and s guaranteed to acheve asymptotc coherence under mld assumptons. We also derved an analytcal model that predcts the performance of the algorthm accurately, and offers nsght nto the convergence behavor. Ths paper represents an ntal study nto a new approach to the problem of dstrbuted synchronzaton, and leaves several open ssues. We presented the Laplacan dstrbuton to model the statstcs of the phase angles φ as an emprcal observaton. However the underlyng reason why the Laplacan dstrbuton works so well s not clear. In addton the stablty and convergence behavor of the feedback control algorthm under nondealtes lke tme-varyng channels, and the effects of nose are open ssues for future work. Whle we use the term sensors for the cooperatng nodes performng dstrbuted beamformng, the technque developed here s of more general applcablty. For example, t could be used as the bass for cooperatve communcaton between clusters of nodes n a wreless ad hoc network. In such a context, t would be of nterest to examne how the use of dstrbuted beamformng would mpact the desgn of medum access control and network layer protocols. ACKNOWLEDGMENT The authors would lke to acknowledge a dscusson wth Dr. Babak Hassb whch stmulated detaled nvestgaton of the scalablty of the proposed protocol. REFERENCES [1] A. Sendonars, E. Erkp, and B. Aazhang, User cooperaton dversty. part. system descrpton, IEEE Trans. on Commun., vol. 51, no. 11, pp , Nov 23.

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