Distributed Beamforming for Information Transfer in Sensor Networks

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1 Distributed Beamforming for Information Transfer in Sensor etworks G. Barria Dept. of Eletrial and Computer Engineering University of California Santa Barbara, CA 936, USA usb.edu R. Mudumbai Dept. of Eletrial and Computer Engineering University of California Santa Barbara, CA 936, USA U. Madhow Dept. of Eletrial and Computer Engineering University of California Santa Barbara, CA 936, USA ABSTRACT Energy effiient transfer of data from sensors is a fundamental problem in sensor networks. In this paper, we propose a distributed beamforming approah to this problem, with a luster of sensors emulating a entralized antenna array. While it is well-known that beamforming an provide large performane gains, suh gains presuppose not only aurate knowledge of the hannel, but also time and phase synhronization at the transmitter. We propose expliit methods for ahieving suh synhronization in a distributed fashion, and analyze the effets of various soures of oordination error on the attained performane. We find that, as long as the error in range measurements or plaement of the sensor nodes is within a fration of a arrier wavelength, the proposed distributed beamforming strategies ahieve most of the gains available from a entralized beamformer. Categories and Subjet Desriptors H.. [Information Systems]: MODELS AD PRICI- PLES Systems and Information Theory General Terms Design Keywords sensors, distributed beamforming, synhronization This work was supported by the ational Siene Foundation under grants AI 8 and EIA 834, and by the Offie of aval Researh under grant Permission to make digital or hard opies of all or part of this work for personal or lassroom use is granted without fee provided that opies are not made or distributed for pro t or ommerial advantage and that opies bear this notie and the full itation on the rst page. To opy otherwise, to republish, to post on servers or to redistribute to lists, requires prior spei permission and/or a fee. IPS 4, April 6 7, 4, Berkeley, California, USA. Copyright 4 ACM /4/4...$5... ITRODUCTIO In this work, we propose distributed beamforming tehniques for dramatially inreasing the energy effiieny of ommuniation in sensor networks. This tehnique is omplementary to the methods disussed in [3, 6], whih address the problem of data relay within a sensor network. In our work, we assume effiient loal ommuniation (i.e. ommuniation within the sensor field) and show that neighboring sensors are able to oordinate their transmissions to form a distributed antenna array that direts a beam in the desired diretion of transmission. The gains from idealized distributed beamforming are known to be large. Our ontribution is to provide speifi methods for ahieving the neessary oordination, and to analyze the effets of various soures of oordination error on the gains obtained. While there is a ost assoiated with synhronising the sensors and for the loal exhange of sensor observations, the available SR (and apaity) gains are ompelling. Also one ould easily imagine a senario where individual sensors have a limited transmit power range, so their effetive ommuniation distane is onstrained by path loss. In this ase, the ommuniation distane an be extended onsiderably by beamforming and the distane an be inreased simply by adding more sensors. Figure 8 shows a oneptual view of the sensor field. In addition to the sensor networks appliation disussed here, distributed beamforming may also be appliable in more general ommuniation ontexts for low arrier frequenies (e.g. MHz or lower), sine a standard diretional antenna may be diffiult to implement at large wavelengths (3 meters or more). Prior work on distributed beamforming has foused mainly on reeive beamforming. Perhaps the most dramati example of this is the Very Large Array (VLA) of antennas used for radio astronomy[4]. Other examples of distributed reeive beamforming in the literature inlude [7, ]). An example of distributed transmission is [8], where the authors propose to use distributed transmit beamforming involving multiple ellular Base Stations to improve the mobile s SR. The ruial issues of synhronization required for distributed beamforming are not addressed in this work. We now provide a brief outline of the rest of this paper. In Setion, we desribe the motivation for, and some requirements for distributed beamforming.

2 In Setion 3, we analyze the effet of the phase noise on reeive SR under a stohasti model for the position error. We also derive simple analytial expressions for the mean and variane of reeiver SR and under Central Limit Theorem assumptions for large, and obtain an expression for the distribution of SR as a onvolution of two squared Gaussian distributed random variables. Setion 4 desribes a simulation model for this system and Setion 5 shows results orresponding to the analytial expressions from Setion 3. Setion 6 onludes the paper with a summary of results and a disussion of possible areas of future work.. BACKGROUD In order to establish the requirements for distributed beamforming, we disuss first the operation of entralized beamforming, i.e., the operation of a standard antenna array. For a transmitter A with n antenna elements, suitably weighting the signals sent from eah antenna element an reate beams in a diretion of interest towards reeiver B, leading to a fator of n gain in signal-to-noise ratio. Often, suh transmit beamforming from A to B ould be preeded by training a reeive beamformer at A using a transmission from B to A. If the same frequeny band is used in both diretions, then reiproity an be used to infer the transmit beamforming weights from the reeive beamforming weights. For wideband signaling over ertain types of hannels, it is even possible to employ different frequeny bands in the two diretions, and to use statistial reiproity [] to learn the transmit strategy from A to B, based on what is reeived at A from B. However, in order to employ the reiproity-based method of using weights learnt from reeive beamforming, we must be able to perform distributed reeive beamforming. To understand the issues involved, onsider first the operation of entralized reeive beamforming. Assuming that the hannel from the transmitter to the reeiver is frequeny nonseletive, the RF signal reeived at the ith antenna is given by y i(t) = Re(s(t)w ie j(πf t+θ ) ) () where f, θ, are the arrier frequeny and phase, respetively, of the arriving wave. Assume that a ommon arrier frequeny and phase of f and θ is employed at eah antenna element to demodulate the reeived signal from RF to baseband. In this ase, the omplex baseband signal at the ith antenna element is given by s(t)w ie j(π ft+γ) () where f = f f and γ = θ θ. Assuming that f is small enough, the weights w i an be reovered upto a ommon phase unertainty by filtering and sampling the omplex baseband signals at a ommon time. Thus, while it is not required that the reeiver lok up to the arrier frequeny of the arriving wave, an impliit assumption is the use of a ommon arrier frequeny and phase for demodulation at eah antenna element, and timing synhronization of the samples at eah antenna element used to estimate the reeive beamforming weights. One the spatial hannel {w i} has been estimated, a reeive beamformer orresponds to multiplying the omplex baseband signal for antenna i by w i. This orresponds to a spatial mathed filter. ote that reeive beamformers an also be used to put spatial nulls in the diretion of interferene, e.g., by omputing the beamformer based on a Minimum Mean Squared Error (MMSE) or Minimum Variane Distortionless Response (MVDR) riterion. However, in emulating reeive beamforming in a distributed fashion, our first priority is to ahieve the simpler task of spatial mathed filtering, leaving the issue of distributed interferene suppression for later. One the reeive beamforming weights {w i} have been estimated, if reiproity applies, they an be used for transmit beamforming. Thus, the omplex-valued transmit beamforming weights for a entralized transmit beamformer are given by w,..., w n. If the information to be sent is enoded in the baseband signal q(t), the omplex baseband signal transmitted from the ith antenna element is w i q(t). The radio frequeny (RF) signal transmitted from the ith antenna is Re(w i q(t)e jπf t+θ ) (3) where f is the arrier frequeny and θ is the arrier phase. The impliit assumptions here are as follows. First, the baseband signals for all antenna elements are synhronized in time. Seond, the arrier frequeny and phase used to modulate the baseband signals are the same for all antenna elements. These assumptions are easy to satisfy for entralized beamforming, beause of the ommon iruits used to generate the baseband signals and arriers for all antenna elements, and the tight ontrol on the iruit delays in distributing these signals to the antennas. The preeding disussion reveals that, in order to emulate entralized transmit or reeive beamforming in a distributed manner, the key requirements are arrier frequeny and phase synhronization, and timing synhronization, among the distributed antenna elements. In the next setion, we desribe simple methods for ahieving suh synhronization, and show that the level of preision ahieved is suffiient to ahieve signifiant beamforming gains in the diretion of interest.. Methods for Distributed Beamforming The key onept for ahieving the synhronization required for distributed beamforming using a luster of nodes is to have one node in the luster serve as a master node, broadasting both a arrier and timing signals. Figure 7 shows the funtionality of a sensor node in blok diagram form. Assuming that the slave nodes in the luster know their distane relative to the master node, they an lok up to the arrier and timing signals sent by the master, and ompensate for the delay with whih the master signal arrives, thereby ahieving frequeny, phase and timing synhronization. The preision with whih this synhronization is ahieved depends both on the signal-to-noise ratios for the synhronization iruits employed, and on the auray of the estimates of the delay between the master and slave nodes. Before disussing implementation of this general onept, we onsider a speial senario in whih the implementation is partiularly simple. Suppose that the master and slaves are arranged in a star topology, with the master at the enter. That is, the master is at approximately equal distane from eah slave. This topology ould be ahieved either by initial plaement of the nodes, or, for mobile nodes, by suitable ontrol algorithms that plae the slave nodes at a desired distane from the

3 master. Speifially, slave i is at distane d(i) = d + d e(i) from the master, where d is the nominal distane, and d e(i) is the plaement error. The required tolerane in plaement error is disussed shortly. The master broadasts a arrier signal os(πf t). Let denote the speed of light, and λ = f the arrier wavelength. Assuming LOS, slave i reeives a noisy arrier signal where u i(t) = os(πf t + θ + θ e(i)) + n i(t) θ = πfd = πd λ is the nominal phase offset from the transmitted arrier, and θ e(i) = πfde(i) = πde(i) λ is a phase error resulting from the plaement error, and n i(t) is noise. Eah slave employs a phase loked loop (PLL) to lok on to the arrier, whose output an be approximately written as v i(t) = os(πf t + θ + θ e(i) + θ pll (i)) where θ pll (i) is the phase error due to PLL imperfetions and noise. Slave i will employ the arrier v i(t) for both reeption and transmission, as speified in the following.. Distributed Reeive Beamforming For LOS reeption, the reeived signal at slave i is given by (). Upon demodulation by the arrier v i(t), the omplex baseband signal at slave i is given by where f = f f and z i(t) = s(t)w ie j(π ft+γ(i)) (4) γ(i) = θ θ θ e(i) θ pll (i) Comparing with (), we see that the phase variations aross different slaves our due to plaement error and PLL error. Thus, if these two errors an be ontrolled, we an approximate () using (4). The PLL error an be made small by inreasing the signal-to-noise ratio, whih is not diffiult if the sensors are relatively lose to eah other. We therefore neglet it in the following. To understand the requirements on plaement error, suppose that we wish to bound θ e(i) to within δθ, where we might set δθ =.5 radians, for example. This implies that d e(i) λ δθ π Thus, the smaller the arrier frequeny (i.e., the larger the wavelength), the slaker the requirements on plaement error for satisfying a desired phase synhronization tolerane. For example, for f = MHz and δθ =.5 radians, we obtain that d e(i).4 meters, whih is relatively straightforward to ahieve using either manual plaement or a suitable ontrol algorithm (in onjuntion with a ranging sheme) with mobile sensors. On the other hand, if f is inreased to MHz, then the tolerane on the plaement error for the same phase tolerane beomes -fold more stringent. One the required phase tolerane is ahieved in the baseband models, it remains to make a measurement of the beamforming gains w i. One approah for doing this is for the master to broadast a trigger sequene (possibly in parallel to the arrier it is broadasting). Suppose, for example, that the transmitted baseband signal s(t) is a sequene of pulses. Eah slave passes the demodulated baseband signal z i(t) through a pulse mathed filter, and makes a measurement of the amplitude and phase at the peak of the mathed filter output immediately following detetion of the trigger sequene. ote that the trigger sequene arrival time at different slaves is slightly different due to the plaement error. Further, the next peak of the mathed filter output may our at slightly different times for different slaves beause of the different propagation delays between the transmitter and the slave. However, if the period of the pulses is short enough, and the differene in frequeny f between the transmitted arrier and the master s arrier is small enough (whih is a funtion of osillator toleranes), the differenes in phase between the measurements at different slaves remains small. At the end of this proess, eah slave has an estimate of its own reeive beamforming oeffiient ŵ i. Distributed reeive beamforming an now be performed to enhane the reliability of reeption. The details of how this is done depends on the modulation and oding format of the information being reeived, and we do not disuss this any further here. Instead, we fous attention on how these oeffiients would be used for distributed transmit beamforming by exploiting reiproity..3 Distributed Transmit Beamforming One the reeive beamforming oeffiients have been estimated, transmit beamforming orresponds to implementing a spatial mathed filter. The master sends another trigger sequene to initiate transmission from the slaves. It is assumed that the baseband signal q(t) ontaining the information to be sent has already been agreed upon. Upon reeipt of the trigger sequene, slave i modulates the baseband signal q(t) with the arrier at the output of its PLL, sending the RF signal x i(t) = Re( w i q(t τ e(i))e jπf (t τ e(i)) θ θ e(i) θ pll (i) ) (5) where τ e(i) is the timing synhronization error assoiated with the reeipt of the transmission trigger sequene. The dominant omponent of this is due to plaement error. Letting τ e(i) de(i), and realling that θ e(i) = pif d e(i), (5) an be written as x i(t) = Re( w i q(t τ e(i))e jπf t θ θ e(i) θ pll (i) ) (6) If the plaement error is small enough to permit aurate distributed reeive beamforming, then it is easy to see that it also is small enough for aurate distributed transmit beamforming..4 Beyond the Star Topology The preeding onepts generalize quite easily beyond the star topology. For a general master-slave topology, it is neessary that eah slave has obtained a prior estimate ˆτ(i) of the propagation delay between the master and itself. The required auray of this estimate is proportional to the required auray in the plaement error disussed earlier for the star topology. The master broadasts a arrier and trigger sequenes, and the slaves operate PLLs and detetion iruits, as before. However, upon reeipt of a trigger sequene, slave i takes a delayed ation at time τ ˆτ(i),

4 where τ is an upper bound on the {ˆτ(i)}. This effetively implements a ausal filter whih ompensates for the variations in delay between the master and different slaves. The analysis of soures of synhronization error are now exatly as in the ase of the star topology. 3. AALYSIS In this setion we determine how plaement errors, whih translates into phase errors, affet the gains ahieved by distributed transmit beamforming. Letting P R denote the reeived signal power when the transmit power is kept onstant, and letting denote the number of oordinating sensors, we find that: (a) The expeted value of P R inreases as β θ, where β θ is a funtion of the phase error distribution and β θ. When there is no phase error, E[P R] =, meaning that beamforming with elements gives a power gain of over transmission with a single element. Thus, the degredation aused by phase errors is ontained in the term β θ. (b) The variane of P R also inreases linearly with, for both zero and nonzero phase errors. Of ourse, the existene of phase errors an only inrease the varaine over that of an ideal, error free system. Thus, as long as the distribution of plaement errors is ontained in suh a way as to keep β θ lose to, large gains an be still be realized using distributed beamforming. We model the hannel oeffiients w i, i =..., as irularly symmetri omplex normal random variables with zero mean and unit variane, as denoted by w i C(,). With the assumption that w i w i, we an write (6) as where x i(t) = Re(ŵ i q(t τ e(i))e jπf t θ ) (7) ŵ i = w i e j(θe(i)+θ pll(i)) Sine the phase loked loop error terms θ pll (i), i =..., an be made arbitrarily small by inreasing the signal to noise ratio at the slave nodes when reeiving from the master node, we neglet these terms in the subsequent analysis, giving (8) ŵ i = w i e jθe(i) (9) We also assume that the loss aused by the delays τ e(i) in the omplex baseband signal is muh smaller than the loss aused by the phase errors in ŵ i. This an be justified as follows. Writing q(t) as a pulse train modulated by omplex symbols, we have q(t) = k p(t kt)s k () where {s k } is the symbol stream, and p(t), < t < T is the transmitted pulse. The delays τ e(i), i =..., in (7) ause both intersymbol interferene (ISI) and a redution in the reeive power due to the misalignment of the mathed filter. Sine the plaement errors are (assumed) on the order of.λ, meaning the timing errors τ e(i) are on the order of.λ, and sine the pulse duration T is generally on the order of λ or more, a guard interval on the order of.t between the transmitted pulses in enough to make ISI negligible. We thus assume that there is suh a guard interval and ignore ISI. Also, beause the timing error is small ompared to the pulse duration T, the resulting loss in the reeive signal strength after mathed filtering will be small, on the order of %. We therefore onentrate on the loss aused by phase misalignment, sine this loss an be on the order of 4% for similar plaement errors. The symbol stream {s k } ( see ()) is normalized so E[s k ] =, and the power in eah pulse, p(t), is normalized to P T/, where P T is the total transmit power, so the power transmitted by all sensors remains onstant at P T regardless of. For simpliity of exposition, we set P T =. Thus, the baseband representation for the reeived signal, given that symbol s k was sent from all sensors, an be written as r k = w H ŵ s k + n k () where w is the vetor ontaining the elements {w i}, ŵ is defined similarly, and n k is omplex gaussian zero mean noise. Sine the index k is arbitrary, we will heneforth drop it from the notation. The reeived signal power, our figure of merit, is therefore P R = wh ŵ () Using (9), this an be written as P R = w i e jθ f(i) (3) where θ f = θ e. Proposition : PR E[os(θ f)] a.s. as, where a.s. denotes almost sure onvergene. In other words, when the total transmit power is kept a onstant, the reeived signal power inreases linearly with as tends to. Proof: We an rewrite (3) as follows: P R = w i e jθ f(i) (4) Invoking the law of large numbers, and the fat that the { w i }, are i.i.d. exponential random variables whih are independent from the i.i.d {θ f (i)}, we have w i e jθ f E [ w (os(θ f ) + j sin(θ f )) ] a.s. (5) where w w i and θ f θ f (i), i. The expetation on the RHS of (5) simplifies as follows E [ w (os(θ f ) + j sin(θ f )) ] = E[ w ]E[os(θ f )] = E[os(θ f )] (6) We have assumed that θ f is symmetrially distributed around, and hene E[sin(θ f )] =. Equation (6) results beause w C(,) and hene w is exponential with unit mean. We thus have that w i e jθ f (E[os(θ f )]) a.s. (7) sine funtions of variables whih are onverging almost surely also onverge almost surely, and the desired result follows. ote that when there are no phase errors, i.e. f θ (θ ) = δ(), then PR a.s.

5 Proposition : For finite, E[P R] = +( )E[os(θ f )]. Thus, even for finite, the expeted value of the reeived signal power inreases linearly with. Proof: The expeted value of P R an be written as [ E[P R] = ] E w i e jθ f(i) w l e jθ f(l) l= = ( ) ( + E[ w w.r(e j(θ f() θ f ()) )] (8) = ( + ) ( ) E[os(θ f () θ f ())] = + ( )E[os(θ f () θ f ())] = + ( )E[os(θ f ()) os(θ f ()) sin(θ f ()) sin(θ f ())] = + ( )E[os(θ f )] (9) where we have used the fat that the {w i}, {θ i} are i.i.d. and independent, and that the {θ i} are symmetrially distributed around. In the absene of phase errors, Proposition gives that E[P R] =. Proposition 3: When is large enough for the entral limit theorem to apply, P R X + X s () where X (m, σ ), X s (, σ s), and the parameters m, σ, and σ s, are given as follows: m = E[os(θ f )] () σ = E[os (θ f )] E[os(θ f )] () σ s = E[sin (θ f )] (3) The variane of the reeived signal power is then var[p R] = 4σ m + σ 4 + σ 4 s (4) whih inreases linearly with. Proof: We one again begin with the definition for P R. P R = w i os(θ f (i)) + j w l sin(θ f (l)) = ( w i os(θ f (i)) α) +j w l sin(θ f (l)) + α l= (5) where α = E[ w i os(θ f )] = E[os(θ f )]. Invoking the entral limit theorem, as gets large, the first term in (5) tends to a Gaussian random variable with mean and variane σ var[ w os(θ f )]. Similarly, the seond term tends to a Gaussian random variable with mean and variane σ s var[ w sin(θ f )]. Sine the last term in (5) is real and onstant, it only shifts the mean of the first Gaussian random variable, so we an write P R X + jx s (6) where X ( α, σ ), and X s (, σ s). Making use of the fat that w is a unit mean exponential random variable, and similarly σ = var[ w os(θ f )] = E[ w 4 os (θ f )] E[ w os(θ f )] = E[os (θ f )] E[os(θ f )] (7) σ s = E[sin (θ f )] (8) Letting m α, we have that P R = X + Xs, as given. The variane of P R follows from standard alulations for moments of Gaussian random variables. When there are no phase errors, (4) redues to var[p R] = SIMULATIO MODEL In order to verify the viability of using distributed beamforming, we simulate the sensor system analysed in Setion 3 using MATLAB s SIMULIK software. We now desribe the simulation methodology and all simplifying assumptions. Figure 6 shows the SIMULIK model of the sensor network. Beause the expeted value of the reeived signal strength, P R, as given in Proposition, remains unhanged when the magnitudes of the {w i} are fixed at, and we are interested in finding the empirial value of E[P R], we an, without loss of generality, let w i = i (9) We do not model the timing errors {τ e(i)} in the base band signal (7), sine they ause negligible degradation in P R ompared to the phase errors, as mentioned previously. Also, we assume ideal hannel estimation, so w i = w i i. Finally, sine we are interested in the reeive signal power, and not the reeived noise power, we do not model the AWG in (). Essentially, our simulation modifies (), and models the following baseband system r k = e jθf(i) s k (3) where k is the symbol index and {θ f (i)} are the phase errors. We assume that the {θ f (i)} are uniformly distributed around, i.e. i θ f (i) U[ π, π ] (3) For eah value of and, we run 6 MonteCarlo simulations to determine E[P R]. Though more averaging is desirable, the time requirements of the simulation make larger numbers of runs impratial. The sensor data that is transmitted is just a binary pulse train of random bits. The signalling rate is hosen at about % of the arrier frequeny (i.e. there are about arrier sine wave yles in a bit interval). The arrier is modulated by multiplying the arrier wave (obtained from the VCO output of the PLL in the sensor s synhronization iruits) with the pulse train. This is equivalent to a BPSK modulated signal, sine multiplying the pulse train is the same as a degree phase shift on a bit and a 8 degree phase shift on a - bit. We assume that the reeiver has perfet knowledge of the hannel, whih is a fairly standard assumption in wireless

6 reeiver design. In this ase, this ensures phase synhronization and therefore oherent demodulation is possible. The reeiver first onverts the signal to baseband by using a mixer to multiply the inoming arrier signal with a loal osillator whih is assumed to be frequeny synhronized with the transmitting sensors. The mixer is followed by a lowpass filter to remove the unwanted mixer omponent. Sine we are using a retangular pulse, the baseband mathed filtering operation is equivalent to integrating the signal over a bit interval. The square of the integrator output is proportional to the reeived SR, and is our figure of merit. 5. RESULTS In this setion, we verify that the SIMULIK simulations math the analytial results, and show how the gains in reeived signal power sale with. Using a uniform distribution for the phase errors as given in (3), ( θ f (i) U[ π, π ] i), letting = π, and using the approximation os(θ) = θ, we have from Proposition ( E[P R] = + ( ) ) 6 (3) Values of E[P R] omputed using (3) are plotted in Figure vs. along with the values of E[P R] obtained from the MonteCarlo simulations. The top set of lines is for =., the next set is for =., and the last two sets are for =.3 and =.4, respetively. =. orresponds to a plaement error spread of.5λ, =. orresponds to a spread of.λ, et. This an be seen by realling that and that θ f (i) = θ e(i) (33) θ e(i) = πf o d e(i) (34) where d e(i) is the plaement error. If the {d e(i)} are i.i.d. zero mean uniform random variables over an interval of length λo = f, then the {θ e(i)} are uniform over an interval of π. In other words, so θ e(i) U[ π, π] i (35) θ f (i) U[ π, π] i (36) and thus =. orresponds to a plaement error spread of.5λ, et. The empirial and analytial results in Figure math well, indiating that the phased loked loop does not introdue any signifiant errors. We therefore present further results for the analytial model only. Figure shows how E[P R] (alulated using (3)) sales with for disrete values of between. and.4. The maximum slope is, orresponding to the ase of ideal beamforming. ote that for =.4, at = 4, there is almost a 5% loss ompared to this ideal benhmark. However, the expeted reeived signal power is still over times greater than that for a simgle antenna transmission. Hene, signifiant gains an still be expeted for distributed beamforming. We now onsider how the variane of P R sales with. Using the approximations os θ = θ θ and sin θ = E[P R ]/ analytial simulated Figure : E[P R]/ vs, empirial and analytial results. The four sets of urves are for (top to bottom), =. :. :.4. E[P R ] =. 5 =. =.3 = Figure : The expeted value of the reeived signal power vs. the number of sensor nodes. θ 3 6, as well as the fat that the phase errors are uniformly distributed (36), we an write m, σ, and σs, (defined in Proposition 3), as m = ( ) (37) 6 σ = σs = (38) (39) The variane of P R an then be alulated using (4), and is shown vs. for various values of in Figure 3. Histograms of P R, alulated using the ormal approximations in Proposition 3, are shown in Figure 4 for =. and = : : 4. Most of the variane is due to variations in w i, as an be seen by omparing Figure 4 to Figure 5, where the magnitudes of the {w i} have been set to

7 = = var[p R ] =3 = =. =. =.3 = Figure 3: The variane of the reeived signal power vs. the number of sensor nodes. Figure 5: Histograms of P R where the hannel oeffiients all have unit magnitude. =. i. Thus, Figure 5 more aurately aptures the variation in reeived signal power due to synhronization errors COCLUSIO We have shown that the large potential gains from distributed beamforming an indeed be realized using a masterslave arhiteture. Our analysis aurately predits the performane degradation due to phase noise and range errors. The ruial assumption in our results is that ranging errors between master and slaves are small ompared to the arrier wavelength. An important topi for future work is, therefore to develop arhitetures for realizing this assumption. Another important issue is the integration of suh tehniques with advanes in soure oding and sensor net organization ([5]) whih also point towards a luster based network arhiteture = = =3 =4 5 5 Figure 4: Histograms of P R. =. Figure 7: Blok diagram of sensor node funtionality

8 Figure 6: SIMULIK model of beamforming sensors Sensor e -jθ e -jθ Information soure Loal sensor ommuniation Sensor Sensor 3 Reeiver e -jθ 3 Figure 8: Top level view of sensor field 7. REFERECES [] G. Barria and U. Madhow. Spae-time ommuniation for OFDM with impliit hannel feedbak. In IEEE Global Teleommuniations Conferene, volume 3, pages 3 35, 3. [] J. Chen, L. Yip, J. Elson, H. Wang, D. Maniezzo, R. Hudson, K. Yao, and D. Estrin. Coherent aousti array proessing and loalization on wireless sensor networks. In Proeedings of the IEEE, volume 9, pages 54 6, August 3. [3] C. Intanagonwiwat, R. Govindan, D. Estrin, J. Heidemann, and F. Silva. Direted diffusion for wireless sensor networking. IEEE/ACM Transations on etworking, : 6, February 3. [4] J. D. Kraus. Antennas, Seond Edition. M-Graw Hill, 988. [5] D. Petrovi, R. Shah, K. Ramhandran, and J. Rabaey. Data funneling: routing with aggregation and ompression for wireless sensor networks. In IEEE International Workshop on Sensor etwork Protools and Appliations, pages 56 6, May 3. [6] A. Saglione and S. Servetto. On the interdependene of routing and data ompression in multi-hop sensor networks. In Pro. ACM Mobiom. [7] K. Yao, R. Hudson, C. Reed, D. Chen, T. Tung, and F. Lorenzelli. Array signal proessing for a wireless mem sensor network. In IEEE Workshop on Signal Proessing Systems, pages, Ot 998. [8] M. Yipeng Tang; Valenti. Coded transmit marodiversity: blok spae-time odes over distributed antennas. In Vehiular Tehnology Conferene, volume, pages , May.