Feedback-Based Closed-Loop CarrierSynchronization: A Sharp Asymptotic

Bound, an Asymptotically OptimalApproach, Simulations, and Experiments

Stephan Sigg, Member, IEEE, Rayan Merched El Masri, and Michael Beigl, Member, IEEE

Abstract—We derive an asymptotically sharp bound on the synchronization speed of a randomized black box optimization technique

for closed-loop feedback-based distributed adaptive beamforming in wireless sensor networks. We also show that the feedback

function that guides this synchronization process is weak multimodal. Given this knowledge that no local optimum exists, we consider

an approach to locally compute the phase offset of each individual carrier signal. With this design objective, an asymptotically optimal

algorithm is derived. Additionally, we discuss the concept to reduce the optimization time and energy consumption by hierarchically

clustering the network into subsets of nodes that achieve beamforming successively over all clusters. For the approaches discussed,

we demonstrate their practical feasibility in simulations and experiments.

Index Terms—Analysis of algorithms, wireless communication, wireless sensor networks.

Ç

1 INTRODUCTION

IN recent years, sensor nodes of extreme tiny size havebeen envisioned [1], [2], [3]. In [4], for example,

applications for square-millimetre-sized nodes that seam-lessly integrate into an environment are detailed. At thesesmall form factors, transmission power of wireless nodes isrestricted to several microwatts. Communication between asingle node and a remote receiver is then only feasible atshort distances. It is possible, however, to increase themaximum transmission range by cooperatively transmit-ting information from distinct nodes of a network [5], [6].Cooperation can increase the capacity and robustness of anetwork of transmitters [7], [8] and decrease the averageenergy consumption per node [9], [10], [11].

Related research branches are cooperative transmission[12], collaborative transmission [13], [14], distributedadaptive beamforming [15], [16], [17], [18], collaborativebeamforming [19], or cooperative/virtual MIMO for wire-less sensor networks [20], [21], [22], [23]. One approach is toutilize neighboring nodes as relays [24], [25], [26] asproposed by Cover and El Gamal [27]. Cooperativetransmission is then achieved by multihop [28], [29], [30]or data flooding [31], [32], [33], [34] approaches. Thegeneral idea of multihop relaying based on the physicalchannel is to retransmit received messages by a relay nodeso that the destination will receive not only the messagefrom the source node, but also from the relay. In data

flooding approaches, a node will retransmit a receivedmessage at its reception. It has been shown that theapproach outperforms noncooperative multihop schemessignificantly. In particular, the transmission time is reducedas compared to traditional transmission protocols [35].

In these approaches, nodes are not tightly synchronizedand transmission may be asynchronous. Synchronoustransmission, however, is achieved by virtual MIMOtechniques. In these implementations, identical RF carriersignal components from various transmitters that functionas a distributed beamformer are superimposed. When therelative phase offset of these carrier signal components at aremote receiver is small, the signal strength of the receivedsum signal is improved. In virtual MIMO for wirelesssensor networks, single antenna nodes are cooperating toestablish a multiple antenna wireless sensor network [21],[20], [22]. Virtual MIMO has capabilities to adjust todifferent frequencies and is highly energy efficient [23],[11]. However, the implementation of MIMO capabilities inWSNs requires accurate time synchronization, complextransceiver circuits, and signal processing that mightsurpass the power consumption and processing capabilitiesof simple sensor nodes.

Other solutions proposed are open-loop synchronizationmethods such as round-trip synchronization [36], [37], [38].In this scheme, the destination transmits beacons inopposed directions along a multihop circle in which eachof the nodes appends its part of the overall message to thebeacons. Beamforming is achieved when the processingtime along the multihop chain is identical in bothdirections. This approach, however, does not scale withthe size of a network.

Closed-loop feedback-based approaches include full-feedback techniques, in which carrier synchronization isachieved in a master-slave manner. The phase offset amongthe carrier signals of destination nodes is corrected by a

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. S. Sigg is with TU Braunschweig Muehlenpfordtstrasse 23, Braunschweig38106, Germany. E-mail: sigg@ibr.cs.tu-bs.de.

. R.M. El Masri and M. Beigl are with the Telecooperation Office (TecO),Karlsruhe 76131, Germany. E-mail: {rmasri, michael}@teco.edu.

Manuscript received 28 Oct. 2009; revised 22 Oct. 2010; accepted 15 Dec.2010; published online 2 Feb. 2011.For information on obtaining reprints of this article, please send e-mail to:tmc@computer.org, and reference IEEECS Log Number TMC-2009-10-0463.Digital Object Identifier no. 10.1109/TMC.2011.21.

1536-1233/11/$26.00 � 2011 IEEE Published by the IEEE CS, CASS, ComSoc, IES, & SPS

receiver node. Diversity between RF-transmit signal compo-nents is achieved over CDMA channels [39]. This approachis applicable only to small network sizes and requiressophisticated processing capabilities at the source nodes.

A more simple and less resource demanding implementa-

tion is the 1-bit feedback-based closed-loop synchronization

considered in [39] and [40]. The authors describe an iterative

process in which n source nodes i2½1; . . . ; n� randomly adapt

the phases �i of their carrier signal <ðmðtÞejð2�ðfcþfiÞtþ�iÞÞ.Here, mðtÞ denotes the transmit message and fi denotes the

frequency offset of node i to a common carrier frequency fc.

Initially, i.i.d. phase offsets �i of carrier signals are assumed.

When a receiver requests a transmission from the network,

carrier phases are synchronized in an iterative process.

1. Each source node i adjusts its carrier phase offset �iand frequency offset fi randomly.

2. The source nodes transmit to the destinationsimultaneously as a distributed beamformer.

3. The receiver estimates the level of phase synchroni-zation of the received sum signal (for instance by thesignal-to-noise ratio (SNR)).

4. This value is broadcast as a feedback to the network.Nodes interpret this feedback and adapt the phase oftheir carrier signal accordingly.

These four steps are iterated repeatedly until a stopcriterion is met (e.g., maximum iteration count or sufficientsynchronization). Fig. 1 illustrates this procedure. It hasbeen studied by different authors [41], [42], [43], [13]. Thedistinct approaches proposed differ in the implementationof the first and the fourth step specified above. The authorsof [43] show that it is possible to reduce the count oftransmitters in a random process and still achieve sufficientsynchronization among all nodes.

In [41], [42], and [43], a process is described in whicheach node alters its carrier phase offset �i according to anormal distribution with small variance in Step 1. In [13], auniform distribution is utilized instead, but the probabilityfor one node to alter the phase offset of its carrier signal islow. We show in Section 5 that both approaches achieve asimilar performance. Only in [42] not only the phase, butalso the frequency is adapted.

Significant differences among these approaches alsoapply to the feedback and the reactions of nodes in Step4. In [15], [42], and [43], a 1-bit feedback is utilized. Nodessustain their phase modifications when the feedback hasimproved and otherwise reverse them. In [43], it wasshown that the optimization time is improved by a factor of2 when a node as response to a negative feedback from thereceiver applies a complementary phase offset instead ofsimply reversing its modification. In [13], the authorssuppose to utilize more than 1-bit as feedback so thatparameters of the optimization can be adapted with regardto the optimization progress.

The strength of feedback-based closed-loop distributedadaptive beamforming in wireless sensor networks is itssimplicity and low processing requirements that make itfeasible for the application in networks of tiny sized, lowpower, and computationally restricted sensor nodes.

We study aspects of this transmission scheme and derivesharp asymptotic lower and upper bounds on the expectedoptimization time of a common implementation in Section 2.Together with these bounds, we show that the feedbackfunction is weak multimodal so no local optimum exists. Bysmall modifications of the common algorithm, however,further improvements in the synchronization time can beachieved. In Section 3, we discuss a hierarchical clusteringscheme that exploits that the superimposed signal strengthof a set of nodes is increased at a slower pace than thesynchronization time with increasing node count. Whenadditional information available at a receiver node isutilized, further improvements are possible. In Section 4,we show that by providing more than 1 bit as feedback tothe transmitters, knowledge about the feedback functioncan be derived from measurements of a single node alteringthe phase offset of its carrier signal. We present anasymptotically optimal algorithm that utilizes this knowl-edge and significantly improves the synchronization pro-cess. In Section 5, algorithms are compared for theirsynchronization performance in numerical simulations. Inthese simulations, the impact of various environmentalsettings and algorithmic configurations can be approxi-mated. Finally, in Section 6, we demonstrate the feasibilityof distributed adaptive beamforming in wireless sensornetworks in a near-realistic instrumentation with softwareradios. Section 7 draws our conclusion.

2 SYNCHRONIZATION TIME ANALYSIS

We analyze the process of distributed adaptive beamform-ing in wireless sensor networks, as described in Section 1,and assume that each one of the n nodes decides withprobability 1

n to change the phase of its carrier signaluniformly at random in the interval ½0; 2��. On obtaining thefeedback of the receiver, nodes that recently updated thephase of their carrier signal either sustain this decision orreverse it, depending on whether the feedback hasimproved or not. A feedback function F : ��sum ! IR mapsthe superimposed received carrier signal

�sum ¼ < mðtÞej2�fctXni¼1

RSSiejð�iþ�iþ iÞ

!ð1Þ

to a real-valued feedback score. In (1), RSSi denotes thereceived signal strength (RSS) of the ith signal out of

2 IEEE TRANSACTIONS ON MOBILE COMPUTING, VOL. 10, NO. X, XXXXXXX 2011

Fig. 1. Schematic illustration of feedback-based distributed adaptivebeamforming in wireless sensor networks.

n received signal components. As local oscillators are notsynchronized and nodes are spatially distributed, �i and iaccount for the phase offset in the received signalcomponents due to the offset in the local oscillators anddue to distinct signal propagation times. A possible feed-back function that is proportional to the distance betweenan observed superimposed carrier �sum and an optimumsum carrier signal

�opt ¼ <�mðtÞRSSopte

jð2�fctþ�optÞ�

ð2Þ

is Fð�sumÞ ¼R 2�t¼0 j�sum � �optj. Since this function can be

mapped onto other feedback measures as, for instance, theSNR or the received signal strength, the followingdiscussion remains valid for these feedback measures.While the multimodality of this feedback function isstraightforward, we derive in Appendix A, which can befound on the Computer Society Digital Library at http://doi.ieeecomputersociety.org/10.1109/TMC.2011.21, that itis also weak multimodal so no local optima exists.

Distributed adaptive beamforming in wireless sensornetworks is a search problem. The search space S is givenby the set of possible combinations of phase and frequencyoffsets �i and fi for all n carrier signals. A global optimum isa configuration of individual carrier phases that result inidentical phase and frequency offset of all received directsignal components. For the analysis, we assume that theoptimization aim is to achieve for an arbitrary k a maximumrelative phase offset of 4�

k between any two carrier signals.This means that we can control the quality of thesynchronization achieved by the variable k. An optimumis then reached when the phases of all carrier signalcomponents of a receiver are within an interval of 4�

k in thephase space. When k is increased, this directly translates toan improved phase synchronization among signal compo-nents. Naturally, we can expect that the accuracy of thesynchronization also impacts the synchronization time.

For our analysis, we logically divide the phase space fora single carrier signal into k intervals of width 2�

k . Observethat this is half of the interval that was used to definethe optimum synchronization. Consequently, when theachieved phase offset of each received signal component iswithin a maximum distance of 2�

k to the optimum phaseoffset, all received carrier phase signals are within aninterval of 4�

k in the phase space and the optimum is reached.For a specific superimposed carrier signal � at a

receiver, we represent the corresponding search point s� ¼ð�t; FtÞ� 2 S at iteration t by a specific combination ofphase and frequency offsets with �t ¼ ð�t;1; . . . ; �t;nÞ andFt ¼ ðft;1; . . . ; ft;nÞ. In order to respect neighborhoodsimilarities, we represent search points as Gray-encodedbinary strings s� 2 IBn�logðkÞ so that similar points have asmall Hamming distance [44]. A search point is thencomposed from n sections of logðkÞ bits each. Every blockof length logðkÞ describes one of the k intervals for thephase offset of one carrier signal. For the analysis, weassume that the frequency offset fi is zero for all carriers.Observe, however, that the discussion can be easilyadapted to also cover a simultaneous carrier frequencysynchronization. In [42], the authors demonstrated that thesame random synchronization approach can be utilized tosynchronize carrier frequencies when in each iteration notonly the carrier phase, but also the frequency of the

transmit signals is altered. By this generalization, thesearch space of the algorithm is increased. For each node,not only k distinct possibilities exist, but k � � where �denotes the count of distinct frequencies that can possiblybe applied for each carrier signal. The optimization timethen increases by a factor of �. However, the analyticaldiscussion becomes more complicated as the commonperiod of the received sum signal might be increasedconsiderably.

The optimization problem is denoted as P and TPdescribes the count of iterations required to reach oneoptimum for the problem P.

2.1 An Upper Bound on the ExpectedSynchronization Time

The value of the feedback function increases with the numberof carrier signals �i that share the same interval for their phaseoffset �i at the receiver. Assume that � 2 ½1; k� is the intervalthat contains most of the carrier phase offsets. As worsefeedback values are not accepted, we count the iterationsrequired for all carrier signals to change to interval �. We canroughly divide the values of the feedback function inton partitions L1; . . . ; Ln depending on the number of carriersignals with their phase in the interval �. For each onetransmitter, the probability to adapt its phase to one specificinterval is 1

k . The probability to increase the feedback valueso that at least the next partition is reached is then

1

k� n� Lið Þ � 1

n; ð3Þ

since one carrier signal �i is altered with probability 1n and

the probability to reach any particular of the ðn� LiÞpartitions that would increase the feedback value is 1

k . Inpartition i, a total of

n� i1

� �¼ n� i ð4Þ

carrier signals each suffice to improve the feedback valuewith probability 1

n � 1k . We, therefore, require that at least one

of the not synchronized carrier signals is correctly altered inthe phase while all other n� 1 signals remain unchanged.This happens with probability

n� i1

� �� 1n� 1k� 1� 1

n

� �n�1

¼ n� in � k

� �� 1� 1

n

� �n�1

:

ð5Þ

Since

1� 1

n

� �n<

1

e< 1� 1

n

� �n�1

; ð6Þ

we obtain the probability P ½Li� that Li is left and apartition j with j > i is reached as

P ½Li� �n� in � e � k : ð7Þ

The expected number of iterations to change the layer isbounded from above by P ½Li��1. We, consequently, obtainthe overall expected synchronization time as

SIGG ET AL.: FEEDBACK-BASED CLOSED-LOOP CARRIER SYNCHRONIZATION: A SHARP ASYMPTOTIC BOUND, AN ASYMPTOTICALLY... 3

E½TP� �Xn�1

i¼0

e � n � kn� i

¼ e � n � k �Xni¼1

1

i

< e � n � k � lnðnÞ þ 1ð Þ¼ O n � k � lognð Þ:

ð8Þ

2.2 A Lower Bound on the ExpectedSynchronization Time

After the initialization, the phases of the carrier signals are

identically and independently distributed. Consequently,

for a superimposed received sum signal �, each bit in the

binary string s� that represents the corresponding search

point has an equal probability to be 1 or 0. The probability

to start from a search point s� with Hamming distance

hðsopt; s�Þ not larger than l 2 IN ; l� n � logðkÞ to one of the

global optima sopt directly after the random initialization is

at most

P ½hðsopt; s�Þ � l� ¼Xli¼0

n � logðkÞn � logðkÞ � i

� �� k

2n�logðkÞ�i

� n � logðkÞð Þlþ2

2n�logðkÞ�l :

In this formula,

n � logðkÞn � logðkÞ � i

� �ð9Þ

is the count of possible configurations with i bit errors to a

given global optimum, 12n�logðkÞ�i represents the probability for

all these bits to be correct, and k is the count of global

optima. This means that with high probability (w.h.p.) the

Hamming distance to the nearest global optimum is at least

l. We use the method of the expected progress to calculate a

lower bound on the optimization time.Let ðs�; tÞ denote the situation that search point s� is

achieved after t iterations of the algorithm. We assume a

progress measure � : IBn�logðkÞ ! IRþ0 such that �ðs�; tÞ < �

represents the case that a global optimum was not found in

the first t iterations. For every t 2 IN, we have

E½TP� � t � P ½TP > t�¼ t � P ½�ðs�; tÞ < ��¼ t � ð1� P ½�ðs�; tÞ � ��Þ:

ð10Þ

With the help of the Markov inequality, we obtain

P ½�ðs�; tÞ � �� � E½�ðs�; tÞ��

; ð11Þ

and, therefore,

E½TP� � t � 1� E½�ðs�; tÞ��

� �: ð12Þ

This means that we can obtain a lower bound on the

optimization time by providing the expected progress

after t iterations. The probability for l bits to correctly flip

is at most

1� 1

n � logðkÞ

� �n�logðkÞ�l� 1

n � logðkÞ

� �l

� 1

ðn � logðkÞÞl:

ð13Þ

In this formula, ð1� 1n�logðkÞÞ

n�logðkÞ�l describes the probability

that all “correct” remain unchanged while the remaining

l bits flip with probability ð 1n�logðkÞÞ

l. The expected progress in

one iteration is, therefore,

E½�ðs�; tÞ;�ðs�0 ; tþ 1Þ� �Xli¼1

i

ðn � logðkÞÞi

<2

n � logðkÞ ;ð14Þ

and the expected progress in t iterations is, consequently,

not greater than 2tn�logðkÞ . When we choose t ¼ n�logðkÞ��

4 � 1, the

twice of the expected progress is still smaller than �. With

the Markov inequality, we can show that this progress is not

achieved with probability 12 . Altogether we conclude that the

expected synchronization time is bounded from below by

E½TP� � t � 1� E½�ðs�; tÞ��

� �

� n � logðkÞ ��4

� 1�2�n�logðkÞ4�n�logðkÞ ��

�

!

¼ �ðn � logðkÞ ��Þ:

ð15Þ

With � ¼ k � logðnÞlogðkÞ , we obtain a lower bound in the same

order as the upper bound derived in Section 2.1 and,consequently, an asymptotically sharp bound of

E½TP� ¼ � n � k � logðnÞð Þ: ð16Þ

Note that in [41], an upper bound on the expectedasymptotic synchronization time was derived that scaleslinearly in the number of nodes n when the probabilitydistribution is optimally altered repeatedly during thesynchronization. However, simulation results derived fora fixed uniform distribution in this study also indicate alogarithmic factor in the synchronization time of 1-bitfeedback-based synchronization.

3 HIERARCHICAL CLUSTERING

A further improvement of the synchronization time can beachieved by synchronizing smaller clusters of nodesseparately. Since this bound on the synchronization timegrows faster than linearly with the network size n, but thereceived signal strength RSSsum of the received super-imposed signal grows linearly with n, the overall energyconsumption and synchronization time might be reducedwhen fewer nodes transmit for a shorter time, but with anincreased transmission power. Note that currently mostlow-cost radios are not capable of altering their transmis-sion power and, therefore, are not able to exploit thisproperty. More sophisticated radios could, however,achieve carrier phase synchronization more efficiently whenthis fact is utilized. We propose the following hierarchicalclustering scheme that synchronizes all transmit nodesiteratively in clusters of reduced size.

4 IEEE TRANSACTIONS ON MOBILE COMPUTING, VOL. 10, NO. X, XXXXXXX 2011

1. Determine clusters (e.g., by a random processinitialized by the receiver node).

2. Synchronize clusters successively as described abovewith possibly increased transmit power. Whencluster � is sufficiently synchronized, nodes in thiscluster sustain their carrier signal and stop transmit-ting until all clusters are synchronized.

3. At this stage, carrier signals in all clusters are inphase, but carrier phases of distinct clusters mightdiffer. Determine representative nodes from allclusters and synchronize these.

4. Nodes in all clusters alter their carrier phase by thephase offset experienced and broadcast by thecorresponding representative node (broadcast). Let

�i ¼ <�mðtÞRSSie

j2�fctð�iþ�iþ iÞ�

and �0i ¼ <ðmðtÞRSSiej2�fctð�0iþ�iþ iÞÞ be the carrier

signals of representative node i from cluster � beforeand after synchronization between representativenodes were achieved. A node h from cluster � altersits carrier signal �h ¼ <ðmðtÞRSShe

j2�fctð�hþ�hþ hÞÞ to�0h ¼ <ðmðtÞRSShe

j2�fctð�hþ�hþ hþ�i��0iÞÞ. Under idealconditions, all nodes are now in phase.

5. To account for synchronization errors, a finalsynchronization phase in which all nodes participateconcludes the overall synchronization process.

Fig. 2 illustrates this procedure. The crucial idea of thisapproach is applied in Step 4. Since nodes inside a clusterhave already been synchronized, they are still in phaseafter all applying an identical phase offset. Because thisoffset is the phase alteration the representative nodesexperienced during their synchronization, all nodes shouldbe synchronized after this step.

A potential problem for this approach is phase noise. Sinceonly one cluster is synchronized at a time, phases of nodes inthe inactive clusters experience phase noise and start driftingout of phase due to practical properties of oscillators.However, we show in Section 5 that the sufficient synchro-nization is possible in the order of milliseconds. Therefore,we do not consider phase noise an important issue. Observethat all coordination is initiated by the receiver node sono internode communication is required for coordination.

Depending on the network size, more than one hierarchystage might be optimal for the synchronization time and theenergy consumption. To estimate the optimal hierarchydepth and the optimum cluster size, the count of nodesparticipating in the synchronization must be computed. Weassume that the nodes themselves do not know the networksize. This means that the remote receiver derives thenetwork size, calculates optimal cluster sizes and hierarchydepths, and broadcasts this information. In [45], it wasdemonstrated that the superimposed sum signal fromarbitrarily synchronized nodes is sufficient to estimate thenumber of transmitters. We derive the optimum hierarchydepth and cluster size by integer programming in timeOðn2Þ (cf. Appendix B, which can be found on the ComputerSociety Digital Library at http://doi.ieeecomputersociety.org/10.1109/TMC.2011.21). The expected synchronizationtime is dependent on the cluster count, cluster size, andhierarchy depth. Since for each cluster a small instance ofthe original problem is solved, the synchronization time canbe composed from the synchronization times of individualclusters.

4 AN ASYMPTOTICALLY OPTIMAL ALGORITHM

Since no local optimum exists in the search space due to itsweak multimodality (cf. Appendix A, which can be found

SIGG ET AL.: FEEDBACK-BASED CLOSED-LOOP CARRIER SYNCHRONIZATION: A SHARP ASYMPTOTIC BOUND, AN ASYMPTOTICALLY... 5

Fig. 2. Illustration of the approach to cluster the network of nodes in order to improve the synchronization time of feedback-based closed-loop

distributed adaptive beamforming.

on the Computer Society Digital Library at http://doi.ieeecomputersociety.org/10.1109/TMC.2011.21), the perfor-mance of �ðn � k � logðnÞÞ of the random search seems weak.Previously, only one feedback bit was utilized. Due to thisreduced information, some of the information available atthe receiver is not available by the nodes that actuallyreact on the feedback. When more information is included inthe feedback of the receive node, we are able to design anasymptotically optimal synchronization algorithm.

In every iteration, the receiver provides additionalinformation over a feedback value so that a node i canlearn the optimum phase offset of its own carrier

�i ¼ < mðtÞRSSiej2�fctð�iþ�iþ iÞ

� �relative to the superimposed sum signal

�sumni ¼ < mðtÞej2�fctX

o2½1;n�;o 6¼iRSSoe

jð�oþ�oþ oÞ

0@

1A

of all other nodes, provided that the latter does not changesignificantly. �sumni is a sinusoidal signal. The feedback ismaximal when �i and �sumni have identical phase offset at areceiver. With increasing phase offset

�i þ �i þ ið Þ � �sumni þ �sumni þ sumni� ��� ��;

the feedback value decreases symmetrically. Consequently,the feedback function has the formFð�iÞ ¼ A sinð�i þ �Þ þ c.This is an equation with the three unknowns A (amplitude),� (phase offset of F ), and the additive term c so that a node ican calculate it with three distinct measurements. Fig. 3illustrates the accuracy of this procedure for 100 transmitters.The root of the mean square error (RMSE) is calculated as

RMSE ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiXt¼0

�sum þ �noise � �opt

� �2

n

vuut : ð17Þ

Here, is chosen to cover several signal periods.For the optimization process, a node will during each of

four subsequent iterations either alter the phase offset of itscarrier signal or sustain it for all four iterations. Theprobability to alter the phase offset should be low as, forinstance, 1

n . A node that decides to alter its phase offset will

do this three times to measure feedback values for distinctphase offsets, derive with these measurements the feedbackfunction, alter its phase offset accordingly, and finallytransmit a fourth time to obtain the amount by which theachieved feedback value deviates from the expected value.If the deviation is small, the node will not alter its phasefurther since the current phase offset is considered optimal.All other nodes then adapt the probability to alter theircarrier phase so that one node alters the phase of its carriersignal on average per iteration (for instance, from 1

n to 1n�1 ).

As nodes are chosen according to a random process, it ispossible that more than one node simultaneously alters itsphase offset. In this case, the node’s conclusions on theimpact of their phase alteration on the feedback value arebiased. Therefore, in the fourth measurement, when themeasured value deviates significantly from the expectedfeedback, a node concludes that it was not the only one toalter its phase and reverses its decision.

In our measurements, the deviation of the calculatedfeedback curve did not exceed 0.6 percent when only onenode adapts its phase offset. With two nodes simulta-neously adapting their phase offset, we already experi-enced a deviation of approximately 1.5 percent. As thisprocedure is guided purely by the feedback broadcast bythe receiver, internode communication is not required.

Asymptotically, the synchronization time of this algo-rithm is �ðnÞ since on average the count of carrier signalsthat are in phase increases by 1 in each iteration. Further,performance improvements can be achieved when nodesutilize only three subsequent iterations and acquire the firstmeasurement from the last transmission of the precedingthree subsequent iterations.

The asymptotic synchronization time derived for thisapproach is optimal when we assume that individual nodeshave to compute their optimal carrier phase offset inde-pendently since n carrier signals have to be adapted. When,however, a synchronization scheme is utilized in whichinformation about the optimum relative carrier phaseoffsets of all nodes is provided, as, e.g., in typical open-loop carrier synchronization schemes (cf. [18]), the asymp-totic synchronization time can be further reduced.

This improved carrier synchronization scheme can beapplied in any scenario in which a rich feedback as, forinstance, the SNR can be provided. It is, however, notapplicable when only binary feedback is provided by thereceiver. When, for example, high noise and interferencewould force an impractically complex error correctionscheme, it might be beneficial to utilize the 1-bit feeback-based carrier synchronization instead.

5 SIMULATION STUDIES

We have implemented the scenario of distributed adaptivebeamforming in Matlab to obtain a better understanding ofthe impact of environmental parameters and algorithmicconfigurations. In particular, the effect of distinct prob-ability distributions as well as the count of transmitters andthe transmission distance are considered. In these simula-tions, 100 transmit nodes are placed uniformly at randomon a 30 m 30 m square area. The receiver is located 30 m(100 m; 200 m; 300 m) above the center of this area. Receiverand transmit nodes are stationary. Simulation parametersare summarized in Table 1.

6 IEEE TRANSACTIONS ON MOBILE COMPUTING, VOL. 10, NO. X, XXXXXXX 2011

Fig. 3. Deviation of the feedback curve calculated from threemeasurements to the feedback curve plotted from 100 measurements.

Frequency and phase stability are considered perfect. Wederived the median and standard deviation from 10 simula-tion runs. One iteration consists of the nodes transmitting,feedback computation, feedback transmission, and feed-back interpretation by transmitters. It is possible to performthese steps within few signal periods so that the timeconsumed for 6,000 iterations is in the order of millisecondsfor a base band signal frequency of 2.4 GHz. Signal qualityis measured by the RMSE of the received signal to anexpected optimum signal as detailed in (17). The optimumsignal is calculated as a perfectly aligned and properlyphase-shifted received sum signal from all transmitsources. For the optimum signal, noise is disregarded.

Fig. 4a depicts the optimum carrier signal, the initialreceived sum signal, and the synchronized carrier after6,000 iterations when carrier phases are altered withprobability 1

n in each iteration according to a uniformdistribution. In Fig. 4b, the phase offset of received signalcomponents for an exemplary simulation run with the sameparameters is illustrated. We observe that after 6,000 itera-tions about 98 percent of all carrier signals converge to arelative phase offset of about þ=� 0:1�. The median of allvariances of the phase offsets for simulations with thisconfiguration is 0.2301 after 6,000 iterations. The actualsynchronization time is dependent on the time to complete

a single iteration. In each iteration, a synchronization signal

is transmitted, the received sum signal is analyzed, and

feedback is calculated, broadcast to the network, and

interpreted by transmit nodes. While the processing speed

might be improved with improved hardware, the round

trip time of the signal poses a definite lower bound for the

time a single iteration lasts. At a distance of 30 meters, for

instance, we cannot hope to complete a single iteration in

less than 0:2 s.

5.1 Uniform versus Normal Distribution

Distributed adaptive beamforming in wireless sensor net-

works has been studied in the literature according to

various random phase alteration processes. The authors in

[15], [16], and [17] report good results when the probability

p� to alter the phase of a single carrier signal in one

iteration is 1 for all nodes and the phase offset is chosen

according to a normal distribution. The variance �2� applied

is not reported. In [13] and [14], p� was set to 1n for each one

of the n nodes while the phase is altered according to a

uniform distribution.For both, uniform and normal distributed processes, we

consider several values for p� and �2� . Generally, we

achieved good performance when modifications in one

iteration were small. For the uniform distribution, this

translates to p� ¼ 1n . For the normal distribution, good

results are achieved when �2� and p� are balanced so that the

modification to the overall sum signal is small. With

increasing p� , good results are achieved with decreasing

�2� . Fig. 5 depicts the results for p� ¼ 1

n and �2� ¼ 0:5�.

The figure shows the median RMSE value achieved in

10 simulations by normal and uniform distributed pro-

cesses over the course of 6,000 iterations. For ease of

presentation, error bars are omitted in this figure. However,

the standard deviation is low for both processes (the

standard deviation of this normal distributed process is

depicted in Fig. 9b).The normal distributed process has a slightly improved

synchronization performance. The optimum feedback value

reached is, however, identical.

SIGG ET AL.: FEEDBACK-BASED CLOSED-LOOP CARRIER SYNCHRONIZATION: A SHARP ASYMPTOTIC BOUND, AN ASYMPTOTICALLY... 7

TABLE 1Configuration of the Simulations

Prx is the received signal power, d is the distance between thetransmitter and the receiver, and � is the wavelength of the signal.

Fig. 4. Simulation results for a simulation with 100 transmit over 6,000 iterations of the random optimization approach to distributed adaptivebeamforming in wireless sensor networks.

5.2 Impact of the Network Size

When the count of nodes that participate in the synchro-nization is altered, this also impacts the performance of thisprocess (cf. Section 2). We conducted several simulationswith network sizes ranging from 20 to 100 nodes. Fig. 6depicts the performance of several synchronization pro-cesses with varying network sizes.

In these simulations, we set p� ¼ 0:05 and utilized auniformly distributed phase alteration process. We see thatthe maximum feedback value achieved is lower for smallernetwork sizes. This is due to the RMSE measure thatcompares the achieved sum signal to an expected optimumsuperimposed signal. As the count of participating nodesdiminishes, also the amplitude of the optimum signaldecreases. As expected, the optimum value is reachedearlier for smaller network sizes.

5.3 Impact of the Transmission Distance

We are also interested in the performance of distributedadaptive beamforming when the distance between thenetwork and a receiver is increased. For a uniformlydistributed phase alteration process with p� ¼ 1

n , weincrease the transmission distance successively. Fig. 7depicts the phase coherency achieved and the receivedsum signal for various transmission distances.

Although the noise power relative to the sum signalincreases, synchronization is possible at about 200 metersdistance. Observe that in our model with Ptx ¼ 1 mW, weexpect a signal strength at the receiver of 0:1 W or�40 dBm for each single carrier at this distance.

When the distance is further increased to 300 meters,however, synchronization is not possible with this config-uration, due to the high impact of the noise fluctuation onthe received signal. This has a higher impact on the signalthan the alteration of single carrier signals.

However, when more carrier signals are altered simulta-neously, a weak synchronization is still possible. Fig. 8depicts the received carrier signal after 100 iterations for theuniformly distributed process with p� ¼ 0:2 and p� ¼ 0:6.We see that the synchronization quality is improved withincreasing p� . While the superimposed signal is indistin-guishable for p� ¼ 0:2, the synchronization quality increaseswith p� ¼ 0:6. Although the signal is heavily distorted, thecarrier can be extracted.

5.4 Utilization of Additional Feedback Information

We also conducted simulations in which our implementa-tion of the asymptotically optimal algorithm described inSection 4 is compared to the classical process with normaldistributed phase alterations. When optimum phase offsetsare calculated by solving multivariable equations at thetransmit nodes, the synchronization performance can begreatly improved as detailed in Section 4. Fig. 9 depicts theperformance improvement achieved by solving multivari-able equations to determine the feedback function com-pared to a global random search approach. We observe thatthe global random search heuristic is outperformed alreadyafter about 1,000 iterations and the feedback value reachedis greatly improved.

The phase offset of distinct nodes is within þ=� 0:05�for up to 99 percent of all nodes.

6 NEAR REALISTIC INSTRUMENTATION

We have utilized USRP software radios (http://www.ettus.com) to model a sensor network capable of distrib-uted adaptive transmit beamforming. The software radiosare controlled via the GNU radio framework (http://gnuradio.org). The transmitter and receiver modulesimplement the feedback-based distributed adaptive beam-forming.1 For the superimposed transmit channel and thefeedback channel, we utilized widely separated frequenciesso that the feedback could not impact the synchronizationperformance. We conducted experiments with severaltransmit frequencies of nodes. In these experiments, werepeatedly synchronized the carrier phases of the threetransmit devices with the help of the 1-bit feedback-basedalgorithm described in [16] and [18] with uniform ornormal probability distribution on the phase modulation.Table 2 summarizes the configuration and results of twoexperiments with low and high transmit frequencies of27 MHz and 2.4 GHz, respectively. After 10 experiments atan RF transmit frequency of 27 MHz, we achieved amedian gain in the received signal strength of 3.72 dB for

8 IEEE TRANSACTIONS ON MOBILE COMPUTING, VOL. 10, NO. X, XXXXXXX 2011

Fig. 5. Performance of normal and uniform distributions for a network

size of 100 nodes and p� ¼ 0:01; �2� ¼ 0:5�.

Fig. 6. The synchronization performance for various network sizes in auniformly distributed process with p� ¼ 0:05.

1. The software for our feedback-based closed-loop implementation isconstantly further improved and extended in student projects. It is currentlynot recommended for productive environments. If you are interested toreceive a copy of the code in order to participate in the development andtesting of the implementation please contact sigg@ibr.cs.tu-bs.de.

three independent transmit nodes after 200 iterations. In14 experiments with four independent nodes that transmitat 2.4 GHz, the achieved median gain of the received RFsum signal was 2.19 dB after 500 iterations.

For the transmitters, we utilized the clock of the firstdevice for all transmit nodes. The receive node utilizes itsown clock and is, therefore, not synchronized to any of thetransmit nodes. Apart from this clock synchronization, noother communication or synchronization between trans-mitters was applied. In future implementations, it ispossible to utilize GPS for the clock synchronization.

In the third experiment, we altered the transmissiondistance and the phase alteration variance for a normaldis-tributed random process. Fig. 10 depicts our experi-mental setting. Table 3 summarizes our experimentalconfiguration.

Carrier phases have been adapted for each transmitdevice independently following a normal distributed ran-dom process. We modified the probability to alter the phaseoffset of one device and the variance for its normaldistributed random process as well as the distance betweentransmit and receive devices.

SIGG ET AL.: FEEDBACK-BASED CLOSED-LOOP CARRIER SYNCHRONIZATION: A SHARP ASYMPTOTIC BOUND, AN ASYMPTOTICALLY... 9

Fig. 7. RF signal strength and relative phase shift of received signal components for a network size of 100 nodes after 10,000 iterations. Nodes aredistributed uniformly at random on a 30 m 30 m square area and transmit at PTX ¼ 1 mW with p� ¼ 1

n .

Some results derived are depicted in Fig. 11. In thefigure, the mean gain of the received sum signal over all12 experiments to the initially received sum signal isdepicted. As expected, we observe that the synchronizationprocess differs for different parameter settings. Again, bestresults are achieved when small changes are applied in eachiteration. Therefore, the experiments in which the phasealteration probability and the variance are small achievesuperior results.

7 CONCLUSION

We have considered randomized search approaches tosolve the problem of distributed adaptive transmit beam-forming. In an analytic consideration, an asymptoticallytight bound on the expected optimization time of �ðn � k �logðnÞÞ was derived.

Additionally, a protocol to further reduce the optimiza-tion time and energy consumption of distributed adaptivebeamforming was introduced. In this protocol, the problemwas divided into subproblems that were solved iteratively.Since the decrease in the synchronization time is greater

10 IEEE TRANSACTIONS ON MOBILE COMPUTING, VOL. 10, NO. X, XXXXXXX 2011

Fig. 8. RF signal strength and relative phase shift of received signal components for a network size of 100 nodes after 10,000 iterations. Nodes are

distributed uniformly at random on a 30 m 30 m square area and transmit at PTX ¼ 1 mW.

Fig. 9. Distributed adaptive beamforming with a network size of 100 nodes where phase alterations are drawn uniformly at random. Each nodeadapts its carrier phase offset with probability 0.01 in one iteration. In this case, multivariable equation are solved to determine the optimum phaseoffset of the carrier signal.

TABLE 2Experimental Results of Software Radio Instrumentations

than the increase in transmission power in smaller clusters,this approach can improve the optimization time andreduce energy consumption.

Furthermore, an asymptotically optimal algorithm wasderived. For this approach, we considered the possibility toestimate the unknown feedback function by an individualnode so that an optimization approach is possible thatscales linearly with the network size n. This approach isasymptotically optimal since each carrier signal has to beconsidered at least once individually in order to find itsoptimum phase offset.

In mathematical simulations, we demonstrated the

effect of several configurations for distributed adaptive

transmit beamforming with uniform and normal distrib-uted phase alteration methods. Generally, a low mutation

probability translates to a better performance in the phase

synchronization process. An adaptive probability over the

course of the optimization might further improve theoptimization speed. While a moderate mutation probabil-

ity is beneficial at the beginning of the simulation, a

smaller mutation probability shows an improved optimi-

zation speed later in the process. Also, our implementation

of the asymptotically optimal method greatly outperforms

the global random search approach in the synchronization

achieved and the optimization speed.Finally, in an instrumentation with USRP software

radios, we demonstrated the feasibility of distributed

adaptive transmit beamforming in a concrete implementa-

tion with up to four transmitters.

ACKNOWLEDGMENTS

The authors would like to acknowledge partial funding by

the European Commission for the ICT project “Cooperative

Hybrid Objects Sensor Networks” (CHOSeN) (Project

number 224327, FP7-ICT-2007-2) within the seventh Frame-

work Programme. They would further like to acknowledge

partial funding by the “Deutsche Forschungsgemeinschaft”

(DFG) for the project “Emergent radio” as part of the

priority program 1183 “Organic Computing.”

SIGG ET AL.: FEEDBACK-BASED CLOSED-LOOP CARRIER SYNCHRONIZATION: A SHARP ASYMPTOTIC BOUND, AN ASYMPTOTICALLY... 11

Fig. 10. Experimental instrumentation of distributed adaptive beamform-ing among three transmit USRP devices and one receive USRP device.

TABLE 3Configuration of the Experiment

Fig. 11. Mean gain in the signal strength of three collaboratively transmitting devices.

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12 IEEE TRANSACTIONS ON MOBILE COMPUTING, VOL. 10, NO. X, XXXXXXX 2011

Stephan Sigg received the diploma in computersciences from the University of Dortmund,Germany, in 2004 and the PhD degree incommunication technology at the University ofKassel, Germany, in 2008. Currently, he isworking in the Group for Pervasive ComputingSystems (TecO) at the Karlsruhe Institute ofTechnology (KIT), Germany. His research inter-ests include the analysis, development, andoptimization of algorithms for pervasive comput-

ing systems. He is a member of the IEEE.

Rayan Merched El Masri received the BScdegree in computer science from the Tech-nische Universitat Braunschweig, Germany, in2009. He is a student at the Karlsruhe Institutefor Technology (KIT), Germany. Currently, he isworking in the Group for Pervasive ComputingSystems (TecO) at the Karlsruhe Institute ofTechnology.

Michael Beigl received the MSc and PhD(Dr.-Ing.) degrees from the University of Karls-ruhe in 1995 and 2000, respectively. He is aprofessor of pervasive computing systems at theKarlsruhe Institute of Technology. Previously, hewas a professor of ubiquitous and distributedsystems at the Carl-Friedrich Gauss Faculty,Technische Universitat Carolo-Wilhelmina zuBraunschweig (2005-2009), the research direc-tor of TecO, University of Karlsruhe (2000-2005),

and a visiting associate professor at Keio University (2005). His researchinterests include wireless sensor networks and systems, ubiquitouscomputing, and context awareness. He is a member of the IEEE.

. For more information on this or any other computing topic,please visit our Digital Library at www.computer.org/publications/dlib.

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