Deterministic Bisection Search Algorithm forDistributed Sensor/Relay Networks

Juwendo Denis, Chia-Shiang Tseng, Cheng-Wei Lee, Chia-Yu Tsai, and Che Lin†Institute of Communication Engineering & Department of Electrical Engineering

National Tsing Hua University, Hsinchu, Taiwan (R.O.C)

Abstract—This paper investigates the problem of adaptive dis-tributed beamforming for wireless sensor/relay networks. Ex-isting schemes seek to adaptively achieve distributed phasealignment for beamforming via the help of an error-free, one-bitfeedback link from the receiver to all distributed transmitters.Under such setting, we proposed a Deterministic BisectionSearch Algorithm (DBSA), designed to achieve distributed phasealignment efficiently in a greedy fashion. Extensive simulationsdemonstrated significant improvement in the convergence time ofthe proposed DBSA compared with existing schemes. For wirelesssensor/relay networks, inherent issues of system uncertainty suchas random node addition or removal often occur due to powerdepletion or requests from new nodes to join the network. Toresolve these issues, DBSA was further extended and numericalexperiments have shown that the extended version is indeedrobust against such system uncertainty.

I. INTRODUCTION

There has been a huge demand for improving the reliabilityand energy efficiency of wireless communication systems per-taining to distributed sensor/relay networks. While distributedtransmitters can potentially provide diversity and power gain,several challenges inherent to the system have emerged. Oneof the most important challenges is that distributed trans-mitters are constrained to a limited transmit power sourcethat cannot be recharged consistently. To overcome suchproblem, practical solutions have been proposed for distributedtransmitters to collaboratively send a common informationvia distributed beamforming that aims to exploit the potentialpower gain. Although the use of conventional beamformingrequires perfect knowledge of channel state information (CSI),that may be too expensive to acquire, a low rate feedback linkfrom the receiver can be used to make partial CSI availableto the transmitter end. Consequently, distributed transmitterscan generate beamforming coefficients and make proper phaseadjustment accordingly to achieve phase alignment at thereceiver end.

The process of distributed phase alignment can be achievedby means of deterministic phase adjustment or random phaseperturbation algorithms. The latter has been greatly investi-gated in the literature over the last decade. In [1], a randomizedapproach was proposed in which the receiver uses a feed-back link that allows each transmitter to make proper phase

†Che Lin is the corresponding author. This work is supported by NationalScience Council, Taiwan (R.O.C.), under Grant NSC99-2221-E-007-089-MY3. E-mail: s100064519@m100.nthu.edu.tw, s9964518@m99.nthu.edu.tw,{u9761123,u9761223}@oz.nthu.edu.tw, clin@ee.nthu.edu.tw.

adjustment. In [2], Mudumbai et al investigated an adaptivedistributed beamforming paradigm which requires only a sin-gle bit feedback. The scalability and the rate of convergenceanalysis of that scheme were given in [3]. In [4], the problemof adaptive distributed beamforming was reformulated as alocal random search algorithm and analyzed. An extensiveconvergence analysis of these scheme was presented in [5].A new scheme inspired from a completely different fieldwas proposed in [6], where a bio-inspired robust adaptiverandom search algorithm was presented, and has been provento converge in probability.

Several deterministic algorithms for distributed beamform-ing have also been proposed in the literature. Thibault et alintroduced a deterministic algorithm with individual powerconstraint in [7]. A new algorithm using additive deterministicperturbations and extending to the case of amplify-and-forwardwireless relay networks was presented in [8]. In [9], Fertl etal investigated a multiplicative deterministic perturbations fordistributed beamforming under a total power constraint.

In this paper, we propose a novel algorithm that belongs tothe category of deterministic phase adjustment algorithm: theDeterministic Bisection Search Algorithm (DBSA), which isan iterative method intending to halve the deterministic ad-justment at each iteration whenever an improvement has beennotified in the receiver signal strength (RSS). The proposedalgorithm is much more efficient than existing deterministicalgorithms while sacrificing not much of the complexity.Furthermore, random node addition and removal, which areinherent issues for distributed transmitters in wireless sen-sor/relay networks, are investigated. In fact, at anytime, activenodes may crash due to power depletion or hardware failureand new nodes may also request to join the network. Furtherinvestigation shows that the proposed DBSA can be extendedto handle such random node addition and removal. Numericalresults show promising performance of the extended DBSAcompared to other existing algorithms.

This paper is structured as follows: in section II, wepresent the system model and depict the problem of adaptivedistributed beamforming in wireless networks. The descriptionof our proposed scheme, DBSA, is presented in section III.Section IV introduces how DBSA can be extended to handlerandom node addition and removal in the system. In sectionV, simulations highlight some important features of DBSA.Finally, we conclude the work in section VI, and suggestrelevant directions for future research.

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II. SYSTEM MODEL AND PROBLEM FORMULATION

In this section, we introduce the system model and for-mulate the problem of adaptive distributed beamforming withlimited feedback. We consider a wireless sensor/relay networkconsisting of Ns distributed transmitters. All distributed trans-mitters seek to collaboratively transmit a common messageto a remote receiver via a frequency flat, slow fading channelbetween each transmitter and the receiver. Each transmitter andthe receiver is equipped with a single antenna. The messages ∈ C is subject to an average power constraint E

[|s|2]≤ P .

Furthermore, there exists a one-bit, error-free feedback linkfrom the receiver to all distributed transmitters.

Given the system model, the discrete-time, complex base-band model can be expressed as follow:

y[n] =

Ns∑i=1

higi[n]s+ w[n] =

Ns∑i=1

aibi[n]ej(φi+ψi[n])s+ w[n]

(1)where y[n] denotes the received signal, hi = aie

jφi corre-spond to the channel coefficients, gi[n] = bi[n]eψi[n] corre-spond to the beamforming coefficients, and w[n] is the additivewhite Gaussian noise.

To focus on the problem of distributed phase alignment,we let s =

√P , and a fixed power constraint bi = 1 is

imposed to all transmitters. Let θi[n] = φi + ψi[n] denotethe total aggregate phase of the received signal from the i-th distributed transmitter during the n-th transmission timeslot. The received signal is then given by

√P∑Ns

i=1 aiejθi[n].

Assuming that the RSS of the received composite signal canbe perfectly estimated at the destination, we have:

Mag(θ1[n], · · · , θNs[n]) =

√P

∣∣∣∣∣Ns∑i=1

aiejθi[n]

∣∣∣∣∣ (2)

The primary goal of distributed beamforming schemes isto maximize this magnitude function given in equation (2)in order to exploit the potential power gain. Furthermore, theglobal maximum of the magnitude function is reached whenall phases are completely aligned, i.e, θ1[n] = θ2[n] = · · · =θNs

[n]. Consequently, any relevant algorithm should seek toadaptively adjust the phase of all distributed transmitters inorder to achieve phase alignment with the help of a one-bit,error free feedback link.

III. DETERMINISTIC BISECTION SEARCH ALGORITHM

In the following section, we introduce in details the pro-posed DBSA algorithm, which is inspired by the idea of theBisection Method. To simplify notation, let us assume thatthe phase of each distributed transmitter remains confinedwithin an angle varying betweens [0, 2π]. By viewing thephase alignment problem as a maximization problem within anNs dimensional hypercube with edge length equal to 2π, it isthen suggested that the searching process for global maximumbe done in a bisection fashion.

The phase alignment procedure is done in a greedy fashionwithin the proposed DBSA. In fact, only one distributed

TABLE IALGORITHM I - DETERMINISTIC BISECTION SEARCH ALGORITHM

Initialize parametersrepeatk = 0for i← 1 to Ns do

for j ← 1 to b(360/α)− 1c doθ′i = θi[0] + α · jif Mag(θ′[0]) > Mag(θ[0]) then

θ[0]← θ′[0]end if

end forend forfor i← 1 to Ns dok = k + 1θi[k]← θi[k − 1] + αif Mag(θ[k]) < Mag(θ[k − 1]) thenθi[k]← θi[k − 1]− αif Mag(θ[k]) > Mag(θ[k − 1]) then

θ[k]← θ[k − 1]end if

end ifend forα← α

2until Stopping criteria reached

transmitter is permitted to adjust its phase at any time whileall other transmitters keep their phase unchanged. With thepseudocode given in Table I, we elaborate the algorithm byproviding a detailed step-by-step description as follows:

Step 1 - Initialization: The system is initializedby randomly generating the initial phase θ[0] ={θ1[0], · · · , θNs

[0]} ∈ RNs . The phase adjustment parameterα, known to all transmitters, should be configured to anumber αinit that divides 2π. Distributed transmitters jointlybeamform a fixed symbol s =

√P using the generated phase

configuration. The receiver then records the value of the RSSgiven by (2) corresponding to the initial phase θ[0].

Step 2 - Initial Phase Rotation: The algorithm undergoesNs rounds of phase evaluations corresponding to the Nsdistributed transmitters.

During the i-th round, only the i-th transmitter transmitsfor K − 1 iterations. The i-th transmitter alters its phase byiterating through the elements in the set Θi given by:

Θi = {θi[0] + α, · · · , θi[0] + (K − 1)α} ∀i ≤ Ns

where K is an integer that equals to 2π/α. At each iteration,the receiver compares the RSS with the largest RSS recordedso far, and feedback a single bit that indicates whether thecurrent selected phase configuration by the i-th transmitterexceeds previous recorded RSS. Should the RSS exceed thelargest recorded value, the feedback bit is set to 1, and to 0otherwise.

By the time all phase configurations in Θi are tested,the i-th transmitter should be able to distinguish the phaseconfiguration that correspond to the best RSS performance,and this particular phase configuration should replace its initialphase configuration generated in Step 1, i.e., the i-th element

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of θ[0]. Once all transmitters are done with the evaluationprocedures, the parameter α is halved, and the algorithm entersStep 3.

Step 3 - Forward Adjustment: Before proceeding to Step5, the algorithm undergoes Ns iterations where each iterationinvolves procedures Step 3 and Step 4. At the n-th iteration,the n-th distributed transmitter adjusts its phase by addingup the parameter α to its phase, while all other distributedtransmitters keep their phases unchanged. From the systempoint of view, the adjusted phase vector can be expressed as :

θ′[n] =[θ1[n− 1], · · · , θn−1[n− 1], θn[n− 1] + α,

θn+1[n− 1], · · · , θNs[n− 1]

]T (3)

The receiver compares the RSS corresponding to the ad-justed phase vector θ′[n] with the highest recorded RSS upto the current iteration, and broadcasts a one-bit feedbacksignal BF back to all transmitters. If improvement on RSSis observed, the adjusted phase is kept by the n-th transmitterand the algorithm proceeds to the next iteration that adjuststhe (n+ 1)-th transmitter provided that n < Ns. Otherwise, ifno RSS improvement is observed, the algorithm enters Step4, or if n = Ns, the algorithm enters Step 5.

More precisely, the feedback signal BF is generated as

BF = 1{Mag(θ′[n]) > max

i=1,··· ,n−1Mag(θ[i])

} (4)

Step 4 - Reverse Adjustment: At this stage of thealgorithm, the n-th distributed transmitter adjusts its phaseby adding the inverse of the adjustment parameter to itsphase, where the inversely adjusted phase vector θ′′[n] canbe expressed as

θ′′[n] =[θ1[n− 1], · · · , θn−1[n− 1], θn[n− 1]− α,

θn+1[n− 1], · · · , θNs[n− 1]

]T (5)

As in the previous step, the receiver compares the RSScorresponding to θ′′[n] with the largest recorded RSS andbroadcasts a one-bit feedback signal BR back to transmitters.Upon receiving “1” from the receiver, the n-th transmitterupdates its phase with the inversely adjusted value, whileno action is taken for the cases of feedback being “0”. Thefeedback signal BR is generated as in (4) with θ′[n] replacedwith θ′′[n], and the procedures throughout Step 3 and Step 4can be characterized by the following expression:

θ[n] =

θ′[n], BF = 1

θ′′[n], BF = 0, BR = 1

θ[n− 1], o.w.

(6)

Once completed with the updating process, the algorithmreturns to Step 3 and proceed to the (n+1)-th iteration wherethe phase of the (n+ 1)-th transmitter can be adjusted, or, ifn = Ns, the algorithm enters Step 5.

Step 5 - Parameter Adjustment: Upon entering Step 5,all transmitters scale down the phase adjustment parameter α

TABLE IIALGORITHM II - DBSA FOR RANDOM NODE ADDITION AND REMOVAL

Initialize parametersrepeat

Execute algorithm Iif A node is added to the network at kth iteration then

Evaluate Mag(θ[k]) and postpone node adjustmentswhile αadd > α doθadd[k]← θadd[0] + αaddif Mag(θ[k]) < y thenθ[k]← θadd[k]− αadd

end ifif Mag(θ[k]) > y thenθ[k]← θadd[k] + αadd

end ifαadd ← αadd

2end while

end ifif A node is removed from the network then

Evaluate Mag(θ[k]) and advance node adjustmentsend if

until Stopping criteria reached

to half of its current value, and return to Step 3. For notationalsimplicity, the iteration index is reset to 1 and θ[0] is replacedwith θ[Ns].

Stopping criterion: Given a threshold value for the RSS,if the RSS is greater than or equal to this target value at anystep of the algorithm, the phase alignment process is said to becompleted. The threshold value may be based on any statisticalinformation and has to be available and known to the receiver.

IV. DETERMINISTIC BISECTION SEARCH ALGORITHM INTHE PRESENCE OF RANDOM NODE ADDITION AND

REMOVAL

Due to to hardware failure and power depletion of dis-tributed transmitters, the topology of wireless sensor/relaynetworks can be dynamical in nature. On the other hand,the designer should bear in mind that the network should beflexible enough to accept newly added nodes. Hence, in thissection, we investigate possible modifications or improvementson the proposed DBSA scheme in the presence of random nodeaddition and removal.

We consider the situation where complete information isavailable, that is, the exact time instances of random addi-tion/removal are known to the receiver. Based on this assump-tion, we propose the extended DBSA for random node additionand removal. The extended scheme performs exactly the sameway as algorithm I and invokes procedures stated in algorithmII whenever a node is added or removed from the network.The corresponding pseudocode for the case of single nodeaddition/removal can be found in Table II, and we elaborate theextended algorithm in the detailed descriptions that follows.• For the case of node addition, the following steps are

invoked immediately:Status update: At the instance a new node joins thenetwork, the added node is initialized with a randomly

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generated phase, and the phase adjustment parameter αfor the added node is initialized to αinit. The originalalgorithm then halts and all distributed transmitters, in-cluding the newly added transmitter, jointly transmit withthe known best phase configuration. Note that the newlyadded node transmit using the randomly generated initialphases. This procedure is necessary for the receiver todetect the RSS corresponding to the best known phaseconfiguration after the node addition. Moreover, due tothe increase in the node count, all scheduled phaseadjustments, regarding Step 3 and Step 4 in the originalalgorithm that are yet need to be executed, are postponedafter the following steps are executed. Meanwhile, thestopping criteria should be recalculated based on thenewly estimated maximum achievable RSS as well.Parameter synchonization: While the original algo-rithm remains halted, the newly added transmitter willproceed to a series of phase adjustments in order to “catchup” with the existing nodes of the network. The newlyadded node then proceeds by executing Step 2 to Step5 stated in the original DBSA until the phase adjustmentparameter α has reached the same value as the remainingdistributed transmitters. After the adjustment parameterfor the added node is fully synchronized with the state ofthe network before node addition, the original algorithmcontinues along with the added node.

• For the case of node removal, the situation is relativelysimple and we proceed with the following step:Status update: As in the case of node addition, thealgorithm halts and all distributed transmitters jointlytransmit with the known best phase configuration. Dueto the decrease in the node count, all scheduled phaseadjustments are advanced by the corresponding amountof time slots of nodes removed from the network. Thestopping criteria should be recalculated based on thenewly estimated maximum achievable RSS as well.

V. SIMULATION RESULT

In this section, we investigate the performance of theproposed DBSA by presenting the numerical results and byproviding comparisons with some other existing schemes.In the following simulation results, we set the transmittedsymbol power to be

√P = 1, and unless stated otherwise

the initial phase adjustment parameter αinit is fixed to be πand the number of distributed transmitters Ns = 500 . In oursimulations, all channel realizations are assumed to be zero-mean, unit variance i.i.d Rayleigh flat fading.

As mentioned above, we set the initial value of the phaseadjustment parameter to be π for DBSA, and Figure 1 il-lustrates that this choice is in fact optimal. Indeed, Figure 1depicts the trajectories of the RSS for different initializationof the adjustment parameter αinit. Among the following valuesof αinit = {π/4, π/2, 3π/4, π}, the simulation reveals that theoptimal choice of αinit is indeed π.

In Figure 2, we compare the proposed DBSA with threeother schemes of distributed beamforming. We carefully

0 1000 2000 3000 4000 5000 60000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Time Slot

RS

S

α = π/4

α = π/2

α = 3π/4

α = π

Fig. 1. Performance comparison of different phase adjustment parameter α.

0 500 1000 1500 2000 2500 3000 35000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Time Slot

RS

S

D1BF 90 [4]

BioRARSA [9]

DBSA

ONEBIT [2]

Fig. 2. Performance comparison of DBSA with existing schemes.

choose the optimal parameters for all schemes as stated in[7], [6] and [2], respectively. After careful calibration, Figure2 shows that the proposed scheme performs significantly betterthan the other schemes. It is worth noting that the proposedDBSA essentially does not sacrifice much of the complexity.The major complexity here is the evaluation of the RSS and isdirectly related to the number of transmission required beforethe algorithm converges. DBSA certainly requires minor extramemory storage due to additional steps and manipulations.However, such increase in the computational complexity be-comes negligible when compared with the potential saving ofDBSA in terms of the number of transmissions required.

The hitting time, i.e convergence time performance of theproposed DBSA assuming different number of distributedtransmitters is depicted in Figure 3. In Figure 3, each linecorresponds to different stopping criteria c. In fact, we considerthe case where the algorithm will stop whenever Mag(θ[n]) ≥c ·Mag(θ∗), where c is a constant 0 ≤ c ≤ 1 and Mag(θ∗) is

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50 100 150 200 250 300 350 400 450 5000

500

1000

1500

2000

2500

3000

3500

4000

4500

Number of Transmitters

Hitting T

ime

60%

75%

90%

Fig. 3. Scalability of DBSA with respect to network sizes.

0 1000 2000 30000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Time slot

RS

S

Elimination probability: 0.03

0 1000 2000 30000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Time slot

RS

S

Elimination probability: 0.05

DBSA

Extended DBSA

D1BF

DBSA

Extended DBSA

D1BF

Fig. 4. Performance comparison with existing schemes under the presenceof random node removal.

the maximum achievable RSS. The simulation result demon-strates that the convergence time of the proposed DBSA scalesapproximately linearly with the number of transmitters.

Figure 4 depicts the impact of random node elimination andaddition in distributed sensor/relay networks. In the simulationsetting presented in Figure 4, there exist a non-zero probabilitythat a single node may be removed from the network ateach transmission. Specifically, we evaluate the performanceof the network with node elimination probability equalingto 0.03 and 0.05 for D1BF 90 [7], the proposed DBSAand the extended DBSA. Simulations demonstrate significantperformance differences among different elimination probabil-ities corresponding to different schemes. In other words, theinfluence of having node removed from the networks can besignificant if DBSA is not extended to handle this problem.Once extended, it can be observed that the performance curvesof the extended version with elimination probability being 0.03

and 0.05, almost overlap. This demonstrates that the extendedDBSA is robust to system uncertainty.

For the case where new nodes may randomly request tojoin the network, existing schemes fail to address this issue,hence the newly added node will not be able to cooperate withthe existing transmitters. On the contrary, the extended DBSAallows new node to synchronize with the existing distributedtransmitters, and eventually achieve the new global maximumRSS. Since the performance curve of extended DBSA with orwithout node addition are alike, we then omit the details here.

VI. CONCLUSION

In this work, we proposed DBSA for adaptive distributedbeamforming that is inspired by the Bisection Method. Exten-sive numerical experiments suggest that the proposed DBSAsignificantly outperforms existing schemes in terms of con-vergence time. In the case where random node additionor removal arise in the system due to power depletion orrequests from new nodes to join the network, DBSA wasfurther extended to resolve these issues. The extended DBSAwas shown to be robust against such system uncertainty vianumerical simulations. During our numerical experiments, wehave observed that DBSA and its extended version havealways converged for all our parameter settings. We are henceconvinced that it can be rigorously shown that DBSA and itsextended version converge in theory. Such theoretical analysisis hence a topic of interest for us in the future. Furthermore, indesigning the extended version to handle system uncertainty,we assume that the complete information of random nodeaddition and removal is available, which may be too optimisticin practice. We also hope to investigate further along thisdirection.

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