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Review ArticleA Survey of Sound Source Localization Methods inWireless Acoustic Sensor Networks

Maximo Cobos1 Fabio Antonacci2 Anastasios Alexandridis34

Athanasios Mouchtaris34 and Bowon Lee5

1Department of Computer Science Universitat de Valencia 46100 Burjassot Spain2Dipartimento di Elettronica Informazione e Bioingegneria Politecnico di Milano 20133 Milano Italy3Institute of Computer Science (ICS) Foundation for Research amp Technology-Hellas (FORTH) Heraklion 70013 Crete Greece4Department of Computer Science University of Crete Heraklion 70013 Crete Greece5Department of Electronic Engineering Inha University Incheon 22212 Republic of Korea

Correspondence should be addressed to Maximo Cobos maximocobosuves

Received 18 May 2017 Accepted 28 June 2017 Published 17 August 2017

Academic Editor Alvaro Marco

Copyright copy 2017 Maximo Cobos et alThis is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Wireless acoustic sensor networks (WASNs) are formed by a distributed group of acoustic-sensing devices featuring audio playingand recording capabilities Current mobile computing platforms offer great possibilities for the design of audio-related applicationsinvolving acoustic-sensing nodes In this context acoustic source localization is one of the application domains that have attractedthe most attention of the research community along the last decades In general terms the localization of acoustic sources canbe achieved by studying energy and temporal andor directional features from the incoming sound at different microphones andusing a suitable model that relates those features with the spatial location of the source (or sources) of interest This paper reviewscommon approaches for source localization in WASNs that are focused on different types of acoustic features namely the energyof the incoming signals their time of arrival (TOA) or time difference of arrival (TDOA) the direction of arrival (DOA) and thesteered response power (SRP) resulting from combining multiple microphone signals Additionally we discuss methods not onlyaimed at localizing acoustic sources but also designed to locate the nodes themselves in the network Finally we discuss currentchallenges and frontiers in this field

1 Introduction

With the rapid development in fields like circuit design andmanufacturing wireless nodes incorporating a variety ofsensors communication interfaces and compact micropro-cessors have become economical resources for the designof innovative monitoring systems Networks of such typeof devices referred to as wireless sensor networks (WSNs)[1 2] have been widely spread and used in many fields withapplications ranging from surveillance and military deploy-ments to industrial and health-care systems [3] When thenodes incorporate acoustic transducers and the processinginvolves the manipulation of audio signals the resultingnetwork is usually referred to as a wireless acoustic sensornetwork (WASN) A WASN consists of a set of sensor nodes

interconnected via a wireless medium [4] Each node has oneor several sensors (microphones) a processing unit awirelesscommunication module and sometimes also one or severalactuators (loudspeakers) [5]

During the last decade the use of location informa-tion and its potentiality in the development of ambientintelligence applications has promoted the design of localpositioning systems with WSNs [6] Using WSNs to performlocalization tasks has always been a desirable property sincebesides being considerably cheap they are easily deploy-able Localization and ranging in WSNs have been typicallyaddressed by measuring the received signal strength (RSS)or time of arrival (TOA) of radio signals [7] However theRSS approach while being significantly inexpensive incurssignificant errors due to channel fading long distances

HindawiWireless Communications and Mobile ComputingVolume 2017 Article ID 3956282 24 pageshttpsdoiorg10115520173956282

2 Wireless Communications and Mobile Computing

and multipath In the context of acoustic signal processingWASNs also provide advantages with respect to traditional(wired) microphone arrays [8] For example they enableincreased spatial coverage by distributing microphone nodesover a larger volume a scalable structure and possibly bettersignal-to-noise ratio (SNR) properties In fact since theranging accuracy depends on both the signal propagationspeed and the precision of the TOA measurement acousticsignals may be preferred with respect to radio signals [9]

There are two typical localization tasks in WASNs thelocalization of one or more sound sources of interest and thelocalization of the nodes that make up the network The firstcase may involve locating the position of unknown soundsources for example talkers inside a room or unexpectedacoustic events or sometimes other devices emitting knownbeacon signals The second case is usually related to the self-calibration or automatic ranging of the nodes themselves

To estimate the locations of the sound sources that areactive in an acoustic environment monitored by a WASNdifferent methods exist in the literature Usually a centralizedscheme is adopted where a dedicated node known as thefusion center is responsible for performing the localizationtask based on information it receives from the rest of nodesThe sensor network itself poses many limitations and chal-lenges that must be considered when designing a localizationapproach in order to facilitate its use in practical scenariosSuch challenges include the bandwidth usage which limitsthe amount of information that can be transmitted in thenetwork and the limited processing power of the nodeswhich prohibits them from carrying out very complex andcomputationally expensive operations Moreover each nodehas its own clock for sampling the signals and since the nodesoperate individually the resulting audio in the network willnot be synchronized

A taxonomy of sound source localization methods canbe built up upon the nature of information from the sensorsthat it is utilized in order to estimate the locations Hencethe WASN can estimate the locations of the acoustic sourcesbased on (i) energy readings (ii) time-of-arrival (TOA)measurements (iii) time-difference-of-arrival (TDOA) mea-surements and (iv) direction-of-arrival estimates or (v) byutilizing the steered response power (SRP) function

In DOA-based approaches each node estimates the DOAof the sources it can detect and transmits the DOA estimatesto the fusion center Although such approaches requireincreased computational power and multiple microphonesin each node they can attain very low-bandwidth usage asonly the DOA estimates need to be transmitted Also sincethe DOA estimation is carried out in each node individuallythe audio signals at different nodes need not be synchronizedDOA-based approaches can tolerate unsynchronized input aslong as the sources move at a rather slow rate relative to theanalysis frameThe location estimators are generally based onestimating the location of a sound source as the intersectionof lines emanating from the nodes at the direction ofthe estimated DOA However for multiple sources severalchallenges arise the number of detected sources (and thusthe number of DOA estimates) in each time instant can varyacross the nodes due to missed detections (ie a source is not

detected by a node) or false-alarms (ie overestimation of thenumber of detected sources) and an association procedure isneeded to find the DOA combinations that correspond to thesame source This is known as the data-association problemand is crucial for the localization task

The TDOA is related to the difference in the time of flight(TOF) of the wavefront produced by the source at a pair ofmicrophones in the same node TDOAs can be estimatedat a moderate computational cost through the generalizedcross correlation (GCC) [10] of the signals acquired bymicrophones in the pair The source location estimate isaccomplished by combining TDOA measurements comingfrom multiple sensors Notice that as for the DOA only theTDOA measurements must be transmitted over the wirelessnetwork with clear advantages in terms of transmissionpower and required bandwidth Though suitable for WASNsin practical scenarios (reverberant environments presence ofnoise and interferers) TDOA measurements are prone toerrors which in their turn lead towrong localization In orderto mitigate the impact of these adverse phenomena severaltechniques have been presented with the aim of identifyingand removing outliers in the TDOA set [11ndash13]

A TDOA measurement bounds the source to lie ona branch of hyperbola whose vertices are in microphonepositions and whose aperture is determined by the TDOAvalue When two (three in 3D) measurements from differentpairs are available the source can be localized through inter-section of hyperbolasThe resulting cost function however isstrongly nonlinear and therefore its minimization is difficultand prone to errors Linearized cost functions have beenproposed to overcome this difficulty [14ndash16] It is importantto notice however that the linearized cost functions requirethe presence of a reference microphone with respect towhich the remaining microphones in all the sensors must besynchronized This poses technological constraints that insome cases hinder the use of such techniques More recentlymethodologies that include the synchronization offset in theoptimization of the cost function have been proposed toovercome this problem [12 17]

TOA measurements are obtained by detecting the timeinstant at which the source signal arrives at the microphonespresent in the network Since in passive source localizationthe source emission time is unknown the TOA is notequivalent to the TOF of the signal preventing a directmapping from TOAs to source-to-node distances Whilesome applications involve the use of sound-emitting nodesthat allows performing localization by using trilaterationtechniques TDOA localization methods are usually chosenIn this case although the source emission time does notneed to be known the registered TOAs need to be referencedto a common clock requiring precise timing hardware andsynchronization mechanisms

Energy-based localization relies on the averaged energyreadings computed over windows of signal samples acquiredby the microphones incorporated by the nodes [18]Compared to TDOA and DOA methods energy-basedapproaches are attractive because they do not require theuse of multiple microphones at the nodes and are freeof synchronization issues unlike those based on TOA

Wireless Communications and Mobile Computing 3

m1m1(1)

m2(1)

m3(1)

m1(2)

m2(2)

m3(2)

m1(3)

m2(3)

m3(w)

q1

q2

q3

xs

2

3y1

y2

y3

r1(1)

r3(1)

1(1)3

(1)

13(1)

Figure 1 WASN with119872 = 3 nodes and 119873 = 3microphones per node

However TDOA- and DOA-based methods considered assignal-based approaches offer generally better performancethan energy-based methods This is due to the fact that theinformation conveyed by all the samples of the signal isdirectly exploited instead of their average at the expenseof more sophisticated capturing devices and transmissionresources [18 19]

SRP approaches are beamforming-based techniques thatcompute the output power of a filter-and-sum beamformersteered to a set of candidate source locations defined by apredefined spatial grid Since the computation of the SRPinvolves the accumulation of GCCs from multiple micro-phone pairs the synchronization requirements are usuallythe same as those of TDOA-based methods The set of SRPsobtained at the different points of the grid make up theSRP power map where the point accumulating the highestvalue corresponds to the estimated source location Whenusing unsynchronized nodes with multiple microphones theSRP power maps computed at each node can be used toobtain their corresponding DOAs Alternatively the SRPpower maps from the different sensors can be accumulatedat a central node to obtain a combined SRP power mapidentifying the true source location by its maximum

Besides the localization of acoustic sources approachesfor localizing the nodes in the network are also of highinterest within a WASN context Based on the estimatedTOAs and TDOAs algorithms for self-localization of thesensor nodes usually assume known source positions playingknown probe signals In practical scenarios each sensor nodedoes not have any information regarding other nodes orsynchronization between the sensor and the source Theseassumptions allow all processing to take place on the nodeitself Several issues for example reverberation asynchronybetween the sound source and the sensor poor estimationof the speed of sound and noise need to be considered forrobust self-localization methods In addition the processingneeds to be computationally inexpensive in order to be runon the sensor node itself

Some state-of-the-art solutions for acoustic sensor local-ization detail the challenges facing these algorithms andmethods to tackle such problems in a unified context [2021] Furthermore recent methodologies have been proposedfor probe signal design aimed at improving TOF estimation[22] the joint localization of sensors and sources in an adhoc array by using low-rank approximation methods [23]and an iterative peak matching algorithm for fast nodeautocalibration [24]

The paper is structured as follows Section 2 discussessome general considerations regarding a general WASNstructure and the notation used throughout this paperSection 3 presents the fundamentals of energy-based sourcelocalization methods Section 4 discusses TOA-based local-ization approaches Methods for TDOA-based localizationare presented in Section 5 Section 6 discusses the use ofDOAmeasurements to perform localization of one or severalsound sources In Section 7 the fundamentals of the conven-tional and modified SRP methods are explained Section 8reviews some recent methods for the self-localization of thenodes in the network Some future directions in the field arediscussed in Section 9 Finally the conclusions of this revieware summarized in Section 10

2 General Considerations

In order to clarify the notation used throughout this paperand the type of location cues used to perform the localizationtask Figure 1 shows a general WASN with a set of wirelessnodes and an emitting sound source It is assumed that thenetwork consists of119872 nodes and that each node incorporates119873microphones In the example shown in Figure 1119872 = 3 and119873 = 3The nodes are assumed to be located at positions q119898 =[119902119909119898 119902119910119898 119902119911119898]119879 119898 = 1 119872 while the microphonelocations are denoted asm(119898)

119894 = [119909(119898)119894 119910(119898)119894 119911(119898)119894 ]119879 119894 = 1 119873 where the superscript (119898) identifies the node at whichthe microphone is located The source position is denoted


as x119904 = [119909119904 119910119904 119911119904]119879 while a general point in space is x =[119909 119910 119911]119879 Note that all these location vectors are referencedto the same absolute coordinate system The distance fromany microphone to the sound source is denoted as 119903(119898)119894 whilethe time it takes the sound wave to travel from the source toa microphone that is the time of flight (TOF) is denotedby 120578(119898)119894 The time instant at which the source signal arrivesto a given microphone that is the TOA is denoted as 120591(119898)119894 TDOAs are denoted by 120591(119898)119894119895 and correspond to the observedTOA differences between pairs of microphones (119894119895) It is acommon practice to identify different pairs of microphonesby using an index 119901 = 1 119875 where 119875 is the total numberof microphone pairs involved in the localization task TheDOA corresponds to the angle that identifies the directionrelative to the nodemicrophone array that points to the soundsource and is denoted by 120579119898 Finally the energy values ofthe source signal measured at the nodes are denoted as 119910119898These are negligible formicrophones located at the samenode(especially if the nodes are relatively far from the source) soit is usually assumed that only one energy reading is obtainedfor each node

3 Energy-Based Source Localization

Traditionally most localization methods for WASNs havebeen focused on sound energy measurements This is moti-vated by the fact that the acoustic power emitted by targetsusually varies slowly in time Thus the acoustic energy timeseries does not require as high a sampling rate as the rawsignal time series avoiding the need for high transmissionrates and accurate synchronization The energy-based modelwas first presented in [25] In this model the acoustic energydecreases approximately as the inverse of the square of thesource to sensor distance Without loss of generality it willbe assumed in this section that the node locations determinethe microphone positions and that nodes only incorporateone microphone (119872 = 119873) Note that as opposed to timedelaymeasurements the differences in energymeasurementsobtained from different microphones at the same node arenegligible

31 Energy-Decay Model Assuming that there is only onesource active the acoustic intensity signal received by the 119894thsensor at a time interval 119897 is modeled as [25]

119910119894 (119897) = 119892119894 119868 (119897 minus 120578119894)1199032119894 + 120598119894 (119897) = 119904119894 (119897) + 120598119894 (119897) (1)

where 119904119894(119897) is the source intensity at the sensor location 119892119894 isa sensor gain factor 119868(119897) denotes the intensity of the sourcesignal at a distance of one meter from the source 120578119894 is thepropagation delay from the source to the sensor 119903119894 is thedistance between the sensor and the source and 120598119894(119897) is anadditive noise component modeled as Gaussian noise In

practice for each time interval 119897 a set of 119879 samples is usedto obtain an energy reading 119910119894(119897) at the sensor

119910119894 (119897) asymp 1119879119897+1198792sum119897minus1198792

1199092119894 (119905) (2)

where 119909119894(119905) are the samples obtained from the microphoneof the 119894th node In the case when several microphones areavailable at each node the final energy reading is obtainedby averaging the energies computed from each of the micro-phones

By assuming that the maximum propagation delaybetween any pair of sensors is small compared to119879 and takinginto account the averaging effect 120578119894 can be neglected for theenergy-decay function so that

119910119894 (119897) asymp 119892119894 119868 (119897)1199032119894 + 120598119894 (119897) (3)

32 Energy Ratios Considering the energy measurementsobtained by a group of 119873 sensor nodes the energy ratio 120581119894119895between the 119894th and the 119895th sensors is defined as

120581119894119895 ≜ (119910119894119910119895119892119894119892119895)minus12 = 1003817100381710038171003817x119904 minusm119894

100381710038171003817100381710038171003817100381710038171003817x119904 minusm119895

10038171003817100381710038171003817 = 119903119894119903119895 (4)

where x119904 is the location of the source and m119894 and m119895 are thelocations of the two microphones For 0 lt 120581119894119895 = 1 all thepossible source coordinates x that satisfy (4) must reside on ahypersphere (sphere if x isin R3 or circle if x isin R2) describedby

10038171003817100381710038171003817x minus c119894119895100381710038171003817100381710038172 = 9848582119894119895 (5)

where the center c119894119895 and the radius 984858119894119895 of this hypersphere arec119894119895 ≜ m119894 minus 1205812119894119895m1198951 minus 1205812119894119895 984858119894119895 ≜ 120581119894119895 10038171003817100381710038171003817m119894 minusm119895

100381710038171003817100381710038171 minus 1205812119894119895 (6)

In the limiting case when 120581119894119895 rarr 1 the solution of (4)forms a hyperplane betweenm119894 andm119895

w119879119894119895x = 120595119894119895 (7)

where w119894119895 ≜ m119894 minusm119895 and 120595119894119895 ≜ (12)(m1198942 minus m1198952)By using the energy ratios registered at a pair of sensors

the potential target location can be restricted to a hyperspherewith center and radius that are functions of the energy ratioand the two sensor locations If more sensors are used morehyperspheres can be determined If all the sensors that receivethe signal from the same target are used the correspondingtarget location hyperspheres must intersect at a particularpoint that corresponds to the source location This is thebasic idea underlying energy-based source localization In the


100

200

300

400

500

600

minus5 0 15105minus10

x (m)

minus10

minus5

0

5

10

15

y(m

)

Figure 2 Example of energy-based localization with three nodesThe red circle indicates the true source location Note that due tomeasurement noise circles do not intersect at the expected sourcelocation The center of the contour plots indicates estimated targetlocation according to the nonlinear cost function of (9)

absence of noise it can be shown that for 119873 measurementsonly119873minus1 of the total119873(119873minus1)2 ratios are independent andall the corresponding hyperspheres intersect at a single pointfor four or more sensor readings For noisy measurementshowever more than119873minus 1 ratios may be used for robustnessand the unknown source location x119904 is estimated by solvinga nonlinear least squares problem as explained in the nextsubsection As an example Figure 2 shows a 2D setup withthree sensors and the circles resulting from noisy energyratios

33 Localization through Energy Ratios Given 119873 sensornodes providing 119875 = 119873(119873minus1)2 energy ratios the followingleast squares optimization problem can be formulated

119869(ER)NLS (x) = 1198751sum1199011=1

1003816100381610038161003816100381610038171003817100381710038171003817x minus c119901110038171003817100381710038171003817 minus 9848581199011 100381610038161003816100381610038162 + 1198752sum

1199012=1

10038161003816100381610038161003816w1198791199012x minus 1205951199012

100381610038161003816100381610038162 (8)

x(ER-NLS)119904 = argminx 119869(ER)NLS (x) (9)

where 1198751 is the number of hyperspheres and 1198752 is the numberof hyperplanes (120581119894119895 close to 1) with corresponding indices 1199011and1199012 indicating the associated sensor pairs (119894119895) (119875 = 1198751+1198752)Note that the above cost function is nonlinear with respect tox resulting in the energy-ratio nonlinear least squares (ER-NLS) problem It can be shown that minimizing this costfunction leads to an approximate solution for the maximumlikelihood (ML) estimate A set of strategies were proposedin [26] to minimize 119869(ER)NLS (x) by using the complete set ofmeasured energy ratios A popular approach to solve theproblem is the unconstrained least squares method Sinceevery pair of hyperspheres (with double indices 119894119895 replaced bya single pair index for the sake of brevity) xminus c1198982 = 9848582119898 and

xminusc1198992 = 9848582119899 a hyperplane can be determined by eliminatingthe common terms

(c119898 minus c119899)119879 x = (1003817100381710038171003817c11989810038171003817100381710038172 minus 1003817100381710038171003817c11989910038171003817100381710038172) minus (9848582119898 minus 9848582119899)2 (10)

The combination of (7) with (10) leads to a least squaresoptimization problemwithout inconvenient nonlinear termsknown as the energy-ratio least squares (ER-LS) methodwith cost function

119869(ER)LS (x) = 1198751sum1199011=1

10038161003816100381610038161003816u1198791199011x minus 1205771199011 100381610038161003816100381610038162 + 1198752sum1199012=1

10038161003816100381610038161003816w1198791199012x minus 1205951199012

100381610038161003816100381610038162 (11)

where

u119894119895 ≜ 2m1198941 minus 1205812119894 minus 2m1198951 minus 1205812119895 120577119894119895 ≜ 1003817100381710038171003817m119894

100381710038171003817100381721 minus 1205812119894 minus 10038171003817100381710038171003817m119895

1003817100381710038171003817100381721 minus 1205812119895 (12)

The closed-form solution of the above unconstrainedleast squares formulation makes this method computation-ally attractive however it does not reach the Cramer-Raobound In [27] energy-based localization is formulated asa constrained least squares problem and some well-knownleast squares methods for closed-form TDOA-based loca-tion estimation are applied namely linear intersection [28]spherical intersection [29] sphere interpolation (SI) [30] andsubspaceminimization [31] In [32] an algebraic closed-formsolution is presented that reaches the Cramer-Rao bound forGaussian measurement noise as the SNR tends to infiniteThe authors in [33] formulated the localization problem as aconvex feasibility problem and proposed a distributed versionof the projection onto convex sets method A discussion ofleast squares approaches is provided in [19] presenting alow-complexity weighted direct least squares formulation Arecent review of energy-based acoustic source localizationmethods can be found in [18]

4 TOA-Based Localization

Typically a WASN sound source location setup assumesthat there is a sound-emitting source and a collection offixed microphone anchor nodes placed at known positionsWhen the sound source emits a given signal the differentmicrophone nodes will estimate the time of arrival (TOA) ortime of flight (TOF) of the sound These two terms may notbe equivalent under some situations The TOF measures thetime that it takes for the emitted signal to travel the distancebetween the source and the receiving node that is

120578119894 ≜ 1119888 1003817100381710038171003817m119894 minus x1199041003817100381710038171003817 (13)

In fact when utilizing TOA measurements for sourcelocalization it is often assumed that the source and sensornodes cooperate such that the signal propagation time can


be detected at the sensor nodes However such collaborationbetween sources and sensor nodes is not always availableThus without knowing the initial signal transmission time atthe source fromTOAalone the sensor is unable to determinethe signal propagation time

In the more general situation when unknown acousticsignals such as speech or unexpected sound events are to belocalized the relation between distances and TOAs can bemodeled as follows

120591119894 = 120578119894 + 119905119904 + 120598119894 (14)

where 119905119904 is an unknown transmission time instant and 120598119894 isthe TOA measurement noise Note that the time 119905119904 appearsdue to the fact that typical sound sources do not encode atime stamp in their transmitted signal to indicate the startingtransmission time to the sensor nodes and moreover thereis not any underlying synchronization mechanism Hencethe sensor nodes can only measure the signal arrival timeinstead of the propagation time or TOF One way to tacklethis problem is to exploit the difference of pairwise TOAmeasurements that is time difference of arrival (TDOA) forsource localization (see Section 5) Although the dependenceon the initial transmission time is eliminated by TDOA themeasurement subtraction strengthens the noise To overcomesuch problems some works propose methods to estimateboth the source location and initial transmission time jointly[34]

When the TOA and the TOF are equivalent (ie 120591119894 = 120578119894)for example because there are synchronized sound-emittingnodes the source-to-node distances can be calculated usingthe propagation speed of sound [35] This may require aninitial calibration process to determine factors that have astrong influence on the speed of sound The computation ofthe source location can be carried out in a central node byusing the estimated distances and solving the trilaterationproblem [36] Trilateration is based on the formulation ofone equation for each anchor that represents the surface ofthe sphere (or circle) centered at its position with a radiusequal to the estimated distance to the sound source Thesolution to this series of equations finds the point whereall the spheres intersect For 2D localization at least threesensors are needed while one more sensor is necessary toobtain a 3D location estimate

41 Trilateration Let us consider a set of119873 sensor TOAmea-surements 120591119894 that are transformed to distances by assuming aknown propagation speed

119903119894 = 119888 sdot 120591119894 (15)

where 119888 is the speed of sound (343ms) Then the followingsystem of equations can be formulated

1003817100381710038171003817m119894 minus x11990410038171003817100381710038172 = 1199032119894 119894 = 1 119873 (16)

Each equation in (16) represents a circle inR2 or a spherein R3 centered atm119894 with a radius 119903119894 Note that the problemis the same as the one given by (5) Thus solving the abovesystem is equivalent to finding the intersection pointpoints

of a set of circlesspheres Again the trilateration problem isnot straightforward to solve due to the nonlinearity of (16)and the errors in m119894 and 119903119894 [37] A number of algorithmshave been proposed in the literature to solve the trilaterationproblem including both closed form [37 38] and numericalsolutions Closed-form solutions have low computationalcomplexity when the solution of (16) actually exists Howevermost closed-form solutions only solve for the intersectionpoints of 119899 spheres in R119899 They do not attempt to determinethe intersection point when 119873 gt 119899 where small errorscan easily cause the involved spheres not to intersect at onepoint [39] It is then necessary to find a good approximationthat minimizes the errors contained in (16) considering thenonlinear least squares cost function

119869(TOA)NLS (x) = 119873sum119894=1

(1003817100381710038171003817m119894 minus x1003817100381710038171003817 minus 119903119894)2 x(TOA-NLS)119904 = argminx 119869(TOA)NLS (x)

(17)

Numerical methods are in general necessary to estimatex(TOA-NLS)119904 However compared with closed-form solutionsnumerical methods have higher computational complexitySome numerical methods are based on a linearization of thetrilateration problem [40ndash42] introducing additional errorsinto position estimation Common numerical techniquessuch as Newtonrsquos method or steepest descent have alsobeen proposed [38 40 41] However most of these searchalgorithms are very sensitive to the choice of the initial guessand a global convergence towards the desirable solution is notguaranteed [39]

42 Estimating TOAs of Acoustic Events As already dis-cussed localizing acoustic sources from TOAmeasurementsonly is not possible due to the unknown source emissiontime of acoustic events If the sensors are synchronized thedifferences of TOA measurements can be used to cancel outthe common term 119905119904 so that a set of TDOAs are obtainedand used as discussed in Section 5 A low-complexitymethodto estimate the TOAs from acoustic events in WASNs wasproposed in [43] where the cumulative-sum (CUSUM)change detection algorithm is used to estimate the sourceonset times at the nodes The CUSUM method is a low-complexity algorithm that allows estimating change detectioninstants by maximizing the following log-likelihood ratio

120591119894 = arg max1le120591le119896

119896sum119905=120591

ln(119901 (119909119894 (119905) 1205791)119901 (119909119894 (119905) 1205790)) (18)

The probability density function of each sample is givenby 119901(119909119894(119905) 120579) where 120579 is a deterministic parameter (not tobe confused with the DOA of the source) The occurrenceof an event is modeled by an instantaneous change in 120579 sothat 120579 = 1205790 before the event at 119905 = 120591119894 and 120579 = 1205791 when119905 ge 120591119894 To simplify calculations at the nodes the samplesbefore the acoustic event are assumed to belong exclusively toaGaussian noise component of variance12059020 while the samplesafter the event are also normally distributed with variance


EW event warning

EW

EW

EW

EW

Central node

(a)

Central node

TOA

TOA

TOA

TOATOA

(b)

Figure 3 Node communication steps in CUSUM-based TOA estimation (a) One of the nodes detects the sound event and sends an eventwarning alert to the other nodes (b) The nodes estimate their TOAs and send it to a central node

12059021 gt 12059020 These variances are estimated from the beginningand tail of a window of samples where the nodes have strongevidence that an acoustic event has happenedThe advantageof this approach is twofold On the one hand the estimationof the distribution parameters is more accurate On the otherhand the CUSUM change detection algorithm needs only tobe runwhen an acoustic event has actually occurred allowingsignificant battery savings in the nodes The detection ofacoustic events is performed by assuming that at least oneof the nodes has the sufficient SNR to detect the event by asimple amplitude threshold The node (or nodes) detectingthe presence of the event will notify the rest by sending anevent warning alert in order to let them know that they mustrun the CUSUM algorithm over a window of samples (seeFigure 3) The amplitude threshold selection is carried out bysetting either the probability of detection or the probabilityof false alarm given an initial estimate of the ambient noisevariance 12059020 Note that synchronization issues still persist (all120591119894 must have a common time reference) so that the nodesexchange synchronization information by using MAC layertime stamping in the deployment discussed in [43]

5 TDOA-Based Localization

When each sensor consists of multiple microphones local-ization can be accomplished in an efficient way demandingas much as possible of the processing to each node andthen combining measurements to yield the localization atthe central node When nodes are connected through lowbitrate channels and no synchronization of the internal clocks

is guaranteed this strategy becomes a must Among all thepossible measurements a possible solution can be found inthe time difference of arrival (TDOA)

51 TDOA and Generalized Cross Correlation Consider thepresence of119872 nodes in the network For reasons of simplicityin the notation all nodes are equipped with119873microphonesThe TDOA refers to the difference of propagation time fromthe source location to pairs of microphones If the source islocated at x119904 and the 119894th microphone in the 119898th sensor is atm(119898)

119894 119894 = 1 119873 the TDOA is related to the difference of theranges from the source to the microphones 119894 and 119895 through120591(119898)119894119895 = 119903(119898)119894119895119888 = 10038171003817100381710038171003817x119904 minusm(119898)

119894

10038171003817100381710038171003817 minus 10038171003817100381710038171003817x119904 minusm(119898)119895

10038171003817100381710038171003817119888 119894 = 1 119873 119895 = 1 119873 119894 = 119895 119898 = 1 119872

(19)

Throughout the rest of this subsection we will considerpairs of microphones within the same node so we will omitthe superscript (119898) of the sensor The estimate 120591119894119895 of theTDOA 120591119894119895 can be accomplished performing the generalizedcross correlation (GCC) between the signals acquired bymicrophones at m119894 and m119895 as detailed in the followingUnder the assumption of working in an anechoic scenarioand in a single source context the discrete-time signalacquired by the 119894th microphone is

119909119894 (119905) = 120572119894119904 (119905 minus 120578119894) + 120598119894 (119905) 119894 = 1 119873 (20)

where 120572119894 is a microphone-dependent attenuation term thataccounts for the propagation losses and air absorption 119904(119905)


is the source signal 120578119894 is the propagation delay betweenthe source and the 119894th microphone and 120598119894(119905) is an additivenoise signal In the discrete-time Fourier transform (DTFT)domain the microphone signals can be written as

119883119894 (120596) = 120572119894119878 (120596) 119890minus119895120596120578119894 + 119864119894 (120596) 119894 = 1 119873 (21)

where 119878(120596) and 119864119894(120596) are the DTFTs of 119904(119905) and 120598119894(119905) respec-tively and 120596 isin R denotes normalized angular frequency

Given the pair of microphones 119894 and 119895 with 119894 = 119895 theGCC between 119909119894(119905) and 119909119895(119905) can be written as [10]

119877119894119895 (120591) ≜ 12120587 int120587

minus120587119883119894 (120596)119883lowast

119895 (120596)Ψ119894119895 (120596) 119890119895120596120591119889120596 (22)

where119883119894(120596) is the DTFT of 119909119894(119905) lowast is the conjugate operatorand Ψ119894119895(120596) is a suitable weighting function

The TDOA from the pair (119894119895) is estimated as

120591119894119895 = argmax120591 119877119894119895 (120591)119865119904 (23)

where 119865119904 is the sampling frequency The goal of Ψ119894119895(120596) is tomake 119877119894119895(120591) sharper so that the estimate in (23) becomesmore accurate One of the most common choices is to use thePHAse Transform (PHAT) weighting function that is

Ψ119894119895 (120596) = 110038161003816100381610038161003816119883119894 (120596)119883lowast119895 (120596)10038161003816100381610038161003816 (24)

In an array of119873microphones the complete set of TDOAscounts119873(119873 minus 1)2measures If these are not affected by anysort of measurement error it can be easily demonstrated thatonly119873minus1 of them are independent the others being a linearcombination of them It is common practice therefore toadopt a reference microphone in the array and to measure119873 minus 1 TDOAs with respect to it We refer to the set of119873 minus 1 measures as the reduced TDOA set Without any lossof generality for the reduced TDOA set we can assume thereference microphone be with index 1 and the TDOAs 1205911119895 inthe reduced set for reasons of compactness in the notationare denoted with 120591119895 119895 = 2 119873

52 TDOA Measurement in Adverse Environments It isimportant to stress the fact that TDOA measurements arevery sensitive to reverberation noise and the presence ofpossible interferers in a reverberant environment for somelocations and orientation of the source the peak of the GCCrelative to a reflective path could overcome that of the directpathMoreover in a noisy scenario for some time instants thenoise level could exceed the signal thus making the TDOAsunreliable As a result some TDOAs must be consideredoutliers and must be discarded from the measurement setbefore localization (as in the example shown in Figure 4)

Several solutions have been developed in order to alle-viate the impact of outliers in TDOAs It has been observedthat GCCs affected by reverberation and noise do not exhibita single sharp peak In order to identify outliers thereforesome works analyze the shape of the GCC from which

minus20 minus15 minus10 minus5minus25 50

Range difference (cm)

minus05

0

05

1

GCC

-PH

AT

minus05

0

05

1

GCC

-PH

AT

minus20 minus15 minus10 minus5minus25 5 100


minus20 minus10minus30 10 200


minus05

0

05

1

GCC

-PH

AT

GCC-PHATGTPeak

minus20 minus10 0 2010minus30 30


minus05

0

05

1

GCC

-PH

AT

Figure 4 Examples of GCCs measured for pairs of microphoneswithin the same node It is possible to observe that for most ofthe above GCCs the measured TDOA (Peak) is very close to theground truth (GT)This does not happen for one measurement dueto reverberation Note also that TDOA values have been mapped torange differences

they were extracted In [12] authors propose the use of thefunction

120588119894119895 ≜ sum120591isinD119894119895119877119894119895 (120591)2sum120591notinD119894119895119877119894119895 (120591)2 (25)

to detect GCCs affected by outliers More specifically thenumerator sums the ldquopowerrdquo of the GCC samples withinthe interval D119894119895 centered around the candidate TDOA andcompares it with the energy of the remaining samplesWhen 120588119894119895 overcomes a prescribed threshold the TDOA isconsidered reliable Twometrics of the GCC shape have beenproposed in [13 44] The first one considers the value ofthe GCC at the maximum peak location while the secondcompares the highest peak with the second highest oneWhen both metrics overcome prescribed thresholds theGCC is considered reliable

Another possible route to follow is described in [11] andis based on the observation that TDOAs along closed pathsof microphones must sum to zero (zero-sum condition) andthat there is a relationship between the local maxima of the


autocorrelation and cross correlation of the microphone sig-nals (raster condition)The zero-sum condition onminimumlength paths of three microphones with indexes 119894 119895 and 119896 inparticular states that

120591119894119895 + 120591119895119896 + 120591119896119894 = 0 (26)

By imposing zero-sumand raster conditions authors demon-strate that they are able to disambiguate TDOAs in the caseof multiple sources in reverberant environments

In [17] authors combine different approaches A redun-dant set of candidate TDOAs is selected by identifyinglocal maxima of the GCCs A first selection is operated bydiscardingTDOAs that do not honor the zero-sumconditionA second selection step is based on three quality metricsrelated to the shape of the GCC The third final step is basedon the inspection of the residuals of the source localizationcost function all the measurements related to residualsovercoming a prescribed threshold are discarded

It is important to notice that all the referenced techniquesfor TDOA outlier removal do not involve the cooperation ofmultiple nodes with clear advantages in terms of data to betransmitted

53 Localization through TDOAs In this section we willconsider the problem of source localization by combiningmeasurements coming from different nodes In order toidentify the sensor from which measurements have beenextracted we will use the superscript (119898) From a geometricstandpoint given a TDOA estimate 120591(119898)119894119895 the source is boundto lie on a branch of hyperbola (hyperboloid in 3D) whosefoci are in m(119898)

119894 and m(119898)119895 and whose vertices are 119888120591(119898)119894119895 far

apart If source andmicrophones are coplanar the location ofthe source can be ideally obtained by intersecting two ormorehyperbolas [14] as in Figure 5 and some primitive solutionsfor source localization rely on this idea It is important tonotice that when the source is sufficiently far from the nodethe branch of hyperbola can be confused with its asymptotein this case the TDOA is informative only with respect tothe direction towards which the source is located and not itsdistance from the array In this context it is more convenientto work with DOAs (see Section 6)

In general intersecting hyperbolas is a strongly nonlinearproblem with obvious negative consequences on the compu-tational cost and the sensitivity to noise in themeasurementsIn [12] a solution is proposed to overcome this issue whichrelies on a projective representation The hyperbola derivedfrom theTDOA 120591(119898)119894119895 atmicrophonesm(119898)

119894 andm(119898)119895 is written

as

119886(119898)119894119895 1199092 + 119887(119898)119894119895 119909119910 + 119888(119898)119894119895 1199102 + 119889(119898)119894119895 119909 + 119890(119898)119894119895 119910 + 119891(119898)119894119895 = 0 (27)

where the coefficients 119886(119898)119894119895 119887(119898)119894119895 119888(119898)119894119895 119889(119898)119894119895 119890(119898)119894119895 119891(119898)119894119895 are

determined in closed form bym(119898)119894 m(119898)

119895 and 120591(119898)119894119895 Equation(27) represents a constraint on the source location In the

xs

mk(l)

mm(l)

mj(l)

Figure 5 The source lies at the intersection of hyperbolas obtainedfrom TDOAs

presence of noise the constraint is not honored and a residualcan be defined as

120576(119898)119894119895 (x) = V(119898)119894119895 (119886(119898)119894119895 1199092 + 119887(119898)119894119895 119909119910 + 119888(119898)119894119895 1199102 + 119889(119898)119894119895 119909+ 119890(119898)119894119895 119910 + 119891(119898)

119894119895 ) (28)

where V(119898)119894119895 = 1 for all the TDOAs that have been foundreliable and 0 otherwise The residuals are stacked in thevector 120576(x) and the source is localized byminimizing the costfunction

119869(TDOA)HYP (x) = 120576 (x)119879 120576 (x) (29)

If TDOA measurements are affected by additive white Gaus-sian noise it is easy to demonstrate that (29) is proportionalto the ML cost function

It has been demonstrated that a simplification of thelocalization cost function can be brought if a referencemicrophone is adopted at the cost of the nodes sharing acommon clock Without loss of generality we assume thereference to be the first microphone in the first sensor (ie119894 = 1 and119898 = 1) andwe also setm(1)

1 = 0 that is the origin ofthe reference frame coincides with the referencemicrophoneMoreover we can drop the array index 119898 upon assigning aglobal index to the microphones in different sensors rangingfrom 119895 = 1 to 119895 = 119873119872 In this context it is possible tolinearize the localization cost function as shown in the nextparagraphs By rearranging the terms in (19) and setting 119894 = 1it is easy to derive

1003817100381710038171003817x1199041003817100381710038171003817 minus 119903119895 = 10038171003817100381710038171003817x119904 minusm119895

10038171003817100381710038171003817 (30)

where 119903119895 = 119888120591119895 In [45] it has been proposed to representthe source localization problem in the space-range referenceframe a point x = [119909 119910]119879 in space is mapped onto the 3Dspace-range as |x119879 119903]119879 where

119903 = 1003817100381710038171003817x119904 minus x1003817100381710038171003817 minus 1003817100381710038171003817x1199041003817100381710038171003817 (31)


C1

Cs

x

r

C2

y

[xs rs]T

lowast

Figure 6 In the space-range reference frame the source is a pointon the surface of the cone 119862119904 and closest to the cones 119862119894

We easily recognize that in absence of noise and measure-ment errors 119903 is the range difference relative to the source inx119904 between the reference microphone and a microphone in xIf we replace

119903119904 = minus 1003817100381710038171003817x1199041003817100381710038171003817 (32)

we can easily interpret (30) as the equation of a negativehalf-cone 119862119895 in the space-range reference frame whose apexis [m119879

119895 119903119895]119879 and with aperture 1205874 and the source point[x119879119904 119903119879119904 ] lies on it Equation (30) can be iterated for 119895 = 2 119873times119872 and the source is bound to lie on the surface of all theconesMoreover the range 119903119904 of the source from the referencemicrophone must honor the constraint

119903119904 = minus 1003817100381710038171003817x1199041003817100381710038171003817 (33)

which is the equation of a cone 119862119904 whose apex is in [m1198791 0]119879

and with aperture 1205874 The ML source estimate therefore isthe point on the surface of the cone in (33) closest to all thecones defined by (30) as represented in Figure 6

By squaring both members of (30) and recognizing that

1199092119904 + 1199102119904 minus 1199032119904 = 0 (34)

(30) can be rewritten as

119909119895119909119904 + 119910119895119910119904 minus 119903119895119903119904 minus 12 (1199092119895 + 1199102119895 minus 1199032119895 ) = 0 (35)

which is the equation of a plane in the space-range referenceframe on which the source is bound to lie Under error in therange difference estimate 119903119895 (35) is not satisfied exactly andtherefore a residual can be defined as

119890119895 = 119909119895119909119904 + 119910119895119910119904 minus 119903119895119903119904 minus 12 (1199092119895 + 1199102119895 minus 1199032119895 ) (36)

Based upon the definition of 119890119895 the LS spherical cost functionis given by [46]

119869(TDOA)SP (x119904 119903119904) = 119873119872sum

119895=2

1198902119895 (37)

Most LS estimation techniques adopt this cost function andthey differ only for the additional constraints The Uncon-strained Least Squares (ULS) estimator [16 30 31 47 48]localizes the source as

x(ULS)119904 = argminx119904 119903119904119869(TDOA)SP (x119904 119903119904) (38)

It is important to notice that the absence of any explicitconstraint that relates x119904 and 119903119904 provides inmany applicationsa poor localization accuracy Constrained Least Squarestechniques therefore reintroduce this constraint as

x(CLS)119904 = argminx119904 119903119904119869(TDOA)SP (x119904 119903119904)

subject to 119903119904 = minus 1003817100381710038171003817x1199041003817100381710038171003817 (39)

Based on (39) Spherical Intersection [29] Least Squares withLinear Correction [49] and Squared Range Difference LeastSquares [15] estimators have been proposed which differfor the minimization procedure ranging from iterative toclosed-form solutions It is important to notice howeverthat all these techniques assume the presence of a globalreference microphone and synchronization valid for all thenodes Alternative solutions that overcome this technologicalconstraint have been proposed in [17 50] Here the concept ofcone propagation in the space-range reference has been putat advantage In particular in [50] the propagation cone isdefined slightly different from the one defined in (30) theapex is in the source and 119903119904 = 0 As a consequence in absenceof synchronization errors all the points [x(119898)119879119895 119903(119898)119895 ]119879 119895 =1 119873 119898 = 1 119872 must lie on the surface of thepropagation cone If a sensor exhibits a clock offset itsmeasurements will be shifted along the range axis The shiftcan be expressed as a function of the source location andtherefore it can be included in the localization cost functionat a cost of some nonlinearity The extension to the 3Dlocalization cost function was then proposed in [17]

6 DOA-Based Localization

When each node in the WASN incorporates multiple micro-phones the location of an acoustic source can be estimatedbased on direction of arrival (DOA) also known as bearingmeasurements Although such approaches require increasedcomputational complexity in the nodesmdashin order to performthe DOA estimationmdashthey can attain very low-bandwidthusage as only DOA measurements need to be transmittedin the network Moreover they can tolerate unsynchronizedinput given that the sources are static or that they moveat a rather slow rate relative to the analysis frame DOAmeasurements describe the direction from which sound ispropagating with respect to a sensor in each time instant andare an attractive approach to location estimation also due tothe ease in which such estimates can be obtained a varietyof broadband DOA estimation methods for acoustic sourcesare available in the literature such as the broadband MUSICalgorithm [51] the ESPRIT algorithm [52] IndependentComponent Analysis (ICA) methods [53] or Sparse Compo-nent Analysis (SCA) methods [54] When the microphones


4 3

1 21 2

34

Figure 7 Triangulation using DOA estimates (1205791ndash1205794) in aWASN of4 nodes (blue circles numbered 1 to 4) and one active sound source(red circle)

at the nodes follow a specific geometry for example circularmethods such as Circular Harmonics Beamforming (CHB)[55] can also be applied

In the sequel we will first review DOA-based localizationapproaches when a single source is active in the acoustic envi-ronmentThen we will present approaches for localization ofmultiple simultaneously active sound sources Finally we willdiscuss methods to jointly estimate the locations as well asthe number of sources a problem which is known as sourcecounting

61 Single Source Localization through DOAs In the singlesource case the location can be estimated as the intersectionof bearing lines (ie lines emanating from the locations ofthe sensors at the directions of the sensorsrsquo estimated DOAs)a method which is known as triangulation An example oftriangulation is illustrated in Figure 7 The problem closelyrelates to that of target motion analysis where the goalis to estimate the position and velocity of a target fromDOAmeasurements acquired by a single moving or multipleobservers Hence many of the methods were proposed forthe target motion analysis problem but outlined here in thecontext of sound source localization in WASNs

Considering a WASN of 119872 nodes at locations q119898 =[119902119909119898 119902119910119898]119879 the function that relates a location x = [119909 119910]119879with its true azimuthal DOA estimate at node119898 is

120579119898 (x) = arctan(119910 minus 119902119910119898119909 minus 119902119909119898) (40)

where arctan(sdot) is the four-quadrant inverse tangent functionNote that we deal with the two-dimensional location esti-mation problem that is only the azimuthal angle is neededWhen information about the elevation angle is also availablelocation estimation can be extended to the three-dimensionalspace

4 3

1 21

2

34

Figure 8 Triangulation using DOA estimates contaminated bynoise (1205791ndash1205794) in a WASN of 4 nodes (blue circles numbered 1 to4) and the estimated location of the sound source (red circle)

In any practical case however the DOA estimates 120579119898119898 = 1 119872 will be contaminated by noise and tri-angulation will not be able to produce a unique solutioncraving for the need of statistical estimators to optimallytackle the triangulation problem This scenario is illustratedin Figure 8 When the DOA noise is assumed to be Gaussianthe ML location estimator can be derived by minimizing thenonlinear cost function [56 57]

119869(DOA)ML (x) = 119872sum

119898=1

11205902119898 (120579119898 minus 120579119898 (x))2 (41)

where 1205902119898 is the variance of DOA noise at the 119898th sensorAs information about the DOA error variance at the

sensors is rarely available in practice (41) is usually modifiedto

119869(DOA)NLS (x) = 119872sum

119898=1

(120579119898 minus 120579119898 (x))2 (42)

which is termed as nonlinear least squares (NLS) [58] costfunction Minimizing (42) results in the ML estimator whenthe DOA noise variance is assumed to be the same at allsensors

While asymptotically unbiased the nonlinear nature ofthe above cost functions requires numerical search methodsfor minimization which comes with increased computa-tional complexity compared to closed-form solutions andcan become vulnerable to convergence problems under badinitialization poor geometry between sources and sensorshigh noise or insufficient number of measurements Tosurpass some of these problems some methods form geo-metrical constraints between the measured data and result inbetter convergence properties than the maximum likelihoodestimator [59] or try to directly minimize the mean squared


location error [60] instead of minimizing the total bearingerror in (41) and (42)

Other approaches are targeted at linearizing the abovenonlinear cost functions Stansfield [61] developed aweightedlinear least squares estimator based on the cost functionof (41) under the assumption that range information 119903119898 isavailable and DOA errors are small Under the small DOAerrors assumption 120579119898minus120579119898(x) can be approximated by sin(120579119898minus120579119898(x)) and the ML cost function can be modified to

119869(DOA)ST (x) = 12 (Ax minus b)119879Wminus1 (Ax minus b) (43)

where

A = [[[[[

sin 1205791 minus cos 1205791 sin 120579119872 minus cos 120579119872

]]]]]

b = [[[[[

1199021199091 sin 1205791 minus 1199021199101 cos 1205791119902119909119872 sin 120579119872 minus 119902119910119872 cos 120579119872

]]]]]

W = [[[[[[[

1199032112059021 01199032212059022

d

0 11990321198721205902119872

]]]]]]]

(44)

which is linear and has a closed-form solution

x(ST)119904 = (A119879Wminus1A)minus1 A119879Wminus1b (45)

When range information is not available the weightmatrix W can be replaced by the identity matrix In thisway the Stansfield estimator is transformed to the orthogonalvectors estimator also known as the pseudolinear estimator[62]

x(OV)119904 = (A119879A)minus1 A119879b (46)

While simple in their implementation and computation-ally very efficient due to their closed-form solution theselinear estimators suffer from increased estimation bias [63]A comparison between the Stansfield estimator and the MLestimator in [64] reveals that the Stansfield estimator providesbiased estimates Moreover the bias does not vanish even fora large number of measurements To reduce that bias variousmethods have been proposed based on instrumental variables[65ndash67] or total least squares [57 68]

Motivated by the need for computational efficiency theintersection point method [69] is based on finding thelocation of a source by taking the centroid of the intersectionsof pairs of bearing lines The centroid is simply the meanof the set of intersection points and minimizes the sum ofsquared distances between itself and each point in the set

To increase robustness in poor geometrical conditions themethod incorporates a scheme of identifying and excludingoutliers that occur from the intersection of pairs of bearinglines that are almost parallel Nonetheless the performanceof the method is very similar to that of the pseudolinearestimator

To attain the accuracy of nonlinear least squares estima-tors and improve their computational complexity the grid-based (GB) method [70 71] is based on making the searchspace discrete by constructing a grid G of 119873119892 grid pointsover the localization area Moreover as the measurementsare angles the GB method proposes the use of the AngularDistancemdashtaking values in the range of [0 120587]mdashas a moreproper measure of similarity than the absolute distance of(42)TheGBmethod estimates the source location by findingthe grid point whose DOAsmost closely match the estimatedDOAs from the sensors by solving

119869(DOA)GB (x) = 119872sum

119898=1

[119860 (120579119898 120579119898 (x))]2 (47)

x(GB)119904 = argminxisinG

119869(DOA)GB (x) (48)

where 119860(sdot sdot) denotes the angular distance between the twoarguments

To eliminate the location error introduced by the discretenature of this approach a very dense grid (high 119873119892) isrequired The search for the best grid point is performed inan iterative manner it starts with a coarse grid (low valueof 119873119892) and once the best grid point is foundmdashaccording to(47)mdasha new grid centered on this point is generated witha smaller spacing between grid points but also a smallerscope Then the best grid point in this new grid is foundand the procedure is repeated until the desired accuracy isobtained while keeping the complexity under control as itdoes not require an exhaustive search over a large numberof grid points In [70] it is shown that the GB method ismuchmore computationally efficient than the nonlinear leastsquares estimators and attains the same accuracy

62 Multiple Source Localization When considering mul-tiple sources a fundamental problem is that the correctassociation of DOAs from the nodes to the sources isunknown Hence in order to perform triangulation onemust first estimate the correct DOA combinations fromthe nodes that correspond to the same source The use ofDOAs that belong to different sources will result in ldquoghostrdquosources that is locations that do not correspond to realsources thus severely affecting localization performanceThis is known as the data-association problem The data-association problem is illustratedwith an example in Figure 9in aWASNwith twonodes (blue circles) and two active soundsources (red circles) let the solid lines show the DOAs tothe first source and the dashed lines show the DOAs to thesecond source Intersecting the bearing lines will result in 4intersection points the red circles that correspond to the truesourcesrsquo locations and are estimated by using the correctDOAcombinations (ie the DOAs from the node that correspond


1 2

Figure 9 Illustration of the data-association problem A two-node WASN with two active sound sources The solid lines showthe DOAs to the first source while the dashed lines show theDOAs to the second source Intersecting the bearing lines fromthe sensors will result in 4 possible intersection points the truesourcesrsquo locations (red circles) that are estimated by using the correctcombination of DOAs and the ldquoghostrdquo sources (white circles) thatare the results of using the erroneous DOA combinations

to the same source) and the white circles that are the result ofusing the erroneous DOA combinations (ldquoghostrdquo sources)

Also with multiple active sources some arrays mightunderestimate their number especially when some nodesare far from some sources or when the sources are closetogether in terms of their angular separation [54] Thusmissed detections can occur meaning that the DOAs of somesources from some arrays may be missing As illustrated in[72] missed detections can occur very often in practice Inthe sequel we review approaches for the data-association andlocalization problem of multiple sources whose number isassumed to be known

Some approaches tried to tackle the data-associationproblem by enumerating all possible DOA combinationsfrom the sensors and deciding on the correct DOA com-binations based on the resulting location estimates fromall combinations In general if 119862119904 denotes the number ofsensors that detected 119904 sources the number of possible DOAcombinations is

119873comb = 119878prod119904=1

119904119862119904 (49)

The position nonlinear least squares (P-NLS) estimatordeveloped in [73] incorporates the association procedure inthe ML cost function which takes the form

119869(DOA)P-NLS (x) = 119872sum

119898=1

min119894

10038161003816100381610038161003816120579119898119894 minus 120579119898 (x)100381610038161003816100381610038162 (50)

where 120579119898119894 is the 119894th DOA estimate of sensor119898 To minimize(50)119873comb initial locations are estimated (one for each DOAcombination) using a linear least squares estimator such asthe pseudolinear transform of (46) Then the cost function(50) is minimizedmdashusing numerical searchmethodsmdash119873combtimes each time using a different initial location estimate

Each time for each sensor the DOA closest to the DOAof the initial location estimate is used to take part in theminimization procedure In that way for all initial locationsthe estimator is expected to converge to a location of a truesource However as illustrated in [70] in the presence ofmissed detections and high noise the approach is not able tocompletely eliminate ldquoghostrdquo sources

Themultiple source grid-based method [70] estimates aninitial location for each possible DOA combination from thesensors by solving (47) It then decides which of the initiallocation estimates correspond to a true source heuristicallyby selecting the estimated initial locations whose DOAs arecloser to the DOAs from the combination used to estimatethat location

Other approaches focus on solving the data-associationproblem prior to the localization procedure In this way thecorrect association of DOAs from the sensors to the sourcesis estimated beforehand and the multiple source localizationproblem decomposes into multiple single source localizationproblems In [74] the data-association problem is viewedas an assignment problem and is formulated as a statisticalestimation problem which involves the maximization of theratio of the likelihood that the measurements come from thesame target to the likelihood that themeasurements are false-alarms Since the proposed solution becomes NP-hard formore than three sensors suboptimal solutions tried to solvethe same problem in pseudopolynomial time [75 76]

An approach based on clustering of intersections of bear-ing lines in scenarios with no missed detections is discussedin [77] It is based on the observation that intersectionsbetween pairs of bearing lines that correspond to the samesource will be close to each other Hence intersectionsbetween bearing lines will cluster around the true sourcesrevealing the correct DOA associations while intersectionsfrom erroneous DOA combinations will be randomly dis-tributed in space

Permitting the transmission of low-bandwidth additionalinformation from the sensors can lead to more efficientapproaches to the data-association problem The idea is thatthe sensors can extract and transmit features associated witheach source they detect Appropriate features for the data-association problem must possess the property of beingldquosimilarrdquo for the same source in the different sensors Thenthe correct association of DOAs to the sources can be foundby comparing the corresponding features

In [78] such features are extracted using Blind SourceSeparation The features are binary masks [79] in the fre-quency domain for each detected source that when appliedto the corresponding source signals they perform sourceseparation The extraction of such features relies on the W-disjoint orthogonality assumption [80] which states that ina given time-frequency point only one source is active anassumption which has been showed to be valid especiallyfor speech signals [81] The association algorithm worksby finding the binary masks from the different arrays thatcorrelate the most However the method is designed forscenarios with no missed detections and as illustrated in[72] performance significantly dropswhenmissed detections


0

50

100

150

200

250

300

350

400

y(c

m)

50 100 150 200 250 300 350 4000

x (cm)

(a)

500

400

300

200

100

00

50

100

150

200

250

300

350

400

y(c

m)

50 100 150 200 250 300 350 4000

x (cm)

(b)

Figure 10 Example of (a) narrowband location estimates and (b) their corresponding histogram for a scenario of two active sound sources(the red Xrsquos) The narrowband location estimates form two clusters around the true sourcesrsquo locations while their corresponding histogramdescribes the plausibility that a source is present at a given location

occurMoreover the association algorithm is designed for thecase of two sensors

The design of association features that are robust tomissed detections is considered in [72] along with a greedyassociation algorithm that canworkwith an arbitrary numberof sources and sensorsThe association features describe howthe frequencies of the captured signals are distributed to thesources To do that the method estimates a DOA 120601(120596 119896)in each time-frequency point where 120596 and 119896 denote thefrequency and time frame index respectively Then a time-frequency point (120596 119896) is assigned to source 119904 if the followingconditions are met

119860(120601 (120596 119896) 120579119904 (119896)) lt 119860 (120601 (120596 119896) 120579119902 (119896)) forall119902 = 119904 (51)

119860(120601 (120596 119896) 120579119904 (119896)) lt 120598 (52)

where 120579119904(119896) is the DOA estimate at time frame 119896 for the119904th source at the sensor of interest and 120598 is a predefinedthreshold Equations (51) and (52) imply that each frequencyis assigned to the source whose DOA is closest to theestimated DOA in this frequency as long as their distancedoes not exceed a certain threshold 120598 The second condition(see (52)) adds robustness to missed detections as it rejectsthe frequencies with DOA estimates whose distance from thedetected sourcesrsquo DOAs is significantly large

63 Source Counting Assuming that the number of sourcesis also unknown and can vary arbitrarily in time otherapproaches were developed to jointly solve the source count-ing and location estimation problem In these approachesthe central idea is to utilize narrowband DOA estimatesmdashforeach time-frequency pointmdashfrom the nodes in order to

estimate narrowband location estimates Appropriate pro-cessing of the narrowband location estimates can infer thenumber and locations of the sound sources The location foreach time-frequency point is estimated using triangulationbased on the corresponding narrowband DOA estimatesfrom the sensors at that time-frequency point Figure 10shows an example of such narrowband location estimatesand their corresponding histogram which also describesthe plausibility that a source is at a given location Theprocessing of these narrowband location estimates is usuallydone by statistical modeling methods in [82] the narrow-band location estimates are modeled by a Gaussian MixtureModel (GMM) where the number of Gaussian componentscorresponds to the number of sources while themeans of theGaussians determine the sourcesrsquo locations A variant of theExpectation-Maximization (EM) algorithm is proposed thatincorporates empirical criteria for removing and mergingGaussian components in order to determine the number ofsources as well A Bayesian view of the Gaussian MixtureModeling is adopted in [83 84] where a variant of the 119870-means algorithm is utilized that is able to determine boththe number of clusters (ie number of sources) and thecluster centroids (ie sourcesrsquo locations) using split andmerge operations on the Gaussian components

7 SRP-Based Localization

Approaches based on the steered response power (SRP)have attracted the attention of many researchers due totheir robustness in noisy and reverberant environmentsParticularly the SRP-PHAT algorithm is today one of themost popular approaches for acoustic source localizationusing microphone arrays [85ndash87] Basically the goal of SRPmethods is to maximize the power of the received sound


source signal using a steered filter-and-sum beamformer Tothis end the method uses a grid-search procedure wherecandidate source locations are explored by computing a func-tional that relates spatial location to the TDOA informationextracted from multiple microphone pairs The power mapresulting from the values computed at all candidate sourcelocations (also known as Global Coherence Field [88]) willshow a peak at the estimated source location

Since SRP approaches are based on the exploitation ofTDOA information synchronization issues also arise whenapplying SRP inWASNs As in the case of source localizationusing DOAs or TDOAs SRP-based approaches for WASNshave been proposed considering that multiple microphonesare available at each node [89ndash91] In these cases the SRPmethod can be used for acquiring DOA estimates at eachnode or collecting source location estimates that are mergedby a central node Next we describe the fundamentals of SRP-PHAT localization

71 Conventional SRP-PHAT (C-SRP) Consider a set of 119873different microphones capturing the signal arriving from asound source located at a spatial position x119904 isin R3 in ananechoic scenario following the model of (20) The SRP isdefined as the output power of a filter-and-sum beamformersteered to a given spatial location DiBiase [85] demonstratedthat the SRP at a spatial location x isin R3 calculated over a timeinterval of 119879 samples can be efficiently computed in terms ofGCCs

119869(SRP)119862 (x) = 2119879119873sum119894=1

119873sum119895=119894+1

119877119894119895 (120591119894119895 (x)) + 119873sum119894=1

119877119894119894 (0) (53)

where 120591119894119895(x) is the time difference of arrival (TDOA) thatwould produce a sound source located at x that is

120591119894119895 (x) = 1003817100381710038171003817x minusm1198941003817100381710038171003817 minus 10038171003817100381710038171003817x minusm119895

10038171003817100381710038171003817119888 (54)

The last summation term in (53) is usually ignored sinceit is a power offset independent of the steering locationWhenGCCs are computed with PHAT the resulting SRP is knownas SRP-PHAT

In practice the method is implemented by discretizingthe location space regionV using a search gridG consisting ofcandidate source locations inV and computing the functionalof (53) at each grid position The estimated source location isthe one providing the maximum functional value

x(C-SRP)119904 = argmaxxisinG

119869(SRP)119862 (x) (55)

72 Modified SRP-PHAT (M-SRP) Reducing the compu-tational cost of SRP is an important issue in WASNssince power-related constraints in the nodes may renderimpractical its implementation in real-world applicationsThe vast amount of modified solutions based on SRP isaimed at reducing the computational cost of the grid-searchstep [92 93] A problem of these methods is that they areprone to discard part of the information available leading

to some performance degradation Other recent approachesare based on analyzing the volume surrounding the gridof candidate source locations [87 94] By taking this intoaccount the methods are able to accommodate the expectedrange of TDOAs at each volume in order to increase therobustness of the algorithm and relax its computationalcomplexity The modified SRP-PHAT collects and uses theTDOA information related to the volume surrounding eachpoint of the search grid by considering a modified functional[87]

119869(SRP)119872 (x) = 119873sum119894=1

119873sum119895=119894+1

119871119906119894119895(x)sum120591=119871119897119894119895(x)

119877119894119895 (120591) (56)

where 119871119897119894119895(x) and 119871119906119894119895(x) are the lower and upper accumulationlimits of GCC delays which depend on the spatial location x

The accumulation limits can be calculated beforehandin an exact way by exploring the boundaries separating theregions corresponding to the points of the grid Alternativelythey can be selected by considering the spatial gradient ofthe TDOA nabla120591119894119895(x) = [nabla119909120591119894119895(x) nabla119910120591119894119895(x) nabla119911120591119894119895(x)]119879 where eachcomponent 120574 isin 119909 119910 119911 of the gradient is

nabla120574120591119894119895 (x) = 1119888 ( 120574 minus 1205741198941003817100381710038171003817x minusm1198941003817100381710038171003817 minus 120574 minus 12057411989510038171003817100381710038171003817x minusm119895

10038171003817100381710038171003817) (57)

For a rectangular grid where neighboring points areseparated a distance 119903119892 and the lower and upper accumulationlimits are given by

119871119897119894119895 (x) = 120591119894119895 (x) minus 100381710038171003817100381710038171003817nabla120591119894119895 (x)100381710038171003817100381710038171003817 sdot 119889119892119871119906119894119895 (x) = 120591119894119895 (x) + 100381710038171003817100381710038171003817nabla120591119894119895 (x)100381710038171003817100381710038171003817 sdot 119889119892 (58)

where 119889119892 = (1199031198922)min(1| sin(120579) cos(120601)| 1| sin(120579) sin(120601)|1| cos(120579)|) and the gradient direction angles are given by

120579 = cosminus1 ( nabla119911120591119894119895 (x)100381710038171003817100381710038171003817nabla120591119894119895 (x)100381710038171003817100381710038171003817) 120601 = arctan2 (nabla119910120591119894119895 (x) nabla119909120591119894119895 (x))

(59)

The estimated source location is again obtained as thepoint in the search grid providing the maximum functionalvalue

x(M-SRP)119904 = argmax

xisinG119869(SRP)119872 (x) (60)

Figure 11 shows the normalized SRPpowermaps obtainedby C-SRP using two different grid resolutions and the oneobtained by M-SRP using a coarse spatial grid In (a) thefine search grid shows clearly the hyperbolas intersecting atthe true source location However when the number of gridpoints is reduced in (b) the SRP power map does not providea consistent maximum As shown in (c) M-SRP is able to fix


x (m)

C-SRP (rg = 1 cm)

4353252151050

10

08

06

04

02

000

05

1

15

2

25

3

35

4

y(m

)

(a)

x (m)4353252151050

10

08

06

04

02

000

05

1

15

2

25

3

35

4

y(m

)

C-SRP (rg = 10 cm)

(b)

x (m)4353252151050

10

08

06

04

02

000

05

1

15

2

25

3

35

4

y(m

)

M-SRP (rg = 10 cm)

(c)

Figure 11 Example of SRP power maps obtained by C-SRP and M-SRP for a speech frame in anechoic acoustic conditions using differentspatial grid resolutions Yellow circles indicate the node locations

this situation showing a consistent maximum even when acoarse spatial grid is used

An iterative approach of the M-SRP method wasdescribed in [95] where the M-SRP is initially evaluatedusing a coarse spatial grid Then the volume surroundingthe point of highest value is iteratively decomposed by usinga finer spatial grid This approach allows obtaining almostthe same accuracy as the fine-grid search with a substantialreduction of functional evaluations

Finally recent works are also focusing on hardwareaspects in the nodes with the aim of efficiently computingthe SRP In this context the use of graphics processingunits (GPUs) for implementing SRP-based approaches is

specially promising [96 97] In [98] the performance of SRP-PHAT is analyzed over a massive multichannel processingframework in amulti-GPU system analyzing its performanceas a function of the number of microphones and availablecomputational resources in the system Note however thatthe performance of SRP approaches is also related to theproperties of the sound sources such as their bandwidth ortheir low-passpass-band nature [99 100]

8 Self-Localization of Acoustic Sensor Nodes

Methods for sound source localization discussed in previoussections assume that q119898 for 119898 = 1 119872 the locations of


x2

x1 x3

x4

m

x = 12 my = 25 mz = 07 m

Figure 12 Illustration of a WASN comprised with119872 = 1 node and119878 = 4 loudspeakers for self-localization scenario

the acoustic sensor nodes or m119894 for 119894 = 1 119873 those ofmicrophones are known to the system and fixed in timeIn practical situations however the precise locations of thesensor nodes or the microphones may not be known (egfor the deployment of ad hoc WASNs) Furthermore insome WASN applications the node locations may changeover time Due to these reasons self-calibration for adjustingthe known nodemicrophone locations or self-localization ofunknown nodes of the WASNs becomes necessary

The methods for self-localization of WASNs can bedivided into three categories [101] The first one uses nona-coustic sensors such as accelerometer and magnetometersthe second one uses the signal strength and the third oneuses the TOA or TDOA of acoustic signals In this sectionwe focus on the last category since it has been shown toenable localization with fine granularity centimeter-levellocalization [21] and requires only the acoustic sensors

The TOATDOA-based algorithms can be furtherdivided into two types depending upon whether the sourcepositions are known for self-localization Most of theearly works addressed this problem as microphone arraycalibration [102ndash105] with known source locations Thegeneral problem of joint source and sensor localization wasaddressed by [5] as a nonlinear optimization problem Thework in [106] presented a solution to explicitly tackle thesynchronization problem In [107] a solution consideringmultiple sources and sensors per device was described

This section will focus on algorithms that assume knownsource positions generating known probe signals without theknowledge of the sensor positions as well as the synchro-nization between the sensors and the sources This approachallows all processing to take place on the sensor node forself-localization The system illustration for this problem isgiven in Figure 12The remainder of this section describes theTOATDOA-based methods for acoustic sensor localizationby modeling the inaccurate TOATDOA measurements forrobust localization followed by some recent approaches

81 Problem Formulation Consider a WASN comprised of 119878sources and 119872 nodes The microphone locations m(119898)

119894 for119894 = 1 2 119873 at node q119898 can be determined with respectto the node location its orientation and its microphone

configuration So we consider the sensor localization offinding the microphone location m(119898)

119894 as the same problemas finding the node location q119898 in this section Withoutloss of generality we can consider the case with only onenode and one microphone (119872 = 119873 = 1) because eachnode determines its location independently from others Inaddition we consider that the sources are the loudspeakers ofthe system with fixed and known locations in this scenario

Let m and x119904 be the single microphone position andthe position of the 119904th source for 119904 isin S ≜ 1 2 119878respectively The goal is to find m by means of 119878 receivedacoustic signals emitted by 119878 sources where the location ofeach source x119904 is known

The TOF 120578119904 from the 119904th loudspeaker to the sensor isdefined as

120578119904 ≜ 1119888 1003817100381710038171003817m minus x1199041003817100381710038171003817 (61)

where 119888 is the speed of sound Note that this equation isequivalent to (13) except thatwe consider a singlemicrophone(m) multisource (x119904) case for 119904 isin S thus the TOF isindexed with respect to 119904 instead of 119894 From (61) it is evidentthatm can be found if a sufficient number of TOFs are knownIn practice we need to rely on TOAs instead of TOFs due tomeasurement errors

In order to remove the effect from such unknown factorsthe TDOA can be used instead of TOA which is given by

12059111990411199042 ≜ 1205781199041 minus 1205781199042 =10038171003817100381710038171003817m minus x1199041

10038171003817100381710038171003817 minus 10038171003817100381710038171003817m minus x119904210038171003817100381710038171003817119888 (62)

regarding a pair of sources 1199041 1199042 isin S Please note that thesubscripts for the TDOA indicate the source indexes that aredifferent from those defined for sensor indexes in (19)

Since the probe signals generated from the sourcesalong with their locations are assumed to be known to thesensor nodes the derivations of the self-localizationmethodshereafter rely on the GCC between the probe signal and thereceived signal at the sensor Provided that the direct line-of-sight between the source and the sensor is guaranteed thenthe time delay found by the GCC in (22) between the probesignal and the signal received at the sensor provides the TOAinformation

82 Modeling of Time Measurement Errors Two mainfactorsmdashasynchrony and the sampling frequency mismatchbetween sources and sensorsmdashcan be considered for themodeling of time measurement errors When there existsasynchrony between a source and a sensor the TOA can bemodeled as

120591119904 = 120578119904 + Δ120578119904 (63)

where 120578119904 is the true TOF from 119904th source to the sensorand Δ120578119904 is the bias caused by the asynchrony If there existssampling frequency mismatch then the sampling frequencyat the sensor can bemodeled as119865119904+Δ119865119904 where119865119904 is that of thesource Considering these and ignoring the rounding of the


discrete-time index the relationship between the discrete-time TOA 120591119904 and the actual TOF 120578119904 is given by

120591119904 = 120578119904 + Δ120578119904119865119904 + Δ119865119904 = 120578119904 + Δ120578119904119865119904 ( 11 + Δ119865119904119865119904) (64)

which can be further simplified for |Δ119865119904| ≪ 119865119904 as120591119904 asymp 120572120578119904 + 120573119904 (65)

where 120572 ≜ (119865119904 minus Δ119865119904)119865119904 120578119904 asymp 120578119904119865119904 and 120573119904 ≜ Δ120578119904(119865119904 minusΔ119865119904)1198652119904 If the sources are connected to the playback system

with the same clock such that the sources share the commonplayback delay then 120573119904 = 120573 for all 119904 isin S Therefore1003817100381710038171003817m minus x119904

1003817100381710038171003817 = 119888120578119904 asymp 119886120591119904 + 119887 (66)

where 119886 = 119888120572 and 119887 = minus11988812057312057283 Least SquaresMethod Given a sufficient number of TOAor TDOA estimates the least squares (LS) method can beused to estimate the sensor positionThe TOA-based and theTDOA-based LS methods proposed in [20 21] are describedin this subsection

831 TOA-Based Formulation Motivated by the relationin (66) the localization error 119890119904 corresponding to the 119904thloudspeaker can be defined as

119890119904 ≜ (119886120591119904 + 119887)2 minus 1003817100381710038171003817m minus x11990410038171003817100381710038172 forall119904 isin S (67)

If we define the error vector as e(TOA) ≜ [1198901 1198902 sdot sdot sdot 119890119878]119879 withunknown parameters 119886 119887 andm then the cost function canbe defined as

119869(TOA)LS (m 119886 119887) = 10038171003817100381710038171003817e(TOA)100381710038171003817100381710038172 (68)

Then the localization problem is formulated as

m(TOA)LS = argmin

m119886119887119869(TOA)LS (m 119886 119887) (69)

which does not have a closed-form solution

832 TDOA-Based Formulation We can consider the firstloudspeaker for 119904 = 1 as the reference loudspeaker whoseTOA and position can be set to 1205911 and x1 = 0 Given a set ofTDOAs in the LS framework it can be used with (66) as

119886120591119904 + 119887 = (119886120591119904 minus 1198861205911) + (1198861205911 + 119887) asymp 1198861205911199041 + m (70)

where 1205911199041 is the TDOA between the 119904th and the firstloudspeakers With the first as the reference we can definelength 119878 minus 1 vectors 119890119904 ≜ (1198861205911199041 + m)2 minus m minus x1199042 for119904 = 2 3 119878 and e(TDOA)

LS ≜ [1198902 sdot sdot sdot 119890119878]119879 the cost functioncan be defined as

119869(TDOA)LS (m 119886) = 10038171003817100381710038171003817e(TDOA)

LS100381710038171003817100381710038172 (71)

and the LS problem can be written as

m(TDOA)LS = argmin

m119886119869(TDOA)LS (m 119886) (72)

Note that the TDOA-based approach is not dependent uponthe parameter 119887 unlike the TOA-based approach

833 LS Solutions For both the TOA- and TDOA-basedapproaches the error vector can be formulated as e = HΘminusgwhere the elements of the vectorΘ are unknown and those ofthe matrixH and the vector g are both known to the systemThe error vectors for both approaches in this formulation canbe expressed as follows [21]

e(TOA) = [[[[[

1 2x1198791 12059121 21205911 1 2x119879119878 1205912119878 2120591119904

]]]]][[[[[[

1198872 minus m2m1198862119886119887

]]]]]]minus [[[[[

1003817100381710038171003817x1100381710038171003817100381721003817100381710038171003817x11987810038171003817100381710038172]]]]]

e(TDOA) = [[[[[

212059121 2x1198792 120591221 21205911198781 2x119879119878 12059121198781

]]]]][[[[m 119886m1198862

]]]]minus [[[[[

1003817100381710038171003817x2100381710038171003817100381721003817100381710038171003817x11987810038171003817100381710038172]]]]]

(73)

Although the elements of the vector Θ are not indepen-dent from one another the constraints can be removed forcomputational efficiency [15 21] then the nonlinear problemsin (69) and (72) can be considered as the ULS problem asfollows

minΘ

e2 (74)

which has the closed-form solution given by

Θ ≜ (H119879H)minus1H119879g (75)

It has been shown that it requires 119878 ge 6 to find the closed-form solution for both approaches [21]

For the case when 119886 = 119888 that is no sampling frequencymismatch with known speed of sound the TDOA-basedapproach can be further simplified as

e = [[[[[

211988812059121 2x1198792 21198881205911198781 2x119879119878

]]]]][ m

m] minus [[[[[

1003817100381710038171003817x210038171003817100381710038172 minus 11988821205912211003817100381710038171003817x11987810038171003817100381710038172 minus 119888212059121198781]]]]]

(76)

which is related to the methods developed for the sensorlocalization problem [15 108]

The self-localization results of the LS approaches arehighly sensitive to the estimated values of TOATDOA Ifthey are estimated poorly then the localization accuracy maysignificantly suffer from those inaccurate estimates In orderto address this issue a sliding window technique is proposedto improve the accuracy of TOATDOA estimates [21]


84 Other Approaches More recently several papers havetackled the problem of how to design good probe signalsbetween source and sensor and how to improve TOF estima-tion in [22] a probe signal design using pulse compressiontechnique and hyperbolic frequency modulated signals ispresented that is capable of localizing an acoustic sourceand estimates its velocity and direction in case it is movingA matching pursuit-based algorithm for TOF estimation isdescribed in [109] and refined in [101] The joint localizationof sensor and source in an ad hoc array by using low-rankapproximation methods has been addressed in [23] In [24]an iterative peak matching algorithm for the calibration of awireless acoustic sensor network is described in an unsyn-chronized network by using a fast calibration process Themethod is valid for nodes that incorporate a microphone anda loudspeaker and is based on the use of a set of orthogonalprobe signals that are assigned to the nodes of the net-work The correlation properties of pseudonoise sequencesare exploited to simultaneously estimate the relative TOAsfrom multiple acoustic nodes substantially reducing thetotal calibration time In a final step synchronization issuesare removed by following a BeepBeep strategy [106 110]providing range estimates that are converted to absolute nodepositions by means of multidimensional scaling [104]

9 Challenges and Future Directions

91 Practical Challenges Some real-world challenges arise inthe design of localization systems using WASNs To builda robust and accurate localization system it is necessaryto ensure a tradeoff among aspects related to cost energyeffectiveness ease of calibration deployment difficulty andprecision Achieving such tradeoff is not straightforward andencompasses many practical challenges as discussed below

911 Cost-Effectiveness The potentiality of WASNs to pro-vide high-accuracy acoustic localization features is highlydependent on the underlying hardware technologies inthe nodes For example localization techniques based onDOA TDOA or SRP need intensive in-node processingfor computing GCCs as well as input resources permittingmultichannel audio recording With the advent of powerfulsingle-board computers high-performance in-node signalprocessing can be easily achieved Nonetheless importantaspects should be considered regarding cost and energydissipation especially for battery-powered nodes that aremassively deployed

912 Deployment Issues Localization methods usuallyrequire a predeployment configuration process For exampleTOA-based methods usually need to properly set upsynchronization mechanisms before starting to localizetargets Similarly energy-based methods need nodes withcalibrated gains in order to obtain high-quality energy-ratiomeasurements All these tasks are usually complex and time-consuming Moreover they tend to need human supervisionduring an offline profiling phase Such predeploymentphase can become even more complex in some application

environments where nodes can be accessed by unauthorizedsubjects Moreover WASNs are also applied outside ofclosed buildings Thus they are subject to daily and seasonaltemperature variations and corresponding variations ofthe speed of sound [35] To cope with this shortcomingcalibration needs to be automated and made environmentadaptive

913 System Resiliency Besides predeployment issues aWASN should also implement self-configuration mecha-nisms dealing with network dynamics such as those relatedto node failures In this context system design must take intoaccount the number of anchor nodes that are needed in thedeployment and their placement strategy The system shouldassure that if some of the nodes get out of the network therest are still able to provide location estimates appropriatelyTo this end it is important to maximize the coverage areawhile minimizing the number of required anchor nodes inthe system

914 Scalability Depending on the specific application theWASN that needs to be deployed can vary from a very smalland simple network of a few nodes to very largeWASNs withtens or hundreds of nodes and complex network topologiesFor example in wildlife monitoring applications a very largenumber of sensors are utilized to acoustically monitor verylarge environments while the topology of the network canbe constantly changing due to sensors being displaced bythe wind or by passing animals The challenge in suchapplications is to design localization and self-configurationmethods that can easily scale to complex WASNs

915 Measurement Errors It is well known that RF-basedlocalization in WSNs are prone to errors due to irregularpropagation patterns induced by environmental conditions(pressure and temperature) and random multipath effectssuch as reflection refraction diffraction and scattering Inthe case of WASNs acoustic signals are also subject tosimilar distortions caused by environmental changes andeffects produced by noise and interfering sources reflectedechoes object obstruction or signal diffraction Other errorsare related to the aforementioned predeployment processresulting in synchronization errors or inaccuracies in thepositions of anchor nodes These errors must be analyzedin order to filter measurement noise out and improve theaccuracy of location estimates

916 Benchmarking In terms of performance evaluationso far there are no specific benchmarking methodologiesand datasets for the location estimation problem in WASNsDue to their heterogeneitymdashin terms of sensors number ofsensors and microphones topology and so onmdashworks com-paring different localizationmethods using a common sensorsetup are difficult to find in the literature The definitionof formal methodologies in order to evaluate localizationperformance and the recording of evaluation datasets usingreal-life WASNs still remain a big challenge


92 Future Directions

921 Real-Life Application In our days the need for real-life realizations of WASN with sound source localizationcapabilities is becoming more a more evident An importantdirection in the future will thus be the application of thelocalization methodologies in real-life WASNs In this direc-tion the integration ofmethodologies from a diverse range ofscientific fields will be of paramount importance Such fieldsinclude networks (eg to design the communication andsynchronization protocols) network administration (eg toorganize the nodes of the network and identify and handlepotential failures) signal processing (eg to estimate thesourcesrsquo locations with many potential applications) andhardware design (eg to design the acoustic nodes that canoperate individually featuring communication and multi-channel audio processing capabilities in a power efficientway)While many of these fields have flourished individuallythe practical issues that will arise from their integration inpractical WASNs still remain unseen and the need for bench-marking and methodologies for their efficient integration isbecoming more and more urgent

922 Machine Learning-Based Approaches One of the prac-tical challenges for the deployment of real-life applicationsis the huge variability of acoustic signals received at theWASNs due to the acoustic signal propagation in the physicaldomain as well as the inaccuracies caused at the system leveland uncertainties associated with measurements of TOAsand TDOAs With the help of large datasets and vastlyincreased computational power of off-the-shelf processorsthese variabilities can be learned by machines for designingmore robust algorithms

10 Conclusion

Sound source localization throughWASNs offers great poten-tial for the development of location-aware applicationsAlthough many methods for locating acoustic sources havebeen proposed during the last decadesmostmethods assumesynchronized input signals acquired by a traditional micro-phone array As a result when designing WASN-orientedapplications many assumptions of traditional localizationapproaches have to be revisited This paper has presenteda review of sound source localization methods using com-monly used measurements inWASNs namely energy direc-tion of arrival (DOA) time of arrival (TOA) time differenceof arrival (TDOA) and steered response power (SRP) More-over since most algorithms assume perfect knowledge onthe node locations self-localizationmethods used to estimatethe location of the nodes in the network are also of highinterest within aWASN contextThe practical challenges andfuture directions arising in the deployment of WASNs havealso been discussed emphasizing important aspects to beconsidered in the design of real-world applications relying onacoustic localization systems

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Authorsrsquo Contributions

All authors contributed equally to this work

References

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[2] J Segura-Garcia S Felici-Castell J J Perez-Solano M Cobosand J M Navarro ldquoLow-cost alternatives for urban noisenuisance monitoring using wireless sensor networksrdquo IEEESensors Journal vol 15 no 2 pp 836ndash844 2015

[3] M Cobos J J Perez-Solano and L T Berger ldquoAcoustic-basedtechnologies for ambient assisted livingrdquo in Introduction toSmart eHealth and eCare Technologies pp 159ndash177 CRC PressTaylor amp Francis 2017

[4] A Bertrand ldquoApplications and trends in wireless acousticsensor networks a signal processing perspectiverdquo inProceedingsof the 18th IEEE Symposium on Communications and VehicularTechnology in the Benelux (SCVT rsquo11) pp 1ndash6 IEEE GhentBelgium November 2011

[5] V C Raykar I V Kozintsev and R Lienhart ldquoPosition calibra-tion ofmicrophones and loudspeakers in distributed computingplatformsrdquo IEEE Transactions on Speech and Audio Processingvol 13 no 1 pp 70ndash83 2005

[6] L Cheng C Wu Y Zhang H Wu M Li and C Maple ldquoAsurvey of localization in wireless sensor networkrdquo InternationalJournal of Distributed Sensor Networks vol 2012 Article ID962523 12 pages 2012

[7] H Liu H Darabi P Banerjee and J Liu ldquoSurvey of wirelessindoor positioning techniques and systemsrdquo IEEE Transactionson Systems Man and Cybernetics C vol 37 no 6 pp 1067ndash10802007

[8] H Wang Wireless sensor networks for acoustic monitoring[PhD dissertation] University of California Los Angeles(UCLA) Los Angeles Calif USA 2006

[9] N Priyantha A Chakraborty and H Balakrishnan ldquoThecricket location-support systemrdquo in Proceedings of the 6thAnnual International Conference on Mobile Computing andNetworking (MobiCom rsquo00) pp 32ndash43 August 2000

[10] C H Knapp and G C Carter ldquoThe generalized correlationmethod for estimation of time delayrdquo IEEE Transactions onAcoustics Speech and Signal Processing vol 24 no 4 pp 320ndash327 1976

[11] J Scheuing and B Yang ldquoDisambiguation of TDOA estimationfor multiple sources in reverberant environmentsrdquo IEEE Trans-actions on Audio Speech and Language Processing vol 16 no 8pp 1479ndash1489 2008

[12] A Canclini F Antonacci A Sarti and S Tubaro ldquoAcousticsource localization with distributed asynchronous microphonenetworksrdquo IEEE Transactions on Audio Speech and LanguageProcessing vol 21 no 2 pp 439ndash443 2013

[13] D Bechler and K Kroschel ldquoReliability criteria evaluationfor TDOA estimates in a variety of real environmentsrdquo inProceedings of the IEEE International Conference on Acoustics


Speech and Signal Processing (ICASSP rsquo05) vol 4 pp iv985ndashiv988 March 2005

[14] Y A Huang and J BenestyAudio Signal Processing for Nextgen-eration Multimedia Communication Systems Springer Scienceamp Business Media 2007

[15] A Beck P Stoica and J Li ldquoExact and approximate solutionsof source localization problemsrdquo IEEE Transactions on SignalProcessing vol 56 no 5 pp 1770ndash1778 2008

[16] M D Gillette and H F Silverman ldquoA linear closed-form algo-rithm for source localization from time-differences of arrivalrdquoIEEE Signal Processing Letters vol 15 pp 1ndash4 2008

[17] A Canclini P Bestagini F Antonacci M Compagnoni ASarti and S Tubaro ldquoA robust and low-complexity source local-ization algorithm for asynchronous distributed microphonenetworksrdquo IEEE Transactions on Audio Speech and LanguageProcessing vol 23 no 10 pp 1563ndash1575 2015

[18] W Meng and W Xiao ldquoEnergy-based acoustic source localiza-tion methods a surveyrdquo Sensors vol 17 no 2 p 376 2017

[19] C Meesookho U Mitra and S Narayanan ldquoOn energy-based acoustic source localization for sensor networksrdquo IEEETransactions on Signal Processing vol 56 no 1 pp 365ndash3772008

[20] D B Haddad L O Nunes W A Martins L W P Biscainhoand B Lee ldquoClosed-form solutions for robust acoustic sensorlocalizationrdquo in Proceedings of the 14th IEEEWorkshop on Appli-cations of Signal Processing to Audio and Acoustics (WASPAArsquo13) pp 1ndash4 October 2013

[21] D B Haddad W A Martins M D V M Da Costa L WP Biscainho L O Nunes and B Lee ldquoRobust acoustic self-localization of mobile devicesrdquo IEEE Transactions on MobileComputing vol 15 no 4 pp 982ndash995 2016

[22] R Pfeil M Pichler S Schuster and F Hammer ldquoRobust acous-tic positioning for safety applications in underground miningrdquoIEEE Transactions on Instrumentation and Measurement vol64 no 11 pp 2876ndash2888 2015

[23] L Wang T-K Hon J D Reiss and A Cavallaro ldquoSelf-localization of ad-hoc arrays using time difference of arrivalsrdquoIEEE Transactions on Signal Processing vol 64 no 4 pp 1018ndash1033 2016

[24] M Cobos J J Perez-Solano O Belmonte G Ramos andA M Torres ldquoSimultaneous ranging and self-positioningin unsynchronized wireless acoustic sensor networksrdquo IEEETransactions on Signal Processing vol 64 no 22 pp 5993ndash60042016

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[27] D Li and Y H Hu ldquoLeast square solutions of energy basedacoustic source localization problemsrdquo in Proceedings of theWorkshops on Mobile and Wireless NetworkingHigh Perfor-mance Scientific Engineering ComputingNetwork Design andArchitectureOptical Networks Control andManagementAdHocand Sensor NetworksCompil pp 443ndash446 IEEE MontrealCanada August 2004

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[32] K C Ho and M Sun ldquoAn accurate algebraic closed-form solu-tion for energy-based source localizationrdquo IEEETransactions onAudio Speech and Language Processing vol 15 no 8 pp 2542ndash2550 2007

[33] D Blatt and A O Hero III ldquoEnergy-based sensor networksource localization via projection onto convex setsrdquo IEEETransactions on Signal Processing vol 54 no 9 pp 3614ndash36192006

[34] H Shen Z Ding S Dasgupta and C Zhao ldquoMultiple sourcelocalization in wireless sensor networks based on time of arrivalmeasurementrdquo IEEE Transactions on Signal Processing vol 62no 8 pp 1938ndash1949 2014

[35] P Annibale J Filos P A Naylor and R Rabenstein ldquoTDOA-based speed of sound estimation for air temperature and roomgeometry inferencerdquo IEEE Transactions on Audio Speech andLanguage Processing vol 21 no 2 pp 234ndash246 2013

[36] F Thomas and L Ros ldquoRevisiting trilateration for robot local-izationrdquo IEEE Transactions on Robotics vol 21 no 1 pp 93ndash1012005

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[39] Y Zhou ldquoAn efficient least-squares trilateration algorithm formobile robot localizationrdquo in Proceedings of the IEEERSJInternational Conference on Intelligent Robots and Systems pp3474ndash3479 IEEE October 2009

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[43] M Cobos J J Perez-Solano S Felici-Castell J Segura and JMNavarro ldquoCumulative-sum-based localization of sound eventsin low-cost wireless acoustic sensor networksrdquo IEEEACMTransactions on Speech and Language Processing vol 22 no 12pp 1792ndash1802 2014

[44] D Bechler and K Kroschel ldquoThree different reliability criteriafor time delay estimatesrdquo in Proceedings of the 12th EuropeanSignal Processing Conference pp 1987ndash1990 September 2004

[45] P Bestagini M Compagnoni F Antonacci A Sarti andS Tubaro ldquoTDOA-based acoustic source localization in thespace-range reference framerdquo Multidimensional Systems andSignal Processing vol 25 no 2 pp 337ndash359 2014


[46] Y Huang ldquoSource localizationrdquo in Audio Signal Processing forNext-Generation Multimedia Communication Systems chapter8 pp 229ndash253 Springer 2004

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[48] J Abel and J Smith ldquoThe spherical interpolation method forclosed-form passive source localization using range differencemeasurementsrdquo in Proceedings of the IEEE International Con-ference on Acoustics Speech and Signal Processing (ICASSP rsquo87)vol 12 pp 471ndash474

[49] Y Huang J Benesty G W Elko and R M Mersereau ldquoReal-time passive source localization a practical linear-correctionleast-squares approachrdquo IEEETransactions on Speech andAudioProcessing vol 9 no 8 pp 943ndash956 2001

[50] M Compagnoni P Bestagini F Antonacci A Sarti and STubaro ldquoLocalization of acoustic sources through the fittingof propagation cones using multiple independent arraysrdquo IEEETransactions on Audio Speech and Language Processing vol 20no 7 pp 1964ndash1975 2012

[51] S Argentieri and P Danes ldquoBroadband variations of theMUSIC high-resolution method for sound source localizationin roboticsrdquo in Proceedings of the IEEERSJ International Con-ference on Intelligent Robots and Systems (IROS rsquo07) pp 2009ndash2014 October 2007

[52] R Roy and T Kailath ldquoESPRIT-estimation of signal parametersvia rotational invariance techniquesrdquo IEEE Transactions onAcoustics Speech and Signal Processing vol 37 no 7 pp 984ndash995 1989

[53] F Nesta and M Omologo ldquoGeneralized state coherencetransform for multidimensional TDOA estimation of multiplesourcesrdquo IEEE Transactions on Audio Speech and LanguageProcessing vol 20 no 1 pp 246ndash260 2012

[54] D Pavlidi A Griffin M Puigt and A Mouchtaris ldquoReal-time multiple sound source localization and counting using acircular microphone arrayrdquo IEEE Transactions on Audio Speechand Language Processing vol 21 no 10 pp 2193ndash2206 2013

[55] A M Torres M Cobos B Pueo and J J Lopez ldquoRobustacoustic source localization based on modal beamforming andtime-frequency processing using circular microphone arraysrdquoJournal of the Acoustical Society of America vol 132 no 3 pp1511ndash1520 2012

[56] L M Kaplan Q Le and P Molnar ldquoMaximum likelihoodmethods for bearings-only target localizationrdquo in Proceedingsof the IEEE Interntional Conference on Acoustics Speech andSignal Processing (ICASSP rsquo01) vol 5 pp 3001ndash3004 May 2001

[57] K Dogancay ldquoBearings-only target localization using total leastsquaresrdquo Signal Processing vol 85 no 9 pp 1695ndash1710 2005

[58] L M Kaplan and Q Le ldquoOn exploiting propagation delays forpassive target localization using bearings-only measurementsrdquoJournal of the Franklin Institute vol 342 no 2 pp 193ndash211 2005

[59] A N Bishop B D O Anderson B Fidan P N Pathiranaand G Mao ldquoBearing-only localization using geometricallyconstrained optimizationrdquo IEEE Transactions on Aerospace andElectronic Systems vol 45 no 1 pp 308ndash320 2009

[60] Z Wang J-A Luo and X-P Zhang ldquoA novel location-penalized maximum likelihood estimator for bearing-onlytarget localizationrdquo IEEE Transactions on Signal Processing vol60 no 12 pp 6166ndash6181 2012

[61] R G Stansfield ldquoStatistical theory of DF fixingrdquo Journal ofthe Institute of Electrical EngineersmdashPart IIIA Radiocommuni-cation vol 94 no 15 pp 762ndash770 1947

[62] S C Nardone A G Lindgren and K F Gong ldquoFundamentalproperties and performance of conventional bearings-only tar-get motion analysisrdquo IEEE Transactions on Automatic Controlvol 29 no 9 pp 775ndash787 1984

[63] K Dogancay ldquoOn the bias of linear least squares algorithms forpassive target localizationrdquo Signal Processing vol 84 no 3 pp475ndash486 2004

[64] M Gavish and A J Weiss ldquoPerformance analysis of bearing-only target location algorithmsrdquo IEEE Transactions onAerospace and Electronic Systems vol 28 no 3 pp 817ndash8281992

[65] K Dogancay ldquoPassive emitter localization using weightedinstrumental variablesrdquo Signal Processing vol 84 no 3 pp 487ndash497 2004

[66] Y T Chan and S W Rudnicki ldquoBearings-only and doppler-bearing tracking using instrumental variablesrdquo IEEE Transac-tions on Aerospace and Electronic Systems vol 28 no 4 pp1076ndash1083 1992

[67] K Dogancay ldquoBias compensation for the bearings-only pseu-dolinear target track estimatorrdquo IEEE Transactions on SignalProcessing vol 54 no 1 pp 59ndash68 2006

[68] K Dogancay ldquoReducing the bias of a bearings-only tls targetlocation estimator through geometry translationsrdquo in Proceed-ings of the European Signal Processing Conference (EUSIPCOrsquo04) pp 1123ndash1126 IEEE Vienna Austria September 2004

[69] A Griffin and A Mouchtaris ldquoLocalizing multiple audiosources from DOA estimates in a wireless acoustic sensor net-workrdquo in Proceedings of the 14th IEEEWorkshop on Applicationsof Signal Processing to Audio and Acoustics (WASPAA rsquo13) pp1ndash4 IEEE October 2013

[70] A Griffin A Alexandridis D Pavlidi Y Mastorakis and AMouchtaris ldquoLocalizing multiple audio sources in a wirelessacoustic sensor networkrdquo Signal Processing vol 107 pp 54ndash672015

[71] A Griffin A Alexandridis D Pavlidi and A MouchtarisldquoReal-time localization of multiple audio sources in a wirelessacoustic sensor networkrdquo in Proceedings of the 22nd Euro-pean Signal Processing Conference (EUSIPCO rsquo14) pp 306ndash310September 2014

[72] A Alexandridis G Borboudakis and AMouchtaris ldquoAddress-ing the data-association problem for multiple sound sourcelocalization using DOA estimatesrdquo in Proceedings of the 23rdEuropean Signal Processing Conference (EUSIPCO rsquo15) pp 1551ndash1555 August 2015

[73] L M Kaplan P Molnar and Q Le ldquoBearings-only target local-ization for an acoustical unattended ground sensor networkrdquoin Unattended Ground Sensor Technologies and Applications IIIvol 4393 of Proceedings of SPIE pp 40ndash51 April 2001

[74] K R Pattipati S Deb Y Bar-Shalom and R B Washburn ldquoAnew relaxation algorithm and passive sensor data associationrdquoIEEE Transactions on Automatic Control vol 37 no 2 pp 198ndash213 1992

[75] S Deb M Yeddanapudi K Pattipaii and Y Bar-ShalomldquoA generalized S-D assignment algorithm for multisensor-multitarget state estimationrdquo IEEE Transactions on Aerospaceand Electronic Systems vol 33 no 2 pp 523ndash538 1997

[76] R L Popp K R Pattipati and Y Bar-Shalom ldquoM-best S-Dassignment algorithm with application to multitarget trackingrdquo


IEEE Transactions on Aerospace and Electronic Systems vol 37no 1 pp 22ndash39 2001

[77] J Reed C da Silva and R Buehrer ldquoMultiple-source local-ization using line-of-bearing measurements approaches to thedata association problemrdquo in Proceedings of the IEEE MilitaryCommunications Conference (MILCOM rsquo08) pp 1ndash7 November2008

[78] M Swartling B Sallberg and N Grbic ldquoSource localization formultiple speech sources using low complexity non-parametricsource separation and clusteringrdquo Signal Processing vol 91 no8 pp 1781ndash1788 2011

[79] M Cobos and J J Lopez ldquoMaximum a posteriori binarymask estimation for underdetermined source separation usingsmoothed posteriorsrdquo IEEE Transactions on Audio Speech andLanguage Processing vol 20 no 7 pp 2059ndash2064 2012

[80] S Schulz and T Herfet ldquoOn the window-disjoint-orthogonalityof speech sources in reverberant humanoid scenariosrdquo inProceedings of the 11th International Conference on Digital AudioEffects (DAFx rsquo08) pp 241ndash248 September 2008

[81] S T Roweis ldquoFactorialmodels and refiltering for speech separa-tion and denoisingrdquo in Proceedings of the European Conferenceon Speech Communication (EUROSPEECH rsquo03) 2003

[82] M Taseska and E A P Habets ldquoInformed spatial filtering forsound extraction using distributed microphone arraysrdquo IEEETransactions on Audio Speech and Language Processing vol 22no 7 pp 1195ndash1207 2014

[83] A Alexandridis and A Mouchtaris ldquoMultiple sound sourcelocation estimation and counting in a wireless acoustic sensornetworkrdquo in Proceedings of the IEEE Workshop on Applicationsof Signal Processing to Audio and Acoustics (WASPAA rsquo15) pp1ndash5 October 2015

[84] A Alexandridis N Stefanakis and A Mouchtaris ldquoTowardswireless acoustic sensor networks for location estimation andcounting of multiple speakers in real-life conditionsrdquo in Pro-ceedings of the IEEE International Conference on AcousticsSpeech and Signal Processing (ICASSP rsquo17) pp 6140ndash6144 NewOrleans LA USA March 2017

[85] J H DiBiase A high-accuracy low-latency technique for talkerlocalization in reverberant environments using microphonearrays [PhD dissertation] Brown University Providence RIUSA 2000

[86] J H DiBiase H F Silverman and M S Brandstein ldquoRobustlocalization in reverberant roomsrdquo inMicrophone Arrays Tech-niques and Applications pp 157ndash180 Springer Berlin Germany2001

[87] M Cobos A Marti and J J Lopez ldquoA modified SRP-PHATfunctional for robust real-time sound source localization withscalable spatial samplingrdquo IEEE Signal Processing Letters vol 18no 1 pp 71ndash74 2011

[88] M Omologo P Svaizer and R DeMori ldquoAcoustic transduc-tionrdquo in Spoken Dialogue with Computers chapter 2 pp 23ndash64Academic Press San Diego California USA 1998

[89] S Astapov J Berdnikova and J-S Preden ldquoOptimized acousticlocalization with SRP-PHAT for monitoring in distributed sen-sor networksrdquo International Journal of Electronics and Telecom-munications vol 59 no 4 pp 383ndash390 2013

[90] S Astapov J-S Preden and J Berdnikova ldquoSimplified acousticlocalization by linear arrays for wireless sensor networksrdquo inProceedings of the 18th International Conference onDigital SignalProcessing (DSP rsquo13) pp 1ndash6 July 2013

[91] S Astapov J Berdnikova and J-S Preden ldquoA two-stageapproach to 2D DOA estimation for a compact circular micro-phone arrayrdquo in Proceedings of the 4th International Conferenceon Informatics Electronics and Vision (ICIEV rsquo15) pp 1ndash6 June2015

[92] H Do H F Silverman and Y Yu ldquoA real-time SRP-PHATsource location implementation using stochastic region con-traction (SRC) on a large-aperture microphone arrayrdquo inProceedings of the IEEE International Conference on AcousticsSpeech and Signal Processing (ICASSP rsquo07) vol 1 pp Indash121ndashIndash124 April 2007

[93] HDo andH F Silverman ldquoA fastmicrophone array SRP-PHATsource location implementation using coarse-to-fine regioncontraction (CFRC)rdquo in Proceedings of the IEEE Workshopon Applications of Signal Processing to Audio and Acoustics(WASPAA rsquo07) pp 295ndash298 October 2007

[94] L O Nunes W A Martins M V Lima et al ldquoA steered-response power algorithm employing hierarchical search foracoustic source localization using microphone arraysrdquo IEEETransactions on Signal Processing vol 62 no 19 pp 5171ndash51832014

[95] A Marti M Cobos J J Lopez and J Escolano ldquoA steeredresponse power iterative method for high-accuracy acousticsource localizationrdquo Journal of the Acoustical Society of Americavol 134 no 4 pp 2627ndash2630 2013

[96] T Lee S Chang and D Yook ldquoParallel SRP-PHAT for GPUsrdquoComputer Speech amp Language vol 35 pp 1ndash13 2016

[97] V P Minotto C R Jung L G Da Silveira Jr and B LeeldquoGPU-based approaches for real-time sound source localizationusing the SRP-PHAT algorithmrdquo International Journal of HighPerformance Computing Applications vol 27 no 3 pp 291ndash3062013

[98] J A Belloch A Gonzalez A M Vidal and M CobosldquoOn the performance of multi-GPU-based expert systems foracoustic localization involving massive microphone arraysrdquoExpert Systems with Applications vol 42 no 13 pp 5607ndash56202015

[99] J Velasco C J Martın-Arguedas J Macias-Guarasa D Pizarroand M Mazo ldquoProposal and validation of an analytical genera-tive model of SRP-PHAT powermaps in reverberant scenariosrdquoSignal Processing vol 119 pp 209ndash228 2016

[100] M Cobos M Garcia-Pineda and M Arevalillo-HerraezldquoSteered response power localization of acoustic passbandsignalsrdquo IEEE Signal Processing Letters vol 24 no 5 pp 717ndash721 2017

[101] F J Alvarez T Aguilera and R Lopez-Valcarce ldquoCDMA-basedacoustic local positioning system for portable devices withmultipath cancellationrdquo Digital Signal Processing vol 62 pp38ndash51 2017

[102] V C Raykar and R Duraiswami ldquoAutomatic position calibra-tion of multiple microphonesrdquo in Proceedings of the Interna-tional Conference on Acoustics Speech and Signal Processing(ICASSP rsquo04) vol 4 pp 69ndash72 IEEE May 2004

[103] J M Sachar H F Silverman andW R Patterson ldquoMicrophoneposition and gain calibration for a large-aperture microphonearrayrdquo IEEE Transactions on Speech and Audio Processing vol13 no 1 pp 42ndash52 2005

[104] S T Birchfield andA Subramanya ldquoMicrophone array positioncalibration by basis-point classical multidimensional scalingrdquoIEEE Transactions on Speech and Audio Processing vol 13 no 5pp 1025ndash1034 2005


[105] I McCowan M Lincoln and I Himawan ldquoMicrophone arrayshape calibration in diffuse noise fieldsrdquo IEEE Transactions onAudio Speech and Language Processing vol 16 no 3 pp 666ndash670 2008

[106] C Peng G Shen Y Zhang Y Li and K Tan ldquoBeepBeepa high accuracy acoustic ranging system using COTS mobiledevicesrdquo in Proceedings of the 5th ACM International Conferenceon Embedded Networked Sensor Systems (SenSysrsquo07) pp 1ndash14November 2007

[107] M H Hennecke and G A Fink ldquoTowards acoustic self-localization of ad hoc smartphone arraysrdquo in Proceedings of the3rd Joint Workshop on Hands-free Speech Communication andMicrophone Arrays (HSCMA rsquo11) pp 127ndash132 Edinburgh UKMay-June 2011

[108] P Stoica and J Li ldquoSource Localization from Range-DifferenceMeasurementsrdquo IEEE Signal Processing Magazine vol 23 no 6pp 63ndash66 2006

[109] F J Alvarez and R Lopez-Valcarce ldquoMultipath cancellation inbroadband acoustic local positioning systemsrdquo in Proceedingsof the 9th IEEE International Symposium on Intelligent SignalProcessing (WISP rsquo15) pp 1ndash6 IEEE Siena Italy May 2015

[110] C Peng G Shen and Y Zhang ldquoBeepBeep a high-accuracyacoustic-based system for ranging and localization using COTSdevicesrdquo Transactions on Embedded Computing Systems vol 11no 1 pp 41ndash429 2012

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Active and Passive Electronic Components

Control Scienceand Engineering

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Submit your manuscripts athttpswwwhindawicom

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

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Electrical and Computer Engineering

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



and multipath In the context of acoustic signal processingWASNs also provide advantages with respect to traditional(wired) microphone arrays [8] For example they enableincreased spatial coverage by distributing microphone nodesover a larger volume a scalable structure and possibly bettersignal-to-noise ratio (SNR) properties In fact since theranging accuracy depends on both the signal propagationspeed and the precision of the TOA measurement acousticsignals may be preferred with respect to radio signals [9]

There are two typical localization tasks in WASNs thelocalization of one or more sound sources of interest and thelocalization of the nodes that make up the network The firstcase may involve locating the position of unknown soundsources for example talkers inside a room or unexpectedacoustic events or sometimes other devices emitting knownbeacon signals The second case is usually related to the self-calibration or automatic ranging of the nodes themselves

To estimate the locations of the sound sources that areactive in an acoustic environment monitored by a WASNdifferent methods exist in the literature Usually a centralizedscheme is adopted where a dedicated node known as thefusion center is responsible for performing the localizationtask based on information it receives from the rest of nodesThe sensor network itself poses many limitations and chal-lenges that must be considered when designing a localizationapproach in order to facilitate its use in practical scenariosSuch challenges include the bandwidth usage which limitsthe amount of information that can be transmitted in thenetwork and the limited processing power of the nodeswhich prohibits them from carrying out very complex andcomputationally expensive operations Moreover each nodehas its own clock for sampling the signals and since the nodesoperate individually the resulting audio in the network willnot be synchronized

A taxonomy of sound source localization methods canbe built up upon the nature of information from the sensorsthat it is utilized in order to estimate the locations Hencethe WASN can estimate the locations of the acoustic sourcesbased on (i) energy readings (ii) time-of-arrival (TOA)measurements (iii) time-difference-of-arrival (TDOA) mea-surements and (iv) direction-of-arrival estimates or (v) byutilizing the steered response power (SRP) function

In DOA-based approaches each node estimates the DOAof the sources it can detect and transmits the DOA estimatesto the fusion center Although such approaches requireincreased computational power and multiple microphonesin each node they can attain very low-bandwidth usage asonly the DOA estimates need to be transmitted Also sincethe DOA estimation is carried out in each node individuallythe audio signals at different nodes need not be synchronizedDOA-based approaches can tolerate unsynchronized input aslong as the sources move at a rather slow rate relative to theanalysis frameThe location estimators are generally based onestimating the location of a sound source as the intersectionof lines emanating from the nodes at the direction ofthe estimated DOA However for multiple sources severalchallenges arise the number of detected sources (and thusthe number of DOA estimates) in each time instant can varyacross the nodes due to missed detections (ie a source is not

detected by a node) or false-alarms (ie overestimation of thenumber of detected sources) and an association procedure isneeded to find the DOA combinations that correspond to thesame source This is known as the data-association problemand is crucial for the localization task

The TDOA is related to the difference in the time of flight(TOF) of the wavefront produced by the source at a pair ofmicrophones in the same node TDOAs can be estimatedat a moderate computational cost through the generalizedcross correlation (GCC) [10] of the signals acquired bymicrophones in the pair The source location estimate isaccomplished by combining TDOA measurements comingfrom multiple sensors Notice that as for the DOA only theTDOA measurements must be transmitted over the wirelessnetwork with clear advantages in terms of transmissionpower and required bandwidth Though suitable for WASNsin practical scenarios (reverberant environments presence ofnoise and interferers) TDOA measurements are prone toerrors which in their turn lead towrong localization In orderto mitigate the impact of these adverse phenomena severaltechniques have been presented with the aim of identifyingand removing outliers in the TDOA set [11ndash13]

A TDOA measurement bounds the source to lie ona branch of hyperbola whose vertices are in microphonepositions and whose aperture is determined by the TDOAvalue When two (three in 3D) measurements from differentpairs are available the source can be localized through inter-section of hyperbolasThe resulting cost function however isstrongly nonlinear and therefore its minimization is difficultand prone to errors Linearized cost functions have beenproposed to overcome this difficulty [14ndash16] It is importantto notice however that the linearized cost functions requirethe presence of a reference microphone with respect towhich the remaining microphones in all the sensors must besynchronized This poses technological constraints that insome cases hinder the use of such techniques More recentlymethodologies that include the synchronization offset in theoptimization of the cost function have been proposed toovercome this problem [12 17]

TOA measurements are obtained by detecting the timeinstant at which the source signal arrives at the microphonespresent in the network Since in passive source localizationthe source emission time is unknown the TOA is notequivalent to the TOF of the signal preventing a directmapping from TOAs to source-to-node distances Whilesome applications involve the use of sound-emitting nodesthat allows performing localization by using trilaterationtechniques TDOA localization methods are usually chosenIn this case although the source emission time does notneed to be known the registered TOAs need to be referencedto a common clock requiring precise timing hardware andsynchronization mechanisms

Energy-based localization relies on the averaged energyreadings computed over windows of signal samples acquiredby the microphones incorporated by the nodes [18]Compared to TDOA and DOA methods energy-basedapproaches are attractive because they do not require theuse of multiple microphones at the nodes and are freeof synchronization issues unlike those based on TOA


m1m1(1)

m2(1)

m3(1)

m1(2)

m2(2)

m3(2)

m1(3)

m2(3)

m3(w)

q1

q2

q3

xs

2

3y1

y2

y3

r1(1)

r3(1)

1(1)3

(1)

13(1)















119910119894 (119897) = 119892119894 119868 (119897 minus 120578119894)1199032119894 + 120598119894 (119897) = 119904119894 (119897) + 120598119894 (119897) (1)



119910119894 (119897) asymp 1119879119897+1198792sum119897minus1198792

1199092119894 (119905) (2)



119910119894 (119897) asymp 119892119894 119868 (119897)1199032119894 + 120598119894 (119897) (3)


120581119894119895 ≜ (119910119894119910119895119892119894119892119895)minus12 = 1003817100381710038171003817x119904 minusm119894

100381710038171003817100381710038171003817100381710038171003817x119904 minusm119895

10038171003817100381710038171003817 = 119903119894119903119895 (4)


10038171003817100381710038171003817x minus c119894119895100381710038171003817100381710038172 = 9848582119894119895 (5)


100381710038171003817100381710038171 minus 1205812119894119895 (6)


w119879119894119895x = 120595119894119895 (7)




100

200

300

400

500

600


x (m)

minus10

minus5

0

5

10

15

y(m

)




119869(ER)NLS (x) = 1198751sum1199011=1


1199012=1

10038161003816100381610038161003816w1198791199012x minus 1205951199012

100381610038161003816100381610038162 (8)






119869(ER)LS (x) = 1198751sum1199011=1

10038161003816100381610038161003816u1198791199011x minus 1205771199011 100381610038161003816100381610038162 + 1198752sum1199012=1

10038161003816100381610038161003816w1198791199012x minus 1205951199012

100381610038161003816100381610038162 (11)

where


100381710038171003817100381721 minus 1205812119894 minus 10038171003817100381710038171003817m119895

1003817100381710038171003817100381721 minus 1205812119895 (12)




120578119894 ≜ 1119888 1003817100381710038171003817m119894 minus x1199041003817100381710038171003817 (13)





120591119894 = 120578119894 + 119905119904 + 120598119894 (14)




119903119894 = 119888 sdot 120591119894 (15)


1003817100381710038171003817m119894 minus x11990410038171003817100381710038172 = 1199032119894 119894 = 1 119873 (16)



119869(TOA)NLS (x) = 119873sum119894=1


(17)



120591119894 = arg max1le120591le119896

119896sum119905=120591

ln(119901 (119909119894 (119905) 1205791)119901 (119909119894 (119905) 1205790)) (18)



EW event warning

EW

EW

EW

EW

Central node

(a)

Central node

TOA

TOA

TOA

TOATOA

(b)








119894

10038171003817100381710038171003817 minus 10038171003817100381710038171003817x119904 minusm(119898)119895

10038171003817100381710038171003817119888 119894 = 1 119873 119895 = 1 119873 119894 = 119895 119898 = 1 119872

(19)


119909119894 (119905) = 120572119894119904 (119905 minus 120578119894) + 120598119894 (119905) 119894 = 1 119873 (20)




119883119894 (120596) = 120572119894119878 (120596) 119890minus119895120596120578119894 + 119864119894 (120596) 119894 = 1 119873 (21)



119877119894119895 (120591) ≜ 12120587 int120587

minus120587119883119894 (120596)119883lowast

119895 (120596)Ψ119894119895 (120596) 119890119895120596120591119889120596 (22)



120591119894119895 = argmax120591 119877119894119895 (120591)119865119904 (23)


Ψ119894119895 (120596) = 110038161003816100381610038161003816119883119894 (120596)119883lowast119895 (120596)10038161003816100381610038161003816 (24)






minus05

0

05

1

GCC

-PH

AT

minus05

0

05

1

GCC

-PH

AT





minus05

0

05

1

GCC

-PH

AT

GCC-PHATGTPeak



minus05

0

05

1

GCC

-PH

AT








120591119894119895 + 120591119895119896 + 120591119896119894 = 0 (26)








119894 andm(119898)119895 is written

as

119886(119898)119894119895 1199092 + 119887(119898)119894119895 119909119910 + 119888(119898)119894119895 1199102 + 119889(119898)119894119895 119909 + 119890(119898)119894119895 119910 + 119891(119898)119894119895 = 0 (27)




xs

mk(l)

mm(l)

mj(l)



120576(119898)119894119895 (x) = V(119898)119894119895 (119886(119898)119894119895 1199092 + 119887(119898)119894119895 119909119910 + 119888(119898)119894119895 1199102 + 119889(119898)119894119895 119909+ 119890(119898)119894119895 119910 + 119891(119898)

119894119895 ) (28)


119869(TDOA)HYP (x) = 120576 (x)119879 120576 (x) (29)




1003817100381710038171003817x1199041003817100381710038171003817 minus 119903119895 = 10038171003817100381710038171003817x119904 minusm119895

10038171003817100381710038171003817 (30)


119903 = 1003817100381710038171003817x119904 minus x1003817100381710038171003817 minus 1003817100381710038171003817x1199041003817100381710038171003817 (31)


C1

Cs

x

r

C2

y

[xs rs]T

lowast



119903119904 = minus 1003817100381710038171003817x1199041003817100381710038171003817 (32)



119903119904 = minus 1003817100381710038171003817x1199041003817100381710038171003817 (33)




1199092119904 + 1199102119904 minus 1199032119904 = 0 (34)






119869(TDOA)SP (x119904 119903119904) = 119873119872sum

119895=2

1198902119895 (37)










4 3

1 21 2

34








4 3

1 21

2

34



119869(DOA)ML (x) = 119872sum

119898=1

11205902119898 (120579119898 minus 120579119898 (x))2 (41)



119869(DOA)NLS (x) = 119872sum

119898=1

(120579119898 minus 120579119898 (x))2 (42)







where

A = [[[[[


]]]]]

b = [[[[[


]]]]]

W = [[[[[[[

1199032112059021 01199032212059022

d

0 11990321198721205902119872

]]]]]]]

(44)




x(OV)119904 = (A119879A)minus1 A119879b (46)





119869(DOA)GB (x) = 119872sum

119898=1

[119860 (120579119898 120579119898 (x))]2 (47)


119869(DOA)GB (x) (48)





1 2





119873comb = 119878prod119904=1

119904119862119904 (49)


119869(DOA)P-NLS (x) = 119872sum

119898=1

min119894

10038161003816100381610038161003816120579119898119894 minus 120579119898 (x)100381610038161003816100381610038162 (50)









0

50

100

150

200

250

300

350

400

y(c

m)

50 100 150 200 250 300 350 4000

x (cm)

(a)

500

400

300

200

100

00

50

100

150

200

250

300

350

400

y(c

m)

50 100 150 200 250 300 350 4000

x (cm)

(b)




119860(120601 (120596 119896) 120579119904 (119896)) lt 119860 (120601 (120596 119896) 120579119902 (119896)) forall119902 = 119904 (51)

119860(120601 (120596 119896) 120579119904 (119896)) lt 120598 (52)










119869(SRP)119862 (x) = 2119879119873sum119894=1

119873sum119895=119894+1

119877119894119895 (120591119894119895 (x)) + 119873sum119894=1

119877119894119894 (0) (53)



10038171003817100381710038171003817119888 (54)




119869(SRP)119862 (x) (55)



119869(SRP)119872 (x) = 119873sum119894=1

119873sum119895=119894+1

119871119906119894119895(x)sum120591=119871119897119894119895(x)

119877119894119895 (120591) (56)




10038171003817100381710038171003817) (57)





(59)



xisinG119869(SRP)119872 (x) (60)



x (m)

C-SRP (rg = 1 cm)

4353252151050

10

08

06

04

02

000

05

1

15

2

25

3

35

4

y(m

)

(a)

x (m)4353252151050

10

08

06

04

02

000

05

1

15

2

25

3

35

4

y(m

)

C-SRP (rg = 10 cm)

(b)

x (m)4353252151050

10

08

06

04

02

000

05

1

15

2

25

3

35

4

y(m

)

M-SRP (rg = 10 cm)

(c)









x2

x1 x3

x4

m

x = 12 my = 25 mz = 07 m












120578119904 ≜ 1119888 1003817100381710038171003817m minus x1199041003817100381710038171003817 (61)



12059111990411199042 ≜ 1205781199041 minus 1205781199042 =10038171003817100381710038171003817m minus x1199041

10038171003817100381710038171003817 minus 10038171003817100381710038171003817m minus x119904210038171003817100381710038171003817119888 (62)




120591119904 = 120578119904 + Δ120578119904 (63)




120591119904 = 120578119904 + Δ120578119904119865119904 + Δ119865119904 = 120578119904 + Δ120578119904119865119904 ( 11 + Δ119865119904119865119904) (64)




1003817100381710038171003817 = 119888120578119904 asymp 119886120591119904 + 119887 (66)





119869(TOA)LS (m 119886 119887) = 10038171003817100381710038171003817e(TOA)100381710038171003817100381710038172 (68)


m(TOA)LS = argmin

m119886119887119869(TOA)LS (m 119886 119887) (69)



119886120591119904 + 119887 = (119886120591119904 minus 1198861205911) + (1198861205911 + 119887) asymp 1198861205911199041 + m (70)



119869(TDOA)LS (m 119886) = 10038171003817100381710038171003817e(TDOA)

LS100381710038171003817100381710038172 (71)


m(TDOA)LS = argmin

m119886119869(TDOA)LS (m 119886) (72)



e(TOA) = [[[[[

1 2x1198791 12059121 21205911 1 2x119879119878 1205912119878 2120591119904

]]]]][[[[[[

1198872 minus m2m1198862119886119887

]]]]]]minus [[[[[

1003817100381710038171003817x1100381710038171003817100381721003817100381710038171003817x11987810038171003817100381710038172]]]]]

e(TDOA) = [[[[[

212059121 2x1198792 120591221 21205911198781 2x119879119878 12059121198781

]]]]][[[[m 119886m1198862

]]]]minus [[[[[

1003817100381710038171003817x2100381710038171003817100381721003817100381710038171003817x11987810038171003817100381710038172]]]]]

(73)


minΘ

e2 (74)


Θ ≜ (H119879H)minus1H119879g (75)



e = [[[[[

211988812059121 2x1198792 21198881205911198781 2x119879119878

]]]]][ m

m] minus [[[[[

1003817100381710038171003817x210038171003817100381710038172 minus 11988821205912211003817100381710038171003817x11987810038171003817100381710038172 minus 119888212059121198781]]]]]

(76)


















10 Conclusion






References





















































































































RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










m1m1(1)

m2(1)

m3(1)

m1(2)

m2(2)

m3(2)

m1(3)

m2(3)

m3(w)

q1

q2

q3

xs

2

3y1

y2

y3

r1(1)

r3(1)

1(1)3

(1)

13(1)















119910119894 (119897) = 119892119894 119868 (119897 minus 120578119894)1199032119894 + 120598119894 (119897) = 119904119894 (119897) + 120598119894 (119897) (1)



119910119894 (119897) asymp 1119879119897+1198792sum119897minus1198792

1199092119894 (119905) (2)



119910119894 (119897) asymp 119892119894 119868 (119897)1199032119894 + 120598119894 (119897) (3)


120581119894119895 ≜ (119910119894119910119895119892119894119892119895)minus12 = 1003817100381710038171003817x119904 minusm119894

100381710038171003817100381710038171003817100381710038171003817x119904 minusm119895

10038171003817100381710038171003817 = 119903119894119903119895 (4)


10038171003817100381710038171003817x minus c119894119895100381710038171003817100381710038172 = 9848582119894119895 (5)


100381710038171003817100381710038171 minus 1205812119894119895 (6)


w119879119894119895x = 120595119894119895 (7)




100

200

300

400

500

600


x (m)

minus10

minus5

0

5

10

15

y(m

)




119869(ER)NLS (x) = 1198751sum1199011=1


1199012=1

10038161003816100381610038161003816w1198791199012x minus 1205951199012

100381610038161003816100381610038162 (8)






119869(ER)LS (x) = 1198751sum1199011=1

10038161003816100381610038161003816u1198791199011x minus 1205771199011 100381610038161003816100381610038162 + 1198752sum1199012=1

10038161003816100381610038161003816w1198791199012x minus 1205951199012

100381610038161003816100381610038162 (11)

where


100381710038171003817100381721 minus 1205812119894 minus 10038171003817100381710038171003817m119895

1003817100381710038171003817100381721 minus 1205812119895 (12)




120578119894 ≜ 1119888 1003817100381710038171003817m119894 minus x1199041003817100381710038171003817 (13)





120591119894 = 120578119894 + 119905119904 + 120598119894 (14)




119903119894 = 119888 sdot 120591119894 (15)


1003817100381710038171003817m119894 minus x11990410038171003817100381710038172 = 1199032119894 119894 = 1 119873 (16)



119869(TOA)NLS (x) = 119873sum119894=1


(17)



120591119894 = arg max1le120591le119896

119896sum119905=120591

ln(119901 (119909119894 (119905) 1205791)119901 (119909119894 (119905) 1205790)) (18)



EW event warning

EW

EW

EW

EW

Central node

(a)

Central node

TOA

TOA

TOA

TOATOA

(b)








119894

10038171003817100381710038171003817 minus 10038171003817100381710038171003817x119904 minusm(119898)119895

10038171003817100381710038171003817119888 119894 = 1 119873 119895 = 1 119873 119894 = 119895 119898 = 1 119872

(19)


119909119894 (119905) = 120572119894119904 (119905 minus 120578119894) + 120598119894 (119905) 119894 = 1 119873 (20)




119883119894 (120596) = 120572119894119878 (120596) 119890minus119895120596120578119894 + 119864119894 (120596) 119894 = 1 119873 (21)



119877119894119895 (120591) ≜ 12120587 int120587

minus120587119883119894 (120596)119883lowast

119895 (120596)Ψ119894119895 (120596) 119890119895120596120591119889120596 (22)



120591119894119895 = argmax120591 119877119894119895 (120591)119865119904 (23)


Ψ119894119895 (120596) = 110038161003816100381610038161003816119883119894 (120596)119883lowast119895 (120596)10038161003816100381610038161003816 (24)






minus05

0

05

1

GCC

-PH

AT

minus05

0

05

1

GCC

-PH

AT





minus05

0

05

1

GCC

-PH

AT

GCC-PHATGTPeak



minus05

0

05

1

GCC

-PH

AT








120591119894119895 + 120591119895119896 + 120591119896119894 = 0 (26)








119894 andm(119898)119895 is written

as

119886(119898)119894119895 1199092 + 119887(119898)119894119895 119909119910 + 119888(119898)119894119895 1199102 + 119889(119898)119894119895 119909 + 119890(119898)119894119895 119910 + 119891(119898)119894119895 = 0 (27)




xs

mk(l)

mm(l)

mj(l)



120576(119898)119894119895 (x) = V(119898)119894119895 (119886(119898)119894119895 1199092 + 119887(119898)119894119895 119909119910 + 119888(119898)119894119895 1199102 + 119889(119898)119894119895 119909+ 119890(119898)119894119895 119910 + 119891(119898)

119894119895 ) (28)


119869(TDOA)HYP (x) = 120576 (x)119879 120576 (x) (29)




1003817100381710038171003817x1199041003817100381710038171003817 minus 119903119895 = 10038171003817100381710038171003817x119904 minusm119895

10038171003817100381710038171003817 (30)


119903 = 1003817100381710038171003817x119904 minus x1003817100381710038171003817 minus 1003817100381710038171003817x1199041003817100381710038171003817 (31)


C1

Cs

x

r

C2

y

[xs rs]T

lowast



119903119904 = minus 1003817100381710038171003817x1199041003817100381710038171003817 (32)



119903119904 = minus 1003817100381710038171003817x1199041003817100381710038171003817 (33)




1199092119904 + 1199102119904 minus 1199032119904 = 0 (34)






119869(TDOA)SP (x119904 119903119904) = 119873119872sum

119895=2

1198902119895 (37)










4 3

1 21 2

34








4 3

1 21

2

34



119869(DOA)ML (x) = 119872sum

119898=1

11205902119898 (120579119898 minus 120579119898 (x))2 (41)



119869(DOA)NLS (x) = 119872sum

119898=1

(120579119898 minus 120579119898 (x))2 (42)







where

A = [[[[[


]]]]]

b = [[[[[


]]]]]

W = [[[[[[[

1199032112059021 01199032212059022

d

0 11990321198721205902119872

]]]]]]]

(44)




x(OV)119904 = (A119879A)minus1 A119879b (46)





119869(DOA)GB (x) = 119872sum

119898=1

[119860 (120579119898 120579119898 (x))]2 (47)


119869(DOA)GB (x) (48)





1 2





119873comb = 119878prod119904=1

119904119862119904 (49)


119869(DOA)P-NLS (x) = 119872sum

119898=1

min119894

10038161003816100381610038161003816120579119898119894 minus 120579119898 (x)100381610038161003816100381610038162 (50)









0

50

100

150

200

250

300

350

400

y(c

m)

50 100 150 200 250 300 350 4000

x (cm)

(a)

500

400

300

200

100

00

50

100

150

200

250

300

350

400

y(c

m)

50 100 150 200 250 300 350 4000

x (cm)

(b)




119860(120601 (120596 119896) 120579119904 (119896)) lt 119860 (120601 (120596 119896) 120579119902 (119896)) forall119902 = 119904 (51)

119860(120601 (120596 119896) 120579119904 (119896)) lt 120598 (52)










119869(SRP)119862 (x) = 2119879119873sum119894=1

119873sum119895=119894+1

119877119894119895 (120591119894119895 (x)) + 119873sum119894=1

119877119894119894 (0) (53)



10038171003817100381710038171003817119888 (54)




119869(SRP)119862 (x) (55)



119869(SRP)119872 (x) = 119873sum119894=1

119873sum119895=119894+1

119871119906119894119895(x)sum120591=119871119897119894119895(x)

119877119894119895 (120591) (56)




10038171003817100381710038171003817) (57)





(59)



xisinG119869(SRP)119872 (x) (60)



x (m)

C-SRP (rg = 1 cm)

4353252151050

10

08

06

04

02

000

05

1

15

2

25

3

35

4

y(m

)

(a)

x (m)4353252151050

10

08

06

04

02

000

05

1

15

2

25

3

35

4

y(m

)

C-SRP (rg = 10 cm)

(b)

x (m)4353252151050

10

08

06

04

02

000

05

1

15

2

25

3

35

4

y(m

)

M-SRP (rg = 10 cm)

(c)









x2

x1 x3

x4

m

x = 12 my = 25 mz = 07 m












120578119904 ≜ 1119888 1003817100381710038171003817m minus x1199041003817100381710038171003817 (61)



12059111990411199042 ≜ 1205781199041 minus 1205781199042 =10038171003817100381710038171003817m minus x1199041

10038171003817100381710038171003817 minus 10038171003817100381710038171003817m minus x119904210038171003817100381710038171003817119888 (62)




120591119904 = 120578119904 + Δ120578119904 (63)




120591119904 = 120578119904 + Δ120578119904119865119904 + Δ119865119904 = 120578119904 + Δ120578119904119865119904 ( 11 + Δ119865119904119865119904) (64)




1003817100381710038171003817 = 119888120578119904 asymp 119886120591119904 + 119887 (66)





119869(TOA)LS (m 119886 119887) = 10038171003817100381710038171003817e(TOA)100381710038171003817100381710038172 (68)


m(TOA)LS = argmin

m119886119887119869(TOA)LS (m 119886 119887) (69)



119886120591119904 + 119887 = (119886120591119904 minus 1198861205911) + (1198861205911 + 119887) asymp 1198861205911199041 + m (70)



119869(TDOA)LS (m 119886) = 10038171003817100381710038171003817e(TDOA)

LS100381710038171003817100381710038172 (71)


m(TDOA)LS = argmin

m119886119869(TDOA)LS (m 119886) (72)



e(TOA) = [[[[[

1 2x1198791 12059121 21205911 1 2x119879119878 1205912119878 2120591119904

]]]]][[[[[[

1198872 minus m2m1198862119886119887

]]]]]]minus [[[[[

1003817100381710038171003817x1100381710038171003817100381721003817100381710038171003817x11987810038171003817100381710038172]]]]]

e(TDOA) = [[[[[

212059121 2x1198792 120591221 21205911198781 2x119879119878 12059121198781

]]]]][[[[m 119886m1198862

]]]]minus [[[[[

1003817100381710038171003817x2100381710038171003817100381721003817100381710038171003817x11987810038171003817100381710038172]]]]]

(73)


minΘ

e2 (74)


Θ ≜ (H119879H)minus1H119879g (75)



e = [[[[[

211988812059121 2x1198792 21198881205911198781 2x119879119878

]]]]][ m

m] minus [[[[[

1003817100381710038171003817x210038171003817100381710038172 minus 11988821205912211003817100381710038171003817x11987810038171003817100381710038172 minus 119888212059121198781]]]]]

(76)


















10 Conclusion






References





















































































































RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation














119910119894 (119897) = 119892119894 119868 (119897 minus 120578119894)1199032119894 + 120598119894 (119897) = 119904119894 (119897) + 120598119894 (119897) (1)



119910119894 (119897) asymp 1119879119897+1198792sum119897minus1198792

1199092119894 (119905) (2)



119910119894 (119897) asymp 119892119894 119868 (119897)1199032119894 + 120598119894 (119897) (3)


120581119894119895 ≜ (119910119894119910119895119892119894119892119895)minus12 = 1003817100381710038171003817x119904 minusm119894

100381710038171003817100381710038171003817100381710038171003817x119904 minusm119895

10038171003817100381710038171003817 = 119903119894119903119895 (4)


10038171003817100381710038171003817x minus c119894119895100381710038171003817100381710038172 = 9848582119894119895 (5)


100381710038171003817100381710038171 minus 1205812119894119895 (6)


w119879119894119895x = 120595119894119895 (7)




100

200

300

400

500

600


x (m)

minus10

minus5

0

5

10

15

y(m

)




119869(ER)NLS (x) = 1198751sum1199011=1


1199012=1

10038161003816100381610038161003816w1198791199012x minus 1205951199012

100381610038161003816100381610038162 (8)






119869(ER)LS (x) = 1198751sum1199011=1

10038161003816100381610038161003816u1198791199011x minus 1205771199011 100381610038161003816100381610038162 + 1198752sum1199012=1

10038161003816100381610038161003816w1198791199012x minus 1205951199012

100381610038161003816100381610038162 (11)

where


100381710038171003817100381721 minus 1205812119894 minus 10038171003817100381710038171003817m119895

1003817100381710038171003817100381721 minus 1205812119895 (12)




120578119894 ≜ 1119888 1003817100381710038171003817m119894 minus x1199041003817100381710038171003817 (13)





120591119894 = 120578119894 + 119905119904 + 120598119894 (14)




119903119894 = 119888 sdot 120591119894 (15)


1003817100381710038171003817m119894 minus x11990410038171003817100381710038172 = 1199032119894 119894 = 1 119873 (16)



119869(TOA)NLS (x) = 119873sum119894=1


(17)



120591119894 = arg max1le120591le119896

119896sum119905=120591

ln(119901 (119909119894 (119905) 1205791)119901 (119909119894 (119905) 1205790)) (18)



EW event warning

EW

EW

EW

EW

Central node

(a)

Central node

TOA

TOA

TOA

TOATOA

(b)








119894

10038171003817100381710038171003817 minus 10038171003817100381710038171003817x119904 minusm(119898)119895

10038171003817100381710038171003817119888 119894 = 1 119873 119895 = 1 119873 119894 = 119895 119898 = 1 119872

(19)


119909119894 (119905) = 120572119894119904 (119905 minus 120578119894) + 120598119894 (119905) 119894 = 1 119873 (20)




119883119894 (120596) = 120572119894119878 (120596) 119890minus119895120596120578119894 + 119864119894 (120596) 119894 = 1 119873 (21)



119877119894119895 (120591) ≜ 12120587 int120587

minus120587119883119894 (120596)119883lowast

119895 (120596)Ψ119894119895 (120596) 119890119895120596120591119889120596 (22)



120591119894119895 = argmax120591 119877119894119895 (120591)119865119904 (23)


Ψ119894119895 (120596) = 110038161003816100381610038161003816119883119894 (120596)119883lowast119895 (120596)10038161003816100381610038161003816 (24)






minus05

0

05

1

GCC

-PH

AT

minus05

0

05

1

GCC

-PH

AT





minus05

0

05

1

GCC

-PH

AT

GCC-PHATGTPeak



minus05

0

05

1

GCC

-PH

AT








120591119894119895 + 120591119895119896 + 120591119896119894 = 0 (26)








119894 andm(119898)119895 is written

as

119886(119898)119894119895 1199092 + 119887(119898)119894119895 119909119910 + 119888(119898)119894119895 1199102 + 119889(119898)119894119895 119909 + 119890(119898)119894119895 119910 + 119891(119898)119894119895 = 0 (27)




xs

mk(l)

mm(l)

mj(l)



120576(119898)119894119895 (x) = V(119898)119894119895 (119886(119898)119894119895 1199092 + 119887(119898)119894119895 119909119910 + 119888(119898)119894119895 1199102 + 119889(119898)119894119895 119909+ 119890(119898)119894119895 119910 + 119891(119898)

119894119895 ) (28)


119869(TDOA)HYP (x) = 120576 (x)119879 120576 (x) (29)




1003817100381710038171003817x1199041003817100381710038171003817 minus 119903119895 = 10038171003817100381710038171003817x119904 minusm119895

10038171003817100381710038171003817 (30)


119903 = 1003817100381710038171003817x119904 minus x1003817100381710038171003817 minus 1003817100381710038171003817x1199041003817100381710038171003817 (31)


C1

Cs

x

r

C2

y

[xs rs]T

lowast



119903119904 = minus 1003817100381710038171003817x1199041003817100381710038171003817 (32)



119903119904 = minus 1003817100381710038171003817x1199041003817100381710038171003817 (33)




1199092119904 + 1199102119904 minus 1199032119904 = 0 (34)






119869(TDOA)SP (x119904 119903119904) = 119873119872sum

119895=2

1198902119895 (37)










4 3

1 21 2

34








4 3

1 21

2

34



119869(DOA)ML (x) = 119872sum

119898=1

11205902119898 (120579119898 minus 120579119898 (x))2 (41)



119869(DOA)NLS (x) = 119872sum

119898=1

(120579119898 minus 120579119898 (x))2 (42)







where

A = [[[[[


]]]]]

b = [[[[[


]]]]]

W = [[[[[[[

1199032112059021 01199032212059022

d

0 11990321198721205902119872

]]]]]]]

(44)




x(OV)119904 = (A119879A)minus1 A119879b (46)





119869(DOA)GB (x) = 119872sum

119898=1

[119860 (120579119898 120579119898 (x))]2 (47)


119869(DOA)GB (x) (48)





1 2





119873comb = 119878prod119904=1

119904119862119904 (49)


119869(DOA)P-NLS (x) = 119872sum

119898=1

min119894

10038161003816100381610038161003816120579119898119894 minus 120579119898 (x)100381610038161003816100381610038162 (50)









0

50

100

150

200

250

300

350

400

y(c

m)

50 100 150 200 250 300 350 4000

x (cm)

(a)

500

400

300

200

100

00

50

100

150

200

250

300

350

400

y(c

m)

50 100 150 200 250 300 350 4000

x (cm)

(b)




119860(120601 (120596 119896) 120579119904 (119896)) lt 119860 (120601 (120596 119896) 120579119902 (119896)) forall119902 = 119904 (51)

119860(120601 (120596 119896) 120579119904 (119896)) lt 120598 (52)










119869(SRP)119862 (x) = 2119879119873sum119894=1

119873sum119895=119894+1

119877119894119895 (120591119894119895 (x)) + 119873sum119894=1

119877119894119894 (0) (53)



10038171003817100381710038171003817119888 (54)




119869(SRP)119862 (x) (55)



119869(SRP)119872 (x) = 119873sum119894=1

119873sum119895=119894+1

119871119906119894119895(x)sum120591=119871119897119894119895(x)

119877119894119895 (120591) (56)




10038171003817100381710038171003817) (57)





(59)



xisinG119869(SRP)119872 (x) (60)



x (m)

C-SRP (rg = 1 cm)

4353252151050

10

08

06

04

02

000

05

1

15

2

25

3

35

4

y(m

)

(a)

x (m)4353252151050

10

08

06

04

02

000

05

1

15

2

25

3

35

4

y(m

)

C-SRP (rg = 10 cm)

(b)

x (m)4353252151050

10

08

06

04

02

000

05

1

15

2

25

3

35

4

y(m

)

M-SRP (rg = 10 cm)

(c)









x2

x1 x3

x4

m

x = 12 my = 25 mz = 07 m












120578119904 ≜ 1119888 1003817100381710038171003817m minus x1199041003817100381710038171003817 (61)



12059111990411199042 ≜ 1205781199041 minus 1205781199042 =10038171003817100381710038171003817m minus x1199041

10038171003817100381710038171003817 minus 10038171003817100381710038171003817m minus x119904210038171003817100381710038171003817119888 (62)




120591119904 = 120578119904 + Δ120578119904 (63)




120591119904 = 120578119904 + Δ120578119904119865119904 + Δ119865119904 = 120578119904 + Δ120578119904119865119904 ( 11 + Δ119865119904119865119904) (64)




1003817100381710038171003817 = 119888120578119904 asymp 119886120591119904 + 119887 (66)





119869(TOA)LS (m 119886 119887) = 10038171003817100381710038171003817e(TOA)100381710038171003817100381710038172 (68)


m(TOA)LS = argmin

m119886119887119869(TOA)LS (m 119886 119887) (69)



119886120591119904 + 119887 = (119886120591119904 minus 1198861205911) + (1198861205911 + 119887) asymp 1198861205911199041 + m (70)



119869(TDOA)LS (m 119886) = 10038171003817100381710038171003817e(TDOA)

LS100381710038171003817100381710038172 (71)


m(TDOA)LS = argmin

m119886119869(TDOA)LS (m 119886) (72)



e(TOA) = [[[[[

1 2x1198791 12059121 21205911 1 2x119879119878 1205912119878 2120591119904

]]]]][[[[[[

1198872 minus m2m1198862119886119887

]]]]]]minus [[[[[

1003817100381710038171003817x1100381710038171003817100381721003817100381710038171003817x11987810038171003817100381710038172]]]]]

e(TDOA) = [[[[[

212059121 2x1198792 120591221 21205911198781 2x119879119878 12059121198781

]]]]][[[[m 119886m1198862

]]]]minus [[[[[

1003817100381710038171003817x2100381710038171003817100381721003817100381710038171003817x11987810038171003817100381710038172]]]]]

(73)


minΘ

e2 (74)


Θ ≜ (H119879H)minus1H119879g (75)



e = [[[[[

211988812059121 2x1198792 21198881205911198781 2x119879119878

]]]]][ m

m] minus [[[[[

1003817100381710038171003817x210038171003817100381710038172 minus 11988821205912211003817100381710038171003817x11987810038171003817100381710038172 minus 119888212059121198781]]]]]

(76)


















10 Conclusion






References





















































































































RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










100

200

300

400

500

600


x (m)

minus10

minus5

0

5

10

15

y(m

)




119869(ER)NLS (x) = 1198751sum1199011=1


1199012=1

10038161003816100381610038161003816w1198791199012x minus 1205951199012

100381610038161003816100381610038162 (8)






119869(ER)LS (x) = 1198751sum1199011=1

10038161003816100381610038161003816u1198791199011x minus 1205771199011 100381610038161003816100381610038162 + 1198752sum1199012=1

10038161003816100381610038161003816w1198791199012x minus 1205951199012

100381610038161003816100381610038162 (11)

where


100381710038171003817100381721 minus 1205812119894 minus 10038171003817100381710038171003817m119895

1003817100381710038171003817100381721 minus 1205812119895 (12)




120578119894 ≜ 1119888 1003817100381710038171003817m119894 minus x1199041003817100381710038171003817 (13)





120591119894 = 120578119894 + 119905119904 + 120598119894 (14)




119903119894 = 119888 sdot 120591119894 (15)


1003817100381710038171003817m119894 minus x11990410038171003817100381710038172 = 1199032119894 119894 = 1 119873 (16)



119869(TOA)NLS (x) = 119873sum119894=1


(17)



120591119894 = arg max1le120591le119896

119896sum119905=120591

ln(119901 (119909119894 (119905) 1205791)119901 (119909119894 (119905) 1205790)) (18)



EW event warning

EW

EW

EW

EW

Central node

(a)

Central node

TOA

TOA

TOA

TOATOA

(b)








119894

10038171003817100381710038171003817 minus 10038171003817100381710038171003817x119904 minusm(119898)119895

10038171003817100381710038171003817119888 119894 = 1 119873 119895 = 1 119873 119894 = 119895 119898 = 1 119872

(19)


119909119894 (119905) = 120572119894119904 (119905 minus 120578119894) + 120598119894 (119905) 119894 = 1 119873 (20)




119883119894 (120596) = 120572119894119878 (120596) 119890minus119895120596120578119894 + 119864119894 (120596) 119894 = 1 119873 (21)



119877119894119895 (120591) ≜ 12120587 int120587

minus120587119883119894 (120596)119883lowast

119895 (120596)Ψ119894119895 (120596) 119890119895120596120591119889120596 (22)



120591119894119895 = argmax120591 119877119894119895 (120591)119865119904 (23)


Ψ119894119895 (120596) = 110038161003816100381610038161003816119883119894 (120596)119883lowast119895 (120596)10038161003816100381610038161003816 (24)






minus05

0

05

1

GCC

-PH

AT

minus05

0

05

1

GCC

-PH

AT





minus05

0

05

1

GCC

-PH

AT

GCC-PHATGTPeak



minus05

0

05

1

GCC

-PH

AT








120591119894119895 + 120591119895119896 + 120591119896119894 = 0 (26)








119894 andm(119898)119895 is written

as

119886(119898)119894119895 1199092 + 119887(119898)119894119895 119909119910 + 119888(119898)119894119895 1199102 + 119889(119898)119894119895 119909 + 119890(119898)119894119895 119910 + 119891(119898)119894119895 = 0 (27)




xs

mk(l)

mm(l)

mj(l)



120576(119898)119894119895 (x) = V(119898)119894119895 (119886(119898)119894119895 1199092 + 119887(119898)119894119895 119909119910 + 119888(119898)119894119895 1199102 + 119889(119898)119894119895 119909+ 119890(119898)119894119895 119910 + 119891(119898)

119894119895 ) (28)


119869(TDOA)HYP (x) = 120576 (x)119879 120576 (x) (29)




1003817100381710038171003817x1199041003817100381710038171003817 minus 119903119895 = 10038171003817100381710038171003817x119904 minusm119895

10038171003817100381710038171003817 (30)


119903 = 1003817100381710038171003817x119904 minus x1003817100381710038171003817 minus 1003817100381710038171003817x1199041003817100381710038171003817 (31)


C1

Cs

x

r

C2

y

[xs rs]T

lowast



119903119904 = minus 1003817100381710038171003817x1199041003817100381710038171003817 (32)



119903119904 = minus 1003817100381710038171003817x1199041003817100381710038171003817 (33)




1199092119904 + 1199102119904 minus 1199032119904 = 0 (34)






119869(TDOA)SP (x119904 119903119904) = 119873119872sum

119895=2

1198902119895 (37)










4 3

1 21 2

34








4 3

1 21

2

34



119869(DOA)ML (x) = 119872sum

119898=1

11205902119898 (120579119898 minus 120579119898 (x))2 (41)



119869(DOA)NLS (x) = 119872sum

119898=1

(120579119898 minus 120579119898 (x))2 (42)







where

A = [[[[[


]]]]]

b = [[[[[


]]]]]

W = [[[[[[[

1199032112059021 01199032212059022

d

0 11990321198721205902119872

]]]]]]]

(44)




x(OV)119904 = (A119879A)minus1 A119879b (46)





119869(DOA)GB (x) = 119872sum

119898=1

[119860 (120579119898 120579119898 (x))]2 (47)


119869(DOA)GB (x) (48)





1 2





119873comb = 119878prod119904=1

119904119862119904 (49)


119869(DOA)P-NLS (x) = 119872sum

119898=1

min119894

10038161003816100381610038161003816120579119898119894 minus 120579119898 (x)100381610038161003816100381610038162 (50)









0

50

100

150

200

250

300

350

400

y(c

m)

50 100 150 200 250 300 350 4000

x (cm)

(a)

500

400

300

200

100

00

50

100

150

200

250

300

350

400

y(c

m)

50 100 150 200 250 300 350 4000

x (cm)

(b)




119860(120601 (120596 119896) 120579119904 (119896)) lt 119860 (120601 (120596 119896) 120579119902 (119896)) forall119902 = 119904 (51)

119860(120601 (120596 119896) 120579119904 (119896)) lt 120598 (52)










119869(SRP)119862 (x) = 2119879119873sum119894=1

119873sum119895=119894+1

119877119894119895 (120591119894119895 (x)) + 119873sum119894=1

119877119894119894 (0) (53)



10038171003817100381710038171003817119888 (54)




119869(SRP)119862 (x) (55)



119869(SRP)119872 (x) = 119873sum119894=1

119873sum119895=119894+1

119871119906119894119895(x)sum120591=119871119897119894119895(x)

119877119894119895 (120591) (56)




10038171003817100381710038171003817) (57)





(59)



xisinG119869(SRP)119872 (x) (60)



x (m)

C-SRP (rg = 1 cm)

4353252151050

10

08

06

04

02

000

05

1

15

2

25

3

35

4

y(m

)

(a)

x (m)4353252151050

10

08

06

04

02

000

05

1

15

2

25

3

35

4

y(m

)

C-SRP (rg = 10 cm)

(b)

x (m)4353252151050

10

08

06

04

02

000

05

1

15

2

25

3

35

4

y(m

)

M-SRP (rg = 10 cm)

(c)









x2

x1 x3

x4

m

x = 12 my = 25 mz = 07 m












120578119904 ≜ 1119888 1003817100381710038171003817m minus x1199041003817100381710038171003817 (61)



12059111990411199042 ≜ 1205781199041 minus 1205781199042 =10038171003817100381710038171003817m minus x1199041

10038171003817100381710038171003817 minus 10038171003817100381710038171003817m minus x119904210038171003817100381710038171003817119888 (62)




120591119904 = 120578119904 + Δ120578119904 (63)




120591119904 = 120578119904 + Δ120578119904119865119904 + Δ119865119904 = 120578119904 + Δ120578119904119865119904 ( 11 + Δ119865119904119865119904) (64)




1003817100381710038171003817 = 119888120578119904 asymp 119886120591119904 + 119887 (66)





119869(TOA)LS (m 119886 119887) = 10038171003817100381710038171003817e(TOA)100381710038171003817100381710038172 (68)


m(TOA)LS = argmin

m119886119887119869(TOA)LS (m 119886 119887) (69)



119886120591119904 + 119887 = (119886120591119904 minus 1198861205911) + (1198861205911 + 119887) asymp 1198861205911199041 + m (70)



119869(TDOA)LS (m 119886) = 10038171003817100381710038171003817e(TDOA)

LS100381710038171003817100381710038172 (71)


m(TDOA)LS = argmin

m119886119869(TDOA)LS (m 119886) (72)



e(TOA) = [[[[[

1 2x1198791 12059121 21205911 1 2x119879119878 1205912119878 2120591119904

]]]]][[[[[[

1198872 minus m2m1198862119886119887

]]]]]]minus [[[[[

1003817100381710038171003817x1100381710038171003817100381721003817100381710038171003817x11987810038171003817100381710038172]]]]]

e(TDOA) = [[[[[

212059121 2x1198792 120591221 21205911198781 2x119879119878 12059121198781

]]]]][[[[m 119886m1198862

]]]]minus [[[[[

1003817100381710038171003817x2100381710038171003817100381721003817100381710038171003817x11987810038171003817100381710038172]]]]]

(73)


minΘ

e2 (74)


Θ ≜ (H119879H)minus1H119879g (75)



e = [[[[[

211988812059121 2x1198792 21198881205911198781 2x119879119878

]]]]][ m

m] minus [[[[[

1003817100381710038171003817x210038171003817100381710038172 minus 11988821205912211003817100381710038171003817x11987810038171003817100381710038172 minus 119888212059121198781]]]]]

(76)


















10 Conclusion






References





















































































































RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












120591119894 = 120578119894 + 119905119904 + 120598119894 (14)




119903119894 = 119888 sdot 120591119894 (15)


1003817100381710038171003817m119894 minus x11990410038171003817100381710038172 = 1199032119894 119894 = 1 119873 (16)



119869(TOA)NLS (x) = 119873sum119894=1


(17)



120591119894 = arg max1le120591le119896

119896sum119905=120591

ln(119901 (119909119894 (119905) 1205791)119901 (119909119894 (119905) 1205790)) (18)



EW event warning

EW

EW

EW

EW

Central node

(a)

Central node

TOA

TOA

TOA

TOATOA

(b)








119894

10038171003817100381710038171003817 minus 10038171003817100381710038171003817x119904 minusm(119898)119895

10038171003817100381710038171003817119888 119894 = 1 119873 119895 = 1 119873 119894 = 119895 119898 = 1 119872

(19)


119909119894 (119905) = 120572119894119904 (119905 minus 120578119894) + 120598119894 (119905) 119894 = 1 119873 (20)




119883119894 (120596) = 120572119894119878 (120596) 119890minus119895120596120578119894 + 119864119894 (120596) 119894 = 1 119873 (21)



119877119894119895 (120591) ≜ 12120587 int120587

minus120587119883119894 (120596)119883lowast

119895 (120596)Ψ119894119895 (120596) 119890119895120596120591119889120596 (22)



120591119894119895 = argmax120591 119877119894119895 (120591)119865119904 (23)


Ψ119894119895 (120596) = 110038161003816100381610038161003816119883119894 (120596)119883lowast119895 (120596)10038161003816100381610038161003816 (24)






minus05

0

05

1

GCC

-PH

AT

minus05

0

05

1

GCC

-PH

AT





minus05

0

05

1

GCC

-PH

AT

GCC-PHATGTPeak



minus05

0

05

1

GCC

-PH

AT








120591119894119895 + 120591119895119896 + 120591119896119894 = 0 (26)








119894 andm(119898)119895 is written

as

119886(119898)119894119895 1199092 + 119887(119898)119894119895 119909119910 + 119888(119898)119894119895 1199102 + 119889(119898)119894119895 119909 + 119890(119898)119894119895 119910 + 119891(119898)119894119895 = 0 (27)




xs

mk(l)

mm(l)

mj(l)



120576(119898)119894119895 (x) = V(119898)119894119895 (119886(119898)119894119895 1199092 + 119887(119898)119894119895 119909119910 + 119888(119898)119894119895 1199102 + 119889(119898)119894119895 119909+ 119890(119898)119894119895 119910 + 119891(119898)

119894119895 ) (28)


119869(TDOA)HYP (x) = 120576 (x)119879 120576 (x) (29)




1003817100381710038171003817x1199041003817100381710038171003817 minus 119903119895 = 10038171003817100381710038171003817x119904 minusm119895

10038171003817100381710038171003817 (30)


119903 = 1003817100381710038171003817x119904 minus x1003817100381710038171003817 minus 1003817100381710038171003817x1199041003817100381710038171003817 (31)


C1

Cs

x

r

C2

y

[xs rs]T

lowast



119903119904 = minus 1003817100381710038171003817x1199041003817100381710038171003817 (32)



119903119904 = minus 1003817100381710038171003817x1199041003817100381710038171003817 (33)




1199092119904 + 1199102119904 minus 1199032119904 = 0 (34)






119869(TDOA)SP (x119904 119903119904) = 119873119872sum

119895=2

1198902119895 (37)










4 3

1 21 2

34








4 3

1 21

2

34



119869(DOA)ML (x) = 119872sum

119898=1

11205902119898 (120579119898 minus 120579119898 (x))2 (41)



119869(DOA)NLS (x) = 119872sum

119898=1

(120579119898 minus 120579119898 (x))2 (42)







where

A = [[[[[


]]]]]

b = [[[[[


]]]]]

W = [[[[[[[

1199032112059021 01199032212059022

d

0 11990321198721205902119872

]]]]]]]

(44)




x(OV)119904 = (A119879A)minus1 A119879b (46)





119869(DOA)GB (x) = 119872sum

119898=1

[119860 (120579119898 120579119898 (x))]2 (47)


119869(DOA)GB (x) (48)





1 2





119873comb = 119878prod119904=1

119904119862119904 (49)


119869(DOA)P-NLS (x) = 119872sum

119898=1

min119894

10038161003816100381610038161003816120579119898119894 minus 120579119898 (x)100381610038161003816100381610038162 (50)









0

50

100

150

200

250

300

350

400

y(c

m)

50 100 150 200 250 300 350 4000

x (cm)

(a)

500

400

300

200

100

00

50

100

150

200

250

300

350

400

y(c

m)

50 100 150 200 250 300 350 4000

x (cm)

(b)




119860(120601 (120596 119896) 120579119904 (119896)) lt 119860 (120601 (120596 119896) 120579119902 (119896)) forall119902 = 119904 (51)

119860(120601 (120596 119896) 120579119904 (119896)) lt 120598 (52)










119869(SRP)119862 (x) = 2119879119873sum119894=1

119873sum119895=119894+1

119877119894119895 (120591119894119895 (x)) + 119873sum119894=1

119877119894119894 (0) (53)



10038171003817100381710038171003817119888 (54)




119869(SRP)119862 (x) (55)



119869(SRP)119872 (x) = 119873sum119894=1

119873sum119895=119894+1

119871119906119894119895(x)sum120591=119871119897119894119895(x)

119877119894119895 (120591) (56)




10038171003817100381710038171003817) (57)





(59)



xisinG119869(SRP)119872 (x) (60)



x (m)

C-SRP (rg = 1 cm)

4353252151050

10

08

06

04

02

000

05

1

15

2

25

3

35

4

y(m

)

(a)

x (m)4353252151050

10

08

06

04

02

000

05

1

15

2

25

3

35

4

y(m

)

C-SRP (rg = 10 cm)

(b)

x (m)4353252151050

10

08

06

04

02

000

05

1

15

2

25

3

35

4

y(m

)

M-SRP (rg = 10 cm)

(c)









x2

x1 x3

x4

m

x = 12 my = 25 mz = 07 m












120578119904 ≜ 1119888 1003817100381710038171003817m minus x1199041003817100381710038171003817 (61)



12059111990411199042 ≜ 1205781199041 minus 1205781199042 =10038171003817100381710038171003817m minus x1199041

10038171003817100381710038171003817 minus 10038171003817100381710038171003817m minus x119904210038171003817100381710038171003817119888 (62)




120591119904 = 120578119904 + Δ120578119904 (63)




120591119904 = 120578119904 + Δ120578119904119865119904 + Δ119865119904 = 120578119904 + Δ120578119904119865119904 ( 11 + Δ119865119904119865119904) (64)




1003817100381710038171003817 = 119888120578119904 asymp 119886120591119904 + 119887 (66)





119869(TOA)LS (m 119886 119887) = 10038171003817100381710038171003817e(TOA)100381710038171003817100381710038172 (68)


m(TOA)LS = argmin

m119886119887119869(TOA)LS (m 119886 119887) (69)



119886120591119904 + 119887 = (119886120591119904 minus 1198861205911) + (1198861205911 + 119887) asymp 1198861205911199041 + m (70)



119869(TDOA)LS (m 119886) = 10038171003817100381710038171003817e(TDOA)

LS100381710038171003817100381710038172 (71)


m(TDOA)LS = argmin

m119886119869(TDOA)LS (m 119886) (72)



e(TOA) = [[[[[

1 2x1198791 12059121 21205911 1 2x119879119878 1205912119878 2120591119904

]]]]][[[[[[

1198872 minus m2m1198862119886119887

]]]]]]minus [[[[[

1003817100381710038171003817x1100381710038171003817100381721003817100381710038171003817x11987810038171003817100381710038172]]]]]

e(TDOA) = [[[[[

212059121 2x1198792 120591221 21205911198781 2x119879119878 12059121198781

]]]]][[[[m 119886m1198862

]]]]minus [[[[[

1003817100381710038171003817x2100381710038171003817100381721003817100381710038171003817x11987810038171003817100381710038172]]]]]

(73)


minΘ

e2 (74)


Θ ≜ (H119879H)minus1H119879g (75)



e = [[[[[

211988812059121 2x1198792 21198881205911198781 2x119879119878

]]]]][ m

m] minus [[[[[

1003817100381710038171003817x210038171003817100381710038172 minus 11988821205912211003817100381710038171003817x11987810038171003817100381710038172 minus 119888212059121198781]]]]]

(76)


















10 Conclusion






References





















































































































RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










EW event warning

EW

EW

EW

EW

Central node

(a)

Central node

TOA

TOA

TOA

TOATOA

(b)








119894

10038171003817100381710038171003817 minus 10038171003817100381710038171003817x119904 minusm(119898)119895

10038171003817100381710038171003817119888 119894 = 1 119873 119895 = 1 119873 119894 = 119895 119898 = 1 119872

(19)


119909119894 (119905) = 120572119894119904 (119905 minus 120578119894) + 120598119894 (119905) 119894 = 1 119873 (20)




119883119894 (120596) = 120572119894119878 (120596) 119890minus119895120596120578119894 + 119864119894 (120596) 119894 = 1 119873 (21)



119877119894119895 (120591) ≜ 12120587 int120587

minus120587119883119894 (120596)119883lowast

119895 (120596)Ψ119894119895 (120596) 119890119895120596120591119889120596 (22)



120591119894119895 = argmax120591 119877119894119895 (120591)119865119904 (23)


Ψ119894119895 (120596) = 110038161003816100381610038161003816119883119894 (120596)119883lowast119895 (120596)10038161003816100381610038161003816 (24)






minus05

0

05

1

GCC

-PH

AT

minus05

0

05

1

GCC

-PH

AT





minus05

0

05

1

GCC

-PH

AT

GCC-PHATGTPeak



minus05

0

05

1

GCC

-PH

AT








120591119894119895 + 120591119895119896 + 120591119896119894 = 0 (26)








119894 andm(119898)119895 is written

as

119886(119898)119894119895 1199092 + 119887(119898)119894119895 119909119910 + 119888(119898)119894119895 1199102 + 119889(119898)119894119895 119909 + 119890(119898)119894119895 119910 + 119891(119898)119894119895 = 0 (27)




xs

mk(l)

mm(l)

mj(l)



120576(119898)119894119895 (x) = V(119898)119894119895 (119886(119898)119894119895 1199092 + 119887(119898)119894119895 119909119910 + 119888(119898)119894119895 1199102 + 119889(119898)119894119895 119909+ 119890(119898)119894119895 119910 + 119891(119898)

119894119895 ) (28)


119869(TDOA)HYP (x) = 120576 (x)119879 120576 (x) (29)




1003817100381710038171003817x1199041003817100381710038171003817 minus 119903119895 = 10038171003817100381710038171003817x119904 minusm119895

10038171003817100381710038171003817 (30)


119903 = 1003817100381710038171003817x119904 minus x1003817100381710038171003817 minus 1003817100381710038171003817x1199041003817100381710038171003817 (31)


C1

Cs

x

r

C2

y

[xs rs]T

lowast



119903119904 = minus 1003817100381710038171003817x1199041003817100381710038171003817 (32)



119903119904 = minus 1003817100381710038171003817x1199041003817100381710038171003817 (33)




1199092119904 + 1199102119904 minus 1199032119904 = 0 (34)






119869(TDOA)SP (x119904 119903119904) = 119873119872sum

119895=2

1198902119895 (37)










4 3

1 21 2

34








4 3

1 21

2

34



119869(DOA)ML (x) = 119872sum

119898=1

11205902119898 (120579119898 minus 120579119898 (x))2 (41)



119869(DOA)NLS (x) = 119872sum

119898=1

(120579119898 minus 120579119898 (x))2 (42)







where

A = [[[[[


]]]]]

b = [[[[[


]]]]]

W = [[[[[[[

1199032112059021 01199032212059022

d

0 11990321198721205902119872

]]]]]]]

(44)




x(OV)119904 = (A119879A)minus1 A119879b (46)





119869(DOA)GB (x) = 119872sum

119898=1

[119860 (120579119898 120579119898 (x))]2 (47)


119869(DOA)GB (x) (48)





1 2





119873comb = 119878prod119904=1

119904119862119904 (49)


119869(DOA)P-NLS (x) = 119872sum

119898=1

min119894

10038161003816100381610038161003816120579119898119894 minus 120579119898 (x)100381610038161003816100381610038162 (50)









0

50

100

150

200

250

300

350

400

y(c

m)

50 100 150 200 250 300 350 4000

x (cm)

(a)

500

400

300

200

100

00

50

100

150

200

250

300

350

400

y(c

m)

50 100 150 200 250 300 350 4000

x (cm)

(b)




119860(120601 (120596 119896) 120579119904 (119896)) lt 119860 (120601 (120596 119896) 120579119902 (119896)) forall119902 = 119904 (51)

119860(120601 (120596 119896) 120579119904 (119896)) lt 120598 (52)










119869(SRP)119862 (x) = 2119879119873sum119894=1

119873sum119895=119894+1

119877119894119895 (120591119894119895 (x)) + 119873sum119894=1

119877119894119894 (0) (53)



10038171003817100381710038171003817119888 (54)




119869(SRP)119862 (x) (55)



119869(SRP)119872 (x) = 119873sum119894=1

119873sum119895=119894+1

119871119906119894119895(x)sum120591=119871119897119894119895(x)

119877119894119895 (120591) (56)




10038171003817100381710038171003817) (57)





(59)



xisinG119869(SRP)119872 (x) (60)



x (m)

C-SRP (rg = 1 cm)

4353252151050

10

08

06

04

02

000

05

1

15

2

25

3

35

4

y(m

)

(a)

x (m)4353252151050

10

08

06

04

02

000

05

1

15

2

25

3

35

4

y(m

)

C-SRP (rg = 10 cm)

(b)

x (m)4353252151050

10

08

06

04

02

000

05

1

15

2

25

3

35

4

y(m

)

M-SRP (rg = 10 cm)

(c)









x2

x1 x3

x4

m

x = 12 my = 25 mz = 07 m












120578119904 ≜ 1119888 1003817100381710038171003817m minus x1199041003817100381710038171003817 (61)



12059111990411199042 ≜ 1205781199041 minus 1205781199042 =10038171003817100381710038171003817m minus x1199041

10038171003817100381710038171003817 minus 10038171003817100381710038171003817m minus x119904210038171003817100381710038171003817119888 (62)




120591119904 = 120578119904 + Δ120578119904 (63)




120591119904 = 120578119904 + Δ120578119904119865119904 + Δ119865119904 = 120578119904 + Δ120578119904119865119904 ( 11 + Δ119865119904119865119904) (64)




1003817100381710038171003817 = 119888120578119904 asymp 119886120591119904 + 119887 (66)





119869(TOA)LS (m 119886 119887) = 10038171003817100381710038171003817e(TOA)100381710038171003817100381710038172 (68)


m(TOA)LS = argmin

m119886119887119869(TOA)LS (m 119886 119887) (69)



119886120591119904 + 119887 = (119886120591119904 minus 1198861205911) + (1198861205911 + 119887) asymp 1198861205911199041 + m (70)



119869(TDOA)LS (m 119886) = 10038171003817100381710038171003817e(TDOA)

LS100381710038171003817100381710038172 (71)


m(TDOA)LS = argmin

m119886119869(TDOA)LS (m 119886) (72)



e(TOA) = [[[[[

1 2x1198791 12059121 21205911 1 2x119879119878 1205912119878 2120591119904

]]]]][[[[[[

1198872 minus m2m1198862119886119887

]]]]]]minus [[[[[

1003817100381710038171003817x1100381710038171003817100381721003817100381710038171003817x11987810038171003817100381710038172]]]]]

e(TDOA) = [[[[[

212059121 2x1198792 120591221 21205911198781 2x119879119878 12059121198781

]]]]][[[[m 119886m1198862

]]]]minus [[[[[

1003817100381710038171003817x2100381710038171003817100381721003817100381710038171003817x11987810038171003817100381710038172]]]]]

(73)


minΘ

e2 (74)


Θ ≜ (H119879H)minus1H119879g (75)



e = [[[[[

211988812059121 2x1198792 21198881205911198781 2x119879119878

]]]]][ m

m] minus [[[[[

1003817100381710038171003817x210038171003817100381710038172 minus 11988821205912211003817100381710038171003817x11987810038171003817100381710038172 minus 119888212059121198781]]]]]

(76)


















10 Conclusion






References





















































































































RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











119883119894 (120596) = 120572119894119878 (120596) 119890minus119895120596120578119894 + 119864119894 (120596) 119894 = 1 119873 (21)



119877119894119895 (120591) ≜ 12120587 int120587

minus120587119883119894 (120596)119883lowast

119895 (120596)Ψ119894119895 (120596) 119890119895120596120591119889120596 (22)



120591119894119895 = argmax120591 119877119894119895 (120591)119865119904 (23)


Ψ119894119895 (120596) = 110038161003816100381610038161003816119883119894 (120596)119883lowast119895 (120596)10038161003816100381610038161003816 (24)






minus05

0

05

1

GCC

-PH

AT

minus05

0

05

1

GCC

-PH

AT





minus05

0

05

1

GCC

-PH

AT

GCC-PHATGTPeak



minus05

0

05

1

GCC

-PH

AT








120591119894119895 + 120591119895119896 + 120591119896119894 = 0 (26)








119894 andm(119898)119895 is written

as

119886(119898)119894119895 1199092 + 119887(119898)119894119895 119909119910 + 119888(119898)119894119895 1199102 + 119889(119898)119894119895 119909 + 119890(119898)119894119895 119910 + 119891(119898)119894119895 = 0 (27)




xs

mk(l)

mm(l)

mj(l)



120576(119898)119894119895 (x) = V(119898)119894119895 (119886(119898)119894119895 1199092 + 119887(119898)119894119895 119909119910 + 119888(119898)119894119895 1199102 + 119889(119898)119894119895 119909+ 119890(119898)119894119895 119910 + 119891(119898)

119894119895 ) (28)


119869(TDOA)HYP (x) = 120576 (x)119879 120576 (x) (29)




1003817100381710038171003817x1199041003817100381710038171003817 minus 119903119895 = 10038171003817100381710038171003817x119904 minusm119895

10038171003817100381710038171003817 (30)


119903 = 1003817100381710038171003817x119904 minus x1003817100381710038171003817 minus 1003817100381710038171003817x1199041003817100381710038171003817 (31)


C1

Cs

x

r

C2

y

[xs rs]T

lowast



119903119904 = minus 1003817100381710038171003817x1199041003817100381710038171003817 (32)



119903119904 = minus 1003817100381710038171003817x1199041003817100381710038171003817 (33)




1199092119904 + 1199102119904 minus 1199032119904 = 0 (34)






119869(TDOA)SP (x119904 119903119904) = 119873119872sum

119895=2

1198902119895 (37)










4 3

1 21 2

34








4 3

1 21

2

34



119869(DOA)ML (x) = 119872sum

119898=1

11205902119898 (120579119898 minus 120579119898 (x))2 (41)



119869(DOA)NLS (x) = 119872sum

119898=1

(120579119898 minus 120579119898 (x))2 (42)







where

A = [[[[[


]]]]]

b = [[[[[


]]]]]

W = [[[[[[[

1199032112059021 01199032212059022

d

0 11990321198721205902119872

]]]]]]]

(44)




x(OV)119904 = (A119879A)minus1 A119879b (46)





119869(DOA)GB (x) = 119872sum

119898=1

[119860 (120579119898 120579119898 (x))]2 (47)


119869(DOA)GB (x) (48)





1 2





119873comb = 119878prod119904=1

119904119862119904 (49)


119869(DOA)P-NLS (x) = 119872sum

119898=1

min119894

10038161003816100381610038161003816120579119898119894 minus 120579119898 (x)100381610038161003816100381610038162 (50)









0

50

100

150

200

250

300

350

400

y(c

m)

50 100 150 200 250 300 350 4000

x (cm)

(a)

500

400

300

200

100

00

50

100

150

200

250

300

350

400

y(c

m)

50 100 150 200 250 300 350 4000

x (cm)

(b)




119860(120601 (120596 119896) 120579119904 (119896)) lt 119860 (120601 (120596 119896) 120579119902 (119896)) forall119902 = 119904 (51)

119860(120601 (120596 119896) 120579119904 (119896)) lt 120598 (52)










119869(SRP)119862 (x) = 2119879119873sum119894=1

119873sum119895=119894+1

119877119894119895 (120591119894119895 (x)) + 119873sum119894=1

119877119894119894 (0) (53)



10038171003817100381710038171003817119888 (54)




119869(SRP)119862 (x) (55)



119869(SRP)119872 (x) = 119873sum119894=1

119873sum119895=119894+1

119871119906119894119895(x)sum120591=119871119897119894119895(x)

119877119894119895 (120591) (56)




10038171003817100381710038171003817) (57)





(59)



xisinG119869(SRP)119872 (x) (60)



x (m)

C-SRP (rg = 1 cm)

4353252151050

10

08

06

04

02

000

05

1

15

2

25

3

35

4

y(m

)

(a)

x (m)4353252151050

10

08

06

04

02

000

05

1

15

2

25

3

35

4

y(m

)

C-SRP (rg = 10 cm)

(b)

x (m)4353252151050

10

08

06

04

02

000

05

1

15

2

25

3

35

4

y(m

)

M-SRP (rg = 10 cm)

(c)









x2

x1 x3

x4

m

x = 12 my = 25 mz = 07 m












120578119904 ≜ 1119888 1003817100381710038171003817m minus x1199041003817100381710038171003817 (61)



12059111990411199042 ≜ 1205781199041 minus 1205781199042 =10038171003817100381710038171003817m minus x1199041

10038171003817100381710038171003817 minus 10038171003817100381710038171003817m minus x119904210038171003817100381710038171003817119888 (62)




120591119904 = 120578119904 + Δ120578119904 (63)




120591119904 = 120578119904 + Δ120578119904119865119904 + Δ119865119904 = 120578119904 + Δ120578119904119865119904 ( 11 + Δ119865119904119865119904) (64)




1003817100381710038171003817 = 119888120578119904 asymp 119886120591119904 + 119887 (66)





119869(TOA)LS (m 119886 119887) = 10038171003817100381710038171003817e(TOA)100381710038171003817100381710038172 (68)


m(TOA)LS = argmin

m119886119887119869(TOA)LS (m 119886 119887) (69)



119886120591119904 + 119887 = (119886120591119904 minus 1198861205911) + (1198861205911 + 119887) asymp 1198861205911199041 + m (70)



119869(TDOA)LS (m 119886) = 10038171003817100381710038171003817e(TDOA)

LS100381710038171003817100381710038172 (71)


m(TDOA)LS = argmin

m119886119869(TDOA)LS (m 119886) (72)



e(TOA) = [[[[[

1 2x1198791 12059121 21205911 1 2x119879119878 1205912119878 2120591119904

]]]]][[[[[[

1198872 minus m2m1198862119886119887

]]]]]]minus [[[[[

1003817100381710038171003817x1100381710038171003817100381721003817100381710038171003817x11987810038171003817100381710038172]]]]]

e(TDOA) = [[[[[

212059121 2x1198792 120591221 21205911198781 2x119879119878 12059121198781

]]]]][[[[m 119886m1198862

]]]]minus [[[[[

1003817100381710038171003817x2100381710038171003817100381721003817100381710038171003817x11987810038171003817100381710038172]]]]]

(73)


minΘ

e2 (74)


Θ ≜ (H119879H)minus1H119879g (75)



e = [[[[[

211988812059121 2x1198792 21198881205911198781 2x119879119878

]]]]][ m

m] minus [[[[[

1003817100381710038171003817x210038171003817100381710038172 minus 11988821205912211003817100381710038171003817x11987810038171003817100381710038172 minus 119888212059121198781]]]]]

(76)


















10 Conclusion






References





















































































































RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











120591119894119895 + 120591119895119896 + 120591119896119894 = 0 (26)








119894 andm(119898)119895 is written

as

119886(119898)119894119895 1199092 + 119887(119898)119894119895 119909119910 + 119888(119898)119894119895 1199102 + 119889(119898)119894119895 119909 + 119890(119898)119894119895 119910 + 119891(119898)119894119895 = 0 (27)




xs

mk(l)

mm(l)

mj(l)



120576(119898)119894119895 (x) = V(119898)119894119895 (119886(119898)119894119895 1199092 + 119887(119898)119894119895 119909119910 + 119888(119898)119894119895 1199102 + 119889(119898)119894119895 119909+ 119890(119898)119894119895 119910 + 119891(119898)

119894119895 ) (28)


119869(TDOA)HYP (x) = 120576 (x)119879 120576 (x) (29)




1003817100381710038171003817x1199041003817100381710038171003817 minus 119903119895 = 10038171003817100381710038171003817x119904 minusm119895

10038171003817100381710038171003817 (30)


119903 = 1003817100381710038171003817x119904 minus x1003817100381710038171003817 minus 1003817100381710038171003817x1199041003817100381710038171003817 (31)


C1

Cs

x

r

C2

y

[xs rs]T

lowast



119903119904 = minus 1003817100381710038171003817x1199041003817100381710038171003817 (32)



119903119904 = minus 1003817100381710038171003817x1199041003817100381710038171003817 (33)




1199092119904 + 1199102119904 minus 1199032119904 = 0 (34)






119869(TDOA)SP (x119904 119903119904) = 119873119872sum

119895=2

1198902119895 (37)










4 3

1 21 2

34








4 3

1 21

2

34



119869(DOA)ML (x) = 119872sum

119898=1

11205902119898 (120579119898 minus 120579119898 (x))2 (41)



119869(DOA)NLS (x) = 119872sum

119898=1

(120579119898 minus 120579119898 (x))2 (42)







where

A = [[[[[


]]]]]

b = [[[[[


]]]]]

W = [[[[[[[

1199032112059021 01199032212059022

d

0 11990321198721205902119872

]]]]]]]

(44)




x(OV)119904 = (A119879A)minus1 A119879b (46)





119869(DOA)GB (x) = 119872sum

119898=1

[119860 (120579119898 120579119898 (x))]2 (47)


119869(DOA)GB (x) (48)





1 2





119873comb = 119878prod119904=1

119904119862119904 (49)


119869(DOA)P-NLS (x) = 119872sum

119898=1

min119894

10038161003816100381610038161003816120579119898119894 minus 120579119898 (x)100381610038161003816100381610038162 (50)









0

50

100

150

200

250

300

350

400

y(c

m)

50 100 150 200 250 300 350 4000

x (cm)

(a)

500

400

300

200

100

00

50

100

150

200

250

300

350

400

y(c

m)

50 100 150 200 250 300 350 4000

x (cm)

(b)




119860(120601 (120596 119896) 120579119904 (119896)) lt 119860 (120601 (120596 119896) 120579119902 (119896)) forall119902 = 119904 (51)

119860(120601 (120596 119896) 120579119904 (119896)) lt 120598 (52)










119869(SRP)119862 (x) = 2119879119873sum119894=1

119873sum119895=119894+1

119877119894119895 (120591119894119895 (x)) + 119873sum119894=1

119877119894119894 (0) (53)



10038171003817100381710038171003817119888 (54)




119869(SRP)119862 (x) (55)



119869(SRP)119872 (x) = 119873sum119894=1

119873sum119895=119894+1

119871119906119894119895(x)sum120591=119871119897119894119895(x)

119877119894119895 (120591) (56)




10038171003817100381710038171003817) (57)





(59)



xisinG119869(SRP)119872 (x) (60)



x (m)

C-SRP (rg = 1 cm)

4353252151050

10

08

06

04

02

000

05

1

15

2

25

3

35

4

y(m

)

(a)

x (m)4353252151050

10

08

06

04

02

000

05

1

15

2

25

3

35

4

y(m

)

C-SRP (rg = 10 cm)

(b)

x (m)4353252151050

10

08

06

04

02

000

05

1

15

2

25

3

35

4

y(m

)

M-SRP (rg = 10 cm)

(c)









x2

x1 x3

x4

m

x = 12 my = 25 mz = 07 m












120578119904 ≜ 1119888 1003817100381710038171003817m minus x1199041003817100381710038171003817 (61)



12059111990411199042 ≜ 1205781199041 minus 1205781199042 =10038171003817100381710038171003817m minus x1199041

10038171003817100381710038171003817 minus 10038171003817100381710038171003817m minus x119904210038171003817100381710038171003817119888 (62)




120591119904 = 120578119904 + Δ120578119904 (63)




120591119904 = 120578119904 + Δ120578119904119865119904 + Δ119865119904 = 120578119904 + Δ120578119904119865119904 ( 11 + Δ119865119904119865119904) (64)




1003817100381710038171003817 = 119888120578119904 asymp 119886120591119904 + 119887 (66)





119869(TOA)LS (m 119886 119887) = 10038171003817100381710038171003817e(TOA)100381710038171003817100381710038172 (68)


m(TOA)LS = argmin

m119886119887119869(TOA)LS (m 119886 119887) (69)



119886120591119904 + 119887 = (119886120591119904 minus 1198861205911) + (1198861205911 + 119887) asymp 1198861205911199041 + m (70)



119869(TDOA)LS (m 119886) = 10038171003817100381710038171003817e(TDOA)

LS100381710038171003817100381710038172 (71)


m(TDOA)LS = argmin

m119886119869(TDOA)LS (m 119886) (72)



e(TOA) = [[[[[

1 2x1198791 12059121 21205911 1 2x119879119878 1205912119878 2120591119904

]]]]][[[[[[

1198872 minus m2m1198862119886119887

]]]]]]minus [[[[[

1003817100381710038171003817x1100381710038171003817100381721003817100381710038171003817x11987810038171003817100381710038172]]]]]

e(TDOA) = [[[[[

212059121 2x1198792 120591221 21205911198781 2x119879119878 12059121198781

]]]]][[[[m 119886m1198862

]]]]minus [[[[[

1003817100381710038171003817x2100381710038171003817100381721003817100381710038171003817x11987810038171003817100381710038172]]]]]

(73)


minΘ

e2 (74)


Θ ≜ (H119879H)minus1H119879g (75)



e = [[[[[

211988812059121 2x1198792 21198881205911198781 2x119879119878

]]]]][ m

m] minus [[[[[

1003817100381710038171003817x210038171003817100381710038172 minus 11988821205912211003817100381710038171003817x11987810038171003817100381710038172 minus 119888212059121198781]]]]]

(76)


















10 Conclusion






References





















































































































RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










C1

Cs

x

r

C2

y

[xs rs]T

lowast



119903119904 = minus 1003817100381710038171003817x1199041003817100381710038171003817 (32)



119903119904 = minus 1003817100381710038171003817x1199041003817100381710038171003817 (33)




1199092119904 + 1199102119904 minus 1199032119904 = 0 (34)






119869(TDOA)SP (x119904 119903119904) = 119873119872sum

119895=2

1198902119895 (37)










4 3

1 21 2

34








4 3

1 21

2

34



119869(DOA)ML (x) = 119872sum

119898=1

11205902119898 (120579119898 minus 120579119898 (x))2 (41)



119869(DOA)NLS (x) = 119872sum

119898=1

(120579119898 minus 120579119898 (x))2 (42)







where

A = [[[[[


]]]]]

b = [[[[[


]]]]]

W = [[[[[[[

1199032112059021 01199032212059022

d

0 11990321198721205902119872

]]]]]]]

(44)




x(OV)119904 = (A119879A)minus1 A119879b (46)





119869(DOA)GB (x) = 119872sum

119898=1

[119860 (120579119898 120579119898 (x))]2 (47)


119869(DOA)GB (x) (48)





1 2





119873comb = 119878prod119904=1

119904119862119904 (49)


119869(DOA)P-NLS (x) = 119872sum

119898=1

min119894

10038161003816100381610038161003816120579119898119894 minus 120579119898 (x)100381610038161003816100381610038162 (50)









0

50

100

150

200

250

300

350

400

y(c

m)

50 100 150 200 250 300 350 4000

x (cm)

(a)

500

400

300

200

100

00

50

100

150

200

250

300

350

400

y(c

m)

50 100 150 200 250 300 350 4000

x (cm)

(b)




119860(120601 (120596 119896) 120579119904 (119896)) lt 119860 (120601 (120596 119896) 120579119902 (119896)) forall119902 = 119904 (51)

119860(120601 (120596 119896) 120579119904 (119896)) lt 120598 (52)










119869(SRP)119862 (x) = 2119879119873sum119894=1

119873sum119895=119894+1

119877119894119895 (120591119894119895 (x)) + 119873sum119894=1

119877119894119894 (0) (53)



10038171003817100381710038171003817119888 (54)




119869(SRP)119862 (x) (55)



119869(SRP)119872 (x) = 119873sum119894=1

119873sum119895=119894+1

119871119906119894119895(x)sum120591=119871119897119894119895(x)

119877119894119895 (120591) (56)




10038171003817100381710038171003817) (57)





(59)



xisinG119869(SRP)119872 (x) (60)



x (m)

C-SRP (rg = 1 cm)

4353252151050

10

08

06

04

02

000

05

1

15

2

25

3

35

4

y(m

)

(a)

x (m)4353252151050

10

08

06

04

02

000

05

1

15

2

25

3

35

4

y(m

)

C-SRP (rg = 10 cm)

(b)

x (m)4353252151050

10

08

06

04

02

000

05

1

15

2

25

3

35

4

y(m

)

M-SRP (rg = 10 cm)

(c)









x2

x1 x3

x4

m

x = 12 my = 25 mz = 07 m












120578119904 ≜ 1119888 1003817100381710038171003817m minus x1199041003817100381710038171003817 (61)



12059111990411199042 ≜ 1205781199041 minus 1205781199042 =10038171003817100381710038171003817m minus x1199041

10038171003817100381710038171003817 minus 10038171003817100381710038171003817m minus x119904210038171003817100381710038171003817119888 (62)




120591119904 = 120578119904 + Δ120578119904 (63)




120591119904 = 120578119904 + Δ120578119904119865119904 + Δ119865119904 = 120578119904 + Δ120578119904119865119904 ( 11 + Δ119865119904119865119904) (64)




1003817100381710038171003817 = 119888120578119904 asymp 119886120591119904 + 119887 (66)





119869(TOA)LS (m 119886 119887) = 10038171003817100381710038171003817e(TOA)100381710038171003817100381710038172 (68)


m(TOA)LS = argmin

m119886119887119869(TOA)LS (m 119886 119887) (69)



119886120591119904 + 119887 = (119886120591119904 minus 1198861205911) + (1198861205911 + 119887) asymp 1198861205911199041 + m (70)



119869(TDOA)LS (m 119886) = 10038171003817100381710038171003817e(TDOA)

LS100381710038171003817100381710038172 (71)


m(TDOA)LS = argmin

m119886119869(TDOA)LS (m 119886) (72)



e(TOA) = [[[[[

1 2x1198791 12059121 21205911 1 2x119879119878 1205912119878 2120591119904

]]]]][[[[[[

1198872 minus m2m1198862119886119887

]]]]]]minus [[[[[

1003817100381710038171003817x1100381710038171003817100381721003817100381710038171003817x11987810038171003817100381710038172]]]]]

e(TDOA) = [[[[[

212059121 2x1198792 120591221 21205911198781 2x119879119878 12059121198781

]]]]][[[[m 119886m1198862

]]]]minus [[[[[

1003817100381710038171003817x2100381710038171003817100381721003817100381710038171003817x11987810038171003817100381710038172]]]]]

(73)


minΘ

e2 (74)


Θ ≜ (H119879H)minus1H119879g (75)



e = [[[[[

211988812059121 2x1198792 21198881205911198781 2x119879119878

]]]]][ m

m] minus [[[[[

1003817100381710038171003817x210038171003817100381710038172 minus 11988821205912211003817100381710038171003817x11987810038171003817100381710038172 minus 119888212059121198781]]]]]

(76)


















10 Conclusion






References





















































































































RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










4 3

1 21 2

34








4 3

1 21

2

34



119869(DOA)ML (x) = 119872sum

119898=1

11205902119898 (120579119898 minus 120579119898 (x))2 (41)



119869(DOA)NLS (x) = 119872sum

119898=1

(120579119898 minus 120579119898 (x))2 (42)







where

A = [[[[[


]]]]]

b = [[[[[


]]]]]

W = [[[[[[[

1199032112059021 01199032212059022

d

0 11990321198721205902119872

]]]]]]]

(44)




x(OV)119904 = (A119879A)minus1 A119879b (46)





119869(DOA)GB (x) = 119872sum

119898=1

[119860 (120579119898 120579119898 (x))]2 (47)


119869(DOA)GB (x) (48)





1 2





119873comb = 119878prod119904=1

119904119862119904 (49)


119869(DOA)P-NLS (x) = 119872sum

119898=1

min119894

10038161003816100381610038161003816120579119898119894 minus 120579119898 (x)100381610038161003816100381610038162 (50)









0

50

100

150

200

250

300

350

400

y(c

m)

50 100 150 200 250 300 350 4000

x (cm)

(a)

500

400

300

200

100

00

50

100

150

200

250

300

350

400

y(c

m)

50 100 150 200 250 300 350 4000

x (cm)

(b)




119860(120601 (120596 119896) 120579119904 (119896)) lt 119860 (120601 (120596 119896) 120579119902 (119896)) forall119902 = 119904 (51)

119860(120601 (120596 119896) 120579119904 (119896)) lt 120598 (52)










119869(SRP)119862 (x) = 2119879119873sum119894=1

119873sum119895=119894+1

119877119894119895 (120591119894119895 (x)) + 119873sum119894=1

119877119894119894 (0) (53)



10038171003817100381710038171003817119888 (54)




119869(SRP)119862 (x) (55)



119869(SRP)119872 (x) = 119873sum119894=1

119873sum119895=119894+1

119871119906119894119895(x)sum120591=119871119897119894119895(x)

119877119894119895 (120591) (56)




10038171003817100381710038171003817) (57)





(59)



xisinG119869(SRP)119872 (x) (60)



x (m)

C-SRP (rg = 1 cm)

4353252151050

10

08

06

04

02

000

05

1

15

2

25

3

35

4

y(m

)

(a)

x (m)4353252151050

10

08

06

04

02

000

05

1

15

2

25

3

35

4

y(m

)

C-SRP (rg = 10 cm)

(b)

x (m)4353252151050

10

08

06

04

02

000

05

1

15

2

25

3

35

4

y(m

)

M-SRP (rg = 10 cm)

(c)









x2

x1 x3

x4

m

x = 12 my = 25 mz = 07 m












120578119904 ≜ 1119888 1003817100381710038171003817m minus x1199041003817100381710038171003817 (61)



12059111990411199042 ≜ 1205781199041 minus 1205781199042 =10038171003817100381710038171003817m minus x1199041

10038171003817100381710038171003817 minus 10038171003817100381710038171003817m minus x119904210038171003817100381710038171003817119888 (62)




120591119904 = 120578119904 + Δ120578119904 (63)




120591119904 = 120578119904 + Δ120578119904119865119904 + Δ119865119904 = 120578119904 + Δ120578119904119865119904 ( 11 + Δ119865119904119865119904) (64)




1003817100381710038171003817 = 119888120578119904 asymp 119886120591119904 + 119887 (66)





119869(TOA)LS (m 119886 119887) = 10038171003817100381710038171003817e(TOA)100381710038171003817100381710038172 (68)


m(TOA)LS = argmin

m119886119887119869(TOA)LS (m 119886 119887) (69)



119886120591119904 + 119887 = (119886120591119904 minus 1198861205911) + (1198861205911 + 119887) asymp 1198861205911199041 + m (70)



119869(TDOA)LS (m 119886) = 10038171003817100381710038171003817e(TDOA)

LS100381710038171003817100381710038172 (71)


m(TDOA)LS = argmin

m119886119869(TDOA)LS (m 119886) (72)



e(TOA) = [[[[[

1 2x1198791 12059121 21205911 1 2x119879119878 1205912119878 2120591119904

]]]]][[[[[[

1198872 minus m2m1198862119886119887

]]]]]]minus [[[[[

1003817100381710038171003817x1100381710038171003817100381721003817100381710038171003817x11987810038171003817100381710038172]]]]]

e(TDOA) = [[[[[

212059121 2x1198792 120591221 21205911198781 2x119879119878 12059121198781

]]]]][[[[m 119886m1198862

]]]]minus [[[[[

1003817100381710038171003817x2100381710038171003817100381721003817100381710038171003817x11987810038171003817100381710038172]]]]]

(73)


minΘ

e2 (74)


Θ ≜ (H119879H)minus1H119879g (75)



e = [[[[[

211988812059121 2x1198792 21198881205911198781 2x119879119878

]]]]][ m

m] minus [[[[[

1003817100381710038171003817x210038171003817100381710038172 minus 11988821205912211003817100381710038171003817x11987810038171003817100381710038172 minus 119888212059121198781]]]]]

(76)


















10 Conclusion






References





















































































































RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation













where

A = [[[[[


]]]]]

b = [[[[[


]]]]]

W = [[[[[[[

1199032112059021 01199032212059022

d

0 11990321198721205902119872

]]]]]]]

(44)




x(OV)119904 = (A119879A)minus1 A119879b (46)





119869(DOA)GB (x) = 119872sum

119898=1

[119860 (120579119898 120579119898 (x))]2 (47)


119869(DOA)GB (x) (48)





1 2





119873comb = 119878prod119904=1

119904119862119904 (49)


119869(DOA)P-NLS (x) = 119872sum

119898=1

min119894

10038161003816100381610038161003816120579119898119894 minus 120579119898 (x)100381610038161003816100381610038162 (50)









0

50

100

150

200

250

300

350

400

y(c

m)

50 100 150 200 250 300 350 4000

x (cm)

(a)

500

400

300

200

100

00

50

100

150

200

250

300

350

400

y(c

m)

50 100 150 200 250 300 350 4000

x (cm)

(b)




119860(120601 (120596 119896) 120579119904 (119896)) lt 119860 (120601 (120596 119896) 120579119902 (119896)) forall119902 = 119904 (51)

119860(120601 (120596 119896) 120579119904 (119896)) lt 120598 (52)










119869(SRP)119862 (x) = 2119879119873sum119894=1

119873sum119895=119894+1

119877119894119895 (120591119894119895 (x)) + 119873sum119894=1

119877119894119894 (0) (53)



10038171003817100381710038171003817119888 (54)




119869(SRP)119862 (x) (55)



119869(SRP)119872 (x) = 119873sum119894=1

119873sum119895=119894+1

119871119906119894119895(x)sum120591=119871119897119894119895(x)

119877119894119895 (120591) (56)




10038171003817100381710038171003817) (57)





(59)



xisinG119869(SRP)119872 (x) (60)



x (m)

C-SRP (rg = 1 cm)

4353252151050

10

08

06

04

02

000

05

1

15

2

25

3

35

4

y(m

)

(a)

x (m)4353252151050

10

08

06

04

02

000

05

1

15

2

25

3

35

4

y(m

)

C-SRP (rg = 10 cm)

(b)

x (m)4353252151050

10

08

06

04

02

000

05

1

15

2

25

3

35

4

y(m

)

M-SRP (rg = 10 cm)

(c)









x2

x1 x3

x4

m

x = 12 my = 25 mz = 07 m












120578119904 ≜ 1119888 1003817100381710038171003817m minus x1199041003817100381710038171003817 (61)



12059111990411199042 ≜ 1205781199041 minus 1205781199042 =10038171003817100381710038171003817m minus x1199041

10038171003817100381710038171003817 minus 10038171003817100381710038171003817m minus x119904210038171003817100381710038171003817119888 (62)




120591119904 = 120578119904 + Δ120578119904 (63)




120591119904 = 120578119904 + Δ120578119904119865119904 + Δ119865119904 = 120578119904 + Δ120578119904119865119904 ( 11 + Δ119865119904119865119904) (64)




1003817100381710038171003817 = 119888120578119904 asymp 119886120591119904 + 119887 (66)





119869(TOA)LS (m 119886 119887) = 10038171003817100381710038171003817e(TOA)100381710038171003817100381710038172 (68)


m(TOA)LS = argmin

m119886119887119869(TOA)LS (m 119886 119887) (69)



119886120591119904 + 119887 = (119886120591119904 minus 1198861205911) + (1198861205911 + 119887) asymp 1198861205911199041 + m (70)



119869(TDOA)LS (m 119886) = 10038171003817100381710038171003817e(TDOA)

LS100381710038171003817100381710038172 (71)


m(TDOA)LS = argmin

m119886119869(TDOA)LS (m 119886) (72)



e(TOA) = [[[[[

1 2x1198791 12059121 21205911 1 2x119879119878 1205912119878 2120591119904

]]]]][[[[[[

1198872 minus m2m1198862119886119887

]]]]]]minus [[[[[

1003817100381710038171003817x1100381710038171003817100381721003817100381710038171003817x11987810038171003817100381710038172]]]]]

e(TDOA) = [[[[[

212059121 2x1198792 120591221 21205911198781 2x119879119878 12059121198781

]]]]][[[[m 119886m1198862

]]]]minus [[[[[

1003817100381710038171003817x2100381710038171003817100381721003817100381710038171003817x11987810038171003817100381710038172]]]]]

(73)


minΘ

e2 (74)


Θ ≜ (H119879H)minus1H119879g (75)



e = [[[[[

211988812059121 2x1198792 21198881205911198781 2x119879119878

]]]]][ m

m] minus [[[[[

1003817100381710038171003817x210038171003817100381710038172 minus 11988821205912211003817100381710038171003817x11987810038171003817100381710038172 minus 119888212059121198781]]]]]

(76)


















10 Conclusion






References





















































































































RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










1 2





119873comb = 119878prod119904=1

119904119862119904 (49)


119869(DOA)P-NLS (x) = 119872sum

119898=1

min119894

10038161003816100381610038161003816120579119898119894 minus 120579119898 (x)100381610038161003816100381610038162 (50)









0

50

100

150

200

250

300

350

400

y(c

m)

50 100 150 200 250 300 350 4000

x (cm)

(a)

500

400

300

200

100

00

50

100

150

200

250

300

350

400

y(c

m)

50 100 150 200 250 300 350 4000

x (cm)

(b)




119860(120601 (120596 119896) 120579119904 (119896)) lt 119860 (120601 (120596 119896) 120579119902 (119896)) forall119902 = 119904 (51)

119860(120601 (120596 119896) 120579119904 (119896)) lt 120598 (52)










119869(SRP)119862 (x) = 2119879119873sum119894=1

119873sum119895=119894+1

119877119894119895 (120591119894119895 (x)) + 119873sum119894=1

119877119894119894 (0) (53)



10038171003817100381710038171003817119888 (54)




119869(SRP)119862 (x) (55)



119869(SRP)119872 (x) = 119873sum119894=1

119873sum119895=119894+1

119871119906119894119895(x)sum120591=119871119897119894119895(x)

119877119894119895 (120591) (56)




10038171003817100381710038171003817) (57)





(59)



xisinG119869(SRP)119872 (x) (60)



x (m)

C-SRP (rg = 1 cm)

4353252151050

10

08

06

04

02

000

05

1

15

2

25

3

35

4

y(m

)

(a)

x (m)4353252151050

10

08

06

04

02

000

05

1

15

2

25

3

35

4

y(m

)

C-SRP (rg = 10 cm)

(b)

x (m)4353252151050

10

08

06

04

02

000

05

1

15

2

25

3

35

4

y(m

)

M-SRP (rg = 10 cm)

(c)









x2

x1 x3

x4

m

x = 12 my = 25 mz = 07 m












120578119904 ≜ 1119888 1003817100381710038171003817m minus x1199041003817100381710038171003817 (61)



12059111990411199042 ≜ 1205781199041 minus 1205781199042 =10038171003817100381710038171003817m minus x1199041

10038171003817100381710038171003817 minus 10038171003817100381710038171003817m minus x119904210038171003817100381710038171003817119888 (62)




120591119904 = 120578119904 + Δ120578119904 (63)




120591119904 = 120578119904 + Δ120578119904119865119904 + Δ119865119904 = 120578119904 + Δ120578119904119865119904 ( 11 + Δ119865119904119865119904) (64)




1003817100381710038171003817 = 119888120578119904 asymp 119886120591119904 + 119887 (66)





119869(TOA)LS (m 119886 119887) = 10038171003817100381710038171003817e(TOA)100381710038171003817100381710038172 (68)


m(TOA)LS = argmin

m119886119887119869(TOA)LS (m 119886 119887) (69)



119886120591119904 + 119887 = (119886120591119904 minus 1198861205911) + (1198861205911 + 119887) asymp 1198861205911199041 + m (70)



119869(TDOA)LS (m 119886) = 10038171003817100381710038171003817e(TDOA)

LS100381710038171003817100381710038172 (71)


m(TDOA)LS = argmin

m119886119869(TDOA)LS (m 119886) (72)



e(TOA) = [[[[[

1 2x1198791 12059121 21205911 1 2x119879119878 1205912119878 2120591119904

]]]]][[[[[[

1198872 minus m2m1198862119886119887

]]]]]]minus [[[[[

1003817100381710038171003817x1100381710038171003817100381721003817100381710038171003817x11987810038171003817100381710038172]]]]]

e(TDOA) = [[[[[

212059121 2x1198792 120591221 21205911198781 2x119879119878 12059121198781

]]]]][[[[m 119886m1198862

]]]]minus [[[[[

1003817100381710038171003817x2100381710038171003817100381721003817100381710038171003817x11987810038171003817100381710038172]]]]]

(73)


minΘ

e2 (74)


Θ ≜ (H119879H)minus1H119879g (75)



e = [[[[[

211988812059121 2x1198792 21198881205911198781 2x119879119878

]]]]][ m

m] minus [[[[[

1003817100381710038171003817x210038171003817100381710038172 minus 11988821205912211003817100381710038171003817x11987810038171003817100381710038172 minus 119888212059121198781]]]]]

(76)


















10 Conclusion






References





















































































































RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0

50

100

150

200

250

300

350

400

y(c

m)

50 100 150 200 250 300 350 4000

x (cm)

(a)

500

400

300

200

100

00

50

100

150

200

250

300

350

400

y(c

m)

50 100 150 200 250 300 350 4000

x (cm)

(b)




119860(120601 (120596 119896) 120579119904 (119896)) lt 119860 (120601 (120596 119896) 120579119902 (119896)) forall119902 = 119904 (51)

119860(120601 (120596 119896) 120579119904 (119896)) lt 120598 (52)










119869(SRP)119862 (x) = 2119879119873sum119894=1

119873sum119895=119894+1

119877119894119895 (120591119894119895 (x)) + 119873sum119894=1

119877119894119894 (0) (53)



10038171003817100381710038171003817119888 (54)




119869(SRP)119862 (x) (55)



119869(SRP)119872 (x) = 119873sum119894=1

119873sum119895=119894+1

119871119906119894119895(x)sum120591=119871119897119894119895(x)

119877119894119895 (120591) (56)




10038171003817100381710038171003817) (57)





(59)



xisinG119869(SRP)119872 (x) (60)



x (m)

C-SRP (rg = 1 cm)

4353252151050

10

08

06

04

02

000

05

1

15

2

25

3

35

4

y(m

)

(a)

x (m)4353252151050

10

08

06

04

02

000

05

1

15

2

25

3

35

4

y(m

)

C-SRP (rg = 10 cm)

(b)

x (m)4353252151050

10

08

06

04

02

000

05

1

15

2

25

3

35

4

y(m

)

M-SRP (rg = 10 cm)

(c)









x2

x1 x3

x4

m

x = 12 my = 25 mz = 07 m












120578119904 ≜ 1119888 1003817100381710038171003817m minus x1199041003817100381710038171003817 (61)



12059111990411199042 ≜ 1205781199041 minus 1205781199042 =10038171003817100381710038171003817m minus x1199041

10038171003817100381710038171003817 minus 10038171003817100381710038171003817m minus x119904210038171003817100381710038171003817119888 (62)




120591119904 = 120578119904 + Δ120578119904 (63)




120591119904 = 120578119904 + Δ120578119904119865119904 + Δ119865119904 = 120578119904 + Δ120578119904119865119904 ( 11 + Δ119865119904119865119904) (64)




1003817100381710038171003817 = 119888120578119904 asymp 119886120591119904 + 119887 (66)





119869(TOA)LS (m 119886 119887) = 10038171003817100381710038171003817e(TOA)100381710038171003817100381710038172 (68)


m(TOA)LS = argmin

m119886119887119869(TOA)LS (m 119886 119887) (69)



119886120591119904 + 119887 = (119886120591119904 minus 1198861205911) + (1198861205911 + 119887) asymp 1198861205911199041 + m (70)



119869(TDOA)LS (m 119886) = 10038171003817100381710038171003817e(TDOA)

LS100381710038171003817100381710038172 (71)


m(TDOA)LS = argmin

m119886119869(TDOA)LS (m 119886) (72)



e(TOA) = [[[[[

1 2x1198791 12059121 21205911 1 2x119879119878 1205912119878 2120591119904

]]]]][[[[[[

1198872 minus m2m1198862119886119887

]]]]]]minus [[[[[

1003817100381710038171003817x1100381710038171003817100381721003817100381710038171003817x11987810038171003817100381710038172]]]]]

e(TDOA) = [[[[[

212059121 2x1198792 120591221 21205911198781 2x119879119878 12059121198781

]]]]][[[[m 119886m1198862

]]]]minus [[[[[

1003817100381710038171003817x2100381710038171003817100381721003817100381710038171003817x11987810038171003817100381710038172]]]]]

(73)


minΘ

e2 (74)


Θ ≜ (H119879H)minus1H119879g (75)



e = [[[[[

211988812059121 2x1198792 21198881205911198781 2x119879119878

]]]]][ m

m] minus [[[[[

1003817100381710038171003817x210038171003817100381710038172 minus 11988821205912211003817100381710038171003817x11987810038171003817100381710038172 minus 119888212059121198781]]]]]

(76)


















10 Conclusion






References





















































































































RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation













119869(SRP)119862 (x) = 2119879119873sum119894=1

119873sum119895=119894+1

119877119894119895 (120591119894119895 (x)) + 119873sum119894=1

119877119894119894 (0) (53)



10038171003817100381710038171003817119888 (54)




119869(SRP)119862 (x) (55)



119869(SRP)119872 (x) = 119873sum119894=1

119873sum119895=119894+1

119871119906119894119895(x)sum120591=119871119897119894119895(x)

119877119894119895 (120591) (56)




10038171003817100381710038171003817) (57)





(59)



xisinG119869(SRP)119872 (x) (60)



x (m)

C-SRP (rg = 1 cm)

4353252151050

10

08

06

04

02

000

05

1

15

2

25

3

35

4

y(m

)

(a)

x (m)4353252151050

10

08

06

04

02

000

05

1

15

2

25

3

35

4

y(m

)

C-SRP (rg = 10 cm)

(b)

x (m)4353252151050

10

08

06

04

02

000

05

1

15

2

25

3

35

4

y(m

)

M-SRP (rg = 10 cm)

(c)









x2

x1 x3

x4

m

x = 12 my = 25 mz = 07 m












120578119904 ≜ 1119888 1003817100381710038171003817m minus x1199041003817100381710038171003817 (61)



12059111990411199042 ≜ 1205781199041 minus 1205781199042 =10038171003817100381710038171003817m minus x1199041

10038171003817100381710038171003817 minus 10038171003817100381710038171003817m minus x119904210038171003817100381710038171003817119888 (62)




120591119904 = 120578119904 + Δ120578119904 (63)




120591119904 = 120578119904 + Δ120578119904119865119904 + Δ119865119904 = 120578119904 + Δ120578119904119865119904 ( 11 + Δ119865119904119865119904) (64)




1003817100381710038171003817 = 119888120578119904 asymp 119886120591119904 + 119887 (66)





119869(TOA)LS (m 119886 119887) = 10038171003817100381710038171003817e(TOA)100381710038171003817100381710038172 (68)


m(TOA)LS = argmin

m119886119887119869(TOA)LS (m 119886 119887) (69)



119886120591119904 + 119887 = (119886120591119904 minus 1198861205911) + (1198861205911 + 119887) asymp 1198861205911199041 + m (70)



119869(TDOA)LS (m 119886) = 10038171003817100381710038171003817e(TDOA)

LS100381710038171003817100381710038172 (71)


m(TDOA)LS = argmin

m119886119869(TDOA)LS (m 119886) (72)



e(TOA) = [[[[[

1 2x1198791 12059121 21205911 1 2x119879119878 1205912119878 2120591119904

]]]]][[[[[[

1198872 minus m2m1198862119886119887

]]]]]]minus [[[[[

1003817100381710038171003817x1100381710038171003817100381721003817100381710038171003817x11987810038171003817100381710038172]]]]]

e(TDOA) = [[[[[

212059121 2x1198792 120591221 21205911198781 2x119879119878 12059121198781

]]]]][[[[m 119886m1198862

]]]]minus [[[[[

1003817100381710038171003817x2100381710038171003817100381721003817100381710038171003817x11987810038171003817100381710038172]]]]]

(73)


minΘ

e2 (74)


Θ ≜ (H119879H)minus1H119879g (75)



e = [[[[[

211988812059121 2x1198792 21198881205911198781 2x119879119878

]]]]][ m

m] minus [[[[[

1003817100381710038171003817x210038171003817100381710038172 minus 11988821205912211003817100381710038171003817x11987810038171003817100381710038172 minus 119888212059121198781]]]]]

(76)


















10 Conclusion






References





















































































































RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










x (m)

C-SRP (rg = 1 cm)

4353252151050

10

08

06

04

02

000

05

1

15

2

25

3

35

4

y(m

)

(a)

x (m)4353252151050

10

08

06

04

02

000

05

1

15

2

25

3

35

4

y(m

)

C-SRP (rg = 10 cm)

(b)

x (m)4353252151050

10

08

06

04

02

000

05

1

15

2

25

3

35

4

y(m

)

M-SRP (rg = 10 cm)

(c)









x2

x1 x3

x4

m

x = 12 my = 25 mz = 07 m












120578119904 ≜ 1119888 1003817100381710038171003817m minus x1199041003817100381710038171003817 (61)



12059111990411199042 ≜ 1205781199041 minus 1205781199042 =10038171003817100381710038171003817m minus x1199041

10038171003817100381710038171003817 minus 10038171003817100381710038171003817m minus x119904210038171003817100381710038171003817119888 (62)




120591119904 = 120578119904 + Δ120578119904 (63)




120591119904 = 120578119904 + Δ120578119904119865119904 + Δ119865119904 = 120578119904 + Δ120578119904119865119904 ( 11 + Δ119865119904119865119904) (64)




1003817100381710038171003817 = 119888120578119904 asymp 119886120591119904 + 119887 (66)





119869(TOA)LS (m 119886 119887) = 10038171003817100381710038171003817e(TOA)100381710038171003817100381710038172 (68)


m(TOA)LS = argmin

m119886119887119869(TOA)LS (m 119886 119887) (69)



119886120591119904 + 119887 = (119886120591119904 minus 1198861205911) + (1198861205911 + 119887) asymp 1198861205911199041 + m (70)



119869(TDOA)LS (m 119886) = 10038171003817100381710038171003817e(TDOA)

LS100381710038171003817100381710038172 (71)


m(TDOA)LS = argmin

m119886119869(TDOA)LS (m 119886) (72)



e(TOA) = [[[[[

1 2x1198791 12059121 21205911 1 2x119879119878 1205912119878 2120591119904

]]]]][[[[[[

1198872 minus m2m1198862119886119887

]]]]]]minus [[[[[

1003817100381710038171003817x1100381710038171003817100381721003817100381710038171003817x11987810038171003817100381710038172]]]]]

e(TDOA) = [[[[[

212059121 2x1198792 120591221 21205911198781 2x119879119878 12059121198781

]]]]][[[[m 119886m1198862

]]]]minus [[[[[

1003817100381710038171003817x2100381710038171003817100381721003817100381710038171003817x11987810038171003817100381710038172]]]]]

(73)


minΘ

e2 (74)


Θ ≜ (H119879H)minus1H119879g (75)



e = [[[[[

211988812059121 2x1198792 21198881205911198781 2x119879119878

]]]]][ m

m] minus [[[[[

1003817100381710038171003817x210038171003817100381710038172 minus 11988821205912211003817100381710038171003817x11987810038171003817100381710038172 minus 119888212059121198781]]]]]

(76)


















10 Conclusion






References





















































































































RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










x2

x1 x3

x4

m

x = 12 my = 25 mz = 07 m












120578119904 ≜ 1119888 1003817100381710038171003817m minus x1199041003817100381710038171003817 (61)



12059111990411199042 ≜ 1205781199041 minus 1205781199042 =10038171003817100381710038171003817m minus x1199041

10038171003817100381710038171003817 minus 10038171003817100381710038171003817m minus x119904210038171003817100381710038171003817119888 (62)




120591119904 = 120578119904 + Δ120578119904 (63)




120591119904 = 120578119904 + Δ120578119904119865119904 + Δ119865119904 = 120578119904 + Δ120578119904119865119904 ( 11 + Δ119865119904119865119904) (64)




1003817100381710038171003817 = 119888120578119904 asymp 119886120591119904 + 119887 (66)





119869(TOA)LS (m 119886 119887) = 10038171003817100381710038171003817e(TOA)100381710038171003817100381710038172 (68)


m(TOA)LS = argmin

m119886119887119869(TOA)LS (m 119886 119887) (69)



119886120591119904 + 119887 = (119886120591119904 minus 1198861205911) + (1198861205911 + 119887) asymp 1198861205911199041 + m (70)



119869(TDOA)LS (m 119886) = 10038171003817100381710038171003817e(TDOA)

LS100381710038171003817100381710038172 (71)


m(TDOA)LS = argmin

m119886119869(TDOA)LS (m 119886) (72)



e(TOA) = [[[[[

1 2x1198791 12059121 21205911 1 2x119879119878 1205912119878 2120591119904

]]]]][[[[[[

1198872 minus m2m1198862119886119887

]]]]]]minus [[[[[

1003817100381710038171003817x1100381710038171003817100381721003817100381710038171003817x11987810038171003817100381710038172]]]]]

e(TDOA) = [[[[[

212059121 2x1198792 120591221 21205911198781 2x119879119878 12059121198781

]]]]][[[[m 119886m1198862

]]]]minus [[[[[

1003817100381710038171003817x2100381710038171003817100381721003817100381710038171003817x11987810038171003817100381710038172]]]]]

(73)


minΘ

e2 (74)


Θ ≜ (H119879H)minus1H119879g (75)



e = [[[[[

211988812059121 2x1198792 21198881205911198781 2x119879119878

]]]]][ m

m] minus [[[[[

1003817100381710038171003817x210038171003817100381710038172 minus 11988821205912211003817100381710038171003817x11987810038171003817100381710038172 minus 119888212059121198781]]]]]

(76)


















10 Conclusion






References





















































































































RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











120591119904 = 120578119904 + Δ120578119904119865119904 + Δ119865119904 = 120578119904 + Δ120578119904119865119904 ( 11 + Δ119865119904119865119904) (64)




1003817100381710038171003817 = 119888120578119904 asymp 119886120591119904 + 119887 (66)





119869(TOA)LS (m 119886 119887) = 10038171003817100381710038171003817e(TOA)100381710038171003817100381710038172 (68)


m(TOA)LS = argmin

m119886119887119869(TOA)LS (m 119886 119887) (69)



119886120591119904 + 119887 = (119886120591119904 minus 1198861205911) + (1198861205911 + 119887) asymp 1198861205911199041 + m (70)



119869(TDOA)LS (m 119886) = 10038171003817100381710038171003817e(TDOA)

LS100381710038171003817100381710038172 (71)


m(TDOA)LS = argmin

m119886119869(TDOA)LS (m 119886) (72)



e(TOA) = [[[[[

1 2x1198791 12059121 21205911 1 2x119879119878 1205912119878 2120591119904

]]]]][[[[[[

1198872 minus m2m1198862119886119887

]]]]]]minus [[[[[

1003817100381710038171003817x1100381710038171003817100381721003817100381710038171003817x11987810038171003817100381710038172]]]]]

e(TDOA) = [[[[[

212059121 2x1198792 120591221 21205911198781 2x119879119878 12059121198781

]]]]][[[[m 119886m1198862

]]]]minus [[[[[

1003817100381710038171003817x2100381710038171003817100381721003817100381710038171003817x11987810038171003817100381710038172]]]]]

(73)


minΘ

e2 (74)


Θ ≜ (H119879H)minus1H119879g (75)



e = [[[[[

211988812059121 2x1198792 21198881205911198781 2x119879119878

]]]]][ m

m] minus [[[[[

1003817100381710038171003817x210038171003817100381710038172 minus 11988821205912211003817100381710038171003817x11987810038171003817100381710038172 minus 119888212059121198781]]]]]

(76)


















10 Conclusion






References





















































































































RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation
























10 Conclusion






References





















































































































RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation
















































































































RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









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