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Robust acoustic source localization in sensor networks

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Robust acoustic source localization in sensor networks

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Content ROBUSTACOUSTICSOURCELOCALIZATIONINSENSORNETWORKS by ChartchaiMeesookho ADissertationPresentedtothe FACULTYOFTHEGRADUATESCHOOL UNIVERSITYOFSOUTHERNCALIFORNIA InPartialFulfillmentofthe RequirementsfortheDegree DOCTOROFPHILOSOPHY (ELECTRICALENGINEERING) August2007 Copyright 2007 ChartchaiMeesookho Dedication tomyparents ii Acknowledgments First of all, I would like to thank my advisor, Prof. Shrikanth Narayanan, for his guid- ance and invaluable advices. I am very grateful for a precious opportunity to start in PhDprogramandcontinuoussupporthegaveme. Hehastaughtmemanyusefulskills including how to do research efficiently. My thesis would not have been finished with- out tremendous support from my co-advisor, Prof. Urbashi Mitra. Her wisdom helps me through all difficulties and improves the quality of work. I thank Prof. Panayiotis Georgiou for helping me get started in the topic and his suggestions to overcome many obstacles. DuringacademicyearsatUSC,Iwouldliketothankallofmyfellowcolleaguesfor helpfuldiscussionsandunforgettablefriendship. SpecialthanksareforJorgeSilvaand Viktor Rozgic who give extensive comments and reviews on my papers. I would also liketothankGloriaHalfacre,TimBoston,andDianneDemetrasfortheiradministrative work. Finally, everyone in my family has been my motivation and inspiration. I cannot achievethisgoalwithoutthem. iii TableofContents Dedication ii Acknowledgments iii ListofTables vii ListofFigures viii Preface xi Chapter1: Introduction 1 1.1 SensorNetworkOverview . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.3 OurContributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.4 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Chapter2: BackgroundandReview 6 2.1 TimeDelayBasedApproach . . . . . . . . . . . . . . . . . . . . . . . 6 2.2 SpatialLossBasedApproach . . . . . . . . . . . . . . . . . . . . . . . 7 2.3 DataFusionProtocolsforSourceLocalizationandTracking . . . . . . 8 Chapter3: DistributedRangeDifferenceBasedAcousticSourceLocalization 10 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 3.2 ClusteringforSourceLocalization . . . . . . . . . . . . . . . . . . . . 12 3.3 RangeDifferenceBasedLeastSquareLocalization . . . . . . . . . . . 14 3.4 DataModel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 3.5 DistributedLocalization . . . . . . . . . . . . . . . . . . . . . . . . . 17 3.6 ExperimentalResults . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 3.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 Chapter4: OnEnergyBasedAcousticSourceLocalization 27 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 4.2 SignalModel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 iv 4.3 LeastSquaresSolutionsforEnergyRatioApproach . . . . . . . . . . . 33 4.3.1 QEvsOneStepLeastSquaresSolution . . . . . . . . . . . . . 35 4.4 WeightedOneStepLeastSquaresSolution . . . . . . . . . . . . . . . 36 4.5 WeightedDirectLeastSquaresSolution . . . . . . . . . . . . . . . . . 39 4.6 CorrectionTechnique . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 4.7 SimulationResults . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 4.8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 Chapter 5: Design Considerations for Acoustic Source Localization with a Grid-BasedSensorField 56 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 5.2 SignalModel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 5.2.1 TimeDelayBasedObservationModel . . . . . . . . . . . . . . 60 5.2.2 EnergyBasedObservationModel . . . . . . . . . . . . . . . . 61 5.3 PerformanceAnalysisforDesignConsiderations . . . . . . . . . . . . 62 5.3.1 Caseoflargenumberofsensors . . . . . . . . . . . . . . . . . 68 5.3.2 PerformanceComparison. . . . . . . . . . . . . . . . . . . . . 73 5.4 Caseofuniformlyrandomsensorfield . . . . . . . . . . . . . . . . . . 75 5.5 SimulationResults . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 5.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 Chapter 6: A Hybrid Maximum-Likelihood Estimator for Acoustic Source LocalizationbasedonTemporalandSpatialAttenuationInformation 87 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 6.2 SignalModel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 6.3 HybridMaximumLikelihoodEstimator . . . . . . . . . . . . . . . . . 90 6.4 PerformanceAnalysisviaCRB . . . . . . . . . . . . . . . . . . . . . . 92 6.4.1 Practicallimitations . . . . . . . . . . . . . . . . . . . . . . . 94 6.5 SimulationResults . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 6.6 FieldExperiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 6.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 Chapter7: ConclusionsandFutureWork 108 7.1 Futurework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 References 111 Appendices 116 AppendixA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 AppendixB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 AppendixC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 AppendixD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 v AppendixE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 AppendixF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 vi ListofTables Table4.1: Biasesofdifferentschemeswhennumberofsensorsisvaried. . . . 50 Table4.2: BiasesofdifferentschemeswhenSNR 0 isvaried. . . . . . . . . . 51 Table4.3: Biasesofdifferentschemeswhennumberofsamplesisvaried. . . 53 Table 4.4: RMS error for the case of bird song: weighted schemes are still betterthannon-weightedschemes(QE,OS). . . . . . . . . . . . . . . . 55 Table6.1: RMSerrors(m)producedbyeachestimatorfordifferenttopologies 105 Table6.2: RMSerrors(m)producedbyeachestimatorfordifferenttopologies 106 vii ListofFigures Figure1.1: Thediagramillustrateshowourcontributionfitsinthebigpicture ofacousticsourcelocalizationinsensornetworks . . . . . . . . . . . . 5 Figure3.1: MSEvs. numberofsensors: Distributedmethodproducessmaller errorthancentralizedmethod. . . . . . . . . . . . . . . . . . . . . . . 23 Figure 3.2: The accuracy of the distributed method is less affected by a low energysourcesignalthanthecentralizedmethod. . . . . . . . . . . . . 24 Figure 3.3: Energy consumed by centralized method is larger than that con- sumedinthedistributedmethod. . . . . . . . . . . . . . . . . . . . . . 25 Figure 3.4: Distance error and distance between consecutive estimates are highlycorrelated. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 Figure 4.1: Acoustic sensors (filled circles) receive signal generated by a source(representedbyastar)andexchangedataforlocalization. . . . . 50 Figure 4.2: Comparison of different weighting schemes: weighted schemes (WOS,WD,WDC) outperform non-weighted schemes (QE,OS) and the differencebecomeslargerwhennumberofsensorsisincreased. . . . . 51 Figure4.3: EffectofSNR 0 : weightedschemes(WOS,WD,WDC)aredegraded withhigherratethannon-weightedschemes(QE,OS)whenSNR 0 becomes lower. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 Figure4.4: Effectofdatasize: WOS,WD,andWDCareconsiderablyaffected bysmallL. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 Figure 4.5: Weighted schemes (WOS,WD,WDC) are still superior to non- weightedschemes(QE,OS)withthepresenceofnoisevariancedeviation. 53 Figure4.6: WOSandWDarelesssensitivetothedecayfactordeviationthan otherschemes.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 Figure4.7: Allmethodsaresimilarlyaffectedbysensorlocationdeviation. . 54 viii Figure4.8: PowerSpectralDensityofbirdsong(Commonloon). . . . . . . . 55 Figure5.1: Definingrelevantdistancesandanglesforthecaseof3participat- ingsensors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 Figure5.2: ThediagrampresentsthedifferencebetweenTOMandEOMthat areassociatedwithCRB t andCRB e . . . . . . . . . . . . . . . . . . . 80 Figure 5.3: The approximated φ t (N t ,¯ z s ) are close to that obtained from the exhaustivesearchforvariedN t . . . . . . . . . . . . . . . . . . . . . . 81 Figure 5.4: The approximated φ e (N e ,¯ z s ) are close to that obtained from the exhaustivesearchforvariedN e . . . . . . . . . . . . . . . . . . . . . . 82 Figure5.5: Thevarioussourcelocationsconsideredinthesensorfield. . . . . 82 Figure5.6: ThePowerSpectralDensity(PSD)ofthesourcewithf max =250 Hz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 Figure 5.7: The performance of EML and TML is sensitive to the source locationswithinthetrianglearea. . . . . . . . . . . . . . . . . . . . . 83 Figure5.8: TMLbecomessuperiortoEMLwhengridsizeisincreased. . . . 84 Figure5.9: ρ min andρ max whengridsizeisvaried. . . . . . . . . . . . . . . . 84 Figure5.10: TMLoutperformsEMLwithhigh-frequencysource(large f s ). . 85 Figure5.11: ρ min andρ max whenf s isvaried. . . . . . . . . . . . . . . . . . 85 Figure 5.12: EML can be better than TML by using more collected samples (increasingL e ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 Figure5.13: ρ min andρ max whennumberofsamplesforEML(L e )isvaried. . 86 Figure6.1: Sourcetrackandsensorlocationsassumedinthesimulation. . . . 100 Figure 6.2: HML outperforms other estimators for all source locations while TMLbecomesworsethanEMLwithfarawaysource. . . . . . . . . . . 101 Figure 6.3: The improvement of HML over TML and EML becomes more conspicuousforfarawayandnearsource,respectively. . . . . . . . . . 101 Figure 6.4: HML outperforms other estimators for all sources with different maximumfrequencieswhileTMLbecomesworsethanEMLwithlow- frequencysource. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 ix Figure 6.5: The improvement of HML over TML and EML becomes more conspicuousforlow-frequencyandhigh-frequencysource,respectively. 102 Figure6.6: HMLforthesourcewithf max =400Hz,usingfewersamples(up to50%less),stilloutperformsTMLandEML. . . . . . . . . . . . . . 103 Figure 6.7: The marks of source and sensor locations where each run is dif- ferentaccordingtoTable6.1 . . . . . . . . . . . . . . . . . . . . . . . 104 Figure6.8: PowerSpectralDensityofthecarsound(Jeep). . . . . . . . . . . 104 Figure6.9: Onaverage,HMLoutperformsTMLandEMLforallscenarios. . 107 Figure6.10: Experimentlocation,AENSboxes,andmicrophones . . . . . . 107 x Preface Our goal is to improve the performance of acoustic source localization in the context of sensor networks. A number of relevant problems are addressed and solutions are proposed. As data is usually collected by sensors with randomly distributed locations, a prob- lemthatemergedforsourcelocalizationishowtoefficientlygatheralltherequireddata and combine them subject to selected localization schemes. In Chapter 3, we propose a distributed algorithm based on a specific method, range difference based localization. Simulation results illustrate that the distributed localization produces smaller error and consumeslessenergythancentralizedmethod. Theadvantageofdistributedprocessing becomes more conspicuous for error considerations when the number of participating sensors is small and obtains more energy saving with the large number of participating sensor. Theproposedmethodisalsomorerobusttodecreasingtargetsignalenergyand theinstantaneouserrorfromthesequenceofestimatescanbeapproximatedandusedto reconcilethecostandthesystemperformance. Consideringrecentlyproposedmethods,energybasedlocalizationattractsourinter- est due to their simplicity and the possibility of earning energy saving while maintain- ing acceptable accuracy. The investigation of existing energy based methods leads to findings that show the possibility of further improvement. In Chapter 4, energy-based localizationmethodsforsourcelocalizationinsensornetworksareexamined. Thefocus xi is on least squares based approaches due to a good trade-off between performance and complexity. A suite of methods are developed and compared. First, two previously proposed methods (Quadratic Elimination and One Step) are shown to yield the same location estimate for a source. Next, it is shown that as the errors which perturb the leastsquaresequationsarenon-identicallydistributed,itismoreappropriatetoconsider weightedleastsquaresmethodswhichisobservedtoyieldsignificantperformancegains over the unweighted methods. Finally, a new weighted direct least squares formulation is presented and shown to outperform the previous methods with much less computa- tional complexity. Unlike the Quadratic Elimination method, the weighted direct least squares method is amenable to a correction technique which incorporates the depen- dence of unknown parameters leading to further performance gains. For a sufficiently largenumberofsamples,simulationsshowthattheWeightedDirectsolutionwithCor- rection (WDC) can be more accurate with significantly less computational complexity thanthemaximumlikelihoodestimatorandapproachesCram´ er-RaoBound(CRB).Fur- thermore,itisshownthatWDCattainsCRBforthecaseofawhitesource. Sincetheconsiderationofsystemdesignisinevitablyimportant,inChapter5,design rules for sensor network deployment for acoustic source localization are determined. In particular, methods based on time-delay information in the received acoustic signal time series and those based on energy readings are examined through the evaluation of Cram´ er Rao bounds on the estimation error variance. Assuming unknown source locationandnearestneighborsensorparticipationinlocalization,theminimumandthe maximum of the CRBs (over source location) for the case of three sensors and the lim- iting case approximation for large number of sensors are derived. The derived limits, which are functions of key design parameters such as number of sensors, grid size, sampling frequency, and number of collected samples, are shown to be good approx- imations as validated by direct numerical evaluation of the true CRBs. Experimental xii simulation results demonstrate that the performances of the Maximum-Likelihood esti- mators using each observation model are consistent with the corresponding CRB, and thus the derived CRBs can be used for performance prediction. The comparison illus- trates that the energy based observation model, generally assumed to be less accurate due to potential information loss in the energy calculation, can be superior for low- frequency sources and small grid size (dense) fields. Design rules, explicitly presented asmathematicalexpressions,andexamplesoftheirapplication,areprovidedforsystem parameterandschemeselection. InChapter6,maximum-likelihoodestimationfornear-fieldacousticsourcelocaliza- tion is examined. Prior work has focused on localization schemes based on time-delay information solely or received signal energy both of which are a function of the source location. Herein,asignalmodelwhichexplicitlycaptureslocationdependenceonsignal delayandsignalattenuationisdevelopedandacorrespondingmaximum-likelihoodesti- mator derived (HML). Analysis of achievable performance for time-delay information only,receivedsignalenergyonlyandthehybridsignalmodelsisdoneviathedetermina- tionoftheCram´ er-RaoBounds(CRB).TheCRBanalysisshowsthatthehybridscheme offers significant performance improvement over time-delay based methods for low- frequency sources; similarly strong performance gains over the received signal energy model is achieved for high-frequency sources. Simulation results confirm these trends fortheassociatedmaximumlikelihoodestimatorsandfurtherindicatethatsourceloca- tion plays a strong role in determining whether energy or time-delay based schemes are superior relative to each other. Energy based schemes incur a smaller communica- tioncostrelativetotheothermethods,suggestingthatHMLshouldalwaysbeusedover time-delaybasedmethods. However,fieldexperimentswiththeAcousticENSBoxshow that HML also exhibits significant robustness over energy-based schemes, suggesting xiii thatHMLshouldalsobeconsideredforlowpowerenvironmentswherecommunication costscanbereducedbyreducingthenumberofobservations. xiv Chapter1 Introduction Even though acoustic source localization is a classical problem in signal processing, it is associated with some new problems when applied in an emerging technology, sensor networks. Sensor network characteristics pose the problems of the efficiency of local- ization systems regarding design trade-offs between accuracy and energy consumption cost. This dissertation focuses on deriving theoretical methods and their experimental evaluationsforperformingacousticsourcelocalizationwithsensornetworks. In this chapter, an overview of sensor networks is first provided. Then, motivations foracousticsourcelocalizationinsensornetworksaredescribed. Finally, ourcontribu- tionsaresummarized. 1.1 SensorNetworkOverview Sensor networks emerged from the rapid development of Micro Electro-Mechanical Systems (MEMS) concurrently with advances in wireless network technologies. Sen- sornetworkstypicallyincludealargenumberofinexpensiveandsmartsensorsthatare expected to provide opportunities for instrumenting, monitoring and controlling target- ing systems, for instance, environment, buildings, and high-risk areas (e.g. battlefields) [EGPS01]. Sensor networks are suitable for applications that require redundant data and close proximity to targets or areas of interest. Despite originally motivated by mil- itary applications such as acoustic surveillance systems and ground target detection, 1 sensornetworkscanbeappliedformanypotentialapplicationsforexamplesinfrastruc- turesecurity,environmentandhabitatmonitoring,industrialsensing,andtrafficcontrol [CK03]. Sensors usually have capabilities for acquiring and embedded-processing of a varietyofdataformssuchasacoustic,seismic,andinfraredsignals. Networkingproto- colsandwirelesscommunicationareusedtomanageinformationexchangeandsharing so that systems can obtain high quality data integration. The ultimate goal of sensor networks is to be deployed in any scenarios that information is demanded. Therefore, autonomousandrobustsystemsarerequiredparticularlyforsomeinaccessibleandhaz- ardous areas. These systems are also preferred to be ad hoc and self configuration, and, in several scenarios, sensors are deployed in random topology. Since sensors are expected to be unmanned and remotely operated, they typically utilize battery power whichislimitedwhenthebatteriescannotbeconvenientlyreplaced. Hence,challenges aremotivatedbyenergyconsumptionconstraints. Relatedprocessesincludingsensing, computing,andcommunicationcarriedoutbythesensorsmustbeefficientlydesigned. Problems mainly relate to how to gain the most meaningful information from the data collected by distributed sensor nodes in efficient and robust manners. These lead to the problems of participating sensors selection, data fusion, and routing etc. An impor- tant design issue is the investigation of how system parameters such as network size andthedensityofsensornodesaffecttrade-offsbetweenlatency,reliability,andenergy consumption. 1.2 Motivation Acoustic sources are typically needed to be observed in many scenarios for examples sounds created by people in a room, animals in their natural habitats, or military vehi- clesinabattlefield. Sensornetworkspotentiallyfacilitateefficientsurveillancesystems 2 for these scenarios since a large number of sensors enable the redundancy of the obser- vations and close proximity to acoustic sources. Acoustic sensors are also appealing becausetheyarepassive,affordable,robust,andcompact. Moreover,thepropagationof sound energy is not limited by obstacles, which block or obscure the clear line of sight thatisrequiredfortheeffectiveoperationofelectromagneticsystems[FL04]. Acoustic source observation, therefore, takes a significant part of a sensor network framework. Some specific applications have been addressed in the literature for instances shooter localization[LVM + 05],habitatmonitoring[WEG + 03a],andmilitaryvehicledetection, classification and tracking [LWHS02a]. One of the significant information needed to be extracted from observations is a source location. To specifically implement acoustic source localization in a sensor network environment, two significant issues need to be considered. The first one relates to specific localization schemes to be suitably applied in a scenario. For example, for acoustic signals, a selected localization scheme should properly work for wideband sources. In addition, the scheme should be applicable for near-field sources which frequently occur because sensors are preferred and likely to locate near sources. Simple schemes can be desirable since low complexity decreases energy consumption and system latency. Relevant existing schemes will be mentioned in Section 2.1 and 2.2. Regardless of the selected localization techniques, the data col- lectedatsensorsneedtobefusedbysomemeansatsomeplaceswhereestimatedsource locations are computed. The second issue, thus, relates to data fusion under power constraints including sensor selection mechanisms, i.e., deciding which sensors should participate and how to efficiently collect the data from such sensors before processing. The efficiency of designs in associated with data fusion and sensor selection, however, dependsonspecificlocalizationalgorithms. Section2.3willdescribepreviousworksin thisissue. 3 1.3 OurContributions The contribution for acoustic source localization in sensor networks can be twofold: first, novel localization approaches, second, design methods to accommodate sensor networkcharacteristicsandconstraints. Ourcontributioninthisscopeissummarizedas follows 1. Proposeadistributedalgorithmbasedonaspecificmethod,rangedifferencebased localization. Simulationresultsillustratethatthedistributedlocalizationproduces smallererrorandconsumeslessenergythancentralizedmethod(seeChapter3). 2. Develop the Weighted One Step (WOS) and the Weighted Direct (WD) least squares solutions for acoustic source localization that significantly outperform existingenergybasedlocalizationmethods(seeChapter4). 3. Suggest design rules for energy based and time delay based localization in grid- based sensor fields based on the analysis of energy based and time delay based observationmodelsanddetermineunderwhatconditionsonemodelisbetterthan another(seeChapter5). 4. Develop a hybrid maximum-likelihood estimator based on both temporal and spatial attenuation information that advantageously balance trade-offs between energybasedandtimedelaybasedmethods. Theproposedestimatorenablescost savingduetomoretolerationinlesscollecteddataanditsrobustnessisvalidated byfieldexperiments. (seeChapter6). 4 Sensor Networks Characteristics Applications Acoustic Source Localizations Localization Methods Data Fusion Protocols Spatial Loss based Time delay based Energy Based Range Difference Based Energy Constraints Grid-based field Hybrid ML Estimator Design Rules Improved Least Squares Solution Distributed Method Figure1.1: Thediagramillustrateshowourcontributionfitsinthebigpictureofacoustic sourcelocalizationinsensornetworks 1.4 Outline All significant artifacts related to our work are outlined and shown in Figure 1.1. The diagram illustrates major components in this area including our contributions. Circu- larobjectswithsolidlinecontaincompleteworksthatweresummarizedintheprevious sectionanddetailscanbefoundinChapter3,4,5,and6. Weproposedanimprovedleast squares solutions for energy based localization method, suggest design rules for local- ization in sensor network scenarios with grid-based topology, introduced a distributed range difference based method that is more accurate and earns energy saving, and a combining scheme based on temporal and spatial attenuation information that is more accurateandmoreenergy-efficient. 5 Chapter2 BackgroundandReview As mentioned in the previous chapter, acoustic source localization in sensor networks relatestotwomajorissuesnamelyselectedlocalizationmethodsanddatafusionproto- colssubjecttosuchlocalizationmethods. Thefirsttwosectionsinthischapterdescribe two important classes of localization techniques. In Section 2.1, a number of existing schemes that are fundamentally suitable for acoustic source localization problems but the consequences of their applications in sensor network context need to be addressed and investigated are summarized. Energy constraints in sensor networks motivated a novel scheme that is less complicated and provides more options for energy manage- ment, these techniques will be described in Section 2.2. Data fusion techniques that have been introduced subject to different localization schemes and system assumptions willbegiveninSection2.3. 2.1 TimeDelayBasedApproach TimeDelayApproach(TDA)usesthetemporalinformationextractedfromsignaltrav- eling time combining with the spatial information of sensor locations. There have been a large number of methods based on TDA proposed during last few decades and these methods are applicable depending on source characteristics such as wideband, narrow- band,near-field,orfar-field. Sinceacousticsourcesarewidebandandareassumedtobe near-field in sensor network scenarios, only suitable methods for these assumptions are considered. Chen et al derived a single-step Maximum Likelihood location estimator 6 forwidebandsourcesinthenearfieldofthesensorarray[CHY02]. Thisalgorithmwas implemented on wireless networked acoustic sensor array testbed to perform acoustic source localization [CYE + 03]. However, the experiments were carried out only in cen- tralizedprocessingscenarioswhicharenotdeemedefficientforpracticalsensornetwork application. Another class of wideband source localization algorithms is based on time delays. Usingtimedelayestimatestocomputerangedifferences,closed-formsolutions can be obtained through two-step methods. These methods will be called Range Dif- ference (RD) based methods which involve our work in chapter 3. Different RD based techniques can be found in [SA87, CH94, HBE01]. To the best of our knowledge, the in-depth study of the implementation of RD based methods in sensor network context has not been presented. Although TDA have already been well developed, its applica- tions in sensor networks are limited by energy constraints. Since time delay estimates are computed from relevant time series data collected at the sensors, it is required to relocate these data from the sensors to appropriate places and, for a wireless network as typically used in sensor network, it is costly. Therefore, the implementation of TDA inrealisticsensornetworkapplicationsrequirestheconsiderationsofthereconciliation betweencommunicationcostandlocalizationaccuracy. 2.2 SpatialLossBasedApproach Recently,energybasedmethods[LH03,SH05]werepurposedandpointedoutthatthey are suitable for sensor network applications. These methods use the property of acous- tic signal intensity attenuation which is inversely proportional to the distance from the source. Sinceenergybasedmethodsusetheknowledgeofspatiallossmodelofacoustic signals, they will be categorized in Spatial Loss based Approach (SLA). Energy based localization can be obtained using Maximum Likelihood Estimation (MLE) [SH05] or 7 least squares estimation [LH03]. MLE, however, includes nonlinear optimization that entails high computational complexity. The advantage of energy based methods over TDAisthattheexchangeoftimeseriesdataacrossthesensorsisnotrequired. Onlythe averageofthereceivedenergyoveratimewindowisneededtobesharedwiththeother sensors. This emphasizes the concept of using local processing and transmitting only meaningful information to be further (globally) processed and enables saving a great deal of communication cost. However, the acoustic signal propagation is sensitive to the environment and, thus, causes the uncertainty for the decay model and would make the system not robust. Moreover, information loss due to averaging process potentially causesSLAtobelessaccuratethanTDA. 2.3 Data Fusion Protocols for Source Localization and Tracking Data fusion protocols in sensor networks relate to an idea called decentralized process- ing. Its main concept is to not rely on only one central processing unit. Since a pro- cessingunitpresumablyrequiressomeinformationfromtheneighboringunitstoobtain the results for the particular task, data transmission is inevitably necessary. However, to globally process the information or measurements gathered from all sensors seems unreasonableparticularlyinalargeanddensesensorfieldduetotheenergyconstraints. Aconceivablesolutionistodividesensorsintoanumberofgroupstooperatetheirtasks using local processing units. These groups of sensors are commonly called clusters and the cluster head typically plays a role of processing unit. Generic energy-efficient clustering protocols for decentralized processing in sensor network can be found in [HCB00, LR02]. Source localization problems using clustering technique, however, 8 may require somewhat specific formulation. Since the ultimate goal is to finally iden- tifyaparticularpointwhichisthemostlikelysourcelocation,theessentialinformation seems available in a distinct area. Hence, only the most informative cluster may be needed and that motivates the design of an efficient cluster formation at the particu- lar time and place. Cluster forming protocols for the purpose of source localization and tracking have been presented in [PJB03, FGJ + 02]. IDSQ introduced by Zhao et al [CHZ02, ZSR02] allows the maximum information gain for the dynamic clustering using an information utility measure. Dynamic clustering for acoustic target tracking was presented in [CHS04]. In [WCZ + 03], quantity-driven approach is implemented in the acoustic tracking system. The idea is to design the communication parameter based on the quality of data acquired from each cluster. Decentralized source localiza- tion based on incremental gradient descent-like optimization was proposed in [RN04]. These methods were proposedand experimented mostly in ad hocscenarios using sim- ple localization schemes. The problems of how to efficiently implement complicated localizationmethodssuchasTDAinthelargescale(densefield)systemarestillopen. 9 Chapter3 DistributedRangeDifferenceBased AcousticSourceLocalization Ofthevariousconventionalmethodscanbeapplied,andhavebeenproposedforacous- tic source localization, the Range Difference (RD) Based method is attractive due to improved accuracy and ease of implementation it affords. While the basic concepts of the RD based method can be adopted to the case of sensor networks, the data acquisi- tion and aggregation procedures need to be formulated and characterized subject to the energy constraint. The challenge is to design an efficient algorithm that is economical and still accurate. In this chapter, based on range difference localization method, we proposeadistributedalgorithmwhichallowsthetimedelayestimationtobecarriedout at each participating sensor. The acquired data is fused using a sequential least squares schemewhichenablestheappropriatesensorselectionbasedonthecurrentestimate. 3.1 Introduction Acousticsourcelocalizationisoneofthekeymotivatingapplicationsforimplementing sensor networks. A large number of sensors enables the redundancy of the observa- tions and close proximity to the source, and thus, improving the chances for improved sourcelocalizationandtrackingperformance. Someexampleapplicationsincludelocal- izingmilitaryvehiclesinabattlefieldandtrackingwildanimalsintheirnaturalhabitat. Recently, conventional source localization methods have been applied to the case of 10 sensor networks. The Range Difference (RD) based method is particularly attractive in this context [CYE + 03, WCZ + 03] since it offers better ease of implementation than the Maximum Likelihood (ML) estimator [CHY02, KS04], is more accurate than energy based localization [SH03], and does not require the prior knowledge of the signal gen- eratedby thesource. While thebasicconceptsof theRDbasedmethod canbeadopted to the sensor networks problem, the data aggregation procedure needs to be developed and characterized. In traditional systems such as radars and microphone arrays, time series data collected from each sensor, the fundamental information needed in the pro- cess, is assumed to be available at the central processing unit without the concern for the cost incurred in gathering such information. However, due to the characteristics of sensornetworks,whicharetypicallybattery-poweredandwireless,theenergyexpendi- ture for time series data exchange between sensors should be taken into account. The challenge is to design an efficient algorithm that is economical and still accurate. In [CYE + 03],localizationwasimplementedonasensorarraytestbedbutthecommunica- tioncostwasnotconsidered. Thecluster-basedarchitectureforacousticsourcetracking wasstudiedin[WCZ + 03]. Nonetheless,thesystemperformancesubjecttothecommu- nication protocol within the cluster was not addressed. We believe that the impact of the designed algorithm on the system efficiency is highly dependent on what specific methodisimplemented. Inthischapter,basedonrangedifferencelocalization,wepro- poseadistributedalgorithmwhichallowstimedelayestimationtobecarriedoutateach participatingsensorsothattheamountofenergyincurredfortimeseriesdatatransmis- sion can be decreased. The acquired data, which are the range differences, are fused using a sequential least squares scheme. The sequential nature allows for efficient sen- sorselectionbasedonthecurrentestimateateachtimestep, thus, enablingaccuracyto beimproved. Theresults,evaluatedusingrealisticmodelsandconditions,illustratethat the distributed localization produces smaller error and consumes less energy than the 11 centralized method. Notably, the advantage of distributed localization in terms of the accuracy becomes more significant when the number of participating sensors is small whiletheenergysavingincreaseswhenthenumberofparticipatingsensorsislarge. The proposedmethodisalsorobustinthatitsaccuracyislessaffectedbyasourcesignalwith lowenergy(lowerSNR)andtheinstantaneouserrorfromthesequenceofestimatescan beapproximatedandusedtoreconcilethecostandthesystemperformance. 3.2 ClusteringforSourceLocalization Since a centralized global processing of information or measurements gathered from all sensors does not seem to be attractive, or may be feasible, especially in a large and densesensorfieldduetothedemandsofhighcommunicationcost,anappropriatesolu- tion is to divide the sensors into a number of smaller groups to operate on the tasks where each group has a local processing unit. The fusion and compression of locally processed data can save the energy used for the transmission of raw data to the base station or the end user. These groups of sensors are commonly called clusters and one sensorisselectedtobeaclusterheadplayingtheroleofalocalprocessingunit. Generic energy-efficientclusteringprotocolsfordecentralizedprocessinginsensornetworkcan be found in [HCB00]. A source localization problem using clustering technique may require a somewhat more specific formulation. Since the ultimate goal is to identify a particularpointwhichisthemostlikelysourcelocation,thecriticalinformationshould beavailableinthecorrespondingsourcearea. Hence,onlythemostinformativecluster maybeneededandthatmotivatesthedesignofanefficientclusterformationatthepar- ticular time and place. Cluster forming protocols for the purpose of source localization andtrackinghavebeenpresentedin[PJB03]wheretheDynamicSpace-TimeClustering (DSTC)algorithmwasproposedtoworkasaclusterformingprotocolbasedonClosest 12 Point of Approach (CPA). Information Driven Sensor Querying (IDSQ) introduced by Zhao et al [CHZ02] allows the maximum information gain for the dynamic clustering using an information utility measure. Dynamic clustering for acoustic source tracking waspresentedin[CHS04]. Asparselyplacedhigh-capabilitysensorscenario(expected to play cluster head roles) is assumed and a cluster is formed when the acoustic signal strength detected by the cluster head exceeds a predetermined threshold. The cluster’s priority is to integrate the measurements collected at each cluster member to represent the earning knowledge from each cluster. It is fairly application specific in order to describethemechanismoftheinformationmanagementassociatedwiththeclusterfor- mation. Generally, all members communicate with the cluster head either by direct or multi-hop communication. The signal processing functions are carried out at the clus- ter head before transferring compressed data to the base station or end user. We will call this a centralized processing scheme. A specific application such as source local- ization using conventional methods, however, requires a circumspect design in order to obtain an accurate and cost-effective system. The main reason is that the observation at each sensor is commonly time series data. To individually transmit such data from all cluster members to be processed at cluster head entails a large amount of overall communication cost particularly when the cluster is designed to be large to reach the localization accuracy requirement. An alternative is to apply distributed processing by some means depending upon the characteristics of the utilized methods for a particular application. We exploit a well-known scheme, the range difference based localization, for distributed processing in sensor networks and demonstrate that the system perfor- mancecanbeimprovedwhencomparedwiththecentralizedmethod. 13 3.3 RangeDifferenceBasedLeastSquareLocalization Range Difference (RD) based localization methods are well-known and applicable for wideband sources such as acoustic signal. RDs can be derived from Time Difference of Arrival (TDOA) estimation through the relationship between distance and travel- ing speed of the signal over a medium. Time delay estimation technique [KC76] is the fundamental tool used to determine TDOAs. We will assume the existence of an optimal time delay estimator producing estimated TDOA perturbed by additive noise to model uncertainty. There have been a number of RD based approaches proposed in the past few decades [HBE01, CH94]. We focus on a closed-form least square method proposed in [HBE01] since it was reported to be more efficient than the other schemes and was shown to approach the Cramer Rao Bound (CRB) in high Signal to NoiseRatio(SNR)environments. LetN sensorsbeassignedtoparticipateinthelocal- ization process located at coordinates {(x 1 ,y 1 ),...(x N ,y N )}. Assuming the source is located at z s = (x s ,y s ), the differences of the distance between sensors i and j where i,j = 1,...,N and the source denoted by d ij can be obtained by the basic relation: d ij = D i − D j where D i = p (x s −x i ) 2 +(y s −y i ) 2 . RDs with respect to one arbitrary reference sensor are typically used. Without the loss of the gener- ality, we select (x 1 ,y 1 ) to be the location of reference sensor. The time series data collected from the other sensors together with the received signal at the reference sen- sor can produce the TDOA estimates and RDs can be derived from TDOAs using the knowledge of signal traveling speed. In the real application, however, the actual RDs are not available since there are some errors from TDOA estimation. Consequently, we have ˆ d i1 = d i1 + n i1 , i = 1...N. The TDOA estimate obtained by general- izedcrosscorrelationwithGaussiandataisasymptoticallynormallydistributedinhigh SNRenvironment[Car81]. Therefore,theRDestimateisalsoGaussianandweassume 14 n i1 ∼ N(0,σ 2 i1 ). The Localization problem can be formulated as a linear least squares problem,Aθ =b,where A =      x 2 y 2 ˆ d 12 . . . . . . . . . x N y N ˆ d N1      θ =      x s y s R s      , b = 1 2      R 2 1 − ˆ d 2 12 . . . R 2 N − ˆ d 2 N1      R i = p (x i −x 1 ) 2 +(y i −y 1 ) 2 These linear least square equations can be solved by a batch approach and the solu- tion, ˆ θ = (A T A) −1 A T b. However,wecanupdate ˆ θ withouthavingtoresolvethelinear equations by a sequential least squares procedure [Kay93] which can be described by letting A[n] = [A[n−1]a T [n]] T and b[n] = [b[1] b[2] ... b[n]] T . The sequential leastsquareestimatorbecomes ˆ θ[n] = ˆ θ[n−1]+K[n] b[n]−a T [n] ˆ θ[n−1] (3.1) K[n] = Σ[n−1]a[n] 1+a T [n]Σ[n−1]a[n] Σ[n] = I−K[n]a T [n] Σ[n−1] (3.2) andindexncorrespondstothen th sensor. 15 3.4 DataModel AstaticacousticsourcegeneratingaWideSenseStationaryGaussianrandomobserva- tion process,s(t), is assumed where the intensity attenuates at the rate that is inversely proportional to the distance from the source. Perturbed by additive Gaussian measure- mentnoisew i (t),thereceivedsignalatthei th sensorisgivenbyx i (t) = s(t−τ i ) D i +w i (t). TheenergycanbecalculatedbyaveragingoveratimewindowT = M fs whereM isthe number of samples and f s is the sampling frequency as y i [k] = 1 M P kM j=(k−1)M+1 x 2 i [j]. Assumings(t)andw(t)areindependent,weget E{y i [k]} = E{s 2 (t)} D 2 i +E{w 2 (t)} (3.3) var{y i [k]} = 2( E{s 2 (t)} D 2 i +E{w 2 (t)}) 2 M (3.4) Let s(t) ∼ N(0,σ 2 s ) and the noise at each sensor has the same distribution so that w i (t)∼ N(0,σ 2 w ). The signal PSD (G s (f)), the noise PSD (G w (f)), and the coherence are assumed to be flat over a bandwidth Δf Hz centered at frequency f 0 . The SNR at each sensor, G s,i (f) G w,i (f) = σ 2 s D 2 i σ 2 w . According to [KC76], CRB of the TDE estimate is the following σ 2 ij ≥ 8Tπ 2 3 C ij 1−C ij (f 0 + Δf 2 ) 3 −(f 0 − Δf 2 ) 3 −1 where C ij = 1 (1+ G s,i (f) G w,i (f) −1 )(1+ G s,j (f) G w,j (f) −1 ) It is simple to derive that the variance of the estimate can be in the form, σ 2 i1 = σ 2 1 +αD 2 i ,where 16 σ 2 1 = 3D 2 1 8Tπ 2 ((f 0 + Δf 2 ) 3 −(f 0 − Δf 2 ) 3 )SNR 0 α = 3(1+ D 2 1 SNR 0 ) 8Tπ 2 ((f 0 + Δf 2 ) 3 −(f 0 − Δf 2 ) 3 )SNR 0 (3.5) and SNR 0 denotes σ 2 s σ 2 w . Please note that σ 2 i1 is the variance of TDE between i th sensor and the reference sensor as assumed in the previous section. Such variance is proportional toD 2 i where the constants, σ 2 1 andα are functions ofD 2 1 . Therefore, with a fixed reference sensor, TDE with respect to the farther sensors from the source is less accurate. 3.5 DistributedLocalization Fromthedescriptionoftherangedifferencebasedlocalizationmethod,wecannotethat therearetwokeystepswhichareTDOAestimationandsourcelocalizationobtainedby solving least square equations. In a Centralized scheme, both steps take place at the cluster head. The cluster head should be a reference sensor and TDOAs with respect to the cluster members can be obtained through time delay estimation. The distributed localization concepts can be adopted by enabling some processes to occur at each par- ticipating sensor, not just at the cluster head. If time series data collected at the cluster head is transmitted to the participating sensors, time delay estimation can be operated there. Broadcasting the data from one reference sensor to many participating sensors isexpectedtorequirelesstotalcommunicationoverheadthanintheoppositedirection. Solving least square equations encompasses two mechanisms depending on whether batch or sequential procedure is applied. Batch estimator requires all measurements availableatthesametimewhereassequentialestimatorneedsonlytheestimateobtained 17 fromthe(n−1) th sensorandaTDOAcorrespondingtothen th sensor. Thelatter,how- ever, demands less computational complexity as it does not have to deal with matrix inversion which might be burdensome when the matrix is large due to a large number of participating sensors. Another advantage is that the current estimate can be used as the prior information to properly select the next participating sensor. According to the data model that the variance of time delay estimation is proportional to the square dis- tance between the sensor and the source, the preferred sensors can be simply selected by considering the nearest sensors to the current estimate. Consequently, combining theideasofdistributedprocessingfortimedelayestimationandsequentialleastsquare localization is expected to improve the localization performance in terms of both com- munication cost savings and accuracy. By using the notations defined in the previous section,weproposethefollowingalgorithm: 1. The sensor which receives the highest average signal energy in a certain time windowisselectedtobeaninitialsensor. Pleasenotethattheterm“initialsensor” isusedtocallthesensorthatstartstheprocessinsteadof“clusterhead”. 2. Theinitialsensorbroadcastscollectedtimeseriesdatatoatmostk nearestneigh- bors within the maximum radio range where k is the initial expected number of participating sensors. There might be a possibility that less than k sensors can be reached depending on the coverage of the radio range and the density of the sensorfield. 3. Each neighbor operates time delay estimation using time series data collected at thesensorandtheonesentfromtheinitialsensortoestimateTDOAs. 18 4. The initial estimate is obtained by using batch estimator based on TDOAs com- puted by the three nearest neighbors. The neighbor might be requested to broad- cast time series data received from the initial sensor if there are less than three sensorsthathavealreadyreceivedit. 5. k−4nearestsensorstotheinitialestimateachievedfromthebatchestimatorare expected to participate in the sequential least square method. It becomes k− i nearest sensors for the following estimate wherei is a number of sensors that are alreadyincludedinthelocalizationprocess. 6. Therouteofsequentialestimatorisconstructedfromthesensorsdescribedinthe previousstepbyconvexhullinsertionalgorithmforTravelingSalesmanProblem (TSP)route[CLRS01]. 7. The estimate is updated in the sequential fashion by using (3.1). Only ˆ θ andΣ areneededtobesentacrossthesensors. 8. Every time the estimate is updated, the route is also updated based on the new estimateandthedecreasingnumberofremainingparticipatingsensors. 9. If the next sensor in the path has not received time series data from the cluster head, the current sensor will broadcast the data to the next sensor including at most k−j nearest sensors located in the radio range where j is the number of sensorswhichalreadyreceivedthedata. 3.6 ExperimentalResults For the experimental simulation, we assume a 100x100 square meter sensor field with 2,500 sensors randomly and uniformly placed. A static acoustic source location is 19 assumed to have a uniform distribution within the sensor field and generates a 3500- 4500 Hz signal. The data observation time for time delay and average energy esti- mate is 1 second where sampling rate is 9000 Hz. σ 2 s and σ 2 w are 100 and 1, thus SNR 0 = 100. The numbers of participating sensors considered are 10,15,20 and 25. Theradiorangeis5meters. TheestimationerrorisevaluatedbytheMeanSquareError (MSE),(ˆ x s −x s ) 2 +(ˆ y s −y x ) 2 . 100MonteCarlotrialswereconductedtoaveragethe effectofsourcelocationandsensortopology. Foreachtopology,100runningsareused to average the noise. We assume that the average energy of the received signal at each sensor has a normal distribution with the mean and variance defined in (3.3) and (3.4). The sensor which receives the highest energy is selected to be the initial sensor for the distributedmethodandtheclusterheadforthecentralizedmethod. Figure3.1illustratestheMSEforbothcentralizedanddistributedlocalizationwithdif- ferent numbers of participating sensors. It is obvious that the error caused by the dis- tributedmethodissmallerthanthecentralizedmethod. Thereasonisthatthedistributed algorithmusestheimmediateestimateasthepriorinformationtoselectthesequenceof theparticipatingsensorsthatincludethenearestsensorstotheestimatedsourcelocation. Since the closer sensors to source provide the more accurate time delay estimation as describedintheprevioussection,thelocalizationaccuracyisimprovedwhencompared withthecentralizedmethodwhichpreselectstheparticipatingsensorsbeforeachieving the result. The difference in the error caused by the two algorithms becomes smaller when the number of participating sensors increases since a large number of sensors makes the coverage of region activated by both algorithms more similar. We also study the performance of both schemes whenσ 2 s is varied which corresponds to the variation of the energy of the signal (SNR) created by the source. Values of σ 2 s = 10,40,70, and 100 are used and the number of participating sensors is 15. The results in Figure 3.2illustratesthatthecentralizedschemesuffersmorethandistributedschemewhenσ 2 s 20 decreases. This is also affected by the different participating sensors selected by both schemes. When σ 2 s is small, SNR 0 is small and it makes α in (3.5) become large. σ 2 i1 , thus, isproportionaltoD 2 i withahigherrate. Thedistancebetweensensorsandsource becomes more influential on the localization performance. The sensors selected by the centralizedschemewhicharelikelytobefartherfromthesourcethanthoseselectedby distributedscheme,therefore,producelargererror. We use the first order radio model described in [HCB00] for the communication over- head evaluation. To exchange k-bit data between a distance d, the radio expends E Tx =E elec ∗k+ amp ∗k∗d 2 andE Rx =E elec ∗k. E Tx andE Rx representtheenergy dissipated by the transmitting and receiving data, respectively. E elec = 50nJ/bit is the energyusedtorunthetransmitterorreceivercircuitryand amp = 100pJ/bit/m 2 isforthe transmissionamplifier. Weassumethateachdatasampleandthedatarepresentingeach element in ˆ θ andΣ requires 8 bits to be encoded. Thus, for time series data exchange, k =(SamplingRate)x(ObservationTime)x(bits/sample)= 9000∗1∗8 = 72000bits. Forthecommunicationrequiredinsequentialleastsquare,k =(Summationofnumber of elements in ˆ θ andΣ) x (bits/sample)= (3+9)∗8 = 96 bits. AsE Tx andE Rx ∝ k, wecannotethattheenergyspenttoexchangethetimeseriesdataismuchmorethanthe amount used for sequential estimate update. Figure 3.3 illustrates the energy dissipated by both schemes. It is can be noticed that the distributed method enables significant amount of energy savings compared to the centralized method, particularly when the number of sensors is large. This can be simply explained by considering, firstly, the communication between reference sensor and L participating sensors within the radio range. The centralized scheme requires L transmissions in order to obtain the time delay estimates with respect to all participating sensors while only one transmission is enough for distributed algorithm. The extra cost for distributed algorithm is what is 21 usedtosequentiallyaggregateTDOAsextractedfromtimedelayestimatesateachsen- sor. Suchcommunicationcostisrelativelysmallaspointedoutabove. Secondly,forthe sensorsoutsidetheradiorangeofthereferencesensor,inthecentralizedscheme,multi- hop communication is required and the data transmission from such sensors still has to finally reach the reference sensor. On the other hand, in the distributed scheme, such sensorscanobtainthedatafromthosewhoalreadyreceivedthedatalocatedwithinthe first hop from the reference sensor. That requires shorter total communication distance and lower energy than what is needed in the centralized scheme. Hence, the advantage ofdistributedmethodbecomesmoreconspicuouswhenmultiplehopsareneededorthe number of participating sensors is increases as illustrated Figure 3.3. We also study the convergence issue for the distributed method by considering the localization error ateachiterationcomparingitwiththedistancedifferencebetweentheconsecutiveesti- mates. Figure3.4showsthatbothamountsarehighlycorrelated. Theadvantageofthis scenario is that we can approximately evaluate the current error from the sequence of estimates. Thisisimportantifwewanttosaveunnecessarycostwhentheaccuracyhas alreadyreachedacceptableorrequiredlevels. 3.7 Conclusions We proposed a distributed algorithm, based on range difference localization method, which allows time delay estimation to be carried out at each participating sensors. TDOAs computed from time delay estimates are fused using a sequential least squares scheme which enables the appropriate sensor selection based on the current estimate. The results illustrate that the distributed localization produces smaller error and con- sumes less energy than centralized method. The advantage of distributed processing becomes more conspicuous for error considerations when the number of participating 22 10 15 20 25 30 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 x 10 −8 Number of participating sensors MSE Distributed Centralized Figure3.1: MSEvs. numberofsensors: Distributedmethodproducessmallererrorthan centralizedmethod. sensors is small and obtain more energy saving with the large number of participating sensor. Theproposedmethodisalsomorerobusttodecreasingsourcesignalenergyand theinstantaneouserrorfromthesequenceofestimatescanbeapproximatedandusedto reconcilethecostandthesystemperformance. 23 10 20 30 40 50 60 70 80 90 100 0 0.2 0.4 0.6 0.8 1 1.2 1.4 x 10 −7 SNR MSE Centralized Distributed Figure 3.2: The accuracy of the distributed method is less affected by a low energy sourcesignalthanthecentralizedmethod. 24 10 15 20 25 30 0.05 0.1 0.15 0.2 0.25 0.3 Number of participating sensors Energy Consumption (Joule) Distributed Centralized Figure3.3: Energyconsumedbycentralizedmethodislargerthanthatconsumedinthe distributedmethod. 25 0 5 10 15 20 25 0 0.5 1 1.5 2 2.5 x 10 −3 Numbers of included sensors Distance Distance to the actual target location Difference distance between consecutive estimate Figure3.4: Distanceerroranddistancebetweenconsecutiveestimatesarehighlycorre- lated. 26 Chapter4 OnEnergyBasedAcousticSource Localization Considered as a suitable method for low-power localization system, a study of energy based localization is worthwhile. In this chapter, energy-based localization methods for source localization in sensor networks are examined with focus on least squares solutions. Asuiteofmethodsaredevelopedandcompared. 4.1 Introduction A common application of a sensor network framework is source localization. Source localization is inherent to many monitoring applications such as those for wildlife or surveillance. We shall assume acoustic sources herein. Localization methods based on direction of arrival or time delay estimation have been previously developed [CYE + 03, CHY02, SA87]. These methods exploited the temporal information con- veyed in multiple collected samples, requiring time series information from multiple sensors. Theacquisitionofthesampledsignalsentailsthetransmissionoftherawtime series(hencewedenotesuchschemessignalbased)whichcanrequiresignificantwire- less resources. A contrasting method is to use energy information, exploiting the fact that acoustic signal intensity attenuates with distance from the source. Least squares solutions for energy based methods can be found in [LH04, LH03] and a maximum- likelihoodestimatorwaspresentedin[SH05]. In[BH06],afastconverginglocalization 27 solution was proposed using projection-onto-convex-sets although the performance is inferiortothemaximum-likelihoodestimatorin[SH05]. Energyestimatesofthesource areobtainedateachsensorviaaveragingofthedatasamples;thesesingleestimatesare fused either in a centralized (transmitted to a fusion center) or decentralized fashion to formthefinallocalizationestimate. Forsomescenarios,signalbasedmethodsmayoffer improved performance versus energy based methods since the information conveyed in allsamplesaredirectlyexploited(withoutaveraging),butattheexpenseoflargertrans- mission resources e.g. wireless bandwidth. We also note that for white sources, the energy observations are a set of sufficient statistics; however for colored sources, they arenot. ThescenarioconsideredhereinisdepictedinFigure4.1,whereallsensorsmeasure the acoustic signal created by the acoustic source. The average energy is computed fromthereceivedsignalbeforebeingtransmitted(viaawirelesschannelrepresentedby dashedarrowpaths)tocombinewiththeenergyreadingsfromothersensorsatafusion center where localization is conducted. The combining procedure can be carried out in acentralizedordecentralizedfashion. Energy based localization capitalizes on ideas similar to Received Signal Strength (RSS)basedmethods,thatis,thesignalenergydecayswithdistance. However,theprior workonRSSbasedmethodshassomesignificantdifferenceswiththeworkconsidered herein: the nodes themselves are to be localized [PAK + 05] versus a source; the sensor nodes cooperate with each other, whereas a target source typically does not, thus the transmittedenergyisoftenassumedknown;andthetransmittedsignalsfromthesensors can be designed to improve localization, whereas a source signal can have arbitrary characteristics. Maximum-likelihood estimators based on RSS measurements can be foundin[DA04,Wei03]andaleastsquaressolutionwaspresentedin[CSMC03]. 28 In the current work, we do not assume access to range information a priori. Unlike the signal models in [SH05, BH06], we consider an arbitrary acoustic source charac- terized by an unknown correlation function and thus derive more generalized statistical properties of energy observations. The source generates a signal experiencing attenua- tionmodeledasin[SH05]. Incontrasttothemaximum-likelihoodestimatorsof[SH05], whichrequireiterativesolutionsandmaydemandhighcomputationalcomplexity(later discussed in Section 4.7), we focus on least squares methods which offer a good trade- off between performance and complexity. The least squares approaches considered herein are limited to the case of single source, but their ease of implementation and feasibility for real-time scenarios motivates the investigation. The least squares solu- tions for a signal model similar to that in [SH05] were previously reported in [LH03]. However, the solutions were based on the assumption that the errors that perturb the leastsquaresequationsarei.i.d,zero-meanGaussianrandomvariables;hereinweshow thisassumptiontobeinaccurateforourmodel. The main contribution of the work has two parts. First, we compare, contrast and improve some existing methods. We prove that the least squares solution recently pro- posed in [LH03] based on energy ratio approach, referred to as the Quadratic Elimina- tion algorithm (QE), yields the same location estimate as the previously proposed One Step (OS) least squares solution [HBE00]. We show that given our signal model, the errors that perturb the equations employed for the least squares methods are not white andthusweightedschemesshouldbeemployed. Wenotethatthecolorationofthenoise occurs even when a white signal source is assumed. The need for a weighted schemes motivates the development of the weighted one step least squares (WOS) method. Sec- ond,weintroduceanewapproachtoformulatetheleastsquaresproblemwhichdoesnot requiretheenergyratiocomputationandproposedweighteddirectleastsquaresmethod (WD) that yields the same the location estimate as WOS; however, WD offers lower 29 computational complexity. WOS and WD can employ a correction technique [CH94] 1 which enables the incorporation of parameter dependencies leading to further perfor- mancegains,whiletheQEcannotmakeuseofthiscorrectionmethod. Simulationresultsdemonstratethat,overall,WDandWOSsignificantlyoutperform OS and QE. WD performance is further improved when combined with a correction technique(WDC).Asthenumberofsamplesemployedincreases,thesimulationsshow that WDC approaches the Cram´ er-Rao Bound (CRB) and can outperform maximum- likelihood estimators while requiring significantly less complexity. Furthermore, we prove that WDC attains the CRB for the case of a white source. The robustness of all schemes is investigated through the variation of system parameters including the consideration of errors in prior information. We note that the trends observed in the simulated acoustic source also hold true when the various methods are compared when usingexperimentalsourcedatabasedonrealbirdsongs. This chapter is organized as follows. In Section 4.2, the signal model is provided. TheexistingleastsquaressolutionsforenergyratiobasedmethodsarereviewedinSec- tion 4.3. The Weighted One Step (WOS) least squares solution is presented in Section 4.4. In Section 4.5, the weighted direct (WD) least squares approach is developed and compared with WOS. The correction technique that can improve WOS and WD solu- tionsisreviewedinSection4.6. InSection4.7,simulationresultsareprovidedwiththe aimofcomparingtheperformanceofdifferentmethodssubjecttosignificantparameter variations. Conclusions are given in Section 4.8. Appendix A shows that QE and OS yield the same location estimate. WOS and WD are shown to yield the same location estimateinAppendixB. AppendixCderivestheCram´ er-RaoBound(CRB)forenergy 1 Althoughtheideaofcorrectiontechniquewaspreviouslypresentedin[CH94],itwasappliedtotime delay based localization where the problem formulation is different from the energy based localization weconsideredherein. 30 based localization, and Appendix D proves that WDC achieves the CRB when a large numberofsamplesandawhitesourceareassumed. 4.2 SignalModel A static acoustic source generating a Wide Sense Stationary (WSS) Gaussian random process,s(t)∼ N(0,σ 2 s ), withthecorrelationfunctionR s (δ)isassumed. Theintensity of the source attenuates at a rate that is inversely proportional to the distance from the source[Kin82]. GivenN sensors,thereceivedsignalatthei th sensorisgivenby x i (t) = s(t−τ i ) kr s −r i k α/2 +w i (t), 1≤i≤N, (4.1) where w i (t) is white Gaussian measurement noise, w i (t) ∼ N(0,σ 2 w i ). The vectorsr i andr s denote the coordinates of thei th sensor and the source, respectively. In practice, σ 2 w i can be estimated based on measurements when the source is absent. For concise notation, σ 2 w i is assumed to be identical for all sensors, σ 2 w i = σ 2 w for 1 ≤ i ≤ N, and is assumed to be known. Note that for the case of different σ 2 w i , the signal model is slightly changed and the considered algorithms require only simple modification. The time required for the acoustic signal to propagate from the source to the i th sensor is denoted by τ i . In [LH03], it was shown that the effective decay factor, α, is approxi- mately 2 and this value is assumed for this chapter. In reality, however, α can deviate fromtheassumedvalueduetotheenvironment. Thesensitivityoftheconsideredalgo- rithmstothisdeviationisinvestigatedviasimulationinSection4.7. Notethatthemodel in Eqn. (4.1) is limited to cases where reflections from the ground and reverberation effects are minimal e.g. the large open space with porous ground such as grassland. 31 The signal energy at the i th sensor can be calculated by averaging over a number of observations(L)sampledatfrequencyf s . y i = 1 L L−1 X n=0 x 2 i (t s + n fs ) (4.2) where t s is the starting time. The sensing process (collecting samples used for energy computation) starts when the presence of source is detected. Source detection schemes canbefoundin[LWHS02a]. Weassumethats(t)andw i (t)areindependent. Expanding Eqn. (4.2)yields y i = 1 Lkr s −r i k 2 L−1 X n=0 s 2 (t s + n f s −τ i )+ 1 Lkr s −r i k L−1 X n=0 2s(t s + n f s −τ i )w i (t s + n f s )+ + 1 L L−1 X n=0 w 2 i (t s + n f s ). (4.3) In[SH05,BH06,HL02],thesecondterminEqn. (4.3)isassumedtobezeroduetothe independencebetweens(t)andw i (t). However,if 1 Lkrs−r i k P L−1 n=0 2s(t s + n fs −τ i )w i (t s + n fs )' 0,theothertermsshouldalsobedeterministici.e. 1 L P L−1 n=0 w 2 i (t s + n fs )'σ 2 w and y i should not be modeled as a random variable. Herein we consider all terms in Eqn. (4.3). Thus,thefirstandsecondmomentsofy i aregivenby E{y i },μ i = σ 2 s kr s −r i k 2 +σ 2 w var{y i },σ 2 i = 1 L 2 2 P L n=1 P L m=1 R 2 s ((n−m)/f s ) kr s −r i k 4 + 4Lσ 2 s σ 2 w kr s −r i k 2 +2Lσ 4 w ! cov{y i ,y j },σ ij = 2 P L n=1 P L m=1 R 2 s ((n−m)/f s +τ j −τ i ) L 2 kr s −r i k 2 kr s −r j k 2 ' 2 P L n=1 P L m=1 R 2 s ((n−m)/f s ) L 2 kr s −r i k 2 kr s −r j k 2 (4.4) 32 We make the assumption that the time delays,|τ j −τ i |, across sensors are small com- pared to the observation time so that the approximation in the last step of Eqn. (4.4) is obtained. WealsoassumethatR s (δ)→ 0asδ→∞sothattheCentralLimitTheorem can be applied when L → ∞ [Ibr75]. Thus y i is approximately normal distributed. 2 Note that, unlike the derivation in [SH05, BH06, HL02], the derived variances and covariances not only are function of the noise variance, σ 2 w , but also the correlation function for the source signal and the distance between sensors and the source. There- fore, the variance of the energy observations at each sensor can be varied significantly depending on the distances to the source. The energy based approaches localize the sourcebasedontheobservations,y i for1≤i≤N. 4.3 LeastSquaresSolutionsforEnergyRatioApproach This section summarizes the least squares solution based on the energy ratio approach recently presented in [LH03, HL02] and compares it with the previously proposed method[HBE00]. Theenergyratiobetweenthei th andj th sensorsisdefinedby ˆ K ij = y i −σ 2 w y j −σ 2 w −1/2 . (4.5) In [LH03, HL02], the least squares equation is formulated by initially assuming that y i =μ i ,thuswehavethenoiselessenergyratioasfollows K ij = μ i −σ 2 w μ j −σ 2 w −1/2 = kr s −r i k kr s −r j k . (4.6) 2 Numerical results show that when the number of samples is larger than 2500 and the signal has significantspectralcontentuptof s /4Hz,themodifiedLillieforsnormalitytest(usingMatlab R function) [Sal06]doesnotrejectthenullhypothesisforthesignificancelevel=0.01,i.e. ourapproximationisgood. 33 Note that given N sensors, M ≤ N(N −1)/2 pairs of energy ratios can be computed i.e. when all possible pairs are used; we define the index and index set as follows, the index(i,j)∈IwherethemembersofIare2-subsetsof {1,...,N}. AccordingtoEqn. (4.6), the source location, r s , must reside on the hypersphere described by the set of equationsgivenby kr s −c ij k 2 =ρ 2 ij , ∀(i,j)∈I (4.7) where c ij = r i −K 2 ij r j 1−K 2 ij , ρ ij = K ij kr i −r j k 1−K 2 ij . (4.8) An unconstrained least squares solution is formulated by considering different pairs of hyperspheres;forexample,apairformedbyK ij andK kl : kr s −c ij k 2 = ρ 2 ij , kr s −c kl k 2 = ρ 2 kl . (4.9) Subtractingeachsideofthesetwoequationstocancelthetermkr s k 2 yields 2(c T ij −c T kl ) | {z } a T n r s = (c 2 ij −ρ 2 ij )−(c 2 kl −ρ 2 kl ) | {z } bn (4.10) where1≤n≤P,andP =M(M−1)/2isthemaximumnumberofpairsofequations (fromatotalofM equationsformedbyM energyratios). Inthepresenceofobservation noise, the source location estimate can be obtained by minimizing the cost function as follows ˆ r s = argmin rs ( J(r s ) = P X n=1 ka T n r s −b n k 2 ) = (G T G) −1 G T b (4.11) 34 where whereG = [a 1 ,a 2 ,...,a P ] T andb = [b 1 ,b 2 ,...,b P ] T . If N ≥ 3, P ≥ 2,G is full column rank, and a uniqueˆ r s can be obtained. However, when noise is present,G andbarecomputedfrom ˆ K ij insteadofthenoiselessK ij whichisnotavailable. Thus, there is a possibility of ill-conditioned cases i.e. G is not full column rank. Such cases are very unlikely particularly whenN > 3. 3 We assumeN ≥ 3 and the cases are not ill-conditioned through out the chapter. The solution in Eqn. (4.11) was denoted the Quadratic Elimination (QE) in [LH04] and we label the location estimate obtained via QEbyˆ r s,QE . 4.3.1 QEvsOneStepLeastSquaresSolution TheQEsolutionisnowcomparedwithanotherleastsquaressolution[HBE00],denoted the One Step (OS) solution, which was proposed for range difference based localiza- tion. The solution in [HBE00] can be adapted to the energy ratio approach as follows. Expandingahypersphereequation,kr s −c ij k 2 =ρ 2 ij gives kr s k 2 −2c T ij r s +kc ij k 2 = ρ 2 ij 2c T ij −1 | {z } ´ a T m   r s kr s k 2   = kc ij k 2 −ρ 2 ij | {z } ´ bm (4.12) where 1 ≤ m ≤ M. A set of equations can be formed and written in matrix form as follows ´ Gθ OS = ´ b (4.13) 3 Theill-conditionedcasesneverhappenedinoursimulationswhere N >6 35 where ´ G = [´ a 1 ,´ a 2 ,...,´ a M ] T , θ OS = [r s ,kr s k 2 ] T and ´ b = [ ´ b 1 , ´ b 2 ,..., ´ b M ] T . The parameterestimate, ˆ θ OS ,isgivenby ˆ θ OS = ( ´ G T ´ G) −1 ´ G T ´ b (4.14) and the location estimate using OS is denoted by ˆ r s,OS . Note that under non-ill- conditionedcases, ´ Gisofrank3,andthusisfullcolumnrank. Therelationshipbetween QE and OS can be seen through the conversion of equations used by both methods i.e. each equation used by QE can be constructed from a pair of equations used by OS. Note that the number of equations used for OS is M = N(N − 1)/2 while it is P = M(M −1)/2 for QE whereN is the number of sensors. Appendix A shows that QE and OS yield the same location estimate. 4 However, since QE does not estimate kr s k 2 , the correction technique [CH94] (explained in Section 4.6) that incorporates the dependencebetweenr s andkr s k 2 inordertoimprovethelocalizationperformancecan not be applied and thus QE can be less accurate than a corrected OS. Hence, we will focusonOSandshowthatitsperformancecanbeimprovedfurther. 4.4 WeightedOneStepLeastSquaresSolution In this section, we derive an optimal weighting for a weighted least squares solution to improve the originally proposed OS method [HBE00] since we found that, given our signal model in Eqn. (4.1), the equation noise (later called the LS equation error) perturbingEqn. (4.13)isnotwhite. SquaringbothsidesofEqn. (4.5)gives ˆ K 2 ij = y j −σ 2 w y i −σ 2 w = μ j −σ 2 w μ i −σ 2 w +e ij = kr s −r i k 2 kr s −r j k 2 +e ij (4.15) 4 Note that OS does estimatekr s k 2 as well as r s while QE estimates only r s thus we show the equiv- alenceofthelocationestimate,ˆ r s,OS =ˆ r s,QE withoutconsideringtheestimateofkr s k 2 . 36 wheree ij is the error due to the deviation ofy i andy j fromμ i andμ j . Eqn. (4.15) can berearrangedasfollows 2r T i −2 ˆ K 2 ij r T j 1− ˆ K 2 ij | {z } 2c T ij r s −kr s k 2 = kr i k 2 − ˆ K 2 ij kr j k 2 1− ˆ K 2 ij | {z } kc ij k 2 −ρ 2 ij + e ij kr s −r j k 2 1− ˆ K 2 ij | {z } ij . (4.16) Eqn. (4.16) is the same as Eqn. (4.12), but the error that perturbs the equation ( ij ) whenformedbytheenergyratioscomputedfromobservations,isexplicitlyshown. This error term is referred to as the LS equation error. We make the following observations on ij . Firstly, the variance of ij is dependent on the distance between sensors and the source,kr s −r j k 2 . Secondly, energy ratios computed from common sensors, e.g. ˆ K ij and ˆ K kj , are correlated and that causes the ij and kj to be correlated. Thus we consider a weighted least squares solution [Kay93]. The weighting applied to improve OS should be C −1 where C is the covariance matrix for = [ (1) , (2) ,..., (M) ] T [Kay93] and the subscript, (m),1≤ m≤ M, sequentially represents that appears in the equation formed by the different energy ratios, i.e. (1) = ij if Eqn. (4.16) is the first equation in the set of the equations. C is determined as follows. Let (u) = u 1 u 2 and (v) = v 1 v 2 betheLSequationerrorsthatperturbtheequationformedbytheenergy ratio ˆ K 2 u 1 u 2 = yu 2 −σ 2 w yu 1 −σ 2 w and ˆ K 2 v 1 v 2 = yv 2 −σ 2 w yv 1 −σ 2 w , respectively. For convenience,kr s −r i k is denotedasd i . Insertinge u 1 u 2 = ˆ K 2 u 1 u 2 − d 2 u 1 d 2 u 2 ande v 1 v 2 = ˆ K 2 v 1 v 2 − d 2 v 1 d 2 v 2 (seeEqn. (4.15)) intotheexpressionsfor u 1 u 2 and v 1 v 2 (seeEqn.(4.16))gives u 1 u 2 = d 2 u 2 ˆ K 2 u 1 u 2 −d 2 u 1 1− ˆ K 2 u 1 u 2 = d 2 u 1 y u 1 −d 2 u 2 y u 2 +(d 2 u 2 −d 2 u 1 )σ 2 w y u 2 −y u 1 (4.17) v 1 v 2 = d 2 v 2 ˆ K 2 v 1 v 2 −d 2 v 1 1− ˆ K 2 v 1 v 2 = d 2 v 1 y v 1 −d 2 v 2 y v 2 +(d 2 v 2 −d 2 v 1 )σ 2 w y v 2 −y v 1 (4.18) 37 Expanding u 1 u 2 and v 1 v 2 intoaTaylorseriesabout{μ u 1 ,μ u 2 }and{μ v 1 ,μ v 2 },andusing thefirsttermtoapproximatethemeanyield E{ u 1 u 2 }'E{ v 1 v 2 }' 0. (4.19) Byneglectingmomentsofordershigherthan2,thecovarianceof u 1 u 2 and v 1 v 2 canbe approximatedasfollows [C ] uv ' cov{y u 1 ,y v 2 } ∂ u 1 u 2 ∂y u 1 ∂ v 1 v 2 ∂y v 2 +cov{y u 1 ,y v 1 } ∂ u 1 u 2 ∂y u 1 ∂ v 1 v 2 ∂y v 1 +cov{y u 2 ,y v 1 } ∂ u 1 u 2 ∂y u 2 ∂ v 1 v 2 ∂y v 1 +cov{y u 2 ,y v 2 } ∂ u 1 u 2 ∂y u 2 ∂ v 1 v 2 ∂y v 2 (4.20) wherethederivativesareevaluatedat{μ u 1 ,μ u 2 ,μ v 1 ,μ v 2 }. Eventhough[C ] uv arefunc- tionsof {μ u 1 ,μ u 2 ,μ v 1 ,μ v 2 } which are dependent on r s and thus are not available in practice, theyareapproximatedby{y u 1 ,y u 2 ,y v 1 ,y v 2 }toobtainanestimateofC , ˆ C . Thecom- mon scaling of the elements of the weighting can be neglected without affecting the locationestimate. Thus,theweightingbecomes ˆ C −1 where ˆ C isdefinedasfollows [ ˆ C ] uv =                              2yu 1 −σ 2 w (yu 1 −yu 2 ) 2 (yu 1 −σ 2 w ) 2 + 2yu 2 −σ 2 w (yu 1 −yu 2 ) 2 (yu 2 −σ 2 w ) 2 ,u 1 =v 1 ,u 2 =v 2 2yu 1 −σ 2 w (yu 1 −yu 2 )(yu 1 −yv 2 )(yu 1 −σ 2 w ) 2 ,u 1 =v 1 ,u 2 6=v 2 2yu 2 −σ 2 w (yu 1 −yu 2 )(yv 1 −yu 2 )(yu 2 −σ 2 w ) 2 ,u 1 6=v 1 ,u 2 =v 2 2yu 1 −σ 2 w (yu 1 −yu 2 )(yu 1 −yv 1 )(yu 1 −σ 2 w ) 2 ,u 1 =v 2 ,u 2 6=v 1 2yu 2 −σ 2 w (yu 1 −yu 2 )(yv 2 −yu 2 )(yu 2 −σ 2 w ) 2 ,u 1 6=v 2 ,u 2 =v 1 0 otherwise (4.21) 38 TheWeightedOneStep(WOS)solution,thus,isgivenby ˆ θ WOS = ( ´ G T ˆ C −1 ´ G) −1 ´ G T ˆ C −1 ´ b. (4.22) It should be pointed out here that ˆ C is shown to be singular (see Appendix B) and the pseudoinverseof ˆ C , ˆ C † ,isusedinsteadof ˆ C −1 . Notethatthesingularityoccursdueto the approximation in Eqn. (4.20), thus it does not truly represent the characteristics of C . TheWOSisbasedonanenergyratioapproachandwillbeshowntobeequivalentto asimple,directlocalizationmethodthatdoesnotrequiretheenergyratiocomputations inthenextsection. 4.5 WeightedDirectLeastSquaresSolution The idea of the direct approach is to localize the source directly based on the measured energy (y i in Eqn. (4.2)) without computing the energy ratios (see Eqn. (4.5)). The observationy i canbewrittenas y i = σ 2 s kr s −r i k 2 +σ 2 w +ε i (4.23) whereε i is the LS equation error and is approximately normal distributed as explained in Section 4.2 where its first and second moments are shown in Eqn. (4.4). Eqn. (4.23) canberearrangedasfollows, 2(σ 2 w −y i )r T i r s +(y i −σ 2 w )kr s k 2 −σ 2 s = (σ 2 w −y i )kr i k 2 +ε i kr s −r i k 2 . ConsideringtheN sensorstogetherandwritingtheseequationsinmatrixformyields Aθ WD =h+ε (4.24) 39 where A =      −2(y 1 −σ 2 w )r T 1 (y 1 −σ 2 w ) −1 . . . . . . . . . −2(y N −σ 2 w )r T N (y N −σ 2 w ) −1      , θ WD = [r T s ,kr s k 2 ,σ 2 s ] T h = [−(y 1 −σ 2 w )kr 1 k 2 ,...,−(y N −σ 2 w )kr N k 2 ] T , ε = [ε 1 kr s −r 1 k 2 ,...,ε N kr s −r N k 2 ] T . Note that σ 2 s (the variance of the source signal (see Section 4.2)) is also an unknown parameter which, for the energy ratio approach, is canceled during the energy ratio computation. The estimate can be obtained using the weighted least squares solution, (the Weighted Direct (WD) solution), where the covariance matrix forε needs to be determined. According to Eqn. (4.4) and Eqn. (4.23), the covariance matrix forε is givenby C ε = 2 L 2 Q+β1 N 1 T N (4.25) where Q = 2Lσ 2 w diag kr s −r 1 k 4 2σ 2 s kr s −r 1 k 2 +σ 2 w ,... ...,kr s −r N k 4 2σ 2 s kr s −r N k 2 +σ 2 w β = L X n=1 L X m=1 R 2 s ((n−m)/f s ), 1 N = [1,...,1] | {z } N T . (4.26) Using the Woodbury matrix identity [GvL96] to expandC −1 ε , neglecting the scaling, and approximating Q using μ i ' y i , ˆ Q = diag n 2y 1 −σ 2 w (y 1 −σ 2 w ) 2 ,..., 2y N −σ 2 w (y N −σ 2 w ) 2 o , yields the WDsolution: ˆ θ WD = (A T ˆ Q −1 A) −1 A T ˆ Q −1 h. (4.27) 40 Similar to the comparison between QE and OS presented in Appendix A, the rela- tionship between WOS and WD can be analyzed based on the conversion between the set of equations used by both methods. Appendix B shows that WD and WOS, in fact, yieldthesamelocationestimate. Regardingconsiderationsoncomplexity,O(N 5 )oper- ations are required to construct the SVD in order to obtain the weighting for WOS [GvL96]. The number of multiplications and additions to multiply matrix for WOS is O(N 4 ). Hence, the computational complexity demanded for WOS isO(N 5 ). For WD, theweightingisdiagonalwithdimensionN×N,thus,toobtaintheweightingrequires only O(N) operations. The multiplications and additions used for WD are O(N 2 ). In total, the computational complexity required for WD is O(N 2 ). Therefore, although yielding the same location estimate, WD is more computationally efficient than WOS; andWDshouldbeappliedinsteadofWOSinallscenarios. WD can be implemented in a distributed fashion by developing a sequential least squares solution similar to what was proposed in [MN05] wherein such a solution was used in the context of time delay based localization. The sequential solution allows the information of sensor locations and energy observations to be kept at each sensor and usedtoupdateestimatedsourcelocationwhichispassedthroughtheselectedroute. 4.6 CorrectionTechnique Formulationof ˆ θ WOS and ˆ θ WD ismadeundertheassumptionthatkr s k 2 isascalarthat is statistically independent ofr s ; clearly this is not true. The relationship can be incor- porated in the formulation and used to improve the estimate derived by the approach developed in [CH94]. The steps for applying the correction technique to WOS and WD are similar. Since both methods yield the same location estimate where WD is more computationally efficient, the required steps for WD will be explained and the 41 correctedsolutioniscalledWeightedDirectsolutionwithCorrection(WDC).Let ˆ θ WD be expressed as ˆ θ WD = θ WD + Δθ. When y i is of low variance (having sufficiently largenumberofobservationsamples,L,seeEqn. (4.4)), ˆ Q(seeEqn. (4.27))iscloseto Q in Eqn. (4.26) without the scaling 2Lσ 2 w σ 4 s ; the bias and covariance matrix of ˆ θ WD canbeapproximatedby[CH94] Δθ' ( ¯ A T C −1 ε ¯ A) −1 ¯ A T C −1 ε ε, C Δθ ' ( ¯ A T C −1 ε ¯ A) −1 (4.28) where ¯ A represents E{A} i.e. y i in A is replaced by μ i . Since ε has zero mean, E{Δθ} =0 and ˆ θ WD can be considered as a random vector with its mean centered at the true value and the covariance matrix given by Eqn. (4.28). Thus, the elements of ˆ θ WD canbeexpressedas ˆ θ WD,1 = x s +e 1 , ˆ θ WD,2 = y s +e 2 , ˆ θ WD,3 = kr s k 2 +e 3 , ˆ θ WD,4 = σ 2 s +e 4 (4.29) wheree 1 ,e 2 ,e 3 ,ande 4 areestimationerrorsof ˆ θ WD andr s = (x s ,y s )isthetruesource location. Squaring ˆ θ WD,1 and ˆ θ WD,2 gives ˆ θ 2 WD,1 = x 2 s +2x s e 1 +e 2 1 ' x 2 s +2x s e 1 (4.30) ˆ θ 2 WD,2 = y 2 s +2y s e 1 +e 2 2 ' y 2 s +2y s e 2 (4.31) 42 where the approximation is valid when e 1 and e 2 are small. A set of equations ´ Aθ 2 = ´ h+´ isobtainedwhere ´ A =          1 0 0 0 1 0 1 1 0 0 0 1          , θ 2 = [x 2 s y 2 s σ 2 s ] T ´ h = [ ˆ θ 2 WD,1 ˆ θ 2 WD,2 ˆ θ WD,3 ˆ θ WD,4 ] T ´ = [2x s e 1 2y s e 2 e 3 e 4 ] T . (4.32) Since Δθ = [e 1 e 2 e 3 e 4 ] T (see Eqn. (4.29)) and ´ = BΔθ where B = diag{2x s ,2y s ,1,1} (see Eqn. (4.32)), ´ ∼ N(0,Ψ) and Ψ = BC Δθ B. Therefore, theMLestimatefor ´ Aθ 2 = ´ h+´ isgivenby ˆ θ 2 , [ ˆ x 2 s ˆ y 2 s ˆ σ 2 s ] T = ( ´ A T Ψ −1 ´ A) −1 ´ A T Ψ −1 ´ h = ( ´ A T B −1 ¯ A T C −1 ε ¯ AB −1 ´ A) −1 ´ A T B −1 ¯ A T C −1 ε ¯ AB −1 ´ h = ( ´ A T B −1 ¯ A T Q+β1 N 1 T N −1 ¯ AB −1 ´ A) −1 ´ A T B −1 ¯ A T Q+β1 N 1 T N −1 ¯ AB −1 ´ h = ( ´ A T B −1 ¯ A T Q −1 ¯ AB −1 ´ A) −1 ´ A T B −1 ¯ A T Q −1 ¯ AB −1 ´ h ' ( ´ A T ˆ B −1 A T ˆ Q −1 A ˆ B −1 ´ A) −1 ´ A T ˆ B −1 A T ˆ Q −1 AB −1 ´ h (4.33) The second to last step in Eqn. (4.33) is obtained by using Woodbury matrix identity and the approximation in the last step uses the results of WD to estimate the elements inB to get ˆ B, andA and ˆ Q were defined in Section 4.5. The final location estimate incorporated with the parameter dependence is ˆ θ WDC = [± q ˆ x 2 s ± q ˆ y 2 s ] T . The propersolutionisselectedtobetheonewhichliesintheregionofinteresti.e. closestto 43 ˆ θ WD . The performance of WDC compared with other schemes including the Cram´ er- Rao Bound (CRB) will be shown via simulations in Section 4.7. However, for the case of white source and large number of samples, we show that the WDC attains the CRB in Appendix D. As pointed out earlier, the previously proposed QE method, although yielding the same location estimate as OS, can not be improved by the correction tech- nique since the unknown parameter,kr s k 2 , is eliminated. Consequently, the corrected OScanbemoreaccuratethanQEwhenthecorrectiontechniqueisappropriatelyapplied e.g. whenthenumberofsamplesislarge. 4.7 SimulationResults The performance of the One Step (OS), Weighted One Step (WOS), Weighted Direct (WD),WDwithcorrection(WDC),andMaximum-Likelihood(ML)methodscompared totheCram´ er-RaoBound(CRB)(SeeAppendixC)wasinvestigatedthroughaseriesof MonteCarlosimulations. Anumberofsensors,N,wereplacedrandomlyanduniformly in the region of interest sized 25×25 square meters. The location of a static acoustic sourcewasassumedtohaveauniformdistributionwithinthesensorfield. 100different sensor topologies with 1000 trials per topology were used to average the noise and the effect of sensor locations. The quality of the estimators were evaluated by computing theRootMeanSquareError(RMSE)definedasfollows. RMSE = v u u t Nt X i=1 Ns X j=1 kˆ r s,ij −r s k 2 N t N s (4.34) 44 wheretheindexiandj correspondtotheresultsfromthei th trialandthej th topology, N t = 100,andN s = 1000. TheCRBwascomputedasfollows. CRB = v u u t Nt X i=1 [F −1 ] 11 +[F −1 ] 22 N t (4.35) whereF −1 isderivedinAppendixC. ForOSandWOS,allpossibleN(N−1)/2energy ratioswereusedforthelocalization. TheMLestimatewasobtainedbymaximizingthe followinglikelihoodfunction max ϑ {L(ϑ)} = max ϑ ln|detC(ϑ)|+(y−μ(ϑ)) T C −1 (ϑ)(y−μ(ϑ)) (4.36) where ϑ = [x s ,y s ,σ 2 s ,ν] T , ν is defined in Eqn. (C.1), Appendix C, and C is the covariance matrix for energy measurements, y = [y 1 ,...,y N ] T , whose means are [μ 1 ,...,μ N ] T (see Section II). This non-linear optimization was iteratively solved until convergence by using the Nelder-Mead non-linear minimization [WPMD91] Matlab R function where each iteration requires O(N 3 ) operations. Note that this maximum- likelihood estimator is somewhat different from the one proposed in [SH05] due to the different assumptions in the signal model as explained in Section II (between Eqn. (3) and(4)). In[SH05],theunknownparametersare[x s ,y s ,σ 2 s ] T andthecovariancematrix is a function of only the noise variance, σ 2 w . Herein, there is an additional unknown parameter, ν, which is dependent on the autocorrelation of the source. Moreover, the covariancematrix,C(ϑ), isafunctionofnotonlyσ 2 w butalsounknownparameters,σ 2 s and ν. This likelihood function is non-convex, and thus the optimization may reach a localoptimaifinitialestimatesarefarawayfromthetruevalues. Weusedtwodifferent gridresolutions,1×1and5×5squaremeterslabeledasML-1andML-5,respectively, toperformtheexhaustivesearchfortheinitialestimates. Weobservethattheresolution 45 ofthegridsearchwillimpactthequalityoftheestimates. σ 2 s /σ 2 w ,definedasSNR 0 ,was fixedtobe20dBandthenumberofsamplesusedtocomputetheenergy(L)was5000. Note that the Signal to Noise Ratio (SNR) at each sensor attenuates from SNR 0 due to thedistance(meters)betweenthesourceandthesensorsaccordingtothesignalmodel. In other words, SNR 0 is SNR measured at a distance of 1 meter from the source. The samplingfrequencywassettobe5000Hzandthebandwidthofthesourcewas0-1250 Hz. Theperformanceofeachmethodwhendifferentnumbersofsensorsweredeployed, 6≤ N ≤ 10, is illustrated in Figure 4.2. The results show that WOS and WD perform equivalently and significantly outperform OS and QE. The improvement of WOS and WD over QE and OS becomes larger when the number of sensors is increased. This is because the variance and covariance of the energy ratios can be very large when the means of the received energy at sensor pairs are close (see Eqn. (4.21) for the approx- imated expressions) and the possibility of this occurrence is higher when the number of sensors gets larger. Thus, some energy ratios may produce very inaccurate obser- vations and significantly degrade the performance of OS and QE that equally rely on each energy ratio. The results from WD are improved when the correction technique is applied (WDC). ML-1 performs better than WDC but the difference becomes smaller when the number of sensors is increased. ML-5 becomes worse than other schemes except for QE and OS with number of sensors more than 7. This is because the perfor- manceofML-5islimitedbytheresolutionoftheexhaustive. Otherthanthenumberofsensors,theperformancewiththevariationofothersignif- icantsystemparameterswasexplored. SincetheweightingsforbothWOSandWDare computedfromtheobservations,theerrorsoftheenergyreadingsmightleadtoinaccu- rate weightings that can cause degradation instead of improvement. Also, WDC can be affected by the inaccurate estimate obtained from WD. According to the signal model, 46 SNR 0 and L are the parameters that influence the accuracy of the measured energy. Experiments were set up to investigate the robustness of each method with respect to variation in these parameters. SNR 0 was set to be varied from 5,10,...,30 dB where L = 5000 and N = 10. The results in Figure 4.3 illustrate that even though WDC per- forms best when SNR 0 is high, it becomes worse than WD and WOS when SNR 0 is lessthan10dB.Therefore,thebadenergyreadingsduetolowSNR 0 candeterioratethe correction. The performance of WD and WOS is better than QE and OS for the entire range, but is degraded with a higher rate and finally converges to that of QE and OS with low SNR 0 (5 dB). ML-1 outperforms other schemes while ML-5 gains the least improvementoverincreasingSNR 0 . The number of samples used to compute the energy (L) is important for the sys- tem design since requiring fewer samples can save sensing power. However, a small L increases the variance of the observed energy as described in Eqn. (4.4). Figure 4.4 illustratesthelocalizationperformancewhenLisvaried,SNR 0 isfixedtobe20dB,and N = 10. WOS, WD, and WDC are shown to be better than OS for a wide range of L. However, when L is smaller than 10, OS becomes superior to weighted schemes. This demonstratesthattheweightingsforWOSandWDaresignificantlyaffectedbysmallL andtheyarenotappropriateforthecaseofverysmallnumberofsamples. However,the caseofusinglessthan10samplesforenergycomputationinthisscenarioisnotpracti- calsinceitproducestoolargeanerror(RMSE>10mfor25m×25mfield). WhenLis verylarge,WDCapproachestheCRBandismoreaccuratethanML-1sinceML-1,for some trials, reaches the local maxima due to the non-convex likelihood function. Con- sidering the complexity whenL = 10000, ML-1 requires 26×26 = 676 trials (O(N 3 ) operations/trial) for the initial estimate and uses an average of 54 iterations (O(N 3 ) operations/iteration)toconvergewhileWDCusesonly2×O(N 2 )forthestepsofWD and the correction. Thus, WDC is preferable to ML estimators under this condition. 47 Table 4.1, 4.2, and 4.3 report the biases of all schemes with different parameters. The biases shown in the tables are P Nt i=1 P Ns j=1 (ˆ r s,ij −rs) T NtNs . It is shown that the biases for all estimators are fairly small particularly with large number of sensors and samples, and ML estimators are less biased than the least squares methods. We also found that if the source is located in outside the area that the sensors are located, the sensors (using all schemes)willbiasthesolutiontowardthesensorarea. Theerrorofpriorinformationcanalsoaffecttherobustnessofthelocalization. The deviationofthetruevaluesofnoisevariance(σ 2 w ),sensorlocations,anddecayfactor(α) from the assumed values is the parameter of interest for this experiment. The deviation is assumed to be uniformly distributed within the interval 2δ centered at the assumed values. δ is set to be 0,0.2,...,1 for noise variance, 0,0.2,...,1 for decay factor, and 0,1,...,5 for sensor locations. The results in Figure 4.5, 4.6, and 4.7 show that WOS and WD are still better than QE and OS with the parameter deviation where WDC is degradedtobeinferiortoWDwhenthedeviationgetslarger. MLsufferslessthanother schemes under the deviations of noise variance and sensor locations but is degraded morethanWOSandWDwiththedeviationofdecayfactor. Theaforementionedsimulationresultswereobtainedwhenanartificiallygenerated source signal is used (pre-determined PSD). Next, we investigate the performance of eachschemewhenarealacousticsource, apre-recordedbirdsong(Commonloonbird inJasperNationalPark,Canada[FP]),isused. Thecharacteristicsofthissourcedeviate from the assumptions previously made e.g. it may not be zero-mean WSS Gaussian random process. The sampling frequency for this recording is 11 KHz. The power spectral density of the bird song is shown in Figure 4.8. The number of sensors is set to be 10 with uniformly random locations over 1000 trials, number of samples is 5000, and SNR 0 is 20 dB. For each trial, the interval of collected samples is randomly selectedfromthetotal109052samplesoftheentirerecord. TheRMSerrorproducedby 48 eachmethodispresentedinTable4.4. Theresultsshowthatdespitethedeviationfrom the assumptions, WOS and WD still significantly outperform QE and OS. Although ML estimators perform better, ML-5 and ML-1 require 73 and 56 iterations on average (O(N 3 )operations/iteration),respectively. Thisisveryhighcomplexitycomparedwith 2×O(N 2 )operationsdemandedbyWDC. 4.8 Conclusions In this chapter, a set of source localization methods based on energy measurements are developed, compared and contrasted. Two previously proposed methods, the Quadratic Elimination (QE) least squares solution for the energy ratio approach and the One Step (OS) least squares solution are shown to yield the same location estimate. In the pres- ence of additive Gaussian noise, it is shown that OS can be improved by the weighted least square solution (WOS) due to the fact that the errors that perturb the least squares equations are not white. We introduce a new least squares method: the weighted direct (WD) approach; WD is shown to yield the same location estimate as WOS with much lower computational complexity. The simulation results illustrate that WD and WOS significantly outperform QE and OS in most scenarios. WD can be further improved whencombinedwithcorrectiontechnique(WDC).Asthenumberofsamplesemployed increases, WDC approaches the Cram´ er-Rao Bound and can be more accurate than maximum-likelihood estimators, but requires less computation. We also proved that WDCattainstheCRBforthecaseofwhitesource. 49 Fusion Center Figure4.1: Acousticsensors(filledcircles)receivesignalgeneratedbyasource(repre- sentedbyastar)andexchangedataforlocalization. Table4.1: Biasesofdifferentschemeswhennumberofsensorsisvaried. Numberofsensors 6 7 8 9 10 QE/OS [-0.12-0.20] [-0.14-0.24] [-0.09-0.14] [0.01-0.13] [-0.06-0.11] WOS/WD [0.00-0.11] [-0.01-0.05] [-0.01-0.01] [0.00-0.01] [-0.01-0.01] WDC [0.08-0.05] [0.04-0.01] [0.030.02] [0.040.02] [0.020.02] ML-5 [0.02-0.03] [-0.10-0.15] [0.01-0.07] [0.02-0.13] [-0.110.04] ML-1 [-0.050.03] [-0.030.00] [0.000.08] [-0.020.11] [0.000.13] 50 6 7 8 9 10 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 Number of sensors RMS error (m) QE / OS WOS / WD WDC ML−5 ML−1 CRB Figure 4.2: Comparison of different weighting schemes: weighted schemes (WOS,WD,WDC) outperform non-weighted schemes (QE,OS) and the difference becomeslargerwhennumberofsensorsisincreased. Table4.2: BiasesofdifferentschemeswhenSNR 0 isvaried. SNR 0 (dB) 5 10 15 QE/OS [0.12-0.31] [0.09-0.27] [0.04-0.17] WOS/WD [0.19-0.02] [0.17-0.12] [0.07-0.07] WDC [0.37-0.05] [0.24-0.05] [0.09-0.05] ML-5 [-0.040.15] [-0.12-0.03] [-0.080.04] ML-1 [-0.07-0.10] [0.02-0.07] [0.000.00] SNR 0 (dB) 20 25 30 QE/OS [0.03-0.12] [0.04-0.04] [0.01-0.00] WOS/WD [0.02-0.04] [0.01-0.03] [0.01-0.03] WDC [0.03-0.02] [0.01-0.02] [0.01-0.02] ML-5 [-0.060.02] [-0.040.00] [-0.040.02] ML-1 [0.000.00] [0.000.00] [0.000.00] 51 5 10 15 20 25 30 10 −1 10 0 10 1 SNR (dB) RMS error (m) QE / OS WOS / WD WDC ML−5 ML−1 CRB Figure 4.3: Effect of SNR 0 : weighted schemes (WOS,WD,WDC) are degraded with higherratethannon-weightedschemes(QE,OS)whenSNR 0 becomeslower. 10 0 10 1 10 2 10 3 10 4 10 5 10 −1 10 0 10 1 Number of Samples RMS error (m) QE / OS WOS / WD WDC ML−5 ML−1 CRB Figure4.4: Effectofdatasize: WOS,WD,andWDCareconsiderablyaffectedbysmall L. 52 Table4.3: Biasesofdifferentschemeswhennumberofsamplesisvaried. Numberofsamples 1 10 10 2 QE/OS [0.24-0.17] [0.22-0.17] [0.22-0.24] WOS/WD [3.500.51] [0.27-0.02] [0.32-0.14] WDC [15.0211.18] [1.801.13] [0.45-0.03] ML-5 [0.37-1.08] [0.42-0.10] [0.020.02] ML-1 [-0.16-0.85] [0.47-0.09] [0.03-0.01] Numberofsamples 10 3 10 4 10 5 QE/OS [0.12-0.14] [0.14-0.09] [0.06-0.01] WOS/WD [0.16-0.09] [0.05-0.07] [0.02-0.07] WDC [0.20-0.04] [0.06-0.05] [0.02-0.05] ML-5 [0.03-0.03] [0.000.01] [0.000.01] ML-1 [-0.010.00] [0.000.00] [0.000.00] 0 0.2 0.4 0.6 0.8 1 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 Noise variance deviation RMS error (m) QE / OS WOS / WD WDC ML−5 ML−1 Figure 4.5: Weighted schemes (WOS,WD,WDC) are still superior to non-weighted schemes(QE,OS)withthepresenceofnoisevariancedeviation. 53 0 0.2 0.4 0.6 0.8 1 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Decay factor deviation RMS error (m) QE / OS WOS / WD WDC ML−5 ML−1 Figure 4.6: WOS and WD are less sensitive to the decay factor deviation than other schemes. 0 1 2 3 4 5 0 1 2 3 4 5 6 7 8 Sensor location deviation (m) RMS error (m) QE / OS WOS / WD WDC ML−5 ML−1 Figure4.7: Allmethodsaresimilarlyaffectedbysensorlocationdeviation. 54 0 1000 2000 3000 4000 5000 −20 −10 0 10 20 30 40 Frequency (Hz) Power Spectrum Magnitude (dB) Figure4.8: PowerSpectralDensityofbirdsong(Commonloon). Table 4.4: RMS error for the case of bird song: weighted schemes (WOS,WD,WDC) arestillbetterthannon-weightedschemes(QE,OS). LocalizationMethods QE/OS WOS/WD WDC ML-5 ML-1 RMS(m) 5.26 1.52 1.50 1.32 0.99 55 Chapter5 DesignConsiderationsforAcoustic SourceLocalizationwithaGrid-Based SensorField Oneofthesignificantissuesinsensornetworkcontextissystemdesigni.e. howtoselect parametersandschemestoachievedesiredperformance. Inthischapter,designrulesfor sensornetworkdeploymentforacousticsourcelocalizationaredeterminedconsidering twopopularobservationmodels. 5.1 Introduction Acoustic source localization has emerged as an important utility for sensor net- works with applications as diverse as surveillance [SML + 04] to habitat monitoring [WEG03b, WCA + 05]. Much of the existing literature on acoustic source localiza- tion falls within one of two approaches. The first employs the emitted acoustic signal and exploits the time delay information embedded within the time series (time-delay based) and the second pre-processes the time series data to provide energy measure- ments (energy based). Thus, we categorize the various estimators based on their effec- tive observation models: Time delay based Observation Model (TOM) or Energy based Observation Model (EOM). In EOM, the received samples are squared and averaged, leading to a potential information loss with respect to location information (e.g. for 56 colored sources). A disadvantage of the TOM methods is that more system resources (e.g. wirelessbandwidthorpower)arerequiredtotransmitthereceivedacousticsignals from different sensors to a fusion center for localization. While EOM and TOM meth- odshavebeenextensivelystudiedoflate,therehasbeennocomprehensivecomparison of the two strategies. Our objective herein is to provide such a comparison and to offer design rules for sensor network deployment to optimize localization performance. It is generally assumed that EOM-based localization is inferior to TOM-based localiza- tion;however,weshowthatthereexistscenariosinwhichEOM,infact,offerssuperior performance. In[CHY02],maximum-likelihood(ML)basedlocalizationfornear-field,wideband signalswasexamined,offeringimprovedperformanceoveraleastsquaresbasedmethod [HBE00] and MUSIC [TYC + 99]. ML acoustic localization using energy observations was studied in [SH05]. We view ML methods as optimal and thus dub the estimators in[CHY02]and[SH05]asTMLandEML,respectivelyduetothedifferentlyassumed observationmodels. WhileCram´ erRaobounds(CRB)werederivedin[SH05,CHY02], the calculations were topology dependent (e.g. 5 sensors located in pentagon shape and 12 source locations along a linear track in [CHY02]) and thus do not offer much insightintothenetworkdesignproblem. Networkdesignisthemotivatorforourcurrent study: howdenseshouldthesensorfieldbe,howmanysensorsshouldparticipateinthe localization, and how many observations should be collected to ensure specified levels of localization accuracy and energy constraint satisfaction. All such questions must be answered in light of the facts that EML can be potentially less accurate than TML due topre-processing;whileTMLincurshighercoststhanEML. Thus,weseekdesignruleswhichcanbeappliedtodeterminedeployment. Ourmet- ric of interest will be the CRB as it is estimator independent, although we observe that, 57 for many cases, ML based methods are asymptotically efficient [Kay93] 1 . To avoid the computationallyexpensiveexerciseofevaluatingCRBsforallpossiblesourcelocations andsensordeployments,weinvestigateboundsontheCRBsforTOMandEOMunder theassumptionofagrid-basedsensorfield. ThatistheminimaandmaximaoftheCRBs oversourcelocationaredetermined. In this chapter, the performance of the location estimators using TOM and EOM is analyzed via the CRB. Assuming nearest neighbor sensor participation in localization, weapproximatetheminimumandthemaximumoftheCRBsforthesimplethreesensor caseandgeneralizetothelimitingcaseoflargenumberofsensors. Thederivedbounds, which are functions of key design parameters such as number of sensors, grid size, sampling frequency, and number of collected samples, are shown to be good approxi- mations when compared with direct numerical evaluation of the true CRBs. With the samenumberofcollectedsamples,theCRBforEOMcanbesmallerthanthatforTOM for the case of low-frequency source and small grid size (dense) field. Increasing the numberofsamplescandecreasetheCRBforbothmodels,althoughitismorepractical for EOM since additional measurements do not translate to an increased communica- tion cost. Simulation results demonstrate that the performances of TML and EML are consistent with the corresponding CRBs, and thus the derived CRB can be used for localizationperformanceprediction. Weprovidedesignrulesforparameterandscheme selection,explicitlypresentedasmathematicalexpressions,andillustrativeexamplesof theirapplication. Thechapterisorganizedasfollows. InSection5.2,thesignalmodelissummarized; TOMandEOMareintroducedinSection5.2.1and5.2.2,respectively. Theperformance 1 In[CHY02],theproposedMLestimator(TML)wasshowntoapproachtheCRBforcertainspecific topologies. 58 of the observation models is analyzed and compared in Section 5.3. Simulation results aregiveninSection5.5andconclusionsareprovidedinSection5.6. 5.2 SignalModel Then th sampleoftheacousticdatacollectedbythei th sensorisgivenby r i (n) = s(t 0 +n/f s −τ i ) d i +w i (t 0 +n/f s ) (5.1) for 1 ≤ i ≤ N {t,e} and 0 ≤ n ≤ L {t,e} − 1 where N {t,e} is a number of participat- ing sensors and L {t,e} is the number of collected samples. The subscript t or e repre- sents those associated with time delay or energy based methods, respectively. s(t) is anacousticsourcesignalthatisdelayedbyacoustictravelingtimeτ i ,attenuatedbythe distanced i between the source (located atz s = [x s ,y s ] T ) and thei th sensor (located at z i = [x i ,y i ] T , assumed known), and perturbed by zero-mean i.i.d Gaussian measure- ment noise, w i (t), with variance σ 2 w . The observation is assumed to start at time t 0 with sampling frequencyf s . Note that the model in Eqn. (5.1) was assumed in [HL02] where it was experimentally validated. However, this model is limited to cases where reflections from the ground and reverberation effects are minimal e.g. the large open space with porous ground such as grassland. The number of samples is assumed to be sufficiently large compared with the time delays (|τ i −τ j | fori6= j in samples) across sensors. Sensors are assumed to be deployed on a grid-based field where the intervals betweenadjacentsensorsinbothhorizontalandverticalaxesisg meters(willbecalled grid size in this chapter). N {t,e} nearest sensors to the source are selected to participate inthelocalizationprocess. Onesimplesensorselectionschemeistoselectsensorswith highest signal energy measurements which are inversely proportional to the squared 59 distance from the source. This decision is assumed to be perfect i.e. all participating sensorsaretrulynearesttothesource 2 . 5.2.1 TimeDelayBasedObservationModel Thisobservationmodelexploitstemporalinformationconveyedinthereceivedsamples, τ i inEqn. (5.1). Signalattenuationasafunctionofthedistancebetweenthesensorsand the source is not taken into account. Thus, the signal model becomesr i (n) = a i s(t 0 + n/f s −τ i )+w i (t 0 +n/f s )wherea i isassumedtobeknownortobeestimatedseparately from the location estimates. Since the number of collected samples is assumed to be large compared to the time delay across sensors, the DFT is insignificantly affected by edgeeffects. Therefore,thefollowingfrequency-domainapproximationissatisfied DFT Lt {s(t 0 +n/f s −τ i )}'S 0 (k)exp − j2πkτ i f s L t (5.2) for0≤ n≤ L t −1and0≤ k≤ L t −1whereDFT Lt {•}representstheL t pointDFT andS 0 (k)isthesourcespectrum. Hence, DFT Lt {r i (n)} =R i (k)' S 0 (k) d i exp − j2πkτ i f s L t +η(k), 0≤k≤L t −1 (5.3) whereη(k) is a zero mean complex Gaussian random variable with varianceL t σ 2 w . We defineR i (k)astheobservationobtainedfromtheTimedelaybasedObservationModel (TOM).Notethatτ i = d i c wherecistheacousticspeed(m/s);thusτ i conveysthecoarse information of source location which is used by the estimators based on this model. A set of equations can be formed by considering non-repeated Lt 2 frequency bins from N t sensors (k = 0 is not used since τ i disappears). The Maximum-Likelihood (ML) 2 Anotherpotentialsensorselectionschemecanbefoundin[CHS04](clusteringalgorithm);theopti- malityofthisschemeis,however,beyondthescopeofthischapter. 60 estimator was derived in [CHY02] where it was called Approximated ML estimator (AML).HereinwecallittheTimeDelayBasedMLestimator(TML)toemphasizethat itisderivedfromtheTOM. 5.2.2 EnergyBasedObservationModel This model utilizes the signal attenuation rate to acquire the information of the source location. Energy estimates are obtained by averaging squared signal amplitude over a numberofcollectedsamples,L e ,asfollows e i = 1 L e Le−1 X n=0 r 2 i (n), 1≤i≤N e (5.4) whereN e isthenumberofparticipatingsensors. TobeconsistentwiththeTOM(Section 5.2.1),weassumethatEqn. (5.2)issatisfied. ByusingParseval’stheorem,thefirstand secondmomentsofe i aregivenby E{e i },μ i = β L 2 e d 2 i +σ 2 w var{e i },σ 2 i = 4σ 2 w β +2L 2 e d 2 i σ 4 w d 2 i L 3 e (5.5) where β = P Le−1 k=0 |S 0 (k)| 2 . The number of samples, L e , is assumed to be sufficiently large so that the average energy, e i , is approximately normal due to the Central Limit Theorem[Pap91]. We,thus,have e i =μ i +ε i , 1≤i≤N e (5.6) where ε i ∼ N(0,σ 2 i ). e i is defined as the observation obtained from the Energy based Observation Model (EOM). Letμ = [μ 1 ,μ 2 ,...,μ N ] T ,ε = [ε 1 ,ε 2 ,...,ε N ] T ∼ 61 N(0,Q),Q = diag{σ 2 1 ,...,σ 2 N }, and the unknown parameter vector beθ = [z T s ,β] T . TheEnergybasedMaximum-Likelihoodestimator(EML)isgivenby ˆ θ = min θ ln|detQ|+(e−μ) T Q −1 (e−μ). (5.7) Notethatthedifferencebetweentheestimatorinthisworkandthatin[SH05]isthatour signalmodelassumptionresultsindifferentstatisticalpropertiesoftheenergymeasure- ments and thus a different noise covariance matrix (Q). Herein we derive the estimator basedontheobservationmodelwiththesameassumptionsmadeforTOMsothattheir performancecanbefairlycomparedinsequel. 5.3 PerformanceAnalysisforDesignConsiderations In this section, we develop two CRBs, CRB t and CRB e , with the objective of provid- ing design rules for the deployment of estimators in the grid-based sensor field based on such CRBs. Each CRB provides the theoretical lower bound of the variance of the errorproducedbytheunbiasedestimatorsusingeachobservationmodel. Thecompari- sonbetweentheMaximum-Likelihoodestimators, e.g. TMLandEML,andthederived CRBs will be presented in the simulation section (Section 5.5). Note that although both observation models use the same original collected samples (r i (n) in Eqn. (5.1)), CRB e and CRB t are different due to the fact that there is potential information loss in the EOM as a result of the energy calculation. The preprocessing is depicted in Fig- ure 5.2. These CRBs are dependent on the source location, which can be anywhere in the field. However, the localization system is typically designed before deploy- ment. Thus, we seek a performance metric that does not require the knowledge of the source location. We investigate minimum and maximum possible CRB t and CRB e 62 as a function of source location and use these bounds to evaluate the limits of local- ization performance. Thus, such limits can provide guidelines for the system design. The CRB for the location estimate based on TOM, also derived in [CHY02], is given by [F zs,t ] −1 = [ζ(A t −Z t )] −1 where ζ = 8f 2 s π 2 L 3 t σ 2 w c 2 P Lt/2 k=1 k 2 |S 0 (k)| 2 ,A t = P Nt i=1 u i u T i d 2 i , Z t = 1 P N t i=1 1/d 2 i P Nt i=1 u i d 2 i P Nt i=1 u i d 2 i T , andu i = (zs−z i ) d i . The CRB of distance error is defined as CRB t = F −1 zs,t 11 + F −1 zs,t 22 . CRB t can be written in polar coordinates (withthesourcelocationattheorigin,seeforexample,Figure5.1forthe3-sensorcase) asCRB t =g 2 ζ −1 φ t where φ t = P Nt i=1 1 ¯ d 2 i 2 − P Nt i=1 cosθ i ¯ d 2 i 2 − P Nt i=1 sinθ i ¯ d 2 i 2 P Nt i=1 1 ¯ d 2 i det{A t −Z t } det{A t −Z t } =    Nt X i=1 cos 2 θ i ¯ d 2 i − P Nt i=1 cosθ i ¯ d 2 i 2 P Nt i=1 1 ¯ d 2 i    ×    Nt X i=1 sin 2 θ i ¯ d 2 i − P Nt i=1 sinθ i ¯ d 2 i 2 P Nt i=1 1 ¯ d 2 i    − −   Nt X i=1 sinθ i cosθ i ¯ d 2 i − P Nt i=1 cosθ i ¯ d 2 i P Nt i=1 sinθ i ¯ d 2 i P Nt i=1 1 ¯ d 2 i   2 , (5.8) and ¯ d i = d i g . It can be noted that φ t is dependent on the geometry (source location withrespecttothesensorlocations)whereg 2 ζ −1 isafunctionofsystemparametersand sourcecharacteristics. Wefirstconsiderφ t . Assumingthatthefirstsensoristhenearest sensor to the source (as shown in Figure 5.1), sinθ i ,cosθ i , and ¯ d i can be rewritten as follows. sinθ i = y i1 + ¯ d 1 sinθ 1 ¯ d i , cosθ i = x i1 + ¯ d 1 cosθ 1 ¯ d i ¯ d i = q x 2 i1 +y 2 i1 +2 ¯ d 1 (x i1 cosθ 1 +y i1 sinθ 1 )+ ¯ d 2 1 (5.9) 63 where x i1 = x i −x 1 g and y i1 = y i −y 1 g . Substituting Eqn. (5.9) in Eqn. (5.8) gives φ t as a function of ¯ d 1 , θ 1 , and relative distances, x i1 and y i1 , between the first sensor and the rest. x i1 and y i1 are independent of the source location and are the same for any grid-based sensor field due to the normalization by the grid size. Thus, we will denote φ t as φ t ( ¯ d 1 ,θ 1 ,N t ) to explicitly show that it is characterized by these three parame- ters. Based on the assumption that the nearest sensors are selected to participate in the localization,sourcesappearingindifferentpartsofthefieldcanyieldthesamevalueof φ t ( ¯ d 1 ,θ 1 ,N t ). Duetosymmetryofthegrid-basedfield,itcanbeshownthatthevalueof φ t ( ¯ d 1 ,θ 1 ,N t )repeatsineverytriangleareashownbytheshadedareainFigure5.1. This area will be denoted as the triangle area in this chapter. Thus, investigating the value of φ t ( ¯ d 1 ,θ 1 ,N t ) when the source is located in a single triangle area covers all possible cases by trivial extension. We first consider the minimal case (N t = 3) in detail (see Figure5.1). Weevaluatetheminimumandthemaximumofφ t ( ¯ d 1 ,θ 1 ,3)asafunctionof θ 1 and ¯ d 1 . Itissufficienttoconsideronlytherange0≤ θ 1 ≤ π 4 and0≤ ¯ d 1 ≤ 1 cos(θ 1 ) to coverthetrianglearea. Itcanbeshownthatφ t ( ¯ d 1 ,θ 1 ,3)isconvexwithrespectto ¯ d 1 and θ 1 . The minimizing value of θ 1 is θ 1 = π 4 for all ¯ d 1 3 . Due to convexity, we can deter- minetheminimizingvalueof ¯ d 1 bydeterminingthestationarypointofφ t ( ¯ d 1 ,θ 1 ,3);we denote this value as ¯ d (t) 1,min,3 . Because of the complex nature of this expression, numer- ical methods are needed to find ¯ d (t) 1,min,3 . A lower bound, however, can be analytically obtained. φ t ( ¯ d 1 , π 4 ,3)canbewrittenasfollows. φ t ( ¯ d 1 , π 4 ,3) = 2 ¯ d 2 3 (1−cos( π 4 −θ 2 ))+ ¯ d 2 2 (1−cos( π 4 −θ 3 ))+ ¯ d 2 1 (1−cos(θ 2 −θ 3 )) sin(θ 3 −θ 2 )+sin( π 4 −θ 3 )+sin(θ 2 − π 4 ) 2 (5.10) 3 Thiscanbeanalyticallyshownbuttheproofistrivialsothatwedonotincludeithere. 64 Considering that ¯ d 2 1 (1− cos(θ 2 −θ 3 )) ≥ 0 (the other terms in the numerator can not be zero considering all possible source locations in the triangle area in Figure 5.1), we havethefollowinginequality. φ t ( ¯ d 1 , π 4 ,3)> 2 ¯ d 2 3 (1−cos( π 4 −θ 2 ))+ ¯ d 2 2 (1−cos( π 4 −θ 3 )) sin(θ 3 −θ 2 )+sin( π 4 −θ 3 )+sin(θ 2 − π 4 ) 2 ,f t ( ¯ d 1 ) (5.11) The minimum of f t ( ¯ d 1 ) is given by f t ( 6 √ 2− √ 3 12 ) = 3125 6912 ' 0.4521, and thus φ t ( ¯ d 1 , π 4 ,3) > 0.4521. An exhaustive search with 0.04×0.04 unit grid blocks over the triangleareagivesmin{φ t (d 1 ,θ 1 ,3)} = 0.5861. Therefore,theobtainedlowerboundis approximately 23% smaller than the true minimum, yielding a somewhat loose bound. Hence, we approximate ¯ d (t) 1,min,3 by using the Taylor series expansion with order 2 to approximate φ t ( ¯ d 1 , π 4 ,3). We assume that the unknown ¯ d (t) 1,min,3 has a uniform distribu- tion within the range [0, 1 √ 2 ] (the possible range of ¯ d (t) 1 for θ 1 = π 4 ). The half distance ( ¯ d 1 = 1 2 √ 2 ) is picked to be the center of the expansion since it gives the least mean squaredistancetothetrue ¯ d (t) 1,min,3 . Theapproximatedclosed-formsolutionof ¯ d (t) 1,min,3 is obtainedtobe: ¯ d (t) 1,min,3 ' 609 √ 10+1405 √ 2 2066 √ 5+4730 . (5.12) Substituting ¯ d (t) 1,min,3 inEqn. (5.8)givesmin{φ t (d 1 ,θ 1 ,3)}'φ t (d (t) 1,min,3 , π 4 ,3) = 0.5861 which is very close to the numerical results presented earlier (the difference is less than 0.02%). Since φ t ( ¯ d 1 ,θ 1 ,3) is convex, max{φ t ( ¯ d 1 ,θ 1 ,3)} can be determined by considering the value along the border of the triangle area and it is found that max{φ t ( ¯ d 1 ,θ 1 ,3)} = φ t ( 1 2 ,0,3) = 15 8 = 1.875. Substituting the bounds of φ t in the expression of CRB t yields, for 3 participating sensors, 3125 6912 (g 2 ζ −1 ) < CRB t ≤ 1.875(g 2 ζ −1 ), and approximately 0.5861(g 2 ζ −1 ) ≤ CRB t ≤ 1.875(g 2 ζ −1 ). The derived CRB is based on one particular set of collected samples while in reality the 65 received signal is random, thus S 0 (k) is also random. In such case, we apply Jensen’s inequality[Pap91]andobtainthelowerboundoftheaveragedCRBbyusingtheexpec- tation of|S 0 (k)| 2 ,E{|S 0 (k)| 2 } instead of|S 0 (k)| 2 to compute ζ −1 . With the assump- tionoflargeL t andusingWiener-KhintchineTheorem[Cou01],wehavethefollowing approximation Lt/2 X k=1 k 2 |S 0 (k)| 2 'L 4 t Z 1/2 0 f 2 G(f)df (5.13) where G(f) is the power spectral density of the source signal, and hence, ζ ' 8f 2 s π 2 Lt σ 2 w c 2 R 1/2 0 f 2 G(f)df. Suppose that the G(f) is flat between 0 and f max , ζ ' 4π 2 Ltf 2 max SNR 0 3c 2 ,where SNR 0 = σ 2 s σ 2 w . With prior knowledge or estimates of SNR 0 , acous- tic speed (c), and f max , the number of collected samples L t and the grid size g can be selectedtoachievearequiredaccuracy(basedontheCRB). To derive the CRB for EOM, we consider the Fisher information matrix given by [Kay93] [F] jk = ∂μ ∂θ j T Q −1 ∂μ ∂θ k + 1 2 tr Q −1 ∂Q ∂θ j T Q −1 ∂Q ∂θ k . (5.14) The second term in Eqn. (5.14) is small compared with the first term when the number of samples is large (see Appendix E) and is thus neglected. Therefore, the CRB for EOMisgivenbythefollowing [F zs,e ] −1 = [γ(A e −Z e )] −1 (5.15) 66 where γ = 2β 2 L e σ 2 w A e = Ne X i=1 u i u T i d 4 i (2β +L 2 e σ 2 w d 2 i ) Z e = 1 P Ne i=1 1 d 2 i (2β+L 2 e σ 2 w d 2 i ) Ne X i=1 u i /d 3 i (2β +L 2 e σ 2 w d 2 i ) ! Ne X i=1 u i /d 3 i (2β +L 2 e σ 2 w d 2 i ) ! T . (5.16) The CRB of the distance error is thus CRB e = F −1 zs,e 11 + F −1 zs,e 22 . CRB e can be rewrittenasCRB e = 2βg 4 γ −1 φ e where φ e = P Ne i=1 1 ¯ d 4 i i P Ne i=1 1 ¯ d 2 i i − P Ne i=1 cosθ i ¯ d 3 i i 2 − P Ne i=1 sinθ i ¯ d 3 i i 2 P Ne i=1 1 ¯ d 2 i i det{A e −Z e } det{A e −Z e } =    Ne X i=1 cos 2 θ i ¯ d 4 i i − P Ne i=1 cosθ i ¯ d 3 i i 2 P Ne i=1 1 ¯ d 2 i i    ×    Ne X i=1 sin 2 θ i ¯ d 4 i i − P Ne i=1 sinθ i ¯ d 3 i i 2 P Ne i=1 1 ¯ d 2 i i    − −   Ne X i=1 cosθ i sinθ i ¯ d 4 i i − P Ne i=1 cosθ i ¯ d 3 i i P Ne i=1 sinθ i ¯ d 3 i i P Ne i=1 1 ¯ d 2 i i   2 i = 1+ L 2 e σ 2 w ¯ d 2 i 2β . (5.17) Due to the assumption of large L e , β L 2 e ' σ 2 s where σ 2 s is the variance of the source signal measured at unit distance from the source. Thus, i ' 1 + d 2 i 2SNR 0 where SNR 0 represents the signal to noise ratio at unit distance from the source. We assume large SNR 0 so that i ' 1 4 . Similar to the steps taken for time delay based estimator, for N e = 3,wederiveφ e ( ¯ d 1 ,θ 1 ,3)andfindthat 8(10−2 √ 7) 4 g 4 81 (5 √ 2− √ 14) √ 11−4 √ 7+20 √ 2−4 √ 14 2 LeSNR 0 < CRB e ≤ 2.1875 g 4 LeSNR 0 , and approximately CRB e ≥ 0.2486 g 4 LeSNR 0 . The lower 4 Ifthisassumptionisnotmet,weconsiderthelowerboundofφ e . 67 bound( 8(10−2 √ 7) 4 g 4 81 (5 √ 2− √ 14) √ 11−4 √ 7+20 √ 2−4 √ 14 2 )isfoundtobe18%lessthanfromtheresults obtained by the exhaustive search while it is only 0.2% for the approximated minimum (0.2486 g 4 LeSNR 0 ). It is clearly seen that the value of SNR 0 determines the selection of L e andg inordertoobtainthedesiredlimitsofthelocalizationperformance. Itisnoticeablefromtheobtainedboundsforthe3-sensorcasethattheperformance of the estimators can be significantly varied depending on the source location. The maximum of CRB t is more than 3 times larger than the minimum while the difference is almost 10 times for that of CRB e . This can be applied for the case of small number ofsensorswherethechangingsourcelocationssignificantlyaffecttherelativedistances withrespecttothesensorlocations. 5.3.1 Caseoflargenumberofsensors Similar to the case of minimal number of sensors, we next seek to determine the limits of localization performance (or the minimum and the maximum of the CRBs) of the system with a large number of sensors. In this case, the expressions for φ t ( ¯ d 1 ,θ 1 ,N t ) and φ e ( ¯ d 1 ,θ 1 ,N e ) become more complicated than for the 3-sensor case. We simplify thecomputationbymakingthefollowingapproximations. Thesensorsthatarefurtherfromthesourcethanthefournearestsensors(thoseareat thecornersofsmallestsquareareawherethesourceislocated)areconsideredtobenot impactedbythechangingsourcelocationwithinthetriangleareasi.e. giventhatsensors are ordered by distance to the source from the nearest to the farthest, the derivative of sinθ i ,cosθ i , and ¯ d i for i = 5,...,N {t,e} with respect to the source location is zero. Let ¯ d (t,e) 1,{min,max},N {t,e} and θ (t,e) 1,{min,max},N {t,e} denote the ¯ d 1 and θ 1 that yield minimum or maximum φ t or φ e with N {t,e} . We assume that ¯ d (t,e) 1,{min,max},∞ and θ (t,e) 1,{min,max},∞ are close to those for arbitrary large N {t,e} . Our assumptions are based on the fact that the 68 farther sensors contain less reliable information due to signal attenuation. The results willbevalidatedbycomparisonswithdirectnumericalmethodslaterinthissection. We now consider the case of N {t,e} → ∞. From the law of large numbers, when N {t,e} → ∞, P N {t,e} i=5 f( ¯ d i ,θ i ) → (N {t,e} − 4)E{f( ¯ d i ,θ i )} whereE{•} represents an expectation andf(•) is an arbitrary function. We approximate the grid-based field as a uniformly distributed sensor field and the area covered by the 5 th to N th {t,e} sensor as a ring. Its inner radius is the distance between the source and the 5 th sensor, denoted by R 0 ,whileitsouterradiusis∞(becauseN {t,e} →∞). Thelimitsofthefollowingterms canbeapproximatedasfollows(seeAppendixFforanexamplederivation). lim Nt→∞ Nt X i=5 cosθ i ¯ d 2 i = lim Nt→∞ Nt X i=5 sinθ i ¯ d 2 i = lim Nt→∞ Nt X i=5 sinθ i cosθ i ¯ d 2 i ' 0 lim Ne→∞ Ne X i=5 cosθ i ¯ d 3 i = lim Ne→∞ Ne X i=5 sinθ i ¯ d 3 i = lim Ne→∞ Ne X i=5 sinθ i cosθ i ¯ d 4 i ' 0 lim Nt→∞ Nt X i=5 1 ¯ d 2 i = lim Nt→∞ Nt X i=5 cos 2 θ i ¯ d 2 i = Nt X i=5 sin 2 θ i ¯ d 2 i '∞ lim Ne→∞ Ne X i=5 1 ¯ d 4 i = π R 2 0 , lim Ne→∞ Ne X i=5 cos 2 θ i ¯ d 4 i = lim Ne→∞ Ne X i=5 sin 2 θ i ¯ d 4 i ' π 2R 2 0 (5.18) 69 Based on these limits, we have the following expressions forφ t andφ e whenN {t,e} → ∞. φ t ( ¯ d 1 ,θ 1 ,∞)→ ( 4 X i=1 1 ¯ d 2 i + Nt X i=5 1 ¯ d 2 i ) × × ( 4 X i=1 cos 2 θ i ¯ d 2 i + Nt X i=5 cos 2 θ i ¯ d 2 i ! × × 4 X i=1 sin 2 θ i ¯ d 2 i + Nt X i=5 sin 2 θ i ¯ d 2 i ! − − 4 X i=1 sinθ i cosθ i ¯ d 2 i ! 2    −1 (5.19) φ e ( ¯ d 1 ,θ 1 ,∞)→ ( 4 X i=1 1 ¯ d 4 i + π R 2 0 ) × × ( 4 X i=1 cos 2 θ i ¯ d 4 i + π 2R 2 0 ! × × 4 X i=1 sin 2 θ i ¯ d 4 i + π 2R 2 0 ! − − 4 X i=1 sinθ i cosθ i ¯ d 4 i ! 2    −1 (5.20) It can be shown that ∂φt ∂d 1 is positive for 0 ≤ θ 1 ≤ π 4 . Thus, the minimum of φ t occurs when ¯ d 1 = 0 ( ¯ d (t) 1,min,∞ = 0). We use this location to estimate φ t for arbitrary large N t andobtainthefollowing min{φ t ( ¯ d 1 ,θ 1 ,N t )}'φ t (0,θ 1 ,N t ) = 8 3D(0,θ 1 ,N t ) (5.21) 70 where D(0,θ 1 ,N t ) equals P Nt i=2 1 ¯ d 2 i when ¯ d 1 = 0 and θ 1 is arbitrary (when ¯ d 1 = 0, θ 1 does not affect the value of D). Since φ t ( ¯ d 1 ,θ 1 ,∞) is monotonically increasing with respect to ¯ d 1 for all θ 1 , finding ¯ d (t) 1,max,∞ requires the investigation of the border of the triangleareabetweenlocation( ¯ d 1 = 1 2 ,θ 1 = 0)and( ¯ d 1 = 1 √ 2 ,θ 1 = π 4 ). Itcanbeshown that φ t ( 1 √ 2 , π 4 ,∞) is the maximum along this border. Thus, we have d (t) 1,max,∞ = 1 √ 2 , θ (t) 1,max,∞ = π 4 ,and max{φ t ( ¯ d 1 ,θ 1 ,N t )}'φ t ( 1 √ 2 , π 4 ,N t ) = 4 F( 1 √ 2 , π 4 ,N t ) (5.22) where F( 1 √ 2 , π 4 ,N t ) equals P Nt i=1 1 ¯ d 2 i when ¯ d 1 = 1 √ 2 and θ 1 = π 4 . The normalized dis- tances are such that g = 1, thus the sensor density is 1 sensor per square unit area. The radius of this circular area containing participating sensors is thus approximated by R = q Nt π . Therefore, D(0,θ 1 ,N t ) and F( 1 √ 2 , π 4 ,N t ) can be estimated through a virtuallyconstructedcircularcoverageoftheparticipatingsensorsasfollows D(0,θ 1 ,N t )' R X n=−R √ R 2 −n 2 X m=− √ R 2 −n 2 1 n 2 +m 2 , n6= 0,m6= 0 F( 1 √ 2 , π 4 ,N t )' R X n=−R √ R 2 −n 2 X m=− √ R 2 −n 2 1 (n−0.5) 2 +(m−0.5) 2 . (5.23) For large N t , we approximately get g 2 ζ −1 φ t (0,θ 1 ,N t ) ≤ CRB t ≤ g 2 ζ −1 φ t ( 1 √ 2 , π 4 ,N t ) wherethesimplifiedversionofφ t canbefoundinEqn. (5.21),(5.22),and(5.23). Unlike φ t ( ¯ d 1 ,θ 1 ,∞), we need to determine R 0 in the expression of φ e ( ¯ d 1 ,θ 1 ,∞) (see Eqn. (5.20)). R 0 is dependent on the source location which is unknown. φ e ( ¯ d 1 ,θ 1 ,∞) is shown to be non-convex and thus it is complicated to find the global minima and maxima. It is noticeable from the expression of φ e (Eqn. (5.17)) that the terms associated with the sensor with large distances from the source (large d i ) can be 71 very small and smaller than those in the expression of φ t (Eqn. (5.8)) due to the divi- sion by the higher order of d i . Thus, the farther the sensors, the far less effect they have on the value of φ e , and we simplify the problem by assuming that ¯ d (e) 1,min,∞ and θ (e) 1,min,∞ are close to ¯ d (e) 1,min,3 (= 773 √ 2 2362 ) and θ (e) 1,min,3 (= π 4 ), and R 0 is determined based on ¯ d (e) 1,min,3 andθ (e) 1,min,3 . This assumption may lead to local optima, however, the results fromdirectnumericalmethodsconfirmthatitisaglobaloptimum. Itcanbeshownthat φ e ( 773 √ 2 2362 ,θ 1 ,∞) is still minimum when θ 1 = π 4 . However, it is not a local minimum with respect to ¯ d 1 . We approximateφ e ( ¯ d 1 , π 4 ,∞) using Taylor series expansion order 2 about 1 2 √ 2 andfindthatmin{φ e ( ¯ d 1 , π 4 ,∞)}'φ e (0.4662, π 4 ,∞). Thus,weapproximate min{φ e ( ¯ d 1 ,θ 1 ,N e )} by φ e (0.4662, π 4 ,N e ). To find d (e) 1,max,∞ and θ (e) 1,max,∞ , we also start fromtheresultsobtainedfromthe3-sensorcasewhere d (e) 1,max,3 = 1 2 andθ (e) 1,max,3 = 0. It can be shown thatd (e) 1,max,3 andθ (e) 1,max,3 give the local maxima ofφ e ( ¯ d 1 ,θ 1 ,∞) and thus it is presumed to be ¯ d (e) 1,max,∞ and θ (e) 1,max,∞ as well. Therefore, max{φ e ( ¯ d 1 ,θ 1 ,N e )} is approximatedbythefollowing. max{φ e (d 1 ,θ 1 ,N e )}'φ e ( 1 2 ,0,N e ) = P Ne i=1 1 ¯ d 4 1 P Ne i=1 sin 2 θ i ¯ d 4 1 P Ne i=1 cos 2 θ i ¯ d 4 1 (5.24) where the following approximations can be substituted (the approximation was carried outsimilartoEqn. (5.23)). Ne X i=1 1 ¯ d 4 1 ' R X n=−R √ R 2 −n 2 X m=− √ R 2 −n 2 1 [(n−0.5) 2 +m 2 ] 2 Ne X i=1 cos 2 θ i ¯ d 4 1 ' R X n=−R √ R 2 −n 2 X m=− √ R 2 −n 2 (n−0.5) 2 [(n−0.5) 2 +m 2 ] 3 Ne X i=1 sin 2 θ i ¯ d 4 1 ' R X n=−R √ R 2 −n 2 X m=− √ R 2 −n 2 m 2 [(n−0.5) 2 +m 2 ] 3 . (5.25) 72 Hence, for large L e , we find that approximately 2βg 4 γ −1 φ e (0.4662, π 4 ,∞)≤ CRB e ≤ 2βg 4 γ −1 φ e ( 1 2 ,0,N e )wherethesimplifiedφ e canbefoundinEqn. (5.24)and(5.25). Figure 5.3 and 5.4 illustrate the approximated lower and upper bounds compared with numerical results from an exhaustive search as the number of sensors is varied. It can be seen that the approximations are close (less than 0.01% error) to the results from exhaustive search. φ e is shown to be slightly affected by increasing the number of sensors (less than 1% decrease when the number of sensors is varied from 1000 to 10000). This is because, unlike TOM, EOM uses the attenuation rate of the signal magnitude for the localization. At the farther sensors, the signal magnitude becomes very small and the information is significantly lost. Thus, the performance is not much improved. Theapproximatedlimitsofthelocalizationperformanceforlargenumberof sensors can also be used for design considerations similar to what we described for the 3-sensorcase. However, it is shown that the range of the CRBs are narrower than that for the 3- sensor case. For CRB t , the maximum is about 1.2 times the minimum and it is about 2 times for CRB e . The reason is that, unlike the 3-sensor case, the relative distances betweenthesourcelocationandmostofthesensorsbecomelessaffectedbythechang- ingsourcelocationwithinthetrianglearea. Thedesignis,thus,subjecttotheparameter selectionandthesourcecharacteristicsmorethanthegeometry. 5.3.2 PerformanceComparison Oneimportantdesignissueistheselectionofthelocalizationschemethatisappropriate under different operational conditions. The limiting case of large number of sensors (large N {t,e} ), and the relevant bounds of the CRBs, were considered in Section 5.3.1. To investigate cases when this assumption is relaxed, we use a similar approach as the 73 one used for deriving the CRBs in Section 5.3 for the 3-sensor case. We first evaluate thefollowing. CRB t CRB e = γ 2ζβg 2 φ t ( ¯ d 1 ,θ 1 ,N t ) φ e ( ¯ d 1 ,θ 1 ,N e ) ρ min < CRB t CRB e <ρ max (5.26) where ρ min = α min{φt( ¯ d 1 ,θ 1 ,Nt)} max{φe( ¯ d 1 ,θ 1 ,Ne)} , ρ max = α max{φt( ¯ d 1 ,θ 1 ,Nt)} min{φe( ¯ d 1 ,θ 1 ,Ne)} , and α = c 2 SNR 0 Leσ 2 w L 3 t 4π 2 g 2 f 2 s P L t /2 k=1 k 2 |S 0 (k)| 2 . For the source with a flat spectrum between 0 and f max , α ' 3c 2 Le 4π 2 g 2 Ltf 2 max . The value of min φ t ( ¯ d 1 ,θ 1 ,N t ) , max φ t ( ¯ d 1 ,θ 1 ,N t ) , min φ e ( ¯ d 1 ,θ 1 ,N e ) , and max φ e ( ¯ d 1 ,θ 1 ,N e ) can be estimated as we presented in the previous section while α is a function of system parameters and source character- istics. We can select the parameters to make ρ max < 1 or ρ min > 1 and obtain that CRB t < CRB e or CRB e < CRB t , respectively. Regardless of the value of φ t and φ e whichisrelatedtothenumberofsensorsandgeometricalproperties,largeα,e.g. large L e , makes EOM more preferable while small α, e.g. large L t or large grid size (g), is good for TOM. Although increasing number of samples makes both methods more accurate, TOM requires Lt 2 DFT values from each sensor for the localization process, andthusalargeL t willusealargercommunicationbandwidth. UnlikeTOM,increasing L e does not require more communication bandwidth since the required data that has to be transmitted and fused with observations obtained at other sensors is still an average energywhichisonesinglevalue. Consideringthesourcespectrum,TOMbecomesless accurate with a low-frequency source while EOM is not affected. The extreme case is the constant signal where TOM cannot acquire information from the signal traveling time since the phase is not different. On the other hand, EOM still operates based on the amplitude of the received signal that is used to compute the energy. An example application that EOM may be better than TOM is tracking of military vehicles such as 74 DragonWagonandAmphibiousAssaultVehiclethatgeneratelowfrequencysignals(in the range of 0 - 125 Hz [LWHS02b]). Nonetheless TOM can become much better with high-frequency source (large f max ). An illustrative example of the parameter selection willbeprovidedinthesimulationresultssection(Section5.5). 5.4 Caseofuniformlyrandomsensorfield When the sensor field is assumed to be uniformly and randomly distributed, we have CRB t =ζ −1 ˜ φ t andCRB e = 2βγ −1 ˜ φ e where ˜ φ t and ˜ φ e areφ t andφ e with ¯ d i isreplaced by d i (the grid size no longer exists in the expressions). Considering that the sensor locations are random, the CRBs also become random and thus, what can be evaluated is the expectations of the CRBs. To exactly derive the expectations involves computing means of non-linear function of a large number of random variables which becomes significantly complicated. We use an approximation by first assuming that the area covered by the participating sensors is a ring. Its inner radius is the distance between the source and the nearest sensor, denoted by R 0 . This assumed radius, in fact, helps to avoid infinite values for some steps in the approximation i.e. d i = 0 is assumed not possible. Also, in reality, the sensor is unlikely to locate exactly on the source (due to 75 theareacoveredbythesource)sothatR 0 maybedeterminedbythesizeofthesource. Basedontheseassumptions,thefollowingapproximationsareobtained Nt X i=1 cosθ i d 2 i = Nt X i=1 sinθ i d 2 i = Nt X i=1 sinθ i cosθ i d 2 i ' 0 Ne X i=1 cosθ i d 3 i = Ne X i=1 sinθ i d 3 i = Ne X i=1 sinθ i cosθ i d 4 i ' 0 Nt X i=1 1 d 2 i 'ρπln N t ρπR 2 0 , Nt X i=1 cos 2 θ i d 2 i = Nt X i=1 sin 2 θ i d 2 i ' ρπ 2 ln N t ρπR 2 0 Ne X i=1 1 d 4 i =ρπ 1 R 2 0 − ρπ N e , Ne X i=1 cos 2 θ i d 4 i = Ne X i=1 sin 2 θ i d 4 i ' ρπ 2 1 R 2 0 − ρπ N e . (5.27) Hence, ˜ φ t ' 4 ρπ Nt ρπR 2 0 (5.28) ˜ φ e ' 4 ρπ 1 R 2 0 − ρπ Ne . (5.29) SubstitutingEqn. (5.28)intheCRBexpressionsyieldstheapproximatedexpectationof theCRBs. However,fordesignconsiderationswehavetorealizethattheseexpectations, unlike the CRBs, do not actually represent the lower bounds i.e. for some topologies, thebetterperformancecanbeobtained. 5.5 SimulationResults The goal of this simulation is to investigate the performance of Maximum-Likelihood estimators compared with the derived CRBs. In addition, we present case example results in investigating conditions leading to the design rules for deciding under which 76 specific selection of design parameters, one observation model is better than the other. Six different source locations were evaluated as shown in Figure 5.5. These source locations are chosen to represent possible source locations in the triangle area. For the first simulation, we assume 3 participating sensors. Besides those six locations, the minimizingandmaximizinglocations(forCRB)thatwerepreviouslyderivedinSection 5.3 are investigated (labeled by min and max in Figure 5.7). Figure 5.7 demonstrates the sensitivity of the source location that affects the localization performance. Also, it illustratesthattheperformancelimitationisconsistentwiththeminimumandmaximum CRB. Next, we focus on the large number of sensor case and present how the obtained rules based on the performance comparison in Section 5.3.2 are helpful in determining the3designparameters;gridsize(g),numberofsamples(L t andL e ),andsourcespec- trum. The source spectrum is assumed flat and bandlimited between 0 and f max . The SNR 0 is set to be 10 dB. Figure 5.6 shows the Power Spectral Density (PSD) of the source with f max = 250 Hz and the sampling frequency (f s ) is 2000 Hz. For the first experiment, various grid sizes (g) are considered between 0.2 and 0.45 m. The num- ber of participating sensors is 100 and the number of samples is 250 for both methods (N t = N e = 100,L t = L e = 250). The root mean square error when each estimator is applied over 1000 trials is shown in Figure 5.8 where the square root of CRBs are also included. It is shown that the error produced by the estimators are consistent with the CRBs. When g = 0.2 m, EML outperforms TML. However, when g is increased to be larger than 0.35 m, TML is improved and becomes superior to EML. This is consistent with theplot ofρ min andρ max in Figure5.9 whereρ max is shownto beless than1 forg = 0.2 m andρ min is greater than 1 forg > 0.35 m. There are cases, e.g. 0.2 < g < 0.3, thatρ min <1butEMLisbetterthanTML.Thisisbecausetheruleisobtainedbycom- paringjusttheminimumandthemaximumofφ e andφ t (seeSection5.3.2),notalltheir 77 possible values. Wheng = 0.34m, itis notalways true thatone estimatoris betterthan anotherdependingonthesourcelocation. Similar to the first experiment, we next investigate the effect of the source spec- trumbandwidth(differentf max values). Figure5.10illustratesthatwithhigh-frequency sources, f max ≥ 500 Hz, TML outperforms EML. For this experiment, the grid size is settobe0.2m. Thus,EMLisnotappropriateforhigh-frequencysourceunlessthegrid size is very small (must be smaller than 0.2 m for f max ≥ 500 Hz) corresponding to very dense field which might not be feasible in some scenarios. The method selection can be guided by the plots in Figure 5.11 where it illustrates that for f max ≥ 500 Hz, TOM should become superior. Note that, for f max = 450 Hz, TML should be more accurate than EML since ρ max < 1, however, there are some source locations where EMLisstillbetterthanTML.ThereasonisthatthePSDofthesource,showninFigure 5.6, is slightly different from the assumed ideally flat PSD. An alternative option that can make EOM competitive with TOM is to increase the number of samples (L e ). In reality, this can be feasible since EOM, unlike TOM, does not require the exchange of all samples between sensors, thus increasing number of samples does not significantly affect the total cost. The grid size is now set to be 0.45 m and f max = 600 Hz. L e is varied from 1000 to 8000 whileL t is fixed to be 250. Figure 5.12 explicitly shows that EMLwithincreasingnumberofsamplesbecomesmoreaccuratethanTML.Theresults agree withρ min andρ max plotted in Figure 5.13. These examples verify that the design rules based on the derived limits of the CRBs can guide the selection of appropriate localizationschemetoachievethebestlocalizationperformance. 78 5.6 Conclusions In this chapter, the limits of the performance of the acoustic source location estima- tors using two different observations, time delays and energy readings, were analyzed and compared based on the Cram´ er-Rao Bound (CRB), when considering a grid-based sensor field. Assuming that pre-selected number of nearest sensors participate in local- ization,theminimumandthemaximumoftheCRBsasfunctionsofnumberofsensors, grid size, sampling frequency, and number of collected samples, for the elemental case ofthreesensorsandthelimitingcaseoflargenumberofsensors,werederived. Experi- mentalsimulationresultsdemonstratethatthederivedCRBscanbeusedforlocalization performance prediction. The comparison illustrates that, with the same number of col- lected samples, time delay based model is better than energy based model except for low-frequency sources and small grid size (dense) fields. The design rules for parame- ter and scheme selection were presented as mathematical expressions, and examples of theirapplicationwerealsoprovided. 79 d 2 d 1 d 3 θ 1 θ 2 θ 3 g g (x 2 ,y 2 ) (x 1 ,y 1 ) (x 3 ,y 3 ) (x s ,y s ) (x 4 ,y 4 ) Figure5.1: Definingrelevantdistancesandanglesforthecaseof3participatingsensors. ∑ n=0 L e −1 r 1 2 n {R 1 0,...,R 1 L t −1} ∑ n=0 L e −1 r 2 2 n {R 2 0,...,R 2 L t −1} ∑ n=0 L e −1 r N e 2 n {R N t 0,...,R N t L t −1} CRB e . . . . . . . . CRB t Figure 5.2: The diagram presents the difference between TOM and EOM that are asso- ciatedwithCRB t andCRB e 80 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 0.09 0.1 0.11 0.12 0.13 0.14 0.15 0.16 Number of sensors φ t max φ t − exhaustive search max φ t − approximated min φ t − exhaustive search min φ t − approximated Figure 5.3: The approximatedφ t (N t ,¯ z s ) are close to that obtained from the exhaustive searchforvariedN t . 81 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 0.2 0.25 0.3 0.35 0.4 0.45 Number of sensors φ e max φ e − approximated max φ e − exhaustive search min φ e − approximated min φ e − exhaustive search Figure5.4: Theapproximatedφ e (N e ,¯ z s )areclosetothatobtainedfrom theexhaustive searchforvariedN e . g g #1 #2 #3 #4 #5 #6 Figure5.5: Thevarioussourcelocationsconsideredinthesensorfield. 82 0 100 200 300 400 500 −80 −70 −60 −50 −40 −30 −20 −10 0 10 20 Frequency (Hz) Power Spectrum Magnitude (dB) Figure5.6: ThePowerSpectralDensity(PSD)ofthesourcewithf max =250Hz. min 1 2 3 4 5 6 max 0.5 1 1.5 2 2.5 3 x 10 −3 Source locations RMS error (m) TML CRB t 1/2 EML CRB e 1/2 Figure5.7: TheperformanceofEMLandTMLissensitivetothesourcelocationswith inthetrianglearea. 83 0.2 0.25 0.3 0.35 0.4 0.5 1 1.5 2 x 10 −3 0.2 0.25 0.3 0.35 0.4 0.5 1 1.5 2 x 10 −3 0.2 0.25 0.3 0.35 0.4 0.5 1 1.5 2 x 10 −3 RMS error (m) 0.2 0.25 0.3 0.35 0.4 0.5 1 1.5 2 x 10 −3 0.2 0.25 0.3 0.35 0.4 0.5 1 1.5 x 10 −3 0.2 0.25 0.3 0.35 0.4 0.5 1 1.5 x 10 −3 EML CRB e 1/2 TML CRB t 1/2 Grid size (m) Figure5.8: TMLbecomessuperiortoEMLwhengridsizeisincreased. 0.2 0.25 0.3 0.35 0.4 0.45 0 0.5 1 1.5 2 2.5 3 3.5 Grid size (m) ρ min,max ρ max ρ min Figure5.9: ρ min andρ max whengridsizeisvaried. 84 250 300 350 400 450 500 550 600 0 0.5 1 1.5 2 2.5 3 3.5 f max (Hz) ρ min,max ρ max ρ min Figure5.10: TMLoutperformsEMLwithhigh-frequencysource(large f s ). 250 300 350 400 450 500 550 600 0 0.5 1 1.5 2 2.5 3 3.5 f max (Hz) ρ min,max ρ max ρ min Figure5.11: ρ min andρ max whenf s isvaried. 85 2000 4000 6000 8000 4 6 8 10 x 10 −4 2000 4000 6000 8000 4 6 8 10 x 10 −4 2000 4000 6000 8000 6 8 10 x 10 −4 RMS error (m) 2000 4000 6000 8000 4 6 8 x 10 −4 2000 4000 6000 8000 4 6 8 x 10 −4 2000 4000 6000 8000 4 6 8 x 10 −4 EML CRB e 1/2 TML CRB t 1/2 Number of samples used for EML Figure5.12: EMLcanbebetterthanTMLbyusingmorecollectedsamples(increasing L e ). 1000 2000 3000 4000 5000 6000 7000 8000 0 0.5 1 1.5 2 2.5 3 3.5 Number of samples used for EML ρ min,max ρ max ρ min Figure5.13: ρ min andρ max whennumberofsamplesforEML(L e )isvaried. 86 Chapter6 AHybridMaximum-Likelihood EstimatorforAcousticSource LocalizationbasedonTemporaland SpatialAttenuationInformation In this chapter, a hybrid Maximum-Likelihood estimator for near-field acoustic source localization based on a hybrid observation model, which combines time delay based andenergybasedobservationModels,isdeveloped. Resultsfromfieldexperimentsare reported. 6.1 Introduction Manysensornetworkshavebeendevelopedforsourcelocalizationinwhichanacoustic source is a common target. Acoustic source localization techniques have been stud- ied for the past few decades [CHY02, SA87, HBE01]. They can be categorized into one-step methods that directly estimate the source location [CHY02], and techniques that comprise two independent steps, namely, estimating the time difference of arrival (TDOA) through time delay estimates and source location estimation based on TDOAs [SA87, HBE01]. These previously proposed methods exploit an observation model basedonthetemporalinformationthatisafunctionofsourcelocation. Thisinformation 87 iscommonlyintheformoftimedelayssothatwedenotethisobservationmodelasthe TimedelaybasedObservationModel(TOM). Even though a variety of localization methods based on TOM can be applied in a sensor network context, their utility is limited by characteristics of sensor networks. Most TOM methods rely on the transmission of the time series of collected signals as such,thesemethodsincurhighcommunicationcost,whichinturnconsumessignificant energyforthewirelesscommunication. Alowcommunicationcostalternativeisenergy based localization methods, as in [LH03], which are suitable for sensor network appli- cations. These methods exploit the spatial attenuation of a signal, which is inversely proportional to the signal traveling distance. The associated signal model is termed the Energy based Observation Model (EOM). The energy based methods require lim- ited information exchange since only a single energy measurement is transmitted from a sensor. However, energy based methods can suffer with respect to accuracy since information is lost in the averaging process. In [MNM07], we compared the inherent accuracyofEOMandTOMbasedmethodsviathecomputationofCramer-Raobounds for a grid-based sensor field. In particular, we examined a very small network as well as a very large network. We showed that for the number of collected samples, TOM is betterthanEOMexceptforlow-frequencysourcesandverydensesensorfields. Herein, we develop a localization technique that exploits both temporal and spatial information, as such, our hybrid maximum likelihood method (HML) offers perfor- mance superior to that of the TOM only and EOM only based methods [Kay93]. The combinationoftemporalandspatialinformationforacousticlocalizationhasbeenpre- viously considered in [CS04]. Therein, the combining step is carried out under the assumptionofstatisticalmodelsforreceivedsignalstrength(energy)measurementsand time delay (or difference) estimates; the two sets of observations are also assumed to bestatisticallyindependentleadingtoalinearcombiningofcostfunctions. Incontrast, 88 we develop a signal model which explicitly captures the effect of source location on the signal attenuation and delay jointly. We further quantify the gains due to the hybrid approachandevaluatetheefficacyandrobustnessofthevariousmethodsviafieldtests. Inthischapter,wederiveaHybridMaximum-Likelihood(HML)estimatorfornear- fieldacousticsourcelocalizationbasedonahybridsignalmodelthatcombinestemporal andspatialattenuationinformationtoobtainmorecompleteandaccuratemodel. Based on the Hybrid Observation Model (HOM), the Cram´ er-Rao Bound (CRB) is derived for the cases of known and unknown signal attenuation factors and is shown that the improvement of HOM over TOM and EOM is evident with low-frequency sources and high-frequency sources, respectively. The maximum-likelihood estimators using each model,Hybrid(HML),Timedelay(TML)[CHY02],andEnergy(EML)[MNM07],are compared under different scenarios. Simulation results demonstrate that HML remains the best choice when TML becomes worse than EML in the case of a low-frequency and far away sources and is much more accurate than EML with a high-frequency and nearer source. HML is also shown to be robust under realistic experimental scenarios conductedusingtheAcousticENSBox[Gir05]. Withunknownacousticspeed,onaver- age, the errors produced by TML and EML are 2.11 and 6.13 times as much as that produced by HML, respectively. Results from both simulations and real experiments demonstrate that HML is more accurate than TML, but uses fewer observed samples resulting in less communication cost. Although HML still requires more communica- tion cost than EML, it is considerably more accurate and robust. In sum, this confirms that HML can be a good choice in low-power applications such as sensor networks to providebothcostefficientandrobustperformance. Thischapterisorganizedasfollows. Insection5.2,thesignalmodelissummarized. TheHybridMaximum-Likelihoodestimator(HML)isderivedinSection6.3. TheCRB based on hybrid observation model is derived and discussed in Section 6.4. Simulation 89 results are given in Section 6.5. The results under realistic experimental conditions are presentedinSection6.6andconclusionsareprovidedinSection6.7. 6.2 SignalModel Then th sampleoftheacousticdatacollectedbythei th sensorisgivenby r i (n) = s(t 0 +n/f s −τ i ) d i +w i (t 0 +n/f s ) (6.1) for 1 ≤ i ≤ N and 0 ≤ n ≤ L− 1 where N is a number of participating sensors and L is the number of collected samples. s(t) is a source signal that is delayed by acoustic traveling time τ i , attenuated by the distance d i between the source (located atz s = [x s ,y s ] T ) and the i th sensor (located atz i = [x i ,y i ] T , assumed known), and perturbedbyzero-meani.i.dGaussianmeasurementnoise, w i (t),withvarianceσ 2 w . An observation starts at time t 0 with sampling frequency f s . The number of samples is assumed to be sufficiently large compared with the time delays (|τ i −τ j | for i 6= j in samples) across sensors. Note that τ i = d i c where c is the acoustic speed. Thus, there aretwopartsthatarefunctionsofthesourcelocation,thetimedelays(τ i )andthesignal attenuation( 1 d i ). 6.3 HybridMaximumLikelihoodEstimator Due to a sufficiently large number of collected samples (L) compared to the time delay across sensors, the DFT is insignificantly affected by edge effects. We, thus, assume thatthefollowingfrequency-domainapproximationissatisfied. DFT L {s(t 0 +n/f s −τ i )}'S 0 (k)exp − j2πkτ i f s L (6.2) 90 for0≤n≤L−1and0≤k≤L−1whereDFT L {•}representtheL-pointDFTand S 0 (k)isthesourcespectrum. Hence, DFT L {r i (n)} =R i (k)' S 0 (k) d i exp − j2πkτ i f s L +η(k), 0≤k≤L−1 (6.3) where η(k) is zero mean complex Gaussian random variable with variance Lσ 2 w . Con- sidering only non-repeated L/2+1 frequency bins and stack up the equations fromN sensors,theequationscanbewritteninmatrixformasfollows R =GS+ξ (6.4) where R = [R(0) T ,...,R(L/2) T ] T , G = [D(0) T ,...,D(L/2) T ] T , S = [S(0) T ,...,S(L/2) T ] T , R(k) = [R 1 (k),...,R N (k)] T , D(k) = [ e −j2πkτ 1 /L d 1 ,..., e −j2πkτ N /L d N ] T , S(k) =S 0 (k)1 T , 1 = [1,...,1] | {z } N , and ξ ∼ N(0,Lσ 2 I (L+2)N/2 ). Since ξ is Gaussian and white, the log-likelihood function of ξ, considering only the terms with unknown parameters, Θ = [z s ,S 0 (0),...,S 0 (L/2)] T , is given by L(Θ) = −kR − GSk 2 . The Maximum- Likelihood(ML)estimatoris,thus,givenbythefollowingoptimizationcriterion ˆ Θ h = min Θ L/2 X k=0 kR(k)−D(k)S 0 (k)k 2 . (6.5) Note that the derived ML estimator in Eqn. (6.5) is similar to the solution proposed in [CHY02]. A key difference is that the signal model in [CHY02] was assumed to be r i (t) =a i s(t−τ i )+w i (t)wherea i isaknownsignalgainlevel. Inourmodel,however, 91 the signal gain level, a i , is modeled by signal attenuation, 1 d i (see Eqn. (6.1)), which is also a function of source location. Thus,D(k) is dependent on the source location in both its phase and magnitude. We denote the ML estimator in Eqn. (6.5) as the Hybrid MLestimator(HML).Weusethesubscripthtoidentifytheparameterestimatesthatare obtained via HML. For each k in Eqn. (6.5), there is a relation betweenR(k),D(k), andS 0 (k) asR(k) =D(k)S 0 (k)+ψ whereψ∼ N(0,Lσ 2 w I N ). Therefore,S 0 (k) can be estimated by ˆ S 0 (k) = D † (k)R(k) whereD † (k) = (D(k) H D(k)) −1 D(k) H is the pseudo-inverse of D(k)and thesuperscriptH represents complexconjugate transpose. SubstitutingS 0 (k)inEqn. (6.5)by ˆ S 0 (k)yieldstheMLestimatorforthesourcelocation asfollows. ˆ Θ h,zs = min zs L/2 X k=1 kR(k)−D(k)D † (k)R(k)k 2 (6.6) Theoptimalsolutioncanbeobtainedbyapplyinganiterativesearchalgorithme.g.,the Nelder-Meaddirectsearchmethod[Avr03]. 6.4 PerformanceAnalysisviaCRB The Cram´ er-Rao Bound (CRB) provides a lower bound on the estimation error vari- ance for any unbiased estimator [Kay93]. In this section, we derive the CRB for the Hybrid Observation Model (HOM, see Section 6.2), to compare with those derived in [CHY02] (Time delay based Observation Model, TOM) and [MMN07] (Energy based Observation Model, EOM). The set of subscripts {h,t,e} is used to represent those associated with each signal model, hybrid (h), time delay (t), or energy (e). For HOM and TOM, a vector of unknown parameters is Θ h = Θ t = [z T s ,|S| T ,Φ T ] T where |S| and Φ are the magnitude and phase components of S while it is Θ e = [z T s ,ν] T where ν = P L k=0 |S 0 (k)| 2 for energy model [MNM07]. Let ˆ Θ {h,t,e} be the vec- tors of estimated parameters,F {h,t,e} be the Fisher Information Matrix (FIM) [Kay93] 92 for ˆ Θ {h,t,e} , [F {h,t,e},zs ] −1 be the CRB for the location estimates, and CRB {h,t,e} , [F {h,t,e},zs ] −1 11 +[F {h,t,e},zs ] −1 22 betheCRBforthedistanceerroroftheestimatedlocation fromthetruesourcelocation. AccordingthesignalmodelinEqn. (6.4),F h isgivenby F h = 2Re[H H C −1 ξ H] (6.7) whereH = [∂G/∂z T s ,∂G/∂|S| T ,∂G/∂Φ T ] andC ξ is the covariance matrix ofξ (see Eqn. (6.4)). SubstitutingHandC ξ inEqn. (6.7)yields [F h,zs ] −1 = [ζ(A t −Z t )+γ(A h −Z h )] −1 (6.8) where ζ = 8f 2 s π 2 L 3 σ 2 w c 2 L/2 X k=0 k 2 |S 0 (k)| 2 , γ = 2β Lσ 2 w , β = L/2 X k=0 |S 0 (k)| 2 Z t = 1 P N i=1 1/d 2 p N X i=1 u i d 2 i ! N X i=1 u i d 2 i ! T , Z h = 1 P N i=1 1/d 2 i N X i=1 u i d 3 i ! N X i=1 u i d 3 i ! T A t = N X i=1 u i u T i d 2 i , A h = N X i=1 u i u T i d 4 i , u i = (z s −z i ) d i (6.9) Thederivationin[CHY02]showsthat[F t,zs ] −1 = [ζ(A t −Z t )] −1 . From[MNM07]we havethatforlargeLandthus[F e,zs ] −1 ' [γ(A e −Z e )] −1 where A e ' N X i=1 u i u T i d 4 i ε i , Z e = 1 P N i=1 1 d 2 i ε i N X i=1 u i d 3 i ε i ! N X i=1 u i d 3 i ε i ! T , ε i = 1+ L 2 σ 2 w d 2 i 4β . Note thatζ(A t −Z t ) appears in [F h,zs ] −1 , and the termA e −Z e is similar to the term A h −Z h in [F h,zs ] −1 except for the presence of ε i . It is simple to show thatA h −Z h in Eqn. (6.8) is non-negative definite. Since ε i > 1 ∀i, it can be also shown that 93 (A h −Z h )− (A e −Z e ) is non-negative definite. For large L, 2β L 2 ' σ 2 s , and thus ε p = 1+ σ 2 w d 2 p 2σ 2 s = 1+ d 2 p 2SNR 0 where SNR 0 is the Signal to Noise Ratio measured at 1 m from the source. Hence,A e −Z e is dependent on SNR 0 and it is close toA h −Z h if SNR 0 >>>d 2 i . Asaresult,CRB h isalwayssmallerthanCRB t andCRB e . Define ρ , γ ζ . Regardless of the geometry (related to A t ,Z t ,A h , and Z t ), the decreasing value of CRB h compared with CRB t is increased with large ρ whereas that comparedwithCRB e isincreasedwithsmallρ. Notethatρ = c 2 4π 2 φwhere φ = P L/2 k=0 |S 0 (k)| 2 P L/2 k=0 kfs L 2 |S 0 (k)| 2 . (6.10) Givenasetofcollectedsamples,φandthusρarenotdependentonLandf s . Theterm P L/2 k=0 kfs L 2 |S(k)| 2 inφisalinearcombinationofthesquaredmagnitudeoffrequency components where the higher frequency components have greater weights. For low- frequency sources, φ and thus ρ becomes larger resulting in increasing the difference between CRB h and CRB t . On the other hand, high-frequency sources cause ρ to be smallersothatthedifferencebetweenCRB h andCRB e becomesmoreconspicuous. 6.4.1 Practicallimitations The decay factor (α) can vary due to several factors such as temperature, air pressure, and humidity. Also, the acoustic speed (c) changes depending on temperature. An inaccurately assumed value of these parameters can significantly degrade the local- ization performance. We can cope with this problem by treating them as unknown parameters to be estimated. The relevant steps for both parameters are similar, thus we will show only the case of unknown α. The ML estimator can be formulated sim- ilar to what was used for the case of known α. The vector of unknown parameters becomes[z T s ,|S| T ,Φ T ,α] T . Let[ ´ F {h,t,e},zs ] −1 betheCRBforthelocationestimate,and 94 ´ CRB {h,t,e} ' [ ´ F {h,t,e},zs ] −1 11 +[ ´ F {h,t,e},zs ] −1 22 betheCRBforthedistanceerrorforthecase ofunknownα. [ ´ F h,zs ] −1 = h ζ( ´ A− ´ Z)+´ γ( ´ A h − ´ Z h −κ −1 B h B T h ) i −1 where B h = N X i=1 lnd i d 2α i N X i=1 u i d 2α+1 i − N X i=1 1 d 2α i N X i=1 lnd i u i d 2α+1 i , κ = N X i=1 1 d 2α i    N X i=1 (lnd i ) 2 d 2α i N X i=1 1 d 2α i − N X i=1 lnd i d 2α i ! 2    ´ γ =α 2 2 Nσ 2 w L/2 X k=0 |S(k)| 2 , ´ A = N X i=1 u i u T i d 2α i , ´ A h = N X i=1 u i u T i d 2α+2 i ´ Z = 1 P N i=1 1/d 2α i N X i=1 u i d 2α i ! N X i=1 u i d 2α i ! T , ´ Z h = 1 P N i=1 1/d 2α i N X i=1 u i d 2α+1 i ! N X i=1 u i d 2α+1 i ! T while [ ´ F t,zs ] −1 = h ζ( ´ A− ´ Z) i −1 . It can be shown that ´ A h − ´ Z h −κ −1 B h B T h is non- negative definite. Thus, ´ CRB h is still smaller than ´ CRB t for the case of unknown α. However, the difference between them is less than the case of knownα as exhibited by theadditionalterm,κ −1 B h B T h . [ ´ F e,zs ] −1 isgivenby ´ β( ´ A e − ´ Z e −κ −1 B e B T e )where ´ A e , ´ Z e , andB e are similar to ´ A h , ´ Z h , andB h except for thatd 2α i andd 2α+1 i are substituted by i d 2α i and i d 2α+1 i , respectively. It is simple to show that ( ´ A h − ´ Z h −κ −1 B h B T h )− ( ´ A e − ´ Z e −κ −1 B e B T e ) is non-negative definite and thus ´ CRB h is smaller than ´ CRB e . We conducted experiments where these limitations are considered and the results are reportedinSection6.6. 95 6.5 SimulationResults In this section, a series of Monte Carlo simulations were conducted to evaluate the per- formanceofeachMLestimator. First,weinvestigatethelocalizationperformancewith differentsourcelocations. Theassumedsensorlocationsandthesourcetrackareshown in Figure 6.1. The sampling frequency was set to be 5000 Hz and the acoustic signal speedwas345m/s(correspondingtowhentheairtemperatureis74F).Thesourcesig- nal was generated by passing a zero-mean, Gaussian, random sequence through a FIR lowpassfilterwhosecut-offfrequency, f max ,is400Hz. Theresultswereaveragedover 1000trialsforeachsourcelocationandthenumberofcollectedsampleswas1024. The ratio of the source signal variance and measurement noise variance, σ 2 s /σ 2 w , was set to be 20 dB. The initial estimates used for all maximum-likelihood estimators are the true sourcelocations. Figure6.2illustratestheperformanceofeachestimatoraswellasthe CRBs (the square root of the CRBs are actually shown, to be compared with the RMS errors). ItisshownthatTMLissuperiortoEMLwhenthesourceisclosetothesensors, butitbecomesworsewhenthesourceisfarawayfromthesensors. Asexpected,HML is the best estimator for all source locations. This is consistent with the plot in Figure 6.3wheretheimprovementofHMLoverTMLandEMLislargeforthefarawaysource and near source, respectively. HML, thus, compensates for the lack of performance of each estimator that is affected by the source location as it is shown to be robust for all consideredsourcelocations. Next, we investigate the localization performance when the source is band-limited with different maximum frequencies, f max = 200,400,...,1000 Hz. The source loca- tion is fixed to be location #5. Figure 6.4 illustrates that the performance of EML is notaffectedbysourceswithdifferentmaximumfrequencieswhileTMLbecomesmore accurate with high-frequency sources. The shortfalls of TML are discerned with low- frequency source while HML remains the superior estimator. HML is also still much 96 more accurate than EML (and not worse than TML) for high-frequency source. The improvement of HML over other estimators is shown in Figure 6.5. Similar to what wasshownforthecaseofdifferentsourcelocations,HMLadvantageouslybalancesthe performanceofEMLandTML. Wealsoinvestigatetheperformanceoftheestimatorswithdifferentnumberofsam- ples,L =500,600,...,1000. f max isfixedtobe400Hzandthesourceisatlocation#5. Figure 6.6 illustrates that, when number of of samples is 500, HML is still better than otherestimatorsforallconsideredscenarios. Consideringthatthesesamplesareneeded to be transmitted between sensors for localization, HML saves communication cost by more than 50% in this particular scenario compared to TML. Although, the required cost for HML is still more than that for EML whose samples are used to compute the energybeforebeingfusedwithothersensors,HMLhasanadvantagewithregardstothe accuracy. Moreover,inthenextsection,wewillshowthatEMLhassomelimitationsin realscenarioswheretheperformanceofHMLismuchmorerobust. 6.6 FieldExperiment We conducted experiments using the Acoustic ENSBox developed by Girod formerly oftheCenterforEmbeddedNetworkedSensing(CENS),UCLA[Gir05]. Thelocations of the sensors and the source used in the experiment are shown in Figure 6.7. Each run uses a different source and/or sensor locations (referred to as a topology). 5 sensors (microphones) were used for each topology. The experiment was conducted in front of the Molecular & Computational Biology building on the USC campus (see Figure 6.10). The original sampling frequency (operated by AENSBox) is 48000 Hz. Two different source signals were used. One was an artificially generated Gaussian random signal with the significant energy between 0 and 500 Hz. The other was a pre-recorded 97 static car (jeep) sound clip [FP] whose power spectral density is shown in Figure 6.8. At the beginning of each sound clip, a maximal length sequence [Pet61] of 2-second duration was attached. These sequences were used to estimate the relative time delays and the acoustic speed to be used as ground truth for the estimators (for the case of known acoustic speed). The source signals were played from a loud speaker and 8- seconddata(includingtheMLS)wasrecordedateachsensor. Foreachrecordedframe, 200000 samples were retrieved from the interval that contains the source signal. Dif- ferent frames of 1024 samples were randomly chosen from the 200000-sample frame after downsampling by a factor of 5 (now sampling rate becomes 9600 Hz). 100 trials (using 100 different frames) were conducted and the Root Mean Square (RMS) errors were computed for 23 different topologies (described in Table 6.1 and Table 6.2). The decay factor was assumed to be unknown and, thus needed to be estimated. The initial estimates for the decay factor was 1 while those for the source location were randomly chosenwithinthedistance0.25mfromthetruesourcelocations. Theresultsforthecase of known and unknown acoustic speed were investigated. The initial estimate for the caseofunknownacousticspeedwas345m/s. TheNelder-MeaddirectsearchMatlab R functionwasusedtofindtheoptimalsolution. Weobservethelocalizationperformance basedontheaverageofRMSerrorsproducedbyeachestimatoroveralltopologies. The detailed results including the average errors over all topologies are shown in Table 6.1, andtheaverageerrorsareseparatelyplottedinFigure6.9forcomparison. Notethatfor the case of known and unknown acoustic speed, the results for EML are theoretically not affected, and thus the results are not included in Table 6.1. However, the average errors for EML for these two cases are plotted identically in Figure 6.9 for complete- ness. Itcanbeseenthat,onaverage,HMLismoreaccuratethanTMLandEMLforall considered scenarios. For the generated source signal and known acoustic speed, HML onaverageproduces21%and81%lesserrorthanTMLandEML,respectively. 98 The overall performance of each estimator is degraded when the source is the car sound. There are some topologies for which the results for all trials are not acceptable i.e. the location estimate is always far from the area of interest (> 10 m, shown as N/A in Table 6.1). Excluding these outlier topologies, HML is8% and84% better than TML and EML, respectively. EML performs much worse than other estimators. The reason is that the energy of the original source signal is different at each sensor due to the non-stationary source besides the effect of the attenuation. Also, the estimate of the variance of the measurement noise, which is assumed known in the derivation of energy based methods, may not be accurate and results in the localization performance degradation. Moreover, energy based methods rely only on the signal decay model. When the decay factor is unknown or varied due to the environment, the system may not be robust. This illustrates the limitation of the energy based methods. The results alsoshowthatTMLsuffersmorethanHMLwhentheacousticspeedisunknownasthe difference between HML and TML is increased to 41% and 52% for generated signal and car sound, respectively. We also observed the performance of HML using fewer samples (900 samples) for the case of known acoustic speed and found that the error is degraded to 0.33 m which is still smaller than TML with 1024 samples. Thus, in the realexperiments,HMLcanstillsavecommunicationcostwithbetterperformance. 6.7 Conclusions Inthischapter,wederiveaHybridMaximum-Likelihood(HML)estimatorfornear-field acousticsourcelocalizationbasedonahybridsignalmodelthatincludestimedelaysand signal attenuation rate. The CRB for the hybrid observation model is derived, and its decrease from the CRB for time delay and energy based observation models becomes larger for low-frequency and high-frequency source, respectively. sSimulation results 99 5 6 7 8 9 10 11 5 5.5 6 6.5 7 7.5 8 8.5 9 Sensor locations Source track Figure6.1: Sourcetrackandsensorlocationsassumedinthesimulation. demonstrate that HML compensates for the trade-offs between time delay and energy basedmethods,andthusyieldsthebestperformanceforallconsideredsources. HMLis alsothemostrobustestimatorintherealexperiments,hassignificantimprovedaccuracy, andcanbeappliedwithcostsaving. Acknowledgment The authors would like to thank Lewis Girod, Travis Collier, and Deborah Estrin for technicalhelpandgenerositywithlendingustheirAcousticENSBox. 100 1 2 3 4 5 10 −2 10 −1 Source locations RMS error (m) EML TML HML CRB e 1/2 CRB t 1/2 CRB h 1/2 Figure 6.2: HML outperforms other estimators for all source locations while TML becomesworsethanEMLwithfarawaysource. 1 2 3 4 5 25 30 35 40 45 50 55 60 65 70 75 Source locations Improvement of HML over other estimators (%) EML TML Figure6.3: TheimprovementofHMLoverTMLandEMLbecomesmoreconspicuous forfarawayandnearsource,respectively. 101 200 300 400 500 600 700 800 900 1000 0 0.005 0.01 0.015 f max (Hz) RMS error (m) EML TML HML CRB e 1/2 CRB t 1/2 CRB h 1/2 Figure 6.4: HML outperforms other estimators for all sources with different maximum frequencieswhileTMLbecomesworsethanEMLwithlow-frequencysource. 200 300 400 500 600 700 800 900 1000 0 10 20 30 40 50 60 70 80 90 f max (Hz) Inprovement of HML over other estimators (%) TML EML Figure6.5: TheimprovementofHMLoverTMLandEMLbecomesmoreconspicuous forlow-frequencyandhigh-frequencysource,respectively. 102 500 600 700 800 900 1000 2 4 6 8 10 12 14 x 10 −3 Number of samples RMS error (m) EML TML HML Figure 6.6: HML for the source with f max = 400 Hz, using fewer samples (up to 50% less),stilloutperformsTMLandEML. 103 −1 −0.5 0 0.5 1 1.5 2 2.5 3 3.5 −1 −0.5 0 0.5 1 1.5 2 2.5 3 3.5 1 2 3 4 5 6 7 8 9 10 11 12 17 18 19 13 14 15 16 20 21 22 Figure6.7: Themarksofsourceandsensorlocationswhereeachrunisdifferentaccord- ingtoTable6.1 0 500 1000 1500 2000 2500 3000 3500 4000 −40 −35 −30 −25 −20 −15 −10 −5 0 Frequency (Hz) Power Spectrum Magnitude (dB) Figure6.8: PowerSpectralDensityofthecarsound(Jeep). 104 Table6.1: RMSerrors(m)producedbyeachestimatorfordifferenttopologies Topology GeneratedSourceSignal CarSound Knownc Unknownc Knownc Unknownc TML HML TML HML EML TML HML TML HML EML 1 0.45 0.12 0.42 0.22 0.71 0.42 0.39 0.35 0.36 1.58 2 0.64 0.33 0.66 0.14 0.67 0.60 0.60 0.56 0.62 0.82 3 0.11 0.10 0.06 0.05 0.45 0.09 0.07 0.07 0.05 0.72 4 0.10 0.08 0.10 0.06 0.57 0.85 0.36 N/A 0.32 0.75 5 0.19 0.14 0.17 0.12 0.50 0.04 0.05 0.39 0.10 0.46 6 0.13 0.11 0.20 0.17 1.04 0.14 0.15 0.20 0.16 2.24 7 0.10 0.11 0.14 0.29 1.42 0.21 0.21 0.23 0.31 2.43 8 0.07 0.07 0.97 0.13 0.57 0.15 0.19 3.51 0.27 0.92 9 0.31 0.08 0.48 0.08 0.52 0.39 0.36 0.52 0.35 0.35 10 0.05 0.09 0.29 0.13 0.57 0.11 0.10 0.26 0.21 0.59 11 0.04 0.04 0.19 0.08 0.71 0.06 0.02 0.23 0.08 0.71 12 0.42 0.43 0.38 0.40 2.11 0.46 0.44 0.42 0.28 N/A 13 0.16 0.19 0.31 0.23 3.85 0.24 0.25 0.28 0.25 3.57 14 0.21 0.22 0.29 0.28 0.64 0.13 0.14 0.22 0.23 0.83 15 0.20 0.19 0.12 0.13 2.93 0.38 0.31 3.31 0.29 6.64 16 0.49 0.19 0.88 0.43 1.20 0.27 0.24 0.70 0.72 2.31 17 0.77 0.70 0.84 0.84 1.44 0.63 0.57 0.66 0.67 N/A 18 0.43 0.21 0.84 0.33 1.77 0.52 0.45 0.41 0.78 N/A 19 0.32 0.32 0.51 0.32 1.34 0.21 0.22 0.35 0.22 2.80 20 0.32 0.31 0.39 0.31 1.55 0.24 0.27 0.42 0.26 N/A 21 0.39 0.39 0.35 0.35 0.78 0.78 0.78 1.35 0.76 5.38 22 0.31 0.30 0.44 0.16 0.81 0.71 0.50 0.24 0.65 1.64 23 0.33 0.33 0.41 0.33 0.72 0.35 0.37 1.67 0.24 1.35 Average 0.28 0.22 0.41 0.24 1.17 0.35 0.31 0.74 0.35 1.90 105 Table6.2: RMSerrors(m)producedbyeachestimatorfordifferenttopologies Topology Sensor&Source(*)locations 1 1,2,3,4,9,13 ∗ 2 5,6,7,8,9,13 ∗ 3 9,10,11,12,17,13 ∗ 4 9,11,12,18,19,13 ∗ 5 1,2,5,9,10,14 ∗ 6 3,6,7,10,11,14 ∗ 7 4,8,10,12,19,14 ∗ 8 10,12,17,18,19,14 ∗ 9 2,3,4,6,10,15 ∗ 10 7,8,10,11,12,15 ∗ 11 10,12,17,18,19,15 ∗ 12 1,2,3,4,10,16 ∗ 13 5,6,7,8,10,16 ∗ 14 9,10,11,12,17,16 ∗ 15 10,11,12,18,19,16 ∗ 16 2,3,4,6,16,20 ∗ 17 7,8,10,11,16,20 ∗ 18 7,8,11,12,16,20 ∗ 19 1,2,3,4,17,21 ∗ 20 5,6,7,8,17,21 ∗ 21 9,10,11,12,17,21 ∗ 22 1,2,3,4,17,22 ∗ 23 5,6,7,8,17,22 ∗ 106 TML HML EML 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Estimators Average RMS error (m) Generated signal / Known speed Generated signal / Unknown speed Car sound / Known speed Car sound / Unknown speed Figure6.9: Onaverage,HMLoutperformsTMLandEMLforallscenarios. Figure6.10: Experimentlocation,AENSboxes,andmicrophones 107 Chapter7 ConclusionsandFutureWork This dissertation focuses on acoustic source localization in sensor network context. A main goal is to achieve low power localization system with high performance. This relatestoimprovinglocalizationschemesandappropriatelyapplytheseschemeseffec- tivelyinnetworkenvironments. Weproposedadistributedalgorithm,basedonrangedifferencelocalizationmethod, and simulation results illustrate that the distributed localization produces smaller error and consumes less energy than centralized method. The distributed processing yield increasingimprovementwhenthenumberofparticipatingsensorsissmall,obtainmore energysavingwiththelargenumberofparticipatingsensor,andmorerobusttodecreas- ingsourcesignalenergy. Next, we developed, compared, and contrasted a set of source localization meth- ods based on energy measurements. We introduced a new least squares method: the weighted direct (WD) approach; WD is shown to be significantly more accurate than previously proposed methods with much lower computational complexity. WD, when combined with correction technique, approaches the Cram´ er-Rao Bound with large numberofsamples,andcanbemoreaccuratethanmaximum-likelihoodestimators. We also analyzed and compared two different observation models, based on time delays and energy readings, when considering a grid-based sensor field, and suggested designrulesforsystemparametersandschemeselection. Theanalysisresultsinthepre- diction of what method (associated with each observation model) is better than another underwhatconditions. Theexampleapplicationswereprovided. 108 Finally, we derived a Hybrid Maximum-Likelihood (HML) estimator for near-field acoustic source localization based on a hybrid signal model that includes time delays and signal attenuation rate. Simulation results demonstrate that HML compensates for the trade-offs between time delay and energy based methods, and thus yields the best performance for all considered sources. HML is also the most robust estimator in field experiments,hassignificantimprovedaccuracy,andcanbeappliedwithcostsaving. 7.1 Futurework For issues that directly relate to our work, there are few studies that should be further explored. There is limitation of energy based localization due to the uncertainty of the spatial attenuation factor and reverberation effect. This limitation can be observed in details including the impact on localization performance. More field experiments should validate the assumptions of the signal model and may support the possibility of practicalimplementationoftheproposedschemes. Designproblemscanbegeneralized toothertypesofsensorfielde.g. uniformlyrandomfield. Amorerealisticsignalmodel, e.g. non-stationarysignalwithshorttimeobservation,willbemorechallenging. Design consideration based on least squares methods may be interesting and more practical due to the ease of implementation. The evaluation of the hybrid scheme requires more fieldexperiments,i.e. moredifferenttopologiesunderdifferentenvironments,sincethe uncertainty of the spatial attenuation factor as well as the model mismatch may impact itsperformance. Regarding the direction of the acoustic source localization problems in sensor net- works, the relevant problems should be studied when the localization techniques are alreadyappliedoratleasthavehighpotentialtobeappliedinrealscenariossincethere certainly are many additional factors beyond what we have considered or imagined for 109 instanceamorecomplexsignalmodelandtheabilityofdatagatheringprotocolsinspe- cificapplications. Thestudyshouldbemoreapplicationspecificasintegratinglocaliza- tiontechniquesinthewholesystemwilldefineproblemswell. 110 References [Avr03] M. Avriel. Nonlinear Programming: Analysis and Methods. 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For example, for a given M number of equations used for OS, multiplying a rowvector[1 −1 0 ... 0 | {z } M ]bothsidesofEqn. (4.13)givesoneequationusedfor QE. As many as M(M − 1)/2 equations can be constructed from M equations used by OS by multiplying both sides of Eqn. (4.13) by the matrixP whose rows contain 2 nonzero elements, 1 and−1, and0 otherwise. The columns of the nonzero elements of eachrowaredifferentresultinginM(M−1)/2distinctrows. Let ´ G in Eqn. (4.13) be [C −1 M ] where 1 M = [1,...,1] | {z } M T (see Eqn. (4.12)). MultiplyingbothsidesofEqn. (4.13)byPgives h PC 0 i   r s kr s k 2   =P ´ b (A.1) PCr s =P ´ b (A.2) Hence,theleastsquaressolutionforQEmethodis ˆ r s,QE = ((PC) T PC) −1 (PC) T P ´ b. (A.3) Note that the above solution is in fact Eqn. (4.11) whereG is now shown as a function ofPandCwhichisablockmatrixin ´ GandCisfullcolumnrankM by2matrix. 116 TheOSsolutiongiveninEqn.(4.14)canbewritteninpartitionedmatrixasfollows ˆ θ OS =   C T C −C T 1 M −1 T M C 1 T M 1 M   −1   C T −1 T M  ´ b =   Q 11 Q 12 Q 21 Q 22     C T −1 T M  ´ b =   Q 11 C T −Q 12 1 T M Q 21 C T −Q 22 1 T M  ´ b where Q 11 = (C T (I−( 1 M )1 M 1 T M )C) −1 Q 12 = Q 11 C T 1 M ( 1 M ). Although ˆ θ OS givestheestimateforbothr s andkr s k 2 ,tocomparewithˆ r s,QE ,onlythe locationestimation,ˆ r s ,isconsideredasfollows ˆ r s,OS = (Q 11 C T −Q 12 1 T M ) ´ b (A.4) SubstitutingQ 11 andQ 12 andusingP T P =MI−1 M 1 T M gives ˆ r s,OS = (C T P T PC) −1 C T P T P ´ b =ˆ r s,QE (A.5) and, comparing Eqn. (A.3) and Eqn. (A.5), we see that QE and OS yield the same locationestimate. 117 AppendixB ThisappendixshowsthatWOSandWDyieldthesamelocationestimate. Eachequation used for WOS (Eqn. (4.16)) is obtained from those used for WD (Eqn. (4.24)) through thefollowingconversion: ´ p ij Aθ WD = ´ p ij (h+ε) (B.1) where ´ p ij = h ...0... −1 y i −y j ...0... 1 y i −y j ...0... i . Note that if some energy observations have the same quantities i.e. y i = y j , we can not pair these observations toformequationsforWOS.Weassumethatthissituationdoesnotoccurandit,infact, neveroccurredintheconsideredsimulation. Stackingupallpossible ´ p ij ’sthatconstruct N(N −1)/2 equations for WOS gives a conversion matrix ´ P which is of rankN −1. LettingA = [B −1 N ]where1 N = [1,...,1] | {z } N T gives ´ P h B −1 N i      r s kr s k 2 σ 2 s      = ´ PB   r s kr s k 2   = ´ Ph+ ´ Pε (B.2) andthesolutionforWOSisthefollowing ˆ θ WOS = (B T ´ P T ˆ C −1 ´ PB) −1 B T ´ P T ˆ C −1 ´ Ph. (B.3) 118 Notethattherelationshipbetween ˆ C (definedinEqn. (4.21)and ˆ Q(definedinSection 4.5)isfoundtobe ˆ C = ´ P ˆ Q ´ P T . (B.4) Since the rank of ´ P is N − 1, ˆ C must be singular and the pseudoinverse of ˆ C , ˆ C † , is used instead of ˆ C −1 as pointed out earlier in Section 4.4. By using Singular Value Decomposition, ˆ C canbefactoredinto ˆ C =UΣU T . (B.5) whereΣ isN −1 byN −1 diagonal matrix that contains non-zero eigenvalues of ˆ C and the columns ofU are corresponding eigenvectors. ´ P and ˆ Q can be represented as partitionedmatricesasfollows ´ P = h ´ P 0 − ´ P 0 1 N−1 i (B.6) ˆ Q =   ˆ Q 11 0 0 ˆ Q 22   (B.7) where ˆ Q 11 and ˆ Q 22 are(N−1)×(N−1)and1×1matrices, respectively, and ´ P 0 is N(N−1)/2byN−1matrixwithrankN−1. Thus, ˆ C isgivenby ´ P ˆ Q ´ P T = ´ P 0 Υ ´ P T 0 (B.8) 119 whereΥ , ( ˆ Q 11 +1 N−1 ˆ Q 22 1 T N−1 ) Substituting Eqn. (B.5) and (B.8) in Eqn. (B.4) gives UΣU T = ´ P 0 Υ ´ P T 0 Σ −1 = ( ´ P T 0 U) −1 Υ −1 (U T ´ P 0 ) −1 UΣ −1 U T =U( ´ P T 0 U) −1 Υ −1 (U T ´ P 0 ) −1 U T = ˆ C † (B.9) Theterm ´ P T ˆ C −1 ´ PinEqn. (B.3)canbewrittenasapartitionedmatrixasfollows ´ P T ˆ C † ´ P =   ´ P T 0 −1 T ´ P T 0   ˆ C † h ´ P 0 − ´ P 0 1 N−1 i =   ´ P T 0 ˆ C † ´ P 0 − ´ P T 0 ˆ C † ´ P 0 1 N−1 −1 T N−1 ´ P T 0 ˆ C † ´ P 0 1 T N−1 ´ P T 0 ˆ C † ´ P 0 1 N−1   . (B.10) InsertingEqn. (B.9)inEqn. (B.10)gives ´ P T ˆ C † ´ P =   Υ −1 −Υ −1 1 N−1 −1 T N−1 Υ −1 1 T N−1 Υ −1 1 N−1   . (B.11) Υ −1 canbeexpandedusingtheWoodbury’sidentityasfollows Υ −1 = ˆ Q −1 11 − ˆ Q −1 11 1 N−1 1 T N−1 Q −1 11 Q −1 22 +1 T N−1 Q −1 11 1 N−1 . (B.12) 120 SubstitutingEqn. (B.12)andEqn. (B.11)inEqn. (B.10)gives ´ P T ˆ C † ´ P = ˆ Q −1 − −    ˆ Q −1 11 1 N−1 1 T N−1 ˆ Q −1 11 ˆ Q −1 22 +1 T N−1 ˆ Q −1 11 1 N−1 ˆ Q −1 11 1 N−1 ˆ Q −1 22 ˆ Q −1 22 +1 T N−1 ˆ Q −1 11 1 N−1 1 T N−1 ˆ Q −1 11 ˆ Q −1 22 ˆ Q −1 22 +1 T N−1 ˆ Q −1 11 1 N−1 ˆ Q −2 22 ˆ Q −1 22 +1 T N−1 ˆ Q −1 11 1 N−1    . (B.13) LetusnowconsiderthesolutionobtainedfromWDusingthepartitionedmatrixform. ˆ θ WD = (A T ˆ Q −1 A) −1 A T ˆ Q −1 h (B.14) =     B T −1 T N   ˆ Q −1 h B −1 N i   −1   B T −1 T N   ˆ Q −1 h (B.15) Consideringonlytheestimateof[r T s kr s k 2 ] T ,weobtain ˆ θ WD,r T s ,krsk 2 = (B T HB) −1 B T Hh (B.16) where H = ˆ Q −1 − ˆ Q −1 1 N 1 T N ˆ Q −1 1 T N ˆ Q −1 1 N (B.17) Expandingtheterm ˆ Q −1 1 N 1 T N ˆ Q −1 1 T N ˆ Q −1 1 N inEqn. (B.17)usingthepartitionedmatrixgives ˆ Q −1 1 N 1 T N ˆ Q −1 1 T N ˆ Q −1 1 N =    ˆ Q −1 11 1 N−1 1 T N−1 ˆ Q −1 11 ˆ Q −1 22 +1 T N−1 ˆ Q −1 11 1 N−1 ˆ Q −1 11 1 N−1 ˆ Q −1 22 ˆ Q −1 22 +1 T N−1 ˆ Q −1 11 1 N−1 1 T N−1 ˆ Q −1 11 ˆ Q −1 22 ˆ Q −1 22 +1 T N−1 ˆ Q −1 11 1 N−1 ˆ Q −2 22 ˆ Q −1 22 +1 T N−1 ˆ Q −1 11 1 N−1    (B.18) 121 FromEqn. (B.13),(B.17),and(B.18),wehave ´ P T ˆ C † ´ P =H (B.19) andusingEqn. (B.3)and(B.16)gives ˆ θ WD,rs,krsk 2 = ˆ θ WOS . (B.20) Therefore,weconcludethatWDandWOSyieldthesamelocationestimate. 122 AppendixC ThisappendixderivestheCram´ er-RaoBound(CRB)forenergybasedlocalization. The assumptionofsufficientlylargenumberofsamplesgivesy∼ N(μ,C)(seeSection4.2) wherey = [y 1 ,...,y N ] T ,μ = [μ 1 ,...,μ N ] T , andC is the covariance matrix whose elementsaredeterminedinEqn. (4.4). Notethat L X n=1 L X m=1 R 2 s ((n−m)/f s ) =Lσ 4 s +ν where ν = L X n=1 L X m=1 R 2 s ((n−m)/f s ), n6=m. (C.1) Thus,CisgivenbyΦ+γDD T where Φ = 2σ 2 w L diag 2μ 1 −σ 2 w ,...,2μ N −σ 2 w , γ = 2(Lσ 4 s +ν) L 2 σ 4 s , D = [μ 1 −σ 2 w ,...,μ N −σ 2 w ] T . Lettingϑ = [x s ,y s ,σ 2 s ,ν] T ,theFisherinformationmatrixisgivenby[Kay93] [F] ij = ∂μ ∂ϑ i T C −1 ∂μ ∂ϑ j + 1 2 tr C −1 ∂C ∂ϑ i C −1 ∂C ∂ϑ j (C.2) 123 where ∂μ ∂ϑ 1 = ∂μ ∂x s = [ 2σ 2 s (x 1 −x s ) kr s −r 1 k 4 ,..., 2σ 2 s (x N −x s ) kr s −r N k 4 ] T ∂μ ∂ϑ 2 = ∂μ ∂y s = [ 2σ 2 s (y 1 −y s ) kr s −r 1 k 4 ,..., 2σ 2 s (y N −y s ) kr s −r N k 4 ] T ∂μ ∂ϑ 3 = ∂μ ∂σ 2 s = [ 1 kr s −r 1 k 2 ,..., 1 kr s −r N k 2 ] T ∂μ ∂ϑ 4 = ∂μ ∂ν = [0,...,0] T ∂C ∂ϑ 1 = ∂C ∂x s = 4σ 2 w L diag ∂μ ∂x s +γ ∂μ ∂x s D T +D ∂μ ∂x s T ∂C ∂ϑ 2 = ∂C ∂y s = 4σ 2 w L diag ∂μ ∂y s +γ ∂μ ∂y s D T +D ∂μ ∂y s T ∂C ∂ϑ 3 = ∂C ∂σ 2 s = 4σ 2 w L diag ∂μ ∂σ 2 s − 4ν L 2 σ 6 s DD T ∂C ∂ϑ 4 = ∂C ∂ν = 2 L 2 σ 4 s DD T . (C.3) ByinsertingEqn. (C.3)inEqn. (C.2),FcanbeobtainedandtheCRBisgivenbyF −1 . 124 AppendixD Thisappendixshowsthatwhenthesourceiswhite,WDCattainstheCRB.Thenotations arethesameasinAppendixDunlessdefined. Whenthesourceisassumedtobewhite, ν = 0, theunknown parameters isnow definedas ´ ϑ = [x s ,y s ,σ 2 s ] T , andthe covariance matrix,K, becomes 2 L diag{μ 2 1 ,...,μ 2 N }, ∂K ∂ ´ ϑ i = diag{ 4μ L ∂μ ∂ ´ ϑ i } where the symbol representsSchur(element-by-element)product. TheCRBis,thus,givenby F −1 = (1+ 4 L ) −1 (H T K −1 H) −1 (D.1) where H =      2σ 2 s (r 1 −rs) T krs−r 1 k 4 1 krs−r 1 k 2 . . . . . . 2σ 2 s (r N −rs) T krs−r N k 4 1 krs−r N k 2      . (D.2) To derive the covariance matrix of the WDC solution, we first consider the covariance matrixoftheestimation ˆ θ 2 inEqn. (4.33)whichis Σ = (J T Ψ −1 J) −1 . (D.3) Let the elements of ˆ θ WDC be in the formx s +e x ,y s +e y , andσ 2 s +e σ . The estimates oftheelementsof ˆ θ 2 inEqn. (4.33)canbegivenby ˆ x 2 s = (x s +e x ) 2 ' x s 2 +2x s e x (D.4) ˆ y 2 s = (y s +e y ) 2 ' y s 2 +2y s e y (D.5) ˆ σ 2 s = σ 2 s +e σ . (D.6) 125 The approximation is valid when e x and e y are small. Therefore, [e x e y e ] T = ´ B −1 Δθ 2 where Δθ 2 = ˆ θ 2 −θ 2 and ´ B = diag{2x s ,2y s ,1}. The covariance matrix of ˆ θ WDC is, thus, found to beC WDC = ´ B −1 Σ ´ B −1 where Σ is defined in Eqn. (D.3). SubstitutingΣyields C WDC = ( ´ BJ T B −1 ¯ A T C −1 ε ¯ AB −1 J ´ B) −1 (D.7) In the white source case,C ε becomesDKD whereD = diag{kr s −r 1 k 2 ,...,kr s − r N k 2 }(seeSection4.5). SubstitutingC ε inEqn. (D.7)yields C WDC = ( ´ BJ T B −1 ¯ A T D −1 K −1 D −1 ¯ AB −1 J ´ B) −1 (D.8) Multiplying the matrices gives D −1 ¯ AB −1 J ´ B = −H. Therefore, C WDC = (H T K −1 H) −1 'F −1 whenLisassumedtobesufficientlylarge. 126 AppendixE ThisappendixshowsthatthesecondterminEqn. (5.14)issmallcomparedwiththefirst termwhenthenumberofsamplesislarge. σ 2 i inEqn. (5.5)canberewrittenasfollows. σ 2 i = 4σ 2 w L e μ i − σ 2 w 2 . (E.1) Thus, ∂σ 2 i ∂θ {j,k} = 4σ 2 w L e ∂μ i ∂θ {j,k} (E.2) SubstitutingEqn. (E.2)in ∂Q ∂θ {j,k} yields 1 2 tr Q −1 ∂Q ∂θ j T Q −1 ∂Q ∂θ k = 4 L e " diag 2μ σ 2 w −I −1 ∂μ ∂θ j # T Q −1 ∂μ ∂θ k (E.3) It is simple to show that the diagonal element of diag n 2μ σ 2 w o −I −1 is less than 1. Thus, 1 2 tr Q −1 ∂Q ∂θ j T Q −1 ∂Q ∂θ k < 4 L e ∂μ ∂θ i T Q −1 ∂μ ∂θ j (E.4) ,and thus 1 2 tr h Q −1∂Q ∂θ j T Q −1 ∂Q ∂θ k i is small compared with 4 Le h ∂μ ∂θ i i T Q −1 h ∂μ ∂θ j i when L e islarge. 127 AppendixF This appendix shows an example of how to approximate the limit of the terms in Eqn. (5.27). Weselectoneterm, P Ne i=5 cos 2 θ i ¯ d 4 i ,toshowthestepsindetailwhiletheotherterms canusesimilarsteps. lim Nt→∞ Ne X i=5 cos 2 θ i ¯ d 4 i ' Z 2π 0 Z ∞ R 0 cos 2 θ r 4 rdrdθ = − 1 2r 2 ∞ R 0 × θ 2 + sin2θ 4 2π 0 = π 2R 2 0 128

Abstract (if available)

Abstract Our goal is to improve the performance of acoustic source localization in the context of sensor networks. A number of relevant problems are addressed and solutions are proposed.

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Asset Metadata

Creator Meesookho, Chartchai (author)

Core Title Robust acoustic source localization in sensor networks

School Viterbi School of Engineering

Degree Doctor of Philosophy

Degree Program Electrical Engineering

Publication Date 07/29/2007

Defense Date 06/19/2007

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag acoustic,localization,networks,OAI-PMH Harvest,sensor

Language English

Advisor Mitra, Urbashi (committee chair), Narayanan, Shrikanth S. (committee chair), Sukhatme, Gaurav S. (committee member)

Creator Email chartchai@gmail.com

Permanent Link (DOI) https://doi.org/10.25549/usctheses-m715

Unique identifier UC1467555

Identifier etd-Meesookho-20070729 (filename),usctheses-m40 (legacy collection record id),usctheses-c127-536859 (legacy record id),usctheses-m715 (legacy record id)

Legacy Identifier etd-Meesookho-20070729.pdf

Dmrecord 536859

Document Type Dissertation

Rights Meesookho, Chartchai

Type texts

Source University of Southern California (contributing entity), University of Southern California Dissertations and Theses (collection)

Repository Name Libraries, University of Southern California

Repository Location Los Angeles, California

Repository Email cisadmin@lib.usc.edu