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Time-Based Sampling using the Integrate and Fire Model

Permanent Link: http://ufdc.ufl.edu/UFE0043614/00001

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Title: Time-Based Sampling using the Integrate and Fire Model
Physical Description: 1 online resource (121 p.)
Language: english
Creator: Singh Alvarado, Alexander
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2012

Subjects

Subjects / Keywords: fire
Electrical and Computer Engineering -- Dissertations, Academic -- UF
Genre: Electrical and Computer Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

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Abstract: The digital era is founded on the fact that analog signals have an equivalent representation in the digital domain. Nevertheless, turning to biology there are no systems based on a binary logic. Taking the brain as an example, it is the agglomeration of billions of neurons in a web of trillions of synapses. Considering a signal neuron, its functionality is roughly approximated by assuming an integrate and fire model, although there are several more realistic models this one manages to capture the important nonlinear characteristics. Therefore the brain manages to encode the analog signals into a set of spikes where the information is carried only in the timing of these events. Inspired in the behavior of the brain we first present a simple neuron based sampler, the IF model. Using a single neuron we are able to show it's applicability in compression of neural and ECG signals. This dissertation presents a suite of recovery algorithms which include theoretical results and a set of practical tools that allow us to decode the original input. Furthermore, we present preliminary results toward the encoding of signal in multi-channel, multi-layered feedforward networks of IF. Which provides representation in which the events can on longer be described as linear constraints on the input.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Alexander Singh Alvarado.
Thesis: Thesis (Ph.D.)--University of Florida, 2012.
Local: Adviser: Principe, Jose C.

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Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2012
System ID: UFE0043614:00001

Permanent Link: http://ufdc.ufl.edu/UFE0043614/00001

Material Information

Title: Time-Based Sampling using the Integrate and Fire Model
Physical Description: 1 online resource (121 p.)
Language: english
Creator: Singh Alvarado, Alexander
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2012

Subjects

Subjects / Keywords: fire
Electrical and Computer Engineering -- Dissertations, Academic -- UF
Genre: Electrical and Computer Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: The digital era is founded on the fact that analog signals have an equivalent representation in the digital domain. Nevertheless, turning to biology there are no systems based on a binary logic. Taking the brain as an example, it is the agglomeration of billions of neurons in a web of trillions of synapses. Considering a signal neuron, its functionality is roughly approximated by assuming an integrate and fire model, although there are several more realistic models this one manages to capture the important nonlinear characteristics. Therefore the brain manages to encode the analog signals into a set of spikes where the information is carried only in the timing of these events. Inspired in the behavior of the brain we first present a simple neuron based sampler, the IF model. Using a single neuron we are able to show it's applicability in compression of neural and ECG signals. This dissertation presents a suite of recovery algorithms which include theoretical results and a set of practical tools that allow us to decode the original input. Furthermore, we present preliminary results toward the encoding of signal in multi-channel, multi-layered feedforward networks of IF. Which provides representation in which the events can on longer be described as linear constraints on the input.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Alexander Singh Alvarado.
Thesis: Thesis (Ph.D.)--University of Florida, 2012.
Local: Adviser: Principe, Jose C.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2012
System ID: UFE0043614:00001


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TIMEENCODEDCOMPRESSIONANDCLASSIFICATIONUSINGTHEINTE GRATE ANDFIRESAMPLER By ALEXANDERSINGHALVARADO ADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOL OFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENT OFTHEREQUIREMENTSFORTHEDEGREEOF DOCTOROFPHILOSOPHY UNIVERSITYOFFLORIDA 2012

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c r 2012AlexanderSinghAlvarado 2

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Tomyfamily 3

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ACKNOWLEDGMENTS IthankmyadviserDr.Jos eC.Prncipe,forgivingmetheopportunityandfunds topursuemyPhD.Ispeciallythankmycommitteemembers,who havebeenagreat inspiration.Dr.MuraliRao,helpedmethroughmyrststeps insamplingtheory. Dr.Entezariformallyintroducedmetotheproblemsininter polation,samplingand reconstruction.HehasbeeninstrumentalinmylearningofB -splinesandmultivariate splines.Dr.Banerjeewhowasalwaysoptimistic,providedg reatinsightintothe connectionsbetweenthisworkandtheresultsincomputatio nalneuroscience.Iwould alwaysleavetheirofcesmotivatedandreadytotacklethen extproblem.FinallyIthank Dr.Harris,whoguidedagreatpartofthisworkandwouldkeep megroundedtothe application. IwouldalsoliketothankDr.HansG.Feichtinger,whogaveme theopportunityto spendasummerwithhisgroupinViennaduringthesummerof20 09.Thiswasagreat opportunitytomeetsomeoftheleadingresearchersinthesa mplingeld.Furthermore, IhadthechancetoworkwithhisgroupspecicallyJoseLuisR omero,GinoVelascoand SaptarshiDastowhompartofthisworkisindebtto.Duringth isvisitIalsometoneof myfuturecollaborators,Dr.ChoudurLakshminarayan.Iowe himaspecialthanks,he hastaughtmythetruemeaningofpatienceandattentionto`e xcruciatingdetail',hehas beeninstrumentalinmylearningofstatisticsandplayedaf undamentalroleinthelater chaptersofthisdissertation. IwouldalsoliketothankmyundergraduateadviserDr.JoseP abloAlvarado,his enthusiasmanddrivearewhatinspiredmetopursuemyPhD.Ia mthankfultoDr. FrankBova,ifnotforhimIwouldhavenotcontinuedmyPhDinF lorida,hegaveme theopportunitytoworkathislabbeinganundergraduate.Du ringthistimeIrealizedI wantedtocontinueinterdisciplinaryresearchinthemedic aleld. Thisgroupofmentorshaveplayedafundamentalroleinthede velopmentofmy researchandduringmytimewiththemIalsohadtheopportuni tyofworkingwithagreat 4

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setoffriendsandcolleagueswhohavebeenwithmethroughth eyears.Iwouldlike tothankAtcharSudhyathom,SureshMalakar,RaulRiveros,M emmingPark,Shalom Darmanjian,PuskalPokharel,WeifengLiu,ErionHasanbell iou,SohanSeth,Luis SanchezGiraldo,AbishekSingh,ManuRastogi,VaibhavGarg ,RaviShekhar,XieXu, HectorGalloza,RakeshChalsani,EvanKriminger,AustinBr ockmeir,GoktugCinar, LinLi,KittipatKampa,SavyasachiSingh,StefanCraciun,B ilalFadlallahandAysegul Gunduz. Ialsothankmyfamilyawayfromhome,AuntiePramila,UncleS ubarna,AshaMaya, SheelaMaya,AuntieRadhaandUncleTirthawhoalwaysmadesu remytummywasfull andwerealwaysthereforme.Finally,Ithankmyfamilywhoga vemetheirunconditional supportthroughtheyears,everythingseemedsosimpleafte rtalkingtothem.Iam thankfultoDivyaAgrawal,whohasbeenbymysideforeverysi nglestepinthisjourney andwhowillwalkwithmefortherest. 5

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TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................. 4 LISTOFTABLES ...................................... 8 LISTOFFIGURES ..................................... 9 ABSTRACT ......................................... 11 CHAPTER 1INTRODUCTION ................................... 12 1.1NeuralRecordings ............................... 15 1.2ECGHeartbeats ................................ 16 1.3DissertationOutline .............................. 17 2THEORETICALBACKGROUND .......................... 18 2.1RepresentationSetsinEngineering ..................... 19 2.2DenitionofSampling ............................. 20 2.3UniformSampling ............................... 21 2.4NonuniformSampling ............................. 23 2.4.1DufnandSchaeffer'sWork ...................... 24 2.4.2AbstractFrames ............................ 27 2.4.3AnIterativeReconstructionAlgorithmUsingtheNeum annSeries 28 2.5ReconstructionfromLocalAverages:InniteDimension s ......... 31 2.5.1BandlimitedSpace ........................... 31 2.5.2ShiftInvariantSpace .......................... 33 2.6ReconstructionfromLocalAverages:FiniteDimensions .......... 37 2.6.1FourierSeries .............................. 38 2.6.2SplineSpaces .............................. 42 3THEINTEGRATEANDFIRESAMPLER ...................... 46 3.1EffectoftheIFparameters .......................... 46 3.2SimilarSamplingSchemes .......................... 48 3.2.1TimeEncodingMachine ........................ 48 3.2.2SigmaDeltaConverter ......................... 49 4RECONSTRUCTIONALGORITHMS:INTEGRATEANDFIRESAMPLER ... 53 4.1ApproximateReconstructioninBandlimitedSpaces ............ 54 4.1.1SomeRemarksontheIFOutput ................... 55 4.1.2Reconstruction ............................. 57 4.1.3NumericalExperiments ........................ 61 4.2ReconstructioninFiniteDimensionalSpaces ................ 63 6

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4.3ConstrainedB-splines ............................. 66 5APPLICATION:RECOVERYOFNEURLRECORDINGS ............ 71 5.1CurrentMicro-ElectrodeRecordingSystems ................ 72 5.2DataDescription:NeuralRecordings ..................... 73 5.2.1SpikeSorting .............................. 75 5.3TimeBasedEncodingofNeuralRecordings ................. 76 5.3.1SamplerandInputEffectsonReconstructionAccuracy ....... 78 5.3.2ProcessingTimes ............................ 79 5.3.3SensitivitytoTimeQuantization .................... 80 5.3.4EffectofthePulseRatesontheClassicationError ......... 81 5.4PulseBasedDiscriminationofNeuralActionPotentials .......... 83 5.4.1RelatedWork .............................. 85 5.4.2PulsetrainClassication ........................ 86 5.4.3ResultsandDiscussion ........................ 87 6APPLICATION2:ELECTROCARDIOGRAMRECORDINGS .......... 91 6.1HeartbeatClassication ............................ 93 6.2DataDescription ................................ 95 6.3PulseBasedClassication .......................... 99 6.3.1PulsePre-Processing ......................... 99 6.3.2Classier ................................ 100 6.4PerformanceMetrics .............................. 103 6.5ResultsandDiscussion ............................ 103 7CONCLUSIONSANDFUTUREWORK ...................... 109 REFERENCES ....................................... 111 BIOGRAPHICALSKETCH ................................ 121 7

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LISTOFTABLES Table page 5-1Confusionmatrixbetweenthesortsobtainedfromtheori ginaluniformlysampled neuraldataandthereconstructedsignalatdifferentavera gesamplerates, 19.784Ksamples/s( =1.97 ),9.605Ksamples/s( =0.96 ),5.430Ksamples/s ( =0.54 )and1.281Ksamples/s( =0.12 ).Therowscorrespondtothe sortsonthereconstructeddataandthecolumnstotheorigin aldata.Theaction potentialsmissedintheoriginalrecordingaremarkedunde rM.O.,likewise forthereconstructedaredenotedasM.R.Thisisrepeatedfo reachrecovery method. ........................................ 84 6-1HeartbeatclassesgivenbytheMIT-BIHdatabasealongwi ththeregrouping denedbytheAAMIstandard. ........................... 97 6-2Classicationresultsusingthebinnedpulsefeatures( 20bins)foreachofthe testpatients.Column1referstotherecordnumberasassign edintheMIT-BIH database,columns2-6representthenumberofbeatsperclas s,columns7-12 representtheperformancemetricsandcolumn13showstheav eragesample ratefromthetime-encodedrecord.Theaverageresultsfora llmetricsalong allpatientsarepresentedinTable6-3astherowinbold. ............ 107 6-3Averageclassicationmetricscomparedtothestateoft heart.Thesample ratesintheproposedmethodistheaverageoverallpatients .......... 108 8

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LISTOFFIGURES Figure page 3-1Integrateandreblockdiagram. .......................... 46 3-2Frequencyresponseofthelterfordifferentvaluesof normalizedtounit gainat( f =0 ).Thespectrumoftheoriginalsignalisshowninblack. ...... 47 3-3Integralfunction v fordifferentvaluesof .Eachsignalhasbeennormalized tohavethesamepower. ............................... 48 4-1Reconstructionkernels. ............................... 62 4-2Reconstructionresults. ............................... 62 4-3Variationoftheerrordenedas jj f b f jj 1 ..................... 63 4-4DifferenttypesoflinearsleevesforB-splines. ................... 68 4-5Exponentialchannelandlinearapproximation. .................. 69 4-6Reconstructionexampleusingneuraldata. .................... 70 5-1Simulatedintra-cellularactionpotentialgeneratedb ytheHogkin-Huxelymodeled (Toppanel).Bottompanelcorrespondstothenormalizedsec ondderivative oftheactionpotential,whichmatchesthebasiccablemodel .Comparingthe bottompanelwithFigure5-2Aitisevidentthetheshapesare similar. ..... 74 5-2Examplesoflteredneuralrecordings.Theredlinesden otetheactionpotential regionsinthetimedomain.Thebandwidthoftheneuralsigna listypicallyassumed tobe5KHz. ...................................... 75 5-3Overlaidactionpotentialsforeachoftheneuronclasse s.Themeanisshown asathickblacklineineachcase. .......................... 76 5-4Effectofthesamplerparametersandtheinputpoweronth epulseratevariation intheactionpotentialregions.Eachcolumnrepresentsadi fferentvalueof theleakfactor ,rowscorrespondtovariationintherefractory(fromtopto bottom, =0 s =10 s =20 s ). ........................ 77 5-5Effectofthesamplerparametersandtheinputpoweronth epulseratevariation inthenoiseregions.Eachcolumnrepresentsadifferentval ueoftheleakfactor ,rowscorrespondtovariationintherefractory(fromtopto bottom, =0 s = 10 s =20 s ). .................................... 78 5-6Effectsofthesamplerparametersandtheactionpotenti alinputpoweronthe averagereconstructionaccuracyactionpotentialsfromne uronclass1(Number ofactionpotentialis466). .............................. 80 9

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5-7Reconstructiontimevsthewindowlengthfordifferentn euronclassesand sampleratesachievedbyvaryingthepower,fromtoptodownt hepowervaries as 3 e 9,1 e 9,2 e 8 .Thepoweriscalculatedonlyovertheactionpotentialrang e. Alsothesamplerparametersarexedsuchthat =10 s 1 =0.2 ms .... 81 5-8Effectsofthetimequantizationontheaverageactionpo tentialSERforneuron 1.Allquantizedpulsetimingsareanintegermultipleofthe clockperiod.A 50 ms windowisreconstructed,neverthelesstheaveragepulsera tesandSER valuesareestaimtedovera 2 ms windowaroundtheactionpotentials.The rowscorrespondtodifferentrefractoryperiodsfromtopto bottom =0 s =10 s =20 s ,whilexing 1 =0.2 ms .Thepoweroftheinput increasesinthecolumnsfromlefttoright.Theeffectsofth isincreaseappear intheaveragesamplingratedenotedforeachplot. ................ 82 5-9Overlaidactionpotentialsforeachneuronorclassingr ayalongwiththeirmean inblack. ........................................ 87 5-11LDAprojection:uniformsamplesasfeatures. ................... 89 5-12ROCcurves:ForbothfeaturesIFanduniform. .................. 89 6-1ExampleofheartbeatshapesfromtheMIT-BIHdataset.Ea chcolumnrepresents apatientandeachrowthebeatsforthatspecicclass(using theclassnumbering fromtable6-1).Notethevariationsinthebeatmorphologya crosspatientsas wellaswithinapatient. ............................... 94 6-2ExampleofheartbeatshapesfromtheMIT-BIHdatasetfor patient100.The toppanelshowsboththerawdataandtheprocesseddatafollo wingthelter proposedindeChazalandReilly(2006).Thebottompanelsho wsthepulses generatedbytheIFsampler,theaveragepulseratealongthe entirerecording is 40.59 pulses/second. ............................... 98 10

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AbstractofdissertationPresentedtotheGraduateSchool oftheUniversityofFloridainPartialFulllmentofthe RequirementsfortheDegreeofDoctorofPhilosophy TIMEENCODEDCOMPRESSIONANDCLASSIFICATIONUSINGTHEINTE GRATE ANDFIRESAMPLER By AlexanderSinghAlvarado May2012 Chair:Jos eC.Prncipe Major:ElectricalandComputerEngineering Thisdissertationprovidesasamplingandreconstructionf rameworkforthe IntegrateandFire(IF)model.Theencodingschemetransfor msthecontinuoustime inputintoasetofpreciselytimedevents.Similartime-bas edsamplingschemes arewellknownforthesimplicityoftheirhardwareimplemen tation.Inpractice,the samplersencodetheinputintherateoftheevents,wellabov etheNyquistbound. Incontrast,weproposetousetheIFsampleratsub-Nyquistr ates.Weprovidean approximatereconstructionalgorithmforthecaseofbandl imitedfunctionsalongwith thecorrespondingerrorbounds.Furthermore,perfectreco veryisattainablefornite dimensionalspacesunderspecicsampleconstraints.Thes ereconstructionalgorithms differintheirassumedinputmodel,theirreconstructiona ccuracyandtheirspeed.In ordertomeasuretheperformanceofthesamplingscheme,weu setheIFtoencode neuralrecordingsusedinBrainMachineInterfaces(BMIs). Furthermore,sincethe samplesareinputdependent,compressionwilldependonthe characteristicsofthe inputclass.Wepresentresultsonreconstructionaccuracy andbothdetectionand clusteringofneuralsignalsinrelationtothevariationsi nthesamplerparametersand inputfeatures.Furthermore,insomeapplicationsreconst ructionmaybeavoidedand wecanworkdirectlywiththeoutputsampledistributions.W eshowthisbyproviding compressionandtime-basedclassicationforwirelessele ctrocardiograms(ECG). 11

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CHAPTER1 INTRODUCTION TheShannonWhittaker-Kotelnikovsamplingtheoremisatth eheartoftheeld ofdigitalsignalprocessing Unser ( 2000 ).Itprovidesarepresentationofbandlimited functionsintermsofuniformlydistributeddiscretesampl esandadictionaryofbasis elements.ForsampledensitiesboundedbytheNyquistrate, aonetoonemappingwith astableinversebetweenthesamplesandthecontinuousfunc tionexists.Furthermore, manipulationsonthecontinuousdomaincanbetranslateddi rectlyontothesamples,an advantageinherenttothespecicdictionarychosen. Ingeneralanysamplingandreconstructionframeworkcanbe denedbythree basicblocks.Therstdenestheinputclass,thesecondde nesthenotionofasample (e.g.pointevaluations),andthethirdconsistsofastable reconstructionalgorithmwhich dependsontheprevioustwoblocks.Inthetypicalcaseofban dlimitedfunctions,the inputisassumedtohaveacompactlysupportedFouriertrans formandsamplesare denedbyfunctionevaluationsonanequallyspacedgridove rthedomain.Inthis case,itcanbeshownthattheinputspaceisspannedbyshifte dsincfunctions.Since thebasisareorthonormal,astablereconstructionisattai nedbyalinearcombination ofthebasisfunctionswhoseweightsaregivenbythesamplev alues.Therehave beenanumberofgeneralizationstothissamplingtheorem,i nresponsetoitspractical drawbacks,specicallybecauseoftheinnite,non-causal supportofthebasisandtheir slowdecay.Therearetwomaintrendsintheliterature,the rstimposesstructure apriori onthespacee.g.shift-invariantspaces AldroubiandGr ¨ ochenig ( 2001 )andsparse representation Cand esetal. ( 2006 ).Thesecondlearnsthedictionarydirectlyfromthe datae.g.radialbasisfunctions Wendland ( 2005 ).Bothofthesecanbetreatedunder adeterministicorstochasticframework.Mostofthecurren tadvancesinsamplingare concernedwiththerstcase.Speciallywiththeformalizat ionofcompressivesensing Cand esetal. ( 2006 ).Inthiscase,theconstraintsontheinputhavemovedawayf rom 12

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standardinputvectorspacesandinsteadassumeaninputcla sswhichissparse.A k-sparse vectorisdenedasoneforwhichonly k ofitselementsarenon-zero.The samplesarestillconsideredasprojections,althoughther ecoveryalgorithmsdepend onconstrained L 1 L 0 optimizationstoenforcesparsesolutions.Afurtherabstr action onsparsityisgivenbytheworkonniterateofinnovation Vetterlietal. ( 2002 ),where theinputisassumedtohaveonlyafewdegreesoffreedom.Int hiscase,therecovery algorithmrstdeterminesthelocationoftheunknownspars eelements,reducingthe problemtoastandardleastsquares.Thesetypeofapproache shavebeenwellstudied inspectralestimationandarebasedonProny'smethod deProny ( 1795 ). Thesamplingframeworkalsoextendstorandomprocesswhere thereconstruction algorithmsarebasedonestimatesofthepowerspectraldens ity.Incontrasttothe previousapproaches,thesearenotconcernedwiththerecon structionofaspecic realizationoftheprocessbutratherspecicfeaturesthat deneit(powerspectral density).Intheliterature,thissamplingframeworkiskno wnasalias-freesampling ofstochasticsignals Beutler ( 1966 ); Bilinskis ( 2007 ); Masry ( 1978 ); Shapiroand Silverman ( 1960 ).Traditionally,randomnonuniformsamplesetshavebeenp referred overtheuniformsamplesetssincetheyprovidealias-freer epresentationsevenat averagesub-Nyquistrates Bilinskis ( 2007 ).Specicallyithasbeenshownthatsample locationsdenedbyahomogenousPoissonpointprocess DaleyandVere-Jones ( 2008 ) provideanalias-freerepresentationevenfornon-bandlim itedpowerspectrums Masry ( 1978 ).Howevertheseassumptionsleadstoimpracticalconditio nsonthesampleset, specicallysincetheminimumdistancebetweenthesamples isunconstrained,leading toinconsistentestimatesofthepowerspectraldensity. Thesamplingschemesconsidereduntilnowencodetheinputi ntherangeofthe discretesamples.Nevertheless,therearesamplingimplem entationsthatalsousethe domain.Time-basedencodershavebeenstudiedindetailint hesamplingliterature Bilinskis ( 2007 ).Someofthetypicalschemesincludezerocrossingsampler s Marvasti 13

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( 1987 ),levelcrossing Allieretal. ( 2005 ); AstromandBernhardsson ( 2002 ); Guan andSinger ( 2006 ),referencecrossingsamplers Bilinskis ( 2007 ),delta-modulators Inoseetal. ( 1966 )and modulators Inoseetal. ( 1962 ),includingtheirdiscrete, continuousandasynchronous Ouzounovetal. ( 2006 ); Roza ( 1997a )implementations. Thetraditionalmotivationforusingtime-basedencodings chemesreliesonthe simplicityoftheirhardwareimplementation.Nevertheles s,asinthecaseof modulatorstheinputisencodedintheoutputrates.Through outthisdissertation,we proposetheuseoftime-basedencodersassub-Nyquistsampl ersandshowthatfor certaininputclassesthisprovidesbothanaccuraterepres entationoftheregionsof interestandoverallcompression. Specically,weconsidertheintegrateandre(IF)sampler .TheIFmodeliswell knowninthecomputationalneurosciencecommunity,specia llyinthestudyofneural networks GerstnerandKistler ( 2002 ).TheIFwasoneoftherstmodelsforbiological neuronsbasedonLapique'sworkin1920,althoughtheactual name`integrateandre' wascoinedlateron BrunelandvanRossum ( 2007 ).Nevertheless,itsuseasasampling frameworkhasonlyrecentlybeeninvestigated Alvaradoetal. ( 2009 ); Lazar ( 2004 ); Wei andHarris ( 2004 ). Becauseofthenonlinearbehaviorofthesampler,thestanda rdsamplingframework isnolongerapplicablei.e.itisnotpossibletodeneavect orspaceforwhichevery functioncanberepresentedbyitsIFsamples.Theconventio nalbandlimitedspace isnolongersuitable,sincewecanprovethatrecoveryinthi sspaceisnotpossible. Nevertheless,wecanboundtheapproximationerror.Itcana lsobeshownthat thisencodingisinvertibleandrecoveryoftheoriginalfun ctionispossiblefornite dimensionalspaces. Furthermore,theoutputpulsesfromtheIFcanbetreatedasa spiketrainand wecanrelyonanumberofmethodsfromtheneuraldecodinglit erature.Oneofthe standardmethodsisproposedin Riekeetal. ( 1997 ),whichusesalinearltertosmooth 14

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theoutputpulsetrainsimilartotheradialbasisapproachw ithxedweights.This approachisaccurateonlywhentheIFisusedasarateencoder ,analogoustothe caseofthe modulators.Otherreconstructionmethodsrelyonstochast icmodels fortheobservedsamples Gerwinnetal. ( 2011 ); Pillowetal. ( 2004 )andtherefore implyarandominput.Nevertheless,thesemethodsfallback tothestandardleast squaresolutionspreviouslyproposedundercertainassump tionsortheybecome computationallyandanalyticallyintractableforreal-ti meapplications. Thesetime-basedencodingschemeshaveallowedustotradeo ffrelativelysimple hardwareimplementationformoreelaboratereconstructio nalgorithms,incontrastwith traditionalanalogtodigitalconverters.Ironically,inm ostapplicationthestricthardware constraintsaretypicallyimposedatthesamplingsite,whi lemostoftheresourcesare locatedatthereceiverend.Suchisthecaseforwirelessneu ralrecordingdevicesused inBrainMachineinterfaces(BMI),aswellasheartbeatreco rdingsmonitoredthrough wirelessECGdevices. 1.1NeuralRecordings Themotivatingapplicationforthisworkstemsfromthechal lengesinBMI,wherean efcientencodingofneuralrecordingsisneeded Bergeretal. ( 2008 ).Inthisscenario thesamplinghardwareisconstrainedbypower,area,andban dwidth.Therefore,each sampleshouldbecarefullychosen.Thiscontrastswithconv entionalsamplingschemes inwhichthemeasurementsandthefeatureextractionstages aretreatedseparately. ThemainobjectiveofBMIsistogivemeaningtobrainactivit y.Therearedifferent waystointerfacewiththebrain,inthisworkweconsidermic ro-electroderecordings. Thesemicro-electrodearraysareimplantedinspecicstru cturesofthebraindepending ontheBMItask Bashirullahetal. ( 2007 ).Thesignalsofinterestaregeneratedbythe neuronsandareknownasactionpotentialsorspikes.Theseh aveastereotypicalshape forallneuronsandcanbethoughtofasashort1msburstinamp litudeascoveredin chapter 5 15

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CurrenttrendsinBMIsaremovingtowardwirelesssolutions .Afullyimplantable devicewouldgivethesubjectfreedomtoperformthenecessa rytasksandalsoreduce thenoiseintroducedbythewiringandmovements.Incontras ttothetetheredsolution, strictconstraintsareimposedonthehardwaredesign.Stan darduniformsamplers andcompressionalgorithmsarenolongersuitableforthisa pplicationsgiventhetight constraints.Foran8channelsystemat20Ksamp/secsamplin grate,witha2byte resolution,thedataratesareabove2Mbits/sec.Thisisabo vethehardwarelimitations giventheareaandpowerspecications.Thereforeafundame ntallydifferentapproachis required. Fortunately,neuralsignalshavesparsefeatureswhichcan beexploitedto reducethenecessarymeasurementsforareliabledetection andclusteringofthe actionpotentials.Hereweevaluatetheperformanceinterm sofcompressionand reconstructionoftheactionpotentials.Thesealgorithms differinaccuracyandspeed, crucialinthedesignofreal-timeneuro-prosthetics. 1.2ECGHeartbeats Thesecondapplicationwepresentinthisdissertationavoi dstheneedfor reconstructionanddirectlytriestoextractdiscriminati vefeaturesinthesampleddomain forawirelesselectrocardiogram(ECG)application. HeartfunctionmeasuredbyECGiscrucialforpatientcare.E CGgenerated waveformsareusedtondpatternsofirregularitiesincard iaccyclesinpatients.In manycases,irregularitiesevolveoveranextendedperiodo ftimewhichrequires continuousmonitoring.However,thisrequireswirelessEC Grecordingdevices.These devicesconsistofanenclosedsystemwhichincludeselectr odes,processingcircuitry andawirelesscommunicationblockimposingconstraintson area,power,bandwidth andresolution.Inordertoprovidecontinuousmonitoringo fcardiacfunctionsfor realtimediagnostics,weproposeamethodologythatcombin escompressionand analysisofheartbeats.Thesignalencodingschemeistheti me-basedIntegrateand 16

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Fire(IF)sampler.Thediagnosticscanbeperformeddirectl yonthesamplesavoiding reconstructionrequiredbythecompetingniterateofinno vation(FRI)andcompressed sensing(CS). Asanaddedbenet,theIFschemeprovidesanefcienthardwa reimplementation andacompressedrepresentationfortheECGrecordings,whi lestillpreserving discriminativefeatures.Wedemonstratetheperformanceo fourapproachthrougha heartbeatclassicationapplicationconsistingofnormal andirregularheartbeatsknown asarrhythmia.Ourapproachwhichusessimplefeaturesextr actedfromECGsignalsis comparabletoresultsinthepublishedliterature deChazalandReilly ( 2006 ). 1.3DissertationOutline Chapter2coversthebasictoolsusedininthenonuniformsam plingliterature, specicallythedesignofrecoveryalgorithms.Chapter3in troducestheIFmodeland describesitsparametersandtheireffectsonthesampledis tribution.Chapter4frames thereconstructionintermsofniteandinnitedimensiona lspaces.Specicallyfor bandlimitedspaces,itisshownthattheconventionalShann onsamplingframework isnotdirectlyapplicabletotheIFsampler.Nevertheless, weprovideanapproximate reconstructionalgorithmwithitscorrespondingerrorbou nds.Finitedimensionalmodels ontheotherhandprovideperfectrecoveryunderspecicsam pleconstraints.Here weconsiderthreedifferentmodel,splinespaces,radialba sisfunctionsandaFourier model.Chapter5usestheIFtoencodeneuralrecordings,the performanceinthiscase ismeasuredintermsofthereconstructionaccuracyaswella stheclassicationerror. Chapter6appliestheIFtocompressheartbeatsrecordedbyE CGs.Theperformance oftheencodingismeasuredintermsoftheclassicationerr orandcomparedto standardfeatures.Chapter7summarizesthedissertationi nasetofconclusionsand recommendationsforfuturework. 17

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CHAPTER2 THEORETICALBACKGROUND Typicalengineeringapplicationsassumethesignalsofint erestbelongtoaknown vectorspace.Constraintsareimposedbydeningrulessuch asinnerproducts,norms, andothers.Theseprovidestructureonourspacewhichsimpl iesthemathematical analysisofthesefunctions.Acrucialpropertyistheabili tytobreakdownafunctioninto basicbuildingblocks.Thisleadstotheconceptofabasis. Givenavectorspacewecanextractasubsetoflinearlyindep endentelements thatspantheentirespace;thissetisknownasaHamelbasisi nthecaseofaBanach space.Theexistenceofthebasisdependsontheaxiomofchoi ceandwillnotbe coveredindetailhere.Amoreusefulnotionofbasisforinn itedimensionalspaceswas introducedbySchauder. Denition1. Asequenceofvectors f x 1 x 2 ,... g inaninnitedimensionalBanachspace X issaidtobeaSchauderbasisfor X iftoeachvector x inthespacetherecorresponds auniquesequenceofscalars f c 1 c 2 ,... g suchthat x = 1 X n =1 c n x n (2–1) Theconvergenceoftheseriesisunderstoodtobewithrespec ttothestrong(norm) topologyof X :inotherwords, k x n X i =1 c i x i k! 0 (2–2) Althoughtheexistenceofthebasiscanbeproveditisthestr uctureofthespace thatwilldenethemannerinwhichthecoefcientscanbedet ermined.Orthonormal basisispreferred,sincethecoefcientsreducetoinnerpr oductsbetweenthevector andthebasiselements.Basiselementsthatarenotorthonor malrequireother procedurestodeterminethecoefcients. 18

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2.1RepresentationSetsinEngineering Inpracticethechoiceofanappropriatebasisisnottrivial andmustbejustied bythesignalcharacteristics.Differentexamplesofbasis setsappliedinengineering andthatwillbeusedtomodelourspecicapplicationaresum marizedinthefollowing section. Fourierseries: Thisisanorthonormalbasis,whichconvergesinthemeansqu are senseto f if f 2 L 2 (0,1) ,thismeansthat jj f ( t ) P 1n = 1 a n e i 2 nt jj 2 tendsto 0 as n !1 .Becauseinpracticewecanonlydealwithnitelengthsigna lsthisapproachis theoreticallysound,althoughitconstrainsthereconstru ctionto C 1 (0,1) whichmaynot bethecase. Sincbasis: Thetypicalassumptioninengineeringwhenitcomestounifo rm samplingisthatthefunctionisbandlimitedandthereforer econstructionispossibleby theShannonsamplingtheorem.Thistheoremindirectlystat esthatthesincfunctions areanorthonormalbasisforthisspace. Shiftinvariantspaces: Thesecorrespondtoageneralizationofthebandlimited space.Theyallowforanensembleofgeneratingfunctions.E lementsinthisspaceare givenby: f ( t )= M X j =1 1 X k = 1 c k j k ( t Tk ) (2–3) where T istheunderlyingperiod.Althoughthisbasisallowsthecho iceofthegenerators thedenitionofthesamplingprobleminthisspacebecomesd ifcult,sincethe conditionsonthesamplingsetinordertoensureperfectrec overyarestillunknown totheauthorunlessthegeneratorsareconstrainedtospeci cfunctions. Thereareseveralotherbasissetswedidnotcover.Thewavel etbasisspansa spacebytranslationsandmodulationsofaspecicgenerato r.Polynomialbasissuchas Legendremayalsobeusefulsincepolynomialsaredenseinth espaceofcontinuous functionsdenedoveraninterval.Thedrawbackofpolynomi alspacesisthedecrease innumericalstabilityasthedegreeofthepolynomialincre ase.Animprovementonthe 19

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polynomialbasisisgivenbythepiecewisepolynomialssuch asthosespannedbythe splinebasis. 2.2DenitionofSampling SamplinginengineeringisassociatedwithShannon'ssampl ingtheorem,butwe willnotconstrainourselvestothisnotion.Theprocessofs amplingingeneralcanbe consideredastheencodingofafunctionintoasetofknownat tributesthataresufcient torecovertheoriginalfunction.Thesecharacteristicsco uldrefertopointevaluations, projectionsontootherfunctionsorevensegmentsofthefun ction.Thereforeaprecise frameworkisrequiredunderwhichwecandenethesamplingp rocess. InthecaseofaHilbertspace H thesamplingprocesscanbedenedasalinear mappingfromafunctiontoasequenceofscalars.Thisproces sisinvertibleaslong asthereisaone-to-onemappingbetweenthefunctionandthe sequence.Although thiswouldguaranteeperfectrecoveryitdoesnotguarantee astablereconstruction algorithm.Thereforeforpracticalapplicationsitisnece ssarytodenestablesampling operators.FollowingLuandDo LuandDo ( 2008 ): Denition2. (Stablesampling):Wecall A astablesamplingoperatorfor X ifthere existsconstants 0 < < 1 suchthatforevery x 1 2 X x 2 2 X jj x 1 x 2 jj 2H jj Ax 1 Ax 2 jj 2l 2 jj x 1 x 2 jj 2H (2–4) Notethatwearecomparingthenormsfortwodistinctspaces. Thisrelationimplies thatiftwosamplesequencesareclosethentheircorrespond ingfunctionsarealso close.Intermsofreconstructionthisallowsustointroduc easmallerrorinthesamples whichwilltranslateintoasmallerrorintherecoveredfunc tions,otherwiseitwouldnot beapracticalrepresentation.Wecansimilarlydeneastab lesetofsampling Landau ( 1967 ),inthecaseofthebandlimitedspace: 20

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Denition3. (setofsampling)Auniformlydiscreteset isa setofsampling ifthere existsaconstant K suchthat: jj f jj 2 K X 2 j f ( ) j 2 (2–5) Thisimpliesthesamplescontainsufcientinformationabo utthefunctiontobound itsnorm.Itisevidentthattheconditionofbandlimitednes shastobeaddedinthiscase sinceitboundsthetotalvariationofthefunctionbetweens amples.Thecharacteristics ofthesamplesetcanbeusedtoclassifythedifferentsampli ngschemes,i.e.uniform andnonuniformsampling.Aswellashowthesamplesaregener ated,pointevaluations andlocalaverages. Inthischapterwerstdescribetheclassicaltheoryofunif ormsamplingandlaythe theoreticalfoundationthatwillallowustoextendintothe nonuniformsamplingscenario. Bydoingso,wewilldenetheconceptofframesandsamplingd ensitythatwillprovide conditionsunderwhichwecanrecovertheoriginalfunction s.Thesetoolswilllaterbe usedtodenereconstructionalgorithmsbasedonprojectio nsofthefunction,which weconsideraslocalaverages.Althoughthetheorycannotbe applieddirectlytothe integrateandre,wepresentboundsfortheerrorinthesesp aces.Aswell,asnite dimensionalmodelsthatallowforperfectrecovery. 2.3UniformSampling Asmentionedbeforethesamplingandreconstructionproces sisdenedbythe structureofthevectorspace.Hereweconstrainthefunctio nstoabandlimitedspace B n Denition4. (Spaceofbandlimitedfunctions) B n = f : Z R j f ( x ) j 2 dx 1 2 < 1 and supp ( ^ f ) [n,+n] (2–6) Where ^ f istheFouriertransformof f and L 2 [ a b ] isdenedbyallcontinuous functionsthataresquareintegrableontheinterval [ a b ] .Onemayarguethatthis 21

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modeldoesnottrealitysinceallpracticalsignalsareof nitelength,butthenagainan innitespectrumisalsounrealistic.Authorshavealsoloo kedforalternativenotionsfor band-limitedness Slepian ( 1976 ).Regardless,wewillusethepreviousdenition. Assumingabandlimitedspacewedenedsamplesasfunctione valuationson auniformsampleset.UndertheseassumptionstheShannon-W hittaker-Kotel'nikov theoremprovidesareconstructionalgorithmandthesufci entconditionsonthesample setthatguaranteerecovery. Theorem1. Afunction f thatisbandlimitedto [ n,n] canbecompletelyreconstructed fromitssampledvalues f ( k n ) takenatinstants k n k 2 Z ,equallyspacedaparton R ,in termsof: f ( t )= 1 X k = 1 f k n sinc n t k ,( t 2 R ) (2–7) Theseriesconvergesabsolutelyanduniformlyfor t 2 R Inthiscasetheshiftsofthesincfunctionformanorthonorm albasisaswellas acardinalbasisforthebandlimitedspace.Thefunctioneva luationcanactuallybe rewrittenasinnerproductsbetweenthefunctionsandthesh iftedsincfunctions,this alsoimpliesthatthesincfunctionisthereproducingkerne lforthisHilbertspace.This theoremisbetterunderstoodfromthefrequencyrepresenta tionanditssamples.The samplevaluescanbedenedaspointwisemultiplicationsof thefunctionwithadelta comb,thereforethesamplescanbeexpressedas: comb ( t = T )= T 1 X k = 1 f ( kT ) ( t kT ) (2–8) Where ( t ) denotesthedeltadiracdistribution.Wewillnotgointothe detailsofthe generalizedFouriertransformfordistributionsandwewil lsaythattheFouriertransform ofthedeltacombisgivenby: comb ( t T ) $ Tcomb ( T 2 ) (2–9) 22

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Thereforethespectrumofthesamplesconsistsofrepetitio nsoftheoriginalspectrum. Thedistancebetweenthesereplicasisgivenbytheinverseo fthedistancebetween thedeltadiracs.Thereforeasthesamplingperiodincrease sthereplicascomecloserto eachotheruntiltheyoverlap.Atthispointitisnotpossibl etorecovertheoriginalsignal, butbydecreasingthesampleratethereplicasmovefurthera partandstillretaintheir originalshapeuptosomeconstantfactor.Thereforewecanr ecoverthespectrumofthe originalfunctionbylowpasslteringthesamples.Thisisp reciselythereconstruction suggestedbytheShannon-Whittaker-Kotel'nikovtheorem. Inapplicationsthesampledfunctionsarerarelybandlimit ed.Thereforean anti-aliasinglterisappliedbeforethesampler.Thislow passlterprojectsthefunctions intoabandlimitedspace,forwhichweknowhowtosampleandr ecoverthefunctions. Thissameschemecanbeextendedtomoregeneralsamplingpro cedures Unserand Aldroubi ( 1994a ),thatusedifferentprojectorsandbasiselementsforther ecovery. Nonbandlimitedfunctionscanstillberecoveredfromproje ctionsintoabandlimited spaceaslongaswecanguaranteethatnotwofunctionsfromth einputspaceare mappedontothesamesequenceofpoints Eldar ( 2008 ). Althoughauniformsamplingschemeleadstogreatsimplica tionsinthesampling processitmaynotalwaysbeapplicable.Forexample,astron omicalmeasurements areweatherdependentandingeologysoilprolesarehighly irregular.Ontheother handitmaybemoreefcienttosamplenonuniformlysincehar dwareimplementations ofuniformsamplersarecostlyintermsofpowerandarea.Ana turalquestionarises: is thereaframeworktorecoverfunctionsthathavebeensample donanunequallyspaced sampleset? Thisisthequestionwewillanswerinthenextsection. 2.4NonuniformSampling Thissectionwillcoverthebasicconceptsinnonuniformsam plingastheyappeared throughtime.First,thedensityforasamplingsetisdened .Next,theconceptofframes isintroducedandrelatedtothisdensity.Thesignalsareco nstrainedtoabandlimited 23

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spaceforwhichaframeoperatorcanbedened.Wediscussthe inversionofthisframe operatorwhichwillleadtothereconstructionalgorithm.2.4.1DufnandSchaeffer'sWork Itisawellknownfactincomplexanalysis,thatif f ( z ) isanentirefunction 1 of exponentialtype 2 r 0 r< ,then f ( z ) iscompletelydeterminedbyitsvaluesatany sequence f n g ofuniformdensity1.Theinputhasbeenrestrictedtocomple xfunctions ofexponentialtypeandthesamplingsetisconstrainedbyth edensity.Thedensityis denedasfollows: Denition5. Asequence f n g n =0, 1, 2,... ,has“uniformdensity1”ifthereare constants L and suchthat j n n j L and j n m j > 0 for n 6 = m Denition6. Asequence f n g n =0, 1, 2,... ,ofrealorcomplexnumbershas uniformdensityd, d > 0 ifthereareconstants L and suchthat; n n d L (2–10) n m > 0 for n 6 = m (2–11) Eventhoughasamplingsetwithacertaindensitydenesauni quefunctionof exponentialtype,nothinghasbeensaidonhowtorecoversuc hafunctionfromits samples.Thereforetheconceptofframesisintroducedwhic hisusedtodesigna reconstructionalgorithm. 1 Entirefunctionreferstocomplexanalyticfunctions. 2 Afunction f ( z ) issaidtobeofexponentialtypeifthereexistconstants M and suchthat: j f ( z ) j Me j z j Inthelimitof j z j!1 .If r istheinmumofallsuch thenwe saythatthefunction f isofexponentialtype r 24

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Denition7. Asetoffunctions f e i n t g isaframeovertheinterval ( r r ) ifthereare positiveconstantsAandBsuchthat: A 1 2 P n j R r r g ( t ) e i n t dt j 2 R r r j g ( t ) j 2 dt B (2–12) istrueforall g ( t ) 2 L 2 ( r r ) Forthecase A = B ,thesetofexponentialsturnsintoabasis.Theframeisactu ally anoverdeterminedbasisset,meaningtheelementsinthefra mearenotnecessarily linearlyindependent.Thereareseveralotherfactsaboutt heframesbutthesewillbe treatedinthecaseofabstractFrameslateron. Next,therelationshipbetweenthedensityofthesamplings etandtheframeinterval isstudied.Notethatasthesamplingsetbecomesdenser,the intervaloverwhichthe familyofexponentialsconstitutesaframeincreases. Theorem2. If f n g isasequenceofuniformdensityd,thenthesetoffunctions f e i n t g isaframeovertheinterval ( r r ) where 0
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forpositiveconstants A and C andallvaluesof z ,and Z 1 1 j f ( x ) j 2 dx < 1 (2–15) Thenthereexistsafunction in L 2 [ A A ] suchthat f ( z )= 1 2 Z A A ( ) e iz d (2–16) Notethattherestrictionontheenergyisovertherealaxis x .Thistheoremstates thatthereexistsaFouriertransformforallentirefunctio nsofexponentialtypethat areboundedin L 2 overtheirrealaxis,andmoreimportantlytheFouriertrans formhas compactsupportboundedbytheorderoftheexponentialtype .Thereforeafunction f ( z ) thatsatisesTheorem 3 alsosatisesthePaley-WienerTheorem.Henceifonly realfunctionsareconsidered: f ( x )= 1 2 Z R ^ f ( ) X [ n,+n] ( ) e jx d (2–17) = Z R f ( t ) sin (n( t x )) ( t x ) dt (2–18) = n h f T x sinc n i (2–19) where X [ n,+n] isthecharacteristicfunctionovertheinterval [ n,+n] (denedas oneovertheintervalandzeroeverywhereelse)and T x isatranslationoperatorwhich shiftsthesignalby x .Sincewecansubstitute( 2–19 )in( 2–13 ),itimpliesthattheset f T x n sinc n g isaframeforthebandlimitedspace. Aclearrelationshiphasbeenestablishedbetweenthesampl ingdensity,theframe intervalandthesupportofthefrequencydomainofanentire functionofexponential type.Fromnowon,onlyrealfunctionsareconsidered.Thene xtsubsectiongivesa broadernotionofframeswhichwillbecomeusefulwhendeali ngwithnitedimensional 3 Rememberthat z = x + iy 26

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spaces.ForabetterreviewonFramesandnon-harmonicFouri erseriesconsult Christensen ( 2002 ); Young ( 2001 ),byChristensenandYoungrespectively. 2.4.2AbstractFrames Fortwovectors u v inaHilbertspace,let h u v i = h v u i bethecomplexscalar productwhichdenesthenorm jj u jj = h u u i 1 2 .Aframeisdenedtobeaninnite sequenceofnonzerovectors 1 2 3 ,... suchthatforanarbitraryvector v A jj v jj 2 X n jh n v ij 2 B jj v jj 2 (2–20) AandBarecalledframebounds,andthenumbers n = h v n i n =1,2,3,... ,are calledthemomentsequenceofthevector v relativetotheframe.Wecandenealinear operator S bytherelation Sv = 1 X n =1 h v n i n (2–21) Fromtheframedenition: A jj v jj 2 h Sv v i B jj v jj 2 (2–22) Itcanbeproventhatthisoperatorisone-to-oneandon-toan dthusinvertibleand selfadjoint.Thisstatesthat'S'ispositivedenitewithp ositiveupperandlowerbounds. Let S 1 betheinverseoperator,thenwecandenethedualframeas f S 1 n g .Ifthe dualframeisknownwecanrecover f by: f = X n h f S 1 n i n = X n h f n i S 1 n (2–23) Atthispointaframeforthebandlimitedspacehasbeenident iedandtheframe operatorhasbeenconstructed.Thenextstepistodetermine theinverseofthis operator.Thismaynotbeasstraightforwardandthereforei terativeinversionalgorithms mayhavetobeused.Inoperatortheoryawellknownseriestha tapproximatesthe inverseofcertainoperatorsistheNeumannseries. 27

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2.4.3AnIterativeReconstructionAlgorithmUsingtheNeum annSeries TheNeumannseriescanbeusedtoinvertlinearboundedopera tors. Denition8. (NeumannSeries)Iftheoperator A onaBanachspaceBwithnorm kk B satises k A k B < 1 then ( I A ) isinvertiblewithinversegivenby ( I A ) 1 = + 1 X k =0 ( A ) k (2–24) Notethatanyboundedlinearoperatorcanbeinverted,there foredifferentoperators canbeconstructedfromthesamples.Aslongasthecondition saremet,theoriginal functionscanberecovered.WiththisinmindFeichtingeran dGrochenig Feichtingerand Gr ¨ ochenig ( 1994 )proposedthefollowingmethod. Proposition1. Let A beaboundedoperatoronaBanachspace( B kk B )thatsatises forsomepositiveconstant r< 1 k f Af k B r k f k B forall f 2 B (2–25) then A isinvertibleon B and f canberecoveredfromAfbythefollowingiteration algorithm. Setting f o = Af and f n +1 = f n + A ( f f n ) (2–26) for n 0 ,wehave lim n !1 f n = f (2–27) withtheerrorestimateafter n iterations k f f n k B r n +1 k f k B (2–28) Iftheoperator A isratheramatrix,thisisknownastheRichardsons'smethod .If A isapositivedenitematrixthentheChebysheviterationor conjugategradientiteration yieldbetterconvergencerates,becauseweintroducestruc tureintothematrix. 28

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Usingthisiterativeoperatorinversion,itispossibletor ecovertheoriginalsignal fromtheframeoperatorconstructedformthesamples.Butwe arenotlimitedtothis operatorandanyoperatorthatcanbeconstructedfromaprio riknowledgeandthat complieswiththeconstraintsgivenbytheNeumannseriesca nbeused.Feichtinger proposedusinganapproximationoperatoroftheform: Af = P n h f e n i h n ,andalso provedwhenthesesetswereframesanddeterminedtheframeb ounds. Proposition2. Supposethatthesequences f e n g n 2 N and f h n g n 2 N havethefollowing properties:thereexistconstants C 1 C 2 > 0,0 << 1 ,suchthat. X n jh f e n ij 2 C 1 k f k 2 (2–29) k X n n h n k 2 C 2 X n j n j 2 (2–30) k f X n h f e n i h n k k f k (2–31) forall f 2H and ( n ) n 2 N 2 l 2 .Then f e n g isaframewithframebounds (1 ) 2 C 2 and C 1 ,and f h n g isaframewithbounds (1 ) 2 C 1 and C 2 Theproposedapproximationoperatorconsistsofasimpleze roorderhold interpolationbetweenthesamplepointsfollowedbyaproje ctiontothebandlimited space P : Af = P X n 2 Z f ( x n ) X n (2–32) where X n isthecharacteristicfunctionontheinterval [ y n 1 y n ) ,for y n = x n + x n +1 2 .This operatorisinvertibleunderthefollowingconditions. Proposition3. If = sup n 2 Z ( x n +1 x n ) < ,thenforall f 2 B k f Af k < k f k (2–33) Consequently,Aisboundedandinvertibleon B withbounds k Af k 1+ k f k (2–34) 29

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and k A 1 f k 1 1 k f k (2–35) SincetheoperatorisboundedwecanalsousetheNeumannseri estoinvertitand recovertheoriginalsignal.UsingProposition 2 ,itcanbeshownthat f p n T x n sinc n n 2 Z g and f 1 p n P X n n 2 Z g areframesof B n ,where n = x n +1 x n 1 2 Feichtingerproposedtheadaptiveweightsmethodbasedont heapproximation operatorperviouslydened: Af = P n 2 Z h f p n L x n sinc i 1 p n P X n Inconclusiontworeconstructionalgorithmshavebeenpres entedusingthe Neumannseries.Oneisbasedonthereconstructionfromthef rameoperator(based ontheset f L x n sinc n g )andtheotherusestheapproximationoperatorpreviously described.Thenextstepistoshowhowtheapproximationope ratoractuallyleads toareconstructionalgorithmforlocalaverages.Whenthef ramesintheapproximation operatorareswitchedtheresultis: Af = X n 2 Z h f 1 p n P X n i p n T x n sinc (2–36) Thereforetheinnerproductintheframeoperatoriscalcula tedasfollows. h f 1 p n P X n i = h Pf 1 p n X n i = 1 p n Z y n y n 1 f ( x ) dx (2–37) UsingthisoperatorandtheNeumannserieswearriveatthefo llowingalgorithm. Theorem5. (Reconstructionfromlocalaverages)Let f x n n 2 Z g beasampling sequencewith =sup n 2 Z ( x n +1 x n ) < n y n themidpoints y n = x n + x n +1 2 n = y n y n 1 andlet n bethelocalaverage n = 1 n R y n +1 y n f ( x ) dx forafunction f .Thenevery f 2 B n isuniquelydeterminedbyitsaverages n andcanbereconstructedbythefollowing 30

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algorithm: f o = X n 2 Z n n n T x n sinc (2–38) f k +1 = f k + f o X n 2 Z Z y n y n 1 f k ( x ) dx n T x n sinc n for k 0 (2–39) then f =lim k !1 f k in L 2 ( R ) and k f f k k n k +1 k f k (2–40) 2.5ReconstructionfromLocalAverages:InniteDimension s 2.5.1BandlimitedSpace Beforecoveringthereconstructionalgorithmscertaincon ceptsneedaproper denition: Denition9. ( k th LocalAverageof f ) h u k f i = Z f ( x ) u k ( x ) dx (2–41) Denition10. (AveragingFunctions) f u k g in L 2 ( R d ) arecalledaveragingfunctions centeredat f x k g ofwidth iftheyhavefollowingproperties 1. supp u k [ x k 2 x k + 2 ] ,for > 0 2. u k ( x ) 0 3. R u k ( x ) dx =1 Denition11. Thesamplingdensity foraseparatedsamplingsequence isdened as: =sup i 2 Z ( i +1 i ) (2–42) BeforeFeichtingerandGrochenigpresentedtheorem 5 Grochenig Grochenig ( 1992 ),hadstated: 31

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Theorem6. Assumethat = f i g i 2 Z isa -densesamplingsetandthat u i i 2 Z isacollectionofaveragingfunctions.If < 1 p 2n ,thenevery f 2 B n isuniquely determinedbythelocalaverages h u i f i around i .Moreover, f canbereconstructedby thefollowingiterationscheme: 0 = P X i h u i f iX i (2–43) n +1 = n P X i h u i n iX i (2–44) and f = 1 X n =0 n (2–45) Thisisavalidreconstructionalgorithmsincetheoperator Af = P P i h u i f iX i satisestherequirementsoftheNeumannseries.Inthiscas etheaveragingfunctions overeachintervalcanbearbitraryandarenotxedto 1 asinthetheorem 5 ,butthe samplingpointshavetobecloser.SunandZhou Songetal. ( 2007 ); SunandZhou ( 2002 )improvedGrochenig'sresults: Theorem7. (SunandZhou)Let bedenedasindenition 11 (samplegapsize)and asindenition 10 ,with < n (2–46) 0 << n (2–47) thenthereisaframe f k ( x ) g : k 2 Z for f 2 B n withestimates 1+ n p 2 2 1+ n 2 C L and C U 81 2 1 n 3 (2–48) suchthatforany f 2 B n f ( x )= X k 2 Z x k +1 x k 1 2 1 = 2 h f u k i k ( x ) (2–49) wheretheconvergenceisbothin L 2 ( R ) anduniformon R 32

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If = 2n thenthetheoremstatesthatevery f 2 B n isuniquelydetermineby h f u k i foranysetofaveragingfunctions f u k ( x ) g with suppu k [ x k = 2, x k + = 2] 0 << 2n Because 2n > 1 p 2n ,thisresultcontainsGrochenig'sresult.Eventhoughsuch aFrame exists,identifyingitmaynotbeasstraightforward.Sunan dZhoualsogaveusan iterativereconstructionalgorithm: Let f x k : k 2 Z g and f u k : k 2 Z g besequencesofsamplingpointsandaveraging functions,respectively.Dene ~ u k tobetheprojectionof u k tothebandlimitedspace. Let A and B betheframeboundsandlet 0 << 2 B .Thenevery f 2 B n canbe reconstructedfromitslocalaveragesbythefollowingiter ativealgorithm: f 0 = Sf := X k 2 Z h f u k i ~ u k (2–50) f k +1 = f k + S ( f f k ), k > 0, (2–51) with f = lim k !1 and k f f k k 1 r k +1 k f k 2 (2–52) where r = max fj 1 A j j 1 B jg .Thisimpliesthat ~ u k isactuallyaframefor B n .It canbeshownthatforagivendensityoftheaveragesamplesth eset f u k g satisesthe frameconditionforall f in B n ,eventhoughthe f u k g areinnotin B n .If f u k g satisesthe frameconditionsforall f in B n ,anyorthogonalprojectionof f u k g into B n isaFramefor B n .Therefore, f canbereconstructedfromitslocalaverages.Thisisthesta teoftheart withrespecttothereconstructionfromlocalaveragesforb andlimitedsignals. 2.5.2ShiftInvariantSpace Thebandlimitedassumptiongreatlysimpliesthesampling problem,yetitisnot arealisticassumption.Theconceptofabandlimitedfuncti onisjustamathematical constructthathelpsusdenethesamplingframeworkbutisn otarealisticmodel Slepian ( 1976 ). 33

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WecanimprovethismodelbygeneralizingtheShannon-Whitt aker-Kotel'nikov samplingtheorem.Recallthatabandlimitedfunctionwitha sufcientlydensesample setcanbeexpressedas: f ( t )= 1 X k = 1 f k n sinc n t k ,( t 2 R ) (2–53) Byallowingdifferentgeneratorfunctions ( t ) insteadofthe sinc ( t ) ,weintroducea generalshiftinvariantspace deBooretal. ( 1992 ): V p ( )= f ( )= Xk 2Z c k ( k ),( a k ) 2 l p ( Z ) (2–54) Differentassumptionscanbemadeonthegeneratorfunction swhichallowusto considermorerealisticsamplingmodels.Astheirnameimpl iesshiftinvariantspaces areclosedundertranslations.Forany f 2 S ,where S denotesashiftinvariantspace impliesthatitsintegershiftsalsobelongto S f ( k ) 2 S k 2 Z Thesetypesofmodelshavebeenextensivelyusedinuniforma ndnonuniform samplingliterature Aldroubietal. ( 2004 ); Aldroubi ( 2002 ); AldroubiandGr ¨ ochenig ( 2001 ).Giventhefreedomintheselectionofthegeneratorsthesa mpleconstraints andtherecoveryalgorithmsmustbeadaptedcorrespondingl y.Theliteraturecovers severalspeciccases.Forshiftinvariantsplinespacesth enecessaryconditionson theBeurlingdensityforstablereconstructionaredenedi n AldroubiandGr ¨ ochenig ( 2000 ).Samplingofmultivariatefunctionsinsplinelikespaces wascoveredin Aldroubi andFeichtinger ( 1998 ).Forpracticalapplicationsweassumecompactsupported generators Gr ¨ ochenigandSchwab ( 2003 ).Asurveyonsamplingandreconstructionin shiftinvariantspacescanbefoundin AldroubiandGr ¨ ochenig ( 2001 ). Asmentionedpreviously,therearethreebasicconstraints neededtodenethe samplingproblem.First,theinputspaceconsistingofthef unctionsthatwillbesampled mustprovideastablerepresentationforitselements.Seco nd,thesamplesaredened 34

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andtheconditionsforthemtobeasetofsamplingareimposed .Finally,usingboth constraintsarecoveryalgorithmisdesigned. Althoughshiftinvariantspacesallowustogeneralizethec onventionalsampling problemthegeneratorscannotbeselectedarbitrarily.Iti snecessarytorestrictthem inorderforthesamplingframeworktobemeaningfuli.e.for pointevaluationstobe welldened.ThisisdonewiththehelpofWieneramalgamspac es W ( L pv ) Feichtinger ( 1990 ); FournierandStewart ( 1985 ). Theseconsistoffunctionswhicharelocallyin L 1 andgloballyin L pv .Ameasurable function f belongsto W ( L p ) 1 p < 1 ,ifitsatises: k f k W ( L p ) = X k 2 Z d esssup fj f ( x + k ) j p : x 2 [0,1] d g 1 p < 1 (2–55) If p = 1 ,ameasurablefunctions f belongsto W ( L 1 ) ifitsatises: k f k W ( L 1 ) = sup k 2 Z d esssup fj f ( x + k ) j : x 2 [0,1] d g < 1 (2–56) Inthiscase W ( L 1 ) coincideswith L 1 ( R d ) andthesubspaceofcontinuousfunctionsis denotedby W 0 ( L p ) (functionsintheamalgamspacewhicharealsocontinuous). Itis alsorequiredthatthegeneratorssatisfy: 0 < m ^ a ( )= X j 2 Z d j ^ ( + j ) j 2 M < 1 a e (2–57) bothconditionsimplythereexistsconstants m > 0, M > 0 suchthat Aldroubiand Gr ¨ ochenig ( 2000 ): m k a k p k X k 2 Z a k ( k ) k p M k a k p 8 a 2 l p (2–58) whichimpliesthegeneratorsforaRieszbasisfor V p ( ) .Thesamplesareusually denedaspointevaluationsorlocalaveragesonthefunctio n.Wewillfocusonthe later,thesampleset f g x j g isdenedas R R d f ( x ) u x j ( x ) dx .Reconstructionfromsamples 35

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generatedbyshiftsofthesameaveragingfunction x j = ( x j ) wascoveredin Aldroubietal. ( 2004 ); UnserandAldroubi ( 1994b ). Inorderforthesesamplestoprovidesufcientinformation forrecoverythesample set f g x j g mustbea setofsampling ,thisholdsif: a p k f k L p X x j 2 X j g x j ( f ) j p 1 p A p k f k L p (2–59) with a p and A p positiveconstantsindependentof f .Theconditionsonthesamplingset inordertoguaranteerecoveryareusuallydenedintermsof thedensityofthepoints.A set X = f x j j 2 J g is r 0 dense in R d if: R d = [ j = B r ( x j ) ,forevery r>r 0 ,where B r ( x j ) areballscenteredat x j ,andwithradius r Givenastablerepresentationanda setofsampling recoveryalgorithmscan bedened.Currentreconstructionmethodsarebasedonappr oximation-projection (A-P)schemes Aldroubi ( 2002 ); AldroubiandFeichtinger ( 2002 ).Thesealgorithms consistsoftwosteps.Firstthefunctionisapproximatedby applyinganinterpolator orquasiinterpolatortothesamples,thentheinterpolated signalisprojectedontothe SIStoobtainarstapproximation.Theerrorbetweentheori ginalsamplesandthe approximationiscomputed,theinterpolation-projection procedureisrepeatedonthe errortoobtainacorrectionterm.Repeatingthisprocessle adstoperfectrecoveryofthe originalsignal. Thequasiinterpolationoperator A X a isdenedasfollows: A X f = X j 2 J h f u x j i j (2–60) wheretheset f j g j 2 J formsaboundedpartitionofunitydenedas: Denition12. Aboundeduniformofunity(BUPU)associatedwith f B r ( x j ) g j 2 J isasetof functions f j g j 2 J thatsatisfy: 1. 0 j 1, 8 j 2 J 36

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2. supp j B r ( x j ) 3. P j 2 J j =1 Althoughtheinterpolatedfunctiondoesnotbelongto V p ( ) itisstillpossibleto recoverthesampledfunctionexactlyasfollows Aldroubietal. ( 2004 ): Theorem8. Let bein W 0 ( L 1 ) ,let P beaboundedprojectionfrom L p onto V p ( ) ,and lettheaveragingsamplingfunctionals u x j 2 W ( L 1 ) satisfy R R d u x j =1 and R R d j u x j j < M where M > 0 isindependentof x j .Thenthereexistsadensity r = r ( M ) > 0 and a 0 > a 0 ( M ) > 0 suchthatif X = f x j : j 2 J g isseparatedand r -densein R d andiftheaveragingsamplingfunctionals u x j satisfysupp u x j x j +[ a a ] d forsome 0 < a < a 0 ,thenany f 2 V p ( ) canberecoveredfromitsweightedaveragesamples fh f u x j i : j 2 J g bythefollowingA-Piterativealgorithm: f 1 = PA X f (2–61) f n +1 = PA X ( f f n )+ f n (2–62) Inthiscase,theiterate f n convergesto f uniformly,andalsointhe W ( L p ) -normandthe L p -norm.Moreover,theconvergenceisgeometric, k f f n k L p k f f n k W ( L p ) C 1 n k f f 1 k W ( L p ) (2–63) forsome = ( r a M ) < 1 and C 1 < 1 Oneofthedifcultiesofapplyingthesealgorithmsinpract iceisthatthedensityon thesamplesetisnotknown,ithasonlybeenshownthatitexis tsandisgreaterthan 0 Bothframeworksforsamplinginbandlimitedspacesandshif tinvariantspacesare theoreticallysoundbutcannotbeapplieddirectlytoappli cations,thereforemorerealistic samplingmodelsarenecessary. 2.6ReconstructionfromLocalAverages:FiniteDimensions Inpracticefunctionsareconsideredtobecompactlysuppor ted.Withthisconstraint wecannolongerusethepreviousmodels.Thefollowingsecti onpresentstwotypesof 37

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models.TherstusesaFourierseriestorepresentthefunct ion.Thesebasisfunctions areglobal,sincetheycovertheentireinterval.Incontras t,thesecondmodelbasedon splinesubspacesuseslocalgenerators.2.6.1FourierSeries Letusassumewithoutlossofgeneralitythatourfunction p ( t ) isdenedonlyonthe interval [0,1] .Letusassume f canbeexpressedas: p ( t )= M X n = M a n e i 2 nt (2–64) Thisassumptionhasseveralimplications.Wecanonlyrepre sentfunctionsthatshare thesamepropertiesasthecomplexexponentials;continuit y,smoothness.Inthiscase theinputspaceisreferredtoas P m P m = f f : f 2 span ( e i 2 tn ) g (2–65) Thecoefcientscanberewrittenastheprojectionofthefun ctionontotheFourierbasis as: a n = Z 1 0 p ( t ) e i 2 nt dt (2–66) Dependingonthesymmetriesinthesignalthesecoefcients mayalsoexhibitsome typeofstructure.Bycombiningboththeseequationswecanw rite: p ( t )= M X n = M Z 1 0 p ( ) e i 2 n d e i 2 nt (2–67) p ( t )= Z 1 0 p ( ) M X n = M e i 2 n e i 2 nt d (2–68) Nowweknowthat: M X n = M e i 2 n = sin ( M +1 = 2)2 sin ( ) (2–69) 38

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ThisisreferredtoastheDirichletKernel D M ( t ) Thereforewecanwrite: p ( t )= Z 1 0 p ( ) M X n = M e i 2 n e i 2 nt d (2–70) p ( t )= Z 1 0 p ( ) sin ( M +1 = 2)2 ( t ) sin ( ( t )) d (2–71) Thismeansthat p ( t ) canbeseenastheinnerproductbetween p andashiftedDirichlet Kernel,analogoustothesincfunction. p ( t )= h p L t f D M gi (2–72) where L t isthetranslationoperatorby t FollowingFeichtinger Feichtingeretal. ( 1995 )itiseasytoprovethattheshifted Dirichletfunctionsareactuallyaframeforthisbandlimit edspace. Let 0 t 1 < ... < t r < 1 be r arbitrarydistinctsamplingpointsin [0,1) and p 2P m Then p isuniquelydeterminedbyits r samples p ( t i ) ifandonlyif r 2 M +1 .Inthis casethereexisttwoconstants 0 < A B sothat A Z 1 0 j p ( t ) j 2 dt r X i =1 j p ( t i ) j 2 B Z 1 0 j p ( t ) j 2 dt (2–73) Thisiseasytoprovesincewearedealingwithsignalsin L 2 (0,1) .Observethatthe constraintonthedensityofthesampleshaschangedandnowi tonlydependsonthe numberofsamples.Thismakessensesincetheoriginalsigna lisacombinationof 2 M +1 exponentials.Thereforeifwehavethismanysampleswecani nvertthesystem anddeterminethecoefcients. Theinequalityistruesincethemap p 2P M [ p ( t i ) i =1,..., r ] 2 C r isinvertible. Whichmeansthatasetofsamples f p ( t i ) g ri uniquelydetermineafunctionin p 2P M Thespace P M canbeexpressedasanitedimensionalvectorofthecoefci ents a n sinceany p 2P M canbeexpressedas p ( t )= P Mn = M a n e i 2 nt .Thereforewecandene alinearmapping T : C r C 2 M +1 sincethemappingislinearandbetweentonite 39

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dimensionalspacesweknowthatitisbounded,therefore: k Tv k A k v k (2–74) andsinceitisinvertible k T 1 Tv k C k Tv k (2–75) therefore B k v kk Tv k A k v k (2–76) forconstants A B > 0 .UsingthepreviousresultthismeansthattheDirichletKer nels areactuallyaframeforthebandlimitedspace P m .Nowwecandeneaframeoperator callit S ; Sp = r X i =1 h p L t i f D M gi L t i f D M g (2–77) Theoperatorislinearandboundedandthereforewecanuseth eNeummanseries toinvertit.Inthenitecasewearedealingwithamatrixand thereforelinearalgebra techniquessufce.Similarlytotheinnitecasewecanden eaframeoperatorbased ontheaveragingfunctions u j : Sp = N X j =1 h p u j i ( u j ? D M ( t )) (2–78) where N arethenumberofsamples,and 2 M +1 ,thenumberofcoefcients.Wecan arguethatthesetofaveragingfunctionsactuallysatises theframeconditionsforall elementsin P M andthereforeit'saprojectionintothebandlimitedspacea ndaframefor P M 40

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Substitutingequations 2–69 and 2–64 in 2–78 weobtain: Sp = N X j =1 h p u j i ( u j ? D M ( t )) (2–79) Sp = M X k = M M X l = M N X j =1 Z e i 2 k u j ( ) d Z u j ( ) e i 2 l ( ) d a k e i 2 lt (2–80) Let A bethe (2 M +1) (2 M +1) matrixwithentries: A l k = N X j =1 Z e i 2 k u j ( ) d Z u j ( ) e i 2 l ( ) d (2–81) Thereforethecoefcientsof Sp ,aregivenby A ~ y ,where ~ y isthevectorofcoefcients a k nowlet Sp = N X j =1 h p u j i M X k = M [ u j ? e i 2 kt ] (2–82) Sp = M X k = M N X j =1 h p u j i Z u j ( ) e i 2 k d e i 2 kt (2–83) Let b k be: b k = N X j =1 h p u j i Z u j ( ) e i 2 k d (2–84) andtherefore: A ~ y = ~ b (2–85) thereforetondtheoriginalcoefcientswemustndtheinv erseofthismatrix,thiscan bedoneiterativelyordirectly. ~ y = A 1 ~ b (2–86) Thegreatadvantageofusingframetheorycomesintoplaywhe nwearedealingwith innitedimensionalspaces.Butinthecaseofnitedimensi onalspacesreconstruction basedonframesandtheinversionoftheframeoperatorsimpl yreducestoinverting thenormalequationsforthesamplingproblem.Thereforewe couldhavetakenalinear 41

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algebraapproach: p ( t )= M X n = M a n e i 2 nt (2–87) Fromthesamplerweobtain: j = Z p ( t ) u j ( t ) dt = M X n = M a n Z e i 2 nt u j ( t ) dt (2–88) Wecandeneamatrix U suchthat: U j k := Z e i 2 kt u j ( t ) dt (2–89) andthevectors ~ a :=[ a M ,..., a M ] ~ =[ M ,..., M ] (2–90) Wecannowdenethefollowinglinearsystem: U ~ = U U ~ a (2–91) Thesolutiontothenormalequationsisequivalenttothesol utiongivenbytheframe algorithm.Itisevidentthatinthecaseofinnitedimensio nsitisnotusefultosetupthe problemasaninversionofainnitelylargematrix,whichis whyframetheorybecomes useful.2.6.2SplineSpaces AsstatedbytheWeierestraapproximationtheorem,contin uousfunctionsdened overanintervalcanbeapproximatedtoanyprecisionbyapol ynomialseries.Although, inpracticepolynomialapproximationsbecomeunstableast hedegreeincreases,limiting theiruseinapplications. Splinesovercomethisissuebyapproximatingthefunctionu singpiecewise polynomialswithoverallsmoothnessconstraints.Thesepo lynomialsaredened overintervalsdeterminedbytheknotsequence. Denition13. (knotsequence)Consistsofanon-decreasingsequence t =[..., t i t i +1 ,...] 42

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Inordertodenesplinespreciselywerstintroducethebas icbuildingblocksfrom whichsplinescanbeconstructedknownasB(asic)-splines. Wemakeaclarication thatB-splines,donotrefertobasissplines,thishasbeena commonmisconception intheliterature.B-splinescanbedenedindifferentways ,hereweonlyrefertothe recurrencerelation,whichallowsforastableandefcient evaluationofthesplines. B-splinesoforder1aredenedintermsofthecharacteristi cfunction: B i ,1 ( t ):= 8><>: X i ( t )=1 t i t < t i +1 0 o.w. (2–92) Noticethesefunctionsarechosentoberight-sidecontinuo usandtheyformapartitions unity.Theknotsequencemayhaverepeatedknotsofdifferen tmultiplicity,inthiscase, t i = t i +1 andtherefore B i ,1 =0 .A k th orderB-splinesisdenedbytherecurrence: B i k := i k B i k 1 +(1 i +1, k ) B i +1, k 1 (2–93) where i k ( t ):= 8><>: t t i t i + k 1 t i t i 6 = t i + k +1 0 o.w. (2–94) Asplineisthendenedas: Denition14. Asplineoforder k withknotsequence t isbydenitionalinearcombinationofB-splinesoforder k associatedtotheknots: S k t := X i B i k a i : a i 2 R (2–95) Thefollowingtheoremallowsustocharacterizethepiecewi sepolynomialsthatlive inthesplinespaceandforwhichtheB-splinesformabasis. Theorem9. Thespace S k t coincideswiththespaceofallpiecewisepolynomials functionsofdegree < k ,withbreakpoints t i whichare k 2 timescontinuously differentiableat t i 43

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Theorem10. Let I =[ a b ] beaniteinterval.ThoseB-splineswhichhavesame supportinIformabasisforthespaceofpiecewisepolynomia lsfunctionsofdegree < k on I ,withbreakpoints a < t i < b whichare k 2 timescontinuouslydifferentiableat t i Incasetheknotsareselectedtolieonthe Z theB-splinebecomesacardinalspline (notthesameascardinalbasis).Inthiscasethesplinecanb eexpressedasalinear combinationofasingleshiftedB-spline: B i k = N k ( i ) (2–96) N k = B 0, k (2–97) ( k 1) N k ( t )= tN k 1 ( t )+( k t ) N k 1 ( t 1) (2–98) Inthiscase S k t isactuallyashiftinvariantspace.Inthiscasewecanuseth etheorems denedforsamplingshiftinvariantspaces 2.5.2 byassumingthegeneratorsare compactsupportandbounded.Thisassumptionleadstothefo llowingtheorem Aldroubi andGr ¨ ochenig ( 2001 ): Lemma1. If iscontinuousandsatisesthecondition P k 2 Z max x 2 [0,1] j f ( x + k ) j < 1 (WinerAmalgam),inparticular,if iscontinuouswithcompactsupport,thenforall x 2 R thereexistsafunction K x 2 V ( ) suchthat f ( x )= h f K x i for f 2 V ( ) .Wesay that V ( ) isareproducingkernelHilbertspace. ThestructureofthereproducingkernelHilbertspace(rkhs ),allowsustodene aframe,ifweconsiderthesamplesasprojectionsontothese elementstheinversion oftheframeoperatorsolvesthesamplingproblembutitisal sopossibletoinvertthe innitesamplingmatrix U givenbytheevaluationofthegenerators.Inthiscasesince it isapointevaluation U j k = ( x j k ), k j 2 Z .Bothoftheseapproachesareequivalent: Lemma2. If satisestheconditionofLemma 1 ,thenthefollowingareequivalent: 1. X = f x j : j 2 Z g isasetofsamplingfor V ( ) 44

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2. Thereexists A B > 0 suchthat A k c k l 2 k Uc k l 2 B k c k l 2 8 c 2 l 2 ( Z ) (2–99) 3. Thesetofreproducingkernels f K x j : j 2 Z g isaframefor V ( ) Aswementionedin 2.5.2 itisdifculttodeterminetheconditionsonthedensity ofthesamplingsetsuchthatrecoveryispossible,butinthe casetat isaB-splineof order N thethemaximumgapcondition sup j 2 Z ( x j +1 x j )= < 1 issufcientforlemma 2 tohold AldroubiandGr ¨ ochenig ( 2000 ). Thechapterhascovereddifferentsamplingspaces,includi ngniteandinnite dimensionalmodels.Thequestionnow,ishowdoestheintegr ateandresamplert intothisframework.Thefollowingchaptersintroducethes ampleranddealwiththe recoveryalgorithmsinthesespaces. 45

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CHAPTER3 THEINTEGRATEANDFIRESAMPLER TheIFmodelpresentedhereisslightlydifferentfromtheco nventionalneuron models.ThebasicbuildingblocksareshowninFigure 3-1 .Theinput x isweightedby aleakyfactorandintegrated.Whenthevalueoftheintegral reacheseitherthepositive ornegativethresholds,apulseisgeneratedwiththecorres pondingpolarity f t n n g .The valueoftheintegralisthenheldatzeroforthetimespecie dintherefractoryperiod ,andtheprocessrepeats.Thereforetheinputisencodedent irelyinthetimingofthe pulses. Assumingtheinputexistsfrom t 0 onwardswecandenethesamplesrecursively as: Z t k +1 t k + x ( t ) e ( t t k +1 ) dt = q k (3–1) where q k 2f p n g 3.1EffectoftheIFparameters Thesamplerparametersarethethresholds p n ,theleakyparameter andthe refractoryperiod .Theeffectsofthesamplerparametersareeasilyunderstoo difwe settherefractoryperiodtozero.Thisallowsustoestablis hadirectrelationshipbetween theinput x anditsintegral v ,dening: v ( t ):= Z t t 0 x ( t ) e ( t t k +1 ) dt (3–2) !!" !!" #!" !!! !"#"$% "!! Figure3-1.Integrateandreblockdiagram. 46

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0 2000 4000 6000 8000 10000 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Frequency [Hz]Normalized Magnitude 1ms 0.5ms 0.2ms 0.1ms 0.05ms 0.02ms Original Figure3-2.Frequencyresponseofthelterfordifferentva luesof normalizedtounit gainat( f =0 ).Thespectrumoftheoriginalsignalisshowninblack. Theconditiontorecanberewrittenas: v ( t k +1 ) v ( t k ) e ( t k t k +1 ) = q k (3–3) Therefore,theIFsamplescanbetranslatedintopointevalu ationsoftheintegral v ,and giventheintegralwecandetermine x sincetheyarerelatedby: x ( t )= @ v ( t ) @ t + v ( t ) (3–4) Theintegral v isalowpasslteredversionoftheinput x ,andtheleakyparameter controlsthecutofffrequency.Figure 3-2 showsthenormalizedfrequencyresponseof thelteraswellasthenormalizedinputspectrum.Figure 3-3 showstheoriginalinput x and v fordifferentvaluesof ,allnormalizedtounitpower.Asthevalueof decreases theoutputbecomessmootherandvice-versa. Theeffectsoftheleakparameteralsoextendtotheconditio ntore(eq 3–3 ). As decreases,theexponentialweightgoestounityandtherefo reitbehavesas a -modulator.Inthecasethat increases,thevalueofthepreviousringloses importanceandthesamplerbecomesalevelcrossingsampler SaeedMianQaisar ( 2009 ).Wecouldexpectsimilarbehaviorwithanon-zerorefracto ryperiod.Themain 47

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0 0.002 0.004 0.006 0.008 0.01 -0.05 -0.04 -0.03 -0.02 -0.01 0 0.01 0.02 0.03 Time [s]Normalized Amplitude 1ms 0.5ms 0.2ms 0.1ms 0.05ms 0.02ms Original Figure3-3.Integralfunction v fordifferentvaluesof .Eachsignalhasbeennormalized tohavethesamepower. purposeoftherefractoryperiodistolimitthedatarates,s peciallyinthecaseoflarge amplitudeartifacts.Furthermore,itcanalsobeusedtogen eratenearlyuniformsample distributions,sincebydecreasingthethresholdandxing ,theintegrationintervalwill tendtozeroproducingequallyspacedsamples.Nevertheles s,thiswouldbesensitive totimingjitterandthereforeisnotrecommended.Theeffec tsofthethreshold canbe incorporatedintothescaleoftheinputandthereforeisusu allynormalizedtoone. 3.2SimilarSamplingSchemes 3.2.1TimeEncodingMachine TheTimeEncodingMachine(TEM)proposedby LazarandToth ( 2003 )canbe consideredanasynchronous sampler.TheTEMconsistsofafeedbackloop thatcontainsanadder,anintegratorandanoninvertingSch mitttrigger.Theinputis constrainedtobebandlimitedandboundedsuchthat j f ( t ) j c < b .Theinputtothe integratorisbiasedbytheoutputoftheSchmitttrigger( b ),thereforetheintegralisa monotonicallyincreasingordecreasingfunction.Thisbeh aviorguaranteesaminimum andmaximumsamplingrate. Astheoutputoftheintegralincreasesfrom to theoutputoftheSchmitttrigger remainsat b .Whenthevaluereaches theSchmitttriggertogglesto b andthevalue 48

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oftheintegralstartstodecreasefrom to ,untilthenegativethresholdismet.This oscillationwillcontinueevenwithoutaninputtothesyste m.ThenaloutputoftheTEM isgivenbythetogglingtimesoftheSchmitttrigger.Simila rworkthatusesdutycycle hasbeendownby Roza ( 1997b ). TheTEMcanberelatedtotheIFmodelsincetheintegralofthe inputoverthe togglingtimeoftheSchmitttrigger( t k )canbewrittenas: Z t k +1 t k x ( u ) du =( 1) k [2 B ( t k +1 t k )] (3–5) ThereforethereexistsathresholdfunctionfortheIFthatw ouldproducethesameoutput astheTEM.Bothencodingschemesprovideanaveragevalueof thefunctionsovera setofintervals.Whenseeninthismannerwecanusetheframe workforreconstruction fromlocalaveragesthathasbeendevelopedintheareaofnon uniformsampling. ImplementationofthereconstructionalgorithmfortheTEM appearin Lazar ( 2005 ); Lazaretal. ( 2005 2008 ); LazarandToth ( 2003 2004 ).LazarhasalsoworkedwithIF models,butincontrastwiththeIFtheinputsignalisbiased suchthattheinputisalways positive.Thisapproachisnotfeasibleinourapplications inceweareconcernedwith bandwidthreduction.Buttherehavebeeninterestingexten sionsthatusepopulationsof TEMtoencodesinglechannelandmultichannelsignals,aswe llasrecoveryalgorithms basedonoptimizationresults.3.2.2SigmaDeltaConverter Asstatedbythesamplingtheorem,abandlimitedfunctionsa mpledabovethe Nyquistratecanbeperfectlyrecoveredusingasincinterpo lator.Bythisweimplicitly assumedthatthevaluesatthesamplinglocationsareknownp recisely.Unfortunately, forpracticalimplementations,thevaluesmustbequantize dtoaniteresolutionwhich introduceserror.Although,wecannotcorrectthiserror,w eareabletoreduceitby increasingtheresolutionofthequantizer. 49

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The samplersontheotherhandallowatradeoffbetweenresoluti onin amplitudeandtime.Thefewerbitsusedforquantizationcan becompensatedwitha highersamplingrateinordertoassurethesameaccuracyasa conventionaluniform sampler.Current convertersoversamplethesignalby100timestheNyquistra te. Thistradeoffisbenecialsinceitismoreconvenienttodes ignfastersamplersthan moreaccuratequantizers. AsinthecaseofconventionaluniformADC,afunctionsample dbythe convertercanberecoveredbylowpassltering.Thismethod isnotoptimalbutits simplicityandreliableresults Nguyen ( 1993 )makeitattractive.Thereasonlowpass lteringisanoptionbecomesevidentwhenlinearlymodelin gofthesampler. Itisdifculttoanalyzethesesystemsduetothenonlinearb ehaviorofthequantizer. Thiscanbeavoidedbyreplacingitwithasourceofrandomnoi se.Foradiscretetime usingtheZ-transform,wecanexpresstheoutputasanadditi onoftheinputanda highpassversionofthenoise. Y [ z ]= X [ z ]+(1 z 1 ) E [ x ] (3–6) Thispropertyofthe samplerisknownasnoiseshaping.Higherordersofnoise shapingarepossiblebyincreasingtheorderofthesamplerw hichbasicallyimplies replacingthequantizerwithaanother sampler. Althoughthenoisespectrumisshapedintothehigherfreque ncies,theoverall performanceofthereconstructiondependsonthesamplingr ate.Perfectrecovery usingalinearlterispossibleonlyifthesamplingrateten dstoinnity.Thereforein practicalapplicationsfasterconvergenceratesareprefe rred.Letusassumeafunction f ( x ) ,bandlimitedto hasbeenoversampled.Thisallowsustousemorelocalized reconstructionkernelsthatdecayfasterthan 1 = x asthe`sinc'.Oversamplingrestricts thespectrumof f ^ f to L 2 ( n, n) ,with > 1 .Thereforethereconstructionisgiven 50

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by: f ( x )= 1 X n f n g x n (3–7) with ^ g ( )= 1 p 2 for j j ^ g ( )=0 for j j > and ^ g 2 C 1 .Inrelationtotheoutput ofthe wemayaskthequestion;Howwelldoesthisseriesapproximat e f ifthe coefcientsarelimitedto q n 2f 1,1 g suchthat: f app ( x )= 1 X n q n g x n (3–8) Buthowshouldthecoefcients q n bedenedtoattainthebestapproximation?.One solutionisgivenbythe samplerwhichprovidesarecursivedenitionfor q n althoughnottheoptimalsolutionitprovidesaboundonthea pproximationerror: u n = u n 1 + f n q n (3–9) q n = sign u n 1 + f n (3–10) Theinitialcondition u 0 isarbitrarilychosenin ( 1,1) .IthasbeenshownbyDaubecheis DeVoreandDaubechies ( 2001 )thatthissequenceof f q n g isuniformlyboundedandthat for f bandlimited, > 1 ,and g satisfyingthepreviousconditionsthen: f ( x ) 1 X n q n g x n 1 jj g 0jj L 1 (3–11) Theerrorconvergestozeroas 1 = ,thiscanbeimprovedbyincreasingtheorderof thesampler,thereforethedenitionin 3–10 isnotusedinpractice.Thereconstruction resultscanalsobeimprovedbyusingadifferentrecoveryal gorithms Nguyen ( 1993 ), whicharecomputationallyexpensivebutprovidemoreaccur ateresultsatlower oversamplingrates. The converterhasalsobeenextendedtocontinuostimesignalsw hichallows acontinuousinput,neverthelessthequantizerstillremai nsclocked.Anasynchronous implementationofthe convertercorrespondstotheTEM. 51

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InrelationtotheIFsamplerthereareseveraldifferences. Thefeedbackinthe1-bit samplerisconstantandonlychangesinsignwhilethereseti ntheIFmodel returnstheintegraltoapredenedvalue.Intermsofnoises hapingwecanshowthata singleIFsamplerthatproducessamplerateswellabovetheN yquistboundalsoshapes thenoiseinthehigherfrequencies.Similarworkhasbeendo neonpopulationsofIF samplersandithasbeenshownexperimentallythatnoisesha pingoccursbyadding inhibitionbetweenthesamplers,whichguaranteesaunifor mspreadoftheoverlaid samples CheungandTang ( 1993 ).Furthermore,bothsamplersareusedwithadifferent purpose.The encodesthesignalinthesamplerate,whiletheIFhasbeen designedtoencodetheinformationintheprecisetimingsof thesamples. Whenusingaratecode,alineardecoderissufcienttorecov ertheoriginalfunction fromthe encoding.Therefore,linearoperationsonthesamplescorr espondto linearoperationsonthesignals.Theadditionoftwofuncti onscanbeperformedasan additionofthecorrespondingsamples daFonteDias ( 1995 ).Unfortunately,thislinearity breaksdownasthesampleratedecreasesandwetransitionfr omaratecodetoatime code. 52

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CHAPTER4 RECONSTRUCTIONALGORITHMS:INTEGRATEANDFIRESAMPLER IntuitiveapproachesforthereconstructionfromtheIFsam pleshavebeenproposed inthepast.Inthecasetheaveragingfunctionsareassumedt obeunitythesampled signalcanbeconsideredasastaircasefunction.Thisimpli esthattheareabetweentwo adjacentsamplesisgivenby = h ( t k +1 t k ) .Thisrelationcanbeusedtoestimatethe amplitudeoftheoriginalsignalateachpulsetime.Giventh esmoothnessconstraints theapproximationmustthenbelteredorprojectedintothe appropriatespace.Itisalso possibletoheuristicallydeterminearelationshipbetwee ntheoutputtimeintervalsand theinputamplitudesasproposedby AugustandLevy ( 1996 ). Theseapproachesonlyprovideapproximationstothesoluti on,weareinterested inperfectrecovery. Ferreira ( 2005a b )proposedanalgorithmforasimilarsampler,it wasassumedthethresholdwasvariableandtherewasnorefra ctoryperiod.These constraintscompletelychangethetheoreticalanalysisof theIF.Thedrawbackofhaving aconstantthresholdwillbecomeevident. Anaiverstapproachwouldassumethatlocalaveragesprodu cedbytheIF samplercanbetreatedunderthesameframeworkpresentedin section 2.5 .Thereforeif thesamplesproducedbytheIFaresufcientlydenseanouter frameisinducedandthe functioncanberecovered.Nevertheless,wecanprovethere doesnotexistsafunction inthebandlimitedspacethatwouldproducesuchasampleset whentheaveraging functionsischosentobeunity,theexponentialcaseistrea tedlateron. Theorem11. If f t k k 2 Z g issuchthatthe sup n 2 Z ( t k +1 t k ) < K < 1 then,thereisno function f 2 L 2 : R t k +1 t k f ( t ) dt = > 0 forall k 2 Z Proof1. If: R t k +1 t k f ( t ) dt = C thenbytheCauchy-Schwarzinequality: j j 2 = Z t k +1 t k f ( t ) dt 2 Z t k +1 t k j f ( t ) j 2 dt Z t k +1 t k 1 dt = Z t k +1 t k j f ( t ) j 2 dt ( t k +1 t k ) (4–1) 53

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Nowsince f 2 L 2 thatmeansthat R t k +1 t k j f ( t ) j 2 dt hastogotozeroforsomevalue n ,and therefore doesaswell. Thisimpliesthattheredoesnotexistabandlimitedfunctio nthatwouldproducea innitesamplesetwiththedeneddensityconstraints.The reforewecannotreconstruct abandlimitedsignalorinfactany L p signalfromthesamplesproducedbythetheIF. 4.1ApproximateReconstructioninBandlimitedSpaces Inchapter 3 wedenedtheoutputofthesamplerasanonuniformlyspacedp ulse train.Thepreciseringconditiondeterminingthepulses( WLOGweassume p = n )is givenby: = Z t k +1 t k f ( t ) e ( t t k +1 ) dt =: h f u k i (4–2) Wenowdenepreciselytheintegrateandresamplingscheme Assumption1. Abandlimitedfunction f 2 PW n andnumbers t 0 2 R > 0 aregiven. Here, PW n isthePaley-Wienerspace PW n := f 2 L 2 ( R ) supp( ^ f ) [ n,n] and ^ f ( w ):= R R f ( x ) e 2 iwx dx istheFouriertransformof f .Wecall t 0 the initialtime the ringparameter and the threshold .Usingtheseparametersweformallydene theoutputofthesampler.Werstdenerecursivelyaniteo rcountablesequence t 0 < ... < t j ... called thetimeinstants .Supposethattheinstants t 0 < ... < t j have alreadybeendenedandconsiderthefunction F j :[ t j ,+ 1 ) C givenby F j ( t ):= Z t t j f ( x ) e ( x t ) dx Observethat F j iscontinuousand F j ( t j )=0 .If j F j ( t ) j < ,forall t t j ,thentheprocess stops.If j F j ( t ) j ,forsome t t j ,bythecontinuityof F j ,wecandene t j +1 asthe minimumnumbersatisfyingtheequation Z t j +1 t j f ( x ) e ( x t j +1 ) dx = (4–3) 54

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Clearly,inthiscase t j +1 > t j Inthiswaywehavedenedaniteorcountablesequenceofpoi nts t 0 < ... < t j ... WewillproveinProposition 4 thatthissequenceisinfactnite(thatis,thattheintegra te andreprocesscannotgoonforever.)Letusassumethisfact forthemomentand numbertheinstants f t 0 ,..., t n g .Letusalsodenethe samples f q 1 ,..., q n g by, q j := Z t j t j 1 f ( x ) e ( x t j ) dx ,(1 j n ). (4–4) Observethat,bythedenitionofthetimeintervals, j q j j = Theoutputofthesamplerisformallygivenbythetimeinstan ts f t 0 ,..., t n g andthe numbers f q 1 ,..., q n g .Wesaythatthisoutputhasbeenproducedbythe integrateand rescheme .Inthecaseoftheapplicationthatmotivatedthissampling scheme,the signalisreal-valuedandtheoutputofthesamplerisencode dasatrainofimpulses, whereonlythesignofthesamples q j isstored. 4.1.1SomeRemarksontheIFOutput Wenowprovethatthesetoftimeinstantsisindeedniteandg ivesomebounds onitsdistribution.Tothisendweintroducesomeauxiliary functionsthatwillbeused throughouttheremainderofthearticle. Considerthefunction g : R R givenby g ( x )= e x [0, 1 ] anddene, v ( t ):=( f g )( t ):= Z t 1 f ( x ) e ( x t ) dx (4–5) Since g 2 L 1 ( R ) v 2 PW n .IntheFourierdomain, v and f arerelatedby ^f ( w )= ( 2 iw + ) ^ v ( w ). (4–6) Inthetimedomain,thiscanbeexpressedas f ( t )= @ v ( t ) @ t + v ( t ). (4–7) Observation1. Thefunction v iscontinuousand v ( t ) 0 ,when t !1 55

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Proof2. Wehavealreadyobservedthat v 2 PW n .Since ^ v 2 L 2 and supp(^ v ) [ n,n] wehavethat ^ v 2 L 1 andtheconclusionfollowsfromtheRiemann-LebesgueLemma Thefollowingstraightforwardequationrelates v totheintegrateandreprocess. Z t s f ( x ) e ( x t ) dx = v ( t ) e ( s t ) v ( s ), s t (4–8) WecannowprovethattheoutputoftheIFprocessisnite. Proposition4. UnderAssumption1,thefollowingholds. (a) Thesetoftimeinstantsproducedbytheintegrateandresch emeisaniteset f t 0 ,..., t n g (b) Thenumbersoftimeinstants t j inagivenniteinterval [ a b ] isboundedby k f k 2 ( b a ) 1 = 2 +1. (c) If f isintegrable,thetotalnumberoftimeinstantsisboundedb y k f k 1 +1. Proof3. Werstprove(b)and(c).Let [ a b ] beanintervalandlet f t j ,..., t j + m 1 g be m consecutivetimeinstantscontainedin [ a b ] .If m 1 theboundistrivial,soassumethat m 2 .Foreach 0 k m 2 ,usingEquation ( 4–3 ) wehave, = Z t j + k +1 t j + k f ( x ) e ( x t j + k +1 ) dx Z t j + k +1 t j + k j f ( x ) j dx Summingoverthe m 1 intervalsdeterminedbythepoints f t j ,..., t j + m 1 g wehave, ( m 1) Z b a j f ( x ) j dx (4–9) Letting a = 1 and b =+ 1 yields(b).For(a),H ¨ older'sinequalitygives, ( m 1) k f k 2 ( b a ) 1 = 2 56

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andtheconclusionfollows. Nowweprove(a).AssumeonthecontrarythattheIFprocessgo esonforever producinganinnitesetofinstants f t j : j 0 g .Given s > t 0 ,bypart(b),onlyanite numberofinstants t j belongto [ t 0 s ] .Therefore t n + 1 ,as n + 1 .Using Equations ( 4–8 ) and ( 4–3 ) itfollowsthat, = Z t j +1 t j f ( x ) e ( x t j +1 ) dx = v ( t j +1 ) e ( t j t j +1 ) v ( t j ) j v ( t j +1 ) j + j v ( t j ) j ThiscontradictsObservation 1 4.1.2Reconstruction Wenowaddresstheproblemofapproximatelyreconstructing abandlimitedfunction fromtheintegrateandreoutput.Sincethesamplesaretake ninthehalf-line [ t 0 ,+ 1 ) wewillmakesomeassumptionaboutthesizeof f beforetheinitialinstant.Roughly speaking,thisassumptionmeansthattheintegrateandrep rocesswouldnothave producedanysampleintheinterval ( 1 t 0 ] Assumption2. ThefunctiondenedinEquation ( 4–5 ) satises, j v ( t ) j forall t t 0 NotethatbyObservation 1 ,any t 0 0 satisesthisassumption.Toapproximately reconstruct f wewillrstapproximatelyreconstruct v fromtheintegrateandreoutput andthenderiveinformationabout f bymeansofEquation( 4–6 ).Wewillusethe structureoftheIFprocesstoproduceanumberofapproximat esamplesfor v Firstwearguethat,fromtheoutputoftheIFprocess,wehave enoughinformation toapproximate v onthetimeinstants f t 0 ,..., t n g .RewritingEquation( 4–4 )intermsof v (cf.Equation( 4–8 ))wehave, v ( t j +1 )= e ( t j t j +1 ) v ( t j )+ q j +1 ,(0 j n 1). (4–10) 57

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Sincethevalue v ( t 0 ) maynotbeexactlyknownwecannotdeterminefromthis recurrencerelationallthevalues v ( t j ) .However,wecanconstructanapproximation tothesevalues.Let w 0 :=0 anddenerecursively, w j +1 = e ( t j t j +1 ) w j + q j +1 ,(0 j n 1). (4–11) ObservethatAssumption 2 impliesthat j w 0 v t 0 j .Usingthisestimateasastarting pointwecaniterateonEquation( 4–10 )and( 4–11 )toget, j w j v ( t j ) j ,(0 j n ). (4–12) Consequently,usingonlytheoutputoftheIFsamplingschem e,wehaveconstructeda setofvalues f w 0 ,..., w n g thatapproximates v ontheinstants f t 0 ,..., t n g .Thesecond stepistoapproximate v onanarbitrarypointof R Tothisendobservethat,accordingtothedenitionof t j astheminimumnumber satisfyingEquation( 4–3 ),wehavethat, Z t t j f ( x ) e ( x t ) dx forall t 2 [ t j t j +1 ] Rewritingthisequationintermsof v (cf.Equation( 4–8 ))gives, v ( t ) e ( t j t ) v ( t j ) forall t 2 [ t j t j +1 ] .(4–13) CombiningthislastequationwithEquation( 4–12 )yields, v ( t ) e ( t j t ) w j 2 forall t 2 [ t j t j +1 ] .(4–14) Wenowshowthatthisequationallowsustoapproximate v anywhereontheline. Claim1. Givenanarbitrarytimeinstant t 2 R ,choose x 2 R d inthefollowingway: (a) if t < t 0 ,let x :=0 (b) if t belongstosome(unique)interval [ t j t j +1 ) ,let x := e ( t j t ) w j (c) if t t n ,let x := e ( t n t ) w n 58

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Then, j v ( t ) x j 2 Remark1. Observethattheproceduretoobtain x from t dependsonlyontheoutputof theIFprocess. Proof4. Forcase(a),theconclusionfollowsfromAssumption 2 .Forcase(b),the conclusionfollowsfromEquation ( 4–14 ) .Forcase(c),thefactthatthereconditionis neversatisedafter t n gives, v ( t ) e ( t n t ) v ( t n ) (4–15) CombiningthisestimatewithEquation ( 4–12 ) ,theconclusionfollows. Wewillnowchooseawindowfunction. Assumption3. ASchwarzclassfunction suchthat ^ 1 on [ n,n] ,and, ^ iscompactlysupported, hasbeenchosen. Since v 2 PW n ,theclassicoversamplingtrickforbandlimitedfunctions (seefor example Feichtinger ( 1992 )or FeichtingerandGr ¨ ochenig ( 1994 ))impliesthatthere existsanumber 0 << (2n) 1 suchthat v = X k 2 Z v ( k ) ( k ). (4–16) UsingtheproceduredescribedinClaim 1 ,weproduceaset f s k g k 2 Z suchthat j v ( k ) s k j 2 forall k 2 Z (4–17) Let bethefunctiondenedby, ^ ( w )= ( 2 iw + ) ^ ( w ). (4–18) 59

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Itfollowsthat isalsoaSchwarzfunction.Moreover,usingEquation( 4–6 )wehave that, f = X k 2 Z v ( k ) ( k ). (4–19) Observethat,since v 2 PW n ,thesequence f v ( k ) g k 2 ` 2 andtheseriesinEquation ( 4–19 )convergesin L 2 anduniformly-infact,itconvergesintheWieneramalgamno rm W ( C 0 L 2 ) ,seeforexample Feichtinger ( 1992 ), FeichtingerandGr ¨ ochenig ( 1994 )and AldroubiandGr ¨ ochenig ( 2001 ).) Nowwecandenetheapproximationof f constructedfromtheIFsamples.Let, ~ f := X k 2 Z s k ( k ). (4–20) Since,byEquation( 4–17 ),thesequence f s k g k isboundedand isaSchwarzfunction, itfollowsthatEquation( 4–20 )denesaboundedfunctionandthattheconvergenceis uniform(see Feichtinger ( 1992 )or AldroubiandGr ¨ ochenig ( 2001 ).) Thereconstructionalgorithmconsiststhenofcalculating theapproximated samples f s k g k followingClaim1andthenconvolvingthemwiththekernel ,that canbepre-calculated. Wenowgiveapreciseerrorboundforthereconstruction. Theorem12. UnderAssumptions1,2and3,thefunctiondenedbyEquation ( 4–20 ) satises, k f ~ f k 1 C forsomeconstantCthatonlydependson n andthewindowfunctionchoseninAssumption 3 Proof5. AccordingtoEquations ( 4–19 ) and ( 4–20 ) k f ~ f k 1 supess X k 2 Z j v ( k ) s k jj ( k ) j 2 supess X k 2 Z j ( k ) j 60

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Itsufcestodene C :=2sup P k 2 Z j ( k ) j .Since isaSchwartzfunction, C < + 1 Remark2. Wecurrentlydonotknowwhatchoiceofthewindowfunction minimizes theconstantinthetheorem.Amoredetailedstudyofthechoi ceofthewindowfunction shouldnotonlyconsiderthesizeofthatconstantbutalsoth erateofconvergenceofthe seriesinEquation ( 4–20 ) 4.1.3NumericalExperiments Westudythebehaviorofthereconstructionalgorithmunder variationsinthe thresholdandtheoversamplingperiodforaspecicchoiceo freconstructionkernel Thetestsignal f isofnitelengthandrealvalued,producedasalinearcombi nationof ve`sinc'kernels( sin( x ) = ( x ) )ata1Hzfrequency,withrandomlocationsandweights. Theamplitudeoftheinputhasbeennormalizedto1.Thissign alisencodedbytheIF samplerwith =1 andrecoveredusingtheproceduredescribedinSection 4.1.2 .The reconstructionkernel isaraisedcosine,denedby, ( t )=sinc( t = T s ) cos( r t = T s ) 1 4 r 2 t 2 T s 2 (4–21) where r =0.5 and T s =0.25 aredeterminedbythemaximuminputfrequency n and thedesiredoversamplingperiod (cf.Equation( 4–16 ).)Figure 4-1A showstheraised cosine inthetimeandfrequencydomain.Observethatthespectrumo f isconstant forfrequencieslessthantheinputbandwidthandthendecay ssmoothlytowardszero. Thecorrespondingkernel (cf.Equation( 4–18 ))isshowninFigure 4-1B Using werecover ~ f (cf.Equation( 4–20 ).),anapproximationof f asshownin Figure 4-2A .Asexpectedtheerrordecreasesinregionswithhighdensit yofsamples. ThisbehaviorisevidentfromFigure 4-2B ,thedenseregionsimplythattheuniform sampleswillmostlikelycoincidewiththeestimatedvalues of v ( t ) atthesample locations.Ontheotherhand,forsamplesthatarefarapartt heapproximationfollowsa exponentialdecayfromtheitsoriginalvaluewhichisnotth enaturaltrendinthesignal. 61

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1 1.5 2 2.5 -0.5 0 0.5 1 Time [s] 0 1 2 3 4 5 6 0 0.5 1 1.5 Frequency [Hz]Magnitude A 1 1.5 2 2.5 3 3.5 4 -10 -5 0 5 10 Time [s] 0 1 2 3 4 5 6 0 0.2 0.4 0.6 0.8 1 Frequency [Hz]Magnitude B Figure4-1.Reconstructionkernels. Figure 4-2B shows v ( t ) (solidline),andtheapproximatedsamplesof v onthelattice Z ,constructedusingtheproceduredescribedinClaim 1 called f s k g k andtheenvelope v ( t ) wherethesesamplesareknowntolie(dashedline.) Currentlythereconstructionalgorithmusestheapproxima tedsamplesof v ( t ) atthepulselocationstodenethepiecewiseexponentialbo undandestimatethe reconstructioncoefcientsontheuniformlattice.Basedo nthenumericalexperiments thealgorithmcanbeimprovedbyincludingtheestimatedval ueof v ( t ) atthepulse locationsalthoughitimpliesreconstructiononanonunifo rmgrid. 0 2 4 6 8 10 -3 -2 -1 0 1 2 Original Recovered 0 2 4 6 8 10 -0.05 0 0.05 Time [s] AReconstructionof f ( t ) fromtheimpulsetrain. 0 2 4 6 8 10 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 Time [s] v(t) approx Samples samples v(t) bounds BReconstructionof v ( t ) with =1 = 4 =0.05 Figure4-2.Reconstructionresults. 62

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ForbothcasessimilarerrorboundscanbedenedasinTheore m 12 .Thevariation oftheerrorinrelationtothethreshold(pulserate)isshow ninFigure 4-3A .The errordependsonthechoiceofgeneratorandtheoversamplin gperiod ,asseenin Figure 4-3B .Therelationshipbetweenthekernelsandtheoptimalovers amplingperiod isstillnotevident. 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 ThresholdError 0 50 100 150 200 250 300 350 400 450 0 0.2 0.4 0.6 0.8 1 1.2 1.4 Pulse rate AErrorinrelationtothethresholdwith =0.25 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Oversampling [ b ]Error q = 1e-3 q = 1e-2 q = 5e-2 BErrorinrelationtotheoversamplingfordiffer-entthresholds. Figure4-3.Variationoftheerrordenedas jj f b f jj 1 4.2ReconstructioninFiniteDimensionalSpaces Inthiscaseweassumetheinputlivesinanitedimensionals pace.Recallthatthe IFsamplescanbetranslatedintopointevaluationsofthein tegral v ,whichfurthermore hastobeconstrainedintoapiecewiseexponentialtubewhen weassumea zero refractoryperiod .Furthermore,theintegral v isalowpasslteredversionoftheinput x suchthat: x ( t )= @ v ( t ) @ t + v ( t ) (4–22) Hereweareconcernedwithpracticalrecoveryalgorithms.O nlyanitenumberof samplesareavailabletoapproximatethenitelengthinput signal.Wecomparethree reconstructionmethods.Allthesemethodsrelyona`leasts quares'approximationtothe modelparameters.Thersttwoassumesmodelsontheintegra l v i.e.acubicB-spline modeloraradialbasisfunctionwithcubicB-splinesasitsg enerators.Oncetheintegral 63

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isapproximatedwecaneasilyobtaintheoriginalinput x (usingequation 4–22 ).The thirdapproachusescomplexexponentialstoapproximateth einputdirectly.Thebasic inputmodelforall3methodscanbewrittenas: s ( t )= M X j =0 c j j ( t ) (4–23) Where s ( t ) canrepresentboththeinput x orintegral v .Thecoefcientsarerepresented by c j and j representsthegenerator.InthecaseofFourierbasis j = e i 2 jt = T ,with Tbeingthedurationofthesegmentwewanttorecover.Unifor mCubicB-splinescan beexpressedintermsofashiftinvariantgenerator j = N 3 ( t Tj ) ,where T denotes thedistancebetweenthegeneratorcenters(a.k.aknots).S imilarly,theradialbasis functionsarecubicB-splineshiftedtothesamplelocation s, j = N 3 ( t r j ) .Theuniform cubicB-spline N 3 ( t ) isdenedas: 1 6 8>>>>>>>>>>>>>><>>>>>>>>>>>>>>: u i ( t ) 3 if t 2 [ t i t i +1 ] 3 u i +1 ( t ) 3 + 3 u i +1 ( t ) 2 +3 u i +1 ( t )+1 if t 2 [ t i +1 t i +2 ] 3 u i +2 ( t ) 3 6 u i +2 ( t ) 2 +4 if t 2 [ t i +2 t i +3 ] (1 u i +3 ( t )) 3 if t 2 [ t i +3 t i +4 ] 0 Otherwise (4–24) where u i ( t ):= t t i t i +1 t i Independentofthemodel,eachpairofIFsamplesproducesal inearconstrainon thecoefcients.Theconditionsonperfectrecoveryaredet erminedbytheinvertabilityof thesamplingmatrix S .Whichincludestheeffectsofthesamplersonthegenerator .In vectornotationwecanwrite: = Sc (4–25) Where c =[ c 1 ,..., c M ] T =[ q 1 ,... q N ] T T denotesthetransposeandthe sampling matrix S .Inthecasewemodeltheinput x usingasetofcomplexexponentials S is 64

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denedas: S ( k j ) = Z t k +1 t k + e i 2 jt = T e ( t t k +1 ) dt (4–26) Ontheotherhandifwemodeltheintegral v ,then S fortheuniformcubicB-splinesis denedas: S ( k j ) = ( t k +1 jT ) ( t k + ) e ( t k + t k +1 ) (4–27) andinthecaseoftheradialbasisfunctionsas: S ( k j ) = ( t k +1 r j ) ( t k + ) e ( t k + t k +1 ) (4–28) Inthiscaseweneedtofurtherprocess v toobtain x usingtherelationshipgivenin 4–22 Theconditionsonperfectrecoverydependontheinvertabil ityofthesematrices. Whichinturndependsonthesamplelocationsandtheselecte dbasis.Thecompact supportofthebasisfunctionsintroducesabandedstructur einthematrixwhichallows ustoefcientlygeneratethismatrix.Itiswellknownthatt heinverseofbandedmatrices havethepropertythatitsvaluesdecayexponentiallyoffth ediagonal Demkoetal. ( 1984 ).Thisimpliesthatthereconstructionofanyregiononlyde pendsonthesamples initsvicinity Gr ¨ ochenigandSchwab ( 2003 ).Hereweuseaconjugategradient Hager ( 1988 )approachtosolvethelinearsystem. Incontrastifthegeneratorswereinnitelysupportedthen itwouldbelikely thatallentriesinthematrixbenonzero.Theconditionnumb erofthematrixmainly dependsonthesampledistributionssincelargegapsinthes amplesetleadtorepeated columns.Sincethegeneratorsarecompactlysupported,ino rderforthismatrixtobe wellconditioneditisnecessarythataatleastonesamplefa llinthesupportofeach generator,thisconstrainsthenumberofgeneratorsaswell asthesamplesfavoring uniformdistributions.Nevertheless,thesamplesetsgene ratedbytheneuralsignalwill 65

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producedenseclustersofsampleswithveryfewinter-clust ersamples.Furthermore, itishighlyunlikelythatthenumberofsamplesmatchthenum berofbasisfunctions. Thereforeill-conditionedmatricesareinevitable.Typic allythenumberofunknownswill overwhelmthenumberofconstraintsleadingtoaninnitenu mberofsolutions.Ifthe regionofinterestwereknown apriori thenitismorelikelythatthesamplingmatricesbe overdetermined.Nevertheless,detectingtheseregionsis aprobleminitself. Thesethreereconstructionalgorithmshavebeendescribed intermsofanite lengthinput.Nevertheless,inrealapplicationstheinput dataisstreaminganddecisions needtobemadeonline.Speciallysincethegoalistoapplyth esesamplingschemes inBMIs.Thereforeweuseanoverlappingwindowapproachsim ilartomoving-least squares Levin ( 1998 ); Wendland ( 2005 ).Givenawindow,allsamplesinthesupportare usedtorecovertheinput.Theboundariesoftherecoveredsi gnalareignoredinorder toavoidartifacts.Thewindowisthenshiftedandtheproces srepeats.Theshiftandthe windowsizearecarefullychoseninordertoproduceasmooth reconstruction.Inthis casewecanalsosaveoncomputationsinceconsecutivewindo wssharemostofthe samples,thereforethesamplingmatrixdoesnothavetobere computedentirely.Many oftherowsandcolumnscanbereuseddependingontheoverlap .Weonlyneedto computethecontributionsofthenewsamplesandbasisfunct ionsthatenterthewindow. Itisalsopossibletopre-computethesamplingmatrixifwea rewillingtoquantizethe locationofthesamples.Bydoingsowecandeterminethesamp lingmatrixforall samplelocations.Thengivenaspecicsamplesetwewouldex tractthecorresponding rowsfromthematrixandassumethosetoformthesamplingmat rix.Althoughthisis feasibleitismemoryintensive.Thereconstructionalgori thmthenconsistsofsolvinga leastsquaresproblemateachwindowbyconjugategradients Hager ( 1988 ). 4.3ConstrainedB-splines Althoughperfectreconstructioncanbeguaranteedforanin vertiblesamplingmatrix. Thismatrixistypicallyill-conditioned(speciallyifwea relookingforcompression),inthis 66

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casewecanalsousethepriorinformationthatthesolutionm ustsatisfyboththesample constraintsandboundaryconditions.Inthissectionwesho wthatthiscanbedoneby carefullythreadingB-splines.Wehavechosentousespline functionssincetheycan beeasilyboundedbylinearenvelopes LutterkortandPeters ( 2000 ); Peters ( 2003 )and hencethreadedthroughachannel.Otherwiseourinterpolat ingfunctionwouldhaveto becheckedateverypointwhichwouldleadtoaninnitevaria tionalproblem. RecentworkbyLutterkort Lutterkort ( 2000 )hasaddressedsimilarproblems.In ordertosolvethechannelproblemthesplineisrstconstra inedtoatighterlinear enclosureknownasasleeveorslefe Peters ( 2003 ),althoughtheenclosuredoesnot havetobenecessarilylinear,thisreducesndingthesplin econtrolpointsthatsatisfy thechannelproblemtoalinearprogram.Basedon Lutterkort ( 2000 )theenclosurecan beconstructedbyimplementingthefollowingsteps: 1. Choosetheinputspace B andtheenclosuresspace H 2. Selectadifferenceoperator ,with ker ()= B\H 3. Determine K suchthat K = I 2 R s 4. Computethebounds b K c (lower)and d K e (upper) Althoughthesestepsaregeneral,wewillonlyconsiderB-sp linesandboundtheseby linearfunctions.Incasewechoosetheknotsequenceasunif ormorinvariant,step (4)canbecomputedofineandthereforethesleevecanbecon structedapplying thedifferenceoperatortothespline.Thisoperatorissimp lythesecondderivative,or equivalentlytheseconddifferencesofthecontrolpoints. Dependingonhowtheaniti-differenceoperator K isbounded,differentenclosures canbeobtained.Weillustratetwotypesofenclosure,ther stforcestheenclosureto completelysurroundthespline,wewillrefertothisasthe“ loosesleeve”(Figure 4-4A ) LutterkortandPeters ( 1999 ).Thesecondapproachprovidestighterboundsandis referredtoas“tightsleeve”(Figure 4-4B ) LutterkortandPeters ( 2001 ); Peters ( 2003 ). 67

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Whentheknotsareequallyspacedthetighterboundscanbepa rtiallycomputed ofine.TableswiththeprecomputedboundsforB-splinesar ereferencedin Petersand Wu ( 2004 ).Usingthesepre-computedboundstighterenclosuresarep ossibleforthe B-splinesasseeninFigure 4-4B .Unliketheloosesleeves,inthiscase,bothvaluesof thecontrolpolygonattheGrevilleabscissaearealtered.I nordertoapplythesleeves 0 1 2 3 4 5 6 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 Knots Spline Upper bound Lower bound Control Polygon ALoosesleeve. -1 0 1 2 3 4 5 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 Knots Spline Upper bound Lower bound Control Polygon BTightsleeve. Figure4-4.DifferenttypesoflinearsleevesforB-splines tosolvethechannelproblem,alinearapproximationtothee xponentialboundisneeded (Figure 4-5 ),wehaveusedsimplelooseboundsfortheexponentialfunct ion.Thelower boundtotheexponentialisdeterminedbycombiningtheline 1+ x whichbounds e x withaconstantlineequaltothenalvalueoftheexponentia lduringthatsegment.The upperboundconsistsoftwolines,fromthepointalongtheex ponentialthatmatchesa predenedderivativevaluetobothextremes,ifthepointwe retobeoutsidetherange forthatsegmentthetwoendpointsoftheexponentialarecon nected.Theuseofthis approximationnearlyduplicatestheerrorboundfortherec onstructionintheworst casescenario.Sinceboththechannelandthesleevearepiec ewiselinear,inorder tosatisfythechannelconstraintsweonlyneedtobeconcern edwiththebreakpoints. Notethatboththechannelandthesleeve'sbreakpointsmayn otcoincide.Thereforewe mustensurethatthebreakpointsoftheuppersleevearelowe rthantheupperchannel. Furthermore,thebreakpointsoftheupperchannelmustbeab ovethecorresponding 68

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valueoftheuppersleeve,similarlyforthelowersleeve.Al thoughwehaveasolution forthechannelproblemwemustalsoensurethecurveinterpo latesthesamplepoints. Thereforewemustaddthelinearsetofconstraintsimposedb ythesamples.The 1.8 2 2.2 2.4 2.6 2.8 3 x 10 -3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 x 10 -9 Original U exp chan L exp chan L lin chan U lin chan samp knots Figure4-5.Exponentialchannelandlinearapproximation. reconstructionalgorithmshavebeenappliedtoneuralreco rdingsfrombehavinganimals treatedintheapplicationchapters.Figure 4-6A ,showstheapproximatedlinearbound, alongwiththesamples,theestimatedcontrolpointsandthe solution.Asexpectedthe splineinterpolatesallsamplepointswhileremaininginsi dethechannel.Notethatthe sampleswillalwayslieattheboundaryofthechannelbycons truction.Inthecaseof negativesamplesitwillmatchthelowerchannelboundary.F igure 4-6B ,showsthe originalsignal f andreconstructed ^ f .Inthiscasethesamplerateis 6 Ksps,wellunder theNyquistbound( 10 Ksps)andyetthesignaltoerrorratio(SER 10log( P s = P e ) )is nearly35dBintheregionofinterest,where P s isthepowerinthesignal f and P e isthe powerinthedifference( f ^ f ).Sincetheknotsareconnedtotheintervalinwhich thefunctionisdenedtheedgeswillnotbecoveredbythesup portofallthenecessary B-splinesandthereforetheseregionsarenotconstrainedl eadingtotheoscillationsas seeninFigure 4-6B .Theselectionoftheknotlocationswillalsodependonthei nput signal,inthiscasewechoosetheperiodasathirdofthereci procalofthebandwidth oftheinput.Dependingonthesamplelocationsandtheirden sitytheirmaynotexista 69

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9 9.5 10 10.5 11 x 10 -3 -5 -4 -3 -2 -1 0 1 2 3 x 10 -9 Original L chan U chan rec samp ctrl P AActionpotential 0 0.002 0.004 0.006 0.008 0.01 0.012 -5 -4 -3 -2 -1 0 1 2 3 4 x 10 -5 SER = 35dB Original Reconstructed 0 0.002 0.004 0.006 0.008 0.01 0.012 -1 0 1 x 10 -9 BNeuralrecording Figure4-6.Reconstructionexampleusingneuraldata. solutionthatsatisestheconstraints,althoughfurtherr enementoftheknotsmaylead toasolutionitmayalsoincreasethesmalloscillationsint hesplineasitmovesthrough thechannelsincewehaveincreaseditsdegreesoffreedom.T hebehaviorofthespline canalsobeaffectedbythequantityweareminimizingthroug hthelinearprogram. Differentobjectivefunctionscanleadtosolutionswithle ssoscillationsasseenin Myles andPeters ( 2005 ).Thedrawbackofusingthesereconstructionalgorithmsis theneed forheavycomputationalpower,thereforelimitingtheuseo fthisalgorithminreal-time BMIapplications. 70

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CHAPTER5 APPLICATION:RECOVERYOFNEURLRECORDINGS AsuitablecandidatefortheIFencoding/decodingparadigm istheefcient compressionofneuralrecordings.Crucialforimplantable wirelessrecordingdevices usedinstateoftheartBMIs.Throughoutthischapterweprov ide: 1. Characterizationofthedataratesandreconstructionaccu racyintermsoftheinput featuresaswellasthesamplerparameters. 2. Spikesortingandclassicationresultsundervariationin thesamplerparameters forreconstructedneuraldata,recordedfrombehavinganim als. 3. Classicationresultsonabinnedrepresentationofthepul seswhichavoids recovery. Theminiaturizationrequiredforinterfacingwiththebrai ndemandsnewmethodsof transformingneuronresponses(actionpotentialsa.k.a.s pikes)intodigitalrepresentations. Relevantinformationforbrainmachineinterfaces(BMIs)i sencodedinthethetimings oftheactionpotentialsgeneratedbyneurons.Theseneural patternsaredecodedand relatedtotheintendedactionsbythesubject.Thereforewi thoutareliabledetectionand classicationoftheactionpotentialsthissystemisnotfe asible. Thereareanumberofapproacheswheninterfacingwiththebr ain.Eachhasits tradeoffsintermsoftheinformationweobtainandtheeffec tsonthepatient.These varyfromnon-invasivemethodssuchasEEG(Electroencepha lography),toinvasive techniquesusingmicro-electrodes,whichallowustomeasu retheactionpotentialrings fromeachneuroncell.Itisbelievedthatthetimingofthese actionpotentialencodes informationinthebrain RollsandTreves ( 2011 ).HerewefocusonBMIsbasedon micro-electroderecordings.Theserecordingsareobtaine dbyintroducingmicro-wires intothespecicregionsofthebrain.Typicallythenumbero frecordingelectrodesvaries betweenthetens Leeetal. ( 2010 )andhundreds Azizetal. ( 2009 ); Harrisonetal. ( 2007 ); Shahrokhietal. ( 2009 )andcanuseseveralrecordingsites Azizetal. ( 2009 ). 71

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Themicro-electroderecordingsarecharacterizedbyhigha mplitudetransients(a.k.a actionpotentials)overlaidoveralowamplitudenoiseoor (duetodistantneurons). 5.1CurrentMicro-ElectrodeRecordingSystems AlthoughBMIshavegreatpotentialinhumanhealthcare,the reisaconstant tradeoffbetweentheamountofneuralactivityrecordedand thehardwareand experimentallimitations.Furthermore,mostresultsarec urrentlyobtainedfrombehaving animals,mainlyprimatesandrats Gosselin ( 2011 ).Inordertosatisfysize,powerand datarateconstraints,mostofthesesystemsaretethered SanchezandPrincipe ( 2007 ). Thislimitstherangeofexperimentsthatcanbeperformedon theanimal,aswellas,the qualityofthedatarecordedduetomovementartifacts.Inco ntrast,others Bashirullah etal. ( 2007 ); Harrisonetal. ( 2007 2009 ); YinandGhovanloo ( 2009 )haveproposed fullyimplantablewirelessinterfacesatthecostofreduci ngthenumberofelectrodesand addingseverearea,bandwidth,heat,resolutionandpowerc onstraints. Inordertoincreasethenumberofrecordingchannelsinthes ewirelesssolutions wemustfurthertradeoffbetweenthedelityoftheactionpo tentialrecoveryandthe hardwareresources.Inthecaseofsimplehardwaresolution s,on-chipspikedetectors Chaeetal. ( 2009 ); RogersandHarris ( 2004 )havebeendesignedwhichwillonly transmitthetimingofanactionpotentialandtheneuronlab el.Althoughthisreduces thedataratesitisuncertainweatherthereceiveddataisre liable.Incontrast,others haveuseddigitalsignalprocessorstoperformcompression attherecordingsite.The drawbackisthatthehardwarecannotbefullyimplantedandi sstillsusceptibleto movementartifacts Gohetal. ( 2008 ). Thesecompressiontechniquesareindependentfromthesamp leacquisition.An alternativeapproachcombinesbothcompressionandsampli nginordertosatisfythe designconstraints.Theseadaptivesamplingschemes,such astheIntegrateandFire (IF) Bashirullahetal. ( 2007 ),deltamodulators Tangetal. ( 2010 )orasynchronous modulators(ASDM) Azizetal. ( 2009 ); Senayetal. ( 2009 )usenonuniformsamplesets 72

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thatdependonthespecicfeaturesoftheneuralrecordings .Compressionisattained sinceactionpotentialsarelocalizedintimeandoccurinfr equently. WehaveshownthattheIFsamplerprovidesanaccuraterepres entationofthe actionpotentialsevenatsub-Nyquistrates Alvaradoetal. ( 2009 ); AlvaradoandPrincipe ( 2011 ).Theseresultsareencouraging,neverthelesstheinputch aracteristicswilldene thelimitsoncompression.Themainfactorbeingtheringra teoftheneuronsandthe signaltonoiseratio.Forexample,assumingtheneuronrin grateis 100 Hz andeach actionpotentialisassigned10samplestheaveragesampler ateis 1 Ksamp = s ,well undertheNyquistbound( 10 Ksamp = sec ). Inthissectionwerstdescribethecharacteristicsofneur alrecordings,andthe regionsofinterestdenedasactionpotentials.Thenthebe haviorofthesamplerand thereconstructionalgorithmsforneuralrecordingsisqua ntied.Sincethesampleris inputdependentweprovideadescriptionoftheenvelopeofp erformance(compression, recoveryaccuracy)fortheIFsamplerinrelationtotheconv entionalinputdescriptors suchassignaltonoiseratio(SNR)andtheireffectonthedat arates.Furthermore, theparametersofthesampleralsoplayanimportantrole,wh icharequantied againinrelationtothedataratesandtherecoveryaccuracy .Finallywemeasure thediscriminabilityintheactionpotentialsbytheclassi cationerrorinthesortingofthe spikes. 5.2DataDescription:NeuralRecordings Althoughactionpotentialsproducedattheneuronaxonhill ock DayanandAbbott ( 2005 )arealmostidenticalinshapeacrossdifferentneurons.Th eyaredistortedat therecordingsiteduetotissueeffects SanchezandPrincipe ( 2007 ).Thetoppanelin Figure 5-1 showsanexampleofanactionpotentialgeneratedattheneur on,modeled bytheHodgkinHuxleyequations.Thebottompanelshowsthee xtra-cellularrecording asthesecondderivativeoftheoriginalactionpotential. 73

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10 15 20 25 30 35 40 45 50 -50 0 50 100 Voltage [mV] 10 15 20 25 30 35 40 45 50 -1 -0.5 0 0.5 Time [ms]Normalized amplitude Figure5-1.Simulatedintra-cellularactionpotentialgen eratedbytheHogkin-Huxely modeled(Toppanel).Bottompanelcorrespondstothenormal izedsecond derivativeoftheactionpotential,whichmatchesthebasic cablemodel. ComparingthebottompanelwithFigure 5-2A itisevidentthetheshapes aresimilar. Nevertheless,thereareanumberofdifferentparametersan dartifactsthat affectthenalrecording.Figure 5-2 ,showsrealneuralrecordingsobtainedfroma behavinganimal.Inthiscasethenoiseregionsinbetweenth eactionpotentialsarethe cumulativeeffectofallbackgroundneurons.Furthermore, theelectrodesandrecording hardwareintroducenoiseaswell. Notethatthespectrumoftherecordingsisbandlimitedatap proximately5KHz. WhichdenesthesamplingfrequencyinmostconventionalBM Is.Inaconventional BMItheserecordingsarefurtherprocessedandeachactionp otentialisdetectedand assignedtoaspecicclass(neuron).Thetimeinstantsofth eactionpotentialsare fundamentalinthedecodingprocess RollsandTreves ( 2011 ). Allresultspresentedinthissectionarebasedonneuraldat arecordedbythe formerNeuro-prosotheticsresearchgroupattheUniversit yofFlorida(currently attheUniversityofMiami).Weused 60 secondsofdatarecordedat 12207.03125 samplespersecondwitha16-bitsampleresolutionusingamo dulefromTuckerDavies Technologies. 74

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0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 -6 -5 -4 -3 -2 -1 0 1 2 3 4 x 10 -5 Time [s]Amplitude [Volts] ATimedomain. 0 1000 2000 3000 4000 5000 6000 0 0.5 1 1.5 2 2.5 3 x 10 -3 Frequency [Hz]Spectrum Magnitude BSpectrum. Figure5-2.Examplesoflteredneuralrecordings.Theredl inesdenotetheaction potentialregionsinthetimedomain.Thebandwidthofthene uralsignalis typicallyassumedtobe5KHz. 5.2.1SpikeSorting ThemajorityoftheneuraldecodingalgorithmsusedinBMIsr equiretheprecise ringtimesforeachoftheneurons.Therefore,theactionpo tentialsintherecorded datamustbedetectedandclusteredintogroupscorrespondi ngtotheneuronthat generatedthem.Inthiscasewedetectedthreedifferentneu rons(seeFigure 5-3 ). Althoughthereareanumberofautomaticspikedetectionalg orithms( Lewicki ( 1998 ) andreferenceswithin),Mostoftheneuralphysiologistsst illrecurtohuman-assisted sortingsoftwaresimilarto`spike2,CED,UK'.Theprocedur eisdividedintoboth detectionandclustering.Detectionisbasedonthecrossin gsofthethresholddenedby theuser.Awindowofapproximately2msofdataisextracteda roundeachofthelevel crossings.Allsnippetsareconsideredaspotentialspikes .Templatesaregeneratedby iteratingthroughthesegmentsandclusteringthembasedon thenumberofsamples thatfallwithinadeneddistancefromthetemplate(whichi sasinglesnippetfortherst iteration).Whenaclusterreachesacertainamountofpoint sitisrecommendedasa template,theusercanthendecidewhethertoretainthetemp late,discarditorcombine 75

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itwithothertemplates.Finallytheselectedtemplatesare appliedtotheentiresignal providingalabeledset. 0 1 2 3 4 5 x 10 -3 -10 -8 -6 -4 -2 0 2 4 6 x 10 -5 Time [s]Amplitude [Volts] ANeuron1. 0 1 2 3 4 5 x 10 -3 -10 -8 -6 -4 -2 0 2 4 6 x 10 -5 Time [s]Amplitude [Volts] BNeuron2. 0 1 2 3 4 5 x 10 -3 -10 -8 -6 -4 -2 0 2 4 6 x 10 -5 Time [s]Amplitude [Volts] CNeuron3. Figure5-3.Overlaidactionpotentialsforeachoftheneuro nclasses.Themeanis shownasathickblacklineineachcase. 5.3TimeBasedEncodingofNeuralRecordings SincetheIFsampledistributionisinputdependent,thecom pressionandreconstruction accuracywilldependbothontheinputfeaturesandthesampl erparameters.Neural recordingsconsistsofhighamplitudetransients(actionp otentials)overalowamplitude backgroundnoise.Theregionsinbetweentheactionpotenti alsarereferredtoasnoise. Alsonotethatthesamplesdependonlyonthelocalstructure oftheinput.Therefore,we willassumethesampledistributionintheactionpotential rangeisnotdependentonthe characteristicsofthenoiseregion. ThecompressionratesachievablebytheIFsamplerwilldepe ndbothonthe characteristicsofthenoiseandoftheactionpotentials.T hetwomajorfactorsarethe power,bothintheactionpotentialandnoiseregionsaswell astheneuronalringrates. ThecompressionachievedusingtheIFsamplerwilldecrease astheneuronringrates increase.Nevertheless,BMIapplicationstypicallytarge tthemotorcortex,inwhich neuronringratesrangefrom10-100Hz SanchezandPrincipe ( 2007 ). Assumingaxedringrate,compressionandrecoveryaccura cydependonthe poweroftheactionpotentialsandnoiseregions.Eventhesh apeoftheactionpotential isnotcrucialintermsofthepulserates.Figure 5-4 showstheeffectofthesampler 76

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parametersonthesampleratesfor3differentactionpotent ialclasses(neurons).Inthis casetheneuralrecordingwasprocessedbyanexpertandthea ctionpotentialregions wereextractedforeachneuronclassalongwithitscorrespo ndingpulserepresentation. EachoftherowsinFigure 5-4 correspondtodifferentvaluesoftherefractoryperiod( ) whileeachcolumnrepresentsadifferentvalueoftheleakyp arameter( ).Inthiscase wehaveassumedthethresholdtobeunity( =1 )andthereforethescaleiscontrolled bytheinputpower.Inordertocomparetheeffectoftheparam eters,therangehasbeen selectedsuchthatthecorrespondingpulseratesspanthesa merange.Therefore,for every f g pairthepowerrangeisselectedappropriately. 1 =1 ms 1 =0.2 ms 1 =0.1 ms 1 =0.02 ms 0 5 10 x 10 8 0 1 2 3 x 10 4 PowerAverage Pulse Rate [pulses/s] 0 5 10 x 10 8 0 1 2 3 x 10 4 PowerAverage Pulse Rate [pulses/s] 0 5 10 x 10 8 0 1 2 3 x 10 4 PowerAverage Pulse Rate [pulses/s] 0 1 2 3 4 x 10 9 0 1 2 3 x 10 4 PowerAverage Pulse Rate [pulses/s] 0 1 2 3 x 10 9 0 1 2 3 x 10 4 PowerAverage Pulse Rate [pulses/s] 0 1 2 3 x 10 9 0 1 2 3 x 10 4 PowerAverage Pulse Rate [pulses/s] 0 1 2 3 x 10 9 0 1 2 3 x 10 4 PowerAverage Pulse Rate [pulses/s] 0 5 10 x 10 9 0 1 2 3 x 10 4 PowerAverage Pulse Rate [pulses/s] 0 2 4 6 x 10 9 0 1 2 3 x 10 4 PowerAverage Pulse Rate [pulses/s] 0 2 4 6 x 10 9 0 1 2 3 x 10 4 PowerAverage Pulse Rate [pulses/s] 0 2 4 6 x 10 9 0 1 2 3 x 10 4 PowerAverage Pulse Rate [pulses/s] 0 0.5 1 1.5 2 x 10 10 0 1 2 3 x 10 4 PowerAverage Pulse Rate [pulses/s] N1 N2 N3 Figure5-4.Effectofthesamplerparametersandtheinputpo weronthepulserate variationintheactionpotentialregions.Eachcolumnrepr esentsadifferent valueoftheleakfactor ,rowscorrespondtovariationintherefractory(from toptobottom, =0 s =10 s =20 s ). 77

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1 =1 ms 1 =0.2 ms 1 =0.1 ms 1 =0.02 ms 0 5 10 15 x 10 8 0 1 2 3 x 10 4 PowerAverage Pulse Rate [pulses/s] 0 5 10 15 x 10 8 0 1 2 3 x 10 4 PowerAverage Pulse Rate [pulses/s] 0 5 10 15 x 10 8 0 1 2 3 x 10 4 PowerAverage Pulse Rate [pulses/s] 0 2 4 6 x 10 9 0 1 2 3 x 10 4 PowerAverage Pulse Rate [pulses/s] 0 1 2 3 x 10 9 0 1 2 3 x 10 4 PowerAverage Pulse Rate [pulses/s] 0 1 2 3 x 10 9 0 1 2 3 x 10 4 PowerAverage Pulse Rate [pulses/s] 0 1 2 3 x 10 9 0 1 2 3 x 10 4 PowerAverage Pulse Rate [pulses/s] 0 5 10 x 10 9 0 1 2 3 x 10 4 PowerAverage Pulse Rate [pulses/s] 0 5 10 x 10 9 0 1 2 3 x 10 4 PowerAverage Pulse Rate [pulses/s] 0 5 10 x 10 9 0 1 2 3 x 10 4 PowerAverage Pulse Rate [pulses/s] 0 5 10 x 10 9 0 1 2 3 x 10 4 PowerAverage Pulse Rate [pulses/s] 0 1 2 3 4 x 10 10 0 1 2 3 x 10 4 PowerAverage Pulse Rate [pulses/s] Figure5-5.Effectofthesamplerparametersandtheinputpo weronthepulserate variationinthenoiseregions.Eachcolumnrepresentsadif ferentvalueof theleakfactor ,rowscorrespondtovariationintherefractory(fromtopto bottom, =0 s =10 s =20 s ). 5.3.1SamplerandInputEffectsonReconstructionAccuracy Sinceweonlyrequireanaccuratereconstructionoftheacti onpotentials,the reconstructionalgorithmsaretestedintheseregions.The effectsofthenoisewill primarilyaffectthedataratesthesewillbetreatedinthef uture.Thereconstruction accuracyismeasuredintermsoftheSignaltoErrorRatio(SE R),denedasthe logarithmoftheratiobetweenthepowerinthesignal( P s )andthepowerintheerror ( P e ): SER =10log 10 P s Pe (5–1) 78

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Figure 5-6 ,showstheeffectsoftheparametersonthereconstructiona nalogousto Figure 5-4 .NotethatthereportedSERandpulseratesaretheaveragees timatedover alltheactionpotentialsforthecorrespondingclasses.Th ereforeforeverydatarate wecanndthecorrespondingreconstructionaccuracyforev eryalgorithm.Notethat astheleakparameter, decreases,theshapeof v tendstotheinput x ,andtherefore samplesaremorelikelytooccuratthepeaksoftheactionpot entialsinsteadofthe borders.Thereforethereconstructionerrorsstarttoappe arintheselowamplitude regions.Incontrastas increase, v becomessmootherandthesamplesarespreadout alongtheactionpotential.Nevertheless,thiswillalsore ducetheamplitudedifference betweentheactionpotentialandthenoise.Therefractoryp eriodisabletoincreasethe SERregardlessoftheleakparameter,withanappropriatere scalingofthepower.The increaseintherefractoryperiodforcesthesampledistrib utiontospreadoutandwould alsodecreasethesampleratewhichiswhywemustvarythepow eroftheinputin ordertopreservethesameamountofpulses.Although,incre asingtherefractoryperiod furtherwillalsohavenegativeconsequencessincetherang eoverwhichthesignalis beingintegratedwilldecreasecausingittobesensitiveto timingjitter. 5.3.2ProcessingTimes Herewecomparethethreereconstructionmethodsintermsof theprocessing time.InthiscasethealgorithmswereimplementedinMatlab 2010a64-bitonanIntel XeonCPU2.4GHzwith16GBRAM.Theresultsoftheprocessingt imefordifferent parametersaswellasdifferentwindowlengthsisshowninFi gure 5-7 .Onceagainthe timesareaveragesovereachoftheclasses.Foreachsetofpa rameters,wevarythe datawindowincreasingthenumberofsamplesandnumberofba sisfunctionsforeach algorithm.Realtimereconstructionisachievedforthosep arametersforwhichthecurve liesunderthegrayline.Notethatweareinterestedinsampl ingratesundertheNyquist boundwhichis 10 Ksamp = s .Inmostcasestheradialbasisfunctionprovidesarealtime recoveryalthoughwepayintermsofthereconstructionaccu racy. 79

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1 =1 ms 1 =0.2 ms 1 =0.1 ms 1 =0.02 ms 0 2 4 6 8 10 12 x 10 8 0 10 20 30 40 50 60 PowerAverage SER [dB] 0 2 4 6 8 10 12 x 10 8 0 10 20 30 40 50 60 PowerAverage SER [dB] 0 2 4 6 8 10 12 x 10 8 0 10 20 30 40 50 60 PowerAverage SER [dB] 0 1 2 3 4 x 10 9 0 10 20 30 40 50 60 PowerAverage SER [dB] 0 0.5 1 1.5 2 2.5 3 x 10 9 0 10 20 30 40 50 60 PowerAverage SER [dB] 0 0.5 1 1.5 2 2.5 3 x 10 9 0 10 20 30 40 50 60 PowerAverage SER [dB] 0 0.5 1 1.5 2 2.5 3 x 10 9 0 10 20 30 40 50 60 PowerAverage SER [dB] 0 2 4 6 8 10 12 x 10 9 0 10 20 30 40 50 60 PowerAverage SER [dB] 0 1 2 3 4 5 6 x 10 9 0 10 20 30 40 50 60 PowerAverage SER [dB] 0 1 2 3 4 5 6 x 10 9 0 10 20 30 40 50 60 PowerAverage SER [dB] 0 1 2 3 4 5 6 x 10 9 0 10 20 30 40 50 60 PowerAverage SER [dB] 0 0.5 1 1.5 2 x 10 10 0 10 20 30 40 50 60 PowerAverage SER [dB] Radial Basis Fourier Basis Cubic B-spline Figure5-6.Effectsofthesamplerparametersandtheaction potentialinputpoweron theaveragereconstructionaccuracyactionpotentialsfro mneuronclass1 (Numberofactionpotentialis466). 5.3.3SensitivitytoTimeQuantization Inordertoprocessthesamplesinanydigitalsystemthetimi ngoftheevents mustbequantized.Figure 5-8 ,showstheeffectofthequantizationontheaverage reconstructionSER.EachrowinFigure 5-8 correspondstoadifferentrefractoryperiod (fromtoptobottom =0 s =10 s =20 s ).Aswemovefromlefttorightthe powerisincreasedwhichisreectedontheaveragepulserat esandSERestimated onlyovertheactionpotentialregions.ItisevidentfromFi gure 5-8 thatthefrequencyof thequantizationclockmustbegreaterthan 1 MHz.Thisnumberiscrucialinhardware implementationssinceitconstraintsthecommunicationbl ock. 80

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Neuron1Neuron2Neuron3 0.05 0.1 0.15 0.2 0 0.1 0.2 0.3 0.4 0.5 Average Pulse Rate: 12301.3095 Window length [s]Average Processing time [s] Radial Basis x=y Fourier Basis Cubic B-spline 0.05 0.1 0.15 0.2 0 0.1 0.2 0.3 0.4 0.5 Average Pulse Rate: 11973.425 Window length [s]Average Processing time [s] Radial Basis x=y Fourier Basis Cubic B-spline 0.05 0.1 0.15 0.2 0 0.1 0.2 0.3 0.4 0.5 Average Pulse Rate: 10396.0847 Window length [s]Average Processing time [s] Radial Basis x=y Fourier Basis Cubic B-spline 0.05 0.1 0.15 0.2 0 0.1 0.2 0.3 0.4 0.5 Average Pulse Rate: 6629.3261 Window length [s]Average Processing time [s] 0.05 0.1 0.15 0.2 0 0.1 0.2 0.3 0.4 0.5 Average Pulse Rate: 6406.64 Window length [s]Average Processing time [s] 0.05 0.1 0.15 0.2 0 0.1 0.2 0.3 0.4 0.5 Average Pulse Rate: 5346.9487 Window length [s]Average Processing time [s] 0.05 0.1 0.15 0.2 0 0.1 0.2 0.3 0.4 0.5 Average Pulse Rate: 1831.0138 Window length [s]Average Processing time [s] 0.05 0.1 0.15 0.2 0 0.1 0.2 0.3 0.4 0.5 Average Pulse Rate: 1729.07 Window length [s]Average Processing time [s] 0.05 0.1 0.15 0.2 0 0.1 0.2 0.3 0.4 0.5 Average Pulse Rate: 1287.9387 Window length [s]Average Processing time [s] Figure5-7.Reconstructiontimevsthewindowlengthfordif ferentneuronclassesand sampleratesachievedbyvaryingthepower,fromtoptodownt hepower variesas 3 e 9,1 e 9,2 e 8 .Thepoweriscalculatedonlyovertheaction potentialrange.Alsothesamplerparametersarexedsucht hat =10 s 1 =0.2 ms 5.3.4EffectofthePulseRatesontheClassicationError Althoughthereconstructionaccuracyisimportantinprese rvingthequalityof therecordings,discriminabilitymustalsobeassessed.He rewepresentthespike sortingresultsforthe 1 minutesegmentofneuralactivityatdifferentsamplerates .The variationintheratesisachievedbychangingthepowerofth einput,whilexingthe otherparameters =10 us and =0.1 ms = 1 RC ,where R =5 M n, C =20 pF .The lasttwocorrespondtothevaluesofresistanceandcapacita nceusedinthehardware implementation.Table 5-1 showstheconfusionmatricesbetweentheoriginalsignal andtherecovered.Nevertheless,thenumberofmiss-detect ionsbothintheoriginal 81

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10 -8 10 -6 10 -4 0 10 20 30 40 50 60 Quantization period [s]Average SER [dB]Average Pulse Rate: 20462.0172 10 -8 10 -6 10 -4 0 10 20 30 40 50 60 Quantization period [s]Average SER [dB]Average Pulse Rate: 17575.2146 10 -8 10 -6 10 -4 0 10 20 30 40 50 60 Quantization period [s]Average SER [dB]Average Pulse Rate: 14131.1159 10 -8 10 -6 10 -4 0 10 20 30 40 50 60 Quantization period [s]Average SER [dB]Average Pulse Rate: 9333.9056 10 -8 10 -6 10 -4 0 10 20 30 40 50 60 Quantization period [s]Average SER [dB]Average Pulse Rate: 21629.9356 10 -8 10 -6 10 -4 0 10 20 30 40 50 60 Quantization period [s]Average SER [dB]Average Pulse Rate: 19596.7811 10 -8 10 -6 10 -4 0 10 20 30 40 50 60 Quantization period [s]Average SER [dB]Average Pulse Rate: 16836.9099 10 -8 10 -6 10 -4 0 10 20 30 40 50 60 Quantization period [s]Average SER [dB]Average Pulse Rate: 12607.618 10 -8 10 -6 10 -4 0 10 20 30 40 50 60 Quantization period [s]Average SER [dB]Average Pulse Rate: 20143.5622 Radial Basis Fourier Basis Cubic B-spline 10 -8 10 -6 10 -4 0 10 20 30 40 50 60 Quantization period [s]Average SER [dB]Average Pulse Rate: 18795.1717 10 -8 10 -6 10 -4 0 10 20 30 40 50 60 Quantization period [s]Average SER [dB]Average Pulse Rate: 16853.1116 10 -8 10 -6 10 -4 0 10 20 30 40 50 60 Quantization period [s]Average SER [dB]Average Pulse Rate: 13691.4163 Figure5-8.Effectsofthetimequantizationontheaveragea ctionpotentialSERfor neuron1.Allquantizedpulsetimingsareanintegermultipl eoftheclock period.A 50 ms windowisreconstructed,neverthelesstheaveragepulse ratesandSERvaluesareestaimtedovera 2 ms windowaroundtheaction potentials.Therowscorrespondtodifferentrefractorype riodsfromtopto bottom =0 s =10 s =20 s ,whilexing 1 =0.2 ms .Thepowerof theinputincreasesinthecolumnsfromlefttoright.Theeff ectsofthis increaseappearintheaveragesamplingratedenotedforeac hplot. andrecoveredsignalsmustalsobeconsidered.Thecolumnla beled`Missedinoriginal' correspondstotheactionpotentialsthatwerenotdetected intheoriginalrecordingbut wereinthereconstructed.Likewise,therowslabeled`Miss edinRec'arethoseaction potentialslabeledintheoriginalandmissedinthereconst ructed.Notethatmostof theerrorsstarttoappearbetweenneurons1and2,sincethey arebothatasimilar amplitudescalewhileneuron3islarger.Furthermore,then umberofmiss-detections increaseasthesampleratesdecrease.Thesemiss-detectio nareconsequenceof thedistortionintroducedbythesampler,whichappearatth eboundariesoftheaction 82

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potential.Theseresultsarebasedonthetemplatescreated formtheoriginalrecording. Nevertheless,ifthesereconstructedrecordingsweretobe resortedthedetectionand classicationaccuracywouldimprove,althoughwewouldin troducehumanerror.The variabilityofhumanspikesortingisnearly 20% Woodetal. ( 2004 ). 5.4PulseBasedDiscriminationofNeuralActionPotentials InthissectionweshowevidencethattheIFsamplescontaind iscriminative informationabouttheinputclasses.Hence,weoperatedire ctlyonthesamples avoidingtheconventionalframeworkofsamplingandrecons truction.Wecompare ourencodingwiththeconventionaluniformsamplingmethod sonneuraldata.We showdiscriminabilityintermsoftheclassicationerrord eterminedontheprojection ofthesamplesbylineardiscriminantanalysis.Resultssho wthattheIFsampler preservesdiscriminabilityfeaturesoftheinputsignalev enatsub-Nyquistsampling rates.Furthermore,theIFencodingperformsatleastaswel lasuniformsamplersatthe samedatarate. Mosteffortshavebeengearedtowardsthedesignoftheserec onstruction algorithms,inordertousethestandardsignalprocessingm achinerydeveloped foruniformrepresentations.Forexample,theIFencodedne uralsignalsmustbe reconstructedinordertodetectandsortallactionpotenti als.Incontrast,wepropose workingdirectlyonthesamples,avoidingrecoverycomplet ely.Wewillnottacklethe entireproblemofdetectionandclusteringinthesampleddo main,butstartbyproviding evidencethatthesamplespreservediscriminativeinforma tionabouttheinputclasses evenatsub-Nyquistrates.Inthispaperwetreattheclassi cationproblemfortwoaction potentialclassesencodedthroughtheIF. Severaldifcultiesarise,sincethesamplesetforeachinp utislikelytobedifferent intermsofthenumberofsamplesandtheirlocations,contra rytotheconventional uniformsamplingschemes.Therefore,acarefullychosenem bedding(binning)is needed.Weshowdiscriminabilityintermsoftheclassicat ionerrorofaLinear 83

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Table5-1.Confusionmatrixbetweenthesortsobtainedfrom theoriginaluniformlysampledneuraldataandthe reconstructedsignalatdifferentaveragesamplerates,19 .784Ksamples/s( =1.97 ),9.605Ksamples/s ( =0.96 ),5.430Ksamples/s( =0.54 )and1.281Ksamples/s( =0.12 ).Therowscorrespondtothesortson thereconstructeddataandthecolumnstotheoriginaldata. Theactionpotentialsmissedintheoriginal recordingaremarkedunderM.O.,likewiseforthereconstru ctedaredenotedasM.R.Thisisrepeatedforeach recoverymethod. SamplingfactorCubicB-splinesFourierRadialBasis N1N2N3M.O.N1N2N3M.O.N1N2N3M.O. =1.97 N143176234474211397102136N23119046129031412108N3402274312371602219 M.R.2899101349410 =0.96 N14136642433431640191442N2711901561250412117010N35122982123303022116 M.R.419725565097 =0.54 N1406137693915631378131938N2511001741180610109014N3101219144023055021315 M.R.45111667126731310 =0.12 N1324274173145132225217173134N22568015839161565010N333321433611106601509 M.R.843724307821092285361 84

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DiscriminantAnalysis(LDA)classier JohnsonandWichern ( 1988 ).Althoughthese methodscanbeappliedtoanyofthetimebasedsamplers,wewi llfocusontheIF sampleranditsapplicationtoneuralencoding.5.4.1RelatedWork TheIFmodelhasbeenextensivelyusedinthecomputationaln euroscience literaturetostudythedynamicsofneuronpopulations GerstnerandKistler ( 2002 ). Informationintheselargesystemsisencodedinthetimingo ftheevents,also knownasspikes.Themainapproachinthestudyandanalysiso fthesetime-based codesassumestheyarerealizationsofastochasticpointpr ocessmodel Daleyand Vere-Jones ( 2002 ).Inthiscase,theoutputoftheIFisconsideredarealizati onfroma stochasticpointprocess.Themeasureofsimilaritybetwee ntwospiketrainsisgiven bythestatisticsofthegeneratingpointprocesses.Apoint processcanbecompletely describedbyitsconditionalintensityfunction ( t j H t ) DaleyandVere-Jones ( 2002 ), where H t denotesthehistoryoftheprocessuntiltime t .Intuitivelyitdenesthe instantaneousrateofoccurrencegivenallpreviousevents .Nevertheless,sincethe intensityfunctionisconditionedontheentirehistory,it cannotbeestimatedinpractice asthedataisnotavailable.Atypicalassumptionisthat ( t j H t ) onlydependsonthe currenttimeandthetimetothepreviousevent t ( t j H t )= ( t t t ) (5–2) Furthermore,mostBMIrelyontheestimateofthemeancondit ionalintensity functiondeterminedbyaveragingoverthebinnedspiketrai ns.Wewillalsousebinned vectorsofourpointprocessrealizationsasourfeatures. Anumberofdifferentsimilaritymeasuresforspiketrainan alysishavebeen proposedintheliteratureintermsofpopulations,aswella scomparisonsbetween singlerealizations Gr¨unandRotter ( 2010 ).Basedonthesesimilaritymeasures, clusteringandclassicationalgorithmscanbeimplemente d.Inthispaperweare 85

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concernedwithclassication,sinceweshowdiscriminabil itybetweentwoclassesin theencodedspikerepresentationbasedontheclassicatio nerrorinthisdomain.The drawbackofthesemeasuresisthattheydonotdescribethecl assicationerror.Onthe otherhand,mutualinformation CoverandThomas ( 1991 )canbeshowntobeabound ontheclassicationerrorbyFano'sinequality CoverandThomas ( 1991 ).However,itis verydifculttoestimateinpracticegiventheamountofdat aitrequires Paninski ( 2003 ). Thereforeinsteadofusingoneofthesemeasureweusetheper formanceoftheLDA classierasameasureofdiscriminability.5.4.2PulsetrainClassication Inordertoclassifytheencodedsignals,wemustrstdenet hefeaturespace todescribethesamples.Inthiscasethefeaturespaceconsi stsofbinningthedata andcreatingavectorwiththesamplecountswhichisknownas ringrate Dayanand Abbott ( 2001 ).Binningimplieswedividethetimedomainintoequalsizeb insand simplycountthenumberofeventsthatfallwithineachinter val.Ingeneral,determining thenumberofbinsisnotclearandsoitmustbeestimated.How ever,weassumeour inputisband-limitedandtheIFsamplingrateisnearNyquis t,andthusthebinsizeis determinedinrelationtothemaximuminputfrequency. Theobjectiveofthissectionistocomparethefeaturevecto rsderivedfromthe originalinputseriesfortwosignalclassesanddistinguis hbetweentheclassesusing LDA.Inthissettingoftwoclassclassication,LDAallowst oassignafeaturevectorto agivenclassbymaximizingtheposteriorprobabilityofcla ssicationorexpectedcost ofmisclassication.Inthiscasetheoptimalprojectionve ctor a ischosensuchthatit maximizestheseparabilitybetweentheclassmeansandmini mizestheclassspread a = 1 p ( 1 2 ) intheprojectedspace(1D),where p isobtainedbypoolingthe covariancematrices 1 and 2 ofthetwoclasses.LDAassumesthatthedistribution ofthefeaturevectorismultivariateNormal.Whilethisass umptionmayappeartobe 86

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restrictive,itdoesnotaffectthediscriminabilitycapac ityoftheclassieranditiswell knownthatLDAisresistanttodeparturesfromnormality.5.4.3ResultsandDiscussion WeshowthattheIFencodingpreservesdiscriminativefeatu resoftheinputclasses. Inthiscasetheinputsareneuralactionpotentialsoverape riodof 2 millisecondsthat havebeensortedbyanexpert.Althoughactionpotentialsar eingeneralsimilar,the geometryoftherecordingsetupinducesdistortionsinthei rshapes.Itisthisdistortion thatallowsustogrouptheactionpotentials.Figure 5-9 showtheoverlaidaction potentialsforthecorrespondingclasses.Eachvoltagetra ceingraycorrespondstoa singlerealizationwiththeaverageoftherealizationssho wninblack.Thebandwidth forneuralrecordingsistypicallysetat5Khz,inthiscaset hesamplingratewasnearly 12Ksamples/s.InordertogeneratetheIFsamplestheinputw asup-sampledbya factorof50toreducethetimingquantizationofthesamples .Eachofthesesegments 0 0.5 1 1.5 2 x 10 -3 -1 0 1 x 10 -4 N1Amplitude [v]Time [s] 0 0.5 1 1.5 2 x 10 -3 -1 0 1 x 10 -4 N2 Time [s] Figure5-9.Overlaidactionpotentialsforeachneuronorcl assingrayalongwiththeir meaninblack. isthenencodedthroughtheintegrateandresampler.Eacht rialinFigure 5-10 representsasinglepulsetrain(singlerowinthetopgraphs ),thecolorsencodethe polarityofthepulses,thebottomplotshowsthemeanringr ateestimatedbybinning 87

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eachrealizationbycountingthenumberofeventsineachbin andaveragingoverall trials.Notethatweareignoringthepolarityofthepulses. 0 0.5 1 1.5 2 2.5 x 10 -3 0 2 4 x 10 4 Sample RateTime [s] 0 0.5 1 1.5 2 2.5 x 10 -3 0 500 1000 1500 Trials AClass1 0 0.5 1 1.5 2 2.5 x 10 -3 0 2 4 x 10 4 Sample RateTime [s] 0 0.5 1 1.5 2 2.5 x 10 -3 0 200 400 600 Trials BClass2 Figure5-10.TopgraphscorrespondtotheIFencodingofeach actionpotential realizationinthecorrespondingclass(trials).Thecolor sindicatethe polarityofthesamplesinthepulsetrain(blackpositivean drednegative). Thebottomplotshowsthemeanringratewiththethreshold xedat 2 e 9 WewillcomparethetimebasedencodingprovidedbytheIFtot heconventional uniformsampler.Inordertoshowdiscriminabilitywewillu setheclassicationerrorfrom theLDAbasedclassierasourcriterion.Inotherwords,wea recomparingtwodifferent featurerepresentationsforthecontinuousinput.Therst simplyusesauniformsample distributionfromtheoriginalsignal,whilethesecondisb asedontheIFencoding. SinceweareusingLDAandthisisatwoclassproblem,thefeat uresdetermined fromthesamplesareprojectedontoaline.Figure 5-11 showsthedistributionofthe projectionsofthetwoclasseswhenweuseauniformsamplera t30Ksamples/s(above theNyquistrate10KHz).Thelargewithinclassvariancecau sesthedistributionofthe twoclassestooverlap.Notethatthedistributionhavebeen approximatedbyaGaussian function.Theperformanceoftheencodingforbothsamplers ispresentedoverarange ofdecisionboundariesgivenbytheReceiverOperatingChar acteristic(ROC)curves, whichrelatethetruepositiverates(TPR)andthefalseposi tiverates(FPR).Figure 5-12 showstheROCcurvesatdifferentsamplerates.Sincewearei nterestedinthebest 88

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-8 -6 -4 -2 0 2 4 0 0.1 0.2 0.3 0.4 0.5 LDA projection N1 N2 Figure5-11.LDAprojection:uniformsamplesasfeatures. sampledistributiontorepresentthedata,wecomparethema tdifferentsamplerates. Aswecansee,theIFfeaturesslightlyoutperformtheconven tionaluniformsamplerat 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.75 0.8 0.85 0.9 0.95 1 True positive rateFalse positive rate Uniform: 3Ksamp/s IF: Avg 3Ksamp/s Uniform: 12Ksamp/s IF: Avg 10Ksamp/s Figure5-12.ROCcurves:ForbothfeaturesIFanduniform. sampleratesneartheNyquistboundaryandalsolowerrates. Itisinterestingtosee thattheimprovementisgreaterasthesamplerthresholdinc reases.Notethatasthe thresholdincreasesthesampleratedecreases.Intuitivel y,theIFsamplesareplacedin thediscriminativeregionsbetweenthetwoclasseswhichar erelatedtohighamplitude. ItisclearthattheIFencodingnotonlyprovidesdiscrimina tioninthesampleddomain butalsodoessoatsub-Nyquistrates.Incomparisontotheco nventionalapproachof reconstructingtheinput,wehaveshownthatIFtogetherwit hbinningoverthesamples retainsimportantfeaturesoftheinputtoprovidediscrimi nabilitybetweentwosignal 89

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classes.Wehaveshownthatthetime-basedencodingprovide dbytheIFsampler conservesdiscriminativefeaturesoftheneuronactionpot entials.Discriminabilityis measuredinrelationtotheclassicationerrorusingLDA.N otethattheIFsamples outperformedtheuniformsamplerandthatthedifferencein theclassicationerror increasedasthesampleratesdecreased.Althoughthetimeb asedencodingisa competitivesolutiontoconventionalADCs,ingeneral,due tothenonlinearnatureof thetransformation,conventionalwisdomisthatitisdifc ulttodescribethefeatures thatareretainedintheoutputsamples,unlesswemovetowar dsratecodeswherethe relationshipissimplylinear.Thispapershowsthattime-e ncodingschemescanindeed carrydiscriminativefeaturesintotheoutputdomain.Howe ver,furtherexplorationofthe relationshipbetweentheparametersthatcontroltheencod ingandtheinformationthat ispreservedisneeded.Thisalsoemphasizestheneedforana lysisandprocessing toolsthatapplydirectlytothepointprocesses.Finally,i tisconceivabletoextendthis approachtomulti-channelIFthatconsistsofanetworkofIF samplers-inthespiritof biologicalconstructs-isrelevant. 90

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CHAPTER6 APPLICATION2:ELECTROCARDIOGRAMRECORDINGS AccordingtotheWorldHealthOrganization(WHO)cardiovas culardiseases(CVD) aretheleadingcausesofdeathanddisabilityintheworld.A nestimated17.3million peoplediedfromCVDsin2008,representing30%ofallglobal deaths Organization ( 2008 ).Furthermore,in2010,heartrelateddiseasescostthehea lthcareindustryin theUnitedStates$316.4billion.Thistotalincludestheco stofhealthcareservices, medications,andlostproductivity forDiseaseControlandPrevention ( 2010 ).Although alargeproportionofCVDsarepreventable,theycontinueto risemainlybecause preventivemeasuresareinadequate.Electrocardioagram( ECG)isastandardtool tomonitorheartfunction.AlthoughaccesstoECGisreadily available,patientswith compromisedheartfunctionrequirecontinuousmonitoring .RegularmonitoringofECG helpsinearlydetectionofirregularitiesintheheartbeat s.Someheartbeatarrhythmias, althoughnotimmediatelylifethreatening,mayneedtimely detectionandattentionto preventfuturecomplications.Inordertoprovidecontinuo ushealthmonitoring,devices mustintegrateseamlesslyintothepatientslifeandnotint erferewithdailyactivities. Towardsthisgoal,wirelessbodyareanetworks(WBAN)haveb ecomeanemerging technology Lupranoetal. ( 2006 ); VolmerandOrglmeister ( 2008 ).Thesenetworks consistofanumberofphysiologicalsensorswhichmonitort hepatientsvitalsigns. UnlikeconventionalHoltermonitorswhichrecordtheheart beatrhythmsandprocess thedataofine Irhythm ( 2011 ),WBANaredesignedtoproviderealtimediagnostics. CurrentwirelessECGsystemscanbegroupedintotwobroadca tegories Mamaghanian etal. ( 2011 ); SapioandTsouri ( 2010 ):thosewithwiredsensorsandthosewithwireless sensors.Therstgroup SapioandTsouri ( 2010 )connectsasinglewirelesstransmitter toallthesensorsbyasetofwires.Thisapproachstillrequi resbulkyequipmentand introducesmovementartifactsandnoise.Thesecondgroupi ntegratesthesensor andthewirelesstransmitter,providingthepatientgreate rmobilityandpotentially24/7 91

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monitoring Corventis ( 2011 ).Themainchallengesofthesesystemsincludepower, area,bandwidthandqualityoftherecording. Inordertosatisfythebandwidthconstraintsanumberofcom pressionmethods havebeenproposedintheliterature(waveletdecompositio ns,spectralcoefcients andothers).Thesecompressionalgorithmsareappliedtoda tathathasbeensampled uniformlyatarateabovetheNysquistbound.Thesamplesare thencompressedto reducethedatarates.Thedownsidebeingthatthesesolutio nsleadtobulkycircuits whichconsumeareaandpower. Novelsamplingmethodsmergeboththecompressionandsampl ingstagesby liftingthebandlimitednessassumptionandworkingdirect lywithsparserepresentations oftheinput.ThisincludestheworkinCompressiveSensing Candesetal. ( 2006 ) andFiniteRateofInnovation Vetterlietal. ( 2002 ).Boththesemethodsassumea specicsparsityordegreesoffreedomontheinput,whichma ychangethroughtime dependingontheshapeoftheheartbeatcycles.Furthermore thebasisfunctionsare stochastic,thesignalstructureisspreadacrossthefulls pace,whichmeansthatitis veryunlikelytoextractfeaturesinthisspacethatwilllea dtoclassierswithoutdoing thereconstruction.Lastly,theyhavenotbeendesignedwit hlowpowerhardware constraints. Inordertosatisfyallconstraints(bandwidth,area,power ,resolution)wepropose atime-basedencodingschemeforECGrecordings,basedonth eIntegrateandFire sampler Feichtingeretal. ( 2010 ).Thissamplerhasbeenshowntoprovideaccurateand compressedrepresentationsofasimilarsignalclass(neur alrecordings Alvaradoetal. ( 2009 )),alongwithanefcienthardwareimplementation Rastogietal. ( 2011 ). InthispaperweusetimebasedencodingsoftheECGrecording stoperformthe classicationofnormalheartbeatsandirregularheartbea tsknownasarrhythmias.We usethetime-basedrepresentationdirectlytoextractdisc riminativefeatures,avoiding reconstructionalltogether.Inordertocomparetheperfor manceofourmethodwith 92

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thoseinthepublishedliterature,weusetheclassierdesc ribedby deChazalandReilly ( 2006 ).Wedifferfrom deChazalandReilly ( 2006 )inthatthefeaturesareextracted fromtheoutputofanIFsampler.While,theactivelearninga pproachpresentedin WiensandGuttag ( 2010 )outperformsourmethod,thepulsebasedfeaturescanbe usedintheirframework. Thepaperisorganizedasfollows,rstwedescribetheclass icationproblemfor ECGs(SectionII)alongwiththedescriptionoftherecordin gs(SectionIII).SectionIV introducestheIFsamplerandsectionVdescribestheclassi erbasedontheoutput pulsesfromtheIF.SectionVIdenestheperformancemetric susedandnally,Section VIIpresentsclassicationanddatarateresultsbasedonth eMIT-BIHarrhythmia database. 6.1HeartbeatClassication ClassicationofECGsignalsisaverydifcultproblemandc urrentresearchis focusedoncarefulextractionofheartbeatfeatures.Amajo rproblemencountered bymachinelearningtechniquesforclassifyingECGsignals isduetothelarge inter-patientandintra-patientvariabilityinthetiming prolesandmorphologyof damagedcardio-vascularprocesses(seeFigure 6-1 ).Theeffectofthisbehavioris thatclassierstrainedusingtraditionalmethodsfailwhe nappliedtonewpatients. Severalapproacheshavebeenproposed. Huetal. ( 1997 )usedamixtureofexperts approachwhichcombinesglobalandlocalclassiers.Thegl obalclassieruses heartbeatsignaturesfromavastcollectionoflabeleddata .Thelocalclassiertrains onpatient-specicECGrecordings.Agatingfunctionweigh tstheclassicationresults oftheglobalandlocalexpertsandcombinesthemtomakeadec ision. deChazaland Reilly ( 2006 )proposedanapproachbasedonheartbeatmorphologyfeatur es,heartbeat intervalfeatures,andRRintervalfeaturesandutilizesth elineardiscriminantclassier (LDA)forclassication. Inceetal. ( 2009 )proposedapatient-specicclassication methodologyforaccurateclassicationofheartbeatpatte rns. WiensandGuttag ( 2010 ) 93

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Patient119Patient106 C:1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 -0.5 0 0.5 1 1.5 2 2.5 3 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 -1 -0.5 0 0.5 1 1.5 2 2.5 3 C:5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3 3.5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3 Figure6-1.ExampleofheartbeatshapesfromtheMIT-BIHdat aset.Eachcolumn representsapatientandeachrowthebeatsforthatspecicc lass(usingthe classnumberingfromtable 6-1 ).Notethevariationsinthebeatmorphology acrosspatientsaswellaswithinapatient. proposedactivelearningwhichreducestheamountofpatien t-specic(annotated) dataneededforclassication.Itisthereforeevidentthat heartbeatclassication researcheshavefocusedexclusivelyonpatient-specicEC Gsignaturesandfeatures. OurapproachistoanalyzetheECGsignalsusingthestreamof pulsesgeneratedby theIFsampler,extractthepulsefeaturesbasedonthepreci setimingoftheseevents, andevaluateclassierperformance.ThereforetheECGsign aldoesnotneedtobe reconstructed,atleastforclassication.Wechoosetheli neardiscriminantclassier (LDA)following deChazalandReilly ( 2006 ).Needlesstosay,ourapproachachieves thedualpurposeofcompressingthedataintensiveECGsigna ls,andperforming classicationinthepulsedomain.While,ECGdatacompress ionhasbeendonein 94

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thepastusingFRIandCS,theseapproachesrequireknowledg eofthesparsityof thesignal.Heartbeatcycles,howeverareregularlypaced, butarrhythmiaconditions induceirregularitiessuchthatknowledgeofsignalsparsi ty apriori maybeunknown. Furthermore,signalanalysisstillrequiresreconstructi on.WeproposetheIFsampler becausetheheartbeatcyclesarelocalizedintimeandampli tudeandmanyforms ofarrhythmiasarediscernibleirregularitiesneartheQRS complexintheheartbeat cycle.Furthermore,theoutputpulsedistribution(genera tedbyIF)isinputdependent anditadaptstochangesintheheartbeatsshapesorrhythms. Thus,ourproposalisa holisticapproachtotackletheproblemofcompressionandc lassicationofheartbeats whichwebelievecancircumventthebandwidthandreal-time analysisbottlenecksto improveon-timedeliveryofhealthcaretomitigateheartre latedfatalities.Resultsfrom ourapproacharecomparabletostateoftheart. 6.2DataDescription Inordertotestbothcompressionanddiscriminabilityofhe artbeatsfromECG recordingsofourmethod,weleveragedthestandardMIT-BIH arrhythmiadatabase Goldbergeretal. ( e13 ).Thedatabaseconsistsof48fullyannotated,30minute, two-leadECGrecordingsfrom47differentpatients(record ings201and202arefrom thesamepatient).Theleadsusuallyinvolvethemodiedlim bleadII(MLLII)andone ofthemodiedleadsV1,V2,V4orV5 Goldbergeretal. ( e13 ).Sincethesecondlead usuallyvariesforeachrecording(patient),alltheresult sinthispaperarebasedonthe MLLII.Thedataissampledat360Hzandthe 5 mVrangeisquantizedto11-bits. Unfortunatelyanumberofartifactssuchaspowerlineinter ference,contactnoise, motion,electromyographicnoiseandbaselinedriftcompro misethequalityofrecordings Cliffordetal. ( 2006a ).Inordertoattenuatetheeffectoftheseartifactstherec ordings werepreprocessedasdescribedin deChazalandReilly ( 2006 ); WiensandGuttag ( 2010 ): 95

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1. Firstthedataispassedthroughamedianlterwithwindowsi ze200mswhich removestheP-wavesandQRScomplexes Cliffordetal. ( 2006b ). 2. Asecondmedianlterwithwindowsize600ms,removestheT-w aves. 3. Thelteredsignalrepresentsthebaselinewhichisthensub tractedfromthe originalrecording. 4. Finallyanotchltercenteredat60Hzisimplementedthroug ha60tapnite impulseresponselter,inordertoremovepowerlineinterf erence. ThetoppanelinFigure 6-2 showsashortsegmentoftherawECGrecordingsandthe processedversion.TheMIT-BIHdatabaseincludesanumbero fexamplesfordifferent rhythms,QRSmorphologyandsignalquality,forareviewont hefundamentalsofECG recordingsreferto Cliffordetal. ( 2006b ).Inthispaperweareonlyconcernedwiththe heartbeatannotations.Mostoftheselabelsareplacedatth epeakoftheR-waveof eachbeat,althoughsomearemisaligned.Whenthepeakofthe R-waveismisaligned intheheartbeatcycle,wesearchforthelocalmaximaandadj ustfortheR-wavetiming. Weextractwindowsof700msaroundthebeat,300msbeforeand 400msafterthe R-peakannotations. ThebeatsintheMIT-BIHdatabaseareclassiedinto20class es.Theliterature usesanumberofdifferentclusteringsofthedatainorderto testheartbeatclassiers. However,wewillusethestandarddenedbythe fortheAdvancementofMedicalInstrumentation andInstitute ( 1999 )(AAMI).Table 6-1 showsthelabelsprovidedbytheMIT-BIH databaseandhowtheyarereclusteredaccordingtotheAAMIs tandard. AccordingtotheAAMIstandard,althoughtherearevedesig natedclasses,the classicationproblem,isdecomposedintotwobinaryclass icationstasks.Thersttask consistsofdistinguishingVentricularEctopicBeat(V)ag ainsttheremainingclasses(N, S,F,Q).Thesecondclassicationtaskconsistsofdetectin gSvs.theclasses(N,V,F, Q). InordertoperformtheclassicationusingLDAinasupervis orymode,wecreate trainingandtestingdatasets.Asmentionedintheintroduc tion,patientspecicheartbeat 96

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Table6-1.HeartbeatclassesgivenbytheMIT-BIHdatabasea longwiththeregroupingdenedbytheAAMIstandard. MIT-BIHMIT-BIHNumberofAAMINumberofclassnumberSamplesgroupssamples Normalbeat175052 N:BeatsnotfoundintheclassS,V,FandQ.90631 Leftbundlebranchblockbeat37259Rightbundlebranchblockbeat28075atrialescapebeats3416Nodal(junctional)escapebeat11229Atrialprematurebeat82546 S:Supraventricularectopicbeat.2781 Aberratedatrialprematurebeat4150Nodal(junctional)prematurebeat783Supraventricularprematurebeat92Prematureventricularcontraction57130 V:Ventricularectopicbeat.7236 Ventricularescapebeat10106Fusionofventricularandnormalbeat6803Fusionbeat(F)80 3 Pacedbeat127028 Q:Unknownbeat.8043 Fusionofpacedandnormalbeat38982Unclassiedbeat.1333 97

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958 958.5 959 959.5 960 960.5 961 961.5 962 -1 -0.5 0 0.5 1 1.5 raw data processed data 958 958.5 959 959.5 960 960.5 961 961.5 962 -5 0 5 x 10 -4 Time [s]Threshold Value Figure6-2.ExampleofheartbeatshapesfromtheMIT-BIHdat asetforpatient100.The toppanelshowsboththerawdataandtheprocesseddatafollo wingthelter proposedin deChazalandReilly ( 2006 ).Thebottompanelshowsthe pulsesgeneratedbytheIFsampler,theaveragepulserateal ongtheentire recordingis 40.59 pulses/second. signalshavebeenshowntoimproveclassicationaccuracy. Followingtheapproach presentedin deChazalandReilly ( 2006 ),theclassicationisbasedonaglobalanda localclassierdiscussedinsection 6.3 .Thetrainingdatarelativetotheglobalclassier istheset T consistingofpatientrecordsgivenbelow: T := f 101,106,108,109,112,114,115,116,118,119, 122,124,201,203,205,207,208,209,215,220,223,230 g Eachitemintheset T isapatientidenticationnumberdesignatedintheMIT-BIH arrhythmiadatabase.Thetestingdatasetconsistsofthere mainingpatientsrecords fromtheMIT-BIHarrhythmiadatabase. V := f 100,103,105,111,113,117,121,123,200,202, 210,212,213,214,219,221,222,228,231,232,233,234 g 98

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Sinceeachpatient'suniquebeatmorphologycontributesto predictionaccuracy, asampleofheartbeatsofapatientinthetestingset V isappendedtotheset T .For example,inordertolabelasequenceofheartbeatsofpatien t100in V ,therstve hundredbeatsareaddedtotheset T .Inthenextstep,theglobalandlocalclassiers trainedwiththedatain T andthepatient-specicsetof500beatsrespectively.Fina lly, theremainingheartbeats(excludingtherst500)forthepa tientareclassiedusing theglobal-localclassierpair.Thesestepsarerepeatedf orallthepatientsin V to determineoverallaccuracyofthemethod. 6.3PulseBasedClassication TheoutputpulsesfromtheIFdonotfallintostandardrepres entationsusedin signalprocessing,mainlybecausepulsetrainsarenoteasi lyrepresentedinvector spaces.Anumberofdifferentrepresentationsforpulsetra inshavebeensuggested inthecomputationalneuroscienceliterature,theseinclu destochasticpointprocess models DaleyandVere-Jones ( 2002 ),projectionsintoreproducingkernelHilbertspaces Paivaetal. ( 2009 ),linearltering Riekeetal. ( 1997 ),timeembeddingsbasedonthe interpulseintervals Sauer ( 1994 1997 )andothers Rossum ( 2001 ); VictorandPurpura ( 1997 ).Throughoutthispaperweuseasimplebinningapproach,wh ichiscommonly usedinthecomputationalneuroscienceliterature SanchezandPrincipe ( 2007 ).We willshowthatthissimpleandcomputationallyeffective(i ntegeraddition)approachis sufcienttoobtainclassicationresultscomparedtothos ereportedin deChazaland Reilly ( 2006 ).Nevertheless,inourfutureworkwewillexploreotherrep resentationsfor pulsetrainsthatwouldallowustoclassifythem.6.3.1PulsePre-Processing Assumingapulsetraindenedontheinterval [0, T ] ,binningconsistsofdividing thedomainintoasetofnon-overlappingintervalsandcount ingthenumberofeventsin eachinterval.Thesecountsrepresentthebinnedvector.Th isrepresentationmaintains thetemporalstructureofthepulsewhileprovidingavector spacerepresentation. 99

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Nevertheless,theprecisetimingislost.Asmallerbinsize wouldreducethetimejitter butresultsinahigherdimensionalspacewhichwouldrequir emoresamplesina classicationtask.Hereweuseabinsizeof 35 ms ,whichwaschosenexperimentally fromFigure 6-3 Sincetheheartbeatsarelimitedindurationwecanrepresen tthemasan N dimensionalvector,where N isthenumberofbins.Figure 6-1 showstheoverlaid heartbeatsforclasses1(N)and5(V)fortwodifferentpatie nts(119,106).Thevariability intheshapesforthesamepatientandacrosspatientsisevid entspeciallyinthecase ofclass(V).Nevertheless,theshapeoftheheartbeatsisno tsufcientinsomecases todiscriminatetheclasses.Therefore,informationonthe preandpostRRintervalsis included.Inordertousethisfeatureacrosspatientsanorm alizationstepisrequired, assuggestedin deChazalandReilly ( 2006 )weuseboththepreandpostRRintervals aswellasthepreandpostRRintervalsnormalizedbythemean RR-intervalforthe specicpatient.Nevertheless,thispaperisonlyconcerne dwiththediscrimination poweroftheproposedpulsebasedrepresentationandtheref oreweassumethe locationofthebeatsisgiven.Theproblemofdetectionwill betreatedinfuturework. Thefeaturevectorforanygivenbeatconsistsof N countsfollowedbypreandpost RRintervalswithandwithoutnormalization.Thisproduces a N +4 dimensionalfeature vectorwhichisthenfedintotheclassier.6.3.2Classier Anumberofdifferentclassiershavebeenusedinthelitera tureincluding,SVM M.R.Homaeinezhad ( 2012 ),post-classicationfollowingclustering,lineardiscr iminant analysis(LDA) deChazalandReilly ( 2006 ),fuzzynetworks,Phasespacereconstruction Chanetal. ( 2010 ); Nejadgholietal. ( 2011 )andothers. deChazalandReilly ( 2006 ) proposedclassicationbasedontheLDA Bishop ( 2006 )classier.Weimplementthe LDAusingfeaturesextractedfromthepulsesgeneratedbyth eIFsampler.Classication byLDAisbasedontheposteriorprobabilityofclassmembers hipofanewexample. 100

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Also,LDAassumesthattheunderlyingprobabilitydensityf unctionofthedatais Gaussian.Moreformally,theposteriorprobabilityofanex ample x belongingtoclass k is; P ( k j x )= P ( x j k ) P ( k ) P ( x ) (6–1) where P ( x j k ) istheclassconditionaldistributionofthefeaturevector x P ( k ) istheprior distributionofclass k ,and P ( x ) isthemarginaldistributionofthesamplefeatures.Since themarginaldistribution P ( x ) iscommonacrossallclassesitisoftendroppedfrom thedecisionfunction.Here k 2f 1, c g indexestheclass, c thenumberofclasses, and x representsthefeaturevector.Inthefollowing,wewillpar ametrizethedifferent componentsinvolvedintheposteriordensityfunction P ( k j x ) .UndertheGaussian assumptionofthefeaturevector x ,thesamplingdistribution P ( x j k ) isrewrittenas; P ( x j k )= 1 j j 1 2 (2 ) p 2 exp( 1 2 ( x k ) T 1 p ( x k ) T ) (6–2) where k k 2f 1, c g aretheclassconditionalmeans,and p isthepooled covariancematrixandisgivenby; p = 1 + + c c (6–3) Thepriordistribution P ( k ) is k ,andthedistribution p ( x ) isgivenby: p ( x )= c X k =1 P ( x j k ) P ( k ) (6–4) Thedecisionrule d k ( x ) todeterminemembershipofanewvector x isgivenby: d k ( x )= argmax| {z } k P ( k j x ) (6–5) Sincethemarginaldistribution P ( x ) iscommonacrossallclasses,itisdroppedfrom P ( k j x ) andthereforefromthedecisionrule.As P ( k j x ) ispositive,wemayaswelluse 101

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thelogarithmofthe P ( k j x ) sinceitdoesnotaffectthedecision.Therefore,usingthe Gaussianprobabilitydensityfunctionandtakinglogarith ms,wehave: log( P ( k j x )) C 1 2 ( x k ) T 1 p ( x k ) T +log k (6–6) whereCisaconstant.Thequantities k k and k arelearnedfromthetrainingset apriori andarepluggedinto( 6–6 )toevaluatethedecisionrule.Givenanewexample x new ,itsmembershipisdeterminedbyndingthelargest d k k 2f 1, c g .Iftheprior probabilities k areassumedtobeequal,the log k terminthedecisionrulemaybe dropped. Intheclassicationofpatientheartbeats,duetotheinher entvariabilityacrossand withinpatients,atestpatient'suniqueheartbeatdataisc ombinedwiththeglobalsetof featuresofallpatientstoperformclassication.Therefo rethetrainingsetisdependent onthetestpatient.Intuitivelytrainingsamplesfromthet estpatient(a.k.alocaltraining set)shouldbegivenhigherimportancethanthoseintheglob altrainingset.Inorder todoso deChazalandReilly ( 2006 )suggestslearningtheparametersfortwoLDA classiers,basedonlocal( lk lk )andglobal( gk gk )parametersestimatedfromthe trainingsets.Theestimatedparametersarethencombinedl inearly,suchthat: k = K k gk +(1 K k ) lk (6–7) k = K k gk +(1 K k ) lk (6–8) GiventhatweonlyhaveaccesstoashortsegmentoftheECGiti sconceivable thatcertainbeatclassesmaynotappearinthelocaltrainin gset.Insuchascenario thelocalsetisnotusefulandmaybeignored.Operationally ,weuseaparameter K k whichweighsthelocalandglobalclassiersrespectivelyg iveninequation( 6–9 ).The parameter K k isdeterminedinrelationtothenumberofsamplesinagivenc lassas 102

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suggestedin deChazalandReilly ( 2006 ): K k =min N g k 10 W (6–9) Where N g k isthenumberofsamplesinclass k andtheparameter( W )isusuallysetto 0.7. 6.4PerformanceMetrics Wemeasuretheperformanceofourmethodologyintermsofove rallsamplerates andtheclassicationerror.Thedataratesarevariedbycha ngingthethresholds( p n ) oftheIFsampler.Foreachthreshold,all48recordsaretran sformedintopulsetrains andtheaveragepulserateisreported.Theclassicationre sultsaretabulatedinterms ofthesensitivity( SE i ),specicity( SP i ),positivepredictivevalue( PPV i )andaccuracy ( ACC i )whicharerespectivelydenedas: SP i = TN i TN i + FP i PPV i = TP i TP i + FP i (6–10) SE i = TP i TP i + FN i ACC i = TP i + TN i TP i + TN i + FP i + FN i (6–11) where TP i (truepositive)isthenumberofbeatsofthe ith classcorrectlyclassied; TN i (truenegative)isthenumberofbeatsnotbelongingtothe ith classandnotclassied inthe ith class; FP i (falsepositive)numberofbeatserroneouslyclassiedint othe ith class;andnally FN i (falsenegatives)isthenumberofbeatsofthe ith classclassied intoadifferentclass.Theaccuracy ACC i ,denotestheratiobetweenallcorrectlyand incorrectlyclassiedbeats. 6.5ResultsandDiscussion EvaluationofclassierperformanceisreportedusingtheM IT-BIHarrhythmia databaseandinaccordancewiththeAssociationfortheAdva ncementofMedical Instrumentation(AAMI)standard.Theresultsofbinarycla ssicationtests-VEB(V)vs (N,S,FandQ)andSVEB(S)vs(N,V,FandQ)arereportedandtab ulatedinTable 6-2 .Eachrowcorrespondstoatestpatientwhoserecordnumberi sgivenincolumn1, 103

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columns2-6correspondtothenumberofbeatsineachclass,c olumns7-12presentthe classicationresultsintermsofaccuracy(ACC),sensitiv ity(SE)andpredictivepositive value(PPV).Finally,column13displaystheaveragesample rate.Asthefeaturevector forclassicationisbasedonpulsecountsineachbin,wedet erminedthenumberthe numberofbinstobe20.Figure 6-3 showstheeffectsofvaryingthenumberofbinsasa functionoftheclassicationmetrics.Itisnoteworthytha tsizeofthebindoesnotplaya crucialrole,whichsuggeststhatthenerdetailsinthehea rtbeatshapesdonotprovide extrainformationfordiscriminabilitypurposes. 0 20 40 60 80 100 50 55 60 65 70 75 80 85 90 95 100 Number of Bins SE avg PPV avg SP avg AVEB(V). 0 20 40 60 80 100 20 30 40 50 60 70 80 90 100 Number of Bins SE avg PPV avg SP avg BSVEB(S). Figure6-3.PlotoftheclassicationmetricsforbothVEBan dSVEBclassesinrelation tothenumberofbinsusedtocreatethefeaturevector.Thesa mpler parameterswerechosentomatchthoseusedtogeneratethero w highlightedinboldfaceinTable 6-3 .Throughoutthepaperwexedthe numberofbinsequalto20,whichcorrespondstoabinwidthof 35 ms and theheartbeatrangeparameter( W )issetto 0.7 s InourexperimentalsetuptheIFsamplerparametersweresel ectedsuchthat =0.1, p =0.0005, n =0.0005, =1 ms .Theseparameterswerexedthroughall theexperiments.Fromthelastcolumn(averagesamplingrat e)inTable 6-2 itisevident thatthereisalargevariationinthesamplingrates.Noneth eless,forpatientrecords 100 and 221 weobtaincompressionanddiscriminabilityfortheSVEBand VEBclasses respectively.Furthermore,Table 6-3 ,showstheaverageresultsoverallthetestpatients fordifferentaveragesamplerates.Thesesamplerateswere obtainedbyvaryingthe 104

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thresholds( p and n ).SpecicallywecompareourresultstothosefromChazalet al deChazalandReilly ( 2006 )sincebothpreprocessingandclassierareidentical. Thereforeanydiscrepancyisduetothefeaturesthatdescri betheheartbeatshapes. Comparingtherowinbold(Table 6-3 )forwhichtheaveragesamplerateis117.86(note thatChazaletal deChazalandReilly ( 2006 )uses360samples/s).Alsonotethatthe classicationresultsarecomparabletothosein deChazalandReilly ( 2006 ); Huetal. ( 1997 ); Inceetal. ( 2009 ).WehavealsoaddedtheresultsfromWienetal Wiensand Guttag ( 2010 )forcompleteness.Theseresultscannotbedirectlycompar edtoours sincetheclassicationmethodologyiscompletelydiffere nt.In WiensandGuttag ( 2010 ) anactivelearningapproachispresentedinwhichtheclassi erhasaccesstoallthe testbeats(withoutthelabels).Incontrastweonlyusethe rst500beats(withlabels) following deChazalandReilly ( 2006 ).Wemaypointoutthatthepulse-basedfeatures canbeusedinanactivelearningframework.Inordertoimple ment WiensandGuttag ( 2010 )requiresanelaborateexperimentalsetupandresultswill bereportedinafuture submission. Ourgoalisherewastoshowthatthecompressedrepresentati oncontains sufcientdiscriminativeinformationandatthesametimew ecanaccessdiscriminative informationdirectlyinthepulsedomainwithouthavingtor econstructthesignalwhich isthetypicalapproachwhencombiningcompressionandclas sication.Intermsof compressionweseethatwehavereducedthesampleratesforc ertainpatients.Itmay benotedthatcomparisonofourmethodologytoexistingmeth ods,suchasCS Kanoun etal. ( 2011 ); Polaniaetal. ( 2011 )orFRI Haoetal. ( 2005 ),isnotpossiblesincetheir approachesaredesignedtowardsignalreconstruction.Fur thermore,inthecaseofCS itisnotclearhowonecoulddesignaclassiersbasedonrand omprojectionsusedfor compression.ThismaybefeasibleinthecaseofFRI,wearein vestigatingmethodsto combineFRIbasedcompressionandclassication. 105

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Wehaveshownthattimebasedrepresentationsgeneratedbyt heIFsampler providecompressionwhilestillpreservingdiscriminabil ityinheartbeatshapesin patients.Theproposedfeaturesareobtaineddirectlyfrom thecompressedrepresentation avoidinganyreconstruction.Thisapproachtocompression andclassicationfor heartbeatshasnotbeentreatedintheliteraturetotheauth orsknowledge. Thefeaturespresentedherearebasedonsimplepulsecounts inthebins.Inthe futurewemaydrawfromcurrentresultsinpointprocesstheo rywhichwillallowusto modeleachclassbasedonaclassconditionalintensityfunc tionconstructedfromthe precisetimingsoftheeventswhichmaybeconstruedasevent sfromarenewalprocess. Furthermore,thesefeaturescanalsobeusedincombination withtheactivelearning frameworkwhichhasbeenshowntoreducethenumberoflabele dheartbeatsrequired fromanexpertcardiologist. 106

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Table6-2.Classicationresultsusingthebinnedpulsefea tures(20bins)foreachofthetestpatients.Column1refers to therecordnumberasassignedintheMIT-BIHdatabase,colum ns2-6representthenumberofbeatsperclass, columns7-12representtheperformancemetricsandcolumn1 3showstheaveragesampleratefromthe time-encodedrecord.Theaverageresultsforallmetricsal ongallpatientsarepresentedinTable 6-3 astherow inbold. Rec NumberofBeatsSVEBVEB SampleRate NSVFQACCSEPPVACCSEPPV 10017442810096.5696.4331.03100.00100.00100.0040.591031582200098.860.000.0099.18-0.0093.4410520410260589.29-0.0091.5196.1512.50157.001111623010099.63-0.0099.940.00-101.701131293200099.92100.0066.6799.77-0.00118.401171034100099.81100.0033.3399.81-0.0089.281211361110097.36100.002.7099.93100.0050.0047.231231016020099.90-0.00100.00100.00100.0078.792001390286812092.3460.7110.1898.4895.4599.85144.70202156954121056.3094.4406.6899.7691.6778.5781.252101959201629094.7085.0013.2898.0591.9883.7193.872122248000099.02-0.0099.87-0.00165.60213225227199273098.733.7010.0096.6996.4869.57327.80214156401971095.29-0.0099.8399.4998.99172.7021916043470096.370.000.0099.5282.98100.00133.00221162603010098.70-0.0099.9599.67100.0093.48222177420900086.4371.7741.6799.85-0.0045.25228130032500097.420.000.0099.8198.80100.00100.502311071000099.81-0.00100.00--104.80232271100900099.1499.4199.50100.00--69.99233187346966094.77100.002.8899.8499.8699.57243.1023422005030098.2250.0062.5099.78100.0037.5089.52 107

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Table6-3.Averageclassicationmetricscomparedtothest ateoftheart.Thesample ratesintheproposedmethodistheaverageoverallpatients Methods SVEBVEB SampleRate SESPPPVSESPPPV Huetal. ( 1997 )78.996.875.8N/AN/AN/A180.0 deChazalandReilly ( 2006 )87.796.247.094.399.796.2360.0 Inceetal. ( 2009 )63.599.053.784.698.787.4360.0 WiensandGuttag ( 2010 )92.0100.099.599.699.999.3128.0 Proposed 72.389.320.795.871.897.329.780.695.039.396.191.099.363.890 194 940 497 399 188 4117 8 88.695.242.897.898.986.4223.3 108

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CHAPTER7 CONCLUSIONSANDFUTUREWORK Wehavepresentedasamplingandreconstructionframeworkf ortheIntegrateand Firemodel.Wehaveshownthatperfectrecoveryisnotpossib leinthebandlimted space.Nevertheless,weprovideanapproximatereconstruc tionalgorithmwith itscorrespondingerrorbounds.Inapracticalscenariowea reonlyconcerned withreconstructioninnitedimensionalspaces.Underthi sassumption,recovery algorithmsbasedontheFourierseries,B-splinesandradia lbasishavebeenpresented. Furthermore,B-splinescanbeusedtosatisfythelinearcon straintsimposedbythe samplesandalsothevariationalrestrictionsontheintegr aloftheinput.Thisprior knowledgefurtherconstraintstheinputwhenthesamplingm atrixisill-conditioned.The algorithmsproposedtradeoffreconstructionaccuracyand speed,acrucialfactorin real-timeBMIs.AvenuewedidnotexploreistheuseofVolter rakernelsasmodelfor non-linearoperators,suchastheIFtransformation.Thebe netofthisapproachisthat oncethekernelsarelearnedtheycanbequicklyappliedtoth esamplestorecoverthe input.Thedrawbackistheexponentialincreaseintheparam etersasweusehigher orderapproximations.Thisapproachcanalsobeusedtostud yhowthebehaviorofthe IFencodingvariesfromlineartononlinearasthesamplerat esshiftfromratecodesto timecodes. TheproposedrecoveryalgorithmsandIFsamplerweretested intwoapplications. Therst,encodedneuralactionpotentialsrecordedfrommi cro-electrodesinaBMI setting.Thesevereconstraintsonarea,bandwidthandpowe r,makethisacandidate applicationfortheIF.ItwasshownthattheIFrepresentati onprovidedsub-Nyquist dataratesandyetanaccuratereconstructionoftheactionp otentials.Theresultswere notonlymeasuredintermsofthereconstructionaccuracybu talsointermsofspike sortingerrorandbothinrelationtothevariationofthesam plerparametersandthe inputcharacteristics.Furthermore,wealsopresentedpre liminaryresultsshowedthat 109

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discriminabilitybetweentheactionpotentialswasretain edinthepulsedomain,by simplybinningthesamples.Therefore,pulse-basedfeatur esmaybeabletoreplacethe traditionalrepresentationandatthesametimeavoidthene edforreconstruction. ThesecondapplicationusedtheIFrepresentationtoencode ECGsignals.Similarly theperformancewasmeasuredintermsofthesampleratesand theclassicationerror whencomparingheartbeats.Wehaveshownthatforcertainpa tientsweareableto preservethecurrentclassicationaccuracieswhilereduc ingthedatarates. Theseapplicationsshowcasetheadvantagesofusingasingl eIFsamplerin encodingsignalclasseswhoseregionsofinterestarelocal izedbothintimeand amplitude.Thenextstep,isbeingabletoprovidecompresse dpulserepresentations forarbitraryinputspaces.Oneoftheapproachesistochang ethedesignoftheneuron modelandaddadaptivefeaturesthattracknoveltyintheinp ut.Furthermore,wecan alsousenetworksofIFsamplersandadapttheirparametersa nalogoustotheworkin thepulsedneuralnetworkandcomputationalneurosciencel iterature. 110

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BIOGRAPHICALSKETCH AlexanderSinghAlvaradoreceivedhisBachelorofSciencei nElectricalengineering in2005fromInstitutoTecnol ogicodeCostaRica,andhisMasterofSciencedegreein electricalandcomputerengineeringin2008fromtheUniver sityofFlorida,Gainesville, Florida.Hisinterestincludemachinelearning,adaptives amplingandcomputational neuroscience. 121