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Mathematics for Industry 3 Mikio Tohyama Waveform Analysis of Sound Mathematics for Industry Volume 3 Editor-in-Chief MasatoWakayama(KyushuUniversity,Japan) ScientificBoardMembers RobertS.Anderssen(CommonwealthScientificandIndustrialResearchOrganisation, Australia) HeinzH.Bauschke(TheUniversityofBritishColumbia,Canada) PhilipBroadbridge(LaTrobeUniversity,Australia) JinCheng(FudanUniversity,China) MoniqueChyba(UniversityofHawaiiatMaNnoa,USA) Georges-HenriCottet(JosephFourierUniversity,France) JoséAlbertoCuminato(UniversityofSãoPaulo,Brazil) Shin-ichiroEi(HokkaidoUniversity,Japan) YasuhideFukumoto(KyushuUniversity,Japan) JonathanR.M.Hosking(IBMT.J.WatsonResearchCenter,USA) AlejandroJofré(UniversityofChile,Chile) KerryLandman(TheUniversityofMelbourne,Australia) RobertMcKibbin(MasseyUniversity,NewZealand) GeoffMercer(AustralianNationalUniversity,Australia)(Deceased,2014) AndreaParmeggiani(UniversityofMontpellier2,France) JillPipher(BrownUniversity,USA) KonradPolthier(FreeUniversityofBerlin,Germany) OsamuSaeki(KyushuUniversity,Japan) WilSchilders(EindhovenUniversityofTechnology,TheNetherlands) ZuoweiShen(NationalUniversityofSingapore,Singapore) Kim-ChuanToh(NationalUniversityofSingapore,Singapore) EvgenyVerbitskiy(LeidenUniversity,TheNetherlands) NakahiroYoshida(UniversityofTokyo,Japan) Aims&Scope Themeaningof“MathematicsforIndustry”(sometimesabbreviatedasMIorMfI)isdifferent fromthatof“MathematicsinIndustry”(orof“IndustrialMathematics”).Thelatterisrestric- tive:ittendstobeidentifiedwiththeactualmathematicsthatspecificallyarisesinthedaily managementandoperationofmanufacturing.Theformer,however,denotesanewresearch field in mathematics that may serve as a foundation for creating future technologies. This conceptwasbornfromtheintegrationandreorganizationofpureandappliedmathematics in the present day into a fluid and versatile form capable of stimulating awareness of the importance of mathematics in industry, as well as responding to the needs of industrial technologies.Thehistoryofthisintegrationandreorganizationindicatesthatthisbasicidea willsomedayfindincreasingutility.Mathematicscanbeakeytechnologyinmodernsociety. The series aims to promote this trend by (1) providing comprehensive content on applicationsofmathematics,especiallytoindustrytechnologiesviavarioustypesofscientific research,(2)introducingbasic,useful,necessaryandcrucialknowledgeforseveralapplica- tionsthroughconcretesubjects,and(3)introducingnewresearchresultsanddevelopments for applications of mathematics in the real world. These points may provide the basis for opening a new mathematics-oriented technological world and even new research fields of mathematics. Moreinformationaboutthisseriesathttp://www.springer.com/series/13254 Mikio Tohyama Waveform Analysis of Sound 123 MikioTohyama Tokyo,Japan ISSN2198-350X ISSN2198-3518(electronic) ISBN978-4-431-54423-4 ISBN978-4-431-54424-1(eBook) DOI10.1007/978-4-431-54424-1 SpringerTokyoHeidelbergNewYorkDordrechtLondon LibraryofCongressControlNumber:2014955147 ©SpringerJapan2015 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped.Exemptedfromthislegalreservationarebriefexcerptsinconnection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Duplication of this publication or parts thereof is permitted only under the provisions of the Copyright Law of the Publisher’slocation,initscurrentversion,andpermissionforusemustalwaysbeobtainedfromSpringer. PermissionsforusemaybeobtainedthroughRightsLinkattheCopyrightClearanceCenter.Violations areliabletoprosecutionundertherespectiveCopyrightLaw. Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant protectivelawsandregulationsandthereforefreeforgeneraluse. While the advice and information in this book are believed to be true and accurate at the date of publication,neithertheauthorsnortheeditorsnorthepublishercanacceptanylegalresponsibilityfor anyerrorsoromissionsthatmaybemade.Thepublishermakesnowarranty,expressorimplied,with respecttothematerialcontainedherein. Printedonacid-freepaper SpringerispartofSpringerScience+BusinessMedia(www.springer.com) About the Author MikioTohyamaholdstheDr.Eng.fromWasedaUniversityinTokyo,Japan. Dr. Tohyama was involved in research projects in acoustics and signal processing at the NTT (Nippon Telegraph and Telephone) research laboratories for 18 years beginning in 1975. He was a professor at Kogakuin University (1993–2003) and v vi AbouttheAuthor Waseda University (2003–2011) in sound and perception. Since 2012 he has operated his research consulting firm, Wave Science Study (WSS). His present interest is sound signature expression in the temporal domain oriented to sound perception.Dr.Tohyamaenjoysplayingthepianoeveryday. E-mailaddress:[email protected] Preface What is this sound? What does that sound indicate? These are two questions frequentlyheardindailyconversation.Soundresultsfromthevibrationsofelastic media, and in daily life provide informative signals of events happening in the surroundingenvironment.Ininterpretingauditorysensations,thehumanearseems particularly good at extracting the signal signatures from sound waves. Although exploring auditory processing schemes may be beyond our capabilities, source signatureanalysisisaveryattractiveareainwhichsignalprocessingschemescan bedevelopedusingmathematicalexpressions. This book is inspired by such processing schemes, and oriented to signature analysis of waveforms. Most of the examples in this book are taken from data of sound and vibrations; however, the methods and theories are mostly formulated usingmathematicalexpressionsratherthanbyacousticalinterpretation.Thisbook might therefore be attractive and informative for scientists, engineers, researchers, and graduate students, who are interested in the mathematical representation of signalsandtheapplicationsofFourieranalysis. The book can be described as being nearly self-contained, but does assume readerstobefamiliarwithintroductorytopicsindiscretesignalprocessingasinthe discreteFouriertransform.Hencethisbookmightbealsousableasatextbookfor useingraduatecoursesinappliedmathematicsontopicsasincomplexfunctions. Almostallscientificphenomenaaresensedaswavespropagatinginsomespace. Overtheyears,waveformanalysishasthereforebeenoneoftheresilientacademic areas of study, and still is seen as fertile ground for development. In particular, waveformanalysisbasedonthetheoryoflinearsystemswouldbeagoodexample whereaphysicalinterpretationcanbegiventothemathematicaltheoryofcomplex functions in terms of magnitude, angle, poles, and zeros of complex functions. Readers who are interested in the physical aspects of sound and vibration data or elementaryformulationofwaveequationsandtheirsolutionsarerecommendedthe book, M. Tohyama, Sound and Signals, Springer 2011, which can be used as a complementarycompaniontothisbookoragoodreference. vii viii Preface Thisbookisorganizedasfollows. The sinusoidal representation of discrete signals dealt with in this book is fundamental, and still newly developing from both a theoretical and a practical perspective in signal analysis. Sinusoidal functions are the conceptual roots from which most, if not all, signals are generated. Therefore sinusoids are inescapable whenanalyzinglinearsystems.Incontrast,discreteFourieranalysisisformulated to develop representations of discrete sequences of finite record length, assuming sequencescanbeperiodicextendedoutsidethefinite-lengthinterval.Consequently, extrapolation (or prediction) even for a sinusoidal sequence is problematic when a sequence observed in a finite interval is represented by its Fourier transform. Fromtheperspectiveoftheoryandpractice,thisissueinFourieranalysistypically motivatestheintroductionofnewschemesforsignalprocessing. Following an elementary introduction into the manipulation of discrete sequences and polynomials, this book is mostly devoted to the analysis and representationofcompoundsinusoidalsequencesandtheirrelatedpolynomials.The compound sinusoidal functions are representatives of auditory sound waveforms, whichrangefromperiodictoalmostperiodicwaves.Thealmostperiodicfunctions might form the conceptual basis for discrete analysis of sequences; however, the functions would be suitable in representing non-periodic waves in a deterministic sense. This book develops spectral peak selection from the interpolated spectral function so that the spectral components might be estimated, and thus a finite- lengthsequencecouldbeextrapolatedoutsideitsintervalofobservation. Signalanalysisasinpowerspectralanalysisismostlyperformedinthefrequency domain.Nevertheless,signalsignaturesinthetemporaldomainaswellasthosein thefrequencydomainaredealtwithinthisbook.Narrow-bandtemporalenvelopes and auto-correlation functions characterize the signal signatures in both these domains. The envelopes are sensitive to phase spectral properties and spectral correlationsinthefrequencyplaneratherthanthepowerspectralfunctions,whereas the auto-correlation functions are essentially independent of phase properties; however,botharefunctionsoftimeandfrequency. Thisbooktakesaframe-wiseapproach tothetimedomaininsteadoftheusual mathematicalformulationofthetemporalenvelopesandauto-correlationfunctions of the time and frequency. The reason is that an open question remains at present astowhichwindowing(orframing)functionsareadequateinmakingthefunctions informative.Triangularwindowingsequences,developedforcalculationsofgroup- delayfunctionswithoutexplicitlyincludingthederivatives,areappliedtotheframe- wise approach. The frame-wise auto-correlation analysis of a reverberant speech samplewouldbeagoodexamplethatillustrateswhyaspeechsamplecouldbestill recognizableevenunderadversereverberantconditions.Theexampleillustratesthe benefits of the frame-wise approach to the time domain in preference to the long- termanalysisinwhichtemporalinformationislost. The envelopes are decomposed into clustered line-spectral components around a central frequency. The clustered components are too closely distributed (with a very narrow spacing) over the frequency interval to be separately picked up by spectral peak selection. This indicates that a sinusoidal compound sequence Preface ix characterized by the envelope might not be well represented by such a selection. Thisbookdevelopsclusteredline-spectralmodeling,calledCLSMforshort,sothat the sequence, including its envelope, can be extrapolated outside the interval of observation. CLSM is formulated using the least-squares error solution of a set of simultaneousequationsoverthefrequencydomain. The complementarity between complex time and frequency planes indicates interestingpropertiesstemmingfromthezerosforatimesequenceinthecomplex time planes, as well as the zeros of the transfer function (or z-transform of a time sequence)inthecomplexfrequencydomain.Thezerosofthecomplextimeplane givesaclearinterpretationoftherelationshipbetweentheinstantaneousmagnitude (ortheenvelope)andtheinstantaneousfrequency(orthefrequencyofthecarrier), aszerosinthecomplexfrequencydomainidentifytheminimum-phasepropertyfor thespectralfunction.Therelationshipbetweenthemagnitudeandphaseproperties thatisexpectedforminimum-phasesystemsmightbeformallyrealizedbetweenthe temporalenvelopeandcarrierfrequencyforminimuminstantaneous-phasesystems. Themethodsofanalysisarebasedonthecomplexcomplementaritybetweenthe timeandfrequencyplaneswherethesignalsarerepresentedusingsinusoidalfunc- tions or discrete sequences. Following the complex complementarity, the analytic signalscanbeinterpretedintermsofthecomplexspectralproperties,and,clustered time-sequence modeling (CTSM) is developed based on the complementary with CLSM. In contrast to CLSM which characterizes the slowly changing temporal envelopes in the time domain, CTSM identifies pulse-like but frequency-band limited waveforms (sequences) by solving the least-squares error solution in the time domain. The Fourier transform of a sequence of finite record length is also asinusoidalcompoundfunctiononthefrequencyplane.Thereforefrequency-band limitingofthesequenceisformallyequivalenttonarrowingtheobservationinterval ofthecompoundsinusoidalfunctiononthetimedomain. Asunderstoodthroughthespectralpeakselection,thepeaksidentifyresonantor eigenfrequencies, the spectral functions corresponding to those eigenfrequencies, and the effect of poles. However, troughs and zeros are also significant signatures that characterize the phase properties of spectral functions, in particular, short waveformsasinpulse-likesoundwaves.Troughsorzerosofthespectralfunctions are strongly sensitive on the waveforms of brief sequences, even if the envelopes ofthemagnitudespectralfunctionslookmoreorlessidentical.Indeed,theresidual of the CLSM (modeling error) for piano-string vibration yields a brief impulsive sequencecorrespondingtothesourcewaveformexcitedbytheactionofthehammer onthepianostring.Itisanexampleofhowthezeroortroughofthespectralfunction uniquelycharacterizesthesourcewaveformbyseparatingthecharacteristicsofthe transferfunction. Thezerosareformallyidentifiedfromthefactorizationofthez-transformgiven asapolynomialoffiniteorder,subjecttotheconstraintthatthesequencehasfinite length in the temporal domain. Every zero is, in principle, identified by a pair of adjacent time pulses composed of a single unit pulse followed by a single pulse withaone-unitsampleofdelayandcomplexmagnitude.Thetimesequencecanbe formallyreconstructedbyconvolutionofallthepairsofadjacenttimepulses.