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PartI
Fourieranalysisand
applicationstosound processing 8
Chapter1
Sound formofaudio. Mostoft hesesoundsaretransferr eddirectlyfrom thesource tooure ars,likewh enwehaveafacet ofaceconvers ationwith someon eor listentothesou nds inaforest orastreet.However,ac onsid erable partofthe soundsare generatedbyl oudspeakersinvari ouskindsof audiomachineslikecell phones,digi talaudioplayers,homecinemas,ra dios,televisio nsetsandsoon. Thesoun dsproducedbythese machinesareeithergenera tedfro minformation storedinside,ore lectromagneticwavesare pick edupbyanantenna,processed, andthenc onvertedtoso und.Itisthiskindofsoundwear egoing tostudyin thischapter .Thesoundthatisstoredins idethe machinesorpick edupbythe antennasisusuallyrepresentedas digitalsound.Thi shascerta inlimitat ions, buta tthesamet imemakesi tveryeasy tomanipulatea ndprocessthesoun d ona computer. Whatwepe rceiveass oundcorrespondstothephysic alph enomenonofslight variationsinairpressurenear ourear s.Larger variationsmeanloudersoun ds, whilefasterva riationscorrespon dtosoundswithahigherpitch.Theai rpres- surevaries continuouslywitht ime,butatagivenpointintimei thasaprecise value.Thismeanstha tsoundcan beconsideredto beamathemati calfunction. withtimeas thefreevaria ble.W henafunct ionrepresen tsasound,itisoften referredtoasacontinuoussignal. Inth efollowin gwewillbrieflydiscusstheb asicpr opertieso fsound:firstthe significanceofthesizeof thev ariations,and thenhow manyv ariationst here areper second,the frequencyofthe sound.We alsoconsidertheimportant factthata nyreasonableso undmaybeconsidered tobebuiltfromverysimple basissounds.Sinceaso undmaybev iewedasafunct io n,themathema tical equivalentofthisisthatany decent functionmaybeconstr uctedfromver y simplebasisfunctions.F ourier-analy sisisthetheoret icalstudyofthis,andin thenext chapterswearegoin gtostudythisfromapractical andcom putation al 9
0.20.40.60.81.0101324101325101326101326
(a)Asoundsh own intermsofair pressure
0.51.01.52.0?1.0?0.50.51.0
(b)Asoundsh own intermsofthe di ff er- encefromthea mbientair pressure
Figure1.1:Tw oexam plesofaudios ignals.
perspective.Towardstheendofthis chapterwealsoconsider thebasic sof digitalaudio,andillustr ateitspowerby performi ngsomesimpleoperationson digitalsounds.
1.1Loudness:So undpressureanddecibels
Anexam pleofasimplesoundiss howninF ig ure1.1(a)wheretheos cillations inair pres sureareplottedagainstime.Weobserve thatth einitialairpress ure hastheva lue101325 (wewillshortlyr eturntowha tunitisus edhere), and thenthepres surestartsto varymoreandmoreuntilitoscillatesregu larly betweenthevalues10132 3and1013 27.Intheareawhereth eairpress ure iscons tant,nosoundwillbeheard,but asthe variationsincre aseinsiz e,the soundbecomes louderandlouderun tilaboutt ime t=0.6wherethesizeo f theoscill ationsbecomesconstant.Thefollowingsum marisessomebasicfact s aboutairpressure. Fact1.2(Airpre ssure).Airpressu reismeasuredbytheSI-unit Pa(Pas cal) whichisequiv alentt oN/m 2 (force/area).Inotherwords ,1P acorresponds tothe forceexe rtedonanareaof1m 2 bythea ircolumn abovethis area.The normalairpressurea tsealevelis10 1325Pa. Fact1.2expla instheval uesontheverticalaxisinF igure1.1 (a):Thesound wasreco rdedatthenormalairpress ureof1 01325Pa.On cethesoundstarted , thepress urestartedtovarybothbelowandab ovethisvalue,and after ashort transientphasethepress urevariedsteadilybe tween101 324Paand101326Pa, whichcorrespo ndstovariationsofsize1Paaboutthe fixedv alue.Everyday soundsty picallycorrespondtovariationsin airpressureofabout0.00002-2P a, whileajeteng ine maycausev ariationsaslargea s200Pa.Sh ortexposure tovari ationsofabout20Pamayinfactlea dtohea ringdamage.Thev olcani c eruptionatKrakatoa,Indon esia,i n1883,producedasoundwavewit hvariations aslargeas almost100 000Pa, andth eexplosion couldb eheard5000kma way. 10 Whendiscuss ingsound,oneisusuallyonlyinterestedin thevariationsin airpressu re,sotheambientair pressure issubtractedf romthemeasurement. Thiscorres pondstosubtracting101325fromthev alues ontheverticalaxisin Figure1.1(a).InFig ure1.1(b)thesubtract ion hasbeenperformedfo ranother sound,andweseethat thesound hasasl ow,cos-like,variationin airpressure, withsomes mallerandfas tervariationsimposed onthis.Thi scombination ofseveral kindsofsystematic oscillationsinairpressureist ypicalf orgeneral sounds.Thesizeoftheos cillat ionsisdirec tlyrel atedtothe loudnessofthe sound.We haveseent hatforaudiblesoundsthev ariations mayrangefrom
0.00002Paallthewayupt o1000 00Pa.Thisi ssuchawide rangetha titis
commonto measuretheloudness ofasound onalogarit hmicscale.O ftenai r pressureisnorma lizedsot hatitliesbetween-1and1:The value0thenrepre- sentstheambient airpressure,w hile-1and1representthelowestandhighes t representableairpressure,respective ly.T hefollowingfactboxsummarises the previousdiscussionofwhat asoundis,andintroducestheloga rithmi cdecibel scale. Fact1.3(Soundpressureand decibels).Thephy sicaloriginofsoundisvar i- ationsinairpressur enear theear.The soundpressure ofasou nd isobtained bys ubtractingtheaverageairpressureo verasuita bletimeintervalfromt he measuredairpressurewithi nthetimei nterval.Asquareofthisdi ff erence isthe naveragedove rtime,andthesoundpre ssureisthesquareroot ofthis average. Itis commo ntorelateagivensou ndpres suretothesmallest sou ndpressure thatcanb eperceiv ed,asale velonadecibelscale, L p =10l og 10 p 2 p 2 ref =20l og 10 p p ref
Herepisthem easuredso undpressurewhilep
ref isthe soundpre ssureofa justper ceivablesound,usuallyconsideredtobe0 .00002 Pa. Thesqua reofthesoundpress ureapp earsinthe definitionofL p sincethis representsthepowerofthes oundwhichi srelevantforwha tweperceivea s loudness. Thesoun dsinFigure1.1ares ynthet icinthattheywerec onstructe dfrom mathematicalformulas(seeExercises1. 4.2and1.4.3).Thesoundsin Figure1.2 ontheo therhand showthevariatio ninairpress urewhentherei snomath- ematicalformulainvolved ,suchasisthecaseforas ong.In(a)thereareso manyoscillationsthatit isimpossibletosee thedetails,but ifw ezoom inas in(c)w ecanseeth atther eisacont inuousfun ctionbehind alltheink.Itis importanttorealisethat inrealit ytheairpress urevariesmore thanth is,even overthes horttimep eriodin(c).Howe ver,themeasur ingequipmentwasno t abletopickup thosev ariations,a nditis alsodoubtfulwhe therwewouldbe abletoperceive such rapidvariations. 11
0.10.20.30.40.5?0.2?0.10.00.10.20.3
(a)0.5seco ndsoftheso ng
0.0050.0100.015?0.2?0.10.00.10.2
(b)thefirst 0.015secon ds
0.00050.00100.0015?0.2?0.10.00.1
(c)thefirst 0.002secon ds Figure1.2:Variations inairp ressureduring parts ofas ong. 12
1.2Thepitch ofasound
Besidesthesize ofthev ariationsin airpr essure,a soundhasanotherimpor- tantcharacte ristic,namelythefrequency(spe ed)ofthevariations .Formost soundsthefrequency ofthevar iationsvarieswithtime,butif wearet operceive variationsinairpressureassou nd,the ymustfall withinacertainrange. Fact1.4.Forahuman with goodhearingtop erceiveva riationsina irpressure assoun d,thenumberofv ariationspersecond mustbeintherange20-20000. Tomake theseconcept smoreprecise,w efirstrecallwhatitmeansfo ra functiontobeperi odic. Definition1.5.Arealfunctionfissaid tobeperiodic withper iodτif f(t+τ)=f(t) forallr ealnumberst. Notethat allthevalue sofa periodicf unctionfwithperiod τarekno wnif f(t)isknown foralltinthe interval[0,τ).The proto typesofperiodicfunctions arethe trigonometricones,and particularlysintandcostareofin teresttou s. trueforcost. Thereisasimp lewayt och angetheperiodofap eriodicfu nction ,namely bymu ltiplyingtheargumentbyaconstant. Observation1.6(Frequency).Ifνisanin teger, thefunctionf(t)= sin(2πνt)isperi odicwithperiodτ=1/ν.Whentvariesintheinter val [0,1],thisfunctioncoversatotalofνperiods.Thisisexpressedby saying thatfhasfrequencyν. Figure1.3illustrat esobserv ation1.6.Thefunctionin(a)isthepla insint whichcoverso neperiodwhentvariesintheinter val[0,2π].Bymultiplyingthe argumentby2π,theperiodissqueezedintotheinterval[0,1]sothe function sin(2πt)hasfrequencyν=1.The n,byalsomult iplyin gtheargumen tby2, wepush twowholeperi odsintoth einterval[0,1],sothefunctionsin(2π2t)has frequencyν=2.In( d)t heargumen thasbeen multipliedby5 - hencethe frequencyis5andt herearefiv ewho leperi odsintheint erval[0,1].Notethat anyfun ctionontheformsin(2πνt+a)hasfrequencyν,regardlessofthevalue ofa. Sincesoundcan bemodelled byfunctions, itisr easonabletosaythat a soundwithfreque ncyνisatri gon ometricfunctionwithfrequencyν. 13
123456?1.0?0.50.51.0
(a)sint
0.20.40.60.81.0?1.0?0.50.51.0
(b)sin(2πt)
0.20.40.60.81.0?1.0?0.50.51.0
(c)sin(2π2t)
0.20.40.60.81.0?1.0?0.50.51.0
(d)sin(2π5t) Figure1.3:Vers ionsof sinwithdifferentfrequencie s. 14 Definition1.7.Thefunc tionsin(2πνt)representswhatwewillcallapure tonewithfrequ encyν.Fr equencyismeasuredinHz(Herz )whichi sthesame ass -1 (thetimetismeas uredinseconds). Apu retonewithfre quency440Hzsoun dslikethis,andapur etonewith frequency1500Hzs oundslikethis. Anysound maybeconsidered tobeafun ction. Inthenextchapterweare goingtoseethata nyreas onablefu nctionmaybewr ittenasa sumofsimple sin-andcos-fu nctionswithintegerfrequenci es.Whenth isistranslatedinto propertiesofsound,weobtainan importa ntprincipl e. Observation1.8(Decompositionofsoundintopure tones ).Anysound fis asu mofpureto nesatd i ff erentfrequenci es.Theamountofeachfrequencyre- quiredtoformfisthe frequenc ycontentoff.Anysoundcanbereconstructed fromitsfrequen cycontent. Themost basicconsequ enceofobservat ion1.8isthatitgivesusanun- derstandingofhowanysound canbebuil tfromt hesimplebuilding blockso f puretones.Th ismeansthatwecanst oreasoun dfbyst oringitsfrequency content,asanalternativet ostor ingfitself.Thisalsogivesu sapossib ilityfor lossycompressi onofdigitalsound:Itturnsoutthatin atypic alaudiosignal therewillbemosti nformati oninth elowerfreque ncies,andsomefrequencies willbealm ostcom pletelyabsent.This canbeexploitedforcompressioni fwe changethefrequenci eswithsma llcontributionalittlebitandsetth emto0,and thenstorethesign albyonlys toringthenon zeropartofthe frequenc ycontent. Whenthesound istobepla yedback,wefirst conv ertth eadjuste dvaluesto theadjus tedfrequencycontentbackt oanormalfunctionrepresent ationwith aninv ersemapping. Fact1.9(Basicideab ehindaudiocomp ression).Supposeanaudiosi gnalf isgive n.Tocompressf,pe rformthefollowingsteps :
1.Rewritethesignalfinanew for matwherefr equencyinformatio nbe-
comesaccessi ble.
2.Removethosefrequen ciesthatonly contributemarginallytohumanper-
ceptionofthesound.
3.Storetheresulti ngsoundb ycodingtheadjustedfrequenc ycontentwith
somelossl esscodingmethod. Thislossy compressionstra tegyisessentiallywhatisusedinpra cticeby commercialaudioformats.Thedi ff erenceisthatcomme rcials oftwaredoes everythinginamoresophisticatedw ayandthereb ygets bettercompressi on rates.Wewillreturnt othisi nlaterch apters. 15 Wewills eelaterth atObserva tion1.8alsoist hebasisforma nyoperationson sounds.Thesameobservat ional somakesitpos sibletoexplainmoreprecisely whatitmean sthatw eonlyperceives oundswithafre quenc yintherange2 0-
20000Hz:
Fact1.10. Humanscanonlyp erceivev ariationsinairpre ssureassoundif theFourier seriesofthesounds ignalcontainsatleaston esu ffi cientlylarge termwithfrequ encyinthe range20-20000Hz. Withapp ropriatesoftwareitiseasy togenerateasoundfromam athem atical function;wecan'play'thefun ction .Ifweplaya functionlikesin(2π440t), wehear apleasantso und withaverydistinctpitch,a sexpected.There are, however,manyotherwaysin whichafuncti oncanoscillater egularly.The functioninFigure 1.1(b)fo rexample,definitelyoscillat es2timeseverysecond, butitdoe snotha vefrequency2 Hzsincei tisnotapuret one.Thissoundisalso notthatpl easanttolist ento.Wewillconsidertwomo reimp ortantexamples ofthis ,whichareverydi ff erentfromsmooth ,trigonomet ricfunctions. Example1.11.Wed efinethesquarewaveofperio dTasthef unctionwhi ch repeatswithperiodT,andis1onthe firsthalfofeac hperiod,an d-1onthe secondhalf.Thismea nsthatwecandefi neitasthe function f(t)= (1.1) InFi gure1.4(a)wehavep lottedthesquarewa vewhenT=1/440.Thi speriod isch osensothatitcorrespond stothepu retonewealre ady havelistenedto, andyoucan listentot hissquar ewavehere(inExerci se5yo uwilllearnhow togene ratethissound).Wehearas oundwiththesa mepitchassin(2π440t), butnoteth atthesquarew aveislesspl easanttol istento:Thereseems tobe somesharp cornersint hesound,translating intoa rathershriek ing,piercing sound.Wewilllatere xplain thisbyth efactthatthesqu arewavecanbeviewed asasu mofman yfrequencies,an dthat allthedi ff erentfrequen ciespollutethe soundsot hatit isnotpleasant tolisten to. Example1.12.Wed efinethetrianglewaveofperio dTasthe functionwh ich repeatswithperiodT,andincreaseslinearlyfrom-1to1onthe firsthalfof eachperio d,anddecreaseslinearlyfrom1to-1onthe secondhalfof each period.Thismeansthatw ecandefin eitasthefunctio n f(t)= (1.2) InFi gure1.4(b)wehavepl ottedthetrianglew avewhen T=1/440.Again, thissamech oiceofperio dgivesusanaudib lesou nd,andyoucanlisten tothe trianglewavehere(inExer cise5youwilll earnhowtogen eratethi ssound). Againyouwillnote thatthet rianglewa vehasthesame pitchassin(2π440t), 16
00.0020.0040.0060.0080.01
1 0.5 0 0.5 1 (a)Thefir stfiveperiod softhesqu arewave
00.0020.0040.0060.0080.01
1 0.5 0 0.5 1 (b)Thefirstfi veperiodso fthet rianglewa ve Figure1.4:Thesquare waveandthet riangl ewave,twofunction swithregular oscillations,butwhicharenotsimple,trigo nometric functions. andislesspleasan ttoli stentoth anthispuretone. Howev er,onecanargue thatitissome whatmor eplea santtolistentothanasq uarewave.Thi swill alsobeexplai nedint ermsofpollutionwithotherfre quenc ieslater. InSe ction2.1wewillbeginto peekbehin dthec urtainsas towhythese wavessounds odi ff erent,eventhoughw erecogniz ethemashavingtheex act samepit ch.
1.3Digita lsound
Inth eprevious sectionweconsideredsome basicpropertiesofsou nd,but it wasalli ntermsoff unctio nsdefinedforall timein stancesinsomeinterval.On computersandvariousk indsof mediaplayersthesoundi susuallydigitalwhich meansthatthe soundis repr esentedbyalar gen umberoffunctionvalues,and notbyafunct iondefinedfor all timesinsomeinter val . Definition1.13(Digitalsou nd).Adigitalsoundisasequencex={x i N i=0 thatcorresp ondstomeasurementsoftheairpressu reofa soundf,recorded ata fixedrate off s (thesampling frequencyorsampling rate)measurements persec ond,i.e., x i =f(i/f s ),fori=0,1;...,N. Themeas urementsareoftenreferredtoassamples .Thetimeb etweensucces- sivemeasurementsis calledthesamplingperio dandisusual lydenotedT s .If thesound isinstereothere willbe twoarr aysx 1 andx 2 ,oneforeachchannel. Measuringthesoundisal soreferredt oassa mplingtheso und,oranalogt o digital(AD)conversion. 17 Inmo stcases,adig italsoundissample dfr omananalog(continuou s)audio signal.Thisisusuallydo newithate chniqu ecalledPulseCodeModul ation (PCM).Theaudiosi gnalis sampledatregul arintervalsandthe sample dval-quotesdbs_dbs17.pdfusesText_23
[PDF] Applications of Fourier Analysis to Audio Signal Processing - CORE