Tutorial for Frequency Modulation Synthesis.docx
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Tutorial for Frequency Modulation Synthesis.docx
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TutorialforFrequencyModulationSynthesis
TutorialforFrequencyModulationSynthesis
Note:
thistutorialwiththeappropriatesoundexamplesisavailableontheHandbookforAcousticEcologywhichisalsoincludedinthe2ndeditionofAcousticCommunication
FrequencyModulation(orFM)synthesisisasimpleandpowerfulmethodforcreatingandcontrollingcomplexspectra,introducedbyJohnChowningofStanfordUniversityaround1973.Initssimplestformitinvolvesasinewavecarrierwhoseinstantaneousfrequencyisvaried,i.e.modulated,accordingtothewaveform(assumedheretobeanothersinewave)oftheso-calledmodulator.ThismodelthenisoftencalledsimpleFMorsine-waveFM.OtherformsofFMareextensionsofthebasicmodel.ThesystematicpropertiesofFMwereusedtocomposeBarryTruax'stapesoloworksArras,Androgyny,WaveEdge,SolarEllipse,SonicLandscapeNo.3,andTapeVIIfromGilgamesh,aswellasthoseinvolvingliveperformersorgraphics(Aerial,LoveSongs,Divan,SonicLandscapeNo.4).
Whenthefrequencyofthemodulator(whichwe'llcallM)isinthesub-audiorange(1-20Hz),wecanhearsiren-likechangesinpitchofthecarrier.However,whenweraiseMtotheaudiorange(above30Hz)thenwehearanewtimbrecomposedoffrequenciescalledSIDEBANDS.Todeterminewhichsidebandsarepresent,wehavetocontroltheratiobetweenthecarrierfrequency(C)andthemodulatingfrequency(M).InsteadofdealingwiththesefrequenciesinHz,we'llrefertothisrelationshipastheC:
MRATIO,keepingCandMasintegers.
PropertiesofC:
MRatios
First,aboutthepropertiesofratios,andsomeconventionswe'reusing:
Wewillonlydealwithratiosthatarecallednon-reducible,thatis,thoseinvolvingintegersevenlydivisibleonlyby1,andnotbyanyotherinteger.Forexample,theratio2:
2isthesameas1:
1andcanbereducedtoitforallpracticalpurposes.Likewise,10:
4isthesameas5:
2,and9:
6isthesameas3:
2,andsoon.
Secondly,we'regoingtodivideallpossibleratiosintosomesubgroupsforeaseofhandling.Onegroupwillbethosedescribedasthe1:
Nratios.Thismeansratioslike1:
1,1:
2,1:
3,1:
4etc.Theywillbefoundtohaveparticularproperties.
AnothergroupwillbethosedescribedasN:
MwhereN,Marelessthan10.Inthisecase,therestrictiontosingledigitnumbersispurelyforeaseofarithmeticcalculation.Thelastgroupiscalled'largenumberratios',andthisinvolvesnumbers10andup.Again,thedivisionisarbitrary.Wewon'tdealwithratioslike100:
1or100:
99.ThosearelegitimateonesandyoucandiscovertheirpropertiesthroughPODsynthesis.C:
Mratiosaresometimesexpressedwithrealnumbers,e.g.theratio1:
1.4,butthesecanbeapproximatedbyintegers,inthiscase5:
7.InFM,asetofsidebandsisproducedaroundthecarrierC,equallyspacedatadistanceequaltothemodulatingfrequencyM.Therefore,weoftenrefertothesidebandsinpairs:
1st,2nd,3rd,andsoon.
CalculatingSidebands
Theso-calleduppersidebandsarethoselyingabovethecarrier.Theirfrequenciesare:
C+MC+2MC+3MC+4MC+5M....
Forexample,ifC:
Mis1:
2,thatis,themodulatoristwicethefrequencyofthecarrier,thenthefirstuppersidebandis:
C+M=1+2=3.Theseconduppersidebandis:
C+2M=1+(2x2)=1+4=5.AnotherwaytogetthesecondsidebandistoaddM=2tothevalueofthefirstsidebandwhichis3;i.e.(C+M)+M=3+2=5.Itquicklybecomesclearthattheuppersidebandsinthisexamplearealltheoddnumbers,andsincethecarrieris1,theuppersidebandsarealltheoddharmonics,withthecarrierasthefundamental(i.e.thelowestfrequencyinthespectrum).
However,ifourC:
Mwere2:
5,thefirstuppersidebandwouldbe2+5=7.Since7isnotamultipleof2,itwouldbetermedinharmonic.Buttheseconduppersidebandwouldbe7+5=12,andthatisthe6thharmonic.Therefore,wecanseethatsidebandscanbeharmonicorinharmonic.
Thelowersidebandsare:
C-MC-2MC-3MC-4MC-5M...
Whenthesidebandisapositivenumberitwillliebelowthecarrier,butatsomepoint,itsvaluewillbecomenegative.Itisthensaidtobereflectedbecausewesimplydroptheminussignandtreatitasapositivenumber,e.g.thesideband-3appearsinthespectrumas3.Acoustically,however,thisreflectedprocessinvolvesaphaseinversion,i.e.thespectralcomponentis180degreesoutofphase.Mathematically,weexpressthisreflectionbyusingabsolutevaluesignsaroundtheexpression:
/C-M/toindicatethatwedroptheminusandtreatthenumberaspositive.
Forexample,for1:
2,the1stlowersidebandis:
/C-M/=/1-2/=/-1/=1.
Thesecondlowersidebandis:
/C-2M/=/1-(2x2)/=/1-4/=/-3/=3.However,tomakethingseasier,wecouldhaveadded2tothefirstlowersideband
(1),whichisalreadyreflected,andhaveobtained3.
Fortheratio1:
1,the1stlowersidebandis0(inaudible)andthe2nd,3rdand4thlowersidebandsare1,2,3,respectively.
For7:
5,thelowersidebandsare:
23813where3isthe1streflectedone.OneconcernwehaveinusingagivenC:
Mratioiswhetherthecarrierfrequencyisthelowestfrequencyinthespectrum,i.e.isitthefundamental?
Ifitis,wecantreatthecarrierfrequencyastheprincipalpitchthatwillbeheardintheresultingtimbre.
Firstwehavethecaseofthe1:
1ratiowhoseuppersidebandsare2,3,4,...andwhoselowersidebandsare0,1,2,3,...Clearlythecarrieristhelowestnon-zerocomponent,andallthesidebandsareharmonics,i.e.multiples.Because1:
1istheonlyratiowithazerolowersideband,itisaspecialcase.Itisalsotheonlyratioproducingtheentireharmonicseries.Forotherratios,wecouldworkouttheirsidebandsanddecideifanyofthemwerelowerthanthecarrier.Thatisfine,buttedious,andwe'dliketoknowinadvancewhattoexpect.First,wemightnotethatinthe1:
2casethe1stlowersidebandis/-1/=1;thereforeitfellagainstthecarrier.Further,ifMislargerthantwiceC,e.g.2:
5,thenthe1streflectedsidebandwillalwaysbegreaterthanthecarrier.
Workoutafewexamplesandyou'llfindthattheruleis:
FORTHECARRIERTOBETHEFUNDAMENTAL:
MMUSTBEGREATERTHANOREQUALTOTWICEC,ORELSEBETHE1:
1RATIO.
Anotherusefulpropertytobefamiliarwithistheoccurrence,asyoumayhavenoticedalready,oflowersidebandscoincidingwiththeupperset.Theyaresaidto'fallagainst'them.
Whatthismeansacousticallyisthattheamplitudesofthetwosidebandswilladdtogether,influencedaswellbythephaseinversionofthereflectedsideband.Sincetheamplitudeofeachsidebandvariesaccordingtothestrengthofthemodulation,asexpressedbytheModulationIndex,thesumofthecontributionsofeachsidebandbecomesquitecomplex!
You'veprobablynoticedthatthesidebandsofthe1:
1ratiohavethispropertyoffallingagainsteachother.You'llfindaverysimilarpatternwithalloftheN:
1ratios,i.e.2:
1,3:
1,4:
1,5:
1...
Thesecondtypeofratioshowingthesamepropertyisthatlike1:
2.Theoddharmonicsarefoundinboththeupperandlowersidebands.Thereforewecanextrapolatethesecondcase,namelyoddN:
2,thatis,theratios1:
2,3:
2,5:
2,7:
2...
NootherratiosexceptN:
1andoddN:
2havethisproperty.AllratiosotherthanN:
1andoddN:
2haveanasymmetricalspacingoftheirsidebands.Thatis,thedistanceinfrequencybetweenadjacentsidebandsisunequal.Forinstance,theratio2:
5withitssidebands237812....However,thereisstillapatterntothespacing.
Wewantthedistancebetweenthefirstuppersideband(C+M)andthefirstreflectedlowersideband(C-M)whichisnegativebydefinitionandthereforecanbemadepositivetogivetheexpression(M-C).Subtractthesetwofrequenciestoget:
(C+M)-(M-C)=2CWefindthattheansweris2C.SincethespacingofallupperorlowersidebandsisM,thentheremainingspacingisM-2C.
Thereforetheruleis:
WHENCISTHELOWESTFREQUENCYINTHESPECTRUM,THESPACINGOFSIDEBANDSABOVEIT(EXCEPTFOR1:
1)WILLBE:
M-2.C&2.C
Fortheaboveexample(2:
5),thespacingis2x2=4and5-4=1.NotethatwehavetostartwithCasthefundamental.ThiswillbereferredtointhenextsectionastheNormalFormoftheRatio.
NormalFormoftheC:
MRatio
Definition:
aC:
MratioisinNormalForm(N.F.)whenthecarrieristhefundamentalinthespectrumitproduces.
Rule:
foraratiotobeinNormalForm,MmustbegreaterthanorequaltotwiceC,orelsebetheratio1:
1
WhatwearedoingwiththeconceptofNormalFormisprovidingthebasisofaclassificationschemeforallratios,andatthesametimecodifyingourruleofthumbabouttheconditionsforthecarriertobethefundamental.Ifweconsideronlyratiosinvolvingintegersupto9,wecanlistallthoseinNormalForm:
1:
11:
24:
93:
72:
53:
81:
32:
71:
42:
91:
51:
61:
71:
81:
9
Thesehavebeenlistedinanorderrelatedtowhat'scalledtheFareySerieswherethevalueofM/Cisincreasing,orC/Misdecreasing.EachN.F.ratiowillhaveassociatedwithitafamilyofratios,asshownbelow.
ReducingaNonNormalFormRatiotoitsNormalForm
WhentheMvalueinaratioislessthantwicetheCvalue,itisnotinNormalForm,butcanbereducedtoitbyapplyingtheoperation:
C=/C-M/
WhatthismeansisthatyousubtractMfromC(ignoringanyminussign)andtreattheresultasthenewCvalue.YoukeepdoingthisuntiltheratiosatisfiestheNormalFormcriterion.
Consider,forexample,theratio3:
1.Wefirsttake3-1=2andgettheratio2:
1,whichisstillnotinNormalForm.Anotherapplicationoftheoperationgivesus2-1=1andtheratio1:
1whichisinNormalFormbydefinition.
Atrickierexampleis8:
5.Firstwereduce8-5=3andget3:
5(notN.F.)Reducingagain,weg
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