Digital compensation of nonlinear distortion in loudspeakers

DigitalCompensationofNonlinearDistortioninLoudspeakers

H.Schurer,C.H.Slump,O.E.Herrmann

UniversityofTwente,DepartmentofElectricalEngineering

LaboratoryforNetworkTheoryandVLSIDesign

P.O.box217NL-7500AEEnschedeTheNetherlands

phone:+31-53-892673

fax:+31-53-334701

E-mail:hans@nt.el.utwente.nl

Practicalloudspeakersarenonlineartransducers,i.e.theacousticoutputisnotalinear

representationoftheelectricalinputsignal.Inthispapertwoloudspeakersareconsidered;

aloudspeakerforreproductionoflowfrequencies(woofer)andahornloudspeaker.Instead

ofimprovingthebehaviorofthetwotransducersbyconstructionalmeansweapplya

feedforwardnonlineardigitalinversecircuit.ResultsofaVolterracompensatorforthe

woofershowasignificantreductionofthesecondorderharmonicsatlowfrequencies,

leavinghigherorderdistortionsunchanged.Tothehornloudspeakeramoredirectmethod

isapplied.Thisloudspeakersuffersfromdistortionscausedbytheadiabaticbehaviorof

theairvolumesinthecompressiondriver.Withthiscompensatorweobtainareductionof

harmonicdistortion,inthefrequencyspanfrom600to1100Hz.

1Introduction

Withtheaudioreproductionchainbecomingmoreandmoredigital,theloudspeakerhasbecome

theweakestlinkbetweenrecordingandreproduction.Amethodtoimprovethetransferbehavior

ofelectro-acousticaltransducersisbymeansofmagneticormechanicaldesignchanges.Disad-

vantageofthiskindofchangesisamorecomplexproductionandthusanincreasingprizeofthe

transducer.Inthispaperweimprovethebehavioroftheloudspeakerbymeansofanalgorithmon

aDigitalSignalProcessor(DSP).Withthedecreasingprizesofprocessinghardwarethismaybea

realisticsolutiontothementioneddesignproblem.Thecompensatorsarerealizedasfeedforward

controllers.Thereforeanadditionalsensorlikeamicrophoneoranaccelerometerissuperfluous.

Suchsensorsmakethesystemmoreexpensive,andtheirqualitydeterminestheeffectivenessof

thecontroller.

Inthispapertwotypesofloudspeakersareconsidered,anelectrodynamicloudspeakerforlow

frequencies(woofer)andamidrangeloudspeaker,ahornloudspeaker.Firsttransducersuffers

fromdistortionscausedbythedisplacementdependentparameters.Inordertohaveagood

bassreproductionalargeconeexcursionisneeded,thisincreasesthealreadyinherentnonlinear

distortion.Hornloudspeakersarewidelyusedintheareawherehighsoundpressurelevelsand

gooddirectivityofsoundisneeded.Theyhaveamuchhigherefficiencythandirectradiator

loudspeakers.Disadvantagesareitshigherprize,greatersizeandhighernonlineardistortions.

Onepartofthehornloudspeakernonlinearityiscausedbyadiabaticcompressionofairinthe

driver.

2Modeling

Startingpointfordesignofacontrollertocompensatefornonlinearityinasystem,isagood

model.Fromthetechniquesavailablewechoosetomodelthetransducersbylumpedparameter

electricalequivalentcircuits.Suchcircuitshaveafullconnectiontowardsphysicalpropertiesof

thetransducerfromwhichwecanuseasmuchexpertknowledgeaspossible.Thisisthecontrary

withothermodelingtechniqueslikeblack-boxmodeling(e.g.NARMAX),althoughgray-box

1modeling(amixtureofdeterministicandstatisticalmodeling)isapromisingtechniqueforthe

future.InFigure1(a)and(b)bothtransducersareschematicallydepicted.Firstwewillconsider

rimconedustcapspider

voice coiliron coremagnet

(a)

magnetvoice coildiaphragmphase plug

throatof thefirst hornfirst hornsecond horn(only first part shown)compression drivermouth of thefirst horn

(b)

i

++

––v

u=Bl(x)vF=Bl(x)iCm(x)mtRmLe(x)Re

ue+

–Gy

(c)L1L2L2

C1C2C3CbLbRbRemtCm

ueBl:1Sd:1ZmouthRm

Cg(pg)

(d)

Figure1:Thetransducersunderconsideration;electrodynamicaldirectradiator(a)andhorn

loudspeaker(b)withtheirequivalentcircuitsinrespectively(c)and(d).

thewoofer.Nonlinearitiesofthistransducerareallcausedbythedisplacementofthecone.

Threenonlinearitiesarefoundtobeofmajorinfluence[1]:

thetransductionbetweenelectricandmechanicdomain

thestiffnessofthespidersuspension

theselfinductanceofthecoil

TheelectricalequivalentcircuitofawooferinsideaclosedboxisdepictedinFigure1(c).It

consistsintheelectricaldomainoftheseriescircuitofthevoicecoilresistanceanddisplacement

dependentselfinductance.Couplingtowardsthemechanicaldomainisformedbya

gyratorwithdisplacementdependentcouplingfactor,alsoknownastheforcefactor.Inthe

mechanicaldomainwehavetheelectricalanalogueofadampedmass-springsystem,formedby

theresistanceofthesuspension,thedisplacementdependentcomplianceofsuspensionand

cabinetcompliance,andthetotalmassofthemovingconeandmovingairinfrontofit

.Fromstraightforwardnetworkanalysisoftheequivalentnetworkforthewooferwefindtwo

couplednonlineardifferentialequations[1]:

(2)

Thedisplacementdependentparameters,andaredescribedbyatruncated

Taylorseriesexpansion.Thetotalnonlineardifferentialequationisobtainedfromsubstitutionof

Eq.(2)intoEq.(1).

Theelectricalequivalentcircuitofthehornloudspeakerismorecomplexcomparedtothewoofer.

Althoughtheelectricandmechanicdomainarethesameaswiththewoofer,theacoustical