use of random noise for transducer modeling in an adaptive active attenuation system 1989

8/6/2019 Use of Random Noise for Transducer Modeling in an Adaptive Active Attenuation System 1989

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Use of random noise for on-line transducer modeling in an adaptiveactive attenuation systema)L.J. Eriksson and M.C. AllieCorporateesearchepartment,elsonndustries,nc.,P.O. Box600,$toughton, isconsin3589-0600(Received January 987; cceptedor publication6 October 988)Activesound ttenuation ystems aybe described sing systemdentificationrameworknwhichan adaptive ilter s used o model he performance f an unknownacoustical lant.Anerrorsignalmaybeobtainedroma locationollowing n acousticalummingunctionwherethe undesired oise s combined ith the outputof a secondaryound ource. or themodeloutput o properly onvergeo a value hat will minimize he errorsignal, t is frequentlynecessaryo determinehe transfer unction f the secondaryound ource nd he path o theerrorsignalmeasurement.ince hese ransfer unctions re unknown ndcontinuouslychangingn a real system,t is desirableo performcontinuous n-linemodeling f the outputtransducernderrorpath. n thisarticle, he useof an auxiliary andom oise eneratororthismodelings described. ased n a Galoissequence,his echniques easy o implement,providesontinuousn-linemodeling,ndhasminimal ffect n he inalvalueof theerrorsignal.PACS numbers:43.60.Gk, 43.50.Ki

INTRODUCTIONActive soundattenuations a relativelyold idea hat hasreceived onsiderablettention n recentyears.This is pri-marily due to the development f improvedsignal-process-ing theoryand hardware hat enablemore sophisticatedp-proacheso this problem.Many of the traditionalproblemswith this technology an now bc treated more effectivelywith proper signalprocessingather than with the directacoustical pproachesf the past.This articledescribes complete ctiveattenuation ys-tem that functions orrectly n the presencef acousticeed-back as well as nonideal nput microphone, rror micro-phone, oudspeaker,nderror path ransfer unctions.t iscompletely daptive nd responds utomaticallyo changesin input signal,acoustic lant, error plant, microphone, ndloudspeaker haracteristics.

I. SYSTEM IDENTIFICATIONActivesound ttenuation ystemsmaybe described s-inga systemdentificationrameworkn whichan adaptivefilter sused o model heperformance f an unknown cous-

ticalplant, sshownnFig.1 a). 1-s n nputmicrophonesused o measureheundesired oise pstream f theacousti-calplant.Thissignalsused s he nput o anadaptiveilterthat generatesn output o a loudspeaker hich s used oproducea secondary ound hat is acoustically ombinedwith the undesired oise.An error signalmeasured own-stream from the acousticalsumming unction is used toadapt he coefficientsf the adaptive ilter to minimize heresidualnoise.When fully adapted, he adaptive ilter re-sponsen serieswith the responsef the nputmicrophoneand loudspeakermatches the responseof the acousticalplant.' Anearlier ersionf this rticlewaspresentedt the112thMeeting f theAcoustical ociety f America,8-12 December 986 n Anaheim,CA [J.Acoust.Soc.Am. Suppl. 80, SII (1986)].

Although this system dentificationproblemhas beenintensively tudied n the controland signal-processingiter-ature, the activeattenuation pplication s complicated ythe presence f acoustic eedback rom the loudspeakerothe input microphone. n the past,a variety of solutions othis problemhavebeenproposedhat utilize either direc-tional ransducer rraysor incorporate compensatingixed

INPUT ERRORMICROPHONE MICROPHONE


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feedbackath hat sdeterminednanoff-line asishroughcalculations r useof a trainingsignal.Erikssonhas presented new technique or active at-tenuation hat effectively tilizesadaptive ignalprocessingto solve heproblem f acousticeedbackrom he secondarysound ourceo the nputmicrophone.This echniqueti-lizesa recursive-least-mean-squaresRLMS) algorithmde-velopedyFeintuch o provide completeole-zero odelof the acousticalplant. The acoustic eedback s consideredpart of the adaptivemodel used o model the plant. Fromthisperspective,he acousticeedbackntroducesixedpolesinto the overall response f the model, which may be re-movedwith the pole-zero esponse f the RLMS algorithm.Using the configuration hown in Fig. l(b), the directacoustical athP and eedback coustical athFare simulta-neouslymodeledby the RLMS model usingadaptive iltersA and B, in serieswith the loudspeaker . Perfectcancella-tion is obtainedwhen the overall model responsematchesthe response f the plant or using transforms:

M r = MS/( 1 + FMS) = AS/( 1 -- B + FAS) = P,(1)where

M r = overallmodelresponse,M = responseof pole-zero recursive filter structureused n RLMS algorithm,A = responsef all-zero east-mean-squaresLMS) ele-ment used n directpath of pole-zero tructure,B = response f all-zeroLMS elementused n recursivepath of pole-zerostructure,P = direct path acousticplant,F--- feedback ath acoustic lant, andS = response f loudspeaker.One solution to this equation is for 4 = P/S andB = PF. However, he actualmodelresponses a complexfunctionof thespectral ontent f thesource nd he acousti-cal plant of the system. he RLMS algorithmprovidesfully adaptivemeans o simultaneouslymodel the directplantand eedback lantwitha given ourcen such wayasto minimize the residual noise.

II. TRANSDUCER MODELINGOne of the problemswith this technique s that theRLMS algorithm equires nowledge f thespeakerransferfunction and error path transfer function for proper conver-gence.Widrow hasshownhat heLMS algorithmanbeusedwith a delayederror signal f the input to the errorcorrelatorss alsodelayedby the sameamount.Similarly,Morgan hasstated hat, with propercompensation,he LMSalgorithmcanalsobe used fa transfer unction,suchas hatdue o the loudspeaker,s in the auxiliarypath following he

adaptive ilter. Propercompensationequires he additionofa transfer unction n the nput o theerror correlators r theaddition of an inverse transfer function in series with theerrorpath. Burgessasdiscussedimilar esultsor the

LMS algorithmwhenboth auxiliarypath and error pathtransferunctionsrepresent.0A transferunctionsaddedto the input to the error correlators,which representsheproductof the auxiliarypathand errorpath ransfer unc-tions.WidrowandStearns.2havesimilarly iscussedhe"filtered-X"LMS algorithmor usewitha plant n theauxil-iarypath. .2These esultshavebeenextendedo an infinite mpulseresponseIIR) adaptive ilter using he RLMS algorithmbyEriksson.he speakerransferunction anderrorpathtransfer unctionE mustbe known o compensateor theireffecton the convergencef both the directand recursireelements f the IIR filter. This can be done hrougheitherthe additionors andEinto the nput ines o the errorcorre-latorsor the additionof the nverse ransfer unctions, - and E-, into the error path,as shown n Fig. 1 b). Asdiscussedbove, he former echnique asbeendescribed yWidrow andBurgess or theLMS algorithm ndassuresthat the error signaland input signalwill have the samerelationshipn time.The latter technique asbeendescribedbyMorgan or heLMS algorithm ndeliminatesheneedfor a modification o the input signal n principle,but, inpractice, he lack of causality or the inverse ransfer unc-tions, - andE- , requiresompensationf the nputsig-nal to the error correlators y delay, as will be discussednthe following.

Unfortunately,both S and E are unknownand are timevaryingdue to effects uchas heat and agingon the loud-speaker nd due o changesn temperature nd flow n theerror path. Thus it is necessaryo obtain either direct orinversemodels fS andE onanon-line asis. lthoughheyarenot shown xplicitly n Fig. 1 b), the errormicrophonemaybeconsideredspartof theerrorpath ransferunctionE, and the input microphone implyaddsan additionaltransfer function in series with the RLMS model and loud-speaker . Since he nput microphone ccurs rior to theadaptivemodel, t does otneed o becompensatedor n thesame manner as S and E.Poole t al. 3havedescribedsystemsingheLMSalgorithm n which a fixedcompensatingnverse ransferfunctionsaddedo theerrorpath.However, ince - andE- are noncausal,n off-linemodelof a delayednversemodelof the oudspeakernderrorpath AS- ]E- I is deter-mined where A is the delay necessaryo make the inversemodelcausal.4The useof this delayednversemodel e-ducesheerrorpath ransfer unction o a fixedpuredelayA.As notedabove, his approach hen requires he additionofthe samedelayA to the input to the error correlators f theLMS algorithmas described y Widrow.8 The primary dis-advantage f this techniques that it doesnot usean on-line,continuouslydaptivemodelof the loudspeakernd errorpath.Erikssonhasdescribedthree-microphoneystem s-ing the RLMS algorithm n which he errorplant s modeledon line using either a direct or inverse model while thespeaker s modeledoff line. However, there have not beenany previous pproaches escribedhat providean on-linemodel of the speakerand the error path that respondsochangesn their response ver ime.

798 J. Acoust.Soc. Am., Vol. 85, No. 2, February1989 L.J. Eriksson nd M. C. Allie:Randomnoise or on-linemodeling 798


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III. MODELING APPROACHESThere are two basic echniques vailable or use n sys-tem modeling singadaptive ilters.The directapproachplaceshemodel n parallelwith theunknown lantand sadapted uch hat thedifferenceetweenheoutputs f theplantandmodel s minimizedor the same ignal. he in-verse pproach laceshe model n serieswith theunknownplantsuch hat the differenceetweenhe outputof thisse-

riescombination nd a delayed ersionof the nput signal sminimized. n this case, he response f the adaptivemodelbecomes delayedversionof the inverseof the unknownplant response.As shown n Fig. 2(a), to determine he speaker nderror path response,he directmodelapproach laces headaptivemodeln parallelwith thespeakernderrorpath.An error signal ormedby subtractinghe adaptivemodeloutput rom hemicrophoneutput s multiplied y the n-putsignalo form heupdateerms or thecoefficientsf theadaptivemodel.The inversemodel approach laces headaptivemodel n serieswith the speaker nderrorpath,asshownn Fig. 2(b). In thiscase, he errorsignal ormedbysubtractinghe adaptivemodeloutput rom a delayedver-sionof the noise nput s multipliedby the nput o the adap-tivemodel o form heupdate erms or thecoefficientsf theadaptivemodel.Thus he adaptivemodel ormsa delayedinversemodelof the speaker nd the error path while at-tempting o match he responsef the delayed oise nput.

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The traditional solution is then to use either the direct orinversemodeling pproach hown n Fig. 2(a) and (b), re-spectively,nanoff-line asis itha broadbandoise ourceN. Sincet isanoff-line rocess,heplantoutputy ndmodeloutput arenot present. he noise ource thusallowsprecise eterminationf either hespeaker nderrorpathresponse,fiE,or the delayednversemodelof the speakeranderrorpath,AS- E . In the directapproach f Fig.2 a), the responseE is fixedafter convergencend thenused n the inputs o the error correlators f the LMS orRLMS algorithms.n the nverse pproach f Fig. 2(b), theresponseS- E- is also ixedafterconvergence,nd hemodeling elayA isusedn the nputso theerrorcorrelatorsof the LMS or RLMS algorithms. oth techniquesssumethe useof a large-amplitude,roadband oise ource n anoff-linebasis o avoidcontamination f the modelingprocessbyy or. and o avoid headdition f undesiredoise uringon-lineoperation y the noise ourceN.IV. CONTINUOUS MODELING SYSTEM

A new approacho the on-linemodeling f S and E isshown n Fig. 3. An uncorrelatedandomnoisesource sused o excite he series ombination f the speakerollowedby the error plant as well as adaptivemodelC while thesystems operating.5 This andom oise ource ill ulti-matelybecomehesource f theresidual oise f thesystem.The direct adaptivemodelC is used o obtaincoefficientsdescribinghe responsef $ andE that canbe used n theinput ines o the error correlatorsor the primaryRLMSalgorithm. he generalized odeloutputandweightupdateequationsor the recursive daptive ilter may be writtenfollowinghenotationf Widrow ndStearns2as_ TYk WkUk , (2)

Wk+ = W + 2MU;ek , (3)U= [u,u_, .... , (4)

=cuk , (5)where

' )+N

FIG. 2. (a) Directmodeling pproachor thedeterminationf the ransferfunction f thespeakernderrorpathwithadaptivemodel E. (b) Inversemodeling pproachor the determinationf the delayednverseransferfunctionf thespeakernd rrorpathwithadaptive odel (SE)

RANDOM

FIG. 3. Newapproacho on-linemodelingf speaker anderrorpathEandusing esultsn RLMS modelwithacousticeedbacko forma fullyadaptive ctive ttenuation ystem.799 J. Acoust.Soc. Am., Vol. 85, No. 2, February 1989 L.J. Erikssonand M. C. Allie: Random noise or on-linemodeling 799


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Yk = scalarmodeloutputat discrete ime k,Wk = generalizedweight vector (includesdirect andrecursire coefficients),W r= transposefWk + = updated eneralized eightvector,M = convergenceactor matrix,e = scalarerror signal,U = generalizednput vector includes irectand re-cursive nput vectors),U , = compensatedeneralizednputvector,u;, = firstcomponentf compensatednputvector, ndC r= transposef weight ector f model ssociatedwith transfer unctionsn auxiliarypath.

The weightvectorof the adaptivemodelC is obtainedon an on-linebasisusingan adaptivealgorithmsuchas theLMS or RLMS algorithm with the independent andomnoise ource san nputand heerrorsignal sshownn Fig.3. The amplitude f thenoise ources keptvery ow so hatthe finaleffecton the residual oise s small.The plant noisey and model output are not presentat the input to theadaptivemodelC andsowill not affect he finalvalues f themodelweights.The use of an uncorrelated random noise source that isindependent f the input signalensureshat the speaker nderror path will be correctlymodeled.The signals rom theplant (y) andmodel .) represent oise n the "plant" sideof the speaker/error ath modelingprocesshat will not af-fect the weightsof the direct model C used o determineSE.2 hismodels hen opiedo he nputines f heerrorcorrelatorsof the RLMS algorithm.It shouldbe noted hat, although he delayedadaptiveinversemodelshown n Fig. 2(b) couldbe used n a similarfashion, his will result n decreased erformance ince he"noise" n the auxiliary path and error path due to y andalso appearsat the input of the adaptive ilter due to theseries rrangement. hus he autocorrelationunctionof thefilter input is adversely ffected, nd the filter weightsaremodified s described y Widrow and Stearns.2 If this"noise" s argeenough, he adaptivemodelmay fail to con-verge. hus hedelayed daptivenverse pproach equiresmuch argeramplitude andomnoisesource hat increasesthe residualnoiseand decreases verall systemquieting.In the directmodelsystem, hown n Fig. 3, the "noise"due o y and. doesnot affect he final weights n the adaptivemodel. n addition, he convergence f the SE model s as-sured s ongas he nitialamplitudes rewithin hedynamicrange and signal-to-noiseatio constraints f the system.Thus, with $E accurately determined, the overall systemmodel will converge, esulting n minimum residual noise.The randomnoisesourceused o modelSE may be read-ily obtained hrough he useof a variety of methods.Onesimple approach s to generatea Galois sequenceusingmethods escribedy Schroeder.6A Galois equences apseudorandomsequence hat repeats after 2'"- 1 points,where n s henumber f stagesn a shift egister.t iseasy ocalculate ndcaneasily avea periodmuch onger han heresponseime of the System.n this'study, 31 stages(m = 31 } were used.

V. RESULTSThe resultsof a computersimulationof the systemshown n Fig. 3 confirmed hat the algorithmproperlycon-verges or either narrow-bandor broadband nput signals.The coefficients f the SE model properly describe he SEplant, and the coefficientsf the overallsystemmodelprop-erly describe , F, and S.The approach hown n Fig. 3 hasalsobeen mplemen-

tedon complete coustical ystems sing heTMS320 familyof digitalsignalprocessing icroprocessors,ith input mi-crophones, canceling loudspeakers,and error micro-phones."?'anitially,oneof these ystems asutilized ocancelelectroacoustically eneratednoise n a 12-in.-diamcircular duct. The duct was about 25 ft long and unlinedexcept or a short4-ft-long adsorptive ilencernear the pri-mary noise source. Typical results after adaptation areshown n Fig. 4. The noise eductionobtainedwith the sys-tem operating or a broadband oise nput is shown n Fig.4(a). This curvewasobtainedby subtractinghe canceledspectrumrom the uncanceledpectrum. he maximaandminima in the spectrumare due to acoustical esonances.The convergedweightstructure or the4,B, and C elementsof Fig. 3 is shown n Fig. 4 (b). The decayof the coefficientsconfirms hat the filter engthchosenwasadequate. he sys-tem is effective on broadband as well as narrow-band noiseand requires o calibration r trainingof any kind.Performancen an actual ndustrial an or heating, en-tilating,and air conditioning uct s mademuchmorediffi-cult by the turbulentairflowand argeduct dimensionshatare usually equired.Goodsystem erformanceequires n-titurbulencemicrophoness well as large,powerful, ow-frequencyources.9Typical esults btainedreshownnFig. 5. The uncanceledutospectrumn a linedsupplyduct(34 X 44 in. ) approximately 0 ft from a centrifugal an is

-10

(a)

0 HZ 2OO()

A B CFIG. 4. (a) Noise eductionwith activeattenuation ystem n for bandlim-ited (15-200 Hz) pink noise nput signal no flow--128 averages). b)Filter coefficientssed o obtain he results hown n (a) for adaptive ilters.4 (32 taps), B (64 taps), and C (64 taps).

800 J.Acoust.ec.Am., ol. 5,No. , February989 L.J.ErikssonndM.C.Allio: andomoiseoron-line odeling 800


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use of random noise for transducer modeling in an adaptive active attenuation system 1989

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