Use of random noise for on-line transducer modeling in an adaptive
active attenuation systema)
L.J. Eriksson and M.C.
Allie
Corporate
Research
Department,
NelsonIndustries,
Inc.,P.O. Box600,$toughton,
Wisconsin
53589-0600
(Received9 January1987;accepted
forpublication
26 October1988)
Activesoundattenuationsystems
maybedescribed
usinga systemidentification
frameworkin
whichan adaptivefilter is usedto modelthe performanceof an unknownacousticalplant.An
errorsignalmaybeobtainedfroma locationfollowingan acoustical
summing
junctionwhere
the undesired
noiseiscombinedwith theoutputof a secondary
soundsource.For themodel
outputto properlyconverge
to a valuethat will minimizetheerrorsignal,it is frequently
necessary
to determinethetransferfunctionof thesecondary
soundsourceandthepathto the
errorsignalmeasurement.
Sincethesetransferfunctionsareunknownandcontinuously
changingin a real system,it is desirable
to performcontinuous
on-linemodelingof theoutput
transducer
anderrorpath.In thisarticle,theuseof an auxiliaryrandomnoisegenerator
for
thismodelingisdescribed.
Basedon a Galoissequence,
thistechniqueiseasyto implement,
provides
continuous
on-linemodeling,
andhasminimaleffectonthefinalvalueof theerror
signal.
PACS numbers:43.60.Gk, 43.50.Ki
INTRODUCTION
Although this systemidentificationproblemhas been
intensively
studiedin thecontrolandsignal-processing
literActivesoundattenuationis a relativelyold ideathat has
ature,
the
active
attenuation
application
is
complicated
by
receivedconsiderable
attentionin recentyears.This is prithe
presence
of
acoustic
feedback
from
the
loudspeaker
to
marily due to the developmentof improvedsignal-processthe
input
microphone.
In
the
past,
a
variety
of
solutions
to
ing theoryand hardwarethat enablemoresophisticated
apthis
problem
have
been
proposed
that
utilize
either
direcproachesto thisproblem.Many of the traditionalproblems
arraysor incorporatea compensating
fixed
with this technologycan now bc treated more effectively tionaltransducer
with propersignalprocessing
rather than with the direct
acousticalapproaches
of the past.
INPUT
ERROR
This articledescribes
a completeactiveattenuationsysMICROPHONE
MICROPHONE
temthat functionscorrectlyin the presence
of acousticfeed<")
PLANT
• ERROR
back as well as nonidealinput microphone,error micro-
phone,loudspeaker,
anderror pathtransferfunctions.It is
completelyadaptiveandresponds
automaticallyto changes
in input signal,acousticplant, error plant, microphone,and
loudspeakercharacteristics.
I. SYSTEM
ACOUSTIC
FEEDBACK '
•SPEAKER
IDENTIFICATION
Activesoundattenuationsystemsmaybe describedusinga systemidentification
frameworkin whichan adaptive
filterisusedto modeltheperformance
of anunknownacous-
ticalplant,asshown
inFig.1(a). 1-sAninputmicrophone
is
usedto measuretheundesirednoiseupstreamof theacousti-
03)
+
+
calplant.Thissignalisusedastheinputto anadaptive
filter
that generates
an outputto a loudspeaker
whichis usedto
producea secondarysoundthat is acousticallycombined
with the undesirednoise.An error signalmeasureddownstream from the acousticalsummingjunction is used to
adaptthe coefficients
of the adaptivefilter to minimizethe
residualnoise.When fully adapted,the adaptivefilter response
in serieswith the response
of the inputmicrophone
and loudspeakermatchesthe responseof the acoustical
plant.
'• Anearlierversion
of thisarticlewaspresented
at the112thMeetingof the
AcousticalSocietyof America,8-12 December1986in Anaheim,CA [J.
Acoust.Soc.Am. Suppl.I 80, SII (1986)].
797
J. Acoust. Sec. Am. 85 (2), February 1989
FIG. 1. (a) Schematicdiagramof activeattenuationproblem.(b) Block
diagramof activeattenuationproblemshowinggeneralsolutionto compensationfor transferfunctions
dueto loudspeaker
SanderrorpathE through
the additionof duplicatetransferfunctionsor inversetransferfunctions.
0001-4966/89/020797-06500.80
•) 1989 Acoustical Society of America
797
feedback
paththatisdetermined
onanoff-linebasisthrough
calculationsor useof a trainingsignal.
LMS algorithmwhenboth auxiliarypath and error path
transfer
functions
arepresent.
•0A transfer
function
isadded
Erikssonhas presenteda new techniquefor active atto the input to the error correlators,whichrepresents
the
tenuationthat effectivelyutilizesadaptivesignalprocessing productof the auxiliarypathanderrorpathtransferfuncto solvetheproblemof acousticfeedbackfromthesecondary tions.WidrowandStearns
••.•2havesimilarlydiscussed
the
soundsource
to theinputmicrophone.
6Thistechnique
uti- "filtered-X"LMS algorithmfor usewitha plantin theauxillizesa recursive-least-mean-squares
(RLMS) algorithmdeiarypath.••.•2
veloped
byFeintuch
7to provide
a complete
pole-zero
model
Theseresultshavebeenextendedto an infiniteimpulse
of the acousticalplant. The acousticfeedbackis considered response(IIR) adaptivefilter usingthe RLMS algorithmby
part of the adaptivemodel usedto model the plant. From
Eriksson.•
The speaker
transferfunctionS anderrorpath
thisperspective,
the acousticfeedbackintroducesfixedpoles transferfunctionE mustbe knownto compensate
for their
into the overall responseof the model, which may be reeffecton the convergence
of both the directand recursire
movedwith the pole-zeroresponseof the RLMS algorithm.
elementsof the IIR filter. This can be donethrougheither
Using the configurationshown in Fig. l(b), the direct theadditionors andEinto theinputlinesto theerrorcorreacoustical
pathP andfeedbackacoustical
pathFare simulta- latorsor the additionof the inversetransferfunctions,S- •
neouslymodeledby the RLMS modelusingadaptivefilters
andE-•, into the errorpath,as shownin Fig. 1(b). •As
A andB, in serieswith the loudspeaker
S. Perfectcancella- discussed
above,theformertechniquehasbeendescribedby
tion is obtainedwhen the overall model responsematches WidrowaandBurgess
•øfor theLMS algorithmandassures
the response
of the plantor usingz transforms:
that the error signaland input signalwill have the same
relationshipin time.The lattertechniquehasbeendescribed
M r = MS/( 1 + FMS) = AS/( 1 -- B + FAS) = P,
byMorgan
9fortheLMS algorithm
andeliminates
theneed
(1)
for a modificationto the input signalin principle,but, in
practice,the lack of causalityfor the inversetransferfunc-
tions,S- • andE- •, requires
compensation
of theinputsig-
where
M r = overallmodelresponse,
M = responseof pole-zero recursive filter structure
usedin RLMS algorithm,
A = response
of all-zeroleast-mean-squares
(LMS) elementusedin directpath of pole-zerostructure,
B = response
of all-zeroLMS elementusedin recursive
path of pole-zerostructure,
P = directpath acousticplant,
F--- feedbackpath acousticplant,and
S = response
of loudspeaker.
One solution to this equation is for •4 = P/S and
B = PF. However,the actualmodelresponseis a complex
functionof thespectralcontentof thesourceandtheacousti-
nal to the error correlatorsby delay,as will be discussed
in
the following.
Unfortunately,bothS and E are unknownand are time
varyingdue to effectssuchas heat and agingon the loudspeakerand dueto changesin temperatureand flowin the
error path. Thus it is necessaryto obtain either direct or
inverse
models
ofS andE onanon-linebasis.Althoughthey
arenot shownexplicitlyin Fig. 1(b), the errormicrophone
maybeconsidered
aspartof theerrorpathtransferfunction
E, and the input microphonesimplyaddsan additional
transfer function in series with the RLMS model and loud-
speakerS. Sincethe inputmicrophone
occursprior to the
adaptivemodel,it doesnotneedto becompensated
for in the
same manner as S and E.
Pooleet al.•3havedescribed
a system
usingtheLMS
cal plant of the system.The RLMS algorithmprovidesa
fully adaptivemeansto simultaneouslymodel the direct
plantandfeedback
plantwitha givensourcein sucha wayas
algorithmin which a fixedcompensating
inversetransfer
functionisaddedto theerrorpath.However,sinceS- • and
to minimize the residual noise.
E- • arenoncausal,
an off-linemodelof a delayedinverse
II. TRANSDUCER
MODELING
One of the problemswith this techniqueis that the
RLMS algorithmrequiresknowledge
of thespeakertransfer
function and error path transfer function for proper conver-
gence.
• WidrowshasshownthattheLMS algorithm
canbe
usedwith a delayederror signalif the input to the error
correlatorsis alsodelayedby the sameamount.Similarly,
Morgan hasstatedthat, with propercompensation,
the LMS
algorithmcanalsobe usedifa transferfunction,suchasthat
dueto the loudspeaker,is in the auxiliarypath followingthe
adaptivefilter. Propercompensation
requiresthe additionof
a transferfunctionin theinputto theerrorcorrelatorsor the
addition
of an inverse transfer function
in series with the
errorpath.9 Burgess
hasdiscussed
similarresultsfor the
798
J. Acoust.Soc. Am.,Vol. 85, No. 2, February1989
modelof theloudspeaker
anderrorpathAS- ]E- I is determinedwhere A is the delay necessary
to make the inverse
modelcausal.
•4The useof thisdelayedinversemodelreducestheerrorpathtransferfunctionto a fixedpuredelayA.
As notedabove,thisapproachthenrequiresthe additionof
the samedelayA to the input to the error correlatorsof the
LMS algorithmasdescribed
by Widrow.8The primarydisadvantageof thistechniqueis that it doesnot usean on-line,
continuously
adaptivemodelof the loudspeaker
and error
path.
Eriksson
• hasdescribed
a three-microphone
system
usingtheRLMS algorithmin whichtheerrorplantismodeled
on line using either a direct or inverse model while the
speakeris modeledoff line. However, there have not been
any previousapproaches
describedthat providean on-line
model of the speakerand the error path that respondsto
changesin their response
overtime.
L.J. Erikssonand M. C. Allie:Randomnoisefor on-linemodeling
798
III. MODELING
The traditional solution is then to use either the direct or
APPROACHES
inversemodelingapproachshownin Fig. 2(a) and (b), reThere are two basictechniques
availablefor usein sysonanoff-linebasiswitha broadband
noisesource
tem modelingusingadaptivefilters.The directapproach spectively,
N. Sinceit isanoff-lineprocess,
theplantoutputyandmodel
placesthemodelin parallelwiththeunknownplantandis
output.• arenotpresent.The noisesourceN thusallowsa
adaptedsuchthatthedifference
between
theoutputsof the
plantandmodelis minimizedfor thesamesignal.The inverseapproachplacesthemodelin serieswith theunknown
plantsuchthatthedifference
betweentheoutputof thisseriescombinationanda delayedversionof the input signalis
minimized.In thiscase,the response
of the adaptivemodel
becomesa delayedversionof the inverseof the unknown
plant response.
As shownin Fig. 2(a), to determinethe speakerand
errorpathresponse,
the directmodelapproachplacesthe
precise
determination
of eitherthespeakeranderrorpath
response,
,fiE,or the delayedinversemodelof the speaker
anderrorpath,AS- •E •. In the directapproach
of Fig.
2(a), the response
SE is fixedafter convergence
and then
usedin the inputsto the error correlatorsof the LMS or
RLMS algorithms.In the inverseapproachof Fig. 2(b), the
adaptivemodelin parallelwiththespeaker
anderrorpath.
An errorsignalformedby subtracting
the adaptivemodel
outputfromthemicrophone
outputismultipliedby theinputsignalto formtheupdatetermsfor thecoefficients
of the
adaptivemodel.The inversemodelapproachplacesthe
adaptivemodelin serieswith thespeakeranderrorpath,as
shownin Fig. 2(b). In thiscase,the errorsignalformedby
subtractingthe adaptivemodeloutputfrom a delayedversionof the noiseinputismultipliedby the inputto theadaptivemodelto formtheupdatetermsforthecoefficients
of the
adaptivemodel.Thusthe adaptivemodelformsa delayed
inversemodelof the speakerand the error path while attemptingto matchthe response
of thedelayednoiseinput.
response
AS- •E- • is alsofixedafterconvergence,
andthe
modeling
delayA isusedin theinputsto theerrorcorrelators
of the LMS or RLMS algorithms.Bothtechniques
assume
the useof a large-amplitude,
broadband
noisesourceon an
off-linebasisto avoidcontaminationof the modelingprocess
byy or.• andto avoidtheadditionof undesired
noiseduring
on-lineoperationby the noisesourceN.
IV. CONTINUOUS
MODELING
SYSTEM
A new approachto the on-linemodelingof S andE is
shownin Fig. 3. An uncorrelatedrandomnoisesourceis
usedto excitethe seriescombinationof the speakerfollowed
by the error plant as well as adaptivemodelC while the
system
is operating.
'5 Thisrandomnoisesource
will ultimatelybecomethesourceof theresidualnoiseof thesystem.
The directadaptivemodelC is usedto obtaincoefficients
describing
the response
of $ andE that canbeusedin the
input linesto the error correlatorsfor the primaryRLMS
algorithm.The generalized
modeloutputandweightupdate
equationsfor the recursiveadaptivefilter may be written
following
thenotation
ofWidrowandStearns
j2as
T
Yk •_ WkUk
,
(2)
Wk+• = W• + 2MU;•ek ,
(3)
U•= [u•,u•_, ....] ,
=cuk ,
(4)
(5)
where
^
+
RANDOM
' )+
N
FIG. 2. (a) Directmodelingapproachfor thedetermination
of thetransfer
functionofthespeaker
anderrorpathwithadaptive
modelSE. (b) Inverse
modelingapproach
for the determination
of the delayedinversetransfer
FIG. 3. Newapproach
to on-linemodeling
of speaker
S anderrorpathE
andusingresults
in RLMS modelwithacoustic
feedback
to forma fully
function
ofthespeaker
anderrorpathwithadaptive
modelA(SE)
adaptiveactiveattenuation
system.
799
J. Acoust.Soc. Am., Vol. 85, No. 2, February 1989
L.J. Erikssonand M. C. Allie:Random noisefor on-linemodeling
799
Yk = scalarmodeloutputat discretetime k,
Wk = generalizedweight vector (includesdirect and
V. RESULTS
recursire coefficients),
W •r= transpose
of
Wk+ • = updatedgeneralized
weightvector,
M = convergence
factormatrix,
e• = scalarerror signal,
U• = generalized
input vector(includesdirectand recursiveinput vectors),
U •, = compensated
generalized
inputvector,
u;, = firstcomponent
of compensated
inputvector,and
C •r= transpose
of weightvectorof modelassociated
with transferfunctionsin auxiliarypath.
The weightvectorof the adaptivemodelC is obtained
on an on-linebasisusingan adaptivealgorithmsuchas the
LMS or RLMS algorithm with the independentrandom
noisesourceasaninputandtheerrorsignalasshownin Fig.
3. The amplitudeof thenoisesourceis keptverylow sothat
the finaleffecton the residualnoiseis small.The plantnoise
y and model output• are not presentat the input to the
adaptivemodelC andsowill not affectthe finalvaluesof the
modelweights.
The use of an uncorrelated random noise source that is
independentof the input signalensuresthat the speakerand
error path will be correctlymodeled.The signalsfrom the
The resultsof a computersimulationof the system
shownin Fig. 3 confirmedthat the algorithmproperlyconvergesfor either narrow-bandor broadbandinput signals.
The coefficientsof the SE model properly describethe SE
plant,andthe coefficients
of the overallsystemmodelproperly describeP, F, and S.
The approachshownin Fig. 3 hasalsobeenimplementedoncompleteacoustical
systems
usingtheTMS320 family
of digitalsignalprocessing
microprocessors,
with inputmicrophones,canceling loudspeakers,and error micro-
phones."•?'•a
Initially,oneof thesesystems
wasutilizedto
cancelelectroacoustically
generatednoisein a 12-in.-diam
circular duct. The duct was about 25 ft long and unlined
exceptfor a short4-ft-longadsorptivesilencernear the primary noise source. Typical results after adaptation are
shownin Fig. 4. The noisereductionobtainedwith the system operatingfor a broadbandnoiseinput is shownin Fig.
4(a). This curvewasobtainedby subtractingthe canceled
spectrumfrom the uncanceled
spectrum.The maximaand
minima in the spectrumare due to acousticalresonances.
The convergedweightstructurefor the•4,B, and C elements
of Fig. 3 is shownin Fig. 4 (b). The decayof the coefficients
confirmsthat the filter lengthchosenwasadequate.The system is effective on broadband
plant (y) andmodel(.•) represent
noiseon the "plant"side
of the speaker/errorpath modelingprocessthat will not affect the weightsof the direct model C usedto determine
SE.•2Thismodelisthencopied
totheinputlinesoftheerror
correlatorsof the RLMS algorithm.
It shouldbe notedthat, althoughthe delayedadaptive
inversemodelshownin Fig. 2(b) couldbe usedin a similar
fashion,this will resultin decreasedperformancesincethe
"noise"in the auxiliary path and error path due to y and
also appearsat the input of the adaptivefilter due to the
seriesarrangement.
Thusthe autocorrelation
functionof the
filter input is adverselyaffected,and the filter weightsare
as well as narrow-band
noise
and requiresno calibrationor trainingof any kind.
Performance
in an actualindustrialfanor heating,ventilating,and air conditioningductis mademuchmoredifficult by the turbulentairflowandlargeductdimensions
that
areusuallyrequired.Goodsystemperformance
requiresantiturbulencemicrophones
as well as large,powerful,low-
frequency
sources.
•9Typicalresultsobtained
areshownin
Fig. 5. The uncanceled
autospectrum
in a linedsupplyduct
(34 X 44 in.) approximately40 ft from a centrifugalfan is
(a)
modifiedas described
by Widrow and Stearns.
•2 If this
"noise"is largeenough,the adaptivemodelmayfail to converge.Thusthedelayedadaptiveinverseapproachrequiresa
much largeramplituderandomnoisesourcethat increases
the residualnoiseand decreasesoverall systemquieting.
In the directmodelsystem,shownin Fig. 3, the "noise"
duetoy and.• doesnot affectthe final weightsin the adaptive
model.In addition,the convergence
of the SE model is assuredaslongastheinitialamplitudesarewithinthedynamic
range and signal-to-noise
ratio constraintsof the system.
Thus, with $E accurately determined, the overall system
-10
0
HZ
2OO
(•)
model will converge,resultingin minimum residual noise.
The randomnoisesourceusedto modelSE may be read-
ily obtainedthroughthe useof a varietyof methods.One
simple approach is to generatea Galois sequenceusing
methods
described
by Schroeder.
•6A Galoissequence
is a
pseudorandomsequencethat repeats after 2'"-
A
1 points,
wherernisthenumberof stages
in a shiftregister.It iseasyto
calculateandcaneasilyhavea periodmuchlongerthanthe
response
time of the System.In this'study,31 stages
(m = 31 } were used.
800
J.Acoust.
Sec.Am.,Vol.85,No.2,February
1989
B
C
FIG. 4. (a) Noisereductionwith activeattenuationsystemon for bandlimited (15-200 Hz) pink noiseinput signal(no flow--128 averages).(b)
Filter coefficients
usedto obtaintheresultsshownin (a) for adaptivefilters
.4 (32 taps),B (64 taps), and C (64 taps).
L.J.Eriksson
andM.C.Allio:
Random
noise
foron-line
modeling
800
- 40 ('
20 (<
WV VTM
-90
-10
o
HZ
200
0
-40
HZ
200
HZ
200
1.00 (d)
-90
t
0
HZ
200
0.80 .......
0
FIG. 5. (a) Relativesound-pressure
spectrumin discharge
duct of centrifugalfan with activeattenuationsystemoff [Mach number(M): 0.04-128
averages].(b) Relativesound-pressure
spectrumin discharge
ductof centrifugalfanwith activeattenuationsystemon (M = 0.04-128averages
). (c) Noise
reductionobtainedfromFig. 5(a) and (b). (d) Typicalcoherence
betweeninputmicrophone
and errormicrophonebeforethe cancellation
usedto obtain
the resultsof Fig. 5(c) (M = 0.04-128 averages).
shownin Fig. 5(a). In additionto a very-low-frequency
peak
at about8 Hz, there is a broadpeakof noisefrom about40
is essentially
no noiseaddedat any frequency.Additional
Hz to about 140 Hz. With the active soundcontrol system,
Although theseresultsare substantiallybetter than
typicalpassivesilencers,
theredo not appearto beany reasonswhy the performanceof this systemcannotbe increased.The primaryareasof potentialimprovementare to
obtainbettercoherencebetweenthe input and error microphonesthroughthe useof improvedantiturbulence
microphones
andto increase
thespeed
ofcomputation
throughthe
useof more powerfulmicroprocessors.
this broadpeak was reduced,as shownin Fig. 5(b). The
noisereductionisplottedin Fig. 5(c). Thereisa broadrange
of attenuationfrom about40 to 140 Hz peakingat about 18
dB. Systemperformance
is limitedby theeffectiveness
of the
antiturbulencemicrophones.There is minimal attenuation
in the8- to 40-Hz rangeduetOthelackof coherence
between
the input and error microphonesat thesefrequencies,as
shownin Fig. 5(d). It shouldbenotedthat it hasbeenfound
that themagnitudeof the coherence
mustbeon the orderof
0.95 or greaterfor cancellationto beeffective.The excellent
coherence from about 40-140 Hz is consistent with the at-
noise and narrow-band electronic tone is shown in Fig. 6.
The tone is reducedabout 30 dB, while the broadband noise
is reducedabout 15 dB. It is importantto note alsothat, in
additionto the attenuationshownin Figs. 5(c) and 6, there
J. Acoust.Soc. Am., Vol. 85, No. 2, February 1989
Vl. CONCLUSIONS
A completeactive attenuationsystemhas been describedin which acousticfeedbackis modeledas part of an
tenuationshownin Fig. 5(c). It wouldbe difficultto obtain
this performanceusinga conventionalpassivesilencer.In
addition,the activeattenuationsystemresultsin essentially
no restrictionto the flow, thus avoidingthe needto modify
thefandrive.As before,thesystemis fully adaptive,andthe
resultsshownwere obtainedwith no training or calibration
beforeoperatingthe system.
To demonstratethe effectiveness
of the systemon narrow-bandaswell asbroadbandnoise,an electronictonegenerator was usedwith a loudspeakerto introducea tone at
about70 Hz at the fan while the HVAC systemwasoperating. The noisereductionon this combinedbroadbandfan
801
performance
results
havebeenpresented
elsewhere.
3'2ø
40
-1
0
HZ
200
FIG. 6. Noisereductionfor an input signalconsistingof broadbandnoise
from a centrifugalfan combinedwith a tone generatedby a speaker
(M = 0.04-128 averages).
L.J. Erikssonand M. C. Allie: Random noisefor on-linemodeling
801
7p.L. Feintuch,
"An adaptive
recursive
LMS filter,"Proc.IEEE 64 ( 11),
adaptive
filterbasedontheRLMS algorithm.Theeffects
of
soundsourceanderrorpathtransferfunctionsareadaptively determined
onlinethroughtheuseof a secondLMS algorithm that usesan independentlow-level random noise
sourceto modelthe soundsourceand error path while the
systemis operating.The combinedsystemis fully adaptive,
compensates
for changesin all transducers,
the source,and
acousticalelements,is effectiveon broadbandas well asnar-
1622-1624 (1976).
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ofNetworkandSystemTheory,
editedby R. E. Kalman and N. Declaris(Holt, Rinehart,and Winston,
New York, 1971).
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ofmultiplecorrelation
cancellation
loopwitha
filter in the auxiliarypath," IEEE Trans.Acoust.SpeechSignalProcess.
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sound
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of the 15th ,4silomarConference
on Circuits,Systems
and
Computers,
9-11 Nov. 1981,PacificGrove,CA, pp. 185-189.
row-bandnoise,and requiresno calibrationor trainingpro-
ceduresprior to operation.
•2B.WidrowandS.D. Stearns,
•4daptiue
SignalProcessing
(Prentice-Hall,
ACKNOWLEDGMENTS
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of
digitalfiltersusinga modifiedWidrow-Hoffalgorithmfor the adaptive
The authorsgratefullyacknowledgethe assistance
of
Cary Bremigan,JamesGilbert, andPatriciaSteaffens
of the
NelsonIndustries,Inc. CorporateResearchDepartment.
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for activeattenuation
in ducts," Sound Vib. 22(2), 30-34 (1988).
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speaker,errorpath,andfeedbackpath,"U.S. PatentNo. 4,677,676(30
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(Springer,Berlin, 1984).
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digitalsoundcan-
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