Auditory filter bandwidths in binaural and monaural listening conditions Birger Kollmeierandlnga Holube DrittesPhysikalisches Institut, Universit?it Gi3ttingen, Biirgerstrasse 42-44, W-3400Gi3ttingen, Germany

(Received25 July 1991;revised28 January1992;accepted19May 1992) The shapeand the effectivebandwidthof the auditoryfilter at 500 Hz wasexaminedfor binauralandmonauraltone-in-noise detectionexperiments in four normallisteners.In the binauralcondition,a broadbandnoisewith an interauralphasedifferenceof 0 belowand an interauralphasedifferenceof •r abovea certain"edgefrequency"wasemployedto maska 500Hz probetonewith an interauralphase•r (denotedasNozrS•r).The thresholdof the probe toneas a functionof the edgefrequencyin this configurationand in a configurationwith an invertedinterauralphaseof the masker(denotedasN•roS•r) wasfittedby assumingdifferent filter shapesandoptimizingtheir respectiveparameters.In an analogousmonaural experiment,the spectralpowerdensityof the maskerwas 15 dB lower belowthe "edge frequency"or 15 dB lower abovethis frequency,respectively.Severalfilter characteristics with two free parametersdescribethe data almostequallywell. Their equivalentrectangular bandwidths(ERB) showconsiderably morevariationsbetweenfilter shapesthan the 10-dB bandwidthand the 90% bandwidthvalues(i.e., the bandwidthsencompassing 90% of the integratedareaaboveand belowthe centerfrequency).This indicatesthat eitherof thesetwo bandwidthparametersis moreappropriatefor comparingauditoryfilter bandwidthsthan the ERB. For the roundedexponentialfilter, the 90% bandwidthaveragesto 147Hz in the binauraland to 125Hz in the monauralcondition.Thesevaluesare up to 12% higherif offfrequencydetectionis accountedfor. Our generalfindingof auditoryfilter bandwidthsin the binauralconditionsexceeding the monauralbandwidthsby approximately20% may be caused by two factors:First, off-frequencydetectionmay be performedin monaural,but not in binaural detectionstasks and second,the random interaural mismatch in binaural noise

reductionprocesses fluctuatesslowlyand thusmodulatesand spectrallysmearsthe output signalof the binauralnoisereductionprocess. PACS numbers: 43.66.Ba, 43.66.Pn, 43.66.Dc

INTRODUCTION

The first descriptionof the ear'spropertyto integrate overadjacentspectralregionsin a way similarto a bank of bandpassfilterswasgivenby Helmholtz (1870). Sincethen, differentpsychophysical methodsfor estimatingthe bandwidths (i.e., the "critical bandwidth" in certain experiments) and the transfer characteristicsof these auditory filters were investigated(Fletcher, 1940; Gfissler, 1954; Zwicker, 1954; Greenwood, 1961; Patterson, 1974; Houtgast,1977;Glasberget al., 1984). As an importantparameter of monauralfrequencyresolutionand integration,the auditory filter bandwidth limits the masking effect of a broadbandsignal(e.g.,noise)ona narrow-bandsignal( e.g., a pure tone or a spectralcomponentof a speechvowel.) A further reductionof the maskingeffectmight be performed by binaurallisteningif the narrow-bandsignalarrivesat the listener's head from a different direction than the broadband

noise.In everydaylife, both effectsthereforecontributeto the astoundingperformanceof the auditorysystemin separatinga desiredspeechsignalfromanundesiredbackground noise. For this reason,the auditory filters under binaural listeningconditionshave to be investigatedif we want to 1889

understand humanlistening performance in everyday life.1 Spectralaspectsof binaural signaldetectionwere already examinedby Bourbonand Jeffress(1965), Severand Small (1979), Hall et al. (1983), and Zurek and Durlach (1987), who conductedbinauralmaskingexperimentswith

a bandpass-filtered noisemaskerthat was interaurallyin phase(No) and a probetone that was presentedwith an interauralphasedifference of 0 (So) or rr (Srr), respectively. Their generalfindingscanbe summarizedasfollows:If the bandwidth of the masker is increased while the total masker

poweris held constant,the maskeddetectionthresholdof theprobetonedecreases steadilyassoonasthemaskerbandwidth exceeds the "critical"

bandwidth. While for the situais obtained as for

tion NoSo the same critical bandwidth

monauralexperiments, thebandwidthfor the configuration NoSrris considerably larger.Specifically,the maskinglevel difference (i.e., the difference between the NoS•r and the NoSo threshold) decreaseswith increasingmasker bandwidth even if the monaural critical bandwidth is exceeded. The differences between the critical bandwidths in binaural and monaural conditions were attributed to the different

cuesin both conditionsusedby the auditory systemfor detectingthe probetone (Zurek and Durlach, 1987).

J. Acoust.Soc. Am. 9• (4), Pt. 1, October 1992 0001-4966/92/101889-13500.80

@ 1992 AcousticalSocietyof America

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However, the estimatesof the auditory filter bandwidth derived from binaural band-narrowingexperimentsdiscussed so far are not consistent with broadband

noise masker

experimentswhere either a spectralnotch or a frequencyvaryinginterauralcorrelationis used:Sondhiand Guttman (1966) determinedthe binaural masking level difference (MLD) of a probe tone that was masked by two noise maskers:The "inner" maskerband was spectrallycentered aroundthe probetone and waspresentedwith the reversed interauralphasedifferenceof the "outer" band that waslocatedbelowand aboveit. By measuringthe MLD asa function of the bandwidthof the innerband,they originallyestimated a critical

bandwidth

of about 200 Hz for a center

frequencyof 500Hz. However,their resultscanbepredicted by using a similar width of the auditory filter as found in monaural experiments(Kohlrausch, 1988). Hall et al. (1983) also found approximatelythe sameauditory filter bandwidthsas in monauralexperimentswhen usingan S•r and an So probetoneat 500 Hz centeredin a spectralnotch of a diotic (No) noisemasker.However,they observedsubstantiallylarger bandwidthsin the binauralconditionsfor a bandlimitingmethod.Kohlrausch (1988) useda broadband noisemaskerwith an interauralphasedifferenceof 0 below

Wightman ( 1971) showedthat off-frequencydetectionin the NoSoconditionasopposedto on-frequency detectionin the NoS•r conditionwas the explanationfor unexpectedly low MLD valuesat smallmaskerbandwidths.Sinceshifting the centerfrequencyof the auditoryfilter only yieldsan increasein signal-to-noiseratio if the central portion of the filteris flat, the roleof off-frequency detectionis closelyconnectedwith the selectionof an appropriatefilter shape.In this study,we thereforeuseddifferentshapesof the auditory filter and studiedthe role of off-frequencydetectionfor the mostcommonlyusedroundedexponentialfilter. The binaural conditionemployedis similar to the one describedby Kohlrausch (1988). For comparison,a monaural experimentis performedwherethe maskinglevelof a broadbandnoisemaskeris decreasedby 15 dB above (or below) a certain"edge"frequency.A thresholddifferenceof 15 dB is expectedto occurwhenthe probetonefrequencyis above the edge frequencyas comparedto the condition where it is below this edge frequency.This differenceis causedby the differencein maskinglevelbetweenfrequenciesbelowand andabovethe edgefrequency.In the binaural case,however, a similar threshold differenceof about 15 dB

is expectedwhichis causedby the subject'sabilityto exploit 500 Hz and •r above 500 Hz (or vice versa). From the freinteraural differencesfor noise suppression.Both experiquency-dependent thresholdof an S•r or So probetone he ments are analogousbecausethe same frequencydepenestimatedauditoryfilterbandwidths.They werecomparable denceof the thresholdis expectedif the sameauditoryfilters are effectivein monauraland binaural listeningconditions. to bandwidths observed in monaural experiments for 500 Hz. The assumptionsusedto fit his as well as several While the edgefrequencywasfixedand the probetonefreother authors'binauraldata werethat the frequency-depen- quencyvariedin Kohlrausch'sstudy,in thisexperimentthe dent interaural correlation of the masker is filtered to obtain edgefrequencywasvariedand the probetonefrequencywas an "effective" interaural correlation by roughly the same kept constantat 500 Hz. Sincethe experimentalconditions auditory filter as in monaural conditions.Sincethe effective in this studyare analogousto the conditionsin the time domain describedby Kollmeier and Gilkey (1990), many of interauralcorrelationis convertedinto the MLD by a highly nonlinearrelation,the frequencydependencyof the MLD the argumentsare on the sameline as givenin that paper. appearsto be generatedby a broaderfilter than wasactually used.

This overestimationof the auditoryfilter bandwidtheffectivein binauraldetectioncanexplainat leastpartsof the observed differences between monaural and binaural band-

width estimates.However,it is not yet clearif both monaural and binauraldata canbe completelyalignedusingthe same auditory filter functionsor if additional factorshave to be

takeninto account.This is partly dueto largeintersubject and interstudydifferencesand the usageof differentauditory filter functionsassumedto explainthe data in different studies.Another reasonis the lack of a direct comparison betweenbinauraland analogousmonaurallisteningsituationswith the samesubjectsto the degreethat an analogy betweenmonauraland binaural auditorytasksexistsat all. The presentstudy therefore examinesthe degreeto whichthe sameauditoryfilterscanbeassumedfor a binaural andan analogous monauralsituation.In addition,theshape of the auditory filter and the possiblecontributionof "offfrequency"detectionboth in binaural and monauralconditions are examined."Off-frequency"detectiontherebyrefers to the effect observedby Patterson (1976) that a maximum signal-to-noiseratio can be attained for certain maskersif the auditoryfilteris shiftedawayfrom the probe tone frequency.In a binaural bandlimiting experiment, 1890

J. Acoust.Soc. Am.,Vol. 92, No. 4, Pt. 1, October1992

I. METHOD

A. Subjects

One femaleand three male graduatestudents,agedbetween23 and29, participatedvoluntarilyin the experiments. All had clinicallynormalheatingand wereexperienced listenerswith similartone-in-noisedetectionexperiments.

B. Apparatus

Signal and noise stimuli were generatedon a TMS 32010-basedsignal processingsubsystemconnectedto a Digital EquipmentPDP 11/73 computer.Three 16-bitdigital-to-analogconverterswereusedto producethe binaural noisemaskerand the probetone separatelyat a sampling rate between 10 and 30 kHz.

The level of the stimuli was

controlledby separateprogrammableattenuatorswith a resolutionof 0.1 dB. The noisestimuliwerebandpass filtered between 0.1 and 1 kHz with two balanced Krohn-Hite

3343

filterswith a slopeof 48 dB per octave.The probetonewas alsobandpassfilteredbetween0.1 and 1 kHz by an Ithaco 4302 filter with slopesof 24 dB per octave.The probetone andnoisewaveformswerethenaddedwith an analogmixer B. Kollmeierand I. Holube:Auditoryfilterbandwidths

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and presentedto the subjectthroughDT-48 headphones in an sound-insulated

anechoic chamber.

Short instructions

monaural conditions, the second stereo channel was atten-

uated and added (or subtracted,respectively)to the first stereo channel in order to obtain a 15-dB decrease (or in-

andmessages for thesubjectweredisplayedona videomonitor in front of the subject.During practicerunsat thebeginningof eachsession, feedbackinformationwasprovidedon the monitorandby illuminatingthe respective buttonof the responsebox. No feedbackwas providedduring the runs

level of 76 dB $PL for the binaural conditions, 61 and 76 dB

used for data collection.

SPL for the monaural

C. Stimuli

and conditions

A sketchof the maskingnoiseand the probetone employedis givenin Fig. 1. The maskerwasa random750-ms segmentof a Gaussiannoisebandpassfilteredfrom 0.1 to 1 kHz. In the binaural conditions, the masker exhibited an

interauralphasedifferenceof 0 belowa certainedgefrequency and •r abovethis frequency[denotedas Noir, cf. Fig. 1(a)] or an interaural phasedifferenceof •r below and 0 abovethe respectiveedgefrequency[denotedas N•ro, cf. Fig. 1(c) ]. In the monauralconditions,the spectraldensity of the maskerwas decreasedby 15 dB below the edgefrequency[ cf. Fig. 1(b) ] or decreasedby 15 dB abovethe respectiveedgefrequency[ of. Fig. 1(d) ]. The noisewasgenerated by first computing 32 768 samplesof a normally distributedrandomvariableand retainingthissignalfor the

crease,respectively)of the spectraldensityabovethe 819th sample.The noisesignalswere D?A converted,bandpass filtered,and presentedto the subjectat a constantoverall reference conditions and between 69

and 75 dB SPL for differentedgefrequenciesin the monaural conditions.The edgefrequencyof the maskerwasvaried between250 and 750 Hz by varyingthe samplingfrequency between10and 30 kHz andcorrectingthe spectraldensityto a constantvalueby an appropriateattenuationor amplification of the wholesignal.For eachpresentationof the noise masker,a 750-mssegmentof the cyclicallycontinuedtime sequencewas randomlyselectedand D?A convertedwithout shapingthe envelope. The probetonewasa 500-Hz sinusoidwith a total duration of 250 ms, including20-msraised-cosine onsetand offsetrampswhichwastemporallycenteredwith respectto the masker.In the binauralexperiment,it waspresentedwith an interaural phasedifferenceof 180ø (S•r). In the monaural experiment,both signaland noisemaskerwere presented monaurally (condition NmSm). Four

individual

reference

thresholds

were

obtained

first stereo channel. The second stereo channel was obtained

with the unprocessed time sequence of the noisemasker.For

by transformingthe time sequence in the frequencydomain with an FFT, invertingthe phaseof the frequencysamples above(or below,respectively)the 819th sampleand transforming the sequenceback into the time domain. For the

thebinaural condition, theN•rS•randNoS•rthreshold was obtainedby presentingthe noiseinteraurallyphase-inverted or in-phase,respectively.For the monaural condition,the NmSm referencethreshold was obtained at the original maskingleveland the attenuatedmonauralreferencethreshold (- 15 dB) NmSm was obtained by attenuating the maskerby 15 dB.

D. Trial structure and measurement procedure

,eff I••

A three-intervalforced-choice(3IFC) procedurewas employed.The three750-msobservationintervalswereseparatedby two 500-msinterstimulusintervals.At the end of

L••••

the third observation interval, unlimited time was allowed

righ'10;O,..,• O0•Hz) 10.•.• 5001000 f•-Hz)

tevet

S edge

(c) tevet

teff '",•'••

•

edc•e

(d)

•///• ''"'""'

for the subjectto respondand a pauseof 400 ms was provided beforethe next trial began.In all three observation intervals,a 270-msmarkinglight wasturnedon 20 msbefore the time whenthe signalmight occur.An additional400-ms feedbacklight wasoptionallyprovidedto mark the interval that actually containedthe probe tone. In addition, a message"correct" or "miss" wasdisplayedon the videoscreen. Trial-by-trial feedbackwas only providedduring training runsthat werenot includedin the data presentedbelow. An adaptivestaircasealgorithmwasusedto controlthe probetone level, followingthe recommendations of Kollmeier et al. (1988). At the beginningof a track, the signal levelwassetwell abovethe expectedthresholdand lowered by 2.0 dB after each correct response.As soonas the first incorrect responsewas recorded,the signal level was increasedby 2.0 dB and a "oneup/two down" rule wasadopted (Levitt, 1971), which loweredthe signallevel after two successive correctresponses at the samesignalleveland increasedthe levelafter oneincorrectresponse. After the third

ß'10,0•_-//•/.5•.0'•/1.•0/0• f(Hz) '10,• 5001000 f(Hz)

FIG. 1. Sketchof the maskersandsignalsemployedin theexperiments. In eachpanel,thespectralpowerdensityat theleftandrightear,respectively, is plottedas a functionof frequency.In the binauralconditions[ (a) and (c) ], the interauralphaseof the noisemaskeris changedfrom 0ø (No) to 180ø(N•r) abovea certain"edgefrequency"[ (a), denotedasNoir] or below the edgefrequency[ (c) denotedas N•ro], respectively. The masked thresholdof a interaurallyphase-inverted probetone (S•r) at 500 Hz is determined.In thecomparablemonauralconditions[ (b) and (d) ], the spectral powerdensityis increasedby 15dB abovethe edgefrequency[ (b) ] or decreased by 15dB[ (c) ], respectively, andthemaskedthresholdof a monaural probetone at 500 Hz is determined.

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B. Kollmeierand I. Holube:Auditoryfilter bandwidths

1891

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ß I

o

i

I

i j

o

-5 -lO

-15

(a) -20

, i

i

I

5

I

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I

I

1

I

I

I

!

SSO

650

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-5 -lO -15

--

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(b)

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edge frequency {Hz]

edge frequency (Hzl FIG. 2. Maskedthresholds(medianvaluesandinterquartileranges)in the

binauralconditions No•rS•r[V, cf. Fig. l(a)] andN•roS•r[/•, cf. Fig. 1(c) ] asa functionof theedgefrequency of themasker.(a)-(d) showthe individualresultsfor subjectsRN, IH, MK, and SU, respectively. In each panel,0 dBat theordinatedenotes theindividualN•S•r reference threshold andthe arrowat the right-handsidedenotethe individualNoS•rreference

threshold. Bothreference thresholds wereobtained withouta frequencydependent interauralphaseof the masker.Solidlinesindicatetheoretical thresholdfunctionsobtainedwith a double-sidedexponentialfilter charac-

FIG. 3. As Fig. 2 for the monauralconditionswith a 15-dB increasein spectralpowerdensityof the maskerabovethe edgefrequency[ V, cf. Fig. 1(b) ] anda 15-dBdecrease in spectralpowerdensityof themasker[•x, cf. Fig. 1(d) ]. (a)-(d) showthe individualresultsfor subjectsRN, IH, MK, andSU, respectively. At theordinate,0 dB denotesthe individualmonaural thresholdfor the referencemaskinglevel and the arrow at the right-hand sidedenotesthe individualmonauralthresholdafter reducingthe masking levelby 15 dB. Solidlinesindicatetheoreticalthresholdfunctionsobtained with a double-sidedexponentialfilter characteristic.

teristic.

reversal, thestepsizewasdecreased to 1.0dB.Subsequently,

II. RESULTS

at least20 trialswereperformedandthetrackendedafteran

The individual resultsfor the binaural experimentare displayedin Fig. 2(a)-(d) for all four subjectswhile the resultsfor the respectivemonauralexperimentare givenin Fig. 3 (a)-(d). The abscissadenotesthe edgefrequencyof the masker.The probetone level at thresholdis plotted on

even number of reversals. The threshold

estimate was ob-

tainedasthe averageof the levelspresentedon all trials after the third reversal.Each data point representsthe median thresholdestimateof threeindependenttracksfor eachsubject plottedtogetherwith the respectiveinterquartileranges. For subjectMK, additionaltwo trackswereperformedif the resultsfrom the initial three tracks differedby more than 4 dB. 1892

J. Acoust. Soc. Am., Vol. 92, No. 4, Pt. 1, October 1992

the ordinate.

For the binauralcondition(Fig. 2), the 0-dB point correspondsto the individualN•rS•r threshold.The arrowsin the fight-hand cornerdepictthe individual NoS•r reference B. Kollmeier and I. Holube: Auditory filter bandwidths

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threshold. Theuprighttriangles(/•) denotemedianthresholdsfor the N•roS•rcondition(maskerphaseinvertedbelow theedgefrequency)whereas theinvertedtriangles(W) denotethe No•rS•rcondition(maskerphaseinvertedabovethe edgefrequency).For the monauralcondition(Fig. 3), the 0-dB point correspondsto the monaural NmSm reference thresholdwhereasthe arrowsin the fight-handcornerdepict the individual ( -- 15 dB) NmSm thresholdwhich was obtainedafterattenuatingthemaskerby 15dB. The upright triangles(/•) denotemedianthresholds for the condition with a decreased maskinglevel abovethe edgefrequency whereastherestingtriangles(W) denotetheconditionwith a decreased maskinglevelbelowthe edgefrequency. In general,the resultsshow smooththresholdtransitionsasthe edgefrequencyof themaskerisincreasedbeyond the probetonefrequencyof 500Hz. The slopesof thesetransitionsare shallowerin the binauralconditionshownin Fig. 2 than in the monauralconditionshownin Fig. 3. For edge frequencieswell below or abovethe probetone frequency, the obtainedthresholdvaluesare expectedto approachthe

The first-orderapproximationof the effectiveauditory filter usesa rectangulartransferfunctionlocatedsymmetrically or asymmetricallyaround the probe tone frequency thusrequiringoneor two freeparameters,respectively, that have to be fitted to the data (Fletcher, 1940; G•issler, 1954;

Zwicker, 1954). In manypsychophysical experiments,however,morecomplexfilter shapesare requiredto accountfor the data (e.g., Patterson,1974;Houtgast,1977;Glasberget

al., 1984;Kohirausch, 1988).Ontheotherhand,if toomany free parametersare neededto describea particular filter shape,their fittedvaluewill fluctuateconsiderablyand thus will have little explanatoryutility. For this reason,we restrictedthe filter characteristics employedto functionswith only two free parameters.For comparison,we includeda five-parameter filter functionemployedby Glasberget al. (1984). Off-frequencydetectionwas optionallytaken into accountfor the roundedexponentialfilter.The detailsof the filter characteristics and the fitting procedureemployedare givenin AppendixA. TableI givesfor eachsubjectandeachfilter characteris-

reference thresholds. The difference between thresholds ob-

tainedfor the lowestand highestedgefrequencies employed shouldthereforeequalthe NoSwbinauralmaskingleveldifference (MLD) similar threshold

at 500 Hz in the binaural conditions and a difference of about 15 dB for the normal

andthe 15-dBattenuatedmaskinglevelin the monauralconditions.While the observedthresholddifferencesin Figs. 2 and 3 arein theexpectedrange,theabsolutethresholdvalues for low and high edgefrequencies differup to 2 dB from the comparablereferencethresholdswithout a spectraltransition of the interauralphaseor maskinglevel.A comparable threshold shift was observed by Kollmeier and Gilkey (1990) in an experimentin the timedomainwhichis analogousto the onereportedherein the frequencydomain.Parts of the threshold

shift observed here can be attributed

to a

confusionbetweenthe probe tone and the "binaural edge pitch" (Klein and Hartmann, 1981), i.e., a tone-likesensation producedby the frequency-dependent interauralphase differencein the binaural conditionsand a monaural edge pitch in the monauralconditions.Sincethe frequencyof this sensationis closeto the edgefrequencyof the masker,the confusionis mostlikely to appearfor edgefrequencies close to the probetonefrequencyof 500 Hz, thusyieldinga rela-

tivelyhighthreshold variabilityin thisregion. 2 In thebinauralconditionsof Fig. 2, the thresholdtransitions appearto be shallowerand more symmetricthan the transitionsin the monauralconditions(Fig. 3) evenif the smallerrangeof the transitionsin the binaural caseis taken into account (i.e., the MLD beingsmallerthan 15 dB). In order to quantify these differences,theoretical threshold transitionswerefitted to the data basedon differentshapes of the effectiveauditory filter. In the binaural case,the effec-

tive interauralcorrelationat the outputof corresponding auditory filters at both ears is first computed.The masked thresholdfor the S•rprobetoneis thenobtainedusinga formula from Durlach's EC theory for the MLD as a function of interaural correlation. In the monaural case,the masker

powerat the output of the auditoryfilter is usedto predict

TABLE I. Frequencyconstants% and% (in Hz) fittedto thebinauraldata from Fig. 2 and the monauraldata from Fig. 3 for four subjectsand five differentfilter characteristics.The last three rows give the valuesfor the roundedexponentialfilter if off-frequencylisteningis accountedfor. As an indicatorof the "goodnessof fit," the nonlineardeviationmeasurer,! is providedfor eachsetof fitted parameters(cf. AppendixA). Subjects RN

Filter shape Rectangular % vI r.• Asymm. Gauss v,

vt r,!

IH

MK

SU

bin/mon

bin/mon

bin/mon

bin/mon

53/54 97/105 0.939/0.936

54/47 94/104 0.955/0.928

48/43 76/72 0.944/0.914

51/35 93/75 0.937/0.956

35/22 68/57 0.948/0.972

25/22 67/80 0.973/0.977

22/21 46/45 0.957/0.951

52/18 16/51 0.935/0.980

Double-sidedexp. % 29/19 28/21 23/16 19/17 % 47/36 47/48 33/27 38/33 r•! 0.957/0.980 0.976/0.978 0.965/0.958 0.947/0.986 Rounded exp. •'u

20/13

19/15

15/11

13/12

•/!

33/25

32/34

22/19

26/23

r,•

0.959/0.978

0.976/0.975

0.961/0.956

0.949/0.985

Roundedexp., five parameters 17/13 % 11/12 25/18 v• 31/14 22/13 $t• 28/13 47/68 /• 31/42 c

rnl

0.44/0.33

0.960/0.981

0.26/0.23 0.976/0.983

10/10

6/11

18/12

24/19

30/8

24/14

36/60

7/44

0.24/0.1 0.961/0.963

0.16/0.16 0.952/0.986

Roundedexp.with off-frequencydetection % 21/14 20/16 15/12 14/13 v• 35/27 34/38 23/22 28/25 • 0.951/0.976 0.972/0.977 0.972/0.954 0.945/0.980

the monaural masked threshold.

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B. Kollmeierand !. Holube: Auditoryfilter bandwidths

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tic the obtainedfitted parametersVu and ¾1for the binaural and monauralconditions.Their values(expressedin Hz) relate to the effective bandwidth

above and below the center

frequency,respectively. For mostsubjects andfilter characteristics,the fittedparametersexhibita wider andmoresymmetrical auditory filter for the binaural conditionsthan for

themonauralconditions. 3 In orderto estimate the "goodnessof fit," the normalized nonlinear correlation coefficient

/•nlis alsoincludedin Table I whichapproaches unity asthe discrepancybetweendata pointsand fitted curvedecreases (cf. AppendixA). Sincein most casesthe value of/•nl is greaterthan0.9, all filtershapes underconsideration yielda satisfactoryfit to the data. For mostsubjectsand conditions, the bestfit with two parametersis obtainedwith a doublesidedexponentialfilter shape,i.e., a triangularshapeon a logarithmiclevelscale.For thisreason,the curvesplottedin Figs. 2 and 3 were obtainedusing this filter shape.The rounded-exponential (roex) shapeonlydiffersslightlyfrom the double-sidedexponentialfilter. In few cases,the Gaussian filter yieldsa better predictionthan the double-sided exponentialfilter but a substantially worsefit is obtainedin mostcases.The rectangularfilter yieldsthe worstprediction of the data.

TABLE II. Equivalentrectangularbandwidths(ERB), 10-dBbandwidths (10dB-BW) and 90% bandwidths(BW9o, i.e., bandwidthwhich encompasses 90% of the integratedareaaboveandbelowthe centerfrequencyof the respectivefilter characteristic)for the binauraland monauraldata for four subjectsand for fivedifferentfilter characteristics.

RN Filter shape bin/mon

IH bin/mon

Subjects MK bin/mon

SU bin/mon

Average bin/mon

,

,

Rectangular ERB 10 dB-BW

150/159 150/159

148/151 148/151

124/115 124/115

144/110 144/110

142/134 142/134

BW9o

135/143

133/136

112/103

130/99

128/120

Asymm. Gauss ERB 10 dB-BW

128/98 220/169

115/128 198/220

85/83 146/142

86/86 147/147

104/99 178/170

BW9o

168/129

151/168

112/109

113/112

136/130

Double-sidedexp. ERB 10 dB-BW

77/55 176/126

75/68 172/158

56/43 129/100

57/50 130/115

66/54 152/125

BW•

176/126

172/158

129/100

130/115

152/125

ERB 10 dB-BW

106/77 205/149

101/97 196/189

75/61 146/119

76/70 148/136

90/76 174/148

BW•

173/126

165/159

123/100

125/114

147/125

Roundedexp.

Roundedexp., five parameters

The rounded-exponential filter with five parameters ERB 99/72 98/84 74/54 60/69 83/70 10 dB-BW 202/154 196/182 154/104 125/137 169/144 yieldsthe highestvaluesof/•nl and thus the bestfit to the BW9o 179/152 174/210 157/151 131/128 160/160 data.While the parametersVuand Vl correspondto the two Roundedexp. with off-frequencydetection parametersof the double-sided exponentialand the roundERB 113/83 109/107 75/69 84/76 95/84 ed-exponentialfilter discussed above,the remainingthree 10dB-BW 219/161 211/207 146/133 163/148 185/162 parameters exhibitlargevariationsof theirfittedvalueseven BW9o 184/136 177/175 123/112 137/125 155/137 if only minor changesof their initial valuesareintroducedin the fitting procedure.Thesefluctuationsare a consequence of the limited measurementaccuracyand the limited dynamicrangeof the thresholdtransitionsin our experiments. For this reason, a reliable estimate of the three additional for mostsubjects andfilter shapes. Unfortunately,the ERB valuesdiffer considerablybetweendifferentfilter functions parameterscannotbe derivedfrom our dataalthoughthey fitted to the sameset of data. Thus a comparisonof ERB slightlyimprovethe "goodness of fit." The roundedexponentialfilter includingoff-frequency valuesbetweendifferentstudiesis only valid if the sametype detectionyieldsaboutthe same"goodness of fit" values/•nl of filter characteristicis used.For the roundedexponential window,our averagevalueof 76 Hz in the monauralcondithanwithoutoff-frequency detection.As expected,the fitted parameters VuandVl areconsistently higherif off-frequency tion and 90 Hz in thebinauralconditioncompareswell with themonauralvalueof 82 Hz for a signalfrequencyof 500Hz detectionis accountedfor. Averagedacrosssubjects,the increase is 5% for the binaural and 9% for the monaural conobtainedby Moore and Glasberg(1983). When datafrom differentsignalfrequencies are extrapolatedto 500 Hz by dition.This comparativelysmalleffectis dueto the restrictroughlyassuming a linearrelationbetweencenterfrequency ed dynamicrangeof the maskerin the frequencydomain and filter bandwidth, valuesof 61-101 Hz are obtainedfrom whichwas15dB in the monauralconditions.Hence,shifting the auditory filter away from the probe-tonefrequency the data providedby Glasberget al. (1984) and approximately80 Hz fromthedataof DreschlerandFesten(1986). yieldsconsiderablylessincreasein detectabilitythan in conThe fact that different ERB values result from the same ditionsusedby, e.g., Patterson(1974) wherehigh-passor data if differentfilter characteristics are employedclearly low-passnoisemaskerswith a higherdynamicrangein the limitsthe utility of ERB valuesfor comparingthe "effective" frequencydomainwereemployed. bandwidthbetweenstudiesusingdifferentfilter shapes.As In orderto comparethe "effective"bandwidthof differan alternativeapproach,we thereforeemployedthe 10-dB ent filter functions,the "equivalentrectangularbandwidth" bandwidth,i.e., the frequencydifferencebetweenthe points (ERB) is mostcommonlyused,i.e., the bandwithof a recwherethefilterfunctionhasdroppedby 10dB fromitsmaxitangularfilter functionwhich has the sameintegratedarea and the same maximum

mum value, and the 90% bandwidth of the filter function,

functionsand conditionsemployedfor eachindividualsubject. Obviously,thevaluesof theERB in thebinauralconditionssystematically exceedthosein the monauralconditions

i.e., the bandwidthwhich encompasses 90% of the integratedareaunderthe respective filterfunction.In orderto faciliate the calculationof the 90% bandwidthfor asymmetrical filters,it is definedasthe sumof the independent contribu-

value as the filter function under consideration. Table II lists the ERB values for all filter

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B. Kollmeierand I. Holube:Auditoryfilter bandwidths

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tionsfrom aboveandbelowthe centerfrequency.Specifically, it encompasses the lower 90% of the integratedarea abovethe centerfrequencyas well asthe upper90% of the integratedarea below the centerfrequency.The detailsof computingthesevaluesfor differentfilter functionsare ineludedin AppendixB. The outcomeof the 10-dBbandwidth and90% bandwidthfor eachsubject,filter functionand condition is included in Table II. Obviously,the deviationbetween the values for different filter characteristics

fitted to

the samedata is significantlyhigherfor the respectiveERB valuesthan for both alternativebandwidthparameters.Of these,the 10-dB bandwidth, which is a common measureof

frequencyselectivityin neurophysiology, exhibitsslightly more variationsbetweenfilter shapesthan the 90% bandwidth.Therefore,the 90% bandwidthispreferableif a bandwidth comparisonbetweendifferentfilter shapesis desired. Note, however,that the roundedexponentialfilter with five parametersyields considerablyhigher 90% bandwidths than the filterswith two parameterswhichis dueto the relatively largefractionof the filter characterizedby the larger frequencyconstants/•, and/•. Averagedacrossall filter shapesemployedin thisstudy, the average90% bandwidthof 132 Hz for the monaural conditionsand 145 Hz for the binaural conditionsagain comparewell with a monauralvalue of 134 Hz for Moore and Glasberg (1983), 131 Hz for Dreschler and Festen (1986) and 128 Hz for Glasberget al. (1984). In addition, the polynomialfilter shapeof Patterson(1976) yieldsa valueof 122Hz. For thebinauralcondition,the computationof the 90% bandwidthfor the trapezoidalfilter characteristic employedby Kohlrausch(1988) yieldsa valueof 151 Hz. This valueis slightlyhigherthan the monauralvalueslisted aboveand compareswell with the valuesfrom the binaural conditionsin this study.

might be more precisethan ours.Basedon their resultsobtained

with

several

filter

characteristics

and

different

maskers,the roundedexponentialfilter appearsto havethe bestproperties,althoughan unambigousdecisionaboutthe exactfilter shapein the monauralconditionis still not possible.

On the other hand, a further increaseof precisionby decreasing the measurement error of the thresholdestimates is not realistic,sincethe accuracyof psychophysical measurementsis restrictedby the maximum time allowablefor data collectionas well as by fluctuationsof each subject's performance(cf. Kollmeier et al., 1988). Therefore,it appearsto beimpossible inprincipleto derivetheexactshapeof the auditory filter from psychophysical data and other factors (suchasnumberof freeparameters,computationalsimplicity, and universality)shouldbe taken into account,instead.In addition, the shapeof the auditory filter fitted to our data might be distorteddue to incorrectmodelassumptions.Ia particular, we assumedfor the binaural conditions that the filtered "effective" interaural

correlation

is convert-

ed into binauraldetectionthresholdsby a nonlinearrelation. Any variationin thisrelationinfluencesthe predictionof the data by a given filter shape.In the monaural case,the assumptionthat the maskedthresholdcorresponds to a constantsignal-to-noise ratio might alsobe inaccurate. If we accept that different filter characteristicsem-

ployedin differentstudiesare equivalentin describingthe shapeof the peripheralauditoryfilter with a certaindegree of inaccuracy,a measureof the "effective"auditory filter bandwidthshouldbe providedthat is asindependentof the exactfilter shapeas possible.As can be seenfrom Table II, the 90% bandwidthfulfills this specificationmuch better than the equivalent rectangularbandwidth (ERB) and shouldthereforebe adoptedfor comparisons betweenstudies. The reason for this difference is the fact that the 90%

III. DISCUSSION

A. Effective shape of the auditory filter

From the data presentedhere, no definiteconclusion can be drawn about the exact shapeof the auditory filter effective in monaural and binaural conditions since most fil-

ter functionsemployedyieldedan adequatefit of the data. The discrimination

between different filter functions is limit-

bandwidthis (inversely) relatedto the averagevalue of the filter characteristicin the regioncloseto the centerfrequency whereasthe ERB relates (inversely) to the maximum value of the filter characteristic(see Appendix B for the details). Thus filter functionsexhibitinga prominentmaximum value (e.g., the double-sidedexponentialfunction) yielda substantiallylowerERB valuethan filterswith a less

prominentmaximumvalue (e.g., the roundedexponential function) evenif their respectiveintegratedarea as well as

edby the ratio betweenmeasurement error of the thresholds (which is approximately1 dB in our experiment) and dynamicrangeof the thresholdtransitions(which is approximately 15 dB in our experiments).A better discrimination betweendifferentfilter functionswould thereforerequire a substantiallylarger dynamicrangeof the thresholdtransitionsor an increasein measurementaccuracy.Unfortunately, due to the limited maximum effectof binaural unmasking, a furtherincreaseof the dynamicrangeis impossiblein

their 90% bandwidths are identical and both of them de-

the binaural

ERB, it is dependenton the maximumvalue of the filter. Similar to the 90% bandwidth, however, only the central portion of the filter functioncontributesto its calculated

conditions.

In the monaural

conditions

em-

ployedin our study,the dynamicrangewaslimitedin order to faciliatecomparisons with the binauralconditions.Since most studiesof monaural auditory filters employedlarger dynamic ranges (e.g., Patterson, 1976; Glasberget al., 1984), their estimatesof the monaural filter characteristics 1895

J. Acoust.Soc. Am., Vol. 92, No. 4, Pt. 1, October 1992

scribethe data almost equally well. Using the 90% bandwidth is therefore in accordance with the notion that not

only the componentat the centerfrequencyof the auditory filter, but all componentsin a certainregionaroundthe center frequencycontributeto maskingin the respectiveauditory filter. In a certainsense,the 10-dBbandwidthisa compromise between the ERB and the 90% bandwidth.

Similar to the

value. In addition, the observed variations of the 10-dB

bandwidthbetweendifferentfilter shapesfitted to the same set of data are higherthan for the 90% bandwidthbut still B. Kollmeierand I. Holube:Auditoryfilter bandwidths

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considerablylower than for the ERB. Sincethe 10-dBbandwidth is more commonlyused,e.g., in physiologicalacoustics,it shouldbe the preferedbandwidthparameterif a computation of the 90% bandwidthis not possible.

derivedfrom our binaural experimentsare slightly higher than estimatesfor the sameperipheralfiltersderivedfrom the monauralconditions,two additionalassumptionsappear reasonable:(a) "Off-frequency-detection" is possible in the monauralconditions,but not in the respectivebinaural conditions,thus yieldinglower effectivebandwidthsfor the monauralconditions.(b) The amplitudeand timejitter B. Binaural versus monaural auditory filter bandwidth encounteredin binaural noisereductionprocesses (e.g., the equalization and cancellation mechanism by Dufiach, In our experimentsa small,but significantincreasein the width of the auditory filter was found in the binaural 1972) producean effectivebroadeningof the auditoryfilter. conditionascomparedto an analogousmonauralcondition. With respectto assumption(a) it shouldbe notedthat (i.e., an improvementin signal-toFor the double-sidedexponentialand rounded-exponential off-frequency-detection fitted filter function, this increase amounts to a factor of noiseratio by shiftingthe centerfrequencyof the auditory approximately1.2. Although the monauralexperimentwas filter away from the probetone frequency)is only efficient for filter characteristics havinga shallowerslopecloseto the designedto matchthe binauralexperimentascloselyaspossible, it is not clear whether the effectiveauditory filter centerfrequencythan far off (i.e., the roundedexponential, shapesderived from both experimentsdescribethe same rectangular,and Gaussianfunction,but not the double-sided exponentialfunction). We therefore usedthe rounded property of the auditory system.This is enhancedby the notion that probe tone detectionin the binaural condition exponentialfunction to checkthe changesof the fitted pamight rely on differentcuesavailableto the binaural system rametersif off-frequencylisteningis accountedfor. The results are included in Tables I and II. For the binaural condi(i.e., interauraldifferencesin instantaneous amplitudeand phasecausedby the presenceof an S•r probe tone) than tion, the increasein the fitted bandwidthparametersaverprobetonedetectionin the monauralcondition(i.e., energy agesto approximately5% which might be consistentwith changesin the spectralregionof the probetone). This differassumption(a) that off-frequencydetectiondoesnot occur in binaural conditions. In the monaural condition, an inencein detectionstrategiesmight evenbe more prominent for experimentsemployingdifferent maskersthan in this creaseof approximately9% is observed.Assumption(a) study.In band-narrowingexperiments,for example,the efcouldthereforeexplainat leasthalf of the observeddiscrepfective auditory bandwidthsin the dichotic conditionsexancyof about 18% betweenmonaural and binaural auditory filter widths. ceedthe monaural auditory bandwidth by a factor of 1.0 to 6.1 (Hall et al., 1983) or 4.2 to 35.7 (rough estimatefrom While off-frequencydetectionhas beenshownto be an Zurek and Durlach, 1987). If a broadband masker is emimportant effect in monaural and diotic detectionexperiployedwith frequency-dependent interauralphase,the facments(e.g.,Patterson,1976), it isnotyetclearwhy it should tor varies between about 2.0 (Sondhi and Guttman, 1966 in not be possiblein binaural experiments,especiallyif the comparisonto Patterson,1976) and 1.0to 1.4 (Kohlrausch, sameperipheralauditoryfiltersare assumedto be involved 1988 in comparisonto Patterson, 1976) and between1.1 to in both cases.The absenceof off-frequencydetectionin a 1.3 if a spectralnotchis introducedin the broadbandnoise NoS•r conditionin comparisonto its presencein the NoSo (Hall et al., 1983). In addition,sincethe auditory systemis conditionwereattributedby Wightman ( 1971) to the differhighly nonlinear,the estimatesof auditoryfilter bandwidths ent detectioncuesoperativein dichoticversusdiotic situaderived from broadbandexperimentsfor a single subject tions. In our experiment,the differentrole of off-frequency detectionfor the binauralandmonauralconditionsmightbe vary considerably,if maskersare usedwhichare shapeddifferentlyin the frequencydomain (i.e., sinusoidal,rectangu- explainedas follows:In our binaural conditions,a steplike lar, or step-shaped variation of the spectralpower density, variation of the masker'sspectralpower densityoccursat Holube et al., 1991). For this reason, we restrict the discusmore centralauditoryprocessing stages(i.e., after comparsion of auditory filter widths in binaural versusmonaural ing the inputsfrom both ears) wheredifferentspectralproconditionsto the data providedby this study. cessingmechanismsmight occur than at the periphery.In The auditoryfilter bandwidthsin the monauralandbinthe monauralconditions,however,the steplikevariation of aural conditionsare comparedin this study under the asthe masker'sspectralpowerdensityis alreadypresentat peripheral auditory stages where off-frequency detection sumptionthat the binaural input signalsare first filtered might occur.In addition,the effectivefilter shapesfitted to with the respectivemonaural (i.e., peripheral) auditory eachsubject'sdata exhibit somedifferencesbetweenthe binfilters. From both filter outputs, the "effective" interaural aural and the monaural conditions: The filters were more correlationfor the auditoryfilter under considerationis obsymmetricin the binaural caseand showedlessincreasein tainedwhichdeterminesthe MLD througha nonlinearrelation which might be obtainedfrom Durlach'sequalization bandwidth if off-frequencydetection was accountedfor. Thesedifferences might be due to the differentcuesusedin andcancellation(EC) theory (Durlach, 1972). Usingsimimonaural and binaural detectionalready mentionedabove lar assumptions,Kohlrausch (1988) demonstratedthe coand the nonlinear relation between the effective interaural herencebetweenhis binaural data and the auditory filter correlationat the outputof the auditoryfilter andthe binaubandwidthsrequiredto fit the data from severalother studral maskedthreshold.The mechanismsresultingin different ies employingbroadbandmaskersin binaural or monaural conditions.To explainthat the auditoryfilter bandwidths observedauditory filter bandwidthsfor binaural and mon1896

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aural conditionswere alsodiscussed by Zurek and Durlach (1987) who foundthat the insensitivityof the binauralsystem to rapidchangesin interauralcueslimitsthe bandwidth on which a binaural noise reductioncan be performed. Therefore,the effectiveshapesof the auditory filters assumedfor the monauraland binaural experimentare not necessarily equalevenif the sameperipheralauditoryfilters areassumed in bothcases.Sincesmallchanges in thecentral portionof the effectivefilter shapecandramaticallychange the benefitobtainablefrom off-frequencydetection,these differences in the effectivefilter shapebetweenbinauraland monauralconditionsmight be anotherreasonfor the lack of off-frequencylisteningin binaural conditions.To summarize, off-frequency detectionmay not be possiblein binaural listeningconditionsbecausethe spectralchangeoccursat a more centralauditoryprocessing stageand becausethe effectiveauditoryfilter shapein binaurallisteningmight not yield as much benefit from off-frequencydetectionas in monaural

conditions.

With respectto assumption(b) we estimatedthe bandwidth of a slowly statisticallyvaryingamplitudemismatch in the binaural noise reduction scheme which would be re-

quiredto explainthe differencebetweenmonauraland binaural auditory filter bandwidth on the basisof a slightly modifiedEC theory.The detailsof thisapproacharegivenin AppendixC. The resultimpliesthat a low-passnoise,with a bandwidthof approximately50 Hz, modulatingthe performanceof a binaural noise-reductionprocessmight explain the observedeffectivebroadeningof the binaural versus monaural auditory filter bandwidthsat 500 Hz. Since this estimatedbandwidthlieswithin a physiologicallymeaningful range, the sameperipheralfilter characteristicsare assumedfor monauraland binaural situationsand only minor modificationshave to be introduced in the EC theory or othermodelsof binauralinteraction,assumption(b) isquite appealing. On the basisof the availabledata, neither assumption can be ruled out. Probably, the factorsdescribedby both assumptionscontributeto the apparentbroadeningof the auditory filter in binaural detectiontasks.

IV. CONCLUSIONS

( 1) The auditory filter bandwidthsestimatedfrom our experimentsare about 1.2timeslargerandlessasymmetrical in the binaural than in the monaural

500 Hz averagedacrosssubjectswas 125Hz in the monaural and 147Hz in the binauralconditionfor the roundedexponential filter. As an alternative, the 10-dB bandwidth can be

employedthat yieldsnearly as little variationsbetweendifferent filter shapesas the 90% bandwidth. (4) Two assumptionswere qualitativelyshown to be major factorsin the effectivebroadeningof binaural auditory filters:First, off-frequencydetectionmay be performed in monaural, but not in binaural detectiontasksand second, the random interaural

mismatch in binaural noise reduction

processes fluctuatesslowlyandthusmodulatesandspectrally smearsthe output signalof the binaural noisereduction process.

(5) The resultsof thisstudyarein qualitativeagreement with some of the literature

where broadband

maskers were

usedanddisagreewith findingsof substantialwiderauditory filters in binaural than in monaural conditionsprimarily foundin bandlimitingexperiments. Besidesthe difficultiesin comparingresultswith differentfilter characteristics pointed out above,this disagreementmight be due to the very roughdescriptionof the ear'sspectralintegrationproperties by assuming(linear) auditory filters.Instead,the complex and nonlinear nature of the auditory systemwill require more sophisticated linear and nonlinearmodelsof spectral integration.

ACKNOWLEDGMENTS

Supported by Deutsche Forschungsgemeinschaft. Thanks to M. R. Schroeder, M. Kinkel, and A. Kohlrausch

for helpfulhints and discussions and to H. S. Colburn, B.C. J. Moore, and three anonymousreviewersfor improvinga previousdraft of the paper.Technicalassistance by L. Martens and G. Kirschmann-SchrSderis gratefully acknowledged.

APPENDIX

FILTER

A: FITTING

THE DATA

WITH

DIFFERENT

CHARACTERISTICS

To fit theoretical

threshold curves to the binaural

data

presentedin Fig. 2, we usedthe relationbetweenthe masked thresholdof an S•rprobetone (denotedasL) and the effective interaural correlationof the noisemaskerat the probe tonefrequency(denotedasr) givenby the EC theory (Durlach, 1972):

detection tasks.

(2) No definitejudgementaboutthe exactshapeof the

L = Lu -- 10 log[ (K + 1)/(K-

r) ].

(A1)

auditory filter characteristiccan be made from our data, sinceseveralfilter functions(includingthe rounded-exponentialand double-sided exponentialfunctionwith two parameterseach) yieldedcomparablegoodfitsto the data. (3) When comparingthe "effective"bandwidthof dif-

Here, LM denotesthe monauralreferencemaskedthreshold whichis assumedto equalthe NoSothreshold.The variable K is determinedby the individual'sperformancein binaural detectiontasksand canbe computeddirectlyfrom the indiferent filter characteristics fitted to similar (or even the vidual NoS•rmaskedthresholdsinceEq. (A 1) shouldpresame) setsof data, the equivalentrectangularbandwidth dict thisthresholdifris setto q- 1.The valuerin Eq. (A1) is computedby averagingthe interauralcorrelationof the bin(ERB) shouldbeavoidedbecause it depends stronglyonthe filter characteristicemployed.Instead,the 90% bandwidth aural noisemasker (which is characterizedby an edgefreis recommended,i.e., the bandwidth which encompasses quencyfe and extendsover a sufficientlywide freqencyre90% of the integratedfilter functionbelow and abovethe gion from F• to F2 ) after havingpassedthe auditoryfilter centerfrequency.In our experiment,the 90% bandwidthat centeredaroundthe frequencyfm: 1897

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with

r= r(fmfe ) =

w(fm

)'p(f'fe )df'.

(A2)

d=2[(1-c)(vu

+ Yl) -}-C(/-Iu q-t/-//)] ß

Here,p( f'f• ) denotes thefrequency-dependent interaural NotethattheparametersV•/l andla•/lfor therounded-expocorrelation of the binaural noise masker that is + 1 above

and - 1 belowthe edgefrequency f• (or viceversa).The auditoryfilterfunctionwith centerfrequency fm is denoted by w( f• --f'). Generally, f• isassumed to equalthesignal frequency of 500Hz. If off-frequency detection isaccounted for,fm is shiftedto yieldthe optimumdetectability of the signal.The followingfilterfunctions havebeenused,each beingdescribed bytwoparameters v• and¾1andnormalized suchthattherespective integralofthefilterfunctionoverthe frequencyequalsunity. ( 1) Rectangularwindow:

to(f ) : (Vl q-Vu) -1, for -- Vl q-fm