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Suggested Formulae for Calculating Auditory-filter Bandwidths and ...

Suggested Formulae for Calculating Auditory-filter Bandwidths and ...

Suggested Formulae for Calculating Auditory-filter Bandwidths and

Suggested formulae for calculating auditory-filter bandwidthsand excitation patternsBrian C. J. Moore and Brian R. GlasbergDepartment of Experimental Psychology, Unioersitj• of Cambridge, Downing Street, Cambridge CB2 3EB,England{Received 3 December 1982; accepted for publication 22 April 1983)Recent estimates of auditory-filter shape are used to derive a simple formula relating theequivalent rectangular bandwidth (ERB) of the auditory filter to center frequency. The value ofthe auditory-filter bandwidth continues to decreasecenter frequency decreases below 500 Hz.A formula is also given relating ERB-rate to frequency. Finally, a method is described forcalculating excitation patterns from filter shapes.PACS numbers: 43.66.Ba, 43.66.Dc [JH]INTRODUCTIONThe peripheral auditory system has often been modeledas a bank of overlapping bandpass filters (Fletcher, 1940; seePlomp, 1976 and Moore, 1982 for reviews). This concept isclosely related to the concept of the critical band {Zwicker etal., 1957}, and many workers would consider that the criticalbandwidth as measured in a variety of experiments {Scharf,1970}, is related to the bandwidth of the auditory filter at agiven center frequency (e.g., Plomp, 1976, p. 10; Moore,1982}. However, the critical bandwidth may also be consideredas a purely empirical phenomenon, for example as "thatbandwidth at which subjective responses rather abruptlychange" (Scharf, 1970, p. 159}. Classical estimates of the valueof the critical bandwidth as a function of frequency (thecritical-band function} indicate that its value is approximatelyconstant below 500 I-Iz, but increases above that frequencyroughly in proportion with center frequency {Zwicker et aL,1957; Scharf, 1970; Zwicker and Terhardt, 1980}. However,there have been comparatively few direct estimates of criticalbandwidth below 500 Hz, and indirect estimates based onthe critical ratio may be in error, since the efficiency of thedetector mechanism following the auditory filter appears tochange with center frequency {Patterson et aL, 1982; Fidellet al., 1983}.Recently several groups of workers have estimated theshape of the auditory filter, using the notch-noise methoddescribed by Patterson { 1976) or the rippled-noise method ofHoutgast {1977). The filter shape has been estimated using alinear model of masking in which it is assumed that thethreshold corresponds to a constant signal-to-masker ratioat the output of the auditory filter. In this paper we present asummary of these results which shows that.the equivalentrectangular bandwidth {ERB) of the auditory filter continuesto decrease below 500 Hz. If the critical bandwidth isrelated to the ERB of the auditory filter, this may indicate aneed to revise the classical critical-band function.I. ESTIMATES OF AUDITORY FILTER SHAPE AND ERBA. The notch-noise methodIn this method the threshold of a sinusoidal signal ismeasured as a function of the width of a spectral notch in anoise masker; the notch is arithmetically centered at the signalfrequency (Patterson, 1976). This method has been usedby Patterson {1976), Weber (1977), Moore and Olasberg(1981}, Patterson et al. (1982), Fidell et al. (1983), Glasberg etal. (1983}, and Shailer and Moore (1983). Patterson et al.(1982) suggested a method of deriving the filter shape whichassumed that the filter had the formW(g) = {1 -- r)(1 +pg•e -• + r, (1)where g is the deviation in frequency from the filter centerfrequency divided by the center frequency, and p and r areadjustable parameters. The parameter p determines theshape of the passband of the filter, while ß determines thepoint at which the passband gives way to the shallower"tails" of the filter. The filter shape is derived by fitting theintegral of Eq. (1) to the data relating threshold to notchwidth. For full details see Patterson et al. {1982}. For thisfilter shape the ERB expressed as a proportion of center frequencyis approximately equal to 4/p.We have derived values ofp and r, and hence estimatesof ERB, for all studies where more than one frequency wasused, and more than one subject was tested. Results areshown in Fig. 1. All the data presented in the figure wereobtained at a noise spectrum level of 40 dB excepthose ofFidell et al. { 1983) which were obtained at a spectrum level of60 dB. At middle to high frequencies the ERB tends to increasewhen the spectrum level is increased above a certainlevel (about 40 dB; see Weber, 1977), but we do not knowwhether this happens at low frequencies, or at what noisespectrum level it occurs; the level would probably be higherthan at high frequencies, owing to the transmission characteristicsof the middle ear. The continuing decrease in theERB as frequency decreases below 500 Hz would, if anything,be more marked if data at low spectrum levels wereavailable. The data shown in Fig. 1 cover a range of frequefi-cies from 0.124 to 6.5 kHz.B. The Hppled-noise methodThis method is fully described by Houtgast (1977).Glasberg et al. (1983), in a direct comparison of the notchnoisemethod and the rippled-noise method, found thatERBs estimated from the latter (using Houtgast's method of750 J. Acoust. Soc. Am. 74 (3), September 1983 0001-4966/83/090750-04500.80 (D 1983 Acoustical Society of America 750nloaded 10 May 2010 to 200.39.9.109. Redistribution subject to ASA license or copyright; see http://asadl.org/journals/doc/ASALIB-home/info/terms.j

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