Effective masking bandwidths at low frequencies

Effective masking bandwidths at low frequencies Sanford Fidell, Richard Horonjeff, Sherri Teffeteller, and David M. Green Bolt Beranek and Newman Inc., P.O. Box 633, Canoga Park, California 91305

(Received 10 April 1981; accepted for publication 26 October 1982)

Masking of low-frequency sinusoids of varying durations by Gaussian noise of varying spectral shape was •measured in three different studies. Common solutions to technical problems associated with control and specification of low-frequency signals were used in the three studies. If interpreted in terms of Fletcher's critical ratio assumptions, data from the first study lead to the inference that the masking bandwidth is inversely related to signal frequency below about 200 Hz. Data from the second study rule out the likelihood that the apparent increase in masking bandwidth is attributable to changes in auditory integration times at low frequencies. The data of the third study suggest that the apparent increase in auditory filter bandwidth at low frequencies is more adequately explained by a decrease in observers' detection efficiency than by broadening of the filter bandpass.

PACS numbers: 43.66.Dc, 43.66.Ba [JH]

INTRODUCTION

Masking phenomena are often accounted for theoreti- cally by assuming that human observers detecting acoustic signals embedded in wideband noise employ a hypothetical first stage bandpass filter. This filter's limited bandwidth im- proves the effective signal-to-noise ratio of the detection pro- cess by allowing observers to restrict their attention to a narrow band of frequencies in the vicinity of the signal. Thus noise energy in other spectral regions does not degrade detection performance.

Two sets of estimates of the effective' width of this hypothetical filter may be distinguished at frequencies above several hundred Hertz. As discussed by Fidell et al. (1979}, the "German" estimates of "critical bands" [e.g., those of Zwicker (1952), Gassler {1954), Zwicker •1954), Zwicker and Feldtkeller (1955}, and Zwicker et al. {1957}] are about three times wider than the "American" estimates [such as those of Fletcher (1940}, Schafer et al. {1950}, Swets et al. {1962}, and Patterson (1974}, inter alia]. There is, however, reasonable understanding of the origins of these differences in terms of experimental tasks and procedures, and fair agreement among similar studies in their estimates of effective masking bandwidths.

At higher frequencies {above 1000 Hz}, the apparent bandwidth of the hypothetical first stage auditory filter seems to increase as a constant percentage of frequencies (constant Q system}. However, at frequencies below several hundred Hertz, there is much greater uncertainty about the relationship between the filter's bandwidth and signal frequency. Measurements of the masking of sinusoids by noise at frequencies below 400 Hz are sparse, as seen in Fig. 1. The empirical investigation of Hawkins and Stevens (1950}, Fletcher {1953}, and Green et al. {1959} have concentrated their measurements at frequencies above 400 Hz. In the spectral region below 400 Hz, the bandwidth of the first stage filter does not appear to remain a constant percentage of signal frequency, and may actually increase, as the two measurements at 100 Hz in Fig. 1 seem to indicate. Other interpretations of these data may also be made, as discussed later.

Unfortunately, estimates of the detectability of yet lower frequency signals must be based on extrapolation of the limited data from the region of 100 to 400 Hz. Since these limited data are the basis of inferences about the low-fre-

quency bandwidth of the hypothetical first stage auditory filter, there is considerable uncertainty about bandwidths at very low frequencies.

In addition to this empirical uncertaintly, there is also some diversity of theoretical opinion about how to model human frequency selectivity in detection tasks. Most re- searchers attribute to Fletcher the view that detection occurs

when signal power is directly proportional to the power pass- ing through the hypothetical first stage auditory filter, as described in Eq. (1):

Ps = K N (f)lH (f)12 df, (])

where Ps is the signal power at detection, N (f) is the noise power entering the ear, H {f} is a weighting function repre- senting the attenuation characteristic of the auditory filter, and K is the reciprocal of the observer's efficiency as a power detector.

Several technical obstacles make the study of masking

[ ,'- FLETCHER (1953)

O GREEN, McKEY AND LICKLIDER o HAWKINS AND STEVENS (1950)

_

--

_

o --

o o o

o o

•m 400 ß

• •00• _

15 lOO 400 1030 4000

Frequency, Hz

FIG. 1. Critical ratio estimates of effective masking bandwidths as a function of frequency.

628 J. Acoust. Soc. Am. 73 (2), February 1983 0001-4966/83/020628-11500.80 @ 1983 Acoustical Society of America 628

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at low frequencies difficult. These include the irregular low- frequency response of headphones and loudspeakers, and the need for high intensity signal presentation levels due to {1) reduced human auditory sensitivity at low frequencies, and (2) the presence of low-frequency ambient noise in most testing environments.

This paper presents results from three studies of the audibility of low-frequency acoustic signals embedded in Gaussian noise. In all three studies, signals and noise were presented to observers via a single loudspeaker in an anechoic listening environment at relatively high spectrum levels, well above the low-frequency ambient noise in the anechoic chamber.

The first study produced data on the detectability of 750-ms sinusoids at six frequencies (40, 63, 125, 250, 500, and 1000 I-Iz} from which relative masking bandwidths could be inferred by Fletcher's (1953} critical ratio method. In the second study, the sensitivity of the critical ratio assumptions to signal duration was studied by replicating the first study at six different signal durations from 50 to 1000 ms. In the third study, the masking passband shape at four frequencies (124, 252, 500, and 1000 Hz} was explored by notched noise techniques similar to those of Patterson { 1976} and Patterson and Nimmo-Smith {1980}, using a digitally generated, spectrally notched Gaussian noise masker.

Two alternative forced choice (2AFC} trial procedures were used in all testing. All observers had hearing levels within 10 dB of ISO audiometric zero. Table I compares procedural details of the three studies.

I. STUDY 1

A. Method

The goal of this study was to produce data from which apparent masking bandwidths at frequencies as low as 40 Hz

could be inferred through applications of Fletcher's (1953) critical ratio assumptions. This was done by determining signal-to-noise ratios necessary to sustain constant detection performance (d' = 1, or 76% correct in 2AFC trials} at frequencies of 40, 63, 125, 250, 500, and 1000 Hz.

Ten observers were paid an hourly rate and a bonus to detect sinusoids in blocks of one-hundred 2AFC trials. Ten

practice trials preceded each block. The signal occurred with equal probability in either the first or second interval. The 750-ms observation intervals were separated by an intratrial interval of 500 ms, and followed by response and intertrial intervals of 100 and 500 ms, respectively. All signals were gated through an electronic switch with 100-ms rise and decay times. All testing at each frequency was completed be- fore testing at another frequency. The order of presentation of different frequencies was randomized over observers. All data at a given frequency were collected from each observer during the same day.

Detection performance at each frequency was assessed at signal-to-noise ratios selected on an individual basis for each observer to span the linear portion of the psychometric function {about 60% to 90% correct detection}. Data were collected in an anechoic chamber with a cutoff frequency of approximately 100 Hz. Care was taken to fix the seated observer's head position to minimize varibility in absolute signal and noise levels associated with standing waves at lower frequencies. Measurements confirmed that head movements 'through low-frequency standing waves did not affect signal levels by more than 1 dB. Relative levels of signal and noise were little affected by spatial variability of absolute values, since both signal and noise were produced by a single loudspeaker.

Testing at four frequencies (1000, 500, 250, and 125 Hz} was conducted in continuous {analog} Gaussian background

TABLE I. Summary of experimental conditions.

Masker

Signal spectrum Signal Masker Masker frequencies level duration bandwidth duration (Hz) (dB) (ms) (Hz) (ms)

40 60 11-400

63 60 11-400 125 60 11--400

Study 1 125 40 750 11-2000 Continuous 250 40 11-2000 500 40 11-2000

1000 40 11-2000

40 60 17--400

63 55 50,100, 17--400 Study 2 125 60 200,500, 17--400 Continuofis

125 40 750,1000 17-2000 250 40 17-2000

1000 40 17-2000

124 60 35-244

252 60 42-462 Gated Study 3 500 60 250 98-902 with

1000 60 498-1502 signal 1000 40 498-1502

Number

of 2 AFC trial

observers procedure

10

Fixed blocks

of 100 trials

of constant

level

Trialwise

adaptive a

Fixed blocks

of 50 trials

of constant

level

Interleaved 2-down, 1-up and 3-down, 1-up [Levitt and Rabiner {1967) and Levitt {1970)].

629 J. Acoust. Sec. Am., Vol. 73, No. 2, February 1983 Fidell et al.: Masking bandwidths at low frequencies 629


%=••arrowband

• Wideband

7O

m 40

-• •o

•- 2o

o

0 300 600 900 1200 1500 1800 2100 2400

Frequency, Hz

FIG. 2. Spectrum level of masking background noise environments.

noise with band limits of 11 to 2000 Hz(_+ 2 dB) and a nominal spectrum level of 40 dB. Testing at three frequencies (125, 63, and 40 Hz) was conducted in Gaussian noise with band limits of 11 to 400 Hz ( q- 2 dB) and a nominal spectrum level of 60 dB• The overall level of the latter "narrow-band" background noise was approximately 87 dB, to minimize heating damage risk and observer fatigue. The overall level of the "wideband" noise was approximately 76 dB.

Figure 2 plots the spectrum levels (dB/Hz) of the two background noise conditions. Note that the two back- grounds differ only in level below 300 Hz, and that fluctu- ations in level are on the order of _ 2 dB over the frequency ranges of interest.

Figure 3 shows the low-frequency portion of the masking noise on an expanded frequency scale. Only the narrow- band (nominal 60-dB spectrum level) masking noise is shown in the figure; the wideband masking noise was identical in spectral content over the frequency range shown, but presented at a level 20 dB lower. The figure also shows that the narrow-band noise was presented at a level at least 20 dB greater than the ambient noise in the anechoic chamber at frequencies down to 15 Hz. In the wideband (nominal 40-dB spectrum level) masking noise condition, the chamber ambient may have influenced the masking noise environment by 1 to 5 dB at frequencies below 25 Hz, but the lowest frequency signal presented in this background was 125 Hz. Further detail of experimental procedures may be found in Fidell et al. (1979).

B. Results

The raw data consisted of hundreds of individual deter-

minations of detection performance (percent correct) at different signal-to-noise ratios in different signal frequency and noise bandwidth conditions. Best fitting lines to these data (percent correct detection per block of 100 trials) were ob-

Masking Noise

I

•Chamber Ambient Noise

,I

-io

-2O

0 100 200 300 Frequency, Hz

FIG. 3. Spectrum level of narrow-band masking noise and anechoic chamber ambient.

tained by calculating regression equations for points in the linear portions of the psychometric functions (between 60% and 90% correct detection) for each observer. Slopes and intercepts of the regression equations were averaged across observers at each signal frequency to yield best fitting lines.

These averaged regression equations were then used to predict the observer-averaged 76% correct performance level at each frequency. These estimated signal-to-noise ratios may be seen in Table II and Fig. 4. The 1.6-dB offset at 125 Hz between the two sets is discussed later. The general shape of the relationship between signal-to-noise ratio and frequency is consistent with that seen in Fig. 1.

II. STUDY 2

A. Method

The goal of this study was to investigate potential frequency dependence of the auditory integration time, by de-

TABLE II. Observed signal power-to-noise power density ratios needed to maintain 76% correct detection performance--study 1.

Ps /No at Ps /No at Frequency No = 40 dB No = 40 dB (Hz) (dB) (dB)

40 ß .. 14.9

63 -- ß 13.5

125 12.2 13.8

250 11.9 ...

500 14.1 .. ß

1000 15.6 .. ß

630 J. Acoust. Sec. Am., Vol. 73, No. 2, February 1983 Fidell eta/.: Masking bandwidths at low frequencies 630


19

13

12-

11--

10--

I I I I I I

O Observer Mean, NoS 40 dB El Observer Mean, N o = 60 dB I 95% Confidence Interval

40 63 125 250 500 1000

Signal Frequency in Hz

FIG. 4. Summary of signal-to-noise power ratios required for 76% correct detection performance as a function of frequency in 2AFC task--study 1.

termining whether the general shape of the signal-to-noise ratio curve in Fig. 4 depends on signal duration. Green et al. (1957) has shown that for short signal durations (less than 200 to 300 ms) the human auditory system is an equal energy detector, for which equal products of signal power and duration (Es = Ps X T) produce equal detection performance. As Es is held constant, detection performance deteriorates at longer signal durations. This suggests that above some critical duration, the auditory system is no longer capable of completely integrating all of the signal power, but must re- sort to a less efficient detection strategy. One possible strategy that can account for the data of Green is that observers divide the total signal duration into a number of consecutive subintervals. Each of these subintervals is equal in length to the maximum auditory integration time, except the last interval which is some fraction thereof. The observer assesses

the detectability (d ') of the signal in each interval indepen- dently and vector sums (d 'sum = [ •d :2 ] •/2) these detectabi- lities to make his final detection decision. To the extent that

the auditory integration time is longer at some frequencies than others, detection performance will improve. This condition would, in turn, be interpreted through the critical ratio model as a narrowing of the auditory filter.

An examination of the presence or absence of such an effect was accomplished by establishing signal-to-noise ratios for constant detection performance at frequencies of 40, 63, 125, 250, and 1000 Hz, at three durations shorter than an assumed integration time of 250 ms, and at three durations longer than this time. Signal rise and decay times were 10 ms. Data were collected in the presence of continuous Gaussian noise at nominal spectrum levels of 40 and 60 dB, as in study 1.

Five observers were paid an hourly rate plus a bonus to detect the sinusoids in 2AFC trials. A trialwise adaptive procedure [a combination of the "2 down, 1 up" and "3 down, 1 up" block up and down methods described by Levitt and

Rabiner (1967) and Levitt (1970)] was used to estimate the signal-to-noise ratio associated with 75% correct detection performance in a 2AFC trial procedure. Individual determinations of the 75% correct detection point continued until the 95% confidence interval was smaller than 0.5 dB. Sever- al such determinations were made for each observer in each

experimental condition. Further details of this study are described by Horonjeff et al. (1981).

B. Results

Figure 5 summarizes the effects of signal duration on signal-to-noise ratio necessary to maintain 75 % correct performance at different frequencies. Signal-to-noise ratios are calculated as the decibel difference between the sound pres- sure level of the sine wave at each frequency and the average spectrum level (dB per Hz)of the background noise. The data are plotted in terms of signal energy rather than power for ease of interpretation.

The data at each frequency are fitted by two regression lines of forced slope: one for the temporal region shorter than an assumed time constant of the ear of approximately 250 ms, and one for the temporal region longer than the assumed time constant. The zero slope in the temporal region below the assumed time constant was picked to reflect the belief that people act as perfect energy detectors in this temporal region. The slope of 2 dB per doubling of signal duration was

_ --

_ _•_•_1ooo .z, , = ,o d8 • / _ _.•__ o_.•.._ _/

• • = 250 Hz, N ø e -

I I I I I I

_

/ • fo = a0 H z, N ø = 60 dB /

c /

z o / m 10 - /

m // / A / • • = G3 Hz, N ø • 55 dB/ /

8 -

• • = 12S Hz, N ø = ;0 dB 7 - -- .......... • .... & -

fo = 125 Hz, N ø • •0 dB . I I I I I I ,

50 100 200 500 750 1000

Signal Duration in milliseconds

•G. 5. Ob•• me• •i• energy-to-noise •wer r•tios r•uir• For 75% ••t det•tion •o•an• • a Function of si•l duration in 2A•C •k--study 2.

631 J. Acoust. Soc. Am., Vol. 73, No. 2, February 1983 Fidell et al.: Masking bandwidths at low frequencies 631


picked on empirical grounds for the temporal region longer than the assumed time constant. The average slope of least- square regressions at each frequency in the temporal region above 250 ms was 2 dB. No slope differed significantly from 2 dB per doubling of duration, nor was any dependence of slope on signal frequenct apparent; hence, a pooled estimate of the slope was used.

A slope of 2 dB per doubling of duration implies that people are neither perfect energy detectors (which would have a slope of zero in Fig. 5), nor are they completely insen- sitive to signal duration (as would be reflected by a slope of 3 dB per doubling of duration). Instead, people can make par- tial use of duration information. One model of human use of

duration information which would predict a slope similar to that observed is that people base their overall detection decision for long duration signals on a summation of successive independent decisions. A statistical summation (square root of the sum of the squares of independent decisions made at regular intervals equal to the ear's time constant) would produce a slope of 1.5 dB per doubling of duration.

An estimate of the integration time for each fo, No condition may be made by examining the signal duration at which the line segments for the two listening regimes inter- sect. These durations, plotted as a function of frequency in Fig. 6, show no apparent dependence on signal frequency (r 2 = 0.073). The range of 200 to 400 ms, however, is comparable to that observed by Green et al. (1957) at a frequency of 1000 Hz. There is some evidence in Fig. 6 to suggest that integration time is inversely proportional to No---the mean estimated integration times in the high (No = 55-60 dB) and low (No = 40 dB) masker levels are 273 and 355 ms, respectively, which are significantly different at the 95% confidence level.

Figure 7 displays the results of the second study in an identical format to that of Fig. 4. Each data point represents an average over all observers and three signal durations (500, 750, and 1000 ms), and was obtained by reading the predicted value from the dashed line in Fig. 5 at 750 ms and adding a constant (10 log [1/0.75]) to convert from E•/No to P•/No. The inclusion of the data from the additional signal durations is responsible for reducing the 95% confidence intervals over those shown in Fig. 4 from study. 1. The larger confidence interval in Fig. 7 at 125 Hz (No = 60 dB) results from testing only at 100- and 750-ms signal durations for this

• 4oo ._

• •300

"' 200

'o

i I i 40 63 125

O •

•] N = 60 dB o

(• N = 55 dB o

I O N = 4o dB o

250 1000

Frequency (Hz)

FIG. 6. Estimated auditory integration time as a function of frequency.

I • I i I i

• 18 ß Observer Mean, No• 40 dB _

• ß Observer Mean, NoW 55 dB ._

• ß Observer Mean, N • 60 dB • 17 o _

-o I 95% Confidence Interval e- 16

.;

z o ß ,

•n 12

• lO

I I I I

,/

I

1000 40 63 125 250 500

Signal Frequency in Hz

FIG. 7. Summary of signal-to-noise power ratios required for ?6% correct detection performance in 2AFC taskmstudy 2.

condition. The similar general shape of the relationships shown in Figs. 4 and 7 is apparent from inspection.

III. STUDY 3 A. Method

The goal of this study was to determine whether the equivalent rectangular auditory filter bandwidths inferred from the critical ratio experiments were consistent with those derived from direct measurements of auditory filter shape, particularly at lower frequencies. Inferences of bandwidth from the critical ratio experiments through the model of Eq. (1) rest entirely on the assumption that K, the reciprocal of processing efficiency, is invariant with frequency. By the very nature of the critical ratio experiment, processing efficiency and bandwidth (as well as filter shape) are inher- ently confounded. At high frequencies the critical ratio data are orderly and monotonic with frequency, thus providing little evidence to either support or reject the constant K hy- pothesis. At low frequencies, however, the inferred bandwidths (assuming constant K ) become implausibly large; by some estimates, the passband centered at 40 Hz extends down to dc.

If one holds steadfast to the constant K position, there is little alternative [under the model of Eq. (1)] to accepting a highly asymmetric low-frequency passband. If, on the other hand, passband asymmetry at low frequencies is as small as that observed at mid frequencies (Patterson and Nimmo- Smith, 1980), then one must favor the notion that K is frequency dependent. To resolve this issue two experiments in- volving the detection of sinusoids in a spectrally notched noise masker were performed. In the first experiment, observers were asked to detect sinusoids at 124, 252, 500, and 1000 Hz in the presence of symmetrically notched noise {a condition in which the upper and low band edges of the notch are equidistant from the tone). The shape of an assumed symmetric auditory filter may be determined {Patter- son and Henning, 1977) by obtaining signal levels for con-

632 d. Acoust. Sec. Am., Vol. 73, No. 2, February 1983 Fidell eta/.: Masking bandwidths at low frequencies 632


stant detection performance for a range of masker notch widths.

To investigate filter asymmetry, a second experiment was performed using only the 124-Hz signal. In this experiment, additional data were collected with the masker notch center frequency offset from the frequency of the tone. An observer in an offset-notch condition is encouraged to listen at a frequency not exactly coincident with the signal frequency because in so doing, the signal-to-noise ratio at the output of the auditory filter can be maximized. Although some signal power is lost by listening at a frequency between that of the signal frequency and that of the center of the notch, the noise power in the same band is reduced to an even greater extent, depending on the shape of the filter skirts.

The principal interest in the current study was the degree of potential asymmetry of the auditory filter, a matter affecting estimates of equivalent rectangular bandwidths. This issue was studied by making determinations of masked detection levels with the notch in the masking noise offset the same number of Hertz lower and higher in frequency from the signal frequency. If the measured detection levels are constant when the notch in the masking noise is displaced equally (both lower and higher} in frequency from the signal frequency, it can be concluded that the filter is symmetric. Limited amounts of data of this sort, although suffi- cient to test for the symmetry of the auditory filter, do not fully define its shape.

In both experiments, band limited Gaussian noise bursts of 250-ms duration were generated digitally by con- verting 12-bit signal magnitudes to voltages at fixed periodic rates, and low-pass filtering the time varying voltage at a frequency equal to 0.32 that of the conversion rate. Since the voltage values were precomputed by summing closely spaced sine and cosine waves of random amplitude and spaced at intervals given by the reciprocal of signal duration, the edges of the resulting noise bands were virtually infinite- ly sharp. The same software also digitally added the sinusoi- dal signal to the noise waveform to ensure that the signal frequency was precisely positioned with respect to the masker notch. Corrections for the frequency response of the entire reproduction system were included in the software that generated the noise bursts.

Data were collected in blocks of 50 fixed signal-to-noise ratio, two alternative forced choice trials, preceded by ten practice trials per block. Low level (approximately 10 dB/ Hz), broadband Gaussian noise was continuously present throughout the experiment. The digital noise bursts were identical in duration to the two observation intervals, which were separated by 500-ms intratrial intervals and followed by 1000-ms response and feedback intervals. Figure 8 shows a representative frequency spectrum (1-Hz resolution} for a typical notched noise burst and signal. The two 100-Hz-wide noise sidebands are displaced 50 Hz in frequency from the tone center frequency of 400 Hz. As seen in the figure, the spectrum level of the noise bursts in this experiment was nominally 60 dB. B. Results

Figure 9 displays the results of the first experiment in this study in a format similar to that of Figs. 4 and 7. The

633 J. Acoust. Soc. Am., Vol. 73, No. 2, February 1983

7O

• 50

c 40

=• 30

20

! I I I I

ß Signal Power T Signal-

to- Noise ' Ratio

• • NoWise I I Spectrum

I I i I i

200 400 600

Frequency in Hz

FIG. 8. Representative frequency spectrum of notched noise burst and sinu- soidal signal.

data points represent signal-to-noise ratios necessary to maintain 85% (d' - 1.5)correct detection performance in an unnotched masking noise for a single observer. Limited data from other observers (not shown) at several frequencies agreed closely with those seen in Fig. 9. The general shape of the relationship between signal-to-noise ratio and frequency in Fig. 9 resembles those of Figs. 4 and 7, but the absolute signal-to-noise ratios are far higher in Fig. 9. Some of the difference in signal-to-noise ratios needed to maintain con-

25

= 24

+

c o• 21

.--

' 20

.• 19

-o 18

.c_

z ø ... 17

Observer 2, N O -•60 dB

I I

I I

I I

I I

40 63 125 250 500 1000

Signal Frequency in H z

FIG. 9. Summary of signal-to-noise power ratios required for 85% correct detection performance as a function of frequency in 2AFC task--study 3.

Fidell eta/.: Masking bandwidths at low frequencies 633


2C

'10

ø

25 I I I

0 Observer 2

Observer 6 ...... /• Observer 7

I I I I

0 0.1 0.2 0.3 0.4 0.5

Relative Notchwidth (A f/fo)

FIG. 10. Detection performance as a function of noise notch width for sinu- soidal tone at 124 Hz.

stant detection performance in study 3 and in the other two studies is attributable to differences in the criterion level of

performance, some is due to differences in signal duration, and some is due to differences in trial procedures (continuous versus gated noise maskers).

Figures 10 through 13 display the detection data from the notched noise masker conditions of the first experiment in study 3. The figures plot the signal power to noise power density ratio as a function of normalized notch width (half

I I I I

Observer 2 Observer 5

Observer 6 Observer 7

i I I i

0 0.! 0.2 0.3 0.4 0.5

Relative Notchwidth (A f/fo)


20

._

• 10

o

o•O 5

o 0 Z

0 Observer 2

I I i I 0 0.! 0.2 0.3 0.4 0.5

Relative Notchwidth (z• f/fo )


the notch width divided by the center frequency), with all notches symmetrically positioned about the frequency of the signal to be detected. The empirical findings summarized in Figs. l0 through 13 support estimates of auditory filter shapes at low frequencies derived by assumptions and procedures noted by Patterson (1976).

Figure 14 plots the assumed symmetric filter shapes inferred from symmetric notch detection data. A rounded ex- ponential of the form

IH(g)l 2 -(1 - r)(1 +œg)e -• + r (2) was used to approximate the shape of the auditory filter, where the filter gain IH ( g)l 2 is described as a function of the

I I I I

• 20

o--

E •0

o,o

"' 0

Observer 2, N O = 60 dB

Observer 2, N O = 40 dB

0 0. l 0.2 0.3 0.4 0.5

Relative Notchwidth ( A f/f o)


634 J. Acoust. Soc. Am., Vol. 73, No. 2, February 1983 Fidell eta/.' Masking bandwidths at low frequencies 634


i F ø = 12q Hz - 252 Hz 0 z - 5 •

• J Observer •1•'• • •3 'ø I- • /.•-• Observer 6 '•,•,. - ,o -

15 15 •

20 20

.6 .5 .q .3 .2 .1 0 .1 . 2 .3 .q . 5 . 6 .q .3 .2 .1 0 .1 .2 .1 .q .3 .2 .1 0 .1 .2 .3

Normalized Frequency f/fo Normalized Frequency •/fo Normalized Frequency f/fo FIG. 14. Inferred auditory filter shapes.

normalized frequency deviation from the passband center, g = IAf/fol . The two-parameter fit {p,r} of Eq. (2} is a simpli- fication of the three-parameter expression employed by Pat- terson and Nimmo-Smith ( 1980}. Patterson's use of an additional parameter helped control the slope of the filter skirts below the 20-dB down point--a region not explored in this study. In Eq. (2), the parameter p determines the passband and is inversely proportional to the equivalent rectangular bandwidth by

BWer = 4(fo/p). (3)

Use of the normalized frequency g implies that filters of equal "p" are of constant percentage bandwidth. As a point of comparison the value ofp for a filter of equivalent 1/3- octave bandwidth is 17.33. The term r puts a lower limit on the filter skirt• and effectively establishes the dynamic range of the filter. At g = 0 the filter has unity gain.

In the power spectrum model of masking, the masker power (to which the signal power at detection is assumed to ' be proportional) may be determined by integrating Eq. (2) over the unnotched portion of the masking noise. The integral of Eq. (2) is

-- (r -- l)p- '{2 + pg)e - •'• + rg. (4)

With Eq. (4) substituted for the integral in Eq. (1), and divid-

ing both sides by No, the symmetric threshold data may be fitted with a curve of the form

b

Ps( g)/No = 2Kf0 [ -- (r -- 1 }p-](2 + pg)e -•'g + rg], (5) a

where a and b are the normalized lower and upper frequency bounds of the flanking bands of noise. As a practical matter the upper bound b is limited to 0.8 to keep the integral bounded. Otherwise an infinite amount of noise power would be passed by the filter skirts (dictated by r).

An iterative least-squares-fit method was used to determine values ofp and r for individual observer's data at each frequency. The optimalp and r values were those that minimized the sum of the squared residuals between observed and predicted signal-to-noise ratios over all notch conditions. Residuals were then minimized with respect to K to obtain a value for reciprocal detection efficiency. Table III summarizes the results of the fitting procedure.

Figure 15 displays data for the second experiment of study 3: signal detection in notched noise asymmetrically positioned with respect to the signal frequency { 124 Hz). The ordinate of the figure is identical to those of Figs. 10 through 13. The abscissa differs slightly in that data points in the two halves represent experimental conditions in which the center frequency of the notch is less than or greater than the signal

TABLE lll. Filter fit parameters--study 3.

1-parameter fit Frequency No Subject p K BWer

2-parameter fit r K BWer (dB) (dB) (Hz)

124 60 2 11.171 2.678

6 11.190 3.520

7 7.978 3.430

252 60 2 18.661 1.546 5 16.506 2.114

6 20.482 --0.278

7 12.033 --1.262

500 60 2 24.761 -- 0.043

1000 60 2 30.102 3.149

1000 40 2 36.479 1.782

44.4

44.3

62.2

54.0

61.1

49.2

83.8

80.8

132.9

109.7

11.170 -- 161.38 2.678 44.4

12.387 --21.76 4.003 40.0

12.421 --13.45 4.975 39.9

18.661 --143.89 1.546 54.0 16.487 --134.30 2.101 61.1

23.662 -- 27.21 0.710 42.6

19.856 --16.42 0.861 50.8

24.642 --122.49 --0.10• 81.2

30.149 -- 123.44 3.173 132.7

635 J. Acoust. Soc. Am., Vol. 73, No. 2, February 1983 ' Fidell et aL: Masking bandwidths at low frequencies 635


frequency. The points connected by solid lines replot the average signal-to-noise ratios for observers 2 and 6 in Fig. 10. The points connected by the dashed lines are also the averaged values for observers 2 and 6, collected with the notch center displaced from the signal frequency by 12 Hz.

(I AfL -- Afu I/fo - o. 194).

It is apparent from the figure that the differences along the' ordinate between the points connected by solid and dashed lines are slightly greater on the right half of the abscissa then on the left half. This observation suggests a minor asymmetry of the auditory filter passband. For most practical purposes this degree of asymmetry is small, but its direc- tion is consistent with that observed by Patterson and Nimmo-Smith (1980)at 2 kHz.

IV. DISCUSSION

Comparison of Figs. 1, 4, 7, and 9 reveals a general similarity of all low-frequency detection data. In particular, the relationship between signal-to-noise ratio and frequency is nonmonotonic in all data sets, with a broad minimum in signal-to-noise ratio in the vicinity of 250 Hz. The difference in signal-to-noise ratio necessary to maintain constant detection performance at frequencies of 1 kHz and 250 Hz is approximately 3.5 dB in all of the current data, a figure some- what greater than that of the older studies summarized in Fig. 1. The comparable difference between 1 kHz and 40 Hz is less than 1 dB.

Assuming {as Fletcher did) that K in Eq. {1) is unity at all frequencies, and that the "critical band" at 1 kHz is roughly 63 Hz {the antilogarithm of the signal-to-noise ratio of 18 dB at 1 kHz}, the critical ratio method leads to the inference that the critical band is almost as wide at 40 Hz as

at 1 kHz. If one accepts the more direct, notched noise bandwidth estimates [Patterson {1976}, Patterson and Henning (1977}, Weber (1977}] of 100 Hz or wider at 1 kHz, then the

critical ratio inferences of bandwidth at 40 Hz become even wider.

Clearly, the critical ratio method (with constant K} leads to unreasonable critical bandwidths at 40 Hz if the

shape of the auditory filter is even approximately symmetric (as supported by Fig. 15}. Furthermore, noise in the frequency region below 20 Hz at spectrum levels of 60 dB cannot credibly be claimed to contribute to masking at frequencies an octave higher. Thus, to maintain a belief in the power spectrum model of masking [cf. Eq. (1}], one is forced to accept that K is frequency dependent.

Figure 16 contains information which suggests that K is in fact frequency dependent. The middle panel of the figure plots the inferred equivalent rectangular bandwidth of each observer's auditory filter as a function of frequency. These data were plotted from the results of the third study (see Table II}. Note that the data are orderly, and that the relationship between inferred bandwidth and frequency decreases monotonically, with no suggestion of a minimum in the region of 250 Hz. The lowermost panel of Fig. 16 plots K of Eq. (1 } in decibels, as a function of frequency. These data were also derived from the results of study 3 {see Table III}. The top panel of Fig. 16 contains three sets of estimates of signal-to-noise ratios needed to maintain constant detection performance as a function of signal frequency. Two of the sets are simply replottings of Figs. 4 and 7 {results of studies 1 and 2, respectively}. In the 40-dB masking noise spectrum level conditions, the shapes of these two data sets are nearly identical { -I- 0.2 dB}. The small differences {less than 2 dB) in absolute levels of signal-to-noise ratios in the two data sets are almost certainly attributable to minor differences in the data collection procedures of the two studies.

The other curve in the top panel of Fig. 16 is a prediction derived by adding the curves in the two lower panels together. The shape of the fit of the predicted to the observed data is very close, except for an anomalous point at 500 Hz. The displacement of the predicted from the observed data is

25

o 0.4

A•LI afu llfL gfu

I

o 0.0 _

• O. 194

Normalized Frequency Displacement of Nearest Noise Band Edge to Tone Center Frequency

0.4

FIG. 15. Average detection results for two observers for 124-Hz sinus-

oid in bandlimited Gaussian noise

(spectrum level = 60 dB).

636 J. Acoust. Sec. Am., Vol. 73, No. 2, February 1983 Fidell et aL: Masking bandwidths at low frequencies 636


2O

o z

16

• 15

,• 20

!

#!

OBSERVER 2 •O OBSERVER 5 / • OBSERVER 6 •

OBSERVER 7 • 0 •

OBSERVER

OBSERVER

OBSERVER

OBSERVER

0 • 0

• ••Oll • 124 252 500

Auditory Filter Center Frequency in Hertz

FIG. 16. Auditory filter bandwidth and reciprocal detection efficiency as a function of frequency.

attributable in large part to differences in the detection tasks in studies 1 and 2 and in study 3' notably, different criterion levels of detection performance, continuous versus gated presentation of signals and maskers, and signal duration. It is believed that these differences do not materially affect the estimates of the auditory filter bandwidth at low frequencies that are found in Table IlI,'but do in fact have considerable impact on the derived value of K.

To avoid complicating the discussion of Fig. 16, no mention has been made of data collected at the nominal 60-

dB masking spectrum level (as plotted in Figs. 4 and 7). These low-frequency data were not discussed because the 20- dB difference in masking spectrum level was associated with a 1.5-dB difference in signal-to-noise ratio necessary to maintain constant detection performance at 125 Hz. One plausible explanation for this difference may be a widening of the auditory filter bandwidth (or perhaps a decrease in efficiency) with absolute level, as noted at higher frequencies by Weber (1977) and Reed and Bilger (1973). The magnitude of the effect observed in studies 1 and 2 is, however, some-

what greater than the observed by Weber (1977) and Reed and Bilger (1973).

V. CONCLUSIONS

The present data support the following summation: (1) Earlier findings of a nonmonotonic relationship

between frequency and critical ratio (and hence, inferred masking bandwidth) are confirmed.

(2) No evidence was found to suggest that the auditory integration time is a function of frequency.

(3) Detection performance of human observers listening for acoustic signals embedded in noise can be modeled as a simple energy integration for signal durations less than about a quarter of a second, even at frequencies as low as 40 Hz.

(4) The shape of the critical ratio versus frequency relationship is not a function of signal duration (for signals less than 1000 ms).

(5) The symmetric shape of the hypothetical auditory filter is similar at low and high frequencies.

(6) The equivalent rectangular bandwidth of the auditory filter, as inferred from detection performance in spectrally notched noise, decreases monotonically to at least 124 Hz.

(7) The nonmonot0nicity in critical ratio data is prob- ably attributable to a loss of processing efficiency at low frequencies.

ACKNOWLEDGMENTS

Portions of the research and analyses described in this article were sponsored by the U.S. Army Research Office, Tank-Automotive Research and Development Command, and Aviation Research and Development Command. The article is based in part on BBN Reports 4006, 4421, and 4435. The authors thank Suyeo Tomooka for his assistance in all aspects of laboratory procedures and instrumentation. Dr. Roy Patterson's comments on an earlier draft of this manuscript were responsible for reinterpretation of the present findings. His assistance and that of Dr. I. Nimmo-Smith in preparing the filter fit parameters of Table III are grateful- ly acknowledged.

:The terms "effective" and "apparent" bandwidth are used to refer to obser- vable characteristics of the auditory system, without regard to hypothetical constructs of auditory signal processing that may account for detection performance.

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Effective masking bandwidths at low frequencies