predicting frequency selectivity in forward masking from simultaneous masking

Predicting frequency selectivity in forward masking from simultaneous masking

Robert A. Lutfi b) Auditory Research Laboratory, Northwestern University, Evanston, Illinois 60602

{Received 12 December 1983; accepted for publication 9 April 1984}

Measures of frequency selectivity from forward masking suggest sharper tuning than those from simultaneous masking. To account for this result, various interpretations involving additional tuning mechanisms have been propose d. In the present study, it is shown that a simple multiplicative relation between on-frequency forward and off-frequency simultaneous masking predicts this result quite well. The relation assumes that changes in masking produced by separating the masker from the signal in frequency, in time, and in both frequency and time are related to one another by Weber's law. The accuracy of the predictions suggests that the limits of auditory frequency selectivity are already established in simultaneous masking and that special interpretations involving additional tuning mechanisms are not required to account for the difference between simultaneous and forward measures. Implications for a dB scale of masking are discussed.

PACS numbers: 43.66.Ba, 43.66.D½, 43.66.Fe [FLW]

INTRODUCTION

What are the limits of auditory frequency selectivity? Since the earliest quantitative study by Wegel and Lane { 1929}, investigators have refined and quantified measures of simultaneous masking in an attempt to answer this question. Now there is some question as to whether or not these measures reveal the full extent of auditory tuning; measures of frequency selectivity from forward masking suggest that the system is more sharply tuned (Houtgast, 1974; Rodenburg et al., 1974; Wightman et al., 1977; Vogten, 1978; Moore, 1978; O'Loughlin and Moore, 1981; Moore and Glasberg, 1982; Weber and Patterson, 1983}. In all of these studies, the common measure of frequency selectivity is the psychophysical tuning curve {PTC}. This curve gives the level of the masker at each frequency required to mask a fixed-level signal. The finding that PTCs are sharper in forward masking than in simultaneous masking is largely responsible for a contempo- rary reevaluation in psychoacoustics of both the limits and mechanisms of auditory frequency analysis.

Various qualitative interpretations of the apparent difference in tuning have been proposed. Some of these interpretations involve additional tuning mechanisms such as a second filter (Duifhuis, 1976}, suppression {Houtgast, 1974; Wightman et al., 1977}, or additional filtering over time {El- liot, 1969; Weber, 1983}. Others involve processes such as off-frequency listening {O'Loughlin and Moore, 1981; Moore and Glasberg, 1981} or cueing (Terry and Moore, 1977; Moore, 1978} which, in effect, act as additional tuning mechanisms. At their current stage of development, a defini- tive test between one or the other of these interpretations would be quite difficult. Indeed, confusion can arise simply because terms, such as "suppression" or "off-frequency listening" are often used interchangeably to describe both the effect and its interpretation {e.g., O'Loughlin and Moore,

An earlier version of this manuscript was presented at the 105th meeting of the Acoustical Society of America [R. A. Lutfi and D. J. Kistler, J. Acoust. Soc. Am. Suppl. 1 73, S44 {1983}]. Current address: Signal Detection Laboratory, Central Institute for the Deaf, 818 South Euclid Avenue, Saint Louis, MO 63110.

1981; Fastl and Bechly, 1983}. Nonetheless, all of these interpretations contain an implicit assumption; they all assume that at least two separate frequency selective processes are necessary to account for the difference between PTCs obtained in forward and simultaneous masking. If it were shown that a single frequency selective process could quanti- tatively account for this difference, then special interpretations involving additional tuning mechanisms might be con- sidered premature.

In the present study, we replicate the result of sharper PTCs in forward masking and show that a simple multiplica- rive relation between forward and simultaneous masking predicts this result quite well. The degree of sharpening is determined by a single frequency selective function obtained in simultaneous masking. This suggests that the limits of auditory frequency selectivity are already established in simultaneous masking and that additional tuning mechanisms are not required to account for sharper PTCs in forward masking. I. THE RELATION AND ITS DERIVATION

Consider the masking of a 2.0-kHz sinusoidal signal by a simultaneous narrow-band noise masker centered at 2.0

kHz. We refer to this as on-frequency simultaneous masking. Here, masked threshold {Ps} is related to masker intensity {M} over a large range, by a constant Weber's fraction, Ps = kM, where k is approximately 1. Thus, if the masker is attenuated by 3 dB, it produces one-half as much masking. If it is attenuated by 5 dB, it produces one-third as much masking. And overall, if it is attenuated by 3 + 5 = 8 dB, it produces one-sixth as much masking; the product of 1/2 and 1/3. Now, consider the effect of moving the masker away from the signal so as to precede the signal in time. The new masker might produce some fraction g of the original masking. Similarly, if the masker is moved away from the signal in frequency, it might produce some fraction h of the original masking. What is the overall reduction in masking when the masking is moved away from the signal both in frequency and time? For a constant Weber fraction, we might expect that, as before, the fractional amount of masking will be giv-

1045 J. Acoust. Soc. Am. 76 (4), October 1984 0001-4966/84/101045-06500.80 ¸ 1984 Acoustical Society of America 1045

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en by the product ofg and h. This is precisely the reasoning that underlies the predicted relation between forward and simultaneous masking.

In our example, masked threshold for the signal is a function of the time interval (t) between the signal and the masker, the frequency interval {f) between the signal and masker, and masker intensity (M ), Ps - F (tf , M ). For f- 0, we can express masked threshold as a function of t and M alone, Ps = G (t•/}. We refer to this as the on-frequency forward masking function. Similarly, for t = 0, masked threshold can be expressed as a function of f and M alone, Ps = H {f,M }. This is referred to as the off-frequency simulta-

neous masking function. Returning to our previous logic, we wish to express the effect of moving the masker away from the signal in frequency and/or time as a fraction of the on- frequency simultaneous masking, kM. Thus, for any given masker level, the effect of moving the masker away from the signal in time is G/kM. The effect of moving the masker away from the signal in frequency is H/kM. And, for a constant Weber's fraction, the effect of moving the masker away from the signal both in frequency and time is

F G H ß

kM kM kM

Multiplying through by kM, the resulting prediction for the off-frequency forward masking function is

F (tf•l) = G (t,M)H (f,M)/kM. (1)

Quite simply, Eq. (1) predicts that for a given t and M, the off- frequency forward masking is a constant fraction of the off- frequency simultaneous masking. Another way to say this is that, for a given masker level, the dB difference between off- frequency forward and off-frequency simultaneous masking should be equal to the dB difference between on-frequency forward and on-frequency simultaneous masking.

This is certainly true of the PTC since each curve re- flects only one level of masking, that is, one value on F. How then might Eq. (1) predict sharper PTCs in forward masking? The answer lies in the fact that, for the PTC, masker level covaries with masker frequency. According to Eq. (1),

M cc G (t,M )H ( f ,M ).

Since masker level is also an argument of G, we suspect that forward masking PTCs reflect a conjoint dependence of G on M and ofH onf Note that H is the only function of frequency in this expression. The dependence of H on f is the frequency selectivity we wish to measure, but, it is the dependence of G on M that allows for a sharpening effect in forward masking.

II. METHOD

In this experiment the signal is a 10-ms, 2.0-kHz sinus- oid. The masker is a 200-ms, 50-Hz wide narrow-band noise with variable center frequency. The difference between the signal frequency and the center frequency of the masker defines f The time interval between the offset of the masker and the offset of the signal defines t. Masked thresholds for the signal were measured as a function ofmasker intensity to obtain four types of masking functions; (1) on-frequency simultaneous, t = 0 ms, f= 0 Hz, (2) on-frequency forward,

t = 15 ms, f= 0 Hz, (3) off-frequency simultaneous, t = 0 ms,variable f, and (4} off-frequency forward, t = 15 ms, vari- abler To obtain predictions, we estimate the on-frequency forward (G} and off-frequency simultaneous (H} masking functions by linear least squares fits to mean threshold estimates plotted on dB-by-dB coordinates. That is,

G = cM d and H--aM b,

where, the values of a and b are determined for each value of f, and the values c and d are determined by the value oft = 15 ms. The estimated functions are then substituted into Eq. ( 1 } to obtain the resulting prediction for the off-frequency forward masking function,

F = cM • aM •/kM = (ac/k)M • + •- •.

Expressing this as a prediction of masked threshold in dB, we have

101ogPs=(b+d- 1}X 101ogM +(•1 +C--K},

where, on the selected coordinates

•1 - intercept of off-frequency simultaneous masking function, 10 log a

b = slope of off-frequency simultaneous masking function

C- intercept of on-frequency forward masking function, 10 log c

d = slope of on-frequency forward masking function

K = Weber's fraction in dB, 10 log k.

A. Subjecta

A total of eight university students between the ages of 18 and 30 years participated in the experiment. All reported normal hearing and all received at least 2 h of training prior to data collection.

B. Procedure

Signal level was varied adaptively in a one-up, two- down, 2IFC procedure (Levitt, 1971). Initial step size (6 dB) was changed to 2 dB after the first two reversals. Each threshold estimate is the average of the last 16 of 18 reversals in a run. Masker configuration and masker level were fixed for each threshold estimate. An entire growth of masking function was obtained for a single masker configuration before proceeding to the next. For six of the subjects, a single threshold estimate was obtained in each condition of the ex-

periment; for the remaining two subjects, two estimates were obtained, one for each ear. Thus altogether ten threshold estimates were obtained for each condition. Each datum rep- resents the mean of eight threshold estimates after the high- est and lowest estimate for each condition were excluded.

Note, all estimates contribute to this datum even though some are not averaged in the mean. The largest standard error was less than 3 dB. Further details of the procedure can be found in Lutfi (1983).

1046 J. Acoust. Soc. Am., Vol. 76, No. 4, October 1984 Robert A. Lutfi: Predicting forward masking 1046


C. Stimuli

The signal and masker were digitally generated and played over separate 14 bit DACs each at a 10-kHz rate. The noise maskers were computed by convolving a file of Gaus- sian noise with the impulse response of a O-phase, 50-Hz wide rectangular filter (variable center frequency). The noise maskers were randomly sampled, for each observation interval, from a 3-s segment of this file. The center frequencies of the noise maskers were 1.4, 1.6, 2.0, 2.2, and 2.4 kHz. All stimuli were shaped with 5-ms Kaiser onset and offset ramps. The output of each DAC was low-pass filtered (120 dB/oct) at 4 kHz. Programmable attenuators permitted computer control of the levels of all stimuli. After mixing, the stimuli were amplified and presented to subjects over TDH-49 headphones. Subjects listened in a IAC double-wall sound attenuation chamber.

III. RESULTS

Figures 1 and 2 give the simultaneous (squares) and forward (circles) masked thresholds for the different values of• the actual masker frequencies are indicated in the upper left- hand corner of each plot. The on-frequency masked thresholds are given in Fig. 1 and the off-frequency masked thresholds in Fig. 2. The solid lines represent the corresponding masking functions. To avoid floor effects that might affect the goodness of a linear fit, we only included masked thresholds that were at least 6 dB above quiet threshold. The linear fits provided an excellent approximation to these data; r e was never less than 0.96.

The dashed lines give the predictions generated from Eq. {2). The value K in Eq. {2) was estimated by setting it equal to the intercept of the on-frequency simultaneous masking function when this function was forced to have a slope of 1. Thus the prediction for the on-frequency forward masking function is in error only to the extent that the slope of the on-frequency simultaneous masking function deviates from 1 when it is allowed to vary. Clearly, this error is very small. Figure 1 makes it quite easy to derive a prediction for any off-frequency forward masked threshold in Fig. 2. For a given masker level, we simply take the dB difference between the on-frequency masking functions of Fig. 1 and subtract

8O

• 70

•4o

3O

30 40 50 60 70 80

101og(M) SPL

FIG. 1. On-frequency simultaneous (squares) and forward (circles) masked thresholds as a function ofmasker level. Solid lines are linear fits to the data.

Dashed lines are predictions derived from Eq. (2), see text.

7O

6O

5O

4O

=. 30

•7o

6O

5O

4O

3O

I I I I I

1.6 kHz -

--

-

I I I I i

' i i i i i

1.4 kHz

I I I I I 40 50 60 70 80

I I I I I

2.2 kHz

I I I I I

2.4 kHz

' I I I I 40 50 60 70 80

101og(M) SPL

FIG. 2. Same as Fig. 1, except data are off-frequency masked thresholds.

this value from the corresponding off-frequency simultaneous masked threshold in Fig. 2. For example, the 2.4-kHz masker produces an off-frequency simultaneous masked threshold of nearly 50 dB when the masker level is 70 dB. For a 70-dB masker, the difference between the on-frequency masking functions in Fig. 1 is 10 dB. Thus for the 70-dB, 2.4-kHz masker, the predicted off-frequency forward masked threshold is 50 - 10 = 40 dB. Overall, the predicted functions account for 97% of the total variability in the forward masked thresholds. This value might be misleading, however, since the simultaneous masking functions them- selves account for 64% of this variability. Therefore, Fisher's z to r transformation (n = 22) was used to test the hypothesis that t a = 0.97 amounts to a significant increase over t a = 0.64. The outcome of the test was positive at the p < 0.0001 level.

The masking functions of Figs. 1 and 2 were used to obtain PTCs at three signal levels by interpolation. The obtained PTCs and corresponding predictions {dashed lines) are shown in Fig. 3. The obtained PTCs do not appear signif- icantly different from previously obtained PTCs. Each has the familiar V shape with a steep high-frequency branch and a shallower low-frequency branch. The most commonly used measure to describe the sharpness of PTCs is Q 10. It is defined as the center frequency of the curve divided by its bandwidth at the 10 dB down points. The Q 10 values for these curves are given in Table I. These values are well within the range of Q 10 values obtained in previous studies, but the values for the forward masking PTCs are slightly smaller than average (see Moore, 1978). This is probably due to the absence of data points near the tips of the forward masking PTCs where tuning is usually sharpest (Moore, 1978). The Q 10 values for the predicted PTCs are in substantial agreement with those of the forward masking PTCs consistent with the difference in tuning indicated by forward and simultaneous measures.



/ 90 / _

80 ••1 _ •7o

•5o

4O

3O

1.6 2.0 2.4 1.6 2.0 2.4 1.6 2.0 2.4

Ma'sker Frequency in kHz

IV. DISCUSSION

Off-frequency forward masked thresholds are predicted well by a simple multiplicative relation between on-frequency forward and off-frequency simultaneous masked thresholds. The relation assumes only that Weber's law holds for changes in masking produced by separating the masker from the signal in frequency and/or time. Thus, for any given masker level, the dB difference between off-frequency forward and simultaneous masking is predicted to be equal to the dB difference between on-frequency forward and simultaneous masking. The predictions yield PTCs- which, in agreement with the data, are more sharply tuned in forward masking than in simultaneous masking. We now consider the implications of these results for interpreting measures of auditory frequency selectivity.

A. Implications for previous interpretations

As noted in the Introduction, numerous qualitative interpretations have already been proposed to account for the difference between forward and simultaneous PTCs. To at-

tempt a detailed description of these interpretations would only belabor the point we wish to make here. Indeed, within any one of these interpretations there is no general agreement exactly as to how the proposed mechanisms operate (for example, see Weber, 1983). Therefore, our approach is to evaluate these interpretations on the basis'of the general assumptions that are common to all of them.

For simplicity, let us generally refer to the process in- volved in each of these interpretations as P. Thus, P may be replaced with whatever term best describes the process in

TABLE I. Predicted and obtained Q 10 values for PTCs at three levels of the fixed level signal.

lO log(P•)SPL 30 40 50

Simultaneous 4.4 4.7 4.6 Forward 6.7 5.7 5.7 Predicted 6.7 5.7 5.7

FIG. 3. Obtained (solid lines) and predicted (dashed lines) PTCs interpolated from the linear fits to the data of Figs. 1 and 2. As before, simultaneous masking is indicated by squares and forward masking by circles. The three plots give PTCs for three levels of the fixed-level signal as indicated in the upper left-hand comer of each plot.

question, e.g., "suppression," "cueing," etc. We may now identify three assumptions that are common to all of the previous interpretations:

(1} The auditory input passes through an initial filtering stage.

(2} At or subsequent to this stage, P has the effect of further filtering the input. This is literally a filtering effect because P is assumed to be different for different masker frequencies.

(3} PTCs in forward masking are sharper than those in simultaneous masking because the full extent of the additional filtering effect produced by P is only made evident in forward masking.

Taken together, assumptions ( 1} and (2} make clear that all of the previous interpretations involve two separate frequency selective processes. Thus, if a mathematical expression were used to describe the predictions of these interpretations, it would include two separate frequency selective functions: one corresponding to the effects of the initial filtering stage and one corresponding to the effects of P. More- over, assumption {3} would require that the latter function include some information regarding the effects of off-frequency forward maskers.

We have shown that a single frequency selective function H, estimated exclusively from simultaneous masked thresholds predicts the off-frequency forward masked thresholds quite well. According to these predictions, a non- frequency selective function G, interacts with H to produce the sharpening effect seen in forward masking PTCs. The important distinction between this and all previous interpretations is that G, unlike P, is not dependent on masker frequency. To emphasize this distinction, we recall from Eq. (1} that for any given masker level, the dB difference between forward and simultaneous masked thresholds is predicted to be independent of masker frequency. Thus a sharpening effect in forward masking should not be seen if the data are plotted as masking functions of frequency (fixed masker level} rather than PTCs. The data agree to the extent that Eq. ( 1} does indeed provide a Very good fit. In contrast, any of the existing interpretations of P would predict a sharpening effect for both measures. Moore and Glasberg ( 1981 } obtained



masking functions of frequency in simultaneous and forward masking using a notched-noise masker. In their Fig. 8, the forward masking function appears sharper, a result they at- tribute to suppression. However, for their forward masking condition, "threshold" on the abscissa actually refers to the level of a flat noise (on-frequency) forward masker necessary to produce the same amount of masking as the notched- noise. Since the slope of the growth of masking function for the flat noise was most likely less than 1, one could predict from Eq. (1) that their forward measure would be sharper. Thus there is no obvious discrepancy between our data and that of Moore and Glasberg ( 1981).

The results suggest that the limits of auditory frequency selectivity are already established in simultaneous masking, and that additional frequency selective mechanisms are not required to account for what might appear to be evidence from PTCs for sharper auditory tuning in forward masking. At best then, previous interpretations of the difference between forward and simultaneous PTCs are not the most

parsimonious interpretations that might be offered.

B. A measurement approach: dB or not dB?

We are interested in masking to the extent that orderly relations among variables reflect the operation of mechanisms underlying auditory function. Often decisions as to which combinations of variables to examine and what opera- tions to perform on these variables are suggested by a suc- cessful paradigm in a related area of investigation. Indeed, the popularity of the PTC derives, in large part, from its similarity to physiological measures used to describe the frequency response of single fibers in the auditory nerve (Kiang et al., 1965). A second approach is to consider the study of masking as a measurement problem. In this case, an empiri- cal structure is chosen to test formal axioms that, if satisfied, may yield methods for constructing valid numerical repre- sentations (see Coombs et al., 1970, for review). In this sec- tion, we consider the results of the present experiment within the framework of the measurement model in an attempt to show that, whether or not the approach is ultimately suc- cessful, it can potentially provide insights into the mechanisms of masking.

The focus is on the conjoint effects of signal-masker frequency separation and signal delay, so it is not overly restrictive to suppose that masker level is fixed. This will simplify the discussion. The measurement problem can be stated as follows: find transformations •,a, and y, such that for all t and f,

• IX {t•f)] = a{t) + y{f}. (P1)

In words, we wish to find a transformation of the data which can be expressed as some additive combination of signal- masker frequency separation and signal delay.

Using Eq. (1), it is easy to find one set of transformations that satisfy (P 1 ) for the restricted case in which masker level is fixed. To denote the special case, we rewrite F_xl. (1} as

X (tf) = Gm (t)Hm Oe)/kM,

where Gm and H,, describe the functions G and Hat a given masker level. The expression can be reduced further by not- ing that the term H• (f)/kMrepresents a masking function of

ß

frequency where values are scaled relative to the amount of masking produced at f= 0. For instance, in the auditory filter model of Patterson (1974), this function would corre- spond to the attenuation characteristic of the auditory filter. • Replacing this term with the function W(f), and taking the log of both sides of the new expression yields

logX(tf) = log G,,(t)+ log W{f). (3)

Equation {3) can be seen to provide one solution to {P1) where, • = log, a = log G,,, and •, = log W. The general solution derives from the particular. That is, for arbitrary con- stants s, u, v, and w = u + v, it can be verified that any • ' = s• + w, a' = sa + u, and •,' = sy + v is also a solution {Luce and Tukey, 1964). Thus, according to Eq. {3), any dB {s -- 10) transformation of X scaled by an additive constant can be expressed as the sum of any other dB transformation of G,,, and of W.

This result suggests a fundamental feature of the processes underlying off-frequency forward masking. If G,, is identified with the recovery from masking and W with the frequency selectivity of the system, the ability to find transformations preserving additivity of G,, and W imply that these two processes contribute separately and independently to off-frequency forward masking. Moreover, the valid transformations indicate that the combined effect of the two

processes can be described as the dB sum of their separate effects. Similar proposals have been made for the combined effects of two different maskers separated in frequency {Lutfi, 1983)and two maskers separated in time {Penner and Shiffrin, 1980). Although, in these studies, the additivity is within rather than across stimulus dimensions, and the ap- propriate transformations are quite different. A stronger test of independence and the dB summation rule will require a larger scale version of the present experiment including, in particular, a greater number of signal delays. The prelimi- nary data are encouraging inasmuch as they offer the possi- bility of predicting masking for various combinations of masker frequency and signal delay.from measures that con- centrate on these variables separately.

V. SUMMARY

Off-frequency forward masked thresholds are predicted well by a simple multiplicative relation between on-frequency forward and off-frequency simultaneous masking. Predictions yield PTCs with Q 10 values in excellent agreement with those of forward masking PTCs. Thus it is still unclear to what extent additional tuning mechanisms are required to account for diserepant measures of frequency selectivity obtained in forward and simultaneous masking. Future studies may determine the generality of the multipli- eative relation in summarizing the combined effects of masker frequency, masker level, and signal delay.

ACKNOWLEDGMENTS

I would like to thank Tom Hanna and Fred Wightman for helpful discussions during the writing phase. Comments on an earlier version of the manuscript were gratefully received from Bob Gilkey, Dave Green, Don Sinex, Dan We- ber, and an anonymous reviewer. This research was support-

1049 J. Acoust. Sec. Am., Vol. 76, No. 4, October 1984 Robert A. Lutfi: Predicting forward masking 1049


ed by grants from the National Institutes of Health (1F32 NS 06489-01) and the National Science Foundation (BNS 83- 08498).

•Actually, in the filter model of Patterson (1974), Hm {f)/kM would corre- spond to the integral of the filter evaluated over the frequency band of the masker. However, since the frequency band of the masker is small, H• {f)/ kM can be taken directly to represent the value of the filter attenuation characteristic atf.

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predicting frequency selectivity in forward masking from simultaneous masking

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