INTRODUCTION J. Acoust. Soc. Am. 103 (5), Pt. 1, May /98/103(5)/2539/12/$ Acoustical Society of America 2539

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1 Auditory filter nonlinearity at 2 khz in normal hearing listeners Stuart Rosen, Richard J. Baker, and Angela Darling Department of Phonetics & Linguistics, University College London, 4 Stephenson Way, London NW1 2HE, England Received 2 January 1997; revised 29 October 1997; accepted 26 January 1998 Auditory filters broaden with increasing level. Using a recently developed method of fitting filter shapes to notched-noise masking data that explicitly models the nonlinear changes in filter shape across level, results at 2 khz from 9 listeners over a wide range of levels and notch widths are reported. Families of roex( p,w,t) filter shapes lead to models which account well for the observed data. The primary effect of level is a broadening in the tails of the filter as level increases. In all cases, models with filter parameters depending on probe level fit the data much better than masker-dependent models. Thus auditory filter shapes appear to be controlled by their output, not by their input. Notched-noise tests, if performed at a single level, should use a fixed probe level. Filter shapes derived in this way, and normalized to have equal tail gain, are highly reminiscent of measurements made directly on the basilar membrane, including the degree of compression evidenced in the input output function Acoustical Society of America. S X PACS numbers: Ba, Dc, Sr, Bt WJ INTRODUCTION A fundamental property of the peripheral auditory system is that it operates as a frequency analyzer. This processing mechanism can be conceptualized as a bank of overlapping bandpass filters, often referred to as auditory filters. Some of the applications exploiting the concept of auditory filters and requiring accurate auditory filter shape characterization include the design of realistic front ends for automatic speech recognition Zwicker et al., 1979, modeling the effects of normal and impaired frequency selectivity on the perception of speech or music Patterson et al., 1982; Rosen and Fourcin, 1986, and modeling loudness perception e.g., Moore et al., It has also been suggested that auditory filter shape measurement may be a useful diagnostic tool in the detection of preclinical noise-induced hearing loss West and Evans, The most well developed psychoacoustic technique for measuring auditory filter shapes is the notched-noise simultaneous masking method, supported by an explicit analysis algorithm for a review, see Patterson and Moore, Almost always, thresholds are found for a tonal signal at a particular frequency in the presence of fixed-spectrum-level band-stop noises notched noises, with the varying-width notch placed both symmetrically and asymmetrically around the probe frequency e.g., Dubno and Dirks, 1989; Fidell et al., 1983; Moore et al., 1990; Niemiec and Yost, 1992; Patterson, 1976; Patterson et al., 1982; Weber, 1977; Wright et al., 1997; Shailer et al., 1990, not to mention many others. The dependence of probe threshold on the notch configuration is exploited to estimate the auditory filter shape. The implicit assumption in such studies is that by fixing the spectrum level of the masker, the properties of the auditory filter being measured are also being fixed. Of course, we could also fix the level of the probe, and vary the masker level for a particular notch. Were auditory filters linear, it would not matter which level was fixed and which was varied. But it is now well known that auditory filters are highly nonlinear, broadening with increasing level. Therefore it is crucial to decide whether to fix probe level or masker level in such experiments. At a more fundamental level, we need to determine what feature of the stimulus configuration controls filter shape. If filter shapes are somehow controlled by probe level, then fixing the masker level will distort considerably the shape of the derived auditory filters, and vice versa. The question of which, if either of these parameters, actually controls filter shape can also be related to more complete models of auditory filtering. For example, Glasberg and Moore 1990 provide an algorithm for calculating excitation patterns based on the notion that auditory filter shape is determined by the level of the input to the filters. Their computations imply that fixing the level of the masker in a notched-noise experiment leads to an approximation of filter shape for a particular level see below. On the other hand, Carney 1993 has proposed a model in which the output of the filter is rectified and smoothed, then used to control filter shape. With the additional assumption that the efficiency of the post-filtering detection process does not vary with level, such an output-based model implies that a fixed probe level leads to appropriate filter shapes. For an input-based model it seems unlikely that the spectrum level of the masker per se controls the filter shape. Lutfi and Patterson 1984 suggested that the controlling factor is more likely to be the level of the masker in the region of the filter. However, if the region of the filter is restricted to one equivalent rectangular bandwidth ERB around the filter s center frequency and, given the above assumption about post-filtering detection, this restricted input is very closely correlated with filter output. The excitation pattern model of Glasberg and Moore 1990; and Moore and Glasberg, 1987 implements another input-based model in which it is the level of an input com J. Acoust. Soc. Am. 103 (5), Pt. 1, May /98/103(5)/2539/12/$ Acoustical Society of America 2539

2 ponent integrated over an ERB that controls the amount of attenuation of that particular component by the filter concerned. 1 However, for a broadband signal such as those used in notched-noise masking, the ERB integrated input varies little over the width of the masking noise. For example the ERB is 111 Hz at 800 Hz and 241 Hz at 2 khz. Accordingly, for a probe frequency of 2 khz and a fixed masker spectrum level, the ERB integrated input level would vary only by 3.4 db between masker components at an 800-Hz notch edge and those at the probe frequency in a no-notch condition. On the high-frequency side the filter shape is independent of signal level and so only signal components below and immediately around the probe frequency affect the filter shape in Glasberg and Moore s model. Thus in a notched-noise masking task, fixing the masker spectrum level would be a good approximation to the fixed ERB-integrated-input level that would be required to ensure a constant filter shape under Glasberg and Moore s assumptions. While there is some debate as to which aspect of the signal controls the filter shape, it is the level of the probe tone or the spectrum level of the masker that are the ones most readily fixed in empirical studies and in fact the only ones which have been. Given also that there are theoretical proposals to justify both, we focus solely on them. It is, perhaps, a little surprising that this crucial issue has received relatively little attention until recently. Although two previous studies mentioned above have claimed to come to decisive conclusions Lutfi and Patterson, 1984; Moore and Glasberg, 1987, we have recently shown that such claims cannot be upheld Rosen and Baker, Instead, we have developed quite a different approach, which is to explicitly model the changes in filter parameters as a function of level, and to determine whether a dependence of these parameters on masker level masker-dependent or probe level probe-dependent better accounts for the data. To illustrate this idea in a relatively simple case, consider a notched-noise masking experiment performed with symmetric notches only, at a fixed masker level of N 0 db SPL/Hz. Let us assume the symmetric roex( p,w,t) filter shape to be appropriate Patterson et al., 1982, so off-place listening can be ignored: 1 w 1 pg e pg w 1 tg e tg where g is normalized frequency ( f f 0 / f 0, f 0 filter center frequency, p and t are parameters determining the sharpness of the filter in the passband and tail, respectively, and w is a weight determining where the tail slopes take over from the passband slopes. For such a shape, the tail slopes can only be shallower than those in the passband. Under certain simplifying assumptions Rosen, 1989, the power spectrum model of masking Patterson and Moore, 1986 leads to the level of the probe (P s db SPL being given by P s k N 0 10 log 2 f 0 10 log 1 w 2 pg e pg p w 2 tg e tg t, where g is now the particular symmetric notch width in question normalized as above, and k is a measure of the detector efficiency at filter output signal-to-noise ratio in db. Our Polynomial Fitting PolyFit procedure 2 might have the analogous equation: P s k 0 k 1 N 0 N 0 10 log 2 f 0 10 log 1 w 0 w 1 N 0 2 p 0 p 1 N 0 g e p 0 p 1 N 0 g p 0 p 1 N 0 w 0 w 1 N 0 2 t 0 t 1 N 0 g e t 0 t 1 N 0 g, t 0 t 1 N 0 where each estimable parameter has been replaced by a linear function depending on masker level (N 0 ) with two coefficients to be estimated the slope and intercept. Note the distinction made here between a parameter such as p or k whose single value determines the auditory filter shape, and a coefficient whose value is used in an equation to calculate the value of any particular filter parameter. Such an equation would correspond to a family of roex( p,w,t) filters whose shapes would be given by: 1 w 0 w 1 N 0 1 p 0 p 1 N 0 g e p 0 p 1 N 0 g w 0 w 1 N 0 1 t 0 t 1 N 0 g e t 0 t 1 N 0 g. PolyFit allows the polynomial terms to be of any degree although terms of a higher order cannot be included in the model without all the terms of lower order, for parameter values to be dependent upon the level of the masker as above or of the probe, and for parameters describing the high and low frequency sides of the filter to be specified independently. Thus instead of estimating coefficients for a parameter p, separate sets of coefficients are estimated for a lower p (p l ) and upper p (p u ), and similarly for w and t. The calculations also take into account the possibility of offplace listening, where a listener is supposed to attend to the filter which has the greatest signal-to-noise ratio at its output Patterson and Moore, More generally, our implementation is a superset of the procedures described by Glasberg and Moore When analyzing a notched-noise experiment at a single fixed level, it is thus possible to select a model using PolyFit that will give identical fits to theirs. In this way, it is possible to construct a single model, which, after appropriate coefficients are chosen, can predict thresholds for any combination of notch widths and level. While we have so far concentrated on whether the filter shape varies with N 0 or P s, there is the separate, and somewhat unrelated, issue of whether P s should be predicted from N 0 or vice versa when investigating the goodness of fit of a given model to the data. We have used the root-mean-square rms residual as our measure of goodness of fit. For example, we can take the value of N 0 from a given data point and use the model under investigation to predict P s and then compare this predicted P s to the empirical value given by the data point. Or we can, in the same way, predict N 0 given the value of P s. Because the mathematical formula for predicting thresholds can be written as P s N 0 other terms, or conversely as N 0 P s other terms, predicting probe thresholds (P s ) from the masker spectrum levels (N 0 ), as in the first example, or masker levels from tone levels, as in the 2540 J. Acoust. Soc. Am., Vol. 103, No. 5, Pt. 1, May 1998 Rosen et al.: Auditory filter nonlinearity 2540

3 TABLE I. Summary data concerning the nine listeners who participated in the study, including the range of conditions over which they participated. Maximum notch widths are shown separately for symmetric and asymmetric notches. Absolute thresholds were obtained using standard pure-tone audiometry. Absolute Number of threshold at 2 Range of levels notch Maximum notch Number of Number of Listener Age khz db HL P s N 0 widths/level width conditions thresholds AMD , 0.5, 0.4, CT , 0.5, 0.4, JD , 0.5, 0.4, LS , 0.4, 0.3, MB , 0.4, 0.3, RC , 0.5, 0.4, RJB , 0.5, 0.4, SK , 0.4, 0.3, WC , 0.5, 0.4, second, leads to identical residuals. Therefore it does not matter which way around the prediction is made. In fact, because we use an additive power term to include the effects of absolute threshold see below, it is intuitively more appealing to predict probe levels, even when the probe level itself is fixed in the experiment. 3 Fitting the model to the data proceeds in essentially the same way as for the single level case. Instead of searching the space of the parameter values directly for the best-fitting values, the space of the coefficients making up the polynomials is searched. 4 Note that this fitting technique is quite different to the typical procedure used for studies across level, in which filter parameters are estimated separately for each of a number of masker levels, and polynomial fits are made to the values of the estimated parameters as a function of level as in Moore and Glasberg, We have, for example, fitted as many as 158 mean data points in a single analysis, instead of doing ten separate analyses five conditions in which the probe level is fixed, and five in which the masker level is fixed. Thus results from the fixed-probe and fixed-masker experiments can be analyzed together, putting stronger constraints on the way filter parameters can change with level. A primary advantage of the PolyFit procedure is one of stability of the model fit, because such a large number of conditions can be fit by relatively few free coefficients. For example, in fitting one set of fixed-masker data involving data points with a roex(p,w,t) filter shape in which p, w, and t are independently estimated on the upper and lower frequency side, seven free parameters must be estimated including k). To describe the filter shape as a function of level requires multiplying the number of parameters by, for example, five levels of N 0 and five levels of P s 70 parameters for 160 data points. Using PolyFit to fit the same data with a roex(p,w,t) model in which each of the seven parameters is allowed to take on the form of a linear function of N 0 or P s ) results in a model with 14 coefficients (7 2) for the same number of data points. This large reduction in the number of free variables allows the stable fitting of complex models such as the roex( p,w,t) shape, something which was not possible using the standard techniques Patterson et al., PolyFit also provides a principled way of comparing results across fixed-masker and fixed-probe conditions. Individual fits leave these results incommensurable. One further extension to typical roex fitting procedures was implemented. Because absolute thresholds can place a lower limit on masked thresholds in some conditions for fixed noise maskers at low levels with wide notches, a single estimated parameter corresponding to the absolute threshold of the probe was incorporated. This was done by adding the value of the estimated threshold in power terms to the probe level predicted by the model for any particular condition. For probe levels more than 10 and 20 db above the estimated absolute threshold, this term changes the initially predicted value by less than 0.5 and 0.05 db, respectively. Rosen and Baker 1994 report an application of PolyFit using a roex(p,r) shape equivalent to a roex(p,w,t) shape with t 0] to a set of data obtained from two normal-hearing listeners. They showed that models which had filter parameters depending upon the level of the probe were considerably more successful than models in which filter parameters depended upon the level of the masker. They therefore argued that probe level should be fixed for simple measures of auditory filter shape at one level. More importantly, auditory filter shape appeared to be controlled by probe level, or something closely related to it. Here we extend those findings in a number of ways: 1 applying them to measurements made over a considerably wider range of notch widths and stimulus levels; 2 using the more complex roex(p,w,t) shape in place of the roex(p,r); 3 analyzing both group and individual results in a total of nine normally hearing listeners; 4 developing a technique for normalizing filter shapes so as to estimate changes in gain as well as shape across level, thus making our shapes comparable to direct measurements of basilar membrane vibration. I. METHODS A. Listeners Table I presents summary data concerning the nine listeners who participated in the study. All had hearing thresholds within normal limits ( 20 db HL. All testing was 2541 J. Acoust. Soc. Am., Vol. 103, No. 5, Pt. 1, May 1998 Rosen et al.: Auditory filter nonlinearity 2541

4 done monaurally, and in a single ear per listener. Mean results averaged over listeners LS and MB have appeared in a previous report Rosen and Baker, B. Threshold estimation Masked thresholds were determined for sinusoidal probe tones of 2 khz in the presence of notched-noise maskers with variable notch widths. The notches were placed both symmetrically and asymmetrically about the probe frequency and either the probe level or the noise level could be varied to determine the thresholds. A two-interval, two-alternative forced-choice paradigm with feedback was used to estimate the 79% point on the psychometric function. Listeners responded on a button box, with illuminated buttons indicating presentation intervals and providing feedback. From a starting level at which the probe was clearly audible, the varying sound, either probe or masker, was initially changed in 5 db steps, with step-size decreasing by 1 db after each turnaround. Once the step-size reached 2 db, it remained constant for a further eight turnarounds, the mean of which was taken as the threshold. For each particular combination of notch-width and fixed probe or fixed masker level, two thresholds per listener were typically obtained. Threshold measurements where the standard deviation of the last eight turnarounds exceeded 3 db were rejected and the measurement repeated. Also, where two measurements of the same condition in the same listener differed by more than 3 db, a further measurement was taken and the average of all measurements used. C. Stimulus configurations The outside edges of the masker noise were fixed at 0.8 f and 3600 Hz. A maximum of 16 different notch conditions were used, six symmetric and ten asymmetric. The frequencies of the edges of the notch are specified in normalized frequency units relative to the probe frequency ( f 0 ) as given by ( f f 0 )/f 0. In the symmetric conditions, both notch edges were placed at normalized values of 0.0, 0.1, 0.2, 0.3, 0.4, and 0.5. In the asymmetric condition one of the notch edges was set at a normalized value of 0.0, 0.1, 0.2, 0.3, and 0.4, while the other was set to 0.2 normalized units further away 0.2, 0.3, 0.4, 0.5, and 0.6. When the masker level was fixed, a subset of noise-spectrum levels (N 0 ) was chosen, ranging from 20 to 60 db re:10 12 W/Hz the units hereafter referred to as SPL/Hz in 10-dB steps. When the probe level was fixed, a subset of probe levels (P s ) was chosen, ranging from 30 to 70 db SPL, again in 10-dB steps. Table I details the experimental conditions for each of the listeners. D. Stimulus generation All the stimuli were computer generated at a sampling frequency of 20 khz. The time waveform of the probe was calculated independently of the masker and consisted of a steady state portion of 360 ms plus 20-ms raised-cosine onsets and offsets. The probe was temporally centered within the masker which consisted of a 460-ms steady-state portion with 20-ms raised-cosine-squared 5 onset and offsets. To generate the masker, the desired frequency spectrum was defined by setting all the spectral components spaced at intervals of 0.61 Hz within the appropriate frequency limits to have equal amplitudes while those outside were set to zero. Nonzero components had their phases randomized uniformly in the range of 0 2 radians. An inverse FFT was then applied to generate the time waveform. At the start of each threshold determination, a s buffer of noise was generated for use during that test. On each trial, a 500-ms portion of the buffer was chosen randomly for each of the two masker intervals within each trial. The probe and masker were played out through separate channels of a stereo 16-bit D-A converter PCLX TM, Laryngograph Ltd. and attenuated independently under computer control before being electrically mixed PA4 and SM3 from Tucker-Davis Technologies. The signal was then sent via a balanced line to a final amplifier in a sound-treated room where it was presented monaurally to the right ear via Etymotic ER2 insert earphones. Calibrations were done using a B&K 4157 ear simulator conforming to IEC 711 and ANSI S3.25/1979 ASA 39/179 with a B&K DB 2012 ear canal extension. Because the experiments took place over a number of years, two distinct sets of apparatus were used. These differed in detail, but not in essentials. The later experiments involving listeners AMD, JD, RJB, and WC were run as described above. The earlier experiments involving listeners CT, LS, MB, RC, and SK had the following differences: 1 The 2-kHz sinusoidal probe was hardware generated, and gated by multiplying it with a computer-generated envelope with 10-ms raised-cosine onsets and offsets, and a steadystate portion of 380 ms. The probe was temporally centered within the masker, which consisted of a noise burst with the same gating envelope, but a 480-ms steady state. 2 A 500-ms portion of the buffer was chosen randomly for the masker burst on each trial, but the same masker burst was used for the two intervals of the trial. 3 The masker bursts were output from 12-bit D-A converters, simultaneously with the appropriate gating envelope for the probes. Probe and masker bursts were controlled independently in level by two digitally controlled attenuators Charybdis, at the output of which they were mixed. We have no reason to suspect that any of these differences affect the results to any significant degree. E. Analyses All analyses were performed on means. When averaging over listeners, the contribution of each listener to the mean was kept equal by taking means within a listener before averaging across listeners. If there was no data for a particular condition for any one of the listeners in the mean, that condition was excised. In particular, it was occasionally not possible to present the masker at a sufficiently high level to mask the probe for the widest notch widths and higher probe levels. Most of the analyses were done individually for each listener. However, in order to simplify the presentation, the first, and most extensive analyses concern the mean of the three listeners who experienced the greatest range of levels 2542 J. Acoust. Soc. Am., Vol. 103, No. 5, Pt. 1, May 1998 Rosen et al.: Auditory filter nonlinearity 2542

5 FIG. 1. A comparison of the goodness-of-fit of PolyFit models of identical structure, one of which has filter parameters depending upon probe level abscissa, and one of which has filter parameters depending upon masker level. The goodness-of-fit measure is the root-mean-square residuals; hence smaller numbers indicate better fits. Only parameter structures resulting in models with a goodness-of-fit less than 2.5 db are included a total of 63 comparisons. The solid line indicates equal goodness-of-fit for the two possible dependences. and notch widths AMD, JD, and RJB. Comparisons are also made with the mean results of LS and MB, as used by Rosen and Baker Because listeners do vary somewhat in their performance, we only took means across listeners with a common set of conditions. Otherwise, we felt it would have been necessary to implement some kind of weighting to ensure listeners responses were given equal weight across all conditions. A variety of models were fitted to each data set, using Polyfit. All of the models were variants of the asymmetric roex( p,w,t) model. These included simplified models in which, for example, the upper half of the filter was described with a roex(p) shape whereas the lower half was a complete roex( p,w,t) shape. It is also necessary to estimate k, the signal-to-noise ratio necessary for detection at the output of the filter. All of these parameters can be arbitrary polynomial functions of the level of the masker or the probe, but we have never investigated models with more than a quadratic dependence on level that is, three coefficients per parameter to be estimated. Finally, we also used Polyfit s ability to estimate an absolute threshold that is never allowed to vary with level described above. II. RESULTS AND DISCUSSION A. Normal listeners: Mean of 3 This data set consists of mean thresholds obtained in 158 distinct conditions as described above, excluding the 0.6, 0.4 notch at fixed probe levels of 60 and 70 db SPL, based on a total of 1154 separate thresholds from the three listeners AMD, JD, and RJB. We fitted 73 models, which differed in parameter structure, assuming filter parameters to vary with probe level or masker level a total of 146 distinct models. Figure 1 shows that, for models that depend on level, probe-dependent models always fit the data better than masker-dependent models. Although the fits obtained from the masker-dependent models might be considered adequate in other circumstances, the FIG. 2. Summary measures of the goodness-of-fit of PolyFit models which fit the data best with a given number of coefficients, for models in which filter parameters depend upon the level of the probe, and of the masker. rms residual is typically 70% larger than that obtained from the corresponding probe-dependent model a statistic based on the median ratio of rms residuals. Another way to demonstrate this is to compare the fits from the best-fitting probe- and masker-dependent models for fixed numbers of coefficients. Figure 2 shows, again, the consistently better fit obtained by making filter parameters depend upon probe rather than masker level. Although it is quite clear that probe-dependent models give better fits than masker-dependent models, it is much more difficult to choose a particular probe-dependent model out of the many possible. Clearly, for a fixed number of coefficients, one would typically choose the best-fitting model. The difficult issue is the choice of the number of coefficients, and how this trades off against the goodness-offit. To aid us in this task, we have adopted a heuristic approach based on those common in statistical model building e.g., see Aitkin et al., Starting off with a model with more coefficients than we think are necessary, we then determine which can be eliminated by looking at changes in the goodness-of-fit as they are excised. Unfortunately, for the type of nonlinear model employed here, there is no statistical theory which can assess the statistical significance of any given numerical change in the rms residual. Often, however, excising some coefficients hardly changes the rms residual, whereas excising others changes the fit of the model dramatically. In our earlier publication Rosen and Baker, 1994, we had the good fortune for this to be true consistently. Here the results are not as clear cut, but as we shall see, models with similar goodness-of-fit lead to filter shapes that are very similar. Therefore it is not particularly important which model is chosen from the better-fitting ones. The relatively large number of good-fitting filter shapes is also an indication that the roex( p,w,t) shape may be too flexible. There are likely to be other adequate functional forms with fewer controlling parameters e.g., Irino and Patterson, 1997; Lyon, For the moment, however, we concentrate solely on the roex family, as they are the only filter shapes which have been shown to be adequate for dealing with a wide variety of results. First note that a quadratic dependence of all parameters 2543 J. Acoust. Soc. Am., Vol. 103, No. 5, Pt. 1, May 1998 Rosen et al.: Auditory filter nonlinearity 2543

6 TABLE II. Goodness-of-fit for the best-fitting model containing 5 22 estimated coefficients. All are probe dependent models. The first seven columns contain the number of polynomial coefficients used for each filter parameter. The last column indicates the % change in the goodness-of-fit measure between the model given by that row, and the model on the row immediately above. Number of rms residual (n 158 p l p u k w l w u t l t u coefficients db % increase x 2 x x 2 x x 1 x x 1 x x 1 x x 1 x x x on probe level leads to little improvement on the fits obtainable from a linear dependence. The quadratic model, with 22 coefficients to estimate three for each of the six filter parameters and k, plus one for absolute threshold leads to an rms residual of db, whereas the linear model, with 15 parameters, leads to a residual of db. Thus a loss of seven coefficients worsens the goodness-of-fit by only 0.03 db. Table II shows the best-fitting models for a particular number of estimated coefficients. Models are generally described by a letter (p or m indicating probe or masker level dependence followed by a string of seven digits indicating the number of coefficients used for each of the parameters p l, p u, k, w l, w u, t l, and t u, respectively. An x indicates a parameter that is not needed, for example, when only a roex(p) shape is needed on the high-frequency side of the filter. Thus p2212x1x indicates a model in which both lower and upper p depend upon probe level in a linear way, as does w l, with a simple roex(p) shape on the high-frequency side. All other parameters are invariant across level. A 0, used only for describing filter parameters for the upper half of the filter, indicates that the filter shape has identical values on its high- and low-frequency sides for that particular parameter. Thus p1012x1x indicates a model with a simple roex( p) shape on its high-frequency side, but symmetric in its passband. Only one filter parameter varies with level (w l ). Note that it is possible to account reasonably well for the data with as few as 6 coefficients. Of these, two concern absolute threshold and detection efficiency (k), while only four describe filter shape assuming symmetry in the passband, by setting p u p l, did not harm the fit and allowed the loss of one parameter. Only one filter parameter need depend on level (w l ). Also of note is the fact that k can generally be assumed to be invariant across level without affecting the fits obtained. This empirical fact supports a simpler interpretation of our results than would otherwise be the case see the discussion below. Figure 3 shows the filter shapes, as a function of level, derived from four of the models described in Table II. Although the goodness-of-fit varies across these four models by about 35%, there is little change evident on the lowfrequency sides of the filter. Even the high-frequency sides which are known to be somewhat difficult to pin down in notched-noise experiments as thresholds are largely determined by the shallower lower side of the filters, Glasberg and Moore, 1990 differ little unless they are not permitted to change with level. So, although there is some uncertainty in choosing a particular model, many of the models which fit the data reasonably well lead to similar conclusions about filter shapes. We have chosen to focus on p1312x2x, as a further reduction in the number of coefficients leads to a relatively large increase in terms of proportions in the rms residual, considerably larger than reductions in models with more coefficients. This model appears to be a good compromise between the number of parameters used and the goodness-offit, but other choices would lead to conclusions that are little different. The quadratic term for p u allows the upper slope of the filter to remain relatively constant for low levels, and then to become shallower with increasing level once the probe level reaches about 50 db SPL. Models in which p u is constant, or a linear function of level, lead to odd behavior in the upper tails of the filter. In particular, the tail starts just a few db down from the peak. Clearly, the roex(p,w,t) was not designed with such a possibility in mind, so it seems far preferable to allow a quadratic dependence in one parameter. That the models do fit the data quite well can be appreciated from Fig. 4, which shows the entire data set, plotted as growth-of-masking functions, along with the predictions from our chosen model. Note that the ordinate here is signal-to-noise ratio as opposed to the probe level more usu J. Acoust. Soc. Am., Vol. 103, No. 5, Pt. 1, May 1998 Rosen et al.: Auditory filter nonlinearity 2544

7 FIG. 3. Filter shapes for four different probe-dependent models. Each plot shows the filter shapes calculated for probe levels of db SPL in 10-dB steps. ally plotted in such curves. The advantage of this measure is that departures from a linear growth of probe level with masker level, indicated by a horizontal line, are more easily assessed visually. All predictions are within 2.8/ 2.0 db of the obtained data points and the rms residual is only db. Contrast this with the predictions from the best maskerdependent model with the same number of estimated coefficients in Fig. 5 (m2222x1x). Although in many situations this would be considered quite a good fit, note the many regions in which the data are consistently predicted poorly. The predictions are only within 5.7/ 3.3 db of the obtained data points with the rms residual being 1.41 db, some 64% worse. There is another way of plotting filter shapes which also points to probe-dependent models being superior. The filter shapes shown above are all normalized to have unity gain at their peak, as a consequence of the assumptions involved in fitting roex filter shapes. Yet we know from direct measurements of basilar membrane vibration that peak gain varies monotonically with level, being greatest at lowest levels Ruggero et al., 1992; Ruggero et al., Such measurements also show basilar membrane response to be linear for frequencies sufficiently below the best frequency of the place on the membrane being investigated, as shown in Fig. 6. While Fig. 6 shows basilar membrane transfer functions for an isointensity stimulus paradigm, the same general properties variation in peak gain and linearity below CF can be seen in isoresponse measurements of basilar membrane motion Ruggero et al., Note too the compressive response at the peak of the filter, with a change in gain of about 20 db for a 40 db input range. Working on the hypothesis that our behavioral results FIG. 4. Masked thresholds expressed as signal-to-noise ratios probe level in db SPL minus the masker level in db SPL/Hz as a function of masker level. Such curves are known as growth-of-masking functions although more typically with the level of the probe as the ordinate. The results from symmetric notches and the two types of asymmetric notches are shown in separate graphs. Each symbol indicates a particular pair of notches. The lines are predictions from a model which assumes filter parameters to depend upon probe level. The diagonal line at left indicates absolute threshold J. Acoust. Soc. Am., Vol. 103, No. 5, Pt. 1, May 1998 Rosen et al.: Auditory filter nonlinearity 2545

8 FIG. 7. Filter shapes for a p1312x2x probe-dependent model, calculated for probe levels of db SPL in 10-dB steps, and normalized to have equal gain a little more than one octave below their center frequency. FIG. 5. As for Fig. 4, but with predictions from a masker-level-dependent model. Arrows indicate regions in which the fit of the model is consistently poor. reflect basilar membrane filtering in a fairly direct manner, we assume that the auditory filter is linear a little more than one octave below its characteristic frequency, thus tacking together the shapes at this point. The resulting curves are highly reminiscent of filtering functions measured on the basilar membrane. In particular the filter sharpness and peak filter gain both increase with decreasing level Fig. 7, and there is a tendency for the filters to become linear again at frequencies high above CF. The 2:1 compression ratio at the peak is also similar to that evidenced on the basilar membrane, although not too much should be made of this. Other basilar membrane studies indicate greater compression ratios e.g., Nuttall and Dolan, 1996, but against this must be set the probability that compression ratios vary significantly FIG. 6. The frequency response isointensity of a single place on the basilar membrane as a function of level redrawn from Ruggero et al., with frequency, decreasing apically i.e., at lower frequencies, Cooper and Yates, 1994; Wilson, Note too that the filters plotted in Fig. 7 change shape right down to within 15 db of absolute threshold. This is to say that the nonlinearity extends to levels as low as it is possible for us to measure. There is some controversy about this, with claims that the basilar membrane is linear even at its peak response for stimulus levels as high as db SPL. Recent measurements by Nuttall and Dolan 1996 support our view in showing that the response of the basilar membrane does indeed become linear at low enough levels, but only... for basilar membrane velocities below afferent neural thresholds based on discharge rates p The same manipulation leads to a much less orderly picture for the masker-dependent models. Figure 8 shows the filter shapes normalized in both ways. Although the filter does become shallower with level on its low-frequency side in the passband, the tail sharpness appears to increase with increasing level. Also, filter sharpness does not turn out to be linked to peak filter gain in the way suggested by basilar membrane experiments. We have also found that the filter shapes obtained from masker-dependent models change much more with changes in parameter structure, as can be seen in Fig. 9. In the m model, the change in gain is even greater than the change in input level. Therefore according to this model, output level would decrease as input level increases. In short, there are strong reasons to prefer models of auditory filtering which make filter parameters depend upon probe level rather than masker level: 1 The probedependent models predict the data more accurately over the entire stimulus space, with the root-mean-squared residuals 40% 70% larger for masker-dependent models; 2 Probedependent models lead to filter shapes much more in keeping with physiological measures. 3 Filter shapes derived from masker-dependent models change greatly with small changes in the parameter structure assumed, while filter shapes derived from probe-dependent models change little even with large changes in the assumed parameter structure. B. Normal listeners: Mean of 2 Rosen and Baker 1994 reported similar analyses to those above on a set of mean data from two different listen J. Acoust. Soc. Am., Vol. 103, No. 5, Pt. 1, May 1998 Rosen et al.: Auditory filter nonlinearity 2546

9 FIG. 8. Filter shapes for a m2222x1x masker-dependent model, calculated for masker levels of db SPL in 10-dB steps, normalized both to have equal gain at their tips, and about an octave below their center frequency. ers. It is interesting to compare the filter shapes from the two sets of mean data given the differences in listener, apparatus, and particular conditions tested. Figure 10 corresponding to Fig. 7 shows the filter shapes derived from the same probedependent model focused on above. It is clear that the two sets of data lead to highly similar derived filter shapes across level, in spite of all the differences between them. C. Normal listeners: Individual results Similar analyses were performed for the individual data sets obtained from each of the listeners. Of some interest is the relationship between the absolute thresholds obtained with standard audiometery Table I and the threshold estimated from the fits. Comfortingly, the estimated absolute threshold varied little with the parameter structure of the fitted model e.g., at most 1.5 db for the four probe-dependent models specified in Table III. Also, there was typically a close relationship between the two absolute thresholds within 5.5 db for six of the nine listeners. Only one listener SK had a large discrepancy ( 8 db, with the threshold estimated from the fits about 23 db greater than her threshold measured audiometrically. Due to the fact that this listener did not participate in the lowest level fixed-masker condition, and that her signal detection efficiency as measured by k) was the highest of any listener 7.5 db re: the mean level of the other eight normal listeners, no probe level even approached her actual absolute threshold. As it is thus clear FIG. 9. Normalized filter shapes for three different masker-dependent models. Each plot shows the filter shapes calculated for masker levels of db SPL in 10-dB steps. Note the strong variation in shapes for different parameter structures J. Acoust. Soc. Am., Vol. 103, No. 5, Pt. 1, May 1998 Rosen et al.: Auditory filter nonlinearity 2547

10 FIG. 10. Filter shapes from the data obtained by two normal listeners in Rosen and Baker The curves show a p1312x2x probe-dependent model, calculated for probe levels of db SPL in 10-dB steps, and normalized to have equal gain at about one octave below their center frequency. Compare this to Fig. 7. that the masked thresholds could not possibly constrain an estimate of her absolute threshold, no fits for SK incorporated an absolute threshold term. Table III shows the goodness-of-fit measures for a selection of models for each individual data set. These measures vary significantly across listeners for at least two possible reasons. First, listeners differ in their inherent consistency, and we would expect more consistent listeners to have better fits. Second, given that there is some individual variation in filter shape, it may be that the roex(p,w,t) shape just happens to fit some listeners better than others. Of more interest, in any case, is the pattern of results across conditions within an individual listener. Note first that probe-dependent models always fit the data better than masker-dependent models, again with the rms residual typically more than 50% greater for the masker-dependent models. In fact, we have never found a case in which a maskerdependent model fits the data better than a probe-dependent model of the same structure in literally hundreds of comparisons. Second, for the probe-dependent model in which all parameters vary linearly with level, there is typically little loss of predictive power in assuming k signal detector efficiency to be constant across level. Figure 11 shows the individual normalized filter shapes arising from the p1312x2x model we used above. Again, however, there are a wide variety of parameter structures FIG. 11. Filter shapes for nine normal-hearing listeners using a p1312x2x probe-dependent model, calculated for probe levels of db SPL in 10-dB steps, and normalized to have equal gain at just less than one octave below their center frequency. The curves have been shifted along the logarithmic frequency axis for clarity. which lead to similar filter shapes. Clearly, individual listeners differ in their frequency selectivity, but the essential pattern is strikingly uniform. In fact, for a number of listeners, there were ten parameter models that led to fits considerably better than the p1312x2x used here. This was investigated further by finding the best ten-parameter model for each listener. If this model led to a goodness-of-fit that was more than 15% better than that from p1312x2x in terms of the sum of the squared residuals, a new model was chosen for that listener. All the other listeners had improvements of fit ranging from 0% to 6.5%. One other constraint influenced the selection of the model. No model with a varying k was used, because the evidence from the Mean-of-3 data set so strongly supported the notion of a constant k. In those cases, another model was chosen, either one with fewer parameters, or with one additional parameter, depending upon the goodness of fit in each case. In all the listeners for whom a new model was chosen, the fit was improved by allowing p l to vary with level, as shown in Table IV. Figure 12 shows the new set of filter shapes derived from the models. As compared to Fig. 11, the main change is seen in listener RC, whose filters were considerably narrower than those of the other listeners. This appears to arise from a poorly fitting model, and indeed, it was TABLE III. Individual measures of goodness-of-fit from a number of Poly Fit models. Also shown are results from the Mean-of-3 and Mean-of-2 data sets described in the text. Listener p p p1312x2x p2222x1x m m m1312x2x m2222x1x Mean of Mean of AMD CT JD LS MB RC RJB SK WC J. Acoust. Soc. Am., Vol. 103, No. 5, Pt. 1, May 1998 Rosen et al.: Auditory filter nonlinearity 2548

11 TABLE IV. Models chosen for listeners whose results were not fit adequately by the p1312x2x model. Listener Chosen model Number of parameters rms residual rms residual for p1312x2x CT p2312x2x LS p2112x1x RC p RJB p2312x1x WC p2312x2x the fit to RC s results that was improved the most by selecting a different model than our standard p1312x2x. Contrast the filter shapes based on probe-dependent models with those that arise from the m2222x1x maskerdependent model discussed above Fig. 13. These are much more variable from listener to listener, both in the degree of compression or even expansion across level, and even in which level leads to the highest gain. III. SUMMARY AND FINAL REMARKS We have shown that it is possible to accurately account for the pattern of results across level in a notched-noise masking experiment with filter shape models that explicitly depend upon probe level. Such models have four advantages over models in which filter parameters depend upon the level of the masker: 1 They always fit the data better; 2 Filter shapes change relatively little with changes in the number of coefficients controlling each parameter, and in parameter structure; 3 Different listeners give results that are much more similar; 4 Appropriate normalization of tail gains lead to filter shapes that are highly reminiscent of vibration patterns observed directly on the basilar membrane. Therefore notched-noise measurements that are made only at one level should be performed with a fixed probe level. Fixing the masker level leads to a derived filter shape that is some kind of average of a number of shapes, caused by the change in probe level as notches are varied. In general, such filter shapes will be too narrow, simply because the filter is becoming sharper as the probe level decreases with increasing notch width. FIG. 12. Same as for Fig. 11, but with different filter shape models for the five listeners specified in Table IV. FIG. 13. Filter shapes for nine normal-hearing listeners using an m2222x1x masker-dependent model, calculated for masker levels of db SPL in 10-dB steps, and normalized to have equal gain at just less than one octave below their center frequency cf. Fig. 11. Our technique is also likely to clarify the effects of hearing impairment on auditory nonlinearity, especially as it is currently believed that many of the concommitants of sensori-neural hearing loss can be attributed relatively straightforwardly to a loss of nonlinearity see Moore 1995 for a review. In particular, damage to the outer hair cells where the motor for the nonlinearity is presumed to reside would be expected to manifest itself in four ways relevant to the discussion here: 1 a loss of absolute sensitivity; 2 smaller or no changes in filter shape across level; 3 degraded selectivity at low levels; and 4 normal selectivity at sufficiently high levels. Using the PolyFit procedure to analyze notched-noise masking data, we have found evidence for all of these four features in two hearing-impaired listeners with quite minor impairments Rosen et al., One had a hearing loss sloping from essentially normal at low frequencies to 35 db HL at 8 khz. Although she used hearing aids, her loss at 2 khz was only 30 db. Strikingly, her auditory filter shapes did not change at all with level. Filtering on the upper side of the filter was essentially normal, while that on the lower side could only be considered normal at relatively high levels. For low measurement levels, her auditory filter slopes were considerably shallower than normal on the low-frequency side. The other listener, with a subclinical impairment of 20 db at 2 khz attributed to frequent exposure to amplified music as a musician, also showed a reduced but not absent nonlinearity. Thus at least in these cases, hearing loss appears to consist of a loss of the nonlinear properties of the auditory periphery, most likely arising from damage primarily to the outer hair cells. On the theoretical and practical side, there is still much to be done regarding the computational implementation of nonlinear filters, in order to produce a general-purpose nonlinear auditory filter bank. In terms of the distinction between input versus output control of filter shape Lutfi and Patterson, 1984; Verschuure, 1981, our results clearly support the notion that the filter shape is controlled by its output level. This arises from the finding that k appears to be constant across level, so that fixing the probe level also fixes 2549 J. Acoust. Soc. Am., Vol. 103, No. 5, Pt. 1, May 1998 Rosen et al.: Auditory filter nonlinearity 2549