on the nature of two-tone aural nonlinearityaudres/publications/humes/papers/18_humes.pdfon the...

On the nature of two-tone aural nonlinearity Larry E. Humes

Hearing Research Laboratory. Division of Hearing and Speech Sciences, Vanderbilt University School of Medicine, Nashville, Tennessee 37232

(Received 23 February 1979; accepted for publication 18 December 1979)

The present manuscript begins with a review of the literature on two-tone nonlinearity and proceeds to the description of an ad hoc model of auditory nonlinearity which can account for several of the features of the psychophysieal data on various two-tone nonlinearities, in particular, (f2 -f0, (2f• -f0, and two- tone suppression. The two basic components of the model are (1) a nonlinearity which is a series combination of the classical power-series and compressire p-law nonlinearities and (2) an intensity- dependent (nonlinear) filtering scheme. The conceptual model proposed here not only describes the psychophysieal data accurately, but also offers possible explanations for some of the apparent discrepancies between psyehophysical and physiological data on two-tone nonlinearities.

PACS numbers: 43.66.Ki, 43.66.Ba [DM]

INTRODUCTION

Auditory nonlinearities have been of interest to hearing researchers for well over a century (e.g., Helm- holtz,1954). It has only been in the past few decades, however, that amplitude-distortion processes operating within the hearing mechanism have become a focal point of physiological and psychoacoustic research in and of themselves (e.g., Goldstein and Kiang, 1968; Dallos, 1973; Pfeiffer and Kim, 1973; Green, 1976; Plomp, 1976). A variety of models of aural nonlinearity, both physiological and psychophysical, have emerged from this extensive pool of data. None of the physiological models that have been derived, however, provides an adequate description of amplitude distortion (e.g., Gold- stein, 1967b; Smoorenburg, 1974; Sachs, 1975).

The present manuscript describes an ad hoc psychophysical model of auditory nonlinearity that, unlike earlier models, can account for the salient features of several two-tone amplitude-distortion processes. The distortion phenomena focused on here are the simple difference tone [(f2-f•),f2>fx], the cubic difference tone [(2f• -f2)], and two-tone suppression forf2>f•. These three nonlinearities are probably the most prominent perceptually and have been investigated-most thoroughly, both psychophysically and physiologically.

The manuscript proceeds as follows. First, in the section to follow, salient features of various classes of nonlinearity are reviewed. Next, a composite nonlinearity is described. Finally, the nonlinearity is then incorporated into an appropriate filtering scheme to make the model more complete. Although the model to be described is developed from psychophysical data, compar- isons to physiological data are made where appropriate.

I. THE NONLINFARITY

Perhaps the earliest attempt to formalize the nonlinear behavior of the hearing mechanism was made •by Helmholtz over a century ago (Helmholtz, 1954, p. 412). His original model was later simplified by Fletcher (199.9, p. 312). Fletcher described a classical power- series or polynomial nonlinearity of the general form

y = ao + a•x•+ aa x 2+ aa x a+...+ anx, ' (1)

where y is the output of the system, x is the input to the

system, and a n are constants. The power-series nonlinearity provided in Eq. (1) is generally considered to be linear at low input levels but becomes increasingly nonlinear as input level increases. Another key feature of this type of nonlinearity is that it is memoryless. That is, the output of the system at some time t is determined solely by the value of the input at this time [i.e.,x(t)].

Equation (1) allows some predictions to be made for the dependence of L(2f• -f2) and L(f2 -f•) on primary level [where L(f2 -fx) and L(2f• -f2) refer to the level of these distortion products as measured psychophysically]. These predictions are shown in Table I. Briefly, for an input, x, comprised of two sinusolds of amplitude A• and A 2 (i.e., x = A• sin2•f•/. A 2 sin2•fzt) , the ampli- tudes of (fz -f•) and (2f• -f•) are determined by A•Az and and A•Az, respectively. The slopes in Table I were then derived using these amplitude valuesJ

Goldstein (1967b) was one of the first to confirm earlier observations made by Hermann (1891) and Zwicker (1955) that the dependence of L(2f• -f2) on primary level predicted by the classical power-series was not realized in the data. Consequently, Goldstein developed a so-called "normalized" version of the power-series nonlinearity which appeared to fit the data for (2ft better. The nonlinear system described by Goldstein was normalized to the peak amplitude of the input in the following manner:

y = a= (xp)--•+ a 2 (x•+ a a (x-•--•-+...+ a, (x-•-• , (2) where xp is the peak amplitude of the input signal x. For an input comprised of two sinusoids with ampli- tudes of A'x and A2, xp assumes a value of Ax+A=.

TABLE I. Slopes of the function relating distortion-product level to primary level in dB/dB as predicted by the classical power-series nonlinearity (see footnote 1).

Lt varied. L2 varied, L 1 =L 2 L 2 fixed L 1 fixed

(2fi- h) +3 +2 +1 • -•) •2 •1 •1

2073 J. Acoust, Soc. Am, 67(6), June 1980 0001-4966/80/062073-11500.80 ¸ 1980 Acoustical Society of America 2073

TABLE H. Slopes of the function relating distortion-product level to primary level in dBAIB as predicted by the normalized power-series nonlinearity.

L1 varied, L2 varied, L 2 fixed œ1 fixed

L 1 =L 2 L 1 <L 2 L! >L 2 L 2 <•L! L 2 >L!

(2fl-•) +i +2 0 +1 -1 • -•) +1 +1 0 +1 0

Whereas the amplitude of (2f• -f2) predicted by the classical power series was A•A2, Goldstein's normalized model predicts an amplitude of (A•A2)/(Ax+A2) •. In a similar manner, the classical power-series nonlinearity predicted an amplitude of A•A2 for the simple difference tone while Goldstein's formula suggests an amplitude of (A•A•)/(A•+A•). The resulting level dependence of L(2f• -f2) and L(f• -f•) predicted by the normalized power-series is shown in Table IL It is apparent from Eq. (2) and Table II that the normalized power-series describes an"essential" nonlinearity. That is, the relative amount of distortion is constant even for low input levels as is reflected in the slope value of 1.0 dB/dB for L•=L2. For example, if L(f• -f•) or L(2f• -f2) is 20 dB down from the primaries at high levels, it is also 20 dB down at lower primary levels.

Although Goldstein's normalized power-series nonlinearfry appears to provide a much better description of data for the behavior of L(2f, -f•) than does the classical power series, some objections to this model have nonetheless been raised. The first objection to Gold- stein's normalized power-series nonlinearfry arises from the fact that since Goldstein's nonlinear mecha-

nism incorporates a normalizing factor equai to the peak amplitude of the input signai, some time must be required for the system to accomplish this normalization (Smoorenburg, 1974). Consequently, the response of the system may no longer be instantaneous as was the case for the classical power series. Smoorenburg (1974) observed, however, that L(2f• -f2) (as determined with a forward-masking pz. radigm) was not affected by the duration of the primaries for durations as short as 24 msec. Although this finding may be interpreted as support for the existence of a memoryless nonlinearfry, it is also possible that the time needed for normalization in Goldstein's system is simply less than 24 msec (Smoorenburg, 1972b, 1974). Support for the latter pos- sibility has been provided by Crane (1972) whose medel- ing results suggest a normalization time constant on the order of 5 msec. Thus, a nonlinear system with memory cannot be completely ruled out by existing psychophysical data.

The second objection to Goldstein's conceptualization of "auditory nonlinearfry" is that it does not provide an accurate description of his own data, nor those of other investigators, for L(f• -f•). In four of the six conditions shown in his Figs. 19 and 20, for instance, "square law" (i.e., classicai power-series) behavior obtains for the simple difference tone. Further, of the remaining two conditions in those figures, the data ob-

rained for L([• -f•) are not in accord with the predictions of the normalized power-series nonlinearity nor the classical power series. Indeed, both Zwicker (1955) and Goldstein (1967b) concluded from their data that L(2f• -f•) and L(f• - f•) follow different distortion laws. Thus, Goldstein's medel might be classified more ap- propriately as a model of L(2f• -f•) behavior rather than as a general model of "auditory nonlinearity."

Smoorenburg (1972a, 1972b, 1974) also recognized the inadequate description of the dependence of L(2fx -fz) on primary level provided by the classical power-series nonlinearity. This investigator, however, proposed that a compressive nonlinearity, referred to as a p-law device, might provide a more accurate description of aural nonlinearity. The original description of the p-law device incorporated full-wave rectification which yielded only odd-order distortion products, such as (2/• -f2). Incorporating a p-law device with either asymmetric rectification or half-wave rectification, however, allows for the generation of even-order nonlinearities, of which (fz-f,) is a member (Duifhuis, 1974, 1976). Since the analysis is not substantially different, a half-wave rectified version of the p-law device is described here. The half-wave rectified p-law device is of the following form:

y=x • for x> O,

y=O for x•< O.

Some attractive features of the p-law device in general include its instantaneous response (memoryless) and its ability to also account for many properties of the nonlinear phenomenon of two-tone suppression when incorporated into an appropriate filtering scheme (Pfeiffer, 1970; Smoorenburg, 1972b, 1974; Duifhuis, 1976, 1977).

The dependence of distortion-product level on primary level predicted for psychophysical data by the hail-wave rectified version of the p-law device is shown in Table III. From consideration of Eq. (3) alone one might expect L(2f• -f2) and L(f• -f•) to grow at a rate ofp dB/dB for L•= L•. In predicting slope values for psychophysical data, however, one must consider that the p-law de-

TABLE Iii. Slopes of the function relating distortion-product level to primary level in dB/dB as predicted by the half-wave rectified p-law nonlinearity.

varied, L 2 varied, L2 fixed Lt fixed

<L2 Lt>L2 L2<L I L2

General

Psychophyalcal Nonsimultaneous

(2fl -f2) + 1 •f• -h)

Simultaneous

(2• -h) {f• -h) +1

+2 p-1 +1 p-2 +1 p-1 +1 p-1

-x/p

+2 0 +1 -1

+1 0 +1 0

2074 J. Acoust. Soc. Am., Vol. 67, No. 6, June 1980 Larry E. Humes: Two-tone nonlinearitv 2074

TABLE IV. Predicted and observed slopes for the dependence of L(2ft-f2) on primary level (n.d. =no data available).

varied L2 varied

Predictions

Classical power series +3.0 +2.0 +2.0 +1.0 +1.0 Norm. power series or

HW p-law (simultaneous) +1.0 +2.0 0.0 +1.0 -1.0 HW p-law device

(nonsimultaneous, p = 0.8) +1.0 +2.5 -0.25 +1.25 -1.5

Observations

Zwicker (1955, 1968) +1.0 +1.5 to +2.0 --1.0 +1.0 -0.5 to -1.0 Goldstein (1967b) +1.0 n.d. n.d. +1.0 0.0

Smoorenburg (1972b) +0.8 na/. na/. +1.0 -0.7 to -1.4 Hall (1972) +0.7 n.d. n.d. n.d. n.d. Sachs (1975) e +0.7 to +1.0 +0.8 to +1.2 -0.5 to -1.5 n.d. n.d. Erdreich (1977) +0.7 to +1.1 n.d. n.d. n.d. n.d.

Srnoore•burg. (1974) a .e Smallf2/fl +0.6 n.d. n.d. +0.5 to +1.0 -0.3 to -1.0 Large f2/fi h +2.7 nxi. n.d. n.d. n.d.

Humes (1980b) •'e Small f•/fi + 0.7 n .d. n.d. n.d. n .d. Large f2/fi d +2.9 n.d. n.d. n.d. n.d.

aN=l, Fig. 5, fl = 1000 Hz, f2/f!=l.ll and 1.41. bL i =L2 >65 dB 8L. oN=2, fi=1550 Hz, f2/fl=l.08 and 1.41, Lt=L2=35 to 85 dB SL. dL t =L2 • 60 dB SL. eStudies employing a nonsimultaneous paradigm.

vice also exerts an effect on the probe tone used to measure the level of the various distortion products (Smoorenburg, 1974; Duifhuis, 1976, 1977). Thus, although the distortion product arising from the p-law nonlinearfry is predicted to grow at a rate of p dB/dB for Li=L•, the probe tone also grows at the same rate, such that a slope of 1.0 dB/dB is expected for psychophysical data. Consequently, three sets of predictions are provided in Table III. The first set of predictions apply to the p-law device in general. The second set applies to psychophysical data obtained with nonsimultaneous methods. The latter values were derived by

multiplying the general slope estimates by (Duifhuis, 1976,1977). Finally, the third set of predictions hold for psychophysical data obtained with simultaneous paradigms, such as the cancellation procedure (Smoorenburg, 1972, 1974; Duifhuis, 1976). The latter values are seen to be identical to those predicted by the normalized power-series nonlineariky (Table II).

Having reviewed the three major classes of nonlinearity that have been developed from psychophysical data during the past decade, a question to be asked is whether avaihble data argue in favor of one class of nonlinearity to the exclusion of the others. To assist in an- swering this question, data for L(2f• -f2) have been summarized in Table IV. Two sets of data are provided in this table. Those investigations not in italics com- prise one set in which measurement conditions were restricted primarily tof2/f•< 1.3 and primary levels lOw 65 dB SL. Initial discussion is restricted to these

studies. Of these studies, only one employed a nonsimultaneous paradigm (Sachs, 1975). The majority of data have been obtained for the condition L•=L.,. For

the most part, regardless of the type of psychophysical paradigm used, slope estimates for L• = L• are fairly consistent across studies, varying between ñ 0.7 and ñ 1.1 dB/dB. Some amount of interstudy variability is to be expected in view of the different values off•,f•/fl , and the range of primary levels used in the various studies. Still restricting the discussion to the studies in Table IV that are not in italics, it is evident that most of the slope values depicted definitely do not agree with those predicted by the classical power-series nonlinearity.

The two studies in italics in Table IV differ from the

other studies of L(2fl -f•) in that a wide range off,/f, values and primary levels were studied within the same subject or set of subjects. -In addition, a nonsimultaneous measurement technique was employed in both studies. The slope values for smallfz/• were obtained for f•/f•< 1.3 and primary levels less than 65 dB SL. It is not too surprising, then, that the slope estimates derived are in good agreement with the results of the non- italicized studies which were obtained under identical

conditions. At largef•/f•, on the other hand, L(2f• -f•) slope estimates were derived typically for fz/fl> 1.3 and primary levels above 60-65 dB SL. Under these conditions, a quite different picture emerges. Specifically, slope values not inconsistent with those predicted by the classical power-series nonlinearity were obtained. It should be noted, hOwever, that only data for L•Lz ax• available at these wider primary separations and higher levels. One can summarize the data in Table IV, then, by stating that either the normalized power- series nonlinearity or the half-wave •ectifled p-law device appears to hold fo• •/ft < 1.3 and primary levels

2075 J. Acoust. Soc. Am., Vol. 67, No. 6, June 1980 Larry E. Humes: Two-tone nonlinearity 2075

TABLE V. Predicted and observed slopes for dependence of L {f2-fi) On primary level (n.d. =no data available).

L! varied L2 varied

L! =L 2 L!<L 2 L! >L 2 L 2 <L! L 2 >L!

Predictions

Classical power series +2.0 +1.0 +1.0 +1.0 +1.0 Norm. power series or HW p-law (simultaneous) +1.0 +1.0 0 +1.0 0

HW p-law device (nonsimultaneous, p = 0.8) +1.0 +1.25 -0.25 +1.25 -0.25

Observations

Bekesy (1960) +1.0 n.d. n.d. n.d. n.d. Zwicker (1955) +2.0 +1.0 n.d. n.d. n.d. Goldstein (1967b) +1.5, +2.0 +1.0 n.d. +1.0, +1.5 n.d. Wenner (1968) +1.0, +1.6 n.d. n.d. n.d. n.d. Hall (1972) +0.7 n.d. n.d. n.d. n.d. Greenwood (1972) + 1.2 + 1.0 n.d. + 0.7 n.d.

Itumes (1979) a Small f2/fl (<1.2) +0.9 +1.1 +0.2 + 0.8 -0.4 Largef2/ft (½1.41) +2.4 +1.1 +1.1 +1.0 +1.0

aStudies employing a nonsimultaneous paradigm.

less than 65 dB SL. At wider primary separations ()• /ft > 1.3) or higher primary levels, on the other hand, the psychophysical data for the cubic difference_tone are best described by the classical power-series nonlinearity .

Table V provides a similar summary of the data for L(f2 -f•) as a function of primary level. The data for this distortion product, however, are even more limited than for L(2fx-f2). Most data were again obtained for the condition L•=L 2. In addition, only one study has ex- amined the conditions where L•> L2or L2> L•, and it is only in this same study that a nonsimultaneous paradigm was employed (Humes, 1979). The variability between studies in the slope estimates for L•=L 2 is con- siderably greater than that observed for the cubic difference tone. The experimental conditions for the measurement of L(f2 -ft), however, were much less bonsis- tent across studies than for L(2f• -f•). The foregoing treatment of the data for the cubic difference tone suggests that the particular value of f•/f• investigated may play a large role in the interstudy discrepancies in slope estimates for the simple difference tone. This has been confirmed recently by Humes (1079) as is evident in the italicized entry in Table V and in Fig. 1. The figure illustrates the effect thatf•/f• exerts on the slope of the function relating L(f2 -fx) to L• = L•. The circles represent individual data points (N= 4), the solid line connects the group medians, the letters represent similar estimates from four other studies, and the horizontal dashed lines indicate the slope values predicted by the various nonlinearities reviewed above. The trend apparent in Fig. 1 for L(f2 -f•) is similar to that noted above for L(2f• -f2). That is, for smallf•/f• the half- wave rectified p-law nonlinearity or Goldstein's nor- realized polynomial nonlinearity describes the data best; whereas, at higherf•/f• the classical power-series nonlinearity appears to describe the data better than the other two classes of nonlinearity. Unlike the situation for the cubic difference tone, moreover, this trend was

confirmed not only for L• = L2, but also for varying the level of each primary independently (Table V).

The foregoing analysis for L(2f• -f•) and L(f• -f•) suggests that any one of Eqs. (1) through (3), taken separately, cannot describe the data in and of itself. Rather, some combination of nonlinearities appears to be operating within the ear. It is maintained here that cascading the classical power-series nonlinearity and the half-wave rectified p-law nonlinearity in the following manner provides the best description of the data on two-tone amplitude distortion reviewed thus far:

y = a•x}+ a2x2}+ a3x3•+ ß ß ß + anx • for x> 0, (4)

y=0 forx•< 0,

where 0.0<p•<l.0 andp-l.0 asf2/f• increases?

The predictions of Eq. (4) for small f2/f• and low intensities are those of the half-wave rectified p-law device as provided previously in Table III. That is, at low input levels and smallf2/f•, the linear term of the classical power-series nonlinearity predominates which re- suits in Eq. (4) becoming y = a•x • (for x> 0). Compari-

i 2œ

o

o

I.I 1.3 1.5 I.?

FIG. 1. The dependence of the slope of the function relating L(f 2 --fl) to L 1 = L 2 on f2/ft from Humes (1979). Circles are median values

for each of four subjects. Solid lines connect group medians. Letters represent results obtained from earlier

investigators (Bekesy, 1960; Wenner, 1968; Goldstein, 1967b; Zwicker, 1955). Dashed horizontal lines are

slope values predicted for L(f2-f0 by the various classes of nonlinearity.

2076 J. Acoust. S øc. Am., Vol. 67, No. 6, June 1980 Larry E. Humes: Two-tone nonlinearity 2076

son of observed slopes for the dependence of L(f 2 - f•) and L(2f• -f2) on primary level (Tables IV and V) to the predictions shown previously in Table III suggests that the appropriate value of p lies somewhere between 0.6 and 0.8 for srnallf2/f • and low input levels.

For high input levels, the nonlinear terms of Eq. (4) begin to play a greater role. That is, one is no longer dealing with just a•x •, as was the case for low input intensities. The second and third terms of Eq. (4) predict slopes of 2/) and 3/) dB/dB for L(/2 -f•) and L(2f• -/2), respectively (for L•=L2). Recall, however, that the probe tone used to obtain estimates of L(f2 -f•) and L(2f, -f2) in psychophysical experiments is also sub- jected to the nonlinearity of Eq. (4). Since only low probe levels are typically employed in such investigations, the probe tone would grow at a rate of/) riB/riB according to Eq. (4) (i.e., y =a•x • for the probe tone). Thus, when the effect of the nonlineartry on the probe signal is considered, the predicted slopes of 2/) and 3/) riB/riB become 2 and 3 dB/dB, respectively. Generally speaking, then, the predictions made by Eq. (4) for psychophysical data obtained at high intensities are identical to those described previously for the classical power-series nonlinearity (Table I).

Available psychophysical and physiological data support the notion that behavior consistent with the predictions of the classical power-series nonlinearity is observed at high intensities. First of all, for fixed f2/f• values, some investigators have reported a change in the slope of the function relating L(f• -f•) and L(2f• -f•) to L•= L2, such that slopes are in closer agreement with the predictions of the classical power-series nonlineartry at high intensities (L• = L2> 80-85 dB SL; Wennet, 1968; Greenwood, 1972; Smoorenburg, 1974; Weber and Mellert, 1975, their Fig. 7; Humes, 1979). Secondly, there is some physiological evidence from cochlear. microphonic and single-unit studies to support this notion of change in slope at high input levels (e.g., Worth- ington and Dallos, 1971; Smoorenburg et al., 1976, their Figs. 17 and 18). Third, at high intensities physical . measurements of distortion products present in the response of the basilar membrane are consistent with the prediction of the classical power-serie's nonlinearity. In particular, measurement of the .(2f• -f2) component in the bas ilar membrane at high intensities (101-111 dB SPL) in the guinea pig shows a growth rate of 3 riB/riB (Wilson and Johnstone, 1973, Animal 149L) which is predicted by the power-series nonlinearity. Fourth, the relative levels of (2f• -f•) and (re -f•) measured in the basilar membrane (at high intensities) are consistent with the predominance of the classical pOWer-series nonlineartry (Wilson and Johnstone, 1973; Rhode, 1977). Fifth, the decrease in the relative level of (2f• -f•) with increasing input level observed in the cat's cochlear nerve by Buunen and Rhode (1976) is also consistent with the emergence of the classical power-series at high intensities. Finally, Zwicker (1976) obtained psychophysical masking period patterns at high intensities which show some possible evidence of power- series distortion. Thus, there is ample psychophysical and physiological support for classical power-series behavior at high intensities as.predicted by Eq. (4).

An attractive feature of the combination of nonlin-

entities described by Eq. (4) is that incorporation of the p-law device also permits predictions to be made regarding the phenomenon of psychophysical two-tone suppression. As p increases, suppression decreases, such that for p = 1.0, no suppression results. Thus, Eq. (4) maintains that essentially no suppression should be observed for large fe/f• (P- 1.0 as fe/f• increases). By and large, existing psychoacoustic data support this supposition (Houtgast,1974; Shannon, 1976; Duifhuts, 1977). The solid lines in Fig. 2, for instance, depict the psychophysical suppression predicted by Eq. (4) for two differing values ofp (0.55 and 0.7). The slope for these functions for L2> L• is described generally for psychophysical data as (1 - l/p) dB/dB (Duiffiuis, 1976, 1977). First of all, this figure demonstrates that as p decreases (from 0.7 to 0.55), the amount of suppression for a given suppressor/suppressee ratio (L2 •L•) increases. Recall from the preceding paragraphs that for smallfe/f• , ap value of 0.6 to 0.8 fits the data for L(2f, -fz) and L(f• -f•) best. Consequently, suppression data obtained at small f2/f• should approximate the solid curves shown in Fig. 2. Two sets of data on psychophysical two-tone suppression (Houtgast, 1974, his Fig. 5.3; Shannon, 1976, his Fig. 5) have been replotted in this figure. Houtgast's results were obtained from seven subjects for f• = 1000 Hz, L,= 40 dB SL, andf•/f• = 1.2 and 1.5. Shannon's data s were obtained from three

subjects at 1000 Hz for 40 dB SPL andfe/f•= 1.1. Note, first of all, that ap value' of approximately 0.6 appears to fit the data for smallf•/f• (fe/f•= 1.1 or 1.2). This range of values is not at odds with the 0.6 to 0.8 range discussed above for (f2-f•) and (2f• -f2). Secondly, note that for largerf2/f• (f2/f• = 1.5), data from Houtgast suggest ap value approaching 1.0 (p--' 0.9). As mentioned previously, these data support the notion that suppression typically is not observed for large fe/f•. Hence, as re/it increases, p -- 1.0.

Physiological data on two-tone suppression also reveal a similar dependence onfe/f•. Rhode (1977), for instance, measured suppression present in the basilar membrane of the squirrel monkey with the MSssbauer technique. His data (in particular, his Fig. 3B) indicate that the slope of the function relating amount of suppression to suppressor/suppressee ratio varied as a function of f2/f•, such that as f•/fx increased, the slope

,.• 20Jr • •P=O. -•o eo -,o o ,o 20 •o

L•-L,•D

FIG. 2. Suppression as a function of (L 2-LI). Solid lines are values predicted by Eq. (4) for p = 0.55 and 0.7. Key: Open circles: suppression data from Houtgast (1974) for f2/f! = 1.2; Solid circles: same as open circles but for f2/ft = 1.5; Squares: unmasking results converted to suppression data for f2/f! = 1.1 from Shannon (1976). s

2077 J. Acoust. Soc. Am., Vol. 67, No. 6, June 1980 Larry E. Humes: Two-tone nonlineariW 2077

decreased. This, then, is entirely consistent with the findings displayed in Fig. 2 of this paper. Such behavior has also been observed in single units of the cat auditory nerve (Sachs, 1969, his Figs. 7 and 8) and is predicted by a nonlinear model of the basilar membrane derived by Kim, Molnar, and Pfeiffer (1973).

The foregoing analysis indicates that as fa/f• increases, the value of/) also increases. Regarding effects of input level, existing data on psychophysical two-tone suppression suggest that p decreases (suppression increases) with increase in input intensity up to moderate levels (L•--50 tb 70 dB SPL; Shannon, 1976; Duifhuis, 1977; Humes, 1980a). For further increments in input level, existing data indicate that the amount of suppression decreases (Shannon 1976; Humes, 1980a). This decrease in suppression at high intensities is explained here by assuming that the auditory structures associated with the p-law nonlinearity are extremely vulnerable physiologically so as to be rendered ineffec- tive at high intensities (Humes, 1980a). A high susceptibility of the nonlinearity (presumably the p-law nonlinearity) to reversible injury from exposure to intense (90 dB SPL) pure tones has been documented recently by Siegel (1978) and Kim, Siegel, and Molnar (1980). Such vulnerability, moreover, can also be invoked to explain the emergence of the classical power-series behavior at high intensities as described previously.

To summarize briefly, the nonlinearity formalized in Eq. (4) appears to provide an accurate description of the level dependence of two-tone aural nonlinearity. In particular, the equation predicts many of the features in- herent in the psychophysical data for L(fa -f,), L(2f• -f•), and two-tone suppression lorry>f,, as well as some physiological data.

II. SELECTION OF AN APPROPRIATE FILTERING SCHEME

Thus far the emphasis has been placed on various level effects apparent in the dataforL (f• -fx), L (2f• - f•), and two-tone suppression and the manner in which the observed level effects correspond to predictions made by various classes of nonlinearity. The focus in this section is shifted to effects of frequency that are apparent in the psychophysical data on auditory nonlinearity. This set of data has implications regarding the manner in which Eq. (4) is incorporated into the filtering function of the ear. Some mention of this has already been made in the general discussion of "small" and "large" fa/f•. Here, however, these concepts are treated in more detail. In this section, moreover, discussion is restricted momentarily to mid and high frequencies (/•> 750 Hz).

Psychophysical data obtained recently make it possible to address the issue of peripheral filtering. To de- rive valid estimates of the filter shape in psychophysical studies it is necessary to utilize nonsimultaneous techniques so as to avoid the confounding influences of detectable distortion products (e.g., Greenwood, 1971) and two-tone suppression (Houtgast, 1974). 4 Houtgast •1973), for example, obtained excitation patterns from seven subjects (12 ears) at 1000 Hz using the nonsi-

multaneous pulsation-threshold technique. The low- frequency slope of the pulsation-threshOld pattern increased from roughly 150 dB/octave to about 350 dB/ octave as the input level was raised from 40 to 80 dB SL. The high-frequency slope changed from -105 dB/ octave to -40 dB/octave over the same range of intensities. Verschuure (1977) has since confirmed these findings using the same pulsation-threshold technique.

In addition to putsation-threshold patterns, however, Verschuure (1977) also obtained what he terms "filter functions" for a fixed input using the pulsation-threshold technique. These functions were obtained by sweep- ing a fixed-level tbne through a range of frequencies above and below the fixed-frequency signal and mea- suring pulsation •threshold for the signal. Filter functions derived in this manner reveal dual high-frequency slopes at input levels of 60 and 70 dB SPL. At 60 dB SPL, for instance, the high-frequency slope of the filter function was approximately -75 riB/octave for < 1.4. Forfa/f•> 1.4, on the other hand, a slope of -25 tO -30 dB/octave was revealed. At 80 dB SPL, however., a single broad filter function was observed for /a//j> 1.0 (-35 to -40 dB/octave). Although the emphasis here has been placed on results of various tone-on- tone paradigms, it is noteworthy that some effects, such as the broilcuing of the excitation pattern at high intensities, have been observed recently in related psychophysical studies (Scharf and Meiselman, 1977; Weber and Green, 1978).

Aside from the psychophysical evidence reviewed above, there are additional supportive data regarding the high-frequency slopes. These data, moreover, are related more directly to measures of the frequency dependence of auditory nonlineartry. Various investigators have noted previously, for example, that the slope of the function relating L(2f,--/2) to fa/fx is in good agreement with the high-frequency slope of the

ß underlying excitation pattern associated with f, (Gold- stein, 1967,1970; Zwicker, 1968; Sachs, 1975). Probably the most detailed data in this regard have been obtained by Goldstein (1967b, 1970) who measured the level of the cubic difference tone for f, values ranging from 250 through 8000 Hz, and f z/f, values varying from 1.1 through 1.35. All data were gathered for Lx=La= 50 dB SL. For 500-<f•- < 4000 Hz a slope of -100 dB/octave was observed. The latter value is in good agreement with the high-frequency slope values of psychophysical excitation patterns obtained at comparable intensities (50 dB SL).

Pure-tone masking patterns reviewed above suggested a broadening of'the excitation pattern at higher intensities. A question to be considered, then, is whether the slope of the function relating L(2f, -fa) and L(f• -f•) tofz/f• also shows a similar effect of level. In an attempt to answer this question, data on the dependence of L(2f• -f•) onfz/f• have been replotted in Figs. 3 and 4 from Hall (1972, his Figs. 1-3) and Smoorenburg (1974, his Fig. 5), respectively. Both sets of data are from one subject. The parameter in each figure is input level (L•=L•). Whereas Hall's da• were obtained with the qimultaneous cancellation para-

2078 J. Acoust. Soc. Am., Vol. 67, No. 6, June 1980 Larry E. Humes: Two-tone nonlinearity 2078

FIG. 3. œ(2•-•) in dB SL as a function off2/fl for differing values of œt= œ2 in dB SL. Re- drawn from Hall (1972, Figs. 1, 2, and 3).

digre, Smoorenburg's data were collected with the nonsimultaneous pulsation-threshold method. Note that in both investigations a broadening of the slope is apparent as input intensity is raised. Hall's data suggest a slope of about -100 dB/octave at the lowest primary level and this decreases to -45 dB/octave at high intensities. Smoorenburg's functions change from roughly -75 dB/octave to -35 dB/octave as primary level increases from 50 to 80 dB SL. Finally, it should be mentioned that Zwicker (1968) has also reported a similar level-dependent decrease in the slope of the function relating L(2f] -fz) to f2/f,.

Recent data obtained for L(fz -fx), furthermore, confirm the general conclusions re•ehed above for L(2fx -]z)' This is exemplified by way of Fig. 5. Shown in this figure are the slopes for the median data (N- 4) from a recent psyehophysical study of the dependence of L(fz-fx) øn/z/fx (Humes, 1979). The L(f 2 data obtained for f, = 1550 Hz are provided simply to demonstrate their computability with the data obtained at 800 Hz. The slopes exhibited for f• = 800 Hz are -125 riB/octave at 70 dB SL, -75 and -35 dB/octave at 80 dB SL, and -45 and -25 dB/octave at 90 dB SL. The median data for the highest intensity were fitted (by eye) nearly as well as by a single slope of roughly -35 riB/ octave. Thus, psychophysicai data on the dependence of L(f 2 -f•) on f 2/f, are consistent with the high-frequency slopes of pure-tone masking patterns reviewed earlier. In addition, the decrease in slope at high input levels is also consistent with data on pure-tone masking patterns.

The functions in Fig. 5 reflect two slopes m•ch more so than was apparent in the previous two figures for L(2fx -f:). Recall, however, that the nature of the non- linearSty described by Eq. (4) is such that at wide primary separations (largef2/fx) the classical power-series nonlinearSty holds (p = 1.0). It is generally believed to be the case for this nonlinearSty that the relative level of the simple difference tone /s greater than that

i i I 1.3 1.5

FIG. 4. L(2f i -f2) in dB SL as a function of f2/fl for differing values of L l- L 2 in dB SL. Data were obtained with the

r artsimultaneous pulsation- threshold paradigm for f•

1000 Hz. Redrawn from

Smcorenburg (1974).

I

•11 = 155o 14=---

60

5O

t

20-

t

LcL2-75 I0-

7O

I.I I.• 1.5 I.?

FIG. 5. L(f 2 -f•) in dB SL as a function off2//! for two values off 1 (800 and 1550 Hz) and varying values of Lt= L 2. Func- tions represent group medians OV-- 4) for data from Humes (1979).

of the cubic difference tone. Consequently, at large f•/f• one would expect measurable levels of (/z-fx) prior to being able to measure the cubic difference tone. Said otherwise, for large,f•/f], the aimpie difference tone will appear at lower •nput hatensities than L(2fx -f:).- This notion is made more clear by way of

I• ß SOOHI

: I ! I O I0 40 &O eO

L, = Lz (dB SL)

FIG. 6. Existence region for L(f 2 -fl)- 10 dB SL from Humes (1979). Solid data points are medians for individual subjects (iV- 4) and dashed lines •onnect group medians. Circled data points represent comparable data (f•= 1000 Hz) for L(2f! -f2) from Smoorel•nL•g (1974, N-l). The"X" represents a com- binatloll of L 1- L 2 andf2/f ! values which were unsuccessful in eliclting a sufficient • 10 dB SL)L(2fl --f2) component in two subjects.

2079 J. Acoust. Soc. Am., VoL 67, No. 6, June 1980 Larry E. Humes: Two-tone nonlinearity 2079

Fig. 6 which provides the 10 dB SL "existence region" for L(fz-jf•). These data represent estimates obtained from four subjects by Humes (1979) using a nonsimultaneous gap-masking paradigm. The dashed lines connect group medians. The 10 dB SL existence region specifies the combination of fz/fx and Lx= L 2 that will yield a difference tone of 10 dB SL. The circled data points in the upper portion of Fig. 6 (fx=800 Hz) represent a comparable 10 dB SL existence region for L(2f• -fz) at 1000 Hz obtained from one subject by Smoorenburg (1974, his Fig. 5) using a nonsimultaneous paradigm. For small/z//x , the existence regions are similar for both L(fz -f•) and L(2f• -/z)- Unfortunately, no published data are available for L(2f• -f2) at anfz/f, value of 1.6 to 1.7 that can be compared to the data available for L(f2 -f•). Attempts to measure a cubic difference tone of 10 dB SL for/j= 800 Hz andfz/f• = 1.67 in two listeners have not met with success for

primary levels as high as 90 dB SL (L• = Lz). This is represented by the "X" in Fig. 6. This level represents the maximum intensity that could be used without con- taminating the data with physically generated distortion products. Yet, under these same conditions, (fz-f•) is elicited with primary levels of 75 dB SL (Fig. 6). Hence, these data on the existence region for L(2f• -f=) and L([z -fx) offer an explanation as to why a two- sloped function is not present in the L(2f• -fz) data. They indicate that at small/•//•, L(2f, -f•) is the pre- dominant distortion product; whereas, (fz -f•) predominates at largefz/f•. Decreasing relative levels for (2f• -f2) as L•--L z and j•//• increase have also been observed physiologically (Bunhen and Rhode, 1978). Such a decrease in relative levels for (2f• -fz) may be considered as further support for a change to the classical power-series nonlinearity for high input intensities or large S/L.

The filter functions depicted in Fig. 7 have been de- veIoped from the preceding synopsis of existing psychophysical data on both pure-tone "masking" patterns and the dependence of L(2f• -fz) and L(/z -f•) on f•/f•. The filter functions depicted in this figure indicate that the auditory filter broadens at high intensities. This basic notion has been incorporated in previous models of psychoacoustic filtering processes (e.g., Goldstein, 1967a; Maiwald, 1967; Zwicker, 1970; Ehret, 1977). It is main-

FIG. 7. Schematic represenf•tion of the nonlinear intensity- dependent filtering scheme suggested from various sets of psychophysical d•ta obtained at mid frequencies.

tained here, however, that the hypothetical functions in Fig. 7 represent composite functions resulting from the series combination of two nonlinear filters. The mech-

ansim underlying the first filter is the basilar membrane into which the classical power-series nonlinearity is incorporated, possibly in the mechanics of basilar membrane motion (e.g., Schroeder, 1973,1975; Hall, 1974). Nonlinearity has been observed directly in the basilar membrane or relegated to this structure through indirect means in a variety of laboratory ani- mals, including squirrel monkey (Rhode, 1971, 1978), rat (Mffiler, 1978), and cat (Smoorenburg and Linschoten, 1977), although no such noalinearity has been observed in the guinea pig to date (Wilson.and Johnstone, 1975; Rhode, 1978). The second filter is hypothesized here to be the directional sensitivity of the hair-cell cilia, into which the half-wave rectified p-law nonlinearity is incorporated(Duifhuis, 1976, 1977)? Thus, the sharpening of the filter functions in Fig. 7 at low intensities may be attributable to both basilar membrane nonlinearity (e.g., Rhode, 1978) and the second filter (Duifhuis, 1976). In addition, the high susceptibility of the hair-cell cilia to intense pure-tone stimulation (e.g., Hunter-Duvar, 1977) offers an explanation as to the physiological vulnerability of the second filter and associated p-law nonlinearity needed to explain decreasing suppression at high intensities, as well as physiological data on the high vulnerability of frequency tuning curves to a variety of otolytic agents (Evans, 1972,1975a, 1975b). Further speculations on the physiological mechanisms underlying aural nonlinearity are beyond the scope of this paper.

To review briefly, the proposed conceptualization of anral nonlinearity incorporates two cascaded nonlinear filters, with each of the filters incorporating one of the types of nonlinearity included in Eq. (4). The filter functions depicted previously in Fig. 7 are believed to represent the composite function of both filters. The observed sharpening as input level decreases from high to low intensities is explained via both a nonlinear damping of the first filter and the inclusion of a second filter. Finally, it is suggested that the second filter and associated nonlinearity are physiologically vulnerable (Evans, 1972,1975a, 1975b; Siegel, 1978; Kim et al., 1980). The consequences of such vulnerability are decreased suppression and decreased tuning at high intensities. In addition, the nature of the nonlinearity described by Eq. (4) is such that behavior consistent with the predictions of the classical power series obtains at high intensities.

As mentioned previously, the foregoing conceptualization of aural nonlinearity is applicable only for f, >• 750 Hz. The following section'extends this conceptualization to low frequencies (f•< 750 Hz).

Ill. FURTHER CONSIDERATIONS

Thus far, the conceptual model of aural nonlinearity proposed here has been restricted in scope to mid and high frequencies (fi '-- 750-4000 Hz). For this range of frequencies, anf•/f, boundary value of 1.4 to 1.5 foF Eq. (4) appears to describe the data best. That is, p-- 1.0 atf•/f,= 1.4 to 1.5 for mid frequencies. There is some evidence, both psychophysical and physiological, that this '%riticaF' fz/f, value may change with frequency.


4O $5 70 85

L• m d8 $PL

FIG. 8. Two-tone suppression as a function of L 1 for three values offt and two value• of/2//1 from Humes (1980a). D•ta represent medians for three s•bjects. Note that data f•r low frequencies (• = 500 Hz) show continual increase In suppression over entire range of L l values studied. A: f2/•- 1.1; B:

Beginning first with the psychophysical data, observations regarding the dependence of L(f= -f,) on L L = L 2 at various frequencies suggest that asfl decreases, the criticalf•_/f: value increases (Hall, 1972; Zurek and Leshowit•, 1976; Humes, 1980c). Specifically, for fx < 750 Hz, some investigators have reported slopes less than or equal to 0.8 dB/dB for the function relating L(f2 -fl) to L i L 2 for f2/f, values greater than 1.4 (Ha11,1972; Humes,1980c). Such slopes are consistent with the operation of the p-law nonlinearity rather than the classical power-series nonlinearfry at these wide primary separations (as high as f2/f, - 1.6 for f•-425 Es; Humes, 1980c). Thus, psychophysical data suggest that only the hallways rectified p-law nonlinearity may

be operating at low frequencies, even for la•ge f2/fr In addition, there is some physiological evidence on

two-tone suppression that suggests that as f, decreases, the suppression region (essentially, the region over which p-law nonlinearfry is operative) increases or broadens (Sachs and Kiang,1968; Dallos, Cheatham, and Ferraro, 1974). Dallos et al. (their Fig. 9), for instance, noted that maximum cochlear microphonic suppression occurred at anf2/f, value of approximately 2.0 in the guinea pig furl, - 5 kHz and at•2/f•= 1.15 for f• = 20 kHz. There is also some limited psychophysical evidence to support this supposition (Shannon, 1976, his Fig. 6). This relative broadening of the suppression region for low frequencies implies a broader frequency region over which the half-wave rectified p-law nonlinearfry is operational. Thus, the critical value of

f2/f, for Eq. (4), above which p - 1.0, may vary system- atically with frequency--being greater than approximately 1.5 for low frequencies and less than this at high frequencies.

In addition to a broadened frequency region (.fro//,) over which the p-law nonlinearity predominates at low frequencies, some recent psychophysical data suggest that the intensity range over which p decreases extends to high intensities for low frequencies (Humes, 1980a). These data are summarized by way of Fig. 8 in which psychophysical data on two-tone suppression ks a function of L, are displayed. Note, as reviewed previously, that two-tone suppression decreases at high intensities for.f,= 1000 and 2000 Hz. Such is not the case, however, for f• = 500 Hz. This suggests that the p-law nonlinearity is not as vulnerable to reversible dysfunction by high intensity acoustic input for 1owf• values. Thus, for low frequencies (f,< 750 Hz), p increases linearly as a function of input level (L,), at least for 40•L•-. < 85 dB SPL (55 •< L 2 •< 100 dB SPL).

IV. CONCLUSION

The ad hoc model of aural nonlinearfry described here appears to provide an adequate description of the most prominent perceptual two-tone distortion processes. It can account for many of the features of two-tone aural nonlinearity, including L(/• -L), L(2f• --f2), and two- tone suppression for/•>fl. One unique feature of the model is the incorporation of two-level-dependent nonlinearities, a half-wave rectified p-law device and the classical power-series nonlinearity. This combination of distortion processes is represented formally by Eq. (4). In that equation, it is noted that p varies as a function offm/fx. In addition, it is suggested that each of the nonlinearfries incorporated into Eq. (4) is associated with an independent intracochlear filtering process.

ACKNOWLEDGMENTS

The author would like to express his sincere appreci- ation to H. Duifhuis for his critical evaluation of an

earlier version of this manuscript. Thanks are also expressed to W. Rhode and G. Smoorenburg for their helpful discussions on this topic. Finally, I thank Barb Coulson for typing the manuscript.

IMoln,ar (1974) has demonstrated that the classical power series can predict almost any slope value by selecting an appropriate number of terms and arbitrary coeffioients. It is perhaps most appropriate,therefore, to refer to this as the "classical t•unc-ted power-series" nonlinearfry in that only the first few terms of the series are usually considered.

mir the classical power-series nouliuearity follows the p-law nonlinearfry, then Eq. (4) follows directly from substitution of Eq. (3) lute Eq. (1). It is suggested later, however, that the classical power-series uonlinearity precedes the p-law device.

aShannon (1976) suggests that his unmasking data can be converted to suppression data by utilizing the following relation: 1 dB of ,,-,masking corresponds to 3 dB of suppression. This assumes that suppression in au -urnasking paradigm may be interpreted as a simple reduction in masker level, which would appear to hold for the paradigm used by Shannon (Weber and Green, 1979).

CAlfhough the data on peripheral filtering to be reviewed here have all been obtained with nonsimultaneous techniques, it should be noted that q•litatiwlv similar findings had been obtained by many other investigators using simultaneous paradigms.


sir, in fact, the locus of the second filter is at the level of the hair cell, then it is quite probable that the asymmetric full~ wave rectified p-law noniinearity is more appropriate than the half-wave rectified version of this noniinearity (e.g., Hudspeth and Corey, 1977).

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2083 J. Acoust. Soc. Am., Vol. 67, No. 6, Ju•e 19l]0 Larry E. Humes: Two-tone nonlinearity 2083

on the nature of two-tone aural nonlinearityaudres/publications/humes/papers/18_humes.pdfon the...

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