Temporal Modulation Transfer Functions for Tonal Stimuli: Gated versus Continuous Conditions

Transcription

1 Auditory Neuroscience, Vol. 3(4), pp Reprints available directly from the publisher Photocopying permitted by license only 1997 OPA (Overseas Publishers Association) Amsterdam B.V. Published in The Netherlands by Harwood Academic Publishers Printed in Malaysia Temporal Modulation Transfer Functions for Tonal Stimuli: Gated versus Continuous Conditions WILLIAM A. YOST* and STANLEY SHEFT Parmly Hearing Institute, Loyola University, 6525 N. Sheridan Rd Chicago IL (Received 12 January 1996; Accepted 2 September 1996) Temporal Modulation Transfer Functions (TMTFs) were obtained for tonal carriers with frequencies of 500, 1000, and 4000 Hz. The carrier was either on continuously or gated. In the continuous-carrier condition the duration of the modulation was 125 or 500 msec, and in a gated-carrier condition the entire stimulus duration was 125 or 500 msec. Data were collected from six listeners using an adaptive, two-alternative, forced-choice procedure. The lowest thresholds were obtained for modulation rates between 4 and 8 Hz in the continuous-carrier condition and between 16 and 64 Hz in the gated-carrier condition. Several different processing strategies were used to try to account for the shape of the tonal TMTFs. The threshold changes with carrier frequency in the tonal TMTFs at high rates of modulation were best accounted for by assuming that listeners can detect the sidebands which fall outside of the critical band centered at the carrier frequency. The low- and mid-rate regions of the tonal TMTF were best described in a qualitative manner by a process that accumulates information based on the number of cycles of envelope fluctuations. However, this process could not quantitatively explain the tonal TMTF data in the low- to mid-rate regions nor could it account for the differences between tonal and noise TMTFs. Keywords: Temporal modulation transfer function (TMTF), sinusoidal amplitude modulation (SAM), envelope fluctuations, adaptation, multiple looks INTRODUCTION The function relating thresholds for detecting sinusoidal am plitude m odulation (SAM) to the rate of modulation has been called a Temporal Modulation Transfer Function (TMTF). A number of psychophysical studies have measured the TMTF for noise carriers (e.g., Rodenburg, 1977; Viemeister, 1979; Sheft and Yost, 1990; and Eddins, 1993 for reviews), but there are fewer data which describe TMTFs for tonal carriers (see Kohlrausch, 1993 for a discussion of tonal TM TF studies). In a study by Zwicker (1952), the modulated tones were presented continuously and a method-of-adjustment procedure was used to estimate SAM detection thresholds. These data suggest that the auditory system is most sensitive to SAM when the Corresponding author. Tel.: (312) Fax: (312) WYOST@LUC.EDU. 401

2 402 W. A. YOST and S. SHEFT tones are m odulated at approxim ately 4 Hz. Reisz (1928), also using a continuous stimulus, established that for two beating tones the perception of best- beats occurs at approximately 3 to 4 Hz. Results from these two studies have been used to suggest that the auditory system is most sensitive to modulation rates of 3 to 4 Hz (see Viemeister, 1979). Although we are unaware of any recent studies of tonal TMTFs, several different lines of research have used SAM o f tonal carriers. In recent studies of M odulation Detection Interference (e.g., Yost and Sheft, 1989; Hall and Grose, 1990; M oore et al., 1991; Y ost and Sheft, 1994), the SAM thresholds for gated tones are higher than those reported by Zwicker for continuous tones. In other studies using gated tonal carriers, modulation thresholds from only some of the listeners were greater than the Zwicker results (e.g., M oore and Sek, 1992 and 1995; Edwards and V iem eister, 1994a and b). Viemeister (1979), Forest and Green (1987) and Sheft and Yost (1990) have all shown that SAM detection thresholds at low modulation rates are higher for gated than for continuous noise carriers. The loss in sensitivity at low modulation rates in the gated-noise conditions results in the lowest m odulation detection thresholds being obtained at rates considerably higher than 3-4 Hz. Sheft and Yost (1990) also showed that the shape of the TMTFs for noise depends on stimulus duration. There is no current study comparing tonal TMTFs with gated and continuous carriers and as a function of duration. Since both of these variables affect noise TMTFs and there is some evidence of an effect o f carrier gating on tonal TM TFs, w e determ ined tonal TM TFs as a function of duration and gated versus continuous presentation. In general the TMTFs for tones have three regions based on modulation rate. In the high-rate region, the sidebands due to SAM become detectable leading to m odulation thresholds decreasing as rate increases. The rate at which sidebands are just detectable depends on the carrier frequency. Because the width of the critical band increases with frequency, a higher SAM rate is needed for sideband detection as the carrier frequency is increased. In the m id-rate region several percepts occur. For rates above about 50 Hz at high modulation depths, a complex pitch equal to the SAM rate can be detected. Also, in this region the roughness of the tem poral fluctuations is detectable but disappears at low modulation depths and high rates. In the m id-rate region, thresholds tend to increase as rate increases. In the low-rate region at high modulation depths there is a clear waxing and waning of loudness. That is, the percept is no longer that of a stable sound but one of a sound of a particular pitch whose loudness is changing over time. At these low rates, thresholds either remain somewhat constant as a function of modulation rate or increase as rate decreases, especially when the rate is less than 4 Hz. Some of the literature cited above suggests that the modulation rates defining these three regions of the TM TF depend on whether the tones are gated or continuous. A m plitude m odulated tones are often used to obtained neural TMTFs for different locations in the auditory nervous system (e.g., Schreiner and Urbus, 1988; Jorris and Yin, 1992; Frisina et al., 1996). Most of the physiological studies of tonal TM TFs use gated signals, yet com parisons are often m ade to psychophysical data obtained from studies using continuous carriers. Thus, docum entation of differences between TMTFs obtained with gated and continuous carriers would provide useful inform ation when com paring physiological and psychophysical measures. METHODS The stimuli are described by the following equation: [1 + msin(27tfm)][cos(27ifc)]/[(l + m2/2 )^ ], (1) where m (0<m <l) is the depth of modulation, fm is the modulation rate (2, 4, 8, 16, 32, 64, and 128 Hz), and fc is the carrier frequency (500, 1000, and 4000 Hz). The carriers were presented at 70 db SPL. The carrier was either gated on and off in each observation interval or was on continuously during a block of trials. Stimulus durations of 125 and 500 msec were used in the gated condition. The modulation had a duration of either 125 or 500 msec in the continuous condition. In the gated conditions, the waveforms were shaped with a 5-msec raised cosine. Table I describes the basic stimulus conditions.

3 TONAL TMTFs 403 TABLE I Stimulus conditions and use of listeners. Labels SI to S6 identify subjects who participated in each condition Duration Carrier Frequencies 500 Hz 1000 Hz 4000 Hz Gated 125 msec Modulation rates of 8,16,32,64,128 Hz Carrier (S1-S3) (S1-S3) (S1-S3) 500 msec Modulation rates of 2,4,8,16,32,64,128 Hz (S1-S3) (S1-S3) (S1-S6) 125 msec Modulation rates of 8,16,32,64,128 Hz Continuous (S1-S3) (S1-S3) (S1-S3) Carrier 500 msec Modulation rates of 2,4,8,16,32,64,128 Hz (S1-S3) (S1-S3) (S1-S6) The thresholds (expressed as 20 log m) were obtained with a two-alternative, forced-choice adaptive procedure. The two observation intervals followed a 500-msec warning interval and were separated by 500 msec. Feedback was provided after each trial in the 50-trial block. In the gated conditions, the signal interval contained the m odulated stimulus and the non-signal interval contained the unmodulated carrier. In the continuous condition, the carrier was on continuously and was modulated only during the signal interval. The Levitt (1971) two-down, one-up method was used to estimate the 70.7% point on the psychometric function. The step size in terms of 20 log m was 4 db until the second reversal and then it was reduced to 2 db. At least ten reversals of modulation depth (20 log m) were required before a threshold for a block was calculated. Excluding the first four reversals, the threshold for any 50 trial block was estimated by averaging the modulation depths of the remaining reversals. At least four thresholds were estimated for each condition. If the four threshold estimates were within ±3 db of each other, then the final threshold was the average of these four thresholds. If the thresholds were not within ±3 db of each other, then additional thresholds were obtained until four consecutive thresholds were within the ±3 db range. The four that were within this range were then averaged for the final threshold estimate. For the majority of the conditions a final threshold was obtained with six or fewer threshold estimates. Six listeners with normal thresholds of hearing were used in the study. Data for all conditions were obtained from three listeners (SI, S2, S3). Data for the 4000-Hz carrier, the gated and continuous conditions at 500 msec were obtained for the three other listeners (S4, S5, S6). See Table I for a description of the assignment of listeners to the various conditions. The stimuli were presented diotically over TDH-49 headphones mounted in MX/41 AR cushions to the listeners seated in a soundproof booth. RESULTS The average TMTFs are shown in Figures 1-4. Figure 1 shows the data for the gated conditions, Figure 2 the data for the continuous conditions, Figure 3 a replotting of the data for the 500-msec conditions, and Figure 4 a replotting of the data for the 125-msec conditions. W ith the exception of many of the data for the 125-msec condition, the TMTFs can be divided into the three general regions described in the Introduction. The divisions are general descriptions of the TMTFs and often depend on carrier gating and duration. For the 500-msec conditions, the low-rate region extends up to 8 Hz for continuous carriers and up to 64 Hz for gated carriers. In the low-rate region thresholds decrease with increasing modulation rates. The low- rate region is more evident with the 500-msec than with the 125-msec durations. For the 500-msec conditions, somewhat independent of whether or not the tones were gated or continuous, the mid-rate region extended up to the highest rate (128 Hz) tested for the 4000-Hz carrier, to about 64 Hz for the 1000-Hz carrier, and to about 32 Hz for the 500-Hz carrier. There

4 404 W. A. YOST and S. SHEFT Gated so o o (N Modulation Rate (Hz) msec, 500-Hz 125-msec, 1000-Hz 125-msec, 4000-Hz A 500-msec, 500-Hz 500-msec, 1000-Hz - F I 500-msec 4000-Hz FIGURE 1 Average AM detection thresholds for the gated conditions as a function of modulation rate (in Hz). Data are shown for 125-msec (closed symbols) and 500-msec conditions (open symbols) for three carrier frequencies (500 Hz, triangles; 1000 Hz, circles; and 4000 Hz, squares). was little evidence of a mid-rate region for 125-msec durations except perhaps for the 4000-H z carrier, where the mid-rate region again extended to 128 Hz. In this mid-rate region, thresholds increase with increasing m odulation rate. The high-rate region, where thresholds decrease with increasing rate, was clear for the 500 and 1000-Hz carriers, but did not occur for the 4000-Hz carrier. For the 125-msec gated conditions, there is no break in the shape of the TMTFs as a function o f rate, except for the 4000-Hz carrier. In addition: 1) thresholds are lower for the continuous than for the gated conditions, especially at low modulation rates; 2) the thresholds are lower for the 500-msec conditions than for the 125-msec conditions, especially at low modulation rates; 3) in the low- and mid-rate regions, thresholds are lowest for modulation rates between 4 and 8 Hz in the 500-msec continuous conditions, and they are lowest for modulation rates of Hz for the 500-msec gated conditions; 4) although the slope of the TM TF from mid to high rates is difficult to estimate for many conditions, it is between 2.5 and 4.5 db/octave in the 500-msec conditions, while the slope on the low-rate side of the 500-msec TMTFs is less than 3 db/octave for the continuous conditions and closer to 3 db/octave for the gated conditions. For all conditions shown in Figures 1-4 the standard error of the mean computed for the six listeners ranged from 0.9 to 1.7 db. For each condition all listener s

5 TONAL TMTFs 405 Continuous 125-msec, 500-Hz 125-msec, 1000-Hz 125-msec, 4000-Hz r- 500-msec, 500-Hz - Q 500-msec, 1000-Hz n 500-msec, 4000-Hz FIGURE 2 Average AM detection thresholds for the continuous conditions as a function of modulation rate (in Hz). Data are shown for 125-msec (closed symbols) and 500-msec conditions (open symbols) for three carrier frequencies (500 Hz, triangles; 1000 Hz, circles; and 4000 Hz, squares). thresholds changed in approximately the same way with modulation rate. The major difference among listeners was in their overall sensitivity for detecting modulation. DISCUSSION As pointed out by many others (e.g., Zwicker, 1952 and Viemeister, 1979), the drop in thresholds at high rates of modulation is most likely the result of sideband detection. Listeners can discriminate the modulated from the unmodulated stimulus by listening for the presence of the sidebands rather than detecting the temporal fluctuations in the envelope. If the critical band at 500 Hz is slightly narrower than 100 Hz, then thresholds for both 64-Hz and 128-Hz rates of modulation should reflect sideband detection. A critical band at 1000 Hz of approximately 120 Hz rate allows for sideband detected with 128-Hz SAM. If the critical band at 4000 Hz is much wider than 128 Hz, then sideband detection would not be expected at any of the rates used. These predictions are reflected in the data of Figures 1-4. The rise in thresholds with increasing modulation rate in the mid-rate region of the TMTF has usually been attributed to a lowpass filter stage for processing temporal envelopes (see Viemeister, 1979; Sheft and Yost, 1990 for a complete discussion of this lowpass filter assumption). The integration time of the lowpass

6 406 W. A. YOST and S. SHEFT 500-msec. 10 -J r : ' i t * : i n Modulation Rate (Hz) -A Cont., 500-Hz Cont., 1000-Hz Cont, 4000-Hz A Gated, 500-Hz O Gated, 1000-Hz Gated, 4000-Hz FIGURE 3 Average AM detection thresholds for the 500-msec conditions as a function of modulation rate (in Hz). Data are shown for gated (open symbols) and continuous conditions (filled symbols) for three carrier frequencies (500 Hz, triangles; 1000 Hz, circles; and 4000 Hz, squares). filter sets the m odulation rate at w hich threshold begins to rise with increasing m odulation rate; the number of filter stages or the order of the filter determines the rate at which threshold increases. Since the TMTFs are clearly not like the output of a lowpass filter, but more like the output of a bandpass filter, it is clear that either a lowpass filter is an inappropriate model for these tonal TMTFs or that processes in addition to lowpass filtering are involved. Sheft and Yost (1-990) and, earlier, V iem eister (1979) discussed a few possible explanations for the low-rate region of noise carrier TM TFs. However, none of these explanations could account for their data. Less attention has been paid to the low-rate region for tonal carriers. The data from Figure 1 clearly indicate a low-rate region for 500-msec gated tonal carriers. For the 4000-Hz carrier, where sideband detection does not occur, the TMTF is clearly bandpass. We will next investigate a few mechanisms that might account for the bandpass shape of the TMTFs: adaptation of various forms, and lateral inhibition. Adaptation Viemeister (1979) suggested that losses in modulation sensitivity at low AM rates with a gated noise carrier might be due to short-term neural adaptation. He proposed that the modulation of the neural response by adaptation might interfere with the detection of m odulation. W ith the largest change in the neural response

7 TONAL TMTFs msec Modulation Rate (Hz) A Cont., 500-Hz 9 Cont., 1000-Hz Cont., 4000-Hz A Gated, 500-Hz O Gated, 1000-Hz Gated, 4000-Hz FIGURE 4 Average AM detection thresholds for the 125-msec conditions as a function of modulation rate (in Hz). Data are shown for gated (open symbols) and continuous conditions (filled symbols) for three carrier frequencies (500 Hz, triangles; 1000 Hz, circles; and 4000 Hz, squares). by adaptation occurring near stimulus onset, the interference w ould be greatest at low m odulation rates where few cycles of modulation are present. Sheft and Yost (1990) evaluated the effect of adding a 500-msec unmodulated forward carrier fringe on the detection of low rate SAM with a noise carrier. The addition of the 500-msec fringe did not elim inate the highpass segment of the TMTF. Since 500 msec is much longer than the time course of neural measures, these results suggested that m echanism s other than short-term neural adaptation are required to account for the bandpass characteristics of the TMTF. Sheft and Yost (1990) suggested that the highpass segment of the gated noise-carrier TM TF may reflect involvem ent o f an adaptation process slow er than short-term neural adaptation. Neural channels selective to AM rate may be the site of one such process. However, attempts by Sheft and Yost (1990) failed to confirm this concept. In addition, there is no convincing argument to limit the effect of this form of adaptation to low AM rates. In general, the fact that tonal TMTFs from both gated and continuous carrier conditions are bandpass leaves consideration of adaptation by itself an inadequate explanation for the bandpass shape of the tonal TMTF. Lateral Inhibition Procedures similar to the ones used to obtain auditory TMTFs have been used in vision to obtain temporal

8 408 W. A. YOST and S. SHEFT modulation transfer functions (see Comsweet, 1970). Gated conditions are almost always used to estimate the visual TMTFs, and the shape of the modulation transfer functions are bandpass. Lateral inhibition is a common process used to explain the bandpass characteristics of the visual TM TFs. The TMTF is treated as a transfer function and its inverse Fourier transform (assuming in our case a zero phase-function) is used to generate a temporal weighting function (an impulse response). Figure 5 shows the tem poral weighting function generated from the inverse Fourier transform of the TMTF obtained with the 500-msec, 4000-Hz tonal carriers in the gated condition (see Fig. 1). The 4000-Hz carrier TM TF was used since it appears to be unaffected by sideband detection. The negative portion of the tem poral weighting function is viewed in vision as indicating a lateral inhibitory mechanism in temporal processing. By convolving a stimulus with this temporal weighting function, one can m ake predictions regarding tem poral resolution. Although such temporal weighting functions have been suggested for visual processing, their use in audition is more limited. Plack and M oore (1990) have assumed a temporal weighting function derived from a variety of different tem poral procedures, but these temporal weighting functions do not show evidence of a lateral inhibitory mechanism. Gilkey and Robinson Temporal Weighting Function 1 <D = 0.8 f- 0.6 < o 0.4 o> N E o 0 z Time - msec. FIGURE 5 Temporal weighting function obtained for the inverse Fourier Transform of the tonal TMTF for the 4000-Hz carrier in the gated condition. See Text for additional information. (1986) derived tem poral weighting functions for detecting tones in frozen noise, and some of these temporal weighting functions did go negative indicating the possibility of a lateral inhibitory mechanism. Since the shape of the TMTFs differ as a function of carrier frequency, duration, and with carrier gating, each condition would require a different weighting function. In vision, a single w eighting function is usually m easured, and there is no a priori reason why different conditions in audition should yield different temporal weighting functions. In the auditory system, unlike in vision, there is no known physiological mechanism for the form of lateral inhibition illustrated in Figure 5. In applying the weighting function, one would also have to deal with the issue of causality, since the weighting function extends to both positive and negative time. M ultiple-looks Perhaps listeners use the modulation cycle as the basis for a decision that modulation has occurred (as described by Viemeister,1979). The more cycles that occur, the more looks the listener has at this decision variable, and perform ance improves as the num ber of looks (number of cycles) increases. Thus, with only a few looks the performance is poor and it improves as the number of modulation cycles increases, explaining the drop in thresholds in the low-rate region as modulation rate increases. Using the number of envelope periods is also one way o f equating the 125-msec and 500-msec conditions. If number of envelope periods is a reasonable way of accounting for the TMTF then one might expect TMTFs for the 125- and 500-msec durations to be similar at least in the low- and mid-rate regions (where sideband detection is not possible). Figure 6 shows the TMTFs for the gated (on top) and continuous (on bottom) conditions plotted as a function of the number of envelope periods (that is, dividing rate by 2 for the 500-msec data and dividing rate by 8 for the 125-msec data of Figures 1 and 2). In the low-rate region the TMTFs tend to align fairly well for the gated conditions, but with more spread for the continuous conditions. Thus, there might be some validity in attributing the shape of the TM TF in the low-rate region, at least

9 TONAL TMTFs 409 Gated I i I r Number of Modulation Periods 125-msec, 500-Hz 125-msec, 1000-Hz 125-msec, 4000-Hz -A 500-msec, 500-Hz O 500-msec, 1000-Hz Q 500-msec, 4000-Hz Continuous A 125-msec, 500-Hz 125-msec, 1000-Hz 125-msec, 4000-Hz A 500-msec, 500-Hz O 500-msec, 1000-Hz -^3 500-msec, 4000-Hz FIGURE 6 The TMTFs of Figures 1 and 2 are replotted as a function of the number of modulation periods. That is, modulation rate was divided by 2 for the 500-msec conditions and by 8 for the 125-msec conditions. The gated conditions are shown on top and the continuous conditions are shown on the bottom.

10 410 W. A. YOST and S. SHEFT for the gated condition, to a process based on accumulating inform ation from each cycle of m odulation. However, such a simple multiple-looks explanation would not predict the bandpass shape of the TMTF, since performance should continue improving as the number of looks increases. To use such an explanation to describe the low-and m id-rate regions of the TM TFs, we have adopted aspects of the model proposed by H after and his colleagues (see Yost and Hafter, 1987 or Hafter et al., 1988). Hafter et al. (1988) developed the model to account for click-lateralization thresholds. We are adopting the model to account for the TMTFs. The m ajor assumption of our im plementation of this m odel is that when m odulation rate increases, the listener has a greater number of looks to determine if the tone was modulated. For the present derivation we assume that each observation of the envelope period is an independent event and the measure of the listener s performance is proportional to the independent variable (e.g., M oore and Sek, 1992, have shown that d' is proportional to m2 for SAM tones). W ith these assum ptions, m odulation thresholds in terms of m are predicted to decrease at a rate equal to the square root of the number of observations of the envelope period (1/N0 5, where N is the num ber of observations). If we assum e, as H after has, that there is a com pressive property operating such that each successive observation contributes progressively less to the decision, then the decline in threshold is proportional to 1/N 5k, where 0 < k < 1. W e assume further (as has Hafter) that k is a function of the rate of modulation, in that k becomes smaller as the rate increases, such that at fast rates the observations come so close together there is no accum ulation of information due to multiple observations. The assumption of the compressive function appears reasonable in that if the period of modulation is too short then some form of integration might make it difficult to determine the envelope period. This is similar to the rationale given by Hafter in his application o f the model to interaural difference thresholds for repeated binaural clicks. This function for accum ulating inform ation as modulation rate increases has a bandpass shape, first increasing and then decreasing as N increases. In order to provide some insight about this relationship, we used a sim ple exponential to define the relationship between k and the modulation rate; that is, k = exp(-cr), where c is a constant 0 < c > 1, and R is the modulation rate [R = fm in equation 1)]. Therefore, this suggests that thresholds would change as: l/n '5[exP(~cR)l (2) If we assume that performance (e.g., d') is proportional to m2 as suggested by Moore and Sek, (1992), then 20 log m is related to N by 10 log(l/na5[exp(-cr)]). Figure 7 shows a family of curves (with c as the parameter from.005 to.03 in.005 steps) plotted in terms of 20 log m versus N. The curves for 500-msec duration are on top and those for 125-msec duration are on the bottom. These curves have several qualitative similarities to the tonal TMTFs. The curves are bandpass in shape as are the TMTFs. For both the predictions and the actual TMTFs there is a larger change in performance as a function of N for 500 msec than for 125 msec. For any one value of c, the peak in the predicted function shifts to a lower number of modulation periods as the duration is lowered from 500 to 125 msec. This is like the peak sensitivity in the TMTF shifting to a lower modulation rate when the duration is lowered. The slopes on the low- and high rate sides are 1,5-dB per octave (or slightly less) for both the 500-msec predictions.1 Even at a qualitative level this approach cannot account for the differences in the TMTFs due to carrier frequency and carrier gating. However, there is one assum ption that can be m ade that will qualitatively m ake the predictions fit with some of the changes in the TMTFs between gated and continuous presentations. One might assume that in the continuous conditions the change from the continuous to a modulated tone and the change back from modulation to a continuous tone serve as two more transitions in 1 Moore and Sek (1992) showed that d' is proportional to m2 for gated tones in the low-rate region of the TMTF. If performance is also proportional to 1/N0'5, then one might predict that the rise in sensitivity with increasing rate in the low-rate region would be 1.5 db per doubling of the number of observations. However, if performance is proportional to 1/Nk05, as we suggest, then the rise in sensitivity with number of observations depends on k as well, and would be between 1.5 and 3 db per doubling of observations. Thus, there is a qualitative agreement between these calculations and the obtained TMTFs.

11 TONAL TMTFs m sec. 125 m sec. FIGURE 7 The predictions based on the equation 201og(l/Na5"xp<-cR)1) plotted against N, where N is the number of modulation periods, R is modulate rate, and c a constant which is the parameter on the figure with c ranging from (top curve) to 0.03 (bottom curve) m steps. Functions for the 500-msec duration are shown on top and those for 125-msec duration are shown on the bottom.

12 412 W. A. YOST and S. SHEFT level that might be used to define two more modulation cycles. By adding 2 to each N of the predictions shown in Figure 7, the peak will shift to a lower number of modulation periods as it does in the TMTFs when the conditions change from gated to continuous. Even though there are a number of qualitative similarities between the curves of Figure 7 and the data of Figure 6, there are many quantitative discrepancies. The maximum change in performance for the predictions is less than 6 db for the 500-msec conditions and less than 4 db for the 125-msec conditions. This is 10 or more db less than obtained in the data. As N increases the predicted perform ance can never be worse than that at N = 1. The TMTFs clearly show that this is not true in the mid-rate region. The slopes of the low- and high-rate regions are never more than 1.5-dB per octave in the predicted functions, yet these slopes can be more then 3-dB per octave in the obtained TM TFs. The shape of the predicted functions with 125-msec duration is always bandpass, whereas the TMTFs for 125 msec are not always bandpass. Also, for the continuous conditions for 125-msec stimuli, the different thresholds for the different carrier frequencies in the low rate region of the TM TF is not accounted for by this model. As Figure 6 indicates, performance is not exactly the same for both durations when considered on the basis of the number of modulation periods. Thus, the model based on accumulating information over the num ber o f m odulation periods does not appear to quantitatively account for the shapes of the TM TFs. However, the qualitative com parisons suggest that there is some validity to a consideration that processing may be partially based on accumulating information over m odulation periods in the manner suggested, especially in the low-rate region. In addition, we assumed that k = exp(-cr ) to describe how k might change with modulation rate. W e have no prior reason for using this exponential relationship expect for the fact that k decreases as R increases which is the desired relationship. O ther relationships m ight provide a better account of the data, but at the moment we have no basis for choosing any particular one. W e have been unable to develop a single model which accounts for the tonal TMTFs and which would be compatible with noise-carrier TMTFs. This may be because the TMTF, especially the tonal TMTF, may not reflect a single process. It is already acknowledged that there are at least two processes involved with tonal TMTFs, one that operates in the high-rate region that takes into account sideband detection, and another that covers the low- and mid-rate regions. It might also be that the low- and mid-rate regions depend on more than one process. After all, as was discussed in the Introduction, the percept varies with modulation rate. These percepts also interact with carrier frequency and duration. For instance, the saliency of the pitch of SAM tones depends on the carrier frequency (see Cohen et al., 1995, for a recent discussion o f the laws of complex pitch) and duration. Roughness depends on duration (see Terhardt, 1968). If these percepts are used by listeners for AM detection, then one might expect that more than one process might be involved in determining a tonal TMTF. ADDITIONAL DISCUSSION The TMTFs obtained in this study differ from those reported by Zwicker (1952) and to some extent from those reported by Viemeister (1979). There appear to be two major differences: the thresholds obtained by Zwicker are 5 to 10 db lower than those obtained in this study (and also by Viemeister, 1979) at the comparable sound levels; and the TMTFs obtained by Zwicker and Viemeister (1979) clearly have their minimum at a modulation rate of 4 Hz, with the minimum for our data are at a slightly higher modulation rate. There are major procedural differences between Zwicker s and our study which might help explain the differences in the TMTFs. Zwicker used a method of adjustment procedure (Bekesy tracking) with continuously modulated tones. That is, the listeners adjusted the depth of a continuously modulated tone until they no longer detected modulation. Perhaps Zwicker s procedure is a more sensitive psychophysical procedure for measuring tonal TMTFs than the method of constant stimulus used in this study (see Viemeister, 1979). Our method was very close to that used by Viemeister (1979) and he found the peak to be at 4 Hz for the continuous condition, not at 8 Hz. There were twelve TMTFs obtained in this study for the continuous condition at 500 msec (3 listeners for 500-Hz

13 TONAL TMTFs 413 carrier, 3 for 1000-Hz carrier, and 6 for 4000-Hz carrier, see Table I). Of these twelve TMTFs, six had best sensitivity at 4 Hz, four had best sensitivity at 8 Hz, and two had equal thresholds at 4 and 8 Hz. The peak sensitivity for two of TMTFs was much greater at 8 Hz than it was for the other TMTFs. Thus, on average the best sensitivity in the TMTF was at 8 Hz, but more listeners generated TMTFs in the continuous case (at 500 msec) with a peak at 4 Hz. A major empirical finding of this study is that the modulation rate at which modulation-detection thresholds are lowest is higher in the gated than in the continuous condition and the rate for best sensitivity depends on the duration of the carrier. This finding of best sensitivity at higher rates for the gated than for the continuous condition for tonal TMTFs has also been found for the TMTFs with noise carriers (Viemeister, 1979; Sheft and Yost, 1990). There is evidence of this same trend in the literature on tonal modulation (e.g., Yost and Sheft, 1989; Green and Nguyen, 1988; Hall and Grose, 1990; Moore et al., 1991; Yost and Sheft, 1994), but other studies have found that this trend is listener dependent (Moore and Sek, 1992, 1995; Edwards and Viemeister, 1994a,b). It is difficult to compare across studies since tonal TM TFs depend on many stimulus conditions (modulation rate, carrier frequency, duration, gated versus continuous presentation and overall level are the major variables for tonal TMTFs). Also Moore and Sek (1995) suggested that detecting modulation might depend on different strategies adopted by listeners in determining what to listen for in a modulation detection task. The large effects many of these variables have on tonal TMTFs was one of our motivations for the present study comparing several of these variables with the same set of listeners. This comparison mirrors the work with noise carriers showing that the rate at which best sensitivity occurs is higher for gated than for continuous presentations. As such, the dictum that best modulation detection occurs at 3-4 Hz may only apply to'continu- ous stimuli (see Zwicker and Fasti, 1990). A comparison of the data from this and other studies of AM detection with tonal carriers (e.g., Yost and Sheft, 1989; Green and Nguyen, 1988; Hall and Grose, 1990; Moore et a l, 1991; Yost and Sheft, 1994) with the data from the study of AM detection with noise earners (Rodenburg, 1977; Viemeister, 1979; Forest and Green, 1987; Sheft and Yost, 1990) suggests that the form of the TMTF for the two types of carriers is different. The noise carrier TMTFs appear to be much more lowpass than the bandpass characteristics of the tonal TMTF, although the noise carrier TMTFs is bandpass in shape for gated conditions (Viemeister, 1979; Sheft and Yost, 1990). Viemeister (1979), Forest and Green (1987), and Sheft and Yost (1990) have all offered explanations of the noise-carrier TMTF that involve assumptions of lowpass filtering. Since the tonal-carrier TMTFs are more bandpass than lowpass, processes in addition to lowpass filtering appear needed to account for the tonal data. However, none of the models discussed in this paper appear capable of accounting for all of the tonal TMTF data and for the differences between tonal and noise TMTFs. Thus, a single comprehensive model of TMTFs is still not available. Acknowledgements We would like to thank the faculty of the Parmly Hearing Institute for their many valuable comments on this work and paper. This research was supported by a Program Project Grant from the National Institute on Deafness and Other Communication Disorders, NIH and a grant from the Air Force Office of Scientific Research. References Cohen, M. A., Grossberg, S. and Wyse, L. L. (1995) A spectral network model for pitch perception, J. Acoust. Soc. Am., 98, Cornsweet, T. (1970) Visual Perception, Academic Press, New York. Eddins, D. A. (1993) Amplitude modulation detection of narrowband noise: Effects of absolute bandwidth and frequency region, J. Acoust. Soc. Am., 93, Edwards, B. W. and Viemeister, N. E. (1994a) Modulation detection and discrimination with 3-component signals, J. Acoust. Soc. Am., 95, Edwards, B. W. and Viemeister, N. E. (1994b) Frequency modulation versus amplitude modulation: Evidence for a second frequency modulation coding mechanism, J. Acoust. Soc. Am., 96, Forest, T. G. and Green, D. M. (1987) Detection of partially filled gaps in noise and the temporal modulation transfer function, J. Acoust. Soc. Am., 82, Frisina, R. P., Karcich, K. J., Tracy, T. C., Sullivan, D. M. and Walton, J. P. (1996) Preservation of amplitude modulation coding in the presence of background noise by the Chinchilla auditory nerve fibers, J. Acoust. Soc. Am., 99,

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