Psychometric functions for pure-tone frequency discrimination

The form of the psychometric function (PF) for auditory frequency discrimination is of theoretical interest and practical importance. In this study, PFs for pure-tone frequency discrimination were measured for several standard frequencies (200-8000 Hz) and levels [35-85 dB sound pressure level (SPL)] in normal-hearing listeners. The proportion-correct data were fitted using a cumulative-Gaussian function of the sensitivity index, d', computed as a power transformation of the frequency difference, Δf. The exponent of the power function corresponded to the slope of the PF on log(d')-log(Δf) coordinates. The influence of attentional lapses on PF-slope estimates was investigated. When attentional lapses were not taken into account, the estimated PF slopes on log(d')-log(Δf) coordinates were found to be significantly lower than 1, suggesting a nonlinear relationship between d' and Δf. However, when lapse rate was included as a free parameter in the fits, PF slopes were found not to differ significantly from 1, consistent with a linear relationship between d' and Δf. This was the case across the wide ranges of frequencies and levels tested in this study. Therefore, spectral and temporal models of frequency discrimination must account for a linear relationship between d' and Δf across a wide range of frequencies and levels.     

Psychometric functions for frequency discrimination from listeners with sensorineural hearing loss

Psychometric functions for pulsed pure-tone frequency discrimination were obtained from hearing-impaired listeners at frequencies with normal hearing and at frequencies with mild or moderate hearing losses. The general form of psychometric functions at hearing-impaired frequencies was found to be the same as at normal-hearing frequencies, i.e.,d' was linear with the frequency difference between tones, in Hz. For all but one psychometric function, the addition of an intercept term to the fitting equation did not account for significantly more variance than did the slope term alone. Therefore, it was concluded that psychometric functions for frequency discrimination can be adequately described with only one parameter: the slope of the psychometric function. Deficits in discrimination at hearing-loss frequencies were manifested by more gradual slopes of psychometric functions. Procedures for normalizing psychometric functions are presented, which facilitate comparisons of normal and impaired frequency discrimination data across studies and frequencies. Comparisons of dlf's (difference limen for frequency) obtained with adaptive and fixed procedures show a bias toward larger dlf's with adaptive procedures, but only at higher frequencies. A discussion of equal-interval and equal-ratio adaptive stepping rules indicates that an equal-ratio rule may be preferable.     

Psychometric functions for level discrimination and the effects of signal duration in the goldfish (Carassius auratus): psychophysics and neurophysiology.

Classical conditioning of respiration was used to obtain psychometric functions for pulsed tone level discrimination in the goldfish (Carassius auratus). Conditioned respiratory suppression is a graded response that has some properties of a confidence rating measure. These properties were used to obtain receiver operating characteristics (ROC) and psychometric functions using a blocked method of constant stimuli. Empirical ROCs and neurometric functions were also obtained for single auditory-nerve fibers using spike count as the decision variable in order to evaluate a simple rate code for level discrimination. Psychometric and neurometric functions for level discrimination are similar in showing the same general form (summarized by Weibull functions) that is independent of signal duration. The lower slope of neurometric functions compared with behavioral functions for level discrimination is in accord with similar data on sound detection and vision in nonhuman mammals. Both neural and psychophysical level discrimination thresholds decline with increasing duration (20 to 320 ms), with similar slopes except at short signal durations (20 to 50 ms). At these durations, the animal's use of a channel-selection strategy and neural information following stimulus offset could reduce the difference between neural and psychophysical thresholds. The slopes of the neural and psychophysical duration functions are similar to those for human observers, but the majority of auditory-nerve fibers sampled have lower level discrimination thresholds than the behaving animal. Since human observers perform better than the majority of neurons in level discrimination, well-trained human listeners may be able to select channels with superior information, or to combine information across channels in ways that the goldfish and other animals do not. In general, one is encouraged to believe that neural mechanisms need not be more complex or sensitive than those considered here to account for pure-tone level discrimination in fishes, humans, and other vertebrates.     

Psychometric functions for discrimination of two-component complex tones in listeners with normal hearing and listeners with hearing loss

This study compared the ability of 5 listeners with normal hearing and 12 listeners with moderate to moderately severe sensorineural hearing loss to discriminate complementary two-component complex tones (TCCTs). The TCCTs consist of two pure tone components (f1 and f2) which differ in frequency by delta f (Hz) and in level by delta L (dB). In one of the complementary tones, the level of the component f1 is greater than the level of component f2 by the increment delta L; in the other tone, the level of component f2 exceeds that of component f1 by delta L. Five stimulus conditions were included in this study: fc = 1000 Hz, delta L = 3 dB; fc = 1000 Hz, delta L = 1 dB; fc = 2000 Hz, delta L = 3 dB; fc = 2000 Hz, delta L = 1 dB; and fc = 4000 Hz, delta L = 3 dB. In listeners with normal hearing, discrimination of complementary TCCTs (with a fixed delta L and a variable delta f) is described by an inverted U-shaped psychometric function in which discrimination improves as delta f increases, is (nearly) perfect for a range of delta f's, and then decreases again as delta f increases. In contrast, group psychometric functions for listeners with hearing loss are shifted to the right such that above chance performance occurs at larger values of delta f than in listeners with normal hearing. Group psychometric functions for listeners with hearing loss do not show a decrease in performance at the largest values of delta f included in this study. Decreased TCCT discrimination is evident when listeners with hearing loss are compared to listeners with normal hearing at both equal SPLs and at equal sensation levels. In both groups of listeners, TCCT discrimination is significantly worse at high center frequencies. Results from normal-hearing listeners are generally consistent with a temporal model of TCCT discrimination. Listeners with hearing loss may have deficits in using phase locking in the TCCT discrimination task and so may rely more on place cues in TCCT discrimination.     

Psychometric functions for gap detection in a yes-no procedure

To examine models of temporal resolution and to investigate the decision processes underlying the detection of a brief pause in a bandpass noise, psychometric functions for gap detection were measured at octave frequencies from 0.25 to 8 kHz. Three normal listeners were tested using a constant-stimulus procedure with a cued Yes-No paradigm. The Minimum Detectable Gap (MDG) estimated from the midpoint of the psychometric functions decreased systematically with increasing frequency. The slopes of the psychometric functions generally increased as the test frequency increased up to 2 kHz, but remained constant at the higher frequencies. Two models were investigated: an energy-detector model and a loudness-detector model. Both consisted of auditory filtering, a nonlinearity, and short-term integration. In the energy-detector model, the nonlinearity was a square law. In the loudness-detector model, it was a compressive power law. Using the usual Gaussian approximations, the energy-detector model fails at low frequencies because the probability distributions of short-term energy differ from Gaussian distributions. The probability distributions of short-term loudness closely follow Gaussian distributions. The loudness-detector model predicts the frequency dependence of the MDG quite accurately, except at 0.25 kHz. It also predicts psychometric functions that resemble the data at low frequencies, but the predicted slopes increase much less with frequency than the measured slopes. This result may indicate that the onset response to the trailing marker of the gap provides an important cue for detection of gaps with durations exceeding the MDG.     

Monaural level discrimination under dichotic conditions

The ability to make judgments about the stimulus at one ear when a stimulus is simultaneously presented to the other ear was tested. Specifically, subjects discriminated the level of a 600 Hz target tone presented at the left ear while an identical-frequency distractor was simultaneously presented at the other ear. When there was no distractor, threshold was 0.7 dB. Threshold increased to 1.1 dB when a distractor with a fixed phase and level was introduced contra-aurally to the target. Further increases in threshold were observed when an across-presentation variability was introduced into the distractor phase (threshold of 1.6 dB) or level (threshold of 5.8 dB). When both the distractor level and phase varied, the largest threshold of 7.3 dB was obtained. These increases in threshold cannot be predicted by common binaural models, which assume that a target stimulus at one ear can be processed without interference from the stimulus at the nontarget ear. The measured thresholds are consistent with a model that utilizes two binaural dimensions that roughly correspond to the loudness and the position of a fused binaural image. The results show that, with binaurally fused tonal stimuli, subjects are unable to listen to one ear.     

Observer weighting strategies in interaural time-difference discrimination and monaural level discrimination for a multi-tone complex

Two experiments measured listeners' abilities to weight information from different components in a complex of 553, 753, and 953 Hz. The goal was to determine whether or not the ability to adjust perceptual weights generalized across tasks. Weights were measured by binary logistic regression between stimulus values that were sampled from Gaussian distributions and listeners' responses. The first task was interaural time discrimination in which listeners judged the laterality of the target component. The second task was monaural level discrimination in which listeners indicated whether the level of the target component decreased or increased across two intervals. For both experiments, each of the three components served as the target. Ten listeners participated in both experiments. The results showed that those individuals who adjusted perceptual weights in the interaural time experiment could also do so in the monaural level discrimination task. The fact that the same individuals appeared to be analytic in both tasks is an indication that the weights measure the ability to attend to a particular region of the spectrum while ignoring other spectral regions.     

The measurement problem in level discrimination

There is disagreement among theorists over the exact measure to be used to quantify auditory level discrimination. It has been proposed that, for level discrimination tasks, the measure that is most linearly related to the sensitivity index, d', will be the correct measure. The level difference (deltaL) and the Weber fraction (theta) are both candidates, though the latter is sensitive to the physical unit in which it is expressed (e.g., pressure or intensity) while the former is not. Psychometric functions for level discrimination were obtained at a number of pedestal levels for 10-ms sinusoids (either 1000 or 6500 Hz) and broadband noise bursts. These functions were used to assess which of three measures: deltaL, theta = deltap/p, or theta = deltaI/I, is most nearly linearly related to d'. The results suggest that deltap/p is the measure that comes closest to being linearly related to d'.     

Perceptual weights in auditory level discrimination

Perceptual weights in level discrimination (also called intensity discrimination) were determined for 3-, 7-, 15-, and 24-component tone complexes with flat spectral envelopes using a correlational paradigm. Each frequency component was randomly and independently perturbed in level oneach presentation. For the target interval, frequency-component levels were additionally increased by the level increment to be detected, deltaL [= 201og10((p + deltap)/p), where p is pressure]. Weights were calculated from the across-trial correlation between the level perturbations for each frequency component and the interval chosen by the listener. Two conditions were investigated: (1) deltaL was equal across frequency components, and (2) deltaL increased progressively across frequency components. For both conditions, data for four listeners usually showed the greatest weight for the highest frequency component. The two-to-four highest frequency components generally were most important for level discrimination. The effect of increasing deltaL progressively with frequency was small and inconsistent. Additional measurements showed that flanking noise maskers designed to mask spread of excitation caused only small and generally unsystematic changes to the weights. Overall, these results indicate that listeners combine information across a wide range of auditory channels to arrive at a decision for level discrimination, but the weighting of channels appears to be suboptimal.     

Psychometric functions for gap detection in noise measured from young and aged subjects.

Psychometric functions for gap detection of temporal gaps in wideband noise were measured in a "yes/no" paradigm from normal-hearing young and aged subjects with closely matched audiograms. The effects of noise-burst duration, gap location, and uncertainty of gap location were tested. A typical psychometric function obtained in this study featured a steep slope, which was independent of most experimental conditions as well as age. However, gap thresholds were generally improved with increasing duration of the noise burst for both young and aged subjects. Gap location and uncertainty had no significant effects on the thresholds for the young subjects. For the aged subjects, whenever the gap was sufficiently away from the onset or offset of the noise burst, detectability was robust despite uncertainty about the gap location. Significant differences between young and aged subjects could be observed only when the gap was very close to the signal onset and offset.     

Power series for level spacing functions of random matrix ensembles

A remarkable set of identities among (odd and even) spheroidal functions and their eigenvalues is indicated. This is then used to derive explicitly the general term in the power series expansion ofE (r, s), the probability that a randomly chosen interval of lengths contains exactlyr levels. The parameter takes the values 1, 2 or 4 according as the ensemble considered is orthogonal, unitary or symplectic.Laboratoire de la Direction des Sciences de la Matière du Commissariat à l'Energie Atomique     

Rollover effect of signal level on vowel formant discrimination

The goal of this study was to measure the ability of normal-hearing listeners to discriminate formant frequency for vowels in isolation and sentences at three signal levels. Results showed significant elevation in formant thresholds as formant frequency and linguistic context increased. The signal level indicated a rollover effect, especially for F2, in which formant thresholds at 85 dB SPL were lower than thresholds at 70 or 100 dB SPL in both isolated vowels and sentences. This rollover level effect could be due to reduced frequency selectivity and forward/backward masking in sentence at high signal levels for normal-hearing listeners.     

Temporal weights in the level discrimination of time-varying sounds

To determine how listeners weight different portions of the signal when integrating level information, they were presented with 1-s noise samples the levels of which randomly changed every 100 ms by repeatedly, and independently, drawing from a normal distribution. A given stimulus could be derived from one of two such distributions, a decibel apart, and listeners had to classify each sound as belonging to the "soft" or "loud" group. Subsequently, logistic regression analyses were used to determine to what extent each of the ten temporal segments contributed to the overall judgment. In Experiment 1, a nonoptimal weighting strategy was found that emphasized the beginning, and, to a lesser extent, the ending of the sounds. When listeners received trial-by-trial feedback, however, they approached equal weighting of all stimulus components. In Experiment 2, a spectral change was introduced in the middle of the stimulus sequence, changing from low-pass to high-pass noise, and vice versa. The temporal location of the stimulus change was strongly weighted, much as a new onset. These findings are not accounted for by current models of loudness or intensity discrimination, but are consistent with the idea that temporal weighting in loudness judgments is driven by salient events.     

The effect of masker level uncertainty on intensity discrimination

Thresholds were measured for detection of an increment in level of a 60-dB SPL target tone at 1 kHz, either in quiet or in the presence of maskers at 0.5 and 2 kHz. Interval-by-interval level rove applied independently to remote masker tones substantially elevated thresholds compared to intensity discrimination in quiet, an effect on the order of 10+dB [10 log(DeltaII)]. Asynchronous onset and stimulus envelope mismatches across frequency reduced but did not eliminate masking. A preinterval cue to signal frequency had no effect, but cuing masker frequency reduced thresholds, whether or not masker level was also cued. About 1 to 2 dB of threshold elevation in these conditions can be attributed to energetic masking. Decreasing the overall presentation level and increasing masker separation essentially eliminates energetic masking; under these conditions masker level rove elevates thresholds by approximately 7 dB when the target and masker tones are gated synchronously. This masking persists even when the flanking masker tones are presented contralateral to the target. Results suggest that observers tend to listen synthetically, even in conditions when this strategy reduces sensitivity to the intensity increment.     

The comparative effects of signal sensation level and sound-pressure level on interaural time discrimination

A series of experiments was performed to examine the extent to which precision of interaural time discrimination depends on the sound-pressure level (SPL) and/or sensation level (SL) of the signal. All experiments used a tone burst signal and a continuous white noise masker, which was either diotic or interaurally phase reversed. Results of the first experiment indicate that (1) at equal signal SLs, interaural time and intensity discrimination is more precise when measured with the added diotic noise, and (2) addition of the phase reversed noise, previously shown to cause less precise interaural time discrimination, has a similar effect on interaural intensity discrimination. In the second experiment, interaural time JNDs for a signal of constant SPL were measured as a function of noise level. Results show that a low-level diotic noise can benefit interaural time discrimination, particularly at 500 Hz. The third and fourth experiments were performed to measure interaural time discrimination as a function of increasing signal SPL but constant signal-to-noise ratio. The data show the JND decreasing with increasing signal SPL at nearly the same rate with or without the added noise, indicating that an increase in signal-to-noise ratio is not necessary for improved discrimination.     

Three-channel electronic instrument for the measurement of metastable level excitation functions

A three-channel electronic instrument for the measurement of the excitation function f₁₂(U₁) of forbidden transitions is described. The instrument consists of two electron guns in antiparallel arrangement and a mobile metallic detector. It is shown that in order to determine f₁₂(U¹), it is sufficient to measure either the current from collisions of the second kind of excited atoms with slow electrons, the current of secondary electrons being emitted during the bombardment of the detector surface by metastable atoms, or the radiation intensity of the spectral line emitted during step-by-step excitation. Formulas are also proposed for the determination of the absolute magnitudes of the effective cross sections of step-by-step excitation and collisions of the second kind.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 12, pp. 61–65, December, 1977.The author wishes to thank N. I. Petrov for valuable assistance in the preparation of the instrument.     

Effects of level and background noise on interaural time difference discrimination for transposed stimuli

A method-of-adjustment procedure was used to measure thresholds for detecting a continuous sequence of brief 2-kHz tonal pulses in the presence of random-frequency masking sequences. Masker pulses consisted of either one or eight sinusoidal components and were either synchronous or asynchronous with the signal pulses. Effects of pulse rate and asynchronous gating were generally consistent with a reduction in informational masking due to segregation of the signal and masker streams. Despite use of continuous stimulus presentation to encourage stream segregation, masking was still obtained from most listeners in most conditions.     

Phases of wave functions and level repulsion

Avoided level crossings are associated with exceptional points which are the singularities of the spectrum and eigenfunctions, when considered as functions of a complex coupling parameter. It is shown that the wave function of one state changes sign but not the other, if the exceptional point is encircled in the complex plane. An experimental setup is suggested where this peculiar phase change could be observed. Received: 7 January 1999 / Received in final form: 15 March 1999     

Level discrimination as a function of level for tones from 0.25 to 16 kHz

Difference limens for level (delta L in dB = 20 log [(p + delta p)/p], where p is pressure) were measured as a function of level for tones at 0.25, 0.5, 1, 2, 4, 8, 10, 12, 14, and 16 kHz. At each frequency, test levels encompassed the range from near threshold to 95 dB SPL in steps of 10 dB or smaller. The stimulus duration was 500 ms and the interstimulus interval was 250 ms. An adaptive two-alternative forced-choice procedure with feedback was used. Results for six normal listeners show individual differences among listeners, but the general trends seen in the average data clearly are present in the individual data and show the following. First, the delta Ls at all but the highest frequencies are generally smaller at high levels than at low levels. Second, the delta Ls at equal SPLs are largely independent of frequency up to about 4 kHz, but increase with frequency above 4 kHz. Third, at 8 and 10 kHz, the delta Ls are clearly nonmonotonic functions of level, showing consistent deterioration in the mid-level delta Ls relative to the low- and high-level delta Ls. The present data are discussed qualitatively in terms of current models of level discrimination.