Modeling the relation between the intensity just-noticeable difference and loudness for pure tones and wideband noise

Abstract

A classical problem in auditory theory is the relation between the loudness L(I) and the intensity just-noticeable difference ͑JND͒ ⌬I(I). The intensity JND is frequently expressed in terms of the Weber fraction defined by J(I)ϵ⌬I/I because it is anticipated that this ratio should be a constant ͑i.e., Weber's law͒. Unfortunately, J(I) is not a constant for the most elementary case of the pure tone JND. Furthermore it remains unexplained why Weber's law holds for wide-band stimuli. We explore this problem and related issues. The loudness and the intensity JND are defined in terms of the first and second moments of a proposed random decision variable called the single-trial loudness L(I), namely the loudness is L(I)ϵEL(I), while the variance of the single trial loudness is L 2 ϵE(LϪL) 2 . The JND is given by the signal detection assumption ⌬LϭdЈ L , where we define the loudness JND ⌬L(I) as the change in loudness corresponding to ⌬I(I). Inspired by Hellman and Hellman's recent theory ͓J. Acoust. Soc. Am. 87, 1255-1271 ͑1990͔͒, we compare the Riesz ͓Phys. Rev. 31, 867-875 ͑1928͔͒ ⌬I(I) data to the Fletcher and Munson ͓J. Acoust. Soc. Am. 5, 82-108 ͑1933͔͒ loudness growth data. We then make the same comparison for Miller's ͓J. Acoust. Soc. Am. 19, 609-619 ͑1947͔͒ wideband noise JND and loudness match data. Based on this comparison, we show empirically that ⌬L(L)ϰL 1/p , where pϭ2 below Ϸ5 sones and is 1 above. Since ⌬L(I) is proportional to L , when pϭ2 the statistics of the single-trial loudness L are Poisson-like, namely L 2 ϰL. This is consistent with the idea that the pure tone loudness code is based a neural discharge rate ͑not the auditory nerve͒. Furthermore, when pϭ1 ͑above about 5 sones͒, the internal loudness signal-to-noise ratio is constant. It is concluded that Ekman's law (⌬L/L is constant͒ is true, rather than Weber's law, in this loudness range. One of the main contributions of this paper is its attempt to integrate Fletcher's neural excitation pattern model of loudness and signal detection theory.

The Dynamic Range Paradox: A Central Auditory Model of Intensity Change Detection

“…This assumption gives rise to the intuitive anticipation of a relationship between loudness and the intensity JND, such that one may be predicted from the other and vice versa. However, previous workers [1]–[3] were not able to provide a unified model due to the apparently paradoxical observation that loudness growth, beyond a certain level, is not reflected in improvement in intensity discrimination performance [3], [4]; the large dynamic range implied by loudness data is in contradiction of the relatively small dynamic range implied by intensity JND data.…”

“…The work of Hellman and Hellman [1], [2] and Allen and Neely [3] resulted in the theoretical construct of the loudness JND, which represents the just-noticeable change in loudness that corresponds to the intensity JND, and the assumption that a reciprocal relationship between loudness and loudness change detection should exist. Focusing on the intensity discrimination paradigm, Hellman and Hellman [1] predicted loudness functions for pure tones from intensity JND data, following the suggestion of McGill and Goldberg [5], [6] that the loudness JND is the square root of loudness (Δ L jnd  =  L 0.5 ).…”

“…Focusing on the intensity discrimination paradigm, Hellman and Hellman [1] predicted loudness functions for pure tones from intensity JND data, following the suggestion of McGill and Goldberg [5], [6] that the loudness JND is the square root of loudness (Δ L jnd  =  L 0.5 ). Allen and Neely [3] tested this for tones and noise using equivalent loudness and intensity JND (Δ I jnd ) data as follows: …”

Systematic covariation of the parameters in the near-miss to Weber's law, pointing to a new law

“…1 These examinations have contributed to the regnant explanation for the near-miss, which is that, while Weber's law may apply for a single auditory 'channel,' an increasing spread of excitation to channels tuned to frequencies above that of the stimulus occurs as stimulus intensity is increased, resulting in improved discriminability at higher intensities (Florentine & Buus, 1981; see also Florentine, 1986;Schroder et al, 1994;Viemeister & Bacon, 1988). A corpus of recent work incorporates the results of these investigations, along with the spread-of-excitation explanation, in the construction of models of the neural activity driving loudness coding (see Allen & Neely, 1997;Hellman & Hellman, 1990; and the references therein).…”

Recruitment compensation as a hearing aid signal processing strategy

“…Today, any power-law relation between the intensity of the physical stimulus and the psychophysical response is referred to as Stevens' law [23,4]. Fletcher's 1923 loudness growth equation established the important special case of loudness for Stevens' approximate, but more general, psychological "law."…”