The Rôle of the Bandwidth-Duration Product WT in the Detectability of Diotic Signals

Abstract

The bandwidth-duration product, WT, is a fundamental parameter in most theories of aural amplitude discrimination of Gaussian noise. These theories predict that detectability is dependent on WT, but not on the individual values of bandwidth and duration. Due to the acoustical uncertainty principle, it is impossible to completely specify an acoustic waveform with both finite duration and finite bandwidth. An observer must decide how best to trade-off information in the time domain with information in the frequency domain. As Licklider (1963) states, "The nature of [the ear's] solution to the time-frequency problem is, in fact, one of the central problems in the psychology of hearing." This problem is still unresolved, primarily due to observer inconsistency in experiments, which degrades performance making it difficult to compare models. The aim was to compare human observers' ability to trade bandwidth and duration, with simulated and theoretical observers. Human observers participated in a parametric study where the bandwidth and duration of 500 Hz noise waveforms was systematically varied for the same bandwidth-duration products (WT=1, 2, and 4, where W varied over 2.5-160 Hz, and T varied over 400-6.25 ms, in octave steps). If observers can trade bandwidth and duration, detectability should be constant for the same WT. The observers replicated the experiments six times so that group operating characteristic (GOC) analysis could be used to reduce the effects of their inconsistent decision making. Asymptotic errorless performance was estimated by extrapolating results from the GOC analysis, as a function of replications added. Three simulated ideal observers: the energy, envelope, and full-linear (band-pass filter, full-wave rectifier, and true integrator) detectors were compared with each other, with mathematical theory and with human observers. Asymptotic detectability relative to the full-linear detector indicates that human observers best detect signals with a bandwidth of 40-80 Hz and a duration of 50-100 ms, and that other values are traded off in approximately concentric ellipses of equal detectability. Human detectability of Gaussian noise was best modelled by the full-linear detector using a non-optimal filter. Comparing psychometric functions for this detector with human data shows many striking similarities, indicating that human observers can sometimes perform as well as an ideal observer, once their inconsistency is minimised. These results indicate that the human hearing system can trade bandwidth and duration of signals, but not optimally This accounts for many of the disparate estimates of the critical band, rectifier, and temporal integrator, found in the literature, because (a) the critical band is adjustable, but has a minimum of 40-50 Hz, (b) the rectifier is linear, rather than square-law, and (c) the temporal integrator is either true or leaky with a very long time constant.