Auditory Filter Behavior and Updated Estimated Constants

TL;DR

This work challenges the long-standing practice of fixing auditory filter constants based on historical psychoacoustic data by introducing a sharp-filter approximation that unifies several Gammatone-family filters and exposes how constants $A_p$, $b_p$, and $B_u$ shape peak-centered behavior. It develops closed-form expressions linking filter characteristics (e.g., $eta_{peak}$, $Q_{ ext{erb}}$, $S_eta$, $N_eta$) to the constants and provides a systematic, characteristic-based method to estimate $B_u$, $A_p$, and $b_p$ from observed characteristics and their ratios. The approach is shown to extend to realizable filter classes (GEFs/P, V, GTFs), with validation that peak-region behavior is well captured by the sharp-filter model, enabling accurate design of human-like auditory filterbanks and analysis of perceptual model dependencies on filter characteristics. By incorporating ratio-based constraints (e.g., $rac{Q_{ ext{erb}}}{N_eta}$ and $rac{Q_{ ext{erb}}}{Q_{10}}$) and cross-species observations, the work offers updated estimates for $B_u$ (including around $7.2$ from physiological data) and practical estimates for $A_p$ across CF, while outlining limitations and uncertainties inherent in cross-species and paradigm comparisons. Overall, the framework supports tailored filterbank design with arbitrary characteristic specifications and provides a principled pathway to study how variations in filter characteristics influence auditory models and hearing-based technologies.

Abstract

Filters from the Gammatone family are often used to model auditory signal processing, but the filter constant values used to mimic human hearing are largely set to values based on historical psychoacoustic data collected several decades ago. Here, we move away from this long-standing convention, and estimate filter constants using a range of more recent reported filter characteristics (such as quality factors and ratios between quality factors and peak group delay) within a characteristics-based framework that clarifies how filter behavior is related to the underlying constants. Using a sharp-filter approximation that captures shared peak-region behavior across certain classes of filters, we analyze the range of behaviors accessible when the full degrees of freedom of the filter are utilized rather than fixing the filter order or exponent to historically prescribed values. Filter behavior is characterized using magnitude-based and phase-based characteristics and their ratios, which reveal which characteristics are informative for constraining filter constants and which are only weakly constraining. We show that these insights and estimation methods extend to multiple realizable filter classes from the Gammatone family and apply them, together with recent physiological and psychoacoustic observations, to derive constraints on and estimates for filter constants for human auditory filters. More broadly, this framework supports the design of auditory filters with arbitrary characteristic-level specifications and enables systematic assessment of how variations in filter characteristics influence auditory models, perceptual findings, and technologies that rely on auditory filterbanks.

Figures (10)

• Figure 1: The left and middle panels show the Bode plots as a function of normalized frequency, $\beta$, for the following transfer functions: $H_P$ (blue solid lines), $H_V$ (red dash-dotted lines), and the approximation, $H_\textrm{sharp}$ (yellow dotted lines), which we use to derive our expressions for filter characteristics in terms of filter constants and based upon which the characteristics-based filter design methods are developed. The left and middle panels are generated using two different sets of filter constant values (shown in the panel titles) chosen from the range we deem appropriate in humans. The left panel is more appropriate for filters with larger CFs and the middle panel is for lower CFs. The left and middle shows that $H_\textrm{sharp}$ is a good approximation for the transfer functions for the realizable filters, $H_P$ and $H_V$. In the rightmost panel, we show the validity of this approximation more clearly for the transfer functions from the middle panel by shifting the $x$ axis by $\beta_{peak}$, which takes on the following values for each of the curves: $\beta_{\textrm{peak}, P} = 0.9798$, $\beta_{\textrm{peak}, V} = 0.9852$, and $\beta_{\textrm{peak, Sharp}} = 1$.
• Figure 2: We plot various filter characteristics of $H_\textrm{sharp}$ as a function of filter constants $A_p$ and $B_u$ using the closed-form expressions in equation \ref{['eq:const2chars']}. For the quality factors, $\beta_{peak}$ and $\beta_{maxN}$, which depend on $b_p$, we used $b_p = 1$ (i.e. $f_{peak} = \text{CF}$). These plots (along with plots of the characteristic ratios in figure \ref{['fig:ratios']}) may be used to understand the effects of varying the filter exponent and pole on behavior, to understand the relationship between various characteristics, and as a look-up table to quickly estimate filter constants or serve as a stepping stone by providing initial guesses for the characteristics-based filter design methods.
• Figure 3: We plot the dependence of several filter characteristics ratios that depend solely on a single filter constant - $A_p$ or $B_u$, as derived using the expressions from equation \ref{['eq:const2chars']} based on $H_\textrm{sharp}$. The top panel reveals that certain $B_u$-dependent ratios such as $\frac{Q_{\textrm{erb}}}{Q_{10}}$ are not particularly useful for determining specific values of $B_u$ due to their slow variation with respect to $B_u$ in the parameter region of interest. We note that for larger values of $B_u$, $\frac{g}{b_p}$ and $\frac{Q_{\textrm{erb}}^2}{\beta_{peak}^2 S_{\beta}}$ also exhibit shallow dependence. In contrast, both ratios that are purely functions of $A_p$ - i.e. $\frac{S_{\beta}}{N_\beta}$ and $\frac{\phi_{accum}}{N_\beta}$, exhibit a strong dependence on $A_p$ and hence - when available, may reliably be used to determine this filter constant.
• Figure 4: We show the relative errors (as a function of filter constants $A_p$ and $B_u$ with a fixed $b_p = 1$) of the values filter characteristics obtained using our closed-form expressions (that are based on the sharp-filter approximation) compared to those computed numerically from $H_P$. The relative errors are small as may be inferred from the colorbars, indicating the accuracy - for the case of GEFs, of our expressions for filter characteristics in terms of filter constants, and consequently, the accuracy of filter design methods that depend on it. The errors typically increase with $A_p$ - i.e. as the sharp-filter approximation starts to break down.
• Figure 5: Relative errors in filter characteristics are small for the case of V filters as seen from the colorbars, indicating the accuracy of our expressions and methods for this class of filters.