Eardrum sound pressure prediction from ear canal reflectance based on the inverse solution of Webster's horn equation

TL;DR

This work addresses predicting eardrum sound pressure from individualized ear canal reflectance by solving Webster's horn equation in one dimension from time-domain reflectance and integrating the derived area function into a compact electro-acoustic model. It introduces a refined procedure with upsampling to $ ext{f}_{sup}=3.5$ MHz (≈0.1 mm spatial resolution), a data-driven linear model linking $f_{cut}$ to $f_{lim}$, and termination-length strategies grounded in simulations, which collectively yield substantial improvements over prior methods. Validation against 3D FEM and measurements shows accurate transfer impedance predictions (low $L_{rmse}$ and small phase errors) across 1–10 kHz, with residual errors largely attributed to measurement artifacts and boundary-condition assumptions. The results support using the inverse Webster solution as a robust, fast initialization step for individualized equalization of eardrum sound pressure in in-ear devices.

Abstract

To derive ear canal transfer functions for individualized equalization algorithms of in-ear hearing systems, individual ear canal models are needed. In a one-dimensional approach, this requires the estimation of the individual area function of the ear canal. The area function can be effectively and reproducibly calculated as the inverse solution of Webster's horn equation by finite difference approximation of the time domain reflectance. Building upon previous research, the present study further investigates the termination of the approximation at an optimal spatial resolution, addressing the absence of higher frequencies in typical ear canal measurements and enhancing the accuracy of the inverse solution. Compared to the geometric reference, more precise area functions were achieved by extrapolating simulated input impedances of ear canal geometries up to a frequency of 3.5 MHz, corresponding to 0.1 mm spatial resolution. The low pass of the previous work was adopted but adjusted for its cut-off frequency depending on the highest frequency of the band-limited input impedance. Robust criteria for terminating the area function at the approximated ear canal length were found. Finally, three-dimensional simulated and measured ear canal transfer impedances were replicated well employing the previously introduced and herein validated one-dimensional electro-acoustic model fed by the area functions.

Figures (24)

• Figure 1: Blackman window as designed by rasetshwane2011inverse and also found in rasetshwaneSourceCode2012, and characteristic frequencies.
• Figure 2: One-dimensional electro-acoustic model used to calculate the transfer impedance $Z\rm{_{trans}}$ from the input impedance $Z\rm{_{ec}}$ of the ear canal (defined as ratios of the drum pressure $p\rm{_{d}}$ or the pressure at the lateral end of the ear canal $p\rm{_{ec}}$ to the volume velocity at the lateral end of the ear canal $q\rm{_{ec}}$, Eqs. (\ref{['eq:Zec']}) and (\ref{['eq:Ztrans_pd']}).
• Figure 3: Right outer ear of subject 3 in the IHA database with the extracted centerline and landmarks (left) and with final cut planes at the entrance, the first and the second bend and at the eardrum (right).
• Figure 4: Area functions of right ear canals cut at the entrance plane for IHA database subjects 1 - 21, and corresponding data from rasetshwane2011inverse, stinson1989specification, balouch2023measurements.
• Figure 5: Mesh of the right residual ear canal for subject 5 of the IHA database cut at the entrance plane (left), at the first bend (mid), and at the second bend (right).