Causal limits and optimal allocation for passive frequency-selective hearing protection

Geonhwi Hwang; Jaewoo Lee; Hyunjun Kim; Hyeonwoo Na

cs.SDSep 18, 2026

Causal limits and optimal allocation for passive frequency-selective hearing protection

Authors: Geonhwi Hwang, Jaewoo Lee, Hyunjun Kim, Hyeonwoo Na

Organizations: Department of Applied Physics, Hokkaido University, Sapporo 060-8628, Japan · CHA University School of Medicine, Gyeonggi-do 13488, Republic of Korea · School of Computing, Korea Advanced Institute of Science and Technology, Daejeon 34141, Republic of Korea · Division of Mechanical and Space Engineering, Hokkaido University, Sapporo 060-8628, Japan

Abstract

Passive Helmholtz-resonator earplugs aim to attenuate hazardous noise while keeping speech audible; the air volume of an ear-sized shell is finite, and we treat it as the resource a design spends. For a vented bore transparent at zero frequency and loaded by reactive side branches, causality fixes a conserved integral of transmission loss. We derive it in device variables, add viscous and flanking constraints, and allocate the budget by hazard-weighted water-filling with a continuous speech-cost price. The audio-band budget B_ad = (20/ln10) pi^2 V / 2cS is set by the ratio of total cavity volume V to bore area S; occluding plugs are exempt. The f^2 price signal favours frequencies well above the source peak. Replacing strict transparency by a speech-cost price gives an exchange rate of about 1.3 dB of protection per decibel of speech cost under the present model, with the budget setting where it saturates. Across three form factors, low-frequency-selective designs stay near a decibel while hazard-matched mid/high-frequency designs reach twelve. The binding variables are bore area and speech allowance, not resonator count.

Figures & tables

	class	$V$ ( ${\mathrm{mm}}^{3}$ )	$N$	bore $\varnothing$ ( $\mathrm{mm}$ )	$S$ ( ${\mathrm{mm}}^{2}$ )	$\mathcal{B}_{\mathrm{ad}}$ ( $\mathrm{dB}$ )
F1	canal plug	150	3	2.0	3.14	0 5.97
F2	deep canal plug	400	4	2.0	3.14	15.91
F3	concha+canal mould	1200	6	2.0	3.14	47.73

Table 1: Representative device classes. No shell geometry is assumed; a class is fixed by its total cavity volume

V

, the number of cavities

N

it is divided into, and the open bore area

S

. The tabulated budget is the audio-band value

\mathcal{B}_{\mathrm{ad}}

of Eq. ( 5 ).

Figure 1: (a) The audio-band budget

\mathcal{B}_{\mathrm{ad}}

of Eq. ( 5 ) over the design plane, with the three representative classes of Table 1 marked; contours are labelled in

\mathrm{dB}

. Because

\mathcal{B}_{\mathrm{ad}}\propto S^{-1}\propto d^{-2}

, reducing the bore diameter raises the budget as effectively as enlarging the volume, at no cost in packaging space. (b) Transmission loss of four dissimilar allocations of the same

400\text{\,}{\mathrm{mm}}^{3}

. (c) Their running integrals

\int_{0}^{f}\mathrm{TL}f^{\prime-2}\mathrm{d}f^{\prime}

, all converging to

\mathcal{B}_{\mathrm{ad}}

(dotted). The

220\text{\,}\mathrm{Hz}380\text{\,}\mathrm{Hz}

array spends its budget below

100\text{\,}\mathrm{Hz}

, in an overdamped and inaudible tail.

band	F1	F2	F3
$150\text{\,}\mathrm{Hz}400\text{\,}\mathrm{Hz}$	0 1.4	0 3.8	0 11.5
$300\text{\,}\mathrm{Hz}1000\text{\,}\mathrm{Hz}$	0 2.6	0 6.8	0 20.5
$1\text{\,}\mathrm{kHz}8\text{\,}\mathrm{kHz}$	0 6.8	18.2	54.6
$2\text{\,}\mathrm{kHz}6\text{\,}\mathrm{kHz}$	17.9	47.7	143.2
$2\text{\,}\mathrm{kHz}4\text{\,}\mathrm{kHz}$	23.9	63.6	190.9
$3\text{\,}\mathrm{kHz}6\text{\,}\mathrm{kHz}$	35.8	95.5	286.4

Table 2: Flat transmission loss sustainable on the audio-band budget

\mathcal{B}_{\mathrm{ad}}

, Eq. ( 6 ). Entries in italics exceed the flanking floor of Sec. 3.3 and are not physically realisable; they indicate only that the causal budget has ceased to be the binding constraint.

$f_{0}$ (Hz)	$a_{\mathrm{req}}$ ( $\mathrm{\SIUnitSymbolMicro m}$ )	$\delta_{v}$ ( $\mathrm{\SIUnitSymbolMicro m}$ )	$a_{\mathrm{req}}/\delta_{v}$	F1 (dB)	F2 (dB)	F3 (dB)
150	22	179	0.12	0.00	0.00	0 0.01
300	45	127	0.35	0.01	0.02	0 0.10
500	75	98	0.77	0.05	0.20	0 0.79
800	123	78	1.58	0.32	0 1.28	0 4.38
1200	189	63	2.98	0 1.57	0 4.96	11.52
2000	331	49	6.74	0 6.84	14.18	23.67

Table 3: Viscous accessibility.

a_{\mathrm{req}}

is the neck radius that places the resonance of one F2 cavity (

V/N=

100\text{,}{\mathrm{mm}}^{3}

) at $f_{0}$ , computed with the flanged effective length $L_{\mathrm{eff}}=L+1.7a_{\mathrm{req}}$ for a geometric neck length $L=$2\text{\,}\mathrm{mm}

(solved self-consistently);

\delta_{v}

is the viscous boundary-layer thickness there. The last three columns give the peak transmission loss attainable from a single cavity of

V/N

in each class.

Figure 2: (a) Constraints 1–3. Solid: the causal ceiling of Eq. ( 6 ) for a one-octave band, for the three classes of Table 1 . Dashed: the viscously attainable peak transmission loss of a single F2 cavity. Shaded: the flanking floor of Sec. 3.3 . The

150\text{\,}\mathrm{Hz}400\text{\,}\mathrm{Hz}

band targeted by low-frequency-selective designs lies beneath both lower constraints. (b) Recovering the thermally stranded budget: transmission loss of the same four-resonator array with adiabatic cavities and with fibre-filled cavities of effective thermal radius

40\text{\,}\mathrm{\SIUnitSymbolMicro m}

. The strict dc-anchored budget is

\mathcal{B}_{\mathrm{dc}}=\gamma\mathcal{B}_{\mathrm{ad}}

; filling converts part of that excess from inaudible frequencies into the audio band.

	unweighted	A-weighting	damage-risk
value-density peak ( $\mathrm{kHz}$ )	2.03	2.21	2.68
$R_{\infty}$ , $[300,3400]$ $\mathrm{Hz}$ (dB)	0.71	0.51	0.33

Table 4: Robustness of the headline results to the hazard weighting

W(f)

, for the

0.3\text{\,}\mathrm{ms}

blast. The damage-risk surrogate is an outer/middle-ear pressure transfer peaking near

3.5\text{\,}\mathrm{kHz}

times the A-weighting skirt; its exact form is in the released code, so a reviewer may substitute a validated model. In every row the weighting moves the result in the same direction, so the conclusions do not depend on the choice.

Figure 3: (a) Source spectra (dotted) and hazard-weighted value densities

f^{2}P(f)W_{A}(f)

(solid), each normalised to its own maximum; triangles mark the value-density peaks. (b) Budget-optimal allocations from Eq. ( 11 ) for class F2 and the

0.3\text{\,}\mathrm{ms}

blast, at four speech allowances; the legend gives the realised A-weighted protection. The narrow spikes at the edges of the speech-importance support are the residue of the strict constraint: budget accumulates where speech importance vanishes, and buys almost nothing there.

Figure 4: Protection–speech-cost Pareto fronts under Eq. ( 11 ) with the flanking floor of Eq. ( 7 ), for (a) the

0.3\text{\,}\mathrm{ms}

blast and (b) continuous broadband noise. Stars mark the strict-transparency ceiling

R_{\infty}

of Eq. ( 9 ). Dotted lines mark the saturation set by each class’s budget. The near-common initial slope is specific to the present source and speech-cost model (see text).

class	archetype	peak $\mathrm{TL}$ (dB)	$\Delta L_{A}$ blast	$\Delta L_{A}$ cont.	$\mathcal{C}$ (dB)
F1	A low-frequency selective	0 0.1	0.11	0.09	0 0.10
F1	B speech band-pass	0 2.7	1.30	0.88	0 1.02
F1	C protection notch	14.9	2.66	3.44	0 4.78
F1	D mid+high protection	0 8.9	3.16	3.17	0 3.82
F1	E hazard-optimal	0 7.7	3.10	2.89	0 3.39
F2	A low-frequency selective	0 1.0	0.46	0.34	0 0.45

Table 5: Design atlas.

\Delta L_{A}

is the A-weighted energy reduction against the

0.3\text{\,}\mathrm{ms}

blast and against continuous noise;

\mathcal{C}

is the articulation-weighted speech cost of Eq. ( 10 ). All entries are realised finite-

Q

designs including the flanking path.

Figure 5: (a) Realised transmission loss of the five archetypes in class F2 (four resonators,

400\text{\,}{\mathrm{mm}}^{3}

), including the flanking path; the dotted line is the flanking floor. (b) Realised A-weighted protection against the

0.3\text{\,}\mathrm{ms}

blast for each archetype and class; the small figures above each bar give the articulation-weighted speech cost incurred.

Sep 1, 2026

ABSE-NET: A Lightweight Neural Model for Active Binaural Speech Enhancement in Open-Fit Hearing Aids

Open-fit hearing aids have attracted growing attention due to their superior wearing comfort. However, the open-fit design inevitably causes acoustic leakage into the ear canal, degrading the performance of existing binaural speech enhancement (BSE). To this end, we propose ABSE-NET, an active BSE framework integrating active noise control (ANC) with BSE to jointly enhance target speech and suppress acoustic leakage. The ABSE-NET pipeline cascades a binaural MVDR (BMVDR) with a lightweight neural network (LNN). The former achieves a coarse BSE, whereas the latter simultaneously cancels acoustic leakage and compensates for BMVDR-induced distortion. The LNN uses an encoder-decoder with a feature fusion module, which includes frequency-time dependency learning and convolutional attention blocks. Unlike traditional BSE+ANC solutions via adaptive filtering, ABSE-NET needs no in-ear microphone in practical deployment. Experiments validate its superiority over state-of-the-art methods. Code repository: https://github.com/Bream101/ABSE-NET.

De Hu, Xue Du, Qingying Zhao +1

College of Computer Science, Inner Mongolia University, China · College of Electronic and Information Engineering, Inner Mongolia University, China

Speech Enhancement

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Jul 21, 2026

CAPS: A Cascaded Reconstruction Model to Power Saving in Hearables Using Sub-Nyquist Sampling with Bandwidth Extension

Hearables are wearable computers worn on the ear. Bone conduction microphones are used with air conduction microphones in hearables for multimodal speech enhancement in noisy conditions. Despite this potential, current models largely fail to explore how jointly reducing sampling bit resolution and sampling frequency in analog-to-digital converters (ADCs) of hearables impacts both power usage and audio quality. Furthermore, current frameworks cannot do sub-Nyquist sampling in hearables because they lack a method to reconstruct wideband signals from narrowband components. We therefore propose CAPS, which (i) intentionally employs sub-Nyquist sampling and low bit resolution in ADCs, achieving a 3.3x reduction in power consumption in hearables, and (ii) supports streaming operation on mobile platforms with an inference time of 1.36 ms and a memory footprint of 11.04 MB. CAPS ensures robust speech intelligibility in real-world settings, bridging the gap between efficiency and power savings.

Tarikul Islam Tamiti, Sajid Fardin Dipto, Luke Baja-Ricketts +2

Cyber-Security Engineering, George Mason University, USA

Energy-Efficient ML Speech Enhancement

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Mar 20, 2026

End-to-End Multi-Task Learning for Adjustable Joint Noise Reduction and Hearing Loss Compensation

A multi-task learning framework is proposed for optimizing a single deep neural network (DNN) for joint noise reduction (NR) and hearing loss compensation (HLC). A distinct training objective is defined for each task, and the DNN predicts two time-frequency masks. During inference, the amounts of NR and HLC can be adjusted independently by exponentiating each mask before combining them. In contrast to recent approaches that rely on training an auditory-model emulator to define a differentiable training objective, we propose an auditory model that is inherently differentiable, thus allowing end-to-end optimization. The audiogram is provided as an input to the DNN, thereby enabling listener-specific personalization without the need for retraining. Results show that the proposed approach not only allows adjusting the amounts of NR and HLC individually, but also improves objective metrics compared to optimizing a single training objective. It also outperforms a cascade of two DNNs that were separately trained for NR and HLC, and shows competitive HLC performance compared to a traditional hearing-aid prescription. To the best of our knowledge, this is the first study that uses an auditory model to train a single DNN for both NR and HLC across a wide range of listener profiles.

Philippe Gonzalez, Vera Margrethe Frederiksen, Torsten Dau +1

The authors are with the Department of Health Technology, Technical University of Denmark, 2800 Lyngby, Denmark.

Speech Enhancement Multi-Task Learning

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