Mode Matching and Infinite-Dimensional Harmonic Analysis: An Experimental Evaluation Shoichi Koyama, Keisuke Kimura, and Natsuki Ueno The University of Tokyo, Tokyo, JAPAN [email protected]
reproduction (SFR) with multiple loudspeakers • Focusing on numerical SFR based on minimization of error between synthesized and desierd sound fields – Flexible loudspeaker array geometry – Cancelling reverberation • Two numerical SFR methods are compared in practical environment – Pressure matching (PM) – Weighted mode matching (WMM) • Infinite-dimensional harmonic analysis for estimating expansion coefs is applied in WMM
region Secondary source • Goal is to synthesize desired sound field inside with secondary sources (loudspeakers) • Optimization problem is formulated as ( is omitted) Synthesized sound field Transfer functions: Driving signals: Difficult to solve because of regional integration
Target region Secondary source • Discretize target region into control points • Optimization problem of PM: Transfer function matrix Desired pressure vector Closed-form solution is obtained as Regularization term – Simple implementation – Fine discretization of is necessary
• Based on spherical wavefunction expansion of sound field [Ueno+ IEEE/ACM TASLP 2019] • and are approximated by spherical wavefunction expansion up to order : Spherical wavefunction Expansion coefs around Spherical wavefunction vector Expansion coefs of Expansion coefs
approximated as Weighted Mode Matching (WMM) [Ueno+ IEEE/ACM TASLP 2019] – Each element of is obtained as • Optimization problem of WMM Closed-form solution is obtained as Weighting factor for each expansion coef is determined by
[Ueno+ IEEE SPL 2018] • Estimation of and/or using mic array of arbitrary geometry by infinite-dimensional harmonics analysis – Reproduction of captured sound field – Using measured transfer functions for cancelling reverberation • Mic signal vector and expansion coefficients around are related as – is translation operator Expansion coef of directivity pattern of each mic Expansion coef
[Ueno+ IEEE SPL 2018] • Expansion coefs are estimated as – th element of is obtained as • Truncation of expansion order is unnecessary • Independent of expansion center
planewave field – : Square region of 1.0 m x 1.0 m – # of loudspeakers: 32 – Uniform distribution of mics on – # of evaluation points: 21 x 21 = 441 – Lowpass filtered pulse (up to 700 Hz) – Compared PM and WMM – Evaluation measure: Evaluation
of two SFR methods: PM and WMM • WMM is based on spherical wavefunction expansion – Weighting factor for each expansion coef is determined by the setting of target region • Infinite-dimensional harmonic analysis is combined – Estimating expansion coefs using mic array of arbitrary geometry – Truncation of expansion order is unnecessary – Independent of expansion center • Experimental comparison using MeshRIR dataset • WMM outperforms PM in the case of small number of mics Thank you for your attention!
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