Generalization of a 3-D Acoustic Resonator Model for the Simulation of Spherical Enclosures
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eneralization of a 3-D Acoustic Resonator Model for the Simulation of Spherical Enclosures Davide Rocchesso Università di Verona, Dipartimento Scientifico e Tecnologico, Strada Le Grazie 15, I-37134 Verona, Italy Email: [email protected]
Pierre Dutilleux Zentrum für Kunst und Medientechnologie (ZKM), Institute for Music and Acoustics, Lorenzstrasse 19, D-76135 Karlsruhe, Germany Email: [email protected] Received 20 April 2000 and in revised form 9 October 2000 A rectangular enclosure has such an even distribution of resonances that it can be accurately and efficiently modelled using a feedback delay network. Conversely, a nonrectangular shape such as a sphere has a distribution of resonances that challenges the construction of an efficient model. This work proposes an extension of the already known feedback delay network structure to model the resonant properties of a sphere. A specific frequency distribution of resonances can be approximated, up to a certain frequency, by inserting an allpass filter of moderate order after each delay line of a feedback delay network. The structure used for rectangular boxes is therefore augmented with a set of allpass filters allowing parametric control over the enclosure size and the boundary properties. This work was motivated by informal listening tests which have shown that it is possible to identify a basic shape just from the distribution of its audible resonances. Keywords and phrases: physically-based sound modelling, spherical resonators, feedback delay networks.
1. INTRODUCTION The feedback delay network (FDN) of order N , as depicted in Figure 1 for N = 4, is the multivariable generalization of the recursive comb filter, and it has been widely used to simulate the late reverbation of an enclosure [1, 2, 3, 4]. The FDN with a diagonal matrix can be used to simulate a box with perfectlyreflecting walls. Energy absorption at the walls and in air can be accounted for by cascading the delay lines with lowpass filters. In regular rooms, some energy gets transferred from one mode to another due to nonmirror reflection at the walls. This effect is called diffusion [5] and is encompassed by the nondiagonal elements of the feedback matrix. As observed in the time domain, diffusion produces a gradual increase in density of the impulse response. The delay lengths of an FDN are sometimes chosen with number-theoretic considerations in order to minimize the overlapping of echoes, as it was done with classic reverberation structures [6]. A more physical criterion is based on the following observation: in a rectangular enclosure, the distribution of normal modes can be obtained as the composition of (infinite) harmonic series, each series being associated with the spatial direction of propagation of the plane wave fronts supporting the modes. For instance,
the longitudinal size of a rectangular box is associated with a low-pitch mode and with all its multiples. Since any harmonic series of resonances can be reproduced by means of a recursive comb filter, a reference FDN can be constructed as a
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