Spherical harmonics, singular-value decomposition and the head-related transfer function

Nelson, P.A. and Kahana, Y. (2001) Spherical harmonics, singular-value decomposition and the head-related transfer function. Journal of Sound and Vibration, 239 (4), 607-38. (doi:10.1006/jsvi.2000.3227).

Abstract

The singular-value decomposition (SVD) can be used to analyze the matrix of Green functions relating the acoustic pressure at a number of field points to the strengths of a number of point sources on the surface of a body which radiates or scatters sound. The left and right singular vectors of the resulting decomposition yield, at a given frequency, two sets of orthogonal basis functions describing "field mode shapes" and "source mode shapes" respectively. This paper attempts to make the connection between this decomposition and the basis functions of classical acoustics. In particular, it is found that for a spherical co-ordinate system, when the source points and field points are chosen in order to sample the source and field appropriately, then the matrices of left and right singular vectors are related to the sampled spherical harmonics by a unitary transformation. Preliminary investigations are presented of the sensitivity of this relationship to the manner in which the source and field are sampled. In addition, the great utility of the method is illustrated with some new results of numerical analysis of the scattering of sound by the outer ear. "Pinna resonances" and their associated "mode shapes" are identified at certain frequencies where high values of the dominant singular values indicate a strong coupling between source and field.

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Contributors

Author: P.A. Nelson

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