On the Conditioning of the Spherical Harmonic Matrix for Spatial Audio Applications

Abstract: In this paper, we attempt to study the conditioning of the Spherical Harmonic Matrix (SHM), which is widely used in the discrete, limited order orthogonal representation of sound fields. SHM's has been widely used in the audio applications like spatial sound reproduction using loudspeakers, orthogonal representation of Head Related Transfer Functions (HRTFs) etc. The conditioning behaviour of the SHM depends on the sampling positions chosen in the 3D space. Identification of the optimal sampling points in the continuous 3D space that results in a well-conditioned SHM for any number of sampling points is a highly challenging task. In this work, an attempt has been made to solve a discrete version of the above problem using optimization based techniques. The discrete problem is, to identify the optimal sampling points from a discrete set of densely sampled positions of the 3D space, that minimizes the condition number of SHM. This method has been subsequently utilized for identifying the geometry of loudspeakers in the spatial sound reproduction, and in the selection of spatial sampling configurations for HRTF measurement. The application specific requirements have been formulated as additional constraints of the optimization problem. Recently developed mixed-integer optimization solvers have been used in solving the formulated problem. The performance of the obtained sampling position in each application is compared with the existing configurations. Objective measures like condition number, D-measure, and spectral distortion are used to study the performance of the sampling configurations resulting from the proposed and the existing methods. It is observed that the proposed solution is able to find the sampling points that results in a better conditioned SHM and also maintains all the application specific requirements.