The Ill-Conditioning Problem in Sound Field Reconstruction
A method for the analysis and reconstruction of a three dimensional sound field using an array of microphones and an array of loudspeakers is presented. The criterion used to process the microphone signals and obtain the loudspeakers signals is based on the minimisation of the least-square error between the reconstructed and the original sound field. This approach requires the formulation of an inverse problem that can lead to unstable solutions due to the ill-conditioning of the propagation matrix. The concepts of generalised Fourier transform and singular value decomposition are introduced and applied to the solution of the inverse problem in order to obtain stable solutions and to provide a clear understanding of the regularisation method.
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