Paillasseur S., Pascal J., Thomas J.

Acoustic holography is a measuring process to localise sound sources from measurements made with an antenna of microphones localised near the acoustic source plane. This method called Near-field Acoustic Holography (NAH) was introduced in the heighties [Maynard et Williams, 1985] and is used for stationary sources. In order to characterise non-stationary sources, a new formulation has been introduced [Grulier, 2004] to propagate signals on a forward plane using a convolution product with a transfer function in the time-wavenumber domain. The purpose of this study is to solve the deconvolution problem in order to improve the Real-Time Near-field Acoustic Holography method and to test its accuracy. The first step is to simulate measurements made on a microphone matrix in the near field of the sources defined by non-stationary punctual sound sources. A spatial two-dimensional Fourier transform is applied to each instantaneous pressure field yielding to a time-dependent wavenumber spectrum. Taking the evanescent waves into account improves the spatial resolution of our solution but makes the deconvolution problem ”ill-posed”. It means that the inverse of the transfer function is neither unique nor stable. To solve this kind of problem, we studied regularisation methods that consist of giving an other constraint to the solution we are seeking. In our case the standard Tikhonov regularisation is used, which is based on the minimisation of the solution’s energy, combined with generalised cross validation to estimate the regularisation parameter. In order to apply this method we first use a formulation that allows us to transform the time-wavenumber domain convolution into a matrix product. Once the retropropagated wavenumber spectrum is obtained, the inverse bi-dimensional Fourier transform is applied to get a picture of the time-dependent pressure of the source. It is then possible to localise and characterise the different sound sources.