Solution of ill-posed problem in plane wave decomposition for sound field reconstruction

Resumo

Plane-wave decomposition is a recently developed technique in the field of acoustics that stems from
an approximation in the inverse spatial tri-dimensional Fourier transform so that it considers that the sound field
is represented by the superposition of plane waves traveling in well-defined directions. The calculations for the
technique involve the solution of an ill-posed matrix equation, requiring a regularized solution for the least squares
problem. This paper will show the implementation of the plane wave decomposition using Tikhonov regularization
in the context of a simulated and a measured impedance tube. The classical L-curve algorithm and a fixed-point
algorithm for the calculation of the regularization parameter were investigated and compared with the goal of
defining which technique produces the least error in this scenario. The reconstructions of the transfer functions
in both the simulated case and the measured case were realised, with the fixed-point algorithm displaying an
advantage over the L-curve with respect to reconstruction errors in both scenarios.