Weyl-Galileo transformations and related Wigner function

The present work concerns the phase space representation of special signals depending on two variables. These signals are of acoustic origin and correspond either to the passive observation of sources with a linear array or to the active study of evolutive sonar targets. The technique used to set up the phase space representation is founded on a constraint of covariance with respect to transformations coming from changes of reference systems. Due to the physical context these transformations correspond to changes of origin, scalings and Galilean boosts and make up the Weyl-Galileo group. A Wigner function affiliated with that group and depending on four variables is effectively obtained. It satisfies both localization and unitarity. The associated wavelet transform is recovered by smoothing