The Wigner transform (WT) is a well known quadratic transform, mapping a wavefunction to a position-wavenumber quasi-density. WTs are used in the asymptotic treatment of semiclassical and other wave problems. The smoothed WT (SWT) method is a regularization and extension of WT-based approaches for the homogenization of wave propagation, which decouples homogenization from asymptotics: exact equations for the evolution of a given problem at coarse scale can be obtained, instead of merely looking at small parameter limits. In this work we present the derivation of the SWT equations.
Athanassoulis, A. G. (2007). Smoothed Wigner transforms and homogenization of wave propagation. In Days on Diffraction: 2007 Proceedings of the International Conference (pp. 13-18). IEEE. https://doi.org/10.1109/DD.2007.4531981