  We construct a distorted Fourier transformation associated with the
multi-dimensional quantum walk. In order to avoid the complication of
notations, almost all of our arguments are restricted to two dimensional
quantum walks (2DQWs) without loss of generality. The distorted Fourier
transformation characterizes generalized eigenfunctions of the time evolution
operator of the QW. The 2DQW which will be considered in this paper has an
anisotropy due to the bias of the shift of the free QW. Then we define an
anisotropic Banach space as a modified Agmon-H\"{o}rmander's $\mathcal{B}^*$
space and we derive the asymptotic behavior at infinity of generalized
eigenfunctions in these spaces. The scattering matrix appears in the asymptotic
expansion of generalized eigenfunctions.
