A method is described for validating fast Fourier transforms (FFTs) based on the use of simple input fiinctions whose discrete Fourier transforms can be evaluated in closed form. Explicit analytical results are developed for one-dimensional and two-dimensional discrete Fourier transforms. The analytical results are easily generalized to higher dimensions. The results offer a means for validatin...

We construct a distorted Fourier transformation associated with the multi-dimensional quantum walk. In order to avoid complication of notations, almost all our arguments are restricted two dimensional walks (2DQWs) without loss generality. The characterizes generalized eigenfunctions time evolution operator QW. 2DQW which will be considered in this paper has an anisotropy due definition shift f...

It is shown that a lossless first-order optical system whose real symplectic ray transformation matrix can be diagonalized and has only unimodular eigenvalues is similar to a separable fractional Fourier transformer in the sense that the ray transformation matrices of the unimodular system and the separable fractional Fourier transformer are related by means of a similarity transformation. More...