Methods for numerically generating a centered discrete fractional Fourier transform matrix on a computer, the centered discrete fractional Fourier transform matrix of size N by N where N is an odd integer. Centering is obtained by simple barrel roll operations on eigenvectors. High-accuracy is obtained by numerically calculating the eigenvectors of the discrete fractional Fourier transform matrix from a closed-form mathematical formula and then iteratively performing a Gram-Schmidt orthogonalization procedure until a resulting set of improved-orthogonal eigenvectors is produced that is sufficiently orthogonal.