There are many ways to choose eigenbasis for the Discrete Fourier Transform matrix since it has only $4$ distinct eigenvalues taken from $\{\pm 1,\pm i\}$.

Has there been any refereed work that provides a sparse eigenvector basis for the Discrete Fourier Transform matrix?

non-refereedarticle available on ArXiv: Sparse Eigenvectors of the Discrete Fourier Transform. – Bill Bradley Mar 12 at 14:18