Closed-form eigenvectors of the discrete Fourier Transform

Properties of eigenvectors and eigenvalues for discrete Fourier transform (DFT) are important for defining and understanding the discrete fractional Fourier transform (DFRFT). In this paper, we first propose a closed-form formula to construct an eigenvector of N-point DFT by down-sampling and then folding any eigenvector of (4N)-point DFT. The result is then generalized to derive eigenvectors of N-point DFT from eigenvectors of (k²N)-point DFT. To show an application of the proposed new closed-form DFT eigenvectors, Hermite-Gaussian-like (HGL) DFT eigenvectors which are much closer to the continuous Hermite-Gaussian functions (HGFs) are computed from existing HGL DFT eigenvectors of larger sizes with computer experiments.