Eigenvectors and functions of the discrete Fourier transform

Abstract

Abstract-A method is presented for computing an orthonormal set of eigenvectors for the discrete Fourier transform (DFT). The technique is based on a detailed analysis of the eigenstructure of a special matrix which commutes with the DFT. It is also shown how fractional powers of the DFT can be efficiently computed, and possible applications to multiplexing and transform coding are suggested.

Digital computation of the fractional Fourier transform

“…The need to limit the extent of vertical or horizontal shearing of the Wigner distribution has also been discussed in [28] some time ago [44], [45]. According to their definition, the athorder discrete fractional transform matrix is found by taking the ath power of the DFT matrix F .…”

“…The ath power of the DFT matrix F a is found by standard procedures, and the ambiguity in taking fractional powers is resolved by choosing the principal roots. Then, one finds that where explicit expressions for the a3(a) are given in [44].…”

Computation of a real eigenbasis for the Simpson discrete Fourier transform matrix

“…In contrast, the DFT matrix possesses an orthogonal set of eigenvectors since it is unitary. Much research has focused on generating such an orthogonal set by using matrices that commute with the DFT matrix 9 and more recently a method based on complete generalized Legendre sequence has been proposed 10 .…”

A Low-Complexity Approach to Computation of the Discrete Fractional Fourier Transform

“…This approach is still used and developed [11]. The second approach relies on a linear combination of ordinary Fourier operators raised to different powers [4,24]. However, as emphasized in [3], these realizations often produce a result that does not match the result of the continuous FRFT.…”