On discrete Gauss–Hermite functions and eigenvectors of the discrete Fourier transform

The problem of furnishing an orthogonal basis of eigenvectors for the discrete Fourier transform (DFT) is fundamental to signal processing and also a key step in the recent development of discrete fractional Fourier transforms with projected applications in data multiplexing, compression, and hiding. Existing solutions toward furnishing this basis of DFT eigenvectors are based on the commuting matrix framework. However, none of the existing approaches are able to furnish a commuting matrix where both the eigenvalue spectrum and the eigenvectors are a close match to corresponding properties of the continuous differential Gauss–Hermite (G–H) operator. Furthermore, any linear combination of commuting matrices produced by existing approaches also commutes with the DFT, thereby bringing up issues of uniqueness.In this paper, inspired by concepts from quantum mechanics in finite dimensions, we present an approach that furnishes a basis of orthogonal eigenvectors for both versions of the DFT. This approach furnishes a commuting matrix whose eigenvalue spectrum is a very close approximation to that of the G–H differential operator and in the process furnishes two generators of the group of matrices that commute with the DFT.