Evolution of Forward and Inverse Discrete Fourier Transform

The problems of the evolution of the forward and inverse discrete Fourier transform are investigated. Forward and inverse discrete Fourier transform is the basis of the classical discrete spectral analysis of signals. The effectiveness of known methods for modifying the original discrete signal is analyzed to improve the characteristics of classical discrete spectral analysis of signals. A generalization of the discrete Fourier transform (DFT) in the form of a parametric discrete Fourier transform (DFT-P) is proposed, which, in essence, is the evolution of the discrete Fourier transform. A generalization of the inverse discrete Fourier transform (IDFT) is given in the form of a modified parametric discrete Fourier transform (MDFT-P). The basic properties of the bases of DFT-P and MDPF-P are presented. The interrelation between the methods of modifying the original discrete signal and the parametric DFT and the modified parametric DFT is considered.