Method and Algorithm of fast Parametric Discrete Fourier Transform with High Frequency Resolution in the Selected Spectral Region of a Finite Discrete Signal

The article is devoted to the development of the theory of digital signal processing, the development of an effective method for the fast parametric Fourier transform of finite discrete signals with high frequency resolution in the selected region of their full frequency spectrum. The development of the method for the fast parametric Fourier transform with high frequency resolution is based on the analysis of the internal organization (structure) of the basis of the discrete parametric Fourier transform - a system of parametric discrete exponential functions. The analysis of the structure of the basis of the discrete parametric Fourier transform made it possible to eliminate one significant drawback of the algorithms for the fast parametric Fourier transform - the limited durations of finite discrete signals that allow the use of fast procedures. The developed method for the fast parametric Fourier transform of finite discrete signals allows one to significantly expand the number of durations that allow the use of fast procedures, with an insignificant increase in computational costs. The paper considers a generalization of classical discrete Fourier transforms by introducing parameters in the time or frequency domain into their bases. The work is a continuation of the authors' research in the field of digital spectral analysis: in particular, it proposes a new basis for the discrete Fourier transform, which provides for a transition to the frequency domain with a selected parameter of a finite discrete signal in the time domain. The article provides a theoretical and experimental justification for the efficiency and effectiveness of the proposed method and algorithm for the fast parametric Fourier transform of finite discrete signals with high frequency resolution in the selected region of the spectrum of a finite discrete signal. Due to its proven efficiency and effectiveness, the proposed method of the fast parametric discrete Fourier transform significantly expands the capabilities of the methods of digital spectral and vector analysis of finite discrete signals of various structures in many areas of science and technology. The role of the work in such areas as vibroacoustic functional diagnostics of objects in mechanical engineering, medicine, and sonar is emphasized.