Fast Parametric Fourier Transform

The purpose of the study is to the develop new discrete Fourier transform, by generalizing the classical discrete Fourier transform (DFT) in the form of a parametric discrete Fourier transform (DFT-P) and the development of fast algorithms for implementing this type of transform - the fast parametric discrete Fourier transform (FFT-P). The classical Fourier processing of complex vibroacoustic finite discrete signals (VFD signals), based on DFT and fast Fourier transform (FFT) algorithm, is the most powerful method of digital analysis, modeling, optimization, improvement of control and decision making in solving problems of machine dynamics and vibroacoustic. The theoretical basis of the classical Fourier processing of VFD signals is the discrete Fourier transform. The practical base of the classical Fourier processing of VFD signals is the fast Fourier transform algorithms. However, the practice of applying the classical Fourier processing of VFD signals of a complex structure containing both deterministic periodic and random signals, on the one hand, confirming the effectiveness of the classical Fourier processing, on the other hand revealed a number of negative effects inherent in this type of digital signal processing (DSP). The aliasing effect, the scalloping effect, the picket fence effect, significantly negatively affect the effectiveness of analysis, modeling, optimization, improvement of control and decision-making in the study. To improve the efficiency and effectiveness of the Fourier processing of VFD signals, authors of this work developed a generalization of the theoretical basis of the classical Fourier processing - DFT in the form of a parametric DFT (DFT-P). Since the direct application of the parametric Fourier processing of VFD signals (as well as the use of the classical Fourier processing of VFD signals) requires complex multiplications, fast procedures are needed for the practical implementation of this type of VFD signals. We have developed fast procedures for implementing DFT-P by time decimation. Parametric FFT (FFT-P) with replacement (in place) and without replacement (no place) are proposed. The estimation of the efficiency of FFT-P algorithms is given. The practical significance of the work lies in the fact that the developed algorithms for the parametric fast Fourier transform make it possible to reduce the computational costs for performing parametric discrete transformations by three or more orders of magnitude.