Two-Dimensional Discrete Fourier Transform with Variable Parameters in Solving Fundamental Problems of Dynamics and Vibrodiagnostics of Machines

The aim of the work is to develop new two-dimensional discrete Fourier transform with variable parameters (2D DFT-VP) and to develop fast algorithms for its implementation. Classical two-dimensional discrete Fourier transform (2D DFT), which is a special case of 2D DFT-VP, is known to have a number of negative effects arising from the nature and analytical properties of its basis. Such negative effects of 2D DFT as the aliasing effect, scalloping effect, picket fence effect significantly reduce the efficiency and effectiveness of classic 2D DFT in solving fundamental problems of dynamics and vibration diagnostics of machines and mechanisms. Not to a small extent, this is due to the transition in the dynamics and diagnostics of machines to models of a two-dimensional vibroacoustic finite signal (2D VFD signal) of a complex and mixed structure. New 2D DFT-VP allows solving the important and urgent problem of eliminating (reducing the influence) of the negative effects inherent in 2D DFT on the efficiency and effectiveness of solving fundamental problems of dynamics and vibration diagnostics of machines and mechanisms. The current transition from one-dimensional Fourier processing (1D Fourier processing) to two-dimensional digital Fourier processing (2D Fourier processing) of 2D VFD signals has shown that, firstly, such a transition is far from trivial and, in secondly, when moving from 1D Fourier processing to 2D Fourier processing, computational costs increase by several orders of magnitude. And new 2D DFT-VP is not exception from this perspective. It is necessary to develop methods and algorithms for improving the performance of 2D DFT-VP. In this work, 3 kinds of 2D DFT-VP transforms are considered in detail, and the analytical properties of their bases are investigated. Three groups of methods for increasing the performance of two-dimensional discrete fast Fourier transform with variable parameters are proposed and investigated in the work. The 1st group of methods for improving the performance of 2D DFT-VP is based on the separability property of the core of 2D DFT-VP and the use of one-dimensional parametric DFT (DFT-P). The 2nd group of methods for improving the speed of 2D DFT-VP is based on the property of separability of the kernel of 2D DFT-VP and the use of one-dimensional parametric fast Fourier transforms (FFT-P). The 3rd group of 2D DFT-VP performance improvement methods based on 2D Fast Fourier Transform with variable parameters (2D FFT-VP) in vector base 2, with space decimation, with or without replacement.