Fast Algorithms for Two-Dimensional Discrete Fourier Transform of Vibroacoustic Signals in Solving Problems of Control and Technical Condition of Machines and Mechanisms

Abstract

The aim of the work is to develop fast algorithms for two-dimensional discrete Fourier transform of vibroacoustic signals in solving problems of control and technical condition analysis of machines and mechanisms. The development of science and technology, the improvement of digital information technologies, the expansion of the range of their applications has led to the need to describe the characteristics, properties and states of the studied complex objects of dynamics and vibration diagnostics of machines not by one-dimensional, but by two-dimensional vibroacoustic finite discrete signals (2D VFD signals). The performed systems analysis of the transition from one-dimensional Fourier processing (1D Fourier processing) to two-dimensional digital Fourier processing (2D Fourier processing) of 2D VFD signals showed that such transition is far from trivial. Some important properties of 2D Discrete Fourier Transform (2D DFT) underlying 2D Fourier Processing have no analogues at all in the case of 1D Discrete Fourier Transform (1D DFT) and hence some important properties of 2D DFT cannot be obtained by generalizing the properties of 1D DFT for the two-dimensional case. When passing from 1D to 2D Fourier processing of 2D VFD signals, the computational costs also increase by several orders of magnitude. This poses an important challenge in developing fast procedures for implementing 2D DFT of 2D VFD signals. At the same time, 2D VFD signals have a complex, mixed structure and, as a rule, consist of two parts: the sum of periodic deterministic components (most often with incommensurable spatial frequencies) and the sum of random components. The Fourier processing of such a class of 2D VFD signals also requires the development of methods for increasing the speed of 2D DFT. The paper sets and successfully solves the problems of increasing the speed of methods and algorithms for two-dimensional digital Fourier processing (2D Fourier processing) of 2D VFD signals of complex structure. One of the important analytical properties of 2D DFT is the separability of its kernel. Based on the consequences arising from this property of the 2D DFT kernel, two methods have been developed to reduce the number of computational operations in the implementation of 2D DFT of 2D VFD signals. The article considers a fast algorithm of one-dimensional fast Fourier transform with thinning in time, without replacement (no place) for calculating 2D DFT. The paper proves the effectiveness and efficiency of the proposed methods by means of mathematical modeling.