2D Discrete Fast Fourier Transform with variable parameters

Abstract

The rapid development of information technology has significantly expanded the scope of application of digital Fourier processing of finite signals. Among them, the work noted tomography, active and passive sonar, radar, seismology, technical diagnostics, medicine, forensic cybernetics, and artificial intelligence. The development of information technologies, the complication of the tasks being solved stimulated the transition from one-dimensional to two-dimensional digital Fourier processing. System analysis has shown that the transition from the one-dimensional to the two-dimensional case is far from trivial and is primarily of a qualitative rather than quantitative nature. At the same time, the generalization of the results of the two-dimensional case to the multidimensional one, as a rule, does not cause difficulties, since it is mainly quantitative, and not qualitative. As is known, for the practical application of the application of Fourier processing methods, expanding the scope of their application, an important role belongs to the procedures for the rapid implementation of corresponding Fourier transforms (the story of FFT algorithm proposed in 1965), a clear confirmation of this. The paper deals with the solution of an important and urgent problem of developing fast algorithms for new discrete 2D Fourier transform with variable parameters (2D DFT - VP). Three groups of methods for increasing the speed of two-dimensional discrete fast Fourier transform with variable parameters are proposed and studied in the paper. The 1 st group of methods for improving the speed of 2D DFT-VP is based on the separability property of the kernel of 2D DFT - VP and the use of one-dimensional parametric DFTs (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). Group 3 2D DFT-VP performance improvement methods based on 2D Fast Fourier Transform (2D FFT - VP) in vector base 2, with space decimation, with or without replacement.