Methods for Increasing Speed of Two-Dimensional Discrete Fourier Transform of 2D Finite Discrete Signals

Abstract

The paper posed and successfully solved the problems of increasing the speed of methods and algorithms of two-dimensional (2D) digital Fourier processing, analyzing the properties of 2D discrete Fourier transform (2D DFT) and the possibilities of applying methods and algorithms developed in one-dimensional digital Fourier processing. The performed systems analysis of the transition from 1D to 2D digital Fourier processing of 2D FD signals showed that such a transition is far from trivial. Some important properties of 2D DFT have no analogs at all in the one-dimensional case and, therefore, some important properties of 2D DFT cannot be obtained by generalizing the properties of 1D DFT to the two-dimensional case. In the transition from 1D to 2D Fourier processing of 2D FD signals, the computational costs also increase by several orders of magnitude. This posed the challenge of developing fast procedures for 2D DFT implementation of 2D FD signals. One of the important analytical properties of 2D DFT, the theoretical basis of classical Fourier processing of 2D FD signals, 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 FD signals. The basis of these methods is fast one-dimensional Fourier transform algorithm with decimation in time, no place way. The work proved the effectiveness and efficiency of the proposed methods by means of mathematical modeling.