Evolution of One-Dimensional and Two-Dimensional Discrete Fourier Transform

The rapid development of information technology has significantly expanded the scope of application of digital Fourier processing of finite signals. We note tomography, active and passive sonar, radar, seismology, technical diagnostics, medicine, forensic cybernetics, and artificial intelligence among these applications. The complication of the tasks solved by information technologies in these subject areas stimulated, firstly, the transition from one-dimensional to two-dimensional digital Fourier processing, and secondly, it posed an urgent theoretical and applied problem of finding new basic systems, both in one-dimensional and two-dimensional case. Systems 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. A systematic analysis of the actual theoretical and applied problem of searching for new basic systems has shown that the most important requirements for basic systems are: orthogonality, symmetry and multiplicativity. The article provides the detailed analysis of the analytical properties of new two discrete Fourier transforms developed by the authors. These are Parametric Discrete Fourier Transform (DFT-P) for Fourier processing of scalar functions of scalar arguments and 2D Discrete Fourier Transform with variable parameters (2D DFT-VP) for Fourier processing of scalar functions of vector arguments. DFT-P and 2D DFT-VP transforms are based on a generalization of exponential basis systems of DFT and 2D DFT transforms.