Properties of Two-Dimensional Discrete Exponential Functions with Variable Parameter in Spatial-Frequency Domain

Abstract

Fourier-processing of finite discrete two-dimensional signals (FDTD signals) (including images) in informational control systems (IC systems) is the most important method for studying processes and analyzing information. The theoretical basis of Fourier-processing of FDTD signals is two-dimensional discrete Fourier transforms. The practice of using Fourier - processing of finite two-dimensional signals (including images), having confirmed its effectiveness, revealed a number of negative effects inherent in it. In order to solve this problem, the authors have developed new discrete Fourier transforms with variable parameters were developed (2D DFT-VP). The purpose of this work is to study the properties of two-dimensional discrete exponential functions with a variable parameter in the spatial-frequency domain. The introduction of discrete exponential functions with a variable parameter makes it possible to generalize the concept of periodicity of the DEF-VP system. Recall that the periodicity of the DEF system in the classical DFT is understood as a periodic continuation of the DEF system outside the interval of N samples. Moreover, the system of discrete basis functions in the classical DFT does not contain discontinuities. In the case of DFT-VP, for the DEF-VP system to be inseparable, the periodicity should be understood as parametric periodicity. The parametric periodicity of discrete exponential functions with a variable parameter is understood as their periodic continuation with rotation in complex space by a certain angle. Note that the introduced concept of parametric periodicity is valid for 1D and 2D real and complex functions. The theorems of linearity, shift, and correlation are proved for Fourier transforms with variable parameters.