INTRODUCTION TO MELLIN OPERATOR THEORY AND SOME OF ITS APPLICATIONS IN SIGNAL PROCESSING

Abstract

Integral transformations have played an important role in the development of the theory and its applications for processing information-bearing processes. Mathematically, integral transformations map the space of the original variable into a new space of a new variable, that is, they map sets of elements of the space of the "many into one" type. In signal theory, the integral Fourier transform has been widely used not only as a representation of signals, but also in their spectral analysis. The Hilbert integral transformation served as a development of the theory of digital representation of broadband signals. The paper discusses the theory of the integral Mellin transform, which is not as well known as the previous ones, for its use in signal processing and interference, as well as some problems of an applied nature in signal theory. We presented the theory of spectral correlation analysis of random processes in the basis of the integral Mellin transform. In particular, a theorem (analogous to the Wiener-Hinchin theorem for the Fourier transform) is proved on its basis on the relationship of the correlation function of noise in the basis of the Fourier transform with the spectral density of noise power in the basis of the Mellin transform. These results can be used as the basis for the synthesis of signal processing algorithms against interference in the basis of the integral Mellin transform. Based on it, the functional structure of the signal detector has been developed against the background of Gaussian noise with unknown a priori correlation function and signal duration. It should be noted that the authors' work considers rather complex mathematical calculations. For beginners who get acquainted with the integral Mellin transformation, we recommend that you first familiarize yourself with the textbook.