Functional conversion of signals in the study of relaxation phenomena

Abstract

The processing of monotonic signals that appeared in studying the relaxation phenomena is considered. These signals are generalized as relaxation signals and are represented by infinite functions with infinite spectra having derivatives tending to zero, when the argument goes to infinity. A logarithmic sampling where a distance between samples increases according to a geometric progression is found to be optimal for the relaxation signals. A periodic spectrum in the Mellin transform domain is shown to have a logarithmically sampled signal. Conversion of the relaxation signals based on carrying out direct and inverse integral transforms of the first kind with kernels depending on the division or product of arguments is offered by digital filtering on the logarithmically transformed argument domain. A theory of digital functional filters (DFFs) with the logarithmic sampling is developed. Conditions of an exact performance of the integral transform are found. A method of regularization is developed for integral transforms being ill-conditioned where a geometric progression ratio is used as a regularization parameter. A multicriterial optimization method is developed for optimal DFF design based on identification of a discrete system where theoretical functions interrelated with each other by the desired integral transform are used as input and output signals.