Detection of Quasi-Periodic Processes in Experimental Measurements: Reduction to an “Ideal Experiment”

Abstract

In this chapter, a general concept for the consideration of anyreproducible data, measured in many experiments, in one unified scheme is proposed. In addition, it has been demonstrated that successive and reproducible measurements have a memory, and this important fact makes it possible to group all data into two large classes: ideal experiments without memory and experiments with memory. Real data with memory can be defined as a quasi-periodic process and are expressed in terms of the Prony decomposition (this presentation serves as the fitting function for the quantitative description of the data), while experiments without memory are needed to present a fragment of the Fourier series only. In other words, a measured function extracted from reproducible data can have a universal quantitative description expressed in the form of the amplitude-frequency response (AFR) that belongs to the generalized Prony spectrum (GPS). The proposed scheme is rather general and can be used to describe all kinds of experiments that can be reproduced (with acceptable accuracy) within a certain period of time. The proposed general algorithm makes it possible to consider many experiments from a unified point of view. Two real examples taken from physics (X-ray scattering measurements) and electrochemistry confirm this general concept. A unified so-called bridge between the treated experimental data and a set of competitive hypotheses that are supposed to described them is discussed. The general solution of the problem, where the apparatus function can be accurately eliminated and the measured data can be reduced to an “ideal” experiment, is presented. The results obtained in this paper help to formulate a new paradigm in data/signal processing for a wide class of complex systems (especially in cases where the best fit model is absent), and the conventional conception associated with the treatment of different measurements should, from our point of view, be reconsidered. As an alternative approach we considered also the nonorthogonal amplitude-frequency analysis of smoother signals (NAFASS) approach, which can be used for the fitting of nonlinear signals containing different beatings. We justify the general dispersion law that can be used for the analysis of various signals containing different multifrequencies.