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

Abstract

In this chapter, a general concept for the consideration of any reproducible 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.

Mathematical Appendix

Generalization of the functional equation ( 1.11 ) for incommensurable periods.

Let us consider the generalization of the functional equation (1.11) for incommensurable periods:

$$ \begin{array}{rl}\hfill F\left(x+{\alpha}_L{T}_x\right)&={\displaystyle \sum_{l=0}^{L-1}{a}_lF\left(x+{\alpha}_l{T}_x\right)+b,}\hfill \\ {}\hfill {\alpha}_l&=0<{\alpha}_1<\dots <{\alpha}_L,\;\hfill \\ {}{\alpha}_m/{\alpha}_n\ne m/n&=\mathrm{irrational}\kern0.24em \mathrm{number}.\end{array} $$

(1.53)

We search for a solution of Eq. (1.53) in the form

$$ \begin{array}{rl}\hfill F(x)&={\displaystyle \sum_{k=0}^K\left[A{c}_ky{c}_k(x)+A{s}_ky{s}_k(x)\right],}\hfill \\ {}\hfill y{c}_k(x)&={\left(\kappa \right)}^{x/{T}_x} \cos \left(2\pi k\frac{x}{T_x}\right),\kern0.24em y{s}_k(x)={\left(\kappa \right)}^{x/{T}_x} \sin \left(2\pi k\frac{x}{T_x}\right).\hfill \end{array} $$

(1.54)

Here the parameter κ is not known, and the period T _x can be found from the fitting procedure described in Sect. 1.2. Based on expressions (1.23), following some simple algebra it is easy to establish the following relationship:

$$ \begin{array}{l}\begin{array}{rl}\hfill F\left(x+\varDelta \cdot {T}_x\right)&={\left(\kappa \right)}^{\varDelta }{\displaystyle \sum_{k=0}^K\left[A{c}_k \cos \left(2\pi k\varDelta \right)+A{s}_k \sin \left(2\pi k\varDelta \right)\right]}\cdot y{c}_k(x)\hfill \\ {}\hfill &+{\left(\kappa \right)}^{\varDelta }{\displaystyle \sum_{k=0}^K\left[-A{c}_k \sin \left(2\pi k\varDelta \right)+A{s}_k \cos \left(2\pi k\varDelta \right)\right]}\cdot y{s}_k(x).\hfill \end{array}\\ {}\kern3.36em \end{array} $$

(1.55)

Taking into account the fact that functions yc _k (x) and ys _k (x) in (1.54) are linearly independent following the substitution of (1.55) into (1.53) we obtain

$$ \begin{array}{rl} &{\left(\kappa \right)}^{\alpha_L}\left[A{c}_k \cos \left(2\pi k{\alpha}_L\right)+A{s}_k \sin \left(2\pi k{\alpha}_L\right)\right]\\ &\quad ={\displaystyle \sum_{l=0}^{L-1}{a}_l{\left(\kappa \right)}^{\alpha_l}\left[A{c}_k \cos \left(2\pi k{\alpha}_l\right)+A{s}_k \sin \left(2\pi k{\alpha}_l\right)\right]}. \end{array}$$

(1.56)

But the decomposition coefficients Ac _k and As _k form also the couple linearly independent sets, and for any k = 0,1,…,K we obtain two independent relationships:

$$ \begin{array}{c}\hfill {\kappa}^{\alpha_L} \cos \left(2\pi k{\alpha}_L\right)={\displaystyle \sum_{l=0}^{L-1}{a}_l{\left(\kappa \right)}^{\alpha_l} \cos \left(2\pi k{\alpha}_l\right)},\hfill \\ {}\hfill {\kappa}^{\alpha_L} \sin \left(2\pi k{\alpha}_L\right)={\displaystyle \sum_{l=0}^{L-1}{a}_l{\left(\kappa \right)}^{\alpha_l} \sin \left(2\pi k{\alpha}_l\right)}.\hfill \end{array} $$

(1.57)

Multiplying the second relationship by the complex unit $i=\sqrt{-1}$ we obtain

$$ \begin{array}{c}\hfill {\kappa}^{\alpha_L} \exp \left(i2\pi k{\alpha}_L\right)={\displaystyle \sum_{l=0}^{L-1}{a}_l{\left(\kappa \right)}^{\alpha_l} \exp \left(i2\pi k{\alpha}_l\right)},\hfill \\ {}\hfill {\kappa}^{\alpha_L}{\left[ \exp \left(i2\pi k\right)\right]}^{\alpha_L}={\displaystyle \sum_{l=0}^{L-1}{a}_l{\left(\kappa \right)}^{\alpha_l}{\left[ \exp \left(i2\pi k\right)\right]}^{\alpha_l}}.\hfill \end{array} $$

(1.58)

From expression (1.58) it follows that for any k = 0,1,…,K

$$ {\kappa}^{\alpha_L}={\displaystyle \sum_{l=0}^{L-1}{a}_l{\left(\kappa \right)}^{\alpha_l}}. $$

(1.59)

The last coincides with equation (1.13) for the case of proportional periods (α _L  → L, α _l  → l) and, thereby, generalizes the case considered in Sect. 1.2. The last generalization (1.59) makes the approach considered in this chapter very flexible and general and merits special research.