Description of multi-periodic signals generated by complex systems: NOCFASS - New possibilities of the Fourier analysis

Author:

Nigmatullin Raoul R.1,Alexandrov Vadim S.1,Agarwal Praveen234,Jain Shilpi5,Ozdemir Necati6

Affiliation:

1. Kazan National Research Technical University, A. N. Tupolev (KNRTU-KAI), Radioelectronics and Informative-Measurements Technics Department, Kazan, Tatarstan, K. Marx str. 10 (420111), Russian Federation

2. Anand International College of Engineering, Near Kanota, Agra Road, Jaipur 303012, Rajasthan, India

3. International Center for Basic and Applied Sciences, Jaipur 302029, India

4. Nonlinear Dynamics Research Center (NDRC), Ajman University, Ajman, UAE

5. Poornima College of Engineering, Jaipur, India

6. Department of Mathematics, Balikesir University, Balikesir, Turkey

Abstract

<p style='text-indent:20px;'>Here, we show how to extend the possibilities of the conventional F-analysis and adapt it for quantitative description of multi-periodic signals recorded from different complex systems. The basic idea lies in filtration property of the Dirichlet function that allows finding the leading frequencies (having the predominant amplitudes) and the shortcut frequency band allows to fit the initial random signal with high accuracy (with the value of the relative error less than 5%). This modification defined as NOCFASS-approach (Non-Orthogonal Combined Fourier Analysis of the Smoothed Signals) can be applied to a wide class of different signals having multi-periodic structure. We want to underline here that the shortcut frequency dispersion has linear dependence <inline-formula><tex-math id="M1">\begin{document}$ \Omega_{k} = c.k+d $\end{document}</tex-math></inline-formula> that differs from the conventional dispersion accepted in the conventional Fourier transformation <inline-formula><tex-math id="M2">\begin{document}$ \omega(k) = \frac{2\pi k}{T} $\end{document}</tex-math></inline-formula>. (<i>T</i> is a period of the initial signal). With the help of integration procedure one can extract a low-frequency trend from trendless sequences that allows to applying the NOCFASS approach for calculation of the desired amplitude-frequency response (AFR) from different "noisy" random sequences. In order to underline the multi-periodic structure of random signals under analysis we consider two nontrivial examples. (a) The peculiarities of the AFR associated with Weierstrass-Mandelbrot function. (b) The random behavior of the voltammograms (VAGs) background measured for an electrochemical cell with one active electrode. We do suppose that the proposed NOCFASS-approach having new attractive properties as the simplicity of realization, agility to the problem formulated will find a wide propagation in the modern signal processing area.</p>

Publisher

American Institute of Mathematical Sciences (AIMS)

Subject

Applied Mathematics,Control and Optimization,Algebra and Number Theory,Applied Mathematics,Control and Optimization,Algebra and Number Theory

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