# American Institute of Mathematical Sciences

doi: 10.3934/naco.2022008
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## Description of multi-periodic signals generated by complex systems: NOCFASS - New possibilities of the Fourier analysis

 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

*Corresponding author: Raoul R. Nigmatullin

This paper is handled by Burcu Gürbüz as guest editor.

Received  January 2022 Revised  March 2022 Early access April 2022

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 $\Omega_{k} = c.k+d$ that differs from the conventional dispersion accepted in the conventional Fourier transformation $\omega(k) = \frac{2\pi k}{T}$. (T 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.

Citation: Raoul R. Nigmatullin, Vadim S. Alexandrov, Praveen Agarwal, Shilpi Jain, Necati Ozdemir. Description of multi-periodic signals generated by complex systems: NOCFASS - New possibilities of the Fourier analysis. Numerical Algebra, Control and Optimization, doi: 10.3934/naco.2022008
##### References:
 [1] L. Cohen, Time-Frequency Analysis, PTR E.M Prentice Hall: Englewood Cliffs, N.J., 1995. [2] F. Cupertino, E. De Vanna, L. Salvatore and S. Stasi, Comparison of spectral estimation techniques applied to induction motor broken bars detection, Proc. of SDEMPED 2003 - 4th IEEE Inter-national Symposium on Diagnostics for Electric Machines Power Electronics and Drives, 11 (2003), 129-134. [3] F. Cupertino, G. Martorana, L. Salvatore and S. Stasi, Diagnostic startup test to detect induction motor broken bars via short-time music algorithm applied to current space-vector, Proc. of EPE 2003 10th Eur. Conf. Power Electron. & App., (2003), 2–4. [4] F. Cupertino, E. De Vanna, G. Forcella, L. Salvatore and S. Stasi, Detection of IM broken rotor bars using MUSIC pseudo-spectrum and pattern recognition, Proc. of IECON '03 The 29th Annual Conf. IEEE Ind. Electron, (2003), 2829–2834. [5] P. A. Delgado-Arredondo, D. Morinigo-Sotelo, R. A. Osornio-Rios, J. G. Avina-Cervantes, H. Rostro-Gonzalez and R. D. J. Romero-Troncoso, Methodology for fault detection in induction motors via sound and vibration signals, Mechanical Systems and Signal Processing, 83 (2017), 568-589. [6] E. Elbouchikhi, V. Choqueuse, M. E. H. Benbouzid and J. F. Charpentier, Induction machine fault detection enhancement using a stator current high resolution spectrum, IECON 2012-38th Annual Conference on IEEE Industrial Electronics Society, (Montreal, Canada), (2012), 3913–3918. [7] J. Feder, Fractals, Plenum Press, NY & London, August 1988. doi: 10.1007/978-1-4899-2124-6. [8] P. Flandrin, G. Rillinga and P. Goncalves, Empirical mode decomposition as a filter bank, IEEE Signal Process. Lett., 11 (2004), 112-114. [9] A. Garcia-Perez, O. Ibarra-Manzano and R. J. Romero-Troncoso, Analysis of partially broken rotor bar by using a novel empirical mode decomposition method, Proceedings of the IECON 2014 - 40th Annual Conference on IEEE Industrial Electronics Society, (2014), 3403–3408. [10] N. Huang, Z. Shen, S. Long, M. Wu, H. Shih and Q. Zheng, The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time series analysis, Proc. R. Soc. A Math. Phys. Eng. Sci. 474, 1971 (1998), 903-995.  doi: 10.1098/rspa.1998.0193. [11] R. G. Lyons, Understanding Digital Signal Processing, A Prentice Hall. PTR Publication, Upper Saddle River, 2001. [12] R. R. Nigmatullin and I. A. Gubaidullin, NAFASS: Fluctuation spectroscopy and the Prony spectrum for description of multi-frequency signals in complex systems, Commun. Nonlinear Sci. Numer. Simul., 56 (2018), 1263-1280.  doi: 10.1016/j.cnsns.2017.08.009. [13] R. R. Nigmatullin and V. A. Toboev, Non-orthogonal amplitude-frequency analysis of the smoothed signals (nafass): dynamics and the fine structure of the sunspots, Journal of Applied Nonlinear Dynamics, 4 (2015), 67-80. [14] R. R. Nigmatullin and W. Zhang, NAFASS in action: how to control randomness?, Commun. Nonlinear Sci. Numer. Simul., 18 (2013), 547-558.  doi: 10.1016/j.cnsns.2012.07.008. [15] R. R. Nigmatullin, D. Striccoli, G. Boggia and C. Ceglie, A novel approach for characterizing multi-media 3D video streams by means of quasiperiodic processes, Signal, Image and Video Processing, (2016), 1–6. [16] R. R. Nigmatullin, V. A. Toboev, P. Lino and G. Maione, Reduced fractal model for quantitative analysis of averaged micromotions in mesoscale: Characterization of blow-like signals, Chaos, Solitons & Fractals, 76 (2015), 166-181. [17] A. Sović and D. Seršić, Signal decomposition methods for reducing drawbacks of the DWT, Eng. Rev., 32 (2012), 70-77. [18] E. M. Stein and G. L. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton University Press, 1971. [19] M. E. Torres, M. A. Colominas, G. Schlotthauer and P. Flandrin, A complete ensemble empirical mode decomposition with adaptive noise, Proceedings of the 2011 IEEE International Conference on Acoustics Speech and Signal Processing. IEEE, (2011), 4144–4147. doi: 10.1109/ICASSP.2011.5947265. [20] Z. Wu and N. E. Huang, Ensemble empirical mode decomposition: a noise-assisted data analysis method, Adv. Adapt. Data Anal., 1 (2009), 1-41.  doi: 10.1142/S1793536909000047.

show all references

##### References:
 [1] L. Cohen, Time-Frequency Analysis, PTR E.M Prentice Hall: Englewood Cliffs, N.J., 1995. [2] F. Cupertino, E. De Vanna, L. Salvatore and S. Stasi, Comparison of spectral estimation techniques applied to induction motor broken bars detection, Proc. of SDEMPED 2003 - 4th IEEE Inter-national Symposium on Diagnostics for Electric Machines Power Electronics and Drives, 11 (2003), 129-134. [3] F. Cupertino, G. Martorana, L. Salvatore and S. Stasi, Diagnostic startup test to detect induction motor broken bars via short-time music algorithm applied to current space-vector, Proc. of EPE 2003 10th Eur. Conf. Power Electron. & App., (2003), 2–4. [4] F. Cupertino, E. De Vanna, G. Forcella, L. Salvatore and S. Stasi, Detection of IM broken rotor bars using MUSIC pseudo-spectrum and pattern recognition, Proc. of IECON '03 The 29th Annual Conf. IEEE Ind. Electron, (2003), 2829–2834. [5] P. A. Delgado-Arredondo, D. Morinigo-Sotelo, R. A. Osornio-Rios, J. G. Avina-Cervantes, H. Rostro-Gonzalez and R. D. J. Romero-Troncoso, Methodology for fault detection in induction motors via sound and vibration signals, Mechanical Systems and Signal Processing, 83 (2017), 568-589. [6] E. Elbouchikhi, V. Choqueuse, M. E. H. Benbouzid and J. F. Charpentier, Induction machine fault detection enhancement using a stator current high resolution spectrum, IECON 2012-38th Annual Conference on IEEE Industrial Electronics Society, (Montreal, Canada), (2012), 3913–3918. [7] J. Feder, Fractals, Plenum Press, NY & London, August 1988. doi: 10.1007/978-1-4899-2124-6. [8] P. Flandrin, G. Rillinga and P. Goncalves, Empirical mode decomposition as a filter bank, IEEE Signal Process. Lett., 11 (2004), 112-114. [9] A. Garcia-Perez, O. Ibarra-Manzano and R. J. Romero-Troncoso, Analysis of partially broken rotor bar by using a novel empirical mode decomposition method, Proceedings of the IECON 2014 - 40th Annual Conference on IEEE Industrial Electronics Society, (2014), 3403–3408. [10] N. Huang, Z. Shen, S. Long, M. Wu, H. Shih and Q. Zheng, The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time series analysis, Proc. R. Soc. A Math. Phys. Eng. Sci. 474, 1971 (1998), 903-995.  doi: 10.1098/rspa.1998.0193. [11] R. G. Lyons, Understanding Digital Signal Processing, A Prentice Hall. PTR Publication, Upper Saddle River, 2001. [12] R. R. Nigmatullin and I. A. Gubaidullin, NAFASS: Fluctuation spectroscopy and the Prony spectrum for description of multi-frequency signals in complex systems, Commun. Nonlinear Sci. Numer. Simul., 56 (2018), 1263-1280.  doi: 10.1016/j.cnsns.2017.08.009. [13] R. R. Nigmatullin and V. A. Toboev, Non-orthogonal amplitude-frequency analysis of the smoothed signals (nafass): dynamics and the fine structure of the sunspots, Journal of Applied Nonlinear Dynamics, 4 (2015), 67-80. [14] R. R. Nigmatullin and W. Zhang, NAFASS in action: how to control randomness?, Commun. Nonlinear Sci. Numer. Simul., 18 (2013), 547-558.  doi: 10.1016/j.cnsns.2012.07.008. [15] R. R. Nigmatullin, D. Striccoli, G. Boggia and C. Ceglie, A novel approach for characterizing multi-media 3D video streams by means of quasiperiodic processes, Signal, Image and Video Processing, (2016), 1–6. [16] R. R. Nigmatullin, V. A. Toboev, P. Lino and G. Maione, Reduced fractal model for quantitative analysis of averaged micromotions in mesoscale: Characterization of blow-like signals, Chaos, Solitons & Fractals, 76 (2015), 166-181. [17] A. Sović and D. Seršić, Signal decomposition methods for reducing drawbacks of the DWT, Eng. Rev., 32 (2012), 70-77. [18] E. M. Stein and G. L. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton University Press, 1971. [19] M. E. Torres, M. A. Colominas, G. Schlotthauer and P. Flandrin, A complete ensemble empirical mode decomposition with adaptive noise, Proceedings of the 2011 IEEE International Conference on Acoustics Speech and Signal Processing. IEEE, (2011), 4144–4147. doi: 10.1109/ICASSP.2011.5947265. [20] Z. Wu and N. E. Huang, Ensemble empirical mode decomposition: a noise-assisted data analysis method, Adv. Adapt. Data Anal., 1 (2009), 1-41.  doi: 10.1142/S1793536909000047.
Initial signal obtained with the help of frequency distribution (11) and distributions of amplitudes and phases from (13)
Different F-spectra obtained with the help of the shifting function $a_{s} = \pi(1+s)$, s = 0, 1, 2, 3. This picture helps to identify the location of the leading amplitudes
The right-hand part of the central F-spectrum 2 (marked on the previous figure by red line). The truncated part of the spectra obtained with the help of expression (8) and its sequence of the ranged amplitudes (shown by red line) are used for calculated the shortcut spectrum $\Omega_{k}$
The fit of the initial signal y(t) realized with the help of the shortcut spectrum $\Omega_{k}$. The limiting frequency value is obtained with the help of the previous figure. The value of the relative error equals 4.5%
(a) This plot demonstrates the differences between true spectra (black squares) and the shortcut spectrum (crossed squares) that is obtained in the result of the 4-step algorithm. The last spectrum contains the excess of frequencies that helps to provide the accurate fit of the signal shown on the previous figure. However, this number is much less (in 6.8 times) in comparison with total spectrum (containing 500 frequencies and coinciding with the total number of data points) defined by expression (9)
(b) This is the central figure of the section 3. The NOCFASS approach identifies the desired frequencies that are contained in the initial dispersion law (11). They are identified with the high accuracy in spite of possible distortions evoked by the nearest lobes of the Dirichlet function and the influence of the neigh-boring modes shown in expression (5). The parasite frequencies marked by red squares demonstrate the strongest influence of the neighboring modes
(c) Phase distribution characteristic $\phi_{k}(\Omega c_{k})$ related to y(t). The sequence of the ranged amplitudes (SRA) (marked by red rhombs) shows that the distribution $\phi_{k}(\Omega c_{k})$ is close to the uniform distribution
The behavior of the WM-function at the limiting values of the fractal dimension D = 1.1 (black squares) and D = 1.99 (blue triangles)
(a) The spectra for the limiting values of the WM-function. One can notice the appearance of the additional components with increasing of the value of D from 1.1. up to 1.99 values. These additional components are appeared because of more random behavior in comparison with a "smooth" behavior of this function at D = 1.1. The shortcut spectrum is calculated on the right hand of this total spectra
(b) On the central figure we show the shortcut spectra for D = 1.1 (black squares) and D = 1.99 (blue triangles). On the small figure we demonstrate the calculated shortcut spectrum $\Omega_{k} = a.k+b.$ It is interesting to notice that the spectra for the limiting values practically coincide with each other giving 22 values with $\Omega_{0} = 1.257$ and $\Omega_{K} = 62.228.$
(c) Here we show the fit of the WM-function for three fractal dimensions D = 1.1, D = 1.5 and D = 1.99
(a) Amplitude-frequency responses for three WM-functions with D = 1.1, 1.5 and 1.99. Again, we observe many additional frequencies for the case $D_{5} = 1.99.$ This quasi-chaotic behavior shown in Fig. 6 requires many additional and comparable frequencies in comparison with the case of WM-function having $D_{1} = 1.1.$ We note that this tendency is conserved for many non-periodic curves studied
(b) Distribution of the phases for three WM-functions with dimensions D = 1.1, 1.5 and 1.99. Again, for smooth curve with D = 1.1 we observe monotone phases distribution, while for D = 1.99 this distribution occupies all band between $(\frac{-\pi}{2},\frac{\pi}{2})$
(a) The fit of the initial current with the value of the relative error 0.12%. On the small figure above, we show the dependence $Amd_{k}(\Omega_{k}) = \sqrt{Ac_{k}^{2}+As_{k}^{2}}$
(b) Here we demonstrate the phase distribution $D(\Omega_{k}) = \tan^{-1}( \Omega_{k})$ for the initial current $J_{0}(t)$. This distribution is uniform because the sequence of the range amplitudes (SRA) shown by red small balls corresponds approximately to a straight line
(a) The fit of the final current with the value of the relative error 0.15%. On the small figure above, we show the dependence $Amd_{k}(\Omega_{k}) = \sqrt{Ac_{k}^{2}+As_{k}^{2}}$. As one can notice from comparison of two distributions the spectral band for these two curves is different
(b) Here we demonstrate the phase distribution $D(\Omega_{k}) = \tan^{-1}( \Omega_{k})$ for the final current $J_{f}(t)$. This distribution can be defined as uniform again, because the sequence of the range amplitudes (SRA) shown by red small balls corresponds approximately to a straight line
(a) The fit of the current ranges with the value of the relative error 0.05%. On the small figure above, we show the dependence $Amd_{k}(\Omega_{k}) = \sqrt{Ac_{k}^{2}+As_{k}^{2}}$. Obviously, as one can notice from comparison with two previous distributions the spectral band for this function $Rg(J_{m}(t)) = \max(J_m)-\min(J_m)$ is different
(b) The phase distribution $D(\Omega_{k}) = \tan^{-1}( \Omega_{k})$ for the ranges of the currents. This distribution can be defined as uniform again, because the sequence of the range amplitudes (SRA) shown by red small balls corresponds approximately to a straight line
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