**Figure 11** shows multiresolution analysis of level 12 for sym6. The signal and details *Dj*ð Þ*t* for j = 9, 10, 11, 12 and the approximation A12(t) are plotted. Each detail *Dj*ð Þ*t* has a sample mean of zero, while the sample mean of the smooth A12(t) is equal to the sample mean of *x t*ð Þ.

It is interesting to observe the Fourier spectrum of the DWT D9. While for *x t*ð Þ only three frequencies appear, as can be seen in the inset of **Figure 12** for D9 (curve in blue) a new frequency emerges at 130000 Hz. Fourier transforms of the detail appear to enhance the SO content of *x t*ð Þ.

There are other discrete transforms defined, whose behavior is similar to DWT because it yields decomposition with the property of being shift-invariant for a time series.

The other Discrete Wavelet transform we used gives an orthogonal scaled base additive decomposition. It returns the projections of the signal *x t*ð Þ onto the various wavelet subspaces and final scaling space. This means the original signal can be recovered by adding all the projections.

In **Figure 13**, it is shown how the SO is partially removed from the original signal *x t*ð Þ by subtracting mra9(t), corresponding to the 9th scale component.

#### **8. Filtering methods to separate super oscillations**

Filtering systems can be also applied to analyze SO. Multiresolution analysis allows separating the SO part of a signal and therefore localizing the oscillations in time, that is, local frequencies. This separation takes place not only in the frequency domain but also in the time domain.

*Local Frequencies in Superoscillatory Phenomena DOI: http://dx.doi.org/10.5772/intechopen.112300*

**Figure 13.**

*MODWTMRA decomposition using sym6. In the first panel, the SO signal (red) and mra9(t) (blue) are shown. In the second panel, the SO signal minus mra9(t) shows that it is capable of "filtrate" the superoscillation.*

Multiresolution analysis (MRA), recovers the original version when added back together. Each division of the signal represents its variability and can be associated with a physical meaning. MRA is based on the analysis of wavelets. Additionally, MRA can be based on wavelet packets, or non-wavelet techniques. Real-world signals are a mixture of different components, and only a subset of these components is interesting. Multiresolution analysis allows different resolutions for the separated components of the signals.

#### **8.1 Filter banks**

When filter banks are applied to a signal it effectively allows it to separate or remove part of the signal. Our result shows that after the application of this filtering, the SO signal is removed.

#### **Figure 14.**

*In the upper panel at the left is the original SO signal. In the lower panel at the left signal after filtering. Components were removed in the range of 110–150 kKHz using CWT continuous wavelet transform. At the right panel spectrograph after filtering the signal. Removed components can be observed in the range of 110–150 kHz.*

Filtering can be described as the creation of a continuous wavelet transform (CWT) or filter bank. Signals to be filtered are sampled with 1024 samples whose peak magnitudes are approximately.

Our parameters considering the time axis to design the filter were

T1 = 5e-6; T2 = 5e-6; T3 = 2e-5; T4 = 3e-5; T5 = 4.5e-5; T6 = 5.5e-5; T7 = 7e-5; T8 = 8e-5; F1 = 105e3; F2 = 190e3;

Here [F1, F2] is the region of the spectra where the SO band lies. The filters are normalized so that the peak magnitude is two for all passbands. The highestfrequency passband is designed so that the magnitude falls to half the peak value at the Nyquist frequency (**Figure 14**).

### **9. Conclusions**

We proposed a method to determine the extra-bandwidth of a SO function by determining its temporal variation according to the variation of the FWHM. This value proved to be an efficient parameter to estimate the width of the dynamic-adaptive window. Even so, the influence of the width of the window in the determination of errors or uncertainties of the superoscillatory band has not been initiated at this stage of the investigation. The FWHM is, however, a parameter of the spectrum but other levels are possible and in the future its influence will be explored, for example in signals contaminated with noise, using this criterion. Likewise, other wavelet-based decompositions -including wavelet packets and dual-tree complex wavelet transforms- and the synchrosqueezing transform (SST) [15] could be applied to analyze instantaneous frequencies in SO, and might provide complementary insights.

### **Author details**

Victoria Vampa1 and Fabian Videla2 \*

1 Department of Basic Sciences, Uidet Matemática Aplicada, Faculty of Engineering, National University of La Plata, La Plata, Argentina

2 Department of Basic Sciences, Faculty of Engineering and Optical Research Center (CIOp), CIC CONICET, National University of La Plata, La Plata, Argentina

\*Address all correspondence to<EMAIL_ADDRESS>
© 2023 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

*Local Frequencies in Superoscillatory Phenomena DOI: http://dx.doi.org/10.5772/intechopen.112300*

#### **References**

[1] Aharonov Y, Colombo F, Sabadini I, Struppa D, Tollaksen J. The mathematics of superoscillations. American Mathematical Society. 2017;**247**:1174. DOI: 10.1090/memo/1174

[2] Berry MV. Faster than Fourier quantum coherence and reality. In: Anandan JS, Safko JL, editors. Celebration of the 60th Birthday of Yakir Aharonov. Singapore: World Scientific; 1994. pp. 55-65

[3] Van der Walt MD. Wavelet Analysis of Non-stationary Signals with Applications [Thesis]. St Louis, Missouri: University of Missouri-Saint Louis; 2015

[4] Eliezer Y. Generation, Evolution and Analysis of Temporal Super Oscillatory Optical Signals [Thesis]. Tel Aviv: Tel Aviv University; 2014

[5] Zitto ME. Modelización y análisis de señales de series temporales asociadas a catástrofes naturales [Thesis]. Buenos Aires: National University of Buenos Aires; 2014. DOI: 10.13140/ RG.2.2.18195.76324

[6] Berry MV, Popescu S. Evolution of quantum superoscillations and optical superresolution without evanescent waves. Journal of Physics A: Mathematical and General. 2006;**39**:6965-6977. DOI: 10.1088/0305-4470/39/22/011

[7] Flandrin P, Goncalves P. Empirical mode decompositions as data-driven wavelet-like expansions. International Journal of Wavelets, Multiresolution and Information Processing. 2004; **2**(04):477-496. DOI: 10.1142/ S0219691304000561

[8] Huang N et al. The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time

series analysis. Proceedings of the Royal Society A Mathematical Physical and Engineering Sciences. 1998. DOI: 10.1098/rspa.1998.0193

[9] Mallat S. A Wavelet Tour of Signal Processing. 3rd ed. San Diego, California: Elsevier; 2003. DOI: 10.1016/B978-0- 12-374370-1.X0001-8

[10] Ezra YB et al. Applications of wavelet transforms to the analysis of superoscillations. Wavelet Theory and Its Applications. 2018:195-214. DOI: 10.5772/intechopen.76333

[11] Percival D, Walden A. Wavelet Methods for Time Series Analysis. USA: Cambridge University Press; 2013. DOI: 10.1017/CBO9780511841040

[12] Daubechies I. Ten lectures on wavelets. Society for Industrial and Applied Mathematics. 1992. DOI: 10.1137/1.9781611970104

[13] Daubechies I. Orthonormal bases of compactly supported wavelets II: Variations on a theme. SIAM Journal on Mathematical Analysis. 1993;**24**:499-519. DOI: 10.1137/0524031

[14] Serrano EP. Introducción a la transformada wavelet y sus aplicaciones al procesamiento de señales de emisión acústica. Buenos Aires: Escuela de Ciencia y Tecnología, Universidad Nacional de General San Martín; 2000

[15] Varanis M, Silva AL, Balthazar JM, et al. A short note on synchrosqueezed transforms for resonant capture, Sommerfeld effect and nonlinear jump characterization in mechanical systems. Journal of Vibrational Engineering and Technology. 2023;**11**:429-434. DOI: 10.1007

### **Chapter 4**

Perspective Chapter: Maximizing Energy Collection from Nonlinear Harvesting System through Optimization and Control Techniques with Induced Time Delays

*Zakaria Ghouli*

#### **Abstract**

This investigation explores the utilization of periodic and Quasi-Periodic (QP) vibrations for Energy Harvesting (EH) in a delayed nonlinear oscillator system. The system consists of a delayed Duffing-van der Pol oscillator and a delayed piezoelectric coupling medium, with a focus on the occurrence of delay parametric resonance. This occurs when the frequency of the delay width modulation in the mechanical component is close to twice the oscillator's natural frequency. The double-step stress system is used to approximate the QP delay width, which is then harnessed for power generation. The results suggest that only periodic vibration can be used for energy harvesting with small, unmodulated delay width in the mechanical component. However, larger values result in instability of periodic vibration, and only QP vibration can be used for better performance. The impact of delay in the electrical circuit on the performance of periodic and quasi-periodic vibration-based EH is also analyzed, and it is found that suitable values of delay and delay frequency in the electrical component can improve energy harvesting performance within a specific range of parameters. These findings are supported by numerical simulations.

**Keywords:** energy harvesting, quasi-periodic, duffing-van der pol oscillator, time delay, piezoelectric coupling

#### **1. Introduction**

The limitations of linear attachments in Energy Harvesting (EH) systems can be overcome by using nonlinear stiffness in the mechanical part. This leads to improved EH performance, either in the case of monostable harvesters with hardening characteristics [1, 2] or bistable ones [3]. However, the utilization of nonlinear attachments may

result in instability and unexpected oscillations near the limits of the stable frequency response range, as stated in Ref. [4]. Under self-excitation and linear stiffness, energy harvesting in the range of large-amplitude oscillations can be achieved through Limit-Cycle (LC) oscillations in the harvester. However, these LC oscillations can become unstable through a secondary Hopf bifurcation, leading to Quasi-Periodic (QP) vibrations over a wide frequency range, which are not near resonance [5, 6].

However, in certain Energy Harvesting systems subjected to aerodynamic and base excitations, it has been observed that Quasi-Periodic (QP) vibrations lead to a significant decrease in harvested power beyond the flutter speed. To optimize energy extraction, it is necessary to avoid these QP vibrations [7, 8].

Despite this, research has shown that when there is a time delay, the vibrations from QP result in large, broad-spectrum amplitude swings [9]. Inspired by these results, in [10] a van der Pol-type energy harvester system was created that uses delay amplitude modulation to harness the large-amplitude, broad-spectrum vibrations from QP. This resulted in efficient energy extraction. The study in [10] introduced time delay in the mechanical component, while [11] explored the case where delay was present in both the mechanical and electrical subsystems. The findings in [11] revealed that the highest output power of the harvester does not always coincide with the maximum system response amplitude, depending on the values of time delay frequency and amplitude.

It was emphasized that in a delayed Duffing-type oscillator system, which is subjected to harmonic excitation and connected to a piezoelectric circuit harvester, the largeamplitude QP vibrations generated can be utilized for energy extraction across a broad range of excitation frequencies that are away from the resonance. This approach allows for the avoidance of hysteresis and instability that may occur near resonance [12].

Recently, in [13] it has been shown that a delayed Duffing harvester device, which is connected to an electrical circuit through a piezoelectric mechanism and has a modulated delay amplitude, is capable of producing large-amplitude Quasi-Periodic (QP) vibrations due to the modulation of the delay amplitude. These vibrations can be effectively utilized for energy extraction, away from resonance, with desirable performance.

Building upon previous studies that have explored the use of QP vibrations for energy harvesting in systems with time delay in the mechanical component [10, 12, 13], and those that have included time delay in both the mechanical and electrical components as a control mechanism [11], this study aims to broaden the scope of the investigation by examining the energy harvesting performance of a delayed Duffing-van der Pol oscillator coupled with a delayed piezoelectric harvester circuit. The focus is on the impact of time delay in both the mechanical and electrical components on the system's energy harvesting performance.

Additionally, the time delay in the piezoelectric coupling can be utilized as a means to control and optimize the harvester's output power. On the other hand, the time delay in the mechanical subsystem of the harvester is a common occurrence in milling and turning operations, resulting in inherent delay in the position [14, 15]. As a result, it is not accounted for as an extra source of energy for the harvester.

In the following section, the harvester system will be introduced. The multiple scales method will be used to determine the periodic response and harvested power near a delay parametric resonance. The effect of the delay parameters on the periodic response and resultant power output will also be explored. The second-step multiple scales method will be utilized in Section 3 to determine the QP response and evaluate the harvested power. The impact of different harvester system parameters on the energy harvesting performance will be examined, with a summary of the results presented in the conclusion section.

*Perspective Chapter: Maximizing Energy Collection from Nonlinear Harvesting System… DOI: http://dx.doi.org/10.5772/intechopen.111597*

#### **2. Explanation of the model and energy harvesting utilizing periodic vibrations**

The energy harvesting device under consideration in this study consists of a piezoelectric device connecting a delayed Duffing-van der Pol oscillator to an electrical circuit, as shown in the schematic illustration in **Figure 1**. The mechanics and electronics of the harvester are impacted by a lag in feedback, leading to a system equation that can be expressed dimensionlessly

$$\ddot{\mathbf{x}}(\mathbf{t}) + 6\dot{\mathbf{x}}(\mathbf{t}) + \lambda \dot{\mathbf{x}}(\mathbf{t})\mathbf{x}(\mathbf{t})^2 + \mathbf{x}(\mathbf{t}) + \gamma \mathbf{x}(\mathbf{t})^3 - \chi \mathbf{v}(\mathbf{t}) = \mathbf{a}(\mathbf{t})\mathbf{x}(\mathbf{t} - \tau\_1) \tag{1}$$

$$
\dot{\mathbf{v}}(\mathbf{t}) + \mathfrak{br}(\mathbf{t}) + \kappa \dot{\mathbf{x}}(\mathbf{t}) = \mathfrak{a}\_3 \mathbf{v}(\mathbf{t} - \mathfrak{r}\_2) \tag{2}
$$

The relative displacement of the rigid mass m is represented by x tð Þ, the voltage across the load resistance is v tð Þ, and various system parameters are denoted by δ, λ, γ, χ, κ, β, α3, τ2, αð Þt , and τ1. These parameters correspond to the mechanical damping ratio, stiffness parameter, piezoelectric coupling in the mechanical attachment, piezoelectric coupling in the electrical circuit, reciprocal of the time constant of the electrical circuit, feedback gain and time delay in the electric circuit, and feedback gain and time delay in the mechanical component. It is worth mentioning that the delay in the mechanical component originates from milling and turning operations, as described in Refs. [14, 15]. In contrast, the delay feedback in the electrical circuit is incorporated to enhance the harvester's output power, as indicated in Ref. [11]. The delay control with modulated delay amplitude in the position is represented by αð Þt

$$\alpha(\mathbf{t}) = \alpha\_1 + \alpha\_2 \cos(\text{ot}) \tag{3}$$

The unmodulated delay amplitude is denoted by α1, while the amplitude and frequency of modulation are represented by α2 and ω respectively. The use of modulated delay amplitude has been extensively utilized to enhance energy harvesting performance [10, 11, 13].

This study expands upon previous investigations, where the effect of time delay on the energy harvesting performance was examined. For instance, the impact of a

**Figure 1.** *Schematic description of the EH system.*

uniform delay in both the mechanical and electrical components without a nonlinear stiffness (γ= 0) was analyzed in [11], and the case of a linear damper and unmodulated delay was studied in [12]. Here, the focus is on examining the optimization of the EH performance through the introduction of different time delays and amplitudes in the mechanical and electrical components, as described by Eqs. (1) and (2). In this study, we examine the system's behavior at the delay parametric resonance by applying the resonance condition 1 ¼ ω2 4 þ σ where σ is a detuning parameter. The multiple scales method [16] is applied by introducing a bookkeeping parameter, ϵ, to allow for scaling of parameters as δ ¼ ϵ~δ, λ ¼ ϵ~λ, γ ¼ ϵ~γ, χ ¼ ϵ~χ, α1 ¼ ϵα~1, α2 ¼ α~2, σ ¼ ϵσ~. Consequently, Eqs. (1) and (2) are expressed as

$$\ddot{\mathbf{x}}(\mathbf{t}) + \frac{\mathbf{u}^2}{4}\mathbf{x} = \epsilon \left[ -\ddot{\mathbf{s}}\dot{\mathbf{x}}(\mathbf{t}) - \ddot{\lambda}\dot{\mathbf{x}}(\mathbf{t})\mathbf{x}(\mathbf{t})^2 - \ddot{\mathbf{y}}\mathbf{x}(\mathbf{t})^3 + \ddot{\chi}\mathbf{v}(\mathbf{t}) - \ddot{\sigma}\mathbf{x}(\mathbf{t}) + (\ddot{\mathbf{u}}\_1 + \ddot{\mathbf{u}}\_2\cos(\alpha\mathbf{t}))\mathbf{x}(\mathbf{t} - \tau\_1) \right] \tag{4}$$

$$
\dot{\mathbf{v}}(\mathbf{t}) + \beta \mathbf{v}(\mathbf{t}) + \kappa \dot{\mathbf{x}}(\mathbf{t}) = \mathbf{a}\_3 \mathbf{v}(\mathbf{t} - \pi\_2) \tag{5}
$$

A solution to Eqs. (4) and (5) can be sought in the form

$$\mathbf{x}(\mathbf{t}) = \mathbf{x}\_0(\mathbf{T}\_0, \mathbf{T}\_1) + \epsilon \mathbf{x}\_1(\mathbf{T}\_0, \mathbf{T}\_1) + \mathbf{O}\left(\epsilon^2\right) \tag{6}$$

$$\mathbf{v}(\mathbf{t}) = \mathbf{v}\_0(\mathbf{T}\_0, \mathbf{T}\_1) + \epsilon \mathbf{v}\_1(\mathbf{T}\_0, \mathbf{T}\_1) + \mathcal{O}(\epsilon^2) \tag{7}$$

where T0 ¼ t, and T1 ¼ ϵt. In terms of the variables Ti, the time derivatives become d dt ¼ D0 þ ϵD1 þ O ϵ2 ð Þ and d2 dt2 ¼ D2 0 þ ϵ2D2 1 þ 2ϵD0D1 þ O ϵ2 ð Þ where Dj i ¼ *∂*j *∂*j Ti . By substituting (6) and (7) into (4) and (5) and equating the coefficients of like powers of ϵ, the following hierarchy of equations to second order is obtained:

$$\mathbf{D}\_0^2 \mathbf{x}\_0 + \frac{\alpha^2}{4} \mathbf{x}\_0 = \mathbf{0} \tag{8}$$

$$\mathbf{D}\_0 \mathbf{v}\_0 + \mathfrak{P} \mathbf{v}\_0 + \mathfrak{x} \mathbf{D}\_0 \mathbf{x}\_0 = \mathfrak{a}\_3 \mathbf{v}\_{0\tau\_2} \tag{9}$$

$$\mathbf{D}\_0^2 \mathbf{x}\_1 + \frac{\alpha^2}{4} \mathbf{x}\_1 = -2\mathbf{D}\_0 \mathbf{D}\_1 \mathbf{x}\_0 - \tilde{\boldsymbol{\delta}} \mathbf{D}\_0 \mathbf{x}\_0 - \tilde{\boldsymbol{\lambda}} \mathbf{x}\_0^2 \mathbf{D}\_0 \mathbf{x}\_0 - \tilde{\sigma} \mathbf{x}\_0 - \tilde{\gamma} \mathbf{x}\_0^3 + \tilde{\chi} \mathbf{v}\_0 + (\tilde{\mathbf{u}}\_1 + \tilde{\mathbf{u}}\_2 \cos(\alpha t)) \mathbf{x}\_{0\tau} \tag{10}$$

$$\mathbf{D}\_0 \mathbf{v}\_1 + \beta \mathbf{v}\_1 = -\mathbf{D}\_1 \mathbf{v}\_0 - \kappa \mathbf{D}\_0 \mathbf{x}\_1 - \kappa \mathbf{D}\_1 \mathbf{x}\_0 + \mathbf{a}\_3 \mathbf{v}\_{1\tau\_2} \tag{11}$$

Initially, the solution can be characterized by

$$\mathbf{x}\_{0}(\mathbf{T}\_{0}, \mathbf{T}\_{1}) = \mathbf{A}(\mathbf{T}\_{1})\mathbf{e}^{\mathbf{i}\frac{\mathbf{a}\mathbf{T}\_{0}}{2}} + \overline{\mathbf{A}}(\mathbf{T}\_{1})\mathbf{e}^{-\frac{\mathbf{a}\mathbf{T}\_{0}}{2}} \tag{12}$$

$$\mathbf{v}\_{0}(\mathbf{T}\_{0},\mathbf{T}\_{1}) = \frac{-\mathbf{\dot{x}}\mathbf{\dot{o}}\mathbf{A}(\mathbf{T}\_{1})}{2\mathfrak{P}+\mathbf{i}\mathbf{o}-2\mathfrak{a}\_{3}\mathbf{e}^{-\frac{\mathfrak{a}\mathbf{r}\_{2}}{2}}}\mathbf{e}^{\frac{\mathfrak{a}\mathbf{r}\_{0}}{2}} + \frac{\mathbf{\dot{x}}\mathbf{\dot{o}}\mathbf{A}(\mathbf{T}\_{1})}{2\mathfrak{P}-\mathbf{i}\mathbf{o}-2\mathfrak{a}\_{3}\mathbf{e}^{\frac{\mathfrak{a}\mathbf{r}\_{2}}{2}}}\mathbf{e}^{-\frac{\mathfrak{a}\mathbf{r}\_{0}}{2}}\tag{13}$$

where A Tð Þ1 and A Tð Þ1 are unknown complex conjugates. Substituting eqs. (12) and (13) into (10) and (11) and deleting the secular terms results in:

*Perspective Chapter: Maximizing Energy Collection from Nonlinear Harvesting System… DOI: http://dx.doi.org/10.5772/intechopen.111597*

$$\begin{split} & - \mathrm{i}\alpha(\mathrm{D\_{1}A}) - \frac{\mathrm{i}\tilde{\mathrm{s}}\alpha\mathrm{A}}{2} - \frac{\mathrm{i}\tilde{\mathrm{j}}\alpha}{2} \mathrm{A}^{2}\overline{\mathrm{A}} - \tilde{\mathrm{\sigma}}\mathrm{A} - 3\breve{\mathrm{\gamma}}\mathrm{A}^{2}\overline{\mathrm{A}} - \frac{\mathrm{i}\mathrm{i}\alpha\mathrm{j}\mathrm{A}}{2\mathfrak{P} - 2\alpha\mathrm{\chi}\cos\left(\frac{\alpha\sigma\tau\_{2}}{2}\right) + \mathrm{i}\left(\alpha + 2\alpha\mathrm{\chi}\sin\left(\frac{\alpha\sigma\tau\_{2}}{2}\right)\right)} \\ & + \breve{\alpha}\_{1}\mathrm{A}\mathrm{e}^{-\frac{\mathrm{i}\sigma\_{1}}{2}} + \frac{\breve{\alpha}\_{2}}{2} \overline{\mathrm{A}}\mathrm{e}^{\frac{\mathrm{i}\sigma\_{1}}{2}} \\ & = 0 \end{split} \tag{14}$$

We can obtain modulation equations up to first order by expressing A ¼ 1 2 aeiθ, where a and θ are the amplitude and phase, respectively.

$$\begin{cases} \frac{d\mathbf{a}}{dt} = \mathbf{S\_1a} + \mathbf{S\_2a^3} + \mathbf{S\_3a\cos(2\theta)} + \mathbf{S\_4a\sin(2\theta)} \\\\ \mathbf{a}\frac{d\theta}{dt} = \mathbf{S\_5a} + \mathbf{S\_6a^3} + \mathbf{S\_4a\cos(2\theta)} - \mathbf{S\_3a\sin(2\theta)} \end{cases} \tag{15}$$

where Sið Þ i ¼ 1, … , 6 are given in Appendix. The solution to the first order given by (12) and (13) is as follows x0ð Þ¼ T0, T1 acos ωt 2 þ θ � � and v0ð Þ¼ T0, T1

Vcos ωt 2 þ θ þ arctan 2β�2α3 cos ωτ2 ð Þ 2 ωþ2α3 sin ωτ2 ð Þ 2 � � such that the condition ω þ 2α3 sin ωτ2 2 � � 6¼ 0 is satisfied. Moreover the voltage amplitude V is given by

$$\mathbf{V} = \frac{\mathbf{\kappa}\mathbf{o}}{\sqrt{\left(2\mathfrak{P} - 2\mathfrak{a}\_3 \cos\left(\frac{\mathfrak{a}\mathfrak{r}\_2}{2}\right)\right)^2 + \left(\mathfrak{o} + 2\mathfrak{a}\_3 \sin\left(\frac{\mathfrak{a}\mathfrak{r}\_2}{2}\right)\right)^2}}\,\mathbf{a}\tag{16}$$

To find the steady-state response of system (15), which represents the periodic oscillations of Eqs. (4) and (5), we set da dt and dθ dt equal to zero. After eliminating the phase, we are left with an algebraic equation in amplitude a

$$\left(\mathbf{S\_1a} + \mathbf{S\_2a}^3\right)^2 + \left(\mathbf{S\_5a} + \mathbf{S\_6a}^3\right)^2 = \left(\mathbf{S\_3^2} + \mathbf{S\_4^2}\right)\mathbf{a}^2\tag{17}$$

To find the average power, we integrate the dimensionless form of the instantaneous power (P tðÞ¼ βv tð Þ2 ) over one period of the delay modulation T. This results in the following expression:

$$\mathbf{P\_{av}} = \frac{1}{T} \int\_{0}^{T} \mathfrak{Bv}^{2} \mathbf{d}t \tag{18}$$

where T ¼ 4π ω . Then, the average power expressed by Pav ¼ βV2 2 reads

$$\mathbf{P\_{av}} = \frac{1}{2} \left[ \frac{\hbar \mathbf{\kappa}^2 \omega^2}{\left(2\mathfrak{\beta} - 2\mathfrak{\alpha}\_3 \cos\left(\frac{\mathfrak{a} \mathbf{r}\_2}{2}\right)\right)^2 + \left(\mathfrak{o} + 2\mathfrak{a}\_3 \sin\left(\frac{\mathfrak{a} \mathbf{r}\_2}{2}\right)\right)^2} \right] \mathbf{a}^2 \tag{19}$$

The value of amplitude "a" is derived from Eq. (17). We determine the highest power response by employing a maximizing process, which yields

$$P\_{\max} = \frac{\beta \kappa^2 \alpha^2 \mathbf{a}^2}{\left(2\beta - 2\alpha\_3 \cos\left(\frac{\alpha \sigma\_2}{2}\right)\right)^2 + \left(\alpha + 2\alpha\_3 \sin\left(\frac{\alpha \sigma\_2}{2}\right)\right)^2} \tag{20}$$

The impact of various system parameters on both the steady-state response and maximum power output of the harvester is evaluated using Eqs. (17) and (20).

#### **3. Vibration-based electromagnetic harvesting with quasi-periodic behavior**

We investigate the response of the harvester system under quick perturbations by using the second-step perturbation method [17]. To make the calculation easier, we transform the polar form (15) into a Cartesian system using a variable change u ¼ acosθ and w ¼ �asinθ

$$\begin{cases} \frac{d\mathbf{u}}{dt} = (\mathbf{S}\_{\mathfrak{S}} - \mathbf{S}\_{\mathfrak{A}})\mathbf{w} + \eta \left\{ (\mathbf{S}\_{\mathfrak{I}} + \mathbf{S}\_{\mathfrak{J}})\mathbf{u} + (\mathbf{S}\_{\mathfrak{I}}\mathbf{u} + \mathbf{S}\_{\mathfrak{G}}\mathbf{w}) \left(\mathbf{u}^{2} + \mathbf{w}^{2}\right) \right\} \\\\ \frac{d\mathbf{w}}{dt} = -(\mathbf{S}\_{\mathfrak{I}} + \mathbf{S}\_{\mathfrak{A}})\mathbf{u} + \eta \left\{ (\mathbf{S}\_{\mathfrak{I}} - \mathbf{S}\_{\mathfrak{J}})\mathbf{w} + (\mathbf{S}\_{\mathfrak{I}}\mathbf{w} - \mathbf{S}\_{\mathfrak{G}}\mathbf{u}) \left(\mathbf{u}^{2} + \mathbf{w}^{2}\right) \right\} \end{cases} \tag{21}$$

where η is a record-keeping factor included to account for damping and nonlinearity. The periodic solution of the slow flow (21) can be represented in the form of

$$\mathbf{u}(\mathbf{t}) = \mathbf{u}\_0(\mathbf{T}\_1, \mathbf{T}\_2) + \eta \mathbf{u}\_1(\mathbf{T}\_1, \mathbf{T}\_2) + \mathcal{O}(\eta^2) \tag{22}$$

$$\mathbf{w(t)} = \mathbf{w\_0(T\_1, T\_2)} + \eta \mathbf{w\_1(T\_1, T\_2)} + \mathbf{O}\left(\eta^2\right) \tag{23}$$

where T1 ¼ t and T2 ¼ ηt. In terms of the variables Ti, the time derivatives become d dt ¼ D1 þ ηD2 þ O η2 ð Þ where Dj i ¼ *∂*j *∂*j Ti . By substituting expressions (22) and (23) into (21), and comparing the coefficients of equivalent powers of η, we arrive at the following set of equations

$$\mathbf{D}\_1^2 \mathbf{u}\_0 + \nu^2 \mathbf{u}\_0 = \mathbf{0} \tag{24}$$

$$(\mathbf{S}\_{\\$} - \mathbf{S}\_{4})\mathbf{w}\_{0} = \mathbf{D}\_{1}\mathbf{u}\_{0} \tag{25}$$

$$\begin{aligned} \mathbf{D}\_{1}^{2}\mathbf{u}\_{1} + \nu^{2}\mathbf{u}\_{1} &= -\mathbf{D}\_{1}\mathbf{D}\_{2}\mathbf{u}\_{0} + (\mathbf{S}\_{1} + \mathbf{S}\_{3})\mathbf{D}\_{1}\mathbf{u}\_{0} + (\mathbf{S}\_{2}\mathbf{D}\_{1}\mathbf{u}\_{0} + \mathbf{S}\_{6}\mathbf{D}\_{1}\mathbf{w}\_{0}) \left(\mathbf{u}\_{0}^{2} + \mathbf{w}\_{0}^{2}\right) \\ &- (\mathbf{S}\_{5} - \mathbf{S}\_{4})\mathbf{D}\_{2}\mathbf{w}\_{0} + (\mathbf{S}\_{2}\mathbf{u}\_{0} + \mathbf{S}\_{6}\mathbf{w}\_{0})\mathbf{D}\_{1} \left(\mathbf{u}\_{0}^{2} + \mathbf{w}\_{0}^{2}\right) + (\mathbf{S}\_{5} - \mathbf{S}\_{4})(\mathbf{S}\_{1} - \mathbf{S}\_{3})\mathbf{w}\_{0} \\ &+ (\mathbf{S}\_{5} - \mathbf{S}\_{4})(\mathbf{S}\_{2}\mathbf{w}\_{0} - \mathbf{S}\_{6}\mathbf{u}\_{0}) \left(\mathbf{u}\_{0}^{2} + \mathbf{w}\_{0}^{2}\right) \end{aligned}$$

$$(26)$$

$$(\mathbf{S}\_{\mathbf{S}} - \mathbf{S}\_{\mathbf{4}})\mathbf{w}\_{1} = \mathbf{D}\_{1}\mathbf{u}\_{1} + \mathbf{D}\_{2}\mathbf{u}\_{0} - (\mathbf{S}\_{1} + \mathbf{S}\_{3})\mathbf{u}\_{0} - (\mathbf{S}\_{2}\mathbf{u}\_{0} + \mathbf{S}\_{6}\mathbf{w}\_{0}) \left(\mathbf{u}\_{0}^{2} + \mathbf{w}\_{0}^{2}\right) \tag{27}$$

where ν ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi S2 5 � S2 4 q represents the frequency of the limit cycle of the slow flow, which is equal to the frequency of the quasi-periodic modulation. The initial solution can be represented as

*Perspective Chapter: Maximizing Energy Collection from Nonlinear Harvesting System… DOI: http://dx.doi.org/10.5772/intechopen.111597*

$$\mathbf{u}\_0(\mathbf{T}\_1, \mathbf{T}\_2) = \mathbf{R}(\mathbf{T}\_2)\cos(\nu\mathbf{T}\_1 + \Psi(\mathbf{T}\_2))\tag{28}$$

$$\mathbf{w}\_{0}(\mathbf{T}\_{1}, \mathbf{T}\_{2}) = \frac{-\mathbf{R}(\mathbf{T}\_{2})\nu}{(\mathbf{S}\_{5} - \mathbf{S}\_{4})} \sin(\nu \mathbf{T}\_{1} + \boldsymbol{\Psi}(\mathbf{T}\_{2})) \tag{29}$$

R and ψ represent the amplitude and phase, respectively, of the slow flow limit cycle. By substituting (28) and (29) into (26) and eliminating non-oscillatory terms, we arrive at the following autonomous slow-slow flow system for R and ψ

$$\begin{cases} \frac{d\mathbf{R}}{dt} = \mathbf{S}\_1 \mathbf{R} + \left[ \frac{\mathbf{S}\_2}{2} + \frac{\mathbf{S}\_2 \nu^2}{2\left(\mathbf{S}\_5 - \mathbf{S}\_4\right)^2} \right] \mathbf{R}^3 \\\\ \mathbf{R} \frac{d\boldsymbol{\mu}}{dt} = \left[ \frac{\nu \mathbf{S}\_6}{4\left(\mathbf{S}\_5 - \mathbf{S}\_4\right)} + \frac{3\nu^3 \mathbf{S}\_6}{8\left(\mathbf{S}\_5 - \mathbf{S}\_4\right)^3} + \frac{3}{8\nu} \mathbf{S}\_6 \left(\mathbf{S}\_5 - \mathbf{S}\_4\right) \right] \mathbf{R}^3 \end{cases} \tag{30}$$

The equilibria of this slow-slow flow system determine the QP solutions of the original system (1), (2). To find the non-trivial equilibrium, we set dR dt equal to zero and obtain the following expression:

$$\mathbf{R} = \sqrt{-\frac{2\mathbf{S}\_1(\mathbf{S}\_5 - \mathbf{S}\_4)^2}{\mathbf{S}\_2(\mathbf{S}\_5 - \mathbf{S}\_4)^2 + \mathbf{S}\_2\nu^2}}\tag{31}$$

Therefore, the approximate periodic solution for the slow flow (21) is given by

$$\mathbf{u}(\mathbf{t}) = \mathbf{R}\cos(\theta \mathbf{t})\tag{32}$$

$$\mathbf{w}(\mathbf{t}) = \frac{-\mathbf{R}\nu}{(\mathbf{S}\_{\mathbf{5}} - \mathbf{S}\_{\mathbf{4}})} \sin(\theta \mathbf{t}) \tag{33}$$

The approximate amplitude a tð Þ of the QP response can be expressed as

$$\mathbf{a(t)} = \sqrt{\frac{\mathbf{R}^2}{2} + \frac{\mathbf{R}^2 \nu^2}{2(\mathbf{S}\_\mathbf{S} - \mathbf{S}\_4)^2} + \left[\frac{\mathbf{R}^2}{2} - \frac{\mathbf{R}^2 \nu^2}{2(\mathbf{S}\_\mathbf{S} - \mathbf{S}\_4)^2}\right] \cos(2\theta \mathbf{t})} \tag{34}$$

and the envelope of the QP modulation is defined by amin and amax given by

$$\mathbf{a}\_{\min} = \min \left\{ \sqrt{\frac{\mathbf{R}^2}{2} + \frac{\mathbf{R}^2 \boldsymbol{\nu}^2}{2\left(\mathbf{S}\_5 - \mathbf{S}\_4\right)^2} \pm \left[ \frac{\mathbf{R}^2}{2} - \frac{\mathbf{R}^2 \boldsymbol{\nu}^2}{2\left(\mathbf{S}\_5 - \mathbf{S}\_4\right)^2} \right]} \right\} \tag{35}$$

$$\mathbf{a}\_{\text{max}} = \max \left\{ \sqrt{\frac{\mathbf{R}^2}{2} + \frac{\mathbf{R}^2 \boldsymbol{\nu}^2}{2\left(\mathbf{S}\_{\text{S}} - \mathbf{S}\_4\right)^2} \pm \left[\frac{\mathbf{R}^2}{2} - \frac{\mathbf{R}^2 \boldsymbol{\nu}^2}{2\left(\mathbf{S}\_{\text{S}} - \mathbf{S}\_4\right)^2} \right]} \right\} \tag{36}$$

Therefore, the QP response of the original Eq. (2) can be explicitly expressed as

$$\mathbf{x}(\mathbf{t}) = \mathbf{u}(\mathbf{t})\cos\left(\frac{\alpha\mathbf{t}}{2}\right) + \mathbf{w}(\mathbf{t})\sin\left(\frac{\alpha\mathbf{t}}{2}\right) \tag{37}$$

The QP solution for the voltage v(t) can be found by inserting the expression for x tð Þ from Eq. (37) into the second equation of system (2), and then evaluating the convolution integral with the boundary condition v 0ð Þ¼ v Tð Þ, where T ¼ 2π ν . This results in

$$\mathbf{v}(\mathbf{t}) = -\kappa \mathbf{e}^{\left(\alpha\_{\beta}\mathbf{e}^{\beta\mathbf{r}\_2} - \beta\right)\mathbf{t}} \int\_0^\mathbf{t} \dot{\mathbf{x}}(\mathbf{t'}) \mathbf{e}^{\left(\beta - \alpha\_{\beta}\mathbf{e}^{\beta\mathbf{r}\_2}\right)\mathbf{t'}} \mathbf{dt'} \tag{38}$$

#### **Figure 2.**

*Vibration and power amplitudes as a function of* α1 *for* α2 ¼ 0*:*25*,* τ1 ¼ 5*:*2*,* τ2 ¼ 4*:*2*,* χ ¼ 0*:*05*,* β ¼ 0*:*05*,* λ ¼ 0*:*2*,* δ ¼ �0*:*1*,* γ ¼ 0*:*05*,* ω ¼ 2, *and* κ ¼ 0*:*5*. The black (gray) line represents the delayed (undelayed) electric circuit with* α3 *equal to β (*α3 *equal to 0). The graph shows both the analytical predictions (solid lines for stable and dashed line for unstable) and numerical simulation results (represented by circles).*

*Perspective Chapter: Maximizing Energy Collection from Nonlinear Harvesting System… DOI: http://dx.doi.org/10.5772/intechopen.111597*

As a result, the QP modulation region yields the following values for power: average, maximum, and output power

$$\mathbf{P}\_{\rm QP}(\mathbf{t}) = \mathfrak{P}\left(\kappa \mathbf{e}^{\left(a\_{\beta}\mathbf{e}^{\theta\tau\_2} - \beta\right)\mathbf{t}} \Big|\_{0}^{\mathbf{t}} \dot{\mathbf{x}}(\mathbf{t}') \mathbf{e}^{\left(\beta - a\_{\beta}\mathbf{e}^{\theta\tau\_2}\right)\mathbf{t}'} \mathbf{d}\mathbf{t}'\right)^{2} \tag{39}$$

$$\mathbf{P\_{avQP}} = \frac{\beta \kappa^2 \nu^2}{2\left[\left(\beta - 2\alpha\_3 \cos\left(\frac{\alpha \pi\_2}{2}\right)\right)^2 + \left(\nu + 2\alpha\_3 \sin\left(\frac{\alpha \pi\_2}{2}\right)\right)^2\right]} \mathbf{a}^2 \tag{40}$$

$$\mathbf{P}\_{\text{maxQP}} = \frac{\beta \mathbf{x}^2 \nu^2}{\left[ \left( \beta - 2\mathbf{a}\_3 \cos\left(\frac{\alpha \mathbf{r}\_2}{2}\right) \right)^2 + \left( \nu + 2\mathbf{a}\_3 \sin\left(\frac{\alpha \mathbf{r}\_2}{2}\right) \right)^2 \right]} \mathbf{a}^2 \tag{41}$$

The amplitude a in Eqs. (40) and (41) can be calculated using Eqs. (35) and (36).

In this work, the response of the energy harvester system near the delay parametric resonance is studied. The effect of different system parameters, including the unmodulated delay amplitude, on the steady-state response and the maximum output power of the harvester is examined. To obtain the quasi-periodic response and corresponding power, a second-step perturbation method is used. The variation of the amplitude of the periodic and quasi-periodic responses and the maximum output power is shown in **Figure 2**. The results of the analytical prediction are compared to numerical simulations obtained by using dde23 algorithm. The box inset in the figures presents time histories of the amplitude and power responses.

**Figure 2** displays the change in amplitude of both the periodic and QP responses and the maximum output power (Pmax, PmaxQP) as the unmodulated delay amplitude α1 increases. It compares the analytical predictions (solid lines for stable and dashed lines for unstable) with numerical simulations (represented by circles) from the dde23 algorithm [18]. The plots show that only periodic vibration-based energy can be extracted at a small α1 value, but as it increases, the periodic solution becomes unstable and energy can only be extracted from QP vibration with improved performance compared to the periodic output power. The figure also demonstrates that the introduction of delay in the electrical circuit (α36¼ 0) decreases the amplitude of both the periodic and QP modulation (**Figure 2a**, black line) while increasing the harvested power (**Figure 2b**, black line), indicating that the maximum power output does not always correspond with the maximum oscillation amplitude.

#### **4. Conclusions**

This research evaluated the effectiveness of energy harvesting (EH) in a system comprising a delayed Duffing-van der Pol oscillator and a piezoelectric harvester with a delayed response. The delay in the mechanical and piezoelectric elements was assumed to have differences in both timing and magnitude. The examination centered around the vicinity of delay parametric resonance where the frequency of modulation was close to twice the natural frequency of the oscillator. Approximation methods were used to calculate the periodic and quasi-periodic vibrations for EH purposes. The influence of the delay in the piezoelectric subsystem on the EH performance of the delayed Duffing-van der Pol harvester was analyzed. The findings showed that the presence of a modulated delay in the mechanical subsystem leads to an ideal set of

system parameters that maximizes the quasi-periodic vibration amplitude and corresponding output power. Moreover, the results indicated that the presence of a delay in the electrical circuit improves the output power.

### **Appendices**

$$\begin{split} \mathbf{S}\_{1} &= \frac{-8}{2} - \frac{\chi \mathbf{x} \left(2\mathfrak{\beta} - 2a\_{\mathfrak{\beta}} \cos\left(\frac{w\tau\_{2}}{2}\right)\right)}{\left(2\mathfrak{\beta} - 2a\_{\mathfrak{\beta}} \cos\left(\frac{w\tau\_{2}}{2}\right)\right)^{2} + \left(a + 2a\_{\mathfrak{\beta}} \sin\left(\frac{w\tau\_{2}}{2}\right)\right)^{2}} - \frac{a\_{\mathfrak{\beta}}}{a} \sin\frac{a\tau\_{1}}{2}, \mathbf{S}\_{2} = \frac{-\mathfrak{\beta}}{8}, \mathbf{S}\_{3} = \frac{a\_{\mathfrak{\beta}}}{2a} \sin\frac{a\tau\_{1}}{2} \\ \mathbf{S}\_{4} &= -\frac{a\_{\mathfrak{\beta}}}{2a} \cos\frac{a\tau\_{1}}{2}, \mathbf{S}\_{5} = \frac{\mathfrak{\sigma}}{a} + \frac{\chi \mathbf{x} \left(a + 2a\mathfrak{\beta}\sin\left(\frac{w\tau\_{2}}{2}\right)\right)}{\left(2\mathfrak{\beta} - 2a\mathfrak{\beta}\cos\left(\frac{w\tau\_{2}}{2}\right)\right)^{2} + \left(a + 2a\mathfrak{\beta}\sin\left(\frac{w\tau\_{2}}{2}\right)\right)^{2}} - \frac{a\_{\mathfrak{\beta}}}{a} \cos\frac{a\tau\_{1}}{2}, \mathbf{S}\_{6} = \frac{3\chi}{4a} \end{split}$$

### **Author details**

Zakaria Ghouli Polydisciplinary Faculty of Taroudant, University Ibn Zohr, Taroudant, Morocco

\*Address all correspondence to<EMAIL_ADDRESS>
© 2023 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

*Perspective Chapter: Maximizing Energy Collection from Nonlinear Harvesting System… DOI: http://dx.doi.org/10.5772/intechopen.111597*

#### **References**

[1] Peano F, Coppa G, Serazio C, Peinetti F, D'angola A. Nonlinear oscillations in a MEMS energy scavenger. Mathematical and Computer Modelling. 2006;**43**:1412-1423

[2] Mann BP, Sims ND. Energy harvesting from the nonlinear oscillations of magnetic levitation. Journal of Sound and Vibration. 2009;**319**:515-530

[3] Daqaq MF. Transduction of a bistable inductive generator driven by white and exponentially correlated Gaussian noise. Journal of Sound and Vibration. 2011; **330**:2554-2564

[4] Quinn DD, Triplett AL, Vakakis AF, Bergman LA. Energy Harvesting From Impulsive Loads Using Intentional Essential Nonlinearities. Journal of Vibration and Acoustics. 2011;**133**:011004

[5] Szabelski K, Warminski J. Self-excited System Vibrations With Parametric And External Excitations. Journal of Sound and Vibration. 1995;**187**:595-607

[6] Belhaq M, Fahsi A. 2:1 and 1:1 frequency-locking in fast excited van der Pol-Mathieu-Duffing oscillator. Nonlinear Dynamics. 2008;**53**:139-152

[7] Abdelkefi A, Nayfeh AH, Hajj MR. Design of piezoaeroelastic energy harvesters. Nonlinear Dynamics. 2012; **68**:519-530

[8] Bibo A, Daqaq MF. Energy harvesting under combined aerodynamic and base excitations. Journal of Sound and Vibration. 2013;**332**:5086-5102

[9] Hamdi M, Belhaq M. Quasi-periodic vibrations in a delayed van der Pol oscillator with time-periodic delay amplitude. Journal of Vibration and Control. 2018;**24**:5726

[10] Belhaq M, Hamdi M. Energy harvesting from quasi-periodic vibrations. Nonlinear Dynamics. 2016; **86**:2193-2205

[11] Ghouli Z, Hamdi M, Belhaq M. Energy harvesting from quasi-periodic vibrations using electromagnetic coupling with delay. Nonlinear Dynamics. 2017;**89**:1625-1636

[12] Ghouli Z, Hamdi M, Lakrad F, Belhaq M. Quasiperiodic energy harvesting in a forced and delayed Duffing harvester device. Journal of Sound and Vibration. 2017;**407**:271-285

[13] Ghouli Z, Hamdi M, Belhaq M. Energy harvesting in a duffing oscillator with modulated delay amplitude. IUTAM Bookseries. 2020;**37**:121-130

[14] Kalmár-Nagy T, Stepan G, Moon FC. Subcritical Hopf Bifurcation in the Delay Equation Model for Machine Tool Vibrations. Nonlinear Dynamics. 2001; **26**:121-142

[15] Stépán G, Szalai R, Insperger T. Nonlinear dynamics of high-speed milling subjected to regenerative effect. Nonlinear dynamics of production systems. 2004. pp. 111-128

[16] Nayfeh AH, Mook DT. Nonlinear Oscillations. New York: Wiley; 1979

[17] Belhaq M, Houssni M. Quasi-Periodic Oscillations, Chaos and Suppression of Chaos in a Nonlinear Oscillator Driven by Parametric and External Excitations. Nonlinear Dynamics. 1999;**18**:1-24

[18] Shampine LF, Thompson S. Solving delay differential equations with dde23, 2000. Available online: http: //www.radf ord.edu/-thompson/webddes/tutorial.pdf

### **Chapter 5**

## A Single-Layer Flat-Coil-Oscillator-Based Technology as a Highly Sensitive Promising Detector for State-of-the-Art "Cognitive Radio Systems"

*Aleksandr S. Khachunts, Gevorg S. Gevorgyan, Anush A. Tumanian, Vardan S. Gevorgyan, Bilor K. Kurghinyan, Sergey A. Khachunts, Narine E. Tadevosyan and Samvel G. Gevorgyan*

#### **Abstract**

A low-power stable self-oscillator with a single-layer flat coil was proposed by us in the late 1990s as a sensitive measurement method. It soon became a technological platform called a single-layer flat-coil-oscillator technology (**SFCO** technology). Two classes of sensors were designed based on SFCO platform: with and without the mechanical vibrating system. Both of them have already demonstrated their capabilities to solve problems of low-temperature experimental physics. This method also helped to increase the resolution of detectors in seismology/geophysics and develop diagnostic techniques for use in physiology and biophysics. We also showed the feasibility of using these novel SFCO sensors for noncontact far-field nondestructive scanning of various structures and media. SFCO sensors are characterized by high sensitivity and the ability to detect mechanical vibration signals in an unprecedentedly wide frequency range – from quasistationary movements to ultrasonic frequencies. Physical principles of operation of SFCO sensors are based on the change in frequency and/or amplitude of the measuring oscillator. In this Chapter, we discuss the principles of operation of SFCO sensors and numerous measurement data obtained by these sensors in different fields of science and technology. The wide potential of the novel SFCO sensors for medical diagnostics will also be discussed.

**Keywords:** single-layer flat-coil oscillator (SFCO), vibration or seismic sensor, vibroacoustic sensor, radiofrequency 'magnetic field' probe (RF MF probe), microseismic signals, magnetic field's power absorption, "foot-step" detection, biomedical applications, tissue and media dielectric properties, nondestructive scanning

#### **1. Introduction**

A stable low-power radiofrequency (**RF**) self-oscillator with an unusual single-layer flat pick-up coil activated by a tunnel diode (**TD**) was proposed by us in the late 1990s in Armenia as a highly sensitive advanced measuring method [1]. Low-temperature meters based on this method have helped us to solve topical problems of experimental physics and engineering. Particularly, they made it possible to discover and study some subtle physical properties [2–5] and refine the "magnetic-phase" diagram [6], as well as the mechanism of superconductivity [7] in high-*T*c superconductors. Soon this method was developed into a powerful technology platform called a **s**ingle-layer **f**lat-**c**oil-**o**scillator (**SFCO**) technology. It helped to increase the resolution of detectors in seismology/geophysics [8–11] and develop research and diagnostic techniques for use in physiology and biophysics [12–15]. We have also recently shown the feasibility of using SFCO method-based sensors for noncontact far-field nondestructive scanning of various structures and media [16].

**Figure 1.** *Classes of the SFCO sensors and their main varieties.*

*A Single-Layer Flat-Coil-Oscillator-Based Technology as a Highly Sensitive Promising… DOI: http://dx.doi.org/10.5772/intechopen.112305*

#### **Figure 2.**

*Principle of operation of SFCO sensors with mechanical vibration system (***Class I** *sensors): A – Deformation of* RF *magnetic field lines around the coil when metallic plate approaches/moves away; B – Schematic graph of the frequency shift of the self-oscillator as distance,* d*, between the coil and metallic plate changes; C – Dependence of the sensitivity of* Class I *SFCO sensors on the distance between the coil and the metallic plate of the vibrating system (more info see in [3]).*

#### **2. A single-layer flat-coil-oscillator sensors and their classification**

All the varieties of SFCO sensors which have been created so far can be divided into **two** main classes (**Figure 1**). Their basic difference is the presence in the sensors of the first class, the mechanical vibrating system with a normal-conducting plate attached to it, suspended above the flat surface of the measuring (pick-up) coil.

The **class I** is the sensors with a mechanical vibrating system [8–13, 15] and includes several types: the *position sensor* (position, speed, and acceleration sensors), *vibration sensor* (vibraphone and seismic sensor), the *acoustic sensor* (microphone and hydrophone) and the vibroacoustic *SFCO magnetometer*. The **class II** involves the sensors without the vibrating system and currently is presented by four types: the radiofrequency *'magnetic field' probe* (**MF** probe) [14, 16], the noninertial (due to low mass) *thermal sensor* (with 1 μK extremely-high resolution) [2, 4], the nanowatt *absorption meter* (quality-factor meter – *Q*-meter) [5, 6], and the MF probebased *SFCO magnetometer* [3, 7]*.*

#### **3. Physical principles of operation of SFCO sensors: Their advantages**

Physical principles of operation of the SFCO sensors are based on changes in frequency and/or amplitude of the measuring oscillator. In sensors with the mechanical vibration system, the main measured quantity is the frequency shift (see **Figure 3** in Ref. [10] or **Figure 1** in Refs. [3, 4]) caused by deformation of the *RF* electromagnetic field lines (*F* 10–50 MHz) near the flat surface of the measuring oscillator coil (*in response to the approach/removal of the normal-metallic* (e.g., *copper*) *plate of the vibrating system* [8–13, 15]) due to shielding *Eddy* currents induced by the same *RF* field in the plate (see **Figure 2A** and **B**). Note that the sensitivity (or resolution) of SFCO sensors exponentially drops from the rising distance between the coil plane and the metallic plate of the vibrating system [3] (**Figure 2C**).

In order to simplify the understanding of drawings in the further presentation, it is just the right moment to note that, as a rule, the SFCO technology-based devices consist of two self-oscillators (**Figure 3** in Ref. [10], or **Figure 1** in Refs. [3, 4]). One of them is the measuring oscillator (*at the flat-coil surface of which an object under study is located*), and the other the reference one. Such a design of the sensors "avoids" *device* effects and correlated noises and simplifies the electronics, increasing the measurement accuracy. Based on the above, the ordinates of the Figures here and below will show not the frequency shift of the measuring oscillator but the shift in the difference between the frequencies of the reference and measuring oscillators, namely

#### **Figure 3.**

*Changes in frequency of the measuring oscillator for the I-st and II-nd class SFCO sensors: A – Demonstration of the principle of operation of the I-st class SFCO sensors. Copper plate approaches the coil, then moves away, being in its* RF *magnetic field; B – Demonstration of the principle of operation of the II-nd class SFCO sensors (e.g., "magnetic field" probe). Placing/removal of the empty spectrophotometric plastic cuvette into/from the* RF *magnetic field of the coil (solid red curve), and the same experiment with distilled water (dashed blue curve).*

*A Single-Layer Flat-Coil-Oscillator-Based Technology as a Highly Sensitive Promising… DOI: http://dx.doi.org/10.5772/intechopen.112305*

**δ**(Δ*F*) = **δ**(*F*ref *F*meas). At the same time, the parameters of the reference oscillator remain unchanged during all measurements.

For the SFCO sensors of the *II-nd class* (e.g., the *MF* probe, the *Q*-meter, the *MF* probe-based SFCO *magnetometer*, etc.), the measured values can be both the frequency and amplitude of the measuring oscillator. However, the reasons for changing the frequency and/or amplitude of the oscillator may differ from the case of the *I-st class* sensors. In studies of dielectric media (**Figure 3**), one of the most probable mechanisms should be considered absorption of detecting coil's *RF* field's power by the medium under the study due to the '*imposed'* reorientation of the dipole structures of the medium by the *RF* magnetic field of the oscillator. Another reason is the absorption of the energy of the measuring *RF* field of the coil by induced *Eddy* currents due to setting in motion even the small amount of free charges in a dielectric media. Other factors also can result in changes in the frequency/amplitude of the oscillator, noticeable by so much sensitive technique. And, what is important, even weak absorption of the coil's *RF* power can be detected not only in the "*amplitude*" curves but also at "*frequency*" measurements (although with a noticeably lower resolution). An experimental confirmation of this (with nano-watt resolution) can be found in **Figures 3** and **4** of Ref. [5]. The peculiarities detected there on the "*frequency*" curves are due to known dependence of the radio-technical circuit's resonant frequency on its *Q*-factor value by *F* = 1/(2π√*LC*)[1-ω0*L* / (*Q* |*R*n|) + ] [17]. Here *R*n is the tunnel diode's negative-differential resistance and ω0 = 1/√*LC*.

With a sufficient approaching of the copper plate (with a thickness greater than the depth of the "skin" layer of this material at room temperatures 30–50 μm), oriented parallel to the surface of the *MF* probe, a frequency shift of the measuring oscillator is registered relative to the initial level (that corresponds to the fragment *a*–*b* in **Figure 3A**). At the very beginning of the approach of the copper plate to the surface of the probe (*so far, the plate is at a distance greater than some critical distance*), the shielding power of *Eddy* currents induced by the *RF* field of the probe in the plate is still insufficient to deform force lines of the testing field of the probe [3, 14, 16]. Although the *Eddy* currents form from the very beginning, their density increases as the metallic plate approach the probe, resulting in an inevitable increase in the energy required for their formation, which is absorbed from the *RF* power of the measuring field of the probe. As a result, the frequency of the measuring oscillator (*Fmeas*) decreases in this phase [3] (the *b*–*c* fragment in **Figure 3A**). Then, starting from a certain distance from the probe's surface, a further approach of the metallic plate results in a much stronger effect, in a growing shielding of the flat-coil-based *MF*

#### **Figure 4.**

*Comparison of the SFCO sensors (the blue and yellow curves) with contemporary microphones (gray and magenta curves). This figure demonstrates the huge advantage (in terms of detection bandwidth) of class II SFCO sensors (e.g.,* RF MF *probes) over class I SFCO sensors and the modern microphones.*

probe's *RF* field by the rising *Eddy* currents, resulting in a steep increase in the frequency (*Fmeas*) of the measuring oscillator (the *c*–*d* fragment in **Figure 3A**). When the copper plate vertically moves away from the probe surface, the frequency shifts in the reverse sequence (fragments *e*–*f* and *f*–*g* in **Figure 3A**). The steepness of described changes depends on the speed of the approaching and removal of the plate. The stated laws underlie the principle of operation of the I-st class SFCO sensors [3], in which the metallic plate is a part of the vibrating system and is located at a close distance from the surface of the pick-up coil (**Figure 2A**) (*as a rule vibrating in a relatively narrow corridor not exceeding tens of micrometers*). When the metallic plate is stationary, the measuring oscillator frequency (*Fmeas*) is fixed at a certain value (*d*–*e* section in **Figure 3A**), determined by the distance between the metallic plate and the surface of the measuring flat coil. The described features allow us to consider the SFCO sensor of the I-st class as a position sensor. And therefore, high-resolution *velocity meters* and *accelerometers* can also be successfully created on their basis.

To clarify the possibility of recording the dielectric characteristics of objects and media, the experiments were performed using the plastic cuvettes of spectrophotometers. When an empty cuvette was placed near the surface of the measuring coil (the *probe*), a certain shift of the frequency was registered upward by 6.7 kHz (**Figure 3B**, the solid **red** curve). In the magnetic field of the probe, a same cuvette filled with water for injection resulted in a more than sevenfold increase ( 57 kHz) in the frequency (**Figure 3B**, dashed **blue** curve). In fact, the *RF* magnetic field of the *MF* probe passing through the plastic walls of the empty cuvette, which is a dielectric, results in the reorientations of its dipoles. This process proceeds under conditions of absorption of a certain energy, which determines the frequency shift. A cuvette filled with water absorbs a much larger fraction of the power of the *RF* magnetic field of the probe because there are a huge number of reorienting dipoles in the water column (*each water molecule is an elementary dipole*). The total absorbed power of the *RF* magnetic field is the sum of the energies needed to reorient the dipole structures of the water molecules on the one hand, and the walls of the cell, on the other. As a result, the greater the absorbed energy of the *RF* field, the greater the shift in the frequency. Conventional measurements of dielectric properties of water, aqueous solutions, and body fluids *in vitro* (*outside a living organism*) are known as a dielectrometry [18], or a dielcometry [19], and are performed using *C*and *L*-cells. Our method of registration of signals by the use of a *RF* SFCO probe strongly differs from the above-mentioned methods (the difference was discussed in detail in [14]). In an SFCO measuring method frequency of the oscillator (*probe*) dynamically changes in accordance with a relation *F* = *A* [1-*B* /*Q* + ] (*A* and B are constants defined in the above formula). These changes are due to the absorption of the part of coil's electromagnetic field's probing energy as a result of dynamic processes occurring in the probed medium/structure (*leading to changes in the coil's loaded Q*-factor *value*), accompanied by the changes in dielectric and/or conductive properties of the object under study.

Summarizing the above, it should be emphasized that the highly sensitive SFCO measurement technology is based on a position sensor designed on the base of a lowpower stable self-oscillator of the radiofrequency range (*F* 10–50 MHz) with an unusual single-layer flat receiving coil activated by a TD. Just this circumstance determines the potential ability to detect processes in an ultra-wide frequency range (*from quasi-stationary to ultrasonic frequencies*). However, this statement is true for the SFCO sensors of the **II-nd class** only (the **yellow** curve in **Figure 4**) that do not have a mechanical vibrating system. In contrast to this, the sensitive range (frequency bandwidth) of the **I-st class** sensor is limited by the characteristics of the vibrating system, which is an integral part of this class of sensors (see **Figure 2**).

*A Single-Layer Flat-Coil-Oscillator-Based Technology as a Highly Sensitive Promising… DOI: http://dx.doi.org/10.5772/intechopen.112305*

#### **Figure 5.**

*Power spectra of the SFCO vibration (or seismic) sensor: A – The spectrum of background activity, B – The signal spectrum from the carotid artery (*the spectrum of sphygmogram*). On the spectrum of the sphygmogram: A green square mark is the peak of the respiratory activity, and the blue round marks are four harmonics of the heart's rhythm.*

To assess the ratio of the recorded signals and the level of intrinsic noise of the SFCO sensor (**Figure 5**), the power spectra of useful signals and background curves obtained using the Fourier transform were compared. The power of the background recording spectrum is negligibly small as compared with the power spectrum of the useful signal recorded at the carotid artery projection point (*the ratio is approximately* 1:500 *for frequencies below 0.2 Hz, and* 1:5000 *for frequencies above 1 Hz*). Peaks of respiration activity with subcomponents, as well as 1–4 harmonics of cardiac activity with subcomponents, are clearly distinguished in the presented fragment of the useful signal's power spectrum.

#### **4. SFCO sensors with a mechanical vibration system**

An important feature of SFCO sensors is the ability to combine several detectors in one case, which provides, for example, the detection of microseismic vibrations in *three* mutually perpendicular planes practically from the "same" point. In this regard, electro-motive force (**EMF**)-based sensors ("*geophones*") have significant limitations that exclude the development of small-size *3D* models in the same housing. Our previous works and related publications have evidently demonstrated the perspectives of using the first class SFCO sensors in various fields of science and technology. Below, in this section, we present and discuss the most important applications of the class I vibration and vibroacoustic SFCO sensors [8–13, 15].

#### **4.1 Human "foot-steps"detection by SFCO seismic sensors**

One of the most promising applications of SFCO seismic sensors is the protection of borders, objects, and territories, as well as localization of the sources of the microseismic signals. Recently, we have published data on the dependence of the currently achieved maximum detection range of human "foot-steps" on the ground level of the environmental activity (**GLEA**) [11]. These results were achieved by the today's SFCO seismic sensors. Future improvements may enhance the detection range even more.

#### **Figure 6.**

*Human steps and jumps registered by the* **3D** *and* **1D** *SFCO sensors: A – RAW signals of steps at different distances from the sensors' location. B – Jumps and steps registered at distances of 340–440 m from the sensors' location and ground level of environmental activity (GLEA) (green tracks). The* **X***-axis is the time in seconds. The* **Y***-axis is the frequency shift of the measuring oscillators in kilohertz.*

**Figure 7.**

*Signals registered by the SFCO seismic sensors during driving a car (A) and flying a helicopter (B). The* **X***-axis is the time in seconds. The* **Y***-axis is the frequency shift of the measuring oscillator in kilohertz.*

As seen in **Figure 6**, at enough low GLEA levels, both our *3D* and *1D* seismic SFCO sensors could reliably detect human "foot-steps" starting at a distance of 440 meters from the location of the sensors. Since the *X*-axis corresponds to the direction of movement and the *Y*-axis is perpendicular to it, the difference in measurement data for these axes of the *3D* sensor becomes clear. The *X*-axis clearly reflects the steps (28 steps correspond to 20 meters of movement) and jumps of a person, while the *Y*axis carries information regarding the GLEA. The *Z*-axes of both *3D* and *1D* sensors are the most sensitive for reliable detection of human "foot-steps" activity.

**Figure 7A** shows car movement signals. The high sensitivity of our SFCO seismic sensors also ensured the detection of the soil micro-vibrations (indicated by arrows in **Figure 7B**) caused by a helicopter flying at a height of 50–70 meters. The closest distance in a straight line from the sensors' location was 200 meters. The possibility of detecting soil anisotropy by the SFCO sensors was also convincingly demonstrated [11], which confirms the feasibility of their worldwide use in geology, geophysics, oil and gas exploration, etc.

#### **4.2 Vibration sensors in biomedical investigations**

Vibration sensors can also be effectively used in studying various problems in biology and medicine. Determination of the stiffness of the vascular wall or the socalled "biological age of the blood vessels" is of them. Currently, a number of methods are available to determine the state of the arterial wall: augmentation index evaluation, measurement of the pulse wave velocity, etc. [20–24]. Evaluation of the stiffness/elasticity of the walls of blood vessels and carotid pulse wave analysis is important in predicting possible problems in the cardiovascular system [25–29]. One of the most frequently used methods for assessing the features of the pulse wave is the curve fitting method by three Gaussians [30, 31]. **Figure 8** shows the results of fitting of the carotid artery pulse curve (recorded by the SFCO vibration sensor) in norm (*P1*), in the presence of mild (*P2*) and more pronounced (*P3*) age-related changes in the vascular wall. As is seen, the normal pattern (*P1*) of the ratio of amplitudes and half-widths of the Gaussians changes with age, sex, and other anthropometric data. Both the amplitude and half-width ratios of all Gaussians of the *P2* type pulse wave are changed due to the dominance of the second Gaussian, the reduction of the

#### **Figure 8.**

*The results of the left carotid artery pulse wave fitting by three Gaussians registered by the new SFCO vibration sensor.* **P1***–***P3** *– Pulse waves of three persons with different age, sex, and anthropometric data. The ratio of the amplitudes and half-widths of the Gaussians clearly reflects the elastic properties of the vascular wall reflecting the "*biological age of the vessels*".* **P1** *– Normal pattern and* **P2** *– Abnormal pattern: Reversed ratio of the first and second Gaussians and/or increased half-width of the second and/or third Gaussian. Person 3 (***P3***) had a complicated medical history of hypertension, diabetes mellitus, and myocardial infarction. The* **X***-axis is the time in seconds. The* **Y***-axis is the frequency shift in kilohertz.*

amplitude of the first, and an increase in the half-width of the third. However, in case of *P2,* we are still dealing with preclinical changes. The last case (*P3*) has a complicated medical history of hypertension, type II diabetes mellitus, and myocardial infarction followed by coronary stenting of the vessels. Only this case is distinguished by the most pronounced changes in the form of a reversal ratio of the amplitudes of the first and second Gaussians due to a decrease in the first and an increase in the second, a decrease in the half-width of the first and an increase in the half-widths of the second and third Gaussians.

#### **4.3 Vibroacoustic sensors in biomedical investigations**

The imposition of a vibration sensor on the projection area of the common carotid artery makes it possible to register not only the arterial sphygmogram but also vibrations associated with the activity of the vocal cords during vocalization [13]. Such a possibility is also preserved when signals are detected by the SFCO vibroacoustic sensor (SFCO microphone). Since the frequency limits of these signals do not overlap (**blue** curves on the left and in the center of **Figure 9**), it becomes easy to separate them using digital filtering methods. Combined recording of voice parameters and cardiovascular activity may be of interest to specialists in human bio-identification. Due to the fact that a "*stethophonendoscope*" is not a precision measuring device (since its acoustic characteristics are not standardized and significantly depend on the manufacturer), the study of the features of conduction of endogenous sound vibrations to the surface of certain areas of the body are no less topical. For example, investigation of the conduction of vibrations to the surface of the chest may give us a more reasonable approach to vibroacoustic phenomena in lung and heart diseases [32, 33]. It should be emphasized that the signals associated with vocalization mainly reflect the fundamental frequency, which reflects the vibrations of the vocal cords and, accordingly, can serve as a tool for studying laryngeal problems in ENT practice for correction of certain disorders of sound production (vocalization) function and for objective classification of voice data, etc.

The reproducibility of patterns during the verbalization of the first letters of the English alphabet is shown in the curves on the left side of **Figure 10**. At the same time,

#### **Figure 9.**

*Voice spectra of the letters "***A***", "***B***", "***C***", and "***D***" were obtained by the SFCO vibroacoustic sensor (SFCO microphone) for two persons. Signals were detected from the projection area of the left common carotid artery. There are clear differences in the characteristics of the spectra. The example on the left also shows the presence in the low-frequency range of the spectrum (up to 2.5 Hz) of both the peak of respiratory activity and the harmonics of cardiac activity. The* **X***-axis is the frequency in Hertz. The* **Y***-axis represents counts.*

*A Single-Layer Flat-Coil-Oscillator-Based Technology as a Highly Sensitive Promising… DOI: http://dx.doi.org/10.5772/intechopen.112305*

#### **Figure 10.**

*Reproducibility of the "***A***", "***B***", "***C***", and "***D***" letters pronunciation signals obtained by the vibroacoustic SFCO sensor (SFCO microphone) (left side) and the patterns ("*images*") corresponding to each letter (right side).*

the patterns ("*images*") of certain letters are quite specific (*A* and *C*), while the patterns of other letters (*B* and *D*) show similarity (see **Figure 10**, right column).

The parameters of the sphygmogram of the common carotid artery detected by the SFCO vibroacoustic sensor (**SFCO** *microphone*) show a high sensitivity of the method to certain factors. **Figure 11** clearly demonstrates the changes in the *"Amplitude"* of characteristic peaks after a five-minute load (the **stress** test) (**Figure 11A**, **right**) compared with the initial (*baseline*) parameters (**Figure 11A**, **left**). The **red** circles correspond to the rapid ejection phase and reflect ejection volume, while the **blue** circles correspond to the closure of the aortic valve at the end of the systolic phase and the beginning of diastole [34, 35]. If at rest, the range of vascular wall vibrations during one contraction was 10 kHz, then in 30 s after the stress test, this parameter was almost three times more ( 29 kHz). In this case, as expected, the increase in a swing is associated with the first component of the pulse wave corresponding to the phase of rapid blood ejection at the beginning of systole. At the same time, a remarkable increase in heart rate also takes place.

**Figure 11B** shows regular changes in the amplitude-time parameters of the carotid artery sphygmogram in extrasystolic heart rhythm disturbances registered by the **SFCO** vibroacoustic sensor (**upper** curve). The electrocardiogram (**bottom** curve) is also presented to demonstrate the correlation of the data. For a better understanding of the specific changes in the recorded data, let us briefly dwell on the features of the physiology of such a state. Ventricular extrasystole (*heart contraction outside the normal rhythm*) is a type of cardiac arrhythmia with premature contractions of the heart ventricles [36]. The trigger for extrasystole is an ectopic focus of excitation in any area of the myocardium, which leads to an abnormal spread of this excitement and, accordingly, to a desynchronous, more or less ineffective involvement of the muscle fibers of the heart ventricles in the contraction process. As a result, the efficiency of blood ejection from the ventricles into the vascular bed sharply decreases, which is expressed in the *"Amplitude"* decrease of the first peak of the carotid artery pulsogram recorded by the vibroacoustic **SFCO** sensor (**red** circular marks in **Figure 11B**). In comparison with the *"Amplitude"* registered at normal systolic volume (**green** marks on **upper** graph), extrasystole on the **ECG** corresponds to a pulse wave with reduced amplitude due to the premature systolic volume (**red** marks on **upper** graph), while the pulse wave after extrasystole has a higher *"Amplitude"* (compare to normal), which reflects the

#### **Figure 11.**

*Carotid pulse waves correlate with the functional state of the circulation system. A – The carotid pulse wave morphology (*"Amplitude" and timing*) at rest (left graph) and 30 s after the stress test (right graph). The* **X***-axis is the time in seconds. The* **Y***-axis is the frequency shift in kilohertz. B – Correlation between the data obtained by the SFCO vibroacoustic sensor (SFCO* microphone*) from the left carotid artery and ECG recorded simultaneously in case of a ventricular extrasystole. Upper graph – Carotid pulse wave registered by the SFCO vibroacoustic sensor. Bottom graph – The ECG. On both graphs, green marks correspond to normal contraction, red marks – To the extrasystole and blue marks – To the activity after* "compensatory pause" *(black lines under data curves on both graphs). Green lines under the upper data curve correspond to the normal diastolic phase after overloaded systolic volume. The* **X***-axis is the time in seconds. The* **Y***-axis is the frequency shift in kilohertz (upper graph) and the amplitude in microvolts (bottom graph).*

overloaded systolic volume (**blue** marks on **upper** graph) due to so-called *"compensatory pause"* (**black** lines under data curves on both graphs) after extrasystole. The **green** lines under data curves correspond to a pause after normal systole.

Taking into account that the *"Amplitude"* of pulse curves reflects the pulse pressure in the vessel, the above-mentioned results allow us to state that long-term monitoring of carotid artery pulse activity by the vibroacoustic SFCO sensor (**SFCO** *microphone*), with appropriate signal processing, can effectively reflect the dynamics of central arterial pressure in relative units, the assessment of which is of great importance in cardiology practice [37–39]. In fact, these data serve as the basis for the development of the "cuff*-*less blood pressure monitor".

#### **5. Radiofrequency far-field "magnetic field" (***MF***) probe as a typical example of the SFCO sensor without the mechanical vibration system**

As stated above, the uniqueness of SFCO *RF MF* probes is in the absence of the mechanical vibrating system in their design. This allows far-field, nondestructive, noncontact, noninvasive probing of structures and media [14, 16]. Probes created to *A Single-Layer Flat-Coil-Oscillator-Based Technology as a Highly Sensitive Promising… DOI: http://dx.doi.org/10.5772/intechopen.112305*

date detect the presence of dielectric structures and/or media already at a distance of 15–20 cm, and there are many opportunities for their further improvement. Another advantage of SFCO *MF* probes is that they do not limit the frequency range of the sensor. Therefore, the spectral characteristics of SFCO probe signals reflect the real physical features of the studied structures and media.

#### **5.1 Two components of the signals registered by the** *RF* **far-field** *MF* **probe**

The signals recorded by the SFCO *"magnetic field"* probe have two components (**Figure 12**). As the probe approaches the surface of the body, a pronounced frequency shift of the measuring oscillator is observed due to the absorption of the energy of the field of the pick-up coil. From a certain level (value) of frequency, further changes are associated with physiological processes occurring in the probed area of the organ and/or tissue (blood circulation, pulsation of the tissue under study, etc.). Thus, by the use of the SFCO *MF* probe, the *constant* (*a*) and *pulsatile* (*b*) components of the signal are separated. The first mainly reflects the biophysical characteristics, while the second is associated with the functional features of the object under study.

#### **5.2 Radiofrequency far-field** *MF* **probes for biomedical investigations**

#### *5.2.1 Study of dielectric properties of different tissue and media*

A typical example of how much informative the constant (*steady*) component of *MF* probe's signal is shown in **Figure 13**. It demonstrates the results of measuring the energy absorption of the probe magnetic field when various media are placed in front of the pick-up coil of the probe: an empty plastic spectrophotometric cuvette, a cuvette filled with distilled water and injection solutions of vitamins *B12* (0.5 mg/ml) and *C* (1 ml of its 5% solution contains 50 mg ascorbic acid, 23.85 mg sodium bicarbonate, and 2 mg anhydrous sodium sulfite). Each tested material has its own magnitude of *MF* probe's power absorption. The above data indicate the

#### **Figure 12.**

*Two main components of the signal are registered by the SFCO* "magnetic field" *probe. The* steady *component (*a*) mainly reflects the biophysical properties of the tissue and/or structure under study, while the* pulsatile *components (*b*) are associated with physiological processes occurring in the scanning organ/region. The* **X***-axis is the time in seconds. The* **Y***-axis is the frequency shift in kilohertz.*

#### **Figure 13.**

Steady *part of the SFCO* MF *probe's signal as a tool for* "in vitro" *study of the probe's electromagnetic field's power absorption by the distilled water and some vitamin solutions (***B12** *– 0.5 mg/ml,* **C** *– 5% solution). The first column shows the energy absorption level by the empty spectrophotometric plastic cuvette. At the top of each column, the values of mean and standard deviation of the power absorption are indicated. The* **Y***-axis is the frequency shift in kilohertz.*

possibility of studying the dielectric and shielding characteristics of biological media "*in vitro*".

Another example is shown in **Figure 14**. It relates to the detection of various pathological processes *"in vivo"*. Normally, there should be a symmetrical absorption of the testing electromagnetic field's power on both sides of the probing (**Figure 14A**). In case of volumetric formations (3D lesions), an asymmetry in absorption levels of the testing *RF* field's energy is revealed (**Figure 14B**). In a particular case, hearing impairment of the type of sensorineural hearing loss and right-sided acoustic neuroma were clinically verified by audiometry & subsequent MRI examination (**red** arrows). Histograms (**bottom** left) clearly show the asymmetry in pathology and its absence in the normal case. It is important to emphasize that with this method of examination, **the asymmetry** of the absorption of the *RF* testing field's energy itself is **more important** than the value of the frequency shift. However, as enough data become available, the value of the frequency shift can also be important parameter, as the limit values of frequency shift for specific verified pathological processes (inflammatory processes, cysts, hematomas, abscesses, bone and glial tumors, etc.) will be clarified. Taking into account the short duration of the examination by SFCO *MF* probes, the safety, and informativeness of results, we can talk about a new valuable method for screening and assessing the condition of patients with craniocerebral injuries, volumetric processes (hematomas, tumors, abscesses, etc.).

#### *5.2.2 Study of cardiovascular system*

The informativeness of the pulsatile component of the SFCO *MF* probe signal is shown in **Figure 15**. It is important to emphasize that the registration of *MF* probe

*A Single-Layer Flat-Coil-Oscillator-Based Technology as a Highly Sensitive Promising… DOI: http://dx.doi.org/10.5772/intechopen.112305*

#### **Figure 14.**

*The constant (*steady*) part of the SFCO* MF *probe's signal as a tool for* "in vivo" *screening of tissue and structure. The Probe's* RF *field power absorption level in a healthy person (A) and in the case of right acoustic neuroma (B). On the middle top is the location of the probing zone (green square).On the bottom left is the diagram of probe electromagnetic field's power absorption levels in norm (red columns) and right neuroma (blue columns). On the bottom right are MRI images. Red arrows show the location of acoustic neuroma.*

signals does not require careful preparation of the sensor application area (degreasing the surface, ensuring reliable contact, etc.). It is enough to fix the probe in the required area. In the simplest case, the subject can be seated on a chair with a mounted probe (lay on a mattress), and signals can be registered even through clothes (**Figure 15**). As can be seen from the Figure, three probes are placed in different areas: in the popliteal fossa (**green** mark and time and frequency tracks corresponding to it in color), on the back surface of the mid-thigh (**blue** mark and tracks) and in the buttock area (**red** mark and tracks). The signals registered in the first two areas by the *MF* probe with a flat-coil diameter of 8 mm correspond to the pulsation signals of the femoral artery, which can be seen both on the time curves (signal shape) and on the frequency spectrum of the signal (typical peaks of spectra corresponding to harmonics of heart contractions). Time graphs of signals from the buttocks area have a less clear pattern, although there is some regularity of fluctuations. This is due to the absence of large main vessels in this area and the presence of an extensive network of small vessels that provide blood supply to a sufficiently large muscle mass, as well as the use of an *MF* probe with a larger flat-coil diameter of 25 mm, covering a wider scan area with a corresponding moderate phase desynchronization of the signals. However, even with such a "degraded" signal pattern in the time domain, the frequency characteristics remain stable (all six harmonics of the heart beats are traced), and even a respiratory peak (0.2 Hz) appears in the signal spectrum. Moderate phase desynchronization is indicated by wider frequency ranges of half-widths of heart harmonics and their reduced power.

#### **Figure 15.**

*Pulsatile component of the SFCO* "magnetic field" *probe signals registered from the thigh and buttock. A – Scheme of the sensor location and approximate diameter of the probes' sensitive part (flat-coil size). B – Signals in the time domain and C – Signals in the frequency domain. The* X*-axis is the time in seconds (B) and frequency in Hz (C). The* Y*-axis is the frequency shift in kilohertz (B) and counts (C).*

One of the most promising applications of SFCO *RF MF* probes is the ability to study the brain. For the electromagnetic field of a radiofrequency range, the bones of the skull are not an obstacle. They just absorb part of the testing field's energy. Consequently, the perspectives of brain probing in any area of the convexital surface of the skull without the need for probing through the natural openings of the cranium are opening up – in contrast to ultrasound research methods. Registration of SFCO probe signals by a probe with a sensitive element diameter of 35 mm located in the area of the left auricle is shown in **Figure 16A**. For comparison, synchronous recordings of signals from the ipsilateral and contralateral carotid arteries by vibration and vibroacoustic SFCO sensors, respectively, as well as an electrocardiogram in the standard lead are presented. The shape of the probe signal correlates with the shape of the left common carotid artery signal recorded by the vibration SFCO sensor, differing only in a regular time delay due to the registration of pulse oscillations of the walls of the internal carotid artery, which lies rather deep in the temporal bone pyramid. The significant role of the internal carotid artery in the formation of the probe signal is confirmed by its topography in the temporal bone pyramid, where the artery forms a knee (at a depth of 60–70 mm from the surface of the auricle) and enters the cranial cavity. The probe signal represents the total activity of the filled triangular zone on MRI images. The probable deepest zone of signal registration is indicated by a **blue** mark.

In the second case, the native and filtered signals of the SFCO *MF* probe located on the vertex and probing quite deep brain structures are presented (**Figure 16B**). Since the diameter of the measuring probe (∅ 35 mm) covers a rather large scanning area with no large-caliber arteries, the signals have a flattened, smoothed shape due to the phase asynchrony of the recorded pulse waves in small arterial vessels. Volumetric pulsation of the brain tissue itself also contributes to the formation of such a signal

*A Single-Layer Flat-Coil-Oscillator-Based Technology as a Highly Sensitive Promising… DOI: http://dx.doi.org/10.5772/intechopen.112305*

#### **Figure 16.**

*Pulsatile activity registered by the SFCO* MF *probe with a large sensitive element ( 35 mm) from different areas of the head. A – Scanning from the external ear. The signal pattern (red trace) is very similar to the signal registered by the vibration SFCO sensor from the left carotid artery (green trace). For comparison, synchronously recorded signals from the right carotid artery (blue trace registered by the vibroacoustic SFCO sensor) and ECG (orange trace) are also presented. B – Scanning from vertex. The top 2 traces are RAW and smoothed signals registered by 35 mm* MF *probe. For comparison, the blue trace reflects the activity of the left common carotid artery registered by the vibroacoustic SFCO sensor.*

pattern. For comparison, the sphygmogram of the left common carotid artery (**blue** track) registered by the vibroacoustic sensor is also shown.

For convincing demonstration of the possibility of recording processes occurring in the depth of the cranium, the signals obtained from vertex by SFCO *"magnetic-field"* probe with a smaller sensitive element (coil ∅ 8 mm) are presented (**Figure 17**). The **upper** row of curves represents the recorded signals and the corresponding spectra after removing the constant component. The next row shows **Signals** reconstructed by adding some *Intrinsic Mode Functions* (**IMF**) obtained after *Empiric Mode Decomposition* (**EMD**) [40, 41] and their corresponding spectra. The recorded curves reflect the state of blood flow in the region of the brain tissue covered by the far-field *MF* probe (**Figure 17**, **Signals**). The modulating effect of respiratory (**Figure 17**, IMF 13, 14) and slower (**Figure 17**, IMF15) waves on the recorded activity should be pointed out. Reconstruction of activity by summing the IMF7–IMF12 clearly highlights vascular activity (**Figure 17**, curves IMF 7–12), reflecting the state of blood flow in the area under study. The spectra of the IMFs (**Figure 17**, IMF 7–15) allow tracking the frequency peaks of activity and, as seen from the Figure, also include slow components below respiratory modulations (below 0.2 Hz). This fact gives grounds to expect that a prolonged (about 5–7 minutes) registration of activity can provide information regarding the average characteristics of microcirculation in the scanned area of the brain tissue.

#### **Figure 17.**

*Transcranial registration of signals from the vertex during probing by the SFCO* MF *probe with a small measuring flat-coil diameter flat-coil ( 8 mm). The upper row of curves shows the recorded signals and their corresponding spectra (Signal). The next row shows the signals reconstructed by summing the* intrinsic mode functions *(IMF 7-12) obtained after the* empirical mode decomposition *and their corresponding spectra. Below are some informative IMFs (IMF 7-15) and their spectra. The* **X***-axis is the time in seconds for time domain graphs and frequency in hertz for frequency domain graphs, respectively. The* **Y***-axis is the frequency shift in kHz for time domain graphs and counts for frequency domain graphs, respectively.*

#### *5.2.3 Study of functional state*

The SFCO *"magnetic-field"* probe also allows monitoring the human functional state (**Figure 18**). In a calm waking state, pulse waves are registered in the rhythm of heartbeats ("Wakefulness" mark on the **green** background). With the onset of drowsiness, the pattern of the signals changes, and high-amplitude peaked oscillations of longer duration appear on the curve ("Drowsiness" mark on the **orange** background). Finally, with the onset of the light sleep phase, the amplitude and duration of the recorded oscillations increase sharply ("Light Sleep" mark on the **red** background). Slow waves during the last two phases reflect a progressive weakening of the control of muscle tone from the central nervous system. It is easy to imagine what valuable information can be obtained when monitoring a person during the execution of responsible work (air traffic controllers, nuclear power plant operators, machine operators, drivers, etc.), in intensive care and resuscitation units, or when detecting the veracity of the information provided, etc. At the same time, the lack of preparation stage for monitoring (skin treatment and application of sensors) provides stability to the psychoemotional state and prevents frustration.

#### **Figure 18.**

*Monitoring of the human (*animals*) functional state by the* radiofrequency *SFCO* "magnetic-field" *probe. The explanations are in the above text.*

*A Single-Layer Flat-Coil-Oscillator-Based Technology as a Highly Sensitive Promising… DOI: http://dx.doi.org/10.5772/intechopen.112305*

#### **5.3 Radiofrequency far-field** *MF* **probes as a tool for nondestructive scanning of objects and structures**

Scanning of research objects was carried out by the SFCO *"magnetic-field"* probe with receiving flat coil (∅ 3 mm) fixed on a stable tripod, which allowed to adjust the sensor position in three coordinates to ensure parallelism of planes of the pick-up coil and the surface of the scanned object, as well as set the required distance (*h*) between the scanned object and the sensor. The distance *h* from the plane of the *MF* probe coil to the surface of the scanned object was 3–7 mm (depending on the size of the scanned object). The scanned object was fixed on a two-coordinate stage driven by two stepper motors with program control. In fact, the assembled scanning setup provided the movement of the scanned object along the *"X"* and *"Y"* axes, while the scanning detector (SFCO *MF* probe) was fixed along the *"Z"* axis at the required height from the surface of the object under study (**Figure 19**). The movement speed of the object relative to the scanning probe along the scanning track was 2 mm/sec, which, taking into account the sampling rate (1000 Hz), provided a resolution of 500 measurements in one millimeter. The consecutive scanning tracks were soft-shifted by 0.2 mm, forming a resolution of the scanned structure/object along the transverse axis of five measurements per 1 mm. The signals from the probe were fed to a special high-speed eight-channel frequency meter (SFFM-8, *"PSI"* LLC, Armenia). Then the measured data were transferred to a PC and recorded by a virtual instrument (*program*) developed by *PSI* in the LabView environment (*NI*, USA).

The scanning of a complex structure sample (**Figure 20**) shows the high spatial resolution of the method used. Since the scanning area exceeded the width of the object during the lateral scanning, there were some scanning areas of a much less dense medium ("air") on both sides of the object, corresponding to lower levels of the testing RF field's power absorption. The "zoom-in"/exit of the *"needle-shaped"* sensitive zone of the *MF* probe scanner on the test object is accompanied by a sharp increase/decrease in the level of testing field power absorption, respectively, forms the side walls of the "well" in the image of the initial scanning signals. The passage of the sensitive zone ("tip") of the scanner over the surface of the test-object forms the "bottom of the well". Against this background, the structural features of the scanned area of the test sample are clearly traced. Both a large through hole with a diameter of 10 mm and 4 through holes with a diameter of 3 mm are visualized, as well as nonthrough holes with a diameter of 2.5 mm located between them (**Figure 20**). As

**Figure 19.**

*Experimental setup for scanning of research objects.*

#### **Figure 20.**

*Maps of values of absorption of testing electromagnetic field's power of the SFCO* MF *probe when scanning a sample of complex structure. On the top – A sample of complex structure, side view. On the bottom right – View the upper surface of a sample of complex structure. On the bottom left – Maps of values of* RF *field energy absorption. Ring marks indicate the points of through holes, circular marks indicate places of nonthrough (*blind*) holes. The* **X***-axis represents the sample number. The* **Y***-axis is the track number. The* **Z***-axis is the energy absorption levels in arbitrary units. The vertical bar on the left shows the energy absorption levels of the measuring field in arbitrary units.*

can be seen, the level of probing *RF* field energy absorption is different for different holes. It is minimal for a large through hole with a diameter of 10 mm, significantly more for four through holes with a diameter of 3 mm (**large** triangular marks), and maximum for three nonthrough holes with a diameter of 2.5 mm (**small** triangular marks), but less than the level of field power absorption in intact (*untreated*) areas of the test-object corresponding to the *"bottom of the well"*. It follows from these data that both the diameter of the holes and the presence of a certain inhomogeneity in the scanned structure of the test object (*in the case of scanning three nonthrough holes with a diameter of 2.5 mm, there is a four-millimeter layer of composite plastic at the bottom of the holes*) significantly affect the level of energy absorption of the *MF* probe's *RF* field.

In the previous work [14], the possibility of detecting the dielectric, magnetic, and conductive (*shielding*) properties of biological tissues and media by the SFCO *MF* probe was shown, and the prospects of using these probes for biomedical research were justified. Based on these results, we attempted to scan biological tissues and media. The convenient place for scanning was the terminal and middle phalanges of II-IV fingers. The results of scanning after the digital signal processing are shown in **Figure 21**. The passage of the *MF* probe over the bone structures of the fingers is accompanied by the maximum energy absorption of the probe testing electromagnetic field. On the other hand, scanning the interdigital zones, in which the thickness of the soft tissue layer is small even when the fingers are close to each other, shows a low level of power absorption of the measuring probe. These data allow to state, that scanning by the SFCO *MF* probe is a new method for visualization of biological media and structures, based on the assessment of their dielectric, magnetic, and conductive characteristics at each scanning point due to changes in a frequency and/or amplitude of the scanning self-oscillator caused by the specific level of the energy (*power*) absorption of the radiofrequency SFCO *"magnetic-field"* probe.

*A Single-Layer Flat-Coil-Oscillator-Based Technology as a Highly Sensitive Promising… DOI: http://dx.doi.org/10.5772/intechopen.112305*

#### **Figure 21.**

*The 3D display of the dynamics of absorption of testing electromagnetic field's power (energy) of the measuring SFCO* MF *probe when scanning the terminal phalanges of II-IV fingers (*"magneto-densitometry"*). The* **X***-axis represents sample number. The* **Y***-axis is the track number. The* **Z***-axis is the power absorption levels in relative units. The vertical bar on the right represents absorption levels of the probing field in arbitrary units.*

#### **6. Conclusion**

A low-power stable self-oscillator with a single-layer flat coil was proposed by us in the late 1990s as a sensitive measurement method. It soon became a Technological Platform called a single-layer flat-coil-oscillator technology (SFCO technology). Two classes of sensors were designed based on the SFCO platform: with and without the mechanical vibrating system. Both of them have already demonstrated their capabilities to solve problems of experimental physics. Particularly, they allowed studying fine properties and clarifying the mechanism of superconductivity in high-*T*c superconductors. This method also helped to increase the resolution of detectors in seismology/geophysics and develop diagnostic techniques for use in physiology and biophysics. We also showed the feasibility of using these novel SFCO sensors for noncontact far-field nondestructive scanning of various structures and media. SFCO sensors are characterized by high sensitivity and the ability to detect mechanical vibration signals in an unprecedentedly wide frequency range – from quasi-stationary movements to ultrasonic frequencies. Physical principles of operation of SFCO sensors are based on the change in frequency and/or amplitude of the measuring oscillator. In SFCO sensors with a mechanical vibrating system, the measured effect is determined by the distortion of the *MHz*-range testing field's configuration near the coil plane, leading to changes in the oscillator frequency. For SFCO sensors without the vibrating system the measured values can be both the frequency and amplitude of the measuring self-oscillator.

The results and data presented and discussed in this Chapter suggest that

1.SFCO measurement technology makes it possible to create sensors with unprecedentedly high resolution due to the unusual flat shape of the pick-up coil and high stability of its low-power measuring self-oscillator.

Finally, in summary, we would like to emphasize that the mentioned numerous advantages of the Single-layer Flat-Coil-Oscillator-based innovative measurement technology may permit to apply it as a highly sensitive promising detector for many state-of-the-art "cognitive radio systems" in the future.

### **Acknowledgements**

The authors express their deep gratitude to the management of PSI"(Precision Sensors & Instruments, LLC, Armenia — https://psi.am/) for providing SFCO sensors, counting and signal processing units with the appropriate software for the implementation of measurements and experiments presented in this Chapter.

*A Single-Layer Flat-Coil-Oscillator-Based Technology as a Highly Sensitive Promising… DOI: http://dx.doi.org/10.5772/intechopen.112305*

#### **Author details**

Aleksandr S. Khachunts1 , Gevorg S. Gevorgyan2,3, Anush A. Tumanian1 \*, Vardan S. Gevorgyan2,4, Bilor K. Kurghinyan2 , Sergey A. Khachunts2 , Narine E. Tadevosyan1 and Samvel G. Gevorgyan2,3,4

1 L.A Orbeli Institute of Physiology, National Academy of Sciences (NAS), Yerevan, Armenia

2 Yerevan State University, Yerevan, Armenia

3 Precision Sensors and Instruments ("PSI" LLC), Yerevan, Armenia

4 Institute for Physical Research, NAS, Ashtarak, Armenia

\*Address all correspondence to<EMAIL_ADDRESS>
© 2023 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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#### **Chapter 6**

## Signal Analysis in Chaotic Systems: A Comprehensive Assessment through Time-Frequency Analysis

*Marcus Varanis, Jose M. Balthazar, Angelo M. Tusset, Mauricio A. Ribeiro and Clivaldo De Oliveira*

#### **Abstract**

Non-stationary and nonlinear signals, which can bring important applications in chaotic dynamics, and are found in several scientific and engineering fields. Several processing techniques have been used to understand and extract information from these signals, and the literature shows that time-frequency analysis techniques are suitable tools for this characterization. They allow to examine the time-varying characteristics of the signals. In this chapter, we will explore time-frequency methods applied especially to nonlinear signals. First, we discuss the diverse range of dynamical systems. Then, we introduce the classical time-frequency methods, including the Short-Time Fourier Transform, the Wavelet Transform, the Hilbert Transform, and the Wigner-Ville distribution. These methods have been widely used in the literature in the study of non-stationary operations. Thus, we present emerging methods of time-frequency analysis, taking advantage of post-processing and synchrosqueezing techniques to improve the accuracy and resolution of the time-frequency representation. We present a comprehensive analysis of these emerging methods, comparing them with classical approaches to show their contributions. Our main goal is to highlight the capabilities of these emerging time-frequency analysis methods in capturing and understanding chaotic patterns in signals.

**Keywords:** signal processing, time-frequency analysis, nonlinear dynamics, Chaos, wavelet analysis

#### **1. Introduction**

The analysis of non-stationary and non-linear signals is fundamental in understanding various phenomena in scientific and engineering domains. Chaotic behavior, observed in nonlinear systems such as vibrations in mechanical systems, electronic circuits, biological processes and weather patterns, has been widely studied using signal processing and time-frequency analysis (TFA) techniques. However, the analysis of this class of signals presents challenges due to their non-stationary and non-linear nature.

Traditional signal processing methods, based on the Fourier Transform (FT), have been used to analyze stationary and deterministic signals. However, these methods are not suitable for non-stationary and non-linear signals [1, 2]. One of the main limitations of TF-based TFA methods, such as the Short Time Fourier Transform (STFT), is the fixed window size constraint. STFT represents the signal in the timefrequency domain by computing the Fourier transform of fixed-size windows with overlap. This choice of window size directly affects the trade-off between time and frequency resolution. Larger windows provide better frequency resolution but sacrifice time resolution, while smaller windows improve time resolution but offer less accurate frequency localization. In addition, fixed window sizes may not capture the rapid spectral changes often observed in chaotic signals. In this context, the wavelet transform emerges as a more suitable tool to capture time-varying behavior, transients, and abrupt variations in the frequency spectrum.

The Continuous Wavelet Transform (CWT) is a TFA method that overcomes the window size limitation of the STFT. CWT uses wavelet functions, known as mother wavelets, to analyze signals at multiple scales and resolutions simultaneously [1, 3]. It is an adaptive approach that allows high time resolution analysis for transient events and fine frequency resolution for temporal variations. Simultaneous localization in the time and frequency domains makes CWT suitable for non-stationary analysis. In addition, CWT offers flexibility in choosing the appropriate wavelet function, allowing the selection of a parent wavelet that best captures the signal characteristics. CWT in diverse fields including mechanical systems [4–8], biomedical signal analysis [9–11], geophysics [12, 13] and financial data analysis [14–16].

Another important TFA is the Wigner-Ville decomposition. In [17] it is described that one of the main challenges is the interference between terms, which occurs when different frequency components overlap in time and become difficult to distinguish in the time-frequency representation. In addition, WVD is sensitive to noise, which can affect the accuracy of energy estimation at different frequencies. Another consideration is that the Wigner-Ville transform is computationally intensive [2, 17]. It requires performing convolutions for all possible combinations of frequencies and time instants, which results in a high computational cost. This can be a problem when analyzing long duration signals or in real-time applications where computational efficiency is a concern. WVD has advantages in terms of time-frequency resolution and preservation of phase information, it is necessary to consider the disadvantages related to cross-term interference, sensitivity to noise and the computational cost associated with its implementation.

Emerging time-frequency analysis methods, such as the Synchrosqueezing Transform (SST), MultiSynchrosqueezing Transform (MSST), and Synchroextracting Transform (SET), have gained attention in recent years. These methods are known as post-processing techniques and they can be improve the TFR obtained from classical methods, such as CWT and the STFT [18, 19]. These classical methods are the foundation for modern time-frequency analysis but may have limitations in terms of resolution, adaptability, or noise sensitivity. The emerging methods, including the synchrosqueezing-based techniques, offer improved accuracy, enhanced timefrequency localization, and better noise robustness [20, 21]. They complement and enhance the results obtained from classical methods, providing more detailed and informative time-frequency representations of complex signals.

SST aims to improve the time-frequency localization of signal components with variable IFs [18]. By applying reassignment techniques, it concentrates the energy around the true instantaneous frequencies, providing a clearer time-varying behavior in the TFR [21]. Building on this approach, MSST extends the feature by considering multiple reference frequencies, allowing simultaneous extraction of multiple modes or

#### *Signal Analysis in Chaotic Systems: A Comprehensive Assessment through Time-Frequency… DOI: http://dx.doi.org/10.5772/intechopen.114199*

components of a signal [22]. In addition, SET focuses on isolating specific modes or components of interest from a signal by separating them from background and noise [23, 24]. These methods have demonstrated a success in the analysis of mechanical systems, structural health monitoring, fault detection and a wide variety of biological signals, including electrocardiograms (ECG), electroencephalograms (EEG) and blood pressure signals.

It is important to emphasize that these methods are known as post-processing techniques as they refine and enhance the results obtained from classical methods such as STFT and CWT [24, 25]. They extract more accurate information from the original TFR, improving the understanding and enabling the analysis of signals with rich dynamics [20]. Post-processing-based TFA methods have shown to be promising tools for the analysis of non-stationary and nonlinear signals. The concentration of energy in the time-frequency representation allows for a better understanding of strong variations of IF's in the frequency spectrum. The post-processing operations of TFA methods are being increasingly used in various science and engineering applications, improving the results obtained through classical methods.

In this chapter, we present a review of the application of TFA in the analysis of non-stationary systems. We will present a strongly nonlinear system, with cubic nonlinearity and fractional damping, and from which we obtain signals for analysis. This system was studied in depth in [26]. In this way, we present the classic TFA methods, including the STFT, CWT and WVD.

Despite the wide range of applications of classical TFA methods, they have some limitations in terms of resolution and accuracy when analyzing non-stationary and nonlinear signals. Therefore, our study focuses in application of emerging timefrequency techniques based on post-processing operations, which have shown superior performance in some analyses. We will compare these emerging techniques with classical methods and illustrate their applications in characterizing nonlinear and chaotic patterns in signals. The main objective of this work is to highlight the potential of post-processing operations in characterizing of nonlinear signals. We will discuss the advantages, limitations and applications of each method.

#### **2. Mathematical background: time-frequency methods**

Frequently used technique for analyzing non-stationary signals is the Short Time Fourier Transform (STFT). Gabor introduced this method by adapting the Fourier Transform to focus on specific segments of the signal in the time domain. With STFT, we can observe the signal in two dimensions: time and frequency. The expression for STFT, as presented in Eq. (1), involves windowing the signal *x*(*t*) with a function *w*(*t* � *t*0) centered around a specific time *t* [2, 7].

$$X(\boldsymbol{w}, t\_0) = \int\_{-\infty}^{\infty} \boldsymbol{\kappa}(t) \, \boldsymbol{w}(t - t\_0) e^{-j\omega t} dt \tag{1}$$

The signal in the frequency domain is denoted as *X*(*ω, t*0), *ω* represents the angular frequency and *t*0 indicates the window center time. In the time domain, *x*(*t*) refers to the original signal, while *w*(*t* � *t*0) represents the window function centered at *t*0. The window function *ω*(*t*) determines the properties of the analysis, including time and frequency resolution. Commonly used window functions include the Hamming window, the rectangular window, and the Gaussian window, among others [27]. The

choice of window function is strongly associated with the nature of the signal. Thus, STFT provides a time-frequency representation that shows how the frequency content of the signal changes over time. However, as already mentioned, STFT has limitations in terms of time-frequency resolution due to its fixed window size. a shorter window gives better time resolution but worse frequency resolution and vice versa. A long study and robust formulation of the STFT can be found in [27]. Even with all its known limitations, the STFT remains an important tool for time-frequency analysis, massively used in several applications and serves as the basis for many other techniques, such as the Continuous Wavelet Transform and methods based on synchrosqueezing [7, 19].

The CWT is a TFA technique that provides a representation of the frequency content of a signal over time, very similar to the process presented in STFT. However, unlike STFT, CWT uses a set of wavelet functions at different scales to analyze the signal [1].

In [7], the CWT of a signal *x*(*t*) on a specific scale *a* and time *t* is described as follows:

$$X(a,t) = \int\_{-\infty}^{\infty} x(t) \Psi^\* \; \left(\frac{t-t'}{a}\right) \, dt' \tag{2}$$

where *X*(*a, t*) is the CWT of the signal or the signal represented in the wavelet domain, *ψ.*

represents the mother wavelet function, *a* denotes the scale parameter and *t* represents the time parameter. The wavelet function *ψ* is a scaled and translated version of the parent wavelet and serves as a time-varying analysis function. In *ψ* we have the main element that denotes the variation window in the formulation [7]. CWT decomposes the signal into different frequency components at different scales, providing a localized time-frequency representation. This allows for better time resolution at high frequencies and better frequency resolution at low frequencies. Commonly used wavelet functions include the Morlet wavelet, Mexican hat wavelet, and Daubechies wavelet, among others. The choice of wavelet function depends on the specific characteristics and requirements of the signal being analyzed. CWT provides a flexible and adaptive time-frequency representation that can capture localized variations in the frequency content of the signal, such as transients, spikes and variations in the spectrum.

The symbol *ψ* represents the parent wavelet function in the CWT formulation. The specific form of the mother wavelet function depends on the chosen wavelet family. Below are some examples of commonly used wavelet functions. The Morlet wavelet is frequently used in many applications and frequently found in the literature [21]. It is defined as a complex value Gaussian window modulated by a complex sinusoid. The expression for the Morlet wavelet is given by:

$$
\Psi(t) = \pi^{-\frac{1}{4}} e^{j\alpha\_0 t} e^{-\frac{t^2}{2}} \tag{3}
$$

where *ωo* is the central frequency of the wavelet.

The Mexican hat wavelet, is commonly used for detecting and analyzing transient events. It has a peaked shape resembling a Mexican hat. The expression for the Mexican hat wavelet function is:

$$
\Psi(t) = \left(\mathbf{1} - t^2\right) e^{-\frac{t^2}{2}} \tag{4}
$$

*Signal Analysis in Chaotic Systems: A Comprehensive Assessment through Time-Frequency…  They are widely used in various applications. The expression for a specific Daubechies wavelet depends on the order chosen, such as Daubechies-4, Daubechies-6, etc. Daubechies wavelets have great application in signal analysis of vibrations, signals of turbulent phenomena, signals that have great variation over time such as transients and even chaotic signals, since it appears that for the analysis of these types of signals the choice should be the one that less unbalances the energy of the signal, that is, the one that needs the smallest number of coefficients to represent the signal [3].

CWT is a suitable tool for analyzing non-stationary and transient signals, as well as signals with time-varying frequency components.

Another method to characterize non-stationary signals is the Wigner-Ville distribution, which is part of a group of so-called bilinear integral transforms. This technique represented the first attempt to perform a joint analysis in time and frequency [2, 19]. According to [2], the bilinear Wigner-Ville distribution provides better resolution in the joint time-frequency domain compared to any linear transform. However, it suffers from a problem of cross-term interference, which does not represent any signal information. In other words, the WVD of two signals is not the sum of their individual WVDs [27].

For a continuous signal *x*(*t*) the Wigner-Ville distribution is defined as:

$$WVD\_x(t, \ f) = \int\_{-\infty}^{\infty} \infty \ \left(t + \frac{\tau}{2}\right) \propto \left(t - \frac{\tau}{2}\right) \ e^{-j2\pi\frac{f}{\tau}} d\tau \tag{5}$$

A comprehensive study on the method is present in [2, 19].

In this chapter, we focus on utilizing the Synchrosqueezing Transform (SST) based on the CWT. The SST enhances the time-frequency representation obtained from the CWT by concentrating the energy of the signal in the time-frequency plane. However, it's important to note that the SST can also be applied using the STFT as an alternative basis. In our formulation CWT was written in Eq. (2).

The SST involves three steps: First, the CWT is calculated. In the second step, we calculate an initial frequency *f* (*a, b*) by analyzing the oscillatory behavior of *Wx*(*a, b*) with respect to *a* [18, 21]. So that:

$$f(a,b) = -i[\mathcal{W}\_\mathbf{x}(a,b)]^{-1}\frac{\partial \mathcal{W}\_\mathbf{x}}{\partial a}(a,b) \tag{6}$$

In the third step, the transformation from the scale-time plane to the timefrequency plane is performed. Next, we reassign each value of *Wx*(*a, b*) to (*a, f*1), where *f*1 corresponds to the closest frequency to the initial frequency *f* (*a, b*) [18, 21]. This operation is described by Eq. (6).

$$T(a, f\_1) = (\Delta f)^{-1} \sum\_{b\_k: \mathbf{f}(\mathbf{a}\_b, \mathbf{b}\_c) \cdot \mathbf{f}\_- \mathbf{l} \le \Delta f/2}^{\infty} \mathcal{W}\_x(a, b\_k) b\_k^{-3/2} \Delta b\_k \tag{7}$$

In Eq. (7), Δ*f* denotes the width of each frequency band Δ*f* = *flfl* � 1, and similarly for Δ*b*. SST based on CWT can provide a high-resolution time-frequency spectrum by compressing (reassigning) the result of the CWT. However, when the amplitude of high-frequency components is low, it becomes challenging to identify these components in the CWT or the SST spectrum derived from the CWT result. Unlike the CWT, the SST efficiently reveals the low amplitude and high-frequency

components of a signal and allows for an inverse transformation without any loss of information. Nevertheless, the resolution of the SST is still not entirely satisfactory [18, 20, 21]. Initially proposed for wavelets, the SST can also be applied using the SSTF, known as STFT-based SST. The SST based on CWT is a TFA that reallocates the energy of the signal in the frequency domain, compensating for the propagation effects caused by the wavelet mother and avoiding distortions in the TFR [20]. Unlike other methods that perform reassignment in both frequency and time directions, synchrosqueezing reallocates power only in the frequency direction, preserving the signal's time resolution. By preserving time, the inverse synchrosqueezing can reconstruct the original signal more accurately [18].

The SST is a post-processing operation that improves the TFR of a signal. While the CWT provides a global representation of the signal's time-varying frequency content, the SST further refines this process by concentrating the energy in the time-frequency representation. By applying the SST, the energy of the signal is redistributed in such a way that it becomes more localized around its instantaneous frequency. This concentration of energy allows for a clearer and more precise identification of the signal's dominant frequency components and their temporal variations. It helps to distinguish fine frequency modulations, identify transient events, and reveal nonlinear interactions that may be concealed in the original CWT TFR.

The MultiSynchrosqueezing Transform (MSST) is an extension of the Synchrosqueezing Transform (SST) that enhances the time-frequency representation of signals using a multiwavelet frame-work. The MSST overcomes certain limitations of the SST by providing an improved analysis of signals with more complex spectral characteristics [22]. In this chapter we used the MSST based on CWT. The MSST extends the SST to a multiwavelet framework by considering multiple analyzing wavelets. It improves the TFR obtained from the wavelet transform by redistributing the energy of the signal in a more localized manner. The MSST builds upon the STFT formulation, which can be expressed as follows: