**2.1 Wavelet transform**

The conventional Fourier series representation gives the information of frequency components in a periodic signal (inability to provide frequency information over a period of time). The simplest solution, then, is to apply the FT within a limited time interval. Thus, the time window is shifted, and frequency components are obtained using the FT. This is the principal idea of the Short Time Fourier Transform (STFT). However, due to its fixed time window, its capabilities in front of complex nonstationary signals, where frequency components vary widely over a short time interval, are reduced. The wavelet transform overcomes such limitations by introducing a scaling function, which gives a variable time window. The WT provides a variable frequency resolution unlike the FT and STFT which have a constant resolution [13].

The selection of the mother wavelet provides different characteristics of the input signal set that can emphasize certain features at the output. The flexibility of choosing the optimal mother wavelet is one of the advantages of using the WT, since the choice of the mother wavelet for a particular problem improves the signal processing capability of the technique. If the shape of the signal to be detected is known a priori, a replica of the set can be utilized as the mother wavelet function, or the mother wavelet can be chosen from a set of theoretical signals. The Mexican hat, Morlet and Daubechies4 (db4) wavelets have been proven to be efficient in improving the signal strength and reducing the noise, making the WT-based technique extremely useful for flaw detection (**Figure 1**).

The wavelet transform employs a sliding window function that is used to decompose the signal into a sum of wavelets added together. Each wavelet has finite propagation in time determined by the window size. These wavelets are limited in time, whereas sinusoidal functions, which are used for the Fourier series and Fourier transform, are continuous in the whole time range. Hence, we can use these wavelets

**99**

components.

*Wavelet Transform Applied to Internal Defect Detection by Means of Laser Ultrasound*

*k*=1 *K*

that can be stretched/compressed in frequency and shifted in time to correlate them with the original signal under analysis in order to determine the set of frequencies propagating at any instantaneous time to a certain level of accuracy that is still not completely accurate due to the uncertainty principle, but this accuracy is sufficient to acquire enough information about both time and frequency composition of the signal. Assuming that a multicomponent time series signal of interest *v(t)* can be

where *ak(t)* are the time-dependent instantaneous amplitudes, *ϕk(t)* denotes the instantaneous phases, and consequently, *ϕ'k(t)* represents its instantaneous frequencies. The wavelet transform can be represented Eq. (2), where the wavelet

where *ψ\** is the complex conjugation of the mother wavelet (a continuous function in both the time domain and the frequency domain). A scale factor *a* either stretches (*a* is large), or compresses (*a* is small) the signal, where a = *ωo*/*ω*, *ω* is the angular frequency and *ωo* is the angular frequency shift, while *b* is the signal's time shift [14–16]. The main purpose of the mother wavelet is to provide a source function to generate the daughter wavelets which are simply the translated and scaled

Although, in comparison with the time-dependent Fourier transform (i.e., STFT), the achieved resolution of the time-frequency representation (TFR) by means of the wavelet analysis is certainly improved, its use still entails uncertainties on the distribution of energy for the said representation. This becomes particularly evident for nonstationary signals with a higher multimodal complexity. While it is true that these inaccuracies somehow respond to the Heisenberg-Gabor uncertainty principle [17], the fact is that they are heavily related to the choice of the wavelet function with regard

In order to overcome this drawback, alternative TFR strategies have been developed. As is the case of the Wigner-Ville distribution (and their modified alternatives, e.g., Gabor-Wigner, Choi-Williams, Cohen's class, Zhao-Atlas marks, among others), despite accomplishing high resolution TFRs, their use results in additional difficulties as in the case of high computational load and artificial frequency components due to the interference between actual ones (cross-term property).

An additional TFR technique is the Hilbert-Huang Transform (HHT); by means of an Empirical Mode Decomposition (EMD) of the analyzed signal, a collection of Intrinsic Mode Functions (IMFs) is obtained, which, along with the Hilbert spectral analysis, will lead to a time-frequency depiction. Although having been successfully applied in a wide range of fields due to adaptively decomposing the signal of interest, its use also carries some drawbacks. Such is the case of a high computation load, the requirement of a stopping criterion for the EMD, the difficulty for discerning separate frequency components in narrow-band signals and a mix of modal

A more recent TFR framework inspired, by the adaptive approach of the HHT and the redistribution concept of the Wigner-class analysis, is the Synchrosqueezing

√ \_\_\_ |*a*| ∫ −∞ ∞ *v*(*t*)ψ<sup>∗</sup>

*ak*(*t*)expcos(2*i*ϕ*k*(*t*)) (1)

*<sup>a</sup>* )*dt* (2)

(\_\_\_ *<sup>t</sup>* <sup>−</sup> *<sup>b</sup>*

*DOI: http://dx.doi.org/10.5772/intechopen.84964*

expressed in the general form (Eq. 1):

*v*(*t*) = ∑

transform *Wv* from the *v(t)* signal is obtained:

*Wv*(*a*,*b*) = \_\_\_1

versions of the mother wavelet.

**2.2 Synchrosqueezed transform**

to the phenomenon of the application.

**Figure 1.** *Examples of wavelets used for acoustic emission processing. (a) Mexican hat. (b) Daubechies. (c) Morlet.*

*Wavelet Transform Applied to Internal Defect Detection by Means of Laser Ultrasound DOI: http://dx.doi.org/10.5772/intechopen.84964*

that can be stretched/compressed in frequency and shifted in time to correlate them with the original signal under analysis in order to determine the set of frequencies propagating at any instantaneous time to a certain level of accuracy that is still not completely accurate due to the uncertainty principle, but this accuracy is sufficient to acquire enough information about both time and frequency composition of the signal.

Assuming that a multicomponent time series signal of interest *v(t)* can be expressed in the general form (Eq. 1):

$$v(t) = \sum\_{k=1}^{K} a\_k(t) \exp\text{cos}\left(2\pi i \phi\_k(t)\right) \tag{1}$$

where *ak(t)* are the time-dependent instantaneous amplitudes, *ϕk(t)* denotes the instantaneous phases, and consequently, *ϕ'k(t)* represents its instantaneous frequencies. The wavelet transform can be represented Eq. (2), where the wavelet transform *Wv* from the *v(t)* signal is obtained:

$$\mathcal{W}\_v(a,b) = \frac{1}{\sqrt{|a|}} \int\_{-\infty}^{\infty} v(t) \, \Psi^\*\left(\frac{t-b}{a}\right) dt\tag{2}$$

where *ψ\** is the complex conjugation of the mother wavelet (a continuous function in both the time domain and the frequency domain). A scale factor *a* either stretches (*a* is large), or compresses (*a* is small) the signal, where a = *ωo*/*ω*, *ω* is the angular frequency and *ωo* is the angular frequency shift, while *b* is the signal's time shift [14–16]. The main purpose of the mother wavelet is to provide a source function to generate the daughter wavelets which are simply the translated and scaled versions of the mother wavelet.

### **2.2 Synchrosqueezed transform**

*Wavelet Transform and Complexity*

**2. Theoretical considerations**

**2.1 Wavelet transform**

constant resolution [13].

investigation, as the Synchrosqueezed Transform (ST), are considered. Thus, in this chapter, a defective metallic component for damage detection and visualization, through a laser-ultrasonic approach and detection of AE waves TOF, is studied. For this objective, the wavelet transform performance, as a time-frequency processing tool, and its results, are studied, compared with a promising variant called synchrosqueezed transform. This chapter is organized as follows: The theoretical basis and its suitability for the ultrasound processing of the wavelet transform and the synchrosqueezed transform are presented in Section 2. The materials and method, including the experimental setup, are explained in Section 3. The competency of the techniques and the experimental results are presented and discussed in Section 4.

Finally, this chapter shows the conclusion dissemination in Section 5.

The conventional Fourier series representation gives the information of frequency components in a periodic signal (inability to provide frequency information over a period of time). The simplest solution, then, is to apply the FT within a limited time interval. Thus, the time window is shifted, and frequency components are obtained using the FT. This is the principal idea of the Short Time Fourier Transform (STFT). However, due to its fixed time window, its capabilities in front of complex nonstationary signals, where frequency components vary widely over a short time interval, are reduced. The wavelet transform overcomes such limitations by introducing a scaling function, which gives a variable time window. The WT provides a variable frequency resolution unlike the FT and STFT which have a

The selection of the mother wavelet provides different characteristics of the input signal set that can emphasize certain features at the output. The flexibility of choosing the optimal mother wavelet is one of the advantages of using the WT, since the choice of the mother wavelet for a particular problem improves the signal processing capability of the technique. If the shape of the signal to be detected is known a priori, a replica of the set can be utilized as the mother wavelet function, or the mother wavelet can be chosen from a set of theoretical signals. The Mexican hat, Morlet and Daubechies4 (db4) wavelets have been proven to be efficient in improving the signal strength and reducing the noise, making the WT-based

The wavelet transform employs a sliding window function that is used to decompose the signal into a sum of wavelets added together. Each wavelet has finite propagation in time determined by the window size. These wavelets are limited in time, whereas sinusoidal functions, which are used for the Fourier series and Fourier transform, are continuous in the whole time range. Hence, we can use these wavelets

*Examples of wavelets used for acoustic emission processing. (a) Mexican hat. (b) Daubechies. (c) Morlet.*

technique extremely useful for flaw detection (**Figure 1**).

**98**

**Figure 1.**

Although, in comparison with the time-dependent Fourier transform (i.e., STFT), the achieved resolution of the time-frequency representation (TFR) by means of the wavelet analysis is certainly improved, its use still entails uncertainties on the distribution of energy for the said representation. This becomes particularly evident for nonstationary signals with a higher multimodal complexity. While it is true that these inaccuracies somehow respond to the Heisenberg-Gabor uncertainty principle [17], the fact is that they are heavily related to the choice of the wavelet function with regard to the phenomenon of the application.

In order to overcome this drawback, alternative TFR strategies have been developed. As is the case of the Wigner-Ville distribution (and their modified alternatives, e.g., Gabor-Wigner, Choi-Williams, Cohen's class, Zhao-Atlas marks, among others), despite accomplishing high resolution TFRs, their use results in additional difficulties as in the case of high computational load and artificial frequency components due to the interference between actual ones (cross-term property).

An additional TFR technique is the Hilbert-Huang Transform (HHT); by means of an Empirical Mode Decomposition (EMD) of the analyzed signal, a collection of Intrinsic Mode Functions (IMFs) is obtained, which, along with the Hilbert spectral analysis, will lead to a time-frequency depiction. Although having been successfully applied in a wide range of fields due to adaptively decomposing the signal of interest, its use also carries some drawbacks. Such is the case of a high computation load, the requirement of a stopping criterion for the EMD, the difficulty for discerning separate frequency components in narrow-band signals and a mix of modal components.

A more recent TFR framework inspired, by the adaptive approach of the HHT and the redistribution concept of the Wigner-class analysis, is the Synchrosqueezing Transform (ST). This framework was developed with the aim to eliminate distorted interference terms while concentrating the energy on their corresponding modal components. This method, belonging to the family of the time-frequency energy reassignment, has arisen with the advantages of offering a better adaptability with regard to the signal, lesser deformation for the IF profiles, and by preserving the time, it admits an exact reconstruction formula for the constituent modal components (i.e., existence of an inverse transformation). Originally proposed for Daubechies [18] for an auditory application and revised for several authors [19–23], it works by redistributing the misallocated energy on the scale axis (due to the mother wavelet).

As aforementioned, the wavelet analysis leads to a depiction of the instantaneous frequencies *ϕ'k(t)* of each existing component in the signal *v(t)* by a correlation between said signal and a chosen atom (mother-wavelet), thus using a scaled and translated version of the mother-wavelet over *v(t)*. Nevertheless, under this framework is presented energy spreading over the TFR associated due to the selection of the mother-wavelet as well as for the Heisenberg-Gabor uncertainty principle, affecting the intelligibility of the analysis. The aim of the synchrosqueezed wavelet transform is to partially reassign the spread energy that occurred during the wavelet analysis for the frequency dimension only, by analyzing each component of the TFR. Therefore, it is necessary that the modal components are intrinsic mode type functions (IMT). Hence, by preserving the time dimension, it is possible to enable an inverse transformation of the obtained signal toward a time series. The wavelet synchrosqueezed transform (WST) involves the following steps. First, obtaining a wavelet transform *Wv* from the *v(t)* signal following Eq. (2). Thus, *ψ* represents the analytic mother wavelet existing only for positive frequencies, that is, the Fourier transform of the mother-wavelet *F[<sup>ψ</sup>]* given by:

$$
\hat{\boldsymbol{\Psi}}(\xi) = \frac{1}{2\pi\sqrt{d}} \int\_{-\infty}^{\infty} \boldsymbol{\Psi}(t) \, \exp\left(-i\xi t\right) dt = \mathbf{0} \tag{3}
$$

for frequencies *ξ <* 0. By Plancherel's theorem, Eq. 2 can be rewritten as:

$$W\_v(a,b) = \frac{1}{2\pi\sqrt{a}} \int\_{\rightsquigarrow}^{\rightsquigarrow} \hat{\nu}(\xi) \,\hat{\Psi}^\*(a\xi) \, \exp\left(ib\xi\right) d\xi = \frac{A}{4\pi\sqrt{a}} \hat{\Psi}^\*(a\,\omega\_0) \, \exp\left(ib\,\omega\_0\right) \tag{4}$$

where *ω0* = 2π*f0* is the angular frequency of *v(t)*.

Second, extracting the IF from the wavelet transform. As each scale *a* of Eq. (5) corresponds to a natural frequency *ξ*/*ω0*, satisfying the relation *a* = *c*/*ξ* where *c* is the center frequency of the mother-wavelet *ψ\**; it concentrates the energy of the transformation around this frequency. By supposing that the shift time *b* is fixed, and if *ξ* = *c*/*a* is close, but not exactly located at the instantaneous frequency *ϕ'k(t)*, the coefficient *Wv(a,b)* will contain some residual nonzero energy (i.e., |*Wv*(*a*,*b*)|2 > 0), smearing the TFR. The aim of the synchrosqueezing is to remove this residual energy centered around *ξ* and reallocating it to a frequency location closer to its corresponding instantaneous frequency *ϕ'k(t)*. So, it is necessary to compute the instantaneous frequency of the wavelet analysis for which *Wv*(*a*,*b*) ≠ 0, by the phase transformation:

$$\alpha\_v(a,b) = \frac{1}{i\,\mathcal{W}\_v(a,b)} \frac{\partial \{\mathcal{W}\_v(a,b)\}}{\partial b} \tag{5}$$

**101**

**Figure 2.**

*Wavelet Transform Applied to Internal Defect Detection by Means of Laser Ultrasound*

∆ω <sup>∑</sup> *ak*:|ω(*ak*,*b*)−ω*l*|≤\_\_\_

values by means of Eq. (5), to then, for each frequency of interest *ωl*, compute the synchrosqueezing by adding all values *Wv*(*a*,*b*), where the reassigned frequency *ωv*(*a*,*b*) is equal to *ωl*. This is achieved by means of the mapping (for discrete values):

where Δ*ω* = *ω<sup>l</sup>* − *ω(l−1)*, (Δ*a*)*k* = *ak* − *a(k−1)*, *ωl* is the *l*th discrete angular frequency, and *ak* is the *k*th discrete scale point. Finally, the instantaneous angular

In general, the modal components from the synchrosqueezed analysis are separated well enough in the TF plane. For a given signal, if this condition is actually met, their modal components could be treated as intrinsic mode function types and their trajectories (known as wavelet ridges) can be tracked over the TF plane as their energy varies in terms of the function of time, enabling their transformation

In order to analyze the suitability of the wavelet transform and the synchrosqueezing to extract a proper TOF related to defect location, a specific experimental bench has been arranged. The procedure is based on five steps. The first step consists of the caption of the ultrasonic signals received by the ultrasonic sensors from all considered laser scan points. These acquired signals are then processed by a noise filtering algorithm and an interpolation and bandpass filter to remove any unimportant components. The resulting A-scan signals are then ready for the next step of applying the wavelet or the synchrosqueezed transform. These transforms will generate the time frequency maps that are useful for detecting the most important propagating frequencies with respect to their times of flight. In order to further clean the signal, it is proposed to apply a signal contouring algorithm. This will help to identify the areas with uniform intensities, and the signal distribution will become clearer. It should be noted that the most important feature in this kind of algorithm is the expected time of flight for the signal. This time of flight is used later on detecting the distance between the sensor and the defect based on

The distance between the individual laser scan points and the receiving sensor is known a priori. In addition, the dimensions of the object under test are also known. In this regard, the time of flight of the main echoes should be equal to, or greater than, either the distance of the path from the laser direct to the sensor, or from the laser to any object boundary and reflecting back to the sensor whichever found shorter. Thus, the distance between the laser and the object boundaries is larger than the direct distance between the laser and the sensor. In addition, if there is any existing defect inside the material, this would create an internal deflection with a distance shorter than that of the object's boundaries. Hence, it is expected that the first main echo received in the analysis is due to the laser's direct propagation toward the sensor, and the second main

echo, in this case, should be due to the deflections from any existing defect.

∆ω 2

*Wv*(*ak*,*b*)*ak*

−3/2 (Δ*a*)*<sup>k</sup>* (6)

*DOI: http://dx.doi.org/10.5772/intechopen.84964*

*Tv*(ω*l*,*b*) = \_\_\_1

into the time domain.

**3. Method and material**

frequency can be normalized by 2π as the IF *f* = *ω*/2π.

the speed of propagation of the ultrasonic waves (**Figure 2**).

*The sequence flow chart of the signal processing procedure for the analysis.*

Third, "squeezing" the wavelet transform over the regions where the phase transformation is constant.

During the scale-frequency mapping, that is, (*a*,*b*) → (*ωv*(*a*,*b*),*b*), the synchrosqueezing is applied to reassign the time-scale representation of the TF. Thus, for a fixed shift time *b*, the frequency reassignment *ωv*(*a*,*b*) is carried out for all *a* scale

*Wavelet Transform Applied to Internal Defect Detection by Means of Laser Ultrasound DOI: http://dx.doi.org/10.5772/intechopen.84964*

values by means of Eq. (5), to then, for each frequency of interest *ωl*, compute the synchrosqueezing by adding all values *Wv*(*a*,*b*), where the reassigned frequency *ωv*(*a*,*b*) is equal to *ωl*. This is achieved by means of the mapping (for discrete values):

$$T\_v(a\iota\_\nu b) = \frac{1}{\Delta a} \sum\_{a,b:\left|a\iota(a\_b,b) - a\iota\right| \in \frac{\Delta a}{2}} W\_v(a\_k, b) \, a\_k^{-3/2} \, (\Delta a)\_k \tag{6}$$

where Δ*ω* = *ω<sup>l</sup>* − *ω(l−1)*, (Δ*a*)*k* = *ak* − *a(k−1)*, *ωl* is the *l*th discrete angular frequency, and *ak* is the *k*th discrete scale point. Finally, the instantaneous angular frequency can be normalized by 2π as the IF *f* = *ω*/2π.

In general, the modal components from the synchrosqueezed analysis are separated well enough in the TF plane. For a given signal, if this condition is actually met, their modal components could be treated as intrinsic mode function types and their trajectories (known as wavelet ridges) can be tracked over the TF plane as their energy varies in terms of the function of time, enabling their transformation into the time domain.
