**3. Method and material**

*Wavelet Transform and Complexity*

transform of the mother-wavelet *F[<sup>ψ</sup>]* given by:

2*π*√ \_\_ *a* ∫ −∞ ∞

*Wv(a,b)* will contain some residual nonzero energy (i.e., |*Wv*(*a*,*b*)|2

the wavelet analysis for which *Wv*(*a*,*b*) ≠ 0, by the phase transformation:

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

(*a*) exp(*ib*)*d* <sup>=</sup> \_\_\_\_ *<sup>A</sup>*

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

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

*iWv*(*a*,*b*)

Third, "squeezing" the wavelet transform over the regions where the phase

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

4*π*√ \_\_ *a* ψ̂ ∗

∂(*Wv*(*a*,*b*)) \_\_\_\_\_\_\_\_\_\_

ψ(*t*) exp(−*it*)*dt* = 0 (3)

(*a*ω0) exp(*ib*ω0) (4)

> 0), smearing

<sup>∂</sup>*<sup>b</sup>* (5)

ψ̂(ξ) <sup>=</sup> \_\_\_\_ <sup>1</sup>

2*π*√ \_\_ *a* ∫ −∞ ∞ *v*̂ (ξ)ψ̂ ∗

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

ω*v*(*a*,*b*) = \_\_\_\_\_\_\_\_ <sup>1</sup>

*Wv*(*a*,*b*) <sup>=</sup> \_\_\_\_ <sup>1</sup>

transformation is constant.

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

**100**

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

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.

### **Figure 2.**

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

### **Figure 3.**

*Aluminum specimen, internal damage, and laser scan area. All dimensions are in millimeters. (a) Isometric view. (b) Front view. (c) Top view.*

The TOF corresponding to the presence of the defect will be equal to the sum of TOF from source of excitation to the defect scatterer and the TOF from the defect scatterer to the receiving sensor. IF this TOF is converted to distance by multiplying by the longitudinal velocity of sound in the material, we can see that the position of the defect scatter would be any point at the surface of a locus ellipsoid whose two foci are the exciter and sensor positions [24].

An aluminum cube, with dimensions of 200 mm3 , and with an embedded cylindrical defect is considered to investigate the detection capabilities of the wavelet and synchrosqueezed transforms. The sample's structure and the position of the defect are shown in the next figure. The hole under investigation is the one on the top around the scan area of the laser-generated ultrasound (**Figure 3**).
