## 2 Materials and Methods
### Complex Continuous Wavelet Transform  
To increase the reliability of low strain integrity testing, in [29; 30; 31; 32; 33; 34], it was proposed to use the complex continuous wavelet transform to perform a time-frequency analysis of the recorded signals. CCWT converts the studied signal \(f(t)\in L^{2}(R)\) into a function of two variables \(a\), \(b\in R\), \(a>0\):  
\[W(a,b)=\int_{-\infty}^{+\infty}f(t)\frac{1}{\sqrt{a}}\psi^{*}\left(\frac{t-b}{ a}\right)\mathrm{d}t, \tag{3}\]  
where \(\psi(t)\) is the complex mother wavelet, * represents the complex conjugation operation, \(a\) is the scale parameter of the mother wavelet, and \(b\) is the time shift that determines the location of the wavelet.  
To apply the wavelet transform successfully, it is important to select the best-suited mother wavelet. In [36], we have shown that an optimal mother wavelet for low strain test data CCWT analysis is a complex Morlet wavelet:  
\[\psi(t)=\frac{1}{\sqrt{\pi B}}exp^{-\frac{t^{2}}{B}}exp^{j2\pi Ct} \tag{4}\]  
where \(B\) is the bandwidth and \(C\) is the center frequency. By varying the bandwidth and center frequency parameters, we can adjust the mother wavelet for the test data obtained with the initial force pulses of different frequency content.  
The CCWT results can be presented as an image in a time-frequency plane, where the energy distribution of the wavelet transform coefficients \(\left|W(a,b)\right|^{2}\) is shown in gradations of the color scale. The more precisely the local feature of a signal coincides with the wavelet function obtained by scaling and shifting, the more effectively it will stand out in the CCWT energy plot.  
To relate scale parameter \(a\) to frequency \(F_{a}\), we use the following relationship:  
\[F_{a}=\frac{F_{c}}{a} \tag{5}\]  
where \(F_{c}\) is the center frequency of the wavelet.  
The instantaneous phase angle \(\phi(a,b)\) of \(W(a,b)\) can be calculated as:  
\[\phi(a,b)=arctan\ \left(\frac{W_{I}(a,b)}{W_{R}(a,b)}\right), \tag{6}\]  
where \(W_{I}(a,b)\) and \(W_{R}(a,b)\) represent the imaginary part and the real part of \(W(a,b)\), respectively. The phase angles of the wavelet coefficients can be presented as an image in a time-frequency plane (phase spectrum), where the blue color indicates \(-180\) degrees (\(-\pi\)) of a phase angle and the red color denotes 180 degrees (\(\pi\)).  
To obtain the location of the pile toe or defects, [PERSON] et al. (2019) [27] suggested finding the area of concentrated energy of the wavelet transform coefficients, then finding the phase shift from \(\pi\) to -\(\pi\) (or where the phase angle equals 0), which distributes in the form of a straight line from a higher frequency to a lower frequency in the phase spectrum near the corresponding concentrated energy area. If the reflected concentrated energy is not obvious, the phase shift line will be closer to a straight line from a higher frequency to a lower frequency.  
By analyzing the phase shift moments, it is possible to assess the pile integrity and precisely localize the defect or determine the pile length. In this paper, we propose to use the CCWT phase analysis to determine the lengths of the driven precast piles, as well as to localize their splices.