## 3 Development of PT Signal Processing
### Discrete Wavelet Transform (DWT)  
Wavelet transform (WT) is widely applied in stationary and non-stationary signal analysis. This technique removes electrical noise from the signal, detects discontinuities, and compresses large amounts of data. Utilizing the WT, it is possible to decompose a signal into a group of constituent signals known as wavelets, each with a well-defined and dominant frequency, similar to the Fourier transform (FT) [47], which is represented by sine and cosine functions of unlimited duration. In WT, a wavelet is a transient function of limited duration centered on a specific time. The problem with the FT is that information loss occurs in time when moving from the time domain to the frequency domain. It is impossible to infer the time of occurrence or disappearance of the signal components of the acquired frequency spectrum using the FT. Unlike the FT, the WT utilizes both the time and frequency domains to provide frequency information of a signal over time. Because the WT is discrete, it is known as the Discrete Wavelet Transform (DWT) and has significant advantages over the FT. The WT decomposes the signal into multiple scales representing a wide range of frequency bands, and the position of the WT at each scale is determined from the temporal characteristic that identifies and efficiently removes electrical noise [48]. A short-time WT is used to extract high-frequency component data. This is an important characteristic to eliminate high frequency fluctuations due to electrical noise. Long-term WT can acquire low-frequency data. High-frequency and low-frequency data can be used to define thresholds and set electrical noise thresholds below zero. DWT is considered as a new technology to replace electrical noise attenuation processing using low-pass filter of Fast Fourier Transform (FFT) or lock-in amplifier system that can be used independently in environments with very low electrical noise [49]. The overlap of bands that are completely different or completely different from the signal and noise for which filtering methods can be utilized is an important characteristic of the moment of processing digital signals whose information does not change over time. FT and WT applied to infrared thermography technology can extract phase angle and amplitude data from the temperature-time history of each pixel. Analysis image sequences by applying transient signals to detect and characterize defects. However, the FT reconstructs all signals by using the infinite constant base of the sequence, which causes a problem when there is a transient component or a sudden change in the signal. However, the WT can be described as an extension of the FT that preserves the time domain information directly related to the defect depth while maintaining the properties of the FT. As a result, it affects the time frequency and resolution of the signal. Therefore, WT is utilized as a transformation algorithm to extract transient function data. The principle of the WT is that the analyzed function, \(f(t)\), is decomposed into a set of basic functions, \(\Psi_{s,\tau}(t)\), which are referred to as wavelets. The continuous WT is expressed as in Equation (15) [50]. \[\gamma(s,\tau)=\int_{-\infty}^{\infty}f(t)\Psi_{s,\tau}^{*}(t)dt=Re_{n}+jIM_{n} \tag{15}\]  
where \(*\) stands for complex conjugation, \(s\) is the scale and defines the wavelet dilation (\(s>1\)) or contraction (\(1>s>0\)), and \(\tau\) corresponds to the WT along the analyzed signal, \(t\) is the time, \(Re_{n}\) is the real part of the transform, \(Im_{n}\) is the imaginary part of the transform and \(j\) is the imaginary number (\(-1^{1/2}\)). Function \(\Psi^{*}_{s,\tau}(t)\) is generated by scaling and translating the Mother Wavelet (MT) according to Equation (16). \[\gamma(s,\tau)=\frac{1}{\sqrt{s}}\Psi\left(\frac{t-\tau}{s}\right) \tag{16}\]  
\[\Psi(x)=\frac{1}{\sqrt{\pi f_{b}}}e^{\beta\pi f_{b}x_{c}e^{-\frac{x^{2}}{f_{b} }}}\]  
where \(f_{b}\) is a constant that defines the width of the wavelet and \(f_{c}\) is the wavelet center frequency constant.