# Wavelet-Based Estimation of Damping from Multi-Sensor, Multi-Impact Data
## 3 Damping Extraction Methods
### Wavelet-Based Damping Extraction
#### 3.2.1 Introduction to Wavelet  
Wavelet analysis is a well-known signal processing tool enabling the investigation of different properties of signals. This mathematical tool has been used in different areas, such as data compression, denoising, smoothing, communication, etc. Wavelet transform is an integral transform using a family of kernels, the wavelets, parametrized in time and scale. Wavelet transform can be represented as  
\[\mathcal{W}_{\Psi,x}(s,\tau)=\int x(t)\Psi_{s,\tau}^{*}(t)dt, \tag{21}\]  
where \(s\) and \(\tau\) are the scale and translation parameters respectively, \({}^{*}\) denotes complex conjugation, and \(\Psi_{s,\tau}(t)\) called the wavelet. Depending on the mother wavelet, wavelet transform can represent different data characteristics in various patterns.  
The wavelet function is chosen from a subspace such that they are both absolutely integrable and square integrable:  
\[\int_{-\infty}^{+\infty}|\Psi(t)|dt<\infty,\quad\int_{-\infty}^{+\infty}|\Psi( t)|^{2}dt<\infty. \tag{22}\]  
On account of these properties, one can formulate a well-defined wavelet transform so that the mother wavelet fulfills zero mean and square norm conditions:  
\[\int_{-\infty}^{+\infty}\Psi(t)dt=0,\quad\int_{-\infty}^{+\infty}|\Psi(t)|^{2} dt=1. \tag{23}\]  
One of the main benefits of wavelet transformation is the capability of analyzing signals over time. Given a mother wavelet \(\Psi(t)\), the wavelets are generated by  
\[\Psi_{s,\tau}(t)=\frac{1}{\sqrt{s}}\Psi(\frac{t-\tau}{s}). \tag{24}\]  
Researchers have been able to compute the Fourier transform of different parts of signals (over time) by formulating the mother wavelet as a multiplication of a band-pass-like spectrum and a complex exponential. As an example, the Morlet wavelet is composed of a Gaussian window and a complex exponential:  
\[\Psi(t)=C_{\omega_{o}}(\sigma^{2}\pi)^{-\frac{1}{2}}e^{-\frac{t^{2}}{2\sigma^{ 2}}}\Big{[}e^{ia\omega_{o}t}-K_{\omega_{o}}\Big{]}, \tag{25}\] \[C_{\omega_{o}}=(1+e^{-\omega_{o}^{2}\sigma^{2}}-2e^{-\frac{3}{4} \omega_{o}^{2}\sigma^{2}})^{-\frac{1}{2}},\quad K_{\omega_{o}}=e^{-\frac{ \omega_{o}^{2}\sigma^{2}}{2}}.\]  
Similar to a bandpass filter, multiplying the signal by a Gaussian window \(((\sigma^{2}\pi)^{-\frac{1}{2}}e^{-\frac{t^{2}}{2\sigma^{2}}})\) forms a new signal which is behaving the same as the original signal around the mean of Gaussian function and overlooks the remainder of the signal. Then, this method finds the Fourier transform of the new signal. \(C_{\omega_{o}}\) and \(K_{\omega_{o}}\) are some constants added to satisfy the wavelet prerequisite (Equation (23)). The primary purpose of the Morlet wavelet is to locate and analyze the dominant frequencies of signals changing over time. Therefore, one can determine how the energy of various frequencies constructing the signal changes over time.  
Another useful property of Continuous Time Wavelet Transform (CTWT) is its linearity [13]:  
\[\mathcal{W}_{\Psi,\sum_{t=1}^{M}a_{t}\Psi_{\Upsilon}}(s,\tau)=\sum_{t=1}^{M}a _{t}\mathcal{W}_{\Upsilon,x_{1}}(s,\tau). \tag{26}\]  
This property will be used below since the processed signal in this article is a multi-component signal.