## 2 Theoretical Considerations of the Digital Filter
### EMD Algorithm
**Step 2:** Average the maxima and minima in order of the time series, which is represented as  
\[m_{t}(t)=\dfrac{U_{\text{1 int}}+L_{\text{1 int}}}{2},\dfrac{U_{\text{2 int}}+L_{ \text{2 int}}}{2},\dfrac{U_{\text{3 int}}+L_{\text{3 int}}}{2},...,\dfrac{U_{ \text{nint}}+L_{\text{nint}}}{2} \tag{4}\]  
**Step 3:** Subtract the \(x_{i}(t)\) from the average of the local extrema (maxima and minima) \(m_{i}(t)\) in order of the time series, and the new decomposed signal is  
\[h_{i}(t)=x_{k}(t)-m_{i}(t) \tag{5}\]  
**Step 4:** Repeat steps (i) through (iii) \(k\) times until \(h_{\text{1}k}(t)\) equals \(c_{1}(t)\). Using Equation (5) where \(h_{\text{1}(k-1)}(t)-m_{\text{1}k}=h_{\text{1}k}(t)\), we designate \(c_{1}(t)\) as the first IMF. **Step 5:** Find other IMFs by calculating the first residual, which is given by  
\[r_{1}(t)=x_{k}(t)-c_{1}(t) \tag{6}\]  
We derive \(c_{2}(t),c_{3}(t),c_{4}(t),...,c_{n}(t)\) ; thus, \(r_{1}(t)\) is treated as a new dataset in the next loop, which is completes after obtaining \(c_{n}(t)\). This procedure is represented by  
\[r_{2}(t)=r_{1}(t)-c_{2}(t) \tag{7}\]  
\[......\] \[r_{n}(t)=r_{n-1}(t)-c_{\[x(t)=\sum_{i=1}^{n}r_{n}(t)+r_{n}(t) \tag{9}\]  
Since HHT has formed forms another function, HSA, it can be used as a tool for the time-frequency analysis of nonlinear and nonstationary datasets to present the results as a time-frequency-energy distribution [12]. For a comprehensive explanation of the Hilbert transform, refer to [PERSON] [16]. Given a real signal \(x(t)\), the analytic signal \(z(t)\) is defined as  
\[z(t)=x(t)+iy(t) \tag{10}\]  
where \(y(t)\) is the Hilbert transform defined using the Cauchy principle value denoted by p.v. of \(x(t)\) such that  
\[f_{\text{int}}\left\{U_{k}(t),x_{k}(t)\right\}=U_{\text{1 int}}(t),U_{\text{2 int} }(t),U_{\text{3 int}}(t),....,U_{\text{nint}}(t) \tag{2}\]0  
We can define the complex signals, amplitude \(A(t)\) and phase \(\psi(t)\) as follows [8]  
\[f_{\text{int}}\left\{U_{k}(t),x_{k}(t)\right\}=U_{\text{1 int}}(t),U_{\text{2 int} }(t),U_{\text{3 int}}(t),....,U_{\text{nint}}(t) \tag{2}\]1  
\[f_{\text{int}}\left\{U_{k}(t),x_{k}(t)\right\}=U_{\text{1 int}}(t),U_{\text{2 int} }(t),U_{\text{3 int}}(t),....,U_{\text{nint}}(t) \tag{2}\]2  
where the time-varying phase or instantaneous frequency \(\omega(t)\) can be given as \(\omega(t)=\dfrac{\mathrm{d}\psi(t)}{\mathrm{d}t}\). From Equation (9), the original data is rearranged as follows:  
\[f_{\text{int}}\left\{U_{k}(t),x_{k}(t)\right\}=U_{\text{1 int}}(t),U_{\text{2 int} }(t),U_{\text{3 int}}(t),....,U_{\text{nint}}(t) \tag{2}\]3  
where \(\mathrm{Re}\{\cdot\}\) denotes real part of the complex quantity. Using Equation (14) and performing the Hilbert transform on each IMF \(c_{n}(t)\), the analytical data can be expressed in terms of the Hilbert amplitude and instantaneous frequency, as follows:  
\[f_{\text{int}}\left\{U_{k}(t),x_{k}(t)\right\}=U_{\text{1 int}}(t),U_{\text{2 int} }(t),U_{\text{3 int}}(t),....,U_{\text{nint}}(t) \tag{2}\]4  
As the HHT model comprises two main parts: EMD and HSA, the EMD process under a cubic spline and sifting technique decomposes the original signals into IMF. HSA calculates the instantaneous frequency using the Hilbert transform and analyses the entire time-varying instantaneous spectrum. It is crucial to mention that the analysis of the Hilbert spectrum is conducted to view the spectrum after the EMD process is finished.