## 2 Methodology
### Step 2: Modal Parameters Identification
Each individual has a length of 32 bits, the first 16 bits are used for the candidate natural frequency, whereas the last 16 bits contain the candidate damping ratio. The initial values of the population are generated randomly, ensuring that they are in a normal distribution in order to guarantee a diversity of potential solutions. Once generated the initial population, the next step is to evaluate their performance. Recalling Equation (7), it is observed that the produced signal depends of two variables: frequency and damping ratio. The amplitude of each mode is taken as the first value of the recovered signal, which corresponds with the maximum amplitude of signal. Therefore, it is necessary to use different functions to evaluate the impact of the subtle changes in the values of frequency and damping ratio in the final solution. For evaluating the natural frequency value, the following equations are proposed:  
\[f_{1}=corr(y_{m},y) \tag{8}\]  
\[f_{2}=corr(g_{m},g) \tag{9}\] \begin{table}
\begin{tabular}{c c c} \hline \hline
**Modal Parameter** & **Minimum Value** & **Maximum Value** \\ \hline Natural Frequency (Hz) & 0.01 & 100 \\ Damping Ratio (\%) & 0.01 & 20 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Range of values used for every modal parameter. Figure 3: Proposed micro-GA scheme for the modal parameter identification (MPI). \[f_{3}=hist(y)-hist(y_{m}) \tag{10}\]  
where \(f_{1}\) represents the correlation of each mono-component estimated by the micro-GA (\(y_{m}\)) using Equation (7) and by SWT (\(y\)); and \(f_{2}\) is the correlation of the envelopes of each mono-component obtained by the micro-GA (\(g_{m}\)) and by SWT (\(g\)). The envelope of each signal is obtained as follows:  
\[g=\sqrt{y^{2}+y_{h}^{2}} \tag{11}\]  
where \(y_{h}\) is the Hilbert Transform of \(y\)[47]. Finally, \(f_{3}\) represents the histograms difference of the signals \(y\) and \(y_{m}\). It is important to mention that the correlation measures the similarity between two signals and the histogram gives the shape and distribution of the signal. Therefore, the combination of these equations provides useful information for the detection of the subtle changes that are produced when the estimated values are close to their exact counterparts. For this reason, the following objective function is proposed to optimize the natural frequency value:  
\[J_{1}=w_{1}f_{1}+w_{2}f_{2}+w_{3}f_{3} \tag{12}\]  
where \(w_{1}\), \(w_{2}\), and \(w_{3}\) are ponderation weights used to set a level of importance of each function, \(f_{i}\), of the objective function. It was found, from previous experimentations, that 0.5, 0.45, and 0.05 for \(w_{1}\), \(w_{2}\), and \(w_{3}\), respectively, were the best values since they allow estimating the natural frequency value with the lowest error, indicating that correlation between the two signals provides the most information to obtain an accurate frequency value. On the other hand, in order to evaluate the damping ratio value, the envelope of the in-test signal and the one constructed are employed. Thus, the following evaluation function is proposed:  
\[f=corr(g_{m},g) \tag{13}\]  
Therefore, Equation (13) allows setting the following objective function:  
\[J_{2}=f \tag{14}\]  
Once the obtained values with the proposed objective functions in Equations (12) and (14) are evaluated, the next step is to verify the criterion for stopping the generation of new populations. In this regard, it is used the maximization criterion of the objective functions, since it is desired to calculate the highest correlation value, because it will indicate that the signal, constructed with Equation (7), has the most resemblance with the mono-component obtained by SWT. If the criterion is not achieved; then, it is necessary to create a new population based on the current one. To this purpose, the best individual, the one that produces the signal with the highest correlation value with respect of the in-test one, is used directly as the first individual of the new generation.