## 6 Conclusions  
To address the challenge of effectively filtering emitted noise, this paper proposes the FCI-VMD algorithm based on variational mode decomposition and applies it to the study of indoor sound emission signal separation and vibration characteristics. Additionally, multiple fractal spectra are utilized to extract AE spectral features. The main findings are summarized as follows:  
* A weighted frequency index is proposed as the objective function to optimize VMD decomposition level \(k\). The BAS algorithm is used to automatically search for the objective function and obtain the effective signal components. The results show that the proposed method can effectively avoid the mode mixing issues and overcome the need to determine the \(k\) value beforehand when employing VMD for AE signal decomposition.
* In comparison to EMD and EVMD, FCI-VMD proves more suitable for AE signal denoising. By utilizing fuzzy entropy as the basis for selecting the noise components in the four stress stages, the method avoids the poor noise reduction effect caused by filtering out only the components with more noise content.
* The multifractal parameter, \(\Delta\alpha\), exhibits a sharp decrease as coal damage occurs, aligning with the evolution pattern of the AE main frequency. Consequently, \(\Delta\alpha\) can be employed as a quantitative analysis parameter for AE spectra. Moreover, \(\Delta\alpha\) can serve as an early warning indicator for coal damage and instability.  
Y.Z.: Conceptualization, Methodology, Investigation, Resources, Review and Editing, Funding acquisition. G.J.: Writing, Software, Formal analysis, Investigation, Writing--Original Draft. P.M.M. and Y.G.: Visualization, Investigation. G.L.: Investigation, Data Curation, Writing--Review and Editing. All authors have read and agreed to the published version of the manuscript.  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Sample**} & \multirow{2}{*}{**State**} & \multicolumn{6}{c}{**Characteristic Parameter of Multifractal Spectrum**} \\ \cline{3-8}  & & \(\alpha_{min}\) & \(f\) (\(\alpha_{min}\)) & \(\alpha_{max}\) & \(f\) (\(\alpha_{max}\)) & \(\Delta\alpha\) & \(\Delta f\) \\ \hline \multirow{4}{*}{WI-2} & 20\% oc & 2.0795 & 2.0795 & 2.2445 & 1.9967 & 0.1650 & \(-\)0.0828 \\  & 50\% oc & 2.0778 & 2.0778 & 2.2473 & 1.9987 & 0.1695 & \(-\)0.0791 \\  & 80\% oc & 2.0770 & 2.0770 & 2.2555 & 1.9960 & 0.1785 & \(-\)0.0810 \\  & 100\% oc & 2.0786 & 2.0034 & 2.2188 & 1.9901 & 0.1402 & \(-\)0.0751 \\ \hline \multirow{4}{*}{WI-3} & 20\% oc & 2.0823 & 2.0823 & 2.2179 & 2.0081 & 0.1356 & \(-\)0.0742 \\  & 50\% oc & 2.0809 & 2.0809 & 2.2373 & 1.9987 & 0.1564 & \(-\)0.0822 \\ \cline{1-1}  & 80\% oc & 2.0790 & 2.0790 & 2.2561 & 1.9907 & 0.1771 & \(-\)0.0883 \\ \cline{1-1}  & 100\% oc & 2.0824 & 2.0824 & 2.2102 & 2.0126 & 0.1278 & \(-\)0.0698 \\ \hline \hline \end{tabular}
\end{table}
Table 4: The multifractal spectral characteristic parameters of AE spectral images in different stages.  
**Funding:** This research was funded by National Natural Science Foundation of China, grants number 52225402 and U1910206.  
**Institutional Review Board Statement:** Not applicable.  
**Informed Consent Statement:** Not applicable.  
**Data Availability Statement:** The data presented in this study are available on request from the corresponding author.  
**Acknowledgments:** The authors would like to express sincere gratitude to the China Scholarship Council (CSC) for providing financial support for this study. The sponsorship guaranteed with basic research funds provided by Politecnico di Torino, Italy, for its financial aid in this work is also acknowledged.  
**Conflicts of Interest:** The authors declare no conflict of interest.