# Application of Generalized S-Transform in the Measurement of Dynamic Elastic Modulus
## 4 Measurement Based on Generalized S-Transform
These comparative experiments demonstrate that the GST algorithm is capable of mitigating the error in peak frequency calculation more effectively than the FFT algorithm, particularly in the context of the broad resonant frequency range of concrete resulting from curing conditions and cracks [42, 43]. ## 5 Conclusions

This study focuses on addressing the challenge of spectral aliasing in resonance-based measurements of the dynamic elastic modulus, which cracks and variations in the curing conditions of concrete specimens may lead to wide resonance bands, ultimately resulting in measurement inaccuracies. We present a time-frequency tool for accurately calculating the frequency spectrum using the generalized S-transform on the DSP platform. For the measurement of the dynamic elastic modulus, this algorithm can improve the accuracy of the resonant frequency by adjusting the width of the window function with the signal frequency. Experimental results demonstrate that the standard deviation of the resonance frequency using the GST algorithm is below 5 Hz, verifying the advantages of the GST algorithm in dynamic elastic modulus measurement with high accuracy. In further research, the focus will be on enhancing the operational speed of the algorithm, especially on low-cost DSP platforms, and this improvement will enable the measurement of dynamic elastic modulus in industrial settings, providing a convenient method for concrete damage and performance analysis in construction engineering. L.W. : research ideas, writing manuscripts and data analysis; Y.G. : charting, data collection; H.L. : experimental design, data collection; G.F.: document retrieval, data collection; D.L. : experimental design, data collection, data analysis. All authors have read and agreed to the published version of the manuscript. This study was supported by Tianjin Educational Commission Scientific Research Program (2020 KJ004) and in part by the State key laboratory of precision measuring technology and instruments (pilab2104). Not applicable. The data presented in this study are available on request from the corresponding author. The data are not publicly available due to privacy. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & \multicolumn{2}{c}{**FFT**} & \multicolumn{2}{c}{**GST**} & \multicolumn{2}{c}{**Swept-Frequency**} \\ \hline
**Specimen** & **Average (kHz)** & \begin{tabular}{c} **Standard** \\ **Deviation** \\ **(kHz)** \\ \end{tabular} & \begin{tabular}{c} **Average (kHz)** \\ \end{tabular} & \begin{tabular}{c} **Standard** \\ **Deviation** \\ **(kHz)** \\ \end{tabular} & \
\begin{tabular}{c} **Average (kHz)** \\ \end{tabular} &
\begin{tabular}{c} **Standard** \\ **Deviation** \\ **(kHz)** \\ \end{tabular} \\ \hline Brick 1 & 2.315 & 0.0045 & 2.326 & 0.0037 & 2.319 & 0.003 \\ Brick 2 & 2.376 & 0.0047 & 2.368 & 0.0039 & 2.378 & 0.0043 \\ Brick 3 & 2.403 & 0.0028 & 2.399 & 0.0017 & 2.393 & 0.0025 \\ Brick 4 & 2.384 & 0.0052 & 2.382 & 0.0046 & 2.391 & 0.0035 \\ Brick 5 & 2.363 & 0.0041 & 2.371 & 0.0031 & 2.366 & 0.0028 \\ Brick 6 & 2.397 & 0.0039 & 2.405 & 0.004 & 2.395 & 0.0037 \\ Aluminum 1 & 2.913 & 0.0024 & 2.918 & 0.0021 & 2.912 & 0.0019 \\ Aluminum 2 & 2.924 & 0.0023 & 2.921 & 0.0029 & 2.931 & 0.0025 \\ Aluminum 3 & 2.895 & 0.0019 & 2.909 & 0.0015 & 2.903 & 0.0021 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Standard deviation (\(\sigma\)) of resonance frequency measurement. **Acknowledgments:** The authors would like to thank the members of the project team for their dedication and effort, and the teachers and schools for their help. **Conflicts of Interest:** Author [PERSON] was employed by the company China Railway Construction Engineering Group. The remaining authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.