## 7 Conclusions  
Any Structural Health Monitoring system relies on data, almost always pre-processed, and on features extrapolated from them. Mel-Frequency Cepstral Coefficients (MFCCs) have recently been proven to be effective in damage detection, relying on the cepstrum of the recorded structural response, even if margins for improvements were evident. The Teager-Kaiser Energy (TKE) operator has been proposed here as the basis for a similar feature, less subject to false positive mislabeling when used for Machine Learning. That is adherent to what is encountered for speech signals and well-known in the field of speaker and speech recognition. The investigation reported here spans over different structures, with different input and very different setups--varying acquisition parameters, such as the number of the output channels, the sampling frequency, and so on. Moreover, in the first and in the second numerical study, the damage was modelled as a linear reduction of stiffness inserted in a linear system, while the experimental case emulated the damage as a pointwise source of nonlinearity in an otherwise linear system and stiffness local reduction was intended as a change in the operational conditions that were unrelated to damage. This shows the excellent capability of generalisation of this approach. The proposed damage sensitive feature, the Teager-Kaiser Energy Cepstral Coefficients (TECCs), has been benchmarked against the MFCCs and some similar variants. Interesting numerical and experimental results were achieved for both the linear and nonlinear models of damage. The main conclusions are the following:  
1. the TECCs, MFCCs, and similar features perform efficiently both with damage emulated by a reduction of the Young's modulus \(E\) or by the presence of nonlinearities, even if the signal is noisy;
2. the Mel-Scale performed similarly to other nonlinear scales such as the Bark Scale or logarithmic spacing, with no clear best option among all the investigated case studies;
3. the algorithms that resort to a Gammatone filterbank generally produced better results than the ones with triangular filters; even if the TECCs still outperforms the GTCCs; \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & \multicolumn{2}{c}{**Sensor Setup \#0**} & **Sensor Setup \#1** & **Sensor Setup \#2** & **Sensor Setup \#3** \\ \cline{2-5}  & **CPU Time [s]** & **CPU Time [s]** & **CPU Time [s]** & **CPU Time [s]** \\ \hline MFCCs & 1.60 & 1.31 & 1.00 & 1.03 \\ \(f_{\text{cut-off}}=\frac{1}{4}\,f_{\text{s}}\) & 1.63 & 1.31 & 1.01 & 1.03 \\ \(f_{\text{cut-off}}=\frac{1}{8}\,f_{\text{s}}\) & 1.62 & 1.35 & 1.04 & 1.04 \\ Log-Scale & 1.62 & 1.37 & 1.01 & 1.02 \\ Linear & 1.56 & 1.30 & 0.99 & 0.99 \\ ECO-MMFCCs & 1.56 & 1.31 & 1.00 & 1.02 \\ MO-MMFCCs & 1.58 & 1.29 & 1.00 & 0.99 \\ LO-MMFCCs & 1.59 & 1.29 & 1.01 & 0.99 \\ BFCCs & 1.58 & 1.32 & 1.02 & 1.00 \\ GTCCs & 1.78 & 1.38 & 1.06 & 1.07 \\ TECCs & 2.03 & 1.62 & 1.15 & 1.16 \\ \hline \hline \end{tabular}
\end{table} 
Table 11: CPU time accordingly to the sensor setup considered (training phase). \begin{table}
\begin{tabular}{c c} \hline \hline \multicolumn{2}{c}{**\(\boldsymbol{\sigma}\)(DI(n))-Cases 1 to 17**} \\ \hline MFCCs & 0.32, 0.20, 0.28, 0.19, 0.28, 0.32, 0.26, 0.21, 0.22, 6.21, 41.74, 62.64, 52.52, 72.21, 6.16, 8.63, 81.06. \\ Log-Scale & 0.24, 0.29, 0.15, 0.22, 0.28, 0.22, 0.27, 0.22, 0.23, 3.29, 22.23, 33.42, 31.63, 32.26, 3.45, 5.39, 35.00. \\ Linear & 0.29, 0.30, 0.22, 0.19, 0.27, 0.22, 0.24, 0.31, 0.24, 6.55, 39.27, 67.89, 46.22, 52.90, 5.50, 7.85, 72.59. \\ BFCCs & 0.31, 0.20, 0.26, 0.19, 0.26, 0.35, 0.22, 0.21, 0.19, 6.73, 44.16, 68.87, 47.46, 64.38, 6.72, 9.36, 90.09. \\ GTCCs & 0.20, 0.28, 0.15, 0.16, 0.20, 0.31, 0.32, 0.19, 0.24, 4.19, 25.88, 45.77, 51.07, 73.55, 3.80, 6.54, 41.64. \\ TECCs & 0.18, 0.14, 0.17, 0.23, 0.14, 0.17, 0.13, 0.26, 0.22, 0.75, 4.78, 5.57, 4.25, 4.64, 0.84, 1.70, 4.35. \\ \hline \hline \end{tabular}
\end{table} 
Table 12: Standard deviation of Damage Index (DI) (sensor setup #0).