## 2 Theoretical Background
### Cepstral Coefficients (CCs) and Mel-Frequency CCs  
The Cepstral Coefficients (CCs) can be extracted from the cepstrum of a signal. In the field of speech and sound processing, the mel-scale is intensively used for this purpose, by rearranging the cepstrum in a nonlinear log-like scale of frequency. The idea is generally credited to the works of [PERSON] [44] and [PERSON] [45], and it has since then been developed to become a standard in speech processing.  
The mel-scale is a purely empirically definition, being intended to mimic the auditory system of the human hear, which is a nonlinear sensing device with almost logarithmically spaced intervals of equivalent perception. The definition of its base unit, the Mel, is one-thousandth of the pitch (\(\varphi\)) of a simple tone with a 40 dB amplitude above the auditory threshold and a frequency that is equal to 1000 Hz [38; 46]. The reason behind the thresholding lies in several experimental investigations (such as [47] to name one), which proved that speech processing linear-like for small spectral values and logarithmic-like for large ones yield to more robust identifications with data affected by additive or convolutional artificial noise. Obviously, this definition is very specific, perceptual, and suited for the aim of emulating the human ear. Nevertheless, this was found from previous studies to be apt for SHM purposes; this might be explained by its logarithmic-like spacing, which is recurrent in many natural events not limited to speech production and perception [21]. Another point of relevance is that the definition itself is perceptual, psychoacoustical, and based on perceived equidistance between subsequent pitches; thus, it is not directly provided with a mathematical formulation. Of the many concurrent analytical definitions for the Mel-Hertz conversion, the most commonly accepted one is the variant proposed by [PERSON] in [48]. Other alternative definitions exist; a comparative study was performed by [PERSON] et al. in [49].  
The procedure to extract MFCCs from a non-stationary recorded signal of interest are the following: (i) framing of the signal into more stationary tracts; (ii) windowing of the frames; (iii) computation of the Power Spectral Density (PSD) in the frequency domain; (iv) frequency warping to re-order the discrete power spectrum in the melody-frequency scale (Mel-scale); and finally, (v) L-points inverse Discrete Cosine Transform (iDCT). This last passage can be also performed by means of inverse Discrete Fourier Transform (iDFT), but iDCT is more technically convenient as more adherent to the Karhunen-Loeve Transform. When considering (arbitrarily) \(L-1\) coefficients to be extracted and \(M\) critical bands, the resulting array of MFCCs, \(\mathbf{c}\in\mathbb{R}^{L\times 1}\), can be defined as  
\[\begin{split} c[l]=\sum\
olimits_{m=0}^{M-1}\!\!a_{m}\log(H_{ \text{mel}}[m])\cos\Bigl{(}\frac{n(2l+1)m}{2M}\Bigr{)},\\ l=0,1,\ldots,L-1,\;a_{m}=\left\{\begin{array}{l}\frac{1}{M} \;if\;m=0\\ \frac{2}{M}\;if\;m\geq 1\end{array}\right.\end{split} \tag{3}\]  
where \(a_{m}\) is the amplitude multiplying the m-th point of the mel-spectrum \(H_{\text{mel}}[m]\). The procedure is illustrated for the example of frame #5 in Figure 1.