## Materials and Methods
Figure 3a,b shows an example of the autocorrelation function of two signals related to conditions L1 and NL1, respectively, measured by sensor Aax in Test-2 and normalized with respect to its value at zero lag (\(\tau\) = 0):  
\[\hat{R}_{xx,norm}(\tau)=\frac{1}{\left|\hat{R}_{xx}(0)\right|}\hat{R}_{xx}(\tau), \tag{5}\]  
Hence being:  
\[\hat{R}_{xx,norm}(0)\equiv 1. \tag{6}\]  
Figure 3c shows the normalized autocorrelation of a synthesized Gaussian white noise signal having the same number of samples, \(N_{S}\), of the measured signals (namely, \(N_{S}\) = 51,200). It is worth noting that the normalized autocorrelation function of Gaussian white noise is not affected by the signal amplitude. The shape of the function characterizing condition L1 (Figure 3a) appears to be more similar to the one observed in Figure 3c than the shape characterizing condition NL1 (Figure 3b). Hence, the similarity between the shape of the normalized autocorrelation associated with the measured signals and the case of Gaussian white noise may be adopted for assessing the presence of leaks. The observed phenomenon may be qualitatively interpreted with the following considerations. The zero-lag value of the autocorrelation represents the signal energy (as it can be inferred from [PERSON]'s theorem [24]). Energy is higher for the leak-related signals than for the non-leaking condition measurements, and it is distributed over a wider frequency range as well. Hence, the corresponding normalized autocorrelation functions are characterized by lower side bands, which are further reduced by normalization. Figure 3: Normalized autocorrelation function of: (**a**) Condition L1 acceleration signal (Test-2, radial accelerometer sensor (Aax), 2 nd acquisition of the dataset); (**b**) Condition NL1 acceleration signal (Test-2, sensor Aax, 2 nd acquisition of the dataset); (**c**) Gaussian white noise. The kurtosis value (\(K_{R}\)) of the normalized autocorrelation function is reported in the upper right corner. In order to quantitatively estimate such similarity, the kurtosis of the normalized autocorrelation (referred to as \(K_{R}\)) is computed using the following equation:  
\[K_{R}=kurt\big{(}\hat{R}_{xx,norm}(\tau)\big{)}, \tag{7}\]  
where  
\[kurt(\mathbf{y})=\frac{\frac{1}{N}\sum\limits_{i=1}^{N}(y_{i}-\overline{y})^{4 }}{\bigg{(}\frac{1}{N}\sum\limits_{i=1}^{N}(y_{i}-\overline{y})^{2}\bigg{)}^{2 }}, \tag{8}\]  
where \(\mathbf{y}\) is a generic vector of length \(N\) and has a mean value \(\overline{y}\). It is observed that the value of \(K_{R}\) also depends on the length of the autocorrelation function, \(N_{R}\). For example, it can be easily proven that for the limit case of a normalized autocorrelation function, having only null terms except for the zero-lag value, i.e. :  
\[\hat{R}_{xx,\text{lim}}(\tau)=\left\{ \begin{array}{ll}1&if\tau=0\\ 0&if\tau\
eq 0\end{array} \right., \tag{9}\]  
the kurtosis is provided by:  
\[kurt\big{(}\hat{R}_{xx,\text{lim}}(\tau)\big{)}=\frac{N_{R}^{2}-3N_{R}+3}{N_{ R}-1}\cong N_{R}-2. \tag{10}\]  
Therefore, the possible influence of \(N_{S}\) on the leak detection performance was also investigated. For this purpose, for each condition and each transducer, signals with extended duration were generated by aggregating the available acquisitions, i.e., by simply concatenating the arrays of the measured data. In particular, two signals of duration \(T_{s}\) = 30 s were created, each one resulting from the aggregation of three acquisitions. Moreover, a single signal with \(T_{s}\) = 60 s was generated by aggregating all the available acquisitions. Then, the \(K_{R}\) values associated with the extended signals were computed and compared to the values characterizing the original acquisitions. Signal processing was carried out by using Mathworks Matlab R2017b.