## 2 IMS Bearing Dataset and Signal Analysis Techniques
### Signal Analysis Techniques
Moreover, these three parameters can be also derived in the frequency domain, starting from the spectral moments \(m_{n}\) of the time signal \(x(t)\)[36]:  
\[\text{Act}(x)=\sigma^{2}(x)=\lim_{T\to\infty}\frac{1}{T}\int_{0}^{T}x^{2}(t)dt =m_{0}=\int_{-\infty}^{+\infty}S(\omega)d\omega\qquad\left[\text{m}/\text{s}^ {2}\right]^{2} \tag{6}\]  
\[\text{Mob}(x)=\sqrt{\frac{\text{Act}\big{(}\dot{x}\big{)}}{\text{Act}(x)}}= \sqrt{\frac{m_{2}}{m_{0}}}\qquad\quad\left[\text{t}^{-1}\right] \tag{7}\]  
\[\text{Com}(x)=\sqrt{\frac{\text{Mob}\big{(}\dot{x}\big{)}}{\text{Mob}(x)}}= \sqrt{\frac{m_{4}/m_{2}}{m_{2}/m_{0}}}\qquad\quad\left[\ \right] \tag{8}\]  
where \(S(\omega)\) is the power spectral function, defined as the product of the Fourier transform \(F(\omega)\) and its complex conjugate function \(F^{*}(\omega)\):  
\[S(\omega)=F(\omega)\cdot F^{*}(\omega)=\int_{-\infty}^{+\infty}x(t)\cdot x^{* }(t)dt \tag{9}\]  
and the generic spectral moment of order \(n\) can be computed in the form:  
\[m_{n}=F(\omega)\cdot F^{*}(\omega)=\int_{-\infty}^{+\infty}\omega^{n}\cdot S( \omega)d\omega \tag{10}\] \begin{table}
\begin{tabular}{c c c} \hline
**Parameter** & **Abbrev.** & **Formula** \\ \hline Mean & Mean & \(\frac{\Sigma\cdot X}{N}\) \\ \hline Variance & Var & \(\frac{\Sigma(-X)^{2}}{N-1}\) \\ \hline Standard Deviation & STD & \(\sqrt{\text{Var}}\) \\ \hline Root Mean Squared Value & RMS & \(\sqrt{\frac{\Sigma\cdot X^{2}}{N}}\) \\ \hline Skewness & Skew & \(\frac{\text{t}}{\text{s}}\frac{\Sigma(X-X)^{3}}{\text{SID}^{3}}\) \\ \hline Kurtosis & Kurt & \(\frac{\text{t}}{\text{s}}\frac{\Sigma(X-X)^{4}}{\text{SID}^{4}}\) \\ \hline Peak & Peak & Max(\(|X|\)) \\ \hline Crest Factor & CF & Peak \\ \hline Clearance Factor & CL & \(\frac{\text{Peak}}{\left(\frac{\Sigma\cdot X}{N}\right)^{2}}\) \\ \hline Shape Factor & SF & \(\frac{\text{RMS}}{\text{Mean}}\) \\ \hline Impulse Factor & IF & Peak \\ \hline Peak-to-Peak & P2P & Max\((X)-\text{Min}(X)\) \\ \hline \end{tabular}
\end{table} 
Table 4: Statistical parameters. As for their interpretation in the field of predictive maintenance, Activity and Complexity are proportional to the amplitude of the fault frequency components, while Mobility is expected to reduce with fault progression. In order to sum up the informative contribution of [PERSON]'s parameters into a single value, another quantity, called Detectivity (Detect), was introduced in [37] and defined as:  
\[\text{Detect}=\text{Act}_{(dB)}-\text{Mob}_{(dB)}+\text{Com}_{(dB)} \tag{11}\]  
where:  
\[\text{Act}_{(dB)}=10\cdot\log_{10}\bigg{(}\frac{\text{Act}}{\text{Act}_{\text {ref}}}\bigg{)} \tag{12}\]  
\[\text{Mob}_{(dB)}=10\cdot\log_{10}\bigg{(}\frac{\text{Mob}}{\text{Mob}_{\text {ref}}}\bigg{)} \tag{13}\]  
\[\text{Com}_{(dB)}=10\cdot\log_{10}\bigg{(}\frac{\text{Com}}{\text{Com}_{\text {ref}}}\bigg{)} \tag{14}\]  
The three terms in the right member of (11) are expressed in Decibel, after being normalized by computing the ratio between their value on the signal to be diagnosed and a reference value coming from their application to the data collected on the faultless component. Mobility behaves as an attenuator due to its inverse trend with respect to the evolution of the fault, while Activity and Complexity behave as amplifiers. Due to the particular nature of the vibration signals produced by damaged rolling bearings, which are classified as quasi-pseudo-cyclostationary of the second order, it is convenient to use dedicated techniques, specifically formulated for their treatment.