## 2 Materials and Methods
### Damage Detection Analysis
SK (Equation (12)) relies on the use of short-time Fourier transform (STFT) of Equation (13) to give a measure of the signal impulsiveness contained in frequency bands being examined. \[SK=\frac{\left\langle\left|X^{4}(f,t)\right|\right\rangle}{\left\langle\left|X (f,t)\right|^{2}\right\rangle^{2}}-2 \tag{12}\]  
\[X(f,\tau)=\int_{-\infty}^{+\infty}x(t)w(t-\tau)e^{-j2\pi ft}dt \tag{13}\]  
where \(\left\langle\cdot\right\rangle\) is the time-averaging operator, \(X(f,t)\) is the time-frequency transform of the signal \(x(t)\), \(w(t-\tau)\) is a time window and \(j\) is the imaginary unit. The time-frequency representation of the signal is obtained by means of STFT, which, differently from FFT, is capable of localizing the harmonic contribution in time, since it is essentially a FFT that slides in time thanks to the window \(w(t-\tau)\). SK is expected to be null for complex signals \(X(f,\tau)\) originating from gaussian noises, whereas high values of SK suggest nonstationarity or non-gaussian  
Figure 7: FFT raw spectra: (**a**) vibration signal, (**b**) pressure signal, (**c**) vibration signal up to 5 kHz and (**d**) pressure signal up to 5 kHz. properties. The size of the chosen window influences the SK computation, since STFT is affected by the well-known indetermination principle. Namely, the more accurate the time representation (short window), the worse the frequency resolution. For this reason, FK (Figures 8-10) represents SK as a function of the window lengths. Therefore, FK provides information not only on the optimal demodulation band but, also, on the choice of the window length to be applied for the SK computation. Then, FK acts with a simplified algorithm, which avoids complete STFT computation. Once signals were band pass-filtered in the frequency bands suggested by SK, the envelope analysis was applied. The modulus of the analytic signal was then investigated in the frequency domain (Figure 11). The x_BPFI_ harmonic contributions were particularly examined. Unlike raw spectra, x_BPFI_ harmonics were fairly identifiable even in the FE signal (Figure 12). Figure 8: Kurtosis: (a) vibration signal at 0.021 in and 3 hp and (b) pressure signal at 0.021 in and \(P_{u}\)/2. Figure 9: Kurtosis: vibration signals. The three columns refer, respectively, to 0.007, 0.014 and 0.021 in. The four rows refer, respectively, to 0, 1, 2 and 3 hp. \[CF=\frac{x_{peak}}{RMS} \tag{10}\]0 Envelope signal and envelope spectrum: (**a**) vibration signal at 0.021 in and 3 hp and (**b**) pressure signal at 0.021 in and \(P_{ul}\)/2. In red, the first ten x_BPFI_ harmonics. _BPFI_: Ball Passing Frequency on the Inner race. \[CF=\frac{x_{peak}}{RMS} \tag{10}\]1 Envelope signal and envelope spectrum. Fan end (FE) signal with damage on DE bearing. In red, the first ten x_BPFI_ harmonics. \[CF=\frac{x_{peak}}{RMS} \tag{10}\]2 Kurtograms: pressure signals. The three columns refer, respectively, to 0.007, 0.014 and 0.021 in. The three rows refer, respectively, to \(P_{ul}\)/8, \(P_{ul}\)/4 and \(P_{ul}\)/2. It is important to underline that the filtering process reduced the range in which the signals fluctuated. Moreover, this effect was different for each signal, given different filtering bands. Therefore, damage detectability cannot purely be assessed on the basis of the absolute value of the harmonic contributions but, rather, on their identifiability. Envelope spectrum was expected to show x_BPFI_ harmonics, with sidebands spaced at the shaft rotation frequency. Clearly, this was mostly visible in the pressure signal, where no disturbance sources (noise or discrete harmonic components) were introduced. The x_BPFI_ harmonic contributions were analyzed for both signal envelopes and for all the damage levels and loads. However, none of these features for the enveloped and nonenveloped signals enabled a direct comparison between the diagnostic contents of the pressure and of the vibration signal, since these quantities, though linked, were intrinsically different. In an attempt to overcome this limitation, the novel step was introduced.