## 3 Results
### Fault Diagnosis
#### 3.2.8. Cepstrum Pre-Whitening (CPW)  
Unlike the mostly unsuccessful trials just reported, the use of the Cepstrum Pre-Whitening proves to be a powerful resource for the application under examination; it is a relatively new technique that has gained appreciation both in the literature [39] and in industrial practice [40], but it is not yet included in the most traditional and ever-invoked set of diagnostics techniques. It can be employed in two ways: the first one exploits the real cepstrum to perform the \"editing\", removing through liftering the peaks of the periodic components that correspond to impulses at particular harmonics; the second method, far more radical, sets the whole real cepstrum to zero except for the amplitude at the null quefrency, removing both resonances and discrete frequencies, and then resumes to time-domain after a recombination with the unwrapped phase of the original signal, neglected in the real cepstrum. This latter approach is actually easily implemented by dividing the signal Fourier transform by its absolute value and after inverse transforming it back to the time-domain:  
\[x_{CPW}(t)=F^{-1}\bigg{\{}\frac{F(x(t))}{|F(x(t))|}\bigg{\}} \tag{24}\]  
The conceptual and operative simplicity of this technique and its limited computational request are as appreciable as the results it provides. The only drawback is represented by the reduction of various orders of magnitude of all the amplitudes, even those that are useful; however, this aspect does not affect its efficacy since the desired components clearly stand out from the others and from noise, which are both significantly mitigated.  
Envelope spectra are indeed dominated by the frequency lines of the fault harmonic series: as for bearing 3, Figures 37 and 38 confirm the previously stated conclusion that the inner race fault harmonics are not visible directly but indirectly, through the excitation of those of the shaft rotation frequency that are near to them and of its even ones in general thanks to modulation effects; in the spectrum of bearing 4 (Figure 39, the cage frequency harmonics are undoubtedly present, as well as those of the BPFO, until the 11 th one, and those of the BSF are present to a lesser extent (until the 6 th one, barely visible), accompanied by the modulation sidebands at a distance equal to the FTF; it is important to note that the strong presence of the harmonics of the shaft and cage rotation frequencies should not be considered a masking factor to be removed--on the contrary, it is just related to faults and ought to be seen as their indirect sign, not present at the beginning of the experiment (Figure 40).  
Figure 37. CPW + SES applied to the signal from sensor 5 on bearing 3 (last 49 s) with harmonics of the BPFI drawn in magenta (fundamental frequency set to 301 Hz instead of 297 Hz).  
CPW has therefore been capable of deleting all the components unrelated to faults, including the harmonic series based on the fundamental frequencies at 160 Hz and 505 Hz and the peaks at very high frequency coming from structural resonances, leaving the three fault signatures thoroughly visible and unmistakably detectable.