## 3 Results
### Fault Diagnosis
#### 3.2.9 Cyclostationary Analysis  
Given that the analysis conducted so far has allowed us to perform the second phase of condition monitoring, namely the fault diagnosis, without taking into consideration the particular nature of the signal in question, for the sake of completeness it is worth trying to apply some techniques specifically formulated for the study of second-order cyclostationary signals, which are the ones actually produced by the damaged bearings. The purpose is twofold: (i) validating the previous conclusions by taking a substantially different point of view, and (ii) testing the efficacy of these tools, which are potentially more powerful but also more complex to understand, interpret and handle.  
It is well known that there is a point of connection between stationary and cyclostationary techniques, owing to the mathematical identity between the integral of the Spectral Correlation function made along all the spectral frequencies and the Squared Envelope Spectrum. It seems then reasonable to compute the Improved Envelope Spectrum (IES)  
Figure 40: CPW + SES applied to the signal from sensor 6 on bearing 3 (first 49 s).  
Figure 38: CPW + SES applied to the signal from sensor 6 on bearing 3 (last 49 s) with harmonics of the BPFI drawn in magenta (fundamental frequency set to 301 Hz instead of 297 Hz).  
Figure 39: CPW + SES applied to the signal from sensor 7 on bearing 4 (last 163 s) with harmonics of the BPFO drawn in red and of the BSF in green (fundamental frequency set to 282 Hz instead of 278 Hz).  
in order to verify that it offers information similar to that obtained with the SES and to understand the differences between them, if any.  
Three-dimensional chromatic maps displaying Spectral Correlation and Cyclic Spectral Coherence cannot be figured out immediately due to the abundance of information, but seen from above, i.e., projected on the frequency plane, they show a preferential arrangement of the peaks at cyclic frequencies \(\alpha\), which correspond to those of faults and their modulating ones (even the harmonics of the shaft rotation frequency and odd harmonics near to those of the BPFI for bearing 3 in Figures 41 and 42 and harmonics of the BPFO, FTF and BSF for bearing 4 in Figures 43 and 44) and at spectral frequencies \(f\), which are always exact multiples of 160 Hz or 80 Hz (320 Hz, 3120 Hz, 3760 Hz, 3680 Hz, 3840 Hz, 5280 Hz); these last frequencies were already observed as being related to system periodicities, although it is not possible to surely state their physical origin with the information provided on the dataset.  
Coming back to the dependence from a single variable, the IES (Figures 45 and 46), computed as the integral of the Cyclic Spectral Coherence on the spectral frequency band [0 10,000 Hz] (nearly more like an EES than an IES), puts the stress on the same spectral frequencies that represent fault signatures, even if with a bit less emphasis with respect to Figures 37, 38 and 40. This makes a lot of sense because the IES was applied on the raw signal while, in the previous paragraph, the signal was pre-processed via the radical technique of the CPW.  
Applying the previous cyclostationary techniques to excerpts of the original signal all along the experiment time duration (omitting the resulting plots for brevity), the fault history can be traced more precisely than it has been before: the inner race fault on bearing 3 occurs almost at the end of the test (33 rd-34 th day), while the outer race fault on bearing 4 can be detected from day 28 and that on the rolling element is still present from around the 26 th day and dominates the other, but it is already less visible in Figure 46, probably due to the progressive smoothing of the fault pit caused by friction.