## 4 Results and Discussion
### Acquisition IR014_2 (176 FE)  
Acquisition IR014_2 (176 FE) was taken into account as it is a good and clean acquisition and can represent a case well in which it should be relatively easy to find a good demodulation band. In fact, the value of \(dE\) = 0.71 is quite high, as the demodulation is effective in most of the considered bands, as can be observed in Figure 4.  
\begin{table}
\begin{tabular}{l l l l l l} \hline \hline \multirow{2}{*}{**Location**} & \multirow{2}{*}{**Name**} & \multicolumn{3}{c}{**Fault Frequencies (Multiple of Shaft Speed)**} \\ \cline{3-6}  & & **BPFI** & **BPFO** & **FTF** & **BSF** \\ \hline Drive End & 6205-2 RS JEM & 5.415 & 3.585 & 0.3983 & 2.357 \\ \hline Fan End & 6203-2 RS JEM & 4.947 & 3.053 & 0.3816 & 1.994 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Bearing fault frequencies details (BPFI/O: inner/outer race, FTF: cage, BSF: ball).  
Figure 3: Time series of the 4 cases of interest: 176 FE (**top left**), 275 DE (**top right**), 177 FE (**bottom left**) with a zoom over the anomaly, 222 DE (**bottom right**).  
\begin{table}
\begin{tabular}{l l l l l l l} \hline \hline \multirow{2}{*}{**Name**} & \multirow{2}{*}{**Code**} & \multicolumn{2}{c}{**Acc.**} & \multicolumn{1}{c}{**Damage**} & \multicolumn{1}{c}{**Damage**} & \multirow{2}{*}{**Fs**} & \multirow{2}{*}{**Details**} \\  & & **Location** & & **Location** & & **Size** & \\ \hline IR014\_2 & 176 & FE & DE-IR & 0.014\({}^{\prime\prime}\) & 48 ksps & Good Acquisition \\ IR014\_1 & 275 & DE & FE-IR & 0.014\({}^{\prime\prime}\) & 12 ksps & Impulsive noise \\ IR014\_3 & 177 & FE & DE-IR & 0.014\({}^{\prime\prime}\) & 48 ksps & Electrical noise \\ B021\_0 & 222 & DE & DE-B & 0.021\({}^{\prime\prime}\) & 12 ksps & Non-periodic impulses \\ \hline \hline \end{tabular}
\end{table}
Table 2: Details of the selected acquisitions (DE: drive end, FE: fan end, IR: inner race, B: ball).  
Once the exact characteristic frequency of the damage of interest \(f_{d}\) is established according to the procedure in Section 2, the section of the bispectral maps at \(f_{d}\) represent the \(N\)\(A\) and the \(SNR\) (i.e., the two proposed a posteriori indicators). It is now easy to compute the effectiveness of the different band indicators, whose frequency trend is shown in Figure 5. In the legend of the picture, the two ingredients of \(iE\) are in fact reported. Their average, which represents the indicator effectiveness according to Equation (11), are summarized in Table 3. For this particular case it is easy to notice that the spectral kurtosis (i.e., the indicator of the kurtogram, \(K\)), the average negentropy (\(I\), the indicator of the infogram), and the indicator of the autogram (\(AK\)) show the best results, followed by the Spectral Gini Index (\(SGI\)). The protrugram indicator (\(P\)), the \(L2/L1\) norm indicator (\(L2/L1\)), and the modified protrugram indicator (\(MP\)), on the contrary, had poor performance (even if in this simple case a weaker peak could be still found, as visible in Figure 4).  
Figure 4: Bispectral demodulation maps of normalized amplitudes and signal to noise ratio given a fixed \(\Delta f\) while the center frequency is increasing from 0 to half the sampling frequency. In the first picture, the average curve is displayed in red and the resulting exact damage frequency \(f_{d}\) is highlighted by the red triangle.  
Figure 5: 176 FE traditional band indicators as a function of the center frequency \(f\), compared to the proposed \(NA\) and \(SNR\) a posteriori indicators. The maximum of each indicator is highlighted by a circle, and the value of \(NA\) and \(SNR\) at that point are reported in the legend. The average of such two values produces the \(iE\), which is summarized in Table 3.