## 2 Methodology and Band Indicators
* Average negentropy (mean of negentropy and spectral negentropy--infogram [20]): \[I(f,\Delta f)=\tfrac{1}{2}(N(f,\Delta f)+SN(f,\Delta f))=\tfrac{1}{2}\Big{(} ne\Big{\{}SE\Sigma x_{\{\ell,\Delta f\}}[n]\Big{\}}+ne\Big{\{}SESx_{\{\ell, \Delta f\}}[k]\Big{\}}\Big{)}\] (5) \[ne\{y[n]\}=\tfrac{1}{N}\sum_{n=1}^{N}\left[\tfrac{y[n]}{m_{1}\{y [n]\}}\right]\ln\left[\tfrac{y[n]}{m_{1}\{y[n]\}}\right]\]  
where \(ne\{y[n]\}\) is the generic definition of negentropy of a signal \(y[n]\) if the (square of the) instantaneous energy flow in the signal is interpreted as a probability distribution. * Kurtosis of the autocorrelation of the envelope (autogram [26; 27]): \[AK(f,\Delta f)=\tfrac{m_{4}\{R\left(E\Sigma x_{\{\ell,\Delta f\}}[n]\right)\} }{\big{(}m_{2}\big{\{}R\left(E\Sigma x_{\{\ell,\Delta f\}}[n]\right)\big{\}} \big{)}^{2}}\] (6) \[R(y[n])[k]=\tfrac{1}{n-k}\sum_{t=1}^{n-k}(y[t]-m_{1}\{y[n]\})(y[t+k ]-m_{1}\{y[n]\})\]  
where \(R(y[n])[k]\) is the definition of the autocorrelation for a signal \(y[n]\). * Kurtosis of the autocorrelation of the envelope spectrum (modified protrugram [28]): \[MP(f,\Delta f)=\frac{m_{4}\Big{\{}R\Big{(}ESx_{\{\ell,\Delta f\}}[k]\Big{)} \Big{\}}}{\Big{(}m_{2}\Big{\{}R\Big{(}ESx_{\{\ell,\Delta f\}}[k]\Big{)}\Big{\}} \Big{)}^{2}}\] (7)  
In order to evaluate the performance of such indices, two targeted a posteriori band indicators are proposed. These require the knowledge of the theoretical bearing-characteristic frequencies \(\hat{f}_{d}\) reported in Equation (8) from [2; 30], where \(BPFO\)/I indicates the outer or inner race defect frequency. \(BSF\) corresponds to the frequency of a defect on the rolling element, while \(FTF\) points to cage defects. \[BPFO=\tfrac{nf_{r}}{2}\Big{\{}1-\tfrac{d}{D}cos\varnothing\Big{\}} \qquad BPFI=\tfrac{nf_{r}}{2}\Big{\{}1+\tfrac{d}{D}cos\varnothing\Big{\}} \tag{8}\] \[FTF=\tfrac{f_{r}}{2}\Big{\{}1-\tfrac{d}{D}cos\varnothing\Big{\}} \qquad BSF=f_{r}\tfrac{Df}{2d}\bigg{\{}1-\Big{(}\tfrac{d}{D}cos \varnothing\Big{)}^{2}\bigg{\}}\]  
where \(d\) is the rolling element diameter, \(D\) is the mean race diameter, \(f_{r}\) is the shaft speed (relative speed among inner and outer ring), \(n\) is the number of rolling elements, and \(\varnothing\) is the angle of the load from the radial plane. Remembering that variations in the order of 1-2% from such theoretical values are common in actual bearings due to slips of the rolling elements, an interval of interest \(f_{int}\) for the bearing spectral signature can be found to range from 0.95 \(\cdot\)\(FTF\) to 3.05 \(\cdot\)\(BPFI\), as it is always wise to account for the first three harmonics of the defect frequencies due to the intrinsic non-linearities in the bearing response [29]. If the exact characteristic frequency of interest \(f_{d}\), identified within the interval \(f_{d-int}=0.95\cdot\hat{f}_{d}\div 1.05\cdot\hat{f}_{d}\) (N.B., \(\hat{f}_{d}\) is the ideal value), occurs at sample \(k_{d}\), the proposed a posteriori (i.e., derived after the demodulation [31]) indicators become:  
\[NA(f,\Delta f)=\frac{ESx_{\{\ell,\Delta f\}}[\text{k}_{\mathcal{A}}]}{\max_{k \in f_{int}}ESx_{\{\ell,\Delta f\}}[\text{k}]} \tag{9}\]\[SNR(f,\Delta f)=\frac{ESx_{\{\ell,\Delta f\}}[k_{\mathrm{d}}]}{\underset{k\in f_{d- int}}{\mathrm{cdf}}\left(ESx_{\{\ell,\Delta f\}}[k]\right)_{95\%}} \tag{10}\]  
where \(NA\) stands for normalized amplitude index, \(SNR\) means signal to noise ratio, and \(\underset{k\in f_{d-int}}{\mathrm{cdf}}\left(ESx_{\{\ell,\Delta f\}}[k]\right)_{9 5\%}\) represents the 95 th percentile of the amplitudes of the envelope spectrum in the range \(f_{d-int}\). The main issue that remains is related to the identification of the exact \(f_{d}\) in the range \(f_{d-int}\). For this purpose, a sort of stabilization diagram is proposed in this work.