## 2 Methodology and Band Indicators  
In this paper, the bearing diagnostics is regarded as an optimization problem based on the vibration signal \(x[n]\), assumed with a null mean value. The goal was to find the best combination of center frequency and band size (f,Af) for filtering such a signal before envelope demodulation so that the damage-characteristic signature is maximized in the resulting SES. According to the FK and most of the subsequent 2-D color maps found in the literature, the (f,Af) plane is partitioned with a multirate filter bank, so that each part of the paving has a corresponding filtered signal \(x_{\{[\ell,\Delta\ell]\}}[n]\) whose envelope \(Ex_{\{[\ell,\Delta\ell]\}}[n]=|x_{\{[\ell,\Delta\ell]\}}[n]+i\;\mathcal{H}(x_{ \{[\ell,\Delta\ell]\}}[n])|\) can be easily estimated. Note that i stands for the imaginary unit and \(\mathcal{H}\) means Hilbert transform. In most of cases, the Squared Envelope \(SEx_{\{[\ell,\Delta\ell]\}}[n]=(Ex_{\{\ell^{*},\Delta\ell^{*}\}}[n])^{2}\) is actually computed, as this can in principle reduce the demodulation aliasing due to the cusps in the \(|\sin\theta|\) function (i.e., the Fourier basis) [29]. Nevertheless, the optimization is carried out by maximizing a band indicator (i.e., the kurtosis) over the entire paving so as to find the optimal \(\{\ell,\Delta\ell\}=\{\ell^{*},\Delta\ell^{*}\}\). Finally, the envelope demodulation is carried out by computing the spectrum of the signal filtered in the optimal \(\{\ell^{*},\Delta\ell^{*}\}\) to obtain an envelope spectrum: \(ESx_{\{[\ell^{*},\Delta\ell^{*}\}}[k]=|\mathrm{DFT}\{SEx_{\{[\ell^{*},\Delta \ell^{*}\}}[n]\}|\)\) or a squared envelope spectrum: \(ESx_{\{[\ell^{*},\Delta\ell^{*}\}}[k]=|\mathrm{DFT}\{SEx_{\{[\ell^{*},\Delta \ell^{*}\}}[n]\}|^{2}\), as in most of situations this improves the signal to noise ratio (\(SNR\)) [29]. According to the literature review in Section 1, the main blind band-selection indicators are reported:  
* Spectral kurtosis (kurtogram [4; 5; 6]): \[ \begin{array}{l}K(f,\Delta f)=\frac{m_{4}\big{\{}Ex_{\{[\ell,\Delta\ell]\}} [n]\big{\}}}{\big{(}m_{2}\big{\{}Ex_{\{[\ell,\Delta\ell]\}}[n]\big{\}}\big{)} ^{2}}\\ m_{\tau}\{y[n]\}=\frac{1}{N}\sum_{n=1}^{N}(y[n])^{\tau}\end{array} \] (1) where \(m_{\tau}\{y[n]\}\) represents the \(r\)th order moment of a generic discrete signal \(y[n]\). * Protrusion (protrugram [10]): \[ \begin{array}{l}P(f,\Delta f)=\frac{m_{4}\Big{\{}ESx_{\{[\ell,\Delta\ell]\}} [n]\Big{\}}}{\Big{(}m_{2}\Big{\{}ESx_{\{[\ell,\Delta\ell]\}}[n]\Big{\}}\Big{)} ^{2}}\end{array} \] (2)
* Spectral \(L2/L1\) sparsity (sparsogram [12]): \[ \begin{array}{l}L_{2}/L_{1}(f,\Delta f)=\frac{L_{2}\big{\{}ESx_{\{[\ell,\Delta \ell]\}}[k]\big{\}}}{L_{1}\big{\{}ESx_{\{[\ell,\Delta\ell]\}}[k]\big{\}}}= \frac{\sqrt{\sum_{k=1}^{k-\frac{\ell^{*}}{2}}\big{(}ESx_{\{[\ell,\Delta\ell]\}} [k]\big{)}^{2}}}{\sum_{k=1}^{k-\frac{\ell^{*}}{2}}\big{|}ESx_{\{[\ell,\Delta \ell]\}}[k]}\\ L_{\tau}\{y[n]\}=\Big{(}\sum_{n=1}^{N}(|y[n]|)^{\tau}\Big{)}^{1/r}\end{array} \] (3) where \(L_{\tau}\{y[n]\}\) is the mathematical definition of the \(r\)-norm of a signal \(y[n]\). * Spectral Gini sparsity (Gini sparsogram [19]): \[SGI(f,\Delta f)=1-2\sum_{k=1}^{K=\frac{f_{r}}{2}}\frac{\overset{\rightarrow}{E} \Sigma x_{\{\ell,\Delta f\}}[k]}{L_{1}\Big{\{}E\Sigma x_{\{\ell,\Delta f\}}[k] \Big{\}}}\bigg{(}\frac{K-k+1/2}{K}\bigg{)}\] (4) where \(\overset{\rightarrow}{E}\Sigma x_{\{f,\Delta f\}}[k]\) is the envelope spectrum ordered from the smallest to the largest value.