## 1 Introduction  
The adequate condition monitoring of bearing vibrations enables the reliable operation of mechanical equipment under complex working situations. To this end, health-condition monitoring assesses the development of internal fluctuating vibration amplitudes generated by bearing failures that deteriorate rotating machinery [1]. Vibration monitoring is widely employed to detect bearing damage of incipient failures in gas, steam, and wind turbines [2, 3, 4], and electric vehicles [5]. In domain-based vibration monitoring, signal-processing methods that effectively synchronously obtain information from time and frequency domains are preferred to deal with nonstationary data. Among time-frequency methods, the following are the most prominent: Empirical mode decomposition that has some limitations, such as border effect evidence, unreliable stopping criterion, and low spectral resolution (separating closely spaced spectral components) because of mode mixing [6]; wavelet transforms that have an energy-leakage issue for which mother wavelet selection must be appropriate because the contents of the daughter wavelets need to be closely matched with each considered signal [7]; short-time Fourier transform (STFT), with the constraint of nonreachable simultaneous best time and frequency resolution [8]. However, the STFT method is widely adopted in preprocessing nonstationary vibration signals to discard redundant information and present more powerful fault characteristics [9]. To date, several moment-based values were reported to measure the damage degree of rotating machinery to fault spectrum sparsity or envelope spectra exhibiting discrete peaks at corresponding fault frequencies. In particular, kurtosis (also related to \(\ell_{1}/\ell_{2}\) norms [10]) measures the spikiness of a vibration signal [11], better estimating the resonance frequency band of bearing impulsive failures [12; 13]. However, kurtosis is sensitive to impulsive noise structure, so that it tends to decrease in value when the frequency of the impulses increases or impulses overlap [14]. Sparse filtering with generalized \(\ell_{p}/\ell_{q}\) norms [15] was also employed with more flexible ruling parameters (\(p\) and \(q\)) to characterize bearing faults [16]. However, a large value of \(p\) measures sensitivity to outliers, while a small \(p\) makes the measure ineffective in sensing sparsity [17]. Entropy is another useful tool in detecting the dynamic characteristics of nonlinear vibration signals, providing robustness to noise and adequate cluster ability, independent of prior knowledge [18]. Estimation types commonly used in fault diagnosis include approximate energy entropy and sample entropy [19], which may be dependent on intrinsic parameters and have limited ability to capture long-term correlations. To cope with this issue, a multiscale approach is proposed that shortens the time series at higher scales, which results in higher fluctuations of entropy values at scale factors [20]. Permutation entropy can also be applied in the analysis of vibration signals [21]. However, entropy methods tend to exclude high-frequency information, yielding limited diagnostic performance in identifying bearing health states [22]. Generally, the STFT approach depends on the length of the applied window and its shape. In this regard, bearing-fault analysis uses a rectangular window with a fixed length with the limitation of power leakage and insensitive tracking of impact signal components [23]. This difficulty mainly concerns the event identification of highly localized time samples. To cope with this issue, the sliding SFFT approach of bearing-fault waveforms is often computed (sliding entropy [24] or sliding kurtosis [25]). Although the sliding strategy enhances the time resolution of measures, it may compromise the quality of probability function estimation due to fewer data being available within the working window [26]. Here, we introduce an approach to highlighting impulsive bearing failures.