## 1 Introduction  
Many studies in recent years have been devoted to the development of diagnostics techniques able to assess machinery health conditions from vibration data. The past twenty years have seen increasingly rapid advances in this field, hence the needs of the growing interest in condition-based monitoring (CBM) have met an ever wider range of monitoring strategies [1]. CBM plays a key role in the industrial context, where machines run for long periods and the long-term economic advantages related to predictive maintenance are significant, especially if compared to preventive maintenance ones [2]. Indeed, even though predictive maintenance requires additional instrumentation, such as sensors, acquisition systems and data elaboration tools, machine conditions are known at any time by means of health indicators properly constructed from the available data. Therefore, the remaining useful life (RUL) can be estimated on the basis of the current conditions, thus saving costs and time with respect to time-based preventive maintenance strategies that mostly rely on a priori time-to-failure assessments. In these cases, some components such as rolling element bearings may suffer a large statistical spread around the mean fatigue life. Consequently, conservative preventive maintenance strategies lead to an excessive number of replacements of components that could run even two or three times longer [1,3]. Then, the availability of CBM techniques has become ever wider, both in terms of physics-based models and data-driven models. The latter have rapidly risen thanks to the spread of machine-learning and deep-learning algorithms [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18] assisted by the increase of the computational power. Among CBM methodologies, the existing research recognizes the applicability of acoustic emission analysis (AE) [19, 20, 21], vibration analysis [22, 23, 24, 25, 26, 27, 28] and lubricant analysis [1] in the field of bearing diagnostics and prognostics. In particular, a considerable amount of literature has been published on vibration-based condition monitoring (VBCM) for bearings damage detection. This is justified by the fact that bearings are key nodes of complex mechanical systems, since their vibrations are affected by the health state of other components, but also, bearing dynamics influence the whole machine behavior [29, 30]. In this scenario, it is now well-established the effectiveness of envelope analysis for bearing fault detection. Actually, envelope analysis, initially introduced with the name of high-frequency resonance technique [31, 32, 33, 34, 35, 36], is able to emphasize peculiar fault harmonics in the vibration spectrum, taking advantage of Hilbert transform. Basically, the modulus of the complex signals whose imaginary part is the Hilbert transform and real part is the signal itself constitutes the Hilbert envelope. The fundamental hypothesis behind this methodology lies in the fact that the vibration signal generated by damaged bearings is essentially an amplitude modulated signal where characteristic fault frequencies and rotation frequency act, respectively, as carrier and modulating components. Hence, amplitude demodulation and cyclostationary analysis have shown to be successful in many fault detection problems [22, 23, 24, 37, 38, 39, 40, 41, 42, 43]. In the pursuit of these tasks, [PERSON] and [PERSON] suggested the spectral kurtosis (SK) and the fast kurtogram (FK) as powerful tools for the choice of the optimal filtering band for signal demodulation [44, 45, 46, 47]. The underlying assumption is that Kurtosis is a viable yardstick of transient impulses, and it is able, insofar, to detect the frequency bands containing most of the nonstationarity. Actually, the diagnostic content related to damaged kinematic frequencies is hidden in the high-frequency bands of the raw spectra, where the machine structural resonances are excited by impulses related to the striking of rolling elements.