## 1 Introduction  
Condition monitoring is today a crucial and unavoidable practice related to the use of widespread mechanical components as rolling element bearings (REBs), which play a fundamental role within industrial rotating machineries to support moving parts, transmit motion and power and reduce friction among them. As such, they experience constant stress and are seriously prone to suffering from wear, which ultimately results in faults. If neglected, a damaged bearing is likely to worsen and affect adjacent components due to physical contact, affecting the functioning of the whole system and turning into a catastrophic failure, imposing an immediate stop of the production or service with deleterious economical consequences. It is therefore necessary to prevent such a harmful situation, identifying in advance characteristic signs of its incoming occurrence, without waiting for defects to become visible, i.e., already at an advanced stage, as has been the case in the past with the run-to-break approach, or relying on periodic replacements carried out blindly, as in time-based maintenance. Thanks to the use of more powerful sensors and technologies, in recent years predictive maintenance has established itself as an effective discipline to real-time trace the healthy state of mechanical parts and find out the best economical time to execute maintenance intervention. Symptoms of different natures can be taken into consideration, from thermal and pressure fields, noise, and IR emissions to current consumption and lubricant-based analysis, but the most common and effective approach is that based on the study of bearing vibrational signals. Vibration-based condition monitoring is a dynamic research field that is performed with tools susceptible to constant progress; nowadays, besides the mathematical signal-based techniques, new data-driven approaches emerge as promising methods. The latter are particularly appreciated for fault prognosis, which is the third and last phase involved in the process, aiming to predict the future evolution of the vibration signal beyond the present state, and therefore the remaining useful time of the mechanical component; the two phases that precede prognosis are fault detection, which consists in defining the component's level of healthiness, namely the potential presence of faults, followed by fault diagnosis, which allows the specific sub-component affected by faults and thus their particular nature to be properly identified. The main disadvantage regarding Machine Learning techniques is that they require a huge amount of data to be trained on, are not always as generalizable as we would like them to be and may have onerous computational demands. Despite the diffusion of new data-driven techniques to analyze vibrational signals and detect the health state of the system, mathematical tools, deriving more or less directly from the Fourier transform and defined either on the time or the frequency domain or on a combination of them (i.e., signal-based techniques), remain still essential and irreplaceable in light of the deep and detailed information they provide. We refer to dedicated books [1; 2] and articles [3; 4] for an exhaustive description of the main signal-based techniques used for diagnostic purposes, including most of those applied in this work. In particular, in [3], which is devoted to the diagnostics of REBs, it is proven that signal-based techniques, thanks to their analytical definition, which is formulated taking into consideration to some extent the mathematical nature of the signal, allow fault signatures to be clearly identified, and their temporal development to be fully traced, confirming their ability as powerful means to constantly monitor the health state of a system even in real-time applications. Other interesting recent articles on the diagnostics of REBs can be found in [5; 6; 7; 8; 9].