## 1 Introduction  
Rotating machinery are fundamental components of mechanical systems for most of the industrial applications, as mechanical power is commonly obtained in the form of torque at a rotating speed. Electrical motors, internal combustion engines and motors in general, in fact, convert some sort of source energy into mechanical energy which is transferred to the final user through a mechanical transmission (e.g., a gearbox) which provides speed and torque conversion.  
The speed is then a piece of essential system information, usually kept under control to accomplish a determined work. As a general consideration, feedback controllers can easily maintain their process variable (i.e., the speed in this case) close to its setpoint, but noise and disturbances may affect the controlled system instantaneous output causing the variable to depart from and oscillate around its desired value. The appearance of damages in the system has then repercussions on the instantaneous speed, which can be used as a source of information of the machine health condition per se.  
The information about the instantaneous speed is fundamental for machine diagnostics also in other ways. In the field of Vibration Monitoring (i.e., a particularly successful kind of condition monitoring based on vibration records) for example, it is very common to use measured or estimated information about the Instantaneous Angular Speed (IAS) to perform the so-called Order Tracking [1]. In fact, most of rotating machines are affected by phenomena which are locked to particular angular positions (e.g., the intake of a 4 strokes diesel engine lasts from 0 to \(\pi\) radians of the main shaft and is periodic of \(4\pi\); a gear featuring a wheel with \(M\) teeth whose \(N^{th}\) tooth is damaged features an anomalous meshing pattern at angle \(\frac{2\pi N}{M}\) of the wheel's shaft with a periodicity of \(2\pi\)). Furthermore, in the case of non-stationary acquisitions, the spectral signature of the machine can only be found after synchronization of the Fourier analysis on the shaft rotation (i.e., the spectrum is not given as a function of the frequency, but as a function of the orders of a reference shaft). Unfortunately, commercial Data Acquisition Systems (DAQ) are not efficient in sampling at constant angular increments of the reference shaft, so that resampling algorithms are used, based on the additional information of the angular position of the reference shaft over time [2; 3; 4; 5].  
The growing interest of the signal processing community in the IAS information is demonstrated by the increasing number of publications involving the estimation of the instantaneous frequency as well as the creation of a special issue on IAS processing in the Journal of Mechanical Systems & Signal Processing [6] and of a conference dedicated solely to condition monitoring in non-stationary operations [7]. In addition, at the international conferences Surveillance 8 (in 2015) [8] and SURVISHNO (in 2019), contests about IAS estimation were organized. The IAS is in effect spreading in many fields of diagnostics of rotating machinery, especially for motors and gearboxes.  
Particular applications for motors diagnostics involve electrical motors [9; 10; 11], internal combustion engines [12; 13; 14; 15; 16], and hydraulic engines too [17].  
Gearbox diagnostics was also covered from a general point of view [18; 19; 20; 21] as well as considering particular applications (e.g., windmills [22]) or focusing on relevant components such as the bearings [23; 24].  
Machining processes can also be subject to diagnostic analysis through the IAS estimation. The milling cutting force, for example, is proved to be reflected by the IAS [25; 26].