## 1 Introduction
### Instantaneous Angular Speed (IAS) Review  
The IAS information is, in simple terms, a measure of the rotation speed of a rotating component of a machine defined at an angular resolution corresponding to at least one value per revolution [26]. From a physical point of view then, the IAS is defined starting from the angular position \(\alpha(t)\) of a shaft as:  
\[\omega(t)=\dot{\alpha}(t)=\frac{d\alpha}{dt} \tag{1}\]  
A measure of this quantity can be obtained using analog sensors such as the tachometer-dynamo which translates the rotational speed into an electrical signal, or with analog angular position sensors such as the resolver. Nevertheless, such devices can nowadays be considered obsolete and are often substituted by more reliable digital sensors, whose signals can be processed to produce an estimate:  
\[\hat{\omega}(t)\equiv\frac{\Delta\alpha}{\Delta t}\ ;\ \hat{\omega}\rightarrow\omega\ for \ \Delta t\to 0 \tag{2}\]In particular, two strategies can be put in place. One involves the measurement of the angle \(\Delta\alpha\) swept in a constant interval of time \(\Delta t\) by the shaft. Nevertheless, the angular measurements can be affected by larger errors than the time measurements' ones. A second more widespread and accurate method consists of measuring the time elapsed between successive pulses (Elapsed Time \(ET=\Delta t\)) corresponding to a known swept angle \(\Delta\alpha\) which is a characteristic of the sensor (e.g., an encoder). An analysis of the resolution and the speed estimation error can be found in [27]. In any case, all these methods involve the use of an additional sensor (generally referred to as a generic \"tachometer\" but not to be confounded with the tachometer-dynamo sensor). The interest of the scientific diagnostics community then has recently moved to IAS estimation from the easily available accelerometric signals, fundamental for vibration monitoring, going \"encoder-less\" or \"tacholess\". In this regard, several different approaches are possible [7]. In particular, the simplest idea is to track a shaft-speed related harmonic (possibly showing a good Signal to Noise Ratio-SNR) from its corresponding peak in a time-frequency representation of the signal (i.e., a Spectrogram, computed for example via a Short Time Fourier Transform). This can be made more accurate by averaging the tracks from multiple harmonic orders, which can also be automatically selected by the algorithm (e.g., Multi-Order Probabilistic Approach-MOPA, Ceptrsum-based MOPA, or ViBES). A second approach to the IAS estimation is the demodulation of a shaft-speed related harmonic exhibiting good SNR. This follows the idea that vibration signals can be modulated by the revolution of the shaft so that phase demodulation can recover the shaft speed from the band-pass isolated harmonic of interest. Demodulation can also be performed using the Teager-Kaiser Energy Operator. Finally, the two main procedures of tracking and demodulation can be exploited together as implemented in the Vold-Kalman filtering. At any rate, both the measurement and estimation of the IAS have some limitations. In the first case, the use of high-resolution encoders allows to get reliable and accurate estimates of the IAS, but the angular sensors can be very expensive and need to be mounted on a shaft added for the purpose. On the other hand, the second case exploits the cheap and reliable accelerometers which can be added on a machine without special design updates. The estimated IAS, however, is less accurate and less reliable, and needs computational time. In this article then, the measurement of the IAS is tackled at the scope of testing an offline, low-cost video-tachometer approach, as suggested by the SURVISHNO 2019 challenge (see Supplementary Material). The question was \"how far is it possible to carry out relevant analysis from a video or a rotating fan acquired by a smartphone which can reach a maximum rate of 30 frames per second?\".