# An Application of Instantaneous Spectral Entropy for the Condition Monitoring of Wind Turbines
## 2 The Instantaneous (Shannon) Spectral Entropy  
In this research, the Instantaneous Spectral Entropy is proposed as a damage-sensitive index for condition monitoring. Specifically, the Shannon Spectral Entropy is utilised for this aim. The rationale for entropy measurements in SHM is quite straightforward. It derives from the eighth (and last) axiom of Structural Health Monitoring [28], which states that: _\"damage increases the complexity of a system\"_[29] where the definition of 'complexity' (intentionally left vague by the original authors) can be intended from both a geometrical and signal processing standpoint. In the first case, the damage is intended as a localised inhomogeneity, which can be detected from a certain time instant (the 'damage event') onwards. From the vibrational perspective, this is rather intended as the occurring of additional signal components, previously inexistent in the undamaged baseline. A classic example would be the insertion of super- and sub-harmonics due to the presence of a breathing crack in an otherwise linearly behaving structure [30]. In this sense, the concept of an entropic framework for signal analysis has been recently further detailed in [31]. Noteworthy, the concept of an 'increase' in complexity naturally incorporates the pre-existing conditions, that is to say, randomly distributed manufacturing defects in the structure under investigation and measurement noise in the recordings derived from its analysis [32]. These aspects do not affect the entropy variation when damage is inserted in the system; therefore, this framework fits well in the general context of SHM as an outlier (anomaly) detection problem, i.e., as a deviation from a known baseline [33]. As mentioned earlier, this work deals with the instantaneous definition of entropy, aiming at damage event detection. In this sense, it is necessary to recall that vibration time series can be analysed with different signal processing approaches, depending on the intended purposes. These fall into two main categories:  
1. real-time analysis, using a small moving window of recent history over the data stream. 2. retrospective analysis, where the time series data are fully available and analysed a posteriori. In both cases, instantaneous parameters can be used to perform event detection; this can be then applied to define the instant of damage occurrence (see, e.g., [34]). For the sake of this research, the second case (retrospective analysis) will be considered. This is not uncommon for SHM applications, where signals are sampled periodically (in the case study of this research, every hour) and then processed. Under these conditions, the rationale is that a mechanical fault should be detected shortly after its appearance, in a near-real-time fashion (i.e., with only some hours or, in the worst case, some days of delay). Truly real-time SHM is more challenging since it strictly requires an uninterrupted, seamless stream of data, plus computationally efficient routines capable of processing the raw data in a short timeframe. This is generally an unnecessary complication, apart from extremely fragile structures or systems, prone to sudden collapse. In this regard, a notable example comes from the Aerospace Engineering field. In the case of rotorcraft's Health and Usage Monitoring Systems (HUMSs [35]), the real-time condition monitoring of the rotating components is essential due to the rotorcrafts implicit inability to sustain non-propelled flight. This sort of application falls beyond the aims of this study; nevertheless, it will be shown how the proposed approach is suitable for such tasks, thanks to a relatively short execution time. Another important caveat should be addressed before moving on to the proper mathematical definition of SSE and ISE. That is, it is important to recall that the entropy of the recorded output does not depend solely on the system behaviour. Indeed, it reflects the frequency content of the input as well.