# An Application of Instantaneous Spectral Entropy for the Condition Monitoring of Wind Turbines
## 3 The Continuous Wavelet Transform and Generalised Morse Wavelet
### Generalised Morse Wavelet  
For CWT-based signal processing, analytic wavelets are currently considered the best option for precise TF analysis. These can be seen as complex-valued time/frequency localised filters with vanishing support on negative frequencies [46]. In this regard, the complex (or analytic) Morlet wavelet is arguably the most common option adopted. It has been widely applied for the extraction of instantaneous parameters, e.g., from seismic data [47]. Recently, the Morlet wavelet power spectral entropy has been investigated as well, specifically for bearing fault localisation and severity assessment [48].  
However, the complex Morlet wavelet is only approximating analytical for large centre frequency (\(f_{c}\)) values. For small \(f_{c}\) values, it may not meet the admissibility condition and thus potentially lead to a negative frequency [49]. For this reason, the generalised Morse wavelet has been preferred for this application, due to its totally analytic definition and better time resolution.  
The GMW was firstly envisioned by [PERSON] [50], and then further detailed and investigated by [PERSON] [46; 49]. Its formulation can be expressed in the frequency domain (considering the natural pulsation \(\omega=2\pi f\) for simplicity) as [46]  
\[\Psi_{\beta,\gamma}(\omega)=U(\omega)\alpha_{\beta,\gamma}\omega^{\beta}e^{- \gamma\omega}, \tag{7}\]  
where \(U(\omega)\) is the Heaviside (or unit) step function, \(\alpha_{\beta,\gamma}\) is a normalizing constant, and \(\beta\) and \(\gamma\) are the two parameters which govern the specific shape taken by the general formulation of Equation (7). More precisely, these two parameters are known as symmetry (\(\gamma\)) and compactness (\(\beta\)). Furthermore, a third parameter, the time-bandwidth product, can be defined as \(P^{2}=\beta\gamma\) and is used often (but not here) in lieu of \(\beta\).  
The main feature of the GWT is its adaptability, as it can be dilated or contracted in the time-frequency domain to better suit the signal processing aims.  
In detail, \(\sqrt[3]{P^{2}}\) is directly proportional to the wavelet support along the time axis. Therefore, for constant \(\gamma\), increasing \(\beta\) implies a longer time-bandwidth. In turn, this increases the rate of the long-time decay and broadens the central portion of the resulting mother wavelet. For time-frequency analysis, a longer time duration, \(T\), implies more refined frequency resolution, \(\Delta f=1/T\), which is generally useful. This point will be better discussed in the Results (Section 5).  
On the other hand, the gamma parameter controls the symmetry of the wavelet in time through the demodulate skewness [49]. Increasing \(\gamma\) for constant \(\beta\) does not affect the time-bandwidth, but broadens the GWT envelope, making it more or less directional. For instance, if \(\gamma=1\), the zeroth-order GWT corresponds to the Cauchy wavelet [51], which is strictly supported in a narrow convex cone in the time-frequency domain.  
Different GWT shapes can better serve different purposes. These shapes can be grouped in a piecewise fashion as follows:  
1. For constant \(\gamma\leq 3\) (negative skewness), the time decay increases as the time-bandwidth (\(P^{2}\), thus \(\beta\)) increases.
2. For constant \(\gamma\geq 3\) (positive skewness), the mother wavelet becomes more symmetric as \(\beta\) increases.  
Therefore, for \(\gamma=3\), both the time decay and the wavelet symmetry increase with \(\beta\), and the resulting mother wavelet narrows in frequency and enlarges in time, with more oscillations under its envelope. This derives from the demodulate skewness of the GWT being null for gamma equal to 3; this results in a global maximum of the time/frequency concentration [46]--that is to say, it maximises the product of the time-domain and frequency-domain standard deviations, known as the Heisenberg's area [40]. Again, all these aspects will be further investigated in a later Section.