## 3 Signal Processing Techniques Applied to Wind Turbine Failure Detection
When, instead of studying the spectrum at a certain time, one wishes to study a specific frequency, then the windowing process is carried out in the frequency domain. The quality of the results depends on selecting the window that minimizes losses due to spectral leakage and reduces the amplitude of the main and secondary lobes [50]. \[x_{\overline{t}}(m)=x(m)h(\overline{t}-m) \tag{32}\]  
\[X_{\overline{t}}(e^{i\overline{\omega}})=\frac{1}{\sqrt{2\pi}}\int\ e^{-i \overline{\omega}m}x_{m}(m)h(\overline{t}-m)dm \tag{33}\]\[X_{\overline{t}}(e^{i\overline{\alpha}\overline{\beta}})=\sum_{m=-\infty}^{\infty}x (m)h(\overline{t}-m)e^{-i\overline{\alpha}m} \tag{34}\]  
\[X_{\overline{t}}(e^{i\overline{\alpha}\overline{\beta}})=\left.(x[t]e^{-i \omega t})*\omega[t]\right|_{t=\overline{t}} \tag{35}\]  
As described in reference [38], the STFT has resolution problems due to signal segmentation; one of the alternatives to overcome this limitation is the WVD. This distribution is one of the most popular ones since, unlike the STFT, it is not based on signal segmentation, providing better resolution in both the time domain and the frequency domain. According to references [47; 51], given a signal \(s(t)\) in the time domain, the WVD is defined by (36), while starting from the frequency spectrum of \(s(t)\), it is given by (37). It can be assumed that using the WVD means dividing the signal into two equal parts in relation to a time \(t\), with the right part overlaid over the left part, which means that when the signal is null before or after \(t\), then the signal is zero in \(t\). The correct average is obtained only when the signal can be separated into a component that is a function of time only and another function of frequency. A signal that is not zero at time zero or a signal with frequencies without the existence of a spectrum indicates the presence of interference or cross terms. As described in reference [34], a sinusoidally modulated amplitude current signal has the same components as in (30), which means that when \(f=fr\), in the case of faults due to torque oscillations and eccentricity, the use of classic spectral analysis (SA) does not allow us to distinguish between amplitude and phase modulation since the modulation indices are small. However, the use of the WVD does allow us to distinguish these faults using Equation (38). \[W(t,\omega)=\frac{1}{2\pi}\int_{-\infty}^{\infty}s^{*}(t-\frac{1}{2}\tau)s(t+ \frac{1}{2}\tau)e^{-i\tau\omega}d\tau \tag{36}\]  
\[W(t,\omega)=\frac{1}{2\pi}\int_{-\infty}^{\infty}S^{*}(\omega+\frac{1}{2} \theta)S(\omega-\frac{1}{2}\theta)e^{-i\theta}d\theta \tag{37}\]  
\[f_{s}\pm\frac{f_{r}}{2} \tag{38}\]  
The small magnitude of the components associated with the faults makes their extraction difficult. To overcome this drawback, one of the most suggested techniques is to demodulate the signal amplitude. For this, according to reference [52], the best option is to use HT due to its strength to handle noise. According to that report, when there are no faults, the amplitude of the envelope is constant over time, and its variance is zero. Otherwise, if the variance is greater than a pre-established threshold level, some type of asymmetry exists. According to reference [53], the HT of a signal is the relationship between the real and imaginary parts of the FT of said signal. Similarly, the function of time obtained by the Fourier inverse is a complex function called the analytical signal, the imaginary part of which is the HT. The analytical signal can be represented as a phasor whose amplitude and rotation speed vary over time, according to Equation (39), implying that, given a function in the time domain, in addition to the amplitude, one can also obtain the components that modulate the frequency or phase, \(\overline{a}(t)\). In the HT, the amplitude function is the envelope of both the real and the imaginary parts and represents the modulated signal plus \(dc\) compensation.