## 3 Signal Processing Techniques Applied to Wind Turbine Failure Detection
In the case of oscillating functions, the direct analysis in the frequency domain of the periodic variations over time of the components introduced by some type of anomaly does not provide enough information. However, if a bandpass filter is used in the region containing the components that modulate the CS and its envelope is obtained, the frequencies associated with the faults can be easily identified. Additionally, since the magnitude of the envelope can be plotted on a logarithmic scale, exponential decays can be converted into straight lines to detect low-level peaks. This is the reason why the demodulation of the amplitude of the current signal is one of the most used techniques for the detection and diagnosis of faults in rotating electrical machines using SA. \[A(t)e^{i\omega(t)}=a(t)+\overline{ja}(t)=a(t)+\frac{1}{\pi}\int_{-\infty}^{ \infty}a(\tau)\frac{1}{t-\tau}d\tau \tag{39}\]  
According to reference [54], mathematically, the HT along with its FFT are given by (40). The composition of the analytical signal and its amplitude, phase of the envelope, and instantaneous frequency can be obtained from Equation (39). Both positive and negative components have a \(90^{\circ}\) offset, and Equation (41), according to the Park transform, takes the same form after applying HT as Equation (17). From this, it follows that the characteristic frequencies are \(f_{m}\) and \(2f_{m}\), there being a dc component in \(f_{m}\), \(2f_{m}\), \(2(f_{m}+f_{1})\), etc. According to reference [55], when some type of failure occurs in the multipliers, a new impulse appears in the original spectrum or phase spectrum. For this reason, prior to using the FFT to obtain the spectrum, demodulation is applied (using the HT) to the current signal, demonstrating the effectiveness of this technique in the detection and diagnosis of pinions and broken teeth in the gearbox of a WT. \[\overline{x}(t)=\frac{1}{\pi}\int_{-\infty}^{\infty}\frac{x(\tau)}{t-\tau}d \tau=x(t)*h(t)=x(t)*\frac{1}{\pi t} \tag{40}\]  
\[x_{\overline{t}}(m)=x(m)h(\overline{t}-m) \tag{32}\]0  
According to reference [56], due to the operating characteristics of the WTs, the resolution of the STFT, in both the time and the frequency domains, is limited. The wavelet transform has the capacity to analyze variations in the signal in the coupled time-frequency domain. However, this process depends strongly on the chosen function and requires prior knowledge of the signal used, while empirical mode decomposition (EMD) lacks a theoretical foundation and requires extreme interpolation. In this context, the author proposes a new method to detect the failures of the gearbox of the WTs, which is based on first demodulating the stator current signal of a DFIG using the HT and then applying a signal resampling algorithm based on the generator rotation frequency, such that the resampled envelope has a constant phase angle range. While the FT decomposes a signal into a set of waves of different frequencies, the wavelet transform transforms a signal contained in space to a time-scale region using an infinite set of functions called wavelets and defined according to Equation (42), called the wavelet mother. Similar to the FT, the continuous wavelet transform (CWT) represents the sum of the products of the signal multiplied by each of the wavelets, as shown in Equation (43). The characteristics and properties vary according to the type of mother wavelets or wavelet families, among which we mention the Haar wavelet, Daubechies wavelets, symlets, coiflets, biorthogonal wavelets, reverse biorthogonal wavelets, Meyer wavelets, discrete approximations of Meyer wavelets, Gaussian wavelets, Mexican hat wavelets, Morlet wavelets, complex Gaussian wavelets, Shannon wavelets, frequency B-spline wavelets, and complex Morlet wavelets.