## 3 Signal Processing Techniques Applied to Wind Turbine Failure Detection  
Initially, the study of the signals was carried out in the time-amplitude plane and was based on the variation of the waveform in addition to the calculation of parameters such as the average value,peak value, interval between peaks, standard deviation, crest factor, root mean square value, kurtosis, and skewness [35; 36]. According to reference [37], synchronizing the sampling of the vibration signal with the rotation of a gear and evaluating the average of several revolutions provide a signal called the synchronized time average, which is expressed by Equation (31). This equation makes it possible to accurately approximate a periodic signal and obtain the vibration pattern (including any modulation effects) of the gear teeth of a gearbox. However, according to the author, this method requires repeating the analysis for each gear. As described in reference [38], the most advanced proposed methods for the analysis in the time domain apply time series models to the signal, among which are those of auto regression (AR) and the autoregressive moving average (ARMA), which are applied in references [39; 40; 41]. Currently, the parameters used in the analysis of the current signal in the time domain can be used for the detection and diagnosis of faults using artificial intelligence models [42]. Thus, in reference [35], the eight parameters of the current signal in the time domain are the input variables of a three-layer artificial neural network (ANN) used to predict the remaining useful life (RUL) of the bearings of the gearbox of a WT. \[g(t)=\sum_{k_{0}=0}^{k_{0}}A_{k_{0}}(1+a_{k_{0}}(t))\cos\left(2\pi k_{0}f_{te }t+\varnothing_{k_{0}}+b_{m_{0}}(t)\right) \tag{31}\]  
As described in reference [43], the complex sinusoidal and cosine signals in the time domain can be better analyzed using their frequency components obtained with the Fourier transform (FT). According to references [38; 44], the analysis in the frequency domain allows us to obtain information that is not available in the time domain, for example, knowing the origin of the signal, the phase modulation, and the moment at which the components arise. When the signals are stationary, their frequency spectrum is constant over time, so the analysis can be performed using the FT. For example, reference [45] determines the frequencies associated with the DFIG faults of WTs by applying the fast Fourier transform (FFT) to the current signal of the electric generator, but during periods of steady-state, that is, when the speed is constant. When the signals are transient and not periodic, such as during startup or load variation or under wind speeds with stochastic behavior, the spectrum is oscillatory. Under these conditions, the FFT and analysis in the frequency domain are not sufficient, so it is necessary to resort to other techniques that allow analysis in the time-frequency domain, such as the short-time Fourier transform (STFT), wavelet transform, Wigner-Ville distribution (WVD), and Hilbert transform (HT) [42; 46; 47]. One of the first alternatives used to overcome the disadvantages of the FT was the windowing technique proposed in 1946 by [PERSON], which consisted of applying the FT to only a small section of the signal at a time. This methodology is the origin of the STFT, which allows the signals to be represented as a function of time and frequency. For this, the total signal time is divided into shorter time intervals [48; 49]. The signal is multiplied by a window function, Equation (32), and for each resulting interval, the discrete time FT (DTFT) is given by (33) and (34). For a fixed time (\(n\)) of analysis, the DTFT is known as the STFT, where (32), which is a sequence of DTFTs, is a periodic function of frequency \(\omega\) and period \(2\pi\). There is a different spectrum in each window, and by analyzing all the intervals as a whole, the variation in frequency over time can be observed.