## 2 Data and Signal Processing  
The data used in our investigation were recorded on a test bench of an aircraft turbojet engine. Two accelerometers are used to measure vibration in front and rear bearing housing. We used only the signals acquired at the accelerometer mounted on the front bearing housing since the faults to be considered are located at turbojet air compressor bearing. The acquisition of the signals was carried out through 348 experiments on healthy bearings and defective bearings. These experiments are classified into 4 groups:  
* 60 were fault-free
* 96 had inner race bearing fault
* 96 had outer race bearing fault
* 96 had ball bearing fault. All the signals recorded with a sampling frequency equal to 12,000 Hz. To increase the number of elements in each class, we divided each signal into 3 segments. We got 1044 signal overall. An illustration in the time domain of an inner ring faulty bearing signal is illustrated in Figure 1.  
Estimating spectral power density is a very important area in the field of digital signal processing. This study explores the Welch estimation method. It is a nonparametric method which provides a consistent estimator of power spectral density [27]. This method consists to subdividing the records, taking modified periodograms for each subdivision and averaging these modified periodograms. In many cases, this method requires less calculation than other methods because it involves transforming the signal into shorter sequences. For \(X(n)\), a second-order stationary random discrete signal of zero mean \((n~{}=~{}0,~{}1,~{}2,\ldots,~{}N-1)\), we take overlapping segments of length \(L\). The starting points of these segments are multiples of \(D\) as illustrated in Figure 2. We assume that we have \(K\) segments, \(X_{1}(n),~{}X_{2}(n),\ldots,~{}X_{K}(n)\) and that they cover the entire record, as described in Equation (1). \[ \begin{array}{c}(K-1)~{}D~{}+~{}L~{}=~{}N\\ \begin{array}{c}X_{1}(n)=X(n)\\ X_{2}(n)~{}=X(n+D)\\ \ldots\\ X_{K}(n)~{}=X(n+(K-1)D)\end{array}\end{array}  \tag{1}\]  
Figure 1: Time domain of an inner ring faulty bearing signal. To obtain the spectral estimation, we calculate first a dependent frequency (\(f\)) modified periodogram \(\hat{P}_{m}^{(i)}\) for each segment,  
\[\hat{P}_{m}^{(i)}(f)=\frac{1}{LV}\Bigg{\{}\sum_{n=0}^{L-1}W_{S}(n)X(n+iD)\exp(-2 \pi jnf)\Bigg{\}}^{2} \tag{2}\]  
where \(W_{s}(n)\) is the window function and \(V\) is defined in (3). \[V=\frac{1}{n}\sum_{n=0}^{N-1}\left|W_{s}(n)\right|^{2} \tag{3}\]  
The spectral estimate \(\hat{P}_{W_{S}}(f)\) is the average of these periodograms as given in (4). \[\hat{P}_{W_{S}}(f)=\frac{1}{K}\sum_{i=0}^{K-1}\hat{P}_{m}^{(i)}(f) \tag{4}\]  
Finally:  
\[\hat{P}_{W_{S}}(f)=\ \frac{1}{KLV}\sum_{i=0}^{K-1}\Bigg{\{}\sum_{n=0}^{L-1}W_{S}( n)X(n+iD)\exp(-2\pi jnf)\Bigg{\}}^{2} \tag{5}\]  
Thus, the estimation of the spectrum consists in estimating \(\hat{P}_{W_{S}}\) (\(f\)) from a finite number of noisy data. [PERSON]'s method reduces noise at the expense of frequency resolution [28].In our case, we used the hamming window function with a length 100 and an overlap of 30. The representation of the spectral density of 2 faulty bearings signals are illustrated in Figure 3. (Frequency and Amplitude are normalized). After analyzing the 1044 available signals, we extracted the 8 most predominant peaks of the power spectral density (pointed out by triangles in Figure 3). Two values were used for each peak: the characteristic frequency and the value of the spectral density, which gave 16 features for each signal. The data were then organized in a matrix of 16 columns and 1044 rows. If the generated data is used directly without any processing before starting the training phase, the input variables of the highest value may tend to suppress the influence of the most small. Thus, normalized data will be treated with the same priority. We normalized each line of the data matrix. The 16 features of each line were then scaled between \(-1\) and \(1\). Figure 2: Illustration of the record segmentation.