## 2 Materials and Methods
### Signal Processing  
Latent information can be extracted manually or automatically for classification purposes by converting original signals into numerical features, whether in the time, frequency, or time-frequency domain.  
In some cases, extracted statistical features from raw data in the time domain, such as standard deviation and peak values, can result in significant accuracy in the classification algorithms although, on the assumption of signals in limited length and negligible differences of different classes, more sophisticated techniques such as wavelet transformation should be applied either in the classification or feature extraction process [51; 52].  
The Gabor (analytic Morlet) wavelet is utilized to construct a network for a WTS decomposition in MATLAB(r). WTS produces representations that are impervious to input signal translations without compromising class discriminability. The results of WTS typically give a hierarchical representation of the input signal, with each layer capturing different time-frequency features. The first layer records low-frequency information, and the succeeding layers collect information with an increasing frequency. These representations can be used as the input feature matrices of a classifier or regressor algorithm. In this technique, real-valued time series data should pass frequently through three steps consisting of convolution, non-linearity, and averaging using wavelet decomposition, modulus, and low-pass filter, respectively [52; 53].  
The number of filters per octave and the length of translation invariance are the only variables that must be determined for feature extraction and signal processing using WTS. Until the duration-based invariance scale (_IS_) is reached, the scattering framework is translation-invariant. The representation of the signal and the features that are derived from it both heavily depend on _IS_. To distinguish between patterns that are truly similar and those that are similar only by chance, a low degree of _IS_ means that the transform can only detect patterns that are similar at a specific scale. Since the scaling filter is used to apply the invariance in the framework, its time support is not greater than that of the invariant. _IS_ should be greater than the wavelet's time support; _IS_ has impacts on the center frequencies' spacing in the filter banks. This parameter can be calculated by inverting the Fourier transform of the scaling function and centering it at 0 in time when the dynamical variations of the signals are known; otherwise, it can be discovered through trial and error [54; 55].  
In this study, the transverse vibration signal of each sample is recorded for 3 s with a sampling frequency of 4 kHz. To avoid transient behavior, the initial 1.55 s is disregarded, while the final 1.46 s is retained.  
In accordance with the Nyquist-Shannon Sampling Theorem, the sampling frequency (in this case, 4 kHz) should be at least twice the maximum frequency component present in the signal (which is 23.98 Hz). Thus, the sampling frequency is chosen significantly higher than necessary to avoid missing any frequency components within the signals.  
Given that the objective of the present investigation is to analyze the stationary signal of defective rotor systems, there are no specific restrictions on the time-wave range. Anysection of the signals that represents stationary behavior can be appropriately incorporated into the methodology employed in this study.  
The WTS technique is utilized with an _IS_ of 0.5 and the original data's sampling frequency to compute a coefficient matrix for each sample. The resulting matrix has a dimension of 199 \(\times\) 12, representing 199 time-frequency representations generated by the WTS, each with 12 features. The feature matrix for the entire data set has a dimension of 140 \(\times\) 199 \(\times\) 12, where 140 is the total number of sample tests. These matrices will be used in the classification process, where they will be fed into both an LSTM network and SVM in subsequent stages.