## 3 Generating Synthetic Bearing Fault Data via cycleGAN
### Data Pre-Processing and CycleGAN Training  
The simulated and experimental signals were pre-processed to emphasize the existence of bearing damages within the harmonic content. In particular, the 250 sample pairs obtained from the simulated and experimental signals (Table 1) were subjected to normalization based on the mean value and variance of each sample. The implementation of the normalizing procedure aims to ensure that both simulated and actual signals can be effectively characterized using comparable scales. Subsequently, the signals were filtered within the frequency range of 1400 Hz to 2800 Hz; then, envelope [10; 12; 13]\(env(t)\) was extracted. The CWT transform shown in Equation (9) was subsequently utilized. In this equation, operator \(\psi^{*}(\cdot)\) denotes the complex conjugate of the Morse wavelet, while \(a\) and \(b\) are the scaling and translation factors of the wavelet transform, respectively. The selection of the Morse wavelet was based on its widespread application in signal analysis for this particular type of signals, owing to its advantageous properties of temporal and frequency localization. These characteristics render it well-suited for the purpose of identifying sudden pulses due to bearing defects. The Morse wavelet was employed with a symmetry parameter \(\gamma=3\) and a time-bandwidth product \(P^{2}=60\); in addition, the CWT had 24 voices per octave. The CWT images are produced by operation \(W^{2}(a;b)\), which enhances the trace of the rolling element's transition from the defect inside the CWT spectrum. \[W(a;b)=\frac{1}{\sqrt{a}}\int\limits_{-\infty}^{+\infty}env(t)\psi^{*}\left( \frac{t-b}{a}\right)dt. \tag{9}\]  
Figure 4a illustrates an instance of the CWT spectrum associated with a simulated signal operating at a rotational speed of 877 rpm. Conversely, Figure 4b shows the corresponding experimental signal. The cycleGAN model developed in a Matlab(r) environment was trained using a dataset consisting of 250 CWT image pairs. The training lasted 9 h and 26 min, utilizing an NVIDIA(r) T4 GPU that was accessible on the High-Performance Computing (HPC) infrastructure provided by the commercial cloud environment Amazon(r) AWS. The training hyperparameters are reported in Table 2. Given the evident computational cost complexities involved in implementing hyperparameter optimization techniques on such large and complex architectures, the authors use as a starting point the hyperparameters shown in Table 2, which are easily found in the literature inherent to cycleGANs [70]. Appendix A provides the description of the architectures employed in cycleGAN, specifically pertaining to the generators and discriminators. Tables A1 and A2 present detailed information regarding these frameworks. After the completion of the training process, the model was utilized to generate images based on simulated data. The signal generated under the condition of 877 rpm is illustrated in Figure 4c. Striking resemblances to the real signal are observed (Figure 4b). The process of generating the data can be summarized as follows:  
* simulating the accelerometer signal;
* pre-processing the signal through normalization, filtering and envelope extraction;
* applying the continuous wavelet transform and generating 256 \(\times\) 256 images by squaring the CWT coefficients;
* utilizing the images as input for the cycleGAN model previously trained to transform images of simulated signals into their corresponding real counterparts. The resulting output of the cycleGAN model is an image that is a surrogate for a real image from experimental activity. Figure 4 demonstrates the capabilities of the generative model to produce synthetic data that retrace the operational conditions on which the algorithm was trained. The acquisition of this ability can be of great value in augmenting the existing data on operational conditions if damage data are already accessible, hence enabling the adjustment of any class imbalance within the datasets.