## 2 Methodology
### Vibration Imaging Using Discrete Orthonormal Stockwell Transform (DOST)  
In the transfer learning (TL), first, the network is trained with a working condition (source task) and then those learnings are passed to another working condition (target task). Processing the substantial amount of 1D vibrational data from sensors requires massive computational cost and time. Therefore, an adjustable sliding window frame [33] mechanism is adopted with the goal of (a) creating a massive amount of source data to train the network efficiently, and (b) handling the large amount of data for further processing to fit it to the proposed network. If the total length of the acquired vibration signal is \(L_{t}\), then the total number of the samples \(n_{t}\) is:  
\[n_{t}\ =\ \left(\frac{L_{t}\ -\ L_{f}}{L_{s}}\right)\ +\ 1 \tag{1}\]  
In Equation (1), \(L_{f}\) is the length of a single frame, which is selected based on the experimental requirements. The step size is \(L_{s}\). The length of the vibration signal \(L_{t}\) is fixed in this study. The overall process of adjusting the sliding window frame is shown in Figure 2.  
With a 1D signal, it is very difficult to observe identical patterns for different health types. Moreover, in recent studies [33; 34], preprocessing-free transfer learning is claimed. Even with the limited amount of data for mechanical machines, network learning for a transferring scenario remains questionable. Huge chunks of data having a lot of variations can help a network gather some intrinsic information. This kind of information can help to classify other mechanical sensors further. In this scenario, TL is used to resolve the issue of reducing time along with boosting performance. However, TL cannot resolve the challenges of raw signals alone since the open source data are incommensurate. Also, for additional white noise, the time-domain information is not always accurate. Rather, preprocessing can result in more robust performance if it uses the TL-based approach instead of classical deep learning because a limited portion of data facilitated with some additional information can help us adjust the network weights to learn more accurate intrinsic feature information for classification. Current study focuses mainly on this idea. A 1D raw signal suffers from the inability to create invariant identical scenarios. Two-dimensional imaging is developed to achieve ascendable feature engineering that processes heterogeneous data from systems with invariant working principles. In this study, discrete orthonormal Stockwell transform (DOST)-based stacking for vibration imaging is employed as a preprocessing step in the proposed approach. The Stockwell transform (ST) was first proposed by [PERSON] [30]. Using local spectral phase properties, it can represent time and frequency. The ST can distinctively combine a frequency-dependent resolution of the time-frequency space with unequivocally referenced local phase information. In other words, the ST is like a fusion between the Gabor transform and the wavelet transform. In practice, this time-frequency decomposition tool overcomes some of the drawbacks (i.e., provides better time-frequency resolution) of the short time Fourier transform (STFT). In [30], [PERSON] described a method to decompose a signal on a discrete orthogonal basis for the ST (DOST basis). In this study, the DOST basis is considered to form 2D images with identical patterns. To perform the orthogonal transform, this method uses the discrete Fourier transform (DFT) [3]. Figure 2: Adjustable sliding window technique. First, the method performs a fast Fourier transform on each segmented windowed sample. Then, it segments the processed signal via frequency partitioning for the DOST, that is, changing dyadic negative to positive. Next, on each bandwidth partition, it calculates the inverse Fourier transformation. Finally, it combines all these partitions together to form the final DOST output. Figure 3 shows the process of this DOST basis method.