## 2 Related Work
### Techniques for Converting Time Series Data into 2D Representations
#### 2.3.3 Methods Based on Data Domain Transformation  
Although the aforementioned methods are simple and intuitive in their implementation, they tend to overlook the information contained in time series data across both the time and frequency domains. This oversight may lead to the loss of temporal information or disruption of periodic characteristics in the original data. To more effectively capture features that are strongly correlated with time and to preserve important signal information, researchers have adopted feature extraction techniques to enhance the data, providing a more comprehensive foundation for subsequent deep learning model training--especially in scenarios that require precise frequency domain analysis and processing of non-stationary signals. The Fourier transform (FT) is a mathematical method that converts time series data from the time domain to the frequency domain [35]. It is a fundamental tool for analyzing the frequency components of signals, particularly adept at revealing periodic and harmonic features within the data. The central concept is that any periodic function can be expressed as a sum of simple oscillating functions at various frequencies. For non-periodic signals, the Fourier transform represents the signal as a continuous spectrum, encompassing a weighted sum of sine and cosine components across all possible frequencies. The fast Fourier transform capitalizes on the periodic properties of signals, reducing computational complexity from \(O(n^{2})\) to \(O(nlogn)\) through a divide-and-conquer strategy [36]. Its fundamental concept involves dividing a sequence of length \(N\) into two subsequences of length \(N/2\) and recursively applying the Fourier transform to these subsequences. The wavelet transform effectively detects transient events and non-stationary features in fault signals by providing simultaneous time and frequency information. The output of the wavelet transform is typically presented as a two-dimensional data array, where one dimension corresponds to time and the other to frequency or scale. This format enables the wavelet transform to capture local features and frequency variations within the signal. This rich time-frequency representation not only elucidates fault characteristics but also serves as a suitable input for subsequent deep learning models, such as convolutional neural networks, for feature learning and pattern recognition. The implementation of the wavelet transform involves several steps. First, an appropriate wavelet basis must be selected based on the characteristics of the signal, such as Daubechies, Morlet, or Mexican hat wavelets. Second, the range of scales for analysis should be determined. The choice of scales depends on the specific requirements of the analysis; for instance, smaller scales are needed to detect fine details, while larger scales are necessary to capture broader signal trends. This selection is typically guided by experimentation and a thorough understanding of the signal characteristics. Next, either the continuous wavelet transform (CWT) or the discrete wavelet transform (DWT) can be performed. The CWT is suitable for detailed time-frequency analysis, providing wavelet coefficients of the signal at every possible scale and position. In contrast, the DWT is more appropriate for rapid analysis and data compression, decomposing the signal into wavelet coefficients at multiple scales and positions. The CWT can be defined as:  
\[CWT(a,b)=\frac{1}{\sqrt{|a|}}\int_{-\infty}^{\infty}x(t)\cdot\psi\left(\frac{t -b}{a}\right)dt \tag{6}\]  
where \(x(t)\) is the signal, \(\psi\) is the mother wavelet function, \(a\) is the scale parameter, and \(b\) is the translation parameter. Discrete wavelet transform (DWT) can be defined as  
\[DWT[k]=\sum_{n}x[n]\cdot\psi_{a,b}[n] \tag{7}\]  
where \(\psi_{a,b}[n]\) is the wavelet function obtained based on the scale \(a\) and the translation \(b\).