## 2 Basic Theory
### Multi-Domain Information Fusion  
Multi-domain information fusion is a critical approach in fault diagnosis, which integrates time-domain, frequency-domain, and time-frequency domain features to enhance diagnostic accuracy and robustness. Time-domain signals capture the amplitude characteristics of vibration intensity, frequency-domain signals reveal the periodic vibration frequencies, and time-frequency domain signals provide the joint distribution of time and frequency. The signal processing techniques employed in this paper include 2D image transformation in the time domain, frequency-domain analysis using the fast Fourier transform (FFT), and time-frequency domain analysis via the short-time Fourier transform (STFT).  
Time-domain feature analysis transforms a one-dimensional vibration signal into a two-dimensional image, preserving the signal's original amplitude characteristics. We let the vibration signal be represented as \(x[t]\), with the discrete sampled signal denoted as \(x[n]\). The signal length is adjusted to \(N=H\times W\), where \(H\) and \(W\) represent the row and column sizes of the target image, respectively. The signal is then reorganized into a two-dimensional matrix:  
\[X_{\text{raw}}(i,j)=x[n],\ n=i\cdot W+j,\ i=0,1,\ldots,H-1,\ j=0,1,\ldots,W-1 \tag{1}\]  
where zero padding is applied when the signal length is insufficient, and truncation is used when the signal exceeds the target size. This method effectively preserves the original waveform characteristics of the vibration signal, ensuring that its time-domain amplitude distribution is accurately represented in the 2D image.  
Frequency-domain analysis utilizes the FFT to convert a time-domain signal \(x[n]\) into the frequency domain, thereby revealing the frequency components and amplitude characteristics. The mathematical expression for the FFT is given as follows:  
\[X[k]=\sum_{n=0}^{N-1}x[n]e^{-j2\pi kn/N},\ k=0,1,\ldots,N-1 \tag{2}\]  
where \(X[k]\) represents the complex value of the frequency component at index \(k\). To visualize the spectrum, the amplitude is extracted and reconstructed into a two-dimensional representation as follows:  
\[X_{\text{HFT}}(i,j)=|X[k]|,\ k=i\cdot W+j,\ i=0,1,\ldots,H-1,\ j=0,1,\ldots,W-1 \tag{3}\]  
meanwhile, the amplitude matrix is normalized and resized to the target image dimensions \(H\times W\) to generate the spectrum image.  
Time-frequency domain feature analysis captures both the time-varying frequency characteristics and the amplitude distribution of a signal using the short-time Fourier transform (STFT). The mathematical expression for the STFT is given as follows:  
\[X_{\text{SIFT}}(t,f)=\int_{-\infty}^{\infty}x(\tau)w(\tau-t)e^{-j2\pi/\tau}d\tau \tag{4}\]  
where \(w(\tau)\) is the window function that controls the time span of the short-time analysis, and t and f represent time and frequency, respectively. The amplitude is calculated as follows:  
\[|\ X_{\text{STFT}}(t,f)\ |=\sqrt{\text{Re}(X_{\text{SIFT}}(t,f))^{2}+\text{Im}(X_{ \text{SIFT}}(t,f))^{2}} \tag{5}\]  
Similarly, the time-frequency spectrum is normalized and resized to a two-dimensional image of dimensions \(H\times W\) for subsequent model processing.