## 3 Adaptive Data Fusion Method Based on ADCNN for Fault Diagnosis
### Data Fusion and Feature Extraction  
Multi-sensing data fusion can preserve and complement the fault characteristics of different sensing signals at the same time, and its effectiveness has been verified in the literature and improved model prediction performance [14; 15]. In order to preserve and strengthen the two-dimensional feature image input by CNN, the signals are divided into non-stationary, time-varying signals, and stationary signals. Vibration X, vibration Y, vibration Z, sound and electric current belong to the former, and data fusion of non-stationary time-varying signals is performed. The latter performs Gaussian filtering. The Gaussian filter has a good denoising and signal smoothing effect on time-stationary sequence signals, but it will ruin the signal structure weakening characteristics for non-stationary time-varying signals. The Gaussian filter of the experimental case is designed with a standard deviation of 3 and a filter size of 19. Assuming that the sensing data set X has n-dimensional sensing data and m samples, the non-stationary, time-varying signal data fusion matrix equation can be expressed as follows:  
\[\mathrm{X}= \begin{bmatrix}\mathrm{x}_{1}^{(1)}&\mathrm{x}_{2}^{(1)}&\cdots& \mathrm{x}_{n}^{(1)}\\ \mathrm{x}_{1}^{(2)}&\mathrm{x}_{2}^{(2)}&\cdots&\mathrm{x}_{n}^{(2)}\\ \vdots&\vdots&\ddots&\vdots\\ \mathrm{x}_{1}^{(m)}&\mathrm{x}_{2}^{(m)}&\cdots&\mathrm{x}_{n}^{(m)}\\ \end{bmatrix}  \tag{14}\]  
Among them, the case n in this article is five channels, starting from one to correspond to vibration X, vibration Y, vibration Z, sound, and electric current sensing data. Figure 3 shows the process. The feature extractor uses data multi-signal fusion matrix equation and the temperature vector data channels and converts them into multi-time-frequency images and two-dimensional images for feature extraction, fusing them into multi-channel image features. Figure 4 displays the flow chart. Figure 2: Time window sampling. Traditional signal analysis focuses on the time domain and frequency domain. Fourier transform has been applied to various analytical instruments; however, Fourier transform has the disadvantage of fixed time length and resolution and cannot effectively extract features for time-varying signals whose waveform cycles vary. The wavelet transform can dynamically and adaptively adjust the time resolution. When the high-frequency component has a short resolution, and the low-frequency component has a short resolution, the time-frequency domain features can be effectively extracted by adaptively adjusting the time length, increasing the robustness of the model prediction. The experimental case uses Morlet as the mother wavelet function. Morlet has been verified to effectively extract fault features from non-stationary signals and obtain good resolution in the time and frequency domains [28]. Morlet wavelet is defined as:  
\[\varphi(\mathrm{x})=\mathrm{e}^{-\mathrm{x}^{2}}\cos(\pi\sqrt{\frac{2}{\ln 2 }\mathrm{x}}) \tag{15}\]  
Adjust the scale factor \(\alpha\) of Formula (1) to obtain the scale factor of different signals in time. By stretching or shrinking the wavelet, slow or sudden changes in the signal can be recorded. In theory, the time step of the signal is multiplied by the scale factor to obtain the product. The wavelet coefficients of all the wavelet coefficients can be converted into images by converting the absolute values of all wavelet coefficients to obtain a scalogram in the time-frequency domain, as shown in Figure 5.  
Figure 4: Feature extraction flow chart. Figure 3: Data fusion flow chart. The Gaussian filter is used for denoising in order to prevent different noise interference in the production environment from affecting the GASF-converted image. Then GASF converts the temperature vector data into an image and calculates the polar coordinate system with Formula (6) to obtain the converted cosine angle of the temperature series data, switching between \([0,\pi]\) in order to preserve the data correlation of time points.