## 4 Experiment
### Real Signal Experiment
#### 4.3.2 Frequency Domain Quality Analysis of Reconstructed Signals  
Whether the reconstructed signal can retain the characteristic frequency of the original signal is another important indicator to test the reconstruction performance of the algorithm. When a ball in the bearing rolls over the fault location, it will produce an instantaneous pulse, so the reciprocal of the time interval between any two adjacent pulses is the fault characteristic frequency, which is determined by the rotation frequency of the bearing and the nature of the fault type. In general, for angular contact ball bearings with fixed outer raceways and rotating inner raceways, the formulas [66, 67] for the theoretical values of the Ball Passing Frequency Inner (BPFI) and Ball Passing Frequency Outer (BPFO) are (\(f_{r}\) is the rotational frequency) as follows:  
\[f_{\text{BPFI}}=\frac{N}{2}f_{r}\bigg{(}1+\frac{d}{D}\cos\alpha\bigg{)} \tag{35}\]  
\[f_{\text{BPFO}}=\frac{N}{2}f_{r}\bigg{(}1-\frac{d}{D}\cos\alpha\bigg{)} \tag{36}\]  
IR and OR fault signal data from the CWRU and XJTU-SY data sets are taken as experimental objects. Based on the calculation formula, the BPFI of the driving end-bearing signal in the CWRU data set is 160.3 Hz, and the BPFO is 105.5 Hz. The BPFI of the horizontal bearing signal in the XJTU-SY dataset is 196.7 Hz, and the BPFO is 123.3 Hz. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Dataset**} & \multicolumn{6}{c}{**Bearing Parameters**} \\ \cline{2-7}  & **Number of** & **Ball Diameter** & **Diameter** & **Contact** & **Rotational** & **Signal Type** \\  & **Balls (N)** & **d/mm** & **D/mm** & **Angle a/(\({}^{\circ}\))** & **Speed (r/s)** & \\ \hline
**CWRU** & 9 & 7.94 & 38.5 & 0 & 1772 &
\begin{tabular}{c} Normal, IR, \\ OR@6, BA \\ \end{tabular} \\ \hline
**XJTU-SY** & 8 & 7.92 & 34.55 & 0 & 2400 & MIX, IR, OR \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Dataset signal type and bearing parameters. Figure 8: The test bench and the failure bearings: (**a**) CWRU bearing test rig; (**b**) XJTU-SY bearing test rig; (**c**) 6205-2 RS JEM SKF; (**d**) LDK UER204 [64, 65]. Considering the limited number of fault signals collected during equipment bursts, we selected 3072 sampling points from each signal after reconstruction for characterization. Figure 9a,b show the IR and OR fault signal time-domain waveforms of the CWRU dataset, and Figure 9c,d show the IR and OR fault signal time-domain waveforms of the XJTY-SY dataset. Using the ISTA-Net network, ADMM-1 DNet network, and ADMM + K-SVD-based reconstruction algorithm to conduct reconstruction experiments on the above four signals and conduct empirical mode decomposition on the reconstructed signals to further analyze the signal composition and feature information. By comparing Figures 10-13, it is found that the reconstruction effect of the ADMM + K-SVD algorithm is the worst compared with the other two network-based algorithms, and the peak value displayed after empirical mode decom-position (EMD) is far from the theoretical value. These unwanted peaks can lead to a misinterpretation of part failure characteristics during fault diagnosis and classification. As shown in Figures 10a and 13a, the reconstruction results of ISTA-Net detect the characteristic frequency of the approximate  
Figure 9: Time domain waveform of the original signal. Table 3 records the specific \(T_{\text{test}}\) and characteristic frequency values of the above reconstruction results. The results were obtained by the algorithm through 100 iterations at most. It is found that the corresponding theoretical characteristic frequency values are detected no matter whether the reconstruction \(T_{\text{test}}\) is 0.0019 or 0.0675. We further visualize the experimental results corresponding to the above two accuracies. (The optimal reconstruction results of the experiments on the two datasets are selected here for visualization). Figure 14a shows the fitting diagram of the IR signal and reconstructed signal of the CWRU dataset, the \(T_{\text{test}}\) is 0.0019.