## 4 Validation of SRSVD
### Case Study II  
The signal processing method is also widely used in the field of fault signal extraction, so we take the actual bearing fault signal as an example to verify our method. We use the open-access experimental data set of bearings run-to-failure as the signal to be processed. As shown in Figure 8, the test bench [26] consists of an AC induction motor, a motor speed controller, a support shaft, and a hydraulic loading system. Two accelerometers are mounted vertically and horizontally on the housing of the test bearing. Set the sampling frequency as 25.6 kHz, and collect data for 1.28 s every 1 min. We select the outer ring fault as the fault to be detected. As the speed and bearing size are known, the fault frequency can be estimated to be about 109 Hz through the fault bearing physical mode. It should be noted that different from case study I, the bearing fault signal model can often be used as prior knowledge in the process of fault signal extraction. The fault signals of rolling bearings can be expressed as the sum of a series of impulses:  
\[x(t)=\sum\limits_{i=1}^{M}A_{i}s(t-iT-\tau_{i})+n(t) \tag{8}\]  
where \(s(t)=A\sin(\omega_{d}t)e^{-\xi_{i}\omega_{d}t}\), \(A_{i}\) is the magnitude of the \(i\)th impulse, \(T\) is the period of the impulse, \(\tau_{i}\) is the rolling mass slip time of the \(i\)th impulse, which is usually a random small value, \(M\) is the number of impulses, and \(n(t)\) is random noise. Therefore, according to this model, we adopt an analytic dictionary composed of tunable Q-factor wavelet transform  
Figure 7: (**a**) The denoised signal waveform by SRSVD and (**b**) its frequency spectrum. (TQWT). Because TQWT matches well with the characteristics of fault bearings, it is widely used in the field of fault diagnosis. The severity of bearing fault can be roughly judged by the value of the root mean square (RMS), and the development stage of the fault can be divided according to this value, as shown in Figure 9. Figure 9c and 9d respectively show the signal waveforms in the early and medium stages of the fault. It can be seen that the dynamic range of the signal in the medium stage is larger and presents obvious periodic pulse characteristics, while it is difficult to find fault characteristics in the early stage of the signal waveform. We use SRSVD to analyze the early stage signal and try to extract the fault characteristics. This work makes a lot of sense because if early fault characteristics can be detected during the operation of rotating equipment, the loss of economy and life can be greatly reduced. We chose the 60 th minute signal for analysis, with data points of 32,768. The parameters of TQWT are as follows: \(Q=2\), \(r=5\), \(J=10\). In the process of SR, the ISTA algorithm is also used to solve the sparse coefficients, and the evaluation index for SVD is also PMI. The denoised signal obtained by SRSVD is shown in Figure 10g, and the fault signal is  
Figure 8: The test platform system of bearing. Figure 9: (**a**) Distribution of RMS values, (**b**) vibration waveform, fault signals in (**c**) early stage and (**d**) medium stage. demodulated through Hilbert envelope spectrum analysis, as illustrated in Figure 10h. As can be seen from Figure 10h, the fault characteristic frequency (109 Hz) and its 2 nd, 3 rd, and 4 th harmonics can be clearly found. We used three other widely used denoise algorithms to compare with SRSVD, these algorithms were L1-Kurtosis [27], RSVD [25], and variational mode decomposition (VMD) [7]. L1-kurtosis is a kurtosis-based weighted sparse representation denoising model. RSVD algorithm is the secondary filtering algorithm used in SRSVD, i.e., the PMI evaluation index is embedded in SVD so that the restructuring process has the ability to select components according to the information content. VMD is an adaptive, fully non-recursive method for modal variation and signal processing. In this paper, the number of decomposition levels was selected as six, and the first two components with the largest kurtosis were selected for reconstruction.