# Weak Multiple Fault Detection Based on Weighted
## 4 Simulation Signal Analysis  
The actual rolling bearing fault signal model is simplified to verify the effectiveness of the proposed algorithm. Therefore, the simulation model of the rolling bearing multiple fault signal is established:  
Figure 3: The fault feature extraction algorithm based on WOGS. \[\left\{ \begin{array}{l}s(t)=s_{1}(t)+s_{2}(t)+w(t)\\ s_{1}(t)=e^{-80t}\sin(600\pi t)\\ s_{2}(t)=e^{-250t}\sin(2000\pi t)\end{array} \right. \tag{25}\]  
where \(s_{1}(t)\) is the impulse response caused by the first partial fault with feature frequency \(f_{1}=20\) Hz, \(s_{2}(t)\) is the impulse response caused by the second local fault with feature frequency \(f_{2}=70\) Hz, and \(w(t)\) is white Gaussian noise. The MATLAB script function awgn (\(x\), \(SNR\)) refers to the addition of white Gaussian noise to the vector signal \(x\). The scalar snr specifies the signal-to-noise ratio per sample, in dB. Here, the vector signal is set to \(s_{1}(t)+s_{2}(t)\), and the signal-to-noise ratio is set to be \(SNR=-5\). Figure 4a-c correspond to \(s_{1}(t)\), \(s_{2}(t)\), and \(s(t)\), respectively. Seeing the periodic impulse feature from synthesized signal \(s(t)\) is difficult. In order to extract periodic impulse features, spectral analysis is a common type of method; such methods include frequency spectral analysis and envelope spectral analysis. Frequency spectrum analysis is used to perform a Fourier transformation on the original signal. Envelope spectrum analysis performs the Hilbert transformation on the original signal, and then the Fourier transform is applied to the envelope. The amplitude of fault feature frequency in the frequency spectrum is small, while the envelope spectrum is sensitive to impulse components, so the amplitude of fault feature frequency in the envelope spectrum is very high and easy to identify. Figure 5 shows the frequency and envelope spectrum of synthesized signal \(s(t)\). Figure 5a exhibits that the resonance band corresponding to two local faults, which are marked with red lines. In the envelope spectrum, the frequency components at 20 and 70 Hz have the largest magnitude; they represent the feature frequencies of the two impulses, respectively. In addition to the above feature frequencies, significant noise components are available. Figure 4: Simulated signal: (**a**) \(s_{1}(t)\). (**b**) \(s_{2}(t)\). (**c**) \(s(t)\). Figure 6 shows the time-frequency representation of the simulated signal based on the Morlet-wavelet transform. As can be seen from the figure, TF\({}_{1}\) and TF\({}_{2}\) correspond to the time-frequency distribution of two local fault signals \(s_{1}(t)\) and \(s_{2}(t)\) in Equation (25), respectively. It can be seen from Figure 6 that they have periodic impact characteristics along the time axis, but it is difficult to get accurate periodic values. The Morlet-wavelet coefficients are taken as the observed signal \(y\) in Equation (1). Figure 7a-c shows the denoised results based on the WOGS algorithm with a group size \(7\times 2\) (i.e., seven frequency bins \(\times\) two time bins). Comparing Figures 5a and 7a, we find that the periodic impulse features are evident after noise reduction. Figure 7b shows the frequency spectrum of the denoised signal, where the two resonance bands representing local faults are preserved. The dominant frequency components in Figure 7c are 20, 40, 60, and 80 Hz, which represent the feature and harmonic frequencies of the first impulse component, and the frequency components at 70, 140, and 210 Hz represent the feature and harmonic frequencies of the second impulse component. Figure 5: Frequency spectrum and envelope spectrum of the simulated signal: (**a**) Frequency spectrum of the simulated signal. (**b**) Envelope spectrum of the simulated signal. Figure 6: Time-frequency representation of simulated signal. The simulated signal is decomposed by the tunable Q-factor wavelet transform (TQWT), which is similar to the WOGS algorithm.