## 3 Simulation Signal Analysis  
Bearings are mainly used to support rotating parts in mechanical equipment and their vibration signals always contain much information, such as fault characteristics, along with noise. The key step of fault diagnosis is an effective feature extraction of vibration signals. Commonly, the vibration signals contain harmonic components, modulation components and noise components. In order to evaluate the effectiveness of the proposed method for fault diagnosis, the simulation signals are generated as follows:  
\[y(t)=x_{1}(t)+x_{2}(t)+n(t) \tag{14}\]  
\[x_{1}(t)=2e^{(-0.2\pi f_{1}t)}sin(2\pi f_{1}t) \tag{15}\]  
\[x_{2}(t)=sin(2\pi f_{2}t) \tag{16}\]  
where \(x_{1}(t)\) is the shock signal with the frequency of \(f_{1}=50\) Hz, \(x_{2}(t)\) is the harmonic signal with the frequency of \(f_{2}=10\) Hz, and \(n(t)\) is a Gaussian white noise with a variance of 0.5. Thus, \(y(t)\) is a composite single-channel signal, which is combined by the shock signal, the harmonic signal and the noise. The sampling frequency of the signal is chosen as 6000 Hz and the sampling point is set as 4000 N. Original shock signal without noise in the time-domain is shown in Figure 4a and the harmonic signal without noise is shown as Figure 4b. Figure 4c is the composite original single-channel signal without noise in the time-domain, and Figure 4d is composite signal with noise. Figure 4: The time response of different components: (**a**) original shock signal in time-domain; (**b**) original harmonic signal in time-domain; (**c**) composite signal without noise in time-domain; and (**d**) composite signal with noise in time-domain. According to Figure 4d, we can find that the characteristics of two constituted signals in the time-domain cannot be clearly indicated under the strong background noise. In the section of simulation signal analysis, in order to accurately evaluate the proposed method on signal reconstruction under noisy conditions, the proposed method, conventional TSSA based on CP-ALS, the traditional BSS method-Fast Independent Component Analysis (Fast-ICA) [37], and EMD-ICA are employed to the comparative analysis process. IMF\({}_{\text{S}}\) is obtained by EMD to the measured single channel composite signal. Thus, the composite signal and the IMF\({}_{\text{S}}\) of the decomposition can form a new multidimensional observation signal. In this way, the dimension of the observation signal can be increased, so that the new observation signal can be in accordance with the blind source separation condition. Then, we can obtain the correlation matrix about the new observation matrix, and the singular value decomposition of the correlation matrix is performed. Finally, the number of source signals can be judged further by the Bayesian information criterion. The BIC value is shown as Figure 5. According to the Figure 5, when \(k=n=2\), we can obtain the maximum BIC value, which indicates the number of source signals should be 2, thus we achieve the goal of estimating the correct number of source signals. After obtaining the number of source signals, the above-mentioned four different SCBSS methods are used to analyze the composite simulation signal. The result of Fast-ICA is shown in Figure 6, where Figure 6a presents the recovered shock signal in time-domain and Figure 6b presents the recovered harmonic signal. From the figure, it can be seen that the Fast-ICA cannot extract the shock signal and the recovered harmonic signal with noise. Hence, the Fast-ICA method is not suitable to achieve the accurate separation of composite original signal, which contains shock signal and high background noise. The operator of EMD is employed for the simulation signal and the result is shown in Figure 7. Firstly, several IMF\({}_{\text{S}}\) can be obtained from the composite original signal using EMD. Then, we need to calculate the correlation coefficient between each IMFs and the original composite signal. Figure 5: The Bayesian information criterion (BIC) value of composite original.