## 5 Simulation Test Research
### Simulation Study on Separation of Mixed Signals with Different Frequency Carriers  
In the simulation experiment, the vibration of the rotor system is mainly simulated. Rotating parts of the rotor system include the rotor, bearing, and other rotating parts. The vibration signal of the rotor system can be determined from the sinusoidal signal of each frequency and each resonance frequency. The simulation signal can be expressed as  
\[\begin{array}{c}s_{1}=\sin(100\pi t)\\ s_{2}=\cos(200\pi t)+\sin(300\pi t)\\ s_{3}:\text{ random noise}\end{array} \tag{26}\]  
The randomly generated mixed matrix is as follows:  
\[A=\left[\begin{array}{cccc}0.4&0.8&0.3\\ 0.6&0.1&0.9\\ 0.8&0.5&0.4\end{array}\right] \tag{27}\]  
Assume that the rotor rub impact fault frequency is a mixed signal of 100 Hz and 150 Hz, and other source signals are composed of a vibration signal with a base frequency of 50 Hz and Gaussian white noise. The source signal waveform obtained is shown in Figure 2.  
The source signal is randomly and linearly mixed, and the mixing matrix is \(A\). The mixed signal is as shown in Figure 3.  
Figure 3: The time-frequency waveform of mixed signal.  
Figure 2: The time-frequency waveform of source signal.  
In Figure 3, it can be seen that the characteristics of the source signal have been completely hidden in the mixed signal, and the information on the source signal cannot be read from the time domain waveform.  
Figure 4 shows the signal directly separated from the mixed signal by the JADE algorithm. It can be seen, from Figure 4, that the noise signal has not been successfully separated, and the other signals separated contain many harmonics, so the signal characteristics cannot be accurately identified, which indicates that the method is invalid under strong noise.  
Figure 5 is the signal separated by the classical SOBI algorithm. It can be seen from Figure 5 that, although the noise signal has been successfully separated, the other two separated signals also contain many harmonics, so it is impossible to accurately identify the signal characteristics. It means that this method can separate the noise signal, but it cannot effectively separate other source signals.  
By comparing Figures 2 and 4, Figure 5, Figure 6, it can be seen that the separation methods based on the classical JADE and SOBI algorithm have more frequency components marked with red circles. It indicates that the separation result is not accurate. From the evaluation indexes in Table 1, we can also see that the improved algorithm MF-SOBI is better than SOBI and JADE.  
Figure 4: The frequency domain signals of the JADE algorithm.  
Figure 5: The frequency domain signals of the SOBI algorithm.  
Figure 6: The frequency domain signals of the MF-SOBI algorithm.  
By comparing Figures 2-6, it can be seen that the uncertainty of the separated signal compared with the source signal mainly lies in the uncertainty of amplitude and signal sequence brought about by the blind source separation technology itself. In addition to this uncertainty, other characteristics of the signal have been better recovered. Furthermore, the performance index and similarity coefficient of the algorithm are calculated and compared with the results of the SOBI and JADE algorithms. The performance index shows the overall separation ability of the algorithm; the smaller the value is, the better. The similarity coefficient shows the reduction of a single source signal; the closer the absolute value is to 1, the better the separation result. The comparison results are shown in Table 1.  
From the data in Table 1, it can be seen that, under the interference of strong impulse noise, the improved algorithm in this paper effectively separates the source signals, and has a higher separation performance than the classical algorithm.