# Application of a New Enhanced Deconvolution Method in Gearbox Fault Diagnosis
## 4 Simulation Verification  
In order to verify the effectiveness and superiority of the proposed method, the following signals were constructed for simulation experiments. The vibration signal of the bearing fault is usually expressed as a periodic impact. The model is shown in Equation (38):  
\[ \begin{array}{l}x_{1}(t)=A_{m}\times\exp(-\frac{g}{T_{m}})\sin(2\pi f_{a}t) \\ x_{2}(t)=x_{1}(t)+\textit{noise}\end{array}  \tag{38}\]  
Figure 4: The flow chart of autoregressive moving average (ARMA)-MOMEDA method. ACF, autocorrelation coefficient; PACF, partial correlation coefficient. where \(x(t)\) is the periodic impact signal, \(A_{m}\) is the amplitude of the impact, \(g\) is the damping coefficient, \(T_{m}\) is the period of the impact, and \(f_{a}\) is the natural frequency. The parameter is set as follows: \(g=0.2,T_{m}=0.025\) s, \(f_{a}=320\) Hz. The resulting fault frequency is \(1/T_{m}=40\) Hz. Figure 5 shows the time domain waveform of the simulated signal. Figure 5a is the simulated bearing fault impact signal. Figure 5b is the noise of \(-4.36\) dB. Figure 5c shows the signal diagram with \(-4.36\) dB noise added. It can be seen that the impact signal shown in Figure 5a simulating the bearing fault is submerged by random noise, and no periodic impact is observed in the composite signal, shown in Figure 5c. In order to fully illustrate the superiority of ARMA-MOMEDA, the above simulation signals are processed by the AR filter [43], ARMA filter [44], MED filter [38], AR-MED [41], ARMA-MED, and ARMA-MOMEDA; Envelope analysis is performed on the processed result to obtain an envelope spectrum, so that the effects of each method were fully compared. Figure 6 shows the results obtained by the respective methods. Figure 6a is a time-domain diagram and a frequency-domain diagram after AR filter processing. From the time domain diagram, it can be seen that the signal can be noise-reduced, and the partially submerged shock signal appears in the time-domain waveform, but the impact has no obvious periodicity. In the envelope spectrum results, the spectral lines are also cluttered, and the spectral lines corresponding to the obvious faults are not resolved. Therefore, in a noisy environment, AR cannot accurately diagnose the fault frequency. Figure 6b is a time-domain diagram and a frequency-domain diagram after ARMA filter processing. From the time-domain diagram, it can be seen that the impact that is submerged by noise appears, and the impact has a certain periodicity. The envelope spectrum also shows the spectral line of the fault characteristic frequency. It can be seen that the ARMA result is better than the AR result. However, a large amount of noise can still be observed from the time-domain map, and the periodic impact is not obvious. However, although the frequency corresponding to the fault appears in the envelope spectrum, there are fewer spectral lines and the fault characteristics are not obvious. If you increase the noise, ARMA will also not detect the fault information. Figure 5: Simulation signal time-domain diagram. (**a**) The fault signal; (**b**) the noise of \(-4.36\) dB; (c) a composite signal of noise and fault signal. Figure 6c is a time-domain diagram and a frequency-domain diagram after MED filter processing. The impact of being submerged by noise appeared from the time-domain diagram, but no periodicity was observed. The frequency of the fault can be observed in the envelope spectrum, but there is no multiplier and there is a lot of noise. It can be seen that the MED extraction results are better than the AR filter, but not as good as the ARMA filter results. Figure 6d is a time-domain diagram and a frequency-domain diagram after AR-MED filter processing. In the time-domain diagram, the impact of being submerged by noise appears, and a certain periodicity can be observed. The envelope spectrum also shows the fault frequency. It can be seen that the results of AR-MED are better than those of AR and MED, and AR-MED can improve AR and MED filters.