# Application of a New Enhanced Deconvolution Method in Gearbox Fault Diagnosis
## 2 Basic Theory
### ARMA Method  
The parametric ARMA model can accurately describe the objective laws of dynamic systems. The autoregressive parameters are most sensitive to the changes in working conditions and have obvious advantages in small damage identification, noise reduction, and operability. In general, the time series required for prediction is a stationary time series, which is also a precondition for the differential autoregressive moving average model. In reality, most sequences are non-stationary sequences. Therefore, it is smoothed by appropriate differentiation of the sequence or logarithmic transformation, and then the differential autoregressive moving average model is used for modeling. The general formula of the autoregressive moving average model \(ARMA(n,m)\) can be expressed as follows:  
\[x_{t}=\sum_{i=1}^{n}\varphi_{i}x_{t-i}+\alpha_{t}-\sum_{j=1}^{m}\theta_{j} \alpha_{t-j}, \tag{11}\]  
where \(n\) is the autoregressive order; \(m\) is the moving average order; \(\varphi_{i}(i=1,2,\cdots,n)\) is the autoregressive parameter; \(\theta_{j}(j=1,2,\cdots,m)\) is the moving average parameter; and \(\alpha_{t}\) is a Gaussian white noise process with a mean of 0 and a variance of \(\sigma_{\alpha}^{2}\). The _ARMA (n, m)_ model has the tailing properties of both the autocorrelation coefficient and the partial autocorrelation coefficient. Trailing means that, as the independent variable \(k\) increases indefinitely, the function value tends to 0 at a negative exponential speed. When \(m=0\), the \(ARMA(n,m)\) model degenerates into the \(AR(n)\) model. When \(n=0\), the \(ARMA(n,m)\) model degenerates into the \(MA(m)\) model. The choice of ARMA for the prediction of vibration data of rolling bearings is to analyze the partial autocorrelation coefficient and autocorrelation coefficient of the sample to be tested,so as to select the appropriate model to fit the sequence to be tested, and finally use the information criterion to test the applicability of the model. The simple correlation between each sequence value \(x_{t},x_{t-1},\cdots,x_{t-k}\) that makes up the time series is called autocorrelation. The autocorrelation coefficient \(\gamma_{k}\) can be used to measure the degree of autocorrelation. \(k\) is a time interval. \[\gamma_{k}=\frac{E[x_{t}x_{t+k}]}{E\left[x_{t}^{2}\right]}=\frac{\sum\limits_ {t=1}^{a-k}\left(x_{t}-\overline{x}\right)\left(x_{t+k}-\overline{x}\right)} {\sum\limits_{t=1}^{a}\left(x_{t}-\overline{x}\right)^{2}}, \tag{12}\]  
where \(a\) is the sample size, \(k\) is the period of lag, and \(x\) is the sample average mean. It is known from [PERSON]'s inequality \(E[xy]\leq\sqrt{E[x^{2}]E[y^{2}]}\) that \([-1,1]\) is the value field of the autocorrelation coefficient \(\gamma_{k}\), and the absolute value of the autocorrelation coefficient is close to 1, the degree of autocorrelation is higher. Partial autocorrelation is defined as time series \(X_{t}\), which indicates the conditional correlation between time series values under certain conditions. The partial autocorrelation coefficient \(\varphi_{kk}\) is used to measure its correlation, and the value range \(\varphi_{kk}\) is \([-1,1]\). \[\left\{ \begin{array}{l}\gamma_{1}\\ \gamma_{k}-\sum\limits_{j=1}^{k-1}\varphi_{k-1,j}\gamma_{k-j}\\ \frac{1-k-1}{1-\sum\limits_{j=1}^{k-1}\varphi_{k-1,j}\gamma_{j}}\end{array} \right. \tag{13}\]  
The AIC (Akaike's Information Criterion) criterion extracts the maximum amount of information in the observation sequence and, from this point on, is suitable for testing the _ARMA (n, m)_ model order. The criteria function is defined as follows:  
\[AIC(p)=-2\ln L+2p, \tag{14}\]  
where \(p=m+n\), \(L\) is the likelihood function of the time series \(x_{t}\); if the time series \(x_{t}\) is stationary and normal, then  
\[L=\prod\limits_{t=1}^{N}\frac{1}{\sqrt{2\pi}\sigma_{a}}\exp[-\frac{1}{2\sigma _{a}^{2}}(x_{t}-\overset{\wedge}{\mu}_{t})^{2}], \tag{15}\]  
where \(\overset{\wedge}{\mu}_{t}\) is the mathematical expectation estimate of \(x_{t}\) whose time is \(t\), that is, \(E[x_{t}]\).