# Application of a New Enhanced Deconvolution Method in Gearbox Fault Diagnosis
## 2 Basic Theory
### ARMA Method
Then, by substituting \(x_{t}-\overset{\wedge}{\mu}_{t}=a_{t}\) into the above formula and performing the multiplication calculation, Formula (16) is obtained. \[L=(\frac{1}{2\pi\sigma_{a}^{2}})^{\frac{N}{2}}\exp[-\frac{1}{2\sigma_{a}^{2}} \underset{t=1}{\overset{\wedge}{\sum}}a_{t}^{2}], \tag{16}\]  
where \(\sigma_{a}^{2}\) is calculated as follows:  
\[\sigma_{a}^{2}=\frac{1}{N}\underset{t=1}{\overset{\wedge}{\sum}}a_{t}^{2}. \tag{17}\]  
Substituting Equation (17) into Equation (16) and taking the natural logarithm on both sides of the equal sign, there are the following:  
\[\ln L=-\frac{N}{2}(\ln 2\pi+\ln\sigma_{a}^{2})-\frac{N}{2}, \tag{18}\]which is  
\[-2\ln L=N\ln 2\pi+N\ln\sigma_{a}^{2}+N. \tag{19}\]  
Substituting Equation (19) into Equation (13), for a given data type, the length is \(N\), and the latter two of Equation (19) are constant, which has no effect on the comparison result of \(AIC(p)\) and can be omitted, thus obtaining the following:  
\[AIC(p)=N\ln\sigma_{a}^{2}+2p. \tag{20}\]  
Obviously, given a model parameter estimation method,\(AIC(p)\) is a function of \(p\). If \(p\) increases, then \(\ln\sigma_{a}^{2}\) decrease, but the latter \(2p\) increases. Therefore, the model order \(p\) when the value of \(AIC(p)\) is the smallest is the applicable model order. After the order of the ARMA model is determined, the parameters in AR and MA should be estimated. In this paper, sequence estimation method is used in the simultaneous order theory estimation method to first estimate \(\varphi_{i}\), and then estimate \(\theta_{j}\). It can be known from the recursive formula of \(R_{k}\) that, when \(k>m\), the formula of \(R_{k}\) will not contain \(\theta_{j}\). \[R_{k}=\varphi_{1}R_{k-1}+\varphi_{2}R_{k-2}+\cdots+\varphi_{n}R_{k-n}(k>m) \tag{21}\]  
Note the subscript of \(R_{k}\) in the above equation. When \(k<0\), as \(R_{k}\) is the property of the even function, there is \(R_{-k}=R_{k}\), separately \(k=m+1,\cdots,m+n\), the following matrix equation can be seen:  
\[\left[ \begin{array}{c}R_{m+1}\\ R_{m+2}\\ \cdots\\ R_{m+n}\end{array} \right]=\left[ \begin{array}{ccccc}R_{m}&R_{m-1}&R_{m-2}& \cdots&R_{m-n+1}\\ R_{m+1}&R_{m}&R_{m-1}&\cdots&R_{m-n+2}\\ \cdots&\cdots&\cdots&\cdots\\ R_{m+n-1}&R_{m+n-2}&R_{m+n-3}&\cdots&R_{m}\end{array} \right]\left[ \begin{array} []{c}\varphi_{1}\\ \varphi_{2}\\ \cdots\\ \varphi_{n}\end{array} \right]. \tag{22}\]  
If both sides of the equation are divided by \(R_{0}\), then each \(R_{k}\) in the equation is transformed into the autocorrelation coefficient \(\rho_{k}\). This equation is the modified Yule-Walker equation, which can be abbreviated as follows:  
\[R_{A}=R_{B}\varphi. \tag{23}\]  
In Equation (23), \(R_{A}\) is the n-dimensional column vector to the left of the matrix equation, \(R_{B}\) is the n-order order square matrix to the right of the matrix equation, and \(\varphi\) is a n-dimensional column vector composed of autoregressive coefficients. This matrix is not a Toeplitz matrix, but generally has an inverse matrix, so the matrix equation can be solved, and the estimated value of the autoregressive parameter is as follows:  
\[\varphi=R_{B}^{-1}R_{A}. \tag{24}\]  
At this point, the autoregressive parameters \(\varphi_{i}(i=1,2,\ldots,n)\) of the ARMA model have been determined. In the ARMA model in Equation (11), let \(y_{t}=x_{t}-\sum\limits_{i=1}^{n}\varphi_{i}x_{t-i}\), then \(y_{t}=a_{t}-\sum\limits_{j=1}^{m}\theta_{j}a_{t-j}\). This can also be recorded as follows:  
\[ \begin{array}{l}y_{t}=-\sum\limits_{i=0}^{n}\varphi_{i}x_{t-i}(\varphi_{0}=- 1)\\ y_{t}=-\sum\limits_{j=0}^{n}\theta_{j}a_{t-j}(\theta_{0}=-1)\end{array}  \tag{25}\]  
Multiply both sides of the AR model of Equation (25) by \(y_{t-k}\) and find the mathematical expectation, there is  
This formula is a one-dimensional m-order equation for \(1/\eta_{j}\), then \(m\) roots \(1/\eta_{j}(j=1,2,\ldots,m)\) can be solved by this formula; then, \(\eta_{j}\) is substituted into the polynomial.