## 3 Signal Analysis
### Bispectrum Analysis
#### 3.2.1 AR Model and Third-Order Cumulants  
According to the analysis results in Section 3.1, arc fault is an obvious random process. So it is very difficult for us to build a mathematical function to describe the variation rules with time. However, we may introduce time series analysis theory to reveal the correlation among the acquired data from arc faults. Here, the AR model is introduced. As a result of environmental impacts, circuit noise, quantization noise and load disturbances, there is much interference mixed in arc fault signals. The random signals which exist in arc faults are assumed to be interfered by the non-Gaussian white noise \(a(t)\) whose mean value is zero. The zero-mean non-Gaussian colored noise \(u(t)\) containing abundant useful dynamic information of arc faults is assumed to be the output of AR process driven by \(a(t)\)[30, 44]. Therefore, the time series AR model of arc fault can be described as  
\[u(t)+\sum_{k=1}^{q}\varphi_{k}u(t-k)=a(t)\,,\,(t=1,\,2,\,3,\,\ldots) \tag{1}\]  
where \(q\) is the order of model AR(\(q\)), \(\varphi_{k}\)\((k=1,2,\cdots,q)\) is the autoregressive coefficient and \(a(t)\) is the input signal. According to the relationship between cumulants and moments in statistical theory, the third-order cumulants of \(u(t)\) can be expressed as  
Figure 3: The power spectra of vacuum cleaner. (**a**) Normal; (**b**) Arc fault. \[C_{{}_{3u}}(\tau_{1},\tau_{2}) =E\big{[}u(t)u(t+\tau_{1})u(t+\tau_{2})\big{]}-E\big{[}u(t)\big{]}E \big{[}u(t+\tau_{1})u(t+\tau_{2})\big{]}-E\big{[}u(t+\tau_{1})\big{]}E\big{[}u(t+ \tau_{2})u(t)\big{]} \tag{2}\] \[-E\big{[}u(t+\tau_{2})\big{]}E\big{[}u(t)u(t+\tau_{1})\big{]}+2E \big{[}u(t)\big{]}E\big{[}u(t+\tau_{1})\big{]}E\big{[}u(t+\tau_{2})\big{]}\]  
where \(\tau_{1}\) and \(\tau_{2}\) are lags. The mean of \(u(t)\) can be denoted as \(\mu_{u}=E\big{[}u(t)\big{]}\) and the correlation function can be expressed as \(\rho_{u}(\tau)=E\big{[}u(t)u(t+\tau)\big{]}\). Then Equation (2) can be denoted as  
\[C_{{}_{3u}}(\tau_{1},\tau_{2})=E\big{[}u(t)u(t+\tau_{1})u(t+\tau_{2})\big{]}- \mu_{u}\rho_{u}(\tau_{2}-\tau_{1})-\mu_{u}\rho_{u}(\tau_{2})-\mu_{u}\rho_{u}( \tau_{1})+2\mu_{u}^{3} \tag{3}\]  
After the preprocessing, the mean value of \(u(t)\) is zero, namely, \(\mu_{u}=0\). Therefore, Equation (3) can be simplified as  
\[C_{{}_{3u}}(\tau_{1},\tau_{2})=E\big{[}u(t)u(t+\tau_{1})u(t+\tau_{2})\big{]} \tag{4}\]  
Here the third-order cumulants of \(u(t)\) equal to its third-order moments. For the input signal \(a(t)\), it has the following properties:  
\[ \begin{cases}E\big{[}a(t)a(t+\tau_{1})\big{]}=\sigma_{a}^{2}\delta(\tau_{1}) \\ E\big{[}a(t)a(t+\tau_{1})a(t+\tau_{2})\big{]}=\gamma\delta(\tau_{1},\tau_{2}) \\ E\big{[}a(t)u(t-\tau_{1})\big{]}=0\ \ (\tau_{1}\geqslant 0,\tau_{2}\geqslant 0) \end{cases}  \tag{5}\]  
where \(\sigma_{a}^{2}\) is the variance of \(a(t)\), \(\delta(\tau_{1})\) is the one-dimensional unit impulse function, \(\delta(\tau_{1},\tau_{2})\) is the two-dimensional unit impulse function and the coefficient \(\gamma=E[a^{3}(t)]\
eq 0\). For a stationary physical process, Equation (1) can be expressed by using third-order cumulants of \(u(t)\)[30, 31]:  
\[C_{{}_{3u}}(\tau_{1},\tau_{2})+\sum_{i=1}^{q}\!\!  
where the frequency transfer function \(H\left(\omega\right)\triangleq\sum\limits_{\beta\sim\cdots}^{n}h(\beta)e^{-i\omega \beta}\) and \(H^{*}\left(\omega_{1}+\omega_{2}\right)\) is the complex conjugate of \(H\left(\omega_{1}+\omega_{2}\right)\)[31,46].The frequency transfer function of the \(q\)-th order AR process in Equation (1) can be further expressed as  
\[H\left(\omega\right)=\frac{1}{1+\sum\limits_{k=1}^{q}\omega_{k}e^{-i\omega }} \tag{16}\]  
To reduce the computational complexity of the bispectrum, bispectrum slice will be introduced. The one-dimensional slices of the bispectrum are the reconstructed spectra.