# Fault Inference of Electronic Equipment Based on Multi-State Fuzzy Bayesian Network
## 4 Example Analysis  
A digital super heterodyne receiver is taken as an example and its functional block diagram is shown as Figure 4. Owing to the faults of power supply unit, microcomputer unit, and panel unit belong to hard faults, the multi-state fuzzy sub-frequency-doubled unit, short-wave mixing unit, frequency synthesizer unit, and medium- and low-frequency units are mainly considered. Of them, the frequency synthesizer unit provides three local oscillator signals, i.e., local oscillator signals 1, 2, and 3 required by receiver. Local oscillator signals 1 and 2 act on the first and second frequency mixing modules of short-wave mixing unit, while local oscillator signal 3 acts on the third frequency mixing module of medium- and low-frequency units. Therefore, local oscillator signals 1 and 2 commonly acting on short-wave mixing unit are regarded as a node \(x_{2}\) of BN, while the local oscillator signal 3 is seen as a node \(x_{3}\) of BN. Based on SDT and FMECA, BN of leaf nodes is established by selecting faults of medium- and low-frequency units, as shown in Figure 5.  
Figure 4: Block diagram of principle of the super heterodyne receiver. Where, \(x_{1}\), \(x_{2}\), \(x_{3}\), \(y_{1}\), and \(T\) represent the sub-frequency doubled unit, the local oscillator signals 1 and 2 of frequency synthesizer, local oscillator signal 3 of frequency synthesizer, short-wave mixing unit and medium- and low-frequency units, respectively. In BN, \(x_{1},x_{2}\), and \(x_{3}\) are regarded as variables of root nodes; \(T\) indicates the variable of leaf nodes and \(y_{1}\) denotes the variable of intermediate node. Each module shows three operating states--i.e., normal state, slight (sub-health) fault, and fault--and they are supposed to be relatively independent. In order to simplify the process of fault inference, the faults induced by parameters smaller or larger than the normal ones are uniformly defined as sub-health faults, while the open-circuit and short-circuit faults are defined as faults. Moreover, \(u_{0}(\mathrm{t})\), \(u_{0.5}(\mathrm{t})\) and \(u_{1}(\mathrm{t})\) are used as the membership degrees in normal state, sub-health state and fault state. The corresponding relationship of these membership degrees with Formula (3) is shown as  
\[u_{0}(\mathrm{t})=r_{3}(\mathrm{t})\]  
\[u_{0.5}(\mathrm{t})=r_{2}(\mathrm{t})\text{ or }r_{4}(\mathrm{t})\]  
\[u_{1}(\mathrm{t})=r_{1}(\mathrm{t})\text{ or }r_{5}(\mathrm{t})\]  
In Table 2, \(\widetilde{P}(y_{1}=0|x_{1}=0,x_{2}=0.5)+\widetilde{P}(y_{1}=0.5|x_{1}=0,x_{2 }=0.5)=0.5\), \(\widetilde{P}(y_{1}=0|x_{1}=0.5,x_{2}=0)+\widetilde{P}(y_{1}=0.5|x_{1}=0.5,x_{ 2}=0)+\widetilde{P}(y_{1}=1|x_{1}=0.5,x_{2}=0)=1\) and \(\widetilde{P}(y_{1}=0|x_{1}=0.5,x_{2}=0.5)+\widetilde{P}(y_{1}=0.5|x_{1}=0.5,x _{2}=0.5)=0.3\). In Table 3, \(\widetilde{P}(T=0|y_{1}=0,x_{3}=0.5)+\widetilde{P}(T=0.5|y_{1}=0,x_{3}=0.5)=0.8\), \(\widetilde{P}(T=0|y_{1}=0.5,x_{3}=0)+\widetilde{P}(T=0.5|y_{1}=0.5,x_{3}=0)+ \widetilde{P}(T=1|y_{1}=0.5,x_{3}=0)=1\) and \(\widetilde{P}(T=0.5|y_{1}=0.5,x_{3}=0.5)+\widetilde{P}(T=1|y_{1}=0.5,x_{3}=0.5) =0.7\). \begin{table}
\begin{tabular}{c c c c c} \hline \(x_{1}\) & \(x_{2}\) & \(P(y_{1}\)=0|x_{1},x_{2})\) & \(P(y_{1}\)=0.5|x_{1},x_{2})\) & \(P(y_{1}\)=1|x_{1},x_{2})\) \\ \hline
0 & 0 & 1 & 0 & 0 \\ \hline
0 & 0.5 & \(\widetilde{P}(y_{1}=0|x_{1}=0,x_{2}=0.5)\) & \(\widetilde{P}(y_{1}=0.5|x_{1}=0,x_{2}=0.5)\) & 0.5 \\  & & [0.1, 0.2, 0.3] & [0.2, 0.3, 0.4] & \\ \hline
0 & 1 & 0 & 0 & 1 \\ \hline
0.5 & 0 & \(\widetilde{P}(y_{1}=0|x_{1}=0.5,x_{2}=0)\) & \(\widetilde{P}(y_{1}=0.5|x_{1}=0.5,x_{2}=0)\) & \(\widetilde{P}(y_{1}=1|x_{1}=0.5,x_{2}=0)\) \\  & & [0.3, 0.4, 0.5] & [0.3, 0.4, 0.5] & [0.1, 0.2, 0.3] \\ \hline
0.5 & 0.5 & \(\widetilde{P}(y_{1}=0|x_{1}=0.5,x_{2}=0.5)\) & \(\widetilde{P}(y_{1}=0.5|x_{1}=0.5,x_{2}=0.5)\) & 0.7 \\  & & [0.1, 0.2] & [0.1, 0.2, 0.3] & \\ \hline
1 & 0.5 & 0 & 0 & 1 \\ \hline
1 & 1 & 0 & 0 & 1 \\ \hline \end{tabular}
\end{table} 
Table 2: CPT of intermediate node \(y_{1}\).