# A Wear Debris Signal Processing Method Based on Inductive Monitoring for Aero-Engine
## 3 Time Domain Analysis  
The overall process of this method is illustrated in Figure 4, including time domain analysis, histogram in the sliding window, and debris information storage.  
In order to enhance the difference between debris signature and noise signature, time domain analysis is carried out firstly. A time interval is set firstly to separate the raw monitoring signal into a number of bins, then an index can be calculated by the data in each bin, so that the raw monitoring signal is converted to the index value signal, in which the difference between debris signature and noise signature has been enhanced. The time interval is equal to the time span of the debris characteristic signature, which is related to  
Figure 3: The demodulated signal in a test.  
Figure 2: (**a**) Triple-coil inductive sensor and (**b**) its signal-conditioning box.  
the axial length of the sensitive space of the inductive sensor and the velocity of the oil flow and can be expressed as Equation (1).  
\[t_{c}=\frac{L_{s}}{v} \tag{1}\]  
where \(t_{c}\) denotes the time span of characteristic signature, \(L_{s}\) denotes the axial length of the sensitive space of the inductive sensor, and \(v\) denotes the velocity of the oil flow. For the signal illustrated in Figure 3, the time interval is solved to 0.004 s.  
In order to compare the enhancement capability of the difference between the debris signature and noise signature, five typical time domain indices are selected: (i) energy; (ii) root mean square (RMS); (iii) standard deviation; (iv) kurtosis; and (v) peak value. The indices mentioned above describe different signal properties from different aspects, whose definition are given below. Where \(x_{i}\) represents the discrete signal, \(n\) is the total data number of the discrete signal, \(I_{1}\) to \(I_{5}\) denote the time domain indices (i) to (v), correspondingly.  
\[I_{1}=\sum\limits_{i=1}^{n}x_{i}^{2} \tag{2}\]  
\[I_{2}=\sqrt{\frac{1}{n}\sum\limits_{i=1}^{n}x_{i}^{2}} \tag{3}\]  
\[I_{3}=\sqrt{\frac{1}{n-1}\sum\limits_{i=1}^{n}\left(x_{i}-\bar{x}\right)^{2}} \tag{4}\]  
\[I_{4}=\frac{\frac{1}{n}\sum\limits_{i=1}^{n}\left(x_{i}-\bar{x}\right)^{4}}{ \left(\frac{1}{n}\sum\limits_{i=1}^{n}\left(x_{i}-\bar{x}\right)^{2}\right)^{ 2}} \tag{5}\]  
Figure 4: The overall process of debris signature detection method.  
\[I_{5}=\max(x_{i}) \tag{6}\]  
The results of the five indices are illustrated in Figure 5. Conclusions can be drawn from the comparison of each subplot in Figure 5 with Figure 3: (i) kurtosis cannot enhance the difference, even debris signatures are lost in the background; (ii) energy is the most effective index among these indices, the debris signature is more outstanding from the background, and the background level is low and stable, not affected by aero-engine's working conditions.  
Figure 5: Results of different time domain indices: (**a**) energy, (**b**) RMS, (**c**) standard deviation, (**d**) kurtosis, and (**e**) peak value.  
To verify the impact of the time interval on time domain analysis, set \(t_{1}=10\cdot t_{c}=0.04\) s and \(t_{2}=0.1\cdot t_{c}=0.0004\) s, the time domain index still has energy. The comparison is illustrated in Figures 6 and 7. Conclusions can be drawn that there are disadvantages when the time interval is too large or too small. If the time interval is excessively large, the difference between debris signature and noise signature will be reduced, and the background can be affected by the working conditions. If the time interval is excessively small, a complete debris signature may be split into several parts, which introduces unnecessary troubles in debris signature detection. Therefore, it is reasonable to set the time interval as the time span of the debris characteristic signature.