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The HT method is applied to perform the transient time-frequency analysis so as to characterize different faults in this system. The five reconstructed results in Figure 8 are transformed by the HT method and shown in Figure 9, where each sharp peak corresponds to the moment of the pressure wave passing through the specific pipe fault. Based on Equation (7), the total number of pipe faults and the locations of the faults in the system can then be predicted and the results are given in Table 3. Meanwhile, the relative errors of the prediction, which is defined by the difference between the predicted and real values normalized by the real value, are calculated and also listed under the location results in Table 3.
Table 3

Location results of the pipe faults by the time-frequency analysis

Reconstructed signalDetection results
Number1st2nd3rd4th
Fault typeBlockage1Leakage1Junction1Leakage2
Real distance7.05 m16.13 m30.22 m45.30 m
(a) IMF4 + IMF5 + IMF6 + IMF7 + IMF8 + IMF9 7.02 m 16.28 m 30.80m 46.75 m 
0.4% 0.9% 1.9% 3.2% 
(b) IMF5 + IMF6 + IMF7 + IMF8 + IMF9 7.44 m 16.08 m 30.13m 43.99 m 
5.5% 0.3% 0.3% 2.9 % 
(c) IMF6 + IMF7 + IMF8 + IMF9 6.62 m 16.54 m 29.21 m – 
6.1% 2.5% 3.3% – 
(d) IMF4 + IMF5 + IMF6 + IMF7 + IMF8 6.36 m 16.33 m 29.21 m 46.23 m 
9.8% 1.2% 3.3% 2.1% 
(e) IMF4 + IMF5 + IMF6 + IMF7 6.36 m 17.05 m – – 
9.8% 5.7% – – 
Reconstructed signalDetection results
Number1st2nd3rd4th
Fault typeBlockage1Leakage1Junction1Leakage2
Real distance7.05 m16.13 m30.22 m45.30 m
(a) IMF4 + IMF5 + IMF6 + IMF7 + IMF8 + IMF9 7.02 m 16.28 m 30.80m 46.75 m 
0.4% 0.9% 1.9% 3.2% 
(b) IMF5 + IMF6 + IMF7 + IMF8 + IMF9 7.44 m 16.08 m 30.13m 43.99 m 
5.5% 0.3% 0.3% 2.9 % 
(c) IMF6 + IMF7 + IMF8 + IMF9 6.62 m 16.54 m 29.21 m – 
6.1% 2.5% 3.3% – 
(d) IMF4 + IMF5 + IMF6 + IMF7 + IMF8 6.36 m 16.33 m 29.21 m 46.23 m 
9.8% 1.2% 3.3% 2.1% 
(e) IMF4 + IMF5 + IMF6 + IMF7 6.36 m 17.05 m – – 
9.8% 5.7% – – 
Figure 9

Instantaneous frequency analysis by different combinations of IMF signals.

Figure 9

Instantaneous frequency analysis by different combinations of IMF signals.

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