# Intelligent Fault Diagnosis of Rotating Machinery Using Hierarchical Lempel-Ziv Complexity
## 2 Proposed Hierarchical Lempel-Ziv Complexity
### Simulated Impulsive Signal  
In this subsection, one impulsive signal was adopted to verify the advantage of proposed HLZC in detecting various fault types. Three different bearing fault types were utilized in the simulated signal: bearing with ball fault, bearing with inner race fault, and bearing with outer race fault. The number of sample points of the synthetic signal was 24,600, which was cut out by a sliding window of 2048 points with a step length of 256 points. Three simulated faulty bearing signals in time domains are shown in Figure 5a. For comparison purposes, LZC, MLZC, and HLZC were all utilized to process the impulsive signals. In this paper, three commonly used distance measures--Euclidean distance (ED), Chebychev distance (CD), and Minkowski distance (MD)--were all applied to verify the advantage of our proposed method in tracking the impulses. Note that we averaged the first 10 samples as normal samples. The distance value between each sample and normal samples was computed for comparisons. Here, we set the scale \(\tau=\) 1:20 for MLZC and the number of hierarchical layers \(k=4\) for HLZC methods. The obtained results are shown in Figures 5-7. Two conclusions can be drawn from Figures 5-7 as follows. First, it can be observed from (b) and (c) of three figures that the original LZC and MLZC could not detect the impulses derived from three different bearing fault types, resulting in failure of bearing feature extraction. In contrast, our proposed HLZC method not only identified three different bearing types by tracking the impulse, but also generated less fluctuation and high accuracy, as shown in (d) of the three figures. Second, the results using the three distance measures had a high consistency, and thus the fault detection ability of three methods can be listed as HLZC \(>\) MLZC \(>\) LZC. This further demonstrated that our proposed HLZC method has a significant advantage in fault feature extraction. To explain the advantage of our proposed HLZC in detecting the impulses, we enlarged the bearing with outer race fault signal and conducted the fast Fourier transform (FFT) analysis. The time domain waveforms and its corresponding frequency spectra are shown in Figure 8a,b, respectively. As can be seen, the fault information was mainly located in the high-frequency component (1000-1500 Hz). The fault information embedded in the high-frequency component was ignored using the traditional MLZC method, resulting in worse impulsive detection performance. Compared with the MLZC method, HLZC utilized both averaging and differencing process to extract the fault information hidden in both low-frequency and the high-frequency components. Therefore, the proposed HLZC method generates the best performance in fault feature extraction. Figure 5: Results of sliding window analysis of Lempel-Ziv complexity (LZC), MLZC, and HLZC methods: (**a**) the simulated impulsive signal, (**b**) the Euclidean distance (ED) value of LZC, (**c**) the ED value of MLZC, and (**d**) the ED value of HLZC. Note that the fault frequency (BPFO) obtained using the envelope spectrum analysis was equal to the side band interval frequency of bearing intrinsic frequency in the conventional FFT method. First, the frequency spectrum using FFT could not detect the bearing fault frequency (BPFO) directly. It can be seen in Figure 8b that the bearing faults could be diagnosed by observing the intrinsic frequency and its harmonics. Second, we also conducted the envelope spectrum analysis. The fault frequency (BPFO) and its harmonics can be clearly observed in Figure 8c. Because the bearing fault can generate repetitive impulses, the measured vibration signal is thus a typical amplitude- and frequency-modulated (AM-FM) signal. The amplitude-modulated (AM) signal can be obtained by using envelope demodulated analysis, and then the fault frequency (BPFO) can be obtained by conducting the FFT on the AM signal.