Another simulated signal is discussed. It is the combination of a sine wave with unit amplitude and 1
2π Hz and a sine wave with 0.1 unit amplitude and 5
π
Hz. At six middle crests of the sine wave with unit amplitude, the cosine wave with a 0.1 unit amplitude and 20π
Hz are riding on it as Fig.4.7. The function of the simulated signal is sint + 0.1sin10t otherwise., 0 ≤ t ≤ 6π
(4.1)The simulated signal is decomposed by EEMD with a white noise level at 10% of the standard deviation of the signals and ensembling 30 times. In Fig. 4.8, the frequency of intermittent signal in IMF 2 is the same to the frequency in IMF 3. In addition IMF 4 and IMF 5 have the same period. There are two events of multi-mode problem. That is the multi-mode problem could happen not only one time. We cluster the IMFs to get the dendrogram (Fig. 4.9) and give 5 which is the number of clusters. All IMFs are combined into CIMFs according to the dendrogram. CIMF 1 is a white noise from EEMD algorithm. CIMF 2 is the intermittent signal. CIMF 3 is the sine wave with 0.1 unit amplitude and 5
π
Hz. CIMF 4 is the main sine wave of the simulated signal, and CIMF 5 is the numeric error signal. CIMFs are shown in Fig. 4.10. The results demonstrate that the proposed method is better than only EEMD algorithm for extract-ing useful information.Figure 4.7: sin(t) + 0.1 sin(10t) and the intermittent signal
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Figure 4.8: Results of the signal: sin(t) + 0.1 sin(10t) and intermittent signal which is decomposed by EEMD. The last means the residue.
Figure 4.9: The figure is the dendrogram with complete linkages for IMFs of sin(t) + 0.1 sin(10t) and the intermittent signal when the number of clusters is 5
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−0.05 0 0.05
CIMF1
0 2 4 6 8 10 12 14 16 18 20
−0.2 0 0.2
CIMF2
0 2 4 6 8 10 12 14 16 18 20
−0.2 0 0.2
CIMF3
0 2 4 6 8 10 12 14 16 18 20
−2 0 2
CIMF4
0 2 4 6 8 10 12 14 16 18 20
0 1
2x 10−5 CIMF5
Figure 4.10: CIMFs of sin(t) + 0.1 sin(10t) and the intermittent signal
4.3 The practical signal: a voice from a wind turbine at Chunan Miaoli
This subsection presents an application for a voice signal of a wind turbine at Chu-nan Miaoli. The wind turbine has 67 meters high, with three rotors diameter of 70 meters, as shown in Fig.4.11a(The figure is fromhttp://www.bhes.ntpc.edu.tw/
~bhesnet9/windpower3.html). The voice signal was recorded with sampling fre-quency 1087 Hz in 5 minutes, and we take the 30 second
∼ 90 second part of it as Fig.
4.11b. We decomposed it into 14 IMFs and a trend through EEMD algorithm with a white noise level at 10% of the standard deviation of the signals and ensembling 30 times, as Fig. 4.12 and 4.13.
(a) The wind turbine at Chunan Miaoli
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(b) The voice signal from The wind turbine at Chunan Miaoli
Figure 4.11: The wind turbine and the voice signal
In Fig.4.11b, we find that the recorded signal has two patterns. One happens about 1 second,and is from a rotor pass through the voice recorder since the period of this turbine is 3 second, and it is also shown in IMF 6. The other happens about 20 second, it likely the main wave of the original signal, and is also recorded in IMF 12. According to the significant test of IMFs, IMF 1 and IMF 2 are close to the white noise.
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−2 0 2
IMF 1
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−1 0 1
IMF 2
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−2 0 2
IMF 3
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−5 0 5
IMF 4
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−5 0 5
IMF 5
Figure 4.12: These figures are IMF 1
∼ 5 of the voice signal.
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Figure 4.13: These figures are IMF 6
∼ 15 of the voice signal. The last IMF means the
trend.We produce the dendrogram as Fig.4.14 through clustering analysis with complete linkage. According to Fig. 4.14, we find that there is a multi-mode problem between IMF 13 and IMF 14. In addition, we also find that IMF 7 and 10 relative to the others are self-reliant, and this indicates that these play an important role in the signal. This dendrogram presents some characteristics of IMFs from EEMD in the view of correla-tion coefficients.
Figure 4.14: The figure is the dendrogram with complete linkages of the wind turbine when the number of clusters is 8.
According to the dendrogram (Fig.4.14), when the number of clusters 8 is given, IMF 1 and 2 are combined into CIMF 1. As we discuss before, CIMF 1 is like white noise. IMF 3 and 4 are combined into CIMF 2; 5 and 6 into CIMF 3; 8 and 9 into CIMF 5; 11 and 12 into CIMF 7 ;13 and 14 into CIMF 8. Especially, IMF 7 is CIMF 4 and IMF 10 is CIMF 6. For CIMFs, the peak which happens about 1 second is presented in CIMF 3, and the other which happens about 20 second is also shown in CIMF 8. All CIMFs are shown in Fig. 4.15. The results of this subsection indicate that the method which we suggest improves the multi-mode problem of the signal and also provides more information from the results of EEMD through presenting the correlations between IMFs.
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Figure 4.15: CIMFs with complete linkage of the wind turbine
4.4 The practical signal: an seismic signal from station CHY080 at Chi-Chi
In 1999, the Chi-Chi earthquake triggered a major landslide in Tsaoling area, with a source volume 125
× 10
6m3. Near the scar area, there is a station, CHY080, recording the seismic signals during the earthquake and the recorded data exhibit some distinc-tive signatures. The location of CHY080 is shown in Fig.4.16. The record has three directions: east-west, north-South, and vertical. This section presents the analysis of its east-west direction. Before analysing the east-west direction signal, we change the sample frequency of it from 200 points per second to 100 with applying an anti-aliasing FIR filter and take frequencies it contains below 20 with a quadric low-pass filter. The resampled and filtered signal is shown in Fig. 4.17.Figure 4.16: The location of CHY080 in Tsaoling area. The NE-SW profile is defined through the gravity center of the slid mass and is in parallel to the slid direction.
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−40
−30
−20
−10 0 10 20 30 40
resampled and filtered chy080e
Figure 4.17: The resampled and filtered signal from the east-west direction of station CHY080
Applying EEMD to this signal with a white noise level at 10% of the standard devi-ation of the signals and ensembling 30 times, we find that the signal is decomposed into 9 IMFs, see Fig. 4.18. The main earthquake contents are likely to be classified into the third to the four IMF because they total contain more than 81% of the spectral energy of the acceleration. Another distinctive characteristic is that outstanding wave packets are found between 76 and 78 second in IMF 2. IMF 2 is likely from the landslide. The significance of IMF white noise is tested. IMF 1 is close to the margin of white noise and hence is likely a white noise.
We get the dendrogram (Fig. 4.19) through clustering analysis with complete link-age. When the number of clusters, 5, is given, we combine IMFs into CIMFs, as shown in Fig. 4.20. CIMF 1 is the IMF 1, and CIMF 2 is the IMF 2 which contains landslide messages, and other CIMFs are from IMFs which contain the main earthquake sig-nal. The results, as described above, show the method which we suggest decomposes this earthquake signal into some components appropriately. In addition, we can find characteristics of this earthquake signal easily according to the dendrogram.
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Figure 4.18: Results of the east-west earthquake signal of CHY080 which is decom-posed by EEMD. The last means the trend.
IMF5 IMF6
Figure 4.19: The figure is the dendrogram with complete linkage for IMFs of the east-west signal when the number of clusters is 5.
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Figure 4.20: CIMFs of the east-west signal