Benchmark examples

We use a simple synthetic signal to test the proposed method. Figure 1 is a synthetic signal from Hou and Shi (2013). The three components of the signal are shown in Figure 2. Figure 3 and Figure 4 show the intrinsic mode functions extracted respectively by ensemble empirical mode decomposition and NPM methods. From the figures, we see that the NPM method accurately identifies the three components that the signal has. The intrinsic mode functions derived by the NPM are more smooth with respect to amplitudes and frequencies compared with the intrinsic mode functions obtained by ensemble empirical mode decomposition. For ensemble empirical mode decomposition, we repeat the empirical mode decomposition 25 times with different level of noises to generate the ensemble empirical mode decomposition results. The time-frequency distributions of the input signal are the Hilbert transform of the intrinsic mode functions. Figure 5a, 5b and 5c are respectively the time-frequency distributions using local attribute (Liu et al., 2011), ensemble empirical mode decomposition (Wu and Huang, 2009) and the proposed method for the synthetic signal of Figure 1. Figure 6 is an another synthetic signal. Figure 7a, 7b and 7c are respectively the time-frequency maps using local attribute (Liu et al., 2011), ensemble empirical mode decomposition (Wu and Huang, 2009) and the proposed method. From the figures, we see that the energies compactly spread over the instantaneous frequencies for the ensemble empirical mode decomposition. However, the energies are not steadily distributed for the ensemble empirical mode decomposition. The proposed method provides a steady and compact energies distribution, which sharpen the time-frequency distribution.

hsig
Figure 1. Synthetic signal.

sigs
Figure 2. Components of the synthetic signal of Figure 1 .

emd
Figure 3. Components of the synthetic signal of Figure 1 using ensemble empirical mode decomposition.

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Figure 4. Components of the synthetic signal of Figure 1 using NPM.

htf,htfemd,htfnar
Figure 5. (a) Time-frequency map for synthetic signal of Figure 1 using local attribute. (b) Time-frequency map for synthetic signal of Figure 1 using ensemble empirical mode decomposition. (c) Time-frequency map for synthetic signal of Figure 1 using the proposed method.

msig
Figure 6. Synthetic signal.

mtf,mtfemd,mtfnar
Figure 7. (a) Time-frequency map for synthetic signal of Figure 6 using local attribute. (b) Time-frequency map for synthetic signal of Figure 6 using ensemble empirical mode decomposition. (c) Time-frequency map for synthetic signal of Figure 6 using the proposed method.