## 4 Signal Feature Extraction Method Based on Intrinsic Mode Function Energy Permutation Entropy
### Permutation Entropy Analysis of Intrinsic Mode Functions  
To validate the algorithm's effectiveness, this section simulates three types of vessels using equivalent electric field dipole sources with three different connection methods. The experiments use the same frequency and amplitude signal as input, and the time-domain signals of the three electric fields are measured, as shown in Figure 8.  
Figure 9 shows the EMD algorithm decomposing the electric field signals of the three vessel axes frequencies into ordered IMF results from high to low frequency. An analysis of the graph reveals that for Vessel A, the amplitude of the IMF1, IMF2, and IMF3 component signals is significantly higher than that of the other components, indicating the primary energy components of the signal features. Next, the primary energy of Vessel B is concentrated in IMF2 and IMF3. For Vessel C, the signal energy is concentrated in the IMF1, IMF2, and IMF3 components. Figure 8: Time-domain signals of electric fields at different vessel axes frequencies. Figure 9: _Cont._  
The PE values are calculated for each IMF decomposition to extract the signal characteristics of different components further. Two parameters are required for calculating PE: time delay \(\tau\) and embedding dimension \(m\). Considering the aim of retaining as many abrupt anomalies and feature parameters of the actual signal as possible, the time delay \(\tau\) is set to 1, and the embedding dimension \(m\) is set to 4 [31]. The PE calculation results for different vessels are shown in Figure 10. The X-axis represents the first to eighth IMF components after decomposing the electric field signal, while the Y-axis represents the PE values for each element. As demonstrated in Figure 10, the permutation entropy (PE) values of intrinsic mode function (IMF) components for the shaft-rate electric field signals of the three vessel types (Vessel A, B, and C) exhibit a monotonic decline with increasing decomposition order. This trend indicates a progressive reduction in signal complexity at higher IMF orders, reflecting diminished discriminative information content pertaining to shaft-rate characteristics. Notably, the PE values across distinct vessels (A, B, and C) display insufficient statistical divergence at identical decomposition orders, thereby limiting the efficacy of conventional empirical mode decomposition (EMD) in differentiating vessel-specific signatures. Due to the slight differences in PE values for same-order components among different vessels, the IMFs are reorganized according to the energy sequence. Assuming the \(j\)-th IMF component  
Figure 10: PE values of electric field signal components for different vessels. Figure 9: EMD of electric field signals for different vessels: (**a**) EMD of Vessel A; (**b**) EMD of Vessel B; and (**c**) EMD of Vessel C.  
after the decomposition of the original electric field signal has \(N\) sampling points, where the instantaneous amplitude of the \(n\)-th sampling point is \(c_{j}(n)\), its instantaneous energy is the following:  
\[Q_{j}=c_{j}^{2}(n) \tag{9}\]  
Therefore, the average energy of the \(j\)-th IMF component is the following:  
\[\hat{E}n_{j}=\sum_{n=1}^{N}Q_{j}(n)/N \tag{10}\]  
The energy values of each IMF component are calculated according to Equations (9) and (10), as shown in Figure 11. The X-axis represents the eight IMF components sorted in descending order of energy, and the Y-axis represents the PE values for each component. The experimental results reveal no discernible systematic correlation between the signal energy levels and permutation entropy (PE) values of the intrinsic mode function (IMF) components. However, significant inter-vessel discrepancies in PE values are observed specifically within the first four energy-dominant IMF ranks. For distinct vessel types, the PE values of these primary energy tiers (IMF1-IMF4) exhibit marked differentiation, with higher energy levels corresponding to richer signal feature compositions.