## 2 Local Oscillatory-Characteristic Decomposition and its Defects  
Local oscillatory-characteristic decomposition is a new self-adaptive time-frequency analysis method based on the local extreme points of the signal, which can decompose a multi-component signal into a plurality of MOCs and a residue. Three operations are adopted in this method: (1) differential,which increases the amplitude ratio of the high- and low-frequency components of the signal and amplifies the local oscillation characteristics of the signal, thereby effectively eliminating the occurrence of modal mixing; (2) coordinate domain transformation, which transforms the signal from the original coordinate domain to the sawtooth domain. By increasing the number of sampling points close to the local extreme points of the signal without changing the total number of sampling points, it improves the decomposition accuracy of LOD based on the local extreme points; and (3) piecewise linear transformation, which can process the data between two adjacent extreme points of the signal in parallel without simultaneously knowing the complete local extreme point information of the signal, thereby improving the calculation efficiency and reducing the endpoint effect. A study [21] described the decomposition process of LOD in detail, where LOD was used to decompose a multi-component AM-FM simulation signal \(x(t)\):  
\[x_{1}(t)=(1+0.6\mathrm{cos}(8\pi t))\times\mathrm{cos}(180\pi t+3\mathrm{cos}( 8\pi t)), \tag{1}\]  
\[x_{2}(t)=\mathrm{sin}(25\pi t), \tag{2}\]  
\[x_{3}(t)=\mathrm{sin}(2\pi t), \tag{3}\]  
\[x(t)=x_{1}(t)+x_{2}(t)+x_{3}(t), \tag{4}\]  
where \(t\in[0,5]\), setting sampling frequency 1000 Hz and the simulation signal consists of a modulated signal and two sine signals. The time domain waveforms of \(x(t)\) and its three components \(x_{1}(t)\), \(x_{2}(t)\), and \(x_{3}(t)\) are shown in Figure 1. This simulation signal \(x(t)\) is decomposed by LOD method and the results are presented in Figure 2.  
As shown in Figure 2, the original signal \(x(t)\) is decomposed into three MOC components and a monotonic residue, and the three MOC components are corresponding to the \(x_{1}(t)\), \(x_{2}(t)\), and \(x_{3}(t)\), respectively. Therefore, LOD, as a self-adaptive time-frequency analysis method based on the characteristics of the signal itself, can reflect the essential characteristics of the signal. However, as illustrated in Figure 3, the locally enlarged view of \(\mathrm{MOC}_{3}(t)\) shows that the waveform of \(\mathrm{MOC}_{3}(t)\) is evidently jagged and exhibits poor smoothness. The main reason is that the use of piecewise linear transformation leads to an unsmooth local mean curve, which causes the gradual accumulation of errors with an increasing number of iterations, thereby affecting the decomposition accuracy. The above decomposition error can be eliminated or reduced if a suitable interpolation method is identified to replace piecewise linear transformation and to obtain a local mean curve with enhanced smoothness. Previous studies [25; 26] proposed the adoption of the rational spline interpolation instead of cubic spline interpolation and moving average to calculate local mean curves in EMD and LMD, respectively, which achieved good results. Accordingly, this study intended to introduce the rational spline interpolation to replace the piecewise linear transformation, so as to improve the poor smoothness of the LOD decomposition.  
Figure 1: Time domain waveform of the simulation signal \(x(t)\) and its components.