## 3 Design Implementation of the Rational Spline Based-LOD Method
### Rational Spline Based-LOD Method  
Based on the above discussion about the interpolation method, it is not difficult for us to conclude that rational spline interpolation is available for precisely fitting discrete points. Furthermore, the mean function algorithm plays a crucial role in the decomposing results with respect to LOD. Hence, the rational spline-based LOD method (RS-LOD) is proposed in this paper. For a given a signal \(x(t)\), the algorithmic procedure of the RS-LOD can be described as follows:  
1. [label=()]
2. Finding out all local extrema \(X_{k}\), \(k=1\), \(2\), \(\ldots\), \(M\) and corresponding time values \(\tau_{k}\), \(k=1\), \(2\), \(\ldots\), \(M\) of original signal \(x(t)\). In order to reduce the decomposed error, the function with respect to coordinate domain transform is defined by performing a transformation from original data  
Figure 4: Local mean curves corresponding to different values of the pole parameter \(p\).  
domain coordinate (\(t\), \(x\)) into sawtooth domain coordinate (\(u\), \(s\)). The transformation formula can be expressed as:  
\[u(t)\ =\ \tau_{k}+\ (x(t)\ -\ X_{k})(t\ -\ \tau_{k})/(X_{k+1}\ -\ X_{k}),\ t\in[\tau_{k}, \tau_{k+1}),k=1,\ 2,\ \ldots,\ \text{M} \tag{11}\]  
\[s(u)=x(t), \tag{12}\]  
From the Formulas (11) and (12), we can see that the coordinate transformation does not change the amplitude of the signal (the direction of ordinate), but only compresses or expands the time axis (the direction of abscissa). Hence, the data point density of different signal positions can be changed without changing the amount of data, and thus to highlight the local characteristic information.  
* Making a differential operation for the original signal \(x(t)\) to get \(x^{\prime}(t)\) and finding out the time values \(\tau_{k}\)', \(k=1,2,\ldots,\) N corresponding to the extrema of \(x^{\prime}(t)\). At the same time, finding out function values \(X_{k}\)', \(k=1,2,\ldots,\) N corresponding to the time values \(\tau_{k}\)' in original signal \(x(t)\). Then the local mean function \(m_{1}(u)\) can be derived by using rational spline function to fit the (\(\tau_{k}\)', \(X_{k}\)') in sawtooth domain.
* Subtract the local mean function \(m_{1}(u)\) from the sawtooth domain function \(s(u)\) of the original signal, and then the high-frequency oscillatory function \(c_{1}(u)\) can be obtained: \[c_{1}(u)=s(u)-m_{1}(u),\] (13)
* Restore the coordinates of \(c_{1}(u)\) from the sawtooth domain to the original data domain \(c_{1}(t)\) using the inverse transformation. The inverse transformation formula can be written as: \[t(u)\ =\ \tau_{k}\text{'}\ +\ (X_{k+1}\ -\ X_{k})(u\ -\ \tau_{k}\text{'})/(x(t)\ -\ X_{k}),\ u\in[\tau_{k}\text{'},\tau_{k+1}\text{'}),k=1,\ 2,\ \ldots,\ \text{N}\ -1\] (14) \[c_{1}(t)\ =\ c_{1}(\tau_{k}\text{'})\ +\ (c_{1}(u)\ -\ c_{1}(\tau_{k}\text{'}))( \tau_{k+1}\text{'}\ -\ \tau_{k}\text{'})/(t\ -\ \tau_{k}\text{'}),\ t\in[\tau_{k}\text{'},\tau_{k+1}\text{'}),k=1,\ 2,\ \ldots,\ \text{N}\ -1\] (15)  
Ideally, \(c_{1}(t)\) is the first component \(\text{MOC}_{1}(t)\) of original signal \(x(t)\) when \(c_{1}(t)\) is a mono-component signal of instantaneous frequency with physical meaning.  
* If \(c_{1}(t)\) is not a mono-component signal of the instantaneous frequency with a physical meaning, we should repeat the above procedure \(m\) times until \(c_{m}(t)\) is a mono-component signal of instantaneous frequency with physical meaning. In other words, in this situation, \(c_{m}(t)\) will be the first component \(\text{MOC}_{1}(t)\) of the original signal.
* This derived \(\text{MOC}_{1}(t)\) is then subtracted from the original signal \(x(t)\), obtaining a new function \(r_{1}(t)\). Regard \(r_{1}(t)\) as a new original signal and repeat the procedure (i)-(v) \(n\) times until \(r_{n}(t)\) is a monotonic function. Thus, \(x(t)\) may be decomposed into the sum of \(n\) MOCs and residue \(r_{n}(t)\). i.e., \[x(t)\ =\ \sum\
olimits_{i=1}^{n}\text{MOC}_{i}(t)\ +\ r_{n}(t),\] (16)  
The flowchart of the RS-LOD method is illustrated in Figure 5.