## 3 Design Implementation of the Rational Spline Based-LOD Method  
### Rational Spline Interpolation  
Since the piecewise linear transformation lacks enough smoothness, a new alternative interpolation method named rational spline interpolation is introduced in this paper. When using the rational spline interpolation, the shape of the fitting curve will change as the pole parameter \(p\) varies. Hence, it is easy for us to determine which curve is more suitable to fit the discrete points. Here the definition of rational spline interpolation function is outlined in some detail [24]. **Definition 1**. : _Given a node sequence \(x_{i}\) (\(i=1,\,2,\,\dots\) ) in interval [\(a,\,b\)] and the corresponding function values are \(y_{i}\) (\(i=1,\,2,\,\dots\) ). The rational spline function \(s(x)\) satisfies: (1) \(s(x_{i})=y_{i}\), \(i=1,\,2,\,\dots\), \(n\), (2) has continuous second derivative in interval [\(a,\,b\)]._  
Then in the fragment interval [\(x_{i}\), \(x_{i+1}\)] (\(i=1,\,2,\,\dots\), \(n-1\)), the rational spline can be described as follows:  
\[s_{i}(x)=A_{i}u+B_{i}t+C_{i}u^{3}/(1+pt)+D_{i}t^{3}/(1+pu),p\geq-1 \tag{5}\]  
where \(t=(x-x_{i})/t_{i}=(x-x_{i})/\left(x_{i+1}-x_{i}\right)\), \(u=1-t\). In Formula (5), \(p\) is the pole parameter used to control the degree of tightness of the spline curve. \(A_{i}\), \(B_{i}\), \(C_{i}\) and \(D_{i}\) are the coefficients that make \(s(x)\) have continuous first and second derivatives, which can be described as the second derivative value at the node:  
\[C_{i}=y_{i}\text{\"}h_{i}/q, \tag{6}\]  
\[D_{i}=y_{i+1}\text{\"}h_{i}/q, \tag{7}\]  
\[A_{i}=y_{i}-C_{i}, \tag{8}\]  
Figure 3: The enlarged image of the third MOC component \(\text{MOC}_{3}(t)\). Figure 2: Local oscillatory-characteristic decomposition (LOD) results of the simulation signal \(x(t)\). \[B_{i}=y_{i+1}-D_{i} \tag{9}\]  
**Definition 2**. : _The second derivative at the node can be obtained by the following linear equations:_  
\[h_{i-1}y_{i-1}{}^{\prime\prime}/(h_{i-1}+h_{i})+(2+p)y_{i}{}^{\prime\prime}+h_{i} y_{i+1}{}^{\prime\prime}/(h_{i-1}+h_{i})=q(d_{i}-d_{i-1})/(h_{i-1}+h_{i}),i=2,\,3,\, \ldots,\,n-1 \tag{10}\]  
_where, \(d_{i}=(y_{i+1}-y_{i})/h_{i}\), \(q=2(p^{2}+3p+3)\). Equation set (10) contains \(n-2\) equations, plus boundary conditions \(s^{\prime\prime}(x_{1})=m_{1}\) and \(s^{\prime\prime}(x_{n})=m_{n}\), can form a system of \(n\) linear equations with \(n\) unknowns. When \(p\geq-1\), the system of equations is a strictly diagonally dominant matrix, and \(y_{i}{}^{\prime\prime}\) (\(i=1,\,2,\,\ldots,\,n\)) can be solved by chasing method. Then the results of \(A_{i}\), \(B_{i}\), \(C_{i}\), \(D_{i}\), and \(s_{i}(x)\) in the interval \([x_{i},\,x_{i+1}]\) can be obtained by taking \(y_{i}{}^{\prime\prime}\) into Formulas (6)-(9)._  
From Formula (5), it can be seen that the pole parameter \(p\) can control the shape of the spline curve \(s(x)\). When \(p=-1\), \(s(x)\) becomes a quadratic spline function showed relatively larger volatility and generally did not apply to strive for fitted curve of discrete points; when \(p=0\), \(q=6\), it can be seen from Equation (10) that \(s(x)\) is a cubic spline function, so the cubic spline function is a special case of the rational spline function. In addition, as \(p\) increases, the last two terms of Formula (5) will gradually decrease, and \(s(x)\) will continue to tighten. Furthermore, when \(p\) tends to infinity, the last two terms of Formula (5) will become zero, and \(s(x)\) will become a piecewise linear interpolation function. Therefore, the piecewise linear transform is also a special case of rational spline interpolation. From the above, we can know that, by adjusting the pole parameter \(p\), the desired fitting curve shape can be obtained theoretically, and the decomposition error can be reduced.