## 2 Basic Theories  
### Review of LMD Method  
An LMD method can decompose a complicated signal into a set of product functions (PFs), each of which is the product of an amplitude envelope signal and a purely frequency-modulated signal. Specific decomposition steps can be summarized as follows:All of the local extrema \(n_{i}\) and the time \(tn_{i}\) of the original signal \(x(t)\) are determined, and the mean value \(m_{i}\) of the two successive extrema \(n_{i}\) and \(n_{i+1}\) is calculated as follows: \[m_{i}=\frac{n_{i}+n_{i+1}}{2},\] (1) All mean values \(m_{i}\) are connected by straight lines between the corresponding time \(tn_{i}\) and \(tn_{i+1}\), to generate the local mean segments. Then, the local mean segments are smoothed by amoving averaging method, to form a continuous local mean function \(m_{11}(t)\).
2. The local amplitude \(a_{i}\) can be given as follows: \[a_{i}=\frac{|n_{i}-n_{i+1}|}{2},\] (2) The moving averaging method is also used to smooth the local amplitude segments, in order to derive the envelope estimate function \(a_{11}(t)\).
3. The local mean function \(m_{11}(t)\) is subtracted from the original signal \(x(t)\), and the remnant signal, denoted by \(h_{11}(t)\), can be given as follows: \[h_{11}(t)=x(t)-m_{11}(t),\] (3)
4. The signal \(h_{11}(t)\) is demodulated by the envelope estimate function \(a_{11}(t)\), and the result, denoted by \(s_{11}(t)\), can be calculated as follows: \[s_{11}(t)=h_{11}(t)/a_{11}(t),\] (4) The envelope estimate function \(a_{12}(t)\) of \(s_{11}(t)\) is calculated. If \(a_{12}(t)\) is not equal to one, \(s_{11}(t)\) is not a purely frequency-modulated signal, and the above procedure for \(s_{11}(t)\) should be repeated until a purely frequency-modulated signal \(s_{1n}(t)\) is obtained. This is obtained when the envelope estimate function meets the condition that \(a_{1(n+1)}(t)\) of \(s_{1n}(t)\) is equal to one. Therefore, \[\left\{\begin{array}{l}h_{11}(t)=x(t)-m_{11}(t)\\ h_{12}(t)=s_{11}(t)-m_{12}(t)\\ \vdots\\ h_{1n}(t)=s_{1(n-1)}(t)-m_{1n}(t)\end{array}\right.,\] (5) where \[\left\{\begin{array}{l}s_{11}(t)=h_{1n}(t)/a_{1n}(t)\\ s_{12}(t)=h_{12}(t)/a_{12}(t)\\ \vdots\\ s_{1n}(t)=h_{1n}(t)/a_{1n}(t)\end{array}\right.,\] (6) An envelope signal \(a_{1}(t)\) can be derived by the product of all of the envelope estimate functions obtained during the iterative process described above. \[a_{1}(t)=a_{11}(t)a_{12}(t)\cdots a_{1n}(t),\] (7) The first product function \(PF_{1}\) of the original signal can be generated by the product of the envelope signal \(a_{1}(t)\) and the purely frequency-modulated signal \(s_{1n}(t)\). \[PF_{1}(t)=a_{1}(t)s_{1n}(t),\] (8)\(\mathrm{PF}_{1}\) is then subtracted from the original signal \(x(t)\), generating a new signal \(u_{1}(t)\). The whole process is repeated \(k\) times, until \(u_{k}(t)\) is a constant or monotonic function. \[\left\{\begin{array}{l}u_{1}(t)=x(t)-PF_{1}(t)\\ u_{2}(t)=u_{1}(t)-PF_{2}(t)\\ \vdots\\ u_{k}(t)=u_{k-1}(t)-PF_{k}(t)\end{array}\right.,\] (9) Finally, the original signal \(x(t)\) is decomposed into \(k\) PFs and a residual \(u_{k}(t)\), and \(x(t)\) can be reconstructed as follows: \[x(t)=\sum_{i=1}^{k}PF_{i}+u_{k}(t),\] (10) According to above algorithm, we can realize that the LMD method is an adaptive signal decomposition method, based on the local extrema information of the signal itself.