## 2 Optimized EEMD Based 2D-MUSIC Method
### Fast EEMD for Impact Signal Extraction  
In 1998, [PERSON] et al. introduced the EMD, which is able to adaptively and effectively decompose a complicated signal into a collection of stationary IMFs [12]. Therefore, it has often been used in nonlinear and non-stationary signal processing. In the EMD method, the data \(x(t)\) is decomposed in terms of IMFs \(c_{j}\) as  
\[x(t)=\sum_{j=1}^{n}\ c_{j}+r_{n} \tag{3}\]  
where \(r_{n}\) is the residual of data \(x(t)\), after \(n\) number of IMFs are extracted. IMFs are simple oscillatory functions that have the following two properties:  
Figure 1: Near-field impact signal model.  
1. Throughout the whole length of a single IMF, the number of extrema \(N_{e}\) (maxima and minima) and the number of zero-crossings \(N_{Z}\) must either be equal or differ at most by one, i.e., \[(N_{z}-1)\leq N_{e}\leq(N_{z}+1)\] (4)
2. At any time point \(t_{i}\), the mean value of the envelope \(f_{\max}(t_{i})\) and \(f_{\min}(t_{i})\) respectively defined by the local maxima and the local minima are zero, i.e., \[[f_{\max}\;(t_{i})+f_{\min}(t_{i})]/2=0t_{i}\in[t_{a},\;t_{b}]\] (5) where \([t_{a},\;t_{b}]\) is the time interval. In practice, the EMD decomposition procedures (sifting procedure) are as follows:
1. Identify all the local maxima and minima and connect all of them using a cubic spline as the upper and lower envelope \(f_{\max}(t)\) and \(f_{\min}(t)\), respectively. Then, calculate the mean value \(m(t)\) of \(f_{\max}(t)\) and \(f_{\min}(t)\) as \[m(t)=[f_{\max}\;(t)+f_{\min}(t)]/2\] (6)
2. Obtain the first component \(h_{1}\) by taking the difference between the data \(x(t)\) and the local mean \(m(t)\) as \[h_{1}(t)=x(t)-m(t)\] (7)
3. Treat \(h_{1}(t)\) as the data and repeat steps 1 and 2 as many times as is required until the two properties of IMF as shown in Equations (4) and (5) are satisfied. Then, the final \(h_{1}(t)\) is designated as an IMF \(c_{1}(t)\).
4. Treat \(r_{i}(t)=x(t)-c_{i}(t)\), \((i=1,2,\cdots,n-1)\) as the data and repeat steps 1-3. Finally, we obtain additional IMFs \(c_{2}(t),c_{3}(t),\cdots c_{n}(t)\) and the final residual \(r_{n}(t)\), which are represented by Equation (3). Generally, the stop criterion of the sifting procedure is restrained by \[S_{d}=\sum_{t=0}^{T}\frac{|h_{k-1}(t)-h_{k}(t)|^{2}}{h_{k}^{2}(t)}\] (8)  
where \(T\) is the signal length; \(h_{k-1}(t)\) and \(h_{k}(t)\) are the neighbor components in sifting procedures for one IMF; and \(S_{d}\) is the standard deviation, which is suggested to be 0.2-0.3.  
However, mode mixing appears to be the most significant drawback of EMD [19]. Therefore, in 2009, a new artificial noise-excited EMD method was proposed by [PERSON] and [PERSON], which was called EEMD [19]. The procedures are similar to EMD, except only one series of white noise with a finite amplitude are added into the original signals and the procedures are summarized as follows:  
1. Add a white noise \(n_{i}(t)\) series (noise level is \(N_{l}\)) to the targeted data and decompose the data with added white noise into IMFs as \[x(t)+n_{i}(t)=\sum_{j=1}^{n}c_{j}^{i}(t)+r_{n}^{i}(t)\] (9) where \(i=1,2,\cdots,q\) and \(q\) is the average time (ensemble number).
2. Repeat step1 \(q\) times with different white noise series \(n_{i}(t)\).
3. Obtain the (ensemble) means of corresponding IMFs of the decompositions as the final result, that is \[x(t)+n_{i}(t)=\frac{1}{q}\sum_{i=1}^{q}\sum_{j=1}^{n}c_{j}^{i}(t)+\frac{1}{q} \sum_{i=1}^{q}r_{n}^{i}(t)=\sum_{j=1}^{n}d_{j}(t)+r_{n}(t)\] (10)where \(d_{j}(t)\) is the \(j\)th IMFs of EEMD decomposition as  
\[d_{j}(t)=\frac{1}{q}\sum_{i=1}^{q}\ c_{j}^{i}(t) \tag{11}\]  
and \(r_{n}(t)\) is the final residual of EEMD decomposition as  
\[r_{n}(t)=\frac{1}{q}\sum_{i=1}^{q}\ r_{n}^{i}(t) \tag{12}\]