*2.1. 2D-MUSIC Algorithm for Impact Localization*

In this section, the 2D-MUSIC algorithm in [10,11] is briefly introduced. As seen in Figure 1, a uniform linear array (ULA) consists of M piezoelectric (PZT) sensors on the structure, which are arranged uniformly along the *x* axis and asymmetric with the *y* axis. The distance between two sensors is *d.*

**A**(*r*, *θ*) is defined as the array steering vector

$$\mathbf{A}(r,\theta) = \left[a\_1(r,\theta), a\_2(r,\theta), \dots, a\_M(r,\theta)\right] \tag{1}$$

where

$$a\_i(r, \theta) = \frac{r}{r\_i} \exp\left(-j\omega\_0 \tau\_i\right)$$

$$\tau\_i = \frac{(-d\sin\theta)}{c\_{AV}}(i-1) + \left(-\frac{d^2}{c\_{AV}}\cos^2\theta\right)(i-1)^2$$

*r* is defined as the distance between the impact source and the ULSA, which is the distance from the source to the reference sensor labeled as 1 in the sensor array. *θ* denotes the wave propagating direction caused by the impact with respect to the coordinate *y* axis. *cAV* is the average velocity of the Lamb wave.

**Figure 1.** Near-field impact signal model.

The basic idea of the MUSIC algorithm is to obtain the signal subspace and noise subspace through eigenvalue decomposition of the signal covariance matrix, and then to estimate the signal parameter using the orthogonality of two spaces. To describe the orthogonality between the signal subspace and noise subspace, the spatial spectrum is used, which can be calculated by

$$\mathbf{P\_{MUSIC}}\left(r,\theta\right) = \frac{1}{\mathbf{A}^{\mathrm{H}}(r,\theta)\mathbf{U\_{N}U\_{N}{}^{\mathrm{H}}\mathbf{A}(r,\theta)}}\tag{2}$$

where **U**<sup>N</sup> denotes the noise subspace spanned by the eigenvector matrix corresponding to those small eigenvalues. Based on Equation (2), by varying *r* and *θ* to realize a scanning process, **A**(*r*, *θ*) is steered to scan the whole structure area. The peak point on the spatial spectrum corresponds to the impact source point. Both the distance and direction of the impact source can be obtained.

#### *2.2. Fast EEMD for Impact Signal Extraction*

In 1998, Huang 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 *cj* as

$$\mathbf{x}(t) = \sum\_{j=1}^{n} c\_j + r\_n \tag{3}$$

where *rn* 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:

1. Throughout the whole length of a single IMF, the number of extrema *Ne* (maxima and minima) and the number of zero-crossings *NZ* must either be equal or differ at most by one, i.e.,

$$(N\_z - 1) \le N\_c \le (N\_z + 1)\tag{4}$$

2. At any time point *ti*, the mean value of the envelope *f*max(*ti*) and *f* min(*ti*) respectively defined by the local maxima and the local minima are zero, i.e.,

$$[f\_{\max}(t\_i) + f\_{\min}(t\_i)]/2 = 0t\_i \in [t\_{a\prime}, t\_b] \tag{5}$$

where [*ta*, *tb*] 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

$$m(t) = \left[f\_{\text{max}}(t) + f\_{\text{min}}(t)\right]/2\tag{6}$$

2. Obtain the first component *h*<sup>1</sup> by taking the difference between the data *x*(*t*) and the local mean *m*(*t*) as

$$h\_1(t) = x(t) - m(t) \tag{7}$$


Generally, the stop criterion of the sifting procedure is restrained by

$$S\_d = \sum\_{t=0}^{T} \frac{\left| h\_{k-1}(t) - h\_k(t) \right|^2}{h\_k^2(t)} \tag{8}$$

where *T* is the signal length; *hk*−1(*t*) and *hk*(*t*) are the neighbor components in sifting procedures for one IMF; and *Sd* 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 Wu and Huang, 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 *ni*(*t*) series (noise level is *Nl*) to the targeted data and decompose the data with added white noise into IMFs as

$$x(t) + n\_i(t) = \sum\_{j=1}^{n} c\_j^i(t) + r\_n^i(t) \tag{9}$$

where *i* = 1, 2, ··· , *q* and *q* is the average time (ensemble number).


$$\mathbf{x}(t) + \boldsymbol{n}\_{i}(t) = \frac{1}{q} \sum\_{i=1}^{q} \sum\_{j=1}^{n} c\_{j}^{i}(t) + \frac{1}{q} \sum\_{i=1}^{q} r\_{n}^{i}(t) = \sum\_{j=1}^{n} d\_{j}(t) + r\_{n}(t) \tag{10}$$

where *dj*(*t*) is the *j*th IMFs of EEMD decomposition as

$$d\_{\dot{j}}(t) = \frac{1}{q} \sum\_{i=1}^{q} c\_{\dot{j}}^{\dot{i}}(t) \tag{11}$$

and *rn*(*t*) is the final residual of EEMD decomposition as

$$r\_n(t) = \frac{1}{q} \sum\_{i=1}^{q} r\_n^i(t) \tag{12}$$
