# Adaptive Sensor Array Error Calibration Based Impact Localization on Composite Structure
## 2 Signal Modal of Imprecise Sensor Array  
Assuming that a uniform linear sensor array consisting of \((2\times M+1)\) piezoelectric (PZT) sensors is arranged on the structure. The distance between two sensors is \(d\). Considering the impact induced elastic waves \(\mathbf{s}(t)\) with a certain frequency component of \(\omega_{0}\) arriving at the sensor array, and \(\theta\) denotes the wave propagating direction caused by impact with respect to the coordinate \(y\) axis. The observed data output from \(q\)th PZT under the near-field situation \(\mathbf{x}(t)\) can be presented as  
\[\mathbf{x}_{q}(t)=\frac{r}{r_{q}}s(t)e^{-j\omega_{0}\tau_{q}}+\mathbf{n}_{q}(t ),\;q=-M,\cdots,0,\cdots,M \tag{1}\]  
where \(r\) and \(r_{q}\) denotes the distance between impact source and PZT \(0\) and PZT \(i\), respectively, \(\mathbf{n}_{q}(t)\) is the output corresponding to the background noise, \(\tau_{q}\) is the arriving time difference among different sensor elements in the sensor array, which can be expressed as  
\[\tau_{q}=\frac{(-d\sin\theta)}{c}(q-1)+(-\frac{d^{2}}{cr}\cos^{2}\theta)(q-1) ^{2}+O(\frac{d^{2}}{r^{2}}) \tag{2}\]  
where \(c\) is the elastic wave average velocity. Denotes the array steering vector of precisely sensor array as \(\mathbf{A}(r,\theta)\), and can be presented as  
\[\mathbf{A}(r,\theta)=[a_{-M}(r,\theta),\cdots,a_{q}(r,\theta),\cdots a_{M}(r,\theta)] \tag{3}\]  
where  
\[a_{q}(r,\theta)=\frac{r}{r_{q}}\exp(-j\omega_{0}\tau_{q})\]In fact, array signal vectors have various error forms including array element position error, gain-phase error caused by structural propagation path deviation, etc. For an imprecise PZT sensor array seen in Figure 1, the observed data output from \(q\)th PZT should be rewritten as  
\[\mathbf{x}(t)=\alpha_{q}\mathbf{s}(t-\psi_{q})+\mathbf{n}_{q}(t) \tag{4}\]  
where \(\alpha_{q}\) and \(\psi_{q}\) is the \(q\)th sensor-related gain and delay due to element position error of the imprecise array. Defined \(\Gamma_{q}\) as the gain and delay the \(q\)th sensor, and  
\[\Gamma_{q}=\alpha_{q}\exp(-jw_{0}\psi_{q}),q=-M,\cdots,M \tag{5}\]  
Moreover, array sensor position error matrix could be obtained as  
\[\mathbf{\Gamma}(r,\theta)=diag[\Gamma_{-M},\cdots,\Gamma_{M}] \tag{6}\]  
By introducing the element position error matrix, the array steering vector for the near-field impact signal could be represented as  
\[\mathbf{\tilde{A}}(r,\theta)=\mathbf{\Gamma}(r,\theta)\mathbf{A}(r,\theta) \tag{7}\]  
For the whole sensor array, the observed input signal vector can be represented as  
\[\mathbf{x}(t)=\mathbf{\tilde{A}}(r,\theta)\mathbf{s}(t)+\mathbf{n}(t) \tag{8}\]  
where  
\[ \begin{array}{l}\mathbf{x}(\mathbf{t})=[\mathbf{x}_{-\mathbf{M}}(\mathbf{t} ),\cdots,\mathbf{x}_{0}(\mathbf{t}),\cdots,\mathbf{x}_{\mathbf{M}}(\mathbf{t} )]^{\text{T}}\\ \mathbf{\tilde{A}}(r,\theta)=[\mathbf{\tilde{a}}_{-M}(r,\theta),\cdots, \mathbf{\tilde{a}}_{0}(r,\theta),\cdots,\mathbf{\tilde{a}}_{M}(r,\theta)]^{ \text{T}}\\ \mathbf{n}(t)=[\mathbf{n}_{-M}(t),\cdots,\mathbf{n}_{0}(t),\cdots,\mathbf{n}_ {M}(t)]^{\text{T}}\end{array} \]  
Adaptive Piezoelectric Sensor Array Error Calibration Based 2D-MUSIC  
The basic idea of the MUSIC algorithm is obtaining the signal subspace and noise subspace through eigenvalue decomposition of the signal covariance matrix, and then estimating the signal parameter using the orthogonality of two spaces. Figure 1: Array element position error schematic diagram. The covariance matrix of the observed signal vector from the imprecise sensor array is  
\[\mathbf{R}_{x}=E[\mathbf{x}\mathbf{x}^{H}]=\mathbf{\Gamma}\mathbf{A}\mathbf{R}_{ \mathbf{s}}\mathbf{A}^{\mathbf{H}}\mathbf{\Gamma}^{H}+\delta^{2}\mathbf{I} \tag{9}\]  
where,\(\mathbf{x}\) the narrow band signals extracted from the sensor outputs using the Gabor wavelet transform [17]. \(\mathbf{R}_{\mathrm{s}}\) is the covariance matrix of signal and \(\delta^{2}\) is noise power.