## 2 Impact Detection and Localization  
Considering a plate-like structure, a point impact on the surface will determine a mechanical response in the form of elastic waves of various natures propagating outward from the point of contact. Our work focused on detecting low-energy impacts (from 35 to 600 mJ) that did not result in permanent damage to the aluminum plate. Low-energy impacts are of interest for laboratory testing of system prototypes because they are non-destructive, while larger energy (>1 J) impacts are used for real-life SHM applications. The implication of sensor choice, regarding sensitivity and impact energy, is discussed in the next section.  
In general, it is possible to detect the location of an impact event by measuring the DToAs of the elastic response wave front to a set of three or more sensors with knowledge of their position (x\({}_{i}\), y\({}_{i}\)) on a planar structure. Using these data, the impact location can be numerically triangulated. Our system estimated the DToA of impact-generated elastic waves using a simple amplitude threshold thanks to the remarkable (better than 70 dB) signal-to-noise ratio (SNR) of the analog front-end.  
The developed system performed impact detection by running in passive (listening) mode, i.e., by continuously sampling the transducer signals in a circular buffer, while waiting for a triggering event (a threshold-crossing) to occur. This threshold level (six times the root-mean-square noise level) had to be verified during the experimental tests to avoid spurious triggering, while maintaining a good sensitivity to low-energy impacts. As the data acquisition hardware detected an impact threshold-crossing event, it immediately froze the circular buffer and transferred the data to the acquisition system.  
We defined a region of interest (ROI) of the aluminum plate equal to 250 mm \(\times\) 240 mm (see Figure 1). This area was divided into a uniform grid of points spaced 1 mm apart with coordinates (x\({}_{p}\), y\({}_{p}\)) for the subsequent analysis. For each of those points, the impact localization algorithm processed the received signals to extract the differential DToAs and calculated the value of the error function E(x\({}_{p}\), y\({}_{p}\)) [15; 16; 17]. Once the error function (1) was calculated for the whole grid, the impact point was assigned by the absolute minimum error value criterion. The error function E(x\({}_{p}\), y\({}_{p}\)) presents a minimum and has a monotonic behavior. In such cases, one can extrapolate a best-guess position of the impact by calculating the centroid of the points having a value within 1% of the absolute minimum.  
The choice to use a grid of points spaced 1 mm apart was a good compromise between the accuracy of estimation of the impact point and the necessity to elaborate signals in real-time and immediately show the results on a PC display.  
\[E\big{(}x_{p},y_{p}\big{)}=\sum_{i=1}^{N_{T}-1}\sum_{j=i+1}^{N_{T}}\left|\frac{ \left(t_{i}-t_{j}\right)v-\left(\sqrt{\big{(}x_{i}-x_{p}\big{)}^{2}+\big{(}y_{i }-y_{p}\big{)}^{2}}-\sqrt{\big{(}x_{j}-x_{p}\big{)}^{2}+\big{(}y_{j}-y_{p}\big{)} ^{2}}\right)}{\sqrt{\big{(}x_{i}-x_{j}\big{)}^{2}+\big{(}y_{i}-y_{j}\big{)}^{2} }}\right| \tag{1}\]  
Here \(N_{T}\) is the total number of installed sensors whose positions are defined by the coordinates \(P_{i}=(x_{i},y_{i})\) and \(P_{j}=(x_{j},y_{j})\). The corresponding measured time of flights for each sensor are \(t_{i}\) and \(t_{j}\), respectively.