## 4 Numerical Results and Discussion  
In order to validate the design procedure and the proposed shaped sensors cluster, as represented in Figure 5, multiple impacts occurring on an aluminium plate (3 mm thick, [PERSON]'s modulus 70 GPa, Poisson's coefficient 0.3 and material density 2700 Kg/m\({}^{3}\)) were simulated. The response of the shaped piezo-patches to impacts generating the fundamental antisymmetric mode (A\({}_{0}\)) was computed using  
Figure 4: (**a,b**) Imposed RTs. (**c,d**) IRTs. (**e,f**) binary quantization of IRTs (Threshold values to binary quantization equal to 30% and 25% of maximum values of IRTs for sensors P2 and P3 respectively). (**g,h**) actual RTs. P2 and P3 are related to first and second column, respectively. Figure 5: Designed cluster of three sensors: a reference circular sensor P1 and two reshaped sensors, P2 and P3. the Greens function formalism adopted in [14]. An impulse response of a band pass Batterworth filter (10 th order) with narrow bandwidth (i.e., 7.5 [KHz]) was used in order to emulate the actuated pulse. Subsequently, the estimated DtoAs between P1 and P2, and the one between the P1 and P3 were computed by the locating the peak of the cross-correlation of acquired signals (i.e., the Maximum Likelihood (ML) estimator of DtoA for measurements affected by additive white Gaussian noise (AWGN) with zero mean [11]), and their ratio \(\Delta t_{12 est}/\Delta t_{13 est}\) was used to estimate the DoA (\(\theta\)). The ideal estimation function, linearly dependent with respect to the ratio \(\Delta t_{12 est}/\Delta t_{13 est}\), could be obtained by inverting (5) with respect to \(\theta\). However, due to binary quantization procedure, the actual values of parameters \(\alpha,\rho_{0}\) and \(\rho_{3}c\) might be slightly different with respect to the imposed ones (6). In Figure 6, it can be seen that, when a narrow-band Lamb wave mode (e.g., A\({}_{0}\)) is simulated, the relationship between the DtoA ratio and the DoA (\(\theta\)) is still linear. The parameters of the best-fit line, i.e., y-intercept \(m\) and slope \(q\), can be computed by applying conventional linear regression procedures. The estimated DoA (\(\theta_{est}\)) as a function of its actual value (i.e., \(\theta\)) for different working conditions is shown in Figures 7 and 8, when the A\({}_{0}\) mode propagation is simulated with r = 0.4 [m] as impacts distance. In particular, the two figures report the results at different values of sampling frequency (Fs) and at different values of Peak Signal to Noise Ratio (PSNR) when the measurements are affected by additive white Gaussian noise (AWGN) with zero mean, respectively. As can be seen, the DoA estimation is very accurate, with a standard deviation (SD) of DoA below three degrees, even when the measurements are affected by Gaussian noise (up to the PSNR = 18 dB) and the simulated 'physical\" sampling frequency is low (down to Fs = 100 KHz). It is worth noting that:  
1. In order to generate the Figure 7, the acquired signals were resampled (upsampling factor equal to 10) to improve the estimation accuracy of DtoAs and so of DoA. The sampling rate conversion is the optimal method to reduce the uncertainty on DtoAs (i.e., using \(sinc\) functions to interpolate), but the computational cost can be significant. In embedded systems, such onerous processing may be unfeasible. Possible alternatives consist in: (i) transmitting the acquired signals to a central processing unit (CPU) or (ii) using different interpolation techniques with reduced computational complexity, such as those proposed in [24; 25]. 2. The upper- and lower- bound errors that are displayed in Figure 7 are calculated by applying the propagation of uncertainty theory [26]. 3. In the Figure 8, to numerically assess the SD of DoA, 1000 simulations were performed with measurements being affected by AWGN with zero mean, for each value of PSNR.