## 2 Minimization of Uncertainty in Doa Estimation  
The simplest sensor configuration to perform the wave DoA estimation is based on a couple of isotropic transducers, i.e., the piezopatches P1 and P2 illustrated in Figure 1. Assuming a planar wavefront, the Difference in Time of Arrival (DToA, \(\Delta t\)) is related to the wave DoA (\(\theta\)) by this formula:  
\[\Delta t(\theta)=\frac{d\cdot\cos\theta}{v} \tag{1}\]  
where \(d\) is the distance among the sensors and \(v\) is wave velocity. Consequently, the estimation of the DoA can be performed from the measured DToA (\(\Delta t_{est}\)) by applying the inverse formula:  
\[\theta_{est}=\arccos\frac{v\cdot\Delta t_{est}}{d} \tag{2}\]  
The Propagation of Uncertainty theory [20] that is applied to this case reveals that the error in the estimation can be very large for low values of \(\theta\). This is due to the non-linear characteristics of the _arccos_ relation. To counteract this problem, in [14], a sensor reshaping procedure was proposed in order to achieve a linear dependence between the DToA and the DoA, so that:  
\[\Delta t_{12}(\theta)=\frac{\alpha\cdot\theta+\rho_{0}}{v}=\frac{\rho_{2}( \theta)}{v} \tag{3}\]  
where \(\rho_{0}\) and \(\alpha\) can be chosen by the designer. Thanks to new linear relation, the worst-case error is constant over the considered range of angles (\(\theta\in[0^{0},90^{0}]\)), and it is considerably lower than the maximum error that is achieved using the conventional sensors. It is worth noting that the worst case error is inversely proportional to the selected value of \(\alpha\). For this reason, the larger \(\alpha\) the lower is the error in the DoA estimation; however, there is a trade off involved in the selection of \(\alpha\), because, the larger is \(\alpha\), the larger are the physical dimensions of the piezoelectric patches and the maximum value of the DToA, which defines the time-window length of the acquired signal (i.e., the information to be stored and processed). It is also worth noting that a cluster that is composed by two sensors can be used to estimate \(\theta\) just if the actual wave velocity \(v\) is assumed to be known. Therefore, if the velocity is not known a  
Figure 1: A basic cluster of two circular piezo-sensors spaced apart by a distance \(d\). priori, a third sensor P3 is needed in order to estimate the DoA. In the proposed approach, the sensor P3 is shaped, so that it establishes a constant reference to estimate the wave velocity, i.e.,  
\[\Delta t_{13}(\theta)=\frac{\rho_{3}(\theta)}{v}=\frac{\rho_{3c}}{v} \tag{4}\]  
(where \(\rho_{3c}\) is a constant), the relation between the DToAs ratio and DoA is:  
\[\frac{\Delta t_{12}}{\Delta t_{13}}=\frac{(\rho_{0}+\alpha\theta)}{\rho_{3c}}= \frac{\rho_{2}(\theta)}{\rho_{3c}} \tag{5}\]  
In the last equation, it can be observed that the DoA \(\theta\) is linked to the ratio of DToA by a simple linear relation. Moreover, in order to estimate the DoA, the knowledge of the wave propagation velocity is not necessary, because such quantity does not appear in (5). However, the third sensor P3 introduces an additional uncertainty in the DoA estimation due to \(\Delta t_{13-est}\). In particular, the worst case error can be very large when the DoA is around 90 [deg], if \(\rho_{0}\) is set to 0. The Figure 2a shows the upper and lower bounds of worst case error for the shaped sensors cluster proposed in [15]. It was designed for a work-range of [0, 90] [deg] and it was defined by the following parameter values: \(d=20\) [mm], \(\alpha=d/90\) [deg] \(\simeq 0.22\) [mm/deg], \(\rho_{0}=0\) [mm], \(\rho_{3c}=-7\) [mm]. The value of \(d\) defines the maximum absolute value of DToA, equal to \(d/v\) (in this case, obtained when \(\theta=90\) [deg], for piezo P2). Therefore, the value of \(d\) is redefined as the Maximum Difference in Distance of Propagation (MDDP). It is worth noting that the MDDP, \(d\),in general, does not correspond to the physical distance between the shaped sensors.