## 2 Minimization of Uncertainty in Doa Estimation
The error bounds were computed and illustrated for all of the clusters here considered by setting the value of the wave velocity \(v=2000\) [m/s] and DToAs Standard Deviation equal to 2 [us]. The worst case error can be reduced by selecting an appropriate value for \(\rho_{0}\) (i.e., \(\rho_{0}\) so that \(\rho_{2}(45^{\circ})=0\)), or by working on a shifted range of angles, centered on the zero angle, i.e., \(\theta\in[-45^{\circ},45^{\circ}]\). In this case, the parameters \(\alpha,\rho_{0}\) and \(\rho_{3}\) can be used in order to minimize the uncertainty in DoA estimation. In particular, \(\rho_{0}\) should be equal to 0, while \(\alpha\) and \(\rho_{3c}\) are defined by selecting the maximum absolute values of \(\Delta t_{12}(\theta),\Delta t_{13}(\theta)\), i.e., \(d/v\), equal to the maximum absolute values of DToAs for the conventional cluster. Limiting \(\Delta t_{12},\Delta t_{13}\) implies the limitation of the maximal dimensions of the transducers, and also the relaxation of memory requirements for the storage and subsequent  
Figure 2: Upper and lower error bounds of two clusters of three shaped sensors to estimate the direction of arrival (DoA), when the wave mode velocity is unknown: (**a**) the case of the cluster presented in [15]. (**b**) The new cluster of three shaped sensors. cross-correlation of the acquired time-waveforms. To summarize, the values of the parameters \(\alpha,\rho_{0},\rho_{3c}\) and the functions \(\rho_{2}(\theta),\rho_{3}(\theta)\) are:  
\[ \begin{split}&\alpha=d/45[\text{deg}],\ \rho_{0}=0,\ \rho_{3c}=d\\ &\rho_{2}(\theta)=\alpha\cdot\theta,\ \rho_{3}(\theta)=\rho_{3c} \end{split}  \tag{6}\]  
In particular, the new shaped sensors cluster was designed, so that the MDDP is again \(d=20\;[\text{mm}]\). Figure 2b illustrates the worst-case error for this configuration, in terms of upper and lower bounds. Figure 3b shows the upper and lower bounds of the conventional cluster with three circular sensors [9; 10] with the MDDP equal to \(d=20[\text{mm}]\) (this distance corresponds, for the circular sensors, to the physical distances between sensors in Figure 3a). The cluster of shaped sensors that implements the relations (3) and (4) with the parameters being detailed in Equation (6) allows for a theoretical average reduction of the worst case error of \(11\%\). It is important to stress the fact that the comparison is provided with the same maximum absolute value for the DToAs (i.e., \(d/v\)). This means that the two clusters have the same memory requirements and same computational cost to estimate the time delays. It is also worth highlighting that the two clusters differ considerably in terms of DoA estimation function: the conventional cluster requires the evaluation of the _atan_ function of the DToAs ratio, while the estimation function of the new cluster is simply linear w.r.t. the DToAs ratio. It is given by inverting (5) with respect to \(\theta\). Therefore it is compatible with embedded systems that are equipped with micro-controller units. Unfortunately, implementing a cluster of piezo-patches that are characterized by the DToA-DoA relationships in (5) and (6) is not possible with simple piezo-disks. Shaped sensors have to be used, but this introduces an additional complication: in fact, shaped sensors are generally characterized by an angle-dependent wavenumber tuning effect [19], i.e., the frequency response of the transducer depends on the direction of propagation of the incident wave. In turn, this effect may hamper the possibility to extract the time difference of arrival with conventional cross-correlation procedures, in which the response among different transducers is assumed to be the same. In the next section, the method that was proposed in [14] to tackle this problem is reviewed and applied to the considered three-sensor cluster case. Figure 3: (**a**) The conventional cluster of 3 circular piezo-sensors spaced apart by a distance \(d_{CC}=20\;[mm]\). (**b**) Upper and lower error bounds of the conventional cluster.