## 1 Introduction  
Structural Health Monitoring (SHM) systems [1], i.e., inspections systems based on permanently installed sensors, have been intensively investigated in recent years, due to the high potential that they have in reducing maintenance costs in a wide range of industrial application fields. In particular, an increasing demand to develop SHM systems comes from the automotive and aerospace industries [2]. In these fields, open challenges are: (i) the reduction of the monitoring system weight (including the associated cabling and circuitry), the minimization of both (ii) the power consumption, and (iii) the amount of data that are to be transferred and processed  
Among the various sensor technologies proposed for SHM, those that are based on the usage of ultrasonic guided waves (GWs), are considered to be particularly attractive because of the GW property to travel across long distances with low attenuation. In particular, passive GW methods can be used to detect and locate acoustic sources (AS) in plates or shells, such as those that are generated by impacts, or acoustic emissions generated by cracks and corrosion, by using piezoelectric sensors [3; 4]. The approaches presented in literature for AS localization with GW can be divided in three main groups: (i) inverse methods [5; 6], (ii) hyperbolic location methods [7], and (iii) methods that use Direction of Arrival (DoA) estimations [8; 9]. However, if the first two methods are well validated at laboratory scale, these approaches are rarely adopted for field deployment, because of the stringent hypotheses that they rely on. In particular, the first group implies onerous calculations because it requires a very accurate modeling of the guided ultrasonic propagation, whereas the second group requires a large number of sensors in order to reduce the estimation uncertainty. Moreover, this latter approach is unsuitable when the wave propagation characteristics vary in the monitored area. For these reasons, the methods belonging to the third group, which estimate DoAs by using clusters of closely located sensors that do not rely on the knowledge of the characteristics of propagation, are often the only viable strategy to locate AS with embedded systems. [PERSON] et al. in [9; 10] proposed a simple cluster of 3 circular sensors. In this work, this solution will be referred to as \"conventional cluster\". The DoA is performed by estimating the Difference in Time of Arrival (DToA) of wave-fronts between sensors. The DToAs, i.e., the time delays, can be estimated with conventional cross-correlation procedures (i.e., the optimal method for measurements that are affected by additive white Gaussian noise \(N_{W_{G}}\) with zero mean [11]). This cluster typology allows for estimating the DoA with minimum amount of data to acquire (only three signals) and reduced computational cost (only two time delays estimations). Such an approach was proven to be suitable for real-field applications to perform real-time impact detection and localization [10]. It is worth noting that [PERSON], [PERSON], [PERSON], and [PERSON] in [12; 13] showed that it is possible to estimate the AS localization even in the case of heavily anisotropic structures (i.e., characterized by elliptical or rhomboidal wave fronts). In this work, we further investigate the concept of DoA detection via DToAs estimations, by analyzing how beneficial the adoption of clusters constituted by irregularly shaped piezoelectric transducers can be. In particular, in the proposed approach, the sensors are shaped with a procedure based on the Radon Transform (RT) and its Inverse (IRT). This procedure allows to define the DToA between two sensors as an arbitrary DoA function, divided by the wave velocity \(v\), as better detailed in Section 3. The RT approach was firstly presented in [14] in order to estimate the DoA by using two shaped sensors when the actual wave velocity is assumed to be known.