Optimal Array Design and Directive Sensors for Guided Waves DoA Estimation

Abstract

The estimation of Direction of Arrival (DoA) of guided ultrasonic waves is an important task in many Structural Health Monitoring (SHM) applications. The aim is to locate sources of elastic waves which can be generated by impacts or defects in the inspected structures. In this paper, the array geometry and the shape of the piezo-sensors are designed to optimize the DoA estimation on a pre-defined angular sector, from acquisitions affected by noise and interference. In the proposed approach, the DoA of a wave generated by a single source is considered as a random variable that is uniformly distributed in a given range. The wave velocity is assumed to be unknown and the DoA estimation is performed by measuring the Differences in Time of Arrival (DToAs) of wavefronts impinging on the sensors. The optimization procedure of sensors positioning is based on the computation of the DoA and wave velocity parameters Cramér-Rao Matrix Bound (CRMB) with a Bayesian approach. An efficient DoA estimator is found based on the DToAs Gauss-Markov estimator for a three sensors array. Moreover, a novel directive sensor for guided waves is introduced to cancel out undesired Acoustic Sources impinging from DoAs out of the given angles range. Numerical results show the capability to filter directional interference of the novel sensor and a considerably improved DoA estimation performance provided by the optimized sensor cluster in the pre-defined angular sector, as compared to conventional approaches.

An Optimal Shaped Sensor Array Derivation

“…It is interesting to compare the cluster optimization of this work with the one in [ 24 ], where just disk sensors were considered. In [ 24 ], the optimal disks positioning, for an unknown wave velocity, is defined with the constraint to lie in a circle domain of a fixed radius.…”

“…It is interesting to compare the cluster optimization of this work with the one in [ 24 ], where just disk sensors were considered. In [ 24 ], the optimal disks positioning, for an unknown wave velocity, is defined with the constraint to lie in a circle domain of a fixed radius. In order to compare the numeric results in DoA estimation obtained via the Disk Sensors Optimal Cluster (DS-OC), defined in [ 24 ], considering a circle radius equal to 2 cm, it is important to highlight that in that case, the maximum DDoAs between two disks is about and , whose average value is .…”

“…Observe that the worst-case variance in Equation ( 20 ) was reduced four times with respect to the worst-case variance of Equation ( 18 ) achievable by a SC. Under a realistic hypothesis, such as signal and noises mutually independent, zero mean stationary Gaussian random processes and equal noise spectra, the time delay variances are equal, when an efficient estimator is used for the DToAs estimation (see [ 24 ]). Therefore, .…”

“…However, in that work, the wave velocity is supposed to be known. Therefore, in a previous work [ 24 ], a modification of the approach presented in [ 23 ] was proposed, which avoids the previous assumption, in order to find the disk sensor positioning for the optimal DoA estimation, when, at the same time, the accuracy loss due to the unknown wave velocity is minimized.…”

“…The numeric results are examined according to the employed estimator, the sensor quantization non-idealities, the signal bandwidth and the wave velocity. In Section 6 , the numeric results are discussed in terms of comparison between the SC and the Disk Sensors Optimal Cluster (DS-OC), which was proposed in [ 24 ]. The conclusions are given in Section 7 .…”