A Systematic Framework to Test the Resilience of Three-Fold Redundant Sparse Arrays Against Two Sensor Failures and Some Never-Before Findings

As the field of sparse arrays progressed, numerous array designs have been introduced with a focus on larger apertures and higher degrees of freedom (DOFs), resulting in maximally economic sparse arrays (MESAs) that operate with the least number of sensors required to provide a given aperture while ensuring a hole-free difference coarray (DCA). Consequently, MESAs are least robust to sensor failures and cannot afford the failure of even a single sensor. Multifold redundant sparse arrays (MFRSAs) provide a practical solution to the problem of sensor failures in sparse arrays by making sure that the array contains enough sensor pairs necessary to produce each spatial lag multiple times. Owing to this property, a \b{eta}-fold redundant array can withstand simultaneous failure of at least \b{eta}-1 sensors without losing the hole-free DCA property. Nevertheless, MFRSAs are also prone to hidden dependencies that prevent them from being fully robust. In this work, we present a systematic framework to evaluate the robustness of triple redundant sparse linear arrays (TRSLAs) against all possible two-sensor failures. After detailing the proposed approach, we present the failure analysis of representative TRSLAs available in existing literature. It is found that existing TRSLAs have some hidden vulnerabilities against the failure of some peculiar sensor pairs. Corresponding MATLAB programs and numerical simulations are provided for evaluation and use by the array processing community. The proposed approach has a great archival value as it can evaluate the robustness of any present or future TRSLAs through objective means.