Discovering Optimal Robust Minimum Redundancy Arrays (RMRAs) through Exhaustive Search and Algebraic Formulation of a New Sub-Optimal RMRA

Modern sparse arrays are maximally economic in that they retain just as many sensors required to provide a specific aperture while maintaining a hole-free difference coarray. As a result, these are susceptible to the failure of even a single sensor. Contrarily, two-fold redundant sparse arrays (TFRSAs) and robust minimum redundancy arrays (RMRAs) ensure robustness against single-sensor failures due to their inherent redundancy in their coarrays. At present, optimal RMRA configurations are known only for arrays with sensor counts N=6 to N=10. To this end, this paper proposes two objectives: (i) developing a systematic algorithm to discover optimal RMRAs for N>10, and (ii) obtaining a new family of near-/sub-optimal RMRA that can be completely specified using closed-form expressions (CFEs). We solve the combinatorial optimization problem of finding RMRAs using an exhaustive search technique implemented in MATLAB. Optimal RMRAs for N = 11 to 14 were successfully found and near/sub-optimal arrays for N = 15 to 20 were determined using the proposed technique. As a byproduct of the exhaustive search, a large catalogue of valid near- and sub-optimal RMRAs was also obtained. In the second stage, CFEs for a new TFRSA were obtained by applying pattern mining and algebraic generalizations to the arrays obtained through exhaustive search. The proposed family enjoys CFEs for sensor positions, available aperture, and achievable degrees of freedom (DOFs). The CFEs have been thoroughly validated using MATLAB and are found to be valid for $N\geq8$. Hence, it can be concluded that the novelty of this work is two-fold: extending the catalogue of known optimal RMRAs and formulating a sub-optimal RMRA that abides by CFEs.

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.