Robustness of Coarrays of Sparse Arrays to Sensor Failures

Sparse arrays can identify $\mathcal{O}(N^{2})$ uncorrelated sources using $N$ physical sensors. This property is because the difference coarray, defined as the differences between sensor locations, has uniform linear array (ULA) segments of length $\mathcal{O}(N^{2})$. It is empirically known that, for sparse arrays like minimum redundancy arrays, nested arrays, and coprime arrays, this $\mathcal{O}(N^{2})$ segment is susceptible to sensor failure, which is an important issue in practical systems. This paper presents the $(k-$)essentialness property, which characterizes the combinations of the failing sensors that shrink the difference coarray. Based on this, the notion of fragility is proposed to quantify the reliability of sparse arrays with faulty sensors, along with comprehensive studies of their properties. It is demonstrated through examples that there do exist sparse arrays that are as robust as ULA and at the same time, they enjoy $\mathcal{O}(N^{2})$ consecutive elements in the difference coarray.