Robustness of Difference Coarrays of Sparse Arrays to Sensor Failures—Part II: Array Geometries

In array processing, sparse arrays are capable of resolving <inline-formula><tex-math notation="LaTeX">$\mathcal {O}(N^2)$</tex-math></inline-formula> uncorrelated sources with <inline-formula><tex-math notation="LaTeX">$N$</tex-math></inline-formula> sensors. Sparse arrays have this property because they possess uniform linear array (ULA) segments of size <inline-formula><tex-math notation="LaTeX">$\mathcal {O}(N^2)$</tex-math></inline-formula> in the difference coarray, defined as the differences between sensor locations. However, the coarray structure of sparse arrays is susceptible to sensor failures and the reliability of sparse arrays remains a significant but challenging topic for investigation. In the companion paper, a theory of the <inline-formula><tex-math notation="LaTeX">$k$</tex-math></inline-formula>-essential family, the <inline-formula><tex-math notation="LaTeX">$k$</tex-math></inline-formula>-fragility, and the <inline-formula><tex-math notation="LaTeX">$k$</tex-math></inline-formula>-essential Sperner family were presented not only to characterize the patterns of <inline-formula><tex-math notation="LaTeX">$k$</tex-math></inline-formula> faulty sensors that shrink the difference coarray, but also to provide a number of insights into the robustness of arrays. This paper derives closed-form characterizations of the <inline-formula><tex-math notation="LaTeX">$k$</tex-math></inline-formula>-essential Sperner family for several commonly used array geometries, such as ULA, minimum redundancy arrays (MRA), minimum holes arrays (MHA), Cantor arrays, nested arrays, and coprime arrays. These results lead to many insights into the relative importance of each sensor, the robustness of these arrays, and the DOA estimation performance in the presence of sensor failure. Broadly speaking, ULAs are more robust than coprime arrays, while coprime arrays are more robust than maximally economic sparse arrays, such as MRA, MHA, Cantor arrays, and nested arrays.