A General Framework for the Robustness of Structured Difference Coarrays to Element Failures

Sparse arrays have received attention in array signal processing since they can resolve $\mathcal{O}\left( {{N^2}} \right)$ uncorrelated sources using N physical sensors. The reason is that the difference coarray, which consists of the differences between sensor locations, has a central uniform linear array (ULA) segment of size $\mathcal{O}\left( {{N^2}} \right)$. From the theory of the k-essentialness property and the k-fragility, the difference coarrays of some sparse arrays are not robust to sensor failures, possibly affecting the applicability of coarray-based direction-of-arrival (DOA) estimators. However, the k-essentialness property might not fully reflect the conditions under which these estimators fail. This paper proposes a framework for the robustness of array geometries based on the importance function and the generalized k-fragility. The importance function characterizes the importance of the subarrays in an array subject to some defining properties. The importance function is also compatible with the k-essentialness property and the size of the central ULA segment in the difference coarray. The latter is closely related to the performance of some coarray-based DOA estimators. Based on the importance function, the generalized k-fragility is proposed to quantify the robustness of an array. Properties of the importance function and the generalized k-fragility are also studied and demonstrated through numerical examples.