Sparse arrays have received attention in array signal processing since they can resolve \(O(N^2)\) 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 \(O(N^2)\). 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.

Sparse arrays; difference coarrays; robustness; the importance function; the generalized $k$-fragility