Novel algorithms for analyzing the robustness of difference coarrays to sensor failures

Abstract Sparse arrays have drawn attention because they can identify O ( N 2 ) uncorrelated source directions using N physical sensors, whereas uniform linear arrays (ULA) find at most N − 1 sources. The main reason is that the difference coarray, defined as the set of differences between sensor locations, has size of O ( N 2 ) for some sparse arrays. However, the performance of sparse arrays may degrade significantly under sensor failures. In the literature, the k-essentialness property characterizes the patterns of k sensor failures that change the difference coarray. Based on this concept, the k-essential family, the k-fragility, and the k-essential Sperner family provide insights into the robustness of arrays. This paper proposes novel algorithms for computing these attributes. The first algorithm computes the k-essential Sperner family without enumerating all possible k-essential subarrays. With this information, the second algorithm finds the k-essential family first and the k-fragility next. These algorithms are applicable to any 1-D array. However, for robust array design, fast computation for the k-fragility is preferred. For this reason, a simple expression associated with the k-essential Sperner family is proposed to be a tighter lower bound for the k-fragility than the previous result. Numerical examples validate the proposed algorithms and the presented lower bound.