Robustness of Difference Coarrays of Sparse Arrays to Sensor Failures-Part I: A Theory Motivated by Coarray MUSIC

In array processing, sparse arrays are capable of resolving O(N 2 ) uncorrelated sources with N sensors. Sparse arrays have this property because they possess uniform linear array (ULA) segments of size O(N 2 ) 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. Broadly speaking, ULAs whose difference coarrays only have O(N) elements are more robust than sparse arrays with O(N 2 ) coarray sizes. This paper advances a theory for quantifying such robustness by introducing the k-essentialness of sensors and the k-essential family of arrays. The proposed theory is motivated by the coarray MUltiple SIgnal Classification (MUSIC) algorithm, which estimates source directions based on difference coarrays. Furthermore, the concept of essentialness not only characterizes the patterns of k faulty sensors that shrink the difference coarray, but also leads to the notion of k-fragility, which assesses the robustness of array geometries quantitatively. However, the large size of the k-essential family usually complicates the theory. It will be shown that the k-essential family can be compactly represented by the so-called k-essential Sperner family. Finally, the proposed theory is used to provide insights into the probability of change of the difference coarray, as a function of the sensor failure probability and array geometry. In a companion paper, the k-essential Sperner family for several commonly used array geometries will be derived in closed form, resulting in a quantitative comparison of the robustness of these arrays.