On the Size and Redundancy of the Fourth-Order Difference Co-Array

In array processing, the fourth-order difference co-array of sparse arrays with N sensors admits to resolveO(N⁴)source directions with proper assumptions. ThisO(N⁴)property is relevant to the size of the central uniform linear array (ULA) segment of the fourth-order difference co-array. However, among the existing sparse arrays, the fundamental limits for the sizes of the fourth-order difference co-array remain a topic for further study. This paper characterizes the lower and upper bounds of the size of the fourth-order difference co-array. It is proved that the ULA and the exponential array achieve the lower and upper bounds, respectively. We also propose the fourth-order redundancy to quantify the efficiency of the ULA segment in the fourth-order difference co-array. The fourth-order redundancy owns a provable lower bound depending only on N, providing further insights into the size of the central ULA segment of the fourth-order difference co-array. These bounds are validated through numerical examples.