Third-Order Sum-Difference Expansion: An Array Extension Strategy Based on Third-Order Cumulants

Recently, numerous design schemes for high-order sparse linear arrays (SLAs) have been introduced for underdetermined direction-of-arrival (DOA) estimation based on high-order cumulants, which utilize both difference co-array (DCA) and sum co-array (SCA) of the generator arrays to construct a large consecutive virtual co-array, achieving a significant increase in the number of uniform degrees-of-freedom (uDOFs). However, this processing places high demands on the generator arrays, which require both long consecutive DCA and SCA. In addition, the robustness of the derived array is prone to deterioration, due to reduced redundancy between DCA and SCA. To that end, in this paper, an alternative design scheme for third-order SLAs termed third-order sum-difference expansion (TO-SDE) is proposed, which no longer separates DCA and SCA by a shift factor, but considers them as a unified whole. In so doing, most desirable characteristics of the generator array are preserved, such as the size of consecutive virtual co-array, resistance to mutual coupling, and robustness against sensor failures, while the mapping from the sum-difference co-array based second-order SLAs to the third-order is achieved. By selecting the appropriate generator array, excellent DOA estimation performance can be attained in various scenarios.