Fourth-Order Hierarchical Array: A Novel Scheme for Sparse Array Design Based on Fourth-Order Difference Co-Array

Conventional array designs based on circular fourth-order cumulant typically adopt a single expression form of the fourth-order difference co-array (FODCA), which limits the achievable degrees of freedom (DOFs) and neglects the impact of mutual coupling among physical sensors. To address above issues, this paper proposes a novel scheme to design arrays with increased DOFs by combining different forms of FODCA while accounting for mutual coupling. A novel fourth-order hierarchical array (FOHA) based on different forms of FODCA is constructed using an arbitrary generator set. The analytical expression between the coupling leakage of the generator and the resulting FOHA is derived. Two specific FOHA configurations are presented with closed-form sensor placements. The arrays not only offer increased DOFs for resolving more sources in direction of-arrival (DOA) estimation but also effectively suppress mutual coupling. Moreover, the redundancy of FODCA is examined, and it is shown that arrays based on the proposed scheme achieve lower redundancy compared to existing arrays based on FODCA. Meanwhile, the necessary and sufficient conditions for signal reconstruction by FOHA are derived. Compared with existing arrays based on FODCA, the proposed arrays provide enhanced DOFs and improved robustness against mutual coupling. Numerical simulations verify that FOHAs achieve superior DOA estimation performance compared with other sparse linear arrays.