Fourth-order shift expansion: An optimized array design scheme with enhanced degrees of freedom for DOA estimation

Recently, fourth-order cumulant (FOC)-based direction of arrival (DOA) estimation methods have garnered significant interest from numerous scholars within the academic community. However, the majority of extant advanced fourth-order sparse arrays have not sufficiently exploited the characteristics of the second-order sum-difference co-array to their fullest potential. Motivated by this, a fourth-order sparse array design scheme, called fourth-order shift expansion (FO-SE), is proposed, which is formed of a shifted sparse array and an extended sparse array. Specifically, this scheme first determines the shifted coefficient by utilizing the second-order difference co-array and sum co-array of the first subarray, and then the extended coefficient is derived by combining the shifted coefficient with the sum-difference co-array of the first subarray. Once the first subarray is determined, we can obtain explicit formulations for the sensor locations of the FO-SE and its uniform degrees of freedom (uDOFs). Theoretical analysis proves that FO-SE has enhanced uDOFs. Numerical simulations demonstrate that the FO-SE has excellent performance with regard to uDOFs and angular resolution. Furthermore, with or without the presence of the mutual coupling (MC) effect, the FO-SE exhibits superior DOA estimation performance in comparison to competing fourth-order and second-order sparse arrays.