Analysis of the fourth-order co-array for a mixture of circular and noncircular signals

A detailed analysis of the degrees of freedom (DOFs) (and therefore the maximum number of signals to be estimated) of the fourth-order sum and difference co-arrays for direction of arrival (DOA) estimation in the presence of circular, strictly noncircular and nonstrictly noncircular signals is presented. There are different ways in combining noncircularity, fourth-order cumulants and sparse arrays to increase the DOFs of the system for DOA estimation. However, there are some confusions or a lack of clarity in the combination. In this work, we aim to fill the gap and clarify some relevant issues by providing a detailed analysis for the fourth-order co-array for a mixture of circular, strictly noncircular and nonstrictly noncircular signals based on a general signal model, including consideration of the noncircular phases of signals in DOA estimation, the fourth-order co-array aperture by considering all the fourth-order cumulants, the difference in the number of signals to be resolved for the strictly and nonstrictly noncircular signals, and the general analysis of DOFs for a mixture of circular, strictly noncircular and nonstrictly noncircular signals based on either uniform or sparse arrays. Furthermore, the expansion and shift scheme with one sub-array being a nested array and another one being a stamp array is proposed, which provides the most DOFs among considered sparse array construction schemes.