Third-Order Nested Array: An Optimal Geometry for Third-Order Cumulants Based Array Processing

Several sparse linear array structures have recently been proposed to enhance the identifiability of direction-of-arrival estimation by using second-order or higher even-order statistics under the co-array equivalence. This paper aims to explore non-conventional odd-order statistics, namely third-order cumulants, to design a sparse linear array under the co-array equivalence for enhancing the identifiability of a given sensors array. This paper presents a mathematical framework to devise a third-order exhaustive co-array equivalence associated with third-order cumulants of the data. A novel sparse linear array geometry, namely the third-order nested array, is proposed by maximizing the third-order exhaustive co-array’s virtual uniform linear array aperture to enhance the identifiability of the given sensors array. The proposed third-order nested array offers a significantly higher virtual uniform linear array aperture than other existing similar sparse linear array structures designed using the second-order statistics of the data for any given number of physical sensors. In addition, the proposed third-order nested array offers a relatively higher virtual uniform linear array aperture even than that of a sparse linear array structure designed using the fourth-order statistics for the number of physical sensors less than equal to eight. The enhanced virtual uniform linear array aperture associated with the proposed third-order nested array increases the identifiability along with the accuracy of direction of arrival estimation for a given number of physical sensors. However, the proposed third-order nested array needs relatively more data than the second-order statistics based sparse linear array, as higher-order statistics based estimators entail a relatively higher variance.