Multiple Level Nested Array: An Efficient Geometry for $2q$ th Order Cumulant Based Array Processing

Recently, direction-of-arrival estimation (DOA) algorithms based on arbitrary even-order (2q) cumulants of the received data have been proposed, giving rise to new DOA estimation algorithms, namely the 2q MUSIC algorithm. In particular, it has been shown that the 2q MUSIC algorithm can identify O(Nq) statistically independent non-Gaussian sources. However, in this paper, it is demonstrated that the processing power of the 2qth-order cumulant based methods can potentially be even larger. It will be shown that the 2qth-order cumulant matrix of the data is directly related to the concept of a 2qth-order difference co-array which can potentially have O(N²q) virtual sensors, leading to identification of O(N²q) statistically independent non-Gaussian sources using 2q th-order cumulants. However, the number of actually realizable virtual elements in the 2q th-order difference co-array depends on the geometry of the physical array. In order to ensure that the co-array indeed has the desired degrees of freedom, a new generic class of linear (one dimensional) nonuniform arrays, namely the 2qth-order nested array, is proposed, whose 2q th-order difference co-array is proved to contain a uniform linear array with O(N²q) sensors. In order to exploit these increased degrees of freedom of the co-array, a new algorithm for DOA estimation is also developed, which acts on the same 2q th-order cumulant matrix as the earlier methods and can yet identify O(N²q) sources. It is proved that the proposed method can identify the maximum number of sources among all methods that use 2q th-order cumulants.