Nested Arrays in Two Dimensions, Part II: Application in Two Dimensional Array Processing

This paper explores the practical application of a new class of two dimensional arrays, namely, the two dimensional nested arrays, in array processing problems like two dimensional direction of arrival estimation. Nested arrays constitute a class of two dimensional arrays with physical sensors on lattice(s), whose difference co-array gives rise to a virtual two dimensional array with
O
(
MN
) elements, although the number of physical sensors used is
M
+
N
. The structure of the two dimensional nested array was proposed in a companion paper and several issues related to the geometrical orientation of the difference co-array in the two dimensional space were addressed. In this paper, the practical aspects of the application of the nested array in two dimensional array processing are considered, which can enable us to exploit the increased degrees of freedom offered by the co-array. Under the constraint of a fixed total number of physical sensors, the optimal structure of the 2D nested array is solved which maximizes the number of elements in the virtual co-array. This provides closed form expressions for the sensor locations and the exact degrees of freedom obtainable from the proposed array as a function of the total number of sensors. It is shown that one can obtain
O
(
N
²
) virtual elements in the co-array using only
N
physical sensors. To exploit the increased degrees of freedom (large number of virtual sensors) offered by the virtual array for two dimensional DOA estimation of more sources than sensors, a novel algorithm based on the concept of two dimensional spatial smoothing is proposed and several issues related to identifiability in two dimensional DOA estimation problem (which is a fundamental issue in two dimensions), are addressed. The validity of the proposed methods is verified through several numerical examples.