Two dimensional nested arrays on lattices

In this paper, we develop the theory of a new class of two dimensional arrays with sensors on lattice(s) which can be used to construct a virtual array of much larger size through passive processing. This structure is obtained by systematically nesting two arrays, one with sensors suitably placed on a sparse lattice and the other on an appropriately chosen dense lattice. The difference co-array of such an array with M and N elements respectively on the two lattices, is proved to be a larger two dimensional array with O(MN) sensors present contiguously (without holes) on the dense lattice. It will be shown that there is complete freedom to choose the sparse and dense arrays as long as they are related by an integer matrix (which can also be arbitrarily chosen). To exploit the increased degrees of freedom offered by the array for two dimensional DOA estimation of more sources than sensors, a novel algorithm based on the concept of two dimensional spatial smoothing is also proposed. The validity of the proposed methods is verified through numerical examples.