Nested Arrays in Two Dimensions, Part I: Geometrical Considerations

A new class of two dimensional arrays with sensors on lattice(s) is proposed, whose difference co-array can give rise to a virtual two dimensional array with much larger number of elements on a “dense” lattice. This structure is obtained by systematically nesting two arrays, one with sensors on a sparse lattice and the other on a dense lattice where the lattices bear a certain relation with each other. The difference co-array of such an array with M and N elements respectively on the two lattices, is a two dimensional array with O(MN) elements present contiguously (without holes) on the dense lattice. The difference co-array can be realized on any arbitrary lattice by choosing the dense and sparse lattices appropriately. The generator matrices of the sparse and the dense lattices are related by an integer matrix. The Smith form of this integer matrix is shown to provide a very insightful perspective which is exploited heavily in the construction of nested arrays. The design of the two dimensional nested array gives rise to several interesting geometrical orientations of the co-array which are addressed in detail, and it is shown how the orientations can be manipulated to yield more virtual sensors in a continuum on the dense lattice. The increased number of elements in the virtual difference co-array can be exploited to perform two dimensional direction of arrival (DOA) estimation of many more sources than what traditional methods can achieve. A novel algorithm for application of the nested array in two dimensional direction of arrival estimation is reported in the accompanying part II of the paper.