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Published September 2012 | public
Journal Article

Nested Arrays in Two Dimensions, Part I: Geometrical Considerations

Abstract

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.

Additional Information

© 2012 IEEE. Manuscript received October 10, 2011; revised March 08, 2012 and May 20, 2012; accepted May 21, 2012. Date of publication June 08, 2012; date of current version August 07, 2012. The associate editor coordinating the review of this manuscript and approving it for publication was Prof. Joseph Tabrikian. This work was supported in part by the ONR grant N00014-11-1-0676, and the California Institute of Technology. The authors would like to thank an anonymous reviewer who suggested the "Offset Configuration" in Section V-C.

Additional details

Created:
August 19, 2023
Modified:
October 19, 2023