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Published August 2011 | public
Journal Article

Theory of Sparse Coprime Sensing in Multiple Dimensions

Abstract

Coprime sampling and coprime sensor arrays have been introduced recently for the one-dimensional (1-D) case, and applications in beamforming and direction finding discussed. A pair of coprime arrays can be used to sample a wide-sense stationary signal sparsely, and then reconstruct the autocorrelation at a significantly denser set of points. All applications based on autocorrelation (e.g., spectrum and DOA estimation) benefit from this property. It was also shown in the past that coprimality can be exploited in the frequency domain by using a pair of coprime DFT filter banks, to produce the effect of a much denser tiling in the frequency domain, compared to what the two filter banks can individually achieve. This paper extends these ideas to multiple dimensions. In the 1-D case the samples or sensors lie on a pair of uniform grids, whereas in the multidimensional case, they lie on a pair of multidimensional lattices, not necessarily rectangular. This makes the developments mathematically more intricate. First several properties of coarrays of lattices are derived. It is shown how one can get dense coarrays from sparse arrays on non rectangular lattices. This requires that the lattice generating matrices M and N be commuting and coprime (to be defined). Multidimensional DFT filter banks for applications such as beamforming, with commuting coprime lattice arrays, are then described, and it is shown that a very dense tiling of the frequency plane can be obtained from the two sparse lattice arrays. A particular family of commuting coprime matrices called adjugate pairs are considered in some detail, and shown to have attractive properties. A brief review of the 1-D case is included at the beginning for convenience.

Additional Information

© 2011 IEEE. Manuscript received July 10, 2010; revised January 26, 2011; accepted March 19, 2011. Date of publication April 05, 2011; date of current version July 13, 2011. The associate editor coordinating the review of this manuscript and approving it for publication was Dr. Arie Yeredor. This work was supported in parts by the ONR Grant N00014-08-1-0709, and the California Institute of Technology.

Additional details

Created:
August 22, 2023
Modified:
October 24, 2023