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Nearly optimal resolution estimate for the two-dimensional super-resolution and a new algorithm for direction of arrival estimation with uniform rectangular array

by P. Liu and H. Ammari

(Report number 2022-17)

Abstract
In this paper, we develop a new technique to obtain nearly optimal estimates of the computational resolution limits for two-dimensional super-resolution problems. Our main contributions are fivefold: (i) Our work improves the resolution estimates for number detection and location recovery in two-dimensional super-resolution problems to nearly optimal; (ii) As a consequence, we derive a stability result for a sparsity-promoting algorithm in two-dimensional super-resolution problems (or Direction of Arrival problems (DOA)). The stability result exhibits the optimal performance of sparsity promoting in solving such problems; (iii) Our techniques pave the way for improving the estimates for resolution limits in higher-dimensional super-resolutions to nearly optimal; (iv) Inspired by these new techniques, we propose a new coordinate-combination-based model order detection algorithm for two-dimensional DOA estimation and theoretically demonstrate its optimal performance, and (v) we also propose a new coordinate-combination-based MUSIC algorithm for super-resolving sources in two-dimensional DOA estimation. It has excellent performance and enjoys many advantages compared to the conventional DOA algorithms. The coordinate-combination idea seems to be a promising way for multi-dimensional DOA estimation.

Keywords: two-dimensional super-resolution, direction of arrival algorithms, resolution estimates, stability results, sparsity-promoting algorithm, model order detection, MUSIC algorithm

@Techreport{LA22_1005,
  author = {P. Liu and H. Ammari},
  title = {Nearly optimal resolution estimate for the two-dimensional super-resolution and a new algorithm for direction of arrival estimation with uniform rectangular array },
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2022-17},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2022/2022-17.pdf },
  year = {2022}
}

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