Efficient two-dimensional line spectrum estimation based on decoupled atomic norm minimization

Abstract

This paper presents an efficient optimization technique for gridless 2-D line spectrum estimation, named decoupled atomic norm minimization (D-ANM). The framework of atomic norm minimization (ANM) is considered, which has been successfully applied in 1-D problems to allow super-resolution frequency estimation for correlated sources even when the number of snapshots is highly limited. The state-of-the-art 2-D ANM approach vectorizes the 2-D measurements to their 1-D equivalence, which incurs huge computational cost and may become too costly for practical applications. We develop a novel decoupled approach of 2-D ANM via semi-definite programming (SDP), which introduces a new matrix-form atom set to naturally decouple the joint observations in both dimensions without loss of optimality. Accordingly, the original large-scale 2-D problem is equivalently reformulated via two decoupled one-level Toeplitz matrices, which can be solved by simple 1-D frequency estimation with pairing. Compared with the conventional vectorized approach, the proposed D-ANM technique reduces the computational complexity by several orders of magnitude with respect to the problem size. It also retains the benefits of ANM in terms of precise signal recovery, small number of required measurements, and robustness to source correlation. The complexity benefits are particularly attractive for large-scale antenna systems such as massive MIMO, radar signal processing and radio astronomy.

Underdetermined DOA Estimation for Wideband Signals via Focused Atomic Norm Minimization

“…In the literature [ 28 , 34 ], the ANM principle is that the signal is expressed as a linear combination of a few atoms over a known atom set, and then, the structural information of these atoms, i.e., Vandermonde structure, is utilized for signal reconstruction from measurements.…”

“…Remark on targets angle separation condition : Here, targets angle separation means the required angle separation between neighbouring signal source. According to the IV.2 theorem [ 34 ], the sufficient separation condition is …”

Super-Resolution Channel Estimation for Arbitrary Arrays in Hybrid Millimeter-Wave Massive MIMO Systems

“…To estimate the 2D channel through gridless CS, a straightforward way is to vectorize the 2D formulations and then to cast the 2D Vandermonde structure into a vectorized SDP formulation via a two-level Toeplitz structured matrix, a.k.a., vectorization based ANM (V-ANM) [32], [52], [53]. However, the V-ANM leads to a high computational complexity on the order of O(N 3.5 BS N 3.5 MS ) [54], because of the huge problem scale resulting from the vectorization operation.…”

“…To solve this problem of V-ANM, we develop an efficient 2D channel estimation at much lower computational cost, by using a decoupled-ANM (D-ANM) technique [54], [55]. Different from the V-ANM, we introduce a matrix-form atom set A M as…”

“…Noticeably, since the PSD constraint in (61) is of size (N BS + N MS ) × (N BS + N MS ), the D-ANM allows a reduced computational complexity on the order of O (N BS + N MS ) 3.5 [54], which is much smaller than that of the V-ANM with large arrays.…”