Three-Dimensional Decoupled Atomic Norm Minimization

This paper focuses on the problem of three-dimensional (3D) frequency retrieval from a set of time samples. The assumption made in this study is that the signal being analyzed consists of K continuous-valued 3D complex sinusoidal signals. The main objective is to identify all frequency components when only one snapshot of regularly-spaced time samples is available. The vectorized version of atomic norm minimization (ANM) method could accurately solve the problem. Nonetheless, this technique demands significant computational resources. In this paper, we propose a computationally-efficient method based on atomic norm minimization known as three-dimensional decoupled ANM (3D-DANM). This approach transforms the estimation problem into a semi-definite programming (SDP) problem. By performing Vandermonde decomposition on the Toeplitz matrices resulted from the SDP problem, the frequencies of all sources will be estimated. The extracted parameters are then paired using a specific approach. Simulation results demonstrate that the proposed method closely approaches the Cramér-Rao lower bound (CRLB) for sufficient signal-to-noise ratio (SNR) values. Additionally, it offers significant improvements in computational cost compared to other existing approaches.