On Gridless Sparse Methods for Line Spectral Estimation From Complete and Incomplete Data

TL;DR

This paper develops gridless sparse methods for line spectral estimation that work with both complete and incomplete data, overcoming grid mismatch issues and providing robust, noise-agnostic algorithms.

Contribution

It introduces a gridless SPICE method and extends atomic norm techniques to incomplete data, offering a unified framework for model order selection and robust frequency estimation.

Findings

Proposed a gridless SPICE (GLS) applicable to complete and incomplete data.

Proved equivalence between GLS and atomic norm methods under noise.

Validated methods through numerical simulations showing improved performance.

Abstract

This paper is concerned about sparse, continuous frequency estimation in line spectral estimation, and focused on developing gridless sparse methods which overcome grid mismatches and correspond to limiting scenarios of existing grid-based approaches, e.g., $\ell_1$ optimization and SPICE, with an infinitely dense grid. We generalize AST (atomic-norm soft thresholding) to the case of nonconsecutively sampled data (incomplete data) inspired by recent atomic norm based techniques. We present a gridless version of SPICE (gridless SPICE, or GLS), which is applicable to both complete and incomplete data without the knowledge of noise level. We further prove the equivalence between GLS and atomic norm-based techniques under different assumptions of noise. Moreover, we extend GLS to a systematic framework consisting of model order selection and robust frequency estimation, and present feasible algorithms for AST and GLS. Numerical simulations are provided to validate our theoretical analysis and demonstrate performance of our methods compared to existing ones.