Mathematical Foundation of Sparsity-based Multi-snapshot Spectral Estimation

TL;DR

This paper establishes a mathematical foundation for sparsity-based super-resolution in multi-snapshot spectral estimation, analyzing resolution, stability, and the impact of incoherence in Fourier measurements.

Contribution

It provides the first stability analysis in the super-resolution regime for the sparse MMV problem in DOA estimation, emphasizing the role of amplitude incoherence.

Findings

Resolution improves with higher incoherence of amplitude vectors.

Stability depends on cut-off frequency, noise level, and sparsity.

First stability result for super-resolution sparse MMV in DOA estimation.

Abstract

In this paper, we study the spectral estimation problem of estimating the locations of a fixed number of point sources given multiple snapshots of Fourier measurements in a bounded domain. We aim to provide a mathematical foundation for sparsity-based super-resolution in such spectral estimation problems in both one- and multi-dimensional spaces. In particular, we estimate the resolution and stability of the location recovery when considering the sparsest solution under the measurement constraint, and characterize their dependence on the cut-off frequency, the noise level, the sparsity of point sources, and the incoherence of the amplitude vectors of point sources. Our estimate emphasizes the importance of the high incoherence of amplitude vectors in enhancing the resolution of multi-snapshot spectral estimation. Moreover, to the best of our knowledge, it also provides the first stability result in the super-resolution regime for the well-known sparse MMV problem in DOA estimation.