Generalized Line Spectral Estimation via Convex Optimization

TL;DR

This paper introduces a convex optimization approach for generalized line spectral estimation, enabling accurate recovery of frequencies and amplitudes from severely undersampled linear measurements under certain conditions.

Contribution

It reformulates generalized spectral estimation as a sparse recovery problem over a continuous dictionary and proves perfect recovery with minimal observations.

Findings

Exact recovery of frequencies and amplitudes is possible under separation conditions.

The method works with a near-minimal number of measurements.

The approach applies to various practical imaging and signal processing scenarios.

Abstract

Line spectral estimation is the problem of recovering the frequencies and amplitudes of a mixture of a few sinusoids from equispaced samples. However, in a variety of signal processing problems arising in imaging, radar, and localization we do not have access directly to such equispaced samples. Rather we only observe a severely undersampled version of these observations through linear measurements. This paper is about such generalized line spectral estimation problems. We reformulate these problems as sparse signal recovery problems over a continuously indexed dictionary which can be solved via a convex program. We prove that the frequencies and amplitudes of the components of the mixture can be recovered perfectly from a near-minimal number of observations via this convex program. This result holds provided the frequencies are sufficiently separated, and the linear measurements obey natural conditions that are satisfied in a variety of applications.