Generalized Line Spectral Estimation via Convex Optimization

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.