Generalized line spectral estimation for radar and localization

A variety of problems arising in imaging, radar, and localization amount to recover a signal that is sparse in a continuous indexed dictionary. Line spectral estimation theory considers a particular instance of this problem, namely that of estimating the frequency and magnitude components of a sum of complex sinusoids from uniformly spaced samples. In this work we consider the significantly more general class of continuous sparse signal recovery problems where only an severely under-sampled version of theses uniformly spaced samples is observed through linear measurements. Applications that fall within this framework are, amongst others, the recovery of the delay-Doppler shifts in radar, and the angle-delay-Doppler triplets in multiple input multiple output (MIMO) radar. We prove that the frequencies and amplitudes of the sinusoids can be recovered perfectly by solving a convex program, from a near-minimal number of observations. This result holds provided the frequencies are sufficiently separated, and provided that the linear measurements obey mild conditions that are satisfied in a variety of applications.