(CCS), DOA estimation, frequency estimation, gridless sparse method, high resolution, reweighted atomic norm minimization (RAM).

Introduction
============

Compressed sensing (CS) [@candes2006compressive; @donoho2006compressed] refers to a technique of reconstructing a high dimensional signal from far fewer samples and has brought significant impact on signal processing and information theory in the past decade. In conventional wisdom, the signal of interest needs to be sparse under a finite discrete dictionary for successful reconstruction, which limits its applications, for example, to array processing, radar and sonar, where the dictionary is typically specified by one or more continuous parameters. In this paper, we are concerned about a compressed sensing problem with a continuous dictionary which arises in line spectral estimation and array processing [@krim1996two; @stoica2005spectral]. In particular, we are interested in recovering $L$ discrete sinusoidal signals which compose the data matrix $\m{Y}^o\in\bC^{N\times L}$ with its $(j,t)$th element (corrupted by noise in practice) where $i=\sqrt{-1}$, $f_k\in\bT\triangleq\left[0,1\right]$, $s_{kt}\in\bC$ and $\mbra{N}=\lbra{1,2,\dots,N}$. This means that each column of $\m{Y}^o$ is superimposed by $K$ discrete sinusoids with frequencies $\lbra{f_k}$ and amplitudes $\lbra{s_{kt}}$. To recover $\m{Y}^o$ (and the frequencies in many applications), however, we are only given partial/compressive samples on its rows indexed by $\m{\Omega} \subset \mbra{N}$ (of size $M < 