N$), denoted by $\m{Y}_{\m{\Omega}}^o$. This problem is referred to as off-grid or continuous compressed sensing (CCS) according to [@tang2012compressed; @yang2014continuous] differing from the existing CS framework in the sense that every frequency $f_k$ can take any continuous value in $\bT$ rather than constrained on a finite discrete grid.

The CCS problem in the case of $L=1$ (a.k.a. the single-measurement-vector (SMV) case) is usually known as line spectral estimation in which frequency recovery though is of main interest. The use of compressive data can lead to efficient sampling and/or energy saving. It also can be caused by data missing due to adversary environmental effects. The multiple-measurement-vector (MMV) case with $L>1$ is common in array processing where one estimates directions of a few narrowband sources using outputs of an antenna array. Readers are referred to [@krim1996two; @stoica2005spectral] for derivation of the model in . Therein $\m{Y}^o$ consists of outputs of a virtual $N$-element uniform linear array (ULA), in which adjacent antennas are spaced by half a wavelength, over $L$ time snapshots. In particular, each column of $\m{Y}^o$ corresponds to one snapshot of the ULA and each row consists of outputs of a single antenna. The fact that we have only access to 