## 1 Introduction
### Off-Grid Problem in CS-Based Methods  
In the CS processing procedure, the first necessary step is to design a dictionary through a discretization operation with the assumption that the elements of unknown \(\mathbf{x}\) lie exactly on those pre-defined grids corresponding to \(\mathbf{A}\). Obviously, this is practically impossible since the continuous nature of the unknowns of \(\mathbf{x}\), such as the unknown directions, may not fall into the predefined angular grids in DOA, and the scatterers of the target may not locate exactly on the pre-discretized scene grids in radar imaging. Hence, once faced with a continuous signal, the off-grid problem in using the conventional sparse recovery techniques is inevitable, no matter how densely we grid \(\mathbf{x}\). Previous researches [15, 16, 17] have demonstrated that the traditional CS-based methods would be severely affected when the off-grid problem emerges, and [18] also claimed that the off-grid problem is one of the major constraints in popularizing CS-based methods in practice. It is worth noting that there may be other factors leading to basis mismatch, for example, in the radar imaging field, unsatisfactory system parameters (e.g., position errors of transceivers [19] and phase synchronization mismatch [20]) are also likely to degrade the performance of conventional CS-based methods from our previous researches, however this paper only focuses on the off-grid problem, and assumes the system errors are small enough.  
The solutions for off-grid problem have been broadly studied in previous literatures [21, 22, 23, 24, 25, 26, 27, 28, 29, 30]. So far, main research topics include three types as follows:* _Direct method according to theories of Xampling and the finite rate of innovation (FRI)_. This scheme is first introduced by [PERSON] _et al._ as a general framework within various solutions for analog signals [21]. The significant advantage of this method is its direct solution, without any pre-discretization of the continuous signal at first. Thus it is less sensitive to off-grid problem compared with traditional CS-based methods. However, this method is mainly based on spectral estimation algorithms (e.g., ESPIRT [22], MUSIC [6], Matrix Pencil [23]), which needs special signal expression in linear measurement model and its performance may suffer from low SNR and few snapshots [24].
* _Off-grid CS method_. This kind of method is under a first order Taylor expansion model, and thus is quite sensitively depended on the model's accuracy, and cannot give a thorough solution when higher order approximations are significant [23]. Furthermore, it is likely to involve highly-computational burden [25, 26] for alternatively finding the sparse solution and estimating the off-grid error, especially in large scale problem.
* _Grid refinement approach_. The idea of the grid refinement technique was firstly introduced by [PERSON] _et al._[27] to mitigate the effect of limiting estimates to a grid of spatial locations in source localization problem. Then it is generalized by [PERSON] _et al._[28] for locating two-dimensional multiple underwater acoustic sources. Recently, [PERSON] _et al._[29] have proposed a novel grid refinement algorithm to alleviate off-grid problem by using a particle swarm optimization (PSO) and orthogonal matching pursuit (OMP), which makes use of the PSO to perturb the location of each grid point.  
Since the \"grid refinement approach\" is iteratively refined to match with the desired resolution of the off-grid components by \"coarse and fine grid partition\" operations, and does not have the drawbacks of (a) and (b), this paper follows the idea of \"grid refinement approach\" which makes a coarse grid first instead of having an universally fine grid to reduce the complexity, then achieves fine grids around the peaks using more refinement levels. In this way, after a few iterations of refining process, it becomes fine enough that off-grid problem effect is negligible.