Super-Resolution Compressed Sensing: A Generalized Iterative Reweighted L2 Approach

Conventional compressed sensing theory assumes signals have sparse representations in a known, finite dictionary. Nevertheless, in many practical applications such as direction-of-arrival (DOA) estimation and line spectral estimation, the sparsifying dictionary is usually characterized by a set of unknown parameters in a continuous domain. To apply the conventional compressed sensing technique to such applications, the continuous parameter space has to be discretized to a finite set of grid points, based on which a "presumed dictionary" is constructed for sparse signal recovery. Discretization, however, inevitably incurs errors since the true parameters do not necessarily lie on the discretized grid. This error, also referred to as grid mismatch, may lead to deteriorated recovery performance or even recovery failure. To address this issue, in this paper, we propose a generalized iterative reweighted L2 method which jointly estimates the sparse signals and the unknown parameters associated with the true dictionary. The proposed algorithm is developed by iteratively decreasing a surrogate function majorizing a given objective function, resulting in a gradual and interweaved iterative process to refine the unknown parameters and the sparse signal. A simple yet effective scheme is developed for adaptively updating the regularization parameter that controls the tradeoff between the sparsity of the solution and the data fitting error. Extension of the proposed algorithm to the multiple measurement vector scenario is also considered. Numerical results show that the proposed algorithm achieves a super-resolution accuracy and presents superiority over other existing methods.