Dealing with frequency perturbations in compressive reconstructions with Fourier sensing matrices

Abstract In many applications in compressed sensing, the measurement matrix is a Fourier matrix, i.e., it measures the Fourier transform of the underlying signal at some specified ‘base’ frequencies { u i } i = 1 M , where M is the number of measurements. However due to system calibration errors, the system may measure the Fourier transform at frequencies { u i + δ i } i = 1 M that are different from the base frequencies and where { δ i } i = 1 M are unknown frequency perturbations. Ignoring such perturbations can lead to major errors in signal recovery. In this paper, we present a simple but effective alternating minimization algorithm to recover the perturbations in the frequencies in situ with the signal, which we assume is sparse or compressible in some known basis. In many cases, the perturbations { δ i } i = 1 M can be expressed in terms of a small number of unique parameters P ≪ M. We demonstrate that in such cases, the method leads to excellent quality results that are several times better than baseline algorithms (which are based on existing off-grid methods in the recent literature on direction of arrival (DOA) estimation, modified to suit the computational problem in this paper). Our results are also robust to noise in the measurement values. We also provide theoretical results for (1) the conditional convergence of our algorithm, and (2) the uniqueness of the solution for this computational problem, under some restrictions.