  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 $\{u_i + \delta_i\}_{i=1}^M$
that are different from the base frequencies and where $\{\delta_i\}_{i=1}^M$
are unknown. Ignoring perturbations of this nature 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 \emph{in
situ} with the signal, which we assume is sparse or compressible in some known
basis. In many cases, the perturbations $\{\delta_i\}_{i=1}^M$ can be expressed
in terms of a small number of unique parameters $P \ll M$. 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 convergence of our algorithm, and (2) the uniqueness of its
solution under some restrictions.
