SIGNAL RECOVERY IN PERTURBED FOURIER COMPRESSED SENSING

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 $\left\{ {{u_i}} \right\}_{i = 1}^M$, where M is the number of measurements. However due to system calibration errors, the system may measure the Fourier transform at frequencies $\left\{ {{u_i} + {\delta _i}} \right\}_{i = 1}^M$ that are different from the base frequencies and where $\left\{ {{\delta _i}} \right\}_{i = 1}^M$ 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 in situ with the signal, which we assume is sparse or compressible in some known basis. In many practical cases, the perturbations $\left\{ {{\delta _i}} \right\}_{i = 1}^M$ 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.