Blind phase calibration in sparse recovery

We consider a blind calibration problem in a compressed sensing measurement system in which each sensor introduces an unknown phase shift to be determined. We show that this problem can be approached similarly to the problem of phase retrieval from quadratic measurements. Furthermore, when dealing with measurements generated from multiple unknown (but sparse) signals, we extend the approach for phase retrieval to solve the calibration problem in order to recover the signals jointly along with the phase shift parameters. Additionally, we propose an alternative optimization method with less computation complexity and memory requirements. The proposed methods are shown to have significantly better recovery performance than individual recovery of the input signals when the number of input signals is sufficiently large.