Sensor Calibration for Off-the-Grid Spectral Estimation

This paper studies sensor calibration in spectral estimation where the true frequencies are located on a continuous domain. We consider a uniform array of sensors that collects measurements whose spectrum is composed of a finite number of frequencies, where each sensor has an unknown calibration parameter. Our goal is to recover the spectrum and the calibration parameters simultaneously from multiple snapshots of the measurements.In the noiseless case, we prove uniqueness of this problem up to certain trivial, inevitable ambiguities with an infinite number of snapshots as long as there are more sensors than frequencies based on an algebraic method. We then analyze the sensitivity of this approach with respect to the number of snapshots and noise. We next propose an optimization approach that makes full use of the measurements and consider a non-convex objective over all calibration parameters and Toeplitz matrices. This objective is non-negative and continuously differentiable. We prove that, in the case of infinite snapshots of noiseless measurements, the objective vanishes only at the equivalent solutions to the true calibration parameters and the measurement covariance matrix with all calibration parameters being 1 which exhibits a Toeplitz structure. The objective is minimized using Wirtinger gradient descent which we prove converges to a critical point. We show empirically that this critical point provides a good approximation of the true calibration parameters and the underlying frequencies.