  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 with an infinite number of snapshots, we prove uniqueness of this problem
up to certain trivial, inevitable ambiguities based on an algebraic method, as
long as there are more sensors than frequencies. We then analyze the
sensitivity of this algebraic technique with respect to the number of snapshots
and noise.
  We next propose an optimization approach that makes full use of the
measurements by minimizing a non-convex objective which is non-negative and
continuously differentiable over all calibration parameters and Toeplitz
matrices. We prove that, in the case of infinite snapshots and noiseless
measurements, the objective vanishes only at equivalent solutions to the true
calibration parameters and the measurement covariance matrix. The objective is
minimized using Wirtinger gradient descent which is proven to converge to a
critical point. We show empirically that this critical point provides a good
approximation of the true calibration parameters and the underlying
frequencies.
