Sensor Calibration for Off-the-Grid Spectral Estimation

TL;DR

This paper addresses the challenge of calibrating sensors in spectral estimation, proposing algebraic and optimization methods to recover frequencies and calibration parameters simultaneously, with theoretical guarantees and empirical validation.

Contribution

It introduces a novel algebraic approach for unique calibration in noiseless, infinite snapshot scenarios and an optimization method with convergence guarantees for practical noisy cases.

Findings

Algebraic method guarantees uniqueness with enough sensors.

Optimization approach converges to true parameters in noiseless, infinite data.

Empirical results show effective calibration and frequency recovery.

Abstract

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.