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.

本文研究了频谱估计中的传感器校准，其中真实频率位于连续域中。我们考虑一个统一的传感器阵列，该传感器阵列收集的测量频谱由有限数量的频率组成，其中每个传感器都有未知的校准参数。我们的目标是从多个测量快照中同时恢复光谱和校准参数。在无数个快照的无噪声情况下，只要传感器比频率多，我们就可以根据代数方法证明这个问题的唯一性，直到某些琐碎的，不可避免的歧义。然后，我们分析该代数技术相对于快照数量和噪声的敏感性。

接下来，我们提出一种优化方法，该方法通过最小化在所有校准参数和Toeplitz矩阵上都是非负且连续可微的非凸物镜来充分利用测量结果。我们证明，在无限快照和无噪声测量的情况下，物镜仅在对真实校准参数和测量协方差矩阵的等效解中消失。使用经证明可以收敛到临界点的Wirtinger梯度下降可将目标最小化。我们凭经验表明，该临界点提供了真实校准参数和基础频率的良好近似值。