A Closed-Form Solution With Asymptotic Performance for Gain-Phase Error Self-Calibration Exploiting Toeplitz Structure

Abstract

This article addresses the self-calibration of gain-phase errors in a uniform linear array. Existing closed-form calibration methods are primarily developed for signal models with external noise and cannot be directly applied to the traditional direction-of-arrival estimation problem with internal noise. To address this limitation, an internal noise signal model is adopted. By exploiting the Toeplitz structure of the covariance matrix, a closed-form solution is derived in the form of a two-step weighted least squares (TWLS) estimator. This solution requires no multidimensional optimization or iterative parameter updates while maintaining high estimation accuracy. In the TWLS formulation, an initial estimate is first obtained using a weighting matrix constructed from the asymptotic covariance and subsequently refined through a decorrelation procedure based on a Taylor series approximation. In addition, the constrained Cramér–Rao lower bound (CCRLB) is derived. Simulation results show that the proposed method achieves accuracy close to maximum likelihood estimation and asymptotically approaches the CCRLB while maintaining high computational efficiency and strong robustness.