Dimension-Reduction Maximum Likelihood Sensor Array Calibration Using Inaccurate Cooperative Sources

The state-of-the-art auxiliary calibration algorithms can perform comprehensive calibration of various array nonideal characteristics, such as mutual coupling, gain/phase uncertainties, and sensor position errors, employing a set of cooperative calibration sources with known direction of arrival (DOA). However, the task of deploying calibration sources at precisely measured DOAs is complex. Otherwise, the calibration source DOA errors would seriously degrade the performance of these algorithms. In this article, a dimension-reduction maximum likelihood calibration algorithm (MLCA) with inaccurate cooperative sources is proposed to overcome this issue. First, a maximum likelihood (ML) calibration model is established including both the unknown array of nonideal parameters and 2-D DOAs of all calibration sources. Next, the ambiguity of sensor position estimation caused by inaccurate 2-D DOAs of calibration sources is analyzed. Furthermore, a dimension-reduction ML calibration model is proposed to resolve the ambiguity under the zero mean Gaussian distribution assumption of the calibration source elevation errors. Then, since the proposed multiparameter dimension-reduction model is nonconvex and multimodal, a new filled function method (FFM) is proposed to cope with its local extrema attractors. The proposed single-parameter filled function (SPFF) has a single form without an exponential term and is second-order continuously differentiable, which is stable for numerical calculations and easy to optimize by local optimization tools. Finally, the closed-form hybrid Cramer–Rao lower-bound (CRB) expressions of array parameters under unknown source DOAs are derived in detail. Numerical results verify the effectiveness of the proposed algorithm.