Improved Active Calibration Algorithms in the Presence of Channel Gain/Phase Uncertainties and Sensor Mutual Coupling Effects

This paper deals with the problem of active calibration when channel gain/phase uncertainties and sensor mutual coupling effects are simultaneously present. The numerical algorithms used to compensate for array error matrix, which is formed by the product of mutual coupling matrix and channel gain/phase error matrix, are presented especially tailored to uniform linear array (ULA) and uniform circular array (UCA). First, the array spatial responses corresponding to different azimuths are numerically evaluated using a set of time-disjoint auxiliary sources at known locations. Subsequently, a least-squares (LS) minimization model with respect to array error matrix is established. To solve this LS problem, two novel algorithms, namely algorithm I and algorithm II, are developed. In algorithm I, the array error matrix is considered as a whole matrix parameter to be optimized and an explicit closed-form solution to the error matrix is obtained. Compared with some existing algorithms with similar computation framework, algorithm I is able to utilize all potentially linear characteristics of ULA’s and UCA’s error matrix, and the calibration accuracy can be increased. Unlike algorithm I, algorithm II decomposes the array error matrix into two matrix parameters (i.e., mutual coupling matrix and channel gain/phase error matrix) to be optimized and all (nonlinear) numerical properties of the error matrix can be exploited. Therefore, algorithm II is able to achieve better calibration precision than algorithm I. However, algorithm II is more computationally demanding as a closed-form solution is no longer available and iteration computation is involved. In addition, the compact Cramér–Rao bound (CRB) expressions for all array error parameters are deduced in the case where auxiliary sources are assumed to be complex circular Gaussian distributed. Finally, the two novel algorithms are appropriately extended to the scenario where non-circular auxiliary sources are used, and the estimation variances of the array error parameters can be further decreased if the non-circularity is properly employed. Simulation experiments show the superiority of the presented algorithms.