Closed-form geometry-aided direction estimation using minimum TDOA measurements

•In the case of the minimum number of TDOA measurements, a closed-form method is proposed to determine the source direction conditionally.•A geometry-aided scheme of direction estimation is proposed to eliminate the GDOP effect when using more than two TDOA measurements at a sensor array.•When using the sub-minimum TDOA measurements, a direction estimation method and the corresponding weighted-sum scheme are also developed.•The theoretical estimation errors of the above two methods are analyzed. The paper focuses on the time-difference-of-arrival (TDOA)-based direction estimation of plane wave from a signal-emitting source by spatial array. In order to avoid the potential impact of geometric dilution of precision (GDOP) on estimation accuracy, we propose that the minimum measurements (i.e. only two TDOAs) are used to deterministically calculate the source direction without involving numerical techniques. It is however challenging to algebraically obtain the common solution of two measurement equations in original nonlinear form. By making use of the TDOA invariance to coordinate system rotation, a novel closed-form method is proposed to tackle the above problem via a well-ordered coordinate rotation. Then, we develop a weighted-sum scheme to combine the minimum measurements based individual direction estimates with their weights determined by GDOP. The theoretical estimation mean and variance of the proposed geometry-aided scheme have been derived. The equation curve of TDOA measurement is also geometrically interpreted in the paper. Computer simulations demonstrate the superior performance of this scheme over other similar estimators in the literature, which involve the grid-scan technique, the sub-minimum-measurement scheme, and the linear least squares method. It is validated that the proposed scheme considerably outperforms the other closed-form estimators and achieves estimation accuracy similar to numerical technique but with significantly reduced computational complexity.