Approximate Closed-Form TDOA-Based Estimator for Acoustic Direction Finding via Constrained Optimization

This paper focuses on estimating the azimuth and elevation of a sound emitter in 3-D space based on time-difference-of-arrival (TDOA) measurements with an array of acoustic sensors. The TDOA-based direction finding problem is significant because in a range of scenarios the source only emits a transient signal and only TDOA measurements can be used to find the direction. The linear least squares estimator provides a suboptimal solution, since there is nontrivial information loss in the linearization of the nonlinear observation equation. To avoid the information loss, the Lagrange multiplier method is usually used to realize the constrained optimization, but the computational complexity is rather high. This paper proposes a constrained least squares estimator to deal with the direction finding problem. The proposed method makes use of both Lagrange multiplier and quadratic constraints to form the cost function. The resultant estimator is shown to be approximate closed-form so that the computational complexity is reduced greatly, but contributes little under a small noise level. Mathematical formula is derived to evaluate the theoretical accuracy of the proposed estimator in terms of mean square error. Both simulation and field experimental results demonstrate that the proposed estimator can outperform the traditional linear and nonlinear estimators.