Time difference of arrival based source localization within a sparse representation framework

The problem addressed is source localization from time differences of arrival (TDOA). This problem is also referred to as hyperbolic localization and it is non-convex in general. Traditional solutions proposed in the literature have generally poor robustness to errors in the TDOA estimates. More recent methods, which relax the non-convex problem to a convex optimization by applying a semi-definite relaxation (SDR) method, were found to be more robust to TDOA errors than the traditional methods. However, the SDR methods are not optimal in general. In this paper, three convex optimization methods with different computational costs are proposed to improve the hyperbolic localization accuracy. The first method takes an SDR approach to relax the hyperbolic localization to a convex optimization. The second method follows a linearized formulation of the problem and seeks for a biased estimate of improved accuracy. The first two methods perform comparably when the source is inside the convex hull of the sensors. When the source is located outside, the second approach performs better, at the cost of higher computation. A third method is proposed by exploiting the source sparsity. With this, the hyperbolic localization is formulated as an ℓ₁-regularization problem, where the ℓ₁-norm is used as source sparsity constraint. Computer simulations show that the ℓ₁-regularization can offer further improved accuracy, but at the cost of additional computational effort.