A Simple Closed-Form Linear Source Localization Algorithm

A closed-form linear solver is proposed to estimate a source location based on time difference of arrivals from the source measured by two neighboring sensors in a linear array. Those sensors are shown to be located at different foci of paired hyperbolas. Using a geometric representation of a hyperbola that is defined as the locus of points whose distance from the focus is proportional to the distance from a directrix, a set of linear equations for the x-coordinate of the source are derived, and subsequently the y-coordinate is solved from a set of equations describing Cartesian coordinates of source and sensors. The algorithm is applicable to both near-field and far-field scenarios without any approximation. It also enjoys simplicity and computational efficiency without the nonlinearity inherited from hyperbolic equations. The estimation error is characterized by the estimation performance of the time differences. Simulation results show that the proposed algorithm has a similar performance with the maximum-likelihood estimator at high signal to noise ratio.