Mathematical analysis of line intersection algorithm for TOA trilateration method

Time of arrival (TOA) technique based on the trilateration method is one of the representative location detection technologies (LDT), which have various applications in the wireless communication system. It utilizes signals from three base stations (BS) for estimating the location of a mobile station (MS). Since the location of MS is estimated using the distance between the MS and BS and the distance is estimated using the number of the time delay, three circles based on three radii between MS and three BSs should not meet at a point. In order to solve this problem, the shortest distance algorithm which selects three nearest intersection points among six intersection points from three circles was proposed. However, it sometimes has high location estimation errors in the extreme cases due to the selection of three intersection points. In order to overcome this problem, we consider the line intersection algorithm which does not require selecting interior intersection points. Although this algorithm has good performance of the location estimation, it must assume that three lines, connecting two intersections based on two circles, meet at a point. In this paper, we also mathematically prove that these three intersection lines based on three circles meet at a point. In this algorithm, we determine this intersection point based on three lines as the location of MS.