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The time of arrival (TOA) trilateration is one of the representative location detection technologies (LDT) that determines the true location of a mobile station (MS) using a unique intersection point of three circles based on three radii corresponding to distances between MS and base stations (BSs) and center coordinates of BSs. Since the distance between MS and BS is estimated by using the number of time delays, three circles based on the estimated radii are generally increased and they may not meet at a single point, resulting in the location estimation error. In order to compensate this estimation error and to improve estimation performance, we present two advanced TOA trilateration localization algorithms with detail mathematical expressions. The considered algorithms are the shortest distance algorithm, which calculates an average of three interior intersection points among an entire six intersection points from three intersecting circles, and the line intersection algorithm, which calculates an intersection point of three lines connecting two intersection points of two circles among the three circles, as the estimated location of the MS. In this paper, we present both algorithms with detailed mathematical expressions. The computer simulation results are provided to compare the location estimation performance of both algorithms. In addition, in this paper, mathematical analysis is provided to indicate the relation between the line intersection algorithm and the shortest distance algorithm. In this analysis, we verify that line equations based on the intersection points obtained from the shortest distance algorithm are identical to those obtained from the line intersection algorithm.

The ever-growing mobile station (MS) localization technologies provide an increasingly important role in all aspects of the wireless cellular systems and Internet of Things (IoT). The accurate MS location information is the basis in connection of different devices in IoT. The MS localization techniques based on time of arrival (TOA) trilateration algorithm, which determines the location of MS using an intersection point of three circles based on distances between MS and base stations (BS) and coordinates of BSs, have been actively studied. In general, the distance between the MS and BS is calculated by counting the number of delay samples or measuring the power of the received signal. Since the estimated distance (radius of a circle) between MS and BS is commonly increased, three circles may not meet at a single point, resulting in the estimation error of MS localization. In order to improve this problem, in this paper, we propose the hybrid TOA trilateration algorithm based on the line intersection algorithm for the general case for intersection of three circles and the comparison approach of intersection distances for the specific case where a small circle is located inside the area of two large circles. The line intersection algorithm has an excellent location estimation performance in the general case, but it does not work in the specific case. The comparison approach of intersection distances has good performance only for the specific case. In addition, we propose the mode selection algorithm to efficiently select a proper mode between the general and specific cases. The representative computer simulation examples are provided to verify the localization performance of the proposed algorithm.