## 2 System Model  
We consider a two-dimensional (2-D) localization scenario. The task is to locate a source signal, e.g., an MS from the measurements of sensors, e.g., BS, and moreover, MS and BS operate in an environment where NLOS signals may exist. For TOA-based localization algorithm, three BSs are used to locate the MS, the corresponding three BSs are referred to as a BS combination. The geometric position relationship of three BSs and MS in absence of distance error, including measurement noise error and NLOS error, is depicted as Figure 1. The circle constructed with BS as the center and the distance between the MS and the BS as the radius, is referred to as the localization circle. The connecting line between the intersection points of two localization circles is referred to as the common chord of the two localization circles. Obviously, in environments without measurement noise error and NLOS error, for the same MS and three BSs, the three localization circles intersect at a point, and the corresponding three common chords intersect at a point too. Furthermore, the two intersection points (denoted by point P) all overlap with the position of MS, which is shown in Figure 1.  
Figure 1: Demonstration of MS and intersection point (P) in absence of distance error. However, in practical environments, measurement noise error is inevitable, and NLOS error may occur due to the presence of obstruction. Therefore, the three localization circles no longer intersect at a point although the corresponding three common chords still intersect at a point, and the intersection point of three common chords no longer overlaps with the position of MS, which is depicted as Figure 2.  
For the condition of Figure 2, let \(r_{i}\) is the actual distance between \(i\)th BS and MS. The distance error corresponding to \(r_{i}\), which results from measurement noise error or NLOS error, is denoted as \(\Delta r_{i}\). Then, the measurement of the distance between \(i\)th BS and MS, \(r_{i}^{\prime}\), can be written as  
\[r_{i}^{\prime}=r_{i}+\Delta r_{i}=r_{i}+n_{i}+e_{i}, \tag{1}\]  
where \(n_{i}\) denotes measurement noise error satisfying Gaussian distribution, and \(e_{i}\) denotes NLOS error with positive value if the \(i\)th BS received signal is due to NLOS transmission. Clearly, when the \(i\)th BS received signal is due to LOS transmission, \(e_{i}=0\), and the corresponding \(i\)th BS is referred as LOS BS, otherwise, the \(i\)th BS is referred as NLOS BS. So, the identification of LOS/NLOS signals can be converted into the identification of LOS/NLOS BSs. When the received signals at the three BSs of BS combination are all due to LOS transmission, i.e., \(e_{i}=0\ (i=1,\ 2,\ 3)\), \(\Delta r_{i}(i=1,\ 2,\ 3)\) only results from measurement noise error, \(\Delta r_{i}\) is small or not being, thus the distance between MS and the intersection point of three common chords (referred as common chord intersection point for simplicity in the following) is small or not being, the BS combination is referred as LOS BS combination. Accordingly, each BS of LOS BS combination is LOS BS, because it only receives LOS signal from MS. However, if any received signal at the three BSs of BS combination is due to NLOS transmission, i.e., \(e_{i}\
eq 0\ (i=1\ \text{or}\ 2\ \text{or}\ 3)\), corresponding \(\Delta r_{i}\ (i=1\ \text{or}\ 2\ \text{or}\ 3)\) is much greater, the distance between MS and common chord intersection point increases greatly, corresponding BS combination is referred to as NLOS BS combination. Among the three BSs of NLOS BS combination, there must be NLOS BS, but the three BSs are not necessarily all NLOS BS. Therefore, we can identify a BS combination is LOS BS combination or not, based on the distance between MS and common chord intersection point.