## 4 Second Step of Identification Algorithm  
With identification Inequality (20), LOS BS combinations can be identified roughly. Note that Inequality (20) may have a small probability in presence of NLOS error. Hence, identification Inequality (20) can only roughly identify LOS BS combination, but cannot eliminate all NLOS BS combinations, the remaining will be eliminated by the second step. For LOS BS combination, the coordinate deviation of common chord intersection point from MS, only results from measurement noise errors. Therefore, corresponding common chord intersection points are randomly distributed near the position of MS, these intersection points are close to each other and are all close to MS. Nevertheless, for the NLOS BS combination, the NLOS error may lead to much greater coordinate deviation between common chord intersection point and MS, which causes corresponding intersection points are far away from that of LOS BS combinations and MS. After the first step, most NLOS BS combination are eliminated, so the intersection points corresponding to the NLOS BS combinations are outliers in all common chord intersection points in the second step. Thus, we can furtherly identify the LOS BS combinations based on the distance between these common chord intersection points. The schematic diagram of distribution of common chord intersection points after the first step is shown in Figure 4. Most of the common chord intersection points are close to each other and near MS, the other few common chord intersection points, which correspond to NLOS BS combinations, are far away from that of LOS BS combination and MS.  
To make full use of the characteristic that the common chord intersection points of LOS BS combinations are all near MS and reduce the computational complexity, in this paper, instead of adopting traditional clustering algorithm to identify common chord intersection points corresponding to LOS combination, for every two common chord intersection points, we introduce mutual distance threshold to judge whether the two intersection points are close to each other and whether they are near MS at the same time. For any two intersection points, only the mutual distance is less than the mutual distance threshold, corresponding two intersection points are judged as being close to each other and being near MS. Onthis basis, for all common chord intersection points in second step, threshold matrix and discrimination result matrix are introduced, then the judgement of whether common chord intersection points are close to each other and whether they are all near MS is achieved. Now we discuss how to choose the proper mutual distance threshold for judgement of whether two intersection points are close to each other and near MS. The illustration as to how to determine the mutual distance threshold is shown in Figure 4. Point \(P\) denotes the position of MS, point \(O\) denotes common chord intersection point of a BS combination, \(|PO|\) denotes the distance between point \(O\) and MS. Point \(Q\) and \(|PQ|\) is similar. Based on (17), we can get  
\[\Delta\theta=A^{-1}\cdot R\cdot\Delta R=\left(A^{T}\cdot A\right)^{-1}\cdot A ^{T}\cdot R\cdot\Delta R \tag{25}\]  
let  
\[K=\left(A^{T}\cdot A\right)^{-1}\cdot A^{T}\cdot R \tag{26}\]  
(25) would become  
\[\Delta\theta=K\cdot\Delta R \tag{27}\]  
According to Equation (27), we can obtain the coordinate deviation of common chord intersection point from MS under the condition of giving \(\Delta R\),  
\[\Delta x=k_{11}\cdot\Delta r_{1}+k_{12}\cdot\Delta r_{2}+k_{13}\cdot\Delta r_{3} \tag{28}\]  
\[\Delta y=k_{21}\cdot\Delta r_{1}+k_{22}\cdot\Delta r_{2}+k_{23}\cdot\Delta r_{3} \tag{29}\]  
where \(k_{ij}\) is the element of matrix \(K\), \(\Delta x\) and \(\Delta y\) is coordinate deviation of common chord intersection point from MS on X axis and Y axis, respectively. Figure 4: Geometric position relationship of intersection points of common chords after first step.