## 4 Second Step of Identification Algorithm
Taking absolute value in both sides of (28) and (29) and using the inequality relation yields  
\[|\Delta x|=|k_{11}\cdot\Delta r_{1}+k_{12}\cdot\Delta r_{2}+k_{13}\cdot\Delta r_{ 3}|\leq\]  
\[|k_{11}\cdot\Delta r_{1}|+|k_{12}\cdot\Delta r_{2}|+|k_{13}\cdot\Delta r_{3}|\]  
Figure 4.0  
Figure 4.1  
For the case of LOS BS combination, the distance error of each LOS BS, i.e., \(\Delta r_{i}(i=1,\ 2,\ 3)\) only results from measurement noise error. Similar to first step, in Inequality (30) and Inequality (31), \(r_{i}\) in \(R\) can be approximated by \(r_{i}\)', and let \(\Delta r_{i}=\sigma\), thus, the upper bound of \(|\Delta x|\), denoted by \(|\Delta x_{max}|\), and upper bound of \(|\Delta y|\), denoted by \(|\Delta y_{max}|\), can be expressed as  
Figure 4.2  
Figure 4.3  
Thus, for the \(i\)th LOS BS combination, the upper bound of distance between common chord intersection point and MS can be calculated as  
Figure 4.4  
where \(|\Delta x_{max}|_{i}\) and \(|\Delta y_{max}|_{i}\) is upper bound of coordinate deviation between common chord intersection point of \(i\)th BS combination and MS on X axis and Y axis, respectively. As shown in Figure 4, according to the triangle edge-length relation, we can have  
Figure 4.5  
Based on Inequality (35), for the case of LOS BS combination, the upper bound of the mutual distance between common chord intersection point of \(i\)th BS combination and that of \(j\)th BS combination, \(t_{ij}\), can be expressed as  
Figure 4.6  
The \(t_{ij}\) can be used as the mutual distance threshold to identify whether the two common chord intersection points are close to each other and whether the two common chord intersection points are near MS.  
For each pair of BS combinations, there is a corresponding mutual distance threshold. Suppose that there are \(n\) BS combinations and corresponding \(n\) common chord intersection points in the second step, the corresponding mutual distance threshold matrix is given by  
\[threshold= \begin{bmatrix}t_{11}&\cdots&t_{1n}\\ \vdots&\ddots&\vdots\\ t_{n1}&\cdots&t_{nn}\end{bmatrix} \]  
Let the common chord intersection point distance matrix of \(n\) BS combinations is  
\[dis= \begin{bmatrix}d_{11}&\cdots&d_{1n}\\ \vdots&\ddots&\vdots\\ d_{n1}&\cdots&d_{nn}\end{bmatrix} ,\]  
where \(d_{ij}\) is the distance between common chord intersection point of \(i\)th BS combination and that of \(j\)th BS combination. Subtracting (37) from (38), let \(RE_{ij}=d_{ij}-t_{ij}\), and adopt following discrimination expression  
\[re_{ij}=\left\{ \begin{array}{ll}1\;\;if\;RE_{ij}\textless 0,\;i.e.\;t_{ij}>d_{ij}\\ 0\;if\;RE_{ij}>0,\;i.e.\;t_{ij}<d_{ij}\textgreater{,}\end{array} \right. \tag{39}\]  
we can obtain the discrimination result matrix  
\[resu= \begin{bmatrix}re_{11}&\cdots&re_{1n}\\ \vdots&\ddots&\vdots\\ re_{n1}&\cdots&re_{mn}\end{bmatrix}  \tag{40}\]  
In the discrimination result matrix \(resu\), the elements of row \(i\) demonstrate the distance between common chord intersection point of of \(i\)th BS combination and that of other BS combinations. \(re_{ij}=1\) illustrates that common chord intersection point of \(i\)th BS combination and that of \(j\)th BS combination are close to each other and the two intersection points are all near MS. The elements equaling to 1 in row \(i\)th form an intersection point set, in which all intersection points are close to each other and near MS. For instance, suppose that in 5 th row, \(re_{51}\), \(re_{52}\), \(re_{54}\) and \(re_{55}\) equal to 1 and the other elements equal to 0, this denotes that the common chord intersection points of 1 st, 2 nd, 4 th and 5 th BS combination are close to each other and are all near MS, and the other intersection points are far away from the above intersection points and MS.  
For \(n\) rows discrimination result matrix, there are \(n\) intersection point sets.