# Target Localization Based on Bistatic T/R Pair Selection in GNSS-Based Multistatic Radar System
## 3 Selection Methods of Bistatic T/R Pairs
### Convex Hull Optimization Method  
Convex hull problem is a classic problem in the computer geometry. In a real vector space \(V\), for a given set \(X\), the intersection \(S\) of all convex sets containing \(X\) is called the convex hull of \(X\)[37]. The convex hull concept provides another algorithm to solve the problem in the paper. The main idea to formulate the T/R pair selection criteria as a convex hull problem bases on resolution capability. First, we define a vector \(v_{i}\) for the \(i-\)th bistatic T/R pair to represent its capability of the range resolution. The vector is along the direction of \(M_{xi}\) and its modulus value is \(1/\sqrt{\Delta_{i}^{2}}\), where \(\Delta_{i}^{2}\) being the \(i-\)th bistatic range variance as discussed in the last subsection. \(P_{i}\), \((i=1,2,\ldots,N)\) represent the endpoints of all vectors \(v_{i}\), and let the point set \(P_{j}=\{P_{i},-P_{i}\}\), \((j=1,2,\ldots,2N)\), then, the smallest convex polygon (polyhedron) that encloses all vertices in \(P_{j}\) is called the convex hull of \(P_{j}\). On 2-D plane, minimizing the positioning error is equivalent to maximizing the area of the polygon expanded by the vector endpoints. Correspondingly, in 3-D space, it is equivalent to maximizing the volume of the polyhedron with the vector endpoints expanded. Figure 4 is demonstrated in 2-D for visualization, and the target localization problem of this paper is solved in 3-D. The red dot in the figure indicates starting point of \(v_{i}\), and the black dots and the gray dots indicate endpoints, which are also the data set of the convex hull method. The dotted arrow is to more intuitively reflect the modulus of \(v_{i}\), that is, to reflect positioning error of each bistatic T/R pair. It can be seen that the smaller the positioning error, the more likely it is to be selected as vertex of the convex hull. These vertices (the black dots in Figure 4) form the selected bistatic T/R pair combination. In 3-D space, the specific coordinate position of \(P_{i}=(x_{i},y_{i},z_{i}),(i=1,2,\ldots,N)\) is calculated as follows:  
\[\left\{ \begin{array}{l}x_{i}=\frac{sim\phi_{1}}{\sqrt{\Delta_{i}^{2}}}\\ y_{i}=\frac{sim\phi_{1}}{\sqrt{\Delta_{i}^{2}}}\\ z_{i}=\frac{cos\phi}{\sqrt{\Delta_{i}^{2}}}\end{array} \right.\text{ and }\left\{ \begin{array}{l}x_{i}=-\frac{sim\phi_{2}}{\sqrt{\Delta_{i}^{2}}}\\ y_{i}=\frac{sim\phi_{2}}{\sqrt{\Delta_{i}^{2}}}\\ z_{i}=\frac{\Delta_{0}\phi}{\sqrt{\Delta_{i}^{2}}}\end{array} \right. \tag{16}\]  
where \(\phi\) is the angle between \(\Delta_{i}^{2}\) and the Z axis, \(\theta_{1}\) is the angle between the projection of \(\Delta_{i}^{2}\) on the XOY plane and the positive half axis of X axis, and \(\theta_{2}\) is the angle between the projection of \(\Delta_{i}^{2}\) on the XOY plane and the positive half axis of Y axis. The presentation diagram of the proposed method is shown in Figure 5, and the specific description is as follows. Figure 4: Schematic diagram of 2-D convex hull. First, we define a new vector that takes \(P_{i}\) as the starting points and \(P_{q}=\{P_{i},-P_{i}\}\), \((q=1,2,\ldots,2N\)_and_\(q\
eq i\)) as the endpoints. Next, if there is a vector \(\lambda_{i}\) that satisfies (17), then, when \(\lambda_{i}=0\), it indicates that the \(i-\)th bistatic vector is not on the convex hull, so its corresponding T/R pair is not selected. When \(\lambda_{i}\
eq 0\), the \(i-\)th bistatic T/R pair is selected. In this way, we get the optimal combination of bistatic T/R pairs. \[\left\{ \begin{array}{c}argmax\|\lambda_{i}\|_{2}\\ s.t.\ \lambda_{i}\big{(}P_{q}-P_{i}\big{)}\geq 0\end{array} \right.. \tag{17}\]  
The computational complexity of the proposed method is \(O(n^{2})\), and \(n\) is the number of bistatic T/R pairs. Compared with other 3-D convex hull algorithms, the computational complexity of proposed method is equivalent to that of the incremental method \((O(n^{2}))\), and is less than the extreme point method \((O(n^{4}))\) and the extreme edge method \((O(n^{3}))\)[38].