# Target Localization Based on Bistatic T/R Pair Selection in GNSS-Based Multistatic Radar System
## 3 Selection Methods of Bistatic T/R Pairs  
### Covariance Matrix Fusion Method  
The principle of this method is to select \(m(m<N)\) bistatic T/R pairs from a multistatic radar system based on resolution unit, to minimize the overlap area of the elliptical rings, equivalent to minimizing the positioning error. The presentation diagram of this method is shown in Figure 3.  
Figure 2: (**a**) Schematic diagram of Bistatic Range (BR) positioning error, (**b**) Particular section.  
First, the range information is used to construct the covariance matrix \(e_{i}(i=1,2,\ldots,N)\), which describes the positioning error of the bistatic T/R pair. The matrix \(E\) obtained by the weighted fusion of \(e_{i}\) is the inscribed ellipsoid covariance matrix of the overlap area. The volume of the overlap area is proportional to the determinant of \(E\)[36]. So, the criterion for selecting bistatic T/R pairs is equivalent to optimizing \(E\) with the smallest determinant after the matrix fusion. Next, for easy understanding, we directly construct the inverse matrix of \(e_{i}\), and express it as follows.  
\[e_{i}^{-1}=\left[\begin{array}{ccc}0&0&0\\ 0&0&0\\ 0&0&\frac{1}{\Delta_{i}^{2}}\end{array}\right], \tag{12}\]  
\[\Delta_{i}^{2}=q_{1}\Delta_{i1}^{2}+q_{2}\Delta_{i2}^{2}+q_{3}\Delta_{i3}^{2}, \tag{13}\]  
where \(\Delta_{i}\) represents the bistatic range measurement error, which is mainly related to the multistatic geometric configuration, the signal bandwidth and the sampling frequency. As shown in (13), \(\Delta_{i1}^{2}=L_{i}^{2}\), where \(L_{i}\) is the modulus of bistatic range vector \(M_{xi}\). \(\Delta_{i2}^{2}=\left[c/(2 Bcos(\phi_{i}/2))\right]^{2}\) and \(\Delta_{i3}^{2}=\left[c/f_{si}\right]^{2}\), with \(c\) being the speed of light (about \(3\times 10^{8}m/s\)), \(B\) being the signal bandwidth, \(f_{si}\) being the sampling frequency of the \(i-\)th satellite signal and \(q_{i}(i=1,2,3)\) being the weighting factor.  
Equation (12) is obtained in the target local coordinate with the \(i-\)th bisector as the \(Z\) axis. Different bistatic geometries correspond to different target local coordinates, hence, while we conduct the following covariance matrix fusion, the covariance matrix must be transformed into the same coordinate, defined as the local coordinate. The classic 7-parameter coordinate rotation method is used (see (A1) (A2) in the Appendix A) to realize the coordinate transformation, and the \(i-\)th covariance matrix in the local coordinate can be written as:  
\[E_{i}^{-1}=R_{i}^{T}*e_{i}^{-1}*R_{i}, \tag{14}\]  
with \(R_{i}\) being the \(i-\)th rotation matrix from the \(i-\)th bistatic coordinate into the local coordinate. Finally, the optimization problem can be obtained as follows:  
\[\begin{split}\underset{k_{i}}{argmaxdet}\left(\sum\limits_{i=1 }^{n}k_{i}E_{i}^{-1}\right)\\ s.t.\;k_{i}=\left\{\begin{array}{cc}1,&if\;k_{i}\;is\; included\;in\;the\;solution\;set\\ &0,\;others\end{array}\right.,\end{split} \tag{15}\]  
where, \(k_{i}\) is the weighting factor.  
By solving the above question, the optimal combination of \(4,5,\ldots,m\) bistatic T/R pairs can be obtained. It can be seen that the method can select the optimal combination of  
Figure 3: Presentation diagram of Covariance Matrix Fusion Method.  
a given number of T/R pairs. Although the result may not be the most effective, for the multistatic radar system, we can use the method to do a reasonable rough selection of multiple bistatic pairs to reduce the amount of system calculations.