# Target Localization Based on Bistatic T/R Pair Selection in GNSS-Based Multistatic Radar System
## 2 Target Localization Scheme with Multistatic Radar  
### System Geometry and Positioning Method Description  
As shown in Figure 1, a multistatic radar system is composed of more than one transmitter or receiver or both. A multistatic geometry containing \(N(N\geq 3)\) transmitters and one receiver can be easily extended to multiple-receiver cases, with all proposed methods in this paper still effective.  
Without the loss of generality, it is assumed that the receiver is located at the coordinate origin (0,0,0). The positions of target and transmitters are defined as:  
\[T_{g}=x=(x,y,z)^{T}, \tag{1}\]  
And  
\[T_{xi}=x_{i}=(x_{i},y_{i},z_{i})^{T}, \tag{2}\]  
where \(i=1,\ldots,N\).  
The multistatic radar system can be regarded as \(N\) bistatic T/R pairs. For visibility, only three of \(N\) bistatic T/R pairs are illustrated in Figure 1. \(\phi_{i}\) is the bistatic angle of the \(i-\)th bistatic radar. \(M_{xi}\) is the bistatic angle bisector of the \(i-\)th bistatic radar and the direction of bistatic positioning error is along the direction of \(M_{xi}\). \(M_{xi}\) is expressed as:  
\[T_{1}=T_{xi}-T_{g}, \tag{3}\]  
\[T_{2}=-T_{g}, \tag{4}\]  
\[M_{xi}=\frac{T_{1}}{\|T_{1}\|}+\frac{T_{2}}{\|T_{2}\|}. \tag{5}\]  
In Figure 1, the sum of the transmitter-target and the target-receiver distances is defined as the Bistatic Range (BR) [34]. In the bistatic radar system, a BR measurement can determine an ellipsoid whose focal points are the transmitter and receiver locations, while in practical cases, usually an elliptical ring is obtained, considering the BR measurement error. Based on the multiple BR measurements of all T/R pairs, the intersection points of ellipsoids, or the intersection area of the elliptical rings indicates the target position [35].  
At the output of the radar processor for each T/R pair, the BR may be written as:  
\[r_{i}=D_{i}+R_{0}-B_{i}, \tag{6}\]  
with \(D_{i}=\|x_{i}-x\|=\sqrt{\left(x_{i}-x\right)^{2}+\left(y_{i}-y\right)^{2}+ \left(z_{i}-z\right)^{2}}\), \(R_{0}=\|x\|=\sqrt{x^{2}+y^{2}+z^{2}}\) and \(B_{i}=\|x_{i}\|=\sqrt{x_{i}^{2}+y_{i}^{2}+z_{i}^{2}}\) being the range between the \(i-\)th satellite and the target, the range between the receiver and the target and the baseline between the \(i-\)th satellite and the receiver.  
Then, the equation of combining all T/R pairs can be expressed as:  
\[Ax=K+C\|x\|, \tag{7}\]  
where, \(A\) is the transmitter position matrix, \(K\) is constant vector, and \(C\) is the sum of the radar measured bistatic ranges and baselines, which can be written as:  
\[A=\left[\begin{array}{c}x_{1}\\ x_{2}\\ \vdots\\ x_{N}\end{array}\right]_{N\times 3},\ K=\frac{1}{2}\left[\begin{array}{c}B_{1}^ {2}-\left(r_{1}+B_{1}\right)^{2}\\ B_{2}^{2}-\left(r_{2}+B_{2}\right)^{2}\\ \vdots\\ B_{N}^{2}-\left(r_{N}+B_{N}\right)^{2}\end{array}\right]_{N\times 1},\ C=\left[ \begin{array}{c}r_{1}\\ r_{2}\\ \vdots\\ r_{N}\end{array}\right]_{N\times 1}+\left[\begin{array}{c}B_{1}\\ B_{2}\\ \vdots\\ B_{N}\end{array}\right]_{N\times 1}. \tag{8}\]  
Figure 1: Geometric structure of the multistatic radar system (for visibility, only three of \(N\) bistatic T/R pairs are illustrated).  
The solution of (7) is [29]:  
\[x=a+\frac{-ba^{T}b\mp b\sqrt{\left(a^{T}b\right)^{2}-\left(b^{T}b-1\right)a^{T}a} }{b^{T}b-1}, \tag{9}\]  
with two introduced variables \(a\) and \(b\) being:  
\[a=\left(A^{T}A\right)^{-1}A^{T}K, \tag{10}\]  
\[b=\left(A^{T}A\right)^{-1}A^{T}C. \tag{11}\]  
This is the classic multi-lateration principle, and its detailed derivation can be referred to the literature [29]. As validated in [29], for improving the BR measurement accuracy, the coherent integration was conducted, ensuring a robust target detection.