# Radar Target Localization with Multipath Exploitation in Dense Clutter Environments
## 3 Method
### Target Localization Algorithm  
In this subsection, we proposed a two-step localization algorithm based on the likelihood function derived in the previous subsection. In the first step, the target multipath propagation model is estimated by traversal of the maximum likelihood between measurement set \(\mathbf{Z}\) and all predicted samples \(\mathbf{H}_{ij}\) as  
\[\mathbf{\hat{H}}=\operatorname*{arg\,max}_{1\leq i\leq N_{x},1\leq j\leq N_{y }}p(\mathbf{Z}|\mathbf{H}_{ij}). \tag{41}\]  
The sample location corresponding to the multipath propagation model \(\mathbf{\hat{H}}\) obtained by MLE can be regarded as a rough estimation of the target location, while the uniformly distributed samples lead to off-grid errors. Hence, in the second step, an NLS-based localization algorithm is proposed to obtain the accurate target location based on the roughly estimated target location \(\mathbf{\hat{x}}\). By law of reflection, the reflection path from radar to the target is equivalent to the path propagated to the target from a virtual radar mirroring the real radar with respect to the reflection obstacle as illustrated in Figure 3. We denote the normal vector of the reflecting surface as \(\mathbf{n}=[n_{1},n_{2},n_{3}]^{T}\); therefore, the reflecting plane equation can be expressed as  
\[\mathbf{n}^{T}\cdot[x-x_{0},y-y_{0},z-z_{0}]^{T}=0, \tag{42}\]  
where \(\mathbf{t}=[x_{0},y_{0},z_{0}]^{T}\) is a point on the plane. Then, the distance from the radar \(\mathbf{r}=[x_{r},y_{r},z_{r}]^{T}\) to the plane is given as  
\[d=\frac{\mathbf{n}^{T}(\mathbf{r}-\mathbf{t})}{\|\mathbf{n}\|}, \tag{43}\]  
where \(\|\mathbf{n}\|=\sqrt{n_{1}^{2}+n_{2}^{2}+n_{3}^{2}}\) is the length of the vector. The virtual radar position with the same distance as \(\mathbf{r}\) in the opposite direction is derived as  
\[\mathbf{r}^{1}=\mathbf{r}-2d\frac{\mathbf{n}}{\|\mathbf{n}\|}. \tag{44}\]  
We assume that \(\mathbf{n}\) is a unit normal vector; thus, the mirrored virtual radar position can be rewritten into a matrix form as  
\[\left[ \begin{array}{c}x_{r_{1}}\\ y_{r_{1}}\\ z_{r_{1}}\end{array} \right]=\left[ \begin{array}{ccc}1-2n_{1}n_{1}&-2n_{2}n_{ 1}&-2n_{3}n_{1}\\ -2n_{1}n_{2}&1-2n_{2}n_{2}&-2n_{3}n_{2}\\ -2n_{1}n_{3}&-2n_{2}n_{3}&1-2n_{3}n_{3}\end{array} \right]\left[ \begin{array}[ ]{c}x_{r}\\ y_{r}\\ z_{r}\end{array} \right]+2\mathbf{n}^{T}\!\left[ \begin{array}{c}n_{1}\\ n_{2}\\ n_{3}\end{array} \right]. \tag{45}\]  
Therefore, the mirroring of a radar point \(\mathbf{r}\) with respect to a plane \(p\) is a linear transformation denoted as  
\[\mathbf{r}^{1}=\mathbf{S}_{p}(\mathbf{r}). \tag{46}\]  
The virtual radar position of multi-bounce reflection can be derived by the cascade of multiple single-bounce reflections. Let us denote the single-trip multi-bounce reflection path from target to the target as \(R-O_{1}-\cdots-O_{n}-T\), where \(O_{1},\cdots,O_{n}\) are reflection points with corresponding reflection planes \(p_{1},\cdots,p_{n}\). The virtual radar with respect to \(p_{1}\) is given as \(\mathbf{r}^{1}=\mathbf{S}_{p_{1}}(\mathbf{r})\), and the equivalent path starting from virtual radar is given as \(R_{1}-O_{2}-\cdots-O_{n}-T\). By recursion, the equivalent virtual radar of the single-trip multi-bounce reflection is obtained by  
\[\mathbf{r}^{n}=\mathbf{S}_{p_{n}}(\mathbf{r}^{n-1})=\mathbf{S}_{p_{n}}( \mathbf{S}_{p_{n-1}}(\mathbf{r}^{n-2}))\cdots\equiv\mathbf{S}(\mathbf{r}). \tag{47}\]  
As described in the previous subsection, measurements and predicted paths of \(\mathbf{\hat{x}}\) are associated by the assignment matrices. Let us denote one of the predicted paths \(\mathbf{p_{k}}\) is associated with measurement \(\mathbf{z}\). The round-trip signal received by the radar consists of single-trip paths, the transmitted path and the received path. The two virtual radars \(\mathbf{r_{1}}=[x_{r_{1}},y_{r_{1}}]^{T}\) and \(\mathbf{r_{2}}=[x_{r_{2}},y_{r_{2}}]^{T}\) of the transmitted path and the received path can be obtained by (47).