# Radar Target Localization with Multipath Exploitation in Dense Clutter Environments
## 3 Method
### Approximation of Likelihood  
Let us consider the number of all possible hypotheses in \(\mathbf{\Theta}\). The association variable \(\boldsymbol{\theta}\) is an \(N_{p}\)-dimensional vector; therefore, the number of nonzero values of \(\boldsymbol{\theta}\) does not exceed \(N_{p}\). In addition, each measurement \(\mathbf{z}\) in the measurement set is associated with the target propagation path at most once; thus, the number of nonzero values of \(\boldsymbol{\theta}\) does not exceed \(N_{m}\). Considering the case where \(\boldsymbol{\theta}\) contains \(k\) nonzero values, then the number of associations is expressed as  
\[\binom{N_{m}}{k}\binom{N_{p}}{k}k!, \tag{27}\]  
where \(\binom{N_{m}}{k}\) denotes the number of ways to select \(k\) measurements from \(\mathbf{Z}\), \(\binom{N_{p}}{k}\) denotes the number of ways to select \(k\) paths from \(\mathbf{H_{x}}\), and \(k!\) denotes the number of ways to assign the selected \(k\) elements. Therefore, the number of hypotheses in \(\mathbf{\Theta}\) is given as  
\[N(N_{m},N_{p})=\sum_{k=0}^{min(N_{m},N_{p})}\binom{N_{m}}{k}\binom{N_{p}}{k}k!. \tag{28}\]  
Since \(N(N_{m},N_{p})\) increases exponentially with the number of measurements and target paths, the \(O(n! )\) level time complexity of \(p(\mathbf{Z}|\mathbf{H_{x}})\) is practically impossible to compute. Hence, appropriate approximation and complexity reduction techniques are introduced to reduce the numerical burden. No threshold is set in the previous formula indicating that the measurement can be associated with any target path parameter \(\mathbf{h_{x}}\). However, when the measurement is far away from \(\mathbf{h_{x}}\), the conditional probability density \(p(\mathbf{z}|\mathbf{h_{x}})\) becomes small and can be ignored. Hence, we introduce a distance matrix \(\mathbf{D}\) to describe the distance between each predicted path and measurement, where the normalized scale-invariant metric Mahalanobis distance is applied to obtain the distance between the \(k\)-th predicted path and \(j\)-th measurement as  
\[d_{j,k}=\sqrt{(\mathbf{z}^{j}-\mathbf{h}_{x}^{k})^{T}\mathbf{\Sigma}^{-1}( \mathbf{z}^{j}-\mathbf{h}_{x}^{k})} \tag{29}\]Then, only measurements within a gate around the predicted target path are considered valid, and the association matrix \(\mathbf{G}\) is acquired by a gating threshold \(T\) as  
\[G_{j,k}= \begin{cases}1,&\text{if }d_{j,k}^{2}<T,\\ 0,&\text{otherwise}.\end{cases}  \tag{30}\]  
Without losing generality, we consider two hypotheses \(\mathbf{\theta}\) and \(\mathbf{\theta}^{\prime}\) that differ only in the \(K\)-th association expressed as  
\[\left\{ \begin{array}{ll}\theta_{k}=\theta_{k}^{\prime},&k=1,\cdots,N_{p},k \
eq K,\\ \theta_{k}>0,\theta_{k}^{\prime}=0,&k=K.\end{array} \right. \tag{31}\]  
The threshold \(T\) should satisfy that the probability of a valid association between \(\mathbf{z}^{\theta_{K}}\) and \(\mathbf{h}_{\mathbf{x}}^{K}\) is larger than the probability that they are not associated with; therefore, we obtain the inequality as  
\[p(\mathbf{Z}_{\mathbf{x}}|\mathbf{H}_{\mathbf{x}},\mathbf{\theta})>p(\mathbf{Z}_{ \mathbf{x}}|\mathbf{H}_{\mathbf{x}},\mathbf{\theta}^{\prime}). \tag{32}\]  
By substituting (21) and (24) into (32), we obtain  
\[P_{d}\cdot p(\mathbf{z}^{\theta_{K}}|\mathbf{h}_{\mathbf{x}}^{K})>\frac{ \lambda}{V}(1-P_{d}). \tag{33}\]  
Therefore, the threshold \(T\) is given by substituting (22) as  
\[T=-2\log\biggl{(}2\pi\sqrt{\det(\mathbf{\Sigma})}\frac{\lambda}{V}\frac{1-P_{ d}}{P_{d}}\biggr{)}. \tag{34}\]  
The number of hypotheses decreases significantly after the gating step. All valid hypotheses are enumerated by the recursive method proposed in the next step. Considering the case where two multipath components are associated with three measurements by an association matrix  
\[\mathbf{G}=\left[ \begin{array}{cc}1&1\\ 0&1\\ 0&0\end{array} \right], \tag{35}\]  
we extend the matrix by appending an \(N_{m}\)-dimentional identity matrix to the right as  
\[\mathbf{G}^{\prime}=\left[ \begin{array}{cccc}1&1\\ 0&1\\ 0&0\end{array}