# Radar Target Localization with Multipath Exploitation in Dense Clutter Environments
## 3 Method
### Likelihood Function with Multi-Measurement  
The classic likelihood function describes the probability between target state and measurement, assuming that a target generates at most one measurement in one observation. However, this is not the case in multipath scenarios where a target may generate more than one measurement. Therefore, finite-set statistics (FISST) theory [44] is introduced to describe both the uncertainty of target location and the uncertainty of multipath propagation simultaneously. This subsection derives the likelihood function \(f(\mathbf{Z}|\mathbf{H}_{\mathbf{x}})\) to describe the conditional probability density that the measurement set \(\mathbf{Z}\) is generated by a potential target multipath propagation \(\mathbf{H}_{\mathbf{x}}\).  
We introduce a multipath measurement association hypothesis variable \(\boldsymbol{\theta}=[\theta_{1},\cdots,\theta_{N_{p}}]\) to interpret whether each path \(\mathbf{h}_{\mathbf{x}}^{k}\) is detected in the measurement set \(\mathbf{Z}\) as  
\[\theta_{k}=\begin{cases}j\in\{1,\cdots,N_{m}\},&\text{if $\mathbf{z}^{j}$ is the detection of $\mathbf{h}_{\mathbf{x}}^{k}$},\\ 0,&\text{if $\mathbf{h}_{\mathbf{x}}^{k}$ is not detected}.\end{cases} \tag{19}\]  
So the measurements generated from multipath component detections are given as  
\[\mathbf{z}_{\mathbf{x}}^{\theta_{k}}=\mathbf{h}_{\mathbf{x}}^{k}+\mathbf{w}, k=1,\cdots,N_{p}. \tag{20}\]  
With the assumption that all measurements, including false alarms from clutter and detections from target multipath returns, are independent of each other, the conditional probability density of detected set \(\mathbf{Z}_{\mathbf{x}}\) with the given association hypothesis is derived as  
\[p(\mathbf{Z}_{\mathbf{x}}|\mathbf{H}_{\mathbf{x}},\boldsymbol{\theta})=\prod _{\theta_{k}=0}(1-P_{d})\prod_{\theta_{k}>0}P_{d}\cdot p(\mathbf{z}^{\theta_ {k}}|\mathbf{h}_{\mathbf{x}}^{k}), \tag{21}\]  
where the first product component denotes the probability of undetected paths, and the second component denotes the probability of detected paths, and the conditional probability density of each measurement given the associated path parameter is modeled as a Gaussian distribution  
\[p(\mathbf{z}|\mathbf{h}_{\mathbf{x}})=\frac{1}{(2\pi)^{\frac{N}{2}}\det^{ \frac{1}{2}}(\mathbf{\Sigma})}\exp\left[-\frac{1}{2}(\mathbf{z}-\mathbf{h}_{ \mathbf{x}})^{T}\mathbf{\Sigma}^{-1}(\mathbf{z}-\mathbf{h}_{\mathbf{x}}) \right], \tag{22}\]  
Figure 2: Framework of the proposed target localization algorithm.  
where \(N\) is the dimension of vector \(\mathbf{h_{x}}\), and \(\mathbf{\Sigma}\) is the covariance of measurement noise \(\mathbf{w}\).  
Measurements unassociated with target multipath echos in the hypothesis are false alarms generated from clutter, which can be expressed as  
\[\mathbf{C}=\{\mathbf{z}^{j}|\
exists\theta_{k}=j,k=1,\cdots,N_{p}\}. \tag{23}\]  
Then, the probability function of clutter conditioned on an association hypothesis is given as  
\[p_{c}(\mathbf{C}|\boldsymbol{\theta})=e^{-\lambda}(\frac{\lambda}{V})^{| \mathbf{C}|}, \tag{24}\]  
where \(|C|\) denotes the cardinality of the set.  
The measurement likelihood conditioned on an association hypothesis is given as  
\[p(\mathbf{Z}|\mathbf{H_{x}},\boldsymbol{\theta})=p_{c}(\mathbf{C}|\boldsymbol {\theta})p(\mathbf{Z_{x}}|\mathbf{H_{x}},\boldsymbol{\theta}). \tag{25}\]  
Finally, the likelihood function of measurement set \(\mathbf{Z}\) given target state set \(\mathbf{H_{x}}\) is derived by summing up all association hypotheses as  
\[p(\mathbf{Z}|\mathbf{H_{x}})=\sum_{\boldsymbol{\theta}\in\mathbf{\Theta}}p( \mathbf{Z}|\mathbf{H_{x}},\boldsymbol{\theta}), \tag{26}\]  
where \(\mathbf{\Theta}\) denotes the space of all possible association hypotheses.