# Radar Target Localization with Multipath Exploitation in Dense Clutter Environments
## 2 Model
### Multipath Propagation Model via Ray-Tracing  
Computational electromagnetics is widely used to simulate the propagation of EM waves in the field of communication, chip design and radar. The ray-tracing technique provides an effective approximation of EM waves when the wavelength is small compared to the size of structures in large-scale scenes, where the full-wave numerical method, such as a finite-difference time-domain (FDTD) method, cannot be applied because of the high computational complexity. The ray-tracing method describes the reflection, diffraction, and transmission of EM waves based on PO theory by representing EM waves radiated from the source as rays [41]. Since the implementation of the ray-tracing method is beyond the scope of our paper, details of the ray-tracing technique are not described further. In this paper, reflection paths from a given radar position to the target position are calculated by a commercial software Wireless Insite [42], with environmental information supposed to be prior knowledge. The diffraction paths are not included in the simulation because the irregular propagation geometry of diffraction paths is usually not applicable to localization algorithms. Let us denote the radar position by  
\[\mathbf{r}=[x_{r},y_{r}]^{T}, \tag{1}\]  
and the target position is  
\[\mathbf{x}=[x_{d},y_{d}]^{T}. \tag{2}\]  
Then the detections of multipath propagation parameters can be abstracted into a non-linear function related to radar and target position as  
\[\mathbf{H}(\mathbf{r},\mathbf{x})=\mathbf{H}_{\mathbf{x}}=\{\mathbf{h}_{ \mathbf{x}}^{k}\}_{k=1}^{N_{p}}, \tag{3}\]  
where \(N_{p}\) is the number of propagation paths. and \(\mathbf{h}_{\mathbf{x}}\) is the measurement vector for each path composed of corresponding TOA \(\tau\) and DOA \(\phi\) given by  
\[\mathbf{h}_{\mathbf{x}}^{k}=[\tau_{\mathbf{x}}^{k},\phi_{\mathbf{x}}^{k}]^{T}. \tag{4}\]  
Note that \(N_{p}\) varies when the target location changes, so the set representation \(\mathbf{H}_{\mathbf{x}}\) is used instead of vector representation, indicating that the number of elements is not fixed. Since the multipath propagation \(\mathbf{H}(\mathbf{r},\mathbf{x})\) has no closed form, an intuitive idea is to sample \(\mathbf{H}_{\mathbf{x}}\) with a numerical method. Randomly distributed particles on the target state space sampled by a real-time ray-tracing method in previous studies revealed high-precision positioning results, while real-time ray-tracing calculation is time-consuming for the high computational complexity [43]. Instead, this paper uses fixed uniformly distributed samples which can be computed offline in advance. The position of uniformly distributed samples are indexed as  
\[\mathbf{s}(i,j)=\mathbf{s}_{ij}=[i\cdot\Delta x,j\cdot\Delta y]^{T},1\leq i\leq N _{x},1\leq j\leq N_{y}, \tag{5}\]  
where \(\Delta x\) and \(\Delta y\) are the interval of samples in the x and y direction, and \(N_{x}\) and \(N_{y}\) are the corresponding number of samples. Figure 1 illustrates an example of a 3D ray-tracing result from Wireless Insite software and a 2D simplification ray-tracing result which ignored the altitude in elevation. For ease of simulation and illustration, the paper makes some simplifications, and 2D ray-tracing is adopted for subsequent simulation verification. Meanwhile, the method can be extended easily to 3D scenarios without modifying the localization algorithm. The parameters of single-trip multipath propagation from the radar to the sampled target positions are acquired by a ray-tracing method with proper simulation setups as  
\[\mathbf{H}_{1}(i,j)=\{\mathbf{h}_{1}(i,j)\}_{k_{1}=1}^{N_{1}}, \tag{6}\]  
where \(N_{1}\) is the number of single-trip paths.