# Radar Target Localization with Multipath Exploitation in Dense Clutter Environments
## 4 Simulation Results  
Simulations are presented to evaluate the proposed algorithm in this section. The scenario setup is illustrated in Figure 4, where the dark blocks represent the obstacles in the scenario. A radar locates at \((60,40)\) with the main lobe direction facing north. A total of 100 targets distributed from \((35.5,125.5)\) to \((134.5,125.5)\) along the x-axis direction at intervals of 1 m are simulated to study the robustness of the proposed localization algorithm in different multipath propagation conditions. The sampled locations computed offline are located at grids from \((0,0)\) to \((160,180)\) with intervals of 1 m along the x and y directions. Therefore, the off-grid error of localization caused by the distance between the targets and their nearest sample is \(\sqrt{2}/2\) m.  
Figure 3: A single-bounce path from radar \(R\) to the target \(T\) reflected at position \(O\) and the equivalent path from virtual radar \(R_{1}\) to the target directly. Figure 5 illustrates the number of round-trip multipath components in the study area restricted by a rectangle with length of 160 m and width of 180 m, where the times of bounces for a simulated single-trip path in Figure 5a,b is limited to less than one and two, respectively. The increase in reflection times complicates the multipath propagation model not only in the maximum received multipath components but also in the heterogeneity of multipath components. As is shown in Figure 5, the maximum number of round-trip paths is 25 for the single-bounce case, while the number increases to 144 for the double-bounce case. Moreover, the propagation model varies quickly and becomes more sensitive to target locations as the reflection times increase. However, the increased paths do not contribute to the localization area because they still cannot reach the shadow regions compared to the single-bounce case. Considering the requirement of radar signal receiving energy, the times of bounces are limited to less than one in the following simulations, and single-trip reflection paths of more than one bounce are considered undetectable. The ground truth of target multipath measurements as well as the cluttered measurements are illustrated in Figure 6. In the simulation, the probability of each path being detected \(P_{D}\) is set to 0.8. Since the measurement of distance and angle are independent, zero-mean Gaussian noises are added to the distance and angle measurements separately with the corresponding standard deviation (STD) of the noises as \(std(\tau)\) and \(std(\phi)\). Therefore, the covariance of measurement is given as  
\[\Sigma=\left[ \begin{array}{cc}std^{2}(\tau)&0\\ 0&std^{2}(\phi)\end{array} \right]. \tag{55}\]  
We assume the Poisson rate \(\lambda\) equals 20, which means that each radar scan produces 20 cluttered measurements on average. Each cluttered measurement \(\mathbf{c}=[\tau,\phi]\) is generated by two uniform distributions as  
\[\tau \sim U(0,350), \tag{56}\] \[\phi \sim U(0,180),\]  
where \([0,350]\) and \([0,180]\) are distance and angle range of radar, respectively. Figure 4: Simulation scenario setup. The localization results of the proposed algorithm are illustrated in Figure 7, where the dotted lines denote the grid-based localization error of the first step, and the solid lines denote the NLS-based localization error of the second step. Results from index 38 to 48 are excluded because no path of the target behind the obstacle propagates to the radar as shown in Figure 6b. Localization error is defined as the Euclidean distance between the estimated target location and the ground truth by \(\|\mathbf{\hat{x}}-\mathbf{x}\|\). The red lines present the simulation results that the STD of distance and angle measurement equals one. In the case of low noise level, the grid-based method results are accurate, and the off-grid error is the main cause of localization error. Therefore, the NLS-based method can improve the localization accuracy based on the correct estimation of the multipath propagation model.