# A Sector-Matching Probability Hypothesis Density Filter for Radar Multiple Target Tracking
## 4 Simulations
### Example 2: CA Motion Model  
In this scenario, each target moves according to the following Gaussian linear model:  
\[\mathbf{x}_{k}=\left( \begin{bmatrix}1&T_{s}&{T_{s}}^{2}/2\\ 0&1&T_{s}\\ 0&0&1\end{bmatrix} \otimes\mathbf{I}_{d}\right)\mathbf{x}_{k-1}+\mathbf{w}_{k} \tag{63}\]  
where \(\mathbf{I}_{d}\) is the identity matrix of dimension \(d\), where \(d=2\). \(\mathbf{x}_{k}=\left[\mathbf{x}_{1,k},\mathbf{x}_{2,k},\mathbf{x}_{3,k},\mathbf{x}_{4,k},\mathbf{x}_{5,k},\mathbf{x}_{6,k}\right]^{T}\), \(\left[\mathbf{x}_{1,k},\mathbf{x}_{4,k}\right]^{T}\) represents the target position at the end of the \(k\)th scan, \(\left[\mathbf{x}_{2,k},\mathbf{x}_{5,k}\right]^{T}\) is the target velocity and \(\left[\mathbf{x}_{3,k},\mathbf{x}_{6,k}\right]^{T}\) is the target acceleration. \(T_{s}\) is the sampling period. The actual target trajectories and the initial states of the targets are given in Figure 11 and Table 3, targets are born from the black dot \(\bullet\), and disappear at the end of their respective tracks. The multi-target measurement model is given by Equation (62):  
\[\mathbf{z}_{k}= \begin{bmatrix}\mathbf{r}(\mathbf{H}_{k}\otimes\mathbf{I}_{d})\mathbf{x}_{k})\\ \mathbf{\theta}((\mathbf{H}_{k}\otimes\mathbf{I}_{d})\mathbf{x}_{k})\end{bmatrix} + \begin{bmatrix} \mathbf{r}(\mathbf{e}_{k})\\ \mathbf{\theta}(\mathbf{e}_{k})\end{bmatrix}  \tag{64}\]  
where \(\mathbf{r}((\mathbf{H}_{k}\otimes\mathbf{I}_{d})\mathbf{x}_{k})\) and \(\mathbf{\theta}((\mathbf{H}_{k}\otimes\mathbf{I}_{d})\mathbf{x}_{k})\) denote the range and bearing of state \(\mathbf{x}_{k}\), respectively. \(\mathbf{H}_{k}=\left[1\ 0\ 0\right]\) is the observation matrix and \(\mathbf{e}_{k}\) is the Gaussian white noise with covariance \(diag\ (\left(\left[10,0.1\right]\right)\). The radar divides the detection airspace into 500 parts. A detailed comparison of the performance of the individual filters at \(T_{s}=1\) s, \(T_{i}=2\) ms, \(\lambda=104\), \(p_{D,k}=0.8\) and \(T_{s}=2\) s, \(T_{i}=4\) ms, \(\lambda=52\), \(p_{D,k}=0.96\), and the effect of different sampling times on the filter performance are given in Figures 12-16. Figure 10: The average OSPA of different filters in example 1 (CV motion model) for 100 sample runs, when \(T_{s}=2\) s (\(T_{i}=4\) ms, \(\lambda=52\), \(p_{D,k}=0.96\)). (**a**) Average OSPA distances of different filters. (**b**) OSPA locations of different filters. (**c**) OSPA cardinalities of different filters. As shown in Figures 12 and 13, the sector-matching framework proposed in this paper can achieve radar MTT when considering radar measurement time diversity. Applying the method to the LGM-PHD filter can achieve track maintenance even in the presence of strong clutter interference, dense targets in the surveillance area and long radar scan periods. Similar to example 1, in Figure 12, a low detection probability of \(p_{D,k}=0.8\) causes the trajectory to be broken, and in Figure 12b,d, the reason why the trajectory can be identified as the same target's trajectory instead of a new-born target trajectory after the trajectory is broken is that in the simulation made in this paper, it is assumed that the new-born targets can only appear at the known new born points, i.e., it is assumed that no new-born target will appear at the trajectory breaking points. In Figure 13, due to the extension of accumulation time, the detection probability (\(p_{D,k}=0.96\)) and the probability of track breakage is significantly reduced. In Figure 14, the computation time (CT) of the MATLAB simulation for 100 sample runs is given and the average computation is shown in Table 4. The results in Figure 14 are the same as in Figure 8, which show that the real-time performances of the proposed method for both LGM-PHD filter and the basic GM-PHD filter are similar to that of the basic PHD filter and the LGM-PHD filter, and much better than that of the joint GLMB filter. In addition, extending the accumulation time can reduce the calculation time.