# A Sector-Matching Probability Hypothesis Density Filter for Radar Multiple Target Tracking
## 4 Simulations
### Example 1: CV Motion Model  
In this scenario, each target moves according to the following Gaussian linear model:  
\[\mathbf{x}_{k}=\left( \begin{bmatrix}1&T_{s}\\ 0&1\end{bmatrix} \otimes\mathbf{I}_{d}\right)\mathbf{x}_{k-1}+\sigma_{w}^{2}\left( \begin{bmatrix}T_{s}^{2}/4&T_{s}^{3}/2\\ T_{s}^{3}/2&T_{s}^{2}\end{bmatrix} \otimes\mathbf{I}_{d}\right) \tag{61}\]  
where \(\mathbf{I}_{d}\) is the identity matrix of dimension \(d\), where \(d=2\) in this scenario. \(\mathbf{x}_{k}=\left[\mathbf{x}_{1,k},\mathbf{x}_{2,k},\mathbf{x}_{3,k},\mathbf{x}_{4,k}\right]^{T}\), \(\left[\mathbf{x}_{1,k},\mathbf{x}_{3,k}\right]^{T}\) represents the target position at the end of the \(k\)th scan and \(\left[\mathbf{x}_{2,k},\mathbf{x}_{4,k}\right]^{T}\) is the target velocity. \(T_{s}\) is the sampling period and \(\sigma_{w}=5\) m/s\({}^{2}\). The actual target trajectories and the initial states of the targets are given in Figure 5 and Table 1, targets are created from the black dot \(\bullet\), and disappear at the end of their respective tracks. The multi-target measurement model is given by Equation (60):  
\[\mathbf{z}_{k}= \begin{bmatrix}\mathbf{r}((\mathbf{H}_{k}\otimes\mathbf{I}_{d})\mathbf{x}_{k})\\ \mathbf{\theta}((\mathbf{H}_{k}\otimes\mathbf{I}_{d})\mathbf{x}_{k})\end{bmatrix} + \begin{bmatrix} \mathbf{r}(\mathbf{e}_{k})\\ \mathbf{\theta}(\mathbf{e}_{k})\end{bmatrix}  \tag{62}\]  
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. \(\mathbf{H}_{k}=\left[1\;0\right]\) is the observation matrix and \(\mathbf{e}_{k}\) is a Gaussian white noise with covariance \(diag\left([10,0.1]\right)\). As shown in Figures 6 and 7, the sector-matching method proposed in this paper can achieve radar MTT when considering radar measurement time diversity. Applying the method to the LGM-PHD filter can achieve tracking maintenance even in the presence of strong clutter interference, dense targets in the surveillance area and long radar scan periods. In Figure 6b,d, it can be seen that a low detection probability of \(p_{D,k}=0.8\) causes the trajectory to be broken, and 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 newborn \begin{table}
\begin{tabular}{c c c} \hline \hline
**Target Index** & **Lifetime (s)** &
\begin{tabular}{c} **Initial States** \\ **(m,m/s,m,m/s)** \\ \end{tabular} \\ \hline No. 1 & (1,70) & [0; 0; \(-10\)] \\ No. 2 & (1,100) & [400; \(-10\); \(-600\); \(10\)] \\ No. 3 & (1,70) & [\(-800\); \(20\); \(-200\); \(-10\)] \\ No. 4 & (20,100) & [400; \(-7\); \(-600\); \(-4\)] \\ No. 5 & (20,100) & [400; \(-2.5\); \(-600\); \(10\)] \\ No. 6 & (20,100) & [0; \(7.5\); \(0\); \(-5\)] \\ No. 7 & (40,100) & [\(-800\); \(12\); \(-200\); \(7\)] \\ No. 8 & (40,100) & [\(-200\); \(15\); \(800\); \(-10\)] \\ No. 9 & (60,100) & [\(-800\); \(3\); \(-200\); \(15\)] \\ No. 10 & (60,100) & [\(-200\); \(-3\); \(800\); \(-15\)] \\ No. 11 & (80,100) & [\(0\); \(-40\); \(0\); \(-30\)] \\ No. 12 & (80,100) & [\(-200\); \(30\); \(800\); \(-10\)] \\ \hline \hline \end{tabular}
\end{table} 
Table 1: List of initial target states in example 1 (CV model). Figure 5: Target trajectories in example 1 (CV motion model), targets are created from the black dot \(\bullet\). targets can only appear at the known new-born points, i.e., it is assumed that no newborn target will appear at the trajectory breaking points. In Figure 7b,d, due to the extension of the accumulation time, the detection probability \(p_{D,k}=0.96\) and the probability of track breakage is significantly reduced. As shown in Figure 8, the computation time (CT) of the MATLAB simulation for 100 sample runs is given and the average CT is shown in Table 2. The basic GM-PHD filter and the modified SM-GM-PHD filter have relatively smooth computation times.