# A Sector-Matching Probability Hypothesis Density Filter for Radar Multiple Target Tracking
## 4 Simulations
### Example 2: CA Motion Model
In Figures 15 and 16, the average OSPA distance of the MATLAB simulation for 100 sample runs is given and the average OSPA distances are given in Table 4. Compared with the basic GM-PHD filter, the average tracking performance of the SM-LGM-PHD filter, the IGM-PHD filter, the SM-GM-PHD filter and the joint-GLMB filter in this scenario is improved by 41.57%, 16.40%, 28.78% and 47.67%, respectively, at \(T_{s}=1\) s (\(T_{i}=2\) ms, \(\lambda=104\), \(p_{D,k}=0.8\)), and 57.32%, 29.11%, 34.66% and 37.69%, respectively, at \(T_{s}=2\) s (\(T_{i}=4\) ms, \(\lambda=52\), \(p_{D,k}=0.96\)). Compared with the \(T_{s}=1\) s scenario, the performance \begin{table}
\begin{tabular}{c c c} \hline \hline
**Target Index** & **Lifetime (s)** &
\begin{tabular}{c} **Initial States** \\ **(m,m/s,m/s\({}^{2}\),m/s,m/s\({}^{2}\))** \\ \end{tabular} \\ \hline No. 1 & (1,70) & [0; 0; 0.03; 0; \(-\)37,0.75] \\ No. 2 & (1,100) & [400; \(-\)10; 0.4; \(-\)600; 5;0.01] \\ No. 3 & (1,70) & [\(-\)800; 20; \(-\)0.16; \(-\)200; \(-\)5;0.3] \\ No. 4 & (20,100) & [400; \(-\)7; \(-\)0.37; \(-\)100; \(-\)42;1] \\ No. 5 & (20,100) & [400; \(-\)1.4; \(-\)0.44; \(-\)600; 10.05] \\ No. 6 & (20,100) & [0; 5.5; 0.29; \(0.22;-\)0.55] \\ No. 7 & (40,100) & [\(-\)800; \(32\); \(-\)1.5; \(-\)200; \(11\); \(-\)0.75] \\ No. 8 & (40,100) & [\(-\)200; \(15\); 0.3; 800; \(-\)10; 0] \\ No. 9 & (60,100) & [\(-\)800; \(-\)3; 1.5; \(-\)200; 15; \(-\)0.75] \\ No. 10 & (60,100) & [\(-\)200; \(-\)3; 0; 800; \(-\)15; 0] \\ No. 11 & (80,100) & [0; \(-\)20; \(-\)1.6; 0; \(-\)45; 0.2] \\ No. 12 & (80,100) & [\(-\)200; 15; 0; 800; \(-\)5; 0] \\ \hline \hline \end{tabular}
\end{table} 
Table 3: List of initial target states in the CA motion model. Figure 11: Target trajectories in example 2 (CA motion model), where targets are born from the black dot \(\bullet\). of the SM-LGM-PHD filter and the SM-GM-PHD filter is highly improved in the \(T_{s}=2\) s scenario. The LGM-PHD filter, the basic GM-PHD filter and the joint-GLMB filter are not time-matched, which is why the performance of these three filters is not particularly improved, and the performance of the joint-GLMB filter even decreases. In addition, it can be seen in Figures 15b and 16b that the values of OSPA location of the LGM-PHD filter, the basic GM-PHD filter and the joint-GLMB filter all increase with the accumulation of time. After 80s, with the appearance of two targets, No.11 and No.12, at a higher speed, the targets' position estimation errors of the LGM-PHD filter, the basic GM-PHD filter and the joint-GLMB filter increase at a faster rate, while the SM-LGM-PHD filter and the SM-GM-PHD filter are not significantly affected. This is mainly caused by the additional measurement error caused by sampling time diversity mentioned earlier. Compared with the CV model in example 1, the CA motion model has a quadratic term in time, so the increases in the estimation error due to the time accumulation of the basic PHD filter, the LGM-PHD filter and the GLMB filter are more significant. Figure 12: The estimations of the multi-target trajectories and the actual measurements in example 2 (CA motion model) for a sample run. (**a**) The actual measurements, actual trajectories and estimated trajectories obtained after filtering by the SM-GM-PHD filter for \(T_{s}=1\) s (\(T_{i}=2\) ms, \(\lambda=104\), \(p_{D,k}=0.8\)). (**b**) The actual measurements, actual trajectories and estimated trajectories obtained after filtering by the SM-LGM-PHD filter for \(T_{s}=1\) s, and the different trajectories are denoted by different color dots. (**c**) The actual measurements, actual trajectories and estimated trajectories obtained after filtering by the SM-GM-PHD filter in x and y coordinates for \(T_{s}=1\) s. (**d**) The actual measurements, actual trajectories and estimated trajectories obtained after filtering by the SM-LGM-PHD filter in x and y coordinates for \(T_{s}=1\) s.  
Figure 13: The estimations of the multi-target trajectories and the actual measurements in example 2 (CA motion model) for a sample run.