# Labeled Multi-Bernoulli Filter Joint Detection and Tracking of Radar Targets
## 4 Simulation Results
The birth distribution is an LMB RFS and each track in the birth distribution obeys a Gaussian distribution with mean and variance shown in Table 2. The other basic simulation parameters are also shown in Table 2. The unit of position \(p_{x},p_{y}\) is m, the unit of velocity \(\hat{p}_{x},\hat{p}_{y}\) is m/s and the unit of the turn rate \(w\) is rad/s. The state noise standard deviations \(\sigma_{W_{n}}=\sqrt{5}\) m/s\({}^{2}\) and \(\sigma_{u}=\pi/180\) rad/s\({}^{2}\). The target survival probability is \(P_{s}=0.99\). The number of particles is set to 1000 for each hypothesized track. We prune hypothesized tracks with a weight threshold of \(P_{T}=10^{-3}\). Due to the nonlinearity of the observation model, we use the SMC method to implement the filter. The performance of our S-LMB-GOM filter is evaluated with OSPA metric [30] (\(c=100\) m, \(p=1\)) and compared with G-LMB-GOM filter over 100 Monte Carlo trials. We should point out that there are many papers [20, 21, 22, 23, 24, 25] based on the Bayes filter to solve the TBD problem of radar, but the G-LMB-GOM filter [26] is the new method which adapts to inseparable likelihood and reduces the computational complexity compared with the method in \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Symbol** & **Value** \\ \hline Signal-to-Noise Ratio & SNR & 7 dB \\ Mean of 1 st birth track & \(x_{B}^{1}\) & \([1250,-10,1000,-10,0]^{\mathrm{T}}\) \\ Mean of 2 st birth track & \(x_{B}^{2}\) & \([1000,-10,1250,-10,0]^{\mathrm{T}}\) \\ Mean of 3 st birth track & \(x_{B}^{3}\) & \([1050,-18,1200,-5,0]^{\mathrm{T}}\) \\ Mean of 4 st birth track & \(x_{B}^{4}\) & \([1250,-15,1250,-15,0]^{\mathrm{T}}\) \\ Variances of all birth tracks & \(Q\) & \(diag(15,2,15,2,2\pi/180)^{2}\) \\ Range Resolution & R & 5 m \\ Azimuth Resolution & B & \(\pi/180\) rad \\ Doppler Resolution & D & 1 m/s \\ Time Sampling Interval & \(T_{s}\) & 1 s \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Some Simulation Parameters. Figure 11: The execution time per frame for S-LMB-GOM filter and G-LMB-GOM filter. Figure 10: The OSPA distances for S-LMB-GOM filter and G-LMB-GOM filter. Figure 9: The true and estimated cardinality for S-LMB-GOM filter and G-LMB-GOM filter. Figure 9 shows the estimated cardinality over 100 Monte Carlo trials. The estimated cardinality mean of two filters both converge to the true cardinality. The estimated cardinality variance of the S-LMB-GOM filter is slightly larger than that of the G-LMB-GOM filter, because the negative effect of the separable likelihood assumption cannot be completely eliminated. The average estimated cardinality variances of two filters are 0.28 and 0.224, respectively. Figure 10 shows the OSPA miss distance for the two filters. The OSPA distance will increase with target births and deaths due to the estimated delay. We also give the average OSPAs of two filters in Table 3. The OSPA in the cardinality errors of the S-LMB-GOM filter is larger than that of the G-LMB-GOM filter, because of the larger cardinality variance for the S-LMB-GOM filter. The OSPA in the localization errors of the S-LMB-GOM filter is less than that of the G-LMB-GOM filter, because of less mutual interference in state estimation for the S-LMB-GOM filter. We can obtain that the total OSPA distance of the S-LMB-GOM filter is slightly less than that of the G-LMB-GOM filter. Figure 11 shows the average execution time per frame. The execution time per frame for the S-LMB-GOM filter is less than 1 s. The average time each simulation for the S-LMB-GOM filter is about 29.3 s, while the average time each simulation for the G-LMB-GOM filter is about 323 s. The execution time from the 5 th frame to 10 th frame is significantly increased for the G-LMB-GOM filter because the hypothesized tracks are relative close and the group processing cannot effectively reduce the computation time. Figure 12 shows the OSPA miss distance at different SNR for the S-LMB-GOM filter. It can be seen that the OSPA distance decreases with increasing SNR.