# Labeled Multi-Bernoulli Filter Joint Detection and Tracking of Radar Targets
## 4 Simulation Results  
Consider a nearly constant turn (NCT) model [8] with noisy range, azimuth and Doppler measurements [20; 25]. A target state can be denoted as \(\mathbf{x}_{k}=\left(x_{k},\ell_{k}\right)\), where \(x_{k}=\left[\tilde{x}_{k}^{\mathrm{T}},w_{k}\right]^{\mathrm{T}}\) \begin{table}
\begin{tabular}{c c c} \hline \hline
**Cell Number \(k\)** & \(\begin{array}{c}\beta\text{ {When}}\\ c_{t}\left(\mathbf{x}\right)=c_{i_{0}}\left(\mathbf{x}\right)\end{array}\) & \(\begin{array}{c}\beta\text{ {When}}\\ c_{i_{0}}\left(\mathbf{x}\right)+0.5c_{0}\end{array}\) \\ \hline
1 & 0.0635 & 0.0436 \\
3 & 0.6882 & 0.5794 \\
5 & 0.9728 & 0.9432 \\
7 & 0.9992 & 0.9973 \\
9 & 0.999991 & 0.999951 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The confidence level \(\beta\) under different cell number \(k\) when \(c_{t}\left(\mathbf{x}\right)=c_{i_{0}}\left(\mathbf{x}\right)\) (the corresponded range, azimuth and Doppler cell of the target \(\mathbf{x}\) is is located in the sampling position) and \(c_{t}\left(\mathbf{x}\right)=c_{i_{0}}\left(\mathbf{x}\right)+0.5c_{0}\) (the corresponded range, azimuth and Doppler cell of the target \(\mathbf{x}\) is is not located in the sampling position with a offset \(0.5c_{0}\) in each dimension). Figure 6: For simplicity of drawing, \(C\left(\mathbf{x}\right)\) is a two dimensional matrix and \(k\) equals to 3. \(c_{t}\left(\mathbf{x}\right)\) is the cell of the target \(\mathbf{x}\) indicated by a orange triangle, \(c_{i_{0}}\left(\mathbf{x}\right)\) is indicated by a hollow red pentagram in the center closest to \(c_{t}\left(\mathbf{x}\right)\), \(C\left(\mathbf{x}\right)\) is the measurement cell set indicated by all pentagrams which is the nearest 3 cells around cell \(c_{i_{0}}\left(\mathbf{x}\right)\) in each dimension and \(c_{t}\left(\mathbf{\tilde{x}}\right)\) is indicated by an blue dot which approximates \(c_{t}\left(\mathbf{x}\right)\). \(\hat{x}_{k}=\left[p_{x,k},\hat{p}_{x,k},p_{y,k},\hat{p}_{y,k}\right]^{\mathrm{T}}\) denotes the planar position and velocity of the target at \(k\) and \(w_{k}\) denotes the turn rate. The state transition model can be denoted as  
\[\hat{x}_{k}=F\left(w_{k-1}\right)\hat{x}_{k-1}+GW_{n,k-1} \tag{30}\]  
\[w_{k}=w_{k-1}+u_{k-1}T_{s} \tag{31}\]  
where \(W_{n,k-1}\) and \(u_{k-1}\) denotes the state noises of the velocity and the turn rate, respectively. \(W_{n,k-1}\sim N(\cdot\ ;0,\sigma_{W_{n}}^{2}I_{2})\) and \(u_{k-1}\sim N(\cdot\ ;0,\sigma_{u}^{2}I_{1})\), where \(I_{i}\) denotes the \(i\) order identity matrix. The state transition matrix and noise transition matrix can be denoted as  
\[F\left(w\right)=\left[ \begin{array}{cccc}1&\frac{\sin wT_{s}}{w}&0&-\frac{1 -\cos wT_{s}}{w}\\ 0&\cos wT_{s}&0&-\sin wT_{s}\\ 0&\frac{1-\cos wT_{s}}{w}&1&\frac{\sin wT_{s}}{w}\\ 0&\sin wT_{s}&0&\cos wT_{s}\end{array} \right], \tag{32}\]  
\[G=\left[ \begin{array}{cccc}\frac{T_{s}^{2}}{2}&T_{s}&0&0\\ 0&0&\frac{T_{s}^{2}}{2}&T_{s}\end{array} \right]^{\mathrm{T}}. \tag{33}\]  
The observation region is \(\left[800,1800\right]\) m \(\times\left[20\pi/180,70\pi/180\right]\) rad \(\times\left[-30,0\right]\) m/s. The received power signal can be denoted by Equations (5) to (9). The sampling interval of the range is 5 m, the sampling interval of the azimuth is \(\pi/180\) rad and the sampling interval of the Doppler is 1 m/s. The variance of observation noise is \(2\sigma_{w_{n}}^{2}=2\). \(A\left(\mathbf{x}\right)=\bar{A}e^{i\theta}\) can be modeled as a complex amplitude of the Swerling 0 type [20,25] where the modulus \(\bar{A}\) is a constant and the phase are uniformly distributed on \(\left[0,2\pi\right)\), \(\bar{A}=sqrt(2\sigma_{w_{n}}^{2}\times 10^{\mathrm{SNR}/10})\). The measurements per frame can be denoted as a \(201\times 51\times 31\) matrix and the measurements with a total of 40 observation frames have been recorded.