# Labeled Multi-Bernoulli Filter Joint Detection and Tracking of Radar Targets
## 2 Background
### The Observation Model and Likelihood Function of Radar  
The range-azimuth-Doppler image of the echo power signal is used as the raw radar measurements [20; 21]. The measurements per frame can be denoted as a three dimensional matrix after sampling. Each element in the matrix denotes a measurement and the coordinate of measurement cell \(i\) can be denoted as \(c_{i}\left(\mathbf{x}\right)\triangleq\left[r^{m},b^{n},d^{q}\right]\), where \(r^{m}\), \(b^{n}\) and \(d^{q}\) denote the range, azimuth and Doppler cells, respectively. \(i=\left(m,n,q\right)\) is the index of cell \(i\). The index set of the measurements affected by the target \(\mathbf{x}\) is called the target template and denoted by \(T\left(\mathbf{x}\right)\). \(i\in T\left(\mathbf{x}\right)\) denotes that the measurement in cell \(i\) is affected by \(\mathbf{x}\). The setting of \(T\left(\mathbf{x}\right)\) is detailed in Section 3.4. The measurement matrix can also be denoted as a vector form \(\mathrm{Y}=\left[z^{1},...,z^{i},...,z^{m_{z}}\right]\), where \(m_{z}\) is the number of the measurements, \(z^{i}\triangleq z^{i}\left(\mathbf{X}\right)\) is the power signal in cell \(i\) and \(\mathbf{X}\) is the multi-target state. The measurement \(z^{i}\) can be denoted as [20; 25]  
\[z^{i}=\left|z_{A}^{i}\right|^{2}=\left|\sum_{\mathbf{x}\in\mathbf{X}}1_{T \left(\mathbf{x}\right)}\left(i\right)A\left(\mathbf{x}\right)h_{A}^{i}\left( \mathbf{x}\right)+w_{n}^{i}\right|^{2} \tag{5}\]  
where \(z_{A}^{i}\) is the received complex signal in cell \(i\) and \(w_{n}^{i}\) denotes a zero-mean complex Gaussian white noise with a variance of \(2\sigma_{w_{n}}^{2}\). \(A\left(\mathbf{x}\right)\) is a complex amplitude. \(\sum_{\mathbf{x}\in\mathbf{X}}1_{T\left(\mathbf{x}\right)}\left(i\right)\) denotes that the signal in cell \(i\) is the sum generated by \(\mathbf{X}\), so the signal in each cell may affected by all \(\mathbf{x}\) (\(\mathbf{x}\in\mathbf{X}\)). \(h_{A}^{i}\left(\mathbf{x}\right)\) denotes the signal generated by the point spread function of the target \(\mathbf{x}\). The expression is (according to the point spread function formula of Reference [21], we assume that \(L_{r}=\frac{1}{R}\), \(L_{d}=\frac{1}{D}\), \(L_{b}=\frac{1}{B}\))  
\[h_{A}^{i}\left(\mathbf{x}\right)=\exp(-\frac{\left(r^{m}-r\left(\mathbf{x} \right)\right)^{2}}{2R^{2}}-\frac{\left(b^{n}-b\left(\mathbf{x}\right)\right)^ {2}}{2B^{2}}-\frac{\left(d^{q}-d\left(\mathbf{x}\right)\right)^{2}}{2D^{2}}) \tag{6}\]  
where  
\[r\left(\mathbf{x}\right)=\sqrt{p_{x}^{2}+p_{y}^{2}}, \tag{7}\]  
\[b\left(\mathbf{x}\right)=atan2\left(p_{y},p_{x}\right), \tag{8}\]  
\[d\left(\mathbf{x}\right)=-\left(p_{x}\dot{p}_{x}+p_{y}\dot{p}_{y}\right)/r \left(\mathbf{x}\right), \tag{9}\]  
\(R,B\) and \(D\) denote the range, azimuth and Doppler resolution of radar, respectively. \(\left(p_{x},p_{y}\right)^{\mathrm{T}}\) and \(\left(\dot{p}_{x},\dot{p}_{y}\right)^{\mathrm{T}}\) denote the position and speed of \(\mathbf{x}\) in the state space, respectively. \(r\left(\mathbf{x}\right)\), \(b\left(\mathbf{x}\right)\) and \(d\left(\mathbf{x}\right)\) denote the range, azimuth and Doppler positions relating to \(\mathbf{x}\), respectively. Note that \(r\left(\mathbf{x}\right)\), \(b\left(\mathbf{x}\right)\) and \(d\left(\mathbf{x}\right)\) are continuous while \(r^{m}\), \(b^{n}\) and \(d^{q}\) are discrete sampling in the measurement space. In the following, we construct the likelihood function of the radar observation model. The estimated value of the measurement \(z^{i}\) in cell \(i\) is denoted by \(\hat{z}^{i}\) and \(\hat{z}^{i}\) can be written as [25]  
\[\hat{z}^{i}=\left|\hat{z}_{A}^{i}\right|^{2}\approx\left|\sum_{\mathbf{x}\in \mathbf{X}}1_{T\left(\mathbf{x}\right)}\left(i\right)\tilde{A}h_{A}^{i}\left( \mathbf{x}\right)\right|^{2}. \tag{10}\]When \(\hat{z}_{A}^{i}\
eq 0\), \(z^{i}/\sigma_{w_{n}}^{2}\) subjects to a noncentral \(\chi^{2}\) distribution with 2 degrees of freedom. The probability density of \(z^{i}\) is  
\[P\left(z^{i}|\mathbf{X}\right)=\left\{