# A Sector-Matching Probability Hypothesis Density Filter for Radar Multiple Target Tracking
## 2 Background  
### Gm-PHD Filter  
In RFS-based filters, the multi-target states and multi-target measurements at the \(k\)th time step are defined as finite sets:  
\[\mathbf{X}_{k} =\left\{\mathbf{x}_{k}^{1},\mathbf{x}_{k}^{2},\cdots,\mathbf{x}_{k}^{N_{k,k}} \right\}\in F(\mathbb{X}), \tag{1}\] \[\mathbf{Z}_{k} =\left\{\mathbf{z}_{k}^{1},\mathbf{z}_{k}^{2},\cdots,\mathbf{z}_{k}^{N_{k,k}} \right\}\in F(\mathbb{Z}), \tag{2}\]  
where \(\mathbf{x}_{k}^{1},\mathbf{x}_{k}^{2},\cdots,\mathbf{x}_{k}^{N_{k,k}}\) represent the states of \(N_{x,k}\) targets, \(F(\mathbb{X})\) is the set of multi-target state sets \(\mathbb{X}\), \(\mathbf{z}_{k}^{1},\mathbf{z}_{k}^{2},\cdots,\mathbf{z}_{k}^{N_{k,k}}\) represent the measurements of \(N_{z,k}\) targets and \(F(\mathbb{Z})\) is the set of multi-target measurement sets \(\mathbb{Z}\). The single target motion model and measurement model are defined as:  
\[\left\{ \begin{array}{l}\mathbf{x}_{k}=\mathbf{F}_{k-1}\mathbf{x}_{k-1}+\mathbf{w}_{k-1}\\ \mathbf{z}_{k}=\mathbf{H}_{k}\mathbf{x}_{k}+\mathbf{v}_{k}\end{array} \right., \tag{3}\]  
where \(\mathbf{x}_{k}\in\mathbb{R}^{n}\) and \(\mathbf{z}_{k}\in\mathbb{R}^{m}\) are state vector and measurement vector, respectively, \(\mathbf{F}_{k-1}\) represents the target state transition function, \(\mathbf{H}_{k}\) is the measurement equation of targets, \(\mathbf{w}_{k-1}\) and \(\mathbf{v}_{k}\) are the system process noise and measurement noise, distributed as a Gaussian with variance \(\mathbf{Q}_{k-1}\) and \(\mathbf{R}_{k}\), respectively. The posterior hypothesis posterior PHD at the \(k-1\) moment can be expressed in the form of a Gaussian sum as:  
\[D_{k-1|k-1}(\mathbf{x})=\sum\limits_{i=1}^{J_{k-1}}\omega_{k-1}^{i}N\left(\mathbf{x}; \mathbf{m}_{k-1}^{i},\mathbf{P}_{k-1}^{i}\right), \tag{4}\]  
where \(J_{k-1}\) is the total number of components and the mean and variance of the \(i\)th Gaussian component are \(\mathbf{m}_{k-1}^{i}\) and \(\mathbf{P}_{k-1}^{i}\) respectively. The prediction of PHD can be expressed in a Gaussian mixed form as:  
\[ \begin{array}{l}D_{k|k-1}(\mathbf{x})=D_{S,k|k-1}(\mathbf{x})+D_{\beta,k|k-1}(\mathbf{x} )+\gamma_{k}(\mathbf{x})\\ =\sum\limits_{i=1}^{J_{kk-1}}\omega_{k|k-1}^{i}N\left(\mathbf{x};\mathbf{m}_{k|k-1}^{ i},\mathbf{P}_{k|k-1}^{i}\right),\end{array}  \tag{5}\]  
where \(D_{S,k|k-1}(\mathbf{x})\) is the probability hypothesis density of survival targets, \(D_{\beta,k|k-1}(\mathbf{x})\) is the spawned target probability hypothesis density and \(\gamma_{k}(\mathbf{x})\) is the newborn target probability hypothesis density [19]. The predicted intensity of the survival target, \(D_{S,k|k-1}(\mathbf{x})\), can be derived from the target survival rate as well as the prediction equation:  
\[D_{S,k|k-1}(\mathbf{x})=p_{S,k}\sum\limits_{i=1}^{I_{k-1}}\omega_{k-1}^{i}N\Big{(} \mathbf{x};\mathbf{m}_{S,k|k-1}^{i},\mathbf{P}_{S,k|k-1}^{i}\Big{)}, \tag{6}\]  
\[\mathbf{m}_{S,k|k-1}^{i}=F_{k-1}\mathbf{m}_{k-1}^{i}, \tag{7}\]  
\[\mathbf{P}_{S,k|k-1}^{i}=F_{k-1}\mathbf{P}_{k-1}^{i}{F_{k-1}}^{\prime}+\mathbf{Q}_{k-1}, \tag{8}\]  
where \(p_{S,k}\) denotes the target survival probability and \(\omega_{k|k-1}^{i}=p_{S,k}\omega_{k-1}^{i}\).The spawned targets at time \(k\) are generated from surviving targets, so that \(D_{\beta,k|k-1}(\mathbf{x})\) can be expressed as:  
\[D_{\beta,k|k-1}(\mathbf{x})=\sum\limits_{i=1}^{I_{k-1}}\sum\limits_{j=1}^{I_{ \beta,k}}\omega_{k-1}^{i}\omega_{\beta,k}^{j}N\Big{(}\mathbf{x}\;;\mathbf{m}_{\beta,k| k-1}^{i,j},\mathbf{P}_{\beta,k|k-1}^{i,j}\Big{)}, \tag{9}\]  
\[\mathbf{m}_{\beta,k|k-1}^{i,j}=\mathbf{P}_{\beta,k-1}^{i}\mathbf{m}_{k-1}^{i}+\mathbf{d}_{ \beta,k-1}^{i}, \tag{10}\]  
\[\mathbf{P}_{\beta,k|k-1}^{i,j}=\mathbf{P}_{\beta,k-1}^{i}\mathbf{P}_{\beta,k-1}^{i}{F_{ \beta,k-1}}^{\prime}+\mathbf{Q}_{\beta,k-1}^{j}, \tag{11}\]  
where \(\omega_{\beta,k}^{j}\), \(\mathbf{P}_{\beta,k-1}^{i}\), \(\mathbf{d}_{\beta,k-1}^{j}\), \(\mathbf{Q}_{\beta,k-1}^{j}\), \(j=1,\cdots,J_{\beta,k}\) are all spawned target parameters.