# A Sector-Matching Probability Hypothesis Density Filter for Radar Multiple Target Tracking
## 3 Sector-Matching PHD Filter
### Update  
Considering the sector-matching issues and combining the PHD filter with actual radar, the posterior density, \(D_{k|k}\), at step \(k\) can be described as:  
\[D_{k|k}(\mathbf{x})\mathbf{\&}=\sum_{i=1}^{|Z_{k,cf}|}D_{t_{k}^{i}|t_{k}^{i}}(\mathbf{x}), \tag{40}\]  
\[D_{t_{k}^{i}|t_{k}^{i}}\mathbf{x}\mathbf{\&}=(1-P_{D,k})D_{t_{k}^{i}|k-1}\mathbf{x}+\sum_{ z_{k,cf}^{(0)}\in\mathbf{Z}_{k,cf}}D_{D,t_{k}^{i}}\Big{(}\mathbf{x};\Big{\{}\mathbf{z}_{k,cf}^{i},t _{k,cf}^{i}\Big{\}}\Big{)} \tag{41}\]  
\[D_{D,t_{k}^{i}}\Big{(}\mathbf{x};\Big{\{}\mathbf{z}_{k,cf}^{i},t_{k,cf}^{i}\Big{\}} \Big{)}=\sum_{i=1}^{I_{kk-1}}\omega_{t_{k}^{i}|t_{k}^{i}}^{j}\Big{(}\Big{\{} \mathbf{z}_{k,cf}^{i},t_{k,cf}^{i}\Big{\}}\Big{)}\times\mathcal{N}\bigg{(}\mathbf{x}; \mathbf{m}_{t_{k}^{i}|t_{k}^{i}}^{j}\Big{(}\Big{\{}\mathbf{z}_{k,cf}^{i},t_{k,cf}^{i} \Big{\}}\Big{)},\mathbf{P}_{t_{k}^{i}|t_{k}^{i}}^{j}\bigg{)}, \tag{42}\]  
\[\omega_{t_{k}^{i}|t_{k}^{i}}^{j}\Big{(}\mathbf{z}_{k}^{i}\Big{)}=\frac{P_{D,t_{k}^{i} }\omega_{t_{k}^{i}|k-1}^{j}N\Big{(}\mathbf{z}_{k}^{i};\mathbf{H}_{t_{k}^{i}}\mathbf{m}_{t_{ k}^{i}|k-1}^{j},\mathbf{S}_{t_{k}^{i}}^{j}\Big{)}}{\lambda_{t_{k}^{i}}(\mathbf{z})+P_{D,t_{k}^{i} }\sum_{j=1}^{I_{kk-1}}\omega_{t_{k}^{i}|k-1}^{j}N\Big{(}\mathbf{z}_{k}^{i};\mathbf{H}_{ t_{k}^{i}}\mathbf{m}_{t_{k}^{i}|k-1}^{j},\mathbf{S}_{t_{k}^{i}}^{j}\Big{)}}, \tag{43}\]  
\[\mathbf{m}_{t_{k}^{i}|t_{k}^{i}}^{j}\Big{(}\mathbf{z}_{k}^{i}\Big{)}=\mathbf{m}_{t_{k}^{i}|k- 1}^{j}(\mathbf{z})+\mathbf{K}_{t_{k}^{i}}^{i}\Big{(}\mathbf{z}_{k}^{i}-\mathbf{H}_{t_{k}^{i}}\mathbf{m}_{t _{k}^{i}|k-1}^{j}\Big{)}, \tag{44}\]\[\mathbf{P}_{\hat{t}_{k}^{i}|t_{k}^{i}}^{j}=\left[\mathbf{I}-\mathbf{K}_{\hat{t}_{k}}^{j}\mathbf{H} _{\hat{t}_{k}}\right]\mathbf{P}_{\hat{t}_{k}|k-1}^{j}, \tag{45}\]  
\[\mathbf{S}_{\hat{t}_{k}^{i}}^{j}=\mathbf{R}_{\hat{t}_{k}^{i}}+\mathbf{H}_{\hat{t}_{k}^{i}}\bm {P}_{\hat{t}|k-1}^{j}H_{\hat{t}_{k}^{i}}^{j}, \tag{46}\]  
\[\mathbf{K}_{\hat{t}_{k}^{i}}^{j}=\mathbf{P}_{\hat{t}_{k}^{i}|k-1}^{j}\mathbf{H}_{\hat{t}_{k }^{i}}^{j}\left(\mathbf{H}_{\hat{t}_{k}^{i}}\mathbf{P}_{\hat{t}_{k}^{i}|k-1}^{j}\mathbf{H}_ {\hat{t}_{k}^{i}}^{j}+\mathbf{R}_{\hat{t}_{k}^{i}}\right)^{-1}, \tag{47}\]  
The parameter set \(\left\{\mathbf{m}_{k|k}^{j}\left(\hat{x}_{k,f}^{i}\right),\omega_{k|k}^{j}\left( \hat{x}_{k,f}^{i}\right),t_{k,f}^{i},\mathbf{P}_{\hat{k}|k}^{j}\right\}\) of the updated Gaussian components is obtained, and \(\mathbf{P}_{D,k}\), \(\mathbf{P}_{D,\hat{t}_{k}^{i}}\) and \(\mathbf{\lambda}_{\hat{t}_{k}^{i}}^{j}\) are written, respectively, as:  
\[\mathbf{P}_{D,k}\mathbf{\xi}=\int_{\mathbf{U}_{T}}^{\infty}\mathbf{p}_{\text{d},k}(r)\text{d}r =\int_{\mathbf{U}_{T}}^{\infty}r\exp\left(-\frac{r^{2}+\mathbf{A}_{k}^{2}}{2}\right) \mathbf{I}_{0}(rA_{k})\text{d}r, \tag{48}\]  
\[\mathbf{P}_{D,\hat{t}_{k}^{i}}\mathbf{\xi}=\int_{\mathbf{U}_{T}}^{\infty}\mathbf{p}_{\text{d}, \hat{t}_{k}^{i}}(r)\text{d}r=\int_{\mathbf{U}_{T}}^{\infty}r\exp\left(-\frac{r^{2} +\mathbf{A}_{\hat{t}_{k}^{i}}^{2}}{2}\right)\mathbf{I}_{0}\left(rA_{\hat{t}_{k}^{i}} \right)\text{d}r, \tag{49}\]  
\[\mathbf{\lambda}_{\hat{t}_{k}^{i}}=\frac{1}{t_{\text{fa}}}=\frac{P_{\text{fa}}N_{ TD}}{t_{\text{int}}}, \tag{50}\]  
The undetected targets corresponding to \((1-\mathbf{p}_{D,k})\mathbf{D}_{k|k-1}(\mathbf{x})\) are approximated by the parameter set \(\left\{\mathbf{m}_{k|k-1}^{i},(1-\mathbf{P}_{D,k})\alpha_{k|k-1}^{i},t_{k},\mathbf{P}_{k|k -1}^{i}\right\}_{i=1}^{l_{k|k-1}}\).