# A Sector-Matching Probability Hypothesis Density Filter for Radar Multiple Target Tracking
## 3 Sector-Matching PHD Filter
### Sector-Matching and Sector Division Issues  
In the process of radar dynamic scanning, the direction that the beam is scanned in and the sampling time correspond one-to-one [13].  
For radars in tracking-while-scanning (TWS) mode, the antenna scans at a constant angular velocity, so we can divide the surveillance area into a number of sectors equally. As shown in Figure 3, where \(z_{k}^{1},z_{k}^{2},\cdots z_{k}^{7}\) are the seven measurements obtained in the \(k\)th scan, the surveillance area is divided into \(n\) sectors (\(n=18\) in Figure 2) and the following assumptions are given.  
Figure 3: Radar measurements generated in the \(k\)th scanning cycle and sector division according to the sampling time.  
Figure 2: Diagram of the relationship between the actual position and the measured position of the target during radar two-dimensional scanning.  
**Assumption 1**.: _The true state of each target and the generated measurements are independent of each other [13]._  
**Assumption 2**.: _Each sector is scanned once at most in each scan._  
**Assumption 3**.: _Any target can only appear in one sector during a scanning cycle._  
Assumption 1 is very common in MTT applications. Assumptions 2 and 3 hold in the case of the TWS operating mode and where the targets do not appear to migrate through resolution cells (MTRC). In tracking-and-searching (TAS) mode, Assumptions 2 and 3 do not always hold, which will be developed in later studies.  
To simplify the issue, we consider the case of a radar counterclockwise electronic scan at a uniform angular velocity. The radar scanning period is \(T\) and the left and right boundaries of the detected airspace are \(\theta_{max}\) and \(\theta_{min}\). The detected airspace is divided equally into \(n\) sectors \(S_{m},\ m=1,\cdots,n\), then the time interval between neighboring sectors is \(d_{t}=n/T\), the width of the sector is \(\Delta\theta=(\theta_{max}-\theta_{min})/n\) and the boundary of sector \(S_{m}\) is denoted as \([\theta_{min}+(m-1)\Delta\theta,\ \theta_{min}+m\Delta\theta],\ m=1,\cdots,n\). The angle \(\theta_{m}\) of sector \(S_{m}\) is approximated by the average of the left and right boundaries, i.e., \(\theta_{m}=\theta_{min}+(m-1/2)\Delta\theta\).  
Radar measurements, \(\mathbf{z}_{k}^{i}=\left(r_{k}^{i},\mathbf{b}_{k}^{i}\right),i=1,\cdots,|\mathbf{Z}_{k}|\), in the \(k\)th scanning cycle, contain the range \(r_{k}^{i}\) and bearing \(\mathbf{b}_{k}^{i}\). If \(\left|\mathbf{b}_{k}^{i}-\theta_{m}\right|\) takes the minimum value when \(m=m_{i}^{\prime}\), \(\mathrm{i}=1,\cdots,|\mathbf{Z}_{k}|\), measurement \(\mathbf{z}_{k}^{i}\) falls into the \(m_{i}\)'th sector, and the measurement time is:  
\[\mathbf{t}_{k}^{i}=(k-1)T+m_{i}^{\prime}\times d_{t}-\frac{d_{t}}{2}, \tag{29}\]  
Thus, the multi-target measurement model in Equation (2) is rewritten as:  
\[\mathbf{Z}_{k}=\Big{\{}\Big{(}\mathbf{z}_{k}^{1},\mathbf{t}_{k}^{1}\Big{)},\cdots,\Big{(} \mathbf{z}_{k}^{J_{k}},\mathbf{t}_{k}^{J_{k}}\Big{)}\Big{\}}, \tag{30}\]  
where \(\mathbf{t}_{k}^{1},\cdots,\mathbf{t}_{k}^{J_{k}}\) is the set of times when different targets are detected in the \(k\)th scanning cycle, correspondingly, then we can rewrite Equation (1) as:  
\[X_{k}=\Big{\{}\Big{(}\mathbf{x}_{k}^{1},\mathbf{t}_{k}^{1}\Big{)},\cdots,\Big{(}\mathbf{x }_{k}^{J_{k}},\mathbf{t}_{k}^{J_{k}}\Big{)},\Big{(}\mathbf{x}_{k}^{J_{k}+1},\mathbf{t}_{k }\Big{)}\Big{\}}, \tag{31}\]  
where \(t_{k}\) is the time at the end of \(k\)th scanning cycle and is used to represent the measurement time of the undetected target.