# A Sector-Matching Probability Hypothesis Density Filter for Radar Multiple Target Tracking
## 2 Background
### Radar Detection  
We generally use the method of increasing the beam dwell time (generally considered to be the accumulation time) to improve the signal-to-noise ratio and target detection probability, as the pulse-echo signal power, \(A_{i}^{2}\), received by the radar from an individual target is very low for dim, small targets.  
Consider a pulsed radar with pulse width \(\tau\), pulse repetition frequency (PRF) \(f_{r}\) and peak transmit power \(P_{t}\). \(N\) pulses are accumulated in one frame, then the radar equation can be written as:  
\[SNR=\frac{P_{t}G^{2}\lambda^{2}\sigma T_{i}f_{r}\tau}{(4\pi)^{3}R^{4}kT_{0}FL}, \tag{20}\]  
where \(G\) is the radar antenna gain, \(\lambda\) is the radar operating wavelength, \(\sigma\) is the radar reflection cross section (RCS) of the target, \(R\) is the distance between the target and radar, \(k=1.38\times 10^{-23}\) J/K is the Boltzmann constant, \(T_{0}=290\) K, \(T_{i}=N\cdot\tau\) is the radar beam dwell time, \(F\) is the receiver noise factor and \(L\) is the loss of radar. As the accumulated echo signal power  
\[A^{2}=(N\cdot A_{i})^{2}, \tag{21}\]  
it can be noted that the SNR is directly proportional to beam dwell time, \(T_{i}\), and the amplitude, \(A\), of the accumulated echo signal is also in direct proportion to \(T_{i}\).  
The radar false alarm probability is defined as the probability that the sampling \(r\) of radar echo signal \(r(t)\) exceeds the threshold value \(U_{T}\) when only noise is present in \(r(t)\). The probability density function of the output voltage of Gaussian white noise after passing through the envelope detector obeys the Rayleigh distribution as follows:  
\[p_{\rm fa}(r)=r\exp\left(-\frac{r^{2}}{2}\right),\ r\geq 0, \tag{22}\]  
by integrating the above equation, the false alarm probability, \(P_{\rm fa}\), can be written as:  
\[P_{\rm fa}=\int_{U_{T}}^{\infty}p_{\rm fa}(r)\mathrm{d}r=\exp\left(-\frac{U_{T }^{2}}{2}\right), \tag{23}\]  
then, with a given \(P_{\rm fa}\), the threshold voltage can be found as:  
\[U_{T}=\sqrt{2\ln\frac{1}{P_{\rm fa}}} \tag{24}\]  
The detection probability, \(P_{D}\), is the probability that a sampling value \(r\) of \(r(t)\) exceeds the threshold voltage when the radar echo signal, \(r(t)\), contains both noise and signal. After the Gaussian noise and radar echo signal with amplitude \(A\) pass through the envelope detector, the probability density function of the envelope output by the detector is:  
\[p_{\rm d}(r)=r\exp\left(-\frac{r^{2}+A^{2}}{2}\right)I_{0}(rA),r\geq 0, \tag{25}\]  
where \(I_{0}(\cdot)\) is the first class zero-order modified Bessel function.  
By integrating Equation \((\ref{eq:25})\), the detection probability, \(P_{D}\), can be obtained as follows:  
\[P_{D}=\int_{U_{T}}^{\infty}p_{\rm d}(r)\mathrm{d}r=\int_{U_{T}}^{\infty}r\exp \left(-\frac{r^{2}+A^{2}}{2}\right)I_{0}(rA)\mathrm{d}r, \tag{26}\]  
The relationship between detection probability and false alarm probability is shown in Figure 1. The area in red represents the false alarm probability, \(P_{\rm fa}\), the area in purple represents the detection probability, \(P_{D}\), and the horizontal coordinate of the intersection ofthe noise envelope probability curve and the envelope curve of the noise plus signal is the voltage threshold, \(U_{T}\).  
For a single threshold detector, the average time between false alarms for the system is calculated as follows:  
\[t_{\text{fa}}=\frac{T_{i}}{P_{\text{fa}}N_{TD}}, \tag{27}\]  
where \(N_{TD}\) is the number of threshold detectors. The false alarm rate is the frequency of false alarm occurrence per unit time, which is the reciprocal of the false alarm time, that is:  
\[\lambda=\frac{1}{t_{\text{fa}}}=\frac{P_{\text{fa}}N_{TD}}{T_{i}}, \tag{28}\]