## 3 Hybrid Differential Evolution Particle Filter Track-Before-Detect Algorithm
### Bayesian Particle Filter Track-Before-Detect  
The main problem of TBD can be formulated in the framework of recursive Bayesian estimation in different expansion forms. The target state is not defined when the target does not exist in the data, where \(E_{k}=0\). Then, target detection and tracking can be implemented based on the posterior density \(p(x_{k},E_{k}\mid Z_{k})\) when the target does exist, where \(E_{k}=1\).  
\[\begin{split} p(x_{k},E_{k}=1\mid Z_{k})=&\frac{p(z _{k}|x_{k},E_{k}=1)p(x_{k},E_{k}=1|\mid Z_{k-1})}{p(z_{k}\mid Z_{k-1})}\\ &\propto l(z_{k}|x_{k},E_{k}=1)p(x_{k},E_{k}=1|\mid Z_{k-1}) \end{split} \tag{12}\]  
The predicted target state can be written in terms of the target and existence at the previous time, giving:\[p(x_{k},E_{k}=1/\,Z_{k-1}) = \left[\left[p(x_{k},E_{k}=1/\,x_{k-1},E_{k-1}=1)\,p(x_{k-1},E_{k-1}=1/ \,Z_{k-1})\right.\right. \tag{13}\] \[\left.+\,p(x_{k},E_{k}=1,E_{k-1}=0/\,Z_{k+1})\right]dx_{k-1}\] \[= \left[\,p(x_{k}\,/\,x_{k-1},E_{k}=1,E_{k-1}=1)\,p(E_{k}=1/\,E_{k-1 }=1)\right.\] \[\left.\cdot\,p(x_{k-1},E_{k-1}=1/\,Z_{k-1})dx_{k-1}\right.\] \[+\,\left.\left.p(x_{k}\,/\,x_{k-1},E_{k}=1,E_{k-1}=1,Z_{k-1})\,p( E_{k}=1/\,E_{k-1}=0)\right.\right.\] \[= \left.\int\,p(x_{k}\,/\,x_{k-1},E_{k}=1,E_{k-1}=1)(1-P_{d})\,p(x_ {k-1},E_{k-1}=1/\,Z_{k-1})dx_{k-1}\right.\] \[+\,p_{b}(x_{k})P_{b}\]  
where \(p_{b}(x_{k})\) is the prior density of a target when it has appeared in the data between time \(k-1\) and \(k\). The algorithm proceeds by constructing two sets of particles. The first set, the birth particles, estimate \(p(x_{k}/\,E_{k}=1,E_{k-1}=0,Z_{k})\), that is the case where the target did not exist in the data at time \(k-1\), but does at time \(k\). The second set, the continuing particles, estimates \(p(x_{k}/\,E_{k}=1,E_{k-1}=1,Z_{k})\), which is the case where the target has continued to exist in the data \(k-1\) to \(k\). Both sets of particles require a proposal distribution. For the continuing particles, the target dynamics are used in Equation (4). For the birth particles, a uniform distribution can be used. Then, the former is approximated using interacting sets of particles, and the calculation of the latter is based on the particle weights in these two sets.  
Additionally, the estimate of the probability of existence can be calculated by particle weights in these two sets. State estimation can be obtained after resampling from these combined sets. There are three main resampling methods: multinomial resampling, residual resampling and systematic resampling [17].  
To a certain extent, the resampling process is introduced to improve the degradation phenomenon of the particles. However, it also brought some problems. Particles that have higher weights will be reproduced many times. Additionally, those that have lower weights will fade away. Then, all particles will collapse to a small area, which will lead to the sample point sets not being sufficient after several iterations. The sample point sets are described the a posteriori probability density function. In the PF-TBD algorithm, if the particles collapse into a small area at time \(k\), most of the particles, after the resampling process, are \"the continuing particles\". This leads to these particles, which are produced by the state transition equation at time \(k+1\), also collapsing into a small area. Moreover, if the area deviates from the true location, the target will be lost.