## 3 Hybrid Differential Evolution Particle Filter Track-Before-Detect Algorithm
### Improved PF-TBD-Based on the HDE Algorithm
#### 3.2.2 Proposed Method  
Firstly, the particle updating process is optimized by the HDE algorithm. The importance-sampling particles are regarded as the initial population of the HDE algorithm, and the corresponding weights are treated as the fitness functions of the target vectors, respectively. The HDE resampling scheme recombines the particles by using an iterative process of mutation, crossover and the simulated annealing operator. Then, a new set of diverse particles is propagated. Secondly, the systematic resampling is chosen to be executed.  
The proposed algorithm exploits a hybrid estimation technique, where the state is expanded to include the existence variable \(\mathbf{s}_{k}\!=\!\!\left[\mathbf{x}_{k}^{T}\,E_{k}\right]^{T}\). Given a set of particles describing the joint posterior density at time \(k\!-\!1\), \(\left\{\mathbf{s}_{k-1}^{n},\mathbf{w}_{k-1}^{n}\right\}_{n=1}^{M}\) with uniform weights, \(N\) is the particle number. Then, the proposed algorithm contains the following steps:  
Step 1: Calculate the target existence variable: \(\left[\!\left[E_{k}^{*}\right]\!\right]_{n=1}^{N}\!=\!\!\left[E_{k-1}^{*}\right] _{n=1}^{N}\!,\!\Pi\right]\).  
Step 2: Create a set of particles. These are two possible situations.  
1. Set the 'birth particle' state as a sample from the proposal density \(\mathbf{x}_{k}^{n}\!\sim\!q_{b}\!\left(\mathbf{x}_{k}\,/\,E_{k}^{n}\!=\!1,E_{k-1 }^{*}\!=\!0,\!z_{k}\right)\). The target's location \(\left(x_{k}^{n},y_{k}^{n}\right)\) is uniform over the sensor field-of-view. Its velocity and intensity are assumed to be uniform as follows \(q_{b}\!\left(\!\left(\mathbf{x}_{k}^{n},\mathbf{y}_{k}^{n}\right)\!/\,z_{k} \right)\!=\!U\!\left[v_{\min},v_{\max}\right]\), \(q_{b}\!\left(\!I_{k}\!/\,z_{k}\right)\!=\!U\!\left[I_{\min},I_{\max}\right]\), where \(v_{\min}\) and \(v_{\max}\)are the minimum and the maximum of the target velocity, respectively. \(I_{\min}\) and \(I_{\max}\)are the minimum and the maximum of the target intensity, respectively.
2. Set the \"survival particle\" state sampled from the proposal \(\mathbf{x}_{k}^{n}\!\sim\!q_{b}\!\left(\mathbf{x}_{k}\,/\,E_{k}^{n}\!=\!1,E_{k -1}^{*}\!=\!1,\!z_{k}\right)\). That is, the \"survival particle\" \(\mathbf{x}_{k}\) is sampled according to the target state transition Equation (1).
3. For each particle, compute the un-normalized weight \(\widehat{\mathbf{w}}_{k}^{n}\) by using the likelihood ratio \(\widehat{\mathbf{w}}_{k}^{n}\!=\!I\!\left(\mathbf{z}_{k}\,/\,x_{k}^{n},E_{k}^{ n}\!=\!1\right)\).
4. Optimize the sample particles by HDE. The sampling particles are regarded as the initial population of the HDE algorithm, and the corresponding weights are regarded as the fitness functions. According to the HDE algorithm, the process is iterated until the optimal population is found or a pre-specified condition is reached, then we can get the optimal set of particles \(\left\{\mathbf{x}_  
Step 5: Resample the particles: \(\left[\left\{x_{k}^{n},1/\,N\right\}_{n=1}^{N}\right]=\textit{systematic}- \mathrm{resample}\left[\left\{x_{k}^{\prime n},w_{k}^{\prime n}\right\}_{n=1}^ {N}\right]\).  
Step 6: Output the state estimation: an estimate of the target state can be made from the set of particles resulting from the time algorithm above.  
The posterior density of target existence at time \(k\) is \(\mathrm{P}_{k}\overset{\Delta}{=}P\left\{E_{k}=1/\,Z_{k}\right\}\), and then, the probability of existence is given by \(\hat{\mathrm{P}}_{k}=\sum\limits_{n=1}^{N}E_{k}^{n}\,/\,N,\,\,\,0\leq\hat{ \mathrm{P}}_{k}\leq 1\). Additionally, the target state can be estimated by \(\hat{\mathrm{x}}_{k}=\sum\limits_{n=1}^{N}(x_{k}^{n}E_{k}^{n})/\sum\limits_{n =1}^{N}E_{k}^{n}\).