## 2 Background
### The TPHD Filter  
Imagine a multi-trajectory scenario. Given the posterior PHD at time \(k-1\) in the scenario, some of the trajectories will disappear, the surviving trajectories will join up with new target points, and some independent target points will be generated at time \(k\). The purpose of the TPHD filter is to identify the true multi-trajectory state from numerous measurements accurately and in a timely manner. The TPHD filter is the output of optimal Poisson approximation by minimizing KLD criterion through Bayesian filtering framework under the condition that the multi-trajectory density is assumed to be Poisson distribution. Before the standard TPHD filter prediction step, the following assumptions [23] are made:  
**Assumption 1**. : _Each trajectory evolves independently, so the trajectory set at the next time is the combination of surviving trajectories and the point targets of the new birth. The newborn target density \(\gamma(\cdot)\) is subjected to the Poisson distribution._  
**Assumption 2**. : _However, each trajectory continues to survive with the probability of \(p_{s,k}(\cdot)\) and then obtains a point target with the transition density of \(f(\cdot)\)._  
**Assumption 3**. : _The multi-trajectory density \(\pi_{k-1}(\cdot)\) obeys the Poisson distribution._  
Based on the above assumptions, the prediction step of single trajectory PHD at time \(k\) can be expressed as follows:  
\[D_{\omega_{k}}(X)=\ D_{\xi_{k}}(X)+D_{\gamma_{k}}(X), \tag{10}\]  
where  
\[D_{\gamma_{k}}\Big{(}t,x^{1:i}\Big{)}=D_{\gamma}(t,x)\delta_{k}[t], \tag{11}\]  
\[D_{\xi_{k}}\Big{(}t,x^{1:i}\Big{)}=p_{s,k}\Big{(}x^{i-1}\Big{)}f\Big{(}x^{i}| x^{i-1}\Big{)}D_{\gamma_{k-1}}\Big{(}t,x^{1:i-1}\Big{)}\delta_{\mathbb{N}_{k}}[t], \tag{12}\]  
where \(\delta[\cdot]\) represents the discrete generalized Kronecker delta variable, \(\mathbb{N}_{k}=\ \{1,\ \ldots,\ k\}\) represents the set of the time at which the trajectory is located. State transition density \(f(\cdot)\) and survival probability \(p_{s,k}(\cdot)\) work together in the \(k-1\) time trajectory of posterior PHD \(D_{\gamma_{k-1}}(\cdot)\) to form a surviving trajectory prior PHD \(D_{\xi_{k}}(\cdot)\). According to [24], when \(t+i-1\
eq k\) at the end time of trajectory, the trajectory disappears, which is not considered; only the alive trajectory is considered in Equation (12). Before the standard TPHD filter update step, the following assumptions [23] are made:  
**Assumption 4**. : _The trajectory formed by independent target generates the corresponding observation and the target can be detected with the probability of \(p_{D,k}(\cdot)\) or omitted with the probability of \(1-p_{D,k}(\cdot)\)._  
**Assumption 5**. : _The prior trajectory density \(\omega(\cdot)\) after the prediction step and the clutter of the scenario obey Poisson distribution._  
**Assumption 6**. : _The measurements come from generated targets and clutter independently._  
Based on the above assumptions, the update step of single trajectory PHD at time \(k\) can be expressed as follows:  
\[D_{\pi_{k}}(X)=\ D_{\omega_{k}}(X)L_{z_{k}}(X)=\ D_{\omega_{k}}(X)\times\] \[\left(1-p_{D,k}\big{(}x^{i}\big{)}+p_{D,k}\big{(}x^{i}\big{)} \times\sum\limits_{z\in Z_{k}}\frac{l_{k}\big{(}z^{i}x^{i}\big{)}}{\lambda_{c ^{c}+}\int p_{D,k}\big{(}x^{i}\big{)}l_{k}\big{(}z^{i}\big{)}D_{\omega_{k}}^{ \tau}\big{(}x^{i}\big{)}dx^{i}}\right), \tag{13}\]  
where  
\[D_{\omega_{k}}^{\tau}\big{(}x^{i}\big{)}=\sum\limits_{t=1}^{k}\int D_{\omega_{ k}}\Big{(}t,x^{1:k-t+1}\Big{)}dx^{1:k-t}, \tag{14}\]  
where \(\lambda_{c}\) is the clutter rate of the trajectory space at time \(k\) and \(c\) is the reciprocal of the area of the trajectory space. \(L_{z_{k}}(X)\) indirectly denoted that the TPHD trajectory is associated with the measurement of the pseudolikelihood function and measuring likelihood \(l_{k}(z|x^{i})\) actually represents the similarity degree between the filtering trajectory and measurement. The higher the degree of similarity, the larger measuring likelihood grows.