## 4 Analytical Implementation of VB-TPHD  
### The Implementation of Gaussian Mixture VB-TPHD Filter  
In the previous section, we obtained the theoretical derivation that variational Bayesian can be well applied to the TPHD filter. In this section, we hope that it can be applied to the Gaussian model to deduce the sequence of measurement noise covariance. It can be seen that the measuring likelihood \(l_{k}\Big{(}z|x^{i},R^{i}\Big{)}\) in Equation (20) only involves measurement noise covariance at the present; therefore, we let VB-TPHD filter perform adaptive estimation of the measurement noise covariance at the current time. Consider that if the state and measurement noise covariance at each moment in the extended state space are included in the calculation, it will add a lot of calculation burden to filter. Therefore, in order to simplify the filtering mechanism of VB-TPHD and improve the filtering efficiency, we only handle the measurement noise covariance at the current time in the tracking process. Thus, the long covariance sequence of measurement noise is not needed to be considered in every single step. Therefore, VB-TPHD derived in the previous section can be further simplified. A posterior PHD of the trajectory at time \(k\) can be expressed as \(D_{\pi_{k}}\Big{(}t,x^{1:i-1},R^{i}\Big{)}\). Equation (19) in the prediction step and Equation (21) in the update step can be expressed as follows, respectively:  
\[ \begin{split}& D_{\xi_{k}}\Big{(}t,x^{1:i},R^{i}\Big{)}=\ p_{s,k}\big{(}x^{i-1}\big{)}f\Big{(}x^{i},R^{i}|x^{i-1},R^{i-1 }\Big{)}D_{\pi_{k-1}}\Big{(}t,x^{1:i-1},R^{i-1}\Big{)}\delta_{N_{k}}[t]\\ &\quad=\ p_{s,k}\big{(}x^{i-1}\big{)}\ f\ \Big{(}x^{i}|x^{i-1}\Big{)}g\Big{(}R^{i}|R^{i-1}\Big{)}D_{\pi_{k-1}}\Big{(}t,x ^{1:i-1},R^{i-1}\Big{)}\delta_{N_{k}}[t],\end{split}  \tag{26}\]  
\[D_{\omega_{k}}^{\tau}\Big{(}x^{i},R^{i}\Big{)}=\sum_{t=1}^{k}\int D_{\omega_{k }}\Big{(}t,x^{1:k-t+1},R^{k-t+1}\Big{)}dx^{1:k-t} \tag{27}\]  
In this section, we deduce the concrete implementation of the Gaussian mixture VB-TPHD filter in the scenario of the linear Gaussian model. According to the parameter definition of multi-trajectory PHD, we make the following assumptions:  
**Assumption 7**. : _The given target obeys the Gaussian linear dynamic model, and the measurement model is also a Gaussian linear model:_  
\[f\Big{(}x^{i}|x^{i-1}\Big{)} =\mathrm{N}\Big{(}x^{i};\ \mathrm{F}x^{i-1},\ Q\Big{)}, \tag{28}\] \[h\Big{(}z|x^{i}\Big{)} =\mathrm{N}\Big{(}z;\ \mathrm{H}x^{i},\ R\Big{)}, \tag{29}\]  
_where \(F\ \in\ \mathbb{R}^{n_{x}\times n_{x}}\) represents the state transition function of the single-moment target, \(H\in\mathbb{R}^{n_{x}\times n_{x}}\) represents the measurement transition function of the single-moment target, and \(Q\in\mathbb{R}^{n_{x}\times n_{x}}\) rep resents the known covariance of process noise. \(R\in\mathbb{R}^{n_{z}\times n_{z}}\) represents the unknown covariance of measurement noise._  
Based on Equations (28) and (29), the two densities on the right side of Equation (23) can be set as specified distributions. Let \(D_{X_{k}}\big{(}x^{1:i}\big{)}\) obey the Gaussian distribution. According to inverse gamma distribution being the conjugate prior distribution of the covariance matrix under Gaussian distribution, \(D_{R_{k}}\big{(}R^{i}\big{)}\) is modeled as a random matrix which obeys the inverse Gamma distribution and generally the optimal solution of both is obtained by using the variational Bayesian approximation. Inverse gamma distribution turns out to be the natural approximating distribution to achieve the goal with the simple form.