## 2 Background
### GM-CPHD Algorithm  
Compared with the PHD filter, CPHD not only propagates the intensity function but also the cardinality distribution, which makes the estimation deviation of the target number small and the accuracy improved [25].The relationship between PHD and CPHD is shown in Figure 3.  
To implement the CPHD filters, under linear target systems, the Gaussian mixture (GM) method has been introduced in CPHD. Based on the assumption A.1-A.3, the main prediction and update process of the GM-CPHD filter algorithm can be described as below.  
Figure 3: Relationship between PHD and CPHD.  
Some general expressions are shown as below.  
\[\begin{array}{l}C_{i}^{l}=\frac{l!}{l!(l-i)!}\qquad P_{i}^{n}=\frac{n!}{(n-i)!} \\ \langle\alpha,\beta\rangle=\int\alpha(x)\beta(x)dx\quad\text{or}\quad\langle \alpha,\beta\rangle=\sum\limits_{l=0}^{\infty}\alpha(l)\beta(l)\\ e_{j}(\mathbf{Z})=\left\{\begin{array}{cc}\sum\limits_{S\subset\mathbf{Z},|S|=j }\left(\prod\limits_{\zeta\in S^{\zeta}}\zeta\right)&j\
eq 0\\ 1&j=0\end{array}\right.\end{array} \tag{13}\]  
where \(C_{i}^{l}\) is binomial coefficient, \(P_{i}^{n}\) is the permutation coefficient, \(\langle\cdot,\cdot\rangle\) is the inner product, and \(e_{i}(\mathbf{Z})\) is the elementary symmetric function of order \(i\) for the finite real set of \(\mathbf{Z}\).  
Prediction: suppose at time \(k-1\), the Gaussian mixture form of posterior intensity \(D_{k-1}(x)\) and posterior cardinality distribution \(\rho_{k-1}(n)\) are given. Then, the prediction are described by Equations (14) and (15), respectively.  
\[D_{k|k-1}(x)=D_{S,k|k-1}(x)+\gamma_{k}(x) \tag{14}\]  
\[\rho_{k-1}=\sum\limits_{j=0}^{n}\rho_{\Gamma,k}(n-j)\sum\limits_{l=j}^{\infty} C_{i}^{l}\rho_{k-1}(l)P_{S,k}^{j}(1-P_{S,k})^{(l-j)} \tag{15}\]  
where \(D_{S,k|k-1}(x)\), \(\gamma_{k}(x)\) are the same with which is shown in Equations (7) and (10), respectively.  
Update:  
\[D_{k|k-1}(x)=\frac{\langle\gamma_{k}^{1}\left[\omega_{k|k-1},\mathbf{Z}_{k} \right]\rho_{k|k-1}\rangle}{\langle\gamma_{k}^{2}(x)\left[\omega_{k|k-1}, \mathbf{Z}_{k}\right]\rho_{k|k-1}\rangle}(1-P_{d,k}(x))\
u_{k|k-1}(x)+\sum \limits_{z\in\mathbf{Z}_{k}}\sum\limits_{j=1}^{k-1}\omega_{k}^{(j)}(z)\text{ N}\Big{(}x:\mathbf{m}_{k}^{(j)}(z),\mathbf{P}_{k}^{(j)}\Big{)} \tag{16}\]  
\[\rho_{k}(n)=\frac{\gamma_{k}^{\rho}\Big{[}\omega_{k|k-1},\mathbf{Z}_{k} \Big{]}\rho_{k|k-1}(n)}{\Big{\langle}\gamma_{k}^{\rho}(x)\Big{[}\omega_{k|k-1},\mathbf{Z}_{k}\Big{]},\rho_{k|k-1}\Big{\rangle}} \tag{17}\]  
where the parameters are shown as below in Equation (18).  
\[\begin{array}{l}\gamma_{k}^{u}[\omega,\rho]=\sum\limits_{j=0}^{\min(| \mathbf{Z}_{k}|^{u})}(|\mathbf{Z}_{k}|-j)!\rho_{\kappa,k}(|\mathbf{Z}_{k}|-j) P_{j+u}^{n}\frac{\langle 1-P_{d,k}\omega\rangle^{u-(j+u)}}{\langle 1,\omega\rangle^{u}}e_{j}( \wedge_{k}(\omega,\mathbf{Z}_{k}))\\ \omega_{k}^{(j)}(z)=P_{d,k}\omega_{k|k-1}^{(j)}q_{k}^{(j)}(z)\frac{\langle \gamma_{k}^{u}[\omega_{k|k-1},\mathbf{Z}_{k}\{z\}]\rho_{k|k-1}\rangle}{ \langle\gamma_{k}^{\rho}[\omega_{k|k-1},\mathbf{Z}_{k}]\rho_{k|k-1}\rangle} \frac{\langle 1,\kappa_{k}\rangle}{\kappa_{k}}\\ \omega_{k|k-1}=\left[\omega_{k|k-1}^{(1)},\cdots,\omega_{k|k-1}^{(k_{|k|-1} )}\right]^{\text{T}}\\ q_{k}^{(j)}(z)=\text{N}\Big{(}z:\hat{e}_{k|k-1}^{(j)},\mathbf{S}_{k|k-1}^{(j)} \Big{)}\\ q_{k}(z)=\left[q_{k}^{(1)}(z),\cdots,q_{k}^{(I_{k|k-1})}(z)\right]^{\text{T}} \end{array} \tag{18}\]  
For a more detailed mathematical derivation of the GM-CPHD filter, look to [17]. The computational complexity of the CPHD filtering algorithm is \(O\big{(}Nm_{k}^{3}\big{)}\), where \(N\) is the target number of tracking and \(m_{k}\) is the cardinalized of RFS measurement at each moment. Compared with the GM-PHD filter, using the GM-CPHD filter, the estimation deviation of the target number is small and the tracking performance is improved.