# SMC-CPHD Filter with Adaptive Survival Probability for Multiple Frequency Tracking
## 2 Processing Steps of the SMC-CPHD Filter
The predicted cardinality distribution, \(\rho_{k|k-1}(n)\), can be expressed by a convolution [10]:  
\[\rho_{k|k-1}(n)=\sum_{j=0}^{n}\rho_{S,k|k-1}(j)\rho_{\Gamma}(n-j) \tag{3}\]  
\[\rho_{S,k|k-1}(j)=\sum_{l=j}^{\infty}\frac{l!}{j!(l-j)!}p_{S}^{j}(1-p_{S})^{l -j}\rho_{k-1|k-1}(l) \tag{4}\]  
For the predicted intensity function, the summation of \(D_{k-1|k-1}(\mathbf{y},0)\) and \(D_{k-1|k-1}(\mathbf{y},1)\) is preferentially conducted and can be presented by the union of two particle sets as follows:  
\[\left\{\left(w_{k-1}^{(n)},\mathbf{y}_{k-1}^{(n)}\right)\right\}_{n=1}^{N_{k- 1}}=\left\{\left(w_{k-1,p^{\prime}}^{(n)}\mathbf{y}_{k-1,p}^{(n)}\right)\right\} _{n=1}^{N_{k-1}^{p}}\cup\left\{\left(w_{k-1,b^{\prime}}^{(n)}\mathbf{y}_{k-1,b }^{(n)}\right)\right\}_{n=1}^{N_{k-1}^{b}} \tag{5}\]  
The predicted intensity function, \(D_{k|k-1}(\mathbf{y},0)\), can be approximately calculated by the particle set:  
\[D_{k|k-1}(\mathbf{y},0)\approx\sum_{n=1}^{N_{k-1}}w_{k|k-1,p}^{(n)}\delta_{ \mathbf{y}_{k|k-1,p}^{(n)}}(\mathbf{y}) \tag{6}\]  
\[\mathbf{y}_{k|k-1,p}^{(n)}\sim q_{k}\Big{(}\cdot\big{|}\mathbf{y}_{k-1}^{(n)},\mathbf{Z}_{k}\Big{)} \tag{7}\]  
\[w_{k|k-1,p}^{(n)}=\frac{p_{S}\Big{(}\mathbf{y}_{k-1}^{(n)}\Big{)}\pi_{k|k-1} \Big{(}\mathbf{y}_{k|k-1,p}^{(n)}\Big{|}\mathbf{y}_{k-1}^{(n)}\Big{)}w_{k-1}^ {(n)}}{q_{k}\Big{(}\mathbf{y}_{k|k-1,p}^{(n)}\big{|}\mathbf{y}_{k-1}^{(n)} \mathbf{Z}_{k}\Big{)}} \tag{8}\]  
For simplicity, the importance density is adopted as \(q_{k}\Big{(}\cdot\big{|}\mathbf{y}_{k-1}^{(n)},\mathbf{Z}_{k}\Big{)}=\pi_{k|k- 1}\Big{(}\cdot\big{|}\mathbf{y}_{k-1}^{(n)}\Big{)}\)[25]. For \(\beta=1\), the predicted intensity function, \(D_{k|k-1}(\mathbf{y},1)\), is approximated as:  
\[D_{k|k-1}(\mathbf{y},1)\approx\sum_{n=1}^{N_{k-1}^{b}}w_{k|k-1,b}^{(n)}\delta_ {\mathbf{y}_{k|k-1,b}^{(n)}}(\mathbf{y}) \tag{9}\]  
\[\mathbf{y}_{k|k-1,b}^{(n)}\sim b_{k}(\cdot\left|\mathbf{z}\right.) \tag{10}\]  
\[D_{k-1|k-1}(\mathbf{y},1)\approx\sum_{n=1}^{N_{k-1}^{b}}w_{k-1,b}^{(n)}\delta_{ \mathbf{y}_{k-1,b}^{(n)}}(\mathbf{y}) \tag{2}\]0  
where \(b_{k}(\cdot\left|\mathbf{z}\right. )\) denotes the newborn target density and depends on \(g_{k}(\mathbf{z}\left|\mathbf{y}\right. )\) and prior knowledge. \(N_{k}^{b}=M_{b}\cdot m_{k}\) defines the total number of newborn target particles and \(M_{b}\) refers to the number of particles per newborn target. The updated cardinality distribution, \(\rho_{k|k}(n)\), is calculated by:  
\[D_{k-1|k-1}(\mathbf{y},1)\approx\sum_{n=1}^{N_{k-1}^{b}}w_{k-1,b}^{(n)}\delta_{ \mathbf{y}_{k-1,b}^{(n)}}(\mathbf{y}) \tag{2}\]1\[ \begin{array}{rl}\Upsilon_{k}^{u}\Big{[}D_{k|k-1},\mathbf{Z}\Big{]}(n)=&\sum\limits_ {j=0}^{\min(|\mathbf{Z}|,n)}(|\mathbf{Z}|-j)!\varrho_{k,k}(|\mathbf{Z}|-j)P_{j+ u}^{n}\\ &\times\frac{\left\langle 1-p_{D}(\cdot),D_{k|k-1}(\cdot,0)\right\rangle^{ \color[rgb]{0,0,0}-(j+u)}}{\left\langle 1,D_{k|k-1}(\cdot,1)+D_{k|k-1}(\cdot,0) \right\rangle^{\color[rgb]{0,0,0}\right\rangle}}\\ &\times e_{j}\Big{(}\Xi_{k}\Big{(}D_{k|k-1},\mathbf{Z}\Big{)}\Big{)}\\ \Xi_{k}\Big{(}D_{k|k-1},\mathbf{Z}\Big{)}=\bigg{\{}\frac{\left\langle 1,\kappa_{k} \right\rangle}{\kappa_{k}(\mathbf{z})}\Big{\langle}D_{k|k-1}(\cdot,1)+p_{D}( \cdot)D_{k|k-1}(\cdot,0),g(\mathbf{z}|\cdot)\Big{\rangle}:\mathbf{z}\in \mathbf{Z}\bigg{\}}\end{array}  \tag{13}\]  
\[e_{j}(\mathbf{Z})=\sum\limits_{\mathbf{W}\subseteq\mathbf{Z},|\mathbf{W}|=j} \left(\prod\limits_{\delta\in\mathbf{W}}\xi\right) \tag{15}\]  
where \(\left\langle g,f\right\rangle=\int f(\mathbf{x})g(\mathbf{x})d\mathbf{x}\), \(\kappa_{k}(\mathbf{z})\) defines the PHD of clutter, \(p_{D,k}(\mathbf{x})\triangleq p_{D}(\mathbf{x})\) refers to the detection probability, and \(g_{k}(\mathbf{z}\,|\,\mathbf{x})\) represents the measurement likelihood at time \(k\). \(e_{j}(\mathbf{Z})\) denotes the elementary symmetric function of order \(j\) for a finite set, \(\mathbf{Z}\)[8].