# SMC-CPHD Filter with Adaptive Survival Probability for Multiple Frequency Tracking
## 2 Processing Steps of the SMC-CPHD Filter
The updated intensity function, \(D_{k|k}(\mathbf{y},0)\) and \(D_{k|k}(\mathbf{y},1)\), can be respectively calculated by:  
\[D_{k|k}(\mathbf{y},0)\approx\sum\limits_{n=1}^{N_{k-1}}w_{k|k,p}^{(n)}\delta_ {\mathbf{y}_{k|k-1},p}^{(n)}(\mathbf{y}) \tag{16}\]  
\[w_{k|k,p}^{(n)}=\Big{(}1-p_{D}\Big{(}\mathbf{y}_{k|k-1,p}^{(n)}\Big{)}\Big{)} w_{k|k-1,p}^{(n)}+\sum\limits_{\mathbf{z}\in\mathbf{Z}_{k}}\frac{p_{D}\Big{(} \mathbf{y}_{k|k-1}^{(n)}\Big{)}g_{k}\Big{(}\mathbf{z}\Big{|}\mathbf{y}_{k|k-1,p}^{(n)}\Big{)}w_{k|k-1,p}^{(n)}}{\theta(\mathbf{z})} \tag{17}\]  
\[D_{k|k}(\mathbf{y},1)\approx\sum\limits_{n=1}^{N_{k}^{k}}w_{k|k,p}^{(n)}\delta _{\mathbf{y}_{k|k-1}^{(n)},p}^{(n)}(\mathbf{y}) \tag{18}\]  
\[w_{k|k,p}^{(n)}=\sum\limits_{\mathbf{z}\in\mathbf{Z}_{k}}\frac{w_{k|k-1,p}^{(n )}}{\theta(\mathbf{z})} \tag{19}\]  
\[\theta(\mathbf{z})=\kappa_{k}(\mathbf{z})+\sum\limits_{n=1}^{N_{k}^{p}}w_{k|k -1,p}^{(n)}+\sum\limits_{n=1}^{N_{k-1}}p_{D}\Big{(}\mathbf{y}_{k|k-1,p}^{(n)} \Big{)}g_{k}\Big{(}\mathbf{z}\Big{|}\mathbf{y}_{k|k-1,p}^{(n)}\Big{)}w_{k|k-1,p}^{(n)} \tag{20}\]  
For resampling steps, the estimated number of persistent targets, \(\vartheta_{k}^{p}=\sum_{n=1}^{N_{k-1}^{p}}w_{k|k,p}^{(n)}\), and the expected number of newborn targets, \(\tilde{\upsilon}_{k}^{p}=\sum_{n=1}^{N_{k}^{p}}w_{k|k,p}^{(n)}\), are respectively calculated. The number of particles, \(N_{k}^{p}\), is selected by \(\Big{[}M_{p}\vartheta_{k}^{p}\Big{]}\), where \([\cdot]\) denotes the nearest integer and \(M_{p}\) is the number of particles per persistent targets. Two intensity functions of (16) and (18) are respectively resampled \(N_{k}^{p}\) and \(N_{k}^{b}\) times to eliminate the particles with small weights and multiply the particles with large weights. After resampling steps, the intensity function, \(D_{k|k}(\mathbf{y},0)\) and \(D_{k|k}(\mathbf{y},1)\), can be respectively approximated by:  
\[D_{k|k}(\mathbf{y},0)\approx\sum\limits_{n=1}^{N_{k}^{p}}w_{k,p}^{(n)}\delta_{ \mathbf{y}_{k,p}^{(n)}}(\mathbf{y}) \tag{21}\]  
\[D_{k|k}(\mathbf{y},1)\approx\sum\limits_{n=1}^{N_{k}^{b}}w_{k,b}^{(n)}\delta_{ \mathbf{y}_{k,p}^{(n)}}(\mathbf{y}) \tag{22}\]  
where \(w_{k,p}^{(n)}=\vartheta_{k}^{p}/N_{k}^{p}\) and \(w_{k,b}^{(n)}=\vartheta_{k}^{b}/N_{k}^{b}\). The whole process of the SMC-CPHD filter is presented in detail in [23], and the SMC-CPHD filter's pseudocode at time \(k\) is as shown below [23]:```
Inputs: cardinality distribution: \(\rho_{k-1|k-1}(n)\), intensity function (case \(\beta=0\)) : \(D_{k-1|k-1}(\mathbf{y},0)\): \(\left\{\left(w_{k-1,p^{\prime}}^{(n)}\mathbf{y}_{k-1,p}^{(n)}\right)\right\}_{n =1}^{N_{k-1}^{\prime}}\), Equation (1) intensity function (case \(\beta=1\)) : \(D_{k-1|k-1}(\mathbf{y},1)\): \(\left\{\left(w_{k-1,p}^{(n)},\mathbf{y}_{k-1,p}^{(n)}\right)\right\}_{n =1}^{N_{k-1}^{\prime}}\), Equation (2) measurement set: \(\mathbf{Z}_{k}=\left\{\mathbf{z}_{k,1},\ldots,\mathbf{z}_{k,m_{k}}\right\}\) (\(m_{k}=|\mathbf{Z}_{k}|\)) predicted cardinality distribution: \(\rho_{k|k-1}(n)\), Equation (3) input particle sets: \(\left\{\left(w_{k-1,p}^{(n)},\mathbf{y}_{k-1}^{(n)}\right)\right\}_{n =1}^{N_{k-1}^{\prime}}\), Equation (5) predicted intensity function (case \(\beta=0\)) : \(D_{k|k-1}(\mathbf{y},0)\), Equation (6) for\(n=1,\ldots,N_{k-1}\)do \(\mathbf{y}_{k|k-1,p}^{(n)}\) \(\sim\)\(\pi_{k-1}\) (\(\cdot|\mathbf{y}_{k-1}^{(n)}\)), Equation (7) \(w_{k|k-1,p}^{(n)}\) \(=\)\(p_{S}\left(\mathbf{y}_{k-1}^{(n)}\right)w_{k-1}^{(n)}\), Equation (8) endfor
```

**Algorithm 1**SMC-CPHD filter

## 3 SMC-CPHD Filter with Adaptive Survival Probability

The processing steps of the SMC-CPHD filter [23] were reviewed in the previous section. In this section, how to adaptively apply the survival probability is explained, and the result that changing the survival probability does not affect the stability of the filter is demonstrated through convergence analysis.