# SMC-CPHD Filter with Adaptive Survival Probability for Multiple Frequency Tracking
## 2 Processing Steps of the SMC-CPHD Filter
\begin{split}& E\bigg{[}\left\langle D_{k|k-1}^{N},\varphi \right\rangle^{2}\bigg{]}-\left\langle D_{k-1|k-1}^{N},\pi_{k|k-1}\varphi \right\rangle^{2}\\ &=\left(\hat{\frac{T}{N}}\right)\sum\limits_{i=1}^{2}\left(E \bigg{[}\left(\varphi\left(\widetilde{\mathbf{y}}_{k}^{(i)}\right)p_{s}\big{(} \mathbf{y}_{k-1}\big{)}\right)^{2}\bigg{|}\xi_{k-1}\bigg{]}-\left(\pi_{k|k-1} \varphi\right)\left(\mathbf{y}_{k-1}^{(i)}\right)^{2}\right)\end{split}  \tag{29}\]

where \(\hat{T}\) denotes the estimated number of targets. According to (29), the equation's boundary can be expressed as:

\[\left|E\bigg{[}\left\langle D_{k|k-1}^{N},\varphi\right\rangle^{2}\bigg{]}- \left\langle D_{k-1|k-1}^{N},\pi_{k|k-1}\varphi\right\rangle^{2}\right|\leq \frac{\hat{T}^{2}}{N}\|\varphi\|^{2}\Big{(}\|p_{s}\big{(}\mathbf{y}_{k-1} \big{)}\|^{2}+\|\pi_{k|k-1}\|^{2}\Big{)} \tag{30}\]

Using the inequality of Minkowski, (30) can be rearranged as:

\[ \begin{split}& E\bigg{[}\left(\left\langle D_{k|k-1}^{N}, \varphi\right\rangle-\left\langle D_{k|k-1}^{\prime},\varphi\right\rangle \right)^{2}\bigg{]}^{1/2}\\ &\leq E\bigg{[}\left(\left\langle D_{k|k-1}^{N},\varphi\right\rangle -\left\langle D_{k-1|k-1}^{N},\pi_{k|k-1}\varphi\right\rangle\right)^{2}\bigg{]} ^{1/2}\\ &+E\bigg{[}\left(\left\langle D_{k-1|k-1}^{N},\pi_{k|k-1}\varphi \right\rangle-\left\langle D_{k-1|k-1},\pi_{k|k-1}\varphi\right\rangle\right) ^{2}\bigg{]}^{1/2}\end{split}  \tag{31}\]

If there are no newly created targets at time \(k\), the boundary of (31) can be finally expressed as:

\[E\bigg{[}\left(\left\langle D_{k|k-1}^{N},\varphi\right\rangle-\left\langle D _{k|k-1}^{\prime},\varphi\right\rangle\right)^{2}\bigg{]}^{1/2}\leq\frac{1}{ \sqrt{N}}\|\varphi\|\Big{(}\hat{T}\|p_{s}(x_{k-1})\|^{2}+\|\pi_{k|k-1}\|^{2} \Big{)}^{1/2}+\sqrt{\epsilon_{k-1|k-1}} \tag{32}\]

In (32), \(c_{k-1|k-1}\) defines the MSE's boundary at the previous time step, \(k-1\), and can be obtained from the assumption that:

\[E\bigg{[}\left(\left\langle D_{k-1|k-1}^{N},\varphi\right\rangle-\left\langle D _{k-1|k-1},\varphi\right\rangle\right)^{2}\bigg{]}\leq c_{k-1|k-1}\frac{\| \varphi\|^{2}}{N} \tag{33}\]

Finally, if \(\|\pi_{k|k-1}\|\leq 1+T_{k|k-1}\), where \(T_{k|k-1}\) denotes the number of spawned targets at time step \(k\):

\[E\bigg{[}\left(\left\langle D_{k|k-1}^{N},\varphi\right\rangle-\left\langle D _{k|k-1}^{\prime},\varphi\right\rangle\right)^{2}\bigg{]}\leq\frac{\|\varphi \|^{2}}{N}\bigg{(}\left(\hat{T}\|p_{s}\big{(}\mathbf{y}_{k-1}\big{)}\|^{2}+\|1 +T_{k|k-1}\|^{2}\right)^{1/2}+\sqrt{\epsilon_{k-1|k-1}}\bigg{)}^{2} \tag{34}\]

In the case of the proposed algorithm, we confirmed that the MSE's convergence depends on only the number of targets, \(\hat{T}\), including the spawned targets, \(T_{k|k-1}\), the particles' number, \(N\), and the maximum boundary of the survival probability. Thus, if the maximum range of the survival probability is specified (\(p_{s,\text{max}}\)), as in the previous section, the SMC-CPHD filter will converge as \(\frac{\|\varphi\|^{2}}{N}\bigg{(}\left(\hat{T}\|p_{s,\text{max}}\|^{2}+\|1+T_ {k|k-1}\|^{2}\right)^{1/2}+\sqrt{\epsilon_{k-1|k-1}}\bigg{)}^{2}\). ## 4 Cardinality Compensation with ICI

Information fusion is mainly applied to combine information using various sensors in the sensor fusion field. Typical methods, such as CI, cross covariance, sequential CI,and ellipsoidal intersection methods [26, 27, 35, 36, 37, 38], have been consistently studied. They are classified as ellipsoidal methods and the ICI method [26, 27], which is one of the latest data fusion methods. A data fusion method based on the ICI is proposed to obtain less conservative but still consistent estimates. A major advantage of the ICI method is that its fusion results are more accurate than those of the CI method [27]. Therefore, the ICI method is used for combining two pieces of cardinality information. According to fusion methods, the cardinality, \(C_{f}\), can be determined as:

\[C_{f}=\alpha C_{1}+\beta C_{2} \tag{35}\]

for any \(\gamma\in[0,1]\).