# SMC-CPHD Filter with Adaptive Survival Probability for Multiple Frequency Tracking
## 2 Processing Steps of the SMC-CPHD Filter
When we apply the calculated value \(d_{\max}\) to set the survival probability, the sample impoverishment can occur in the worst case because the survival probability is directly affected by the distance. Therefore, an additional method such as distance leveling for obtaining enough samples may be necessary. Nevertheless, the proposed method is meaningful because the effect of resampling in the SMC-CPHD filter can be present; \(p_{S}\) has an effect on the weights of theparticles because \(p_{S}\) is multiplied to the weights of the particles. Simulations for verifying the performance improvement of the proposed algorithm are shown in Section 5. ### Convergence Analysis

The bounds for the mean square error (MSE) [33; 34] of the SMC-PHD filter depend on some considerations related to the designed parameters of the model of the target system. In order to verify the convergence of the proposed algorithm, those analysis results are extended and applied to the proposed algorithm. In particular, the effect of the character that changed the survival probability in Section 3.1 on the convergence of the filter is analyzed. Some of the assumptions and considerations used in the convergence analysis are given as follows. If \(\theta^{N}\) defines a sequence of measures that rely on the number of particles, \(N\), then \(\theta^{N}\) converges to \(\theta\) when \(\forall\varphi\in B\Big{(}\mathbb{R}^{d}\Big{)}\). \[\lim_{N\to\infty}E\bigg{[}\Big{(}\big{\langle}\theta^{N},\varphi\big{\rangle} -\langle\theta,\varphi\rangle\Big{)}^{2}\bigg{]}=0 \tag{24}\]

where \(B\Big{(}\mathbb{R}^{d}\Big{)}\) denotes the set of bounded Borel measurable function on \(\mathbb{R}^{d}\) and \(d\) refers to the dimension of the space. When the measure in the inner product, \(\langle.,.\rangle\), is discrete, it is the summation inner product:

\[\Big{\langle}D_{k|k}^{N},\varphi\Big{\rangle}=\sum_{i=1}^{N}w_{k}^{(i)} \varphi\Big{(}x_{k}^{(i)}\Big{)} \tag{25}\]

In order to prove convergence of the proposed algorithm, \(D^{\prime}_{k|k-1}\) is defined as the density propagated from the previous time step, and the boundary of \(\Big{|}\Big{\langle}D_{k|k}^{N},\varphi\Big{\rangle}-\Big{\langle}D^{\prime}_{ k|k-1},\varphi\Big{\rangle}\Big{|}\) is analyzed by the triangle inequality:

\[ \begin{array}{l}\Big{|}\Big{\langle}D_{k|k-1}^{N},\varphi\Big{\rangle}- \Big{\langle}D^{\prime}_{k|k-1},\varphi\Big{\rangle}\Big{|}\leq\Big{|}\Big{ \langle}D_{k|k-1}^{N},\varphi\Big{\rangle}-\Big{\langle}D_{k-1|k-1}^{N},\pi_{ k|k-1}\varphi\Big{\rangle}\Big{|}\\ +\Big{|}\Big{\langle}D_{k-1|k-1}^{N},\pi_{k|k-1}\varphi\Big{\rangle}-\Big{ \langle}D_{k-1|k-1},\pi_{k|k-1}\varphi\Big{\rangle}\Big{|}\end{array}  \tag{26}\]

where \(\pi_{k|k-1}\) refers to the transition density and is bounded by the system transition, \(f_{k|k-1}\), and the PHD of spawned targets, \(s_{k|k-1}\), as follows:

\[\pi_{k|k-1}\big{(}\mathbf{y},\mathbf{y}_{k-1}\big{)}=p_{S}\big{(}\mathbf{y}_{ k-1}\big{)}f_{k|k-1}\big{(}\mathbf{y}\big{|}\mathbf{y}_{k-1}\big{)}+s_{k|k-1} \big{(}\mathbf{y}\big{|}\mathbf{y}_{k-1}\big{)} \tag{27}\]

Figure 1: The adaptive survival probability. Let \(\xi_{k-1}\) be the \(\sigma\)-algebra generated by the particles \(\left\{\mathbf{y}_{k-1}^{(j)}\right\}\). Then, the first term of the right side of (26) can be expected as:

\[E\bigg{[}\left(\left\langle D_{k|k-1}^{N},\varphi\right\rangle-\left\langle D_{ k-1|k-1}^{N},\pi_{k|k-1}\varphi\right\rangle\right)^{2}\bigg{|}\xi_{k-1}\bigg{]}=E \bigg{[}\left\langle D_{k|k-1}^{N},\varphi\right\rangle^{2}\bigg{]}-\left\langle D _{k-1|k-1}^{N},\pi_{k|k-1}\varphi\right\rangle^{2} \tag{28}\]

The above equation can be expressed in sum using the independence of each particle as follows:

\[