# SMC-CPHD Filter with Adaptive Survival Probability for Multiple Frequency Tracking
## 2 Processing Steps of the SMC-CPHD Filter
### New Adaptive Survival Probability

As mentioned previously, the existing versions of the SMC-CPHD filter employ a predesigned value for the survival probability, which remains identical for all targets independent of distances between their posterior estimate and particles [7; 8; 10; 23]. Here, we describe a simple survival model shown by \(p_{S,k|k-1}\left(\mathbf{y}_{k-1}^{i}\right)\triangleq p_{S}\left(\mathbf{y}_{ k-1}^{i}\right)\) to adaptively compute the survival probability of the state variable according to the distance from each particle, \(i=1,2,\cdots,N_{k-1}\). In the proposed algorithm, the survival model includes a set of the boundary with their corresponding posterior estimate. The probability illustrates the survival probability with respect to the posterior estimate, which is calculated by:

\[p_{S}\left(\mathbf{y}_{k-1}^{i}\right)=\left\{ \begin{array}{l}p_{S,\max}\\ \left(if\;\;d_{k-1}^{i}=0\right)\\ p_{S,\max}\exp\left(-d_{k-1}^{i}/d_{\max}\right)\\ \left(if\;\;0<d_{k-1}^{i}<d_{\max}andd_{M}^{2}\leq T_{g}\right)\\ p_{S,\min}=0.3679\times p_{S,\max}\\ \left(if\;\;d_{k-1}^{i}=d_{\max}ord_{M}^{2}>T_{g}\right)\end{array} \right. \tag{23}\]

where \(p_{S}\) represents the adaptive survival probability at time \(k\), \(p_{S,max}\) represents the maximum value of \(p_{S}\), and \(\exp(\cdot)\) is the exponential function. \(p_{S,\min}\) indicates the minimum value of \(p_{S}\) and is equal to \(0.3679\times p_{S,max}\) because \(\exp(-1)=0.3679\) in the case of \(d_{k-1}^{i}=d_{\max}\), with \(d_{\max}\) being the maximum distance of a particle to the prior. \(d_{k-1}^{i}=\left|\mathbf{y}_{k-1}^{i}-\overset{\star}{\mathbf{y}}_{k-1}\right|\) denotes the Euclidean distance between the posterior estimate, \(\overset{\star}{\mathbf{y}}_{k-1}\), and the \(i\)-th of each particle \(\mathbf{y}_{k-1}^{i}\) at time \(k\), \(d_{max}\) is the maximum value of the distance, \(d_{k-1}^{i}\), and \(\overset{\star}{\mathbf{y}}_{k-1}\) is the observation location. \(d_{M}^{2}\triangleq\left(\varepsilon_{k}^{(ij)}\right)^{T}\left(p_{\mathbf{z },k|k-1}^{(j)}\right)^{-1}\varepsilon_{k}^{(ij)}\) is the Mahalanobis distance associated with \(\varepsilon_{k}^{(ij)}\) and \(d_{M}^{2}\leq T_{g}\) represents the validation gate area, which is selected to reduce false but apparently possible target measurements in the presence of the clutter or noise [28; 29; 30; 31; 32]. The other related notations are as follows: \(\varepsilon_{k}^{(ij)}\triangleq\mathbf{z}_{k}^{(i)}-h\left(\mathbf{y}_{k|k-1 }^{(j)}\right)\) denotes the innovation vector associated with the \(i\)-th measurement, \(\mathbf{z}_{k}^{(i)}\in\mathbf{Z}_{k}\), and with respect to that, the \(j\)-th predicted observation, \(P_{\varepsilon,k|k-1}^{(j)}=P_{\mathbf{z},k|k-1}^{(j)}=HP_{\mathbf{y},k|k-1}^{ (j)}H^{T}+R\) is the residual covariance matrix of \(\varepsilon_{k}^{(ij)}\), \(R\) defines the measurement noise covariance matrix, \(h\left(\mathbf{y}_{k|k-1}^{(j)}\right)\) refers to the measurement model, and \(T_{g}\) is the threshold of the \(n_{\mathbf{y}}\)-dimensional validation gate and a design parameter. Figure 1 shows the adaptive survival probability based on (23). As shown in Figure 1, the closer the distance, \(d_{k-1}^{i}\), the higher the probability of survival. As mentioned previously and shown in (23), the adaptive survival probability, \(p_{S}\), does not consist of constant values but varying values according to the distance within a validation gate area, and \(p_{S,\min}\) is not zero but a value determining by \(p_{S,\max}\). The parameter \(p_{S,\max}\) is a design parameter and is set to 1 in our simulations. In the setting of the value of \(p_{S,\max}\), the value of 0.98 is often used for the fixed survival probability [23], while the value of one is used for consideration of the attenuation of the survival probability according to the distance [24]. However, an improvement of the proposed method is still needed.